Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media
by Ioan I. Pop, D...
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Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media
by Ioan I. Pop, Derek B. Ingham
• ISBN: 0080438784 • Pub. Date: February 2001 • Publisher: Elsevier Science & Technology Books
Preface Interest in studying the phenomena of convective heat and mass transfer between an ambient fluid and a body which is immersed in it stems both from fundamental considerations, such as the development of better insights into the nature of the underlying physical processes which take place, and from practical considerations, such as the fact that these idealised configurations serve as a launching pad for modelling the analogous transfer processes in more realistic physical systems. Such idealised geometries also provide a test ground for checking the validity of theoretical analyses. Consequently, an immense research effort has been expended in exploring and understanding the convective heat and mass transfer processes between a fluid and submerged objects of various shapes. Among several geometries which have received considerable attention are flat plates, circular and elliptical cylinders and spheres, although much information is also available for some other bodies, such as corrugated surfaces or bodies of relatively complicated shapes. It is readily recognised that a wealth of information is now available on convective heat and mass transfer operations for viscous (Newtonian) fluids and for fluidsaturated porous media under most general boundary conditions of practical interest. The number of excellent review articles, books and monographs, which summarise the stateoftheart of convective heat and mass transfer, which are available in in the literature testify to the considerable importance of this field to many practical applications in modern industries. Given the great practical importance and physical complexity of buoyancy flows, they have been very actively investigated as part of the effort to fully understand, calculate and use them in many engineering problems. No doubt, these flows have been invaluable tools for the designers in a variety of engineering situations. However, it is well recognised that this has been possible only via appropriate heuristic assumptions, see for example the Boussinesq (1903) and Prandtl (1904) boundarylayer approximations. Today it is widely accepted that viscous effects, although very often confined in small regions, control and regulate the basic features of the flow and heat transfer characteristics, as for example, boundarylayer separation and flow circulation. As a result, these characteristics depend on the development of the viscous layer and its downstream fate, which may or may not experience transition to turbulence and separation to a wake. Numerous numerical schemes have been devel
xii
CONVECTIVE FLOWS
oped and these have proved to be fairly reliable when compared with experimental results. However, applications to real situations sometimes brings difficulties. As mentioned before, it is only in the last two decades that various authors have prepared excellent review articles, books and monographs on the topic of convective heat and mass transfer. However, to the best of our knowledge, the last monograph on this topic is that published by Gebhart et al. (1988). Therefore, it is pertinent now to emphasise some of the important contributions which have been published since then, and, indeed, these are very numerous. On studying the published books and monographs on convective heat and mass transfer, we have noticed that much emphasis is given to the traditional analytical and numerical techniques commonly employed in the classical boundarylayer theory, most of which have been known for several decades. In contrast, rather little attention has been directed towards the mathematical description of the asymptotic behaviours, such as singularities. With the rapid development of computers then these asymptotic solutions have been widely recognised. In fact, in the last few years a large number of such contributions have appeared in the literature, especially those concerning the mixed convection flows and conjugate heat transfer problems. Therefore, we decided to include in the present monograph more on the asymptotic and numerical techniques than what has been published in the previous books on convective heat and mass transfer. This book is certainly concerned with very efficient numerical techniques, but the methods p e r se are not the focus of the discussion. Rather, we concentrate on the physical conclusions which can be drawn from the analytical and mlmerical solutions. The selection of the papers reviewed is, of course, inevitably biased. Yet we feel that we may have overemphasised some contributions in favour of others and that we have not been as objective as we should. However, the perspective outlined in the book comes out of the external flow situations with which we are most personally familiar. In fact, we have knowingly excluded certain areas, such as, convective compressible flows and stability either because we felt there was not sufficient new material to report on, or because we did not feel sufficiently competent to review them. However, we have made it clear that the boundarylayer technique may still be a very powerful tool and can be successfully used in the future to solve problems that involve singularities, such as separation, partially reversed flow and reattachment. It should be mentioned again, to this end, that the main objective of the present book is to examine those problems and solution methods which heat transfer researchers need to follow in order to solve their problems. The book is a unified progress report which captures the spirit of the work in progress in boundarylayer heat transfer research and also identifies the potential difficulties and future needs. In addition, this work provides new material on convective heat and mass transfer, as well as a fresh look at basic methods in heat transfer. We have complemented the book with extensive references in order to stimulate further studies of the problems considered. We have presented a picture of the stateoftheart of boundarylayer heat transfer today by listing and com
PREFACE
xiii
menting also upon the most recent successful efforts and identifying the needs for further research. The tremendous amount of information and number of publications now makes it necessary for us to resort to such monographs. It is evident, from the number of citations in previous review articles, books and monographs on the topic of heat transfer that these publications have played a significant role in the development of convective heat flows. The book will be of interest to postgraduate students and researchers in the field of applied mathematics, fluid mechanics, heat transfer, physics, geophysics, chemical and mechanical engineering, etc. and the book can also be recommended as an advanced graduatelevel supplementary textbook. Also the wide range of methods described to solve practical problems makes this volume a valuable asset to practising engineers.
Acknowledgements A number of people have been very helpful in the completion of this work and we would like to acknowledge their contributions. First, we were impressed with the warm interest and meaningful suggestions of Professor T.Y. Na and Dr. D. A. S. Rees, the reviewers of this work. Secondly, the formatting of this book and the preparation of the figures were performed by Dr. Julie M. Harris, and we are very appreciative of her patience and expertise. Thirdly, we are indebted to Mr. Keith Lambert, Senior Publishing Editor of Pergamon, for his enthusiatic handling of this project. Cluj/Leeds Ioan P o p / D e r e k B. Ingham October, 2000
Nomenclature
ac
A AT A b
B C
cp C
Cs, Cs D Dm DT e~ E g Vr
Gr* h(x) h
I2 J
k kf km
ks
kin1 K K*
radius of cylinder or sphere, or major axis of elliptical cylinder, or body curvature, or amplitude of surface wave radius of core region reactant transversal heat dispersion constant amplitude of surface temperature thickness of plate, or minor axis of elliptical cylinder, or thickness of sheet, or width of jet slit, or body curvature product species body shape parameter, or aspect ratio specific heat at constant pressure concentration skin friction coefficients chemical diffusion mass diffusivity of porous medium transversal component of thermal dispersion tensor stress tensor activation energy transpiration parameter magnitude of acceleration due to gravity Grashof number modified Grashof number film thickness constant solid/fluid heat transfer coefficient second invariant of strain rate tensor microinertia density conjugate parameter thermal conductivity of fluid thermal conductivity of porous medium thermal conductivity of solid thermal conductivity of nearwall layer permeablility of porous medium inertial (or Forchheimer) coefficient, or modified permeability for powerlaw fluid
Ki K: l L L~ m
n n N Nu p Pc Pe Pr qs q~ q" Q r ~(~) R T~ Ra
Rah , Ra; Re Re* Reb ReD Re~,, s
permeabilities of layered porous media micropolar parameter length scale, or length of plate convective length scale, or length of vertically moving cylinder Lewis number exponent in powerlaw temperature, or powerlaw heat flux, or powerlaw potential velocity distributions stratification parameter, or powerlaw index unit vector buoyancy parameter Nusselt number pressure characteristic pressure P~clet number Prandtl number energy released from line heat source wall heat flux heat flux per unit area strength of radial source/sink, or total line heat flux, or volumetric flow rate in film radial coordinate axial distance buoyancy parameter, or gas constant temperature or heat flux parameter Rayleigh number for viscous fluid, or modified Rayleigh number for porous medium Rat modified Rayleigh numbers local nonDarcyRayleigh number Reynolds number modified Reynolds number Reynolds number for jet Reynolds number based on the diameter of cylinder Reo~ Reynolds numbers for moving or fixed plate heat transfer powerlaw index
xviii S(x), S(r Sc
Sh t T T*
% %
T~ TS To T~ T~ T~(x) U Uc
u(~) u~ u~ V
V W
w(z) Wc x, y, z
Yc, Zc
CONVECTIVE FLOWS body functions Schmidt number Sherwood number time fluid temperature reference temperature, or reference heat flux boundarylayer temperature core region temperature, or plume centreline temperature temperature at exit temperature in fluid temperature of outside surface of plate or cylinder temperature of solid plate, or of sheet wall temperature stratified temperature fluid velocity along xaxis, or in transverse direction plume centreline fluid velocity velocity outside boundarylayer velocity of moving sheet, or of moving cylinder velocity of potential flow in xdirection characteristic velocity velocity of moving plate fluid velocity along yaxis, or in radial direction fluid velocity vector fluid velocity along zaxis velocity of potential flow in zdirection characteristic velocity Cartesian coordinates characteristic coordinates
e ~0 ( 7/ ~/(~) 8 Ob 0~ 0 t~ A A~ A H It It* it0 u p a a(x) T T('~) Tij V~ ~o r
G r e e k Letters energy activation parameter c~f thermal diffusivity of fluid c~.~ effective thermal diffusivity of porous medium fl thermal expansion coefficient, or FalknerSkan parameter fl* concentration expansion coefficient 7 eigenvalue, or gradient of viscosity "~ shear rate tensor F conjugate parameter boundarylayer thickness, or plume diameter (~T, t~O thermal boundarylayer thicknesses (f~ momentum boundarylayer thickness AC concentration difference, C w  Coo AT temperature difference, T~  To~
r w
small quantity transformed xcoordinate, or elliptical coordinate quantity related to local Reynolds number similarity, or pseudosimilarity variable in ydirection similarity, or pseudosimilarity variable, or elliptical coordinate viscosity function nondimensional temperature, or angular coordinate conjugate nondimensional boundarylayer temperature nondimensional wall temperature characteristic temperature vortex viscosity mixed convection parameter Richardson number inclination parameter configuration function dynamic viscosity consistency index consistency index for nonNewtonian viscosity kinematic viscosity density heat capacity ratio wavy surface profile nondimensional time shear stress strain rate tensor wall skin friction inclination angle, or porosity of porous medium nondimensional concentration, or angular distance stream function vorticity
Subscripts f ref s w x oc
fluid reference solid wall local ambient fluid
Superscripts ' ~" 
dimensional variables, or average quantities differential with respect to independent variable nondimensional, or boundarylayer variables
Table of Contents
Convective Flows: Viscous Fluids. 1. Free convection boundarylayer flow over a vertical flat plate. 2. Mixed convection boundarylayer flow along a vertical flat plate. 3. Free and mixed convection boundarylayer flow past inclined and horizontal plates. 4. Doublediffusive convection. 5. Convective flow in buoyant plumes and jets. 6. Conjugate heat transfer over vertical and horizontal flat plates. 7. Free and mixed convection from cylinders. 8. Free and mixed convection boundarylayer flow over moving surfaces. 9. Unsteady free and mixed convection. 10. Free and mixed convection boundarylayer flow of nonNewtonian fluids. Convective Flows: Porous Media 11. Free and mixed convection boundarylayer flow over vertical surfaces in porous media. 12. Free and mixed convection past horizontal and inclined surfaces in porous media. 13. Conjugate free and mixed convection over vertical surfaces in porous media. 14. Free and mixed convection from cylinders and spheres in porous media. 15. Unsteady free and mixed convection in porous media.
16. NonDarcy free and mixed convection boundarylayer flow in porous media.
C O N V E C T I V E FLOWS: VISCOUS FLUIDS
3
A body which is introduced into a fluid which is at a different temperature forms a source of equilibrium disturbance due to the thermal interaction between the body and the fluid. The reason for this process is that there are thermal interactions between the body and the medium. The fluid elements near the body surface assume the temperature of the body and then begins the propagation of heat into the fluid by heat conduction. This variation of the fluid temperature is accompanied by a density variation which brings about a distortion in its distribution corresponding to the theory of hydrostatic equilibrium. This leads to the process of the redistribution of the density which takes on the character of a continuous mutual substitution of fluid elements. The particular case when the density variation is caused by the nonuniformity of the temperatures is called thermal gravitational convection. When the motion and heat transfer occur in an enclosed or infinite space then this process is called buoyancy convective flow. Ever since the publication of the first text book on heat transfer by GrSber (1921), the discussion of buoyancyinduced heat transfer follows directly that of forced convection flow. This emphasises that a common feature for these flows is the heat transfer of a fluid moving at different velocities. For example, buoyancy convective flow is considered as a forced flow in the case of very small fluid velocities or small Mach numbers. In many circumstances when the fluid arises due to only buoyancy then the governing momentum equation contains a term which is proportional to the temperature difference. This is a direct reflection of the fact that the main driving force for thermal convection is the difference in the temperature between the body and the fluid. The motion originates due to the interaction between the thermal and hydrodynamic fields in a region with a variable temperature. However, in nature and in many industrial and chemical engineering situations there are many transport processes which are governed by the joint action of the buoyancy forces from both thermal and mass diffusion that develop due to the coexistence of temperature gradients and concentration differences of dissimilar chemical species. When heat and mass transfer occur simultaneously in a moving fluid, the relation between the fluxes and the driving potentials is of a more intricate nature. It has been found that an energy flux can be generated not only by temperature gradients but also by a composition gradient. The energy flux caused by a composition gradient is called the Dufour or diffusionthermal effect. On the other hand, mass fluxes can also be created by temperature gradients and this is the Soret or thermaldiffusion effect. In general, the thermaldiffusion and the diffusionthermal effects are of a smaller order of magnitude than are the effects described by the Fourier or Fick laws and are often neglected in heat and mass transfer processes. The convective mode of heat transfer is generally divided into two basic processes. If the motion of the fluid arises from an external agent then the process is termed forced convection. If, on the other hand, no such externally induced flow is provided and the flow arises from the effect of a density difference, resulting from a temperature or concentration difference, in a body force field such as the grav
4
CONVECTIVE FLOWS
itational field, then the process is termed natural or free convection. The density difference gives rise to buoyancy forces which drive the flow and the main difference between free and forced convection lies in the nature of the fluid flow generation. In forced convection, the externally imposed flow is generally known, whereas in free convection it results from an interaction between the density difference and the gravRational field (or some other body force) and is therefore invariably linked with, and is dependent on, the temperature field. Thus, the motion that arises is not known at the onset and has to be determined from a consideration of the heat (or mass) transfer process coupled with a fluid flow mechanism. If, however, the effect of the buoyancy force in forced convection, or the effect of forced flow in free convection, becomes significant then the process is called mixed convection flows, or combined forced and free convection flows. The effect is especially pronounced in situations where the forced fluid flow velocity is low and/or the temperature difference is large. In mixed convection flows, the forced convection effects and the free convection effects are of comparable magnitude. Both the free and mixed convection processes may be divided into external flows over immersed bodies (such as flat plates, cylinders and wires, spheres or other bodies), free boundary flow (such as plumes, jets and wakes), and internal flow in ducts (such as pipes, channels and enclosures). The basically nonlinear character of the problems in buoyancy convective flows does not allow the use of the superposition principle for solving more complex problems on the basis of solutions obtained for simple idealised cases. For example, the problems of free and mixed convection flows can be divided into categories depending on the direction of the temperature gradient relative to that of the gravitational effect. It is only over the last three decades that buoyancy convective flows have been isolated as a selfsustained area of research and there has been a continuous need to develop new mathematical methods and advanced equipment for solving modern practical problems. For a detailed presentation of the subject of buoyancy convective flows over heated or cooled bodies several books and review articles may be consulted, such as ~k~rner (1973), Gebhart (1973), Jaluria (1980, 1987), Martynenko and Sokovishin (1982, 1989), Aziz and Na (1984), Shih (1984), Bejan (1984, 1995), Afzal (1986), Kaka(~ (1987), Chen and Armaly (1987), Gebhart et al. (1988), Joshi (1990), Gersten and Herwig (1992), Leal (1992), Nakayama (1995), Schneider (1995), Goldstein and Volino (1995) and Pop et al. (1998a). Buoyancy induced convective flow is of great importance in many heat removal processes in engineering technology and has attracted the attention of many researchers in the last few decades due to the fact that both science and technology are being interested in passive energy storage systems, such as the cooling of spent fuel rods in nuclear power applications and the design of solar collectors. In particular, for low power level devices it may be a significant cooling mechanism and in such cases the heat transfer surface area may be increased for the augmentation of heat transfer rates. It also arises in the design of thermal insulation, material processing
CONVECTIVE FLOWS" VISCOUS FLUIDS
5
and geothermal systems. In particular, it has been ascertained that free convection can induce the thermal stresses which lead to critical structural damage in the piping systems of nuclear reactors. The buoyant flow arising from heat rejection to the atmosphere, heating of rooms, fires, and many other such heat transfer processes, both natural and artificial, are other examples of natural convection flows. In the ensuing chapters of this book, a uniform format is adopted to present theoretical (analytical and numerical) results for the most important situations of the buoyancy convective flows obtained over the last few years. Most of these results refer to cases which have never, or only partially, been presented in review articles or handbooks. The most important fluid flow and heat transfer results are presented in terms of mathematical expressions as well as in tabular and graphical form to display the general trends. We believe that such tables are very important since they can serve as reference tests against which other exact or approximate solutions can be compared in the future. Due to the vastness of the results presented in this book, computer codes are not presented. However, frequent references are made to papers and/or books which contain extensive numerical methods collected from worldwide sources. We begin by considering a heated (or cooled) body which has, in general, a variable surface temperature or variable surface heat flux immersed in a fluid which has a uniform or variable (stratified) temperature. Apart from any motion that is generated by density gradients, we suppose that the fluid is motionless. The complete dimensional form of the continuity, momentum, thermal energy and mass diffusion (concentration) equations for a viscous and incompressible fluid, simplified only to the extent that we assume that all the fluid properties, except the density, are constant and neglect viscous dissipation, diffusionthermal (Dufour) and thermaldiffusion (Soret) effects, are given by, see Gebhart et al. (1988) or eejan (1995), v.
v

0
(i.a)
I
OV
l_
oT + ( V . V) V 
Poo
OT 
+
(v.

OC + ( V . V ) C  DV2C o~

+
P ~ Poo
+ ~ g
Pc~
(I.2) (I.3) (I.4)
where V is the velocity vector, T is the fluid temperature, C is the concentration, is the pressure, t is the time, g is the gravitation acceleration vector, u is the kinematic viscosity, p is the fluid density, p ~ is the constant local density, cff is the thermal diffusivity, D is the chemical diffusivity and ~2 is the Laplacian operator. For many actual fluids and flow conditions a simple and convenient way to express the density difference (ppoo) in the buoyancy term of the momentum Equation (I.2)
6
CONVECTIVE FLOWS
is given by, see Gebhart et al. (1988), (I.5) when the thermal gradient dominates over the concentration (mass diffusion) gradient and p 

(T

(V
(I.6)
when both the thermal and concentration (mass diffusion) gradients are important. Here fl and fl* are the thermal and concentration expansion coefficients and Too and C ~ are the temperature and concentration of the ambient medium. If the density varies linearly with T over the range of values of the physical quantities encountered in the transport process, ~ in Equation (I.5) is simply ~ p ~o~ ~ and if the density varies linearly with both T and C then p and ~* in Equation (I.6) are given by r176176
~,U and r
0l(a~~ ~ , ' " b ~
the expansion coefficients ~ a n d
~* may be evaluated anywhere in the ranges (To  Too) and (Co  Coo), where To and Co are the other bounding conditions on the flow. Equations (I.5) and (I.6) are good approximations for the variation of the density, especially when (ToToo) and (CoCoo) are small, and they are known as Boussinesq (1903) approximations. The interested reader should also consult Oberbeck (1879). Other recent considerations of these approximations can be found in the book by Gebhart et al. (1988). Itowever, if the density variation is substantially nonlinear in T or both in T and C over the ranges of their values in the buoyancy region, then the expressions for r and ~* must in general be much more complicated to yield an accurate representation in .Equations (I.5) and (I.6). This occurs for large temperature differences in any fluid and it also may arise, for example, in thermally driven motion in cold water, see Gebhart et al. (1988).
Chapter 1
Free convection boundarylayer flow o v e r a v e r t i c a l flat p l a t e 1.1
Introduction
The problem of free convection due to a heated or cooled vertical flat plate provides one of the most basic scenarios for heat transfer theory and thus is of considerable theoretical and practical interest. The free convection boundarylayer over a vertical flat plate is probably the first buoyancy convective problem which has been studied and it has been a very popular research topic for many years. Since the pioneering work of Schmidt and Beckmann (1930) and Ostrach (1952), both the analytical solution and the experimental data of Eichhorn (1961) have been continuously refined and improved. A very long list would be required to exhaust the published literature for this famous problem. However, we shall review in this chapter some of the most recent and novel results which have been recently published on the problem of steady boundarylayer free and mixed convection over a vertical flat plate. We consider a heated vertical flat plate of temperature Tw, or which has a heat flux ~ , oriented parallel to the direction of the gravitational acceleration and placed in an extensive quiescent medium at a temperature Too, as shown in Figure 1.1. If Tw :> Too, or qw > 0, the fluid adjacent to the vertical surface receives heat and becomes hot and therefore rises. Fluid from the neighbouring areas rushes in to take the place of this rising fluid. On the other hand, if T < Too, or ~ < 0, the plate is cooled and the fluid flows downward. It is the analysis and study of this steady state flow that yields the desired information on heat transfer rates, flow rates, temperature fields, etc. In practice the temperature of the ambient fluid far away from the plate, Too, may be taken as constant (isothermal) or variable (stratified). Special attention will be given in this chapter to both these cases because they occur frequently in the natural environment and also in association with numerous industrial processes.
8
C O N V E C T I V E FLOWS
(a)
(b)
ry
m
Figure 1.1" Physical model and coordinate systems for (a) Tw > Too, qw > 0 and (b) T~ O. Here primes denote differentiation with respect to r]. In the VHF case we have 4
1
r  x~ (qw(x))~ f(x, ~),
1
T
4
x~ (qw(x))~ O(x, r]),
~7
(qw(X))t y
Xg
(1.26)
In this case Equations (1.19) and (1.20) become
f,2 x (f,Of' f.Of) ~ 1 ( l + 4 Q ( x ) ) f ' O  x ( f, O0 i O f '~ 5 Ox 0 ~x]
f ' " + ~l (4 + Q(x)) f f .  ~1 (3 + 2Q(x)) 1
0"+
Pr
5
(4+Q(x))fO'
(1.27) (1.28)
along with the boundary conditions (1.21), which become
x~ (x, 0) + 1 (4 + Q(x)) f (x, O)   N ( x ) f' (x, O)  O, O'(x, O)  1 f'+0,
0+0
as
(1.29)
~+oc
for x > 0. The wall temperature functions P(x) and Q(x) and the mass transfer functions M(x) and N(x) are defined as followsX
Q(x)
x dqw qw(x) dx
1
T~,(x)
1
,
N(x)  Vw(X) qj,(Z)
(1.30)
The system of Equations (1.23)  (1.29) are in a very general form. However, for the special case in which all the functions P(x), Q(x), M(x) and N(x) are constant, the problem reduces to the solution of a fifthorder ordinary differential equation with five boundary conditions, i.e. a similarity solution may be obtained.
12
1.3
C O N V E C T I V E FLOWS
Similarity solutions for an i m p e r m e a b l e fiat plate w i t h a variable wall t e m p e r a t u r e
(vw(x) 
We now consider the case of an impermeable flat plate temperature distribution of the form
0) with a wall
(1.31)
Tw(x) = x m
where rn is a given constant. In this situation when P ( x ) =_ rn and M ( x ) = 0, Equations (1.23) and (1.24) reduce to the similarity form 1 f " + ~ (3 + m) f f "  ~ (1l + m )
f ,2 + 0   0
1 0 " + 1 ( 3 + m ) fO' _ m f 10  0
P7
(1.32)
(1.33)
along with the boundary conditions (1.25) which become f(0)  0, f'+O,
f'(0)  0, 0+0 as
0(0)  1 77+oo
(1.34)
These equations were first considered by Sparrow and Gregg (1958), who gave results for values of m between  0 . 8 and 3. The case rn = 0 corresponds to a uniform plate temperature and has been, as is wellknown, considered by Schmidt and Beckmann (1930) and Ostrach (1952). It is worth mentioning that interest in similarity solutions stems from the fact that they provide intermediate asymptotic solutions which are related to more complex nonsimilar solutions. It is in this context that in this book we give great attention to the possibility of similarity solutions to several problems. The similarity solutions of Equations (1.32)  (1.34) were simultaneously studied in more detail by Ingham (1985) and Merkin (1985a) for several values of m, positive or negative, and for different values of P r . Their numerical results for f"(0) and 0'(0) are shown, for P r = 1, in Figure 1.2 by the solid lines. The exact solution 0'(0)  0 for rn  _ 3 is also included in this figure. These quantities are related to the skin friction ~w at the plate and the heat transfer rate ~w from the plate through the relations .
qw   k I (~yr)~=0
 ~UcGr~ x88 l =
~T*a~88
f't(O )
[0'(011
We shall further present results for some special values of m.
(1.35)
FREE CONVECTION
OVER A VERTICAL
13
FLAT PLATE
(~) 4
3.31938 (m  me)88 0.90819  0.28530m + 0.21603m 2 ~~i ~'''"
"""
t
"=.
0.85147m ~ .....
.......
............
~O.90819  0.28530m f
u
mc = 0.9790
'
m
(b) e'(o)
2
0.11534 (m  me) ~ 0.58233 m ~ ] I
rnc = 0.9790
"'"
0.40103  0.31640m + 0.23431m 2 .... i'
=:.1L. . . .
2
m ,.
~'25.4OlOa  o.a164om
Figure 1.2: Variation of (a) f " (0), and (b) 0' (0), with m as obtained from numerical integration (solid lines) and asymptotic solutions for P r = 1. The symbol 9 shows the position of the exact solution 0 ~(0) = 0 for m = 5" 3
14 1.3.1
CONVECTIVE FLOWS m ,,~0
An approximate solution of Equations (1.32)  (1.34) near m  0 can be obtained by expanding f(~/) and 0(7/) in a power series in m of the form f(rl)
fo(r/)

0(?7)


+ mfl(r/) +
rn2f2(r/)
+...
O0(f]) [ ~'t01 (?7) 4 T/%202(7]) ']
99.
(1.36)
Substituting the expansions (1.36) into Equations (1.32) and (1.33) leads to three sets of ordinary differential equations which are to be solved subject to the appropriate boundary conditions, which are obtained from the boundary conditions (1.34). Ingham (1985) has solved these equations numerically and found, for Pr = 1, f"(0)  0.90819  0.28530m + 0.21603m 2 + ... 0'(0)  0.40103  0.31640m + 0.23431 m 2 + . . .
(1.37)
for m ..o 0. This solution is also shown in Figure 1.2. 1.3.2
m >> 1
In this case it is appropriate to make the following transformation 3
fmZF(~),
00(~),
1
~mZ~
(1.38)
This leads to the equations F'"+~
1 ( l +   m3 ) F F , 1 0"+
PW
~ 1( 1+ 1)
F,2 + 0  0
1(1+3) FO'OF'O
~
(1.39) (1.40)
where primes now denote differentiation with respect to f and the boundary conditions to be satisfied by these equations are still those given by (1.34). A solution of Equations (1.39) and (1.40) subject to the boundary conditions (1.34) is sought of the form F  Fo (f) + m  1F1 (f) + . . . (1.41) 0  O0 (~) 1/rt101 (~) } 9 ..
where Fo, 0o and F1,01 are given by the equations
Fg' + 88Fo f g  89f g + 0o  0, F0(0)  0, F~+O,
~~1e~ + ~ Vo 0;  V~eo  0
Fg(0)  0, 0 o  + 0 as
00(0)  1 ~+cr
F~" + 88Zo F~'  Vd Z~ + 88Fd' F~ + 3 Vo Vg  ~ o1~'~+01  0 1r ~ ~ + ~ Fo Ol  F; O~ + ~ Fo O'o + 88F, O'o OoF {  0 F, (0)  0, F{(0) = 0, 0, (0)  0 F{  + 0 ,
01  ' + 0
as
~   } oo
(1.42)
(1.43)
FREE C O N V E C T I O N OVER A VERTICAL FLAT PLATE
15
It is of some interest to note that Equations (1.42) are the same as those which are appropriate for a constant plate temperature, the solution of which is well documented in the papers by Ostrach (1952) and Sparrow and Gregg (1958). The first set of Equations (1.42) has been solved numerically, for P r = 1, by Ingham (1985), whilst Merkin (19853) has solved both sets of Equations (1.42) and (1.43) for both P r = 1 and P r # 1. Thus, Merkin (19853) found, for P r = 1, 1
f"(O)m ~ (0.85150.1579m1+...) 0 ' ( 0 )  m 88 ( 0 . 5 8 2 3  0.0009m1 + ...)
(1.44)
for m >> 1. The large asymptotic values of f"(0) and 0'(0), as given by expressions (1.44), are compared in Table 1.1 with the values obtained by solving Equations (1.32) (1.34) numerically. It is observed that the two values are in good agreement, even at relatively small values of m. Table 1.1 Comparison of f"(O) and 0'(0) for P r  1 as obtained by an exact solution of Equations (1.32)  (1.3~) and the asymptotic solution (1.4~).
m
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.0O 3.25 3.50 3.75 4.00
1.3.3
m
0 and 0 > 0 for all values of r/when m <  1 . The nature of the singularity near m   1 was also discussed by Merkin (1985a) and the mathematical analysis is straightforward and follows closely the prescribed temperature case presented in Section 1.3. Thus, we replace f, 0 and ~ by 4
feix(~,
..v
Oe~Y(~,
1
r/e~r/,
el+m
(1.68)
where e 0 x > 0
(1.80)
We assume now that Tw(x) and Too(x) have the following forms, see Semenov (1984): Tw(x)  (n + 1)T* ( M x + N ) TM + Tc (1 81) Too(x)  nT* ( M x + N ) m + Tc where n is the parameter describing whether the ambient temperature (n = 0) or the wall temperature (n =  1 ) is fixed; M , N , Tc and T* = T w ( O )  Tc~(O) are constant. The environment is stably stratified if dd~ > 0, hence m M n > O. Further, we introduce the new variables ~Mx+N,
7?
u2
IM,)
1
m,
~ 4 y
(1.82)
1
7])


(g/3T.tp2 '~ ~ m+_._.._~3 [M[3] [ f(rl),
T  (n + 0(rl) ) T*[ TM + Tc
Substitution of these variables into Equations (1.77)  (1.79) leads to the following ordinary differential equations for f(r/) and 0(rl)
f'" + sgn (M)[1 (m+3)ff" ~.1 (m + 1)f,2]
+0
o
O"+Prsgn(M)[4 (m+3)fO'm (O+n)f']  0
(1.83)
(1.8a)
and the boundary conditions (1.80) become f ( 0 ) = 0,
f~+0,
f ' ( 0 ) = 0,
0(0) = 1
0+0
77+ec
as
(1.85)
Solutions of these equations have been determined for the following cases: (i) m = n = 0, sgn (M) = 1 by Ostrach (1952); (ii) n = 0, sgn (M) = 1 and for a limited mrange by Sparrow and Gregg (1958); (iii) n =  1 , sgn (M) = 1 and a limited mrange by Cheesewright (1967) and Yang et al. (1972); (iv) n = 0, sgn (M) = 1 and the complete mrange by Merkin (1985a); and
24
C O N V E C T I V E FLOWS
(v) n   1 , sgn (M)   1 and the complete mrange by Henkes and Hoogendoorn (1989). We next present some of the results reported by Henkes and Hoogendoorn (1989) for a variable wall temperature and a stratified environment with sgn (M)  +1. Analogous to the analysis of Merkin (1985a) for n  0 in the limit Im I + oc, the behaviour for n   1 in tile limit Iml + cr can be found by using the transformation 3
1
f ImlZ F (~,
0  G
(~,
On substituting expressions (1.86) into Equations F'" + sgn (M)
[ l ( sgn (m) + ~ 3 ) F F ,
~
 I m l ~ ~
(1.86)
(1.83) and (1.84) we obtain
1( 1 ) ]  ~ ~gn (m) + ~  [ F '~ + a  0 (1.S7)
G" + Prsgn (M) [ 4 ( s g n ( m ) + ~ 3 )
FG' "s g n ( m ) ( G + n ) F ' ]  0(1.88)
with boundary conditions F(0)0, F'+0,
F'(0)0, G~0 as
G(0)=I ~?+cc
(1 89)
where primes denote differentiation with respect to ~. The transformation (1.86) gives the following relations for the reduced skin friction and wall heat flux f"(O)  Iml4 F"(O),
0'(0)  lml 1 G'(0)
(1.90)
~s I~l ~ ~ . The two sets of Equations (1.83)  (1.85) and (1.87)  (1.89) have been solved numerically by Henkes and Hoogendoorn (1989) for Pr = 0.72 and different combinations of sgn (M) and n. The variation of f"(0) and 0'(0) with m for n   1 and sgn (M)  1 is shown in Figure 1.5. Also, some fluid velocity and temperature profiles are given in Figure 1.6 and it can be seen from these figures that the complete mrange is free of singularities. However, a region with a small backflow and a temperature deficit is found in the outer part of the boundarylayer in a stably stratified environment (m < 0) but there is no backflow or temperature deficit in an unstably stratified environment (m > 0). The values of f"(0) and 0'(0) for n =  1 and sgn(M) =  1 are given in Figure 1.7 which shows in the limit Ira] + c~ that the solution with sgn (m) = :t:1 is identical to the solution for sgn (M) = 1 with sgn (m) = T1. Increasing m from  c ~ to 0 (unstable stratification) gives a zero wall heat flux with a temperature identical to unity everywhere, except in a small region at the outer edge, where the temperature rapidly falls to zero. The zero boundary condition for the fluid velocity
FREE CONVECTION
OVER A VERTICAL FLAT PLATE
(~)
25
(b) 1.5
......
0.0
,,,,, ,,,,,,
~
.....
__."_
1.3900m~
o'(o)
f"(O) N
].0 ! o.9o26 I,nl~/~
"'..I
 0.5
~ 0.5 5
1.0 0
m
5
_0.5913 [m[~ i
i
5
Figure 1.5 Variation of (a) f " (0), and (b) O' (0), with m for n   1 and sgn ( M ) 1 when P r  0.72.
is also satisfied in a small region at the outer edge of the boundarylayer, as can be seen from F i g u r e 1.8. A l t h o u g h the negative m  b r a n c h for sgn ( M )   1 describes the similarity solutions of the b o u n d a r y  l a y e r equations, t h e y cannot be p a r t of t h e
(a)
(b) 1.0
1.2
9 772,0(~) O(~.c!~
1.0
i
f'(~) 0.8 0.6
'~
~t= 1
0.4 0.2
~176i 0
O0
m = r,~
5
77
10
"
0
>'
5
77
10
Figure 1.6: (a) The fluid velocity, f ' ( y ) , and (b) the temperature, 0(~), profiles for n1 and sgn ( M )  1 when P r  0.72.
26
CONVECTIVE FLOWS
(~)
(5) 1.5
i
0.0
,,
\
,
,
,,,,,
.
,,,
1
f"(O)
0.0104 lm[Z
o'(0) /
0.5
1.0
: 0.5913m88
1.3900 Im188
0..5 5
~
0 9026m ~ 0
rn
1.0
5
5
0
m
5
Figure 1.7: Variation of (a) f"(O), and (b) O'(O), with m f o r n   1 and s g n ( M )  1 when P r  0.72.
flow along the heated plate: f'(~7) and 0'(r/) do not vanish if r/increases to infinity. This is required for the matching of the boundarylayer solution (inner solution) with the solution in the ambient fluid (outer solution) within the NavierStokes description. On the other hand, the fluid velocity and the temperature profiles, see Figure 1.9, show that this matching condition is satisfied for the solutions on the positive mbranch (stable stratification). Further, Henkes and Hoogendoorn (1989) have solved the full boundarylayer
\
/
/
/
/ i
\
/ t
\\
\ l
//
l l
// / ~
/
l
I
I
!
,,
_
! f .
,,
0
!
5
i
,,
][1 14.9 15.0 1 4.8
Figure 1.8" Fluid velocity, f'(rl), (broken line) and temperature, 0(77), (solid line) profiles for m   1 , n   1 and s g n ( M )   l when P r  0.72.
FREE CONVECTION OVER A VERTICAL FLAT PLATE
27
~/I "\\ I/~ \\\\
1
o
~ ....
1'o
,
15
Figure 1.9: F l u i d velocity, f ' ( r l ) , ( b r o k e n line) a n d t e m p e r a t u r e , 0 ( ~ ) , (solid line) profiles f o r m  1, n   1 a n d sgn ( M )   1 w h e n P r = 0.72.
Equations (1.77)  (1.80) for the stable stratification Too(x) _ 1 
1
T*
for
0~< x < o c
~ t 1 Xo
(1.91)
xo
where xo is a length scale and have calculated the local Nusselt n u m b e r which is defined as "
iu

T*
)

(1.92)
The variation of N u with ~X is illustrated (by solid lines) in Figure 1.10 for n   1 , sgn (M)  1 and P r  0.72. It is observed t h a t the a s y m p t o t i c similarity solution for large values of z is a good a p p r o x i m a t i o n for ~X as small as 3 whereas the solution for small values of x is only a good a p p r o x i m a t i o n up to about 0.2. T h u s for values of ~z in the range 0.2 ~< zo z < 3 the full equations have to be solved. T h e similarity solutions give 1
Gr 88 
for m 
0.3571
(;o)
9
as
xo ~
(1.92a)
0 and 3
Gr~
 0.5592
xo + 1
as
xo ~ c~
(1.92b)
for m   1 (using M  ~1 and N  1) are also included in Figure 1.10 (shown by the broken lines). This figure shows t h a t the nonsimilar solution smoothly matches b o t h a s y m p t o t i c limits given by E q u a t i o n s (1.92a) and (1.92b).
28
CONVECTIVE FLOWS
1.0
ii
i
/ !
~I~
I
! I I I
+
0.5
o
i
5 XO
Figure I " 10: Variation of the local Nusselt number with Xz__ for P r  0.72 and Too(Z) o T* given by Equation (1.91). The numerical solution for s g n ( M )  1 is indicated by the solid line and the similarity solutions (1.92a) and (1.92b), using n   1 and sgn ( M )  1, are indicated by the broken and dotted lines, respectively.
Henkes and Hoogendoorn (1989) have also solved Equations (1.77)  (1.80) for a linear, stable stratification
T*
=xo
for
0~
xo
~1
(1.93)
T h e y have shown t h a t in the limit x + x0, the similarity solution sgn (M)   1 and m  1 (using M 1 and N  1) gives the following a s y m p t o t i c expression for the local Nusselt n u m b e r = 0.7313 Gr~
1 ~ xo
as
~ 1
(1.94)
xo
However for ~ + 0 + the a s y m p t o t i c value of the local Nusselt n u m b e r is given by expression (1.92a). The calculated local Nusselt number, fluid velocity m a x i m u m and m i n i m u m profiles along with the corresponding similarity solutions given by E q u a t i o n s (1.83)  (1.85) are presented in Figure 1.11. These figures clearly show t h a t the numerical solution smoothly matches the similarity solution for small values of ~x, but does not do so for large values of x since in the limit x ~ x0 the wall heat transfer does not follow the similarity relation (1.94). X0
FREE
CONVECTION
OVER A VERTICAL
(~)
29
FLAT PLATE
(b) o 0"00
1.0
Nu Gr88 0.05
0.5
,
0.0 0.0
0'.5
,
x
,,
0.i0
1.0
0.0
0.5
x
1.0
TO
XO
(c) 0.4
/
\
I
\ ",
II
O.2
0.0 0.0
0:5
x
1.0
;gO
Figure 1.11: Variation of (a) the local Nusselt number, (b) the fluid velocity mini
mum, and (c) the fluid velocity maximum, with ~ for P r  0.72 and Tc~!z) given by Equation (1.93). The numerical solution is indicated by the solid lines and the similarity solutions for m = O, s g n ( M ) = 1 (Equation (1.92a) in (a)) and m   1, sgn(M) =  1 (Equation (1.94) in (a)), when n =  1 , are indicated by the broken and dotted lines, respectively.
30
1.6
C O N V E C T I V E FLOWS
Flat plate w i t h a sinusoidal wall t e m p e r a t u r e
Early work on steady free convection boundarylayer flow over a vertical semiinfinite flat plate has been confined almost entirely to the case of a constant wall temperature or constant wall heat flux. For the case of a variable wall temperature, or variable wall heat flux, there are, as we have seen previously, only a few papers available in the literature which are concerned with similarity solutions. Of equal importance is another class of free convection problems which deal with the case of sinusoidal surface temperature variations. Kao (1976), Kao et al. (1977), Na (1978) and "gang et al. (1982) appear to be the first who have considered this class of problems. Very recently, Rees (1999a) has dealt with the case of sinusoidal wall temperature variations about a constant mean value, Tw, which is above the ambient fluid temperature, Too, of the form (1.95)
T  T ~ + (Tw  Too)(1 + Asin (Trx))
where A is the relative amplitude of the surface temperature variations. In what follows we shall present some very interesting results as obtained by Rees (1999a) for this problem. Starting from the nondimensional Equations (1.19) and (1.20), we use the transformation 1 42  x ~f ( x , rl ) , T  0 ( x , rl ) , 77  y x  ~ (1.96) and then f and 0 arc given by the equations
f
,,,
3 1 f,2 +4.ff"~ +0 1 0"+ 3 f O' p;

X
X
f,O.f'
(1.97)
f,,Of )
( f ' O O _ o'Of )
which have to be solved subject to the boundary conditions
f(x,O)O,
f ' ( x , O)  O, O(x, O)  l + A s i n (Trx) f'+O, 0+0 as r/,oc, x>O
for
x>O
(1.99)
The parabolic Equations (1.97) and (1.98) were solved numerically by Rees (1999a) using the Kellerbox method and this is described in detail in the book by Cebeci and Bradshaw (1984). This scheme has the advantage over the other methods in that it is unconditionally stable and it allows the Richardson extrapolation technique to be used, which enables a high accuracy to be obtained when using a relatively crude grid. The results were reported for three values of the Prandtl number, namely P r  0.01 (liquid metals), 0.7 (air) and 7 (water). The variation with x of the reduced skin friction, f " ( x , 0), and the reduced heat transfer, O'(x, 0), for some values of the parameter ,4 are shown in Figures 1.12 and
F R E E C O N V E C T I O N O V E R A VERTICAL FLAT P L A T E
31
1.13. Some aspects of the overall behaviour of these curves may be explained by observing that the boundarylayer is thinner when the surface temperature is relatively high and thicker when it is low. This arises because the relatively high surface temperature induces relatively large upward fluid velocities, with the consequent increase in the rate of fluid entrainment into the boundarylayer. This causes, in turn, a thinning of the boundarylayer. Thus, we should expect high shear stresses and rates of heat transfer at locations just beyond where the surface temperature attains its maximum value. However, there is an obvious qualitative difference between the curves shown in Figure 1.12 and those in Figure 1.13. As x increases, the amplitude
(a)
(b) 1.6
~1.0
~'1.4
1.2
0.8
1.0 0.6 0.8 0.4
0.6 0
5
10 x 15
20
0
5
10 x 15
20
(c) 2.4 ~2.0 v
1.6 1.2 0.8
0
5
10 x 15
20
Figure 1.12: Variation of f " ( x , O) with x for A  O, 0.2, 0.4, 0.6, 0.8 and 1, where the amplitude increases as the value of A increases, when (a) P r  0.01, (b) P r  0.7 and (c) P r  7.
32
C O N V E C T I V E FLOWS
of the oscillation of f " ( x , 0) decays slowly, whereas the amplitude of O'(x, 0) increases with increasing values of x. Indeed, the curves in Figure 1.13 suggest that, whatever the value of A, there will always be a value of x beyond which some part of the function O'(x, 0) between successive surface temperature maxima will be positive. This somewhat unusual phenomena for boundarylayer flow may be explained by noting that when relatively hot fluid encounters a relatively cold part of the heated surface, the overall heat transfer will be from the fluid into the surface, rather than the other way around. However, these arguments are insufficient to explain why the amplitude of the oscillations shown in Figure 1.12 decay, or to give the rate of
(b)
(~) 0.6 S'0.4 0.2 0.0
i
2
llt!
0
0.2
1
0.4
2
0.6 5
10
x
15
20
0
5
0
i
10' x 15
20
(c) 4
~2 0
2 4
10 x 15
20
Figure 1.13: Variation of O'(x, O) with x for A  O, 0.2, 0.4, 0.6, 0.8 and 1, where the amplitude increases as the value of A increases, when (a) Pr  0.01, (b) P r  0.7 and (c) P r  7.
F R E E CONVECTION OVER A VERTICAL FLAT PLATE
33
decay. A very detailed asymptotic analysis was presented by Rees (1999a) in order to explain all these observations from his numerical calculations. In order to do this, the first task is to determine the thickness of the developing inner (nearwall) layer in terms of 77. Thus, we set f  f0(r]) and 0  00(7/) in Equations (1.97) and (1.98), where f0 and 00 are given by f o"' + 3~ f o f ; '  ~J6 1 .,2 + Oo  0 1
.
(1.100)
3
prOo +~foO'o  0
(1.101)
f;(0)0, 00(0)1 00+0 as ~+cr
(1.102)
along with the boundary conditions f0(0)0, f~ +0,
The solution of Equations (1.100)  (1.102) can be expressed, for small values of r/(> 1, where the values of A~(0), Bf(0), aIi~(o) and G~2s(0 ) can be found in Rees (1999a). Figure 1.14 illustrates the comparison between expressions (1.110) and the full numerical results as obtained from Equations (1.97) and (1.98) for P r = 0.7 and
1
~
o)
I
0
1 2 0
2
4
6
X
8
10
Figure 1.14: CompaTison between the numerical solution (solid lines) and the asymptotic solution (1.110) (broken lines) for A  1 and Pr  0.7.
F R E E C O N V E C T I O N O V E R A V E R T I C A L FLAT P L A T E
35
,4  1. The solid lines represent the numerical solution and the dashed lines the asymptotic solution, respectively, and it is seen that the results are in excellent agreement. This confirms the existence of a thinning nearwall layer. Finally, Figures 1.15 and 1.16 show the isotherms for ,4  0.2, 0.5 and 1 with P r   0.7. It is seen from Figure 1.15 that the boundarylayer remains at its overall thickness in terms of r] when x is large, although variations in the thickness are clearly present when x is small. The thickness of the region in which strong surface induced temperature variations are present reduces slowly in size as the value of x increases. The development of a nearwall layer is clearly evident in Figure 1.16
(~)
(b)
(c)
Figure 1.15:
Isotherms
]or Pr 
0.7
when
( a ) ,4 
0.2,
(b) r

0.5
and (c) Jt 
1.
36
C O N V E C T I V E FLOWS
/
Figure 1.16: Perturbation i s o t h e r m s / o r .4 = 1 and P r = 0.7.
where the perturbation of the temperature field from that given by .A = 0 (isothermal flat plate) is presented. It is also worth mentioning that Rees (1999a) has numerically found that no separation, i.e. there is no location where the skin friction is zero, occurs in this problem when j t ~ 1 for the range of values of the Prandtl numbers considered. However, when .4 > 1 it is possible to obtain a negative skin friction and when P r = 0.7 incipient separation first occurs when A ~ 1.915. When P r = 7, the critical value of Jt is 2.005 and this means that the minimum temperature of the 'heated' surface needs to be well below that of the ambient medium before separation can occur.
1.7
Free c o n v e c t i o n b o u n d a r y  l a y e r flow over a v e r t i c a l p e r m e a b l e flat p l a t e
Since Griffith and Meredith (1936) reported what is now referred to as the asymptotic suction profile for viscous adiabatic flow along a flat plate with uniform suction, interest in flows with transpiration (blowing or suction) at solid boundaries has continued to attract the interest of engineers and scientists. Transpiration at solid boundaries has application to boundarylayer control of fluid flow over wings and turbine blades, the cooling of electronic components, the flow past permeable movingbelt surfaces with mass transfer as found in industrial manufacturing devices, sundry chemical engineering processes, etc. (see Weidman and Amberg, 1996). Research work on the effects of blowing and suction on steady free convection boundary layers has been confined almost entirely to the case of a heated vertical plate. Eichhorn (1960) considered the power law variation in the plate temperature and transpiration velocity which enables a similarity solution of the boundarylayer equations to be found. Sparrow and Cess (1961) discussed the case of constant plate
FREE CONVECTION OVER A VERTICAL FLAT PLATE
37
temperature and constant transpiration velocity. They obtained series expansions 1 for the temperature and the fluid velocity in powers of xZ. Merkin (1972) extended this work by obtaining asymptotic expansions for large values of x in both the cases of blowing and suction. Then, Clarke (1973) extended the problem discussed by Eichhorn (1960) by obtaining the next approximation to the solution of the full NavierStokes equations for large, but finite, values of the Grashof number. The effects of blowing and suction on steady free convection boundary layers on bodies of general shape was also considered by Merkin (1975) and other papers which deal with the effects of blowing and suction on free convection boundarylayer flows over a vertical plate are those by Parikh et al. (1974), Na (1978), Vedhanayagam et al. (1980), Kao (1982), Pop and Watanabe (1992), Chaudhary and Merkin (1993) and Merkin (1994a). The last two papers consider the cases when the wall temperature, wall heat flux and transpiration velocity are proportional to some power of x, such that the governing equations reduce to similarity form. The range of existence of solutions, as well as the asymptotic solutions for strong blowing and suction, were discussed and compared with numerical solutions of the similarity equations. These asymptotic solutions assist us in obtaining a fundamental understanding of many complicated fluid flows of practical interest. In this section, we present some of the results reported by Merkin (1994a) and in order to do this we assume that Tw(x) is given by expression (1.31) and Vw(X) has the form 1
Vw(X)  ~fw (3 + m ) x
(m~)
4
(1.111)
where fw is a nondimensional constant which determines the transpiration rate, with fw > 0 for blowing or injection and fw < 0 for suction. Equations (1.23) and (1.24) then reduce to the ordinary differential Equations (1.32) and (1.33) which have to be solved subject to the appropriate boundary conditions which come from Equation (1.25). In order to write these equations in the form given by Merkin (1994a), we take 1 f  2v~F(~), 0  r ~ = ~~ (1.112) Equations (1.32) and (1.33) now become
F'"+(3+m)FF"2(I+m)F I r Pr + (3 + m ) F r
'2+r
0
(1.113)
4mF'r = 0
(1.114)
together with the boundary conditions (1.25) which become F(0) =
F'+O,
F ' ( 0 ) = 0,
r
as
r
~+c~
=
(1.115)
Equations (1.113)  (1.115) were solved numerically by Merkin (1994a) for fw = i 0.5, Pr = 1 and for several values of m. The results for F"(0) and r
38
C O N V E C T I V E FLOWS
(a)
(b) 12
3
r
F"(0) 8
2 \x i


,
io
i
,,
,
,
,,,,
6.5

m
U
,
o.o
,m
,,
6.5
i.o
Figure 1.17: Variation of (a) F"(O), and (b) r lines) and fw  0.5 (broken lines) when P r  1.
m
o.o
with m for fw  0.5 (solid
are presented in Figure 1.17 which shows that the solution becomes singular as m approaches a lower bound m = m c ( P r ) , say, where mc = 0.9790 for fw = 0.5 and P r = 1. The nature of the singularity in the solution as m 9 mc was also studied by Merkin (1994a). Since this analysis closely follows that of Section 1.3 for an impermeable plate (fw = 0) problem, we do not describe it here and merely present some of the results obtained. Thus, it was found for blowing (fw > 0) and P r = 1, that F " (0) and r behave as
F"(0) ~ 0 . 3 0 0 2 0 / ~ (,~  ,,~)3
r
~ 0.26214
f 5w ( m  m e )  5
+"" o
o
(1.116) o
for m + m c ( P r ) . In contrast, for suction (fw < 0) solutions exist only for m > m0, where 4 mo  0.9790  0.5361 If~l~ + . . . (1.117) as Ifw] + 0 and P r = 1. Further, Merkin (1994a) has studied solutions of Equations (1.113) and (1.114) for strong suction and strong blowing, respectively. For strong suction F"(0) and r have the asymptotic forms F"(0) ~
1
(~n + 3) P r
+...,
r
~  (m + 3) P r Ifw[ + . . .
(1.118)
as ]fwl + c~. Variation of F"(0) and  r with [fw[, obtained by numerically integrating Equations (1.113)  (1.115) for P r = 1 and m = 0 and 1, are illustrated (by solid lines) in Figure 1.18 and also shown are the asymptotic expressions (1.118) (by broken lines). It can be seen that these asymptotic forms are rapidly attained
F R E E C O N V E C T I O N O V E R A V E R T I C A L FLAT P L A T E
39
(b) r/2. =
0
0.6 F"(0) 0.4
4
r176 2 0.2 0.0
L
o.o
m1
7
0 , ~ = 0
0:5
i:0 If l 1:5
_ '. . . . . . . .
0.0
0.5
lfwl
,
......
1.0
Figure 1.18: Variation of (a) F"(O), and (b) r with Ifw! for m  0 and m = 1 when P r  1. The numerical solutions are indicated by the solid lines and the asymptotic expressions (1.118) are indicated by the broken lines.
as lfw] increases and there is very good agreement between the two results beyond fw ~  1 . 4 . However, the a s y m p t o t i c solution is approached even more rapidly as the P r a n d t l number increases. In contrast, Merkin (1994a) has shown t h a t for fw (> 0) large (strong blowing) there are two cases to be considered, namely mc < m < 3 and m > 3. In b o t h cases there is an inner inviscid region, of thickness O (fw), made up of fluid blown t h r o u g h the wall. In the case me < m < 3, there is t h e n a thin shear layer, of thickness O
fw~
, centred on the outer edge of the inner region. For m > 3 there
is still an outer region of thickness O
, which is thicker relative to the inner
region. It was found by Merkin (1994a), after some algebra, t h a t for mc < m < 3 we have the following: 1 F " ( 0 ) ~ (m + 3) fw + ' " '
r
4m (0) ~  ( m
+ 3) 3 P r f 3 + " "
(1.119)
as ]w + c~. These relations show t h a t F"(O) is i n d e p e n d e n t of P r , whilst r does depend on P r and r = 0 w h e n m = 0 in which case r  1. As a check of this analysis, F " (0) and  r (0) were obtained numerically from Equations (1.113) (1.115) and are presented (by solid lines) in Figure 1.19 for m = 0 and m = 1 with P r = 1 and in Figure 1.20 for m = 0 and m = 1 w i t h P r = 1 and for m = 0 with P r  7. The asymptotic expressions (1.119) are also included in these figures and are indicated by the broken lines. I t can readily be seen t h a t the asymptotic solutions (1.119) are approached rapidly as fw increases and the difference is negligible for values of ]w beyond fw ~ 1. G r a p h s of the t e m p e r a t u r e profiles r for P r = 1, and
40
CONVECTIVE FLOWS
m=O 0.6
F"(0) 0.4
m
=
l
~
0.2 0.00.0. . 0:2 . . 0:4 . . . 0:6
f~
018
1i0
Figure 1.19: Variation of F"(O) with fw for m  0 and m  1 when P r  1. The numerical solutions are indicated by the solid lines and the asymptotic expression (1.119) is indicated by the broken lines.
1.0 0.8
m = 1, P r = 1
r
m = O, Pr = 1
0.6
m = O, P r = 7
9 \\\\\\\\
0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Figure 1.20: Variation o f  r with fw f o r m  0 and m  1 when P r  1, and f o r m  0 when P r  7. The numerical solutions are indicated by the solid lines and the asymptotic expression (1.119) is indicated by the broken line.
m =  0 . 5 and m = 0, and f w ranging from 0 to 2.5 are also shown in Figure 1.21. It can be seen that for m =  0 . 5 there is a large t e m p e r a t u r e excess as f~ increases, whilst the t e m p e r a t u r e remains in the range 0 ~< r ~< 1 throughout for m = 0. In another paper by C h a u d h a r y and Merkin (1993), the case of a vertical permeable flat plate with a prescribed surface heat flux distribution qw(X) = x m and variable transpiration velocity has also been studied. Similarity equations were derived and they depend on the parameters m , f w and P r . It was shown t h a t solutions
F R E E C O N V E C T I O N OVER A VERTICAL FLAT PLATE
41
(b) r162
6
r162
1.0]'
5
O.8
4
0.6i 15~
3 2
'
0.4
"
1 0,
0.2 ,.
o
.
.
.
r
.
0.0
o
5
4
r
Figure 1.21: Temperature profiles, r162 for Pr = 1 when ( a ) m   0 . 5 m=O.
and (b)
exist only for m >  1 for injection, whereas they exist for all m > mc (fw) for suction, where mc <  1 . The solutions for strong suction and injection were also derived. In the later case (injection), the asymptotic structure was found to be different for m in the three ranges,  1 < m <  ~1,  ~1 < m < 7 and m > 7. Results were also obtained by Chaudhary and Merkin (1993) for the problem of a vertical permeable flat plate with a constant heat flux distribution qw(x) and constant transpiration velocity. This case reduces to the solution of Equations (1.14)  (1.16) with the boundary (transformed) conditions u0,
v=41, 0o _ 1 Oy u+0, 0+0
on as
y  0, y+c~,
x>0 x>0
(1.120)
where + denotes injection and  suction. This problem is nonsimilar and in order to obtain a solution which is valid for all x > 0, the governing equations have to be solved numerically. This was performed by Chaudhary and Merkin (1993) in two steps. The first step is to obtain a solution for x small by using the transformation
r
4
3=x + xg f(x,r/),
1
1
O xgh(x,7?),
77 = y x  g
(1.121)
Using these expressions, Equations (1.14)  (1.16) become f'" +
p~
(4 )f,, 3.,2 g f 3:: x~  ~.t + h  x
~
hx
(f,Of'f.cgfl Ox
~
~
(1.122)
~x
(1.123)
42
C O N V E C T I V E FLOWS
along with the boundary conditions f(x,0)0, f'(x,0)0, f'+0, h+0
as
h'(x, 0 )   1 for r/+~, x>0
x>0 (1.124)
It is worth noting that equations of such a form have been solved very efficiently by Watanabe and his coworkers in a series of papers on forced, free and mixed convection flows past flat plates, cones and cylinders, see for example, Watanabe et al. (1996). In order to do this they have used the differencedifferential method proposed by Hartree and Womersley (1937) in combination with a fourpoint backward difference formula of the GregoryNewton type. Returning now to the problem governed by the two sets of Equations (1.14) (1.16), subject to the boundary conditions (1.120) and Equations (1.122)  (1.124), Chaudhary and Merkin (1993) have solved them numerically by a very efficient finitedifference method as described by Mahmood and Merkin (1988). The solution starts at x = 0 using Equations (1.122)  (1.124) and continues until x = 1 and the fluid velocity and temperature profiles calculated at x = 1. These solutions are then used as the starting profiles for the solution of Equations (1.14)  (1.16) subject to the boundary conditions (1.120) for x > 1. Thus a smooth transition from one solution regime to the other was achieved. The nondimensional skin friction, ~w(x), and the nondimensional wall temperature, 0w(x), have been obtained and they are presented in Figures 1.22 and 1.23 for P r = 1. (b) 1.0 ~w(x) 0.8
1.0 o.s 0.6O.4O.2
f
0.6 0.4 0.2 o
X
/
o
X
Figure 1.22: Variation of (a) Tw(X), and (b) Ow(x), with x for P r  1 in the case of uniform suction. The numerical solutions are indicated by the solid lines and the asymptotic solutions (1.125) are indicated by the broken lines.
Figure 1.22 is for the case of suction, while Figure 1.23 is for the case of injection. Chaudhary and Merkin (1993) also obtained asymptotic solutions for x :>> 1 showing that 1 Tw(x) ..~ p r 2 ~ . . . ,
1 Ow(x) ~ ~r + . . .
(1.125)
F R E E C O N V E C T I O N OVER A VERTICAL FLAT PLATE
43
(b) 20
?
16
25 i I
12
20
.
I0
.
0
ll
15"
.
~
'
,
0.5 0'.0 015 1:0 15 In x
'i
2.0
0

,
"
0.5 0[0
0:5
~
'5
1.0 1. In x
210
Figure 1.23: Variation of (a) Tw(X), and (b) Ow(x), with x for P r  1 in the case o] uniform injection. The numerical solutions are indicated by the solid lines and the asymptotic solutions (1.126) are indicated by the broken lines.
for suction, and 1
1
~w(X) ~ (2Pr)~ x~ + . . . ,
1
1
Ow(x) ~ (2Pr)~ x~ + . . .
(1.126)
for injection. These solutions are also shown in Figures 1.22 and 1.23 (by broken lines), and we can see that the agreement between the numerical and asymptotic solutions is reasonably good over a large range of values of x. The results reviewed in this chapter have been for the free convective boundary layers adjacent to vertical flat plates with thermal conditions that are continuous and well defined. However, practical problems often involve wall conditions that are arbitrary and unknown a priori, and are to be found. A simple model has been proposed by Lee and Yovanovich (1991, 1992) to predict the heat transfer characteristics due to a vertical plate which is subject to a step change in the wall temperature and also when the surface heat flux variation is discretised into a number of step changes. The problem imposes a mathematical singularity and severe nonsimilar conditions at the wall. The analysis is based on the linearised approximations to the boundarylayer equations. The linearisation is performed by introducing an effective boundarylayer velocity, which is subsequently determined by relating the total thermal energy dissipated into the fluid to the effective kinetic energy of the fluid flow. The effective fluid velocity determined in this manner becomes analogous to an externally induced free stream velocity, thereby allowing the analysis to proceed in a way that is similar to a forced convection analysis. The validity and accuracy of the model proposed by Lee and Yovanovich (1991, 1992) was demonstrated by comparison with the known results from the open literature.
44
CONVECTIVE FLOWS
It was recognised from the outset by the pioneering workers in heat transfer that a characteristic common to most analytical studies on convective flow has been the neglect of all fluid property variations, except for the essential density difference which, in the absence of mass transfer, are a consequence of temperature gradients in the fluid. This greatly simplifies the analytical and experimental studies, since the number of variables are greatly reduced. However, in practice, experimental data usually exhibit considerable deviations from the analytical predictions due to, partially, the inadequacy of the constant fluid properties assumptions. Due to the importance of buoyancy convective flows with variable fluid properties in industrial applications, there has been much analytical and experimental work directed towards determining the effects of variable properties which are cited in the review paper by Kaka~ (1987) and in the book by Gersten and Herwig (1992).
Chapter 2
Mixed convection b o u n d a r y  l a y e r flow along a vertical flat plate 2.1
Introduction
Mixed convection flows, or combined forced and free convection flows, arise in many transport processes in engineering devices and in nature. These flows are characterised by the buoyancy parameter ~  RGr where Re is the Reynolds number, Gr e n is the Grashof number and n (> O) is a constant which depends on the flow configuration and the surface heating conditions. The mixed convection regime is generally defined as the range of/kmin ~ A ~ ~max, where )~min and )~max are the lower and the upper bounds of the regime of mixed convection flow, respectively. The parameter A provides a measure of the influence of the free convection in comparison with that of forced convection on the fluid flow. Outside the mixed convection region, ~min ~ ~ ~ ~max, either the pure forced convection or the pure free convection analysis can be used to describe accurately the flow or the temperature field. Forced convection is the dominant mode of transport of heat when ~~ Gr ~ 0, whereas free R e n ~0. Buoyconvection is the dominant mode when Gr + c~ , or alternatively 07ancy forces can enhance the surface heat transfer rate when they assist the forced flow, and vice versa. Buoyancy forces also play a significant role in the incipience of flow instabilities and they can be responsible for either delaying or speeding up the transition from laminar to turbulent flow.
2.2
Basic e q u a t i o n s
Consider an undisturbed uniform free stream of velocity Uoo at large distances flowing along a semiinfinite vertical fiat plate, which is placed in a viscous incompressible
46
C O N V E C T I V E FLOWS
U~,T~
Assisting flow
T,~ > Too q~, > 0 N~Ik
Tw>Too %>0 x
Opposing Flow U~,T~
Figure 2.1: Physical models and coordinate systems.
fluid of ambient temperature T ~ , see Figure 2.1. We assume that the plate is heated to a constant temperature Tw, or to a constant heat flux qw, where T~ > Too and qw > 0. Heat is supplied to the fluid by diffusion and convection from the plate and this heating gives rise to a buoyant body force. There are two cases to be considered, namely one when the plate extends vertically upwards and the other when it extends vertically downwards. In the first case the buoyancy force acts in the direction of the free stream (assisting flow), and in the second case, it acts in the opposite direction of the free stream (opposing flow). In both cases, near the leading edge, there is little chance for the heat from the plate to be taken into the fluid and a boundarylayer is formed chiefly by the retardation of the free stream, but the effect of the buoyancy force increases as the boundarylayer develops. In the absence of heat generation and viscous dissipation, the boundarylayer equations, under the Boussinesq approximation and for steady state flow conditions, are given by
Ou
+
i)v

o
Ou Ou 02u u~~x + V~yy  V ~ y2 =t=gZ (T  Too/ OT 07' v O~T U ~x + v Oy = P r Oy 2
(2.1)
(2.2) (2.3)
MIXED CONVECTION FLOW ALONG A VERTICAL FLAT PLATE
47
where in Equation (2.2) the + sign is taken when the plate is vertically upwards (assisting flow) and the  sign when the plate is vertically downwards (opposing flow). Equations (2.1)  (2.3) have to be solved subject to the boundary conditions" u  Uc~, uO, T Tw (CWT) , u~Uc~, 2.2.1
T   Too vO aogT qw ks (CHF) T+Too
Flat plate with a constant
on
x  0,
y# 0
on
y = 0,
x > 0
as
y+co,
x>0
(2.4)
wall temperature
S m a l l v a l u e s o f x ( 0
{>0
(2.18)
MIXED CONVECTION FLOW ALONG A VERTICAL FLAT PLATE
49
The form of these boundary conditions suggests expansions for f and 0 in a series for large values of ~ (>> 1) in the form
f  70 (~) + ~ 89 (~) + ~1 [F2 (~)ln~ + f2 (~)] +    ~0 (~) + ~  8 9 (~) + ~~ [G2 (~)~1~ + ~ (~)] + . . .
(2.19)
The terms which are O / \(~1) have been included due to the leading edge shift effect %
]
and the necessity for including logarithmic terms (eigenfunctions) in asymptotic expansions in boundarylayer theory was discussed by Stewartson (1957). On substituting the expansions (2.19) into Equations (2.16)  (2.18) we obtain 
iv
•lll + 3 f 0 f o  2 f o
2
l
0
fo(0)0, fo +0, "f Ill
1
If'
fo(0)0, 0 o  + 0 as

1
F~" + 3 f o F~ '
F~+0,
'_H
f l (0)  0, 01 +0 as
frog2 + G2  O,
F2(0)  0,
(2.20)
0o(0) 1
I
~o~ + 3foO'~ + 2f'o~ + YloO  0
+ 3fo?~ + foY~ + e~  o,
fl(0)  0, f l +2,
1 II "=_l pu 0 at 3fo0 o  0
+00=0,
(2.21)
01 (0)  0 r/+cr
prl G ~ + 370G[ + 4foG2'  0'0F2  0
F~(0)  0, G2+0
as
(2.22)
c2(0)  0 ~+c~
It should be noted that the solution (eigensolution) of the system of Equations (2.22) is given by, see Hieber (1974),
F2  Ao (~]'o 3fo)  AoFc (~),
G 2  Ao ~0'o  AoGc (~)
(2.23)
where A0 is, as yet, an undetermined constant. The equations for ]2 (~) and 02 (~) now become
:,,,
, f
_,,_,,_
(
,,
+ 3 f o f2  fo f2 + 02  4Ao 3 fo fo  2 f
;2)
 f l :tt fl
~;02 + 3 fo 02 + 4 fo 02  0o f2   f l 01  2 f'l ~1 ]" 12Ao0o fo f2(0)0, f2~O,
f2(0)0, 0 2  ~ 0 as
(2.24)
02(0)0 ~oe
The method of determining the constant A0 was described by Merkin (1969), who found A0 = 0.015643 for Pr  1. We can still add arbitrary multiples A1Fc (~) and AIGc (~) (A1 being another arbitrary constant) to any solution of the system of Equations (2.24) and this will still satisfy the required boundary conditions. The constant A1 can be determined if the fluid velocity and temperature profiles obtained
50
CONVECTIVE FLOWS
from the series (2.19) are compared with those obtained from the numerical solution of Equations (2.16)  (2.18) and Merkin (1969) obtained A1  0.03+0.01 for P r  1. The sets of Equations (2.20)  (2.24) were solved numerically by Merkin (1969), who found
1 1[ 
1 o.6422 + o.osao

_ _
+ o.o
o5
+ (0.0974  0.6422 A1) ~1 11[
q~ (~)  2  ~ (  z
__!
0.56 71 + 0.0712 ~ 2 __ 0.0089
I
" " "]
In ~ + 0.5671 A1 ~I + .  . ]
for ~ >> 1. Further, Merkin (1969) has matched the series (2.14) and (2.25) for small and large values of ~ by performing a numerical integration of the full boundarylayer Equations (2.1)  (2.4) using a method first proposed by Terrill (1960). Since the details of this method are well described by Merkin (1969, 1972) we do not repeat them here. On the other hand, Merkin (1969) has shown that in the case of opposing flow, the boundarylayer separates from the plate at the point ~  ~s  0.192357 for P r = 1, where 7~ (() + O,
q~ (() + 0.428
(2.26)
but d'rw _
~ co,
d~
dqw _
~ co
d~
(2.27)
which shows that the flow is singular at ( = (s The behaviour of rw (() and qw (~) near the separation point (  ~s  0.192357 is illustrated in Figure 2.2. We note that the structure of the boundarylayer in the vicinity of the fluid separation from the plate has been studied in detail both analytically and numerically by ttunt and Wilks (1980) and this discussion is presented in Section 2.3.
2.2.2
Flat plate with a constant
surface heat flux
We consider now the situation when the heat is supplied to the fluid flow by diffusion and convection due to a uniform heat flux qw at the plate. This heating, relative to the ambient temperature Too, gives rise to a buoyant body force which again aids or opposes the free stream. The boundarylayer equations which govern this problem are given by the Equations (2.1)  (2.3), along with the boundary conditions (2.4) for the CHF case. A dimensional analysis of Equations (2.1)  (2.3) leads naturally
MIXED C O N V E C T I O N F L O W A L O N G A V E R T I C A L FLAT P L A T E
(a)
51
(b) 0.12
~(~
0.55
~
~(~
0.08 
~
0.04 
0.51 0.47 
9
0.00 0.184 0.186 0.188 0.190 0.192
0.43
.
0.184 0.186 0.188 0.190 0.192
Figure 2.2: Variation of (a) Tw (~, and (b) qw (~), with ~ for Pr = 1 near the separation point ~  ~s  0.192357. w
to the following transformation:
N (2392 2q ) ~
9 2rT5
5kf,.,c~
1
1
9, ,7 ~, ( ~ ) ~
(2.28)

1
for small values of ~ (> 1), we introduce the transformation
T  Too  T*xgO
~ ,
 C2 ~ X5
(2,32)
52
CONVECTIVE FLOWS
where T.
__.
1
C1  (2454g~qwU3kf )
qw
kiCk'
1
C2 ( gflqw
(2.33)
lOkfu 2)
Equations (2.1)  (2.3) then become
ox + ~ : ~
 6 N
l O 20
PrO~2
 o + lO~
o(o~ 2
o~
(o7o~
o~ t8fN2
N
+ 10~
07 o~7) 0~ 0~0~ Of
0~0~
0~
 0
(2.34)
 0
(2.35)
with the boundary conditions (2.4) becoming i

0) o,
o,
0__/./
o~ ~ ~g~
,

00 0~
as
o+o
01
~+cc,
for
~>0
(2.36)
~ >0
A s s i s t i n g flow In this case we have to take the minus sign in Equation (2.29). This problem has been studied by Wilks (1974) who followed closely the method as described by Merkin (1969), and results were obtained again for Pr = 1. Thus, Wilks (1974) obtained results for the nondimensional skin friction and heat transfer parameters, which are given by 1
1
~
1 0.46960 + 5.14956~:  19.23852( 3 + . . . )
Qw (~)  (2~") ~ o(g,0) 1
 (2~) ~
1.54064  2.68850 ~"~ + 20.89185 g3 + . . . ) (2.37)
f o r ~ < < l , and
.547
2~
Qw
_
~~
1
_
_
1i~~
1 18168 .
+ 
0.13634~ ~ 
+ _
].
0.00995~ 6g + . . .
)
(2.38) for ~ >> 1. It is worth mentioning that in this problem the precise contribution of the leading edge shift (eigensolutions) is identically zero, see Wilks (1974), and therefore the series (2.38)is exact up to terms which are O ""(~]). Equations (2.29)  (2.31) and (2.34)  (2.36) were also solved numerically by Wilks (1974) using a technique which is an adaptation of the method employed by Terrill (1960) and Merkin (1969). Results for Tw and Qw were obtained
MIXED C O N V E C T I O N FLOW ALONG A VERTICAL FLAT PLATE
53
from the numerical integration of these equations for the case of aiding flow and they are presented as solid lines in Figure 2.3. The results given by the series (2.37) and (2.38) are also included (shown by the broken lines) in this figure. The velocity profiles at various stations along the plate are also given in Figure 2.4 for the case of assisting flow. A high degree of agreement between the threeterm series representations and the exact numerical solutions is noted from these figures. It is f
A
F
A
also seen that the Tw (~') and Qw (~) estimates can be employed over almost the N
2"
whole range of values of ~. Furthermore, the points at which the series (2.37) and (2.38) diverge from the correct solutions are such as to give us some confidence that straightforward extrapolations linking these two asymptotic series representations may well be sufficient for most practical purposes.
(b) 100 I
...........
100 [
. . . . . . . . . . . . .
1~1 1
0.1
! 0.01
,,
,
0.i
,
1
,
.
,, I0
I00
0.1 0.01
0.1
1
:
10
100
Figure 2.3: Variation of (a) Tw ( ~ , and (b) Qw (~), with ~ for Pr   l in the case of assisting flow. The numerical solutions are indicated by the solid lines and the series solutions (2.37), for ~ > 1, are indicated by the broken and dotted lines, respectively.
O p p o s i n g flow In this case Equations (2.29)  (2.31), with the sign + in Equation (2.29), were solved numerically by Wilks (1974) using the same method as that described by Merkin (1969) and the results reported are again for P r  1. The fluid velocity and the temperature profiles at ~ = 0 (initial point) and at ( = {s (separation point) are shown in Figure 2.5 and the behaviour of rw (~) and Qw @~) in the vicinity of the
54
CONVECTIVE FLOWS
(~)
(b) 1.2
0.20 e~
1.o
0.16
0.8 0.120.6 0.08
0.4
0.04 
0.2 0.0
0
i 1
. . . . .
0.00
i
2
3
~l
5
0
1
2
3
_
9
4
)
Figure 2.4" Velocity profiles, o:(~. ~ , f r o m the n u m e r i c a l solution f o r P r  1 in the case of assisting flow for (a) s m a l l values of ~ and (b) large values of ~.
4
1.0
i
,
0.8
3 .' ....
9
2
1
'
'
~
o: (~,~)
,,7
o,//, o~.//, 0.0 , 0
.1
(~,) , oo,
oo
,? ,,'/
/ 11
, (~.,)
,,7
//
/
1.0
j0.8
//,
oo
Figure 2.5" Velocity, ~o,
0
//
,,U ., 2
9
.. , 3
t~ .
, 4
..... .0.0 5
tem, ero,,re. O (~.,)
.ro~,e, ~ m
~he o,
m e r i c a l solution f o r P r  1 in the case of opposing flow at ~  0 (solid lines) and = G (broken lines).
5
MIXED CONVECTION FLOW ALONG A VERTICAL FLAT PLATE
55
separation point is also shown in Figure 2.6. It is thus concluded that v~
+ 0,
Qw
+ 0.951,
d(
~ c~,
d(
+ o0
(2.39)
as ~ + ~s, where ~s  0.141955. Following the same procedure as that proposed by Terrill (1960), Wilks (1974) has analysed numerically the behaviour of the skin friction and the heat transfer in the vicinity of the separation point ~s on a loglog scale and3 he concluded that both the skin friction and the heat transfer behave as ( ~ s  ~)5 close to the separation point ~  ~ . However, there is a discrepancy between this behaviour and the numerical results in the immediate vicinity of the separation point which may be attributed to accuracy limitations of the numerical solution.
(~)
(b) 0.2
1.05
0.1
1.oo
0.0 0.135
0.i37
0.i39
0.i41
01.'43
0.95 0.135
Variation of (a) Tw ( ~ , and (b) Q~ ( ~ , case of opposing flow.
Figure 2.6:
,, 0.137
0.139
0.141
0.143
with ~ for P r = 1 in the
A detailed analysis of the boundarylayer behaviour near the point of separation has been performed by Hunt and Wilks (1980) and this analysis is presented in the next section.
2.3
Behaviour
n e a r s e p a r a t i o n in m i x e d c o n v e c t i o n
In order to analyse the nature of the opposing fluid flow near the point of separation we introduce the following new variables, see Hunt and Wilks (1980), t
u u~,
,
y=Re~ vRe 89
,
~,
r ,
(2.40) O= TToo T*
56
CONVECTIVE FLOWS
where
T*
T*  Tw  Too (CWT), Equations (2.1)


(q~) ~
'
Re ~ (CHF)
(2.41)
(2.3) then become 0~
0~
o~ + ~
 o
(2.42)
u~~ + v O~ = O~2 =[=0
^og
og
(2.43)
1 o~g
U~x + 90~ = Pr O~
(2.44)
where the T signs correspond to the cases of CWT and CHF, respectively. The transformations appropriate to an initial profile displaying a doUble zero at the origin are defined as follows: A
1
(x)Z,
y
r/
A
r
3
~'
00([,77)
(2.45)
289

On substituting the scalings defined by the expressions (2.45) into Equations (2.42) (2.44) gives rise to the governing equations
f,,,_3ff,+2f~2TO_ ~(f,,Ofo~ f~'Of')
(2.46)
1 0 , _ 3 f 0 , _ ~(o, Of _f, O0)
(2.47)
which have to be solved subject to the boundary conditions (2.4) which become f(~,0)0, f'([,0)0 ]( 0(~,0)  1 (CWT), 0'([,0)  1 ( C H F ) ) for f'~l, 0~0 as r/>cx~, ~ > 0
~>0
(2.48)
These equations were studied by Hunt and Wilks (1980) who obtained, for the skin friction and the heat transfer parameters, near the separation point ~ = ~s(Pr) the following expressions:
for the CWT case: r~({)  25
q~(~) 
=
({, 0) = 2~
2a10 ln{ + 2all + 2a12 In Iln{I + 2alain~=: + . . .
2~ ~o' (~, o)
11 [bl  ~K~(O)bl ( 2alo ln~ +
,n,,nr=
T~ 1000). For small values of ,~z (< 0.1), CI and N u decrease monotonically to a constant value, whilst for intermediate and large values of Az (0.4  40) they decrease to a minimum value at some value of Rex as Az increases.
0.7 2.4
fg(o) 0.6
2.0
9~ r 
1.6
1
A~:= 0.4
0.5
A~ =0.1
],,,.. A~ : 0 (Forced Convection)
O'4 t ,\~ = 0 (Forced Convection) 0.4
0.1
.
1
~
10
,,',
Rex
100
,
1000
0.1
i
1'0
Rex
1()0
1000
Figure 2.8" Variation of (a) f~'(O), and (b) Oto(O), with nez for Pr  0.7 and Az  0  1 in the case of assisting flow.
The variation of Cf and N u in the case of opposing flow is shown in Figure 2.10 for some small values of Az. It is seen that when Ax increases the curves for both Cf and N u show a maxima before decreasing to the boundarylayer values at large values of Rex. It should be noted that when using this theory, Afzal and Banthiya (1977) have obtained for the case of pure free convection flow (,kz + oc) and in the boundary
63
MIXED CONVECTION FLOW ALONG A VERTICAL FLAT PLATE
(~)
(b) 1.36 ] .
28
A~= 40
lk
1.281 \ o~(o~.2o~
24
f~'(o) 20
A~ = 40
1.~176 k
16
0.92 t
~_~
A~  10
12
A. 10 o
~=4 4
0.1
'
i
16o'
10
I000
A~=4
o.1
i
1"0
Re x
F i g u r e 2.9: Ax  4  4 0
'
I{}0
1000
Rex
Variation of (a) f~'(O), and (b) fro(O), with Re~ for Pr  0.7 and in the case of assisting flow.
(b)
(~)
0.46
0.6
0.5
~
~
~
=o
0.44
o~(o)
f~'(O) ~
0.4
x
= 0.04
~A~
= 0.1
A~ = 0
0.40
0.3
0.2
0.1
0.42
0.38
'
io
1oo
Re~:
1ooo
0.36
"'"~
1
= 0.1
~. . . . . .
lo
16o
Rex
F i g u r e 2.10: Variation of (a) f~'(O), and (b) 0~o(0), with R e , for Pr  0.7 and Ax = 0  0.1 in the case of opposing flow.
i000
64
CONVECTIVE FLOWS
layer regime (~0 + 0) the asymptotic values, C f R e ~  2.3476 A4,
, = 0.3394 s
(2.70a)
Re~
for P r  0.7, while the exact values obtained by Oosthuizen and Hart (1973) are given by C/Re~
 1.9195 A4,
~ = 0.3532 A4
(2.705)
Re~
However, for pure forced convection flow (s C I R e ~  0.6641,
 0) we have, see Schlichting (1968), Nu
~ = 0.2927
(2.70c)
Re~
It is clearly seen that the two sets of numerical results (2.70a,c) are different. The result for the heat transfer is underestimated by about 4% and that for the skin friction is overestimated by about 22%. However, the analysis by Afzal and Banthiya (1977) predicts the heat transfer very accurately over the entire range of values of Ax, spanning from pure forced convection to pure free convection, while the skin friction is predicted accurately for values of Ax beginning from pure forced convection to moderately large free convection flows. When the buoyancy effects are sufficiently strong, the skin friction is overestimated. Afzal and Banthiya (1977) claimed that this discrepancy may be attributed to the fact that the leading term for f in Equation (2.61) should be ~ for the buoyancy dominated (free convection) limit rather than ~. It is worth pointing out the existence of an excellent paper by Hussain and Afzal (1988) on mixed convection boundarylayer on a vertical flat plate for both the cases of buoyancy assisting and buoyancy opposing flow situations using a computer extension of a perturbation series. The first thirteen terms for the uniform wall temperature case and the first ten terms for the uniform heat flux case were computed. It was shown that the results of the direct coordinate expansion when transformed suitably by the Euler transformation and an extrapolation of the DombSykes plots predict the exact results which are correct to threedigit accuracy for all values of the streamwise coordinate, ~, along the plate.
2.5
E f f e c t of P r a n d t l n u m b e r on t h e m i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow a l o n g a v e r t i c a l p l a t e w i t h a c o n s t a n t wall t e m p e r a t u r e
Consider a vertical semiinfinite flat plate at a constant temperature T~ which is placed in a viscous and incompressible fluid at the ambient temperature Too flowing
MIXED C O N V E C T I O N F L O W ALONG A VERTICAL FLAT P L A T E
65
vertically upward with the uniform velocity U~. It is assumed that Tw > Too (heated plate) or Tw < Too (cold plate). This problem has been studied by Lin and Chen (1987, 1988) who obtained numerical solutions that are uniformly valid over the entire region of mixed convection flows for fluids with Prandtl numbers in the range 0.001 ~ Pr ~ 10000 using a new mixed convection parameter r which replaces the traditional Richardson number, Ax. Both the cases of buoyancy assisting and opposing flow conditions were treated. Thus, the new mixed convection parameter proposed is given by 1

~ (a2Re
with
Pr al  1 + P r '
where
(2.71)
)
Rax
Pr Pr)89
a2  (1 +
(2.72)
is the local Rayleigh number defined as
Ra~: 
I T!
(2.73)
c~fv
The nondimensional parameter ~ not only serves as an index of the relative contributions of forced convection and free convection flows but also represents a stretched 1 streamwise coordinate x since ~ is proportional to x Z. It is also of some importance
Re~Ra~pr~ (Ra~Pr)~(Re~Pr)89 1
to note that the parameter ~ can readily be reduced to
1
and
for
very large and very small values of Pr, respectively. These two nondimensional groups were defined by Bejan (1984) using a scale analysis to indicate the relative importance of the forced and the free convection flows, the former for large and the latter for small values of the P r a n d t l number. Additionally, to facilitate the solution procedure, Lin and Chen (1987, 1988) introduced the following new variable: ~(x) 
r
(2.74)
1+~
which maps the entire mixed convection domain from 0 ~ ~ < cx) to 0 ~ ( ~ 1. Further, they also defined the variables: ( y) 77 A1 x '
r  ~f)~lf(~,~7),
0(~,r]) 
where
T  Too
IATI
(2.75)
1
1( =
1
(1 + r =
(alRax)~
(2.76)
66
CONVECTIVE FLOWS
Using these variables, Equations (2.1) (2.3) can be written as P ~ I ' " + i1 (2 + ~) ff,,  ~1 f ,2 J: (1 + P~)r
 ~1 (1  ~) ( f , Of' o~
0" + ~1 (2 + ~ ) f O ,
~1

(1

~. , ,~0 f ) (277) "
() ( f , ~OO_o, O f )
(2.78)
where again the J: signs in Equation (2.77) apply for the cases of assisting and opposing flows. Equations (2.77) and (2.78) have to be solved subject to the boundary conditions (2.4) which become ( 2 + ~ ) f ( ~ , O )  ~ ( 1  ~ ) O  ( (ol~ , O ) , 0(~,0)1 for 1 f '  + ( l + P r ) ~ ( 1  ~ ) 2, 0+0 as 7/+oc, ~>~0
f'(~,O)O,
~>0
(2.79) It should be noted that for the special case of pure forced convection flow ~ = 0 (Rax = 0), Equations (2.77)  (2.79) reduce to the following similarity form 1 Pr fm + ~f f
,,
0"
_0,
f(0)=0, f'(0)=0, 1 f ' + ( l + P r ) ~ , 0+0
1 , + ~fO 0
(2.80)
0(0)=1
as rl + oo
whilst for the case of pure free convection limit (Rex = 0) where ~ = 1, Equations (2.77) (2.79) become
Pr f'" + 3f f , , _ 89
3 ! 0" + gfO  0
+ (1 + Pr)O  O,
f (o) = o,
f'~O,
f ' ( o ) = o,
o(o) : 1
0~0
77~
as
(2.81)
The local skin friction coefficient and the local Nusselt number, as given by Equation (2.69a), can now be expressed as 1
C f Re~  2
Nu
2
al
(1~) 3 f"(~, 0),
(o2n~)~
1
=  ~0'(~,
1~
O)
(2.82a)
for forced convection dominated regime (0 ~ ~ < 1) and
Nu
1
,
=   ; 0 (~, O)
(2.82b)
(ol Rax) for free convection dominated regime (0 < ~ ~ 1). 1
Tile variations of C I Re~ and
Nu. i with the mixed convection parameter
(a2 Rex )
are shown in Figure 2.11 for both assisting and opposing flow conditions and different
MIXED CONVECTION FLOW ALONG A VERTICAL FLAT PLATE
67
(b) 10
10
...............
1
CiRd
Assisting
~
.~
F l o ~
Pr = 0.001  10000
1
Assisting Flo~ Pr = 0.001, 0.01, 0.1, / ~ . ~
1,1
Opposin~g~ . Pr = 0.001, Flow 0.01, 1, 10 0.1
9
0.1
,
',
9
9 ' , ' , [ "
,
. . . . . . .
r
10
0.1'
.
.
.
0.1
.
.
.
.
1
.... ' ' r
9
9
5
Figure 2.11 Variation of (a) the skin friction coefficient, and (b) the local Nusselt number, with ~ for different values of Pr and in the cases of assisting flow and opposing flow.
values of Pr. As expected, the skin friction increases as the value of r increases for 1
assisting flows. For ~ ~< 0.2, the value of Cf Re~ for all values of P r are very close to a constant 0.6641 for pure forced convection flow (~  0), see transformations (2.71). 1 w
For opposing flows, the value of C:f Re~ for each value of P r decreases from 0.6641 to a small positive value as the value of ~ increases from 0 to a critical value near which the wall skin friction rapidly decreases and the numerical solution diverges. The boundarylayer approximation breaks down at this critical point. Further, we see from Figure 2.11(b) that the rate of heat transfer increases as ~ increases for buoyancy assisting flow and decreases for buoyancy opposing flow, this decrease being very sharp when the point of breakdown of the boundarylayer approximation is reached. Also, the variation of N.____~_with P r over the range 0.001 ~< P r ~ 10000
Ra~
for the buoyancy assisting flow case is depicted in Figure 2.12 for some values of the variable ~. The figure shows that, for each value of ~,  ~ increases approximately
Ra~
linearly with P r for P r ~ 100. Finally, Tables 2.3 and 2.4 compare the numerical results for the local heat transfer rate in both the cases of pure free convection (~  1 ) a n d pure forced convection (~  0) limits. Further, Table 2.4 contains results given from the following
68
CONVECTIVE
Nuru~2
FLOWS
___ ~ = 0 . 2
Ra~ 1.
~.
~ = 0.3
0.5
0.2 0.1 0.;1
1
Pr
1;0
' 10{)00
Figure 2.12: Variation of the local Nusselt number with Pr in the case of assisting J~OW.
Table 2.3: Values of ~
for pure free convection (~  1) as obtained by different
Ra2
authors. Ostrach (1952)
Ede (1967)
Kuiken (1968)
Lin and Chen (1988) 0.10494 0.18017 0.28925 0,40087 0,45793 0.46425 0.48985 0.49854 0.50125
i
0,162
0.180209
0.4010
100 1000 10000
0.401029 0.458276 0.4650 0.464 0.4899 , 0.490012 0.4987 0.499 0.501431
0.4185 0.4658 0.49O04 0.49863
c o r r e l a t i o n e q u a t i o n , as p r o p o s e d by Lin a n d C h e n (1988), 1
Nu
0.33872 P r ~1
Re5
1
(2.83)
(0.05 + P r ) ~
It c a n be seen t h a t the n u m e r i c a l results a n d those given by r e l a t i o n (2.83) are in excellent a g r e e m e n t a n d t h a t t h e c o r r e l a t e d results are w i t h i n a b o u t 5% error for values of t h e P r a n d t l n u m b e r in t h e r a n g e 0.001 ~ P r 0 (assisting flow) or A < 0 ( o p p o s i n g flow). These equations have to be solved subject to the boundary conditions u0,
vO,
OT _ qw(X) 0~ T+O
u~U(x),
on as
y  O, y+co,
x > 0 x>0
(2 88) "
Equations (2.84) (2.86), subject to the boundary conditions (2.88), admit a similarity solution if U(x) and qw(X) take the following forms: U(x)

X TM,
(2.89a)
qw(X)  x 1(5m3)
and then we have r
1 (m~l)
f(r/),
T
x2m1 O(r/), ~7YX~1 (ml)
(2.89b)
Using the transformations (2.89), Equations (2.84) (2.86) become f ' " + ~ 1 (m + 1) f f " + r e ( l 1 0"+ p;
1 (m+l)f
f,2 ) + A 0  0
Ot + ( 1 _ 2m) f ' O 
(2.90) 0
(2.91)
and the boundary conditions (2.88) reduce to f(0),
0, +1, f
f'(0) e 0, 0 ' ( 0 )   1 0+0 as ~  + ~
(2.92)
Equations (2.90)  (2.92) were studied by Merkin and Mahmood (1989) for the Prandtl number unity, i.e. P r = 1. In the following paper by Merkin et al. (1991) they extended this study to include the behaviour of the similarity solution of these equations for both very small and very large values of Pr. They found, by integrating once Equation (2.91) subject to the boundary conditions (2.92), (5m  1)
f'Od~ f0 ~
2
Pr
(2.93)
which shows that Equations (2.90) and (2.91) possess a solution if only if rn > 51and therefore we assume throughout the remainder of this section that this condition is satisfied.
MIXED CONVECTION FLOW ALONG A VERTICAL FLAT PLATE 2.6.1
L a r g e v a l u e s of P r
71
(>> 1)
The problem of free convection boundarylayer over a vertical flat plate at very large and very small values of P r has been studied by Kuiken (1968). In that problem he found that the boundarylayer divides into two distinct regions, namely a very thin inner thermal region and an outer region which is at the same temperature as the ambient fluid. This concept applies also to the present problem. A balance of the terms in Equations (2.90) and (2.91) suggests that in the inner layer we must scale the equations as follows: 2
1
f  Pr~F(~),
0
1
Pr~g(~),
(2.94)
~  Pr~rl
where F and H are given by the equations F ' " + P r 1 ( 1 ( m + 1) F F "  m F '2 \
1
2
 0
(2.95)
1 (m + 1 ) F H ' + (1  2 m ) F ' H  0 H " + ~
(2.96)
+ Pr~m
+ Pr~AH
and primes now denote differentiation with respect to ~. These equations have to be solved subject to the boundary conditions (2.92) which become F(0)0,
F'(0)0,
H'(0)1
(2.97)
and the remaining conditions which are valid as ~ + oo are to be found through a matching with the solution in the outer region. Equations (2.95) and (2.96) suggest that we seek a solution, when P r >> 1, of the following form: F  Fo(~) + P r  8 9 H = Ho(() + Pr 89
(2.98)
+... +...
where Fo, F1 and Ho are given by the equations F o It!
=0,
~1i l l ! 11 +m=0
(2.99)
Ho,, +21 (m + 1)FoH~ + (1  2m)F~Ho  0
(2.100)
The boundary conditions (2.97) then become Fo(0)  0,
F~(0)  0,
FI(0) = 0,
F~(0)  0,
H~(0) =  1
(2.101)
The functions F0 and F1 are easily seen to be given by Fo = 2
FI =
al ~2 _ 6 ~ 3
(2.102)
72
CONVECTIVE FLOWS
where a0 and al are arbitrary constants which have to be determined by matching the solution (2.98) to the solution in the outer region. Using expressions (2.102), Equation (2.100) becomes 1
H~' + ~ (rn + 1)ao~2H~ + (1  2rn)ao~Ho  0
(2.103)
Since in the outer region of the boundarylayer the fluid is isothermal, then 0 = 0 in this outer region and the matching temperature yields H0(oo) = 0. Then Equation (2.103) has a solution which can be expressed in terms of the confluent hypergeometric function, b/, as follows:
Ho
(
12 )  } F [ ~ ( 5 mm+i  : ) ] e _ S l / l [ 2 ( 5 m  1 ) 2 ]s a0 (m + 1) 3F (2) 3 rn + 1 ;3;
(2.104)
where F is the gamma function and s   ~ a o ( m + 1)~ 3. It should be noted that Equation (2.103) requires a0 to be positive, which we will show to be necessary when m > g .1 On the other hand, the governing Equation (2.90) in the outer boundarylayer region, where 0  0, reduces to the wellknown FalknerSkan equation, see Falkner and Skan ( 1931 ), f , , 1+ ~ ( m + l )
f f,,+ m
(f,2) 10
(2.105a)
and the boundary condition (2.92) requires that f' + 1 as
77 + co
(2.105b)
Thus in order to match with the solution in the inner region, we require that
f~(_~r/2
m rl3 "at...) t Pr g (air/2 + . . . ) '~
(2.:0Sc)
for small values of ~ ( 0 and the assisting or opposing flow cases, determined from the free stream (2.138) are given by 10po
p Oz
=
W
(2.140)
aS
y + oo,
x>0
the flow variables (u, v, w , T ) has been dropped 4 signs in Equation (2.137) again designate the respectively. The pressure terms P0 and pl are conditions and on using Equations (2.137) and
dW dz'
~
Pl _ U 2 + w ~dU p dz
(2.141)
Next, Ridha (1996) has shown that Equations (2.136)  (2.140) admit a similarity solution on introducing the stream functions r and r as well as an appropriate
MIXED C O N V E C T I O N F L O W ALONG A VERTICAL FLAT PLATE
81
transformation of the variables, namely uN,
0r
v
(0r
~~+r
)
,
0r
w~~ 1
r 
l+m
1
1
T
(2.142)
r =
Too  T* (})2m1 0(77/,
m1
77 = y (12~v~U~)~ (})
2
With these variables, Equations (2.136)  (2.139) reduce to the form:
S'" + [(2  / ~ ) h + h'" + [(2  ~) h +
S]S" +/~
S] h"
"[1  S '2)" +
+ [2 (1  / ~ ) f ' 
(2
Z) 0
0
(2.143)
(2  / ~ ) h'] h' = fl (3~  4) (2.144) 4 ( 2  fl)
P1r 0" + [(2  fl) h + s] 0 I  ( 3 f ~  2 )
S'00
(2.145)
and the boundary conditions (2.140) become: f(0)=0, h(0)=0, f'(0)=0, h'(0)=0 0 ( 0 ) = 1 (VWT), 0'(0)=1 (VHF) f ' + 1, h ~  2(2~ f~), 0+0 as ~+c~
(2.146)
where fl is the FalknerSkan parameter and A is the mixed convection parameter, and these quantities are defined as
2m  ~ 1 + m'
~
Gr Re 2
(2.147)
It should be noted that the free convection results may be obtained when considering ;~ + c~. To obtain this asymptotic case, we may use the following transformation: S
A 88
h   A 88
0 = 0"(~),
~? A 88
(2.148)
which is similar to the one used by Mahmood and Merkin (1988) for the axisymmetric mixed convection boundarylayer along a vertical circular cylinder. Equations (2.143)  (2.146) were solved numerically by Ridha (1996) for P r  1, f~ _ 2 (uniform wall temperature), ~  3 (uniform wall heat flux) and fl = 1 (threedimensional stagnation point). However, the results presented here are only for the case of a prescribed wall temperature distribution (~  ]). Before presenting these results, it is interesting to note that Equations (2.143)  (2.146) possess dual solutions for both the parameters fl and A, which are referred to as the upper (U) and the lower (L) branch solutions according to whether the skin friction S"(O) has, for a given value of fl or A, the higher or lower value, respectively. Therefore, according to the terminology used by Ridha (1996) it is denoted, for a given fl the upper branch
82
C O N V E C T I V E FLOWS
(U) solution, the upper branch solution obtained when varying A by (UU) and the corresponding lower branch solution by (UL), with the first letter referring to the /~ upper branch solution. Likewise, for a given /~ the lower branch (L) solution, it is denoted by the upper and lower branch solutions on varying A by (LU) and (LL), respectively. However, it is worth mentioning that Ridha (1996) has shown for a number of twodimensional mixed convection examples that dual solutions are associated not only with opposing flow situations, but they exist also for assisting flow regimes. He also showed that dual solutions do not always terminate in the singularity as A + 0 and a similar tendency was also observed for the present threedimensional fluid flow case as described by Equations (2.143)  (2.146). Some characteristic results are given in Table 2.5 for P r  1 when A  As, the values of A where f"(0)  0 and the boundarylayer separates, location of the bifurcation point A  Ac or the critical point, where the dual solutions branch out and beyond which no solutions are obtained, for the singularity point A  At, where the (UL, LL) lower branch solutions may terminate and also for A  A0, where the wall heat flux 0~(0) becomes zero. We notice from this table that separation takes place for smaller values of A on the (LU) branch solutions than for the (UU) branch solutions. Also, the values of A (  Ac) are smaller for the (L) branch solutions than for the (U) branch solutions. The same applies for values of A (= A0) on the (LL) and the (UL) branch solutions. Table 2.5: Characteristic results for the separation, bifurcation, zero wall heat flux and terminal singularity points for the case of the three wall temperature distributions.
Separation 1 Bifurcation I
2i 1
Irmina, ....
Point, f" (0) = 0 or Critical Point Singularity Point UU, A~ LU, A~ U, A~ L, A~ UL, A~ LL, At 1.1589 0.9804 I1.5356 0.0145 1.2620 I0.0390 1.3589 0.1904 1.7510 1.4875 2.1230 2.5155 2.1386 1.9924 2.5640 2.3260,
i

Zero Wall Heat Flux, O'(0) = 0 UL, Ao LL, Ao 0.9950 1.9427
0.9639 1.8297
Variations of the main and secondary skin friction coefficients, f"(0) and h"(0), as well as for the heat transfer parameter, 0'(0), are shown in Figure 2.17 for the case of a prescribed wall temperature distribution and P r = 1. It can be seen from Figure 2.17(a) that, in the assisting flow case, there exists a value of the buoyancy parameter A = Ab, say, in the vicinity of which the (UU) and (LU) curves of f"(0) cross each other. For A > Ab, f"(0) pertaining to the former solutions has a lower value than that belonging to latter solutions; in the case of the prescribed surface temperature this value is Ab ~ 0.3 at /~  32 which increases up to Ab "~ 1.25 for fl = 1. Of course, the reverse situation takes place for A < ~b but the significance of such a point is not clear yet.
MIXED C O N V E C T I O N F L O W A L O N G A V E R T I C A L FLAT P L A T E
83
(~) 7 / ///
6
f"(O)
/
5
//
/
// / /
/ /
4
o
,i
3 2 1
0 1 2.7
I
0.0
2:5
5:0
(b) 0.50
3.
(c) 0'(0)
....................
""
[ "' "~"
_o.5o1 i 0.75 2.6
1().0
1.5
.i] ~.,.,., ,. , , , ''"
7:5
2. . . . .
0.25 h"(o) o.oo     '  ~ ~ 0.25
A
'0.0
o.ol/, 1 V~(
i
~.~a~ ("itS:
~...=............................... ~.'
"''~""'',,~.,,..~.~, \\\~:. 2:5
,,
5:d
7[5
10.0 1.52.6
0.0
2.6
Figure 2.17: Variation of (a) f"(O), (b) h"(O), and (c) 0'(0), with )~ for P r  1 in the case of a prescribed wall temperature. The upper and lower branch solutions are indicated by the solid and broken lines, respectively.
Further, Figure 2.17(b) shows that the secondary skin friction h"(0), i.e. the skin friction in the x direction, takes larger values for the (UL, LL) solutions. This should not be taken to have a stabilising effect on the boundarylayer since the secondary velocity profile hi07) undergoes a reversal in the flow direction. This is expected to influence the flow stability, which depends through a threedimensional boundarylayer upon the secondary as well as on the main flow. As regards the heat transfer rate, we see from Figure 2.17(c) that it decreases towards its zero value after
84
CONVECTIVE FLOWS
bypassing the critical point Ac as )~ increases within the opposing flow regime and always for the (UL, LL) branch solutions. Therefore, the solutions have no longer any physical meaning. Finally, Figure 2.18 shows the fluid velocities and the temperature profiles for fl  32 and P r  1 in the case of a prescribed surface temperature. Both the upper branch (left) and the lower branch (right) solutions are given. It is seen that the main fluid velocity ft(u ) and temperature 0(r/) profiles display similar trends for both branches, with the lower branch giving rise to a thicker boundarylayer. In contrast, the secondary ht(~/) fluid velocity profiles undergo a single reversal in the (U) solutions, while a double reversal is observed in the (L) solutions. Furthermore, we see that the main fluid velocity undergoes a double reversal of direction for both the (UL,LL) solutions, with the reversal flow undershooting the value of  1 for some values of A < 0. Then, as we proceed from the wall out into the boundarylayer it is observed that the secondary flow undergoes a complex sequence of flow reversals for both the upper and the lower branch solutions which are characterised by a twin positive peak profile. Other very interesting flows properties have been described by Ridha (1996, 1997) and the interested reader should consult these papers for further details.
MIXED CONVECTION
FLOW ALONG A VERTICAL FLAT PLATE
Upper Branch . . . . . . . . . 15 10
Lower Branch 25
i
20
:
10
12 lo
5
5
0
1.0
0.0
0.5
0.5
1.0
0
1.0
0.0 f'(r/)
0.5
f'07) 25 15
i
20
,
,.
llll
~
' i
l.O
....
~11~
15
10
10
o
'
0.5
0.2
0.0 0.2 h'(~) ii
,
i
i
0.4
,,,
47
o
0.40.2
5
1
ii
20
12 13
15
1o
0.2 il
i,
0.4 
10
i
o.o
0.0 if(r])
0.2
o.4 o.6 e(,7)
o.s
1.o
0.0
9
0.2
i
014 016 0:8
1.0
Figure 2.18" Velocity, f ' ( y ) and h'(~), and temperature, 0(~), profiles for ~  2 and P r = 1 with A = 0, 1.2, 1.5356, 1.3,  1 , 0.8, 0.6, 0.5, 0.4, 0.3, 0.2, 0.07, 0.056 pertaining to the upper branch solutions (left) and A  O,  1 , 1.262,  1 , 0.8, 0.6, 0.4, 0.2, 0.1, 0.05, 0.03, 0.024, 0.021 pertaining to the lower branch solutions (right). Values of A corresponding to the respective profiles are as per order of numbering.
85
Chapter 3
Free and m i x e d convection boundarylayer flow past inclined and horizontal plates 3.1
Introduction
Free and mixed convection flow adjacent to inclined and horizontal surfaces bounded by an extensive body of fluid are of considerable importance in micrometeorological and industrial applications. Stewartson (1958) and Gill et al. (1965) were the first to give a theoretical description of the boundarylayer flow over a horizontal surface under the action of the buoyancy force. Since there is no component of the buoyancy force along the surface, the accelerating flow must be driven indirectly by a buoyancy induced pressure gradient. Qualitative confirmation of this model is proved by the experiments of Rotem and Claassen (1969). The flow below the heated plate cannot be described on the basis of boundarylayer theory. Stewartson's theory stipulates that the boundarylayer solution is not applicable to the region near the central line of the plate where the flow will turn and proceed to feed a 'theoretical jet' above the plate if the plate is not too wide. On the other hand, if the experimental plate is sufficiently wide, it has been observed by Rotem and Claassen (1969) that the gravitationally unstable layer will separate from the heated plate and give rise to typical eddying convection well ahead of this axis of symmetry. Another type of important convective heat transfer problem is the free and mixed convection boundarylayer flow near a flat plate which is inclined at a small arbitrary angle to the horizontal or vertical. Jones (1973) appears to be the first who has theoretically studied the free convection boundarylayer near a flat plate at small inclinations to the horizontal by taking into account both the parallel and the normal to the plate temperature gradients which drive the fluid flow and both positive and negative inclinations of the plate were considered. When the inclination of the
88
CONVECTIVE FLOWS
plate is positive, both of the mechanisms which drive the flow produce favourable effective pressure gradients, so that the fluid continues to be accelerated along the plate to a final state, far from the leading edge, which is described by the classical free convection boundarylayer solution over a vertical flat plate. For negative inclinations, although the pressure gradient associated with the processes remains favourable, separation of the boundarylayer from the plate eventually occurs, since the buoyancy force now opposes the motion. Important contributions to these convective flow configurations have also been made by several authors, but notably by Yu and Lin (1988), Schneider (1995), Vmemura and Law (1990), Weidman and Amberg (1996), etc. 3.2
Basic equations
Consider the steady free convection flow of a viscous incompressible fluid over a semiinfinite flat plate which is inclined at an angle ~ to the horizontal, see Figure 3.1. We assume that the plate is maintained at the constant temperature Tw and the ambient fluid has the uniform temperature Too, where Tw > Too. For this configuration, with the assumption that the Boussinesq approximation is valid, the basic conservation Equations (I.1)  (I.3) can be written as follows: 0~ 0v + 0~ 0~ 0~
(~)
.

=
(3.1)
o
10~ Jr u
Too /
(02~
(b)
+
02~)
+ gfl ( T  Too)sin 9~
(3.2)
~ g
T~
Figure 3.1: Physical models and coordinate systems for (a) positive inclination and (b) negative inclination.
FREE AND MIXED CONVECTION FLOW c3~
OV
+
o~
10p

o~
(c32V
+
+
89 02V)
+ gfl (T  Too)cos ~o
~, (o~~ o~'~
~b~ +~0~ = P /
(3.3) (3.4)
0~ ~ + ov ~ )
where Cartesian coordinates 5 and y are taken along and normal to the plate, respectively, with the origin at the leading edge, and g and ~ are the fluid velocity components along the 5 and yaxes, respectively. When the inclination of the plate is positive then 99 > 0, while qo < 0 for a negative inclination of the plate. Equations (3.1)  (3.4) have to be solved subject to the boundary conditions: 0, g0, TToo 0, g0, TTw g+0, T+Tcr P~Pcr
~0,
3.3
Free convection small inclinations
over
an
on on as
50, ~0, ~+c~,
~r 5>0 ~>0
isothermal
flat
(3.5)
plate
at
To solve this problem we introduce the nondimensional variables xT, A
P 
/2
~
y~,
[ P  Poo + pgl
u~,
v~A
(~sin ~o+ ~'cos 99)],
0  TToOzXT
(3.6)
Equations (3.1) (3.4) then become O~
O~
0
(3.7)
+ N
O~
O~
~~ +~o~ 0~
0~"
A0~
Off
A
.. O0
02~
 0~ + ~ 02~
02~ + ~
+ ( Grtan~o) O
02~
A
(3.8) (3.9)
o~ + ~~ + g~ + c ~ o
A
O0 =
(3.10)
~~ + Vo~
and the boundary conditions (3.5) are now given by A
0, ~0, 00 0, 90, 01 +0, 9+0, 0+0, p  ~ 0 A
on on as
~0, y0, y+c~,
A
yr ~>0 x>0
(3.11)
Further, we define the following boundarylayer variables A
x = x,
yGray,
A
u
~
Gr~u,
1
v  Grg~,
00,
p  G r  ~
(3.12)
90
C O N V E C T I V E FLOWS
Substituting expressions (3.12) into Equations (3.7)  (3.10), and ignoring terms which are O
Grg
relative to those retained in the limit Gr + oo, we obtain the
following boundarylayer equations for the problem under consideration:
0r 0 2~2
0~2 0 2@
Op
Oy OzOy
Ox Oy 2 = Ox +
+ AO
op
0= 0r Oy Ox
0 3~/)
(3.14)
+0 Oy 1 020 P r Oy 2
0r Ox Oy
(3.13)
(3.15)
where r is the nondimensional stream function defined by Equation (1.18), A is the inclination parameter which is defined as 1
A = Gr~ tan 7)
(3.16)
and A > 0 for positive inclinations and A < 0 for negative inclinations. Also, the boundary conditions (3.11) become r ~Oy
0~ = 0 Oy , 0+0,
01
on
y0,
x>0
p+O
as
y+oo,
x>O
(3.17)
It should be noted that the range of the inclination angle ~v considered here is such that ~o  O
Gr~
with Gr >> 1 and therefore A is O(1). This implies
that the buoyancy force term in Equation (3.13) is formally comparable with the induced pressure gradient along the plate. Also, we see that the horizontal flat plate problem corresponds to A = 0, while the vertical plate problem corresponds to A ~ cxD, in which case the scalings used in expressions (3.12) are inappropriate. It is also important to point out that it was originally established by Stewartson (1958), and later corrected by Gill et al. (1965), that a boundarylayer solution does not exist for the fluid flow below a heated, or above a cooled, isothermal horizontal flat plate. Therefore, we present here results only for the case of an inclined flat plate which is heated on the upper surface. Equations (3.13)  (3.15), which are subject to the boundary conditions (3.17) were first studied by Jones (1973), who obtained solutions which are valid near to the leading edge of the plate, small values of x, and far downstream, large values of x. These solutions were then matched by solving the full boundarylayer Equations (3.13)  (3.15) numerically using the Chebyshev polynomial method. 3.3.1
S m a l l v a l u e s o f x (0
(3.29)
x>0
We now develop series solutions of the problem defined by Equations (3.27)  (3.29) in the form
f (x, ~  fo (~ + x~ f~ (~ + x~ f2 (~ +...
(3.30)
(X, ~  no (~~)4x3hx (~')~x~h2 (~")q... forx>> 1, where (f0, ho) and ( ~ , h l ) are given by the following ordinary differential equations: 4~" + 3fo~'  2 ~ 2 + 4Ah~  0,
4h~' + 3Prfoh~  0
fo(0)0, f~(0)0, h ~ ( 0 )  I f~0, h o  ~ 0 , h ~  ~ 0 as r j ~ ~
(3.31)
4f~" + 3 l o f t '  fDf~ + 4Ah~ = ho  ~h~ +
fl(0)0,
f~ +0,
+
f~(0)0,
hi +0;
h~ 40
o
(3
h1(0)0
as r/+cx:)
However, an eigensolution in addition to the forced terms in the series (3.30) should be included and this appears in the term which is O (x 1). If we assume this to be the case here, then for Pr = 0.72 and A = 1, Jones (1973) found the following expressions:
~(~) 
~ [0.95600 + 0.6516 ~~ + o (xl)]
q~(~)  ~  ~ [0.35682 + o (~~)]
for x >> 1 since hi ~ 0, as can be seen from the system of Equations (3.32).
(3.33)
94
C O N V E C T I V E FLOWS
(a)
(b) 2.4
2.0
1.6 0
5
x
10
00
" '
5
x
10
Figure 3.2: Variation of (a) the skin friction, and (b) the wall heat flux, with x f o r P r  0.72 and A  1 (positive inclination). The numerical solution is indicated by the solid lines and the series solutions (3.25) and (3.33) are indicated by the broken and dotted lines, respectively.
The series solutions (3.25) and (3.33), which are valid only in the region close to the leading edge of the plate and far downstream, respectively, can be matched by solving numerically the full boundarylayer Equations (3.13)  ( 3 . 1 5 ) . Jones (1973) has done this by using the selected points technique of Lanczos (1957) for which the solutions are represented by Chebyshev polynomials. The numerical results for the nondimensional skin friction and the wall heat flux for P r  0.72 and A  +1 are presented in Figures 3.2 and 3.3. The values given by the series expansions (3.25) and (3.33) are also included in these figures for comparison. It was found that for A   1 the boundarylayer separates at x  x s  3.704, where flow reversal has taken place, and that the solution behaves in a regular manner at this point. This can also be seen from Figure 3.4, where the reduced fluid velocity and the t e m p e r a t u r e profiles near this separation point are shown (by the solid lines). The velocity profiles at x s  3.704 (shown by the broken lines) are also included in these figures. It can be seen from Figure 3.4 that no difficulty was encountered in obtaining solutions at values of x downstream from the point of separation. This result is in contrast to that found by Merkin (1969) and Wilks (1974) for the opposing mixed convection flow over a vertical flat plate where the boundarylayer solution exhibits a singular behaviour near the separation point, x  x s , say, and that the skin friction 1 behaves like ( X s  x ) ~ near x~, see Section 2.2. Jones (1973) has noted that his solution was continued as far as x  4.5 with no obvious sign in the instability of the stepbystep numerical scheme used. Itowever, the calculations were terminated at x  4.5 since in order to determine the solution further downstream satisfactorily the use of a large number of Chebyshev polynomials would have been needed and
FREE AND MIXED CONVECTION
FLOW
(a)
95
(b) 1.0 1.6
~q
"~ 1.2 0.5
0.8 0.4 0.0
o
i
~
~
~
x
5
~176
i
~
~
4
X
5
Figure 3.3: Variation of (a) the skin friction, and (b) the wall heat flux, with x for P r = 0.72 and A   1 (negative inclination). The numerical solution is indicated by the solid lines and the series solution (3.25) is indicated by the broken lines.
large c o m p u t i n g facilities would be required. We m e n t i o n to this end t h a t the m e t h o d used by Jones (1973) involves a switch
(a)
(b)
10 "
10
.5
.5 0 . 0.2 0.0 9
.
9
0.2
o
.
0.4
9
.

0.6 Or/
0.8 '
0 0.0








0.5


1.0
o(~,~1
Figure 3.4 (a) The fluid velocity, ~o (x, r/), and (b) the temperature, O(x, rl) , profiles near the separation point for P r  0.72 and A   1 (negative inclination). The profiles at X s  3.704 are indicated by the broken lines.
96
CONVECTIVE FLOWS
between the leading edge and the far downstream systems of Equations (3.19) (3.21) and (3.27)  (3.29), respectively. However, the switch feature of the algorithm may be avoided by using a continuous transformation in the variable x, a technique proposed by Hunt and Wilks (1981), linking the two limiting solutions (3.18) and (3.26) as follows, see Hossain et al. (1996, 1998):
r
3
px~(l+x)~
g(x,Y),
Yx~(l+x)~y
3
(3.34) Thus, Equations (3.13)  (3.15) transform into the following equations with 0 = OH
03F 3 (4 + 5x) F ~ +A OY 3 ! 2 0 ( 1 + x ) OY 2 1 0 ( 1 + x ) flY _(l+z)_~[ 8+5x ( OH) OH] 20(l+x) H  Y o~ + x ff~x 
x
x ~ OH l +x OY OF 02F OYOxOY
1 03H 3 (4 + 5x) OH ( O F 02H Pr OY 3 ~ 20 (1 + x) g 0Y = x OYOxOY
OF 0 2 F ) (3.35) Ox OY 2 OF 02H)(3.36 ) Ox OY 2
with the boundary conditions (3.17) becoming
F(x 0 )  0 OF(x 0 )  0 , og(x 0 )   1 for x > 0 , , ~V , OH ~Y , OF aY ~0, H+0, oY ~0 as Y+c~, x > 0
(3.37)
It is seen that in the limits x + 0 and x ~ co, the Equations (3.19), (3.20), (3.27) and (3.28) for (f, h) and ( f , h) can easily be recovered from Equations (3.35) and (3.36). Using Equations (3.35)  (3.37) we have determined (by the courtesy of Hossain, 1999) the values of the separation point x = xs(Pr) for A =  1 and some values of Pr, and these values are given in Table 3.2. It can be seen from this table that the position of separation of the boundarylayer from the plate decreases with the increase of Pr. Table 3.2: Values of the separation point z~(Pr) for A = 1 (negative inclination).
l Pr. il 0.1 ] 0.3 I 0"5 10"73..11
.1..2 I 5 I XO I [ ~,(Pr)II 7.671 ]5.26314.25613.64513.215 J 2.46311.76311.386]
3.4
Free c o n v e c t i o n b o u n d a r y  l a y e r flow a b o v e i s o t h e r m a l flat plate of arbitrary i n c l i n a t i o n
an
We consider now the free convection above a heated and arbitrary inclined isothermal flat plate to fluids of any Prandtl number. The formulation is based on the new
FREE AND MIXED CONVECTION FLOW
97
variable introduced by Yu and Lin (1988) which permits rigorous solutions for the entire range of inclinations from the horizontal to the vertical and for any Prandtl number. Also negatively inclined plates at small angles to the horizontal are investigated. The physical model and the coordinate system are those illustrated in Figure 3.1 and the governing boundarylayer equations for this problem are those given by Equations (3.13)  (3.151 subject to the boundary conditions (3.17). When they are expressed in physical (dimensional) variables we obtain
or 2r
or 2r
Oy OxOy
Ox Oy2
03r lop .... t V~y3 + g fl (T  Too)sin ~o p Ox
1@
0r OT coy Ox
O .... ~gfl (T  Too) cos ~o p Oy 0r 07' v 02T Ox Oy Pr Oy 2
(3.38) (3.39) (3.40)
where the bar superscripts have been omitted for convenience and for a clearer presentation. The boundary conditions appropriate to these equations are those given by Equations (3.5). To study this problem, Yu and Lin (19881 proposed the use of the following variables: ~(~) = ~~+~, ~(~, y)  ~2 (~), r = ~s~2/(~, ~) paf,~ h (~, ~/), 0 (~, ~/) T~T~ (3.41) P  Pcr where 
"
r
=
1
(0"1Rax Isin ~VJ)~ 1 '
=
~2  ( 0 l R a x
cos 99) ~ 4 (0lRax Isin (p[)88
(3.42)
(alRax cos ~) and (71 is defined as in expression (2.72). Substitution of expressions (3.411 and (3.42) into Equations (3.38)  (3.40), leads to the equations: alf"' if 2~ ( 1  ira)[3 (44~) f f "   2 (2 q3~)f,2] 1 +~~ [(8  3~1 r/h'  4 (2 + 3~r h] 4 ~40 = 2~~
(3.43/
(I~)[(l(zl)(f, Of'_f,,
h'  (1  ~)5/9 O"F ~ (4F ~) fO'  3 ~ ( 1  ~ ) ( f ' O O ~~ o'Of) ~
(3.44) (3.45)
along with the boundary conditions (3.17) which may be expressed in the form f(~,O)=O, f'(~,O)=O, 0 ( ~ , 0 ) = 1 for ~ > 0 f~+O, 0  + 0 , h  + O as r/~cr (>0
(3.46)
98
CONVECTIVE FLOWS
where the + sign in Equation (3.43) pertains to the positive and negative inclination cases, respectively. These equations represent the universal formulation for the free convection above an isothermal flat plate which is inclined at an arbitrary angle 99 from the horizontal for fluids of any Prandtl number. The inclination angle is either positive (0 ~ < ~ ~ 90 ~ or takes small negative values (~ < 0 ~ since in this latter case the boundarylayer separates from the plate. The coordinate ~ measures the downstream distance along the plate from the leading edge for a specified inclination angle ~. It also represents a parameter of inclination since it varies from 0 to 1 when the inclined angle varies from 0 ~ to 90 ~ for fixed Rayleigh and Prandtl numbers. Equations (3.43)  (3.45) are readily reduced to the following similarity equations for tile limiting cases of a horizontal fiat plate (~o  0 and ~  0), namely a : f ' " + g1 (1  ~ r : ) ( 3 f f " 
f,2 ) + ~ 2 (rlh' h)  0 h'" + ~3 f h"  0
(3.47)
and, for a vertical flat plate (qo = 90 ~ and ~ = 1), namely 1
crlf"' q ~ ( 1  a l )
(3f f "  f '2) + 0  0 1
(3.48)
o" +  4 f o '  o tIaving determined the functions f and O, we can express the skin friction coefficient Cf and the local Nusselt number Nu as follows: Cf (cr:Rax cosqo)~  (1 + ~)3 f,, (~,0)
Nu(a:Razcoscp):
i
= (1 + C ) [  0 ' (~,0)]
(3.49)
Equations (3.43)  (3.46) have been integrated numerically by Yu and Lin (1988) by using the Kellerbox scheme for values of Pr between 0.001 and c 0 for assisting flow or A < 0 for opposing flow. It should be noted that m  1 corresponds to the plane stagnation point flow, m = ~ to a constant wall temperature, m  89 to a constant wall heat flux and m  0 to Schneider's (1979) problem of mixed convection boundarylayer flow past a horizontal flat plate with a uniform free stream velocity and a wall temperature distribution given by Equation (3.54). Equations (3.70)  (3.72) have been solved numerically by Ridha (1996) for m = 1, ~ and 89 corresponding to the situations mentioned above, and with Pr = 1 and for different values of A. Variations of f " (0), 0' (0) or 0(0) as a function of A are shown in Figure 3.13. The striking feature of the solutions obtained is their nonuniqueness, as displayed by the existence of two more regular bifurcation (critical) points for > 0 in addition to those that usually appear when ), < 0. Figures 3.13(a,b) for f"(0) and 0'(0) in the case of a uniform wall temperature (m = ~) show that the critical point, where the solution branches out, occurs at )~ = ~c = 0.47118 and the lower branch terminates at )~  0.019707. However, in the case of uniform wall heat flux (m  89 with the results shown in Figures 3.13(c,d,e), the lower branch solution branches out first at ~ = 0.2866, then at A  0.0175 before terminating in a singularity at ,k  )~s  0.0041. Here, three solutions are obtained in the range 0.0041 < /k < 0.0175 for each value of A. The variation of 0(0) near ~  0 is also illustrated in Figure 3.13(e) and shows that multiple solutions exist in this vicinity. It should be noted that values of f"(0) for the lower branch when )~  0 are designated by the symbol 9 in Figures 3.13(a,c). The fluid velocity and the temperature profiles corresponding to the uniform wall temperature (m = 1) and uniform wall heat flux (m  89 are depicted in Figure 3.14. Both the upper and lower branch solutions, as well as solutions near the critical bifurcation points, have been included in these figures. Figures 3.14(a,c) show that a reversed flow (f~ < 0) exists on the lower branch solutions for ,k = 0.1,  0 . 2 and 0.035. However, on these branch solutions the temperature profiles remain positive, as can be seen from Figures 3.14(b,d).
110
CONVECTIVE FLOWS
(a)
(b) 4
o'(o)
f"(O)
3 2 1
O
m
i
.
2

,
2
,
1
1
0
2
3
4
A
(c)
1
(d)
0
//
3
f .
1
,,
2
I
3
A
9
I
4 ,
.
e(o
f"(O) 1
2
/
. . . . . . . . . . . . .
t
i'0
o~
Graph of m  1 Case Magnified 1 2 3 4 A 
1
0
'
'

,,
,
i

I
0
i
l
i
i~
'
i
2
3
'
'
i
4
A
(e)
i
'
!
O
o(o) 5 10 15 20

0.0
0.1
0.2
0.3 A
Figure 3.13: Variation of f"(O), 0'(0) and 0(0) with A for Pr = 1. Figures (a,b) display prescribed wall temperature results, (c,d,e) represent results for the prescribed wall heat flux and (e) gives results for 0(0) near A = O. The solutions for m = ~, 89and 1 are indicated by the broken, dotted and solid lines, respectively.
F R E E AND MIXED C O N V E C T I O N F L O W
(a)
111
(b)
20
20
'A
= 0.035~1
15
15 r/
10
10
A  0.2, 0.4718,
~A = 0.2, O.4718,
/
5
0.0 ' 0:2' 0:4' 0:6' 0:8' 1.0 if(r/)
(c)
0
o.o
0.2
0.4
0.6 o.8 o(77)
1.o
(d) i
5 4

i
1, _ o 2 5 , _ o 2 8 o o , _ o 2 5 , _o
4
3
0
~ _ _01, 0~,
!~..~ ~
'",,"/
0.0
0.2
0.4
0.6
0.8
f'('7)
1.0
0
0 ~,
0.25,0.2,0
'.'.,\~... ii
0
"1
"
1
S"
2



3
4
0(77)
5
Figure 3.14 (a,c) The fluid velocity, f'(~), and (b,d) the temperature, 0(~), profiles for Pr = 1 with m  ~ (case of variable wall temperature) in (a,b) and m  13 (case of variable wall heat flux) in (c,d). The upper branch, lower branch and critical point are indicated by the broken, solid and dotted lines, respectively.
3.6
M i x e d c o n v e c t i o n b o u n d a r y  l a y e r f l o w a l o n g a n inclined permeable plate with variable wall temperature
Consider the steady mixed convection flow of an incompressible fluid along a heated permeable fiat plate which is inclined at a positive angle ~o to the horizontal, see Figure 3.15, where U ( x ) is the free stream velocity, Tw(x) is the variable fiat plate t e m p e r a t u r e distribution, vw(x) is the value of v at the plate and all these quantities
112
C O N V E C T I V E FLOWS
~5
Figure 3.15" Physical model and coordinate system.
are expressed in nondimensional form. The basic equations which govern this flow configuration were given by Weidman and Amberg (1996) in the form:
or 02r Oy OxOy
0r 0o Oy Ox
Ox Oy 2 
~
• Gr
0 sin ~0 + cos ~ o ~x
0 dy
0r 1 020 = Ox Oy Pr Oy 2
(3.73) (3.74)
where the :t: signs correspond to upwardfacing and downwardfacing heated plates, respectively, and the subscript x denotes differentiation with respect to x. Guided by a method proposed by Burde (1994) for determining the similarity solutions of partial differential equations, Weidman and Amberg (1996) assumed that r 0 and 7/take the forms .
r
 a(x) + #(x) f (~l),
O(x, y)  Tw(x)O (r/),
U(x,y)
_
Y
/3(x) +'y(x) (3.75)
The inclination X(X) of the streamlines which enter or leave the plate is determined from the equation t a n x ( x )  Vw(X) uw(x)
(3.76)
where Uw(X) is the value of u along the plate. Blowing or injection occurs when the streamline inclination falls in the region 0 < X(X) < 7r, and when ~r < X(X) < 0 there is a suction type of flow, which were only considered by Weidman and Amberg (1996). These latter flows are further divided into obtuse suction for which Tr < X 0 and A2 > 0. Thus, Weidman and Amberg (1996) have used the following asymptotic solution to implement integration from ~  oo to the plate, ~  0: 0" ,~ C] exp (  1 (~dPr + x / ~ ) ~ ) f'3 ,,~ q + 62 exp (  1 (~ + x / ~ ) ~) _ b2+~+2qC~.exp (1(E+ for A 1 > 0 and A2 > 0, where b =  89 ('dPr + x / ~ ) constants.
v/~)~)
(3.90)
and C1 and (72 are unknown
FREE AND MIXED CONVECTION A
h
FLOW
A
115
A
V a r i a t i o n s of f " ( O ) , 0'(0), f ' (~) a n d 0 (~) as a f u n c t i o n of q are p r e s e n t e d in
(a)
(b) 0.5 15
A
o'(o)
f"(O) 10
0.0 i"~ ""'
.I~ r
.~
0
0.5 F" 0
5 2.0

'
1.5


1.0
1.0
' "
0.5 q 0.0
0.5
1.52
A
90
1.5
1.0
q 0.0
0.5
0.5
A
Figure 3.16: Variation of (a) f"(O), and (b) 0'(0), with q for Pr = 0.71. The value q   0 . 5 7 5 is indicated by the dotted line and the branch 1 and branch 2
solutions are indicated by the solid and broken lines, respectively.
(~)
(b) 6
77 = 0.6, 1, 1.5
4"
I
q = 0.6, \ 2
0 2
 1,  1 . 5 , , , . ~ ' [ \
,,, _
0
Ii\
"'"''~ I\\ 5,1' 1.5
":~~..~_ 0
2
6
.,
4
6
0.5
0.0
0.5
1.0
f'(~ A
A
Figure 3.17: (a) The fluid velocity, f ' (~), and (b) the temperature, 0 (~), profiles for Pr  0.71 and some values of q. The branch 1 and branch 2 solutions are
indicated by the solid and broken lines, respectively.
116
C O N V E C T I V E FLOWS
Figures 3.16 and 3.17. It is seen from Figure 3.16 that two solution branches exist simultaneously in the region of nonoscillatory asymptotic decay. The branch 1 solutions exhibit a singular behaviour as q approaches 0.575 + 0.005 and cannot be continued to higher values of q. In contrast, the branch 2 solutions can be traced smoothly up to the value of q = 0.0591667, very near the value for which A1 = 0, namely q  ~Pr~d 2  0.0591667. Figure 3.17 clearly shows the difference between the branch 1 and branch 2 boundarylayer profiles of f' (~) and 0 (~) at selected values of the radial sink parameter q (< 0). It is seen that on both branch solutions there exists a reverse sink flow (f~ < 0) for admissible values of q. This reverse sink flow is stronger on the branch 1 solution than on the branch 2 solution. However, the temperature becomes negative on the branch 2 solution for the values of q considered, which does not have any physical sense.
Chapter 4
Doublediffusive convection 4.1
Introduction
In many natural and technological processes, temperature and mass or concentration diffusion act together to create a buoyancy force which drives the fluid and this is known as doublediffusive convection, or combined heat and mass concentration transfer convection. In oceanography, convection processes involve thermal and salinity gradients and this is referred to as thermohaline convection, whilst surface gradients of the temperature and the solute concentration are referred to as Marangoni convection. The term doublediffusive convection is now widely accepted for all processes which involve simultaneous thermal and concentration (solutal) gradients and provides an explanation for a number of natural phenomena. Because of the coupling between the fluid velocity field and the diffusive (thermal and concentration) fields, doublediffusive convection is more complex than the convective flow which is associated with a single diffusive scalar, and many different behaviours may be expected. Such doublediffusive processes occur in many fields, including chemical engineering (drying, cleaning operations, evaporation, condensation, sublimation, deposition of thin films, energy storage in solar ponds, rollover in storage tanks containing liquefied natural gas, solution mining of salt caverns for crude oil storage, casting of metal alloys and photosynthesis), solidstate physics (solidification of binary alloy and crystal growth), oceanography (melting and cooling near ice surfaces, sea water intrusion into freshwater lakes and the formation of layered or columnar structures during crystallisation of igneous intrusions in the Earth's crust), geophysics (dispersion of dissolvent materials or particulate matter in flows), etc. A clear understanding of the nature of the interaction between thermal and mass or concentration buoyancy forces are necessary in order to control these processes. The parameters that determine the relative strength of the two buoyancy forces
118
CONVECTIVE FLOWS
are the buoyancy ratio parameters, N and R, which are defined as N = fl*AC ~AT '
R =
I~AT I [~*AC I
(4.1)
It is important to note, for most fluids at normal pressures, that/~ is positive but/3* can be positive or negative depending on the contribution of the diffusing species to the density of the ambient medium. When N = 0 and N = oo, the case in which a single scalar is diffusing is recovered; when N < 0, thermal and concentration forces drive the flow in opposite directions and the flow field can reverse (opposing flow); when N > 0, buoyancy forces are cooperating and drive the flow in the same direction (assisting flow). The character of a doublediffusive convective motion depends upon the orientation of the two density gradients with respect to the gravitational field. Three different cases can be distinguished. Firstly, if both gradients are vertical the configuration resembles the RayleighB~nard stability problem, except that doublediffusive instabilities can develop even when the net density decreases upwards and the system would appear to be statically stable. Secondly, if one density gradient, say concentration (solutal), is vertical and statically stable, the imposition of a horizontal temperature gradient at a vertical or inclined boundary will induce a multicellular intrusive motion along the boundary. Finally, both of the density gradients might be horizontal, resulting in a boundarylayer flow along a vertical or inclined boundary. A substantial literature survey on the subject of doublediffusive convection has been made by Ostrach (1980), Huppert and Turner (1981), Nilson and Baer (1982), Turner (1974, 1985), Nilson (1985), Gebhart et al. (1988), Napolitano et al. (1992), Angirasa and Srinivasan (1992), Mahajan and Angirasa (1993), Bejan (1995), Rahman and Lampinen (1995) and Mongruel et al. (1996). Theoretically, the governing equations of doublediffusion convection are the classical conservation equations (I.1)  (I.4) for mass, momentum, energy and mass (or concentration) species. While these are easy to formulate, the existence of two buoyancy forces results in a complicated nonlinear partial differential problem. Most of the methods developed in the field of boundarylayer theory have also been successfully applied to doublediffusive situations. Among them the search for similarity solutions has attracted much attention, mainly because similarity formulation transform easily the transport equations into a set of ordinary differential equations which can be solved numerically for different values of the parameters involved. Other numerical investigations have solved the basic flow equations, i.e. the full partial differential equations, by the finitedifference techniques. Analytical methods, such as integral methods and asymptotic expansions, have also been used to obtain the transport properties as a function of the different parameters which are involved in the particular problem under investigation. The results contain evidence of many different and complicated fluid flows but, in general, the predictions are scarce and they are restricted to some specific cases. However, the scale analysis proposed by
DOUBLEDIFFUSIVE C O N V E C T I O N
119
Bejan (1984) has also been applied to some doublediffusive convection problems, see Khair and Bejan (1985) and Trevisan and Bejan (1987), with much success in order to determine the heat and mass transfer characteristics. This method is based on the fact that different terms in the equations of motion are approximated from simple order of magnitude arguments and some dominant balances between them are then considered. Recently, Mongruel et al. (1996) have proposed a novel method to study the doublediffusive boundarylayer flow over a vertical flat plate which is immersed in a viscous fluid or in a fluidsaturated porous medium. This method combines the use of the integral boundarylayer equations and the scaling analysis approach. Three different diffusivities appear in the transport equations (I.1)  (I.4): chemical diffusivity D, thermal diffusivity ~ / a n d viscous diffusivity v. Accordingly, there are three different length scales or boundarylayer thicknesses: concentration (solutal) boundarylayer thickness 5c, thermal boundarylayer thickness 5t and viscous (momentum) boundarylayer thickness (~v. In general, the aim of any doublediffusive study is to predict all the solutions of Equations (I.1)  (I.4) when N (or R) and the diffusion parameters are varied. It is convenient to use the Prandtl number, Pr, the Schmidt number, Sc, and the Lewis number, Le, as diffusion parameters. Their values depend on the nature of the fluid and on the physical mechanisms governing the diffusion of the heat and chemical species. In gases, D ~ ( ~ / ~ v, which leads to Pr, Sc and Le being of the order of unity. However, in most liquids P r > 1 and Sc > 1, except in most molten metals where P r < 1. Usually, heat diffusion is more efficient than mass diffusion, yielding a Lewis number which is greater than unity. Typical values of Le in common solutions are about 100; but Le can be very large in complex situations containing macromolecules or colloidal dispersion. Since sc it is concluded that only two physical Pr, Sc and Le are related through Le = Pi, cases fit the requirement Le > 1: P r < 1 < Sc ( molten metals) and 1 < P r < Sc (solutions). When P r = Sc, the governing equations (I.1)  (I.4) reduce to those for a single buoyancy effect. The case when N > 0 (assisting flow) and different combinations of the scales (~c, ~t and ~v, or equivalently of the parameters Pr, Sc and Le, has been extensively studied in the literature (see Khair and Bejan, 1985; Bejan, 1995; and Mongruel et al., 1996) and therefore will not be presented here. However, we will discuss in more detail the case of opposing thermal and chemical buoyancy forces.
4.2
D o u b l e  d i f f u s i v e free c o n v e c t i o n b o u n d a r y  l a y e r flow over a v e r t i c a l flat plate in t h e case of o p p o s i n g b u o y a n c y forces
Consider a vertical impermeable flat plate of finite height, which is immersed in a binary fluid/solute flow, where the temperature and concentration at the wall,
120
C O N V E C T I V E FLOWS
Tw and Cw, respectively, and in the ambient field, Too and C ~ , respectively, are constant. We assume that the combination of the buoyancy forces tend to induce an upward motion near the wall and that the thermal buoyancy tends to induce a downward motion in the far field. It is also assumed that 6u >> 6t >> 5~, where /1 >> ~f >> D or equivalently
Pr>>l
and
Le>>l
(4.2)
In this case the interaction between the chemical and thermal buoyancy mechanisms depends mainly on the parameters R (or N) and Le. As indicated schematically by Nilson (1985) in the flowregime map of Figure 4.1, the following three different situations are possible: (i) unidirectional downflow occurs when R >
11 ," Le~
(ii) unidirectional upflow occurs when R < ~1; (iii) bidirectionai counterflow occurs when ~ee ~< R > 1), the concentration boundarylayer is much thinner than the thermal boundarylayer, as illustrated in Figure 4.3. Thus, in the limit Le + c~, with R L e held fixed, Equations (4.9)  (4.11) can be reduced to the following form: f"'+ r 0r
3fr
0
(4.13)
1
(4.14)
0
(4.15)
with the boundary conditions f(O)  O, f"+0,
f'(O)  O,
r
r
~+c~
as
 1
(4.16a) (4.16b)
Within this inner concentration layer the temperature is essentially uniform and the thermal buoyancy forces can be neglected for the innerdominated conditions of interest, as, for example, in Figure 4.2 where R < 0.01. The thermal buoyancy effects will be later included in the outer equations.
124
C O N V E C T I V E FLOWS
The outer thermal boundarylayer can be described by introducing the new variables, for both Le and P r large, A
1
T]   C1
y
3
P r ~ z,
3
r  4ucl P r  ~ x ~
(4.17)
x~ 1
wherecl
(g~AT) ~ Under this transformation the Equations (4.3)  (4.6) reduce 4v2 to the following formp"
0  0
(4.18)
0 " + 3 fO'  0
(4.19)
r
0
(4.20)
where primes now denote differentiation with respect to ~. The inner/outer matching conditions for these equations are as follows: A
f(0)  0,
0(0)  1
(4.21a)
A
df
1

df
(co) 
(RLe) ~ drl
0.51
~ (RLe) ~
(4.21b)
as well as the farfield boundary conditions ])'+0,
0~0
as
~+cc
(4.21c)
The matching conditions between the inner (7] >> 1) and outer (~ ~c), where only the counterbuoyant thermal forces are active. Further, Figure 4.6 shows that in the outerdominated flow a looplike multiplicity of solutions is found to exist when R becomes sufficiently large in the neighbourhood of insipient reversed flow. Also, as before, there is an extremal value of R (here a maximum value) for which innerdominated flows may exist.
128
CONVECTIVE FLOWS
0.6 '\ \
0.4 F"(0) 0.2
\
0.0 0.2 ' 0.0
014
018'
112
R
116
2.0
Figure 4.6: Variation of F"(O) with R for Le = 4. The outerdominated and innerdominated flows are indicated by the solid and broken lines, respectively.
Finally, it is worth pointing out t h a t Nilson and Baer (1982) have shown that some qualitative results exist for other values of P r and Le. W h e n P r is finite, it is possible to have innerdominated solutions with a reversed flow in the far field. However, as P r + oo, it is impossible, within the context of selfsimilar theory, for an i n n e r  d o m i n a t e d reverse flow to occur. This argument is based on the observation that f'(c0,
x>0
(4.39) Finally, the average Nusselt number, N u , and the average Sherwood number, S h , can be evaluated from the expressions dx,
yO
Sh 

"~Y y=0
dx
(4.40)
The timedependent Equations (4.34)  (4.39) are marched stepbystep in time until a steady state solution is obtained, see Mahajan and Angirasa (1993) who have used the Alternating Direction Implicit (ADI) scheme as proposed by Peaceman and Rachford (1955) which is well described in the book by Roache (1982). The width of the computational domain in the y direction was chosen to be 0.4 after checking that larger values did not alter the solution and this value was estimated from the similarity solutions of Gebhart and Pera (1971). Also, the wall vorticities were evaluated from the values of the stream function at the adjacent grid points. Other details of the computation can be found in the paper by Mahajan and Angirasa (1993). The computed fluid velocity profiles u for P r  0.7, S c  5, G r  105 with N =  1 and 2 are presented in Figure 4.7, while those for N   1 . 6 to  5 are shown in Figure 4.8. Further, the similarity solutions of Gebhart and Pera (1971) are also shown in these figures for comparison. It can be seen that for N  2 (assisting buoyancy forces) that the agreement between the two set8 of results are very good, while for N =  1 (opposing buoyancy forces) there is a difference between the two solutions. The numerical results give a slightly smaller value of the peak vertical fluid velocity, and a flatter fluid velocity profile. This suggests, for N   1 , that although the numerical results for large value8 of G r , i.e. boundarylayer type flow, predict a reasonably accurate solution this solution does 8tart to depart from the 8imilaxity boundarylayer 8olution. Further, for N =  1 . 6 the difference between the numerical and similarity, i.e. the boundarylayer, solution8 is relatively high, see Figure 4.8. This figure shows that the similarity solution underpredicts the flow reversal near the surface but exaggerates the magnitude of the upward fluid velocity in the outer region of the boundarylayer. The flow reversal, with the upward and downward fluid velocities of the same order, cannot be accounted for by
130
CONVECTIVE FLOWS 0.5 0.4 u(0"5' Y)0.3 0.2 0.1 000.0
0.1
0.2 y 0.3
0.4
Figure 4.7: Fluid velocity profiles, u(O.5, y), for both assisting and opposing buoyancy forces when Pr = 0.7, Sc = 5 and Gr = 105. The profiles o/Mahajan and Angirasa (1993) are indicated by the solid lines and the similarity solutions of Gebhart and Pera (1971) are indicated by the broken lines.
.4
__
_
o
o21 ~(~ Y)0.~ ,," 0.2 0 4 0.0
~ 0.1
0.2
Y
0.3
0.4
Figure 4.8: Fluid velocity profiles, u(O.5,y), for opposing buoyancy forces when Pr  0.7, Sc  5 and Gr  105. The profiles of Mahajan and Angirasa (1993) are indicated by the solid lines and the similarity solution of Gebhart and Pera (1971) is indicated by the broken line.
the boundarylayer equations. With increasing negative values of N, the magnitude of the downward fluid velocities increases, and that of the upward fluid velocities
DOUBLEDIFFUSIVE CONVECTION
131
away from the surface decreases. For P r >~ Sc, and INI >> 1 (N < 0), no flow reversal (upward) appears, and v 0 (4.47)
D oe  k0Cexp (  R~T) 0, g~0,
TToo, T+ Too,
on as
CCc~ C ~Coo
50, ~c~,
~>0 ~>0
Further, we introduce the following nondimensional variables
x _ (, z
y _ Gr88
,
u
U~, u
cs
( gg~ ) ,
v=Gr~
O(TTee)
RTs (4.48)
U~
In terms of the nondimensionM stream function r defined in expression (1.18), Equations (4.42)  (4.45) can then be written as follows:
0r 02r
0r 02r
03r
Oy OxOy
Ox Oy 2
Oy 3
+OS(~)
or oo
or oo
Oy Ox
Ox Oy
or or
or or
1 020 P r Oy2 1 02r
Oy Ox
Ox Oy
Sc Oy 2
(4.49) (4.50) (4.51)
and the boundary conditions (4.47) can also be written in the form r = 0, O0    O l l C e x p ( 1 0 0 ) Oy r or ~0, Oy
0r
oy _
0
(
04, :
0
5
on
y0,
x >0
x0, y >0 y + cx~, x > 0 (4.52) where a is the energy activation parameter and c~1 and a2 are the reactant consumption parameters, which are defined as follows: a
RToo E '
al=
o=o, 0+0,
on as
r
r
EQkolCoo kIRT~Gr 88
exp
( E ) RToo
'
a2=
kfRT 2 CooQED
(4.53)
Next, we present some results for the free convection near the lower twodimensional stagnation point of a cylindrical surface and for a vertical flat plate. We then extend these results to the free convection near a threedimensional stagnation point of attachment.
136
CONVECTIVE FLOWS
4.3.1
Twodimensional
stagnation
point
Since in this case S (x) = x then Equations (4.49)  (4.51) become similar. The transformation (4.54) r 0=0(y), r162 reduces these equations to the form /'" +//"
I '2 + o  o
(4.55)
+ f Ot = 0
(4.56)
1
~0" Pr
1 r
Ir
0
(4.57)
Sc
and the boundary conditions (4.52) become
(,;, 0'o~lCexp
if~0 f=0, f~+0,
0
( )
, c~lol2r 0=0, r 0+0, r
i+~
}
on
y  O,
x>0
on as
x=0, y+oo,
y>0 x>0
(4.5s) Equations (4.55)  (4.58) were solved numerically by Chaudhary and Merkin (1994) in two cases, namely c~2  0 (reactant consumption neglected) and c~2 r 0 (reactant consumption included). Although these authors produced numerous results we only very briefly present some of their most important results.
R e a c t a n t c o n s u m p t i o n neglected, c~2   0 Equation (4.57) together with the boundary conditions r only the trivial solution r = 1
= 0 and r
= 1 has (4.59)
and the boundary conditions (4.58) reduce to
f=0,
fl0, 0 t   C ~ l exp [ l + ~ a 0 ) o n f=0, 0=0 on f~+0, 0+0 as
y0,
X~0
x=0, y+cx~,
y>0 x>0
(4.60)
Equations (4.55) and (4.56), along with the boundary conditions (4.60), were solved numerically by Chaudhary and Merkin (1994) for P r = 1 and different values of al when a = 0 and 0.1. The variation of the wall temperature, 0(0), with al is shown, for a = 0, in Figure 4.12, which shows the existence of a turning (critical) point at al = ac ~ 0.1596, whereas the variation of In (0(0)) with al is illustrated in Figure 4.13 in order to show both critical turning points for the case
DOUBLEDIFFUSIVE CONVECTION
137
12
o(o) 8
0
0.00
Figure 4.12:
0.04
Variation of
0(0)
0.08
Ol 1
0.12
0.16
with c~1 f o r c~  0 a n d P r  1.
3 In (O(0)) 2
0 0.00
Figure 4.13:
Variation of
,
9

,
,
0.04 0.08 0.12 0.16 0.20
In (0(0))
with c~1 f o r c~ 
0.1
a n d P r = 1.
c~ : 0.1. This figure clearly demonstrates the existence of critical (turning) points, which represent a change in the steady state flow stability through a saddlenode bifurcation, and this limits the range of existence of the solution of Equations (4.55) and (4.56) subject to the boundary conditions (4.60). Chaudhary and Merkin (1994) have shown analytically that there exists a hysteresis point, i.e. a point where the solution changes from having multiple solutions to a single solution, for c~ ~ !5" They have also studied the behaviour of the solutions of these equations for large and small values of P r , and have determined analytical expressions for the lower and upper turning (critical) points a l  c~!1) and c~1 : a (2), respectively, as a ~ 0.
R e a c t a n t c o n s u m p t i o n i n c l u d e d , a2 ~ 0 In this case, Equations (4.55)  (4.58) have been solved numerically by Chaudhary and Merkin (1994) for 0 ~ a ~ 0.2 and c~2 = 0.1, 0.2 and 0.3 when P r = S c and
138
CONVECTIVE FLOWS
P r ~ Sc. T h e variation of 0(0) as a function of a ] is shown in Figure 4.14 for a   0 , P r  1 a n d Sc  1. For the case a2  0.1 it can be seen from Figure 4.14(a) (~) t h a t there are two welldefined t u r n i n g (critical) points at c~1  ac  0.1840 and ( c~1  a~ 2)  0.0080 and t h a t 0(0) ~ ~1 as Ctl + (x~ on the u p p e r solution branch. For a2  0.2 there are still two t u r n i n g points, as can be seen f r o m Figure 4.14(b), b u t these have become much closer together, n a m e l y they occur at a l  o ~ 1)  0 . 2 2 7 0 a n d ol 1  ol~2)  0 . 2 0 2 9 , a n d again we observe t h a t 0(0) ~ ~1 as Ctl increases on the u p p e r branch. However, Figure 4.14(c) shows t h a t at a2  0.3 the curve of 0(0) as a function of a] is m o n o t o n i c , showing t h a t there is a hysteresis point in the 1 range of values 0.2 < a2 < 0.3 and again we have 0(0) ~ ~ on the u p p e r solution branch. For a  0.1 and P r Sc there are also multiple solutions, b u t they occur at m u c h smaller values of a2, as can be observed from Figure 6 of the paper by C h a u d h a r y a n d Merkin (1994). These a u t h o r s have d e t e r m i n e d explicit expressions for the location of the hysteresis points and they have also shown t h a t 00(0) varies in the ranges
(b) ,,
,
,
9
,
,
,
10 0(0)
4 0(0) 3
8
6 4 2 0 0.00 ' o . b 4 " o . b 8 '
0
0.~12' 0.~16" 0.20
0.0
oz 1
0.1
0.2
0:3
0:4
0:5
0.6
(c) 3
i
o(o) 2
I
0
1
l ill
I
J
I
I
2
'"i'
.
3
ii
'
i
4
Figure 4.14: Variation of 0(0) with al for Pr  1, S c a2  0.2 and (c) a2  0.3.
1 and (a) a2  0 . 1 , (b)
DOUBLEDIFFUSIVE C O N V E C T I O N 2
2v/1 
a
a
o~2
139
< Oo (0)
~ 0 describes the fluid flow around all possible threedimensional blunt body shapes. However, as in Banks (1972), a restriction can be made to consider only the finite range of values of c, namely 0 ~< c ~ 0 then reveals that all critical points occur within the region 0a ~ 89 At a hysteresis bifurcation point (Oa,h, Ow,h) we further require that the critical points coincide to form a point of inflexion at which the following condition is satisfied:
d20a d02
= 0
(4.106)
Such points must therefore additionally satisfy the constraint 3o~30~, h  2 (5 + 4a3 + a.3Oa,h) Ow,h + 4 (1 + 2a3Oa,h)  0
For a given value of Ol3, the parameter value ( a~o l) is achieved is now defined by Equation (4.101).
h
at which a hysteresis bifurcation
o1 > For disjoint bifurcation diagrams we require 35
v ,uo
where
0
(4.107)
( ~Ao )
0 , where the parameter
is such that the upper critical point lies on the Oa  0 axis at (0, Ow,o) 0~,0 : 2c~3 5 + 4c~3 
on using Equation (4.104) and ( ~ ' )
5 + 24c~3 + 16~
(4.108)
is subsequently found from Equation (4.101). It can easily be verified that for all values of ~A0 and c~3 the bifurcation curves approach the origin of the parameter space (0a, 0w) along the line Ow  Oa, whilst all curves in the parameter space (Oa, Cw) approach the point 0a = 0, Cw : 1 along '
_ _XSo 0
148
C O N V E C T I V E FLOWS
the line Cw  1. However, for large values of Ow and Oa, and given values of ~oo a~ and aa, the behaviour of the solution is described approximately by Ow  Oa + A0,
Cw  1  o~3A0
(4.109)
where A0 is defined by the following relation:
(A0)] Ao (1 
(4.110)
a3A0)
Furthermore, it can be seen that as ~oo a~ + oc then A0 + ~1, which is consistent with the condition (4.102) and ensures that all solutions in the parameter space (Oa, Ow) are bounded by the straight lines 0w  Oa and Ow  Oa + 1. ~3 In Figure 4.19 a sequence of bifurcation diagrams, determined by solving the algebraic Equation (4.101), shows the variation of Ow as a function of 0~ for a3  1, but the behaviour shown in this figure is representative of the general a3 case. As discussed above, the increase of ~o a~ from small values shows that the bifurcation curves deviate further from the solution Ow = 0a but remain monotonic until the a) h 2.385011, after which a hysteresis bifurcation is achieved at the value ( aX~o point Oa,h  0 174859, Ow,h  0.309925. By further increasing the value of ~1 the 9 ~~o ~ hysteresis point develops and it is clearly seen that a typical Sshape behaviour occurs, see Figure 4.19. At the value ( ~Ao 1) 0 6.164140theuppercriticalpoint touches the 0a = 0 axis at Ow,o 
0.468871 9
For al ~>
(a~) ,,~ooo j
the bifurcation
~00 = 1.2, 2.385011, 3.6, 6.164140, 9 1.0
84
0.8 O~(0,~).6 0.4 0.2
0o:0
i,
0:1
0:2 " 0:3'
.
0:4
0:5
0a
Figure 4.19: Variation of the wall temperature, Ow(Oa), with Oa for a3  1, obtained by solving Equation (4.101). The small Ow behaviour Ow ..~ Oa is indicated by the broken line.
DOUBLEDIFFUSIVE CONVECTION
149
o~1 diagram is disjoint and as ~o becomes large the upper solution branch tends towards the bounding solution 0w  0a + 1 _ Oa + 1. Figure 4.19 also demonstrates that, for all values of ~ Ao the bifurcation diagrams follow the behaviour 0w ~ 0a for small values of Ow. The variation of the nondimensional surface concentration r with Oa is shown in Figure 4.20 for a3 = 1 and for the same values of ~oo as discussed in Figure 4.19. From Figure 4.20(a), which displays the (Oa, Cw) parameter space for small values of Oa, we observe that the hysteresis point occurs at Cw,h  0.8649934 whilst the d i a g r a m becomes disjoint when the curve touches the vertical axis at Cw,0  0.531129. Again, the curves are seen to approach the behaviour Cw = 1. However, for larger values of Oa, illustrated in Figure 4.20(b), the bifurcation diagrams are seen to approach the asymptotic value Cw = 1 o~3A0 , where the value of A0 is obtained from Equation (4.110).
a)
O/1 1.2, 2.385011, 3.6,
(b)
1.0 . . . . . . . 0.8
r
r176(Oa)o.6
0.6] ~ _~ __ 1.2, 2.385011, 3.6, ~ ~ 6.164140,9
0.4 0.2 0.0 0.0
0.1
0.2
0.3
0a
0.4
0.5
1
10
e~
100
Figure 4.20: Variation of the wall concentration, Cw(0a), with Oa /or a3 = 1 at (a) small values of Oa and at (b) large values of Oa. The small Ow approximation r = 1 is indicated by the broken line and the asymptotic behaviour Cw = 1a3A0 at large values of Oa is indicated by the dotted line.
Finally, we mention several other interesting papers on the topic of convective flows due to the combined buoyancies of heat and mass diffusion. Angirasa and Srinivasan (1989) have studied the doublediffusive free convection flows adjacent to a vertical surface in a stable thermally stratified medium. Further, Angirasa and Mahajan (1993) have presented a numerical study of doublediffusive free convection over a horizontal finite flat plate for both aiding and opposing buoyancies and for equal and unequal Prandtl and Schmidt numbers. The doublediffusive free convection from a vertical flat plate which is situated in a binary mixture was studied by Rahman and Lampinen (1995) using a finite element method, whereas Lin and Wu
150
CONVECTIVE FLOWS
(1995, 1997) have studied the doublediffusive free convection from a vertical fiat plate for any ratio of the solutal buoyancy force to the thermal buoyancy force using a new similarity transformation. Very accurate correlations of the mass transfer and the heat transfer rates have also been developed by the latter authors.
Chapter 5
Convective flow in buoyant plumes and jets 5.1
Introduction
Free and mixed convection flows arising from point and line thermal sources at the leading edge of a vertical surface, also referred to as a wall plume, in an infinite expanse of otherwise quiescent fluid are of considerable interest from both theoretical and practical points of view. Practical interest in plumes occur in many engineering situations including hotwire anemometry, flows that arise in fires, cooling of electronic circuitry, meteorology, industrial processes such as heat treatment, forging, foundry welding, etc. There is a huge quantity of literature pertaining to this topic and much of this has been reviewed by Schneider (1981), Jaluria (1980), Afzal (1986) and Gebhart et al. (1988). However, there has still been an increasing interest in this area of research during the last years and very valuable contributions on this topic have been made by Worster (1986), Joshi (1987), Angirasa and Sarma (1988), Ingham and Pop (1990), Thomas and Takhar (1988a, 1988b), Riley (1988), Hunt and Wilks (1989), Wang (1989), Srinivasan and Angirasa (1990), Jagannadham et al. (1992), Desrayaud and Lauriat (1993), Kay et al. (1995), Vazquez et al. (1996), Lin et al. (1996), Higuera and Weidman (1998), Lifis and Kurdyumov (1998) and Kurdyumov and Lifis (1999). In the following we review some of the most interesting aspects of buoyant plumes which have been highlighted very recently and which arise from heated point or line sources. We also consider the problem of a laminar buoyant jet.
5.2
F r e e c o n v e c t i o n in a w a l l p l u m e
The geometry considered is equivalent to a line thermal source of heat which is embedded at the leading edge of an adiabatic vertical plane surface bounded by
152
C O N V E C T I V E FLOWS
a horizontal insulated wall, which is placed at the level of the heat source and is maintained at the temperature T ~ of the ambient fluid, see Figure 5.1. The Cartesian coordinate system (~, ~) has the origin at the leading edge of the vertical plate with the 5axis being measured along the plate in the upward direction and the ~axis along the horizontal wall. For this twodimensional flow geometry the governing equations, on using the Boussinesq approximation, can be written in nondimensional form as, see Riley (1974),
or o Ox Oy
0r 0 (V2r
oy Ox 0r OT 0r OT Ox Oy
+ Gr_~V4r _ 0
1
Oy Ox ~ PrrG r  ~ V 2 T
(5.1)
(5.2)
0
where T = TToo is the nondimensional temperature with Tref being the reference Tref temperature which is related to the heat flux per unit length qs released from the line heat source and Gr is the Grashof number based o n T r e f. The boundary conditions of Equations (5.1) and (5.2) are as follows: 0r _ 0, 0Y0~ a~ 0,
r r  0, r
on
_ 0 0YOT o~ = 0 OT
on
T+0
as
0  0, 0   ~, ~ r+oo,
0 0
as
Y + cx~,
X > 0
for
all X
(5.12)
It should be noted that these equations are the boundarylayer equations for the plane wall plume as described by Riley (1974). We seek the solution of Equations (5.1.0)  (5.12) in the form r
_ x ~3 f l (r/)
,
T[t)  x  ~ 3 h i (r/),
7? 
Y2
(5.13)
Xs
where the functions fl and h l satisfy f~, + 3 flf~,  ~ f1~ ,2 + h ~  O 1 hit
1[ 3 (flhx ),  0
ft(0)0, f~ +0,
f~(0)0, h~(0) = 0 hi +0 as 77+oo
P r f o f~ h l d~  Q
where primes denote differentiation with respect to rl.
(5.14)
CONVECTIVE FLOW IN BUOYANT PLUMES AND JETS
155
Higherorder solutions To obtain higherorder solutions it is necessary to match the inner boundarylayer solutions (5.8) and (5.9) to the outer solutions (5.5) and this can be done following the procedure as described by Ingham and Pop (1989, 1990). For the outer flow this leads to the need to solve the following problems: V2r ~  O,
r
(x, O)  fl (oo)x 3
r
y)  0
(5.15)
and V2r ~  0,
r
(x, 0)  0,
r
y)  ~2y~o
(5.16)
along with the boundary conditions at infinity (5.7). Solving Equations (5.15) and (5.16), we obtain r
7r
  f 1 ( c r
sin [3 (0  y)] [ 3~r sin t i6)
r
~
3   _a 2 r _1o
38
sin ( ~ ) sin( 3~
~,..j,~.~,
We can now determine the second and thirdorder terms from the inner series solutions (5.8) and (5.9) and after some algebra we obtain the following system of equations: f2m + 3 flf~' + gfaf~ 1 t + h2  O
pr + ~ (f~h'2 + 2f{h2 + h~f~)  0 f2(O)O, f~(O)O, h~(O)=O
 f (oo)cot
]6)   b2,
h2 + 0
as
(5.18)
r/+ c~
f O (f~ h2 + f ~ h l ) dr]  0 m3:tt
(
2  ~ . f 2 f 2 + ~ g~f 2 ( o )  o,
?~)
0 (5.19)
f2(o)  o
f '2  ~ ~3 ~in(~)~" A(cr =b2 
as
~+cx~
f g3 f l f 3 II A l f ~ f 3I  3 f['f3 "4 h3   0 1 h~ 4 3 f l h l 3 4 3 f~h3 A ~ h l f 3  3 h i l l 3  0 p~ f3111
f3(0)  0,
f~(0)  0,
h~(0)  0,
3 ,, ~2 b3 f 3 + 1Osin(3"~6)  
as
h3(cx3)  0
(5.20)
rl+cxD
f o (f~h3 + f~hl) dO = 0 f :3" '  ]"6J2J3 3 u ~ " ~ 10 11 b2b3
:1
f 3 ( O )   O,
f3 +   3 a 2 c~
~! "=! / f2f3  0
(5.21)
f3(O)  0
(31r ~)~3
as
~  ~
156
CONVECTIVE FLOWS
where for the bar functions the primes denote differentiation with respect to ~, which is defined by X ~7 (5.22) ylo
Inspection of Equations (5.18)  (5.21) indicates that the decay of the fluid velocity field is exponential in the plume layer whereas it is algebraic in the horizontal layer, i.e. the functions f2, f2, f3 and f3 behave as
f2(77) ~ b2rl + a2 + 0 (e a~u) f3(r/)
~
b3 r]
(5.23)
4 a3 4 O (e a~176
73 (~) ~ b 3 ~ d  a 3  4  0 ( ~ _ s ) \ /
as ~, ~ + ~ where aoo 3f1(oc). However, the inner expansions (5.8) and (5.9) are not unique. To each of them may be added any one of an infinite set of eigensolutions which have the form 
1
3
Ck CkGr:(l+~k)x:(1?k)Fk(r]) 1 3 Tk  CkGr~kx~(l+~k)Hk(rl)
(5.24)
~k  CkGr~(3+~k)Y ~(3?k)Fk (~)
(5.25)


and where 7k and 7k are the eigenvalues associated with the inner boundary layers while Ck and Ck are multiplicative constants which, in general, are indeterminate. The differential equations for the functions Fk and Ilk are given by 3
K111!
1
Pr
I!
3
1
3
3 (2  37k)f[F~ + 5(1 3
(1 + 7k)flHk + 5(1
)f~'Fk + Ilk  0 3
)h~Fk + ~hlF~  0
(5.26) (5.27)
along with the boundary conditions Fk (0)  0, F~~0,
F i ( 0 )  0, Hi(0)  0 Hk+0 as V  + ~
(5.28)
Numerical integration of Equations (5.26)  (5.28) has been performed by Ingham and Pop (1990) who found that the smallest value of 7k is ~/1   5 for all values of Pr. This eigenvalue introduces a term in the inner expansions (5.8) which in order of magnitude lies between the third and fourth terms in each of the series. The next eigenvalues ~/k, for k  2, 3 , . . . , depend on the value of the Prandtl number Pr, and we have, for example, ")'2  3.231 for Pr  6.7. Thus, the assumed form of the solutions (5.8)is appropriate to O "(Gr89/ and O ( G r  ~ ) , respectively. \ \ ]
CONVECTIVE FLOW IN BUOYANT PLUMES AND JETS
157
The equation satisfied by the function Fk is given by _..:it
y,,,k  NaJ 2 V 'k 
(8 +
w
(3  7k) f 2 F k = 0
(5.29a)
which has to be solved subject to the boundary conditions F_k(0)  0, Fk(O )  0 Fk~O as ~ o c
(5.29b)
A numerical inspection of Equations (5.29) shows that they do not possess a solution for any real ~k > 0 and therefore the expansion (5.9)is correct to O ( G r  ~ ) . On the other hand, Equations (5.14) and (5.18)  (5.21) have been integrated numerically by Ingham and Pop (1990) for P r = 0.72 and 6.7. The distribution of the fluid velocity and the temperature fields are illustrated in Figure 5.2. This figure shows that the second and thirdorder boundarylayer corrections involve flow reversals at some distances from the vertical wall. (a)
(b)
11 ~
1.5 i
,
n ~b
,
f2
1.o
I
0.5
2 3
hi
0.0
,
4
j
1"o
0.5
5
Figure 5.2: (a) The fluid velocity, and (b) the temperature, functions associated with the plume boundarylayer for Pr  0.72.
We can also calculate the skin friction coefficients C / a n d C f and the adiabatic vertical wall temperature Ta from the following expressions:
C/
2Gr~x~
02r , OY2 y = 0
C/
Gr~
oX2
x=o
Tref
(5.30) Using the series (5.8) and (5.9), and the obtained numerical solutions of Equa
158
C O N V E C T I V E FLOWS
tions (5.14) and (5.18) (5.21), we have
I Cf
1 3 2.6201 + 0.8761 Grz ~ + 5.1859 G r z ~o + h.o.t. 1 3 1.8596 + 0.1245 G r x 5 + 2.3932 G r z 10 + h.o.t.
_ I

Gr~ ~
for
P r = 0.72
for
P r = 6.7
(5.31)
_ 1 Cf
_ 1"
Gry ~
I _ 1 Grz 5 Ta
1.7479 + 1.9813 Gry ~ + h.o.t,
~
for
P r = 0.72
for
P r = 6.7
1
0.5918 + 0.9794 Gry ~ + h.o.t,
1 1 + 0.9875 G r z 5 + 5.9166 G r z 1 1 + 0.3895 G r z 5 + 4.3291 G r z
(5.32)
3 ~o + h.o.t,
for
P r  0.72
~o + h.o.t,
for
P r  6.7
3
(5.33)
Also, the equivalent global heat flux, Q, is given by the expression /0 c~ Q  Pr
f~ hi d~? 
{ 1.0915 2.6446
for for
P r  0.72 P r  6.7
(5.34)
We note from the expressions (5.31) and (5.33) that the skin friction and the adiabatic temperature on the axis of the plume (vertical wall) are underpredicted by the firstorder boundarylayer solution. It is also seen that the thirdorder correction terms add to the secondorder terms in the underprediction of the values. ~br values of G r z less than O (105) it is observed that the errors which occur in using the firstorder theory are in excess of the order of 10%. On the horizontal wall the secondorder correction reinforces the firstorder correction term to further increase the magnitude of the skin friction. This increase in the skin friction coefficients and the adiabatic wall temperature implies a decrease in the thickness of the boundary layers for moderately large values of the Grashof numbers. Further, we note from the expressions (5.31) and (5.33) that the effect of the horizontal wall leads to a correction which is O ( G r z  ~ ) . The first eigensolution which modifies these results is O
r~ ~
and therefore expressions (5.31)  (5.33) are correct to the number of
terms quoted. In fact, the eigensolutions form the next correction to the skin friction coefficients and the adiabatic wall temperature. In the work by Afzal (1980) he ignored the effects of the boundarylayer which is formed on the horizontal wall and hence his solution technique is only correct up to, and including, his second term. In turn, this leads to smaller corrections to the skin friction and adiabatic temperature on the axis of the plume (vertical wall) and the presence of the horizontal wall substantially changes the thirdorder boundarylayer correction terms. The fluid flow pattern outside the inner boundary layers is shown in Figure 5.3 for P r = 0.72 and G r = 10 l~ It is seen from this figure that at a small distance from the horizontal wall that the effect of the boundarylayer is to make the streamlines enter the convective boundarylayer such that they are convex upwards, whereas in the
C O N V E C T I V E F L O W IN BUOYANT PLUMES AND J E T S
1.0
r = 0
//
r = 0.01 i o.Io5
0.8
/ /'/
0.6
/
X
///
I~
/
/
!i
I/
,F
= o.o15
.,,_
I/
~'7 ~176
// //
0.4 0.2
159
,,Y
//
/ 7
///_
0.0 0.0 ' 0 : 2 ' 0 : 4 ' 0 : 6 ' 0y: s '  i : 0 Figure 5.3" The streamlines associated with the outer flow at Gr  10l~ for Pr = 0.72. The 1term and 2term forms of expression (5.5) are indicated by the broken and solid lines, respectively.
absence of the horizontal boundarylayer the streamlines enter convex downwards. It transpires that, rather than becoming less important, viscosity becomes more and more dominant as one moves to the outer edges of the inner boundary layers and the outer flow is inviscid. It is thus quite conceivable that the complete NavierStokes equations have to be solved in the outer region. This is also suggested by the algebraic behaviour of the functions ( ] 2 , f 3 ) given by Equations (5.23). This matter has been also discussed by Schneider (1981), who pointed out that wall jets and plumes which are limited by walls induce non potential outer fluid flows.
5.3
Inclined wall p l u m e s
We consider a buoyancyinduced wall plume of heat flux qs along an inclined adiabatic plate, which arises from a line heat source which is embedded at the leading edge of the plate, see Figure 5.4. The flat plate is inclined with an arbitrary tilt angle ~, covering the range from the vertical to the horizontal, i.e. 0 ~ T ~ 5" ~ On the basis of the boundarylayer approximation, the governing equations are given by the Equations (3.38)  (3.40) and they have to be solved subject to the boundary conditions u=0, u+0,
v=0, T+Too,
0_T_T= 0 Oy
P+Poo
on
y0,
x>0
as
y+c~,
x>0
(5.35)
160
C O N V E C T I V E FLOWS
I J I
g
,~
J i
Line Iteat h l Source Figure 5.4:
Physical model and coordinate system.
along with the constraint condition on the energy conservation, namely
pcpl
u (T  Too) dy  qs for x > 0
(5.36)
fO ~
To solve this problem Lin dimensional quantities
et al. (1996) have proposed the following non
~ ( x )  [l ff (alRa'sin~)~] 1 (~1n ~ ~os ~) 1 x~  ( ~ Ra~ ~os ~) ~ + (~l/~a~ ~in ~) ~ } (~lRa~ ~os ~)~ = ~ 1
1
1
( ~ Ra~ sin ~) (5.37) where (71 is defined as in expression (2.72) and the Rayleigh number Raz is based on the temperature T* = p C pq~ It should be noted that for the limiting cases of a ot f l " vertical plate (~o  0) ~  1 and for a horizontal plate (qo  ~) ~  0. In addition to the nondimensional quantities (5.37), the following nondimensional variables are introduced:
]
(5.38)
Using Equations (5.37) and (5.38) the boundarylayer Equations ( 3 . 3 8 )  (3.40) transform to the form:
pr fm + 5 ; ~ ff,,  5 ~1 f ,2 + 5 10~ r / h '  2 ~ h + ( l + P r ) ~ 5 0 __ ~__d((X_ () (f, Of'
o~
h'  (1 + e~)(1  ~)60
f,,Of
Oh)
(5.39)
(5.40)
CONVECTIVE FLOW IN BUOYANT PLUMES AND JETS
0"+
5+~ 10 (fO)'

1
161
( f, OO _ O,i)f )
~(1~)
(5.41)
which have to be solved subject to the boundary conditions (5.35), namely f(~,O)=O, f'(~,O)=O, 0 ' ( ~ , 0 ) = 0 for ~ > / 0 f'+0, h~0, 0+0 as 7 7 ~ c ~ , ~>~0
(5.42)
and the constraint condition (5.36) which can be written as
o~ f ' O d ~  I
for
~0
(5.43)
For the limiting case of a vertical wall plume (~  1) we have h = 0, see Equation (5.40). Thus, Equations (5.39) and (5.41) reduce to the following similar form 3 f f ,  ~J 1,2 + (1 + Pr)O  0 P r f ' " + ~
(5.44)
3 O" + ~(fO)'  0
(5.45)
while for the other limiting case of a horizontal plume (~ = 0), Equations (5.39) (5.41) become
1
P r f ' " + 2 f f " + ~ (1 + P r ) ~ 0  0 1
0"+~(fO)'
(5.46)
0
(5.47)
Both the sets of equations (5.44), (5.45) and (5.46), (5.47) are subject to the boundary and constraint conditions (5.42) and (5.43). Equations (5.39)  (5.42) have been integrated numerically by Lin et al. (1996) using the Kellerbox method with a slight modification to include the integral condition (5.43). The flow characteristics determined are the fluid velocity component, u, and the temperature, T, namely
u
1
f,
,
TToORa~
(_~) = ~~(fflRaz cos ~) ~ (~ ~),
T*

~
(al cos ~)~
0(~ 71) '
(5.48)
as well as the nondimensional skin friction, 7w(~), and the wall temperature, 0~(~), which are given by
Tw(~) = (al cos~)~ /"
~3
,
(~, O)
~
~ 0((, 0)
Ow(~) : (o1 cos V) g
(5.49)
Typical fluid velocity and the temperature profiles for some specified values of the inclined angle ~ are shown in Figure 5.5 for P r = 0.7 and Raz = 105. It can be seen
162
CONVECTIVE FLOWS
(a)
(b) 80
1.2 1.0
u 60
0~ 30~, 45~, 60~, 75~, 80~, 85~
0.8 40
o
o
o
t ~06 0.4
20
0.2 0
0
2
4
6 I 8 Ra~ Xy
10
0.0 1
Ra~ yX
Figure 5.5" (a) The fluid velocity, and (b) the temperature, profiles for Pr  0.72 and Raz  105.
from these figures that the fluid velocity profiles decrease with increasing values of the angle ~, while the t e m p e r a t u r e profiles increase as the angle ~o increases. This is due to a decrease in the component of the buoyancy force along the wall with an increase in the tilt angle cp. For the limiting case of a vertical wall plume (9  0 and ~  1), relations (5.49) reduce to 3
Tw(1)  ~ f " ( 1 , 0 ) ,
1
0w(1)


O'1 5 0(1, 0)
(5.50)
T h e values of T~(1) and 0w(1), as calculated by Lin et al. (1996) from the direct numerical solution of Equations (5.39)  (5.43) for ~ = 1, are compared with the similarity solution obtained by Liburdy and Faeth (1975) in Table 5.1 for several values of P r between 0.001 and 1000. It is easily seen that the agreement between these results is very good and this suggests that the numerical solution obtained by Lin et al. (1996) is uniformly valid over the entire range of the plate inclination from the vertical to horizontal. A simple, but very accurate, correlation equation was also proposed by Lin et al. (1996) for predicting the nondimensional wall t e m p e r a t u r e of the inclined wall p l u m e problem for 0.001 ~< P r u ~ Q ~ (tan qo)5 and thus the boundarylayer takes on a pancakelike structure as shown in Figure 5.7. On substituting expression (5.59) into Equations ( 5 . 5 1 )  (5.55) gives 11
v T . ~ + 6~  0 lu2
_
~ . V T U  9
4
(5.60)
02u 0t9 02w Oz ~ Oy2
14
~" V T ~ +  ~ u ~  8~zu2 =
op Oz ~. v ~ o where
(5.61)
 Oy 2 + 0
7 9 zO
(5.62)
= 0
11
(5.63)
1 020
 ~ 0  p~ Oy2
5  v  ~ yu,
(5.64)
7 ~  w  ~ zu
(5.65)
are the scaled velocity components normal to the curvilinear coordinate surfaces y  constant and z  constant, respectively. Also, 9  (z~,~) and VT is the gradient operator in the transverse plane and is defined as ( o , b~.~) The boundary conditions (5.56) far downstream of the heat source and the constraint relation (5.57) also become u0,
u~O,
~=0
'
~=0,
0Oy~ = 0
~~o, 0~o, p+o f o f~,~ uOdydz  1
on
y0,
z>/0,
as
(I~I,Y)+cr
as
x 4 c~
x + cx:)
x 4 cx~
(5.66)
Equations (5.60)  (5.64), subject to the conditions (5.66), have been solved numerically by Higuera and Weidman (1998) using finite differences in combination with a pseudotransient method that essentially amounts to adding time derivatives to the lefthand sides of Equations (5.61), (5.62) and (5.64) and marching in time until a steady state solution is attained. Figure 5.8 shows the local thermal, 50(z), and local momentum, 5u(z), boundarylayer thicknesses, which are defined as
5o(z)  Omax
O(y, z) dy,
5u(Z) 
1/0
'ttma x
u(y, z) dy
(5.67)
as a function of z for Pr = 0.1, 0.2, 0.5, 1 and 5; here 0 m a x  0 ( 0 , 0 ) and Umax  maxy,z(U). Also presented in this figure is the wall temperature at each
CONVECTIVE FLOW IN BUOYANT PLUMES AND JETS
167
1.0
e (z)
5o(z), 5,,(z)4
< •," h .  , Pr : 0.1, 0.2, 0.5, 1,5
0.5
0.0
O, o
3
z
6
9
Figure 5.8: The thermal, 5o (z) , and momentum, 6u (z) , boundarylayer thicknesses, indicated by the solid and broken lines, respectively, and the wall temperature distribution, o~(z) indicated by the dotted lines, for several values of Pr 0max ~
section, Ow(z) = O(z, 0), divided by 0ma x. As expected, the thicknesses are maximum at the centreplane and decrease with the spanwise distance z. Further, it is seen from Figure 5.8 that the thermal boundarylayer (solid lines) is thicker than the momentum boundarylayer (dashed lines) for some small values of P r (< 1) and vice versa for some large values of P r (>> 1). However, the spanwise width of the momentum layer is larger than that of the thermal layer for all values of thef Prandtl number P r . This may be an effect of the slow decay with z of the term (0 +  ~ ) , J which acts as a forcing term in the momentum Equation (5.61), compared with the fastest decay of the term 189 in Equation (5.64). Results of this problem were also presented by Higuera and Weidman (1998) for both small and large values of the Prandtl number. Appropriate scales from the variables (y, z, u, ~, ~, 0, p) in the thermal layer, involving the thermal diffusion in Equations (5.60)  (5.64) for small and large values of P r are used. 5.4.2
Vertical adiabatic wall
This corresponds to the situation in which a vertical plume (~p = 0) is cut down its centreline by an infinitely thin insulated wall, and a scaling similar to that for a vertical plume which emanates from a point source in an unbounded fluid may be used, namely
168
C O N V E C T I V E FLOWS
1
Uc 
1
,
Oc
=
qs
_,
Pc=
ux
(qs u) ~ _
3
1
u4 x 2
,
1_
x
1
(~su)
,
,
~ s4
~. (5.68)
The fact that diffusion occurs equally in both the spanwise and platenormal directions (Yc  Zc) were used in establishing relations (5.68). On transforming the dependent and independent variables according to the expressions (5.58), with the scales given by expressions (5.68), yields the equations governing the transverse structure of the vertical wall plume, namely VT "~ +u ~" VTu V "" . V T V 
41u2X
 0
(5.69)
 V ~ u + O
(5.70) 2 OX + V~'v +
   V T p 
1 V20
v'VTOuO
(5.71) (5.72)
Pr
where again ~ 
VTU
(~, ~) with ~ and ~ given by _v   v
y2u'
w_  w
z2u
(5.73)
and X = (z, y). The boundary conditions and constraint equation are the same as those given by the conditions (5.56) and (5.57). Equations (5.69)  (5.72), subject to the conditions (5.56) and (5.57), have also been solved numerically by Higuera and Weidman (1998) using the same pseudotransient method similar to the case described in Section 5.4.1. Some isotherm and streamline plots, as well as a fluid velocity vector field, given by arrows, which represent the transverse velocities (w, v) are shown in Figure 5.9 for P r = 1. Here the streamwise velocity u acts as a sink for the transverse flow and this is in agreement with expression (5.69). In contrast to the case of an inclined wall plume, as described in Section 5.4.1, an appreciable fraction of the fluid comes now from the sides of the insulated wall and is affected by the presence of this wall, see the region of low transverse fluid velocity in the lower right region of Figure 5.10. The case of a wall plume far downstream of a point or line heat source on an isothermal vertical wall (~ = 0) has also been treated by Higuera and Weidman (1998). This problem leads to selfsimilar solutions of the second kind, in which the rate of decay of the temperature with the distance from the source has to be determined as an eigenvalue of a nonlinear problem. To this end, a few comments are noteworthy. Using the matched asymptotic technique, Thomas and Takhar (1988a) obtained the first and secondorder boundarylayer equations for the case of free convection due to a point source of heat, the
C O N V E C T I V E F L O W IN B U O Y A N T P L U M E S A N D J E T S
10
169
II l I 1 1 1 1 1 1 t / / l / Z /
I l / l 1 ///l/lt/zz~
t_l] l 1 1 l l / l l / Z z ~ ~ 1 ['t/t~
1 1 t / z z z ....
~./_ 1/'/'K///z.....
i
0
'
0
'
5
1"5
10
Z
Figure 5.9" Isotherm plots (solid lines) between 0  0.02 and 0.16 and streamline plots (broken lines) between u  0.05 and 0.3 for P r = 1. The arrows represent the transverse velocity field (w, v).
(a)
(b)
'51i, 10
/
i'l 
iii111/II i ifj / 0.4
0
;
5
5
o
o
\
\
..
o
~
0.4
Figure 5.10: Fluid velocity, u, (broken lines) and temperature, 8, (solid lines) profiles ]or P r  1. The dotted line in the right hand figure represents the wall temperature Ow profile.
170
C O N V E C T I V E FLOWS
surrounding fluid being bounded by a conical surface, while Riley (1988) considered the case of mixed convection flow above a point heat source in an unbounded fluid. It has become increasingly apparent that mixed convection regimes occur frequently in practice and that an understanding of these flows is crucial in many processes, e.g. in the dispersion of pollutants and the study of chemical reactions in plumes. In the study of this problem, Riley (1988) was motivated by some difficulties which were encountered in an experiment that he had conducted to measure the temperature and the concentration fields above a smouldering substance because of the instability of the plume. However, he observed that this experimental flow situation may be modelled by a mixed convective flow consisting of a vertical uniform stream passing over a fixedpoint heat source. In this respect, Riley (1988) investigated a more accurate representation of the fluid flow and temperature fields due to a point source of heat in order to improve the description of the experiment, and also to facilitate the investigation of nonparallel flow stability. The results obtained by Riley (1988) are restricted to the case of the Prandtl number being unity. It was found that there exists an unchanging balance between the buoyancy and forced convection effects and there are regions where the forced convection dominates and the asymptotic regions where buoyancy dominates. The balance essentially arises because the centreline fluid velocity in the plume above a point heat source does not vary with height, see Fujii (1963) and Fujii et al. (1973), and thus it is in a constant ratio to the imposed flow. In general, the literature on buoyant plumes and jets deals with plumes and wall jets due to fixed (still) surfaces. However, Wang (1988) has investigated the horizontal boundarylayer due to a line heat source on a moving adiabatic flat plate. He has shown that a horizontal boundarylayer does not exist if the line heat source is still, whilst it exists if the line heat source is moving laterally. Therefore, the boundarylayer caused by a horizontal flat plate is distinctly different from that due to a heated line source on a moving adiabatic wall. As is known, the boundarylayer of the former exists only on the top surface, while the boundarylayer of the latter exists on the bottom of the plate, and only when the heat source is moving with some speed. If a cold source is substituted for a heat source then the results of Wang's problem apply to the top surface instead.
5.5
Laminar plane buoyant jets
Hot fluid which discharges from a narrow slot into a large quiescent fluid reservoir of lower temperature is termed either a plane free jet or a wall jet if it propagates tangentially along a flat surface. The theory of viscous buoyant jets has received great attention in the past due to the many applications of these jets in industrial systems and environmental studies, such as mixing, ocean circulations, and air or water pollution. In practice, the jet flows are turbulent and most of the previous
CONVECTIVE FLOW IN BUOYANT PLUMES AND JETS
171
investigations deal with turbulent buoyant jets, see Rodi (1982) and List (1982). However, laminar buoyant jets have also extensively been studied, see Schneider and Potsch (1979), Jaluria (1986), Yu et al. (1992) and Noshadi and Schneider (1999). It is known, see Savage and Chan (1970) or Mollendorf and Gebhart (1973a, 1973b), that buoyant jets behave like a pure forced convection (momentum) jet in the region close to the nozzle, i.e. where the buoyancy force is negligible in comparison to the inertia force. In the far downstream region from the nozzle, where the buoyancy force is dominant, the buoyant jet is equivalent to a buoyant plume arising from a line heat source. The most simple and traditional methods to describe these limiting jets and plume flows are based on the use of selfsimilar solutions. However, these solutions do not, in general, apply to buoyant jets. An attempt has been made by Maxtynenko et al. (1989) to obtain selfsimilar solutions for a class of plane and axisymmetric buoyant jets on the basis of boundarylayer theory and the Boussinesq approximation. Both the cases of linear and quadratic dependence of the density on the temperature were considered. However, these solutions are far from being adequate to comprehensively describe the flow pattern to a high level of accuracy over the entire range of buoyancy intensities. The numerical methods employed by several researchers involve the solution of two systems of partial differential equations. One of these systems describe the perturbation of the purely forced convection (momentum) jet (the case of a weakly buoyant plume) and the other system describes the perturbation of a purely free convection flow (the case of a strong buoyant plume). Results of this kind of simulation have been reported by Rao et al. (1984) and Wilks et al. (1985). However, this procedure is not free from complications and in particular for the case of the opposed wall buoyant jets where singularities may occur, and this is responsible for a reduced rate of the convergence or even the complete loss of stability in the numerical solutions obtained. Yu et al. (1992) have proposed a very effective and accurate method to study the free and the wall buoyant jets for the entire range of intensities of the buoyancy force including the intermediate region where the inertia is comparable with the buoyancy. In addition, simple but very accurate correlation equations for the centreline fluid velocity and the temperature of the free buoyant jet, as well as the skin friction and the wall temperature of the wall buoyant jet, have been proposed by these authors. We next present some results obtained by Yu et al. (1992) for these free and wall buoyant jets. Consider a laminar, plane buoyant jet of an incompressible fluid which emerges vertically from a long, narrow slit of width b and which spreads into a quiescent fluid reservoir of a constant temperature Too and it is assumed that the fluid temperature at the slit (exit) is To, where To > Too. For the free jet the flow is unconfined after discharging from the slit, while for the wall jet the flow develops along an adiabatic vertical fiat wall. Under the boundarylayer and the Boussinesq approximations, the
172
C O N V E C T I V E FLOWS
governing equations for this physical problem are as follows: Ou Ov Ox + Oyy  0 Ou
Ou
U ~ x + v~~v 
OT
OT
~~
(5.74)
02u u~~u2 4 gj~ (T  Too)
(5.75)
~, 02T
+ v 0~ = p~ 0y~
(5.76)
where the x and y axes are measured in the upward and horizontal directions, respectively, and the + signs in Equation (5.75) designate the assisting or opposing flow cases. The boundary conditions appropriate to Equations (5.74)  (5.76) are given by
o__~_u= 0 Oy
v 0 free jet
OT = 0
u+0,
T+Too
Oy
u0
wall jet ~ as
on
y+c~,
y0,
x>0
(5.77)
x>0
In addition to these boundary conditions, three integral constraints should be considered in order to obtain nontrivial solutions, and these constraints can be obtained using the classical procedure first proposed by Glauert (1956). Thus, we have
df
U2 dy
~ u 2dy
) ]
dx d dx
[/0
c~
p cp
u
~
dy


F

g/~(TToo)dy u
for
]
free jet
(5.78a)
g f l ( T  T o o ) dy dy
for
u (T  T ~ ) d y 
x>0
~ Qo
for for
x > o x>0
x>0
wall jet
(5.TSb) (5.78c)
free jet wall jet
where Q0 is the rate of heat flow which discharges from the slit per unit length. In the region very close to the slit, i.e. x + 0, the buoyant forces are negligible and the jets are nonbuoyant. Therefore, we have the classical relations J0  lim
x+0
K0  lim
x+0
/?
c~
p u 2 dy 
pu
constant p u 2 dy
dy = constant
for
a free jet
(5.79a)
for
a wall jet
(5.79b)
By introducing the mean initial velocity u0 of the jets, which is defined as follows, uo 
N
~
for
a free jet
(o~) ~
for
a wall jet
(5.80)
C O N V E C T I V E F L O W IN BUOYANT PLUMES AND J E T S
173
the quantity Qo is then given by Qo = pcpuob ( T o  Too). Further, we introduce the nondimensional variables defined by Yu et al. (1992), namely ~(x)=(l+~)I
(y)x '
r/),4
A4x(TT,
o(~,n)
r
(5.81) where
(RebR~)~
for
a free jet (5.82a)
Gr~
),4 
{
i Gr~
for
a wall jet
(RebRez)~ + Gr~ (Re, Rex) 88+ Gr~
for
a free jet
(5.82b)
for
a wall jet
Here Rex, Reb and Grz are defined as
R e x  uo___x_x R e b  uob v' v'
G r x  g~T*x 3 v2
(5.83)
where T*  pcpv Qo = ( T o  Too)Reb. It should be noted that the coordinate ~ plays the role of a buoyancy or a mixed convection parameter. In the region near to the slit, i.e. x + 0, the buoyancy force can be neglected and the flow configuration corresponds to a pure forced convection (momentum) jet. In this case Rez >> Grx, so that ff + c~ and ~ = 0. However, far downstream from the slit the buoyancy force is dominant and the system behaves like a buoyant plume arising from a point thermal source. Now Grz >> Rex and thus ff + 0 and ~ = 1. 5.5.1
Free jet
Using the variables (5.81), Equations (5.74)  (5.76) become
f"'+
5 + 4~ f f,, +
15
15
1 0" +
Pr
15
4 (50 
((1  ~)
(fO)' =
~(1  ~)
f, Of'
o~
(
if, Of
~
f, O0 _ 0,0/ ~
)
(5.84) (5.85)
and they have to be solved subject to the boundary and integral conditions (5.77) and (5.78), which can be written in nondimensional form as follows: f(~,O)=O, f"(~,O)O, + o c , 040 as f,
0'(~,0)0 r/+oc
(1  ~)~ ( I o I '~ d.) + 3 So S'~ d .  ~ 4 S o Od, lim f : ft2 dzl  1
~+0
f o f'O drI 
1
for
~> 0
(5.86)
174
CONVECTIVE FLOWS
where the primes denote differentiation with respect to ~. For the limiting case of ~  0, Equations (5.84)  (5.86) reduce to the equations which describe a momentum free jet, see Wilks et al. (1985), namely
f"F f(O)O,
~f f " + ~f,2 _ O, aO'F ~fO 0 f'(O)O, f'+O, 0  + 0 as r/+oo oo 1 oo 1 fo f ' 2 d r /  7 , fo f ' O d r l  ~
(5.87)
On the other hand, for ~  1 the Equations (5.84)  (5.86) describe a pure buoyant line plume, see Fujii et al. (1973), namely
f,,, + 3 f f , ,  ~ lf,2 : t : 0  0 , f(0)0,
f'(0)0,
.r 0+0 f'~0, 1 f o f'O dr/  ~
as
r/~~
(5.88)
Equations (5.84)  (5.86) have been solved numerically by Yu et al. (1992) for P r = 0.7 and 7 using the Kellerbox method along with a specialised algorithm which deals with the integral constraints (5.86). Only the case of assisting flow has been considered and the technique employed is described in detail by Yu et al. (1992) and therefore it is not presented here. Figure 5.11 shows the variation of the nondimensional centreline fluid velocity, Uc(X), and temperature, 0c(X), profiles, i.e.
(a)
(b) 101
. . . . . . . Assisting Flow
102
.,
,
.
,.
Assisting Flow 101 o~(x) 100
u~(x) 100
10 1
r
102 .
I0
I
.,
,
,,,,,
X
9
0
.,, ~
.
I I
.~
~ ~
~ ~
~

I I
~
0
l~i .,_4 ~ bD
10 3
11;s i0'2 i(~1 "1~}0 '1~}1 "l~}e "10s X
Figure 5.11" Centreline (a) fluid velocity, Uc(X), and (b) temperature, Oc(X), profiles of a free buoyant jet for Pr = 0.7 (solid line) and Pr  7 (broken line).
C O N V E C T I V E FLOW IN BUOYANT PLUMES AND JETS
175
1
= r
~21,(~, o)
(5.89)
 r r o(~, o) as a function of the vertical distance 1 15
c;o
,
:r
(5.90)
It is seen from these figures that initially the centreline fluid velocity decreases with X to a minimum value and then increases continuously. Physically this means that the flow decelerates near the slit and accelerates far downstream from it. In the region close to the slit, the centreline fluid velocity is retarded by the viscous forces, whilst in the far flow region the buoyancy force increases and drives the fluid to flow faster. However, the centreline temperature decreases monotonically with increasing distance X. On the other hand, Figure 5.11 reveals that there are two different slopes corresponding to the jet region and the plume region, respectively. The region of transition from the momentum free jet to the buoyant plume jet is near X = 0.5. We also note that both the centreline fluid velocity and temperature at Pr = 7 are higher than those for Pr = 0.7. 5.5.2
Wall jet
Again using Equation (5.81), Equations (5.74)  (5.76) take the form f"'+
5 47~ f f,, + ~5  7{ f ,2 • 20
o_
10
7 (1
"26~
5+7~ 7 10"+~(fO)'= P~ 20 ~
(1


~) ( f, Of' \
~)
O~
f,,Of
(O 0 0 f ) 0 ' f' ~
(5.91) (5.92)
and the appropriate boundary and integral conditions are as follows: f(~,O)O, f +0,
ff(~,O)O, 0+0 as
0'((,0)0 r/+~
( 1  ~)~ (So : : '~ d~) + 4 So : : '~ d ~  ~ So :Od~ lim f o f f,2 d~7 
~~0
for ~ > 0
(5.93)
1
This set of equations can be readily reduced to the set of equations for a nonbuoyant wall jet ( ~  0), see Glauert (1956), namely
fro+ 88 f. + 89
_ O.
f(o)  o, f'+O,
0+0
~70'+ 88  0
f"(o)  0 as
f o f f,2 dT1=
r/+cr 1
(5.94)
176
CONVECTIVE FLOWS
and to the buoyant wall plume, see Liburdy and Faeth (1975), namely
fm+~ ff''lf'2+OO,
AOtpr + 3 f 0 ' = 0
f(0)  0, f'(0)  0 f'+0, 0+0 as 77+oo
(5.95)
f o f f'~ d,  ~ f o f o d, Equations (5.91)  (5.93) have been solved numerically by Yu et al. (1992) for Pr  0.7 and 7 for both the assisting and opposing flow cases. The variation of the nondimensional wall temperature
Ow(X)  (Tw  Too
 ( ~ ~ 0(~, 0)
\Grb]
(5.96)
as a function of the nondimensional distance 1 2O
(5.97)
_
is illustrated in Figure 5.12. It can be seen that 0w(X) decreases with X and that there are also three distinct regions, namely the jet region (X < 0.1), the intermediate region (0.1 < X < 5), and the plume region (X > 5). 101
.....
OpposingFiow
o~(x) 100.
10x
102 102
,!
,I
~
101
100
X
1{)1
102
Figure 5.12: Nondimensional wall temperature distribution, Ow(X), of a wall jet for P r  0.7 (solid line) and P r  7 (broken line).
Finally, the skin friction coefficient 
(1  f )  3 f , , ( f ,
O)
(5.98a)
CONVECTIVE FLOW IN BUOYANT PLUMES AND JETS
177
or
Cf (Re3Grb 5) ~  ~ r
0)
(5.98b)
is plotted as a function of r in Figure 5.13(a) and as a function of X in Figure 5.13(b) for both the assisting and opposing flow cases. It is seen that the skin friction coefficient attains a constant value of about 0.13134 for r < 0.2 and then increases or decreases with ~ depending whether the forced flow is assisting or opposing the buoyancy force. However, it decreases with the distance X in both the assisting and opposing flow cases, see Figure 5.13(b). (a) 7
(b) 102
102 g.
Assisting F l o w / / /
q~
.,~ 101
//I/l/l////lrr//l
~3 ~
101,
lO 0
Assisting Flow 100, 101 upposing
upposing
ow
102
10
.
'
l~ 0
r
10
1
10
ow .
150
.
.
X
Figure 5.13: Variation of the skin friction coefficient of a wall jet for Pr  0.7 (solid line) and P r  7 (broken line) as (a) a function of ~ and as (b) a function of X.
Other flow characteristics, such as the fluid velocity and the temperature profiles, as well as some very accurate correlation equations, can be found in the paper by Yu et al. (1992). However, it should be mentioned that only the buoyancy assisting jets and the buoyancy opposing jets with a slight negative buoyancy force can be studied by the method proposed by Yu et al. (1992). A breakdown in the numerical integration of the governing equations occurs as the negative buoyancy force increases to a critical value of the parameter ~, namely ~  ~c  0.31 for P r = 0.7 in the case of a buoyancy opposing wall jet. The numerical integrations show that the slight negative buoyancy force does not reverse the jet flow but only retards it.
Chapter 6
C o n j u g a t e heat transfer over vertical and h o r i z o n t a l flat plates 6.1
Introduction
In the traditional area of convective heat transfer between a solid wall and a fluid flow the wM1 conduction resistance is usually neglected, i.e. the wall is assumed to be very thin. In this case it is usual to prescribe either the wall temperature or the wall heat flux, and a considerable amount of research work has been done in order to understand the heat transfer characteristics over a wide range of flow configurations and fluid properties. However, in many real engineering systems the wall conduction resistance cannot be neglected since conduction in the wall is able to significantly affect the fluid flow and the heat transfer characteristics of the fluid in the vicinity of the wall. In order to take account of physical reality, there has been a tendency to move away from considering idealised mathematical problems in which the bounding wall is considered to be infinitesimally thin. Thus, the conduction in the solid wall and the convection in the fluid should be determined simultaneously. This type of convective heat transfer is referred to as a conjugate heat transfer process and it arises due to the finite thickness of the wall. Conjugate heat transfer effects are of considerable importance in many practical problems, e.g. in ablation or perspiration cooling problems as well as in heterogeneous chemical reaction situations, where information on the interfacial temperature and concentration distributions is essential because the transfer characteristics are mainly determined by the temperature and concentration differences between the bulk flow and the interface. Further, these effects occur also in the design of thermal insulators and in material processing and geothermal systems. In particular, it has been ascertained that free convection can induce the thermal stresses which lead
180
CONVECTIVE FLOWS
to critical structural damage in the piping systems of nuclear reactors, see Hong (1977). It is also worth mentioning that recent demands in heat transfer engineering has led to new types of heat transfer equipment which has a superior performance, especially in compact and lightweight equipment. Increasing the need for smallsize units, research has focused on the effects of the interaction between the development of the boundarylayer near to the wall and on the axial wall conduction, which usually degenerates the performance of the heat exchanger. It appears that the topic of conjugate heat transfer was originated by Perelman (1961), who was the first to study the boundarylayer equations for the fluid flow over a fiat plate of finite thickness with twodimensional thermal conduction taking place in the plate. He derived theoretical expressions for the interfacial temperature and the local Nusselt number. The investigation was then extended by Luikov et al. (1971) and since then various types of conjugate heat transfer problems have been studied. Conjugate problems can be roughly classified into the following three groups from the viewpoint of fluid flow situations: (a) one or two forced convection flows; (b) one or two free and mixed convection flows; and (c) one forced convection flow and one free or mixed convection flow. In the earlier investigations, the first type of problem (a) was intensively studied because the fluid flow was not greatly influenced by the heat transfer in the incompressible fluid and the equations of motion and energy could be solved separately. However, problems of the types (b) and (c) have been more recently investigated because of the difficulties in solving the coupled equations of the fluid flow and the heat transfer simultaneously, but the present computer facilities has lead to detailed solutions of these types of problems. From the class (b), one case which has received much attention in the past has been that of the twodimensional free convective flow of a viscous incompressible fluid heated by a vertical conducting fiat plate of finite thickness. The early theoretical and experimental work for a viscous fluid has been reviewed by Gdalevich and Fertman (1977), Miyamoto et al. (1980) and Martynenko and Sokovishin (1989). However, the most recent contributions on this subject may be found in the papers by Pozi and Lupo (1988, 1989), Vynnycky and Kimura (1996), Merkin and Pop (1996), and Pop et al. (1996b). Important contributions to the cases (b) and (c) of flow t y p e p r o b l e m s were made by Lock and Ko (1973), Anderson and Bejan (1980), Viskanta and Lankford (1981), Sakakibara et al. (1992), Cdrdova and Trevifio (1994), Trevifio et al. (1996), M@ndez and Trevifio (1996), Camargo et al. (1996), Chen and Chang (1996, 1997) and Shu and Pop (1999). In the next two sections, we present detailed results for the conjugate free or mixed convection flow over a vertical conducting flat plate of finite length and thickness, while in the last section of this chapter we give results for the conjugate free convection boundarylayer past a finite horizontal flat plate. It has been concluded from these studies that it is very difficult to obtain analytical solutions of conjugate heat transfer problems due to the matching conditions at the solidfluid interface, but the use of numerical methods, such as finitedifference schemes, is the most
CONJUGATE HEAT TRANSFER
181
promising procedure for performing this matching.
6.2
Conjugate plate
free convection
over a finite vertical flat
Consider the steady free convection flow over a vertical flat plate of length I and thickness b in a viscous and incompressible fluid of ambient temperature Too, see Figure 6.1. It is assumed that heat is transferred from the outside surface of the plate, which is maintained at the constant temperature To, where To > Too. The energy equation in the solid plate is given by
02T~ 02T~
0~ + 0y 2 = 0
for 0 ~ ~ l ,
b~ O,
~Oy2 ~ O,
Of + O,
w +
Of + O,
w +
02r
Oy 2
02r OX 2
1
as a~
y + :t:cx~,
(6.43a) (6.43b)
x>0
and v  + 0, u+0,
02r
Ox~ ~ O,
02r Oy 2
_
~0
'
00__L _ Ox
~ 0,
w +
OOf ~~0, Oy
w+
02r
Oy 2
as
x
02~r) OX 2
as
y++c~
lyl
cr
'
x>0
1
(6.44a) (6.44b)
respectively. Here c = ab denotes the aspect ratio of the plate and k  k~ is known as the conjugate parameter. In this formulation the local and average Nusselt numbers are given by
( bT )==0~176 for
7'
lyl
1
and
Nu =
/_i2 N u d y
(6.45)
The problem posed by Equations (6.38)  (6.41), along with the boundary conditions (6.42)  (6.44), is now treated for the case when R a >> 1, i.e. in the boundarylayer regime, and also for finite, but moderately large values of R a by solving the full Equations (6.38)  (6.41) when expressed in elliptic coordinates. The situation when R a >> 1 assumes that, at the interface between the plate and the fluid, that there are thermal and viscous boundary layers whose convective heat flow is coupled to the conductive heat flow within the solid. Denoting by 00 the value of the constant temperature, Os, at the location ( 0 ,  8 9 where it is evident that 0 ~< 00 0 and Os = 00 = 0. Also, two distinct values of P r are to be considered for an asymptotic solution. For 00 > 0 and P r ~ O(1) the boundarylayer is locally temperature driven, so that the appropriate variables are as follows: r
Ra 88
w  Ra~gt,
x
1
(6.46)
RaZX
On substituting these expressions into Equations (6.38)  (6.40) and letting R a + cx~, then, on eliminating ~t, we obtain the following boundarylayer equations:
Pr
OY OX 2
OX OXOY
OX 3
O~ OOf
O~ OOf
02 0 f
OY O X
OX OY
OX 2
Of
(6.47) (6.48)
192
CONVECTIVE FLOWS
with Y  y + 1, so that the start of the boundarylayer is shifted to the origin. The boundary conditions (6.42a), (6.42b) and (6.43a) become O,

0tI, o x=O
O~Oy ' o~ ~ O, ox
on
XO,
~O x k ~~176 on OX Of ~ 0 as
XO
YO
O~ ~. Further, there is a singularity in the local Nusselt number here, a feature which the boundarylayer theory cannot pick out. The trailing edge features, whilst quite marked for all the plots as regards N u are less severe for Ob for the larger aspect ratio plate, see Figure 6.9(a). Finally, the variation of the average boundarylayer t e m p e r a t u r e , Ob, and the average Nusselt number, N u , with R a is presented in Figure 6.11 for k = 1~ 2.5 and 10 with c = 1 and P r = 0.1. The agreement between the full numerical and analytical solutions is, as we would expect, better for R a  106 t h a n for R a 102, although even for the latter case, the agreement appears to be sufficiently good for the formulation to be reliably used for a wide range of values of Ra. m
C O N J U G A T E HEAT T R A N S F E R
199
(b)
(a)
6
1.0
i
l
0.8
~
3 0.4
2
0.2 0.0
10 2
1 10 3
10 4
Ra
10 5
0 102
10 6
104
Figure 6.11 Variation of (a) the average boundary (b) the average Nusselt number, N u , with Ra for analytical solutions are indicated by the solid lines ane are denoted by the symbols o ( k  1), A ( k  2.5) an,
6.3
Conjugate mixed convection over a vertical flat plate
Consider a vertical fiat plate of length l and thickne of free stream velocity Ucr and temperature Too. The maintained at a constant temperature To, where To (opposing flow), as shown in Figure 6.12. Introducing the nondimensional variables x = L '
y = Re]
L
'
u =
Uoo '
v = Re 89
Ra
105
106
r temperature, Ob, and I a n d P r  0.1. The full numerical solutions
'k = 10).
mdarylayer
flow
which is placed in a fluid fide surface of the plate is (aiding flow) or To < Too
, o=
T 
Tc ~
To  Too
(6.84)
Equations (6.6)  (6.8) can be written as follows:
or 02r
or 02r
03r
Ox Oy 2
Oy 3
.,
Oy OxOy
(~2
where L   ~ \ k~ )
or oo
or oo
1
Oy Ox
Ox Oy
Pr
is a convective length scale a
parameter, which is defined in Equation (2.147), is
(6.85) (6.86)
is the mixed convection ~l on L and T 0  Too and
200
CONVECTIVE FLOWS
Buoyant Force Vectors
t
T~
To
Aiding Flow Opposing Flow
To>Too ~ b
To0
as
y+c~,
x>0
(6.87)
The boundary value problem governed by Equations (6.85)  (6.87) has been solved by Pop et al. (19965) for Pr  0.7 (air) with the parameter A in the range  2 0 0 ~ A ~ 200 In principle, the method of solution is similar to that used for solving Equations (6.11) and (6.12) for the corresponding problem of conjugate free convection from a vertical flat plate. 6.3.1
Small values of x (((1)
In this case we introduce the transformation 1
r  x 89f (x, 77), 0  x5 h(x, ~),
Yy r /  
(6.88)
X2
and the governing Equations (6.85) and (6.86) transform to
l f f,, + Ax _~h f '" + ~
f ' _ f , , O f ) ~x x ( Of ~x
(6.89)
Pr l h ' ' + 2 fh'   ~ f ' h  x l ( f ' O) h  ~h ' O f~x
(6.90)
CONJUGATE HEAT TRANSFER
201
while the boundary conditions (6.87) become
f (x, O)  O,
f ' ( x , O)  O, h'(x, O)  x89h(x, O)  i f'~0, h+0 as r/+cx~, x > 0
for
x>O (6.91)
The solution of Equations (6.89)  (6.91) is sought for x > 1.

h0(0) 4hl (0) X~ ~...
(6.102a) (6.102b)
CONJUGATE HEAT TRANSFER 6.3.3
Numerical
203
solution
A numerical solution of Equations (6.85)  (6.87), which is valid for both small and large values of x, can be obtained using a finitedifference scheme along with the method of continuous transformation, see Hunt and Wilks (1981). Thus, on defining the variables
~~,
1
r
~2 88
)y,
~2
r
)~F(~,r
o
~ H(~,~)
(1 + ~2)~
(6.103) Equations (6.85) and (6.86) can be written as
F'" +
2 I 3~"2 4(1 +~2)
F F "
~2 F '2 2(1 +~2)+A
~3 (1 + ~2)~
1 ( F l OF' _ _
H
2{ \
0~
2 + 3~ 2 1 1 { OH 1~H"+ FH t  2(1 +(2) F'H  ~ [,F' o~ Pr 4 (1 + ~2)
_
F "OF) ~
(6.104)
H IOF) ~
(6.105)
along with the boundary conditions (6.87), which become
F(~, 0)  0, F'(~, 0)  0
/
H'(~,0)
~ H(~ 0 )  ( 1 + ~ 2 ) 8 8 (1+~2188 ' 1 g'+ (1+(2)~, H+0
for
~:>0
as
~+cx~,
(6.106)
~>0
It should be noted that Equations (6.104)  (6.106) reduce to Equations (6.89) (6.91) for small values of x and to Equations (6.96)  (6.98) for large values of x, respectively. The nondimensional skin friction, zw(~), and the nondimensional wall temperature, 0w(~), are in this case given by 3
Tw(~)  ~1 (1 + ~2)z F"(~, 0)
(6.107a)
Ow(~) 
(6.107b)
~ 1 H(~, 0) (1 +~2) ~
Equations (6.104)  (6.106) have been solvec numerically by Pop et al. (1996b) using a modification of the finitedifference scheme as proposed by Merkin (1969). Figures 6.13 and 6.14 show the nondimensional skin friction, ~'w(f), and the wall temperature, 0w(f), respectively, given by the expressions (6.107) as a function of for Pr = 0.7, and A = 1 and  1 . Also included in these figures are the 1, 4 and 7 terms of the small x solution, as given by the expansions (6.94), and the 2 and 4
204
CONVECTIVE FLOWS
(a)
(b) 0.12
1.5
4 terms
~(~)
....
1.0 4 t ~ r m s /
0.0
0.08 
//
0.5
0.04 
/.,if41 term e , f ~ , 7 terms 0
A
7 terms f'~l~l +.~.~.
~(~)
1
,
2
3
....
"1 4
0.00 0.0
i,
I

I
0.2
0.4
O'.6
Figure 6.13: Variation of the skin friction, Tw((), with ~ for Pr = 0.7 when (a) A  1 and (b) A   1. The numerical solutions, namely Equation (6.107a), are indicated by the solid lines, the 1, 4 and 7 terms of the small x expansion (6.94a) are indicated by broken lines and the 2 and 4 terms of the large x solution (6.102a) for A > 0 are indicated by the dotted lines.
(a) 0.9
(b)
4 terms
1 term
0.6
o,~(~)
0.60.30.0
0.4 .~.~4 terms 1 term i
0
,
0.2 ,
~
0.0 . 0.0
/ / ~
''~' 4 terms
J i.
0:2
014

0.6
Figure 6.14: Variation of the nondimensional wall temperature, Ow(~), with ~ for Pr = 0.7 when (a) A  1 and (b) A =  1 . The numerical solutions, namely Equation (6.107b), are indicated by the solid lines, the 1, 4 and 7 terms of the small x expansion (6.94b) are indicated by broken lines and the 2 and 4 terms of the large x solution (6.102b) for A > 0 are indicated by the dotted lines.
t e r m s large x solution for A > 0, as given by the expansions (6.102). As expected, as the n u m b e r of terms taken in the relevant expansions increases t h e n larger is the range of validity of the solution. Further, when A :> 0 the smaller the value of A
C O N J U G A T E HEAT T R A N S F E R
205
the better do the asymptotic solution approximate to the numerical solution over a larger range of values of (. When A < 0 the boundarylayer separates and hence no large x solutions are given in Figures 6.13(b) and 6.14(b). Clearly, the small x solution cannot accurately predict the point of separation, ~  ~s, as it cannot even predict the position of the maximum skin friction, Tw(~). As the parameter A approaches negative infinity then the location of the point of separation appears to approach zero. Thus, several possible variations of ~  ~s(A) have been investigated and it has been found that for  2 0 0 < A <  1 0 , ~ = ~s(A) behaves approximately as (6.108)
~s(A) ~ 1.3 exp (  0 . 3 IAI89
This is confirmed by a plot of ~s(),) as a function o f  A as obtained from the numerical solution of Equations (6.104)  (6.106) and this is illustrated in Figure 6.15. Also shown in this figure is the function (6.108). It can be seen that for ), ~<  2 0 the agreement between the numerical and analytical values of ~s(A) are almost indistinguishable. Hence, the correlation (6.108) can be used, with confidence, for predicting the position at which the solution of Equations (6.104)  (6.106) breaks down for large negative values of A. However, as )~ ~  o c an asymptotic solution of these equations yields 1
~s()~) ~ 0.42S(A)~
(6.109)
but this is only valid for values of A much smaller than those considered here, i.e. 1000 (7.4) u+0, 0+0 as y+co, x>0 where S = sin (I) and (I) is the angle between the outward normal to the cylinder and the downward vertical direction. The nondimensional variables used are defined as xT, __
yGr~
l
,
r T
1
a ~  ~ (~) ~,
~  a ~  ~ (~) ~,
o 
(7.5) ~,~
Next, the nondimensional stream function r is defined as follows:
0r ru 
Oy'
rv =
0r
(7.6)
Ox
The (x, y) coordinates are then transformed into the GSrtlerMeksyn coordinates (~, 77) according to  fo z r 2 U d x ,
r U ~.y r/  (2~)~
where
(7.7a)
1
and the reduced stream function f(~, r/) is defined by (7.8a)
r  (2~)89f (~, 77) Then it follows that u  Uf'
where
'
v =
rU [
(2~)89
f + 2~
Of +( H+ 2~dr ~
sin (I) ~r dU n(~) = 2   = 2~ r2U3 u d~
rdx1
)
rlf'
]
(7.8b)
(7.9)
212
CONVECTIVE FLOWS
is called the configuration function since it is determined completely by the shape of the body and its orientation relative to the vertical direction. Using the aforementioned transformations, Equations (7.2) and (7.3) take the form
f
ill
O~
l on+fO,_2((f, p~
] ~
~O0  v. , O ~ f)
(7.10) (7.11)
and the boundary conditions (7.4) become f((,O)O, f'(~,O)O, 0 ( ~ , 0 )  1 for ~ > 0 f'+0, 0+0 as 77+oo, ~ > 0
(7.12)
The following cases can now be treated: circular cylinders, elliptical cylinders, ellipsoids and spheres. If r is the eccentric angle and e is the eccentricity given by e 2  1  (~,)2, where a and b are the major and minor axes of the elliptical cylinder, respectively, then we have (a) circular cylinders ( a  b), see Figure 7.2,
xC,
S(x)  s i n x
II [
1
( 258
U(x)  2sin7
X
~ _ 8 s i n 2z
at the lower stagnation point at all other ~ stations
(b) blunt elliptical cylinders, see Figure 7.3(a),
~
g
qw T~
0 Figure 7.2: Circular cylinder and coordinate system.
(7.13)
FREE AND MIXED CONVECTION FROM CYLINDERS
(a)
213
(b)
/
0
(b
Figure 7.3: (a) Blunt, and (b) slender, elliptical cylinder orientations and the coordinate system.

y:
b

s(r
1
U(r
sin~_
 z (~~ si.~ ~)89
1
 ( ~ ) ~ (1  cosr II{
 ( ~ ) 89f r (1  cos r189 (1  e 2 sin 2 r189 de at the lower stagnation point at all other ~ stations
1
(7.14) (c) slender elliptical cylinders, see Figure 7.3(b),
(1e 2 cos 2 r 2
V(r
 [2 (1  cosr189 II
[
~  289 f0~ ( 1  cosr 89(1  e2 cos2 r189 de
1
(7.15)
at the lower stagnation point at all other ~ stations
[ ~2~s 
(d) spheres xr U(x)  2 sin 2 2~
rI
0.5
r=sinx, S(x) = sin x 1  289 fo sin2 x (1  cos x)~ dx at the lower stagnation point at all other ~ stations
~
(7.16)
The local Nusselt number, N u , can be expressed as follows: Nu 1
Grz
 ~u (2~)89[o'(~,o)]
(7.17)
214
CONVECTIVE FLOWS
Equations (7.10)  (7.12) have been integrated numerically by Lin and Chao (1974) for circular and elliptical cylinders using series expansions in terms of the parameter H(~) and its derivatives. Later these equations were solved by Kumar et al. (1989) using the Kellerbox method but they have also considered both normal and tangential mass transfer components. However, the effect of both the Prandtl number and the mass transfer on the free convection over a sphere has been studied by Huang and Chen (1987). The local Nusselt number, as given by Equation (7.17), has been obtained by Several authors and the numerical results are summarised in Tables 7.1 to 7.3 for both circular and elliptical cylinders when P r = 1. Further, the experimental results obtained by Hermann (1936) are also included in Table 7.1. It should be noted that the results obtained using the Blasius series and the integral method were obtained by Merkin (1976), whilst those by GSrtlertype series have been obtained by Saville and Churchill (1967). It is seen from Table 7.1 that the numerical results, unlike the experimental results, predict some nonzero value of the heat transfer rate at the top (r  180 ~ of the cylinder. This appears to be due to the collision of the boundary layers from both sides forming a buoyant plume and hence the boundarylayer model is not valid in the vicinity of r = 180 ~ In the case of the blunt elliptical cylinder, the local Nusselt number first increases, attains its maximum value and then decreases with r irrespective of the value of ba, see Table 7.2. As the value of ab increases, the maximum point moves towards the stagnation point and when ab  1 (circular cylinder) the maximum occurs at the stagnation point. For the slender elliptical cylinder, Table 7.3 shows that the local Nusselt number decreases along different r stations for all values of b. However, the heat transfer rate for a slender elliptical cylinder is higher than that for a blunt cylinder, for a given value of a' b and this suggests that a slender body transfers more heat than does a blunt body. It should be noted that in the papers by Badr and Shamsher (1993) and Badr a
Table 7.1 Variation of N~ul a8 a function of r for a circular cylinder when P r  1 G r 4
and as obtained by several authors.
II 0 7r 6 71"
71"
Kumar et al. (1989) 0.4214 0.4163 0.4008 0.3746 0.3364 0.2824 0.1944
Merkin I Blasius (1976) Series 0.4214 0.4214 0.4161 0.4164 0.4007 0.4008 0.3745 0.3755 0.3364 0.3402 0.2825 0.2934 0.1945 0.2378
GSrtler Series 0.4214 0.4165 0.4003 0.3726 0.3299 0.2605 0.00O3
Integral i Elliott Hermann (1936) Method (1970) Experiments 0.386i 0.3960 0.4310 0.3808 0.3895 0.4263 0.3647 0.3709 0.4073 0.3369 0.3426 0.3763 0.2953 0.3105 0.3297 0.2323 0.2841 0.2595 0.2737 0.0000 m
215
FREE AND MIXED CONVECTION FROM CYLINDERS
Table 7.2: Variation of  ~ when P r 
as a function of r for a slender elliptical cylinder
Gr~
1 and as obtained by several authors. _b = I a
.
r
Kumar et al. (1989) (rad.) [i 0.42143 O.07 0.42070 0.2 0.41847 0.4 0.41473 0.6 0.40949 0.8 0.40271 1.0 0.39438 1.2 1.4 0.38445 0.37285 1.6 0.35951 1.8 0.34429 2.0 0.32699 2.2 0.30728 2.4 0.28462 2.6 0.25798 2.8 0.22501 3.0 3.14 0.19485
.....
Merkin (1977a) 0.4212 0.4204 0.4182 0.4145 0.4093 0.4025 0.3942 0.3843 0.3727 0.3594 0.3443 0.3270 0.3073 0.2847 0.2581 0.2252 0.1963
b_ = 0.5 Kumar et al. Merkin (1977a) (1989) 0.5953 o.596oo 0.5826 0.58318 0.5519 0.55233 0.5159 0.51618 0.4819 0.48206 0.4522 0.45231 0.4270 0.42708 0.4O58 0.40583 0.3878 0.38786 0.3724 0.37245 0.3589 0.35894 0.3465 0.34654 0.3342 0.33422 0.3204 0.32031 0.3019 0.30173 0.2731 0.27277 0.2407 0.24094 ~
. . . . . . .
i
b_ = 0.25 Kumar et al. Merkin (1977a) (1989) 0.8359 0.84286 0.7682 0.77226 0.6617 0.66327 0.5788 0.57946 0.51912 0.5187 0.47472 0.4745 0.44109 0.4409 0.41500 0.4149 0.39440 0.3943 0.37793 0.3779 0.36465 0.3646 0.3538 0.35379 0.34466 0.3447 0.33625 0.3363 0.32628 0.3266 0.30702 0.3084 0.27632 0.2785 a
. . . . . . .
(1997), the p r o b l e m of free convection from a horizontal, or inclined to the horizontal, elliptical cylinder has been studied numerically for Rayleigh numbers ranging from R a  10 to 103. 7.2.2
Constant
wall heat
flux
The problem of s t e a d y free convection b o u n d a r y  l a y e r flow past a horizontal cylinder of radius a with a constant heat flux rate qw has also been considered by several authors, b u t n o t a b l y by Wilks (1972), Lin (1976) and Merkin and Pop (1988) and we give some of the results to this p r o b l e m as obtained by Merkin and Pop (1988). The basic equations for this problem, shown in Figure 7.2, are given by (7.18)
o +~~  o g~xx + V~yy  u ~ OT OT U ~ x + ~ 0~ 
+ g/~ ( T 
v 02T P r 0~2
Too)sin
(7.19)
(7.20)
and they have to be solved subject to the a p p r o p r i a t e b o u n d a r y conditions which
216
CONVECTIVE
FLOWS
Table 7.3: Variation of   ~ as a function of r for a blunt elliptical cylinder when Gr~
Pr
1 and as obtained by several authors. b _ 0.25
a
a
,
(rad.) ] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.14
Kumar et al. (1989) 0.29800 0.29946 0.30396 0.31192 0.32416 0.34205 0.36759 0.40122 0.42491 0.40702 0.36685 0.32387 0.28377 0.24737 0.21341 0.17887 0.15067
b
=0.5
Kumar et al. (1989) 0.35438 0.35563 0.35943 0.36584 0.37489 0.38628 0.39871 o.40848 0.40909 0.39592 0.37124 0.34060 0.30798 0.27507 0.24164 0.20543 0.17489
Metkin (1977a) 0.2979 0.2994 0.3039 0.3118 0.3240 0.3418 0.3673 0.4008 0.4244 0.4070 0.3670 0.3241 0.2840 0.2476 0.2136 0.1791 0.1504
0.75
~t
Merkin (1977a) 0.3542 o.3555 0.3593 0.3657 0.3747 0.3861 0.3984 0.4081 0.4088 0.3958 0.3713 0.3407 0.3081 0.2752 0.2418 0.2056 0.1746
Kumar et al. (1989) 0.39218 0.39270 0.39417 0.39635 0.39879 0.40067 0.40080 0.39766 0.38984 0.37665 0.35843 0.33623 0.31110 0.28359 0.25334 0.21832 0.18761
Merkin (1977a) 0.3920 0.3825 0.3940 0.3961 0.3986 0.4004 0.4006 0.3975 0.3897 0.3766 0.3585 0.3364 0.3112 0.2838 0.2535 0.2186 0.1873 . . . .
are given by  O,
OT __ o~T+Too
V   O, +0,
qw kf
__ y 0,
on as
~>0 5>0
~+oc,
(7.21)
In order to solve the s y s t e m of E q u a t i o n s (7.18)  (7.21) we i n t r o d u c e the new variables
x  , a
y  (~r5
,
T  Too  G r  ~ aqw O(x, y) (7.22)
r  Gr~vxf(x,y),
kf
so t h a t the E q u a t i o n s (7.19) a n d (7.20) t r a n s f o r m to
f,,, + f f,,_ f,2 +
sinx
O  x
(/,Of'f
1 0 " + fO'  x ( f ' O 0
Pi
,,Of)
(7.23)
_ O 'cgf )
(7.24)
a n d the b o u n d a r y c o n d i t i o n s (7.21) b e c o m e f ( x , O)  O, f'~O,
O'(x, O)
f ' ( x , O)  O, 0+0
as
= 1 for y + c 0
x>0
(7.25)
F R E E AND MIXED C O N V E C T I O N F R O M CYLI1~
RS
217
Equations (7.23) and (7.24) have been solved n~ (1988) for P r  0.72 using the same method as tha modified slightly to allow for the change from the isot the constant heat flux case given by the conditions ( started at x  0 (where  ~+ s + i n x1 as x 0) and proce top point (x = 180 ~ of the cylinder and it was fore proceeded to this point without encountering a singuL separates from the top surface of the cylinder by a coll plume above the cylinder. The variation of the nondimensional skin frictioi
ically by Merkin and Pop 3cribed by Merkin (1976), hal boundary condition to ). The numerical solution ! round the cylinder to the 1at the numerical solution . Thus the boundarylayer rand then forms a buoyant (x), given by (7.26)
Tw(X)  x f " ( x , O)
and the nondimensional wall temperature, 0~(x), solution of Equations (7.23)  (7.25) for P r  0.72, solid lines. (a)
brained from a numerical indicated in Figure 7.4 by
(b) 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0
2.0
1.51.0 0.5 0.0
o
g~
g~"
X
s
t
.
0
Figure 7.4: Variation of (a) the nondimensional ski7 nondimensional wall temperature, Ow(x), with x for solutions are indicated by the solid lines, the Blasius .~ the broken lines and the GSrtler series (7.28) are ind
On using the Blasius series technique, Koh (1964) imations for Tw(x) and Ow(x), when P r  0.72"
2
.
,,,
.
.
3
4 X
tion, Tw(x), and (b) the  0 . 7 2 . The numerical (7'.27) are indicated by d by the dotted lines.
,ined the following approx
Tw(X)  1.54419 X  0.155001. Ow(X) = 2.19586 + 0.04679 X2
(7.27)
and these results are also included in Figure 7.4 and GSrtlertype series solution has been obtained by Me
icated by broken lines. A and Pop (1988) for T~(x)
218
CONVECTIVE FLOWS
and Ow(x), and they give for P r 
0.72,
" r w ( x )  (5~) ~ (sinx)~
0.88690+0.27833~ ~ + . . .
_ (sinx) 88( 1.66415+0.06402~ ~ + . . . ) where
j/~ x

(7.28)
1
(sin x) z dx
(7.29)
The solution (7.28) is also shown in Figure 7.4 by the dotted lines. It should be noted from these figures that both the solutions (7.27) and (7.28) are in good agreement over most of the range of values of x, being virtually indistinguishable from the numerical solution for the lower half of the cylinder 0 ~< x ~< ~. The Blasius series underestimates both the values of Tw(X) and Ow(x), whilst tt~e Gbrtler series underestimates Tw(X) and overestimates Ow(X). However, the Blasius series is better for estimating 0~(x), whilst the Gbrtler series is better for Tw(X). However, all of the above mentioned papers refer to the free convection from horizontal cylinders under the assumption of very large (infinite) values of the Grashof number, i.e. the boundarylayer approximation has been made, and this implies that the curvature effects and any pressure differences across the boundarylayer are negligible. The plume region above the top surface of the cylinder is of course excluded, since the development of the plume formed by the separation of the boundarylayer from the surface invalidates the basic assumptions. Furthermore, these results do not adequately describe the fluid flow and the heat transfer at low or moderate values of the Grashof number nor the development of the buoyant plume above the cylinder. Kuehn and Goldstein (1980), Farouk and Gii(;eri (1981), Fujii et el. (1982), Qureshi and Ahmad (1987), Wang et al. (1990) and Saitoh et al. (1993) have provided numerical solutions for the complete NavierStokes and energy equations for laminar free convection about a horizontal circular cylinder which is maintained at a constant surface temperature or a constant surface heat flux for values of the Grashof numbers which vary from being very small to being very large. Fujii et al. (1982) have also obtained experimental results which are in good agreement with their numerical solution. Kuehn and Goldstein (1980) have used the finitedifference method, whilst Wang et al. (1990) have used the spline fractional step method. Wang et al. (1990) made a comparison between their results and those obtained by Kuehn and Goldstein (1980) and found good agreement. However, Saitoh et al. (1993) claimed that the results obtained by Kuehn and Goldstein (1980) and Wang et al. (1990) are not very accurate because they did not treat the inflow and outflow conditions correctly. These solutions contain errors in excess of 2% and therefore they cannot be regarded as the standard bench mark solutions. Motivated by the above remarks, Saitoh et al. (1993) attempted to obtain bench mark solutions for the steady free convection heat
FREE AND MIXED CONVECTION FROM CYLINDERS
219
transfer problem around a horizontal circular cylinder under isothermal conditions for Rayleigh numbers ranging from Ra  103 to 105. The steady vorticity, NavierStokes and energy equations, expressed in cylindrical polar coordinates (G 0), see Figure 7.5, can be written as, see Saitoh et al. (1993), vorticity equation: la
0~
( 0~ rr
1 02r ~2 002
+
(7.30)
=~
momentum equation: V
~~r + ~ OO
[10 (0~ 102~] (0T 0Tcos0) r ~ k 0~] ~ ~2o02 +g~ ~rsin0~ 0 0 ~
(7.31)
energy equation:
\~rr ]
~
U~rr + ~ 00 = P r
q ~2 ~ j
(7.32)
where the vorticity ~ is defined as 1[0

_
~
(~)

___~]
(7.33)
aoj
Introducing the nondimensional variables r
r
: a~
U
~,
~,
w
V
d7 ~,
V
T
AT
Symme.try Line 
4 .....
~!
LI [I
.... ..Imaginary Boundary
",,or " Solid Boundary
Cylinder~y
Figure 7.5" Physical model and coordinate system.
(7.34)
220
CONVECTIVE FLOWS
Equations (7.30) (7.32) become (7.35) 003
0T OT cos ~ r sin0~ 00 r
V OW
U~r + r ~O0 = P r V 2 w + R a P r OT v OT U ~ r +  r~ O0 = V2T where U

(7.36) (7.37)
1 0r
0r
v
(7.38)
Or
r 00' and the Laplacian operator is defined as follows: V2 
)
02 1 0 1 02 Or 2 +r ~r +~r2002
(7.39)
Only half of the domain is considered since the flow is symmetrical about the vertical plane through the axis of the cylinder. Thus, the boundary and symmetry conditions are given by u0, vO,
v0,
r
r
w   y ~ 02~ ,
T
o~ _  0 ~ 00
aT _ O 0O 
wO,
1
on
r  1,
on
0   0 , Tr,
0~> 1 We ~fv 9 note for large values of k that Rab is effectively the same as Ra. On the other hand, when the value of k is small then Rab is also small and most of the temperature drop takes place close to the surface of the cylinder and convection plays only a minor role in the determination of Tb Therefore it is conceivable that Tb* is relatively insensitive to the definition of the Rayleigh number and Rab can be replaced by Ra. Using expressions (7.50) and (7.52), where we take Rab  Ra, we obtain the following relation for the nondimensional average boundarylayer temperature1.89 k Tb* = 1.89 k  na 88 In (~)
(7.53)
226
CONVECTIVE FLOWS
forPr>>l andk~l. The average boundarylayer Nusselt number may be expressed as follows: q" a Nu 
*
T b 
Too
1
1
k i ( Tc  Too) ~ Tc  Tc~ RaZ  T~ Ra~
(7.54)
if Equations (7.50) and (7.52) are used. Equations (7.54) suggests that the ratio of 1 N u to T~ R a ~ is always constant, i.e. Nu 1
Tb*
= A
(7.55)
where A is as yet, an undetermined constant. Insertion of expression (7.53) into Equation (7.55) leads to 1
Nu
ARa~ 
(7.56)
1
1
Ra~I
1.89k
In ( ~ )
for P r >> 1 and k ~ 1. Therefore, N u can readily be obtained once the constant A has been determined. 7.3.2
Pr
O x >0
(7.63)
X
where ), is again the mixed convection parameter which is defined as in Equation (2.147). We note that the case of A = 0 is the forced convection solution obtained by Terrill (1960). The nondimensional skin friction, Tw(X), and wall heat transfer, qw(x), can be expressed as follows:
Tw(x) = x f"(x, 0),
qw(x) = O'(x,O)
(7.64)
Equations (7.61)  (7.63) have been solved numerically by Merkin (1977b) for Pr = 1 using a method similar to the one he employed in his papers for the corresponding problem of free convection boundarylayer flow, see Merkin (1976). The variations of Tw(X) and qw(x) as a function of x are shown in Tables 7.5 and 7.6 for Pr = 1 and for different values of A. The results from these tables show that increasing A delays separation and that separation can be suppressed completely in 0 ~ x ~ r for a sufficiently large value of A (> 0), A = Ac, say. For values of greater than A = Ac, the boundarylayer remains attached to the surface of the cylinder up to the upper point (x  ~) of the cylinder, where the boundary layers on each side must collide and leave the surface of the cylinder to form a thin wake above the cylinder. On the other hand, the separation point, x = Xs(A), is brought nearer to the lower stagnation point (x = 0) of the cylinder. The numerical solutions indicate that the value of A which first gives no separation lies between A = 0.88 and A = 0.89. Moreover, Merkin (1977b) has demonstrated that separation does not, in fact, occur for A > 1. The numerical results also show that, in those cases when the boundarylayer separates, Tw(Xs) + 0 and qw(xs) + qs ( ~ O) in a singular way, as we observed for a vertical flat plate in Section 2.2.
232
CONVECTIVE
FLOWS
Values of the nondimensional skin friction, vw(x), for Pr  1 and different values of )~.
Table 7.5
91 :1.751 i.5 o.o I o.oooo I o.oooo 0.2.1..0._0006' ] 0.0533 0.4 0.0741 0.6 0.0026 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
1.0 0.0000 0.1257 0.2266 0.2784 0.2554 0.1069
0..5 0.0000 0.1871 0.3511 0.4706 0.5271 0.5051 0.3890 0.1253
0.0
0.5
0.88
0.89
o:oooo
0.0000 0.2945 0.5662 0.7941 0.9614 1.0561 1.0727 1.0121 0.8814 0.6927 0.4599 0.1842
o.oooo
0.0000 0.3330 0.6429 0.9085 1.1125 1.2430 1.2941 1.2671 1.1695 1.0491 0.8295 0.6103 0.4O33 0.2219 0.0847 0.0149 0.0504
0.2427 0.4627 0.6393 0.7552 0.7982 0.7615 0.6429 O.44O5 0.1069 ,
0.3321 0.6409 0.9057 1.1088 1.2383 1.2886 1.2608 1.1625 1.0072 0.8131 0.6012 0.3936 0.2112 0.0711
71"
l'
1.0
2.0
0.0000 0.3436 0.6639 0.9398 1.1538 1.2938 1.3541 1.3356 1.2459 1.0986 0.9117 0.7063 0.5048 0.3287 0.1979 0.1292 0.1206
0.0000 0.4354 0.8464 1.2106 0.5094 1.7295 1.8637 1.9117 1.8793 1.7781 1.6236 1.4334 1.2248 1.0123 0.8043 0.6002 O.45O8
.,.
Values of the nondimensional heat transfer, qw(X), for Pr  1 and different values of ~.
Table 7.6
x ...... 1.75 0.4199 0.0 0.2 0.4059 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
1.5 0.4576 0.4498 0.4236 0.3373
1.0 0.5067 0.5018 0.4865 0.4594 0.4160 0.3326
0.5 0.5420 0.5380 0.5260 0.5056 0.4760 0.4353 0.3784 0.2736 .
.
.
.
.
.
0.0 0.5705 0.5668 0.5564 0.5391 0.5145 0.4826 0.4426 0.3928 0.3280 0.2114
0.5 0.5943 0.5911 0.5817 0.5661 0.5443 0.5165 0.4828 0.4431 0.3972 0.3444 0.2821 0.1970
. . . . . .
0.88 0.6096 0.6067 0.5979 0.5833 0.5631 0.5375 0.5066 0.4709 0.4307 0.3863 0.3383 0.2871 0.2331 0.1766 0.1162
71" .
0.89 l, 1 . 0 0.6100 0.6156 0.6071 0.6115 0.5983 0.6028 0.5837 0.5885 0.5636 0.5686 0.5380 0.5435 0.5072 0.5133 0.4716 0.4785 0.4314 0.4394 0.3872 0.3967 0.3394 0.3509 0.2885 0.3029 0.2350 0.2540 0.1796 0.2061 0.1227 0.1634 0.0745 0.1354 0.1033 0.1306 .
.
.
2.0 I 0.6497 0.6471 0.6393 0.6264 0.6086 0.5863 0.5597 0.5294 0.4960 0.4601 0.4225 0.3842 0.3460 0.3088 0.2730 0.2381 0.2122
F R E E AND MIXED C O N V E C T I O N F R O M CYLINDERS
233
The variation of the separation point, Xs(A), as a function of )~ for P r = 1 is given in Figure 7.14. This figure shows that there is a value of A = ~0, say, below which a boundarylayer solution is not possible. The reason is that for A < 0, the cylinder is cooled and the free convection boundarylayer would start at x = 7r and for sufficiently small values of ,k there comes a point where the flow of the stream upwards cannot overcome the tendency of the fluid next to the cylinder to move downwards under the action of the buoyancy forces. This is an unstable situation and whether a boundarylayer can exist at all on the cylinder for ~ < ~0 is still an unanswered question.
2.0
.5 lO
2.0
1.0
0.0
'
1:0
Figure 7.14: Variation of the separation point, xs(A), with )~ for Pr = 1.
It is also worth mentioning the work of Cameron et al. (1991) on mixed convection boundarylayer flow from twodimensionM or axisymmetric bodies of arbitrary shape, but they only considered the case of assisting flow. Numerous papers after 1980 have reported numerical results on mixed convection flow from horizontal circular and elliptical cylinders by solving the full NavierStokes and energy equations under various forced flow directions such as, assisting, opposing and inclined. To this end we mention the work of Badr (1983, 1984, 1985, 1994, 1997), Amaouche and Peube (1985), Moon et al. (1988) and Ahmad and Qureshi (1992). Further, we present some results reported by Badr (1984) for the problem of mixed convection from a horizontal circular cylinder which is maintained at a constant temperature, Tw, and it is placed in a uniform forced flow of velocity, Uc~, and temperature, Too, where Tw > Too, see Figure 7.15. The line 0  0 ~ is taken to be the radius through the rearmost point on the cylinder surface viewed from the upstream direction. Using a modified polar coordinate system (~, 0), where ~  In r, the governing equations of vorticity, NavierStokes and energy can be written in the following form, see Badr (1984),
234
C O N V E C T I V E FLOWS
U~,T~
t t
U~,T~ Figure 7.15:
e2~w 0w
02r 0~ 2
!"
Physical models and coordinate systems.
02r (7.65)
002
2 (02W
02wh
0r
0r
Cr e~ ( OT
+ 2R 2
OT 2(02T 02T) e2~ Ot = Pe  ~ + ~
sin e +
OT
cos e
Or Or O00~ ~ O~ O0
)
(7.66) (7.67)
where Re, Gr and Pe are based on the diameter of the cylinder, with Pe (= Re Pr) being the P~clet number. The i signs in Equation (7.66) depends on the flow regime and it is positive for assisting flow and negative for opposing flow. The boundary conditions appropriate to this problem are as follows: r 
0,
e~ 0r oY+ cos 0,
~o ~ _ 0 ,
oo ~, ~ _ 0
e ~~~ 0~ + sin 0,
,
T=I
w + 0,
T ~ 0
on
~0
0 Too). As is well known, for most external flows, the buoyancy force can be neglected in a small pure forced convection region downstream of the leading edge. Beyond that region the effect of buoyancy crossflow increases as the fluid flows downstream and a secondary flow is induced. In general, close to the leading edge the magnitude of the secondary flow is small and the boundarylayer flow is forcedconvection dominant. The secondary flow grows downstream and the interaction of the free and forced convection becomes important and the flow becomes free convection dominant further downstream. Previously this flow configuration has been treated only by Yao and Catton (1977) and Yao et al. (1978). In terms of the nondimensional variables, the boundarylayer equations for this
240
C O N V E C T I V E FLOWS
U~,T~ [ ~~O(a) , _____~ . . . . . . . . . . . .
i~ L"
I
,,
I I 
,
,,
m
x
Figure 7.20" Physical model and coordinate system.
problem can be written in cylindrical coordinates as, see Yao et al. (1978), Ov
Ox + ~ Ou Ou
i)w
+ Or Ou
=0
(7.77) 019 Ox
Or
op
02 u
+
(7.78)
Or 2
= 0
Or Ov Ov Ov u~x + v  ~ + w 0~ 
(7.79)
02v Or 2 + T sin 0
(7.80)
OT OT OT 1 cO2T U ~ x + v ff~ + w 0~ = P r Or 2
(7.81)
where (u, v, w) are the fluid velocity components along the (x, 0, r) directions and the nondimensional variables are defined as follows: m
,

u
uoo ,
v
g),

Uoov ~ '
TToo AT
T
,
w  Gr ~
P
PPoo
(7.82)
pU2
and the mixed convection parameter A is a positive quantity (assisting flow). The boundary and symmetry conditions appropriate to this problem are given by u=0, Ou _ o, oY
v0,
w0,
vO,
Ow 0o _ O ,
u+l,
T+0
T1 OT _ oY
o
on
r0,
0~0
x>O (7.96)
These equations suggest an expansion for the functions f and 0 for x ~ 1 of the form
f  fo(rl) + x89 (rl) + xf2(q) + . . . e  6o(~) + ~~o~ (~) + ~ o 2 ( ~ ) + . . .
(7.97)
where the coefficient functions fi(r/) and Oi(rl), i = 0, 1, 2,..., have been numerically determined by Mahmood and Merkin (1988) by solving a set of ordinary differential equations. Using the determined values of f and 0 it follows that the skin friction coefficient, Cf, and the local Nusselt number, Nu, are given as follows:
cl
= x  ~ 1f , (x, O)
i Ou
Nu  _(lOO
O7)r=l  ~
1
~
(7.98)
Ot
[ (~, o)]
and for Pr = 1 these quantities become
[
,
Cf  x ~ 0.33206 + 0.69432x~ + (0.65658 + 1.14666,~)x + . . .
] (7.99)
Nu  x89 [0.33206 + 0.69432 x~ + (0.65658 + 0.27108 A)x + . . . ] for x ~ 1. Numerical solution To numerically solve the transformed boundarylayer Equations (7.94)  (7.96), the integration should be started from the leading edge of the cylinder (x = 0), where the fluid flow is basically that on a flat plate, with the effects of the curvature and the
FREE AND MIXED CONVECTION FROM CYLINDERS
247
buoyancy force having only small effects near x  0. However, there is a singularit~r in Equations (7.94) and (7.95) at x = 0 which arises from the presence of the 15 term. In order to remove this singularity it is convenient to make the transformation 1 = x~, so that these equations become [(1 + 2r/~) f"] , + 1 f f " + A~20  ~l ( f ,
1 [(1 + 2v )o'] , + 1 f
Of' 0~
f,,Of)
(7.100)
1 (f'OOo'Of)
o' 
(7.101)
and the boundary conditions are those given by the Equations (7.96). Equations (7.100) and (7.101), along with the boundary conditions (7.96), have of been numerically solved by Mahmood and Merkin (1988) for Pr  1 using q = b~ and 0 as the dependent variables, and then replacing the derivatives in the ~ direction by finite differences and all the other quantities by averages. The details are not given here as they can be found in the paper by Mahmood and Merkin (1988). The skin friction coefficient and the local Nusselt number, given by Equations (7.98), can be calculated at each step in the ~ direction. Approximate solution Further, in order to obtain an approximate solution which is valid for all values of x, Mahmood and Merkin (1988) have used an integral form of the energy Equation (7.90), namely d (fl~176
dx
ruOdr
1 (00) Or r=l

(7.102)
Pr
and considered the following approximate forms for the fluid velocity and the temperature profiles u  } ( 1  r 2) + ~ [1 + } (e TM  1)] lnr 0   1 ~ l n r u = 0,
0 =0
I
for
J for
so that
Nu :
l(r<e
M
(7.103)
r~e M
1
(7.104)
M(x)
The function M(x) can be determined analytically by solving an ordinary differential equation, and which, for Pr = 1, gives x  3Ae4M + 88(1 + ~ ) e TM + 8~ [4 (1  e TM)  A (1 + e TM  2e2M)]
13)k fOM (e4Ml) dM + ( 1 _ rg 11~)fo M +64 M
(e2M1) M"
dM
(7.m5)
248
CONVECTIVE FLOWS 1
For small values of x ( 1 and A > 0. Additionally, Mahmood and Merkin (1988) have obtained the following asymptotic expression N u ( x ) ,,~
12z
In t ~
4
(7.107)
2z
In x)
for x >> 1 and A > 0. The variation of 1
1
NuRe2z =  O ' ( x 0) Re '
C f Re2z = / " ( x , O ) Re '
(7.108)
as a function of x as obtained from the numerical solution of Equations (7.100) and (7.101) for P r  1 when A > 0 are shown in Figure 7.25. It is clearly seen from this figure, for a given value of A, that there is a rapid increase in the skin
(a)
A=IO
16
(b)
A=5
,/,
3o 1 A=10
A=5 1
8
1.2 4
0.6
O
0
4
~
8
12
16
x
20
24
0.0
0
,4
8
1'2
1'6
x
20
~4
Figure 7.25: Variation of (a) the skin friction coefficient, and (b) the local Nusselt number, with x in the case of assisting flow.
F R E E AND M I X E D C O N V E C T I O N F R O M C Y L I N D E R S
249
friction and the heat transfer rate as x increases along the cylinder, with this increase being more pronounced for the larger values of A. Further, Figure 7.26 shows the variation of N u ( x ) as a function of In x for A  1 as obtained from the approximate solution (7.107). Included in this figure are also the numerical results obtained from expressions (7.98) by solving numerically Equations (7.100) and (7.101) for step lengths in the z/direction of h  0.05 and 0.1. 1.6
0.8 Nu 0.4
'2
0
89
4
(i
lnx
8
1'0
Figure 7.26: Variation of the local Nusselt number N u with In x. The numerical solution is indicated by the solid line, the results with h  0.1 and h = 0.05 are indicated by the broken and dotted lines, respectively, and the asymptotic solution (7.107) is indicated by the dotdash line.
Further, the variations of f " ( x , 0) and  O ' ( x , 0) as a function of x are illustrated in Figure 7.27 for A   1 and A   1 0 , i.e. opposing flow with P r  1. These figures indicate that the heat transfer decreases slowly, whilst the skin friction goes to zero at a finite value of the separation point x  Xs(A). The variation of x  xs(A) with A (< 0), as obtained from the numerical solution, is shown in Figure 7.28 and also shown is the variation of x = xs(A) as calculated from the series (7.99) for C f = 0 at x = X s(A). This gives
11 xs2 (A)  1 + 1.746
[ 1] 0.529 + (0.280 + 0.506(1 + 1.746 IA[))
(7.109)
It can be seen that the agreement between the numerical values of x = Xs(A) and those given by the analytical expression (7.109) is good. Further, we see from this expression that x  Xs(A) ~ oi~j29o mr ~ IA[ >> 1. Thus, for IAI >> 1, the curvature effect of the cylinder is small and the flow up to separation is basically given by the flat plate solution found by Merkin (1969), from which it follows that }A[ Xs(A) ,.~ 0.192 for IAI >> 1. We also observe from Figure 7.27 that for smaller values of ]AI, the value
250
CONVECTIVE FLOWS
(a)
(b)
0.6
0.4
0.5
I ]
/
0.3
0.4 0.3
0.2
0.2 0.1
0.1 0.0 0.0
0:1
0:2
0.3
0.4
0.5
0.0 0.0
06
0.01
0.02
0.03
X
Figure 7.27: Variation of f"(x, O) (solid line) and O'(x, O) (broken line) with x 1 when (a) ; ~   1 and (b) , ~   1 0 .
for P r 
16t/ 1.4
I\
1.2
0 l/i
0.6 0.4
0.2 0.0
0.0
i.1
89
ft.3 i.4
5.5
Figure 7.28: Variation of the separation point, Xs()~), with )~ for Pr  1. The nu
merical solution is indicated by the solid line and the approximate solution (7.109) is indicated by the broken line.
of the skin friction at x = Xs(X) behaves in a regular way w i t h o u t the a p p e a r a n c e of a singularity. W h e n I,~] is small, the c u r v a t u r e of the cylinder has a significant effect on the flow a n d the numerical solutions of M a h m o o d a n d Merkin (1988) suggest t h a t this has the effect of inhibiting the square root singularity near s e p a r a t i o n t h a t arises in t w o  d i m e n s i o n a l b o u n d a r y  l a y e r flow, see Section 2.2, with the solution
F R E E AND MIXED CONVECTION FROM CYLINDERS
251
being regular at x = Xs(~). In fact, the numerical solution for A = 0.1 and ), =  1 continued past the point x = Xs(%) into a region of reversed flow, where, as expected, it became unstable and broke down. Finally, it should be noted that the problem of free convection boundarylayer along a partially heated infinitely long vertical cylinder which has been disturbed by a steady horizontal flow has been studied theoretically by Yao (1980), Yao and Chen (1981) and Scurtu et al. (2000). The asymptotic solution indicates that the boundarylayer is mainly induced by the buoyancy force in the vicinity of the thermal leading edge. The effect of the horizontal free stream on the boundarylayer gradually increases as one moves upward away from the thermal leading edge along the cylinder. It is found that the boundarylayer separation does not occur in the vicinity of the thermal leading edge but the asymptotic solution shows that the forced convection tends to separate the boundarylayer along the rear stagnation line of the vertical cylinder.
Chapter 8
Free and m i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow over m o v i n g surfaces 8.1
Introduction
During many mechanical forming processes, such as extrusion, meltspinning, etc., the extruded material issues through a slot or die. The ambient fluid condition is stagnant but a fluid flow is induced close to the material being extruded, due to the moving surface. In regions away from the slot or die the fluid flow may be considered to be of a boundarylayer type, although this is not true in the vicinity of the slot or die. Similar situations prevail during the manufacture of plastic and rubber sheets where it is often necessary to blow a gaseous medium through the material which is not, as yet, solid, and where the stretching force may be varying with time. Another example that belongs to the class of boundarylayer flow problems due to moving surfaces is the cooling of a large metallic plate in a bath, which may be an electrolyte. In this case the fluid flow is induced due to the shrinking of the plate. Glass blowing, continuous casting and the spinning of fibres also involve the flow due to a stretching surface. In all these cases a study of the flow field and heat transfer can be of significant importance since the quality of the final product depends to a large extent on the skin friction and the surface heat transfer rate. The first study on the boundarylayer adjacent to a continuous moving surface was conducted by Sakiadis (1961) and since then it has been much generalised and refined. The fluid flow problem due to a continuously moving surface in an ambient fluid differs from that of the fluid flow past a fixed surface. Unlike the flow past a fixed surface, the continuous moving surface sucks the ambient fluid and pumps it again in the downstream direction. However, in all the earlier studies on boundarylayer flows due to a moving surface the effects of the buoyancy force
254
CONVECTIVE FLOWS
was neglected. Griffin and Throne (1967) in their experimental work employed an isothermal belt that moved through the surrounding air which was at 75~ while the surface temperature of the belt was held at 175 ~ Due to the buoyancy effects, the measured Nusselt number values were found to be 10  60% larger than those predicted without including the buoyancy effects as determined by Erickson et al. (1966). The free and mixed convection boundarylayer flow from vertical, inclined and horizontal moving surfaces has drawn considerable attention in recent years and a large amount of literature has been generated on this problem. The problem has been the subject of studies by Moutsoglou and Chen (1980), Lin and Shih (1981a, 1981b), Kuiken (1981), Khan and Stewartson (1984), Ingham (1986b), Merkin and Ingham (1987), Ramachandran et al. (1987), Lee and Tsai (1990), Riley (1992), Daskalakis (1993), Lin et al. (1993), Vajravelu and Nayfeh (1993), Pop et al. (1995b), Hady et al. (1996), Kumari et al. (1996b) and Fan et al. (1997).
8.2
Free c o n v e c t i o n b o u n d a r y  l a y e r flow from a m o v i n g v e r t i c a l sheet
Consider a plane sheet of thickness 2b which moves vertically downwards with a velocity  U s in a quiescent viscous and incompressible fluid of ambient temperature Tcr We assume that below a certain point, that we shall locate at x = 0, the sheet may release its excess heat to the surrounding fluid. It is also assumed that at x = 0, i.e. at the exit, the temperature of the sheet is To, where To > T~. The physical configuration is schematically illustrated in Figure 8.1 together with the coordinate system employed. The coordinate x measures the distance along
2b ~ I
y
I
U
S
Figure 8.1" Physical model and coordinate system.
CONVECTIVE BOUNDARYLAYER FLOW OVER MOVING SURFACES
255
the sheet, x being negative along the conducting portion, and y is normal to the sheet and being negative at one side of the sheet. This interesting problem was first considered by Kuiken (1981) in his investigation of the problem of free convection arising in the manufacture of glass fibre. The fibre is very hot as it leaves the orifice and it then slowly descends vertically losing heat by convection. As a result, the motion of the fluid is mainly towards the point where the sheet enters the system and a 'backward' boundarylayer ensues. Indeed, the exposed portion of the sheet is semiinfinite but the motion of the fluid is towards the finite end of the semiinfinite region. The equations which govern the motion of the fluid near the plane sheet are of the boundarylayer type and are given by Equations (1.77)  (1.79). Kuiken (1981) assumed in the problem he was investigating, that these equations are subject to the boundary conditions: uus,
vO
on
y=0'
x~~ 5. The fluid velocity function ( x o  x ) u ( x , y) and the fluid t e m p e r a t u r e function ( x o  x ) 3 T ( x , y ) , as o b t a i n e d from the numerical solution of Equations (8.16) (8.18), are plotted as a function of XoY_z in Figure 8.2 for various values of X and
260
CONVECTIVE FLOWS
(a) ~_~
0
0
1
2
(b)
y x0  x
3
4
250
2oo
I
150
tq 100
50
0 0.0
0:5
y x0 
1:0 x
Figure 8.2" (a) The fluid velocity, and (b) the temperature, profiles for Pr  1. The numerical solutions are indicated by the solid lines and the limiting solution (8.13) is indicated by the broken line.
the limiting solutions (8.13) are also shown in these figures. It can be seen that the agreement between these solutions is good. However, for the fluid velocity profiles the shapes differ considerably at finite values of xY_x, the numerical solution being smaller t h a n the limiting form, and this feature accounts for the discrepancies in the displacement thickness. The reduced fluid velocity, ~~, OF and the temperature, T, profiles are shown in Figure 8.3 for various values of X. We can see, for any finite value of Y > 0 it appears t h a t ~V OF and T have virtually zero limit values as X + X0. This result is
C O N V E C T I V E B O U N D A R Y  L A Y E R F L O W O V E R M O V I N G SURFACES
(a)
261
(b)
2
~" x.=3. 0.5
5
3.4
0,
0
y
X = 3.4
5
Figure 8.3: (a) The fluid velocity, OF (X, Y) profiles/or Pr = 1.
0
y
5
and (b) the temperature, T(X, Y)
surprising, for it might have been expected that normal diffusive processes would 1 have produced a boundarylayer of thickness ~ x z from the heating near x  0.
8.3
Free c o n v e c t i o n b o u n d a r y  l a y e r zontal moving sheet
flow f r o m a hori
Consider a thin rigid flat sheet of thickness 2b which issues from an adiabatic shroud and it cools as it moves along in the horizontal direction with a constant velocity  u s into an ambient fluid, see Figure 8.4. The Cartesian coordinates are defined with the origin on one side of the sheet at the orifice of the shroud so t h a t the coordinate measures the distance along the sheet, being negative along its conducting portion, and the coordinate y is in the direction of the gravity vector g. This problem has been solved by Pop et al. (1995b) following the similarity solution method of Kuiken (1981) for a particular fluid model ( P r  1), in a very general m a n n e r to describe the nature of the near surface fluid flow for P r ~ 1. Under the steady state flow condition and the boundarylayer and Boussinesq approximations, the governing equations can be written in nondimensional form, see Pop et al. (1995b), as follows:
Ou Ov + ~  0 0~ oy
(8.22)
262
C O N V E C T I V E FLOWS
Adiabatic Shroud , ~ ~ ' ~
....
2b
~4
~
us
Figure 8.4: Physical model and coordinate system.
Ou
Ou
Op
u~x + v Oy =
OT
(8.23)
Op
0 
~~
02u
Ox ! Oy 2
(8.24)
Oy + T
OT
1 02u
+ v o7 = P~ oy~
(8.25)
where the upper sign (+) in Equation (8.24) corresponds to the case when we choose the positive ~axis upwards and consider the flow above the sheet, whilst the lower sign (  ) corresponds to the case when the positive ~axis is oriented downwards and the flow is under the sheet. These equations have to be solved subject to the boundary conditions: u0, OT
v0
OT
}
O~ + 0~   0 u+0, T+0, p  + 0 u0, T1 u0, T0 r
on
y0
x >0
as on on
y+cx~, x0, x0,
x>0 y0 y>0
(8.26)
On seeking a similarity solution of Equations (8.22)  (8.25) it was found that T, p and 77, the similarity variable, should have the following form: r
 C ~ I(T]),
T
 C
0(77)
( x 0  x) 3'
p
•
1
y
h(r/)
(z0
x) 2'
~  C ~ ~X0
 X
(8.27)
where x0 and g are constants to be determined. On substituting expression (8.27) into Equations (8.22)  (8.25) gives h'(~7)  =h0(~/) and the equations satisfied by f and h are then given as follows:
f ' "  I '~ T (2h + ~h')  0 1 ~h'" Pr
 3f'h' 
0
(8.28) (8.29)
C O N V E C T I V E BOUNDARYLAYER FLOW OVER MOVING SURFACES
263
with the boundary conditions (8.26) reducing to f(O)O, f'(O)=O, h+0, h~ 0
f, +0,
h'(0)I as 77+oo
(8.30)
The problem defined by Equations (8.28)  (8.30) can be solved numerically for different values of Pr. Once the numerical solution has been obtained for a given value of Pr, the constants x0 and C can then be determined by using the boundary condition ~~ OT + ~~ OT  0 at y  0 from expression (8.26). Using this condition, along with T and 77 from expressions (8.27), we obtain 3h' (0) + C 1 h" (0)  0
(8.31)
which, in combination with the boundary conditions (8.30) and that T (x, y) = (0, 0), gives
lat
5
C 
h"(0)
'
x0  C89 
h"(0)
(8.32)
As can be seen from Equation (8.29), the coefficient of the highest derivative of h becomes very small when Pr is very large and this may cause difficulties when the problem is solved numerically. Thus, it is convenient to introduce the transformation, for large values of Pr (>> 1), 4
"~
1 "V
f  P r  ~ f ( ~ ,
h Pr~h(~,
y
Pr~
(8.33)
so that Equations (8.28) and (8.29) become f fff
Pr
:F h "  3f'h' = 0
(8.35)
along with the boundary conditions of the form shown in Equation (8.30) and all the primes now denote differentiation with respect to ~. A solution of Equations (8.34) and (8.35) for large values of Pr is sought in the form of the following series
]~(~  fo (~ + P r  l f~ (~ + Pr2f2 (~ + . . . (~  ho (~ t p r  l h l ( ~ Jrpr2h2 ( ~ 4...
(8.36)
N
where f0, h0, etc. are determined from three sets of ordinary differential equations which are given by Pop et al. (1995b) and therefore they are not repeated here. These sets of equations were solved numerically using the RungeKutta method and therefore the functions f, 9 and h have been determined for various large values of
Ft.
264
C O N V E C T I V E FLOWS
However, it should be noted that it was found to be difficult to use the same numerical method, i.e. the RungeKutta method, to solve Equations (8.28)  (8.30) for both cases of flows under and above the sheet. Thus different numerical techniques were employed for different situations. For the flow under the sheet, i.e. the lower sign (+) in Equation (8.28), it was found that the central finitedifference scheme associated with Newton's method was the most robust numerical technique. However, even this technique sometimes failed to give convergent solutions for small values of P r (0
as on on
Y + c~, X = 0, X = 0,
X>0 Y =0 Y > 0
(8.58)
where FI (~) 
2xo+x 3Xo
,
F 2 ( X )  X(XoX) Xo
F~ ( X , Y ) 
,
F3(X, Y)  ( 4 x  x ~
x~ (xox)
Xo+X~(XoXlY Fs(X) xoo3X
(8.59)
Equations (8.55)  (8.58) were solved numerically by Riley (1992) using an adaptation of the CrankNicolson finitedifference method and he obtained X0 = 3.58634,
x0 = 4.11715
(8.60)
The obtained numerical results are summarised in Figures 8.11 and 8.12 for the wall temperature distribution, the skin friction, and the fluid velocity and temperature profiles. Included here (by the broken line) are also the similarity solutions given by expressions (8.13) and (8.19) with xo being given by (8.60). It is clearly seen that the agreement between these solutions is excellent and it confirms that Kuiken's similarity solution (8.13) is indeed the appropriate limiting solution as x ~ xo.
CONVECTIVE BOUNDARYLAYERFLOW OVER MOVING SURFACES 271 (a)
(b) 0.5
0.5
._0.4
0.4
~'~ 0.3
,~ 0.3
v
'~ 0.2
""..~
0.2 0.1
2.0
2.5
3.0
3.5
4.0
s
o.o20
.
2:5
3'.0

3.5
4:5
4.0 x
x
Figure 8.11" Variation of (a) the wall temperature distribution, and (b) the skin friction coefficient, with x ]or P r  1. The numerical solutions are indicated by the solid lines and the similarity solutions (8.19) are indicated by the broken lines.
(a)
(b)
L
"~4~
~ 180
~
o
.~
I 120 60
1
0
0
1
2
Y X 0  X
3
4
0 0.0
=
.
,3.5
ii
0:5
Y x0
1:0
 x
Figure 8.12" (a) The fluid velocity, and (b) the temperature, profiles for P r  1.
The numerical solutions are indicated by the solid lines and the similarity solutions (8.13) are indicated by the broken lines.
8.5
F r e e c o n v e c t i o n b o u n d a r y  l a y e r f l o w d u e to a c o n t i n u o u s l y m o v i n g v e r t i c a l flat p l a t e
T h e p r o b l e m of t h e free c o n v e c t i o n b o u n d a r y  l a y e r flow over a v e r t i c a l flat p l a t e w h i c h is in c o n t i n u o u s u p w a r d , or d o w n w a r d , m o t i o n is also of i m p o r t a n c e in s e v e r a l
272
CONVECTIVE FLOWS
manufacturing processes in industry. Ingham (1986b) has shown that if the plate moves with a constant velocity Uw, and its surface temperature varies according to the relation (3.54), then the governing boundarylayer equations become similar and their solution depends on the mixed convection parameter ;k, which is now defined as Gr A  2 Re5 (8.61) Several flow situations corresponding to A > 0 (assisting flow), A < 0 (opposing flow) and A = 0 (forced convection flow) have been considered in detail by Ingham (1986b). Under the usual Boussinesq and boundarylayer approximations, the governing equations for this problem can be written in nondimensional form as, see Ingham (1986b),
0r 02r
0r 02r
03r
Oy OxOy
Ox Oy 2
Oy3
I
AT
(8.62)
2
or OT
0r OT
1 02T
Oy Ox
Ox Oy
P r Oy 2
(8.63)
and the appropriate boundary conditions are as follows: r
oOy_ r~ ~ T  !0__~r _~ 0, T + 0 Oy
on as
x
y0, y ~ c~,
x>O x>O
(8.64)
If we look for a similarity solution of the form r189
T10(77),
~/
1
(2~)~
x
(8.65)
then Equations (8.62)and (8.63) become, with P r = 1, f'" + f f " + AO  0
(8.66)
0" + f O' + 2 f'O = 0
(8.67)
and the boundary conditions (8.64) reduce to f(0)=0,
f'+0, 8.5.1
f'(0)=l,
0+0
as
0(0) = 1
(8.68a)
U + cr
(8.68b)
A >0
Solving numerically Equations (8.66)  (8.68), Ingham (1986b) has found that dual solutions exist for 0 < A < Am (= 2.531) and that near A = 0 + the solution on the second (upper) branch curve is of the form f(r/)  fo(r/)+ Afl(r/) + . . .
o(,7) = ~,~Oo(,7)+ o~(,7)
+...
(8.69)
C O N V E C T I V E BOUNDARYLAYER F L O W OVER MOVING SURFACES
273
The functions f0, 00, fl and 01 are given by the following two sets of equations /g' + f o f g + Oo  o,
fo(O)O, f(~O,
o8 + foO'o + 2 y Oo  o
f(~(O)l, 0 o  + 0 as
0o(0)0 r/+oo
f~,r + loft' + f l f ~ ~ + O1   0 Oilt + fOOll + 2f~01 + fl Oto + 2 f~ O0  0 fl(0)0,
f~ +0,
f~.(O)O, 01 ~0 as
01(0)1 rl~C~
(8.70)
(8.71)
and the numerical solution of Equation (8.70) gives f ~ r ( 0 )  29.325,
0(~(0)  393.69,
f0(cx3)= 5.733
(8.72)
The values of f"(0), f(c~) and 0'(0), as obtained by Ingham (1986b), for 0 < < 1 on the second (upper) branch curve are given in Table 8.3 and the values 1 t of ~00(0 ) are also included in this table for comparison. These results clearly show that as A ~ 0 + the asymptotic solution (8.69) is being approached. 1 ! Table 8.3 Variation of f"(O), f(oo), 0'(0) and ~0o(0 ) as a function of )~ for P r  1.
e,r 1.0 0.8 0.6 0.4 0.2 0.1 O.05 0.02 0.01 Limit A = 0.0
8.5.2
300.5 399.9 564.7 893.5 1878.4 3847.1 7784.1 19594.0 39279.0
393.7 429.1 656.2 984.2 1968.5 3936.9 7873.8 19684.0 39369.0
,k < 0
In this case it was found by Ingham (1986b) that no solutions of Equations (8.66)  (8.68) are possible for ~ < ~c (= 0.182) and this occurs at f(c~)  0.348. As the value of f(oo) was further reduced towards zero it was shown that a second branch of the solution curve has been developed. It appeared that )~ ,~ 0.174 as f(c~) + 0 +. Further, a search of the asymptotic solution of Equations (8.66) and
274
CONVECTIVE FLOWS
(8.67) as 77 + oo yields that it must have an algebraic decay which is of the form Ao A1 A2 + + +... ( , + ao) ( , + ao) 2 (~ + ao) 3 Bo B1 B2 0 ~ + + +... ( , + ao) 4 ( , + ao) 5 (~ + ao) 6
f~
(8.73)
where ao, Ao, B o , . . . are constants, 10 A0~,
20 9'
AooB0
8A AcoB1   ~ 1~
AooB2
_
3 2A 2~ A2  T~A~ ,al.
X.J
(8.74) and )%0 is the unknown value of A for which f (oo) = 0. A numerical solution of Equations (8.66) and (8.67), subject to the boundary conditions (8.68a) at 77 = 0 and (8.73) at 77 = 7/oo, gives Aoo0.1739,
f"(0)0.8873,
0'(0)0.7467
(8.75)
The variation of f ' ( 0 ) , 0'(0) and A as a function of f(oo) is shown in Table 8.4. This table clearly shows that the solution of Equations (8.66) and (8.67), which at large values of r/ has the algebraic decay given by expressions (8.73), is being approached and that dual solutions exists for Ac < A < Aoo. Ingham (1986b) has demonstrated analytically that the solution of these equations is singular at ,k = Aoo. Therefore, an important and novel outcome of the opposing flow case (A < 0) of this problem is the singular nature of the solution curve of Equations (8.66)  (8.68), which terminates at f (oo) = 0. In all other similar problems, where dual and singular solutions exist for nonlinear ordinary differential equations, the termination of the solution curve usually occurs in a quite predictable manner but this does not occur in the present problem.
Table 8.4: Variation of f"(0),0'(0) and A as a function of f(c~) for P r 
?(ooi [ f" (0) _
0.7 0.6 0.5 0.4 0.3 0.2 0.1 O.O5 0.02 0.01 0.0
0,(0) _.
0.8479 0.8712 0.8861 0.8939 0.8955 0.8927 0.8883 0.8877 0.8874 0.8873 0.8873
0.8383 0.7967 0.7671 0.7486 0.7401 0.7411 0.7448 0.7460 0.7465 0.7467 0.7467
A 0.1514.... 0.1672 0.1770 0.1814 0.1814 0.1783 0.1746 0.1742 0.1739 0.1739 0.1739
1.
C O N V E C T I V E BOUNDARYLAYER F L O W O V E R MOVING SURFACES
275
Typical results for f"(0), f(er and 0'(0) obtained from a direct numerical integration of Equations (8.66)  (8.68) are shown in Figure 8.13 for several values of A of interest. Also shown in these figures are the limiting and asymptotic solutions. It is seen that there is an excellent agreement between these solutions. Further, it is observed from Figure 8.13(a) that f"(0) > 0 on the lower branch of the solution curve for ~ > 0.3817 and on the upper branch curve for X > 0. This shows that for this range of values of A that the maximum fluid velocity occurs within the boundarylayer. This is not surprising since as ~ increases the buoyancy force becomes larger and it will eventually dominate over the motion caused by the plate. Also, 0'(0) > 0 for all values of A for which solutions are possible and this gives rise to temperature profiles in which the maximum temperature occurs at a point within the boundarylayer rather than on the plate. (a)
(b) 30
600
f"(o)
\
o'(o)
\\
450
20
300 10
150
0:5
1:0 1:5 2. 0
2:5
0:5
1:0 1:5 A 2:0 2:5
Figure 8.13: Variation of (a) f"(O), and (b) 9'(0), with )t for P r  1.
The numerical solutions are indicated by the solid lines, the asymptotic solution (8.69) for small values of A is indicated by the broken line and the limiting solutions are denoted by the symbols o.
It should be noted that the corresponding problem of a flat plate which moves horizontally has been studied in a similar way by Merkin and Ingham (1987). It was found that there is a unique solution for all positive values of the buoyancy parameter A and that for negative values of A the solution terminates in a singular manner with algebraic decay.
276
C O N V E C T I V E FLOWS
8.6
M i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow from a m o v ing horizontal fiat plate
Consider the mixed convection boundarylayer flow over a horizontal flat plate which moves continuously from a slot with a constant velocity Uw (>/0). The plate moves in a viscous incompressible fluid parallel to a uniform free stream U~ (~> 0), see Figure 8.14. The mixed convection boundarylayer flow arises due to the interaction of the free stream, the motion of the plate, and the streamwise pressure gradient caused by the buoyancy force from the temperature difference between the uniform surface temperature Tw and the ambient fluid temperature Too, where Tw > Too for a heated plate and Tw < Too for a cooled plate. This general problem was first formulated by Lin et al. (1993) and it includes six subproblems as we will show below.
t~u~
Too_._~,~y ~
~u~
T~ ._~ ''~ Ay
~
i
:k
v~ Figure 8.14: Physical model and coordinate system.
The basic equations governing the boundarylayer flow for this problem are the Equations (3.50)  (3.52) and they have to be solved subject to the boundary conditions u:i=Uw, vO, T=Tw on y=0, x>0 (8.76) u + Uoo, T + Too, P + poo as y + co, x > 0 where t pertain to the case of the plate moving in the same or the opposite direction to that of the free stream. In order to solve this problem Lin et al. (1993) defined the following variables:
r  o~fA6f(~, 7]),
0(~, ~7)  ~ i/kT ,
P  Poo 
x ~ h ( ~ , 71),
x
(8.77) where AS
1 1
Af  (o'3Rew + o'2Reoo) 7 , O'1  
Pr 1+Pr
~ A6  "+~' ~7'  ( ( = ~R~) 1 Am O2  
1 +
al Rew
Pr
(l+Pr) 89 '
~
AI +
, 
Pr 2 l+Pr
An
An  (al Raz)
~
(8.78)
C O N V E C T I V E BOUNDARYLAYER F L O W OVER MOVING SURFACES
277
with Rew = u~oz and Reoo  Uooz being the local Reynolds numbers based on Uw // /2 and Uoo, respectively. It is worth mentioning that the transformations (8.77) and (8.78) convert the entire mixed convection domain from 0 0
(8.82) where the + sign in Equation (8.80) corresponds to the case of assisting flow above the heated plate due to a favourable pressure gradient, whilst the  sign applies to the opposing flow above the cooled plate. The following six problems can be readily obtained from the general Equations (8.79)  (8.82) by setting appropriate values of the parameters ~ and Am, namely (i) free convection on a horizontal flat plate (~  1); (ii) forced convection from a fixed plate (~ = 0, Am = 0); (iii) forced convection from a moving plate in a quiescent ambient fluid (~  0, Am  1); (iv) forced convection from a moving plate in a free stream (~  0, 0 ~< Am ~< 1); (v) mixed convection from a fixed horizontal plate in a free stream (Am  0, 0 ~< ~ ~< 1); and (vi) mixed convection flow from a moving horizontal flat plate in a quiescent ambient fluid (Am  1, 0 ~< ~ ~< 1). Physical quantities of interest are the fluid velocity and the temperature profiles, as well as the local skin friction coefficient and the local Nusselt number. After some manipulations we obtain
C / R e ~  2a} (1  Am)~ (1  ~)3 f,,(~, O) 1
N~ = aft ( 1  Am) 89 ( 1  ~11 [0'(~,01]
R~
1
(8.83)
1
Nu r =a~A m ~(1~) Re~
1
[0'(~,0)]
Equations (8.79)  (8.82) were solved numerically by Lin et al. (1993) using the Kellerbox method for several values of the parameters ~, Am and Pr which are of interest. Figures 8.15 to 8.17 show the reduced fluid velocity and the temperature profiles for Pr  0.7 from the limiting case of a fixed plate (Am  0) to the other
278
CONVECTIVE FLOWS
(a)
(b) 14
2.5
"~ 1.2
2.0
~1.0
1.5
0.8 0.6
1.0
0, 0.1, 0.2,..., 1
0.4 0.2 0.0
9
0
1
2
3
4q5
6
7
0
1



9
2
3
4
5
7
Figure 8.15: Fluid velocity profiles, f'(~,~), for P r  0.7 in the case of assisting flow when (a) )~r~ 0 and (b) Am  1 (parallel to the free stream moving plate).
1.4 1.2 f'((, 77)1.0 0.8
= 0, 0.05, 0.1,
4
0.6 0.4 0.2 0.0
o
i
6
Figure 8.16" Fluid velocity profiles, f'(~,~7), for Pr = 0.7 and )~m  0.5 (parallel to the free stream moving plate) in the case of opposing flow.
limiting case of a moving plate (Am  1) in both assisting and opposing buoyant flow situations. It is from Figure 8.15(b) t h a t for the special case of U~  Uoo (Am 0.3295) the fluid velocity profiles are uniform in the forced convection d o m i n a n t region, ~ < 0.3. Then, Figure 8.16 shows that in the buoyancy opposing fluid flow case, the fluid velocity profiles, as expected, decrease slightly as the p a r a m e t e r increases from 0 to 0.4. It was stated by Lin et al. (1993) t h a t convergent numerical solutions cannot be obtained for higher values of ~, for example, ~ > 0.4 for ~ m  
C O N V E C T I V E BOUNDARYLAYER F L O W OVER MOVING SURFACES
279
(b) 1.0
1.0
0.8
~ 0.8
0.6
0.6
0.4
0.4
0.2
0.2
9
0.0
,
0.0 0
2
4
~7
6
8
0
2
4
71
6
8
Figure 8.17: Temperature profiles, 0(~,r/), for Pr = 0.7 and Am = 0.5 (parallel
moving plate) for (a) assisting flow and (b) opposing flow.
0.5. This is due to the breakdown of the boundarylayer approximation when the unfavourable pressure gradient is larger than a certain critical value. Also, the temperature profiles shown in Figure 8.17 increase as the parameter ~ increases from 0 to 0.5, but decrease as ~ increases from 0.6 to 1 for the case of buoyancy assisting flow. The skin friction coefficient and the local Nusselt number, as given by the relations (8.83), are shown in Figures 8.18 and 8.19 for several values of ~, Am and Pr. Some interesting features of these quantities can be clearly seen in these figures and a good discussion of the flow characteristics can be found in the paper by Lin et al. (1993). Additionally, Lin et al. (1993) have given some comprehensive correlations for the Nusselt number when 0.01 ~< P r 0 for ~ < T*. Equations (9.30)  (9.32) have to be solved along with the boundary conditions that the solution is that as obtained by the stepbystep marching procedure at ~  T*, and that at T  Too (a large value of T) the solution is that given by the steady state analysis obtained in Section 1.3. Also, the boundary conditions at the plate and at infinity are as follows: I(T,O)O, g+0,
9(T,O)=O, 0+0
as
O(T,O)I 1 77+oo
J
for
~'*
T
Too
(9.34)
The numerical finitedifference scheme used, along with other details of the integration of Equations (9.30)  (9.32) subject to the boundary conditions (9.34), is very well described by Ingham (1985) and therefore it is not repeated here. The variation of the nondimensional skin friction, ~ (~',0), and the nonoo (~., 0) as a function of T are shown in Figure 9.1 dimensional heat transfer, ~~ for Pr = 1 and for some values of m. It is seen from this figure that the larger the value of ra, the earlier is the steady state solution achieved. However, the asymptotic analysis, for large z with the values of ")'min given in Table 9.1, suggests that the larger the value of ra the slower is the approach to the steady state solution. It is also interesting to note for ra >/0.5 it was found that the stepbystep numerical procedure of solving Equations (9.30)  (9.32) numerically could be continued up to a fairly large value of z, by which time the steady state had been reached. Therefore, there was no need to solve these equations by forwardbackward differencing. However, for m = 0.1 and 0.025 the stepbystep numerical method used produced both a maximum and a minimum in the heat transfer and the skin friction, as can be seen from Figure 9.1, before the numerical scheme breaks down. The forwardbackward differencing was then used and the unsteady and steady state solutions
292
CONVECTIVE FLOWS
(~)
(b)
~"
1.1 0.8
~   
~P~
~ ~0.9
0.6 0.4
0.7
0.2
0.5
0.0
0
i
~
~
T
~
g
0.3
=0.1,
0
~
2
o, 0.025, 0.1, .
3
7"
4
Figure 9.1" Variation of (a) the nondimensional skin friction, 5~(T, ~ O), and (b) the nondimensional heat transfer, 5~~176 (T, 0), at the plate, with ~" for P r  1
were successfully matched. Further, when m ~< 0 the numerical scheme eventually breaks down and no smooth solution that matches the unsteady and steady state results could be obtained. T h e case 0 < m < 0.1 needs special attention and it was studied in detail by I n g h a m (1985). Results for the nondimensional t e m p e r a t u r e and the fluid velocity profiles were also obtained by I n g h a m (1985) for m =  0 . 1 , 0, 0.001, 0.025, 0.5, 1, 2 and 4 with P r  1. However, we show in Figures 9.2 and 9.3 only the results for m =  0 . 1 , 0, 1 and 4 and again with P r = 1. The steady state results are also indicated by the dots in these figures. These figures clearly show t h a t at the smaller values of m the t e m p e r a t u r e and the fluid velocity profiles overshoot the steady state profiles. Further, we note that as m increases, the smaller is the value of 77 at which the b o u n d a r y conditions 0(77) + 0 and f'(77) + 0 as 77 ~ oo are approached.
UNSTEADY
FREE
AND MIXED
CONVECTION
(a)
293
(b) 0.6
0.6
0.5
~  0.1325, 0.7725,
~
~
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0
4
0.0
8
(c)
T  0.1325, 0.9725,
0
4
8
(d)
0.3
0.1325, 0.5, 1.1325,
~ o~t~\~.
~~
~ o.~.
0.1 0.0
0.3
~
~
~= 0.1325,
0.6325, 1.3025, 4.2
0.1 0
4
rl
8
0.0
0
4
7/
Figure 9.2: Fluid velocity profiles, 5~(T, of 71), for P r  1 when ( a ) m   0 . 1 , (b) m  O, (c) r n  1 and (d) r n  4. The steady state solutions are indicated by the symbols ..
294
CONVECTIVE
(~)
FLOWS
(b) 1.0
1.0
0.4
0.4
0.2
0.2
o oo
~
~
~
9
(c)
oo.
,
0
.

,
4
9
77
8
(d) 1.0 
1.0
0.8
C
~.
"
0.8
C
~" 0.6
~" 0.6
0.4
0.1325, 0.5, 111:::~.
0.4
.
O'Uo
2
4
77
6"
.
.
.
0.0
Figure 9.3: Temperature profiles, 0(% rj), for P r = 1 when (a) m =  0 . 1 , (b) m = O, (c) rn = 1 and (d) rn = 4. The steady state solutions are indicated by the symbols . .
UNSTEADY FREE AND MIXED CONVECTION
9.4
295
T r a n s i e n t free c o n v e c t i o n b o u n d a r y  l a y e r flow over a s u d d e n l y c o o l e d v e r t i c a l plate
The problem considered now is the one in which, for time t < 0, the steady state free convection boundarylayer over a vertical semiinfinite flat plate of constant temperature Tw (> Too) has been set up. Then, at time t  0 the value of Tw is impulsively reduced to Too. The effect of this change in the wall temperature is confined to a thin boundarylayer, within the steady state boundarylayer (inner layer), but growing in size with increasing time. Eventually, for sufficiently large values of time this will not be true. This problem was studied for the first time by Ingham (1978c) following, in principle, the same method as that described in Section 9.3 for the case of an impulsively heated flat plate but with m = 0, which corresponds to an isothermal flat plate. The transformation (9.8) now becomes
q2  x88f (r, 77), T

O(T, 77),
77 
r
yl',
x~
ta
(9.35)
x~
and Equations (9.3) and (9.4) can be written as 03 f 07] 3
(3 ~f
1 0,)02f ~T 1 020
1 (0,) 2

( O f ) 1
02,
1 Of)O0 ( 1 O f ) O0 ~~ 1  f i  r ~ ~
(3
P r O , 2 t ~f  ~T ~r
(9.36) (9.37)
If in these equations we neglect derivatives with respect to T and write f(T, rl) f0(~) and 0(%r/)  00(7/) then we obtain the steady state equations for the free convection flow over an isothermal vertical flat plate, namely
fg' + YoYg ~ao + Oo  0 1 Dtt
~o
(9.38)
+ ~]oO'o  o
These equations are precisely those which govern the flow for ~ < 0, along with the boundary conditions fo(O)  O,
f~O,
A(O)  O,
0o(0)  I
0o+0
r/+c~
as
(9.39)
Therefore, the problem described by Equations (9.36) and (9.37) must be solved subject to the following initial and boundary conditions:
ffo(r/),
O=Oo(r/) for rO,
f(~,o},=o,_+o, f'O)o~o (~, =o,as,~oo~176=o}
all r/ for
T>~O
(9.40)
296
CONVECTIVE FLOWS
Also, for the initial period (T 1, F and G must have the form: F ( T , ~) = ~2H(~) + h.o.t. G(T, r  T2K(~) + h.o.t.
(9.50)
On substituting expressions (9.50) into Equations (9.41) and (9.42) gives the eigenvalue problem H"' + (2~ + 4H) H " + 4 (2  H') H' + 4 K = 0 1 K" ~ (2~ + 4H) K ! + 4 (2  H l ) K  0 p~ H(0)0, U'(0)=0, K(0)0 H'+0, K~0 as r
(9.51)
Ingham (1978c) has found that there are an infinite number of possible solutions of Equations (9.51) because the asymptotic behaviour of these solutions are of the form: r H ' ,,~ A : ~ 3 e ~2 + A2r 4 + A 3 ~ e + A4~ 2 K ~ Alr r + A2~ 4 (9.52) where the constants A i , for i = 1 , . . . , 4, can be determined by solving numerically Equations (9.51). It is worth mentioning that higherorder terms in the expressions (9.50) can be determined and these terms involve eigensolutions. Since the analytical solutions (9.45) and (9.50) for small values of v and large values of ~, respectively, are expressed in terms of the variables T and ~, it is thus reasonable to deal with Equations (9.41) and (9.42). These equations were solved numerically by Ingham (1978c) using a stepbystep procedure as described by Merkin (1969). The numerical solution starts at ~ = e, where e is a small number, with the velocity and temperature profiles as given by Equations (9.48). The boundary conditions enforced at ~ = 0 and at a large value of ~, say ~m, are as follows: F ( T , O)  O,
(OF
OF (% (m)  : dfo (Vf~m) ~ dr/
o) '
o,
o) = o
G (% (m)  00 (~/~m)
(9.53)
298
CONVECTIVE FLOWS
It was found t h a t accurate results could be obtained using e  10 4, a step length 0.05 in the ~ direction and a value of ~m = 10 was found to be sufficiently large. Figure 9.4 illustrates the variation of ~.2 0(r OF and ~2G as a function of ~ for some values of ~ when P r = 1. Also shown (by the dotted lines) are the steady state solutions obtained from expressions (9.50). Figure 9.4(a) shows, as the value of T increases above about 4, that the numerical solution approaches closely the steady state (analytical) solution and it was found for values of T greater t h a n about 15 that b o t h the solutions are almost indistinguishable. Figure 9.4(b) also shows, as T increases, t h a t the steady state solution is being approached and for values of ~greater t h a n a b o u t 15 the numerical and analytical solutions are almost identical. (a)
(b) 0.6
0.6
0.5
~ 0.5 C 0.4
0.4 ~0.3
T= 2
~" 0.3
0.2
0.2
0.1
0.1
0.0 0.0
T = 4
0.5
1.0
1.5
2.0
2.5
0.0 0.0
9
0.5
1.0
1.5
T
2.0
 1
2.5
Figure 9.4" (a) The fluid velocity, and (b) the temperature, profiles/or Pr  1. The steady state solutions obtained from expressions (9.50) are indicated by the dotted lines.
The variation of the nondimensional skin friction and the wall heat transfer are shown in Figure 9.5 for P r = 1. The analytical results for small and large values of ~, as given by Equations (9.48) and (9.50), respectively, are also included in these figures. It is seen that the first two terms in the small time analysis are in good agreement with the numerical solution of Equations (9.41) and (9.42) up to T ~ 0.5 and the large T solution may be taken for values of T greater than about 4.
UNSTEADY F R E E AND MIXED C O N V E C T I O N
(a)
299
(b) 
1.0
2.0 " i
o
0.8
0.7473 T~
,
,
,
,,,,
,,,,,
,,,,
i
1~
1
1.5
0.6 1.0 0.4 \~
%%
0
2
,
0.5
0.2 0.0
I\,.,.
4
0.0
II
o
i
i
g
T
i
Figure 9.5: Variation of (a) the nondimensional skin friction, ~ ( T , 0), and (b) the nondimensional wall heat transfer, ~~ (v, 0), with T for Pr  1. The numerical solutions are indicated by the solid lines, whilst the approximate solutions (9.~9) are indicated by the broken lines.
9.5
T r a n s i e n t free c o n v e c t i o n b o u n d a r y  l a y e r flow over a v e r t i c a l flat p l a t e at s m a l l and large P r a n d t l n u m bers
In a series of papers, Carey (1983, 1984) and Park and Carey (1985) observed that the problem of transient free convection flow from a vertical fiat plate has been, in general, studied for Prandtl numbers both near 1 (gas) and 7 (water). However, very little information was provided about the systematic behaviour of this transient flow at low and large values of the P r a n d t l number. Therefore, the object of Carey's papers was to study the transient free convection from a vertical fiat plate resulting from the sudden increase of its surface temperature, or from the sudden generation of a uniform heat flux at the plate surrounded by a small or high P r a n d t l number fluid. Flows of these types commonly occur in technological applications. A low Prandtl number is appropriate to liquid metals which are used in nuclear reactor heat transfer technology and thermal transport in metallurgical processing, whilst a fluid with a high value of the Prandtl number is sometimes used as a heat sink in electrical transforms. The sudden application of electrical power to the transformer produces a transient buoyancydriven flow. Transient flows at high P r a n d t l number may also result from the sudden addition or removal of heat in the chemical processing of hydrocarbon and silicone polymers and in thermal energy storage devices.
CONVECTIVE FLOWS
300
The governing Equations (9.2)  (9.4) for these problems have to be solved subject to the boundary conditions u=0,
v=0,
T=0
for
y~>0,
all x,
t=0
(9.54a)
u=0,
v=0,
T=0
on
x=0,
y>0,
t>0
(9.54b)
u=0,
v=0,
T=I
on
y=0,
x~>0,
t>0
(9.54c)
u+0,
T~0
as
y+c~,
all x,
t>0
(9.54d)
It can be seen that as Pr + O, or Pr + oc, the energy Equation (9.4) becomes singular and in order to obtain a solution of these equations for small and large values of Pr, a singular perturbation technique is necessary. Therefore it is assumed that through the transient, the flow consists of two regions, namely an inner region near the surface which is dominated by buoyancy and viscous effects, and an outer region where only thermal and inertial effects are important. The analysis proposed by Carey (1983, 1984) and Park and Carey (1985) to solve Equations (9.2)  (9.4) consists of a combination of the matched asymptotic expansion technique with an explicit finitedifference computational scheme. Asymptotic transients were first obtained followed by a numerical computation of the firstorder correction to the Equations (9.2)  (9.4) to predict the fluid flow and heat transfer characteristics at moderate values of Pr. These results provide a more complete picture to the manner in which the heat transfer and flow behaviour changes with Pr. In addition, the results demonstrate the usefulness of the computational techniques. We notice to this end that the corresponding steady state problems for small and large values of the Prandtl number have been considered by Kuiken (1968, 1969). 9.5.1
Pr ~
1
Based on the arguments of the singular perturbation technique, the (y, t) coordinates in Equations (9.2)  (9.4), as well as the fluid velocity and the temperature functions, are scaled for small values of Pr in the inner region as followsy~ tT _
+
+...
+
2
T  1 + er 89
(9.55)
i
Pr~T(i)+...
On substituting these expressions (9.55) into Equations (9.2)  (9.4) gives the following equations for the inner variables: +
og') = o
(9.56)
UNSTEADY F R E E AND MIXED C O N V E C T I O N
301
o~ ~) ~(o~)o~~) 4) o~ ') 0~ +
Ox
+
On
=
T (i) 
o ~ ') + On 2
1
1
(9.58)
Ou~') O@ ) ! =0 Ox 0~? 0~+
Ox
+
+
Ox
Or] +
On
(9.57)
(9.59)
 Or/~+
(9.60)
= 0
(9.61)
02T~ i) 0V2
On the other hand, the outer scalings are chosen such that, to the lowest or2 der, the term ~Ou m the momentum equation (9.3)is eliminated. Thus, the outer stretched coordinates and expansions are of the form: .
y  Pr~
1
tT
_ ~o)+ p i~o)+ p~ ~o)+...
v  P r  ~
+v
(9.62)
+...
T  To(~ ~IT}~
P~T~(~ + ...
where the coefficients functions u~0), v~0), T (0), etc. are given by the equations
0~ +
O~(o~ N
O~o(~
0~
~ +
~o) o~ ~ N
~o)~0 (~ 0~
~ ~
o~?) or~ ~ oU +
~o) O~ ~ u(oo) o ~ ~ o~
+
(o) O~ ~
~(o) O~ ~
~o) Oro(~
4 ) or~ ~
oVF
N
 0
(9.63)
4 ) O~(o~ _ To(O) N
~o)~ro(~ 0r
o~~ Ox
0(
=
O~o(~ 0~ 2
(9.65)
Ova~ +
0r
= 0
(9.66)
~o) O~ ~ _ ~o) ~o) Oro(~ ~
(9.64)
0r
(9.67)
o~r~~ =
or
(9.68)
The initial and boundary conditions at the plate, ~ = 0, for the inner Equations (9.56)  (9.61) are obtained by substituting the inner variables (9.55) into the
302
CONVECTIVE FLOWS
initial and boundary conditions of Equations (9.54a)  (9.54c). Similarly, the initial and boundary conditions for the outer Equations (9.63)  (9.68) are obtained by substituting the variables (9.62) into Equations (9.54a), (9.54b) and (9.54d). Since the flow is split into two regions, each has its own set of equations, the remaining necessary boundary conditions for each set of equations must come from matching the inner and outer variables (9.55) and (9.62), respectively. The outer, 7/ ~ cx~, boundary conditions for the inner Equations (9.56)  (9.61) and the inner, ( + 0, boundary conditions for the outer Equations (9.63)  (9.68) are obtained by matching the inner and outer expansions. The method used, as well as the long expressions for the initial and boundary conditions, can be found in Carey (1983, 1984) and Park and Carey (1985) and therefore they are not repeated here. During the initial transient, small 7, the following closed form solution may be obtained for the zerothorder Equations (9.56) (9.58) and (9.63) (9.65) _
o.
c
+
o.
v~~0, T (i)  1 , where i 2 erfc z is defined as i2
c
v~~ To(0)  erfc ( ~
erfc z  1 [(1 + 2z 2) erfc z
(9.69)
2 _Z2]
~ze
(9.70)
According to the matched asymptotic method, proposed by Van Dyke (1975), the composite solution is the inner solution plus the outer solution minus terms which are common to both the solutions. Thus, the uniformly valid solution for the fluid velocity u and the local temperature T is given by u4t T
[ (1
erfc
i 2erfc
~
y

erfc
~
~
(9.71b)
It should be noted that the solution for u agrees with the corresponding solution obtained by Goldstein and Briggs (1964) for the free convection flow over an infinite vertical flat plate. The local Nusselt number, Nu, is given for Pr ~ ~ ( ~ ) R a  0 . 5 , where 7 denotes the time scale for the formation of the thermal boundarylayer, the tangential component of the fluid velocity continues to increase. This increase in the convection causes the development of the plume region. The upward flow transports hot fluid to the top
(a)
t=0.5
t=6
t = 16
t = 0.0002
t = 0.0012
t = 0.002
(b)
Figure 9.17: S t r e a m l i n e s (left) a n d i s o t h e r m s (right) f o r P r   0 . 7 w h e n (a) R a = 10 (/%r 1, A T  0.1) a n d (b) R a  l0 T ( A r 20, A T  0.1).
UNSTEADY FREE AND MIXED CONVECTION
319
( t
 0.001
t  0.0055
t = 0.002
t = 0.003
t = 0.006
t = 0.0065
F
t = 0.004
t = 0.005
t = 0.007
t  0.0011
Figure 9.18: S t r e a m l i n e s (left) and i s o t h e r m s (right) f o r P r  0.7 and R a  106 (Ar  10, AT  0.2).
surface of the cylinder a n d gradually this forms a distinct t e m p e r a t u r e front between the heated fluid and the u n h e a t e d ambient fluid as shown in Figures 9.17(b) and 9.18. The fluid at the top surface of the cylinder then detaches itself and rises while rotating at the same time because of viscous effects. At larger values of the Rayleigh number, the rotation is quite evident and this leading edge of the heated fluid forms a ' m u s h r o o m ' p a t t e r n which gradually takes the final (steady) state form of a buoyant plume. At very large values of the Rayleigh number, a b o u t R a = 2 x 107, small separation vortices which are symmetrically disposed near the top surface of the cylinder are formed, a n d they grow at a m o d e r a t e rate. Further, they are shed into the plume and then they reform again with this sequence continually repeating itself.
320
CONVECTIVE FLOWS
9.8
Transient m i x e d convection boundarylayer from a horizontal circular cylinder
flow
It appears that the problem of unsteady mixed convection flow from a horizontal circular cylinder has only been studied up to now by Katagiri and Pop (1979) and Ingham and Merkin (1981) using the boundarylayer equations, and by Jain and Lohar (1979) by solving numerically the full governing equations. In order to solve this problem Katagiri and Pop (1979) have used the local nonsimilarity method together with a differencedifferential method. However, this is an approximate method and the results can be obtained only near the front (lower) stagnation point of the cylinder. Further, the results were obtained only for the case of assisting flow. Ingham and Merkin (1981) have obtained an accurate numerical solution of the boundarylayer equations using the method of series truncation, similar to that described by Ingham (1978d) and presented in Section 8.3. Solutions which are valid near both the front (lower) and the rear (upper) stagnation points of the cylinder have been obtained using a standard finitedifference method. A small time solution was also presented with which the accuracy of the numericalsolution can be checked and both the assisting and opposing flow cases were considered. The unsteady mixed convection boundarylayer flow from an isothermal horizontal cylinder in a stream flowing vertically upwards is described by the following equations, written in nondimensional form, as, see Ingham and Merkin (1981),
02r
0r 02r
0r 02r
03r
= 0y 3 + sin x cos x + ~0 sin x OtOy Oy OxOy Ox Oy 2 1 020 oo or oo or I
Ot
f
Oy Ox
Ox Oy
P,
ay2
(9.107) (9.108)
where )~ is the mixed convection parameter defined as in Equation (2.147) and can take positive or negative values. These equations have to be solved subject to the initial and boundary conditions:
r
r
or =0 Oy
~
r
~162 Oy
~
00
for
t0
00
01
on
y0
0~<x~0
on
x  0,~,
y ~> 0,
t ~> 0
as
y + oo,
0 ~< x ~< 7r,
t > 0
02r 00 _ 0 o x 2  0 , o~o~ 07 + sin x, 0  + 0
(9.109)
To obtain a solution of Equations (9.107) and (9.108) for small values of t (0
t>/ 0
as
rl+oo,
O~x~O
f  O,
Ox2 
Of +sinx, O,7
g+O
(9.113)
A solution of Equations (9.111)  (9.113) in power series for small values of t ( A1 t h e n t h e r e will be no s e p a r a t i o n of the b o u n d a r y  l a y e r . However, for t h e calculations p e r f o r m e d w i t h ~  1.1, 1.125 a n d 1.5, a region of reversed flow a p p e a r s first w i t h i n t h e b o u n d a r y  l a y e r , i.e. OF becomes negative over 02F
p a r t of the flow, whilst ~0~ (x  ~, 0, t) is still positive. T h e region of reversed flow t h e n grows r a p i d l y o u t w a r d s a n d m o r e slowly reaches the wall so t h a t eventually the region of reversed flow extends all t h e way to the wall. T h i s is shown in Figure 9.21 where OF is p l o t t e d at various values of t for ) ,  1 25 as a f u n c t i o n of y. 9
1.0 ~
_
~
~
_
_
0.6
& 0.29
.
_ 0 . 2 ~ 3
.
12
.
.
.J,.
16 y 20
.
24
 0 6~ Figure 9.21" Fluid velocity profiles, ~OF(X  7F, y, t), at x  r (the upper stagnation point) for P r  1 and X  1.25.
Finally, we show in Table 9.2 t h e values of the s e p a r a t i o n angles 0  {?s det e r m i n e d by J a i n a n d L o h a r (1979) for t h e p r o b l e m of u n s t e a d y mixed convection flow from a h o r i z o n t a l circular cylinder for R e  100 a n d 200, a n d G r  104 a n d 5 • 104 w h e n P r   0 . 7 3 a n d 0   0 s is m e a s u r e d from the lower s t a g n a t i o n p o i n t of the cylinder. T h e results of J a i n a n d Goel (1976), for t h e case of u n s t e a d y forced convection flow ( G r  0), have also b e e n i n c l u d e d in this table. It can be seen t h a t
Table 9.2" Variation of the angle of boundarylayer separation, 0  Os, with R e and Gr in mixed convection flow past a horizontal circular cylinder for P r = 0.73. Re= 100 Separation Separat, i0n at RightSide at LeftSide
Re  200 Separation Separation at RightSide at LeftSide 9
Forced convection Jain and Goel (1976) Mixed convection Jain and Lohar (1979)
114~ _ 120~ 116
~

241 ~ _ 246~
122~ 238 ~  244~ (Gr = 104) . . . .
105~  111 ~
250~
255~
108~  113~ 246~ 253~ (Gr = 5 x 104)
326
CONVECTIVE FLOWS
in the case of mixed convection flow, the separation of the boundarylayer is delayed as the buoyancy force accelerates the fluid motion in the layer and so reduces the deceleration of the fluid caused by the adverse pressure gradient and this is in agreement with the results obtained by Merkin (1976). It is also worth mentioning that Oosthuizen and Madan (1970) have derived the following correlation equation for the heat transfer rate Nufc
~e~e2
 0.011
~
(9.125)
where N u is the average Nusselt number for mixed convection flow and Nufc is the average Nusselt number for the forced convection flow past a horizontal circular cylinder. Having values of Nufc reported, for example by Jain and Goel (1976), we can calculate N u using the expression (9.125). It can be shown that these values of N u are in good agreement with the numerical results reported by Jain and Lohar (1979), see Table 2 in the paper by these authors. In closing this section, we mention the paper by Kikkawa and Ohnishi (1978) in which the authors have investigated both theoretically and experimentally the unsteady mixed convection flow from an elliptical cylinder which is inclined at an angle 7 to the horizontal free stream. The full NavierStokes and energy equations have been written in elliptical coordinates and then solved numerically using a finitedifference scheme. Developments of the streamlines and isotherms, as well as the variation of the local and average Nusselt numbers, are illustrated in several figures. It is also worth pointing out that the majority of the previously reviewed papers on unsteady free and mixed convection flows have been primarily focused on problems of vertical flat plates and horizontal cylinders. Very little work exists on unsteady free and mixed convection flow along a vertical cylinder. However, Yang (1960) has presented a discussion of the particular wall temperature variations which lead to similarity representation of the unsteady free convection boundarylayer on a vertical cylinder but no solutions were presented. AbdelelMalck and Badran (1990) have used twoparameter group transformations to study the unsteady free convection along a vertical circular cylinder subject to a variable, with time, surface temperature. The three independent variables are reduced to one and, consequently the governing equations reduce to a system of ordinary differential equations with the appropriate boundary conditions. The particular surface temperature that varies exponentially with time, i.e. of the form Tw(t)  ae bt, where a and b are constants, has been found to be appropriate to the study of the boundarylayer characteristic to this problem.
UNSTEADY FREE AND MIXED CONVECTION
9.9
327
U n s t e a d y free convection boundarylayer flow past a sphere
The problem of unsteady free and mixed convection about a sphere has received very little attention in the literature. Brown and Simpson (1982) studied the structure of the singularity which develops in the solution of the unsteady free convection boundarylayer equations near the upper pole of the sphere when the temperature of the sphere is suddenly raised above that of its surroundings. They argue, on the basis of a local solution in the neighbourhood of the upper pole, that the boundarylayer solution breaks down at a finite time following the initiation of the motion. From a detailed analysis of the complicated multilayered structure of this breakdown, and a numerical solution of the local governing equations, they were able to estimate the time at which the boundarylayer solution fails. Physically this breakdown corresponds to an eruption of the boundarylayer to form the free convection plume above the sphere. Sano (1982) studied the unsteady low Grashof number free convection about a sphere which is suddenly heated using the method of asymptotic expansion. It was shown that the solutions for the fluid velocity and temperature fields can be expressed in terms of three series which reflect the existence of three distinct regions in the (r, t)plane, where r and t are the nondimensional radial coordinate and time, respectively. Further, Awang and Riley (1983) have considered the case of the unsteady free convection boundarylayer about a fixed sphere, whilst Hatzikonstantinou (1990) has considered that of a rotating sphere. Then, Riley (1986) considered the unsteady free convection flow over a sphere by solving numerically the NavierStokes and energy equations for finite values of the Grashof number Gr using a splitoperator method, along with a standard alternating direction implicit scheme. It was shown that the occurrence of a singularity in the solution of the boundarylayer equation (Gr ~ 1) that signals the eruption of the flow from the upper pole on the sphere at a finite time, is associated with the boundarylayer approximation and is not a feature of the solution of the NavierStokes equations. The eruption process was illustrated by the isotherms close to the sphere. These are initially spherical in shape, but rapidly becomes distorted over the upper pole to form the mushroomshaped cap of the incipient plume. On the other hand, Nguyen et al. (1993) have studied the conjugate problem of unsteady heat transfer from a sphere under simultaneous free and mixed convection flow by solving the full NavierStokes equations for the external flow and the energy equations for both inside and outside the sphere. The problem was investigated numerically using a combined ChebyshevLegendre spectral method. They obtained results which show that the effects on the free convection are most remarkable in the wake region, i.e. above the upper pole of the sphere, where the flow structure is changed. Also, the average Nusselt number and the skin friction coefficient show a small increase or decrease in magnitude
328
C O N V E C T I V E FLOWS
depending on whether the buoyancy flow aids or opposes the main forced flow. We present here some results reported by Awang and Riley (1983) for the problem of unsteady free convection boundarylayer over a sphere whose temperature is impulsively raised to a constant value which is above that of the ambient fluid. The governing boundarylayer equations can be written in nondimensional form as, see Awang and Riley (1983), 0 0 (usinx)+ (vsinx) = 0 0x Oy Ou Ou Ou 02u Ot + U~x + v Oy = Oy 2 + 0 sinx
(9 126) (9.127)
O0 /90 O0 1 020 at + U~x + v Oy = P r Oy 2
(9.128)
and the appropriate initial and boundary conditions are as follows: u0, v0, O0 u  0, v  0, 0  1 u+0, 0+0
at on as
t0, y  0, y+c~,
0~<x~ 0 t>0
(9.129)
where the coordinates x and y are measured from the lower stagnation point of the sphere and normal to the surface of the sphere, respectively. The solution procedure for the solution of Equations (9.126)  (9.129) is divided into the following three parts:
(a) A solution for small values of t ( 0.9 a n d very large values of P r x . H o w e v e r , t h e a p p r o x i m a t e r e s u l t s of S h e n o y a n d U l b r e c h t (1979) d e v i a t e from t h o s e of H u a n g
F R E E AND MIXED C O N V E C T I O N IN N O N  N E W T O N I A N FLUIDS
341
0.55 Pr, = 1000, 2000 Nu
0.50
1
/~a 2~a n +
1
0.45
J
Tr~ = 10
0.40 0.35 0.5
0.7
0.9
1.1
n
1.3
1.5
Figure 10.2: Variation of the local Nusselt number with the powerlaw index n. The solutions of Huang and Chen (1990) are indicated by the solid line and the integral solutions of Shenoy and Ulbrecht (1979) and Kawase and Ulbrecht (198~) are indicated by the dotted and broken lines, respectively.
and Chen (1990) for all values of n, even at large values of P r x . Consequently, we conclude that the results of Huang and Chen (1990) are reasonably good for P r x ranging from 1 to 1000. Finally, values of the modified average Nusselt number, N u , as defined in Equation (10.28) and those obtained by some other authors are given in Table 10.1 for n = 0.5, 1 and 1.5 when P r x + co. It is observed that the local similarity solutions proposed by Huang and Chen (1990) give very good results for high values of the modified local P r a n d t l number.
10.3
Free convection boundarylayer flow o f n o n Newtonian powerlaw fluids over a vertical wavy surface
The prediction of the heat transfer from irregular surfaces is a topic of fundamental importance for many practical problems. Surfaces are sometimes roughened in order to enhance heat transfer, for example in flatplate solar collectors and flatplate condensers in refrigerators. The presence of roughness elements on a flat surface disturbs the fluid flow and hence changes the rate of heat transfer. Yao (1983) was probably the first who used the Prandtl transposition theorem, see Yao (1988), to analyse the steady free convection boundarylayer of a nonNewtonian fluid over a vertical wavy surface. A simple coordinate transformation was proposed to transform the wavy surface to a simple shape, namely that of a flat plate. The gist of the theorem is that
342
CONVECTIVE FLOWS
the flow is displaced by the irregularities on the vertical surface and the horizontal component of the fluid velocity is adjusted according to the shape of the surface. The form of the boundarylayer equations is invariant under the transformation and the surface conditions can therefore be applied on a transformed flat surface. Moulic and Yao (1989), Chiu and Chou (1993, 1994), Rees and Pop (1994a, 1994b, 1995a, 1995b), Chen et al. (1996), Yang et al. (1996), Kumari et al. (1996a), Kim (1997) and Pop and Na (1999) have used the transformation proposed by Yao (1983) to solve free convection problems associated with Newtonian fluids, micropolar fluids, fluidsaturated porous media and nonNewtonian powerlaw fluids. Consider the steady laminar free convection of a nonNewtonian powerlaw fluid over a wavy vertical surface which is maintained at the constant temperature Tw, where Tw > Too. The physical model and the coordinate system are shown in Figure 10.3, where x and y are the axial and transverse Cartesian coordinates, a is the amplitude of the surface wave and 1 is a characteristic wavelength of the surface waves. In particular, we assume that the surface profile is given by y  a(x)
(10.30)
where or(x) is an arbitrary geometric function.
y = 'tt
T T~ T~
Figure 10.3: Physical model and coordinate system.
The governing equations for this problem, in nondimensional form, are Equations (10.11)  (10.14) and they have to be solved subject to the boundary (nondimensional) conditions =o,
>0,
~+0,
o1
~'+0,
on
0+0
as
~+c~,
~>0
(10.31)
FREE AND MIXED C O N V E C T I O N IN N O N  N E W T O N I A N FLUIDS
343
where 3(~) = a(x) t 9 We now assume that the modified Grashof number is so large that free convection takes place within a boundarylayer, whose crosssection width a is substantially smaller than the amplitude 3 (= i) of the waves on the surface. Thus, invoking the boundarylayer scalings given by the transformations 1
 ~,
~
1
( ~  3) Gr ~(~+~),
~  ~,
Y(10.32) where az   d Y Equations (10.11)  (10.14) reduce, to leading order as Gr + o0 to the following boundarylayer equations: 0~
u~+Voy= O5 (.,,0~
~~2+y~
+ ~ = 0 oy
(10.33)
0~ + (l+a~) N
N
o
+ 9
(10.34)
= a~(o~, o~ + ~ (1 + ~)~ o ( o~ 0~ N N
+v
u ~x
~  ( 9  ~z~) Gr ~(~+~), p  p,
(10.35) .~OO ~09 1 + 2 a z 020 U~x + v o~  Pr O~2
(10.36)
and the boundary conditions (10.31) become =0,
~0, 01 ~0, 9+0
on as
~=0, ~+c~,
5~>0 ~>~0
(10.37)
It is noted from these boundary conditions that the new variable ~, defined in Equation (10.32), transforms the wavy surface into a flat surface. Also it should be noted that this analysis is valid only within the framework of boundarylayer theory with ~ '  O Gr 2(~+~) and ~  O Gr 2(~+~) as Gr + ec, as obtained from Equations (10.30) and (10.32). Further, Equation (10.35) indicates that oo~A _
(
(

0 Gr 2(,~+~) , which implies that the lowest order pressure gradient along the ~ direction is determined from the inviscid (outer) flow solution. However, for the present problem this gives ~o~  0. In order to eliminate the term Gr2('~+l)~ from Equations (10.34) and (10.35), we multiply Equation (10.35) by ~x and the resulting equation is added to Equation (10.34). After a little algebra, we obtain ~
U~x+V~+
1 +~2
0~
~
+ 1 +~2
(10.38)
Equations (10.33), (10.36) and (10.38), along with the boundary conditions (10.37), form the basic equations for the problem of free convection of a nonNewtonian powerlaw fluid along a vertical wavy surface. These equations can
344
CONVECTIVE FLOWS
be solved numerically using the Kellerbox method as described by Kumari et al. (1996a) for a wavy vertical surface which is subject to a constant heat flux rate. However, Kim (1997) has solved Equations (10.33), (10.36) and (10.38), along with the boundary conditions (10.37), using the finite volume method as proposed by Patankar (1980). In order to do this Kim (1997) introduced the parabolic coordinates X

~,
Y

Y [2(n + 1)x']2(+~) 1
U
,
u , [2(n + 1)x']v


V  [ 2 ( n + 1)x~
1
:~(n
+i
)g
(10.39) so that Equations (~0.33), (10.36) and (10.38) become 2,~(,~+~) OV = 0 n + 1 U + [2(n + 1)X] ~OU  Y [2(n + 1)X] (~~)(2,~+~) n OY
1 U.~OU+ { [2(n +
[2(n + 1)X]~
+
+ 1
n
1n
1)X] 2"(~~)  [ 2 ( n + 1)X] ~ ~" Y U
[2(n+l)X]
}OU OY
[2(n + 1)x]
,~ +
(10.41)
1 +a 2
...2 n ~~ 0 (
OU
: (1 + ax)
3n+1
(10.40)
~10U) OY
0 +l+a~
} O0
1 +
O0
[2(n + 1)X] ~'~(n+i) U OX
+ {V

[2(nk1)X] (1n)(1+2") 2n(n+l)
YU
OY
~ 020 (10.42)
Pr
OY 2
which have to be solved subject to the boundary conditions U0, V=0, 0=1 U+0, 0+0
on as
Y=0, Y+c~z,
X>~0 X~>0
(10.43)
The local heat transfer coefficient may be determined from the expression (10.44)
qw =  k i n . V T
where n

a~
(1 +ag)
~,
1
1
)
(10.45)
is the unit vector normal to the wavy surface. The local Nusselt number can then be expressed as follows:
1 :  (1
oo ( x , o )
bV
(10.46)
F R E E AND MIXED C O N V E C T I O N IN N O N  N E W T O N I A N FLUIDS
345
The numerical results reported by Kim (1997) were obtained for a sinusoidal surface ~ ( X ) = 3sin (27vX) (10.47) in order to show the effects of the wavy surface to the free convection flow. The full details of the numerical procedure can be found in Kim and Chen (1991) and Kim (1997) and therefore we do not repeat them here. Kim (1997) obtained the nondimensional axial fluid velocity, U, and the temperature, 0, profiles for P r  10, ~  0.1, n  0.8 (pseudoplastic fluids), n  1 (Newtonian fluids) and n  1.2 (dilatant fluids). He found that the maximum value of U increases, but the boundarylayer thickness becomes thinner as the flow index n increases. However, the thermal boundary layers of pseudoplastic fluids are thinner than those of dilatant fluids. Further, Kim (1997)investigated the axial velocity profiles for P r  10, ~  0.1 and n = 1 and showed that they are sinusoidal along the X direction. The regular nodes along the X direction being at X  1.5 and 2.0, and X  1.75 and 2.25, which represent the troughs and the crests of one wavy segment, respectively. The difference in the axial velocity at the crest and at the trough are almost indistinguishable but the boundarylayer around the nodes is thicker compared to that of the crests or the troughs. Further, it should be noted that the computation domain is not paralleled to the physical surface. Figures 10.4 and 10.5 show the profiles of the local Nusselt number, given by Equation (10.46), for P r  10 and 1000 and for some values of the parameters and n. It can be seen from Figure 10.4 that for n  1 the local Nusselt number for a wavy surface decreases as ~ increases. This is because the buoyancy forces on an irregular surface are smaller than those on a fiat plate (3  0), except at
.5
~
'
' ""
0.05,
" ' '
o. 5, 0.2, o1251
2.0
~ ~ 1.5 1.0
,~..,,. 9,,,, . . . . .
0.5
,I..,,
]
/,..~,, ....
.i
I
WavySurface,~=0.1  [ 0
1
X
2
3
Figure 10.4: V a r i a t i o n o f the local N u s s e l t n u m b e r with X f o r n fluid) and Pr10.
1 (Newtonian
346
C O N V E C T I V E FLOWS
(a)
(b) 2.8
5.5
26
~
_~
2.4

'

o
.
5.0
2.2
4.0
2.O
3.5
1.8
0
1
2
X
o
"~4.5
.
3.0
3
o
i
x
o
o
5
3
Figure 10.5: Variation of the local Nusselt number with X for ~d  0.1 when (a) P r  100 and (b) P r  1000.
the trough and crest points. On the other hand, Figure 10.5 shows that the local Nusselt number increases as n increases. However, it decreases as the axial distance X increases from X  0 (flat plate).
10.4
Free c o n v e c t i o n b o u n d a r y  l a y e r wall p l u m e in nonN e w t o n i a n p o w e r  l a w fluids
Consider the problem of steady, laminar flee convection from a line source of heat positioned at the leading edge of an adiabatic vertical surface which is immersed in an unbounded nonNewtonian powerlaw fluid with the following transport properties as proposed by Shvets and Vishnevskiy (1987) and Gryglavszewski and Saljnikov (1989), ~~j   p 6~j + #o
1
nI 2
eij,
qs   k /
1
s_ 2
IVTI
(10.48)
where 6ij is the unit tensor,/2 is the second invariant of the strain rate tensor and s is the heat transfer index. It can be shown that the boundarylayer equations in nondimensional form for this problem are givenby, see Pop et al. (1993b), Ou
Ov
0xt ~yy  0 u ~~x + v ~y = Oy
(10.49) ~
Oy
+ 0
(10.50)
F R E E AND M I X E D C O N V E C T I O N IN N O N  N E W T O N I A N FLUIDS
1 0 (
O0 O0 U~x + V Oy = P r Oy
347
Ou
(10.51)
which have to be solved subject to the b o u n d a r y conditions uO, vO } 0  ( T  Too) T~e arbf or oo _ 0 0~u+O, 0+0
on
y =0,
x>~0
as
y + c~,
x>~0
(10.52a)
together with the integral condition, see also Section 5.1, (10.52b)
fo ~ uO dy  Q where the nondimensional variables are defined as follows:
x = 2( ,
y_
~l Gr
a,
u ~c'
U
V
V ~c
Gr a '
O
Gr b
TToo Tre f
1
gc
_
pl n G r 
n and b where a  4ri+i are now given by
g ) __
4nJr.l
6n25n2 (4n+l)(n2) "
qs
pl,~ G r , ~  1
P%ITref
~o
(10.53)
,~ 2
The modified Grashof and Prandtl numbers
2qn
Gr  g~Tref t 2. 2
nC2+s)
P r = ~1 ll+sulc_SG r_ 4~§ a/
(10.54)
We now define the following variables 2nI1
42 = x4,~+l f (x, ~71,
2n+1
0  x 4,~+~h(x, rl),
77  x
nq1
,,~+~y
(10.55)
and assume that the wall t e m p e r a t u r e depends on x in the following manner 2n~l
Tw(x)  Too + GrbTrefX ~"+~
(10.56)
Insertion of the variables (10.55) into Equations (10.49)  (10.51) leads to
(
If"l
n
f'
f,,Of)
Uz
(10.57)
"~" 1 f,, S h, ' 2n + 1 ( f , Oh x 4. +~ p r ( [ I ) + 4 n + l ( f h ) '  z ~z
h, Of ) ~z
(10.58)
f"
)'
2n+lff,,_
+ 4n +5
n
f,2+h_x(f,O
4n +i
Ox
The boundary and integral conditions (10.52) become f  0, ft  0 "[ h  1 or h t = 0 f'+0, h+0
f
on
r/=0,
x ~>0
as
r/~,
x~>0
(10.59a)
348
CONVECTIVE FLOWS
fo c~ f'h d~
Q (n, Pr, x)
(10.59b)
It is apparent that Equations (10.57) and (10.58) permit similarity solutions if the exponent of x in Equation (10.58) vanishes, i.e. s = n 1
(10.60)
Under this condition, these equations reduce to the following ordinary differential equations:
( If''ln1 f")' + 4n2n+~1l+f f , , _ 4nn+1f,2 + h  0 1 (f,,n1)'
Pr [ {
2n+l
h' + 4 n + 1
(fh)'O
(10.61) (10.62)
which have to be solved subject to the boundary and integral conditions f(0)=0,
f'(0)=0, h(0)=l or h ' ( 0 ) = 0 f'+O, h ~ O as 7?+oo
f'h drj /o ~176
Q (n, Pr)
(10.63 ) (10.63b)
It should be noted that for n = 1 (Newtonian fluids), Equations (10.61) and (10.62) reduce to those derived by Afzal (1980) and Ingham and Pop (1990). Further, the skin friction coefficient
C I  2Gr 4,~+~x 4,~+~ ~
y=O
/10
can be expressed as
c s (ar )
 2 If"(o)l
(10.65)
if the variables (10.55) are used. Equations (10.61)  (10.63) have been integrated numerically by Pop et al. (1993b) using the RungeKuttaGill method for several values of n and for Pr = 0.72, 1
10 and 100. For a Newtonian fluid (n  1) the present results give for C/(Grz)~ the values 2.62012 for Pr = 0.72 and 1.85964 for Pr = 6.7, whilst the corresponding values obtained by Ingham and Pop (1990) are 2.6201 for Pr = 0.72 and 1.8596 for Pr = 6.7. This shows that the agreement between the two sets of results is excellent. The results for various transport parameters, which are important for representing some heat transfer correlations are given in Tables 10.2 and 10.3 for the flow behaviour index n ranging from 0.2 to 1.5 and for Prandtl numbers 10 and 100, respectively. It is noted from these tables that f ' ( 0 ) decreases as the values of n and Pr increase and this leads to a decrease in the skin friction coefficient as defined by
F R E E AND M I X E D C O N V E C T I O N IN N O N  N E W T O N I A N FLUIDS
349
Table 10.2" Numerical values of the computed parameters for Pr = 10 and the values 0.2 ~ n ~< 1.5.
[
I! s"(o)
0.2 0.4 0.6 0.8 1.0 1.2 1.5
3.21309 1.56230 1.08804 0.95063 0.86123 0.90450 0.82903
f(oo) 3.46476 2.48399 1.52345 1.32087 0.99482 1.62756 0.70861
f~x(O)
.
....
0.70945 0.56938 0.45349 0.43117 0.39276 O.49054 0.40676
.
.
.
.
.
.
.
.
.
.
Q
.
14.67177 9.05100 5.18968 3.96055 3.00209 2.87300 2.12471
....
Table 10.3: Numerical values of the computed parameters for Pr  100 and the values 0.2 ~< n ~< 1.5.
l I1 s"(o) 0.2 0.4 0.6 O.8 1.0 1.2 1.5
f(~)
Q
0.90717 0.73486 0.19407 0.08289 0.12585 0.19320 0.12537
3.33278 2.33788 1.04260 0.53576 0.52386 0.48404 0.31474
,
1.23283 0.22884 0.73858 0.20504 0.50828 0.13315 O.41895 0.09548 O.48O62 0.12145 0.54364 0.16156 0.53094 0.13440 .
.
.
.
,
.
.
.
.
.
.
E q u a t i o n (10.65). It is also seen t h a t the p a r a m e t e r I, which serves to determine the reference t e m p e r a t u r e Tref t h r o u g h the E q u a t i o n (10.56), decreases with an increase in n and Pr. Figures 10.6 a n d 10.7 display results for the fluid velocity a n d t e m p e r a t u r e profiles in the plume. It is seen from these figures t h a t the m a x i m u m fluid velocity decreases with increasing values of the flow behaviour index n a n d this m a x i m u m moves closer to the wall as the value of n increases. We also see t h a t the fluid velocity and the t h e r m a l b o u n d a r y  l a y e r thicknesses decrease as n increases. Further, as the P r a n d t l n u m b e r increases, the t h i n n i n g effect of the t h e r m a l boundarylayer substantially affects the velocity b o u n d a r y  l a y e r region. Also, it can be noted from Figures 10.6(a) a n d 10.7(a) t h a t for n = 1 the solution a p p e a r s to intersect more curves for P r = 100 t h a n for P r = 10. T h e reason for this a p p e a r s to be the dependence of the P r a n d t l n u m b e r on the index n, reference velocity Uc and the reference length l of the plate. The i m p o r t a n t quantities in this flow geometry are the fluid velocity level, the surface t e m p e r a t u r e a n d the size of the b o u n d a r y  l a y e r region. As the flow proceeds downstream from a h e a t e d element which is located on an u n h e a t e d part of the surface, it influences the cooling characteristics of any other element it may encounter.
350
CONVECTIVE FLOWS
(a)
(b) 1.0
1.0
0.8
~
f'(r/)
0.8
/
h(r/)
o~t//~ 9 ,
~
X~o~,o4,oo, ~
ool ~\\\\oo~,o,,o~,o~,
o~, ~, 1~
,~ I / / k ~
0.4
0.4
0.2
0.2
0.0
0
2
4
77
6
0.0
8
0
. 2
~' ~~' 1.~
.
4
.
. 7/
6
Figure 10.6: (a) The fluid velocity, f'(rl) , and (b) the temperature, h(~), profiles for Pr = 10.
(~)
(b)
010 0"201 l// 0 ~>0
(10.75)
F R E E AND MIXED C O N V E C T I O N IN NONNEWTONIAN FLUIDS
353
The coefficients A(~), B(~), C(~) and n(~) in Equations (10.73) and (10.74) are defined as follows" 1
for the cylinder for the sphere 3(1n) n1 C(()
H(r

(10.76)
u~ d~
where the mixed convection parameter A is now given by
Gr2
A 
Re
(10.77)
2n
and Gr, R e and P r are defined as follows: 2
Gr 
P
g13 [AT] a2~
Re
Pr
Re
~+'
(10.78) Finally, the skin friction coefficient, CI, and the local Nusselt number, N u , are given by 2"rw CI

pUs
aqw '
Nu

kr
IAT]
(10.79)
and these can be expressed in the following form:
!CfRen~12=
~  h4Y ~
(~, 0)]
(10.80)
2n
NuRe
~u
 ~~4f
~
[0'(~, 0)]
Equations (10.73) and (10.74), subject to the boundary conditions (10.75), were solved numerically by Wang and Kleinstreuer (1988) for n ranging from 0.52 to 1.6, P r = 10 and 100, and )~ = 0 (forced convection flow), 1 and 2 using the Kellerbox method. Typical results for the skin friction coefficient and local Nusselt number are shown in Figures 10.9 to 10.12. It is observed from Figure 10.9(a) that for assisting flows pseudoplastic fluids (n < 1) generate higher, and dilatant fluids (n > 1) lower, skin frictions than Newtonian fluids (n = 1). However, both the powerlaw index n and the buoyancy parameter A are less influential on the skin friction coefficient for a sphere than for horizontal cylinders, see Figure 10.9. Further, Figure 10.10(a) shows that, as expected, for a Newtonian fluid the local Nusselt number decreases monotonically along the surface of the cylinder. It reaches a maximum for pseudoplastic fluids and then, similar to Newtonian fluids, decreases
354
CONVECTIVE
(a)
FLOWS
(b) 1.25
1.6
1.00
1.2
0.75
/z.** Iz:,' #..'i~:
\'x .
.
,
\\
,9 \,,.9 ~,\ \
/,~,....
\ ~\ x \ \ '~ ~
0.8
\\
,
0.50
z:
",,~
9"
',l'l
0.4
0.25  0 6 ,.
o~
~
1, 1 . 6
n  0.6, 1.6
"~ ~
0.00~
0
2"5
' 50 r
7'5
!1\/
0.0
I00
0
'
ib
"
40
'
go r '
8"0
Figure 10.9: Variation of the local skin friction coefficient with r for P r  100 in the case of assisting flow for (a) a cylinder and (b) a sphere. The solutions for A  0 (forced convection), 1 and 2 are indicated by the dotted, broken and solid lines, respectively.
(a)
(b)
3.6 'I~ _Nu_ i~n : 1.6 Re~~ "~\\ 3.2 ~ ~ ~ _ _ _ _ . _ ~
6 =
.
Nu 5
2.4
3
2.0
2
1.6
1 0
25
50 r 1 7 6 75
100
0
20
40
60 r
80
Figure 10.10: Variation of the local Nusselt number with r for Pr  100 in the case of assisting flow for (a) a cylinder and (b) a sphere. The solutions for A = 0 (.forced convection), 1 and 2 are indicated by the dotted, broken and solid lines, respectively.
'
FREE
AND MIXED
CONVECTION
IN NONNEWTONIAN
(a)
FLUIDS
355
(b) 1.0
1.6"
cg 0.8
1.2
0.6
0.8
.~/,':~'
,,
y"
0.4
\
0.4
0.2
~/~"'~ n = = 0.6, . 1, 1.6 . 0.0
\~[
0.0.
n=0.6,

o
25
1, 1.6 .l
5'o r (o) '75
~do
o
2b
i
ii
" 40" " 6"o r
9
8'0
F i g u r e 10.11 Variation of the local skin friction coefficient with r for A  0.5 and P r  100 in the cases of assisting flows (solid lines) and opposing flows (broken
(~)
(b) 3.6 ~ n = 1.6
Nu 1 R e Z4~
3.0 n
l
Nu 6 ~ n = 1.6
".."~..
2.4
1.8
1.2 0
25
50
r
1
75
100
0
. . . . . . . 14 28 42
56.,~
Figure 10.12: Variation of the local Nusselt number with r /or A  0.5 and P r 100 in the cases of assisting flows (solid lines) and opposing flows (broken lines)
/or (a) a cylinder and (b) a sphere.
~,~ 84
356
CONVECTIVE FLOWS
gradually. In contrast, for dilatant fluids, the local Nusselt number reduces very rapidly in the vicinity of ~ = 0 (the forward stagnation point) and then follows, after a point of inflection, the general trend of the Newtonian fluids. This behaviour can be explained as follows. From Equation (10.80) we have 1
1n ~
0)]
(10.81)
nl
(10.82)
which implies that for ~ + 0
N u Re
,~
f "~ i
0 cx~
for for
provided that 0'(~, 0) is well behaved at the forward stagnation point. On the other hand, Figures 10.11 and 10.12 show that for opposing flows the skin friction coefficient and the local Nusselt number have lower values than for assisting flows. This trend is comparable to the effect of lowering the buoyancy parameter A, cf. Figures 10.9 and 10.10. In these cases, forced convection (A = 0) is either retarded by the opposing buoyancy forces (cooled cylinder/sphere) or relatively less enhanced by decreasing the buoyancy forces (reduction of A). As can be expected, the separation angle for opposing flows, see Figure 10.12, is similar to that of aiding flows.
10.6
F r e e c o n v e c t i o n b o u n d a r y  l a y e r flow of a m i c r o p o lar fluid over a v e r t i c a l flat p l a t e
Convective flow over a fiat plate which is immersed in a micropolar fluid has attracted an increasing amount of attention since the early studies of Eringen (1966, 1972). Results for this generic problem have been reported by several investigators, including Jena and Mathur (1982), Gorla and Takhar (1987), Yiicel (1989), Gorla (1988, 1992), Gorla et el. (1990), Gorla and Nakamura (1993), Chiu and Chou (1993, 1994), Char and Chang (1995, 1997), Wang (1993, 1998), Hossain and Chaudhary (1998) and Rees and Pop (1998). These latter authors have shown, based on work by Rees and Bassom (1996) on the Blasius micropolar boundarylayer flow over a flat plate, that much more information about the solution of free convection boundarylayer flow of a micropolar fluid from a vertical flat plate can be found. A novel feature of these problems is that the boundarylayer develops a twolayer structure far from the leading edge, namely a mean layer and an inner, nearwall, layer. The nearwall layer is of constant thickness and it is the region where the microelements adjust from their natural freestream orientation to that imposed by the presence of the solid boundary. It should be mentioned that the papers by Rees and Bassore (1996) and Rees and Pop (1998) are the most complete papers in the area of micropolar fluids and we shall therefore present here some results of these papers.
FREE AND MIXED CONVECTION IN NONNEWTONIAN FLUIDS
357
Consider a heated semiinfinite vertical flat plate with a constant wall temperature Tw, which is immersed in a micropolar fluid of temperature Too, where T,o > Too. The governing equations for the steady free convection flow of an incompressible micropolar fluid subject to the Boussinesq approximation can be written in the form, see Chiu and Chou (1993),
~+@  o (o~
(10.83)
op
+~~

0N
05
~~xx+ ~ y y
P

(oN
pj \  ~ +~y] OT
OT
(10.84)
~x2 + ~y2 ] + n~y + pg fl (T  Too)
+(,+n)
@ + (, + ~) (0~ _
2,~N+,~ 0e ( 02 T
O2T )
~ ~ + ~ N  '~: ~ +
o~2
~
0g'~
0~/+7
0~ ( 02~
(10.85) 02N)
o~2 +o~2
(10.86)
(10.87)
N
where N is the component of the microrotation vector normal to the (5, ~)plane and j, n and 7 are the microinertia density, vortex viscosity and spin gradient viscosity, respectively. We assume that 7 is constant and is given by
 (. +
s
88)
and this is invoked in order to allow the field equations to predict the correct behaviour in the limiting case when microstructure effects become negligible, and the microrotation, N, reduces to the angular velocity, see Ahmadi (1976). The boundary conditions appropriate to Equations (10.83)  (10.87) are as follows
=0, +0,
~ 0, ~+0,
N   n N+0,
,
T  Tw T+Too
on as
y0, y+ec,
5~>0 x)0
(10.89)
where n is a constant. On using Equation (10.86), and the boundary conditions (10.89), when n  0 we obtain that N  0. This represents the case of concentrated particle flows in which the microelements close to the wall are not able to rotate. The case of n  89results in the vanishing of the antisymmetric part of the stress tensor and represents weak concentration. Ahmadi (1976) suggested that in this case the particle spin is equal to fluid vorticity at the wall for fine particle suspensions. Then, the case of n  1 is representative of turbulent boundarylayer flow, see Peddieson, Jr. (1972).
358
CONVECTIVE FLOWS Next, we introduce the following nondimensional variables I
2 x=[,
~ yi,
~ Uu~ ,
~ Uc'
v
p  p~ pU2c,
p
T  Too O  AT '
N
lN Uc
(10.90) where Uc  (g~AT l)~ and we assume that the length scale is given by j  12. On using the expressions (10.90) in Equations (10.83)  (10.87), we obtain ,
Ou 0~
+
Ov
uy
:
Ou + Ou
 0
(10.91) _ _, O
(02u + 02u)
1C , Gr: ( 0 2 v : + 02v) Ov Ov _ Op ~ 1tK: K, ON : oy Gr: Ox l+ 89 ON ON 2K IC (Ov Ou) ~N+ + : u~z + v O y at:. at: Oz Oy Gr: O0 00_1(020020) __
+
1+/C
1
ON Oy
+0+
(10.92) (10.93)
02N 02N) + 9 Ox2 OY2
(10.94) (10.95)
+
where K:  ~ is the micropolar parameter and Pr and Gr have been defined in the same way as for a standard Newtonian fluid; nonzero values of K: cause coupling between the fluid flow and the microrotation component N. We now invoke the boundarylayer approximation, namely
x _ Gr~,
yy A
u _ Gr89162, v   G r :
:0r
Oy
5zz'
NGr:N :
(1096)
"
which when substituted into Equations (10.91)  (10.95) and formally letting Gr + cx~ leads to the following boundarylayer equations:
o o o 2o
ooo 2o
0~" 0 ~ 0 ~
O~ 0 ~ 2 = (1 + K:)  ~
03o
o.~
(10.97)
+ 0 + K:
(10.98) 0 r 00
0 r 00
1 020
o~ o~
o~ o~
P rOy2
(10.99)
and the boundary conditions (10.89) become 0r   0 , a~
~.0,
N^    n  b 0~g, ~, _N+O,
0+0
01
on
y..= 0 ,
~>~0
as
~oo,
~>10
(10.100)
F R E E AND MIXED C O N V E C T I O N IN NONNEWTONIAN FLUIDS
359
As a prelude to obtaining numerical solutions, the governing Equations (10.97) (10.99) and boundary conditions (10.100) are first transformed into a local nonsimilarity form. In order to do this we introduce the following variables:

r  X~ f(X,~?),
O  g(X,7?),
1 N  X~h(X,~),
A1 X  x~,
77 Y1 (10.101) X~
where the functions f, g and h are given by the following set of partial differential equations
(1 +
3
K:)f"' + ~ f
/,,
lf,2+y_.h,+g_X
(f,O.f'
2
(
1 )h,
1 + ~IC
+
f,,Of)
(10.102)
ox
3
f h'
(1 Oh h ' O f ) + 1CX (2h + ~h f' _ ~X f' ~  ~ f") (10.103)
 1
1
1 g,,+ 3f g , =
( f , Og _ g, O f )
p;
(10.104)
. ox
where primes denote differentiation with respect to 77. The boundary conditions for these equations are given by f0,
f'O,
f,__+0,
h+nf"O,
h~0,
g1
on as
g+0
770, 77+c~,
X~>0 X/>0
(10.105)
At this stage we draw attention to the one case when Equations (10.102) (10.105) reduce to a similarity form. The last term in Equation (10.103) may be regarded as the forcing term in this set of equations and if it were absent then it is possible for the resulting equations to have an Xindependent solution. This one possibility for a similarity solution to exist is that the term (2h + f") is identically zero. However, it can easily be shown that even when n  89then h   I f " does not give a consistent set of equations. Therefore, one cannot obtain a similarity solution in this way. The second possibility is that K: = 0 and in this case the Equation (10.103) is decoupled from the Equations (10.102) and (10.104). The resulting similarity solutions satisfy the following set of ordinary differential equations f " ' + ~3 f
f ,  ~1 f,2 + g  0 3 hi
h" + 4f
1 g.
Pr
1
 4hf  0 3
+ 4fg'

0
(10.106) (10.107)
(10.108)
which have to be solved subject to the boundary conditions f(O)=O, f'(O) = O, f'+O, h  + 0 ,
h(O) + nf"(O) = O, g ( O ) = l g+0
as
r/+oo
(10.109)
360
C O N V E C T I V E FLOWS
and hence the fluid flow and the temperature fields are unaffected by the microrotation of the fluid. It should be noted that Equations (10'106) and (10.108) represent the equations which govern the free convection boundarylayer flow of a Newtonian fluid over an isothermal vertical flat plate and are well known, see Section 1.3. On the other hand, Equation (10.107) has been solved numerically by Rees and Pop (1998) for n  1 and when P r ranges from 0.1 to 10 and the profiles of the angular velocity h are presented in Figure 10.13. As expected, these profiles remain negative and increase from the value f ' ( 0 ) to zero as ~ increases from zero to infinity, see the boundary conditions (10.109). On the other hand, we can see from this figure that h increases with the increase of P r for 0 ~ 77 ~< 2 and decreases for 77 ~> 2 when P r increases. 0.0 O.2 0.4 0.6 0.8 ].0 1.2 0
1
2
3
4
5
6
7
8
Figure 10.13" Profiles, h(y), o/ the reduced angular v e l o c i t y / o r ~  0 and n  1.
Further, the full boundarylayer equations were solved numerically by Rees and Pop (1998) using the Kellerbox method and full details of the numerical procedure can be found in this paper. A selection of some of the numerical results for the nondimensional skin friction, f " ( X , 0), and the rate of the wall heat transfer, g t ( X , 0), are presented (by full lines) in Figures 10.14 and 10.15, respectively, for P r  6.7 (water) and K:  0, 0.25, 0.5, 0.75 and 1 for the respective cases n  0, 0.5 and 1. It 1 should be noted that all these curves are plotted against X ~ in order to more easily resolve the rapid variation near X  0 (singularity) and the slow approach to the asymptotic solutions, which we will develop further. Figure 10.14 shows that the curve corresponding to K: = 0 is a straight line, a result which is in accord with our earlier observation that K:  0 represents the only similarity solution. When the micropolar parameter K: ~= 0 then the form of the skin friction variation depends very much on the values of n and K:. It is always less than the K:  0 value for sufficiently small values of X but when n  0 its
FREE AND MIXED CONVECTION
IN N O N  N E W T O N I A N
(a)
361
FLUIDS
(b) 0.70
0.70 K: = 0
.~ 0.65
0.60 "~ 0.55 "__ . . . . 0.50
K: = 0.25
~,,~___
..... ........ 0.40 k,,.___ 0.35 2'0 4'0
0.45
~, 0.65
K:=0
g 0.60
K:,,= 0.25
L, 0.55
0.50
K: =0.5 K: = 0.75 K:=! 6'0
8'0 x 89
0.35
160
(c)
.... l
0.45 0.40
]C = 0.5, 0.75, 1
0
2'0
4'o
6'0
x89
8'o
16o
(d)
o%t l
~. 0.70
o.65
!
._.~ 0.70
0.60
~ 0.60
0.55
O.55
0.50
0.50
0.45
0
20
40
60 X8980
100
0.45
0, 0.25, 0.5, 0.75, 1
0
2
4
6
X89
8
10
Figure 10.14: Variation of the skin friction, f " ( X , 0), with X 89 for P r  6.7 when (a) n  O, (b) n = 0.5, (c) n  1 and (d) a closeup view of (c) near X = O. The numerical solutions are indicated by the solid lines and the asymptotic solutions (10.133a) for n ~ 1 and (10.135a) for n  89 at large values of X (>> 1) are indicated by the broken lines.
value decreases f u r t h e r as X increases, w h e r e a s w h e n n = 1 it e v e n t u a l l y a t t a i n s a n a s y m p t o t i c value a b o v e t h e K: = 0 result. However, for low values of n t h e s p r e a d of t h e curves for different values of K: is m u c h g r e a t e r t h a n w h e n n = 1. T h e d e t a i l e d e v o l u t i o n of t h e wall h e a t t r a n s f e r s h o w n in F i g u r e 10.15 is a little m o r e c o m p l i c a t e d t h a n t h e skin friction curves. If we refer to t h e wall h e a t t r a n s f e r in t e r m s of its a b s o l u t e value t h e n t h e K: ~ 0 values are a l w a y s less t h a n t h e K: = 0 value a n d i n c r e a s e m o n o t o n i c a l l y w h e n n = 0, i m p l y i n g t h a t t h e presence of t h e m i c r o s t r u c t u r e r e d u c e s t h e wall h e a t t r a n s f e r . However, w h e n n = 1 t h e v a r i a t i o n is not m o n o t o n i c ; t h e wall h e a t t r a n s f e r g e n e r a l l y r e m a i n s below t h e u n i f o r m K:  0
362
CONVECTIVE
(a)
FLOWS
(b) 0.62
0.66 f
0.64
~
E=I
0.68
0
~ 0.66
" \ ~
0.68
K: = 0.75
..... ~ 
1
0.70 ........
0.70
"
.........
0.72
,
2'0
4'0
/C=O 0.74 !
6'0 '"8'0 ' 1130
o
x89
(c)
]C = 0.25
0.72
= 0.25
K=O
0.74
]C = 0.5
&_o.5
2'o
4'0
6'0
8'0
100
x89
(d)
~.~ 0.66 ]
0.70]
~o.68
~ = 1__.
'~ 0.70
z
,'"" . . . . .
K:  0.75
..."
)U = 0.5
, / / , ~
0.72
/
~~,~ 0.71 t
K:=I
:yff
..............
/, ," ._._===__
~
/~......
/
0.74 !
o
~
E 
._.
.
io
4'0
.
0
0.74 I .~1 K: = 0 0 89 ' zi ' 6 X 8 9 1'0 
6'0
go '" 16o
x89
!
Figure 10.15: Variation of the wall heat transfer, g ' ( X , 0), with X 89 f o r P r  6.7 when (a) n  O, (b) n  0.5, (c) n  1 and (d) a closeup view of (c) near X  O. The numerical solutions are indicated by the solid lines and the asymptotic solutions (10.133b) for n ~ ~1 and (10.1355) for n  ~. 1 at large values of Z (>> 1) are indicated by the broken lines.
value b u t can become slightly g r e a t e r locally w h e n K is sufficiently small. T h e variations of f " ( X , 0) and g ' ( X , 0) for P r = 0.7 (air) and the same values of the p a r a m e t e r s n and E can be found in the p a p e r by Rees a n d Pop (1998). It was found t h a t there is little q u a l i t a t i v e difference b e t w e e n the results for w a t e r and for air, a l t h o u g h the detailed q u a n t i t a t i v e results are quite different. F i g u r e 10.16 illustrates the c o n t o u r plots of the function (h + i f , , ) for the case P r = 0.7, n = 1 and K: = 1. It shows the g r a d u a l d e v e l o p m e n t , as X increases, of a thin, nearwall layer e m b e d d e d w i t h i n the m a i n b o u n d a r y  l a y e r . Indeed, for the
F R E E AND MIXED C O N V E C T I O N IN N O N  N E W T O N I A N FLUIDS
363
~5. "~4"
+ 3 Increasing X
0
0
5
10
15
20
25
30
35
Figure 10.16: Contourplots of the function (h(X, r/) + I C  1 and n1.
40
lf"(X, 71))for Pr 
0.7,
micropolar Blasius boundarylayer flow, discussed by Rees and Bassom (1996), it WaS found that (h + 89  0 when n  1 and that (h + ~.if")  0 except in a thin layer near to the flat plate when n # 89 However, for the present problem, h r  8 9 even when n  89 but Figure 10.16 shows a similar development of a nearwall layer as X increases. In order to examine this nearwall layer in more detail, for K: # 0, we make the substitution 1 r h + ~f" (10.110) into Equations (10.102)  (10.104). Then we have
( 1 + ~IC 1 ) fm (1 + K:)r
+
lf'2
1 (f 'Off OX 1 3 1 ( 0r ~r ~fr + ~X f ' o x 1 (f, Og _g, Of)
+ g + 1Cr  ~
2g'1 g,,
2K:Xr 
3 ,
3 f"
 ~f
+ ~X
f,, ~~ Of ) r ~~ of )
(10.11I) (10.112) (10.113)
and the boundary condition (10.105) becomes
f0
,
f'O,
f'~0,
r189
r
f",
g~0
g1
on as
~?0,
~+c~,
X~>O
X)0
(10114) "
It is readily seen that for X >> 1, the term 2K:Xr from Equation (10.112) dominates this equation, unless r is small, since g~ is O(1) as X + c~. Therefore, the
364
CONVECTIVE FLOWS
asymptotic forms of the solution of Equations (10.111)  (10.113), for X >> 1, are given by
f~Fo(~),
g~Go(rl),
r163
1
,
(10.115)
where Fo and Go are given by the following ordinary differential equations ( 1 + 1 / ( : ) ~'''~0 + ~3 FoF~' _ 1F~2 + Go  0 1
.
3
(10.116)
,
praO +~Foa o  0
(10.117)
which have to be solved subject to the boundary conditions F0(0)0, F~ + 0,
F~(0)0, G0(0)I Go + 0 as 77+ oo
(10.118)
We note that Equations (10.116)  (10.118) can ea;sily be written in terms of the classical vertical free convection equations using the transformation 1
F0
I+~E
1
F(~,
G0G(~,
rl
I+~K:
~
(10.119)
where F and G satisfy equations which are identical in form to Equations (10.106) and (10.108), but where tile Prandtl number is replaced by (1 + 89 Further, it is seen from Equation (10.115) that the boundary conditions (10.114) for r are not satisfied since the highest derivative in Equation (10.112) was neglected when forming the solution for r in Equation (10.115), and hence this is a singular perturbation problem. Even without the numerical evidence presented earlier, it is clear that there must exist a thin layer, a nearwall layer, which is embedded within the main boundarylayer. However, it should be pointed out that the value of n plays an important role in determining the size of r in this nearwall layer. When n  1, we have r  0 at ~  0, so that r is O (X 1) in order to match with the form given in Equation (10.115), but when n r 89the boundary conditions (10.114) for r state that r is O(1) at 77  0. Therefore these cases should be treated separately. First, we introduce the nearwall layer variable ( as follows:
~
1


fiX ~
(10.120)
which results from the balancing of the terms 2K:Xr and r in Equation (10.112). It is worth pointing out that the comparison of the definition of ~ given by the Equation (10.120) with the definition of ~ given in Equation (10.101), shows t h a t  ~', and therefore the nearwall layer has a constant thickness. Equations (10.111)
FREE AND MIXED CONVECTION IN NONNEWTONIAN FLUIDS 
365
(10.113) then become
(I + 2K_.)f'" + X} g + K_.XIr = X 89 ( 2 f , 2 _ ~f3 f,,) + 1X}2 (f,O.f'ox
f" ~~fX)(10.121)
+ ~1 x _89  2](:r 
(1 + ~ ) r
X89 Prg .1
+ 4
(1
89fg,
3
hf'
) 1 .}(,0r162 fh' + ~X f  ~
l 89 ( f, 0X Og  ~X
g, OX Of )
(10"122) (10.123)
where primes now denote differentiation with respect to ~. The boundary conditions appropriate to Equations (10.121)  (10.123) at ~ = 0 are given by
f0,
f'0,
gl,
r
(1),, ~n
(10.124)
and the matching conditions as obtained from the small 7/ (> 1 is sought in the following form: f  Fo(rl) + X 1/;'1 (rl) + . . . g  ~0(7]) t X  8 9 G1 (7/) ~ . . .
r  x~r
(10.125a)
+ x  ~ ~ (,7) + . . .
in the main layer, and the asymptotic solution of Equations (10.121)  (10.123) has the form: f  X lf0(~) + X  ~ f l ( ~ ) + .  . g  1 + X89 + Xlgl(~) +... (10.125b) r  r162 + x  ~ r 1 6 2
+...
in the nearwall layer. It should be noted that the equations and boundary conditions for F0 and Go are precisely those given by Equations (10.116)  (10.118), while the functions F1 and G1 satisfy the following ordinary differential equations:
1 ) 1 ~ ~](: 1 G~+ p~
3
F~" tG1
' 1'  FoF~')  1F~)'F1 (10.126)  ~3 (FoF z
FOG'1+~1 FgG1 + ~1 FIG'oO
(10.127)
366
CONVECTIVE FLOWS
which have to be solved subject to the boundary conditions F1 (0)  0,
F~ + 0,
G1 (0)  0
G1 + 0
as
r/+ cr
(10.128)
and the boundary condition for F~ (0) is obtained using the matching procedure. In order to do this, we observe that for X >> 1 and 7/ 0
for
m 0. However, it may be seen immediately, from Equation (11.23a), that this is not possible. Hence it is concluded that f'(77) > 0
for
m < 0
(11.26)
and because f (0) = 0, this condition gives rise to f(~) t>0
for
0~> 1. They found that 3
f"(O)  0.078103
~ + m
+...
(11.32a)
for m near  89and f"(0)  0.44375  0.85665 m + 0.66943 m
2 A...
(11.32b)
for m near 0, while there exists dual solutions for 1 < m < cr If m is very large, Equations (11.23) can be reduced to f ,,, + ~1 f f,,  f,2  0
(11.33)
and this equation has to be solved subject to the boundary conditions (11.23b). Solving this equation numerically it was found that at least two possible solutions exist such that f"(0)0.90638m~,
1
1
f(c~) = 1.28077m 2
(11.34a)
and 1
f"(0)  0.91334m~,
1
f(cr
 0.43365m~
(11.34b)
for m >> 1. Although the values of f"(0) on both solutions (11.34) are very close, the basic difference between them is that the second solution contains a region within the boundarylayer where the fluid velocity f'(r/), or the temperature 0(~), becomes negative. It was also reported by Ingham and Brown (1986) that on the second branch solution, the numerical solution of Equations (11.23) for m + 1+ gives f(c~) + 0 + and f"(0) +  1  . The results obtained by Ingham and Brown (1986)
F R E E AND MIXED C O N V E C T I O N O V E R VERTICAL SURFACES
387
are summarised in Figure 11.1 where the variation of f"(O) as a function of m is presented. The numerical solution of Equations (11.231 on the first solution branch was obtained for  89 < m < 5 and it is shown by the solid line in this figure. The exact solution (11.29), the solution (11.32a) for m ~  ~1, the two and three terms series solution (11.32b) for m ~ 0 and the asymptotic solution (11.34a) for m >> 1 are also shown in Figure 11.1. This figure clearly shows that extreme care should be taken when dealing with free and mixed convection problems for vertical surfaces which are embedded in porous media when the wall temperature varies as a power of x. It appears that not all of the existing published papers in porous media have dealt with the existence of dual solutions correctly. 2 ] f"(o)
+m)~ ,i
1
"% \ ~.\"
ii
"":~ u.44375"0.85 665 m + 0.66943m 2
1
2
,ill
i
oAa75
 o.8s66s m
Figure 11.1: Variation of f"(0) with m as obtained numerically (solid line) and by asymptotic and series solutions. solution (11.291.
The symbol 9 shows the position of the exact
The existence of eigensolutions for the present problem has been investigated by Banks and Zaturska (19861. They treated these solutions analytically when m   ~1 and m  1, for which closed form solutions (11.291 and (11.30) can be seen to exist, while for other values of m the eigensolutions were found numerically. In order to do this, Banks and Zaturska (19861 introduced the transformation 1
[( r
2x 1 +m
)
xm
]~
1
F(x,r])
'
riy
[(l+m) 2x
xm
]~
(11.35 /
where on using Equations (11.131 and (11.14), the function F ( x , ~ ) is given by F ' " + F F "  ~ F ' 2  (2  fl) ( F ' OF'Ox
F" OF
(11.36a)
388
CONVECTIVE FLOWS
along with the boundary conditions (11.15) for the V W T case with vw(x)  0, which can be written as follows:
F(x, O) = O, F'+0
F'(x, O) = l, as
r/+oo,
x~>0 x>/0
(11.36b)
where the parameter ~ is defined as in Equation (2.147). In order to solve Equations (11.36) an initial condition is required and this is taken to be of the form F (x0, r/)  g(~), where g(~) satisfies certain requirements. Here we assume that ~ ~ 2, since the case/3  2 corresponds to m being infinitely large. In a similar manner to the relation (11.28), an integral constraint on F(x, ~) also exists and it is given explicitly as follows, see Banks and Zaturska (1986), (2 +/3) ~oo~176 F F '2 dr/+ (2  / 3 ) x foo~176 F ' ( 2 F OF' ~ x + F, ~OF ) d r /  ~. 1
(11.37)
Similarity solutions of Equations (11.36) may be obtained if we ignore the initial condition at x0 and write g(x, 77) = f(r/), where f(~) satisfies the equation
f,. + f f.
/~f,2 _ O
(11.38a)
and the boundary conditions (11.36b) for f(rl) become f(0)0,
f'(0)l,
f'+0
as
77+~
(11.38b)
It should be noted that Equations (11.38) were established by Banks (1983) in the context of boundarylayer flow due to a stretching sheet in a viscous fluid and he integrated these equations numerically for 1.9999 ~< fl ~< 202. Returning to the problem posed by Banks and Zaturska (1986) for Equations (11.36), it is formulated as follows: given g(~) is such that F(x,r/) differs only slightly from f(rl), as defined by Equations (11.38), determine the leadingorder term for F(x, 77)  f ( r l ) as x ~ c~ depending on the value of the parameter ft. Thus, F(x, ri) is taken to have the form _
.'rk
F(x, 71)  f (71) + x 2~ Gk(rl) + . . .
(11.39)
where Gk(rl) satisfies the equation
a'"
k +f
+(7k2/~)
/' '
Gk+(1Tk)
f
"ak
0
(11.40a)
77 + ~
(11.40b)
together with the homogeneous boundary conditions
Gk(O)  O,
Gtk(O)  0,
G~ + 0
as
It can be seen from expression (11.39) that for values of fl and rn which are of interest, the convective flow is spatially stable as x + c~ if the minimum value in
FREE AND MIXED CONVECTION OVER VERTICAL SURFACES
389
the set ~'k is positive. With f(~) known, Equations (11.40) constitute an eigenvalue problem for the eigenvalues ")'k and eigenfunctions Gk. On the other hand, from Equations (11.37) and (11.39) we obtain, on equating terms which are O(1) and O(x), respectively, 2(2 +/3) (2 + ~  ~/k) f0 cr
f f,2 d r / =
~0(:~
f' (2fC'k + f'Ck)
1
(11.41)
drl = 0
(11.42)
The condition (11.41) shows that a similarity solution for f (~) does not exist when f~   2 (i.e. m  _1). Further, the integrand in condition (11.42) provides a constraint on the eigensolutions. We can infer from this condition that since f(~) and f~ (~]) are nonnegative for each value of f~ in the range of interest, the integrand is of one sign if an eigensolution exists and is also of one sign in the interval 0 ~ 7/< cx). Thus, the first eigenvalue is given by ~/1 = 2 + ~
(11.43)
which is real and positive for each value of/3 >  2 (m >  89 of interest. The determination of the first eigenfunction is reduced to a onepoint numerical integration of Equations (11.40). Banks and Zaturska (1986) also found analytical expressions for the eigensolutions corresponding to some particular values of ft. Thus, for fl  1 (m = 1) the function f(r]) is given by Equation (11.30) and ~1 = 3 with Gl(r]) where
El(x)
vf2tanh ( ~ )
=
e f

1 + f + 2(1  f ) [ E l ( l )  E1(1  f)]
is the exponential integral.
For /~ 
(11.44)
1 (m   89 then fO?) 
and "/1 1 with (11.45)
GI(~)  tanh2 ( ~ 2 ) and ~2 = 6 with
a2(
For~=0
)  5
2 [hsech2(~2)
In sech ( ~ 2 )
+ tanh2 ( ~ 2 ) ]
(11.46)
(m=0) then~l=2and v
(o) = ' i f '  f ' '
f"(o)
(11.47)
390
C O N V E C T I V E FLOWS
It should also be noted that for/~ ~  2 + (or m +  8 9 the eigenvalue problem posed by Equations (11.40)  (11.42) is similar to that of the wall jet, which was studied by Riley (1962) where such a fluid flow may be described as neutrally stable. Typical results of the analytical and numerical investigations carried out by Banks and Zaturska (1986) are given in Table 11.1 and in Figure 11.2. Table 11.1 shows the second eigenvalue 72 for selected values of f~ and it is observed that for values of/~ considered, all the second eigenvalues are positive and it was demonstrated by Banks and Zaturska (1986) to be so for other values of/~ of interest. Further, we note that there is complete agreement between the analytical eigenfunctions (11.44), (11.45) and (11.46) for ~  _+1 and Figure 11.2 shows the results obtained by numerically integrating Equations (11.40). Therefore, it can be concluded that, for m ~  1 , in the regions where the basic boundarylayer solutions are valid, a l l the convective flows are spatially stable, except the limiting flow when ~ =  2 , when such a flow may be described as neutrally stable.
Table 11.1 Second eigenvalues, 72, for some values of/3.
I~ II~1 I~~ II 6 I
0 7.655
i
~ I
3
19.1961,1:2.101
(~)
.
I 8 .... I ! 1~:s96 I
(b) 2.0
0.5~
Gl(f]) 1.5
G2(rl) 0.0
1.0
0.5
0.5
1.0
0.0
2 ,
4 ,
,
6
8 n 10 ,
,
1.5 0
2
4
6
8 rl 10
1"2
Figure 11.2: First and second eigenfunctions, G1 (q) and G2(ri), for some values
of~.
12 ,
F R E E AND MIXED C O N V E C T I O N OVER VERTICAL SURFACES 11.3.2
Permeable
391
surface
This class of free convective boundarylayer flows in porous media is associated with vertical permeable surfaces through which fluids can be injected into the porous medium or withdrawn from it, i.e. sucked through the surface. Problems of such type were initially treated by Cheng (1977a) and this work was continued by Merkin (1978), Minkowycz and Cheng (1982), Govindarajulu and Malarvizhi (1987), Chaudhary et al. (1995b, 1995c) and Magyari and Keller (2000a, 2000b). However, we are interested here to show the existence of similarity solutions for a vertical permeable surface in a porous medium. This situation occurs when, in addition to Tw(x) being given by expression (1.31), we have vw(x)  f~(1 + m)
( 1V+~ )
m1
x 2
(11.48)
where fw > 0 for injection and fw < 0 for suction (or withdrawal). If we take f
v~F(~),
~ = Vr2~
(11.49)
Equation (11.23a) then reduces to F "l + ( m + 1 ) F F "  2 m F '2  0
(11.50a)
and the boundary conditions (11.18b) become F(O) =  fw,
F'+0
as
F'(0)r
1
(ll.50b)
It was shown by Chaudhary et al. (1995b) that Equations (11.50) have solutions only for m >  89 the same as for the case of fw = 0 (impermeable surface). However, an asymptotic solution has been obtained by these authors for fw > 0 (injection) when m .~  89 and they obtained the following result,
F" ( 0 1  0.00119
+m
+...
( l.al)
for m ~  89 The variation of F"(0) with m for fw  1, as obtained from the numerical integration of Equations (11.50) and from the asymptotic expansion (11.51), are shown in Figure 11.3. It is concluded from this figure that the two curves are in good agreement, with the difference becoming smaller, as expected, as m decreases towards the singular value at m   ~1. A numerical solution of Equations (11.50) for fw  0, 4, 8 and 12 with m = 0 and m  1 has been also obtained by Chaudhary et al. (1995b) and a plot of the temperature profiles 0 = F'(r is given in Figure 11.4. We observe from Figure ll.4(a) that when m = 0 a clear tworegion structure emerges as the value of fw increases. There
392
C O N V E C T I V E FLOWS
10.
F"(0) 6
2 0 0.50.4().30.2(}.1
0:0
m
Figure 11.3: Variation of F"(0) with m for fw  1. The numerical solution is indicated by the solid line and the asymptotic expansion (11.51) is indicated by the broken line.
(a)
(b) 1.0
1.0
. . . . . . . .
F'(~ 0.6
=0, 4, 8, 12
0.6 0.4
0.4
0.2
0.2
F'(~
0.0
.
0
4
8
12
16
20
0.0
0
5
' lb
15
2'0
i5
30
Figure 11.4: Velocity or temperature profiles, F'(~), for (a) rn = 0 and (b) m  1.
is a thick inner region, where the temperature is constant (at its surface value) and a thinner shear layer at the outer edge where the ambient temperature is attained. These profiles are reminiscent of the temperature profiles seen at large distances from the leading edge of the plate in the constant surface temperature and fluid injection rate problem described by Merkin (1978). For m  1, we observe from Figure l l.4(b) that, although the boundarylayer becomes thicker as the value of fw increases, no obvious tworegion structure has been set up. These results suggest that the development of the solution for large values of f w could depend on the value of m.
F R E E AND MIXED C O N V E C T I O N OVER VERTICAL SURFACES
393
On the other hand, for fw < 0 (suction), Chaudhary et al. (1995b) have shown that Equations (11.50) have a solution even for values of m <  89 as can be seen in Figure 11.5, where the variation of F"(0) as a function of m obtained from a numerical solution of these equations for fw   1 is given. This figure suggests that a solution exists for m > mc (f~), a critical value of m, where, for fw =  1 , me .~ 0.5619, and that F"(0) approaches a constant value as m + me (shown by the broken line). Detailed solutions for strong suction, fw > 1, and for m >> 1 were also reported by Chaudhary et al. (1995b). 2
1"
F"(0) :
0I I I
1
I
2
"
0.6
'
0.4
'
m

'
'
6.2
i
0:0
Figure 11.5: Variation of F"(O) with m for fw   1 . The numerical solution is indicated by the solid line and the broken line indicates the value me ~ 0.5619.
11.4
S i m i l a r i t y s o l u t i o n s of t h e b o u n d a r y  l a y e r equat i o n s for surfaces w i t h variable wall heat flux
11.4.1
Impermeable
surface
We assume that qw(x) is given by Equation (1.59), and since Vw(X)  O , then N ( x ) given by Equation (11.22) becomes identically zero, i.e. N ( x )  O. Thus, Q ( x ) which is given by Equation (1.30) now becomes Q ( x )  m and Equation (ll.21a) reduces to
f,,, + ~ + 2 mf f,, _ 1 +32mf,2 _ 0
(I 1.52a)
This equation has to be solved subject to the boundary conditions (11.21b), which reduce to, for the present problem, =0,
f"(O)=I
f~?)+ 0 as ,/+oo
(11.52b)
394
CONVECTIVE FLOWS
Integrating Equation (11.52a) once, and using the boundary conditions (11.52b), we obtain (1 + m)
f,2 dr]  1
(11.53)
which shows that in order for Equations (11.52) to have a solution then it is necessary that m >  1 , with the solution becoming singular as m +  1 . The problem governed by Equations (11.52) has been studied by Merkin and Zhang (1992) who determined the following asymptotic solutions for the wall temperature 0(0) = if(0), 2
f'(O)  0.72112 (1 + m )  ~ + . . .
(11.54a)
f'(0)  1.2962  0.5031m + 0.2313m 2 + . . .
(11.54b)
as m +  1 , for m 0 than it is for m < 0. Possibly this is to be expected because of the existence of the singularity at m =  1 . Also, the large m limit, a~ given by Equation (11.54c), is approached closely for even quite moderate values of m. The behaviour of the velocity or temperature profiles
f'(O)
',,%. 9 ,,,.,

,
..
,..,,
,..
Figure 11.6: Variation of f'(O) with m. The numerical solution is indicated by the solid line and the asymptotic expansions (11.54) are indicated by the broken lines.
F R E E AND MIXED C O N V E C T I O N OVER VERTICAL SURFACES
395
f'(y) at m ~  1 is depicted in Figure 11.7 for several values of m. These plots show that the value of f'(0) increases with m (in line with Figure 11.6) and the boundarylayer becomes thinner as m +  1 . 10 8
f'(~)
m  0.8, 0.9, 6
.
0
m
1
.
2
3
Figure 11.7: Temperature profiles, f'(~), for some values of m.
11.4.2
Permeable
surface
In this case, we take
~ ( ~ ) = f ~ ( 2 + m) ~~ vf~ x 3
(11.55)
where fw is again the suction or injection parameter with fw < 0 for suction and fw > 0 for injection. If we now make the transformation 2
f  3~F(~),
~
3 89
(11.56)
then Equation (11.52a) becomes F ' " + (2 + m ) F F " 
(1 + 2 m ) F '2 = 0
(11.57~)
and the boundary conditions (11.21b) become F"(0)  1 F(0) =  f ~ , F '  + 0 as ~+oo
(11.57b)
396
C O N V E C T I V E FLOWS
This problem was considered by Chaudhary et al. (1995c). If we integrate Equation (11.57a) once and apply boundary conditions (11.57b), we obtain 1 + (m + 2)fwF'(O)  3 (m + 1)
F '2 d~
(11.58)
Now for fw > 0 and m >  2 , the left hand side of this relation is positive and it should be noted that F'(O) must be positive. Hence, Equation (11.58) implies that a solution of Equations (11.57) is possible only if m >  2 for fw > 0 (injection). Chaudhary et al. (1995c) showed that Equations (11.57) are singular near m ~  1 and they found that Ft(0) behaves as follows: F'(0) ~
(m + 1) 2 + . . .
(11.59)
as m +  1 . Figure 11.8 shows the variation of (m + 1)2F'(0) for fw  1 and for decreasing values of m obtained by solving Equations (11.57) numerically (shown by the solid line). These results can be seen to be approaching their asymptotic limit (shown by the broken line) as given by expression (11.59), as m +  1 .
~~ 1.2 "~ 0.8 + ~0.4 0.0
,"
'
.
.
.
.
.
.
.
i
m
0.0
Figure 11.8" Variation of (rn + 1)2 F'(0) with m for fw  1. The numerical solution is indicated by the solid line and the asymptotic limit (11.59) is indicated by the broken line.
For fluid suction, fw < 0, it was established by Chaudhary et al. (1995c) that the solution of Equations (11.57) becomes singular as m = mc +  2 . This has been found numerically and the variation of me with fw is plotted in Figure 11.9. This figure clearly shows that mc +  2 as Ifwl + c~ and mc +  1 as Ifwl + 0 (impermeable plate). Solutions for large values of fw were obtained by Chaudhary et al. (1995c), for both fw > 0 and fw < 0. For fw > 0 the form of the asymptotic solution is seen to depend on the value of m. Solutions for large values of m were also derived and these are seen to depend strongly on whether fw is positive or negative.
F R E E AND MIXED CONVECTION OVER VERTICAL SURFACES
397
2.0
f
1.8 wm c
1.6 1.4 1.2 1.0
"
0
i~
5
'
'
i
10
15
2'0
2'5
Figure 11.9: Variation o f  m c w i t h  f , v .
11.5
Combined heat and mass transfer by free convection over a vertical surface
Convective flows due to the combined buoyancy effects of thermal and species (concentration) diffusion in a fluidsaturated porous medium have many applications, such as, soil pollution, fibrous insulation and nuclear waste disposal. As is wellknown, the nature of convection flows in porous media due to thermal buoyancy alone is welldocumented and a large amount of literature exists on this topic. However, comparatively less work has been published on the buoyancy induced convection flows resulting from the combined buoyancy effects. A review of this topic was recently presented by Nield and Bejan (1999). A n g i r ~ a et al. (1997) studied numerically the combined heat and mass transfer due to free convection adjacent to a vertical surface which is embedded in a porous medium with special attention being given to the opposing buoyancy effects which are of the same order of magnitude and unequal thermal and species (concentration) coefficients. The case of aiding boundarylayer flow for this configuration has been studied by several authors and are cited by Angirasa et al. (1997). However, of particular interest is the work by Bejan and Khair (1985) which appears to be the first paper which has considered the free convection boundarylayer along an isothermal vertical surface in a porous medium due to the combined heat and mass transfer effect based on the similarity analysis of Cheng and Minkowycz (1977) with thermal buoyancy alone. Bejan and Khair (1985) also presented an order of magnitude analysis of the boundarylayer equations using the scale analysis which provide functional relations for the Nusselt and Sherwood numbers in various limiting cases. Boundarylayer analysis was shown to be invalid when the two buoyancy mechanisms oppose each other and then they are of the same order of magnitude.
398
CONVECTIVE FLOWS
T~.=0, Q , = 0 , ~ X = 0
(5)
out
0,
~X
out
0
(:)
II
II
CD
II
II
C:)
II
II O
X
(5)
out
:0
out
:0
y T~=0, G~=0, ~ X = 0 Figure 11.10: Physical model and coordinate system.
The physical model and the coordinate system under consideration are illustrated in Figure 11.10, which represents a vertical flat plate of height l which is embedded in a porous medium. The temperature of the plate is Tw and the surface concentration is Cw. Far away from the plate these values are Too and Coo, respectively, where Tw > Too and Cw > Coo. The buoyancydriven Darcy flow and transport, adjacent to the vertical surface due to the combined effects of the thermal and concentration diffusion is described by Equations (II.1), (II.2), (II.5) and (II.8) and can be written in nondimensional form as follows, see Angirasa et al. (1997)"
Ou Ov 0x + Oy = 0
(11.60)
On Ov OT OC = +NOy Ox Oy Oy OT OT OT 1 + u  ~x + y o u  R a V 2 T OT
(11.61) (11.62)
~oOC OC 0T 1 V2 C ~ Ot+ U~x + v Oy = aL~ R
(11.63)
and these equations have to be solved along with the boundary conditions u0, u+0,
v0, v+0,
T=I, T+0,
C=I C+0
on as
y0, y+c~,
0~<x 0
x>0 y>0
(11.71) where c~2 and a are the reactant consumption and activation energy parameters which are defined by Equation (4.53). Further, we shall deal with both the cases of a vertical semiinfinite flat plate, S(x)  1, and a stagnation point, S(x)  x. 11.6.1
Vertical flat plate
We note that at the leading edge of the catalytic surface, the flow develops due to a constant wall heat flux and this suggests the transformation ~ Of
r
71), O  x ~  ~ ,
r
77
y
r
(11.72)
x~
On applying this transformation to Equations (11.68)  (11.70), for which S(x)  1, we obtain
03 f 2 02f g 0~7 + 5 f O~ 2
l(Of) 3 ~
2
( O f c92f  x O~ OxOq
1 02h 2 Oh ( O f Oh Le Orl2 ~ 3f ~  x OrI Ox
02for) O~2 0x
(11.73)
Oh O f ) &l Ox
(11.74)
with the boundary conditions (11.71) becoming
fO,
02f 
h exp
....
Oh
z~
89e x p
b'~
f'+0,
h+l,
as
( ) z~~ l+ax89 Of
on
r/0
~~eo (11.75)
It results easily from these equations that, for L e r
y) = 1  a2 O(x, y)
1, we have (11.76)
and, for convenience, all the results presented in this section will be for this value of
Le.
F R E E AND M I X E D C O N V E C T I O N O V E R V E R T I C A L SURFACES
403
Minto et al. (1998) first obtained solutions of Equations (11.73)  (11.75) in the form of a power series in small x (> 1) there are two cases to be considered, namely when the reaction consumption parameter a2  0 and c~ ~ 0, and when c~2 ~ 0 and a is arbitrary, and this gives rise to two essentially distinct types of asymptotic solutions. Thus, for a2  0 and a ~ 0, we can obtain for Ow(X) and Cw(X), the following asymptotic expressions for large values of x (>> 1), see Minto et al. (1998),
1
Ow(x)  x5 (exp ( ~ ) ) r

1.29618  0.85333 (exp ( ~ ) )
~ a2x~ +...
]
(11.78)
1
However, for a2 # 0 and a arbitrary, we assume t h a t Ow(X) approaches a constant value and that Cw(X) + 0 for x >> 1. This suggests the introduction of the variables
OF O  0~ '
r  x 89
r
y ~  ~=!
H(x'~)'
(11.79)
X2
Equations (11.68)  (11.70) then take the form
03F 1 02F ( O F 02F 0~Y + ~F 0~~ = x O~ OxO~
OF O2F) Ox ~
1 02H 1 OH ( O F OH Le O~2 ~~F O( = x 0r Ox
(11.80)
OF O H ) Ox ~
(11.81)
along with the b o u n d a r y conditions (11.71) becoming
F  O,
x89 02F

 H exp
( ) Of
~~
OF
OF
o~+0,
1 OH
,
X
5
H+I,
~
as
 a2 exp
~
_
O'f
1+ a _5~
on
~
0
~+c~
(11.82) The form of these b o u n d a r y conditions suggests that F and H can be expanded in 1 power series of x 2 and thus, for Le  1, ~2 # 0 and a arbitrary, we have
[
3
Ow(x)  a ; 1 1  0.44375 a 2 ~ exp
3
Cw(x)  0.44375a 2 2 exp for x >> 1.
(
1
a+~2
)
(
) 1
X
ct+ct2
x 89 +
" " "
1]
+""
(11.83)
404
CONVECTIVE FLOWS
On the other hand, Minto et al. (1998), have solved numerically the three sets of Equations (11.68)  (11.71), (11.73)  ( 1 1 . 7 5 ) and (11.80)  ( 1 1 . 8 2 ) using a method proposed by M a h m o o d and Merkin (1988) starting from x  0. However, at x 0 the first two sets of these equations are singular and in order to remove this 1 singularity a new variable ~  x~ was used in all the numerical computations. Figure 11.12 illustrates the variation of 0w(~) for c~2  0 (reactant consumption neglected) and c~  0.0, 0.1, 0.2 and 0.3; here Cw  1, as can be seen from the relation (11.76). This figure clearly shows that as ~ + oc, Ow({) does not approach a constant value, but instead increases towards an infinite value. It is also observed that all the solutions exhibit a twophase type of behaviour. The initial reaction phase starts at the ambient temperature on the surface away from the leading edge. Initially there is a slow rate of increase in 0w(~) as we move along the surface and then 0w(~) suddenly starts to rise sharply. This sudden change in the behaviour of 0w(~) can also be observed in Figure 11.13, where the nondimensional t e m p e r a t u r e profile 0(~, 77) is plotted against 77 for ~  0.6, 0.65, 0.7, 0.71 and 0.72 when a  c~2  0 . 10
4
2 0 0.0
0.4
0.8
1.2
1.6
2.0
Figure 11.12: Variation of the wall temperature distribution, Ow(~), with ~ for c~2  0 at several values o] c~.
Next, we illustrate in Figure 11.14 the variations of Ow(~) and Cw(~) for a2 ~ 0 (reactant consumption included) and c~ = 0, 0.05, 0.1 and 0.2. These figures show that Ow(~) + c ~ 1 and Cw(~) + 0 as ~ + c~. It is also seen that the smaller the value of c~, the higher is the rate of increase or decrease of 0w(~) and Cw(~), after the initial phase of the reaction at low temperatures. Finally, Figures 11.15 and 11.16 compare the numerical solutions of the full boundarylayer Equations (11.68)  (11.70), where S ( x )  1, with those of the asymptotic solutions (11.77), (11.78) and (11.83) in both the cases a2 = 0 and c~2 ~ 0, when (~ r 0. W i t h o u t going into further details, it should be noted that
FREE AND MIXED CONVECTION
OVER VERTICAL SURFACES
405
10 8 6
.
.
.
.
0.71, 0.72
0
"I
0.0
'
0:4
'
0:8
'
1:2
rl'
Figure 11.13: Temperature profiles, ~(~, 77), for c~  22  0
(a)
"" " '1
1:6
2.0
at several values of ~.
(b)
5
1.0
4 e~(~~
0.8 .
.
.
r
.
0.6
2
0.4
1
0.2
0
0
2
4
6
8
10
0.0
, 0.2
0
2
4
6
Figure 11.14: Variation of (a) the wall temperature distribution, ~?w(~), and (b) the wall concentration distribution, Cw(~), with ~ ]or c~2  0.2 at several values of Ol.
b o t h the n u m e r i c a l a n d a s y m p t o t i c solutions are in very g o o d agreement.
11.6.2
Stagnation
point
T h e p r o b l e m of free convection b o u n d a r y  l a y e r flow near the lower s t a g n a t i o n p o i n t of a t w o  d i m e n s i o n a l cylindrical b o d y which is i m m e r s e d in a p o r o u s m e d i u m , where
406
CONVECTIVE FLOWS 2.0
~.5 ~1.0 0 0.5
3
2
.~o"
6 i
i
~.
log10x
1.0
Figure 11.15" Variation of the wall temperature distribution, logloO~(x), with loglo x for a2  0 and a  1. The numerical solution is indicated by the solid line, the asymptotic solution (11.77) is indicated by the dotted line and the asymptotic solution (11.78) is indicated by the broken line.
(a)
(b)
1~I si
0.8, " ' ,
'.,,\
:tX/
,,\
0.4 "
~ ~',~
~
.''~ ,,,
0.2 'a
2
~
6
i'
,,,
o.6 . \
i'
loglo x
'~
,
3
,
2
I
,,
0
"~
1
"
"'~l~ll4eb,,,~ \
2
3
log10 x
Figure 11.16: Variation of (a) the wall temperature distribution, Ow(x), and (b) the wall concentration distribution, r with loglo x for a2 = 0.1 and a = 0.2. The n~merical solution is indicated by the solid line, the asymptotic solution (11.77) is indicated by the dotted line and the asymptotic solution (11.83) is indicated by the broken line.
the flow results from the heat released by an exothermic catalytic reaction on the cylinder surface, has been treated by Merkin and M a h m o o d (1998). On noting that
FREE AND MIXED CONVECTION OVER VERTICAL SURFACES in this case S ( x )  x
407
and taking r  x f (y),
O = O(y),
(11.84)
r162
Equations (11.68) (11.70) reduce to f '  O, r
(11.85)
O" Jr f Ol  0
(11.86)
+ Le f r = 0
while the boundary conditions (11.71) become
f(o)  o ,
0 ' ( 0 )    a 4 r
exp
0~0,
i+ao~
r
,
r
 OZ2C~4r e x p
l+aO~
y ~ c~
as
(11.87)
where the reactant consumption parameter a4 is defined by the relation Ol4  
EQkolCoo ( km R T 2 R a 89exp  RToo
(11.88)
We will now take 1
f0~F(r/),
1
00wG(r/),
r162
H(rl),
r/0~y
( 1.89)
so that Equations (11.85) and (11.86) become F'" + F F "  O,
H " + L e F H '  0
(11.90a)
and the boundary conditions (11.87) give F(0) = 0, F'+0,
F ' ( 0 ) = 1, H ( 0 )  1 H+0 as ~7~c~
(ll.90b)
On integrating numerically Equations (11.90) it was found by Merkin and Mahmood (1998) that  G ' ( 0 )   F " (0)  Co  0.62756,
  q Y ( O ) " C 1 ( L e )
(11.91)
From the boundary conditions (11.87), we now obtain, after a little algebra, the following relation
3(
(1  a5Ow ) a4  CoO ~wexp
where

C1 . Clearly expression (11.92) requires Ow < 1___ O~ 5 "
408
CONVECTIVE FLOWS
For the case when there is no reactant consumption, i.e. Equation (11.92) gives
3(
o~4  CoO~wexp
O~2

Ol 5

0
then
(11.93)
OW )  1 + o~Ow
On differentiating Equation (11.93) with respect to Ow, we find that the critical points (turning points on the bifurcation diagram , where dr d~4 _ 0) are given by (1,2) (c~)  1  3cr • x/1  6~ w 3ol2
(11.94)
and, for example, when c~ = 0.02 these turning points have the values 0O)  1.5974,
0~)
1565.1
(11.95)
Further, for c~2 # 0 the critical points are given by the equation
0150~20w34
3c~2
(20~50~ .
.
2c~5)02 . + (~5. + 2  6c0 0~
3
0
(11.96)
On p u t t i n g (~ = 0 into this equation, we find that there are critical points at _
+ 2 i

4o~5
4
for
c~5 7(: 0
(11.97)
and we note from this expression that there is a hysteresis bifurcation, i.e. coincident critical points, where c~5 = 1 0  4x/6 = 0.2020. To determine where there is a hysteresis bifurcation it is necessary to solve Equation (11.96) simultaneously with the equation 3c~5~20~ + 2 (2(~5(~  3~ 2  2~5) Ow + ~5 + 2  6(~  0
(11.98)
Equations (11.96) and (11.98) were solved numerically by Merkin and M a h m o o d (1998) and the results are shown in Figure 11.17. It can be concluded that for c~ > 0, and for values of c~5 below the curve shown in Figure 11.17(a), that multiple solutions exist in the region between the upper 0(w2) and lower 0O) critical points. Variations of 0O) (c~5) and 0(w2) (a5) with a5 are shown in Figure ll.17(b) for a  0.02, where the upper and lower critical points are given by the values (11.95).
F R E E AND MIXED C O N V E C T I O N O V E R VERTICAL SURFACES
(a)
409
(b) 12
0.15
10 8
0.10
0.05
Solut!ons Po.ssible..
6 xNN,~
0.00 0.(10 0.05 0.10 0.15 0.20
4 2 0.00 0.04' 0.b8 " 0.~2 " 0.i6 "
O~5
~5
Figure 11.17: (a} The solution of Equation (11.96} in the (c~,c~5) plane; (b) Variations of O~ ) (c~5) and O~ ) (c~5) for c~  0.02.
11.7
F r e e c o n v e c t i o n b o u n d a r y  l a y e r flow o v e r a v e r t i c a l s u r f a c e in a l a y e r e d p o r o u s m e d i u m
Rees (1999b) has studied the steady free convection boundarylayer flow from a vertical heated surface which is embedded in a porous medium consisting of multiple sublayers which are aligned such that the interfaces are parallel with the surface. The first consists of one sublayer sandwiched between the surface and the rest of the medium which is uniform and isotropic. The second has two such sublayers and the third is composed of an infinite number of sublayers with alternating properties. In this section we present results only for the first configuration, i.e. the one sublayer configuration. Consider a semiinfinite vertical surface at a constant temperature Tw, which is embedded in a porous medium of ambient temperature Too (< T~). The heated surface is placed at ~   b for ~ > 0, where ~ and ~ are the Cartesian coordinates measured along and normal to the plate, respectively. In the region ~ > 0 the permeability and thermal diffusivity of the medium are constant and equal to K2 and c~m2, respectively, while in the region lying between ~   b and ~ = 0 the porous layer is of permeability K1 and diffusivity C~m~, which are also constant; these quantities may or may not be different from K2 and C~m2, respectively. The steady, twodimensional DarcyBoussinesq equations can be written in nondimensional form, see Rees (1999b), as follows:
02r
02r
Ox5 + Oy 2  Ra
(Ki)OOi ~
Oy
(11.99)
410
CONVECTIVE FLOWS
am;
Oy2
0x 2 +

Oy Ox
Ox Oy
(11.100)
where the subscript i denotes which layer is being considered: i  1 corresponds to the nearwall layer,  b ~ 0, called region 2. The Rayleigh number R a  gK2flATb is based on the permeability P'~m 2
and diffusivity of region 2, and therefore the nearwall layer (region 1 boundarylayer), is regarded as an imperfection to an otherwise homogeneous and isotropic medium. The appropriate boundary and interface conditions for which solutions of Equations (11.99) and (11.100) are sought are given by r
0,
01  1
Oy _~ 0, 0r 02_+K10 002
0r r
 r
K2O~
on
y
1,
aS
y + oo,
x > 0
 c ~ < x < oo (11.101)
]
~
on
OqO1
y
x > 0
0
The boundarylayer approximation is further invoked by assuming that Ra is very large. Thus, the following boundarylayer variables are introduced, for region 1, ~21  R a ~ I F I ( Y , ~ ) , 01  HI(y,~), x  Ra~ (11.102) which on substitution into Equations (11.99) and (11.100), and letting Ra ~ co, then the region 1 boundarylayer equations are found to be of the form: 02F1
OY2 =
(O~mt)O2Hll~OH1 am:
oY 2 + 2 ~ ~F1 c3T
(K1) IOH1 K22 {~ Oy
(11.103)
(OFIOHIOFIOH1)
= ~89
Oy O~
O~ Oy
(11.104)
For sumciently large values of { it is to be expected that the region 1 boundarylayer will extend well into region 2 and its shape will become similar to that of the homogeneous selfsimilar boundarylayer. Therefore, for region 2, the following boundarylayer transformation is introduced: 1
1
r
 Ra ~ F2(~, {),
02  H2(rl, (),
77
Ra ~y 
1
(11.105)
X7
with the region 2 boundarylayer equations becoming
~
0,72
02F2
0H2
O~2
O~
(11.106)
1
OH2
f OF2 c9H2
0F2 0tI2)
z
O?
~, &7 0~
O~ Ov
(11.107)
FREE AND MIXED CONVECTION OVER VERTICAL SURFACES
411
The boundary and interface conditions (11.102) also become FI=0,
Hl=l H2 +0 O,7 F1  F2, K 2 ~OF, _ K I ~  89~OF= OF2
HI = tt2,
~ 0,
aml Oy
on as
7/+oo,
f>0 oo0
}
0,7
(11.108) It is worth mentioning that Equations (11.103), (11.104) and (11.106), (11.107), subject to the boundary conditions (11.108), reduce to ordinary differential equations when K1 = K2 and am1 = am=. The corresponding equations for the twosublayers and multilayer configuration may be derived in a very similar manner, and they are presented in the paper by Rees (1999b). When ~ is sufficiently small, the region 1 boundarylayer is contained well within the nearwall layer, and the flow is essentially selfsimilar, so that F1 and H1 take the forms 1
F1 (y, ~) "~ \
f(r
O~mrefgref
//1 (y, f) ~ h(()
where
(11.109)
1
(Omre K1) y + 1 
with f(~) and h(r
~ml gref
(11.110)
~1
satisfying the equations
f
1
(11.111a)
h" + ~ f h '  0
" = h',
and the boundary conditions f(O) = 0, h(0) = 1 as r f ' ~ O, h  + 0
(ll.lllb)
We can now calculate the wall heat transfer
(OT)
q~(~) =  k ~
~
= _kmi A r ~Ooi ~
~=0
(11.l12a)
  T   \b~y]~=0
which for small values of f becomes 1
krnrefZ~T C~mrefgl qw(~)  b (~mlKref
~ 89
(11.112b)
A suitable scaled rate of heat transfer Q at the plate, can also be defined as
Q w ( ~ )  kmrefAT  
(
c~ml
y=o
412
CONVECTIVE FLOWS
which for small values of ( reduces to 1
Qw
(OmrefK1) Om1
h'(0)
(ll.l13b)
The calculated value of h'(0) in expressions (ll.l12b) and (ll.l13b) is h'(0) f " ( 0 )   0 . 4 4 3 7 5 , see Equation (11.32b) for m = 0. The two systems of Equations (11.103), (11.104) and (11.106), (11.107), subject to the boundary conditions (11.108), were solved numerically by Rees (1999b) using the Kellerbox method for different values of the parameters ~g l and am2" a'~____L~ The complete details can be found in Rees' paper and therefore are not repeated here. The effect of the parameter ~K1 on the streamlines and isotherms is shown in FigK2 ures 11.18 and 11.19 for otto1 = O~m:~. When K1  g, as shown in Figures ll.18(a) and ll.19(a), the fluid is inhibited from moving quickly in region 1, and therefore the heat lost from the surface by advection is reduced. Consequently, the region 1 boundarylayer grows in thickness relatively quickly, as can be seen from Figures li.19(a,b) near ~ = 0. The opposite occurs when K1  5K2 where the greatly increased nearwall permeability causes an enhanced fluid motion, thereby thinning the boundarylayer, as seen by comparing Figures ll.19(b,c). Much further downstream, the two boundary layers are of comparable width and the presence of a nearwall sublayer serves only to perturb slightly the shape of the main region 1 boundarylayer and its rate of heat transfer. We notice that the discontinuous shapes of the streamlines in Figure 11.18(c) are due to the interface conditionsalthough the normal flux fluid velocity must be continuous on physical grounds, the tangential flux fluid velocity cannot be continuous as the order of the governing partial differential equations is insufficiently high to allow this, see Rees (1999b). The variation of the scaled wall heat transfer, Qw(,~), with ~, given by Equations (11.113), is shown in Figure 11.20. It is seen that Qw(~) varies monotonically as ~ increases and it approaches the value  h ' ( 0 ) of the uniform medium when ~ is large given by Equation (ll.l13b), namely the horizontal lines in Figure 11.20. In common with other types of freeconvection boundary layers at large distances from the leading edge, i.e. large values of ~ (>> 1), asymptotic solutions for the present layering flow configuration were provided by Rees (1999b). The rate of heat transfer Qw(~) was found to be given by
Qw(~) ~ 0.44375  0.17094 (Olm2~ ( K 1 Otm 1 ,] K2
C~m2) A0~_ 1 ln~ + O (~1)
(11.114)
O/m 1
for ~ >> 1 where A0 is an undetermined constant and the term In ~ is included due to the leading edge shift effect. The last term in this expression contains a constant which cannot be obtained using asymptotic methods. Therefore, it is difficult to use expression (11.114) to verify the numerical results. However, an examination of Figure 11.20 shows that the deviation of Qw(~) from h'(O) = 0.44375 decays
F R E E AND M I X E D C O N V E C T I O N O V E R V E R T I C A L S U R F A C E S
(a)
(b)
,,
(
Figure 11 9 18: Streamlines corresponding to the cases of (a) ~2 = 0.2, (b) g_t _ 1 K2 and (c) K~2  5, when am1  am2 (Ar  0.2). The broken line denotes the interface between the regions 1 and 2.
413
r~3
0
> r,p
> Z 0 r,p
_ _ . _ .
9
.
,
..
\
,
\
9
\
...

\ \ \
II
~
.~
II
~1~ ~
..

415
F R E E AND MIXED C O N V E C T I O N OVER VERTICAL SURFACES
(a)
(b) 0.6] 0.9
~ ~ z 
0.1, 0.2, 0.5,
0.5
Q~(r 
Q~(~)
. . . .
2, 0.5, 1, 2, 5, 10
t
0.4
0.7
0.3 0.5 0.2 0.3, 0
I
I
1
g
,
2
,
,
i
3
,,
4
i,,,,,
,
5
0.1
,
0
I
2
l

I
4
I
m
i~
im
1
6
]
8
Figure 11.20: Variation of the scaled wall heat transfer, Qw(~), with ~89 for (a) ~ml  am2 and (b) K1  [(2.
approximately as ~1 and this gives some qualitative verification of the theory. A more substantial confirmation lies in the fact that when K1  K2 and C~m~  C~m2, i.e. regions 1 and 2 have identical properties, the logarithmic term is absent in expression (11.114) and therefore Qw(~) reduces to that of the classical boundarylayer theory, see Equation (11.32b) for m  0.
11.8
Free convection boundarylayer flow over a vertical surface in a porous medium using a thermal nonequilibrium model
The subject of convective local flow in a fluidsaturated porous medium when the solid and fluid phases are not in thermal equilibrium has its origin in two papers by Combarnous (1972) and Combarnous and Bories (1974) on the DarcyB~nard problem. The review article by Kuznetsov (1998) gives very detailed information about the research on thermal nonequilibrium effects of the fluid flow through a porous packed bed. However, it appears that the problem of free convection on a surface which is embedded in a fluidsaturated porous medium using a twotemperature model has only been very recently investigated. Rees and Pop (1999) and Rees (1999b) have studied the effect of adopting this model to the problem of free convection boundarylayer flow from a vertical isothermal surface and near the lower
416
CONVECTIVE FLOWS
stagnation point of a twodimensional cylindrical body in a porous medium, respectively. Such a model, which allows the solid and fluid phases not to be in local thermal equilibrium, is found to modify substantially the behaviour of the flow relatively close to the leading edge of a vertical surface where the boundarylayer is comprised of two distinct asymptotic regions. We now report on some results obtained by Rees and Pop (2000) for the case of a vertical surface in a porous medium adopting a twotemperature model of microscopic heat transfer. Consider the steady flow which is induced by a vertical semiinfinite flat plate held at a constant temperature Tw and embedded in a porous medium with ambient temperature Tcr where Tw :> Too. It is assumed that at sufficiently large Rayleigh numbers, and hence sufficiently large velocities, the local thermal equilibrium breaks down, so that the temperatures T I and Ts in the fluid and solid phases are no longer identical. The mathematical formulation of this problem consists then of the following nondimensional equations, obtained from Equations (II.1), (II.2), (II.6) and (II.7), 0O Ra~
V2r
(11.115) A
v2o

~(o  r
v2r  ~k(r

A
0r oo o~ o~
0r oo o~ o~
(11.116)
o)
(11.117)
where the nondimensional variables were defined as m
2
~=['
^
~
Y7'
^
r
(pc) f .... ~kf
~,
o=
T f Too ' AT
r
T~  Too AT
(11.118)
A
with h and k given by "" hi2 h = kl ,
k =
qok:
(l~)ks
(11.119)
The parameter k is a modified conductivity ratio, and its value is in the range 10 5 < k < 10, which covers most practical applications and low values of k generally correspond to a relatively poorly conducting fluid such as air in a metallic porous medium. ^ Next, we introduce the usual boundarylayer scalings ~ = x, ~" = Ra 89 and r  R a : r as R a ~ c~ into Equations (11.115)  (11.117) to obtain the equations 02r Oy 2
O0 Oy
020  H(O Oy 2
02r Oy 2
(11.120)

r q
= H k ( r  0)
or oo
or oo
Oy Ox
Ox Oy
(11.121) (11.122)
FREE AND MIXED CONVECTION OVER VERTICAL SURFACES
417
where H is defined by (11.123)
 R a i l
Such a scaling for h, where H  O(1) as R a + co, allows the detailed study of how the boundarylayer undergoes the transition from being in strong thermal nonequilibrium near the leading edge, to being in thermal equilibrium far from the leading edge. Under the physical assumptions described above, Equations (11.120)  (11.122) have to be solved with the following boundary conditions: r or Oy ~0,
0=1,
r
0+0,
r
on
y=0,
x>f0
as
y+oc,
x)0
(11.124)
To solve Equations (11.120)  (11.122), along with the boundary conditions (11.124), the classical transformation 1
r  x~ f (x, 77),
O  O(x, rl),
dp  r
~),
77
y
~
(11.125)
xg
can be used. Applying this transformation, we obtain f'=
0
(11.126) (11.127)
0" = H k x ( r  O)
1
0" + ~f O'  Hx(O  r + x
(f, Oxx O0 _
o, O f ) ~
(11 128)
together with the boundary conditions f (x, O) = 0 , 0+0,
O(x, O) = l, r =1 r as rl+oc
(11.129)
These equations form a system of parabolic partial differential equations whose solution is nonsimilar due to the xdependent buoyancy force which is induced by the terms proportional to H in Equations (11.127) and (11.128). A nonsimilar set of partial differential equations of this form is normally solved using a marching finitedifference scheme, such as, for example, the Kellerbox method. Beginning at the leading edge (small x), where the system reduces to an ordinary differential equation, the solution at each streamwise station is obtained in turn at increasing distances from the leading edge. However, such solutions are typically supplemented by a series expansion for small values of x and by an asymptotic analysis for large values of x. The former often reveals no further information, except perhaps validating the numerical scheme, but the latter often yields insights that may not be immediately obvious from the numerical solution. However, the present problem is not of this general nature, since when x = 0 then Equation (11.127) cannot be solved with
418
C O N V E C T I V E FLOWS
the boundary conditions (11.129). Therefore, this boundarylayer has a doublelayer structure near the leading edge (small values of x), rather than far from it as is often the case encountered in other situations. This considerably complicates the numerical integration of Equations (11.126)  (11.129) since it is now essential to derive the small x solution very carefully before commencing on the numerical integration of these equations. In order to do this, Rees and Pop (2000) have considered, for x 0
(11.152)
The =t= signs in Equation (11.150) are taken for the aiding (heated plate) and opposing (cooled plate) flow cases, respectively. These equations can be further reduced to the following nondimensional equation, see Merkin (1980),
0 ar
0r 0 2r
Oy2 = Oy OxOy
0r 0 2r Oz Oy2
(11.153a)
along with the boundary conditions (11.152) becoming ~Oy
(CWT) o__~__+ 1 as 0V
02r ~  ~ = T 1 (CHF) on y+c~,  o o < x < c ~
y0,
x>0 (11.153b)
where the mixed convection parameter A is given by
)~_ g K f l A T Uccv
Ra Pe
(11.154)
with A > 0 in the aiding case and A < 0 in the opposing case. 11.9.1
Constant
wall temperature
If we take r = V/2x89f(F/),
F/=
Y
(11.155)
424
CONVECTIVE FLOWS
then Equation (11.153a) reduces to (11.156a)
f " + f f "  0
and the boundary conditions (11.153b) become f(0)0,
If(0)I+AA*, f~+ 1 as 77+oo
say
(11.156b)
For )~ > 0 solutions of Equations (11.156) can be obtained for all values of A, and these were obtained by Cheng (1977b). Then, Merkin (1980) has shown that for < 0, Equations (11.156) have solutions only in the range 1.354 ~ A ~< 0 and for in the range 1.354 < A <  1 the solution is not unique, there being dual solutions fl and f2 for a given value of A. This can be seen from Figure 11.23(a) where the variation of f ' ( 0 ) with A obtained numerically is plotted using a solid line. Also, values of f~'(0) and f~(0) are given in Table 11.3 for 1.354 < ~ <  1 . The set of solutions fl emerges from the Blasius solution of Equations (11.156) for A   1 , where f ' ( 0 )  0.46960. The other set of solutions f2 have, as can be seen from Table 11.3, f~'(O) < f~'(O) for a given A and are such that f~'(O) + 0 as )~ ~  1 , i.e. the boundarylayer separates from the plate.
(a)
(b) 0
 i . o ' 6.5' o 0.5
f"(O) 1.0 1.5
"~".~2b ' 4'0 ' 6'0 A 8'0
1{)0
\\
5
200 f"(O)
\
400
600
Figure 11.23: Variation of f"(O) with ~ (a) in the vicinity of A = 0 and (b) for )~ ~> 1. The numerical solution is indicated by the solid line, the asymptotic solution (11.160) for IA! > 1 is indicated by the dotted line.
However, Harris et al. (1999) have obtained a further analysis of Equations (11.156) for I)~! > 1, where Harris et al. (1999) have found that 7"(0)  0.627555. Equation (11.163) is shown in Figure 11.23(b) by the dotted line. Again, it can be seen that there is excellent agreement between the numerical and asymptotic solution
(11.163). 11.9.2
Constant
wall heat flux
This problem, which does not admit similarity solutions, was first solved by Merkin (1980). Now, Equation (11.153) is nonsimilar and in order to solve it the following transformation is introduced:
r = y :1=4xf (x, r/),
17
Y,
(11.164)
2x.~ where f(x, rl) is given by the equation
f'"+217f"
Of' =k8x89 ( f , Of' 2 f '  +4x~, (S '2 2if") + 4x~~x Ox
f,,Of ~ ) (11.165a)
which has to be solved subject to the boundary conditions
f(x: O)  O, f'+O
f"(x,O)   1 as
(11.165b)
r/~c~
These equations have been solved analytically for small values of x and numerically for both small and large values of x by Merkin (1980). If we define the n~176 wall temperature by Ow(X)  ~KtTAT ~,g~  then this is given by
Ow(x)  2x89f'(x, O)
(11.166)
and the nondimensional fluid slip velocity along the plate, obtained from Equation (11.150), is given by the expression
uw(x)
=
1 :i: Ow(x)
(11.167)
427
F R E E AND MIXED C O N V E C T I O N O V E R VERTICAL SURFACES Merkin (1980) has shown that for x 0 or +0, 0 ~ 0 as y + c o , c~ < x < oo Oy 0Oxo = 0 on y   0 , l~<x~ 0 for 0 ~< x ~< 1. Thus, Equations (12.37) and (12.38) reduce to ordinary differential equations when 5x  i d k  1, d~
6x i d6 k dx =  1
(12.39)
Eliminating k(x) from the Equations (12.39) leads to the following equation for 6(x)
d(
~~z
5xi
xi
(12.40a)
which has to be solved subject to the boundary conditions (12.35), which reduce to d5 dx(0)  0,
5(1)  0
(12.40b)
On solving the system of Equations (12.40), Higuera and Weidman (1995) found for an infinite strip (i = 0), the closed form solutions for 5(x) and k(x) as given by (~(X)  ~0 [2COS (1 COS1 (1  2x2))  1] 1 1 k(x) = 2 ~ [1  cos (] cos 1 (1  2x2))] ~
(12.41)
442
CONVECTIVE FLOWS
where 50 ~ 1.040042. For a circular disk (i  1) no closed form solution of the boundary value problem (12.40) can be found but this problem was solved numerically by Higuera and Weidman (1995). Results for (f(x) and k(x) for both the infinite strip and the circular plate are presented in Figure 12.3. It can be seen from this figure that both the boundarylayer thickness 5(x) and the scaling function k(x) are greater for an infinite strip than for a circular disk. 1.5
1.0
0.5
0.0 . 0.0
.
0.2
.
. 0.4
. . 0.6 0.8
1.0
X
Figure 12.3" Profiles of the boundarylayer thickness, 5(x), and the scaling ]unc
tion, k(x), ]or a strip (solid lines) and a disk (broken lines).
i0
W i t h Equation (12.39) in mind, Equations (12.37) and (12.38) reduce to, for and i  1, f"
~0'  0
(12.42)
0"+ fO'  0
(12.43)
along with the boundary conditions (12.35) which become f(0)  0, 0(0)  1 f'~O, 0~0 as ~  + c c
(12.44)
On numerically solving Equations (12.42)  (12.44), the nondimensional slip velocity, uw(x), the local wall heat flux, qw(x), and the boundarylayer mass flux, rh (x), are given by 
k(x)
f'(O)
1
,
qw(x)  5(x~0'(0),
dn(x)  (27r)ik(x)f (c~)
where f'(0)  0.9592, 0'(0)  0.7103 and f(c~)  1.5496. Nusselt number, Nu, can be calculated from the expression
Nu = 2 na89
]
 ~Y (x, O) x i dx
(12.45)
Also, the average
(12.46)
FLOW OVER HORIZONTAL AND INCLINED SURFACES and this gives N~ _ 1.024 for the strip (i = 0) and  ~ Ra ~
443
 1 538 for the disk (i  1)
Ra
These results show that the average Nusselt number is greater for circular disks than for infinite strips. Free convection flow past a heated upward facing finite horizontal surface in a porous medium has been studied numerically by Angirasa and Peterson (1998) for a range of values of Ra. Consider a horizontal surface of length I embedded in an extensive fluidsaturated porous medium. It is assumed that the depth of the surface is large compared to 1 and the temperature of the surface facing upwards is Tw, which is greater than the temperature of the medium Too. The temperature difference AT (= Tw  T o o ) then induces a buoyancydriven flow which is characterised by horizontal wallbounded flows from both ends of the surface and a vertical plume at the midplane. This twodimensional flow configuration prevails for a specific range of values of the Rayleigh number. To study this problem, Angirasa and Peterson (1998) have used the following full nondimensional equations V2r 
0T Ox
(12.47)
0T 0T 0T 1 Ot + U~x + VOy~  Ra V 2 T
(12.48)
The physical boundary conditions of these equations are as follows: v0, u ~0,
T1 T+0
on as
y0, y+co,
0~<x~~ q0, where q0 = 1.406 and t h a t for q > q0 there are two solution branches for each value of q. On the u p p e r solution branch f'(O) a n d 0(0) behave as, see Merkin and Pop (1997), f'(O) ~ q 
1 ~ +..., ~.q
0(0) ~
1
t ...
(12.63)
as q + c~. The a s y m p t o t i c expansions (12.63) are also shown in Figure 12.10 by the broken lines and we can see t h a t there is good agreement with the numerically d e t e r m i n e d values. G r a p h s of the fluid velocity, f'(~?), and the t e m p e r a t u r e , t?(~), profiles on the u p p e r and lower solution branches of Equations (12.59)  (12.62) are shown in Figure 12.11 for q  2.7. It is seen t h a t there is a drop in the value of f ' ( ~ ) below f~(0) before the a s y m p t o t i c values (12.63) are reached and for the lower solution b r a n c h this is much more pronounced and this leads to a finite region of 77 over which f,(~) < 0.
F L O W OVER HORIZONTAL AND INCLINED SURFACES
(a)
453
(b) 3.0 0.0 0(77) 0.2
2.0 f'(~) 1.0
0.4
0.0
0.6 ,
i
0 ' 4 ' 8 rl' 1'2 ' 1'6
0
i
4
"
9
8 7/ 1'2
i
11
16
Figure 12.11: (a) The fluid velocity, f'(rl) , and (b) the temperature, 007), profiles on the upper and lower solution branches for q  2.7.
12.5
Free c o n v e c t i o n b o u n d a r y  l a y e r flow past an inclined surface
The problem of free convection boundarylayer flow past a surface which is slightly inclined to the horizontal and bounded by a saturated porous medium has been studied by Rees and Riley (1985) and Ingham et al. (1985) for the case of a surface of constant temperature Tw, while Kumari et al. (1990b) considered the case of a surface with a constant heat flux q~o. The case of an arbitrarily inclined plate of constant temperature has been considered by Pop and Na (1996). The governing Equations (12.5) and (12.6) for the nearhorizontal configuration with a constant temperature are such that the parameter A may be scaled out of the problem by the transformation
where A+ = [A[ and A ~ 0. Thus Equations (12.5) and (12.6) reduce to
O =•
OT
Oy 2
Oy
i)y 2
Oy Ox
Ox Ox Oy
(12.65) (12.66)
where, in Equation (12.65), the + sign is to be taken for an upward (positive) inclination and the  sign for a downward (negative) inclination of the plate to the
454
CONVECTIVE FLOWS
horizontal. The boundary conditions for these equations are as followsr or Oy
T1
on
yO,
x>O
T+O
as
y+c~,
cxD<xO
o~,Oy__O, T  O
(12.67)
When the plate is inclined upwards (favourable flow) two series solutions were obtained, namely one which is valid near the leading edge and the other which is valid at large distances from the leading edge. When the plate is inclined downwards (unfavourable flow) the series solution which is valid only near the leading edge was obtained. In this case the boundarylayer separates and a region of reverseflow develops. Ingham et al. (1985) have demonstrated that there is no evidence of a singularity at the separation point and a mathematical explanation of the behaviour at separation was presented. In both the favourable and unfavourable flow cases the fluid slip velocity, Uw(X), and the wall heat transfer, qw(X), may be expressed in the form oo 1
uw(z) = z  ~ E
OO
fnf"(0)' ,, ~n
2
qw(z)  z  ~ E
nO
~ n On, (0)
(12.68)
nO
where f  x3. Values of ftn (0) and 0~n(0) are given in Table 1 of the paper by Ingham et al. (1985). On the other hand, when the plate is inclined upwards, the asymptotic expressions for uw(x) and qw(X) are given by 1
Uw(X)  1 + 0.808x ~: + O (x 1) qw(x)  x
~ [0.4437 + O ( x  l ) ]
(12.69)
for x >> 1. It is worth mentioning that Ingham et al. (1985) have found that using 15 terms in the series (12.68) is sufficient for comparing the results with those given by the asymptotic solution (12.69) valid at large distances from the leading edge (for upward inclinations) and also to investigate in detail the nature of the separation (for downward inclinations). However, Recs and Riley (1985) have matched the asymptotic solutions by using a numerical solution based on the Kellerbox scheme. The variation of uw(x) and qw(X) as a function of x is given by the series (12.68) and they are shown in Figure 12.12 for the downward (unfavourable flow) inclinations of the plate. Figure 12.12(a) indicates that Uw(X) becomes zero, i.e. the flow separates, near x = xs = 10 but a more detailed check of the numerical results reveals that the boundarylayer separates at xs = 9.863. However, qw(X) is nonzero in the vicinity of xs = 10, as can be seen from Figure 12.12(b). Further, it was shown by Ingham et al. (1985) that both Uw(X) and qw(X) are regular at Xs = 9.863. The development of the fluid velocity and temperature profiles, as determined numerically by Rees and Riley (1985), are shown in Figure 12.13 for the case of a downwardly inclined plate. Figure 12.13(a) shows that in this unfavourable flow
455
F L O W O V E R H O R I Z O N T A L AND I N C L I N E D SURFACES
(a)
(b)
0.0 0.48
~(~)
q~(x)0"1
0.32
0.2
0.16
0.3
0.00
0.4
0.16 0
10
2'0
30
X
40
0.5
50
0
10
20
3O
X
4O
50
Figure 12.12 Variation of (a) the wall velocity, Uw(X), and (b) the wall heat transfer rate, qw(x), with x for downward inclination of the plate (A < O) with n  15 in the series (12. 68).
(~,)
(b)
10
•
10 ~ = 0 , 0 . 5 , 1,
8
[~\\\\,~ ~.5, 2, 2.5,
Zl 6
=
4 2 0 0.4
0.0
0.4
0.8
1.2
0
0.0
0.2
0.4
0.6
0.8
1.0
T([, rl) Figure 12.13: (a) The fluid velocity, ~n(~,rl), and (b) the temperature, T(~,~), profiles for downward inclination of the plate (A < 0).
case, the boundarylayer thickens rapidly and the fluid velocity profiles develop an inflexion point as the plate is traversed, suggesting the existence of a region of reversed flow. Rees and Riley (1985) have succeeded in continuing the numerical integration of Equations (12.65) and (12.66) into the region where there is reversed
456
CONVECTIVE FLOWS
flow without encountering any obvious signs of instability in the numerical procedure. This confirms again that separation is, in general, nonsingular for this type of problem. Finally, in Figure 12.14 a plot of the streamlines is presented, which clearly shows the reverse flow region and the fluid entrainment at the edge of the boundarylayer. 10 8
6
0
10
20
30
:t"
40
50
60
Figure 12.14 Streamline plots/or a downward inclination o/the plate (A < 0).
12.6
M i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow a l o n g a n inclined p e r m e a b l e surface
Consider a fiat permeable heated plate which is embedded in a porous medium and inclined at an angle ~ to the horizontal. It is assumed that the nondimensional wall temperature is Tw(x), the nondimensional outer fluid flow velocity is U(x) and the nondimensional mass flux velocity normal to the plate is vw(x). The flow configuration is depicted in Figure 12.15. Under the boundarylayer approximation, the governing Equations (12.1), (12.2) and (12.4) can be written in nondimensional form as follows:
= 7:R
N
0r OT Oy Ox
+
(12.70)
0r OT 02T Ox Oy Oy2
(12.71)
which have to be solved subject to the boundary conditions
r
TT~(x)
o_~ oy __+U(x),
T + 0
on
y=0,
x>0
as
y + c~,
x > 0
(12.72)
FLOW OVER HORIZONTAL AND INCLINED SURFACES
~g
457
:!i!i:
Figure 12.15: Physical model and coordinate system.
Integrating Equation (12.70) across the boundarylayer from y = 0 to y  c~ and taking into account of the boundary conditions (12.72), we obtain
0Oy r = U (x ) • R a
T sin ~ + cos ~~xx
Tdy
(12.73)
In order to obtain similarity solutions of Equations (12.71) and (12.73), subject to the boundary conditions (12.72), Weidman and Amberg (1996) have used the generMised similarity variables for r and ~, given by Equation (3.75) and proposed by Burde (1994). Introducing these variables into Equations (12.71) and (12.73) yields
f ' = flU 4 Ra sin ~o 0  cos ~ tt tt
~
(flTw)x h
#
~x~0 +
O"  /~# (Tw)x f'O /~ (#xf + ~x) 0',
#
(~'y) 0 x (12.74)
(12.75)
h'  0
Tw
with the boundary conditions (12.72) becoming f=_a0 0=1 ,o' ~0 + 0, h  + 0
on as
r/=q0 ~ + cx~
(12.76)
Similarity solutions of these equations are possible when the following quantities are constant:
#
U,
~ Tw #,
fl (~ Tw ) , # x
~ Tw ~ x , #
P Tw (~,~ ) # x,
fl~ ( Tw ) ~ Tw '
flax,,
fl P x
As in the case of a Newtonian fluid as discussed in Section 3.6, Weidman and Amberg (1996) have found that similarity solutions of Equations (12.74)  (12.76)
458
C O N V E C T I V E FLOWS
fall into two distinct categories, namely class I where we have steeply inclined plates, and class II where we have steeply inclined plates for which new similarity solutions are possible. Here we consider the example of a steeply inclined plate: i.e. class I solutions. Setting #  1, the solution of the relevant Equations (12.77) gives c~(x) : g ln(x + 1),
/~(x) = x + 1,
Tw(X)x+l,
1
U(x)
qs x+l

(12 78) "
where g and qs are nondimensional constants. This family of solutions describe the Darcian mixed convection flow over a steeply inclined, downward facing heated plate which forms the upper boundary of a wedge. Equations (12.74) and (12.75) reduce now to the following: f '  qs :1: R a 0 sin ~o
(12.79)
0" + 50' + f'O  0
(12.80)
which have to be solved along with the boundary conditions (12.76) for some specification of 7(x). In this section we consider the exact solution of Equations (12.79) and (12.80) only in the situation in which qs > 0, and this corresponds to planar source flow in a porous medium. If we take the scalings 1
0qsb20(~),
~ ?  q s 5~,
_
qs b2q,
1
 5  b~d,
3'0  qsS~O,
_
1
(Rasinqo) (12.81) then Equations (12.79) and (12.80) reduce to a single equation for the temperature field, namely
~ ' + ~ + ( 1  0 \~ 0   0/
b
(12.82a)
which has to be solved subject to the boundary conditions ~_ql~. ~'0+0
oil as
~~o~. ~+oc
(12.82b)
A
where A c _ ar . qY
Weidman and Amberg (1996) have shown that Equations (12.82)
consist of the dual family of solutions
[
,
0(0)~" = q 1  (1 + q~) exp
(
~"v/6~0
(12.83)
corresponding to radial outflow along a downward facing inclined heated plate forming one boundary of a wedge which is embedded in a porous medium and this has a physically acceptable meaning only for q > 1. This solution is plotted in Figure 12.16 for some values of q with ~0 = 0 and fi"= v~" 5 The branch 1 curves (solid lines) and
F L O W OVER HORIZONTAL AND INCLINED SURFACES
151
459
,
!',, 10 ~\ %. , , ,
1.3, 2, 5
q=l.1,
q = 100, 3, 1 . 0 ~ Q _ _ _ : : : ~ 0
~
0.0
, ,
0.25
:,
' "' "t
..... ~4.~_.
0.5 ^
0.75
I
1.0
o(o) A
,
Figure 12.16: Normalised temperature profiles, ~o(o)' for ~  ~6 and 7o  O. The dotted line represents the limiting solution (12.84), the solid lines indicate the branch 1 solutions and the broken lines indicate the branch 2 solutions.
branch 2 curves (broken lines) in the figure correspond to the solution branches defined by the upper (+) and lower (  ) signs in Equation (12.83), respectively. In the limit q + c~, the two branches merge to the common temperature profile A
~(~)  exp (  ~ )
(12.84)
o(o)
The asymptotic solution (12.84) is also shown in Figure 12.16 by the dotted line. We note that these asymptotic profiles are consistent with a linear asymptotic analysis of Equations (12.82) in the far field, which shows that solutions for 0 (~) exhibit monotonic exponential decay when the discriminant A  (~2 _ 4) 89 1! is positive V~ Weidman and Amberg (1996) have further found for q > 1 that the wall heat transfer coefficient gives monotonic variations along each solution branch over the ranges 4
0'(0) < A
0
These boundary conditions suggest looking for a solution of f ( x , r/) for x > 1.
(13.22)
466
C O N V E C T I V E FLOWS
13.2.3
Numerical
solution
To obtain a solution which is valid for all values of x, Equations (13.3) and (13.4), subject to the boundary conditions (13.5), have to be solved numerically. As for the corresponding viscous (nonporous) fluid case discussed in Section 6.2, Pop and Merkin (1995) have used a finitedifference method in combination with the continuous transformation method proposed by Hunt and Wilks (1981). Thus, introducing the variables 2
1
1
r  xS(l + x)~F(x,~), Equation (13 3) gives 9
oq3F 0r
1 l +
[(~ a
O  x~ (l + x) 89
~),
~_
Y 1 xS(1 + x)~ 1
(13.23)
H  ~r OF and Equation (13.4) then becomes
1~3) + 2
02F F
2
I(0F)2] 3
5(
I(OFO2F

OFO2F) 2
(13.24a) 1 where, to accommodate for the x5 singularity as x + 0, we use ~  x I as the streamwise variable. Then boundary conditions (13.5) become F0,
02F gd = ~ (1 + ~3)~OF  8   (  ( 1 + ~3 ) 89 on OF o~~0 as r ~>0
~0,
~>0
(13.24b)
and the nondimensional wall temperature is given by the expression 1
Ow(x)  ~ (1 + ~3) 3 H(~, 0)
(13.25)
The problem described by the set of equations and boundary conditions (13.24) was solved numerically using a finitedifference scheme as described by Merkin (1969). Details of the numerical procedure are not presented here as they can be found in the paper by Pop and Merkin (1995). The variation of the nondimensional wall temperature 0w(x), as a function of x, given by the numerical solution (13.25), is shown in Figure 13.1. The asymptotic solution (13.22) for large values of x is also included (by broken lines) in this figure. It is seen that Ow(x) increases monotonically with increasing x and there is a strong singularity at x  0. Then, we can see that the asymptotic solution (13.22) gives a good estimate for Ow(X), even at quite moderate values of x with the difference being about 18% even at x = 1, while at x = 2.5 this difference is reduced to approximately 6%. In fact, this agreement can be seen more clearly from Table 13.1, where we have compared the values of Ow(X) given by the numerical solution (13.25) with the asymptotic limits 0~ ) (x) and O(w2)(x). We can see that there is very good agreement between Ow(x) and 0 ~ ) ( x ) u p to ~  1.2 ( x  1.728). However, the difference between Ow(x) and 0~)(x) becomes more pronounced as the value of x
FLOW OVER VERTICAL SURFACES
467
0.8 0.6 0.40.2 0.0
i
o.o
o:5
1:0
1:5
x
2:0
2:5
Figure 13.1" Variation of the wall temperature distribution, Ow(x), with x. The numerical solution (13.25) is indicated by the solid line and the asymptotic solution (13.22) is indicated by the broken line.
1
Table 13.1: Values of ~w(x) as a function of x and ~  x~.
~ Equation 1. (13.25) 0.008! 0.2245 0.064 i 0.3916 0.216 0.5161 0.512 0.6095 1.0 0.6801 1.728 0.7340 2.744 0.7848 4.096 0.8157 5.832 0.8347 8.0 0.8557 15.625 0.8932 27.0 0.9173 42.875 0.9336 64.0 0.9453 ...
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0
..
Equation (13.12)
Equation (13.22)
0: 24 0.3916 0.5161 0.6095 0.6801 0.7341 0.7758 0.8080 0.8303 0.8301
0.5563 0.6624 0.7321 0.7807 0.8162 0.8431 0.8877 0.9146 0.9322 0.9445
increases further, while the values 0 ~ ) (x) give a reliable e s t i m a t e for large values of x, being, for example, only a p p r o x i m a t e l y 1.5% in error at x = 8.0. Finally, F i g u r e 13.2 shows the d e v e l o p m e n t of t h e n o n  d i m e n s i o n a l t e m p e r a t u r e profile O(x, y) for a range of values of x o b t a i n e d n u m e r i c a l l y from E q u a t i o n s (13.24). As expected, these b e c o m e m o r e spread out as t h e value of x increases, w i t h t h e value at t h e wall t e m p e r a t u r e increasing, in line w i t h F i g u r e 13.1, for increasing values of x.
468
C O N V E C T I V E FLOWS
1 . o o ~
_ o
0.75
.008, o.o91, 0.34, 0.85, 1.95, 6.33, 57.1
9(x,y) 0.50 0.25 0.00
0
4
8
Y
12
Figure 13.2 Temperature profiles, O(x, y), as a ]unction of y for several values of X.
13.3
Free c o n v e c t i o n b o u n d a r y  l a y e r flow over a vertical surface w i t h N e w t o n i a n h e a t i n g
We have seen in the previous sections that the free convection boundarylayer flow in porous media is usually modelled by assuming that the flow is driven either by a prescribed surface temperature or by a prescribed surface heat flux. However, Lesnic et al. (1999) have considered a somewhat different driving mechanism for the free convection boundarylayer along a vertical surface in a porous medium in that it is assumed that the fluid flow is set up by the heat transfer from a surface such that the surface heat flux is proportional to the local surface temperature (Newtonian heating), i.e.
OT
0y
:hwT
on
~0,
~>0
(13.26)
where hw is the constant wall heat transfer coefficient. We note that this situation was recently considered by Merkin (1994b) and Merkin and Chaudhary (1996) for the corresponding problem of a viscous (nonporous) fluid. A similar situation to the present problem arises also, as we have seen in the previous section, in conjugate free convection boundarylayer flow over a vertical surface in a porous medium. However, there is, as we will see later, an essential difference between these problems when the solution far downstream is considered. We now introduce the nondimensional variables
x=~,
y:hw~,
U  u c,
v
Lhw~c
O:
Too
(13.27)
where L and Uc are the convective length and velocity scalings which are defined as
F L O W OVER VERTICAL SURFACES
469
follows" gKflT~ g = amuh2,
2 Uc = c~mhwL
(13.28)
Using expressions (13.27) in Equations (11.7)  (11.9), we tions (13.3) and (13.4) and these equations have to be solved conditions r  O, oo _ O) y o~   ( 1 + on  0, 0+0 as y + co,
obtain again the Equasubject to the boundary x
> 0 x>0
(13.29)
The solution procedure for Equations (13.3) and (13.4), subject to the boundary conditions (13.29), follows closely that of the conjugate situation described in Section 13.2. 13.3.1
S m a l l v a l u e s o f x (> 1 it can be seen from Equation (13.58a) that the change of the wall (interface) temperature distribution Ow(X)is of O ((k(1)) 1) and this suggests the expansion
0w  OwO(X) 4r (k(1)) 1
O w l ( X ) I . . .
(13.60)
FLOW OVER VERTICAL SURFACES
477
which when substituted into Equations (13.58) gives OwO(X)  Oo  constant
(13.61)
Thus, the fluid flows in the boundary layers are selfsimilar at this leading order. Now, a consideration of the boundary conditions (13.57b) suggests looking for a solution of Equation (13.57a) of the form
I1
f2
= y_ = F(~) (1  0 0 ) ~ 0~
(13.62a)
where 1
1
 (1  00) ~ ~1 = 0~ ~/
(13.62b)
Substituting expressions (13.62) into Equation (13.58a) and integrating the resulting equation with respect to x, and using the boundary conditions (13.58b), yields 2
2
Oo 
(13.63)
1 + (k(2)) ~ Alternatively, on using expression (13.62) in Equations (13.57) gives 1
II
F "~ + ~F F
F(O)=O,
F'(O)I,
 0
F'+O
as
(~oo
~13.64]
This equation gives F " ( O )  0.444, see Equation (13.16a), so that the average Nusselt number given by expression (13.59) can be expressed as follows: Nu
3
, = 0.sss
(1  e0)
(13.65)
Ra~
It should be noted that the expressions (13.63) and (13.65) represent the leading order terms in the expansion of the solution in powers of (k(1)) 1. It can be seen that 0w0 + 1 and ~R ~ + 0 as k (2) + co, while 0w0 + 0 and ~Nu + 0.888 as k(2) + 0. These two limits correspond to the cases where the thermal resistance of the boundary layers of the two porous media 1 and 2 is negligible and the temperature of the plate is very close to Tloo, or T2cr In the particular case in which the porous media on both sides of the plate are the same, i.e. k (2)  1, then 0w0  1 and   ~ = 0.314. It should be noted that the expression for N u can be further improved by computing more terms in the series (13.60).
478
CONVECTIVE FLOWS
(ii) A s y m p t o t i c limit k (1)  + 0 In this case, the longitudinal heat conduction in the solid plate becomes negligible, and Equation (13.55) reduces to
02r
1 (92//)2
Oy~
(13.66)
k(2) oy~ Yl 0
y20
This relation shows that the heat fluxes from the two porous media are locally equal to each other and thus both must be finite at the ends of the plate. Thus the balance of the convection and conduction in Equation (13.48) implies (1 Ovo, 01) ,'., x 51 for 1 x small and (0~, 02) ', (1  x)5 for (1  x) small, and this suggests the use of the new transformation
r
_
 ~ f~
(~, ~),
_
e~  e~
(~, ~),
~
~:~.~
y~
 v
k'"
'']
W Equations (13.47)and (13.48) now become 03f/ 202f/ o~ + 3 ov~
1 (0f//2 a ~
( 0 f / 02fi 02f/  ~ o~ o~ox~  o~o~i
02f/0f/) 0,7~ o~i
(1368~1
along with the boundary conditions (13.52), namely 
f i ( x i , O)  O,

2_
~of:
+0
as
0w
(Xl) (13.68b)
offi (x2, 0)  x~  (1  x2) ~ 0w (1  x 2 ) ,
0~i
1 .v
(x:, O) = x~  (I  Xl): O~
(xi, O) = k t 2 ) o ~
~7i + ~ ,
(xi, O)
x i>O
(13.68c)
where O~(x) 
O~(x) + x
1
(13.69)
~(1~)~
The average Nusselt number, given by expression (13.54), now becomes dx: Ra~
13.4.2
0~
(13.70)
,:=0
c + 0 w i t h k(1) ~=  0(I)
In this case Ow  O(1) to fulfil the boundary conditions (13.52a,b) for the heat flux at y  +89 The longitudinal heat conduction is again negligible, except in
FLOW OVER VERTICAL SURFACES
479
small regions close to the edges of the wall, and thus the transformation (13.67) remains appropriate. Equation (13.68a) still holds in both porous media, whilst the boundary conditions (13.68b) at ~i  0 take the form f~ 
o,
0~: 
,
o~i

(13.68d)
where
0~: (x)
 1 Ow :(x' 89 ,
O.2(x ) = Ow (x, 89
x:
(13.71)
(1  x ) ~
The solution of Equation (13.68a), subject to the boundary conditions (13.68c,d), determines Ow~,Ow2, Nu and the other flow characteristics in the boundarylayer. There are also two limiting situations to be considered, namely cW >> 1 and ~ > 1, the heat fluxes in both porous media remain finite and from Equation (13.68d) results in that x89 of case (ii).
+ 1  (1  x)89
recovering the limit k (1) + 0
k(:) + 0 (iv) Asymptotic limit ~
Applying this limit to Equations (13.68d), it is readily seen that the heat flux in the porous media tends to zero and therefore Ow (x, 89 + 1 and 0z (x,21) + 0, except in_ smallregions near the lower and upper edges of the plate. The scaled functions f: and f2 should now take the form
.v (k(1) ~ 89
.v (k(1)k(2)) 89
/k(1)~ 89
(k(1)k(2)) 89 (:3.72)
On substituting these expressions into Equation (13.68a) gives
a'"
2 GG"
1 G,,2 = 0
(13.73a)
5
along with the boundary conditions (13.68b,c) which become G(0)0,
G"(0)1,
G'~0
as
~+cr
(13.73b)
Equations (13.73) describe the free convection over a vertical plate with a prescribed constant heat flux in a porous medium, a problem considered by several authors,
480
CONVECTIVE FLOWS
for example Rees and Pop (1995a) and Wright et al. (1996), whose solution has the property GI(0)  1.2947. The nondimensional temperature Os(x, y) inside the plate and the average Nusselt number, for ~k(1) ~ 1, are now given by the expressions 2
1
o~(~,y)y+i
(k(1))SG,
~
[ 1( 1 )
2 ( y+~ + ( k(2)) ~(1~)~ y  ~1 ) ] +...
(o) ~
(13.74) and Nu
_
Ra~ 1
w
k (1)
c2
I ( )2 ( _ 3
1
k (1)
~  ~
2) 1
g
§
G'(O)
(13.75)
k(1)Ra~l 1(ks) 1
It should be noted that, since it can be readily seen that d ~ = c ~ , the leading order term in the expression (13.75) does not depend on the Rayleigh number. Equations (13.57), (13.58) and (13.68) have been solved numerically by Higuera and Pop (1997) using the finitedifference method as described by Trevifio et al. (1996). The nondimensional wall temperature distribution, Ow(X), as obtained from the numerical solution of Equations (13.57) and (13.58), is presented in Figure 13.6 for k (1) = 0.5, 1, 2 and k (2)  0.5, 1. Also included in this figure is the variation of Ow(X) for k (1) + 0, obtained from the solution of Equations (13.68). The streamline and isotherm plots of the hot fluid corresponding to k (1) + 0 are displayed in Figure 13.7 for k (2)  1, and this corresponds to the same porous media on both sides
.0
i
,
,
,
0.8 0.6 0.4 0.2 0 0 ! ....... oo
o4
oo
X
l O
Figure 13.6: Variation of the wall temperature distribution, Ow(x), with x for k (2)  0.5 and 1 at several values of k (1).
F L O W OVER VERTICAL SURFACES
481
y 6
4
2
0
1 SI'tUt o., %\ttUt I %\ttNUl Figure 13.7: Streamlines (solid lines) and isotherms (broken lines) in the boundarylayer of the hot medium from the solution for k (1) ~ 0 and k (~)  1.
of the wall. We observe that as we approach the asymptotic value k (1) ~ c~, which implies that heat conduction along the wall dominates and Ow(x) remains constant, being given by expression (13.63). Figure 13.7 also shows that the longitudinal heat conduction has an important effect, even for moderate values of k (1), rendering Ow(x) almost constant for values of k (1) greater than about 0.5. We observe that the same trends hold for k (2)  1 and k (2)  0 . 5 but Ow(x) increases with k (2). For k (2) >> 1 the problem reduces to that of a uniform wall temperature and is equal to Tlc~, while for k (2) 1 can be obtained from that for k (2) < 1 using the property of the invariance of the problem, see Higuera and Pop (1997), and therefore it is not discussed here. The average Nusselt number given by expression (13.59), as obtained from the numerical solution, is presented in Figure 13.8 for k (2)  0.5 and 1 and the asymptotic solutions (13.65) and (13.70) for large and small values of k (1) are also included (shown by broken lines). It is seen from this figure that the value of N u decreases as k(1) increases, which is to be expected because increasing k(1) implies making the temperature of the solid and each of the fluids the same. Further, Figure 13.9 illustrates the variation of Ow(x) with x on both faces of the plate for k (2)  1. The upper curves correspond to the side facing the hot medium and the lower curves correspond to the side facing the cold medium, respectively.
482
CONVECTIVE
FLOWS
0.55
Nu ~ 0.50 ........... Ra~
,_.__~(2) 0.5 =
0.45 0.40 ..........
k (2) = 1
0.35 0.30 , 0.001
0.{)1
0'.1
i
~0
Variation of  ~ _ with k (1) for k (2)  0.5 and 1. The numerical Ral2 solution is indicated by the solid line and the asymptotic solutions for large and small values of k (1), namely expressions (13.65) and (13.70), respectively, are indicated by the broken and dotted lines, respectively. Figure 13.8:
1.0

0.8 0.6
"~
0.4 0.2 0.0 0.0

0.2
 ,,
0.4


0.6
" X
I
0.8
1.0
Figure 13.9 Variation of the wall temperature distribution, Ow(x), with x for k (2)  1. The upper curves correspond to the side of the plate facing the hot
medium. The solutions for c~k(1) = 0.5, 1 and Wk(1) ~ oo are indicated by the solid, broken and dotted lines, respectively. The asymptotic solutions obtained from Equation (13.74) for k(1)  0 . 1 are indicated by the dotdash lines.
R e s u l t s for t h e t w o  t e r m a s y m p t o t i c e x p a n s i o n (13.74) for k (1~) = 0.1 ( s h o w n by the d o t  d a s h lines) a n d t h o s e for k(1)  ~ >> 1 ( s h o w n by t h e d o t t e d line) are also i n c l u d e d in this figure. We see t h a t t h e r e is a g o o d a p p r o a c h of t h e n u m e r i c a l r e s u l t s to the
F L O W OVER VERTICAL SURFACES
483
asymptotic limits. k(1) Finally, Figure 13.10 displays the variation of N u with ~ for two values of k (2). The numerical results (solid lines), the twoterm asymptotic expansion (13.75) for k(1)
c2 > 1 are all plotted and these show
k(1) very good agreement, in particular for cvvery small.
We note that N u is an
k (1)
k(1)
increasing function of ~, which can be explained by noting that as c~ increases the temperature difference between the sides of the wall decreases, and thus a larger fraction of the total temperature falls from Tloo to T2oo occurs across the boundary layers, leading to larger heat fluxes. From the data shown in Figures 13.8 and 13.10 we conclude that N u is maximum for a value of k (1) verifying the condition c2 ~ k (1) ~ 1. Indeed, values of this maximum are provided by the solution of Equations (13.68). 
0.5 Nu r 0.4 Ra~
0.3 0.2 0.1 0.0 0.01
0:1 k(1)
1
10
c2
Figure 13.10 Variation of N__~u with ~k(~) for k (2)  0.5 and 1. The numerical
na~ solution is indicated by solid lines and the asymptotic expansion (13.75) for ivk(1) small and the asymptotic values for cZ~(1) large are indicated by the dotted and broken lines, respectively.
13.5
C o n j u g a t e m i x e d convection boundarylayer flow along a vertical surface
This model problem is based on a vertical rectangular plate of length I and thickness b, which is embedded in a porous medium and over which is flowing a fluid with a uniform velocity U~. The outside surface of the plate is maintained at a constant temperature To, while the ambient fluid is at a uniform temperature Too, where
484
CONVECTIVE FLOWS
To > Too (aiding flow) or To < Too (opposing flow). We assume that the boundarylayer approximation holds in the convective fluid and that the solid plate is thin relative to its length, i.e. b ~ 1, so that the axial heat conduction in the plate can be neglected. Consequently, the temperature profile in the plate can be assumed to be linear and therefore the heat flux from the plate is given by Equation (6.5). The boundarylayer Equations (11.149)  (11.151) can be written in nondimensional form as followsOu
Ov
o; +

(13.76)
o
(13.77)
ul+AO O0 020
O0 +

oy2
(13.78)
and these equations have to be solved subject to the boundary conditions v0,
o_A0= 0 _ 1 0+0
on as
oy
y0, y + co,
x>0 x>0
(13.79)
where nondimensional variables have been defined as x

~
~ ~
y
Pe 89
~
~ U = uoo,
1 v Pe~~ Uoo '
v
O
TToo To  Too
(13.80)
where A is the mixed convection parameter, which is defined by Equation (11.154) with R a based on To Too. We note that Equations (13.76)  (13.79) involve only the single parameter A. Further, the physical length 1 of the plate does not appear in the nondimensional variables and therefore I only enters the solution through the range of the validity of the solution given by expression (13.6). This basic conjugate mixed convection flow problem described by the boundarylayer Equations (13.76)  (13.79) has been solved by Pop et al. (1995a) for both aiding (A > 0) and opposing (A < 0) flow cases. The strategy consists of providing asymptotic solutions for small and large values of x, as well as numerical solutions of these equations over a wide range of values of A.
13.5.1
S m a l l v a l u e s o f x (
0 x>0
(13.84)
These boundary conditions suggest a solution, for small values of x, of the form OO
Oo
f  ~ fj(~?)x89 j=0
h
E hj(~?)x89 j=0
(13.85)
where the coefficient functions are given by f~  1, 2hg + foh~o  f~ho  0 fo(O)O, h~(O)1, ho+O as 77+oo
(13.86a)
i1
/'

~h~_,,
2h" +/0h~ (i + 1)f~h~  ~ [(j + 1)hi/' j  ( j + 2)yj+,h~_j_~] j=0
fi(O)  O,
h~(O)  hi_l(O),
hi + O as
r/+oo
(13.86b) with i ~> 1. It is worth mentioning that Equation (13.86a) describes the ordinary forced convection flow along a flat plate with a constant heat flux rate in a porous medium. The exact solutions for f0, h0, fl and hi may be obtained in terms of the complementary error function as follows: for]
~0  ~ o~ (~) §
~
2
ex,
(~)
fl  A [1 + ~1r / e x p ( '72 7)  (1 h~1 ~ 2 )erfc (~ )]
~
 (_1 § ~)[(1 § ~)er~c (~) ~ e x ~
(~)]
(13.87) The higher order terms can also be determined analytically but this process is very laborious. The nondimensional wall temperature, for x > 1)
For this case we introduce the variables
r  ~Y(~,~,
o  h (z,~,
Y
77
i X2
(13.89)
so that Equations (13.76)  (13.78) become
?' 1+~ 2h" + f h ' 
2..
(13.90)
~'~
(13.91)
with the boundary conditions (13.79) becoming fO,
x 89 h+0
on as
~0, 7/+oo,
x>O x>O
(13.92)
The solution of these equations for large values of x is sought of the form
?  fo (~ + ~8971 (~ + ~  ~ h (~ + . .  ho (~ + x  ~ (~ + z  ~ ~ (~ +...
(13.93)
where the coefficient functions are determined from the following three sets of equations f;  1 + ~h0, 2hg + f0h~  0 (13.94~) fo(0)0, ho(0) = 1 , ho+0 as r/+cxD f {  Ah l ,
fl (0)  0, F~  ~Hk,
2h'1~ + f oh'1  0
h~ (0)  h0(0),
hi ~ 0
as
r] ~ oo
2H~' + f0H~ + 2~kf~Hk + (1  2~k) h~Fk  0
Fk(0)0,
Hk(O)O, Ilk~O
as
~/~oo
(13.945)
(13.94c)
Again, it is worth noting that Equations (13.94a) are equivalent to Equations (11.156) for the nonconjugate mixed convection over an isothermal vertical flat plate in a porous medium. It is easily verified that the first eigenvalue is 3'1 = 1 and therefore its corresponding eigensolution is given by F1  A0 ( f 0 
~),
H1
A0~
(13.95)
for some constant A0. Hence the usefulness of asymptotic expansion (13.93) is confined to terms up to O ( x  l ) . The nondimensional wall temperature, for x >> 1, is given by
O~)(x) 
1  hl(O)x 1 4...
(13.96)
FLOW OVER VERTICAL SURFACES 13.5.3
Numerical
487
solution
To obtain a solution of Equations (13.76)  (13.791 which is valid for all values of x, Pop et al. (1995a) have used the method of continuous transformation of Hunt and Wilks (19811. This suggests the transformation r
0~(l+~21~H(~,~),
~=7, q
~x~
(13.971
and then Equations (13.771 and (13.781 transform to 1
F '  1 + ~ (1 + ~2)~ H
2H" + FH !
I +~ 2 F ' H

~ (F 'OH 0~
(13.981
H 'OF) ~
(13.99/
~>0 ~>0
(13.100)
and the boundary conditions (13.791 reduce to 1
F0,
H'~H(I+~2)~ H~0
on as
~=0, ~ + c~,
It should be noted that Equations (13.981  (13.1001 reduce to Equations (13.821 (13.84) for small values of x and to Equations (13.90)  (13.92) for large values of x. The nondimensional wall temperature is now given by 1
Ow(~) ~ (1 + ~2)~ H(~, 0)
(13.101)
Equations (13.98)  (13.100) have been solved numerically for different values of ~ and ), (both positive and negative). It was found that for )~ >/  1 , and for sufficiently large values of ~, then Ow(~) ~ 1. As expected, the larger the value of )~, the longer it is before the asymptotic value of Ow(~) is attained. The calculations also showed that, in contrast to the viscous fluid case, see Pop et al. (1996b), at large distances from the leading edge the flow does not separate from the plate when ~/>  1 . This situation is also in contrast to all other nonconjugate problems in porous media, where there is an opposing flow condition, see Ingham et al. (19851 and Rees and Riley (19851. However, for ~ <  1 the fluid always separates from the plate. This can be seen from Figure 13.11, where the variation of 0w(~) with ~, as given by Equation (13.1011 (shown by solid lines) is presented for various values of ), <  1 . Also included in this figure (by broken lines) is expression (13.881 for small values of ~. The end points of the curves correspond to the values of ~  ~s()~), say, where the numerical solution terminates and therefore the boundarylayer separates from the plate. It is seen from these figures that the small ~ solution is a good approximation to the numerical solution but it cannot exactly predict the value of ~s(),). Figures 13.11(c,d) suggest that ~s(),) + 0 as )~ +  c ~ .
488
CONVECTIVE FLOWS
(a)
(b) 1.0
1.0/
0.8
/
..s
.
0.8
~~ "'" 6 terms 1 term, "" 3 terms
/
0.6
0.6.4, 1.3, 1.2
0.4
0.4
0.2
0.2
0.0
(c)
o
i
~
~
o.5! I
,
i
/
I,
0.3
I
/
/
/ / "
0.04 
'
/
,
A =,50
'
0.02 
f,, ]
,
.2"
/
o12
o13
I
I/'
/
0.01 
0.00 0.00
i
o11
9
0.03 
f/~A=10,5,3,2
0.0
100/
e~(r
/u//
I I I I
0.0
0.05 
/
I
0.2
o
(d)
0.4
0.1
0.0
~
o14
0'.5
0.01
0.02
0.03
Figure 13.11: Variation of the wall temperature distribution, Ow(~), with ~ for (a) A  1.1 and (bd) different ranges of values of A. The numerical solution (13.101) is indicated by the solid line. In (a) various terms in the small ~ expansion (13.88) are indicated by the broken lines. In (bd) the 6term small ~ expansion (13.88) is indicated by the broken lines.
Further, we can see from Equation (13.77) t h a t u0
on
0
1 A
(13.102)
and the numerical solution of Equations (13.98)  (13.100) showed that this condition holds for 1 0~ (~s) iX (13.103) Table 13.2 gives the values of .~s(A) and the corresponding values of O~,(~s) given by
FLOW OVER VERTICAL SURFACES
T a b l e 13.2:
EZ3
489
Values of ~s(A) and Ow (~s) for values of A <  1 . Series (13.88)
N u m e r i c a l (13.101)
~
U
S
0.915 0.838 0.771
3.768 1.818 1.213 0.930 0.762 0.423 0.232 0.123 5.74 x 10  2 2.77 x 10  2 1.18 x 10  2 1.08 x 10 2
0.715 0.666 0.499 0.332 0.199 9.98 x 10  2 4.98 x 10  2 3.31 x 10  2 1.97 x 10  2
5.39 x 10  3
0.97 x 10 3
3.875 1.300
0.805 0.738 0.681
0.960 0.805 0.475 0.270 0.147 7.00 x l 0  2 3.30 x 10  2 2.20 x 10  2 1.35 x 10  2 6.60 x 10  3 1.30 x 10  3 6.50 x l O  4
0.633 0.590 0.443 0.295 0.177 8.86 x 10  2 4.43 • 10  2 2.95 x 10  2 1.77 x 10  2 8.86 x 10  3 1.77 x 10  3 8.86 xlO 4
m
1.670 1.195 0.930 0.780 0.450 0.255 0.136 6.35 x 10 2 3.10 x 10 2
2.05 x I0 2 1.20 x 10 2 6.10 x 10 3
1.07 x 10  3
1.98 x 10 3
1.19 x 10 3
5.46 x 10  4
9.95 x 10 4
5.90 x 10 4
0.909 0.833 0.769 0.714 0.666 0.500 0.333 0.200 0.100 5 . 0 0 x 10  2 3.33 x l 0  2 2.00 x l 0  2 1.00 x 10  3 2.00 x l 0  a 1.00 x 10  3
Equation (13.101). The values o f  ~ are also included in this table in order to check the validity of expression (13.103) as to where the flow separates. It is interesting to conclude from this table that the (critical) function t(),), defined as t(A) = A ~s(X)
(13.104)
is a decreasing function of A and it appears as though there is a finite limit for t(A) as A approaches negative infinity, say t~ The values of t~ are presented in Table 13.2 where, for example, for ),  100,  5 0 0 and 1000, we have t~ 0.539, 0.537 and 0.536. Therefore, it appears that 0.536 is very close to the correct asymptotic value of t~ as A +  c o . This leads us to introduce the approximate function ti(A) = A ~ ) ( A ) for i ~> 1 (13.105) as A +  c ~ , where ~!J) denotes the position of the flow separation given by Equation (13.88) for small values of x when using j terms in order to satisfy the condition (13.105). The values of~J)(A) for j  1, 3 and 6 terms are also included in Table 13.2. Again, it can be observed that the values of tj(A) for A +  c ~ are given by t~
t~
t~
(13.106)
These values were checked by Pop et al. (1995a) by solving a system of ordinary differential equations obtained from Equations (13.86) for small values of ~ when A was scaled out by use of the transformation fi  AiFi(~7),
hi
)~iHi(rl)
(13.107)
490
C O N V E C T I V E FLOWS
Thus the following values for tjo were obtained t~ t~0.626,
t~ t~
t~ t~  0.594
(13.108)
as A +  o c , and these values are in very good agreement with those given in expression (13.106). It is clear from expression (13.108) that the larger the number of terms that are used in the series (13.88), the better does the limit t~ approach the exact value t o ~ 0.536 obtained from the numerical solution of Equations (13.98)  (13.100). This convergence can be also observed in Figure 13.12, where ~s(A) is shown as a function of  A as obtained from the numerical solutions of Equations (13.98)  (13.100) and shown by the broken line. Also shown (by solid lines) are ~1), ~3) and ~6), as given in Table 13.2. It is seen that the curves are virtually parallel when  A is large and the full numerical solution is better approached by the inclusion of more terms in the series (13.88), which is valid for small values of ~. Furthermore, we can see that these curves are almost straight lines for large values o f  A , illustrating the convergence of the functions t(A), tl(A), t3(A) and t6(A). Therefore, it may be postulated that the small x solution of this problem for A < 0 is a good approximation almost up to the point of separation of the boundarylayer from the plate. 1
0
o
hO 0
1 2 3 4
logl0(A) Figure 13.12" Variation of loglo ~s(A) as a function of loglo(A ). The numerical solution is indicated by the broken line and the I, 3 and 6term expansions from series (13.88) (in decreasing order) are indicated by the solid lines.
Chapter 14
Free and m i x ed c o n v e c t i o n from cylinders and spheres in p o r o u s media 14.1
Introduction
Heat transfer by free, forced and mixed convection flow from cylinders and spheres embedded in fluidsaturated porous media have received much attention because of their fundamental nature as well as many engineering applications. Cylindrical and spherical geometries arise in power plant stream lines, industrial and agricultural water distribution lines, buried electrical cables, oil and gas distribution lines, storage of nuclear waste, solar collectors, etc. The early studies which have considered the problem of immersed cylinders assumed the surrounding medium to be purely conductive, see Eckert and Drake (1972). However, the assumption that a pure conduction model can be used to calculate the heat losses from an immersed cylinder (or pipe) may not be valid for high permeability saturated soil. If the surrounding medium is permeable to fluid motion, the temperature difference between the cylinder (or pipe) and the medium gives rise to a free convection flow. As a result, the total heat transfer from the cylinder (or pipe) consists of both conduction as well as convection. Generally, the contribution of free convection to the heat loss from immersed cylinders is as large, and in some cases larger than, the contribution of conduction. Fernandez and Schrock (1982), Farouk and Shayer (1988) and Facas (1995) were the first to have studied numerically the steady free convection from a circular cylinder embedded in a semiinfinite, saturated and permeable porous medium, the surface of which is assumed to be horizontal and permeable to fluid flow. Cheng (1984) considered the case of an isothermal cylinder which is embedded in an infinite fluidsaturated porous medium. Approximate closedform solutions were obtained for the local as well as the average Nusselt number by apply
492
CONVECTIVE FLOWS
ing boundarylayer approximation. The results obtained by Facas (1995) in terms of the local Nusselt number are in excellent agreement with the boundarylayer solution obtained by Cheng (1984) and also with the experimental results presented in terms of the average Nusselt number by Fernandez and Schrock (1982). In the present chapter, we review the stateoftheart of the steady free and mixed convection flow from cylinders (horizontal and vertical) and spheres placed in an infinite fluidsaturated porous medium. Although geometrically simple, these configurations play an important role in practice but, more importantly, they are instructional tools to deepen our understanding of the transport mechanisms which could be generalised to complex geometries. The problems description will tacitly show the possible topics which need more attention and focus on the future research aspects that could be pursued, such as, for example, problems related with flow instabilities, transition to turbulence and condensation phenomena in porous media. Of particular interest in this chapter is the sphere which seems, at least in certain aspects, to be somewhat lagging behind its cylindrical counterpart as far as theoretical development is concerned.
14.2
Free c o n v e c t i o n from a h o r i z o n t a l circular c y l i n d e r
There has been an increasing interest over the last few years on convective flow on heated bodies embedded in a porous medium. In many papers it has been assumed that the Rayleigh number is very large and therefore the boundarylayer approximations have been employed. The first similarity solutions for free convection boundarylayer from a horizontal circular cylinder immersed in a porous medium has been obtained by Merkin (1979), while Cheng (1982) considered the case of mixed convection flow. In contrast, several papers have investigated the situation when the Rayleigh number is small, e.g. Yamamoto (1974) and Sano and Okihara (1994) have obtained asymptotic solutions for the free convection about a heated sphere in a porous medium when the Rayleigh number is very small. However, their solutions have the defect that the pressure does not remain bounded at large distances from the sphere. Hardee (1976) used an integral method to study free convection boundarylayer flow about an infinitely horizontal isothermal cylinder placed in a porous medium of infinite extent and predicted that the local Nusselt number varied as 1 N u = 0.465 R a : (14.1) for R a )~ 1, where Ra is the Rayleigh number based on the diameter of the cylinder. However, using the boundarylayer theory, Merkin (1979) has obtained the result 1
N u = 0.565 Ra~
(14.2)
In order to test these theories, Fand et al. (1987) performed an experimental investigation on the heat transfer by free convection from a horizontal circular cylinder
F R E E AND MIXED C O N V E C T I O N FROM CYLINDERS AND SPHERES
493
in a porous medium which consists of randomly packed glass spheres saturated by either water or silicone. They showed that the overall range of the Rayleigh number can be divided into two subregions, called 'low' and 'high', in each of which the Nusselt number behaves quite differently. It was demonstrated that the lowRa region corresponds to Darcy flow but in the highRa region the flow is nonDarcy and the flow model of Forchheimer (1901) is more appropriate. In the limiting case of very high Rayleigh number, in which the boundarylayer equations may be assumed to hold, Ingham (1986a) modified the theory of Merkin (1979) in order to deal with the nonDarcian effects and found 1
N u ..~ R a ~
(14.3)
g * c~m
where D is the diameter of the cylinder and K* is the inertial or Forchheimer coefficient defined in Equation (II.9). This shows that the Nusselt number is proportional to the Rayleigh number raised to the power 88for nonDarcy flow whereas it is to the 1 power for Darcy flow. In a very thorough numerical and analytical paper, Ingham and Pop (1987) have investigated the steady free convection about a heated horizontal circular cylinder of radius a embedded in a fluidsaturated porous medium. The temperature of the cylinder is Tw and that of the ambient medium is Too, where Tw > Too, see Figure 14.1. The governing Equations (II.1), (II.2) and (II.5), expressed in cylindrical polar coordinates (r, O), can be written in nondimensional form, see Ingham and Pop (1987), as follows: 1 0~b 1 02r 0T OTcosO 02r ~ t    sin O ~ Or ~
r~r
r 2 002
Or
O0
r
::iiiiiiiii!:i!ii!iiii!iii::
':i?i??i??l?!ii?i!?::
Figure 14.1: Physical model and coordinate system.
(14.4)
494
CONVECTIVE FLOWS
02T 0 r
+
107' r~r
102T r 2 002

=
Ra ( Or OT r Or O0
Or OT ) O0 Or
(14.5)
which have to be solved subject to the b o u n d a r y and s y m m e t r y (at 0  ~r) conditions r
T1
u~0,
v~0,
r
T+0
0O0T _ 0
on
rl,
0> 1. Further, we note that the expression (14.82c) agrees with expression (14.70b) for the purely forced convection limit (A 0 is held at a constant value of T~ (> Too), whilst for ~ < 0 the cylinder remains
FREE AND MIXED CONVECTION FROM CYLINDERS AND SPHERES

515
a
U~,T~ v
w
W
v
Figure 14.12: Physical model and coordinate system.
unheated, i.e. at the temperature Tc~. We assume that both the Reynolds and Rayleigh numbers of the flow are large so that the boundarylayer approximation is valid. Thus, the density stratification within the boundarylayer induces a vertical flow, whilst the horizontal flow produces a motion within the boundarylayer around the cylinder. This gives rise to a threedimensional boundarylayer flow. Under the DarcyBoussinesq approximation, the governing equations of this problem in cylindrical polar coordinates (~, 9, ~) are given by 0~
~
~++
10~
10~
~
+
0~ 0~ 0~
O'T
0r ~ OT
05 ~
=o
(14.83)
=0
(14.84) g K ~ 071 v
0~ { 02T
(14.85) 10T
1 02T 02T~ F2 002 t~ ] ~~ + ~ N +~~zz = ~m ~~2 + ~~ ~
(14.86)
where u, v and ~ are the fluid velocity components along the ~, 0 and ~ directions, respectively. Outside the boundarylayer (outer flow) we introduce the following non
516
CONVECTIVE FLOWS
dimensional variables r  a,

2
TToo AT

z  a,
T=
a ~ = v R e ~,
'
~ = ~
a
~,
a
~=
~
(14.87)
On substituting expressions (14.87) into Equations (14.83)  (14.86) and assuming that R e is very large (Re + c~), we obtain 0~ ~ i 0~ 0~ Or + r + r00 +  ~ = 0
1 0~
~
0Y
r O0
r
Or
(14.88)
= 0
(14.89) N
_
O5 Or ~ OT
OT
u~r + 
O~ O~ OT
_
+
OT
(14.90)
Or
(14'91)
o
where A (> 0) is again the mixed convection parameter as defined in Equation (11.154). The associated boundary conditions of Equations (14.88)  (14.91) are given by ~=0 vcos0 + g s i n 0 ~ 0 ~ sin 0  ~ cos 0 + 1
_
T + 0, + O,
}
on
r=l,
all 0,~
as
r + cr
all O,
(14.92)
The solution of Equations (14.88)  (14.91) which satisfies the boundary conditions (14.92) is given by
( 1 ) 1~~
cos0,
~
(1) l+~ff sin0,
~0,
T=0
(14.93)
It is seen that the solution (14.93), for the inviscid flow outside the boundarylayer, satisfies the boundary condition on the cylinder for Y < 0 and hence only the boundarylayer which forms on the cylinder for ~" > 0 should be considered. The governing equations for this layer can be obtained by using the following variables: 1
y_(r_l)(GrPr)~
A
1
,
_(GrPr)5 uu " A
,
w=~,
~
~ z~,
TT

(14.94)
Equations (14.83)  (14.86) then reduce to Ou
Ov
Ow
y u + 00 + ~z  0 ~:C~V
= 0
Oy Ow 07' = Oy Oy OT OT OT 02T u~y + v  ~ + w OZ = Oy 2
(14.95) (14.96) (14.97) (14.98)
F R E E AND MIXED C O N V E C T I O N F R O M CYLINDERS AND SPHERES
517
which have to be solved subject to the boundary conditions u=0, w+0,
T=I
T~0,
v~2sin0
on
y=0,
all 0,
z>0
(14.99a)
as
y+c~,
all 0,
z>0
(14.99b)
where the conditions (14.99b) arise from matching v, w and T with the outer solution (14.93). On using the boundary conditions (14.995), Equations (14.96) and (14.97) give w = T,
v = 2 sin 0
(14.100)
and the Equations (14.95) and (14.98) reduce to
Ou OT cgy f ~ + 2cos0 OT U~y
o00T+ T + 2 sin 0:=
0
(14.101)
OfO ' ___~= c92T Oy2
(14.102)
along with the boundary conditions u0,
T=I T + 0
on as
y0, y + oo,
all 0, all 0,
z>0 z :> 0
(14 103) "
Before numerically solving Equations (14.101)  (14.103), we look for asymptotic solutions which are valid for small and large values of z. 14.5.1
Small
values
of z (((1)
The most appropriate variables to use for small values of z are as follows: 1
u  z~U (r/,0, z),
T = T (r/,0, z),
y
r/ i
(14.104)
z~
which on substitution into Equations (14.101) and (14.102) gives
1
V  ~r/T
) OT ~
OT
+ 2z sin0~ +
OT 02T zT Oz = Or/2
OU 0'1' 1 0 T + z~  ~ + 2z cos 0  0 071 Oz 2
(14.105) (14.106)
and the boundary conditions (14.103) become U=0, T1 T + 0
on as
rl0, 7/+ cx),
all O, all 0,
z>0 z > 0
(14.107)
518
C O N V E C T I V E FLOWS
Inspection of Equations (14.105)  (14.107) suggests looking for a solution for z 1)
At large distances along the cylinder the solution becomes independent of z and Equations (14.101) and (14.102) become
Ou
0~ + 2cos0  0
OT
OT
u ~y + 2 sin 0 O0
=
(14.110)
02T Oy2 =
~
which have to be solved subject to the boundary conditions (14.103). (14.110) gives u = 2ycos 0
(14111) Equation (14.112)
and in order to solve Equation (14.111) we write (14.113) where To satisfies the ordinary differential equation T~' + 2~T~ = 0
(14.114a)
along with the boundary conditions T0(0)=l,
T0+0
as
~+~
(14.114b)
F R E E AND MIXED C O N V E C T I O N F R O M CYLINDERS AND SPHERES
519
Hence To = erfc ~ 14.5.3
Large values ofz
(~1)
on00
(14.115) ~
From Equations (14.112) and (14.115), we see that as z + oc for 0 = 0 ~ (the forward generator of the cylinder) then u +  2 y ,
T + erfcy
(14.116)
Assuming that the approach to this solution is of the form
u =  2 y + ealZu I (y) + . . . (14.117)
T = erfcy + e a~zT 1 (y) + . . .
where al is an unknown constant then Equations (14.101) and (14.102) become ?.tll  a l T 1
(14.118)
 0
2 _y2 v~Ule  alerfcy  T i
2yT~
(14.119)
and the boundary conditions (14.103) reduce to ul(0)0,
TI(0)0,
TI+0
as
y+oc
(14.120)
This eigenvalue problem has been solved numerically by Ingham and Pop (1986b) and it was found that the smallest value of a l is given by a l  6 . 8 8 6 6
(14.121)
Thus, N u on 0 = 0 ~ has the form
Nu Pe51
=
2 ~r~
+ A l e 6"8866z
(14.122)
for z >> 1, where A1 is an unknown constant. In order to match the analytical solutions, as presented above for small and large values of z, the threedimensional boundarylayer Equations (14.95)  (14.98) should be solved numerically. However, it was found most convenient, because of the asymptotic forms of the solution, to use Equations (14.105) and (14.106) for z > 1. A very efficient finitedifference method, which is described in detail in the paper by Ingham and Pop (1986b), was used. Comparing values of N u obtained numerically with those given by the expression (14.122) at z  0.4 suggests that A1 ~ 0.7255 on 8  0 ~ Figure 14.13 shows the
520
CONVECTIVE FLOWS
4 Nu Pe~
7
3
1 term ~~/Z
4 terms and numerical
2
1
o.o
0:1
Figure 14.13: Variation of ~
0.2
0:a
z
with z on 0  0 ~
0:4
0:5
The numerical solution and
Pe~
the terms from the series (14.109) are indicated by the solid lines, the asymptotic 2 value  ~ as z ~ oo is indicated by the broken line and the approximation ~2 +
0.7255 e ~ss66~ is indicated by the dotted line.
variation of N u with z (small) on 0 = 0 ~ obtained numerically and also as predicted by E q u a t i o n (14.122) with A~ = 0.7255 and by the series (14.109). T h e asymptotic value N u ~  ~2 as z ~ (x~ is also included in this figure. We can see t h a t E q u a t i o n (14.122) is a good a p p r o x i m a t i o n to N u for values of z >~ 0.2. In order to illustrate t h a t the asymptotic solution presented in the previous section is also very accurate for all values of y, Figure 14.14 compares the variation of T1 (y) as a function of y obtained by solving Equations (14.118)  (14.120) numerically for various values of z with the expression T1 (y)  (T  erfc y) e 6"s866z (14.123) for z >> 1. We see t h a t this expression approaches well the value of T1 (y) d e t e r m i n e d numerically and this confirms the validity of the present method. Further, Figure 14.15 shows the variation of N u as a function of z on 0 = 180 ~ (backward generator of the cylinder) obtained numerically and given by the series (14.109). T h e analysis presented in Section 14.5.2 shows t h a t Re}

~
cos
(0)
(14.124)
for z >> 1. Thus, on 0 = 180 ~ we have that N u + 0 as z + c~ and this is confirmed by the numerical calculations. We also note t h a t the 4 term expansion (14.109) is a very good a p p r o x i m a t i o n for N u up to z ~ 0.5 and it may be used even up to z ~ 1.0 with good accuracy.
F R E E AND M I X E D C O N V E C T I O N F R O M C Y L I N D E R S AND S P H E R E S
o4]/~
T,(y)
' 02]/I ' I/~'
/~~z0.1,
0.1[11
~\", \ \ \\
o oVo
i
0.2, 0.3
.....
3
y
Figure 14.14: Profiles of T1 (y) given by expression (14.123) (broken line) and also determined numerically (solid lines).
3 Nu Per
7
2
1
3 terms
4 terms
4__/ 0. 2 terms 0.0
015
~
1~0
1.5
Figure 14.15: Variation of p~ Nu with z on 0  180 ~ The terms from the series (14.109) are indicated by the solid lines and the numerical solution is indicated by the broken line.
521
522
C O N V E C T I V E FLOWS
We also present in Figure 14.16 the variation of N u with 0 for several values of z. The asymptotic solution (14.124), which is valid for large values of z, is also presented in this figure. It is seen that for z > 0.4 the asymptotic solution is being approached when 0 ~< 90 ~ As the value of z increases, so does the approach to the asymptotic solution for larger and larger values of 0. The fact that Equations (14.101) and (14.102) are parabolic in the 0 direction automatically implies that the asymptotic solution (14.124) is approached faster for decreasing values of 0. z =0.02
3 Nu
P J,
z = 0.04
2
z ~ O_:l
,,
0
i
4'5
,
~l
1
9'0 o (o) i55
Figure 14.16: Variation of ~_ with O. The numerical solutions are indicated by Pe 2
the solid lines and the asymptotic solution (1~.12~), valid as z + oe, is indicated by the broken line.
We can also calculate the average Nusselt number N u from the cylinder using the expression 1 =
Pe~
dOdz
7rz
(14.125)
~Y y=0
It was found from the numerical calculations that N u may be well approximated by the expression Nu 0.01798 = 0.71835 + (14.126a) Pe~ z for z >> 1, and Nu 1
1 (0.44375 + 0.29258 z 2)
(14.126b)
Pe~.
for z 0, with x being the distance measured along the plate. A very consistent theory based on this assumption has been developed by Ingham and Brown (1986), and Merkin and Zhang (1992) who have also derived the eigensolutions appropriate to the departure from the initial unsteady (onedimensional) solution to the final steady state (twodimensional) solution. The recent monograph by Nield and Bejan (1999), and the review articles by Bradean et al. (1998a) and Pop et al. (1998b), give extensive references of the topic of unsteady convective flow in porous media and a number of specific examples, ranging from natural processes to those of technological importance. Motivated by the importance of the transient nature of the transport phenomena in porous media, we shall review here some external convective flow problems in porous media which involve transient responses for bodies such as flat plates, circular cylinders and spheres. The new solutions are presented in detail and a general discussion indicates a variety of estimates of the boundary data which facilitate the identification of the exact numerical solutions.
15.2
Transient free convection b o u n d a r y  l a y e r from a vertical fiat plate s u d d e n l y h e a t e d
flow
Consider a vertical semiinfinite fiat plate which is embedded in a fluid saturated porous medium of uniform ambient temperature Too. We assume that at the time t < 0, both the plate and the porous medium are at the uniform temperature Too. Then at t  0 the temperature of the plate, or the heat flux at the plate, is suddenly increased to the constant value T w ( > Too), or qw > 0, respectively, and are maintained at these values for t > 0. The unsteady boundarylayer equations are, from Equations (II.1), (II.2) and (II.5), given by 0~
0~ + ~_. uy
0
(15.1)
UNSTEADY F R E E AND MIXED C O N V E C T I O N gK~ (TT~) v 02T
OT
OT
535
OT
(~5.2)
~  ~ + ~ ~ + ~ ~  ~.~ ~
(15.3)
which can be written in nondimensional form as 0r = o Oy 0r 020 OX Oy Oy 2
00 0r ! OT Oy OX
(15.4) (15.5)
where the nondimensional variables are defined as 7
o~mRa ) ~, al 2
x = ~ l' 
y   Ra89y ~,
89 r  olmRa r
0
T  Too T*
(15.6)
Here T* = Tw  Too when the plate is suddenly heated to a constant temperature T~ and T *  Ra 89 ( k ~~) when the heat flux at the plate is suddenly changed to the constant value qw. Having in view the physical model considered, the initial and boundary conditions of Equations (15.4) and (15.5) are given by r
00
r
for
~ Ts (steady state flow) , where ~  ~Raz. y Using the expression for qw(x) given by Equation (15.17), we can express the local Nusselt number as follows" Nu 1
= 0.4318
(15.18)
Ra~ which is about 1.2% lower than the exact value of 0.444 based on the similarity solution found by Cheng and Minkowycz (1977) for the corresponding problem of steady state, free convection boundarylayer flow over a vertical plate in a porous medium. Therefore, we can conclude that the agreement between the two solutions is very good. It is worth noting that 0, as given by expression (15.16), is identical to the exact solution for the onedimensional heat conduction equation in a semiinfinite porous medium with an initial temperature Too when its bounding surface is suddenly raised
538
CONVECTIVE FLOWS
to a temperature Tw. Thus, during the initial stage when the leading edge effect is not felt, the solution for the temperature, or for the fluid velocity field, as given by Equation (15.4) is independent of x. Therefore, we identify the lower region (7 ~ 7s) in Figure 15.1 as the transient onedimensional conduction region and on the other hand, Equation (15.17) is valid in the upper region (T ~> Ts). Equations (15.11) have been also solved by Cheng and Pop (1984) using the method of integral relations and the variation of 5(T) with T is shown in Figure 15.2. It is seen that initially 5(T) increases and then it remains constant when the steady state flow has been attained. ,,,
,,,
X=I X = 0.5
/ / /
'
'
_
_
I///
///_
X=0.1
s /
i
/ /
T
Figure 15.2: Variation of the boundarylayer thickness, 5(T), with T in the case of a sudden change in wall temperature. The results based on the method of characteristics and the method of integral relations are indicated by the solid and broken lines, respectively.
Shu and Pop (1998) have studied the transient heat exchange between a vertical flat plate of finite thickness b and the free convection boundarylayer over the plate which is embedded in a porous medium. It is assumed that at a given time t :> 0 the right hand side of the plate is suddenly subjected to a uniform heat flux, namely
qw  ) % ) +
)qll  k s  Oy   k l Oy
(15.19)
whilst the left hand side of the plate is thermally insulated, see Figure 15.3. The governing equations are Equations (15.1)  (15.3) along with the energy equation in the solid plate OTs
02Ts
Ot = (~s O~2
(15.20)
U N S T E A D Y F R E E AND M I X E D C O N V E C T I O N
539
q~ (~>o)
~yO "~ T~
""
~
Porous Medium
"
Figure 15.3: Physical model and coordinate system.
On introducing the nondimensional variables T 
t~
Y
X  bRa ~ T f Too T* '
Of 
,
,
Os ~
a fRau,
Ts Too T* '
Ow
v =
v
Tw Too T*

(15.21)
where T*  bey and the Rayleigh n u m b e r is based on qw Equations (15.1)  (15.3) 
kf
and (15.20) can be written as follows:
(~V
au
Ox
+ ~yy  0
(15.22) (15.23)
u  Of OO f
OO f
OOf 020f +v= Oy Oy 2 OO~ Or
=
(15.24)
02Os
(15.25)
Oy 2
and these equations have to be solved subject to the initial and b o u n d a r y conditions u0, vO,
v=0,
0f=0,
Of  Os  Ow(x,r) Ol+O
00~ =0 Oy kOOs
Oy
00/
Oy = 1
Os=O
on
y0
as on
y ~ cx) y1
on
y=0
for
r < 0,
all x, y
for
T~>0,
X~>0
(15.26)
540
CONVECTIVE FLOWS
where the modified conjugate parameter F is defined as (15.27)
F = aC~s
c~I
To obtain a solution of this transient conjugate free convection problem we use the KgrmgnPohlhausen integral method, as used by Lachi et al. (1997). Thus, on integrating Equations (15.24) and (15.25), and using the boundary conditions O0f ay + 0 and oo, Oy _+ 0 as y + co, we obtain
/ooOO, O/oo,
(oo,)
fOG
1~rdY (OOs~ ~Y J y=o
1181 (15.29)
Further, we assume a secondorder, KgrmgnPohlhausen temperature profile in the fluidporous medium and in the solid plate with the constraints such that the boundary conditions, Equations (15.26), hold. Then we have, 
(1
Os  ~ (1  ~Ow) y2 + 88(1  ~Ow)y + ew
(15.30)
Substituting these expressions into Equations (15.28) and (15.29), we obtain C 0 (aGo)+
10
(aO~) 
0 ( 2 ~ )
Or Ow +  ~
2
0w
(15.31)
1 ( 2 )
 ~
1 60w
(15.32)
along with the boundary conditions (15.26) which reduce to =0,
Ow=O
on
T=O
or
x=O
(15.33)
For the steady state case ( o _ 0), we take 5(x, O)  6o and Ow(x, O)  Ow. Thus, Equations (15.31)  (15.33) give 1
6o(x)  (20x) ~,
Owo(x)
1 3
(15.34)
Therefore, 60 differs from that of an infinitely thin flat plate given by expression (15.12b). Equations (15.31)  (15.33), which are hyperbolic partial quasilinear differential equations, with two characteristic curves, were integrated numerically using the method of characteristics. The characteristic equations are given by
UNSTEADY FREE AND MIXED CONVECTION
541
so t h a t the wave speed in the p o r o u s m e d i u m is given by
9(2 + kS) Ow(X,t) 5r(4 + 3k5)
(15.36)
Ow(x, t)
is illustrated in Figure 15.4 for
The interface temperature distribution
F  1, 5 a n d 10 w i t h k  1, 5 a n d 10. These figures show t h a t a l t h o u g h the value of Ow(x, t) increases c o n t i n u o u s l y w i t h b o t h increasing values of T a n d x, its slope exhibits a d i s c o n t i n u i t y at ~s, where the h e a t t r a n s f e r is s u d d e n l y changed. This d i s c o n t i n u i t y can be a t t r i b u t e d to t h e presence of an essential singularity in the governing E q u a t i o n s (15.22)  (15.25). We also note t h a t Ow(x,t) increases
(a)
(b) 1.2
"',_~,~ 1.0
1.4
x=0.5
f
0.8
f
0.6  f
~
t = 0.5
1.2
x  0.2
1.0
x=O.1
0.8
.
.
.
.
.
0.6 .
0.4
.
t=0.1 .
.
.
0.4
0.2 0.0 0.0
.
t = 0.2
0.2 0.0,0.0
i
o:1
'
o:2
' i '
I"
0.3 t 0.4
0.5

0[2
014
o.'6
x
0:8
(c) .8 ,,
0.6 r
,
,,,
= 1, 5, 10 0.4 0.2
'~
.0
0.0
0.1
"I'
0.2
"~ "
I
'
0.3 t 014
0.5
Figure 15.4 Profiles of the interface temperature, O~(x,t), for (a) F  1, k  10 as a function of t at several values of x, (b) F  1, k  10 as a function of x at several values of t and (c) x  0.1, k = 10 as a function of t at several values of F.
1.0
542
CONVECTIVE FLOWS
continuously with time and for large time approaches the corresponding steady state value OwO(X), as given by expression (15.34). The case of a plate of infinitely small thickness (b  0) was also treated by Shu and Pop (1998), and they found that the results were identical to those found by Cheng and Pop (1984) for the case of a vertical surface whose heat flux is suddenly changed at time t  0. The general situation when the temperature of a vertical surface which is embedded in a porous medium is suddenly raised from a temperature Tcr to a value that is proportional to x TM for time t ~> 0 has received a very detailed treatment by Ingham and Brown (1986). This work was extended by Merkin and Zhang (1992) who considered the case when the surface heat flux is proportional to x m for t >~ 0. Therefore, Equations (15.4) and (15.5) should now be solved along with the initial and boundary conditions r
00
for
r 0
~
X TM
(VWT)
~~0,
T>~O,
x~>0
x~>0 (15.37)
15.2.1
Variable wall temperature
To reduce the number of independent variables in Equations (15.4) and (15.5) from three to two, the following new variables are introduced rnT1
r = x 2 f(r/, z ) ,
rJ
m1 yx
2
,
T

(15.38)
tx m1
so that Equation (15.4) reduces to
o=
of
( 5.39)
On substituting Equations (15.38) and (15.39) into Equation (15.5), we obtain 1(1m)T~~
OVOT +
(1m)TOT
2
f
O~2 + m
N
= 0Y3
(15.40 ) and the initial and boundary conditions (15.37) in the VWT case reduce to fO fO, ~
o/_1 o,
for
zO
(15.40b)
The variables ~/and T are the most appropriate ones to use for studying the final decay to the steady state solution, and by setting o _ 0 in Equation (15.40a) leads
UNSTEADY F R E E AND MIXED C O N V E C T I O N
543
to the steady state similarity Equations (11.23). However, in the initial period of the flow development, the boundarylayer grows as though the wall was infinitely long with the effect of the finite leading edge (at x = 0) being felt only at a later time. In other words, the solution of Equations (15.40) at T  0 describes only the initial phase of the development of the flow past a vertical semiinfinite flat plate in a porous medium since it does not satisfy the boundary conditions at x = 0 and therefore contains no information about the leading edge. This is analogous to the situation in the problem of the impulsively started flat plate studied analytically by Stewartson (1951, 1973) and numerically by Hall (1969) and Dennis (1972). Consequently, more suitable independent variables to use for small values of ~ are as follows: ~=
~ 2r7
and
T
(15.41a)
with the independent variable being given by 1
f  2T~F(~,T)
(15.41b)
Using the variables defined in expression (15.41), Equation (15.40a) becomes 03F
0r 3
[
2~ +
4mTF  4(1  m)T 20F] 02F 
4mT ( O~~) 2
[
 4T 1 
OF] 02F (1  m)'ro~T Or
(15.42a) =0
which has to be solved subject to the initial and boundary conditions (15.40b) which reduce to F=0 F  0,
OF a~
= 1
for
T0
(15.425)
At ~  0, Equation (15.42a) reduces to F o !I
+ 2~F;' = 0
(15.43a)
where F(~, 0)  F0(~) for all values of m, and the boundary conditions (15.42b) may be written as Fo(0)0, Fg(0)I Fg~0 as ~+c~ (15.43b) The solution of Equations (15.43) is given by F(~  erfc (~ Fo  ~ erfc (~ +
1 ( l  e  C 2)
(15.44)
544
C O N V E C T I V E FLOWS
We have seen in Section 11.3 that Equations (11.23) have a solution only for m >  1 , with the solution becoming singular as m ~ mc _ ~. 1 Therefore, for the numerical solution of Equations (15.40) and (15.42) there are two cases to consider, namely m :> 1 and _ 1 < m < 1, respectively. m>l i)2 F
Since for m > 1 the coefficient of the term ~ remains positive, Equation (15.42a) has been numerically integrated by Ingham and Brown (1986) using a stepbystep procedure in T until the steady state solution (11.23) was obtained. Hence, starting from the solution (15.44), which is valid at r = 0, Equations (15.42) were solved until T  1, and then Equations (15.40) were solved from ~ = 1 onwards. It was found that the solution tends to the first steady state solution (~" + c~) as obtained in Section 11.3, and in no circumstances did the fluid velocity or temperature become negative. The approach of the unsteady solution to its steady state form can be found by looking for a solution of the form (15.45)
f (77, ~)  f (r/) + T ~g(r/) + . . .
where f(r/) is the solution of Equations (11.23) and ), is a positive constant to be determined. On substituting the expansion (15.45) into Equation (15.40a) we obtain g
,t
Jr
[l+m 2
l + m g, 2 f [(m1)7+2m]f'g'+
(1m)v
] f,g
0
(15.46a)
which has to be solved subject to the boundary conditions (15.40b) which reduce to (0)  0,
g~+0
(15.46b)
g'(0) = 0
as
rl+ce
It may be easily verified that for ~,  2 ~
3
(15.47)
m1
the solution of Equations (15.46) is given by g(~7)  f '
/o
9
f ~~ exp
(/o) 1+ m 2
n
f d~
dr/
(15.48)
  ~1< m ~ l 02F In this case the coefficient of the term 02~0r in Equation (15.42a) becomes negative for a value of m within the range 1 < m < 1. Thus, for a given value of m,
U N S T E A D Y F R E E AND MIXED C O N V E C T I O N
545
Equations (15.42) can be solved numerically up to a particular value of z, say r*, after which point the numerical m e t h o d breaks down. This solution at time ~  r* is then expressed in terms of the steady state variables rl and T and the method used by I n g h a m and Brown (1986) to match the steady state solution with that which is valid at T = ~*, given by Equations (15.42), was employed using a variation of the numerical method first proposed by Dennis (1972). In order to do this it is convenient to write Equation (15.40a) in the form 05"
09v
0~7   ~ + PO~
025 .
mflZ'2 = q OT
(15.49)
where .T = ~~, p _ l +2m f _ ( l _ m ) Z q = 1  (1m)T~~
(15.50)
The b o u n d a r y conditions for Equation (15.49) are, from Equation (15.40b), that f: 0 and ~  1 at r ]  0, whereas ~~ o l _ 0 at r/  r]oo (a large value of r / t h a t may be varied). The numerical solution of Equation (15.49) is described in all its details in Ingham and Brown (1986) and therefore it is not repeated here. Additionally, they have presented a very detailed analysis of the final decay of the unsteady to the steady state solution as T + C~ for _ 1 < m ~ 1 where it was demonstrated that this decay is exponential in nature.
(a)
(b) 1.0
1.0
0.s
~
0.6 0.4
0.8
0.6 '~ I lll\ \~, 7=O.Ol, o.o4, o.1,
0.0. o
i
~7 g
0.4

6
0.0 0
1
2
3
4
Figure 15.5" Reduced temperature profiles, ~ (r/, v), as a ]unction of r1 f o r (a) m  1 and (b) m = 4. The steady state solution as T + oc is indicated by the broken line.
546
CONVECTIVE FLOWS
OI at Figure 15.5 shows the distribution of the reduced temperature profiles ~~ various values of ~ for m  1 and 4. We note from this figure that the steady state solution is approached very rapidly but the larger the value of m, the earlier the steady state is achieved. The rate at which this steady state solution is reached is determined by the value of 7 as given by expression (15.47), which shows that the algebraic decay of the unsteady solution to the steady state solution is very fast when m + 1+. However, it appears that the steady state solution is reached near
(a) %
(b) 1.0~\ ,, 0.8
1.0
~'
~
o.8
"
0.4 l l/k \ \ \"," = 0.01, 0.04, 0.21 " ~ \\ i i :815
0.4 "ill \ \ \ ~'\ ~= 0.01, 0.05, 0.25, 0.2 1Ill \ \ t~ \ \~ ~ . 0.5, 2 9,50.
o.o
o.o
0
i
89
3
4
(c)
~
.
0
.
1
.
2
.
3
.
4 rl 5

6
ZI,.x.\\ 1.0 ~ ~
~\ \ it, oo
r = 0.01, 0.05, 0.25, 0.5,
0.0, o
Figure 15.6" R e d u c e d m  O, (b) m = 0.2
~
t e m p e r a t u r e profiles,
a n d (c) m is i n d i c a t e d by the broken line.
0.425.
~
r/ ~

g~(y,T),al as
a function
of rI f o r (a)
T h e s t e a d y state s o l u t i o n as r ~
oo
UNSTEADY F R E E AND MIXED C O N V E C T I O N
547
T ~ 1 but this is very difficult to verify. o f with 77 at various values of 1 for Further, Figure 15.6 shows the variation of ~~ m = 0,  0 . 2 and 0.425 and it can again be seen that the steady state solution is achieved very quickly with the larger the value of m then the earlier this solution is obtained. This observation is confirmed by the results presented in Figure 15.7, which shows the variation of the reduced heat flux at the wall, i.e.  ~02 f (0, T) with T for some values of m. In conclusion, these results clearly show that a smooth transition from the unsteady to the steady state solution takes place for all the values of m for which a steady state solution exists. 3.2
~
2.4
m = 0.425, 0.4,  0. 2, ~ 0, 0.25, 1, 2, 4
_
I
1.6 0.8
0.0 0.0
0.5
1.0
1.5
2.0
T
2.5
Figure 15 97: Variation of the reduced wall heat flux, ~~2 ~ (0, r) with T.
15.2.2
Variable wall heat flux
This situation has been treated by Merkin and Zhang (1992), where the variables r 0, rl and ~ are now defined as followsm+2 3 f(~7, ~') ,
r
O  x
2,,,+~Of 3
0~7 ,
7?  y x
,:
3 '
r = tx
2(.,:)
(15 51)
3
"
so that Equations (15.4) and (15.5) become 2
1 5(1 

Of] m)~~
02f cgrlt:9~
[m 3______22 f _ ~2( 1 
f
m)~" ~Tf] 02 O~2
+~2m+l (Of) 2 =
o3f Or]3
(15.52 )
548
CONVECTIVE FLOWS
and the initial and boundary conditions (15.7) become f=0 f  0, ~on2 = O_EF_~ 0
for
1
077
r oo is indicated by the broken line.
UNSTEADY F R E E AND MIXED C O N V E C T I O N
551
state solution.
15.3
T r a n s i e n t free c o n v e c t i o n b o u n d a r y  l a y e r flow over a vertical plate s u b j e c t e d to a s u d d e n c h a n g e in the h e a t flux
Consider a vertical fiat plate embedded in a porous medium of ambient temperature, Too, and assume that the general transient arises from a sudden change in the level of energy input flux on the surface of the plate, i.e. a steady input heat flux qw~ is changed at time t  0 to a new steady level qw2 and is maintained at this value for > 0. Under this assumption, Equations (15.1)  (15.3) have to be solved subject to the initial and boundary conditions 0,   O~
~0,
0"1" __
qw2
T + Too
k,~
TToo
for
t0,
~>~0

on as
y0 y + c 0, the nondimensional stream function f and the nondimensional temperature 0 are defined according to
f(v,~),
r  v~ (~)5 (~)
0(~, ~) =
where
T  Too
T*
(15.59a)
~(~(~))~
(15.59b)
1
,
2 
~
3
~
T*


km
and Rax is the local Rayleigh number based on the flux qw~. The substitution of expressions (15.59)into Equation (15.2) gives 9  ~ and Equation (15.3) becomes
Oaf ~(it2rO])
Or]~ 

~
02f
b~O~_+
(
Of) o2f
2f2r~
0. 2
(Of) 2 ~
0
(15.60a)
which has to be solved for r > 0 subject to the boundary conditions (15.58) which reduce to f ( 0 , T)   0,
02f (0, T)

~ ~077
qw2 " n
q~ as
(15.60b)
r]~oo
At time T = 0 the flow is steady and hence f(r], 0) = f0(~]), say, so that, from Equation (15.60a), f0(r]) satisfies the steady (outer) boundarylayer equation f
oI I I
+ 2fof~'  f~2 _ 0
(15.61a)
552
CONVECTIVE FLOWS
which has to be solved subject to the boundary conditions (15.58), namely f0(0)  0, f~t(0)   1 f ~  + 0 as ~+oc
(15.61b)
For ~ ~< 1 there exists an inner boundarylayer which is described by Equation (15.60a) and outside this layer the flow remains with the initial steady boundarylayer profile as given by Equations (15.61). Since the appropriate scale of the in1 dependent variable z/ is, for small 7 (inner layer), ~~, we introduce the following variables f   T F ( ~ ' T) ' ~__ rl 1 (15.62) 2~~ in the inner layer. Equation (15.60a) then reduces to
103F
IOF
g0~ 3
40s
1( 5) 02F ~2  ~  + r ~ ~ + ~
5 0 F ) O2F
1(3
~T~~
0~ 2 " 0
(15.63a)
which has to be solved subject to the boundary conditions (15.60b), which reduce to
02F
E(0, ~)  0,
0~ 2 (0, 7)   4 n
(15.635)
The solution in this growing inner layer is taken to match the outer steady boundarylayer, which at small values of r/( 0, both the time of the d e v i a t i o n a n d the time at T = T* reduce as A increases. It was found that as A +  1 , where  1 < A < 0, the time at which the local minimum point is achieved and the time interval over which the ~ =  1 solution is traced both increase, whilst the forward integration approach still breaks down at ~* ~ 1, as predicted by the equation T*  
1
for
A:>O
1
for
A 1. Figure 15.24 shows the instantaneous streamlines for Ra = 50 at times t = 1, 3, 6, 10, 15 and 50. In each plot, the lefthand half of each figure is for the C W T case, whilst the righthand half is for the CHF case. It can be seen that the flow fields for the two cases considered are quite different from each other, with the fluid flow in the vicinity of the sphere at a given time t for the C W T case being much stronger than that for the CHF case. In the early stages, the fluid flow is mainly confined to the vicinity of the sphere, whilst at later times, due to the convection from the sphere, the fluid flow spreads outwards and upwards. From about time t = 10, the flow pattern very close to the sphere does not change very much, as can be observed from Figures 15.24(a,e,f), but there is a large difference far away from the sphere, see Figures 15.24(e,f). The steady state may never be reached, but, after a long time, the fluid flow near the sphere approaches its steady state behaviour. In order to demonstrate how the heat conduction develops with time, the isotherm line T = 0.2 for Ra = 1, 5, 10, 50, 100 and 200 at times t = 1, 3, 6, 10 and 15 is plotted in Figure 15.25. Again, the lefthand half of each figure is for tim C W T case, whilst the righthand half is for the CHF case. It is seen that, as for the streamline plots, the convection of heat is stronger for the C W T case than it is for the CHF case. For relatively small values of Ra, for example Ra = 1, even at time t = 15, the isothermal line T = 0.2 still appears approximately circular, as can be seen from Figure 15.25(a). As the vahm of Ra increases, convection starts to become important and for Ra >1 50 we can see a very clear cap of the plume at about t  10, which travels upwards as the convection process continues. For
UNSTEADY
(a)
FREE
AND MIXED CONVECTION
4
,
(b)
579
4
O
4
(c)
!
4
!
4
4
0
4
(d)
.
0
4
4
4 j~ ..........
"''0.09
0.05'~"
0.03
O
0
! oo; ~4
(e)
_
.
.
.
.
.
4
.
0
(f)
..~3.~2
4 j 4
0
4
41
4
/
/
/ I 0
" \ , "~\
I
4
Figure 15.24: Streamlines for R a  50 at various times, namely (a) t = 1, (b) t  3, (c) t = 6, (d) t  10, (e) t  15, and (f) t  50, for the cases of constant temperature (lefthand half) and heat flux (righthand half).
580
CONVECTIVE
(a)
(b)
(c)
(d)
(e)
(f)
FLOWS
) .
.
.
.
.
F i g u r e 15.25: Isothermal line T = 0.2 for t = 15, 10, 6, 3 and 1 (ordered from the top surface of the sphere) when (a) Re = 1, (b) Ra = 5, (c) Ra = 10, (d) Ra = 50, (e) Ra = 100 and (]) Ra = 200.
UNSTEADY F R E E AND MIXED C O N V E C T I O N
581
R a = 1, 5 and 10, the time at which the cap of the plume appears may be greater
than at t = 50 and this value is the largest time for which the calculations have been performed. Figure 15.26 illustrates the variation of N u with t for some values of R a and the small time solutions (15.152) for R a  50 and 200 are also included in this figure. We can see that these small time solutions give good approximations to the full numerical solution obtained from Equations (15.141) and (15.142), or (15.144) and (15.145). In addition, it is observed that the value of N u settles down very quickly after t = 5 and this suggests that the local heat convection has reached its asymptotic steady state value. However, it is interesting to note that, for R a = 10, 50, 100 and 200, the value of N u decreases from its value at time t = 0 and reaches a minimum value which is below its steady state value and then it starts to increase slightly towards the steady state value. However, if R a is sufficiently small, then N u monotonically decreases towards its steady state value and the same situation occurs for the corresponding problem of a viscous (nonporous) fluid, see Section 9.9. 320"
240" NU
160
~\\x~ "',,
Ra =
1, 5, 10, 50,
"',,,
100, 200 .
80 0
~
0.1
~ 0.5
.
.
.
.
" ..... 1
t
5
10
50
Figure 15.26: Variation of the average Nusselt number, N u , with t for some values of Ra. The numerical solutions are indicated by the solid lines and, for the cases Ra  50 and Ra  200, the small time solutions (15.152) are indicated by the broken lines.
Figure 15.27 shows the variation of N u with R a at t  50, together with the steady state results obtained by Yamamoto (1974) for small values of R a and by Pop and Ingham (1990) for very large values of R a (boundarylayer approximation). It can be seen that the numerical results obtained by Yan et al. (1997) are in very good agreement with those reported by Pop and Ingham (1990), and this agreement becomes better as R a increases, and with those of Yamamoto (1974) for small R a . It is interesting to note that although the results of Yamamoto (1974) are valid for
582
CONVECTIVE FLOWS
100
,,
80
Nu 6040200 0.01
9
,
' 0.1
,
,
1' Ra
,
1'o
,
1;o
Figure 15.27: Variation o/ the average Nusselt number, Nu, with Ra for t = 50. The numerical result obtained by Yah et al. (1997) is indicated by the solid line and the steady state results obtained by Yamamoto (197~), for small values o] Ra, and by Pop and Ingham (1990),/or large values o/ Ra, are indicated by the broken and dotted lines, respectively.
Ra ~ 1, these results can be used for values of Ra up to about 2, see Figure 15.28. T h e variation of the surface t e m p e r a t u r e Tw (0, t) as a function of 0 at different values of t is shown in Figure 15.28 for the case of constant surface heat flux. The first thing we observed is t h a t the m a x i m u m and m i n i m u m surface t e m p e r a t u r e is always at the top (0 = 0 ~ and the b o t t o m (0 = 180~ surface of the sphere, respectively. Further, it is seen t h a t for small values of Ra the surface t e m p e r a t u r e settles down much more slowly t h a n t h a t for large values of Ra with Tw(O, t) reaching its asymptotic value at a b o u t t = 25 and 15 for Ra = 10 and 50, respectively, and for Ra = 50 and 100 the wall t e m p e r a t u r e profile at t = 50 is almost identical to that at t = 15. We also observe t h a t the steady state wall t e m p e r a t u r e for small values of Ra is higher t h a n t h a t for larger values of Ra. In order to see details of the t e m p e r a t u r e distribution at large values of t, Figure 15.29 shows the steady isothermal line at t = 50 for Ra = 50 in the vicinity of the sphere for b o t h the C W T and CHF cases. It is observed from the C W T case t h a t even at t = 50 the heat front is still very strong. The small bulge behind the front represents the oscillation phenomenon observed in the numerical c o m p u t a t i o n and this oscillation starts at a b o u t t = 25. On the other hand, in the C H F case, the heat is confined in a much smaller region t h a n t h a t in the C W T case and no oscillations in the t e m p e r a t u r e are detected up to a time t = 50. As we have mentioned before, it was observed during the c o m p u t a t i o n t h a t for a given value of Ra, the t e m p e r a t u r e vertically above the sphere along the line 0 = 0 ~ starts to oscillate as t increases. The time at which the oscillation starts varies with
UNSTEADY FREE AND MIXED CONVECTION
(a)
583
(b) 0.7
0.9
o.8
~,~_~0.6
0.7
0.5
0.6
t = 3, 6, 10, 0.4
0.5
0.3
0.4 84 0.3
t = 3, 6, 10,
,'
,
~_
0.2
~
7r
!
,

0
,
, 
,
8
(c) 0.~ ~.~0. 0.4 1 0"3"1~
t = 3, 6, ,o, 9 15, 25, 50
0.2 0.1+ 0
w
71"
Figure 15.28: Variation o] the wall temperature distribution, Tw(9, t), with 9 in the case of constant wall heat flux (CHF) when (a) Ra  10, (b) R a  50 and (c) R a  100.
the value of Ra; the larger the value of R a , the earlier the oscillation starts. For R a  100, this oscillation s t a r t s at a b o u t t  10 in the c o n s t a n t t e m p e r a t u r e case, see Yan et al. (1997). We believe t h a t this oscillation p h e n o m e n o n is p r o b a b l y due to a physical instability of the p r o b l e m a n d f u r t h e r investigations are required to clarify this behaviour. It should be p o i n t e d out t h a t in a similar investigation of the free convection N e w t o n i a n (nonporous) flow from a sphere by Riley (1986), oscillations in t e m p e r a t u r e vertically above the sphere along 0  0 ~ are also r e p o r t e d for large values of t.
584
CONVECTIVE
(a)
FLOWS
14
10
(b)
2
4
,
6
,
4
Figure 15.29" I s o t h e r m s f o r R a = 50 and t surface t e m p e r a t u r e ( C W T )
2
4
,
b
.
4
50 in the cases of (a) c o n s t a n t and (b) c o n s t a n t surface heat f l u x ( C H F ) .
Chapter 16
N o n  D a r c y free and m i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow in p o r o u s m e d i a 16.1
Introduction
The classical theory of heat and fluid flow within fluidsaturated porous media has been developed from studies based on Darcy's law, which is essentially appropriate only in very low permeability porous media, or when the Reynolds number based on the pore diameter of the media is small. However, the nonDarcy flow situation prevails when the Reynolds number and characteristic fluid velocity of the medium becomes large. Forchheimer (1901) proposed an additional term to the Darcy equation, which is proportional to the square of the fluid velocity, to account for the inertia of the fluid flow as it makes its way through the porous medium. This pioneering work was followed by many proposals for the correct description of nonDarcy flows, such as the work by Ergun (1952), Ward (1969), etc. As a continuing effort towards a complete understanding of the transport phenomena in porous media, a number of studies have considered various nonDarcian effects on forced, free and mixed convection flow in porous media. In particular, Vafai and Tien (1981) have analysed the boundary and inertia effects on forced convection flow and heat transfer characteristics from a flat plate which is embedded in a porous medium. An attempt to obtain a similarity solution for nonDarcian free convection boundarylayer over a vertical flat plate was first made by Plumb and Huenefeld (1981) using the model proposed by Ergun (1952). The limiting conditions where the Darcy term is negligibly small, namely the Forchheimer (1901) model, has been studied by Bejan and Poulikakos (1984) for the free convection boundarylayer over a vertical flat plate, by Ingham (1986a) for isothermal twodimensional and axisymmetric bodies of arbitrary shapes and by Nakayama et al. (1990) for nonisothermal
586
CONVECTIVE FLOWS
bodies immersed in a porous medium. The model of Ergun (1952) was also employed by Vasantha et al. (1986) for a vertical frustum of a cone and by Lai and Kulacki (1987), Kumari et el. (1990a, 1990c), Rees (1996) and Hossain and aees (1997) for a horizontal flat surface in a porous medium in order to investigate the combined effects of the Darcy and the inertia terms. Detailed mathematical relationships on the nonDarcian flow phenomena in a porous media can also be found in an excellent book and a review article by Nakayama (1995, 1998).
16.2
Similarity solutions for free convection boundarylayer flow over a nonisothermal body of arbitrary shape in a porous medium using the DarcyForchheimer model
Consider a heated twodimensional or axisymmetric body of arbitrary shape which is embedded in a fluidsaturated porous medium of ambient temperature Too, as shown in Figure 16.1, where x and y are the coordinates measured along the body surface and normal to it, respectively, and r*(x) is a function which describes the surface of the body. It is assumed that the surface of the body is heated to a variable temperature Tw(x). Under the Forchheimer (1901) model and the Boussinesq approximation, the governing steady boundarylayer equations are obtained from Equations (II.1), (II.3a) and (II.5) and can be written as, see Nakayama et al.
(~990), 0 (r'u)+ 0 (r* ~) oy
bZ
! [[
[ [
~g , , ~ ' ,"~
..~.. !:.. i i . .
(16.1)
 0
Fluid Temperature Profile
%0 BoundaryLayer
!
Figure 16.1: Physical model and coordinate system.
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
u2 = f l x / ~ (T b*
OT
OT
587
Tc~)gx

(16.2)
02T
 (Xm U~X + vac9y2 oy where
(16 3) 1
g~g
1
~
(16.4)
and r* = 1 for a twodimensional body and r* = r(x) for an axisymmetric body, respectively. The appropriate boundary conditions for these equations are
TTw(x)
v0,
on as
T + T~
y0, y + c~,
x~>0 x >/0
(16.5)
According to the technique proposed by Nakayama and Koyama (1987) for the Darcy flow model, the following variables are introduces to solve Equations (16.1) (16.3) 1
r
amr*
( Ra~')' I
1
f(x,q),
TToo
ATwe(x, ~),
y
/,66/
r]  
X
where AT,, = T w ( x )  T ~ ,
Rax  gx~/KflATwx2 *
I(x) = f o (ATw) ~ g ~r*2 dx (16.7)
2
~
b (~m
1
(AT, o) ~ g~xr*2x
with Raz being the modified local Rayleigh number for the Forchheimer model and r is the stream function defined as
u
1 0r , r* Oy
1 0r r* Ox
~ 
(:6.8)
In terms of the new variables, Equations (16.2) and (16.3) are transformed to f,2
: n,) f O' 
0" + (~ 
_
(16.9)
0
n l f ' O = I x ( f ' OO~x o ' Of
(16.10)
and are subject to the boundary conditions (16.5) which become f=0, 0=1 O+0
on as
r/=0, 77+c~,
x>~0 x>~0
(16.::)
588
CONVECTIVE FLOWS
where primes denote differentiation with respect to ~ and the function n(x) is given by n(x)  d (lnATw) (16.12) d (lnx) Similarity solutions of Equations (16.9) and (16.10) are possible when the lumped parameter n I remains constant. One such obvious case is an isothermal body, which has been considered by Ingham (1986a). To seek other possible similarity solutions, Nakayama et al. (1990) have written the parameter nI in the form
nI
d (lnATw) f : (ATw) ~ d~
d (ln )
(ATe)
5
(16 13)
where 1  j~0x g~r*2dx
(16.14)
Equation (16.13) suggests that similarity solutions of Equations (16.9) and (16.10) are possible when the wall temperature distribution ATw varies according to
ATw ~ ~m
(16.15)
Equation (16.13) then gives 2m
nI 
(16.16) 2+5m and Equations (16.9) and (16.10) can be reduced to the following ordinary differential equation 2+m 0'f~ ~ 2m t?~ = 0 (1617a) 0 " + 2 ( 2 + 5 m ) Jo 0~ds2+5~ which has to be solved subject to the boundary conditions 0=1
on
~70
0+0
as
~? ~ c ~
(16.17b)
It should be noted that for rn  0, Equation (16.17) reduces to that derived by Bejan and Poulikakos (1984) for the nonDarcy free convection boundarylayer flow over a vertical flat plate which is embedded in a porous medium. Finally, the local Nusselt number, Nu, may be evaluated as follows: 1
Nu
1
Xqw __ ( 1 + 2 m ) ~ [d(ln~)] kmATw (lnx) ~ Ra~1 [0'(0)]
(16.18)
The present similarity theory with AT~ given by Equation (16.15) has been applied by Nakayama et al. (1990) to the free convection boundarylayer over a vertical flat plate, a vertical cone pointing downward, a horizontal circular cylinder and a sphere. The similarity variable ~ defined by Equation (16.14) for these geometries is given by
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
589
(i) Vertical flat plate (m  n)
~g~l (x) ~ (ii) Vertical cone pointing downward ( m 
(16 19a)
~) ft sin2 (if)
(16.19b)
where ft is the half angle at the apex of the cone. (iii) Horizontal circular cylinder  g89
~0r (sinr 1 de
(16.19c)
 g89
f0r (sine) ~ de
(16.19d)
(iv) Sphere
where l denotes a length scaling, such as a plate length, a cone slant height and the radii of a cylinder or a sphere, whilst r is the peripherical angle measured from the T lower stagnation point such that T  r The nondimensional wall heat flux, q~ (~), for the cases mentioned above can be defined as follows: 1
qw (~) =
kmAT w
(16.2o)
b, a 2
where ATref is the wallambient temperature difference at the trailing edge of the plate or cone, or at the upper stagnation point of the cylinder or sphere. It is easily shown that q~ (~) varies as (i) Vertical flat plate ( m  n) qw. (x)  F ( n )
(ii) Vertical cone pointing downward ( m 
q; (~1 = v h f
g
(x) [
5m2
(16.21~)
~)
T
5rn2 4
cos88ft
(16.21b)
590
CONVECTIVE FLOWS
(iii) Horizontal circular cylinder 1
, (r qw
5
sln~ r 1 6 2
)  F ( m ) ( K.,~, sinfesin~ r 1 6 2
(16.22a)
9 1
f o s,n~ r 1 6 2
when the wall temperature varies as sm~ r 1 6 2 ~r . ~
= ATref
(16.22b)
f0 sln2 r
(iv) Sphere
( sin3 ) ( sin0
, q~ (r
 F(m)
f~r sin~ r 1 6 2
m
(16.23a)
f o sin~ r 1 6 2
when the wall temperature varies as sm~ r 1 6 2 =
~
.
~
(16.235)
fo sm~ r de
ATref with F ( m ) defined as F(m)

Niz [d(in ~) ] ~ A [d(lnx) ] Ra~
(16 24)
~
The variation of q~ (r given by Equations (16.22) and (16.23) for a horizontal circular cylinder and a sphere, with r is shown in Figure 16.2. It is seen that the heat flux becomes infinite at r = 0 (the lower stagnation point of the cylinder or sphere) for m = 0 (isothermal cylinder and sphere). Itowever, the heat fluxes vanish at both r = 0 and r = 7r (the upper stagnation point of the cylinder or sphere) for any values of m ~ 0 (nonisothermal bodies). It is worth mentioning that Fand et al. (1986) have carried out an experimental study for the free convection over an almost horizontal isothermal circular cylinder and reduced the experimental data by considering only the averaged Nusselt number (based on the diameter of the cylinder, D = 21  0.0145 m), which in the above theory proposed by Nakayama et al. (1990) can be written as follows: Nu 
[( )1/0 4D b*
~ 1 ~
7r
q~ (r
de
](
gK/3ATref D a2m
) ( 88 ~
gK/3ATref D a2
)1 (16.25)
and for the case of water and 3ram diameter glass spheres we have K = 5.6 • 10 9 m 2 and b*  0.64 m 1. The values of N u are given in Table 16.1 along with the
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
(a)
591
(b) 1.0]
1.0
m=O
0.8
m
q~(r

0.8
2
q~(r
0.6
0.6
0.4
0.4
0.2
0.2
0.0
m=O
~2
0
r
0.0
7r
,
,
~
'
0
,
,,
=,
~
Figure 16.2: Variation of the local surface heat flux, q~ (r horizontal circular cylinder and (b) a sphere.

,
r
~"
7]"
with r for (a) a
Table 16.1: Values of the average Nusselt number, N u , for a horizontal circular cylinder which is embedded in a porous medium. c~mv 62.24 125.00 197.90
.
.
.
.~ 3.029 2.521 2.205 .
.
.
.
Equation (16.25) 10.4 11.8 12.8 .
.
Fand et al. (1986), Fand et at. (1986), Experimental Equation (16.26) 7.30 4.46 9.83 6.32 11.40 7.95 .
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
experimental data of Fand et al. (1986) and the values as obtained by Fand et al. (1986) from using the Darcy law model, namely 1
C~m~
(16.26)
It can be seen that the theoretical results of Nakayama et al. (1990) are in reasonably good agreement with the experimental data of Fand et al. (1986). We also observe that the results based on the Darcy law model, given by Equation (16.26), are much smaller than those for a nonDarcy fluid flow model.
592
16.3
CONVECTIVE FLOWS
N o n  D a r c y m i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow along a vertical flat plate in a porous m e d i u m
Consider the mixed convection flow along a vertical flat plate which is heated or cooled to a constant temperature Tw in a fluidsaturated porous medium of ambient temperature Tcr and fluid velocity Ucr which is oriented along the plate in the upward direction. Assuming that the Boussinesq and boundarylayer approximations hold and using the DarcyForchheimer extended model, namely Equation (II.3b), the governing boundarylayer equations can be written as follows:
Ou Ov Ox +~y   0 u + b,Ku2
(16.27)
  Uc~ +
b, ~Ku~ :l= g K___flfl(T  T~ )
1]
OT ~~
b,
OT + v ~y
02T 
(16.28)
v
(16.29)
~m oy2
where the :t: signs in Equation (16.28) denote the assisting (Tw > Too) or opposing (Tw < Too) flow, respectively. The appropriate boundary conditions for these equations are as follows:
vO, u+Uc~,
TTw
on as
T4Tc~
y0, y+oo,
x>/0 x/>0
(16.30)
To solve Equations (16.27)  (16.30), Yu et al. (1991) have proposed the following similarity variables
r  am,~T f (~) ,
0(77) T [ATi  T~ ,
Y) ~ ]  )~7 ( x
(16.31)
where ,k7 is given by 1
~7  Pe~ + Rm
(16.32)
and Rm represents a modified nonDarcy mixed convection parameter and is defined as follows: 1 Rm= 1 (16.33)
~+ ~~. Ra~
Ra*~ ~I
Also Rax and Ra* are the local DarcyRayleigh and the local nonDarcyRayleigh numbers, respectively, which are defined as follows
Ra~  g K f l IATI x 
amU
'
R a ~ = gfl IATI x2
b,a 2
(16.34)
NONDARCY C O N V E C T I V E FLOWS IN POROUS MEDIA
593
Further, the following inertial, ~, and mixed convection, ~, parameters are introduced: 1
=
. 88
1
~ =
1 + ~1=
~
(16.35)
I + P _ ~e~ Rm
Ra~
For the limiting case of the Darcy flow model with inertia totally neglected (Ra~: ec), then r = 0, whilst ~ = 1 represents the case of the nonDarcy flow limit for which inertia is completely dominant (Ra~ ~ 0). The mixed convection parameter ~, on the other hand, is a measure of the relative intensity of the free convection to the forced convection. For the case of pure forced convection (Pex ~ c~) then = 0, whereas for the case of pure free convection (Pez = 0) we have ~ = 1. Substituting the variables (16.31) into Equations (16.27)  (16.29), we obtain the following ordinary differential equations 2 ~ 4 f ' f " + ( 1  ~)2(2f, _ ::k:~40, 2 0 " + f 9'  0
(16.36) (16.37)
and the boundary conditions (16.30) become fO, f'~(1~)
00 2, 0>0
on as
~=0 r/~c~
(16.38)
The local Nusselt number for this mixed convection flow model can be expressed as follows: 1 g u  A7 [9'(0)] = ~1 (1  ~)Ra~z [9'(0)] (16.39) = ~l~Rax 88[  0 ' ( 0 ) ]  ( 1  ~1)Pe~x [0'(0)] The equations for pure nonDarcy natural convection boundarylayer flow can be obtained from Equations (16.36)  (16.38) by letting ~ = 1. The local Nusselt number for this natural convection flow is given by
Nun
( 1  ~)Ra~ [  0 ' ( 0 ) ]  ~Ra*z88[9'(0)]
(16.40)
On the other hand, the equations for pure nonDarcy forced convection flow are obtained from Equations (16.36)  (16.38) by letting ~  0. The corresponding local forced convection Nusselt number is now obtained as 1
N u f  Pe~ [0'(0)]
(16.41)
Equations (16.36)  (16.38) were solved numerically by Yu et al. (1991) using a fourthorder RungeKutta integration scheme and also using the Kellerbox method in order to check the accuracy. Values of the local Nusselt number for both Darcy ( ~  0) and nonDarcy ( ~  1) free convection limits (~ = 1) are given in Table 16.2
594
CONVECTIVE FLOWS
Table 16.2 Values of the local Nusselt number for Darcy (~ = O) and nonDarcy ( ~  1) free convection limits.
Cheng and Minkowycz (1977) Plumb and Huenefeld (1981) Bejan and Poulikakos (1984) Yu et al. (!991) ....
A = Nu1 Ra~ Darcy Flow Model " 0.4440 0.44390
B 
Nu ,s l~a x 4
NonDarcy Flow Model
0.494 0.49380
0.44388 ..
. . . . . . .
1
along with some known results which have been given in the open literature. It can be see that all these results are in excellent agreement. The variation of the local Nusselt number, Equation (16.39), with the buoyancy 1
parameter ~ is shown in Figure 16.3 for some values of the inertial parameter ~. Pe~ It can be seen that the local Nusselt number increases from the forced convection limit to the free convection limit for different values of ~ as the buoyancy parameter
102 Pe~ 0.7, 0 . 8 , ~
10~
~~/////AFil;~ng
10o
Opposing Flow 10 1
. . . . .
102 10'1
r
100
101
102 Ral! Pel
13
!
1
Figure 16.3: Variation of Nu with n ~ for different values of ~ in the cases of Pe~
assisting and opposing flow.
Pe~
NONDARCY C O N V E C T I V E FLOWS IN POROUS MEDIA
595
1
Ra{ increases. It is also observed from this figure that for the assisting flow the Pe~
local Nusselt number decreases as the inertial parameter { increases. However, for the opposing flow, the effect is reversed. The decrease of the local Nusselt number with the increase of 4 can also be seen in Figure 16.4. For specific mixed convection intensities, ~  0.1  1, the local Nusselt number decreases from the Daxcy limit through an intermediate region to the nonDarcy flow limit. 101
N}
10o
Ra~ 101 102
103
,, 10 2
I, , 10 1
, I0 ~
,,,, ! 101
10 2
',~
Rax
Figure 16.4: Variation of ~_ with Raf88for different values Ra~
Ra.
of .
Based upon the above theoretical results, and also upon previously published correlation equations for a viscous (nonporous) fluid, for example as reported by Churchill (1977), it has been established by Yu et al. (1991) the following correlation equation for the present problem:
NuA7  { Ca(1  ~)~ • [Am(1  4)m + B  m ~ m ]  ~ ~'"} }
(16.42)
where n and m are positive quantities and the values of A and B are given in Table 16.2 and C is obtained from Equation (16.41) as
C = Nuf =0'(0) Pe~
(16.43)
Equation (16.42) has been written by Yu et al. (1991) in the form Y" = 1 + X n
(16.44)
596
CONVECTIVE FLOWS
where y =
Nu
[Am(1 _ x
r
1
+ r
=
and the 4 signs in Equations (16.42) and (16.44) correspond again to assisting and opposing flows, respectively. Yu et al. (1991) have shown that over the entire regimes of the flow inertia (0 ~< ( < 1) the maximum error between the exact (numerical) values of the local Nusselt number given by Equation (16.39) and the approximate correlation Equation (16.44) with m  n  3 is about 11% for assisting flow (0 ~< ~ ~< 1) and about 8% for opposing flow (0 ~< ~ < 0.3 before the boundarylayer separates). The maximum error in Equation (16.44) can be reduced significantly if different pairs of m and n are taken, see Table 2 in the paper by Yu et al. (1991). The comparison between the correlated and the numerical results is shown in Figure 16.5 for both assisting flow and opposing flow cases with m  n  3 in Equation (16.44). This figure shows good agreement between the correlated and the calculated results.
.
Y
o a v o o
1: 1:.2 1:.3 C.4 I:.5
i
1:.6
9 1:.7
4
9
/~.
c.8
~4,/y3 
1
+
X3
.
_
_
A ~ I '~ _ X
3
,,
0
1
2
3
!

4
5
X
Figure 16.5: Comparison between the numerical results for Y, given by Equation (16.39) and indicated by the symbols, and the correlated results, given by Equation (16.~) and indicated by the solid line.
NONDARCY C O N V E C T I V E FLOWS IN POROUS MEDIA
16.4
597
T r a n s i e n t n o n  D a r c y free, f o r c e d a n d m i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow o v e r a v e r t i c a l s u r f a c e in a p o r o u s m e d i u m
Consider a vertical flat plate which is placed in a nonDarcian fluidsaturated porous medium of ambient temperature Too and fluid velocity Uoo where the velocity is oriented along the plate in the upward direction. It is assumed that the transient convection takes place as the wall temperature is suddenly increased from the ambient temperature Too to the constant value Tw, where Tw > Too (assisting flow). Under the Boussinesq and boundarylayer approximations, along with the DarcyForchheimer extended law, namely Equation (II.3c), the governing equations can be written as follows:
Ou
Ov
o~ + N
(16.46)
 o
u + b* X/~u2 .... = Uoo + b* V~Ku~ + g K ~ ( T  T o o ) V
V
OT
OT
OT
(16.47)
V
02T
(16.48)
which have to be solved subject to the initial and boundary conditions uUoo,
v0,
TToo
v0, TTw u + Uc~, T + Tc~
on as
at
t0
y=0 1 y + oo J
x,y
for all for
t > 0,
x>0
(16.49)
Equation (16.47) can be easily solved for u to give
u
{[
(
(1 + 2Re*)2 + 4Gr * T AT  Too
2b* ~
)] } 89 1
(16.50)
where Gr* and Re* are the modified Grashof and Reynolds numbers which are defined as follows:
Gr * =
b*K~ gflAT l]2
,
Re* =
b*K89Uoo 12
(16.51)
Thus, the slip velocity uw along the wall is given by UW

2b*x/~
[(1 + 2Re*) 2 + 4Gr*] 89 1
(16.52)
In order to obtain local similarity solutions of Equations (16.46)  (16.48), subject to the boundary conditions (16.49), Nakayama et al. (1991) have used the slip
598
CONVECTIVE FLOWS
velocity uw, as given by Equation (16.52) and the thermal boundarylayer thickness 5 as the velocity and length scales. Thus, in this unified treatment of the transient boundarylayer flow, the modified P~clet number 1
[(1 + 2.e.)2 + 4c .] _1 (16.53)
Pe~ = UwX = Pex C~m
2Re*
is introduced to investigate all possible free, forced and mixed convection cases. It can easily be shown that Pe~ transforms into the following limits: Forced convection regime: Darcy free convection regime: Forchheimer free convection regime:
Pe*  Re .2 >/Gr*
Pe~  P e z
for
Pe
for
Re* 0
(16.58b)
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
599
where Ao and Bo are constants which are given by 20'(0)
Bo  f o fOdrl f o Od~
Ao = f ~ Odrl'
(16.59)
Solving Equations (16.58) we obtain 1

uwBot
for
O'X
(7
and
1
(16.60b)
1

for ~t w
O'X
However, since the constant B0 is of the order unity and the fact that any leading edge effect of the plate propagates at the m a x i m u m s p e e d uw within the boundarylayer, the product uwBo may be replaced by uw. Hence, we have
(om,)
1
5~
a
for
T~I
(16.61a)
~ 7> 1
(16.61b)
for the onedimensional transient solution, and 1
for for the steady state solution, where
Uwt
T
(16.62)
O'X
Therefore, the thermal boundarylayer thickness grows as (i ~ t89 for small times 1 (7 > 1). The above analysis shows that the appropriate scales Uw and 5 vary according to the values of the parameters Re*, Gr* and T. Nakayama et al. (1991) have illustrated the corresponding solution regimes in a threedimensional space as shown in Figure 16.6. This figure shows the onedimensional transient solution regime, which is free from Re* and Gr*, and the steady state solution regime, which consists of three distinct subregimes corresponding to Equation (16.54), namely the forced convection regime, the Darcy free convection regime and the Forchheimer free convection regime. We return now to solving Equations (16.46)  (16.49). In order to achieve this Nakayama et al. (1991) proposed the following variables: 1
~_x,~?Y
(OZmtII ~ 1'
(a.~tI) ~
r

UW
'
g
f (~ r, 77) '
'
0(~ ~, ,7) '
TToo AT (16.63)
600
CONVECTIVE FLOWS
Gr*
f l02
t
OneDimensional 101[.10 10~ Re* Transient Regime '~ ] 10 1 1 Gr*2= Re* 210~~~'~ , i 10 '~ /
n\
K/C '0
J
'
Steady State /~0" I / \ Regime Forchheirner~x ' / \ / Free Conv.ectionN. / ~'~ I . ~~~/N~ Forced
,, , nve t,on
Free //I~N. Convection// ~ ~
/~ ~ ~
/
Gr* = Re*
Figure 16.6"
Flow regime map.
where I is defined as follows" I (T) 
(16.64)
1  e r T
so that I ..~ 1 for T > 1 respectively. Using expressions (16.63) in Equations (16.46)  (16.48) we obtain 1
f, =
. ~ [(1 + 2Re*)2 + 4Gr*] ~  1
1 1 0" + 2etriO ' + ~
_
_
(1  e 
r

(16.65)
re ~) fO'
(~__,) [00 ~+~ ( s, ~ oo _ 0 , o f ) _ ~
( , oo
0, o s ) ] ( 16"66)
(16.67)
~s~
and the boundary conditions (16.49) become f0,
0 1
0+0
on
770,
~>0,
i>0
as
77~oo,
~>0,
r>0
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
601
The local Nusselt number can be expressed as follows: Nu r) 89 ,I_ = (1  e[0'(~, r, 0)] Pex 2
(16.68)
From the general form of Equations (16.65)  (16.67) we can derive the following flow situations: (i) Small time solution, for T ~0 5~>0
These
(16.90)
We now define the following nondimensional variables *5
y*Y
x  7'
r
 ['
r
0*
am'
=
TToo AT
(16.91)
Using these variables Equations (16.87)  (16.89) can be written, after some algebra, as follows:
( G1 r+J ]R~a v2r (16.92)
+ JRa
\ Oz*
Oz .2
Ox~Oy*Ox*Oy* + V20, _ 0r
.
00" .
.
.
.
.
Oy* Ox*
.
~ 0r .
.
0y*2J
Raox;
00"
(16.93)
Ox* Oy*
where (16.94) and Gr is the modified Grashof number which is now defined as follows"
Gr  pgKK* flAT #2
(16.95)
The boundary conditions (16.90) become ~/J*  0,
0r
~ 0,
O* ~
OO*Oy+ . 0,
X*m 0* + 0
on as
y*  0, y* + cx~,
x* ~> 0 x* >~ 0
(16.96)
The boundarylayer equations are found using the new variables defined as followsxx*, yRa 89 r O(16.97) Substituting these relations into Equations (16.92) and (16.93), letting Ra + c~, i.e. the boundarylayer approximation, and retaining only the leading order terms of Ra, we obtain 1+
of
(16.98)
o~
o~"
o ~ o ~ o~o~
o~
o~o~
o~o~
(16.99)
NONDARCY C O N V E C T I V E FLOWS IN POROUS MEDIA
607
er where X  1is a constant which can be scaled out of the problem by introducing
Ra
the transformation ~.
3A2m
A
xx12mx,
2m
A
yx~~my,
l+rn
A
r162
(3+2m)m
0=X
~2,~ 0
(16.100)
Thus the problem reduces to solving the equations
or 1 + 2~
02r
oo
Oy 2 =
Ox
020
0r OO
Oy 2
Oy Ox
(16.101)
or oo Ox Oy
(16.102)
subject to the boundary conditions r 9x m or __+ 0, 9 + 0 Oy
on as
y0, y + cr
x>f0 x ~> 0
(16.103)
It should be noted that the transformation (16.100) becomes invalid when m = 0.5. Further it can also be seen from this transformation that the boundarylayer attains a constant thickness for m  2 and this is an upper bound value of m for which Equations (16.101)  (16.103) possess a solution. The boundarylayer analysis for this problem has been performed by Hossain and Rees (1997) in the three distinct cases: 0 ~< m < 0.5, m = 0.5 and 0.5 < m ~< 2. In the first case, where 0 ~< m < 0.5, the fluid inertia dominates near the leading edge (0 ~< x ~ 1), but its effect diminishes far downstream (x ~> 1). The opposite is true when m > 0.5, in that the inertia effect is absent at the leading edge, but this effect becomes stronger further downstream. There is a transition between these two flow regimes for m = 0.5 and the flow is selfsimilar in this case. 16.5.1
0 ~< m < 0.5
Guided by the Darcy flow regime solution, as first considered by Cheng and Chang (1976), we use the following variables: x
~~
3 ,
y
y
2,~, X
r
~+~
f(~,y),
0xmg(~,77)
(16.104)
3
Thus the Equations (16.101) and (16.102) can then be written as follows" (1 + 2~_1f,) f,, __ ( 2  m ) 3
g" +
l+m
3
f g'  m f 'g =
(12m) 3
rig' rng  ( 1  2 m ) 3 ( Og Of) r f ' ~~  g' ~
~_~Og (16.105) (16.106)
608
CONVECTIVE FLOWS
and the boundary conditions (16.103) become f0, f'+0,
91 9+0
on as
r/0, ~+oc,
~>/1 ~)1
(16.107)
where primes denote differentiation with respect to r/. It is seen from Equation (16.105) that the presence of the (1 term leads to the decrease of the fluid inertial effects as ~ becomes large. However, when ~ is close to zero then the fluid inertia is very high and other variables are necessary in this case to avoid the term ~1 from Equation (16.105). Hossain and Rees (1997) have proposed the following variables for Equations (16.101) and (16.102) when 0 ~< x ~ 1: l  2m X

x
y
~ , ~
am, x
2+m
r  x 5 F(X,s
O  xmG(X,~)
(16.108)
5
to obtain
(X + 2F') F"  ( 3  m ) (G' G" +
2+m
5 FG'  mF'G 
mG_(12m)x 5
OG (16.109)
( I  2m) X (F, OG _ G, OF) 5 OX ~
(16.110)
along with the boundary conditions F0, F'+0,
G1 G+0
on as
~0, ~+oe,
0~<X~~ 1
(16.112a)
for
0 ~< X ~< 1
and
g'(~,0)
for
~ >~ 1
(16.112b)
where ~  X~. The form given in Equation (16.112a) allows the behaviour of the heat transfer in the flow inertia dominated regime to be clearly observed, whilst Equation (16.112b) clearly shows the approach towards the Darcyflow regime at large distances from the leading edge. It is observed from Figure 16.10 that the Darcy flow regime is rapidly established as X increases.
NONDARCY C O N V E C T I V E FLOWS IN P O R O U S MEDIA
609
1.0 0.8 0.6 0.4 0.2 ~.
0.0
0
.
5
.
.
10
.
X
.
15
20
Figure 16.10: Variation of the wall heat transfer with X for different values of m. The two different forms expressed in Equations (16.112a) and (16.112b) are indicated by the broken and solid lines, respectively.
16.5.2
m0.5
In this case it is not possible to scale out the parameter X from Equations (16.98) and (16.99). The physical reason for this is that the induced streamwise fluid velocity does not vary with x when m = 0.5 and therefore the fluid inertia may be either strong or weak. However, the flow is selfsimilar in this case and the substitution ' r  x~f(~7) ,
0
x89
,
Y1
~1
(16 113)
X2
into Equations (16.98) and (16.99) leads to the following ordinary differential equations(1 + 2 x f ' ) f " 1 g" + ~ ( f g ' 
1
(16.114)
f'g)  0
(16.115)
which have to be solved subject to the boundary conditions
fO, f ' ~ 0,
g 1 g + 0
on as
770 77 + cx~
(16.116)
Equations (16.114)  (16.116) have been solved numerically by Hossain and Rees (1997) for the fluid inertia parameter X in the range from 0 to 100. The variation
610
C O N V E C T I V E FLOWS
of f'(0) and g'(0) with X is shown in Figure 16.11 where it can be seen that both these quantities decay as X increases. This is due to the fact that fluid inertia serves to thicken the boundarylayer because of the increased effectiveness of the conduction from the heated surface which is caused by the decreased advection of heat downstream. 1.2 1.0 0.8 0.6 0.4 0.2 0.0
o
lo
Figure 16.11" Variation o/ the reduced slip velocity, f'(O), and the rate o/ heat transfer,  g ' ( O ) , with X for m  0.5.
Further, an asymptotic analysis of the solution of Equations (16.114)  (16.116) for very large values of X has been performed by Hossain and Rees (1997) and they found 2 gl f'(0) ~ 1.0231X~, (0) ,~ 0.8239 X~ (16.117) 16.5.3
0.5 < m ~
1.
x
2rn1 ~
2+m
3~m X 5
7/)X
 5F ( X ,
~),
(16.119)
NONDARCY C O N V E C T I V E FLOWS IN POROUS MEDIA
611
Equations (16.101) and (16.102) then become
(1 + 2~f')f.(2m)~ r i g '  mg + (1  2m)3 ~_~cOg(16.120) l+m (2.~1) ( ,0g Of) g" + ~ f g'  m f'g 3 ~ f ~  g'~ (16.121) which have to be solved subject to the boundary conditions f0, f'+0,
g1 g+0
on as
~0, r/+co,
0~~ 1 0~4~1
(16.122)
and ( X 1 [ 2 F ' ) F " 
3 m
~a'
mG+
X
5
a"+
5
2+mFG'_mF'G(2m1)X(F'OG_G'OF) 5 5 ox ~
ox
(16.123) (16.124)
along with the boundary conditions F=0, F ' + 0,
G=I G >0
on as
~=0, ~ + ec,
X/>I X i> 1
(16.125)
Again, Equations (16.120)  (16.122) have been numerically solved by Hossain and Rees (1997) in the same way as those for 0 ~ m < 0.5. The wall heat transfer rates are given as follows: g'(~,0)
for
~~ 1
and
~g'(~,0)
for
~~ 1
and
X89 G'(X, 0)
for
X ~ 1
(16.126a)
for
X ~> 1
(16.126b)
and these are shown in Figure 16.12 for some values of m. Solutions are again plotted as a function of ~ in order to see more clearly both the inertia free and inertia dominated fluid flow regimes. This figure clearly shows that the effects of fluid inertia increase with increasing distance downstream. Thus, far downstream the fluid flow and the temperature profiles, as well as the boundarylayer thickness, have been changed from what they were with the fluid inertia absent.
612
CONVECTIVE FLOWS
2"0t
m'_'i.6, 0.8:'1, 1.2, 1.6, 2 i
15~
'~ J
o.+
13.0,
0
5
10
+
15
I
20
Figure 16.12: Variation of the wall heat transfer with ~ for different values of m. The two different forms expressed in Equations (16.126a) and (16.126b) are indicated by the broken and solid lines, respectively.
16.6
Effects of h e a t d i s p e r s i o n on m i x e d c o n v e c t i o n b o u n d a r y  l a y e r flow past a h o r i z o n t a l surface
A great number of heat transfer applications in porous media have been studied with the help of a constant coefficient heat conduction model. The terms of Forchheimer and Brinkman, and a variable near wall porosity were added to the Darcy law model in order to account for inertia, boundary drag and flow channelling phenomena, which occur at the higher pore velocities in porous media convection. When combined with heat transfer, a constant heat convection coefficient is not appropriate to describe the additional mechanical mixing of fluid particles with different temperatures that takes place at such velocities and Kaviany (1995) has shown that fluid velocity dependent thermal diffusivities are deemed to better describe these processes. There are several theoretical and empirical models which describe the spreading of heat when it is being conducted through an isotropic homogeneous porous medium and simultaneously transported with a carrier fluid. A counterpart for the most popular in groundwater hydrology solute mixing model is represented by the BearScheidegger dispersion tensor presented in the book by Bear (1979). A similar model was presented by Georgiadis and Catton (1988) and it has been applied by Howle and Georgiadis (1994) to free convection predictions. Thiele (1997) has used a model in which the total thermal diffusivity tensor comprises both of constant
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
613
coefficient heat conduction and fluid velocity proportional mechanical heat dispersion for the mixed convection flow past a horizontal surface in a porous medium. In what follows we shall present some of the results reported by Thiele (1997) for this problem. Consider the steady mixed convection flow of velocity Uoo over a heated horizontal surface of temperature T w ( x ) given by Equation (16.86) which is embedded in a fluidsaturated porous medium. Under the usual Boussinesq and boundarylayer approximations, the basic equations which govern this problem are given by, see Thiele (1997), Ou
Ov
Ox + Oyy  0
(16.127)
Ou
g K fl OT
o~ =
(16.128)
o~
.
~ ~ + ~ oy = o~ (~m + Arl~)
(16.129)
and they have to be solved subject to the boundary conditions v = O,
T w ( x ) = Too + A T x m
u + Uoo,
T 4 Too
on as
y=0, y 4 oo,
x>0 x > 0
(16.130)
where A T > 0 (assisting flow) and the yaxis is oriented in the upward direction. Further, we introduce the following nondimensional variables: 
Ra, 3_, Pe~
Y 77Pe?c, x
r = DTooPe~f(~,~7),
0(~,71) 
TToo Tw  Too
(16.131)
where the local P6clet and Rayleigh numbers are defined as follows: U~x Pex
=
DToo
gK~ATxm+ nax =
,
1
(16.132)
~'DToo
and DToo is the transversal component of the thermal dispersion tensor which is given by (16.133)
DToo  O~m + ATIUc~
On substituting Equation (16.131) into Equations (16.127)  (16.129), we obtain
[ ( 1)., i]
f "  ~ mO+
ra  ~
1,,
(I  AT)0" + AT (f'O" + f"O') + ~f
(16.134)
~ 0~  ~~10
mf'O
(m
~ f,O0
o,Of
)
(16.135)
614
C O N V E C T I V E FLOWS
which has to be solved subject to the boundary conditions (16.130) which become f0, f'+l,
0=1 0+0
on as
770, 77+c~,
~>0 ~>0
(16.136)
where ,~T represents the ratio of the mechanical to total heat dispersion and it is defined as follows: ATIUc~ AT P e ~ AT = = (16.137) am + ATlUcc 1 + ATPe~ where Peoo is the P~clet number based on the length scale 1. The values , ~ T   0 and AT = 1 correspond to pure stagnant heat conduction and pure mechanical heat dispersion, respectively. With AT = 0, we also have the case of an arbitrary constant heat conduction coefficient which has been treated by Minkowycz et al. (1984) and Aldos et el. (1993a, 1993b). On the other hand, the nonsimilarity variable ~ is called the buoyancy variable and i t is a measure of the relative importance of free to forced convection, its value ~ = 0 corresponds to the case of purely forced convection and ~ + c~ to pure free convection. The free convection limit cannot be treated with the use of the coordinates (~, ~?). Mixed convection, with only its forced convection limit which is represented by the (~, r/) coordinate system, has been chosen here to elucidate the interdependence of heat conduction and dispersion in a heat transfer configuration initiated by an outer flow of possibly small velocity and not by buoyancy effects alone. Equations (16.134)  (16.136) were numerically solved by Thiele (1997) for some values of m, AT and ~ using the Kellerbox scheme. Fluid velocity and temperature profiles are shown in Figure 16.13 when both the heat transfer mechanisms have different intermediate shares in the oncoming flow, i.e. for different values of AT. It is seen to have a considerable influence on the prevailing mixing mechanism for both the fluid velocity and temperature distributions. The local Nusselt number can be expressed as follows: Nu
x = O'(~,0) Pe~
(16.138)
and its variation with ~ is shown in Figure 16.14 for some values of AT when m  0.5, 1, 1.5 and 2. We see a strong influence of the heat diffusion/dispersion parameter ~T on the rate of heat transfer. We observe that the heat transfer rate increases with increasing ~values in the pure stagnant heat conduction limit (AT  0), whilst it decreases in pure mechanical heat dispersion (AT  1). Hence, if stagnant heat conduction is the only acting mechanism, the heat flux rate is highest in the free convection limit (~ ~ c~), and for pure mechanical heat dispersion it is highest in the forced convection limit ( ~  0). The local Nusselt number as a function of AT is given by 1
=
A~ [0'(r 0)]
(16.139)
N O N  D A R C Y C O N V E C T I V E F L O W S IN P O R O U S M E D I A
615
(a) 
lll l
,.,.
i
,
i
2
1
.1, 2, 5 ~=1,
O
i
,
2, 5 ~
~
0.0
,
i
9
0.5
iwl
i

0(~,~)
~.0
0
o
0
2
4 f,(~, 77) 6
(b) ,
,
,
,,
,
,,
2.
2
"".. ~ = 1 1
',p< ~ = 1 , 2 , 5 0 0.o
.
0.5
o(~,~)
~.0
0
w
2
w
9
~
w
4 f,(~,~) 6
Figure 16.13: Dimensionless temperature, 0(~, 77), and fluid velocity, f'(~, ~7), profiles for (a) m  0.5 and (b) m  1.5. The solutions for AT  0 and 1 are indicated by the broken and solid lines, respectively. The profile at ~  0 is indicated by the dotted line.
is illustrated in Figure 16.15 for some values of m, )~T and ~. The predicted zero value of N u for /~T  0 results from the fact that the free convection limit cannot be properly dealt with in the coordinate system (~, ~?). At higher ~values there is a maximum of the heat flux rate for some intermediate ATvalues between 0 and 1, especially for the larger values of m. The heat transfer grows for stronger density coupling (larger ~) at smaller values of AT, whilst it decreases with an increase in at values of AT closer to unity.
616
CONVECTIVE
o.~51 0.5
.........s
0.751
Nu
111
FLOWS
s
.."
i~.~
~.__~..._..~'"
m2
ml.5 m1 m=0.5
"!
o
5
~
lo
Figure 16.14: Variation o f  ~ _ as given by Equation (16.138) with ~. Pe}
(a)
(b) 80
80
6O
6O
Nu
Nu
4O
4O
20
20
0 0.0
0.5
AT
1.0
0 0.0
0.5
AT
1.0
Figure 16.15" Variation of Nu as given by Equation (16.139) with AT when ~  103 and AT = 0.3 for (a) different values of m at ~  1 (broken lines) and ~ = 20 (solid lines) and (b) different values of ~ when m = 1.
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
16.7
617
Free c o n v e c t i o n b o u n d a r y  l a y e r flow f r o m a p o i n t h e a t s o u r c e e m b e d d e d in a p o r o u s m e d i u m filled w i t h a n o n   N e w t o n i a n p o w e r  l a w fluid
Convective flows resulting from concentrated heat sources which are embedded in fluidsaturated porous media are of great importance in many applications, such as the recovery of petroleum resources, cooling of underground electric cables, environmental impact of buried heat generating waste, hotwire anemometry, volcanic eruptions, etc. The fluid flow phenomena can be grouped into two distinct regimes: (i) low Rayleigh number regime where the temperature distribution is primarily due to the thermal diffusion, and (ii) the high Rayleigh number regime where the fluid flow driven by the heat source is a slender vertical plume such that the boundarylayer approximation holds. The problems of class (i) were considered by Bejan (1978), Hickox and Watts (1980), Hickox (1981), Nield and White (1982), Poulikakos (1985) and Larson and Poulikakos (1986) while those of the class (ii) have been studied by Wooding (1963), Yih (1965), Bejan (1984), Masuoka et al. (1986), Kumari et al. (1988), Ingham (1988), Lai (1990a, 1990b, 1991), Afzal and Salam (1990), Leu and Jang (1994, 1995) and Shu and Pop (1997). However, all these studies assume that the fluid is a Newtonian Darcian fluid or a Newtonian nonDarcian fluid. This assumption is not justified for a large class of fluids, such as, for example, crude oils which saturate underground beds, polymer solutions in chemical engineering applications, etc. Chen and Chen (1988a, 1988b) were the first to consider free convection boundarylayer of a nonNewtonian powerlaw fluid over a vertical flat plate and a horizontal circular cylinder using a powerlaw model proposed by Christopher and Middleman (1965) and Dharmadhikari and Kale (1985). This model was also used by Nakayama and Koyama (1991) and Nakayama and Shenoy (1993) to study the possible similarity solutions for free and mixed convection boundarylayer flow over a nonisothermal body of an arbitrary shape which is immersed in a porous medium saturated with a nonNewtonian powerlaw fluid. The same model has been employed by Nakayama (1993a, 1993b) to study both the free convection boundarylayer from a point and a horizontal line heat source in a porous medium. He showed that the governing equations possess an elegant analytical solution for arbitrary values of the power law index and we present the results obtained by Nakayama (1993a) for the model problem of free convection generated by a point heat source embedded in a saturated porous medium filled with a nonNewtonian powerlaw fluid. Consider a point heat source of strength qs which is embedded in a porous medium saturated with a nonNewtonian powerlaw fluid, see Figure 16.16. Under the Boussinesq and boundarylayer approximations, the basic equations can be written as, see Nakayama (1993a),
618
C O N V E C T I V E FLOWS
).
.)
:it 2:,O~..'Point ]Heat Source Figure 16.16 Physical model and coordinate system.
0
o

o
(16.140)
gK*~ #* ( T  Too)
(16.141)
oz U n
OT
u z + v

OT r Or
r~r
(16.142)
where n is the powerlaw index, #* is the consistency index, K* is the modified permeability for the powerlaw fluid defined as follows
K*
d~o 6 (3n~i) n (3(1~v))n+l C* ( 3n+1) n~o n (3(1~o)) dqo n+l
Christopher and Middleman (1965) Dharmadhikari and Kale (1985) (16.143)
and C* is a constant which is given by
C*
3(1n3)
3 (9n+3)n(6n+l)(16)  4
8n
IOn 3
lO~+Y
75
(16.144)
It should be noted that for a Newtonian fluid (n = 1) C*  6_ and the two 25 expressions (16.143) for K* are identical. The boundary conditions appropriate to Equations (16.140)  (16.142) are as follows: vO, OTor_O on r   O , x>~O (16.145a) T+T~ as r~o~ x>/O '
~
along with the integral constraint condition
27rpcp
u (T  T~) r dr = qs
~0(:x:)
(16.145b)
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
619
A scale analysis of Equations (16.140)  (16.142), such as that proposed by Bejan (1984), reveals that the centreline temperature, Tc, centreline fluid velocity, uc, and the plume diameter, ~, behave as followsI
Tc  Too ,,~ k m x '
uc ~
x Raz ,
~~ [
gK7~s
= R a ~!
(16.146)
where 1
(16.147)
O~m~ km
is the modified local Rayleigh number, which is constant for n  1 (Newtonian fluid). 1+n It is seen from expressions (16.146) that 3 grows in proportion to x 2~ , as graphically illustrated in Figure 16.17, where distinct shapes of the plume can be observed according to the power law index n. It should be noted that the boundarylayer type for slender plumes can be obtained if the heat source is strong and therefore the local Rayleigh number is sufficient large. However, since R a z ~ 0 for x + c~ when n < 1, the boundarylayer analysis for such fluids is valid only in some limited region above the source where R a x is sufficiently greater than unity, see Nakayama (1993a). (a)
(b)
(c)
I
!
1
Figure 16.17 Plume shapes for (a) pseudoplastic fluids, n < 1, (b) a Newtonian fluid, n : 1, and (c) dilatant fluids, n > 1.
Based on the scalings defined in Equation (16.146), the following similarity variables are introduced r
r = ~xf(~),
T
Too =
qs 0(7]), kmx
yna2x
X
1
(16.148)
where the stream function r is defined by Equations (7.93). On substituting these
620
CONVECTIVE FLOWS
expressions into Equations (16.141) and (16.142) we obtain (~)n
= 0
(16.149)
(~0' + fO)'  0
(16.150)
which have to be solved subject to the boundary and constraint conditions (16.145), which become f0, 0'0 on 7 7  0 (16.151a) 0 + Oas rl + ~ and the integral constraint condition
co (f~)lTn 2~/0
~
d~  1
(16.151b)
Integrating Equation (16.150), and imposing the boundary conditions (16.151a), gives ~0' + 10  0 (16.152) which in combination with Equation (16.149) leads to the equation n (~?f" f') + f f'
(16.153)
0
The solution of this equation is given by f0?)(A~v)2
(16.154)
1 + (A'~)2 4n
where the constant An is determined by using Equation (16.151b) and it has the value 1
An 
(16.155)
n~r23+ n
Therefore, the streamwise fluid velocity and temperature distributions inside the plume are given by
T
._.
....
1+
___
(2A )
(16.156)
1+
and they reduce to those obtained by Masuoka et al. (1986) and Lai (1990b) for n = 1 (Newtonian fluid). Typical nondimensional fluid velocity and temperature profiles are displayed in Figure 16.18 for n = 0.5, 1 and 1.5. Figure 16.18(a) shows that the dilatant fluids (n > 1) make the fluid velocity profile somewhat more peaked, whilst the
NONDARCY CONVECTIVE FLOWS IN POROUS MEDIA
(a)
u
(b)
621
T  T~
t
.... 0"1~5~iI ........ / , /" " ,/0.101 ' x\~,,,.\
.........
;:..:. .... ~  . 8
4
0
'"'"'""........//~0,I0]"~\\:...."""""...."'
. ~ .... ::.,.: 4
~7 8
8
4
0
4
T/ 8
Figure 16.18" (a) The nondimensional fluid velocity, and (b) the nondimensional temperature, profiles. The solutions for n  0.5, 1 and 1.5 are indicated by the dotted, broken and solid lines, respectively.
pseudoplastic fluids (n < 1) tend to produce more uniform fluid velocity profiles. On the other hand, Figure 16.18(b) shows that the temperature profiles become flatter and the temperature level is maintained higher as the powerlaw index decreases. Finally, to show the isotherm patterns, we express the temperature distribution (16.156) in the form
T
(2A n)
Tref
(16.157)
(1 +  ~ r * 2 ( x * ) n  3 R a )
2n
where the nondimensionM coordinates (x*, r*) are defined as x
x* =
,
kmTref
r
r* =
(16.158)
k~T~e~
and the modified Rayleigh number is defined as follows: 1 
Olin#*
kmTref
(16.159)
The isotherms T.Tcr 0.1 generated for n  0.5, 1 and 1 5 at R a  500 and Tre f ' 5000, are plotted in Figure 16.19. This figure shows that a high temperature zone expands further for smaller values of n, as may be expected from the fluid velocity and temperature profiles shown in Figure 16.17. It can also be seen from Figure 16.19 that the effect of increasing R a is to make the plume more slender. =
622
CONVECTIVE FLOWS
(a)
(b) X*
..1.6 .......
::"
1.2 ,,
:t
\
1~8~ X't,l I
i:
I
:
l
i
" "
",
'
I
:
I
I ,
tI i
",
'
" ",, \~\ 1 1 / /
0.4 0.2
0.0
: ' : :
/ !
I I
:
/
!,
I I I I
!
:
i
0.44
,
:
i : ,
"
:":1.2 :
o
X*
1.6.,
i
i :
t t I 1
; : : :
t i
o.8,
I!
:~, ",
Ii: l
'1
i
I~
l
,,
'..,
I I
s..'
: ,'~
/
0.2
0.4 T*
0.2
0.0
0.2 r*
Figure 16.19: Isotherms for TToo ~ e f :  0 . 1 when (a) R a  500 and (b) R a  5000. The solutions for n  0.5, 1 and 1.5 are indicated by the dotted, broken and solid lines, respectively.
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Author index AbdelelMalek, 284, 326 Ackroyd, 496 Acrivos, 335, 336 Adams, 131 Afzal, 4, 59, 61, 62, 64, 103, 104, 107, 108, 151,153, 158, 348, 617 Ahmad, 218, 230, 233 Ahmadi, 357 Ahmed, 96 Aihara, 221 Akiyshi, 180, 189 Aldos, 446, 614 Aleksashenko, A. A., 180, 182 Aleksashenko, V. A., 180, 182 Allain, 118, 119 AItenkirch, 37 Amaouche, 233 Amaya, 180 Amberg, 36, 88, 112114, 432, 457459 Anderson, 180, 473 Andersson, 333, 335, 336, 368373 Andre, 107, 108 Angirasa, 22, 118, 128131, 133, 149, 151, 397400, 437, 443, 444, 446 Ariman, 335 Armaly, 4, 171, 230, 245, 254, 339, 446, 614 Armstrong, 333335 Astarita, 333 Awang, 327329, 332
Aziz, 4 Badr, 214, 215, 233236, 238, 239 Badran, 284, 326 Baer, 118, 126, 128 Baikov, 336 Bakier, 254 Banks, 145, 311,317, 387390 Banthiya, 59, 61, 62, 64, 107 Bassom, 356, 363, 504, 505, 507509 Batta, 222 Bear, 612 Beckmann, 7, 12, 285 Bejan, 4, 5, 65, 118, 119, 180, 377, 378, 381, 397, 399401, 431, 435, 441, 473, 526, 534, 585, 588, 594, 617, 619 Bellman, 61
Bershader, 37 Bhattacharyya, 555 Bird, 333335 Bloor, 290 Bond, 134 Bories, 415 Boussinesq, ix, 6 Boutros, 284 Bradean, 377, 436, 534, 561, 567, 569, 570, 574 Bradshaw, 30 BriiFdli, 308 Briggs, 286, 302 Brown, 285, 286, 327, 331,332, 385, 386, 506, 534, 542, 544, 545, 549, 602, 603 Bui, 245 Burde, 112, 457
Camargo, 180 Cameron, 233 Campbell, 311 Carey, 299, 300, 302304 Carnahan, 286, 308
Carslaw, 510 Castellanos, 151 Carton, 239241, 243, 612 Cebeci, 30, 245 Cess, 36 Chan, 171 Chang, C. L., 356 Chang, G., 236 Chang, I.D., 431434, 607 Chang, S.M., 180 Chao, B. H., 401 Chao, B. T., 210, 214 Char, 356 Chaudhary, 37, 4042, 134, 136142, 1 4 5 , 146, 189, 356, 391,393, 396, 468 Cheesewright, 22, 23 CheF, C. K., 214, 336, 338342, 617 Chen, C.C., 65, 68, 69 Chen, F. F., 251 CheF, H.T., 180, 617 Chen, J. L. S., 345 Chen, J.J., 151, 160163
648 Chen, T. S., 4, 105, 106, 171, 230, 245, 254, 339, 446, 614 Cheng, P., 381, 385, 391, 397, 401, 420, 422, 424, 431434, 446448, 491, 492, 494, 500, 504, 524, 533, 535538, 542, 561, 569, 572, 574, 590, 591, 594, 607, 614 Cheng, Y. S., 22, 23 Chhabra, 333 Chi, 336 Chitou, 540 Chiu, 342, 356, 357 Cho, 333 Chou, H. M., 342, 356, 357 Chou, Y. L., 336 Christopher, 617, 618 Chung, 327 Churchill, 214, 286, 595 Claassen, 87, 91, 92 Clarke, 37 Cloitre, 118, 119 Clutter, 241 Collins, 562 Combarnous, 415 C6rdova, 180 Cramer, 334 Crochet, 333 Curtis, 264
Dale, 336 Daniels, 103 Daskalakis, 254 Datta, 555 Davies, 333 Davis, 59, 61 De Hong, 103, 104 De Kee, 333 De Witt, 30, 233 Del Casal, 87, 90, 102, 473 Dennis, 236, 238, 239, 291,543, 545, 562 Desrayaud, 151 Dey, 103 Dharmadhikari, 617, 618 Domoto, 30 Drake, 491 Dring, 285
Duwari, 446 Ebinuma, 602 Eckert, 285, 491 Ede, 68 Eichhorn, 7, 36, 37, 79 Elliott, 214, 306, 307, 310, 311,313, 321 Elrod, Jr., 30 Emery, 336 Ene, 401 Ergun, 379, 585, 586
CONVECTIVE FLOWS Erickson, 254 Eringen, 335, 356 Evans, 69
Facas, 491,492 Faeth, 162, 163, 176 Falkner, 72 Fan, 254 Fand, 492, 500, 590, 591 Farouk, 218, 491 Fen& 533
Fernandez, 491,492 Fertman, 180
Forchheimer, 493, 585, 586 Fox, 254 Fujii, M., 218 Fujii, T., 170, 174, 218 Gdalevich, 180 Gebhart, x, 46, 118, 121, 126, 129131, 151, 171,284286, 308
Genceli, 318 Georgiadis, 612 Gersten, 4, 44 Ghosh, 333 Gill, 87, 90, 102, 473 Glauert, 172, 175 Goel, 325, 326 Goldstein, 4, 209, 218, 221223, 285, 286, 302 Gorla, 254, 346, 348, 356 Govindarajulu, 391 Gray, 134 Gregg, 12, 15, 19, 23 Griffin, 254 Griflith, 36 Grishin, 106
Gr6ber, 3 Gruzin, 106 Gryglavszewski, 346 Giiqeri, 218 Gupta, 310, 311
Hady, 254 ttall, 329, 543 Hardee, 492 Harris, 142, 306, 307, 424, 426, 551,553, 556, 558 Hart, 64 Hartnett, 333
Hartree, 42 Hasan, 79 Hassager, 333335 Hatlon, 236 Hatzikonstantinou, 327 Heckel, 245 Heggs, 377, 436, 534, 567, 569, 570, 574 Heinisch, 285
AUTHOR INDEX Hellums, 286 Henkes, 22, 24, 26, 28 Hermann, 214, 310, 313, 314 Herwig, 4, 44 Hickox, 617 Hieber, 49 Higuera, 151, 164, 166168, 180, 431, 437, 440442, 461,473, 474, 480, 481 Hirschberg, 614 Hob, 254, 276279 Holman, 225 Honda, 218 Hong, 180 Hoogendoorn, 22, 24, 26, 28 Hossain, 96, 356, 504, 586, 607611 Howle, 612 Huang, J. S., 336 Huang, M. J., 214, 336, 338341 Hudson, 238, 239 Huenefeld, 585, 594 Hunt, 50, 5557, 96, 151, 171, 174, 185, 203, 311,466, 470, 487 Huppert, 118 Hussain, 64, 103, 104, 107, 108 Hussaini, 460 Hwang, 22 Illingworth, 285, 288 Ingham, 4, 12, 14, 15, 17, 18, 103, 142, 151, 153, 155157, 180, 200203, 234, 241, 254, 261, 263, 264, 267, 272275, 286, 288292, 295, 297, 306308, 311, 312, 314, 320324, 348, 377, 381, 385, 386, 401404, 424, 426, 432, 436, 453, 454, 461, 463, 468, 469, 471, 472, 480, 484, 487, 489, 493, 495, 497, 498, 500, 506, 518, 519, 524, 528531, 533, 534, 542, 544, 545, 549, 551, 553, 556, 558, 561563, 567, 569, 570, 574, 575, 577, 578, 581583, 585, 588, 602, 603, 617 Irgens, 333, 335, 336, 368372 Irvine, 333 Jacobs, 22 Jaeger, 510 Jagannadham, 151 Jain, 320, 325, 326 Jaluria, x, 46, 118, 151, 171 James, 236 Jang, 617 Jarrah, 446 Jena, 356 Jeng, 30, 233 Johnson, A. T., 209, 233 Johnson, C. H., 533
649 Jones, 87, 90, 9295, 101 Joshi, 4, 151,286, 308 Kahawita, 218, 222, 223, 311,317, 318 Kakaq, 4, 44 Kalaba, 61 Kale, 617, 618 Kao, 30, 37 Kapustin, 106 Karni, 333 Katagiri, 311, 320 Kaviany, 612 Kawase, 336, 339, 341 Kay, 151 Kays, 37 Keller, 308, 391 Khair, 119, 397, 399401 Khan, 254 , 257, 258 Kikkawa, 326 Kim, B. Y. K., 492 Kim, E., 342, 344, 345 Kimura, 180, 189, 193195, 223, 225, 226, 377, 431,435, 441,461,462 Kiwata, 377, 461,462 Kleinstreuer, 107, 336, 352, 353 Ko, 180 Koh, 217 Kokudai, 585, 586, 588, 590, 591 Konishi, 368 Korovkin, 171 Koyama, 585588, 590, 591, 597, 599, 602, 603, 617 Kuehn, 209, 218, 221223 Kuiken, 68, 69, 71, 122, 151, 245, 254258, 261, 269, 270, 300, 303306 Kulacki, 446448, 586 Kulkarni, 22 Kumar, 210, 214216 Kumari, 254, 335, 342, 344, 453, 504, 524, 586, 617 Kung, 151, 160163 Kurdyumov, 151 Kuribayashi, 310 Kuwahara, 597, 599, 602, 603 Kuznetsov, 415 Lachi, 540 Lai, 446448, 586, 617, 620 Lakin, 460 Lain, 492 Laminger, 103, 104 Lampinen, 118, 149 Lanczos, 94 Lankford, 180 Larson, 617 Lauriat, 151 Law, 88, 106
650
CONVECTIVE FLOWS
LeaI, 4 Lee, L., 230 Lee, S., 43 Lee, S. K., 346, 348 Lee, S. L., 245, 254, 339 Lesnic, 461,468, 469, 471,472, 484, 487, 489 Leu, 617 Liburdy, 162, 163, 176 Lin, F. N., 210, 214, 215 Lin, H.T., 65, 68, 69, 88, 97, 98, 100, 149, 151, 160163, 171, 173, 174, 176, 177, 206, 207, 254, 276279, 592596 Lin, M. T., 342 Lin, P. P., 356 LiSdn, 134, 142, 145, 146, 151 List, 171 Liu, 106 Lloyd, 61 Lock, 180 Lohar, 320, 325, 326 Lu, 592596 Luikov, 180, 182 Luna, 180 Lupo, 180, 183, 187 Luther, 286, 308
Madan, 326 Magyari, 308, 391 Mahajan, x, 46, 118, 128131, 133, 149, 151 Mahmood, 42, 69, 70, 73, 74, 77, 78, 81,245248, 250, 401,404, 406408
Malarvizhi, 391 Marchello, 334 Marrucci, 333 Martynenko, 4, 171, 180 Maruhara, 218223 Mashelkar, 333 Masuoka, 617, 620 Mathur, 356 McDonough, 239241 McFadden, 131 Mdndez, 180, 473, 480 Menold, 284 Meredith, 36 Merkin, 4, 12, 15, 1719, 21, 23, 24, 3742, 4850, 52, 53, 59, 69, 70, 7378, 81, 94, 103, 134, 136142, 145, 146, 151, 180, 182, 185, 187, 189, 203, 209, 214217, 228, 230, 231, 245250, 254, 275, 297, 312314, 317, 320326, 377, 391394, 396, 401, 404, 406408, 422424, 426428, 431, 432, 434440, 451454, 461, 463466, 468, 471, 487, 492, 493, 498, 504, 510, 512, 514, 524,
525, 531, 533, 534, 542, 547, 548, 561565, 572, 574 Metzner, 335 MichaeIides, 533 Michiyoshi, 285 Middleman, 617, 618 Minkowycz, 245,381,385, 391,397, 420, 433, 504, 537, 594, 614 Minto, 401404 Mitsotakis, 153 Miyamoto, 180, 189, 285 Moffat, 37 Mollendorf, 171, 285 Mongruel, 118, 119 Moon, 209, 233 Morgan, 209 Mori, 180 Morioka, 170, 174 M6rwald, 153 Moulic, 342 Moutsoglou, 254 Mucoglu, 105, 106, 245 Mulligan, 284 Na, 4, 30, 37, 342, 432, 453
Nachman, 460 Nakamura, S., 254, 356 Nakamura, T., 180, 189 Nakayama, 4, 377, 378, 461, 504, 585588, 590, 591, 597, 599, 602604, 617, 619 Nanbu, 285 Napolitano, 118 Narayanan, 210, 214216 Nath, 254, 453, 504, 524, 586, 617 Nayfeh, 254 Nguyen, D. L., 311,317, 318 Nguyen, H. D., 327, 574 Nguyen, T. H., 218, 222, 223 Nield, 377, 378, 381,397, 534, 617 Nilson, 118, 120, 122, 123, 126, 128 Noshadi, 171 Novotny, 22, 23
Oberbeck, 6 Ohnishi, 326 Okajima, 377, 461,462 Okihara, 492, 524, 530, 574, 575 Oosthuizen, 64, 326 Ostrach, 7, 12, 15, 23, 68, 118, 184, 473 Padet, 540 Paik, 327, 574 Pal, 555
Parikh, 37 Park, 299, 300, 302, 303 Patankar, 196, 344, 447
AUTHOR INDEX
651
Peaceman, 129 Peddieson, Jr., 357
Riley, N., 152, 154, 254, 268270, 285, 286,
Pera, 121, 126, 129131 Perelman, 180 Pereyra, 264 Perez, 151 Peterson, 397400, 437, 443, 444, 446 Peube, 233 Phan, 492 Plumb, 585, 594 Polidori, 540 Polisevski, 401 Polymeropoulos, 285 Poots, 142 Pop, 4, 37, 42, 69, 70, 73, 74, 77, 78, 96, 142, 151, 153, 155157, 180, 182, 185, 187, 200203, 215, 217, 223, 225, 226, 241, 251, 254, 261, 263, 264, 267, 306, 307, 310, 311, 320, 335, 342, 344, 346, 348, 356, 360, 362, 366368, 377, 378, 381, 391, 393, 396404, 415, 416, 418420, 424, 426, 431, 432, 435441, 446448, 451454, 461466, 468, 469, 471474, 480, 481, 484, 487, 489, 493, 495, 497, 498, 500, 504, 510, 512, 514, 518, 519, 524, 528531, 533536, 538, 542, 551, 553, 555, 556, 558, 561563, 567, 569, 570, 574, 575, 577, 578, 581583, 586, 603, 604, 617 Postelnicu, 251 Potsch, 171 Poulikakos, 435, 585, 588, 594, 617 Powell, 264 Pozi, 180, 183, 187 Prandtl, ix, 377
Roache, 129, 399, 444 Rodi, 171 Rosen, 334 Rotem, 87, 91, 92
Prasad, 446448 Qureshi, 218, 230, 233 Rachford, 129 Rahman, 118, 149 Raju, 106 Ramachandran, 254 Rao, 171 Rees, 30, 33, 34, 36, 96, 335, 342, 356, 360, 362, 363, 366368, 377, 378, 409, 411, 412, 415, 416, 418420, 432, 453455, 463, 480, 487, 504, 505, 507509, 586, 607611
Reid, 264 Rhyne, 284 Ridha, 7982, 84, 108, 109 Riley, D. S., 151,170, 171,174, 432, 453455, 487
327329, 332, 390, 460, 583
Rout, 210, 214216 Rudischer, 103, 104
Saito, 221 Saitoh, 218223 Safik, 218223 Sakakibara, 180 Sakiadis, 253 Salam, 617 Saljnikov, 346 Sammakia, x, 46, 118, 151 Sano, 3!0, 327, 492, 524, 530, 561,574, 575
Sarm a, 151 Savage, 171 Saville, 214 Savino, 118 Schlichting, 64 Schmidt, 7, 12, 285 Schneider, 4, 88, 102104, 106109, 151, 153, 159, 171
Schowalter, 333 Schrock, 491, 492 Scott, 134 Scurtu, 251 Semenov, 22, 23 Shamsher, 214 Shang, 368, 373 Shanks, 439, 472 Shayer, 491 Shenoy, 333, 336, 339341, 617 Shi, 254 Shih, H.C., 171, 173, 174, 176, 177 Shih, T. M., 4, 209, 233 Shih, Y.P., 254 Shu, 180, 451, 461, 538, 542, 617 Shulman, 336 Shvets, 346 Siegel, 285 Siginer, 333 Simpson, 311, 314, 316, 317, 327, 331, 332 Singer, 285 Skan, 72
Slaouti, 254 Smith, A. M. 0., 241 Smith, N., 238, 239 Smith, S. H., 555 Sokovishin, 4, 171, 180 Song, 311 Sparrow, 12, 15, 19, 23, 36, 61, 105, 106, 230, 245
Srinivasan, 22, 118, 149, 151
652
CONVECTIVE FLOWS
Steinberger, 500, 590, 591 Steinrfick, 103105, 107, 108 Stewartson, 49, 87, 90, 91,254, 257, 258, 311, 314, 316, 317, 465, 543
Storesletten, 377 Subba Reddy, 151 Sugawara, 285 Sumikawa, 180, 189 Sundfcr, 561, 574 Swire, 236 Sylvester, 335 Takhar, 151, 168, 254, 335, 342, 344, 356 Taniguchi, 42 Tanimoto, 180 Tanner, 333 Terrill, 50, 52, 55, 57, 187, 209, 231 Thiele, 612614 Thomas, 151, 168 Throne, 254 Tien, 335, 336, 585 Tohda, 617, 620 Trevifio, 180, 473, 480 7Yevisan, 119 Tsai, 254 Tsuruta, 617, 620 Turk, 335 Turner, 4, 118 Tyvand, 561, 574
Uehara, 170, 174 Ulbrecht, 336, 339341 Umemura, 88 Upadhyay, 333 Vafai, 585 Va.jravelu, 254 Van Dyke, 153, 302, 561 Vasantha, 586 Vdzquez , 151 Vedhanayagam, 37 Vishnevskiy, 346 Viskanta, 180, 285 Viviani, 118 Volino, 4 Vynnycky, 180, 189, 193195, 446448, 461 Waiters, 333 Wang, C. Y., 151, 170 Wang, H., 401
Wang, P., 218, 222, 223, 311,317, 318 Wang, T.Y., 107, 336, 352, 353, 356 Ward, 585 Wasel, 106 Watanabe, 37, 42, 368 Watts, 617
Weidman, 36, 88, 112114, 151,164, 166168, 431,432, 437, 440442, 457460
Weiss, 103, 104 White, 617 Wickern, 106, 107 Wilcox, 131 Wilkes, 286, 308 Wilks, 50, 52, 53, 5557, 94, 96, 151, 171, 174, 185, 201, 203, 215, 311, 466, 470, 487 Williams, 284
Womersley, 42 Wooding, 377, 494, 617 Worster, 151 Wright, 463, 480 Wu, C.M., 149 Wu, K.Y., 254, 276279 Xu, 254 Yamamoto, 492, 524, 529, 581, 582 Yah, 254, 261, 263, 264, 267, 524, 575, 577, 578, 581583 Yang, A. J., 356 Yang, J., 30 Yang, K. T., 22, 23, 284, 285, 326 Yang, Y. T., 342 Yao, 79, 239241, 243, 251, 341, 342 Yasuda, 617, 620 Yih, 617 Yovanovich, 43 Ytrehus, 370 Yu, 88, 97, 98, 100, 151, 160163, 171, 173, 174, 176, 177, 206, 207, 592596 Yuan, 180, 200203, 487 Yiicel, 356, 504 Yiincfi, 222
Zaltsgendler, 336 Zaturska, 317, 387390 Zeh, 87, 90, 102,473 Zhang, 394, 431,434, 435, 534, 542, 547, 548