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Mixed convection over a horizontal plate: self-similar and connecting boundary-layer flows
FLUID DYNAMICS RESEARCh ELSEVIER
Fluid Dynamics Research 15 (1995) 113-127
Mixed convection over a horizontal plate: self-similar and connecting boundary-layer flows Herbert S t e i n ~ c k Institut fiir StrOmungslehre und Wiirme~bertragung, TU-Wien, Wiedner Hauptstr. 7, 1040 Wien, Austria
Received 22 August 1994; revised 28 October 1994
Abstract
The boundary-layer flow over a horizontal plate is considered. In case that the heat transfer is limited to the leading edge of the plate two similarity solutions exist, if the buoyancy parameter is above a critical value. By an asymptotic expansion with respect to buoyancy parameters near the critical value it is shown that steady-state (non-similar) solutions connecting both similarity solutions exist. Finally a slowly varying transient solution is constructed. The downstream behavior of the perturbation determines which steady flow will be attained.
1. Introduction
Consider a flat horizontal plate aligned parallel to a uniform flow with velocity Uo~ and temperature To~. It is assumed that a heat source (or sink) of strength Q is at the leading edge of the plate and that the plate is adiabatic everywhere else. In the limit of large Reynolds number Re (boundary layer approximation) a similarity solution exists (see Schneider 1979). Originally this similarity solution had been found for a given temperature distribution on the surface of the plate. Besides the Prandtl number Pr a second dimensionless parameter K appears in the dimensionless equations which describes the strength of the buoyancy effects. The limiting cases are the forced convection, where buoyancy effects are negligible, and free or natural convection, where the velocity of the free stream is negligible. We are interested in mixed convection that is when both effects play a role. For the flow above a plate heated at its leading edge (K > 0) the similarity solution is unique. For K = 0 the "Blasius" solution is obtained and for K --+ c~ the similarity solutions tends to the solution of the free convection problem. In case of the flow above a plate cooled at its leading edge (K < 0) two solution branches exist for buoyancy parameters K between some critical value Kc < 0 and K = 0 (Schneider 1977, Afzal and Hussain 1984, de Hoog, Lamminger and Weiss 1984, Merkin and Ingham 1987). Both solution branches are connected at Kc. For K small and negative, one solution can be viewed as a perturbation of the Blasius solution, where the buoyancy effects do not influence 0169-5983/95/$4.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 01 69-5983 (94)00052-2
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H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
the velocity profile significantly, while the other solution branch has a large region with reverse flow. When K approaches zero this second solution becomes singular and thus cannot be continued to K=0. The aim of the present work is to discuss which of the two solutions for K < 0 is of physical relevance and stable with respect to small perturbations. After presenting the basic equations we discuss briefly the similarity solutions. Then steady flows are presented which connect both similarity solutions. Then a restricted stability analysis will be given for buoyancy parameters close to the critical value Kc.
2. B a s i c e q u a t i o n s
The modified boundary layer equations for the mixed convection flow above a horizontal plate in dimensionless form are (see for example Schneider 1979)
tgU
t~U
0U
t~p --~ --02U
at + U-~x + V~y = - O x
(2.1)
Oy2'
0 = ---OP + KO, Oy 9u Ov
(2.2)
Ox + ~yy = O, O0
--
00
(2.3)
00
at + U~x + V~y
-
1 6320 Pr
(2.4)
3y 2'
where the dimensionless coordinate x parallel to the plate is made dimensionless with the (arbitrary) reference length L and the dimensionless coordinate y perpendicular to the plate is referred to L/x/--~, where Re = U~L/v is the Reynolds number and v the kinematic viscosity. The velocity components u, v parallel and perpendicular to the plate are scaled with the velocity of the free stream U~ and Uo~/x/-~, respectively. The difference 0 between the temperature of the disturbed and undisturbed fluid is scaled with Tw(L) - T ~ in the case of a prescribed plate temperature distribution Tw(x). In the case of a given heat source at the leading edge of an adiabatic plate a scaling value for the temperature difference is given by AT I Q o / p ~ c p ~ ] , where Q0 is a reference value for the strength Q(t) of the heat source, p ~ is the density of the unperturbed fluid and Cp the isobaric specific heat capacity. The dimensionless buoyancy parameter K can be referred either to a temperature difference, i.e. Kr = gfl(Tw(L) - T~)x/Lv/-ffU~ 5/z, or to the strength of the heat source at the leading edge of the plate, i.e. K o = gflQo/U3cpp~, where /3 is the volumetric expansion coefficient and g the gravity acceleration. A reference value for the dimensionless skin friction r is p~U2~/Re. Note that according to this scaling we have r = (Ou/Oy)(x, 0). The boundary conditions at the plate are given by the no-slip conditions =
u(x,O,t)=O,
v(x,O,t)=O,
x>0,
(2.5)
and either the temperature difference to the unperturbed fluid, i.e.
O(x,O,t) = Ow(x,t),
(2.6)
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Dynamics Research 15 (1995) 113-127
the velocity profile significantly, while the other solution branch has a large region with reverse flow. When K approaches zero this second solution becomes singular and thus cannot be continued to K = 0. The aim of the present work is to discuss which of the two solutions for K < 0 is of physical relevance and stable with respect to small perturbations. After presenting the basic equations we discuss briefly the similarity solutions. Then steady flows are presented which connect both similarity solutions. Then a restricted stability analysis will be given for buoyancy parameters close to the critical value Kc.
2. Basic equations The modified boundary layer equations for the mixed convection dimensionless form are (see for example Schneider 1979)
flow above a horizontal
plate in
(2.1) 0=-z
+K0,
(2.2)
where the dimensionless coordinate x parallel to the plate is made dimensionless with the (arbitrary) reference length L and the dimensionless coordinate y perpendicular to the plate is referred to L/v%, where Re = U,L/v is the Reynolds number and v the kinematic viscosity. The velocity components U, u parallel and perpendicular to the plate are scaled with the velocity of the free stream U, and U,/&, respectively. The difference t9 between the temperature of the disturbed and undisturbed fluid is scaled with T,,,(L) - T, in the case of a prescribed plate temperature distribution T,(x) . In the case of a given heat source at the leading edge of an adiabatic plate a scaling value for the temperature difference is given by AT = IQO/~oo~pdmlr where Q0 is a reference value for the strength Q(t) of the heat source, poo is the density of the unperturbed fluid and cp the isobaric specific heat capacity. The dimensionless buoyancy parameter K can be referred either to a temperature difference, or to the strength of the heat source at the leading edge i.e. KT = @(T,(L) - T,)fifiU;5f2, of the plate, i.e. Ke = g~Qo/U~cppm, where p is the volumetric expansion coefficient and g the gravity acceleration. A reference value for the dimensionless skin friction r is poopmIRe. Note that according to this scaling we have T = (&/a~) (x, 0). The boundary conditions at the plate are given by the no-slip conditions U(X,O, t) = 0,
U(X,O, t) = 0,
and either the temperature 0(-L 0, t) = &(&
t),
difference
x > 0, to the unperturbed
(2.5) fluid, i.e. (2.6)
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0"3 I 0.2
f"(o)
o.;[___, -0.1
h'~
-0.05 K
Fig. 1. Similarity solution: wall shear stress f"(0) as a function of the buoyancy parameter K for Pr= 1. subjected to the boundary conditions
f ( x , O, t) = f ' ( x , O, t) = f ' ( x , c¢, t) - 1 = O(x, c¢, t) = h(x, oe, t) = O, O(x, O, t) = Ow(X, t),
(3.5)
and the initial conditions
f ( x , rl, O) = ft(x,~7),
O(x, rl, O) = O i ( x , rl).
(3.6)
Here and in the following derivatives with respect to r/ are denoted with a prime. Self-similar solution can be found for Ow(X, t) = 1. Then the boundary layer equations are reduced to the ordinary differential equations
2f'"+ff/'+K~70=O,
2Pr0 ' + f O = O ,
(3.7)
with the boundary conditions f(0) = f'(0) =0,
f ' ( c ¢ ) = 1,
O(0) = 1.
(3.8)
The solution of the similarity equation has been discussed in many papers (Schneider 1979; Afzal and Hussain 1984; de Hoog, Lamminger and Weiss 1984; Merkin and Ingham 1987; Daniels 1992). In Fig. 1 the wall friction coefficient r = f " ( 0 ) versus the buoyancy parameter K for Pr = 1 is shown. In order to determine the critical value Kc, where both solution branches meet, the buoyancy parameter and the similarity solution are parametrized with the wall shear stress ~- = f " ( 0 ) . Since at Kc the buoyancy parameter has a minimum (as a function of r ) , the derivatives Fc = (Of/O~') (zc), Dc =
(aOlaT)
I
Hc = (an~aT)
11! l/ l/ 2Fc + f c C + f'
satisfy
C-KcSc\
(2/Pr)O'c' + (fcDc + OcFc)'] = 0, H'c+ (~TDc)' /
(3.9)
with the boundary conditions Fc(0) = F~'(0) = F~'(oc) = Dc(OC) = Hc(c¢) = De(0) = 0,
(3.10)
11. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
117
and Fc"(0) = 1, where fc, Oc (3.7), (3.8) Following parametrized
(3.11)
is the similarity solution at the critical value Kc. To determine Kc the similarity equations and (3.9)-(3.11) have to be solved simultaneously. Merkin and Ingham (1987) the similarity solution near the critical point K~ can be with e = q - x / K - Kc:
( f , O, h) = (fc, Oc, he) + e ( f l , O1, hi) + e2(f2,192, h2) + O(e3).
(3.12)
The first-order correction f~ satisfies the linear equation (3.9) with the homogeneous boundary conditions (3.10). The linear homogeneous boundary value problem has the non-trivial solution f l = Cfc,
Ol = CDc,
(3.13)
hi = CH~,
with a yet undetermined constant C. For f2 one obtains: ( 2 / P r ) O ~ ' + (fc02 +O~f2)' h~ + (r/O2)'
=-C 2
(FcDc)' 0
+
0 0
,
(3.14)
with the boundary conditions (3.10). Since the corresponding homogeneous problem has a nontrivial solution, a solution of (3.14) cannot satisfy all boundary conditions (3.10) for an arbitrary inhomogeneity. Thus the right side of (3.10) has to satisfy the solvability condition (3.15)
C2-C~=0,
with the solutions -t-C0, (Co > 0). The solvability condition (3.15) is derived in appendix A.1. Thus for e > 0 the upper solution branch is obtained and for e < 0 the lower solution branch is obtained. 3.2. Connecting flows
In case of negative buoyancy parameters K < 0 two different self-similar flows a)e possible. It is of interest if other steady (non-self-similar) flows exist. In this section a certain class of steady (non-similar) flows will be described. It will be shown that steady flows exist, which are near the leading edge close to one self-similar flow and far downstream close to the other self-similar flow. In order to apply asymptotic methods the analysis is restricted to a neighborhood of the critical point
Kc. The main idea is to replace the coordinate x parallel to the plate by a slowly varying variable ~: = e In x and to expand the stream function, temperature distribution and pressure gradient in terms of e: f ( x , r/, e)
O(x, rl, e) h(x, rl, e)
=
Oc(n)
hc (r/)
+e
O~((,n) h 1((, r/)
+e 2
O2((,n)
+..-
(3.16)
h2(sc, r/)
Inserting the expansion (3.16) into the boundary-layer equations yields for the first order terms again Eqs. (3.9)-(3.10). But now C is a yet unknown function of (, i.e. f t ((, T1) = C (s~) Fc(rl).
(3.17)
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H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
For the second order terms (f2, 82, h2) one obtains
(2/Pr)O~+(fcO2+Ocf2)'| =-C 2 + /
(FcD~)' o
+
0 o
+C¢
f~Dc
0'¢F~ , (3.18)
with the boundary condition (3.10). Eqs. (3.18), (3.10) have a solution only if the solvability condition
(2Co/A1)C~ = C 2 - C~,
(3.19)
where A1 is a positive constant, is satisfied. The solvability condition (3.19) and values for A1 and Co are derived in appendix A.1. Numerical computations show that both constants are positive. The general solution of (3.19) is given by 1 - yexp(Al()
1 -- yx ~al
(3.20)
C(¢) = Co 1 + y e x p ( A l ( ) = C°l q-3/Xeal'
where y is an arbitrary positive constant. Thus a steady flow has been determined with the uniformly valid expansion yx ~a' Fc(r/) + O(s2). + yx ~a'
1 -
f ( x , rl,~) = f~(r/) +sC01
(3.21)
For x ,-~ 0 this flow behaves like the self-similar flow corresponding to the similarity solution on the upper branch and for x --+ oc the flow behaves like the self-similar flow corresponding to the similarity solution on the lower branch. A connecting flow, which connects the 'lower' self-similar flow with the 'upper' self-similar flow has not been found. Moreover in the next subsection a mathematical argument will be given why such a 'reverse' connecting flow is impossible. Here the existence of connecting flows is shown only near the critical value Kc of the buoyancy parameter. But there is numerical evidence (see section 3.4) that connecting flows exist for Kc < K<0. For negative values of 3/ the function C(sc) becomes singular at (* = l n ( - y ) / a l and thus the expansion (3.16) is not valid there. Numerical computations show that in case y < 0 the boundarylayer flow field has a singularity where the wall shear stress tends to infinity. This type of singularity for mixed convection boundary layer flows has been analyzed by Daniels (1992). However it is unclear if such singular solutions are of physical significance.
3.3. An eigenvalue problem In this section a necessary condition for the existence of connecting flows will be discussed. Let
f ( x , ~7), O(x, 7) and h(x, ri) be the stream function, temperature distribution and pressure gradient of a connecting flow. Then near the leading edge of the plate f , O and h can be expanded with respect to x << 1: O(x, 77)
= /Os('q)
h(x, 71)
\ h~(rl)
+ x '~
D(,q)
H(rl)
+o(x'~),
(3.22)
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where fs, Os and hs are a similarity solution and A is a positive constant. Inserting into the boundary layer equations and neglecting higher order terms yields the eigenvalue problem (2/Pr)D"+ (fD + OF)' ] = a H' + ( n O ) ' /
I ' D - O'F D
,
F ( 0 ) = F ' ( 0 ) = F ' ( o c ) = D ( e c ) = H(e~) = D ( 0 ) = 0,
(3.23) (3.24)
for ,~ and F, D, H. Thus a connecting flow which connects fs, Os, hs with another self-similar flow exits only if the eigenvalue problem (3.23), (3.24) has a positive eigenvalue. Analytically the existence of a positive eigenvalue can be shown in two cases. 3.3.1. A positive eigenvalue f o r - K << - 1 For IK[ << 1 a positive eigenvalue with the singular expansion
a = -K--5 + ~5 at + . . . .
with
a~- = 0.00378,
(3.25)
exists. All other eigenvalues are negative. Details of the expansion can be found in appendix A.2. 3.3.2. A positive eigenvalue near the critical value For ~2 = K - Kc << 1 there exists an eigenvalue with an expansion
,~= eAl + O ( e Z ) ,
with
~1 > 0.
(3.26)
This eigenvalue is positive on the upper branch and negative on the lower solution branch. Note the value of A1 is the same as in (3.19). Details are deferred to appendix A.3. In Fig. 2 the positive eigenvalue is shown as a function of K. The positive eigenvalue A(K) has in agreement with (3.25) a singularity at K = 0, stays positive on the entire upper solution branch and has a zero at K = Kc. Numerical computations show that all other eigenvalues corresponding to similarity solutions on the upper branch are negative. On the lower solution branch all eigenvalues are negative as long as there is no reverse flow. This has the following consequences (i) A connecting flow can branch off from the self-similar flow corresponding to a similarity solution on the upper solution branch for Kc < K < 0 (ii) No connecting flow can branch off form the self-similar flow corresponding to a similarity solution on the lower solution branch with a positive wall shear stress. (iii) For self-similar flows with a reverse flow region no statement can be made if a connecting flow can branch off. Numerically only connecting flows which connect the upper similarity solution with the lower have been observed. 3.4. Numerical results
The existence of connecting flows has been shown only for values of the buoyancy parameter K close to the critical value Kc. For K - K c not small one has to compute the connecting flow numerically. The equations for steady boundary layer flows can be considered as an evolution problem with respect
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1000.
100. A+I 10.
1 -0.1
K.
-0.05 K
Fig. 2. Eigenvalue of the Eigenvalue problem (3.23), (3.24) as function of the buoyancy parameter K on the 'upper' and 'lower' solution branch.
to the coordinate x parallel to the plate. The initial value problem with initial data given at x = 0 is due to the existence of a positive eigenvalue not uniquely solvable. Therefore the expansion (3.22) for small x is used to formulate an initial condition at a given x0 > 0. Let f ( x 0 , r/) = f~(r/) + rx~F+(rl),
O(Xo, 77) = Os(rl) + rx~O+(rl),
(3.27)
where r is an arbitrary small parameter. The stationary equations can now be solved by a marching technique. One has to take care to resolve the positive eigenvalue correctly. The computations have been performed with Pr = 1 and different values of the buoyancy parameter K. We have chosen r = - 1 0 -5 and x0 = 1. In Fig. 3 the wall shear stress is plotted as a function of lnx for K = - 0 . 0 8 and is compared with the asymptotic solution (3.21). As expected it tends to the value corresponding to the similarity solution on the lower branch for x -* ~ . For K > - 0 . 0 5 2 0 4 the lower solution branch has a region of reverse flow. A connecting solution for K = - 0 . 0 4 has been computed. The wall shear stress has a zero without any singular behavior. Thus the numerical computation can be extended beyond the point of separation as long as the region of reverse flow is not too large. Again the numerical solution connects the two similarity solutions. In Fig. 4 the wall shear stress is shown and in Figs. 5 and 6 the velocity and temperature profiles for K = - 0 . 0 4 and different values of x are shown. The 'initial' function (3.27) can be replaced by any 'initial' function with a negative component in the direction of the eigenfunction (F+, D+). Initial conditions with a positive component in the direction of the eigenfunction (F+, D+) lead to singular flow fields corresponding to the case y < 0 in equation (3.20).
4. A restricted stability analysis After finding steady flows connecting both self-similar flows the stability of these flows might be of interest. Here a restricted stability analysis will be performed, by what we mean that only certain initial perturbations are considered. In order to employ asymptotic methods K - Kc << 1 is assumed. The main idea is to construct, a solution with an asymptotic expansion of the form
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
121
0.3
0.15
0.2 f"(x, o)
f"(x, O) 0.1
o.1
O-
0.0~
1'0
2'0 lnx 3'0
4'0
-0.1
50
I
0.5
lnx
1
1.5
Fig. 3. Dimensionless wall shear stress for a connecting solution for K = -0.08 (s = 0.106) and Pr = 1. Comparison of the asymptotic solution (dashed line) with the numerical solution. Fig. 4. Dimensionless wall shear stress for a connecting solution with K = - 0 . 0 4 and Pr = 1. 0.05
/
1
U
0.5
0 0
2
4
6
8
-o.o
10
015
i
115
i
2.5
Fig. 5. Velocity profiles of the connecting solution with K = - 0 . 0 4 and Pr = 1 for various values of the x-coordinate: lnx = 0.25, 0.5, 0.75, 1., 2., 3. and similarity solutions (dashed lines). For 0 < ~ < 10 (a) and 0 < r / < 2.5 (b).
f ( x , rl, t , e ) = f c ( r l ) + e f l ( ( , ( , r l )
+ eZf2 + . . . .
(4.1)
w h e r e t h e i n d e p e n d e n t v a r i a b l e ~:, ( a r e d e f i n e d b y
~: = e l n ( x ) ,
( = e ln(1 +
t/x).
(4.2)
T h e n t h e p a r t i a l d i f f e r e n t i a l o p e r a t o r s a r e t r a n s f o r m e d to 0
x~
O
t
O
= s-d~ - s x + t - ~ '
3
x
3
1
and then the first-order terms fl,Ol,hl satisfy (3.9)-(3.10)
O1 (se, ~', 7/) h l ~ , ( , r/)
0
x -at- = e - - x + t a ( = e y + t / x a ( '
= C(~:, ( ) { D c ( ~ 7 ) \ Hc(~7)
,
(4.3)
a n d are g i v e n b y
(4.4)
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
122
\\\\\ \\\ \\\\\
\\ \\\
0.I
06
7/ 6
8
10
Fig. 6. Temperature profiles of the connecting solution with K = -0.04 and Pr = 1 for various values of the x-coordinate: In x = 0.25, 0.5, 0.75, 1., 2., 3. and similarity solutions (dashed lines). where C is now a function of ( and (. Obviously the initial conditions (3.6) have to be compatible with the the expansion (4.1) and thus with (4.4). Therefore the initial condition has to be of the form
O(x,~,O;e)
=
Oc(~)
+C1(eln(x))
+O(e2).
\ De(7/)
(4.5)
To determine C ( s q sr) the equation for the second order terms have to be considered. Since the derivative of C with respect to ( is multiplied by a factor depending on o- = t / x the second order correction f2 has to be a function of ~/,s~,( and tr. Thus
( 2 / P r ) G ' + ( f e 0 2 + G f2) h; + ('/']'/~q'2) t + ~
~ f;
= -c 2
c -- O;Fc + De D~
+
(Fe c)'
+
+ (G - G)
02.,~ tr(f;02,,~ - O;f2,,~) -o'02,,~
.
I;Do - GFo Dc (4.6)
This is a partial differential equation for f2, 02, h2. For o- ~ oo all o" dependent terms on the right side of (4.6) have to vanish and an ordinary differential equation for f2, 02, h2 is obtained, which is solvable if the solvability condition (2Co/al)(C• -- C¢) = C 2 -- C 2,
(4.7)
is satisfied. This is a hyperbolic partial differential equation for C with its characteristics pointing upstream. Thus small perturbations can propagate upstream. The general solution of (4.7) is given by C 0 + Ci(~Jf - ()
-
(C o -
Ci(¢--]-
•)) exp(--Al()
C (~, ( ) = Co Co + C, (/j + ( ) + (Co - C I ( ( + ( ) ) exp(-zl, sr) " Using the x, t variables the solution reads as
(4.8)
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H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
f(x,~7, t) = fc(rl) CCt(~ln(x %- t)){1%- [ x / ( x %- t)]*a'} %- C0(1 - [x/(x%- t)]~a') r , , %- eGo It ln(x %- t)){1 -- [ x / ( x 4- t)] ~hl} %- C0(1 ~ [ x / ( x + t ) f ~ rct~7) + O(e2)"
(4.9)
Stability of the "upper' self-similar flow The asymptotic behavior for t ~ c<~ can be discussed easily. If lime__,~ Ct(~:) 4: -Co and C1 ( ( ) > -Co it can be verified that limt__.~ C (x, t; ~) = Co which corresponds to the upper solution branch. Thus the upper similarity solution is stable against small perturbations of the form (4.5).
Stability of the 'lower' self-similar flow and the connecting flows The stability behavior of the lower similarity solution and the connecting solutions is more complicated. Since the characteristics of (4.7) are pointing upstream the time dependent behavior of the solution at a fixed location x will dependent on the downstream behavior of CI (so). Assume that C~ ( ( ) has an asymptotic expansion of the form
CI(() = - C 0 ( 1 - 2 y e x p ( - c e ( ) + O ( e x p ( - a ( ) ) ) ,
for
~: ~ oo,
with
y > 0.
(4.10)
where y and ce are constant. Then inserting into (4.9) yields
lim C ( x, t; e ) =
t---+~
/
c0i
Co
-Co,
X~AI
.
XF,,~I ,
0[ "-~" i~|
upper solution connecting solution .
(4.11)
a > Al lower solution
Thus the 'lower' self-similar flow is stable with respect to small perturbations which decay faster than the connecting flow for x ~ cx~.
5. Conclusions A new type of steady flows for K < 0 connecting the self-similar flow corresponding to the solution on the upper branch with the self-similar flow corresponding to the solution on the lower branch has been described. Discussing the physical relevance of this one parametric family of steady flows a restricted stability analysis has been performed showing that perturbations of a steady flow propagate upstream. The 'upper' self-similar flow turns out to be stable against small perturbations. The lower self-similar solution and the connecting flows are in general unstable with respect to small perturbations. They are only stable with respect to small perturbations which decay fast enough for x --, cx~. Considering the boundary layer flow only over a finite length L a downstream boundary condition is required, since the characteristics of the perturbation equation (4.7) point upstream. This downstream boundary condition selects uniquely one solution of the one parametric family of solutions described in section 3.2 and stabilizes it with respect to perturbations of the form (4.5). The analysis presented here for the case of a controlled plate temperature holds in case of an adiabatic plate with a prescribed heat sink at its leading edge (see appendix A.4) also. Note that the self-similar flows are identical in both cases, but the corresponding buoyancy parameters Kr
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H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
and KQ are not. Let rr,~ , ~'0,c be the dimensionless wall shear stress where Kr and K o resp. have a minimum. Then rr, c is different from To,c and thus fo,c 4: fr,¢. A self-similar flow with ~'o,~ > f " ( 0 ) > ~'r,c corresponds to a similarity solution on the upper branch when referred to a fixed temperature distribution and the same flow corresponds to a similarity solution on the lower branch when referred to a given heat sink at the leading edge of the plate. For Pr = 1. we have 7"o,~ = 0.120 and ~zc = 0.106. To observe the self-similar flow on the lower solution branch in an experiment one can proceed as follows. A plate is needed which can be switched from temperature control to heat flux control. Starting with the Blasius solution Kr is slowly (quasi statically) decreased. When the skin friction ~is between f o x and ~'T,c the plate is switched to adiabatic boundary conditions. Now the self-similar flow corresponds to a 'lower' solution. Then K o can be quasi statically increased. One has to take care of the downstream edge of the plate that no perturbations are produced there. Small perturbations at the downstream edge would change the flow into a connecting flow or into the 'upper' self-similar flow.
Acknowledgements I want to thank Prof. W. Schneider for many stimulating discussions. The support of the Austrian Science Foundation under project no. P9584-TEC is also acknowledged.
Appendix A A.1. The solvability condition
The solvability condition for (3.7) and (3.18) can be obtained as follows. Let (f0, O0, ho), ( f l , 01, hi), (f2, O2, h2), be the unique solution of (3.18) with the boundary conditions fk(0)=f~'(0)=Ok(0)=f~(c¢)
=Ok(cx~)=hk(c~Z)=0,
k=0,1,2
,
(A.1)
with (C 2 = 0, C¢ = 0), (C 2 = 1, C¢ = 0) and (C 2 = 0, Ce = 1) respectively. Then the solution of (3.18) with the boundary condition (A.2)
f 2 ( 0 ) ~- 0 2 ( 0 ) = f ~ ( o O ) = 0 2 ( 0 0 ) = h 2 ( o o ) = 0,
is given by
O2 h2
=
00 ho
+C 2
01 hi
O0 ho
+C¢
02 h2
00 ho
+F
Dc Hc
,
(A.3)
where F is an arbitrary constant. Since F~'(0) = 0, (A.3) is a solution of (3.18) with the boundary conditions (3.10) if and only if f~(O) + C2(f'~(O) - f ~ ( 0 ) ) + C~(f~(O) - f ~ ( 0 ) ) = 0.
(A.4)
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
125
Setting (, f~(O) ),/2 Co= \f~(O) --7~(0)
and A, =
2{f~(O)[f~_f~(O)]},/2 f~(O) - f~(O)
(A.5)
the solvability condition (3.19) is obtained. The similarity solutions on the upper and lower branch correspond to the constant solutions Co and - C o resp. of (3.19). A.2. Positive eigenvalues for - K << 1
For K << 1 the similarity solution (on the upper solution branch) can be expanded in terms of K: f = fB+Kfl+O(K2),
(A.6)
O=OB+KOI+O(K2),
where fB is the Blasius solution and OB the corresponding temperature distribution. Thus the eigenvalue and the eigenfunction are expanded A=
(A.7)
AO 4- K A I 4- . . . .
(A.8)
( F , D , H ) = (Fo, Do, Ho) 4- K(F1,D~,H~) 4- . . . .
in terms of K. Then the eigenvalue problem for the leading order terms are decoupled:
2F~" + fBF~' + f~'= Ao(f~FD-- f~'Fo), (2/Pr)D~o' 4- (fBDo) l = 2of~' D o,
Fo(O) = F~(O) = FD(oo) =O,
Do(0) = 0,
(A.9) (A.10)
Do(c~) = 0.
Both eigenvalue problems can be written in self-adjoint form and thus have only real eigenvalues. In Libby and Fox (1964) it was shown that all eigenvalues of the momentum equation (A.9) are negative. By partial integration it can be shown that the eigenvalues of the energy equation (A.10) are bounded by 1/2. A numerical computation shows that the largest eigenvalue is negative. Besides the eigenvalues with the regular expansion (A.7) an eigenvalue with a singular expansion for K ---, 0 - exists: a
aft at = - K--3-+ -~- + . . . .
(A.11)
Inserting (A.11) yields a singularly perturbed eigenvalue problem and one has to expect a matched asymptotic expansion for the eigenfunctions. The outer expansion is given by F + = Po+ + K P ? . . . .
D+ = b g + K b t . . . .
H+
= g-3/~
- 4- g - 2 / ~ l +
. . .
(A.12)
The leading order terms are obtained as Po+ = A ,
b ~ = a~,,
(A.13)
~o" = a ~ a B .
Obviously not all boundary conditions can be satisfied at r / = 0 and an inner expansion is needed: F + = KP~-(rl/K) + . . . .
D + = KD~(rl/K) + ....
¢ = rl/K.
(A.14)
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
126
The resulting equations for the leading order terms are identical with the equations for the leading order terms of the eigenvalue problem in Steinriick (1994) and thus one obtains
=-2 IAi' o,(iAi s ds) ]=ooo378 oo
-1
3
(A.15)
o
where Ai denotes the Airy function. Thus the eigenvalue A+ is negative for K > 0 and positive for K < 0.
A.3. A positive eigenvalue for K - K~ << 1 Near K~ an eigenvalue and eigenfunction exist with an expansion of the form A=sA1 + . . . ;
F = F~ + sFI + . . . .
D = D¢ + sDI + . . . .
H = Hc + sHI + . . . .
(A.16)
where ,~, and F1 are determined form
2F;" + fcF;' + f~lFi - K~H, ( 2 / P r ) D ' / + (f~D~ + OcD,)' ] = -2C0 H', + (~TD,)' ,/
( ) FoF"
(FcDc)' o
f
+ A1
-
f~ ' D¢'
;o
X tg'~Fc
)
,
(A.17)
with the homogeneous boundary conditions (3.10). Comparing (A.17) with (3.18) yields that Al is given by (A.5).
A.4. The adiabatic plate Here the necessary changes of the previous analysis for the case of an adiabatic plate are presented. The boundary condition O(0) = 1 has to be replaced by oo
f f'Odrl = 1.
(A.18)
0
Let fr, Or be a similarity solution of (3.7), (3.8) for the buoyancy parameter Kr. Then it can be verified that /r
To=iT,
N
--1
OQ= ( / f~Ord~7) \
,t
Or,
(A.19)
/
o
is a solution of (3.7) with the boundary conditions cx)
f(O)
fl(O) =0,
f'(oc) = l,
/ f'Odrl = 1.
(i.20)
0
for the buoyancy parameter K o = KT f o f~Or dl7 • The connecting flows with a prescribed temperature distribution Ow(x) = 1 have a non-zero heat transfer at the plate. Thus the connecting flows of the temperature and the adiabatic case are different.
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
127
But the analysis of the connecting follows the same lines. Only the boundary condition Dc(0) = 0 has to be replaced by
(fc D+Fd '
dr/=0.
(A.21)
0
For non-steady flows the enthalpy flux condition is given by
/ f' O drl o
O, drl dx' = 1. o
(A.22)
o
For a slowly varying transient solution described in section 4 the integral Assuming ~:, ( > 0 one obtains:
Jo Ot dx' has to be estimated.
t/x')) d x ' = eC¢(u, 0) ln(1 + 1/o-) + O(e2).
(A.23)
j~C
(e lnx', e(1 +
0
This term vanishes for o- ~ cxD and thus an equation of the same type as (4.7) is obtained as solvability condition. Thus the results of the restricted stability analysis of section 4 apply in the case of an adiabatic plate also. References
Afzal N. and T. Hussain (1984) Mixed convection over a horizontal plate, ASME J. Heat Transfer 106, 240-241. Daniels, EG. (1992) A singularity in thermal boundary-layer flow on a horizontal surface, J. Fluid Mech. 242, 419-440. de Hoog, ER., B. Lamminger and R. Weiss (1984) A numerical study of similarity solutions for combined forced and free convection, Acta Mech. 51, 139-149. Libby, E and H. Fox (1964) Some perturbation solutions in laminar boundary layer theory. Part I: The momentum equation, J. Fluid Mech. 17, 433-449. Merkin, J.H. and D.B. Ingham (1987) Mixed convection similarity solutions on a horizontal surface, ZAMP 38, 102-116. Schneider, W. (1979) A similarity solution for combined forced and free convection flow over a horizontal plate, Int. J. Heat Transfer 22, 1401-1406. Steimiick, H. (1994) Mixed convection over a cooled horizontal plate: Nonuniqueness and numerical instabilities of the boundary layer equations, J. Fluid Mech. 278, 251-265.
Fluid Dynamics Research 15 (1995) 113-127
Mixed convection over a horizontal plate: self-similar and connecting boundary-layer flows Herbert S t e i n ~ c k Institut fiir StrOmungslehre und Wiirme~bertragung, TU-Wien, Wiedner Hauptstr. 7, 1040 Wien, Austria
Received 22 August 1994; revised 28 October 1994
Abstract
The boundary-layer flow over a horizontal plate is considered. In case that the heat transfer is limited to the leading edge of the plate two similarity solutions exist, if the buoyancy parameter is above a critical value. By an asymptotic expansion with respect to buoyancy parameters near the critical value it is shown that steady-state (non-similar) solutions connecting both similarity solutions exist. Finally a slowly varying transient solution is constructed. The downstream behavior of the perturbation determines which steady flow will be attained.
1. Introduction
Consider a flat horizontal plate aligned parallel to a uniform flow with velocity Uo~ and temperature To~. It is assumed that a heat source (or sink) of strength Q is at the leading edge of the plate and that the plate is adiabatic everywhere else. In the limit of large Reynolds number Re (boundary layer approximation) a similarity solution exists (see Schneider 1979). Originally this similarity solution had been found for a given temperature distribution on the surface of the plate. Besides the Prandtl number Pr a second dimensionless parameter K appears in the dimensionless equations which describes the strength of the buoyancy effects. The limiting cases are the forced convection, where buoyancy effects are negligible, and free or natural convection, where the velocity of the free stream is negligible. We are interested in mixed convection that is when both effects play a role. For the flow above a plate heated at its leading edge (K > 0) the similarity solution is unique. For K = 0 the "Blasius" solution is obtained and for K --+ c~ the similarity solutions tends to the solution of the free convection problem. In case of the flow above a plate cooled at its leading edge (K < 0) two solution branches exist for buoyancy parameters K between some critical value Kc < 0 and K = 0 (Schneider 1977, Afzal and Hussain 1984, de Hoog, Lamminger and Weiss 1984, Merkin and Ingham 1987). Both solution branches are connected at Kc. For K small and negative, one solution can be viewed as a perturbation of the Blasius solution, where the buoyancy effects do not influence 0169-5983/95/$4.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 01 69-5983 (94)00052-2
114
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
the velocity profile significantly, while the other solution branch has a large region with reverse flow. When K approaches zero this second solution becomes singular and thus cannot be continued to K=0. The aim of the present work is to discuss which of the two solutions for K < 0 is of physical relevance and stable with respect to small perturbations. After presenting the basic equations we discuss briefly the similarity solutions. Then steady flows are presented which connect both similarity solutions. Then a restricted stability analysis will be given for buoyancy parameters close to the critical value Kc.
2. B a s i c e q u a t i o n s
The modified boundary layer equations for the mixed convection flow above a horizontal plate in dimensionless form are (see for example Schneider 1979)
tgU
t~U
0U
t~p --~ --02U
at + U-~x + V~y = - O x
(2.1)
Oy2'
0 = ---OP + KO, Oy 9u Ov
(2.2)
Ox + ~yy = O, O0
--
00
(2.3)
00
at + U~x + V~y
-
1 6320 Pr
(2.4)
3y 2'
where the dimensionless coordinate x parallel to the plate is made dimensionless with the (arbitrary) reference length L and the dimensionless coordinate y perpendicular to the plate is referred to L/x/--~, where Re = U~L/v is the Reynolds number and v the kinematic viscosity. The velocity components u, v parallel and perpendicular to the plate are scaled with the velocity of the free stream U~ and Uo~/x/-~, respectively. The difference 0 between the temperature of the disturbed and undisturbed fluid is scaled with Tw(L) - T ~ in the case of a prescribed plate temperature distribution Tw(x). In the case of a given heat source at the leading edge of an adiabatic plate a scaling value for the temperature difference is given by AT I Q o / p ~ c p ~ ] , where Q0 is a reference value for the strength Q(t) of the heat source, p ~ is the density of the unperturbed fluid and Cp the isobaric specific heat capacity. The dimensionless buoyancy parameter K can be referred either to a temperature difference, i.e. Kr = gfl(Tw(L) - T~)x/Lv/-ffU~ 5/z, or to the strength of the heat source at the leading edge of the plate, i.e. K o = gflQo/U3cpp~, where /3 is the volumetric expansion coefficient and g the gravity acceleration. A reference value for the dimensionless skin friction r is p~U2~/Re. Note that according to this scaling we have r = (Ou/Oy)(x, 0). The boundary conditions at the plate are given by the no-slip conditions =
u(x,O,t)=O,
v(x,O,t)=O,
x>0,
(2.5)
and either the temperature difference to the unperturbed fluid, i.e.
O(x,O,t) = Ow(x,t),
(2.6)
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H. Steinriick/Fluid
Dynamics Research 15 (1995) 113-127
the velocity profile significantly, while the other solution branch has a large region with reverse flow. When K approaches zero this second solution becomes singular and thus cannot be continued to K = 0. The aim of the present work is to discuss which of the two solutions for K < 0 is of physical relevance and stable with respect to small perturbations. After presenting the basic equations we discuss briefly the similarity solutions. Then steady flows are presented which connect both similarity solutions. Then a restricted stability analysis will be given for buoyancy parameters close to the critical value Kc.
2. Basic equations The modified boundary layer equations for the mixed convection dimensionless form are (see for example Schneider 1979)
flow above a horizontal
plate in
(2.1) 0=-z
+K0,
(2.2)
where the dimensionless coordinate x parallel to the plate is made dimensionless with the (arbitrary) reference length L and the dimensionless coordinate y perpendicular to the plate is referred to L/v%, where Re = U,L/v is the Reynolds number and v the kinematic viscosity. The velocity components U, u parallel and perpendicular to the plate are scaled with the velocity of the free stream U, and U,/&, respectively. The difference t9 between the temperature of the disturbed and undisturbed fluid is scaled with T,,,(L) - T, in the case of a prescribed plate temperature distribution T,(x) . In the case of a given heat source at the leading edge of an adiabatic plate a scaling value for the temperature difference is given by AT = IQO/~oo~pdmlr where Q0 is a reference value for the strength Q(t) of the heat source, poo is the density of the unperturbed fluid and cp the isobaric specific heat capacity. The dimensionless buoyancy parameter K can be referred either to a temperature difference, or to the strength of the heat source at the leading edge i.e. KT = @(T,(L) - T,)fifiU;5f2, of the plate, i.e. Ke = g~Qo/U~cppm, where p is the volumetric expansion coefficient and g the gravity acceleration. A reference value for the dimensionless skin friction r is poopmIRe. Note that according to this scaling we have T = (&/a~) (x, 0). The boundary conditions at the plate are given by the no-slip conditions U(X,O, t) = 0,
U(X,O, t) = 0,
and either the temperature 0(-L 0, t) = &(&
t),
difference
x > 0, to the unperturbed
(2.5) fluid, i.e. (2.6)
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H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
0"3 I 0.2
f"(o)
o.;[___, -0.1
h'~
-0.05 K
Fig. 1. Similarity solution: wall shear stress f"(0) as a function of the buoyancy parameter K for Pr= 1. subjected to the boundary conditions
f ( x , O, t) = f ' ( x , O, t) = f ' ( x , c¢, t) - 1 = O(x, c¢, t) = h(x, oe, t) = O, O(x, O, t) = Ow(X, t),
(3.5)
and the initial conditions
f ( x , rl, O) = ft(x,~7),
O(x, rl, O) = O i ( x , rl).
(3.6)
Here and in the following derivatives with respect to r/ are denoted with a prime. Self-similar solution can be found for Ow(X, t) = 1. Then the boundary layer equations are reduced to the ordinary differential equations
2f'"+ff/'+K~70=O,
2Pr0 ' + f O = O ,
(3.7)
with the boundary conditions f(0) = f'(0) =0,
f ' ( c ¢ ) = 1,
O(0) = 1.
(3.8)
The solution of the similarity equation has been discussed in many papers (Schneider 1979; Afzal and Hussain 1984; de Hoog, Lamminger and Weiss 1984; Merkin and Ingham 1987; Daniels 1992). In Fig. 1 the wall friction coefficient r = f " ( 0 ) versus the buoyancy parameter K for Pr = 1 is shown. In order to determine the critical value Kc, where both solution branches meet, the buoyancy parameter and the similarity solution are parametrized with the wall shear stress ~- = f " ( 0 ) . Since at Kc the buoyancy parameter has a minimum (as a function of r ) , the derivatives Fc = (Of/O~') (zc), Dc =
(aOlaT)
I
Hc = (an~aT)
11! l/ l/ 2Fc + f c C + f'
satisfy
C-KcSc\
(2/Pr)O'c' + (fcDc + OcFc)'] = 0, H'c+ (~TDc)' /
(3.9)
with the boundary conditions Fc(0) = F~'(0) = F~'(oc) = Dc(OC) = Hc(c¢) = De(0) = 0,
(3.10)
11. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
117
and Fc"(0) = 1, where fc, Oc (3.7), (3.8) Following parametrized
(3.11)
is the similarity solution at the critical value Kc. To determine Kc the similarity equations and (3.9)-(3.11) have to be solved simultaneously. Merkin and Ingham (1987) the similarity solution near the critical point K~ can be with e = q - x / K - Kc:
( f , O, h) = (fc, Oc, he) + e ( f l , O1, hi) + e2(f2,192, h2) + O(e3).
(3.12)
The first-order correction f~ satisfies the linear equation (3.9) with the homogeneous boundary conditions (3.10). The linear homogeneous boundary value problem has the non-trivial solution f l = Cfc,
Ol = CDc,
(3.13)
hi = CH~,
with a yet undetermined constant C. For f2 one obtains: ( 2 / P r ) O ~ ' + (fc02 +O~f2)' h~ + (r/O2)'
=-C 2
(FcDc)' 0
+
0 0
,
(3.14)
with the boundary conditions (3.10). Since the corresponding homogeneous problem has a nontrivial solution, a solution of (3.14) cannot satisfy all boundary conditions (3.10) for an arbitrary inhomogeneity. Thus the right side of (3.10) has to satisfy the solvability condition (3.15)
C2-C~=0,
with the solutions -t-C0, (Co > 0). The solvability condition (3.15) is derived in appendix A.1. Thus for e > 0 the upper solution branch is obtained and for e < 0 the lower solution branch is obtained. 3.2. Connecting flows
In case of negative buoyancy parameters K < 0 two different self-similar flows a)e possible. It is of interest if other steady (non-self-similar) flows exist. In this section a certain class of steady (non-similar) flows will be described. It will be shown that steady flows exist, which are near the leading edge close to one self-similar flow and far downstream close to the other self-similar flow. In order to apply asymptotic methods the analysis is restricted to a neighborhood of the critical point
Kc. The main idea is to replace the coordinate x parallel to the plate by a slowly varying variable ~: = e In x and to expand the stream function, temperature distribution and pressure gradient in terms of e: f ( x , r/, e)
O(x, rl, e) h(x, rl, e)
=
Oc(n)
hc (r/)
+e
O~((,n) h 1((, r/)
+e 2
O2((,n)
+..-
(3.16)
h2(sc, r/)
Inserting the expansion (3.16) into the boundary-layer equations yields for the first order terms again Eqs. (3.9)-(3.10). But now C is a yet unknown function of (, i.e. f t ((, T1) = C (s~) Fc(rl).
(3.17)
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H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
For the second order terms (f2, 82, h2) one obtains
(2/Pr)O~+(fcO2+Ocf2)'| =-C 2 + /
(FcD~)' o
+
0 o
+C¢
f~Dc
0'¢F~ , (3.18)
with the boundary condition (3.10). Eqs. (3.18), (3.10) have a solution only if the solvability condition
(2Co/A1)C~ = C 2 - C~,
(3.19)
where A1 is a positive constant, is satisfied. The solvability condition (3.19) and values for A1 and Co are derived in appendix A.1. Numerical computations show that both constants are positive. The general solution of (3.19) is given by 1 - yexp(Al()
1 -- yx ~al
(3.20)
C(¢) = Co 1 + y e x p ( A l ( ) = C°l q-3/Xeal'
where y is an arbitrary positive constant. Thus a steady flow has been determined with the uniformly valid expansion yx ~a' Fc(r/) + O(s2). + yx ~a'
1 -
f ( x , rl,~) = f~(r/) +sC01
(3.21)
For x ,-~ 0 this flow behaves like the self-similar flow corresponding to the similarity solution on the upper branch and for x --+ oc the flow behaves like the self-similar flow corresponding to the similarity solution on the lower branch. A connecting flow, which connects the 'lower' self-similar flow with the 'upper' self-similar flow has not been found. Moreover in the next subsection a mathematical argument will be given why such a 'reverse' connecting flow is impossible. Here the existence of connecting flows is shown only near the critical value Kc of the buoyancy parameter. But there is numerical evidence (see section 3.4) that connecting flows exist for Kc < K<0. For negative values of 3/ the function C(sc) becomes singular at (* = l n ( - y ) / a l and thus the expansion (3.16) is not valid there. Numerical computations show that in case y < 0 the boundarylayer flow field has a singularity where the wall shear stress tends to infinity. This type of singularity for mixed convection boundary layer flows has been analyzed by Daniels (1992). However it is unclear if such singular solutions are of physical significance.
3.3. An eigenvalue problem In this section a necessary condition for the existence of connecting flows will be discussed. Let
f ( x , ~7), O(x, 7) and h(x, ri) be the stream function, temperature distribution and pressure gradient of a connecting flow. Then near the leading edge of the plate f , O and h can be expanded with respect to x << 1: O(x, 77)
= /Os('q)
h(x, 71)
\ h~(rl)
+ x '~
D(,q)
H(rl)
+o(x'~),
(3.22)
119
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
where fs, Os and hs are a similarity solution and A is a positive constant. Inserting into the boundary layer equations and neglecting higher order terms yields the eigenvalue problem (2/Pr)D"+ (fD + OF)' ] = a H' + ( n O ) ' /
I ' D - O'F D
,
F ( 0 ) = F ' ( 0 ) = F ' ( o c ) = D ( e c ) = H(e~) = D ( 0 ) = 0,
(3.23) (3.24)
for ,~ and F, D, H. Thus a connecting flow which connects fs, Os, hs with another self-similar flow exits only if the eigenvalue problem (3.23), (3.24) has a positive eigenvalue. Analytically the existence of a positive eigenvalue can be shown in two cases. 3.3.1. A positive eigenvalue f o r - K << - 1 For IK[ << 1 a positive eigenvalue with the singular expansion
a = -K--5 + ~5 at + . . . .
with
a~- = 0.00378,
(3.25)
exists. All other eigenvalues are negative. Details of the expansion can be found in appendix A.2. 3.3.2. A positive eigenvalue near the critical value For ~2 = K - Kc << 1 there exists an eigenvalue with an expansion
,~= eAl + O ( e Z ) ,
with
~1 > 0.
(3.26)
This eigenvalue is positive on the upper branch and negative on the lower solution branch. Note the value of A1 is the same as in (3.19). Details are deferred to appendix A.3. In Fig. 2 the positive eigenvalue is shown as a function of K. The positive eigenvalue A(K) has in agreement with (3.25) a singularity at K = 0, stays positive on the entire upper solution branch and has a zero at K = Kc. Numerical computations show that all other eigenvalues corresponding to similarity solutions on the upper branch are negative. On the lower solution branch all eigenvalues are negative as long as there is no reverse flow. This has the following consequences (i) A connecting flow can branch off from the self-similar flow corresponding to a similarity solution on the upper solution branch for Kc < K < 0 (ii) No connecting flow can branch off form the self-similar flow corresponding to a similarity solution on the lower solution branch with a positive wall shear stress. (iii) For self-similar flows with a reverse flow region no statement can be made if a connecting flow can branch off. Numerically only connecting flows which connect the upper similarity solution with the lower have been observed. 3.4. Numerical results
The existence of connecting flows has been shown only for values of the buoyancy parameter K close to the critical value Kc. For K - K c not small one has to compute the connecting flow numerically. The equations for steady boundary layer flows can be considered as an evolution problem with respect
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H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
1000.
100. A+I 10.
1 -0.1
K.
-0.05 K
Fig. 2. Eigenvalue of the Eigenvalue problem (3.23), (3.24) as function of the buoyancy parameter K on the 'upper' and 'lower' solution branch.
to the coordinate x parallel to the plate. The initial value problem with initial data given at x = 0 is due to the existence of a positive eigenvalue not uniquely solvable. Therefore the expansion (3.22) for small x is used to formulate an initial condition at a given x0 > 0. Let f ( x 0 , r/) = f~(r/) + rx~F+(rl),
O(Xo, 77) = Os(rl) + rx~O+(rl),
(3.27)
where r is an arbitrary small parameter. The stationary equations can now be solved by a marching technique. One has to take care to resolve the positive eigenvalue correctly. The computations have been performed with Pr = 1 and different values of the buoyancy parameter K. We have chosen r = - 1 0 -5 and x0 = 1. In Fig. 3 the wall shear stress is plotted as a function of lnx for K = - 0 . 0 8 and is compared with the asymptotic solution (3.21). As expected it tends to the value corresponding to the similarity solution on the lower branch for x -* ~ . For K > - 0 . 0 5 2 0 4 the lower solution branch has a region of reverse flow. A connecting solution for K = - 0 . 0 4 has been computed. The wall shear stress has a zero without any singular behavior. Thus the numerical computation can be extended beyond the point of separation as long as the region of reverse flow is not too large. Again the numerical solution connects the two similarity solutions. In Fig. 4 the wall shear stress is shown and in Figs. 5 and 6 the velocity and temperature profiles for K = - 0 . 0 4 and different values of x are shown. The 'initial' function (3.27) can be replaced by any 'initial' function with a negative component in the direction of the eigenfunction (F+, D+). Initial conditions with a positive component in the direction of the eigenfunction (F+, D+) lead to singular flow fields corresponding to the case y < 0 in equation (3.20).
4. A restricted stability analysis After finding steady flows connecting both self-similar flows the stability of these flows might be of interest. Here a restricted stability analysis will be performed, by what we mean that only certain initial perturbations are considered. In order to employ asymptotic methods K - Kc << 1 is assumed. The main idea is to construct, a solution with an asymptotic expansion of the form
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
121
0.3
0.15
0.2 f"(x, o)
f"(x, O) 0.1
o.1
O-
0.0~
1'0
2'0 lnx 3'0
4'0
-0.1
50
I
0.5
lnx
1
1.5
Fig. 3. Dimensionless wall shear stress for a connecting solution for K = -0.08 (s = 0.106) and Pr = 1. Comparison of the asymptotic solution (dashed line) with the numerical solution. Fig. 4. Dimensionless wall shear stress for a connecting solution with K = - 0 . 0 4 and Pr = 1. 0.05
/
1
U
0.5
0 0
2
4
6
8
-o.o
10
015
i
115
i
2.5
Fig. 5. Velocity profiles of the connecting solution with K = - 0 . 0 4 and Pr = 1 for various values of the x-coordinate: lnx = 0.25, 0.5, 0.75, 1., 2., 3. and similarity solutions (dashed lines). For 0 < ~ < 10 (a) and 0 < r / < 2.5 (b).
f ( x , rl, t , e ) = f c ( r l ) + e f l ( ( , ( , r l )
+ eZf2 + . . . .
(4.1)
w h e r e t h e i n d e p e n d e n t v a r i a b l e ~:, ( a r e d e f i n e d b y
~: = e l n ( x ) ,
( = e ln(1 +
t/x).
(4.2)
T h e n t h e p a r t i a l d i f f e r e n t i a l o p e r a t o r s a r e t r a n s f o r m e d to 0
x~
O
t
O
= s-d~ - s x + t - ~ '
3
x
3
1
and then the first-order terms fl,Ol,hl satisfy (3.9)-(3.10)
O1 (se, ~', 7/) h l ~ , ( , r/)
0
x -at- = e - - x + t a ( = e y + t / x a ( '
= C(~:, ( ) { D c ( ~ 7 ) \ Hc(~7)
,
(4.3)
a n d are g i v e n b y
(4.4)
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
122
\\\\\ \\\ \\\\\
\\ \\\
0.I
06
7/ 6
8
10
Fig. 6. Temperature profiles of the connecting solution with K = -0.04 and Pr = 1 for various values of the x-coordinate: In x = 0.25, 0.5, 0.75, 1., 2., 3. and similarity solutions (dashed lines). where C is now a function of ( and (. Obviously the initial conditions (3.6) have to be compatible with the the expansion (4.1) and thus with (4.4). Therefore the initial condition has to be of the form
O(x,~,O;e)
=
Oc(~)
+C1(eln(x))
+O(e2).
\ De(7/)
(4.5)
To determine C ( s q sr) the equation for the second order terms have to be considered. Since the derivative of C with respect to ( is multiplied by a factor depending on o- = t / x the second order correction f2 has to be a function of ~/,s~,( and tr. Thus
( 2 / P r ) G ' + ( f e 0 2 + G f2) h; + ('/']'/~q'2) t + ~
~ f;
= -c 2
c -- O;Fc + De D~
+
(Fe c)'
+
+ (G - G)
02.,~ tr(f;02,,~ - O;f2,,~) -o'02,,~
.
I;Do - GFo Dc (4.6)
This is a partial differential equation for f2, 02, h2. For o- ~ oo all o" dependent terms on the right side of (4.6) have to vanish and an ordinary differential equation for f2, 02, h2 is obtained, which is solvable if the solvability condition (2Co/al)(C• -- C¢) = C 2 -- C 2,
(4.7)
is satisfied. This is a hyperbolic partial differential equation for C with its characteristics pointing upstream. Thus small perturbations can propagate upstream. The general solution of (4.7) is given by C 0 + Ci(~Jf - ()
-
(C o -
Ci(¢--]-
•)) exp(--Al()
C (~, ( ) = Co Co + C, (/j + ( ) + (Co - C I ( ( + ( ) ) exp(-zl, sr) " Using the x, t variables the solution reads as
(4.8)
123
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
f(x,~7, t) = fc(rl) CCt(~ln(x %- t)){1%- [ x / ( x %- t)]*a'} %- C0(1 - [x/(x%- t)]~a') r , , %- eGo It ln(x %- t)){1 -- [ x / ( x 4- t)] ~hl} %- C0(1 ~ [ x / ( x + t ) f ~ rct~7) + O(e2)"
(4.9)
Stability of the "upper' self-similar flow The asymptotic behavior for t ~ c<~ can be discussed easily. If lime__,~ Ct(~:) 4: -Co and C1 ( ( ) > -Co it can be verified that limt__.~ C (x, t; ~) = Co which corresponds to the upper solution branch. Thus the upper similarity solution is stable against small perturbations of the form (4.5).
Stability of the 'lower' self-similar flow and the connecting flows The stability behavior of the lower similarity solution and the connecting solutions is more complicated. Since the characteristics of (4.7) are pointing upstream the time dependent behavior of the solution at a fixed location x will dependent on the downstream behavior of CI (so). Assume that C~ ( ( ) has an asymptotic expansion of the form
CI(() = - C 0 ( 1 - 2 y e x p ( - c e ( ) + O ( e x p ( - a ( ) ) ) ,
for
~: ~ oo,
with
y > 0.
(4.10)
where y and ce are constant. Then inserting into (4.9) yields
lim C ( x, t; e ) =
t---+~
/
c0i
Co
-Co,
X~AI
.
XF,,~I ,
0[ "-~" i~|
upper solution connecting solution .
(4.11)
a > Al lower solution
Thus the 'lower' self-similar flow is stable with respect to small perturbations which decay faster than the connecting flow for x ~ cx~.
5. Conclusions A new type of steady flows for K < 0 connecting the self-similar flow corresponding to the solution on the upper branch with the self-similar flow corresponding to the solution on the lower branch has been described. Discussing the physical relevance of this one parametric family of steady flows a restricted stability analysis has been performed showing that perturbations of a steady flow propagate upstream. The 'upper' self-similar flow turns out to be stable against small perturbations. The lower self-similar solution and the connecting flows are in general unstable with respect to small perturbations. They are only stable with respect to small perturbations which decay fast enough for x --, cx~. Considering the boundary layer flow only over a finite length L a downstream boundary condition is required, since the characteristics of the perturbation equation (4.7) point upstream. This downstream boundary condition selects uniquely one solution of the one parametric family of solutions described in section 3.2 and stabilizes it with respect to perturbations of the form (4.5). The analysis presented here for the case of a controlled plate temperature holds in case of an adiabatic plate with a prescribed heat sink at its leading edge (see appendix A.4) also. Note that the self-similar flows are identical in both cases, but the corresponding buoyancy parameters Kr
124
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
and KQ are not. Let rr,~ , ~'0,c be the dimensionless wall shear stress where Kr and K o resp. have a minimum. Then rr, c is different from To,c and thus fo,c 4: fr,¢. A self-similar flow with ~'o,~ > f " ( 0 ) > ~'r,c corresponds to a similarity solution on the upper branch when referred to a fixed temperature distribution and the same flow corresponds to a similarity solution on the lower branch when referred to a given heat sink at the leading edge of the plate. For Pr = 1. we have 7"o,~ = 0.120 and ~zc = 0.106. To observe the self-similar flow on the lower solution branch in an experiment one can proceed as follows. A plate is needed which can be switched from temperature control to heat flux control. Starting with the Blasius solution Kr is slowly (quasi statically) decreased. When the skin friction ~is between f o x and ~'T,c the plate is switched to adiabatic boundary conditions. Now the self-similar flow corresponds to a 'lower' solution. Then K o can be quasi statically increased. One has to take care of the downstream edge of the plate that no perturbations are produced there. Small perturbations at the downstream edge would change the flow into a connecting flow or into the 'upper' self-similar flow.
Acknowledgements I want to thank Prof. W. Schneider for many stimulating discussions. The support of the Austrian Science Foundation under project no. P9584-TEC is also acknowledged.
Appendix A A.1. The solvability condition
The solvability condition for (3.7) and (3.18) can be obtained as follows. Let (f0, O0, ho), ( f l , 01, hi), (f2, O2, h2), be the unique solution of (3.18) with the boundary conditions fk(0)=f~'(0)=Ok(0)=f~(c¢)
=Ok(cx~)=hk(c~Z)=0,
k=0,1,2
,
(A.1)
with (C 2 = 0, C¢ = 0), (C 2 = 1, C¢ = 0) and (C 2 = 0, Ce = 1) respectively. Then the solution of (3.18) with the boundary condition (A.2)
f 2 ( 0 ) ~- 0 2 ( 0 ) = f ~ ( o O ) = 0 2 ( 0 0 ) = h 2 ( o o ) = 0,
is given by
O2 h2
=
00 ho
+C 2
01 hi
O0 ho
+C¢
02 h2
00 ho
+F
Dc Hc
,
(A.3)
where F is an arbitrary constant. Since F~'(0) = 0, (A.3) is a solution of (3.18) with the boundary conditions (3.10) if and only if f~(O) + C2(f'~(O) - f ~ ( 0 ) ) + C~(f~(O) - f ~ ( 0 ) ) = 0.
(A.4)
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
125
Setting (, f~(O) ),/2 Co= \f~(O) --7~(0)
and A, =
2{f~(O)[f~_f~(O)]},/2 f~(O) - f~(O)
(A.5)
the solvability condition (3.19) is obtained. The similarity solutions on the upper and lower branch correspond to the constant solutions Co and - C o resp. of (3.19). A.2. Positive eigenvalues for - K << 1
For K << 1 the similarity solution (on the upper solution branch) can be expanded in terms of K: f = fB+Kfl+O(K2),
(A.6)
O=OB+KOI+O(K2),
where fB is the Blasius solution and OB the corresponding temperature distribution. Thus the eigenvalue and the eigenfunction are expanded A=
(A.7)
AO 4- K A I 4- . . . .
(A.8)
( F , D , H ) = (Fo, Do, Ho) 4- K(F1,D~,H~) 4- . . . .
in terms of K. Then the eigenvalue problem for the leading order terms are decoupled:
2F~" + fBF~' + f~'= Ao(f~FD-- f~'Fo), (2/Pr)D~o' 4- (fBDo) l = 2of~' D o,
Fo(O) = F~(O) = FD(oo) =O,
Do(0) = 0,
(A.9) (A.10)
Do(c~) = 0.
Both eigenvalue problems can be written in self-adjoint form and thus have only real eigenvalues. In Libby and Fox (1964) it was shown that all eigenvalues of the momentum equation (A.9) are negative. By partial integration it can be shown that the eigenvalues of the energy equation (A.10) are bounded by 1/2. A numerical computation shows that the largest eigenvalue is negative. Besides the eigenvalues with the regular expansion (A.7) an eigenvalue with a singular expansion for K ---, 0 - exists: a
aft at = - K--3-+ -~- + . . . .
(A.11)
Inserting (A.11) yields a singularly perturbed eigenvalue problem and one has to expect a matched asymptotic expansion for the eigenfunctions. The outer expansion is given by F + = Po+ + K P ? . . . .
D+ = b g + K b t . . . .
H+
= g-3/~
- 4- g - 2 / ~ l +
. . .
(A.12)
The leading order terms are obtained as Po+ = A ,
b ~ = a~,,
(A.13)
~o" = a ~ a B .
Obviously not all boundary conditions can be satisfied at r / = 0 and an inner expansion is needed: F + = KP~-(rl/K) + . . . .
D + = KD~(rl/K) + ....
¢ = rl/K.
(A.14)
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
126
The resulting equations for the leading order terms are identical with the equations for the leading order terms of the eigenvalue problem in Steinriick (1994) and thus one obtains
=-2 IAi' o,(iAi s ds) ]=ooo378 oo
-1
3
(A.15)
o
where Ai denotes the Airy function. Thus the eigenvalue A+ is negative for K > 0 and positive for K < 0.
A.3. A positive eigenvalue for K - K~ << 1 Near K~ an eigenvalue and eigenfunction exist with an expansion of the form A=sA1 + . . . ;
F = F~ + sFI + . . . .
D = D¢ + sDI + . . . .
H = Hc + sHI + . . . .
(A.16)
where ,~, and F1 are determined form
2F;" + fcF;' + f~lFi - K~H, ( 2 / P r ) D ' / + (f~D~ + OcD,)' ] = -2C0 H', + (~TD,)' ,/
( ) FoF"
(FcDc)' o
f
+ A1
-
f~ ' D¢'
;o
X tg'~Fc
)
,
(A.17)
with the homogeneous boundary conditions (3.10). Comparing (A.17) with (3.18) yields that Al is given by (A.5).
A.4. The adiabatic plate Here the necessary changes of the previous analysis for the case of an adiabatic plate are presented. The boundary condition O(0) = 1 has to be replaced by oo
f f'Odrl = 1.
(A.18)
0
Let fr, Or be a similarity solution of (3.7), (3.8) for the buoyancy parameter Kr. Then it can be verified that /r
To=iT,
N
--1
OQ= ( / f~Ord~7) \
,t
Or,
(A.19)
/
o
is a solution of (3.7) with the boundary conditions cx)
f(O)
fl(O) =0,
f'(oc) = l,
/ f'Odrl = 1.
(i.20)
0
for the buoyancy parameter K o = KT f o f~Or dl7 • The connecting flows with a prescribed temperature distribution Ow(x) = 1 have a non-zero heat transfer at the plate. Thus the connecting flows of the temperature and the adiabatic case are different.
H. Steinriick/Fluid Dynamics Research 15 (1995) 113-127
127
But the analysis of the connecting follows the same lines. Only the boundary condition Dc(0) = 0 has to be replaced by
(fc D+Fd '
dr/=0.
(A.21)
0
For non-steady flows the enthalpy flux condition is given by
/ f' O drl o
O, drl dx' = 1. o
(A.22)
o
For a slowly varying transient solution described in section 4 the integral Assuming ~:, ( > 0 one obtains:
Jo Ot dx' has to be estimated.
t/x')) d x ' = eC¢(u, 0) ln(1 + 1/o-) + O(e2).
(A.23)
j~C
(e lnx', e(1 +
0
This term vanishes for o- ~ cxD and thus an equation of the same type as (4.7) is obtained as solvability condition. Thus the results of the restricted stability analysis of section 4 apply in the case of an adiabatic plate also. References
Afzal N. and T. Hussain (1984) Mixed convection over a horizontal plate, ASME J. Heat Transfer 106, 240-241. Daniels, EG. (1992) A singularity in thermal boundary-layer flow on a horizontal surface, J. Fluid Mech. 242, 419-440. de Hoog, ER., B. Lamminger and R. Weiss (1984) A numerical study of similarity solutions for combined forced and free convection, Acta Mech. 51, 139-149. Libby, E and H. Fox (1964) Some perturbation solutions in laminar boundary layer theory. Part I: The momentum equation, J. Fluid Mech. 17, 433-449. Merkin, J.H. and D.B. Ingham (1987) Mixed convection similarity solutions on a horizontal surface, ZAMP 38, 102-116. Schneider, W. (1979) A similarity solution for combined forced and free convection flow over a horizontal plate, Int. J. Heat Transfer 22, 1401-1406. Steimiick, H. (1994) Mixed convection over a cooled horizontal plate: Nonuniqueness and numerical instabilities of the boundary layer equations, J. Fluid Mech. 278, 251-265.