ACTA 5IECHANICA
Aeta Mechaniea 66, 251--262 (1987)
9 by Springer-Verlag 1987
Mixed Convection Boundary-Layer on a Vertical Cylinder Embedded in a Saturated Porous Medium By J . I I . M e r k i n , L e e d s , U n i t e d K i n g d o m , a n d I. P o p , Cluj, R o m a n i a With 2 Figures
(Received .February 2d; 1986; revised April 24, 1986)
Summary The mixed convection boundary layer on a vertical circular cylinder embedded in a saturated porous medium is considered. I t is found t h a t the flow depends on the parameter e = Ra/P e where R a and Pe are the Rayleigh number and Peeler number respectively. gives the ratio of the velocity scale for free convection to that for the forced convection. When e is small the solution is, to a first approximation, obtained by a known heat conduction problem. The flow near the leading edge is considered and it is shown t h a t a solution is possible only for e > e0, e0 ~-- --1.354, and that a stable finite-difference solution away from the leading edge can be obtained only if e > - - 1 ; with e < --:i there is a region of reversed flow near the cylinder. The finite-difference scheme is unable to give a satisfactory solution at very large distances from the leading edge, and to overcome this difficulty a simple approximate solution is developed. This solution shows t h a t at large distances along the cylinder, forced convection eventually becomes the dominant mechanism for heat transfer. This is also confirmed by an asymptotic solution of the full bouhdarylayer problem.
Nomenclature a g K Nu r /~a Pe T Tw To zJT u
radius of cylinder acceleration of gravity permeability of the porous medium non-dimensional Nusselt number radial coordinate non-dimensional r = r/a Rayleigh number = @(gflAT) Ka/~o~ Peeler number = Uoa/a temperature temperature of the cylinder (constant) temperature of the ambient fluid (constant) temperature difference ~ Tw -- To Darcy's law velocity in the x direction
252
~0 X
X (%
0 0
J . H . Merkin and I. Pop: velocity of the outer flow Darey's law velocity in the r-direction coordinate measuring distance along the cylinder non-dimensional x, .~ x(aPe) -1 equivalent thermal diffusivity coefficient of thermal expansion ratio of free to forced convection ~ Ra/P e viscosity of the convective fluid density of the ambient fluid non-dimensional temperature stream function
1. Introduction The problem of heat transfer in flows in saturated porous media has received much attention lately because of its important applications both in technology and geothermal energy recovery. There has been a substantial amount of work on natural convection over the last few years, the earlier work being extensively reviewed b y Cheng [1]. However, the problem of mixed convection in a saturated porous medium has gained much less attention. The mixed convection boundary layer on a vertical wall has been treated b y Cheng [2] and Merkin [3] and on a horizontal surface b y Cheng [4]. More recently, the higher order effects for the vertical wall problem have been looked at by Joshi and Gebhart [5]. I n this paper we consider the mixed convection boundary layer on a vertical circular cylinder. A situation t h a t would arise, for example, on needle-like intrusions into an aquifer in which there was already set up some general background flow. The related problem of the natural convection boundary layer on a vertical cylinder was discussed first b y Minkowycz and Cheng [6] and more recently b y Merkin [7] and Kumari, Pop and N a t h [8]. Here we assume that a uniform stream U0 is flowing along a cylinder of radius a fixed so that its axis is vertical and aligned with the flow. The temperature of the cylinder is assumed to be held at the constant value Tw with the temperature of the ambient fluid being To. We take the Peeler number U0a to be large so the equations governing the cr flow are the boundary-layer equations for flow in a porous medium. We find t h a t the important parameter which determines the relative strengths of the forced to free convection effects is s z R,/P~, where Ra ~
O(g~ A T ) K a
is the Rayleigh
number. First, we consider the case when e is small, so that we have basically forced convection. Here we find that, to leading order, the velocity field is just the free stream, while the temperature field is given b y a heat conduction problem. We then consider the flow near the leading edge, X ~-- 0, of the cylinder. Here we show that, to a first approximation, the solution is the same as on a vertical wall [2], [3], with the consequence t h a t a solution is possible only for ~ ~ e0 (so
Mixed Convection Boundary-Layer on a Vertical Cylinder
253
--1.354), [3], [9]. When we come to attempt a finite-difference solution away from the leading edge, we find that this is possible only for s ~ --1, for with e ~ --1, the tangential velocity component on the cylinder is negative and this makes the numerical scheme unstable. No such problem is encountered for e ~ --1, but we run into the same problem as described in [7], in that the numerical solution becomes increasingly less accurate as we integrate away from the leading edge. This difficulty arises because away from the leading edge the temperature profile has an initial very rapid decay followed b y a long region where it slowly decreases to the ambient conditions. This becomes more pronounced as X increases and leads to loss of accuracy in the solution, so we conclude that this finite-difference solution is useful only up to moderate values of X. To obtain a solution which gives a good estimate for the heat transfer for all X, we use an approximate method based on an integrated form of the energy equation. This method was used in [7] for the related natural convection problem where it was found that it was at worst only about 6% in error and that it gave the correct asymptotic form of the solution as X --~ 0o. On considering the behaviour of the solution of this approximate method for large X, we find that, for all values of s, the forced convection a!ways dominates over the free convection. This is unexpected, as it is not the case for mixed convection in a Newtonian fluid [10], [11] or in mixed convection in a porous medium with a prescribed heat flux boundary condition [3], We then go on to consider the asymptotic solution of the original boundary-layer equations, which confirms this prediction.
2. Equations The equations governing the boundary-layer flow on an impermeable vertical cylinder embedded in a saturated porous medium are from [3], [6], [7] (on making the usual assumptions about the physical properties of the medium and that the flow is given by Darcy's law) ax (ru) + ~ (rv) = o
K
u = Uo + - - e g ~ ( T
~T
aT
(1)
- - To)
(~T
(2)
1 ~T)
with boundary conditions v:0,
T:Tw
on
r:a,
T--~To,
u-->Uo
as
r->c~.
(4)
The coordinates x and r measure distance along the cylinder and radially outwards respectively and u and v are the velocity components in the x and r directions.
254
J . H . Merkin and I. Pop: We m a k e Eqs. (1), (2), (3) non-dimensional b y defining
x = aPeX,
r =- aN,
T - - To = A T O(X, ~)
and
~ = ~aPe~(X,
1 ~
~)
(5)
1 8~
where ~ is the stream function defined so t h a t u = - - - - and v -r 8r r 8x and Pe = U0~ is the Peclet number. E q u a t i o n (1) is satisfied a u t o m a t i c a l l y and Eqs. (2) and (3) become 1 8T - - - -8~ = 1 + eo
1 ( O T 80
~- W a x
~T ~\
8x ~)
820
(6)
1 80
(7)
= --~ 8~ + r ~-~
with b o u n d a r y conditions T=0,
on
0=1
~ = 1,-
~T ~, 8g
0->0
as
~-->~.
(8)
Kegfl A T eK(g~ ~T) a The p a r a m e t e r e -- - = R a / P e (where Ra = is the R a y /zU0 #a leigh number) is the p a r a m e t e r which determines the relative i m p o r t a n c e of forced to free convection. With e ~ 0, b o t h the flow and the b u o y a n c y forces are in the same direction, while with e < 0 t h e y are in opposite directions. F r o m (6) we have tha~, on the cylinder ~ -~ 1, ~ (the non-dimensional u) is given b y = 1 + e. (9) Finally we note t h a t for e large (and positive) the solution when free convection dominates can be recovered b y writing T = - s T , X = s-~. This gives 1 8~P Eq. (7) (with T r e p l a c e d b y T and X byX:) with Eq. (6) becoming r a--~-= 0 ~- e -~. An expansion then in powers of e-~ gives, at leading order, the free convection solution as discussed in [6], [7], [8].
3. Forced C o n v e c t i o n L i m i t
Here we assume t h a t I~i ~ 1 and obtain a solution of Eqs. (6) and (7) b y expanding W and 0 in the form
~= 1
To + s T 1 -}- ...,
~%
E q u a t i o n (6) gives - - - -
0 = 0 o - s 0 1 + ....
(10)
---- 1, from which it follows that 1 ~o = - - ~2 2
(11)
Mixed Convection Boundary-Layer on a Vertical Cylinder
255
which is just the stream function for the free stream u = Uo (or ~ ---- 1). Using (11), Eq. (7) gives 00o
~X
~20o
1 ~0o
---+---~2 ~ ~
(12)
to be solved subject to 0o = 1 on F = 1, 0o -+ 0 as ~ -+ e~. This is mathematically the same problem as the unsteady heat conduction from the surface on an infinite circular cylinder and its solution is given b y Carslaw and Jaeger [12] and Batchelor [13] as oo
0o = 1 -~ 2 s
e_k~x Jo(/cg) Yo(/c)- Jo(/c) Yo(k~) --.dIc
Jo~(k) + Yo2(k)
J
(13)
k
0
1 O}//i
Then, from (6) we have
- - 0o SO t h a t T1 can be solved in terms of 00 as
T1 = f sOo(X, s) ds
(14)
1
01 is then given b y the linear equation ~201 -+ Dg~
1 ~01 ~ ~
~0i ~0o - - 0o - ~X ~X
1 0~1 ~00 g OX ~
--
(15)
with01=0ong= 1,01~0as~->cc. Using the results given in [12], [13] we find t h a t the non-dimensional Nusselt number, defined b y N~ = . (00) ~ ~= 1 is, for small X, ,
N~ - - (~X)I/~ + T -- -i-
+ "'" + 0(~)
(16)
and, for large X, N~
-
-
-
2
log 4X
+
27
-
-
(log 4X) ~
+
...
+
o(~),
(17}
where Y = 0.57722 is Euler's constant.
4. Solution n e a r the Leading Edge Near the leading edge the boundary layer will be very thin, so the flow will be initially the same as on a vertical wall, which suggests making the transformation ~r] ~_ ( 2 X ) 1 / 2 / ( X ,
T]),
0 ~- O ( X , T]),
~ ~--~ ( 8 X ) l ] 2.
(18}
g . H . Merkin and I. Pop:
256
Equations (6) and (7) then become
a/
--=1§ O~
--~ o~
(19)
~(~+I)@-1+7~
@~
(20)
~
with boundary conditions (8) becoming
/=0,
0=1
on
~=0,
a/
---+1, 0~7
0-+0
as
7-+~
(21)
and where we have put $ = (8X) 1/2. Equations (19) and (20) suggest an expansion in powers of ~ in the form 1(~, ~) = I0(7) + ~11(7) + " " ,
0(~, ~) = 00(7) + ~01(~) + "'"
(22)
which, at leading order, leads to the problem
1o"' + 1olo" = 0
(23)
with boundary conditions /o(0) = O,
/o'(0)= 1 @ e ,
Io'-+1
as
7-+~
(24)
(where primes denote differentiation with respect to 7). The solution of Eq. (23) with boundary conditions (24) has been discussed previously in [3], [9], where it was shown that there was a unique solution for e > --1, two solutions for ~0 < < --1 (s0 ~ --1.354) and no solution for s < ~0. Thus, this mixed convection problem has a solution only when ~ > e0, and in this case the solution can be extended b y the series expansion (22) (for small values of ~) or b y a finite-difference solution (for larger values of ~). We consider this latter course. Equations (19) and (20) were solved numerically using the same finitedifference scheme as described in [14], for example. The integration started at = 0 with the solution as given b y (23) for the appropriate value of s and proceeded in a step-by-step manner for increasing ~. This procedure worked well for s > --1, but for s < --1 the scheme rapidly became unstable. The reason for this can be seen from (9), for, with ~ < --1, ~ < 0 on the cylinder. Now, it is wellknown that stable numerical solutions can be obtained for parabolic heat conduction-type equations only when integrated in the forward time direction. Here X plays the role of a time-like variable and with ~ < 0 we are, in effect, integrating backwards in this time-like variable, which quickly leads to instabilities. No such problem is encountered with e > --1 and graphs of N~ against X as calculated from the numerical integration for various values of s are shown in Fig. 1. Here we can already see the tendency for the curves to approach a common limiting form as X increases. A point which will be discussed further in the next section. As in [7] a difficulty is encountered in obtaining numerical solutions for
Mixed Convection Boundary-Layer on a Vertical Cylinder
257
the very large values of X needed to join onto an asymptotic solution (this requires values of X such that log X is large). This problem arises because as the temperature profile develops away from the leading edge, it has a thin region near the cylinder where it changes rapidly and a much longer 'tail' region where it slowly decays to the ambient value, with this effect becoming more pronounced as X increases. So in the numerical scheme we need a small step length in ~/to account for the region near the cylinder, and the long 'tail' region requires that the outer boundary condition be satisfied at increasingly larger values of ~ to keep the overall accuracy of the scheme. This conflict between the need for a small step length and a large range of integration and the need to keep the computation times within reasonable bounds restricts the usefulness of such methods to
2"5
1"5
E 5 E=10
O"
o
~ 0
5
j 10
P
j 15
x
Fig. 1. Graphs of N u plotted against X, as calculated by a numerical solution of Eqs. (19) and (20), for e = --1, 0, 5 and 10 17
Acta
Mech.
66/1--4
258
J . H . Merkin and I. Pop:
moderate values of X. So to proceed further we use an approximate method, based on an integrated form of Eq. (3). This method was found to be satisfactory in the related natural convection problem, [7], and this is what we consider next.
5. Approximate Solution To obtain an approximate solution for N,, which will hold for all X, we use an integrated form of Eq. (3), namely d d-X
If
]
~0(l@e0) dg
-~--
O0 ~-~ 7=1
(25)
()
and, as in [7], the approximating profile for 0 0= 1--A -zlogg,
l=<~
--0,
~
(26)
~ =>e ~
with N~ then given b y 2g~ = A "'1. This leads to the equation for A dA dX
4A ~ A-&A(2A--
1) e2A~-2e(A + I + ( A - -
1) e2A)
(27)
which has to be solved subject to A = 0 at X = 0, giving
2,_1nAn+I
~
X = n = l ( n - ~ 1) ( n + 1)! -~-e =
2~nAn+l
(n @ 1)(n Jr 2)!"
(28)
Using (28) we can obtain plots of N~ against X for various values of e, and these are shown in Fig. 2. This figure shows that, for all the e considered, the curves of N~ appear to be approaching a common limiting form as X --> oz. This becomes more apparent when we consider the behaviour of Eq. (27) for X (and A) large. dA In this ease, Eq. (27) becomes, approximateIy, ~-~ = 2e -~A, from which it follows that, for X large, 2 N~ -- log 4 ~ ~- "'"
(29)
(29) is independent of s, and, to leading order, agrees with the asymptotic form given by (17), Also, from Eq. (27), we find, for small X, that
~v~ =
3 ~ e/1/2 ~,1--~-I +
(30) ...
Mixed Convection Boundary-Layer on a Vertical Cylinder
259
1.2-
Nu 1-0'
10
0'8-
0-6-
0-4-
0-2-
~-=0
0
-~
,
o
i
J
~
Fig. 2. Graphs of N u against
,,
i
6
J
~
~o
~,
1~
' ~'~ log X
X, obtained from the approximate method, for s =
--1, 0,
5 and 10 which gives a reasonable agreement with the values as calculated b y solving Eq. (23) over the range of e considered (from s = --1 to e = 10). All of this leads us to expect that this approximate solution will give a reasonable estimate for N~ for all X, with this estimate getting better as X increases.
6. Asymptotic Solution The approximate solution discussed in the previous section indicates that forced convection dominates over free convection for large X. This suggests t h a t to obtain a solution of Eqs. (6) and (7), valid for large X, we put 1
X
=2~+log~F(X'~)' 17"
1
O--log ~H(X,~),
~2
~=~-~.
(31)
260
J . H . Merkin and I. Pop:
Equations (6) and (7) then become OF
aS $ --~ +
+
1+
~
--
(32)
eH
1[(
+ 2 log-----~ H -~ 1
(~H ~ 1 x H -i-i OH + x "~-$ 0X
log 2 f 0---(
OH 0 ~ ) ] 0X
X ~H 2 0X = 0
(33)
subject to the boundary conditions that F = 0, H = log X on the cylinder OF ($ ---- 1/2X) and that H --~ 0, - ~ --~ 0 as ~ --> oc. A solution og Eqs. (32) and (33) is sought by expanding F and H as
F(X, r = Fo(~) + ~
1
F~(r +
...
i H(X, $) = Ho($) + ~
(34) H~($) - t - ' " .
The equation for H0 is ~H0"+ (1-t-~)H0':0
(35)
(where primes now denote differentiation with respect to ~). The solution of Eq. (35) which satisfies the condition that H0 -+ 0 as $ ~+ oo can be expressed in terms of the exponential integral, [15], as Ho -
ei(r
(36)
From [15], we have that, for small $, H0 ~- --log (~/2) -- y q- ~ + .-. (where y is again Euler's constant), so that on the cylinder ~ = 1/2X (i.e. ~ = 1) (36) gives 0= 1+
2 log 2 -- y +.... log X
(37)
So the boundary condition on the cylinder is satisfied to leading order, with (37) providing the inner boundary condition for the next term in the expansion. Fo is calculated via (32) as 270 = s[2(1 -- e-~[2) + $ei(~/2)].
(38)
The next term in expansion (34) is calculated from the equation ( ~ ) $H1" -I- 1 -[-
1 HI' + -~ No + ~:t EoHo' = 0
(39)
subject to the boundary conditions that H~ --> 0 as r --> ee, and, from (37), that H0 ~ (2 log 2 -- y) log $ for small $. Equation (39) cannot be integrated fully to
~ixed Convection Boundary-Layer on a Vertical Cylinder
261
give H1 in terms of known functions, we can, however, evaluate H i ' . This is
Hi' --
1[
~ ei
]
§ e-~/2 log ~ -- (3 log 2 - - 2~,) e-~12
(4o) §162247247 which has H~' ~ (2 log 2 - - y) ~-1 for small ~, as required. (36) and (40) are sufficient to calculate N~ to this order in the expansion, and we find, after a little calculation, t h a t 2 N~ -- - log X
4 log 2 -- 27 § .... (log X) ~
(41)
(41) shows t h a t the free convection effects do not enter into the expression for the heat transfer to this order in the expansion, though they do enter into the temperature profile (from (40)). Also we note t h a t (41) agrees with (17) for the purely forced convection limit and the first term agrees with (29) from the a p p r o x i m a t e solution. 7. Conclusion We have considered the mixed convection boundary layer on a vertical circular cylinder. We have shown t h a t the flow depends on the parameter ~ = Ra/P~ giving the ratio of the free to forced convection effects. When s is small, the flow is, to a first approximation, purely forced convection, the solution being given b y a known (unsteady) heat conduction problem. Next we considered the flow near the leading edge of the cylinder, finding t h a t there was a lower bound e0 on e for which solutions were possible. When we then came to consider a finite-difference solution of the equations to obtain a solution b y integrating away from the leading edge, we could get satisfactory results only when s > --1. With s0 < s < --1, the solution quickly became unstable, as there was now a region of reversed flow near the cylinder. This, perhaps, indicates that the model of a semi-infinite cylinder is no longer appropriate and, instead, we should in this instance really consider a cylinder of finite length, with forced convection dominating at one end and free convection dominating at the other. Even when the finite-difference scheme remained stable, we found that we could use this method only up to moderate distances from the leading edge, and getting a satisfactory estimate for the heat transfer at very large distances could be achieved b y a simple approximate method. This approximate method indicared that, at very large distances along the cylinder, forced convection would always be more important than free convection, a conjecture t h a t was confirmed by an asymptotic solution of the original boundary-layer equations. This was unexpected, as in all other previous cases of mixed convection flows, it has been found that free convection eventually becomes the dominant mechanism of heat transfer. This points to an essential difference between mixed convection flows
262
J . H . Merkin and I. Pop: Mixed Convection Boundary-Layer
t h a t are a x i - s y m m e t r i c (as i n the p r e s e n t case) a n d mixed convection flows t h a t are t w o - d i m e n s i o n a l (as i n the previous cases [2], [3], [4], [5], [10], [11]). At large distances along the cylinder the b o u n d a r y layer has spread out a n d the b o d y appears t h i n i n relation to the b o u n d a r y - l a y e r thickness. So it can s u p p l y relat i v e l y little heat, a n d affects the flow to a decreasing e x t e n t . This is n o t the case when t h e flow is p u r e l y t w o - d i m e n s i o n a l ; here the b o d y can s u p p l y enough heat to change the character of the b o u n d a r y - l a y e r flow at large distances from t h e l e a d i n g edge.
References [1] Cheng, P.: Heat transfer in geothermal systems. Adv. in Heat Transfer 14, 1--105 (1978). [2] Cheng, P. : Combined free and forced convection flow about inclined surfaces in a porous media. Int. J. Heat Mass Transfer 20, 807--814 (1977). [3] Merkin, J. H. : Mixed convection boundary layer flow on a vertical surface in a saturated porous medium. J. Engng. Math. 14, 301--313 (1980). [4] Cheng, P. : Similarity solutions for mixed convection from horizontal impermeable surfaces in saturated porous media. Int. J. Heat Mass Transfer 20, 893--898 (1977). [5] Joshi, Y., Gebhart, B. : Mixed convection in porous media adjacent to a vertical uniform heat flux surface. Int. J. Heat Mass Transfer 28, 1783--1786 (1985). [6] Minkowycz, W. J., Cheng, P. : Free convection about a vertical cylinder embedded in a porous medium. Int. J. Heat Mass Transfer 19, 805--813 (1976). [7] ]~erkin, J. H. : Free convection from a vertical cylinder embedded in a saturated porous medium. Acta Mechanica 62, 19--28 (1986). [8] Kumari, M., Pop, I., Nath, G. : Finite-difference and improved pertubation solutions ~or free convection on a vertical cylinder embedded in a saturated porous medium. Int. J. Heat Mass Transfer 28, 2171-2174 (1985). [9] Merkin, J. H. : On dual solutions occurring in mixed convection in a porous medium. J. Engng. Math. 20, 171--179 (1986). [10] Merkin, J. H.: The effect of buoyancy forces on the boundary-layer flow over a semi-infinite vertical flat plate in a uniform stream. J. Fluid Mech. 85, 439--450 (1969). [11] Wilks, G. : The flow of a uniform stream over a semi-infinite vertical flat plate with uniform surface heat flux. Int. J. Heat Mass Transfer 17, 743--753 (1974). [12] Carslaw, H. S., Jaeger, J. C.: The conduction of heat in solids. Oxford University Press 1947. [13] Batchelor, G. K. : The skin friction on infinite cylinders moving parallel to their length. Quart. J. Mech. Appl. Math. 7, 179--192 (1954). [14] 1YIerkin,J. H,: Free convection boundary layer in a saturated porous medium with lateral mass flux. Int. J. Heat Mass Transfer 21, 1499--1504 (1978). [15] Jeffreys, H., Jeffreys, B. : Methods of mathematical physics. Cambridge University Press 1962.
J. H. Merkin Department o] Applied Mathematics University o] Leeds Leeds LS2 9 J T United Kingdom
I. Pop Faculty of Mathematics University of Clu] R - 3400 Cluj Romania