- Email: [email protected]
A horizontal flow past a partially heated infinite vertical cylinder embedded in a porous medium
Inl. J. Engng Sci. Vol. 24, No. 8, pp. 1351-1363, Printed in Great Britain
1986
CO20-7225/86 $3.00 + .JO Pergamon Journals Ltd
A HORIZONTAL FLOW PAST A PARTIALLY HEATED INFINITE VERTICAL CYLINDER EMBEDDED IN A POROUS MEDIUM D. B. INGHAM Department of Applied Mathematical Studies, University of Leeds, LEEDS LS2 9JT, U.K.
I. POP Faculty of Mathematics, University of Cluj R-3400, Cluj, Romania Abstract-The boundary-layer flow on a vertical circular pipe which is embedded in a porous media is considered. The flow is driven by the buoyancy effects, due to the cylinder being at a higher temperature than the surrounding media, and a horizontal flow. This results in a three-dimensional boundary-layer being formed on the pipe and the resulting equations were solved using a finitedifference technique. Also asymptotic solutions which are valid at small and large distances along the pipe are given. Expressions are presented for the variation of the Nusselt number as a function of angular and axial distance along the pipe and the leading edge displacement effect is given.
INTRODUCTION THE BOUNDARY-LAYER formulation
of Darcy’s law and energy equation has been used with considerable success in a number of aspects of diffusion in fluid saturated porous media. An analysis of natural convection from a heated impermeable surface which is embedded in the medium has been used by Cheng and Minkowycz [l] to model the heating of groundwater in an aquafier by a dike. Several other boundary-layer natural and mixed convection are described in detail by Cheng [2]. Most of the studies on flows in a porous media have been restricted to two-dimensional geometry. Three-dimensional combined free and forced convection has rarely been studied, see Ingham and Pop [3], by either an analytical approach or by experimental means due to the inherent complexity of a three-dimensional flow geometry. However, in most engineering applications three-dimensional flow geometries are unavoidable. In this paper we are interested in the boundary-layer which exists on a vertical pipe, of a cylindrical cross section, which is partially heated and embedded in a porous media. This situation models a pipe which is inserted vertically into a geothermal reservoir. Frequently there is a horizontal flow within the reservoir and therefore this is modelled by assuming that there is a uniform stream flowing past the cylinder. Here we consider the interaction between the free and forced convection by considering a steady stream passing a partially heated infinite cylinder, see Fig. 1. The cylinder is heated to a constant temperature T, for axial distances z^> 0 whilst for z^< 0 it is maintained at the ambient temperature of the media, T, . An asymptotic solution which is valid near the thermal leading edge of the cylinder (2 < 1) is generated and this solution is then continued by means of a finite difference technique in order to solve the three-dimensional boundary-layer flow. The terminal solution at large distances along the cylinder is found along with a theory as to how this solution is approached at the front stagnation point of the cylinder. The numerical solution is continued for sufficiently large values of z^in order to confirm this asymptotic solution. Although an infinite rather than the more realistic semi-infinite cylinder has been taken in order to simplify the analysis the results should provide in the practical problem useful information in determining quantities such as the axial distance at which for asymptotic solution at large values of i may be taken to be valid. The present problem is analogous to the one treated by Yao [4] for a Newtonian fluid that flows horizontally past a partially heated infinite vertical cylinder. 1351
1352
D. B. INGHAM and I. POP 2. ANALYSIS
The physical model consists of an infinite vertical cylinder of radius a which is embedded in a saturated porous medium, see Fig. 1. A uniform horizontal free stream is flowing past the cylinder with speed U, and the ambient fluid temperature is T, . The temperature of the cylinder for 2 > 0 is held at a constant value of T, (>T,)whilst for z^< 0 the cylinder remains unheated and therefore at a temperature of T,. It will be assumed that both the Grashof number and the Reynolds number of the flow are so large that the boundary-layer approximations can be made. Thus the density stratifications within the boundary-layer induce a vertical flow whilst the horizontal flow produces motion within the boundary-layer around the cylinder. This gives rise to a three-dimensional boundary-layer flow. For simplicity in this paper we will assume that: (a) the convective fluid and the porous medium are everywhere in local thermodynamic equilibrium, (b) the temperature of the fluid is below the boiling point everywhere, (c) the properties of the fluid and the porous medium, such as viscosity, thermal conductivity, specific heats, thermal expansion coefficient and permeability are constant, (d) the Boussinesq approximation is valid. Under these assumptions the governing equations in cylindrical polar coordinates are given by
(1) 1 dti u^ ao __---_= 0 i ad i a?
(2)
a$ ati gprcaT --a? a;= T
(3)
a;
dT v^dT dT a2T 1 aT ~-++-++-=aa++,,+__++ a: a2 ai r dr r a4
(
Fig. 1. Physical coordinates.
i a2T a2 i2 ab2 a22
3
Horizontal flow past a partially heated infinite vertical cylinder
1353
where ?, 4, i are cylindrical polar coordinates with z^measured along the axis of the cylinder with the heating starting at i = 0, li, ri, FCare the velocity components in the f, C#J, z”directions, T is the temperature and the other symbols have their usual meanings. Outside the boundary-layer on the cylinder we introduce the following non dimensional variables r = ?/a,
T-
a=
2 = .$a,
r;,
Tw - Tm
(~2,5, G) = (6, 6, $/(Re
(5)
v/a)
where Re is the Reynolds number of the flow, = U,a/u. Assuming that the Reynolds number is large, then eqns (l)-(4) become
at-2u”
-$f-+-
I
r
aa a@
-+-jg=o
(6)
r a$
1 at2 d aa ----_-_-= r
r arp
0
dr
a$ aa ---=tar a2
a8
(8)
dr
(9) where t: = GrfRe and Gr = g(JK( TW- T,)af v2. The associated boundary conditions of eqns W(9) are u”=O
at
r=l
all #
w B-+0,
ficosd,+u”sinf&--+O
d--c 0,
5sin4-zZcos$-J
as
and r-co
z”
alld,
and
z
1
The solution of eqns (6)-(9) which satisfy the boundary conditions (10) is
(11)
It is seen that the solution (1 l), of the inviscid flow outside the boundary-layer, satisfies the boundary conditions on the cylinder for z -C0. Hence only a boundary-layer forms on the cylinder for z > 0 and it is this boundary-layer that we will now consider for the rest of this paper. The boundary-layer can now be obtained by stretching the coordinate and velocity normal to the surface of the cylinder as y = (r - l)(Gr/e Pr)“*, w = G/c,
u = zZ(Gr/c Pr)li2
2 = 2f’/t
(12)
where Pr is the Prandtl number, =Y/Q. Substitution of expressions ( 12) into eqns (l)-(4) gives (13)
1354
D. B. INGHAM and I. POP dV
-0 ay-
aw a0 aY
u-+v-+-_=-
(14)
atI
dr=i$
(1%
a8 a0 a20 a# a2 ay2*
(16)
These eqns have to be solved subject to the boundary conditions u = 0, w, fl -
0,
e=
1
at
Y = 0,
all 4,
.z>o
(1W
v -
2 sin d,
as
y-m,
all d,
z>o
t17b)
where condition (17b) arises from matching v, w and 6’with the outer solution (11). Using boundary condition ( 17b) eqns ( 14) and ( 15) give w = 8, and then the boundary-layer
v = 2 sin C$
(18)
eqns ( 13) and (16) reduce to
ati a0
-+--_220s#=0 aY
(19)
az
~~+2sin~~+~~=~ 8Y
az
(20)
aY2 -
The boundary conditions (17) become U = 0,
8=1
6-O
at
Y = 0,
ali cp,
z>o
as
Y-
all 6,
z>o 1 ’
3. METHOD
03,
(21)
OF SOLUTION
(a) Asymptotic solution for z 6 1 On examining the governing eqns ( 19) and (20) subject to the boundary conditions (2 1) at very small values of z we find that most appropriate variables to use are
u=2”2u f 0
n = y/zii2.
and
(22)
Substituting expression (22) into eqns (19) and (20) gives (U - $$) t NJ+zz as
+ 22 sin 4 5 1
+ zB E = 3
(23)
do
--T)-+2zcos~=o az 2 dlt
(24)
and boundary conditions (2 1) become
u = 0, 8-O
@=I
on
rl = 0,
all 4,
z>O]
as
17-
all #,
z>OJ
00,
(25)
Inspection of equations (23) and (24) along with boundary conditions (25) give rise to the following form of the solution, near z = 0,
Horizontal flow past a partially heated infinite vertical cylinder 0 =./XV)
f 2 cos #f’l(tl> + z2(sin29VXrl)
cos24fi2(o))
+
+ z3cos
l-l = fw%d
-h(d)
+
$z3cos
%W
4{sin2f$?f&(v) + COS~~~~Z(~)} +. - . >.
- 3f,W4tll
+ tz cos cbwdd
+ Iz2(sin2dW%(r1) -
1355 >
+ cos2#47u’92(9)
#~sin*~~,(~)
-
-
7~1(~))
(26)
5f22123))_ + cos2~(~2(~~
-
71f;2@?))f,
On substitution of expressions (26) into eqns (23) and (24) and collecting up the terms of the same coefficient of z we obtain,
(27b)
f’i, + tfof’l, - v-&f-L + gf6f2, f% + f
fAd-52-
a-cLfs2
+
$f Efi2
=
-2f’l
=
f ‘: -
(27~)
$fifl -
212f;
(27d)
71+ f.hf’;l - 3fbf51+ ff&&, = 3fif'zl + J?fi, - 4f122 - 3.&f;- 27f%- $_hfll(27e) f% +
UJfl;,
-
3fbf 52 + $f 8f32 = 3f ;fL
- ff’;fi2 - 2&
- $5
f 'zz etc.
(27f)
whilst boundary conditions (25) reduce to
“6x0)= 0,
.f@) = 1,
f&(a) = 0
(284
_fm) = 0,
f
X0) = 0,
f;(a) = 0
(28b)
jj(O) =
f b(O) = 0,
0,
f>(co)
= 0
(28~)
for i = 2, 3 and j = 1, 2 etc. The ordinary differential eqns (27), subject to the boundary conditions (28), were solved using a two point boundary method and the Runge-Kutta Merson method on the Amdahl computer at Leeds. Equation (27a) occurs frequently in fluid mechanics and has been solved on many occasions subject to the boundary conditions (28a), see for example Ackroyd f5]. This solution forms a check on the numerical procedure used in solving the other eqns (27), subject to the boundary conditions (28). (b) Asymptotic solution for z % 1 At large distances along the cylinder the solution will become independent of z and eqns (19) and (20) become
au
--t2cos~=O aY
a0
u-+2sind,--7. ay
(29)
a8 a28 a#
ay
(30)
The solution of eqn (29) subject to the boundary condition (17a) gives u = -2ycos&
(31)
In order to solve the differential eqn (30) we write
!T= Y cos 43/2,
@= @0(!3.
(32)
1356
D. B. INGHAM and I. POP
Then eqn (30) becomes (33) and boundary condition ( 17) for B0gives
e,(o) = 1,
eo(oO) = 0.
(34)
Hence e. = 1 - erf({).
(35)
(c) Asymptotic solutionfor z 9 1 on C#J =0 The above theory shows that as z - co for C#J = 0 then u -+ -2y,
e-
1 - erf(y).
(36)
Assuming that the approach to this solution is of the form +. . . ,
u = -2y + e-%,(y)
e = 1 - erf(y) + Pel(y)
+ - . -,
(37)
where X is an unknown constant then the eqns satisfied by uI and 8, are
4 - xe, = 0, -2yt9; -
_2_ e+ G
241
X( 1 - erf( y)) = e;,
(38)
e,(c0) = 0.
(39)
and the boundary conditions (17) give
Ul(O)= 0,
e,(o) = 0,
The differential eqns (38) were solved, subject to the boundary conditions (39) and the smallest value of X found. (d) Numericalsolution In order to match the analytical solutions as presented in sections (a) and (b) the threedimensional boundary-layer eqns were solved using a finite difference technique. It is found most convenient, because of the asymptotic forms of the solution, to use the differential eqns in the form (23) and (24) for z =Z1 and of the form ( 19) and (20) for z >, 1. A finite difference grid is set up such that there is a constant mesh size in the 4 direction of size A& = X/M, where M is an integer. There is also a constant mesh size in the radial direction of An = q,/N, where qco is the value of TJat which the outer value boundary conditions may be assumed to hold and n is an integer. For a continuous change of variable A variable mesh size, AZ, in the z-direction atz=lwetakeAy=Aqandy,=q,. was taken since more detail is required near z = 0 in order to obtain accurate results. The value of M, N, qa, and AZ were varied but the results presented in this paper were for M= N= 180,0, = 50 and AZ = 10P6, lo-‘, 10e4, 10e3 and lo-* for 0 < z -C 10F4, 10e4 < z < 10e3, 10e3 < z < lo-*, lo-* < z < 10-l and z > 10-l respectively. Results obtained using a slightly cruder discretization showed very little difference from the results presented in this paper and hence these results may be taken to the accuracy as claimed. The numerical scheme was such that first order derivatives in the 6 and z were approx-
Horizontalflowpast a partially heated infinite vertical cylinder
1357
imated by backward differences and in the 7 (or y) direction by central differences. The diffusion term was approximated by the implicit scheme. Writing KY’ = K(iA$, jA7, ~Az)
(40)
where K is either U or 8, then eqns (23) and (24) can be approximated
by
eS’> - 2Aqz cos &J
(42)
fori= 1,2; - aMand N = 1,2,* . .N. The finite difference scheme as presented in eqns (41) and (42) are implicit. However given the solution at z = nAz (when n = 0 we know the solution as it is given in expression (26)) we can obtain the solution at (n + 1)Az for 8 by solving the tri diagonal system of eqns (4 1) in the j-direction for each value of i. Having obtained t9at a given i station U can be ex&citly ob-tined using equation (42). This process was repeated step-by-step until z= 1. For z 2 1 we write K& = K(iAd, jAy, nAz)
(43)
where K is either U or e then eqns (19) and (20) become
for i = 1, 2, - - *Mand j = 1, 2; . *N. The method of solution of eqns (44) and (45) is exactly the same as that described for eqns (4 1) and (42). 4. RESULTS
The primary physical quantity of interest is the Nusselt number, Nu, which is defined by
qHa
Nu=
(46)
km(Tw- Tco)
where q,,, is the rate of heat transfer at the surface of the cylinder and is determined from . Using expression (26) we find that
i=a Nu
=
_
“2[fE(O) + z cos c#$~(O>+ z2{sin24&r(0)
+ cos2bf'i2(0)}
+ z3cos +{sin2@,f&(0) +
Es24:8-H
cos2f$f;2(0)} + * - -1.
(47)
1358
D. B. INGHAM and I. POP
The numerical integration of eqns (27) subject to the boundary conditions (28) gives f;(o)
= -0.44375
f’i(0) = -0.67829 f’$r(O) = -0.85822
.
(48)
f;2(0) = -0.31209 f;,(O) = 2.51583 f&(O) = -0.03237 Figure 2 shows the variation of the Nusselt number as a function of z along (b = 0 as obtained from the full numerical solution as described in section 3b. Also shown are the 1, 2 and 4 term solutions as given in eqn (47). The three term solution being almost indistinguishable from the 4 term solution. The four term solution gives results which are graphically indistinguishable from the full numerical solution for z G 0.5. The approach to the asymptotic solution at large values z is given in section 3c. Numerical integration of the differential equations (39) gives x = 6.8866
(49)
which suggests that for large values of z, on Cp= 0, that Nu
= 1
+ ?&6.88662
G
(50)
where y is an unknown constant. Comparing the computed result with expression (50) at z = 0.4 suggest that y w 0.7255. Figure 2 shows the variation of Nu with z as predicted by eqn (50) with y = 0.7255. It is seen that expression (50) is a good approximation to Nu for values of 2 > 0.2. In order to illustrate that the asymptotic solution as presented in section 3c is also very accurate for all values y, Fig. 3 shows the variation of [0 - 1 + erfy]e+x” for various values of z as a function of y with X = 6.8866. In the limit as z - co the solution e,(y) should be approached. Fig. 3 shows the function e,(y) with 6’,(O) = 0.7255 and it is seen that this asymptotic behaviour is being approached for large values of z.
Fig. 2. The variation of the local Nusselt number as a function of the distance along the cylinder for d, = 0”.
Horizontal flow past a partially heated infinite vertical cylinder
1359
Fig. 3. The variation of [@- 1 + erf_v]? as a function of y for various vaiues of z.
Figure 4 shows the variation of Nu as obtained from the full numerical solution as described in section 3b as a function of z on 4 = 180”. Asymptotically for large z we expect that Nu - 0 and this is confirmed by the numerical calculations. Also shown in Fig. 4 is the variation of Nu as calculated from eqn (47), with the unknown constants being as given in eqn (48), using the first 4 terms in the expansion. It is observed that the 4 term expansion is a very good approximation for Nu up to z - 0.5 and may be used even up to z - 1.0 without too much loss in accuracy. The variation of the Nusselt number as a function of C$is shown in Fig. 5 for various values of z. In addition to the numerically obtained solutions the asymptotic solution which
,---
3rdterm :--
4th
series
term
series
Fig. 4. The variation of the local Nusseit number as a function of the diitance along the cylinder for r#~= 180’.
1360
D. d. INGHAM and I. POP
Fig. 5. The variation of the local Nusselt number as a function of 4 for various values of z.
is valid at large values of z as obtained in section 3b is presented. It is seen that for z z 0.4 that the asymptotic solution for C#J < 90” has been approached. As the value of z increases so does the approach to the asymptotic solution for larger and larger values of 4. The fact that the governing equations are parabolic in the 4 direction automatically implies that the asymptotic solution is approached quicker for the smaller the value of 4.
I
8
0‘.a
-
0,6-
\_;--
____z
q
0.02
_. ~ -: \.- \
\
0 d-
__.2=0.04 \.!_!;-_i-= 0.1
_-,\\\
2=0.4 and _--*-__ limit solution
0 2-
0 0
I
2
3 Y
Fig. 6. The temperature profiles at various values of the distance along the cylinder as a function of the radial distance: (a) 6 = O”, (b) 4 = 60”, (c) @J= 120”, (d) Q = 180”.
1361
Horkontal flow past a partially heated infinite vertical cylinder
Fig. 6. (Continued)
Usually one is interested in the total heat flux from the cylinder and this is given by (51) This quantity has been obtained from the numerical solution but because of the close agreement between the numerical solution and the four term series solution, which is valid for z e 1, then we may approximate & by 1
z
[0.44375 + 0.29258~~1
(52)
D. B. INGHAM and I. POP
1362
forz<
1. Whenz>
1 weuse
(53)
2 1.
= Gr ‘I2 0.71835 + !?JE! pr O[
(54)
The second term in eqn (54) represents the leading edge correction to the asymptotic solution. Figure 6 show the temperature profiles as a function of y for various values of z along with the asymptotic profiles as given by eqn (35). Again we observe that the smaller the value of 4 the quicker is the asymptotic solution approached. It is also seen that the boundarylayer increases in thickness as # increases and as # - 180” the boundary-layer is of infinite thickness as is indicated by eqn (32). Because of the rapid growth in the boundary-layer thickness near Q, = 180” when z - 0(l) the numerical calculations cannot be continued for all values of Cpuntil the asymptotic solution has been achieved. 5. CONCLUSIONS
The laminar funds-layer which is induced by a horizontal forced flow along a heated vertical cylinder which is main~n~ at a temperature in excess of the su~oun~ng porous media has been investigated. A numerical solution of the governing three-dimensional boundary layer equation has been presented along with the asymptotic solutions which are valid at small and large distances along the pipe. Close to the leading edge the buoyancy force induces a strong free-convection boundary-layer. The importance of the forced convection increases as one moves vertically upwards from the leading edge. It is found that for axial distances greater than about at that the asymptotic solution has been achieved. Thus increasing the Reynolds number or decreasing the Grashof number has the effect of decreasing the distance before the asymptotic solution is reached-this is exactly as one would expect. If one is only interested in the heat transfer from the cylinder it is seen that the series solution which is valid for small values of distance along the cylinder can be used up to z = 1 whilst the asymptotic solution which is valid for large values of z may be used for z > 1. However, if full details of the temperature profiles etc. are required then one must resort to the full numerical solution. NOMENCLATURE radius of cylinder acceleration due to gravity g Grashof number Gr permeability of porous medium thermal conductivity of porous medium f local Nusselt number N”, average Nusselt number RG Prandtl number Pr rate of heat transfer 4 radial co-ordinate Reynolds number Re temperature T u, u, w boundary layer velocities axial co-ordinate a
Greek ~~~bof~ thermal diffusivity thermal expansion coefficient ; constants YTh ratio (: similarity variables 8, l e boundary layer temperature kinematic viscosity azimuthal co-ordinate :
Horizontal Ilow past a partially heated infinite vertical cylinder
1363
Subscripts w surface freestream m Superscripts Dimensions quantities inviscid flow quantities derivation with respect to n Acknowledgements-Part of this paper was completed while the second author (IP) has been in the institute B of The~l~ynami~ (Chairman Prof. D. Vortmeyer) of the Technical University of Munich. The author expresses his heartiest thanks to the Alexander von Humboldt Foundation for supporting his research in Germany and to Prof. Vortmeyer for the very kind hospitality in his Institute. REFERENCES
(I] P. CHENG and M. J. MINKOWYCZ, J. GeopJzys. Res. %I,2040 (1977). [2] [3] (41 [5]
P. CHENG, In?. J. Heat Mass Tram& 21, 1499 (1978). D. B. INGHAM and I. POP, To be published in W&me Stoffibertrug. L. S. YAO, J. Heat Transfe 105, 101 (1983). J. A, D. ACKROYD, Proc. Cum. Phil. Sot. 63,871 (1967). (Received 22 Augusf 1985)
1986
CO20-7225/86 $3.00 + .JO Pergamon Journals Ltd
A HORIZONTAL FLOW PAST A PARTIALLY HEATED INFINITE VERTICAL CYLINDER EMBEDDED IN A POROUS MEDIUM D. B. INGHAM Department of Applied Mathematical Studies, University of Leeds, LEEDS LS2 9JT, U.K.
I. POP Faculty of Mathematics, University of Cluj R-3400, Cluj, Romania Abstract-The boundary-layer flow on a vertical circular pipe which is embedded in a porous media is considered. The flow is driven by the buoyancy effects, due to the cylinder being at a higher temperature than the surrounding media, and a horizontal flow. This results in a three-dimensional boundary-layer being formed on the pipe and the resulting equations were solved using a finitedifference technique. Also asymptotic solutions which are valid at small and large distances along the pipe are given. Expressions are presented for the variation of the Nusselt number as a function of angular and axial distance along the pipe and the leading edge displacement effect is given.
INTRODUCTION THE BOUNDARY-LAYER formulation
of Darcy’s law and energy equation has been used with considerable success in a number of aspects of diffusion in fluid saturated porous media. An analysis of natural convection from a heated impermeable surface which is embedded in the medium has been used by Cheng and Minkowycz [l] to model the heating of groundwater in an aquafier by a dike. Several other boundary-layer natural and mixed convection are described in detail by Cheng [2]. Most of the studies on flows in a porous media have been restricted to two-dimensional geometry. Three-dimensional combined free and forced convection has rarely been studied, see Ingham and Pop [3], by either an analytical approach or by experimental means due to the inherent complexity of a three-dimensional flow geometry. However, in most engineering applications three-dimensional flow geometries are unavoidable. In this paper we are interested in the boundary-layer which exists on a vertical pipe, of a cylindrical cross section, which is partially heated and embedded in a porous media. This situation models a pipe which is inserted vertically into a geothermal reservoir. Frequently there is a horizontal flow within the reservoir and therefore this is modelled by assuming that there is a uniform stream flowing past the cylinder. Here we consider the interaction between the free and forced convection by considering a steady stream passing a partially heated infinite cylinder, see Fig. 1. The cylinder is heated to a constant temperature T, for axial distances z^> 0 whilst for z^< 0 it is maintained at the ambient temperature of the media, T, . An asymptotic solution which is valid near the thermal leading edge of the cylinder (2 < 1) is generated and this solution is then continued by means of a finite difference technique in order to solve the three-dimensional boundary-layer flow. The terminal solution at large distances along the cylinder is found along with a theory as to how this solution is approached at the front stagnation point of the cylinder. The numerical solution is continued for sufficiently large values of z^in order to confirm this asymptotic solution. Although an infinite rather than the more realistic semi-infinite cylinder has been taken in order to simplify the analysis the results should provide in the practical problem useful information in determining quantities such as the axial distance at which for asymptotic solution at large values of i may be taken to be valid. The present problem is analogous to the one treated by Yao [4] for a Newtonian fluid that flows horizontally past a partially heated infinite vertical cylinder. 1351
1352
D. B. INGHAM and I. POP 2. ANALYSIS
The physical model consists of an infinite vertical cylinder of radius a which is embedded in a saturated porous medium, see Fig. 1. A uniform horizontal free stream is flowing past the cylinder with speed U, and the ambient fluid temperature is T, . The temperature of the cylinder for 2 > 0 is held at a constant value of T, (>T,)whilst for z^< 0 the cylinder remains unheated and therefore at a temperature of T,. It will be assumed that both the Grashof number and the Reynolds number of the flow are so large that the boundary-layer approximations can be made. Thus the density stratifications within the boundary-layer induce a vertical flow whilst the horizontal flow produces motion within the boundary-layer around the cylinder. This gives rise to a three-dimensional boundary-layer flow. For simplicity in this paper we will assume that: (a) the convective fluid and the porous medium are everywhere in local thermodynamic equilibrium, (b) the temperature of the fluid is below the boiling point everywhere, (c) the properties of the fluid and the porous medium, such as viscosity, thermal conductivity, specific heats, thermal expansion coefficient and permeability are constant, (d) the Boussinesq approximation is valid. Under these assumptions the governing equations in cylindrical polar coordinates are given by
(1) 1 dti u^ ao __---_= 0 i ad i a?
(2)
a$ ati gprcaT --a? a;= T
(3)
a;
dT v^dT dT a2T 1 aT ~-++-++-=aa++,,+__++ a: a2 ai r dr r a4
(
Fig. 1. Physical coordinates.
i a2T a2 i2 ab2 a22
3
Horizontal flow past a partially heated infinite vertical cylinder
1353
where ?, 4, i are cylindrical polar coordinates with z^measured along the axis of the cylinder with the heating starting at i = 0, li, ri, FCare the velocity components in the f, C#J, z”directions, T is the temperature and the other symbols have their usual meanings. Outside the boundary-layer on the cylinder we introduce the following non dimensional variables r = ?/a,
T-
a=
2 = .$a,
r;,
Tw - Tm
(~2,5, G) = (6, 6, $/(Re
(5)
v/a)
where Re is the Reynolds number of the flow, = U,a/u. Assuming that the Reynolds number is large, then eqns (l)-(4) become
at-2u”
-$f-+-
I
r
aa a@
-+-jg=o
(6)
r a$
1 at2 d aa ----_-_-= r
r arp
0
dr
a$ aa ---=tar a2
a8
(8)
dr
(9) where t: = GrfRe and Gr = g(JK( TW- T,)af v2. The associated boundary conditions of eqns W(9) are u”=O
at
r=l
all #
w B-+0,
ficosd,+u”sinf&--+O
d--c 0,
5sin4-zZcos$-J
as
and r-co
z”
alld,
and
z
1
The solution of eqns (6)-(9) which satisfy the boundary conditions (10) is
(11)
It is seen that the solution (1 l), of the inviscid flow outside the boundary-layer, satisfies the boundary conditions on the cylinder for z -C0. Hence only a boundary-layer forms on the cylinder for z > 0 and it is this boundary-layer that we will now consider for the rest of this paper. The boundary-layer can now be obtained by stretching the coordinate and velocity normal to the surface of the cylinder as y = (r - l)(Gr/e Pr)“*, w = G/c,
u = zZ(Gr/c Pr)li2
2 = 2f’/t
(12)
where Pr is the Prandtl number, =Y/Q. Substitution of expressions ( 12) into eqns (l)-(4) gives (13)
1354
D. B. INGHAM and I. POP dV
-0 ay-
aw a0 aY
u-+v-+-_=-
(14)
atI
dr=i$
(1%
a8 a0 a20 a# a2 ay2*
(16)
These eqns have to be solved subject to the boundary conditions u = 0, w, fl -
0,
e=
1
at
Y = 0,
all 4,
.z>o
(1W
v -
2 sin d,
as
y-m,
all d,
z>o
t17b)
where condition (17b) arises from matching v, w and 6’with the outer solution (11). Using boundary condition ( 17b) eqns ( 14) and ( 15) give w = 8, and then the boundary-layer
v = 2 sin C$
(18)
eqns ( 13) and (16) reduce to
ati a0
-+--_220s#=0 aY
(19)
az
~~+2sin~~+~~=~ 8Y
az
(20)
aY2 -
The boundary conditions (17) become U = 0,
8=1
6-O
at
Y = 0,
ali cp,
z>o
as
Y-
all 6,
z>o 1 ’
3. METHOD
03,
(21)
OF SOLUTION
(a) Asymptotic solution for z 6 1 On examining the governing eqns ( 19) and (20) subject to the boundary conditions (2 1) at very small values of z we find that most appropriate variables to use are
u=2”2u f 0
n = y/zii2.
and
(22)
Substituting expression (22) into eqns (19) and (20) gives (U - $$) t NJ+zz as
+ 22 sin 4 5 1
+ zB E = 3
(23)
do
--T)-+2zcos~=o az 2 dlt
(24)
and boundary conditions (2 1) become
u = 0, 8-O
@=I
on
rl = 0,
all 4,
z>O]
as
17-
all #,
z>OJ
00,
(25)
Inspection of equations (23) and (24) along with boundary conditions (25) give rise to the following form of the solution, near z = 0,
Horizontal flow past a partially heated infinite vertical cylinder 0 =./XV)
f 2 cos #f’l(tl> + z2(sin29VXrl)
cos24fi2(o))
+
+ z3cos
l-l = fw%d
-h(d)
+
$z3cos
%W
4{sin2f$?f&(v) + COS~~~~Z(~)} +. - . >.
- 3f,W4tll
+ tz cos cbwdd
+ Iz2(sin2dW%(r1) -
1355 >
+ cos2#47u’92(9)
#~sin*~~,(~)
-
-
7~1(~))
(26)
5f22123))_ + cos2~(~2(~~
-
71f;2@?))f,
On substitution of expressions (26) into eqns (23) and (24) and collecting up the terms of the same coefficient of z we obtain,
(27b)
f’i, + tfof’l, - v-&f-L + gf6f2, f% + f
fAd-52-
a-cLfs2
+
$f Efi2
=
-2f’l
=
f ‘: -
(27~)
$fifl -
212f;
(27d)
71+ f.hf’;l - 3fbf51+ ff&&, = 3fif'zl + J?fi, - 4f122 - 3.&f;- 27f%- $_hfll(27e) f% +
UJfl;,
-
3fbf 52 + $f 8f32 = 3f ;fL
- ff’;fi2 - 2&
- $5
f 'zz etc.
(27f)
whilst boundary conditions (25) reduce to
“6x0)= 0,
.f@) = 1,
f&(a) = 0
(284
_fm) = 0,
f
X0) = 0,
f;(a) = 0
(28b)
jj(O) =
f b(O) = 0,
0,
f>(co)
= 0
(28~)
for i = 2, 3 and j = 1, 2 etc. The ordinary differential eqns (27), subject to the boundary conditions (28), were solved using a two point boundary method and the Runge-Kutta Merson method on the Amdahl computer at Leeds. Equation (27a) occurs frequently in fluid mechanics and has been solved on many occasions subject to the boundary conditions (28a), see for example Ackroyd f5]. This solution forms a check on the numerical procedure used in solving the other eqns (27), subject to the boundary conditions (28). (b) Asymptotic solution for z % 1 At large distances along the cylinder the solution will become independent of z and eqns (19) and (20) become
au
--t2cos~=O aY
a0
u-+2sind,--7. ay
(29)
a8 a28 a#
ay
(30)
The solution of eqn (29) subject to the boundary condition (17a) gives u = -2ycos&
(31)
In order to solve the differential eqn (30) we write
!T= Y cos 43/2,
@= @0(!3.
(32)
1356
D. B. INGHAM and I. POP
Then eqn (30) becomes (33) and boundary condition ( 17) for B0gives
e,(o) = 1,
eo(oO) = 0.
(34)
Hence e. = 1 - erf({).
(35)
(c) Asymptotic solutionfor z 9 1 on C#J =0 The above theory shows that as z - co for C#J = 0 then u -+ -2y,
e-
1 - erf(y).
(36)
Assuming that the approach to this solution is of the form +. . . ,
u = -2y + e-%,(y)
e = 1 - erf(y) + Pel(y)
+ - . -,
(37)
where X is an unknown constant then the eqns satisfied by uI and 8, are
4 - xe, = 0, -2yt9; -
_2_ e+ G
241
X( 1 - erf( y)) = e;,
(38)
e,(c0) = 0.
(39)
and the boundary conditions (17) give
Ul(O)= 0,
e,(o) = 0,
The differential eqns (38) were solved, subject to the boundary conditions (39) and the smallest value of X found. (d) Numericalsolution In order to match the analytical solutions as presented in sections (a) and (b) the threedimensional boundary-layer eqns were solved using a finite difference technique. It is found most convenient, because of the asymptotic forms of the solution, to use the differential eqns in the form (23) and (24) for z =Z1 and of the form ( 19) and (20) for z >, 1. A finite difference grid is set up such that there is a constant mesh size in the 4 direction of size A& = X/M, where M is an integer. There is also a constant mesh size in the radial direction of An = q,/N, where qco is the value of TJat which the outer value boundary conditions may be assumed to hold and n is an integer. For a continuous change of variable A variable mesh size, AZ, in the z-direction atz=lwetakeAy=Aqandy,=q,. was taken since more detail is required near z = 0 in order to obtain accurate results. The value of M, N, qa, and AZ were varied but the results presented in this paper were for M= N= 180,0, = 50 and AZ = 10P6, lo-‘, 10e4, 10e3 and lo-* for 0 < z -C 10F4, 10e4 < z < 10e3, 10e3 < z < lo-*, lo-* < z < 10-l and z > 10-l respectively. Results obtained using a slightly cruder discretization showed very little difference from the results presented in this paper and hence these results may be taken to the accuracy as claimed. The numerical scheme was such that first order derivatives in the 6 and z were approx-
Horizontalflowpast a partially heated infinite vertical cylinder
1357
imated by backward differences and in the 7 (or y) direction by central differences. The diffusion term was approximated by the implicit scheme. Writing KY’ = K(iA$, jA7, ~Az)
(40)
where K is either U or 8, then eqns (23) and (24) can be approximated
by
eS’> - 2Aqz cos &J
(42)
fori= 1,2; - aMand N = 1,2,* . .N. The finite difference scheme as presented in eqns (41) and (42) are implicit. However given the solution at z = nAz (when n = 0 we know the solution as it is given in expression (26)) we can obtain the solution at (n + 1)Az for 8 by solving the tri diagonal system of eqns (4 1) in the j-direction for each value of i. Having obtained t9at a given i station U can be ex&citly ob-tined using equation (42). This process was repeated step-by-step until z= 1. For z 2 1 we write K& = K(iAd, jAy, nAz)
(43)
where K is either U or e then eqns (19) and (20) become
for i = 1, 2, - - *Mand j = 1, 2; . *N. The method of solution of eqns (44) and (45) is exactly the same as that described for eqns (4 1) and (42). 4. RESULTS
The primary physical quantity of interest is the Nusselt number, Nu, which is defined by
qHa
Nu=
(46)
km(Tw- Tco)
where q,,, is the rate of heat transfer at the surface of the cylinder and is determined from . Using expression (26) we find that
i=a Nu
=
_
“2[fE(O) + z cos c#$~(O>+ z2{sin24&r(0)
+ cos2bf'i2(0)}
+ z3cos +{sin2@,f&(0) +
Es24:8-H
cos2f$f;2(0)} + * - -1.
(47)
1358
D. B. INGHAM and I. POP
The numerical integration of eqns (27) subject to the boundary conditions (28) gives f;(o)
= -0.44375
f’i(0) = -0.67829 f’$r(O) = -0.85822
.
(48)
f;2(0) = -0.31209 f;,(O) = 2.51583 f&(O) = -0.03237 Figure 2 shows the variation of the Nusselt number as a function of z along (b = 0 as obtained from the full numerical solution as described in section 3b. Also shown are the 1, 2 and 4 term solutions as given in eqn (47). The three term solution being almost indistinguishable from the 4 term solution. The four term solution gives results which are graphically indistinguishable from the full numerical solution for z G 0.5. The approach to the asymptotic solution at large values z is given in section 3c. Numerical integration of the differential equations (39) gives x = 6.8866
(49)
which suggests that for large values of z, on Cp= 0, that Nu
= 1
+ ?&6.88662
G
(50)
where y is an unknown constant. Comparing the computed result with expression (50) at z = 0.4 suggest that y w 0.7255. Figure 2 shows the variation of Nu with z as predicted by eqn (50) with y = 0.7255. It is seen that expression (50) is a good approximation to Nu for values of 2 > 0.2. In order to illustrate that the asymptotic solution as presented in section 3c is also very accurate for all values y, Fig. 3 shows the variation of [0 - 1 + erfy]e+x” for various values of z as a function of y with X = 6.8866. In the limit as z - co the solution e,(y) should be approached. Fig. 3 shows the function e,(y) with 6’,(O) = 0.7255 and it is seen that this asymptotic behaviour is being approached for large values of z.
Fig. 2. The variation of the local Nusselt number as a function of the distance along the cylinder for d, = 0”.
Horizontal flow past a partially heated infinite vertical cylinder
1359
Fig. 3. The variation of [@- 1 + erf_v]? as a function of y for various vaiues of z.
Figure 4 shows the variation of Nu as obtained from the full numerical solution as described in section 3b as a function of z on 4 = 180”. Asymptotically for large z we expect that Nu - 0 and this is confirmed by the numerical calculations. Also shown in Fig. 4 is the variation of Nu as calculated from eqn (47), with the unknown constants being as given in eqn (48), using the first 4 terms in the expansion. It is observed that the 4 term expansion is a very good approximation for Nu up to z - 0.5 and may be used even up to z - 1.0 without too much loss in accuracy. The variation of the Nusselt number as a function of C$is shown in Fig. 5 for various values of z. In addition to the numerically obtained solutions the asymptotic solution which
,---
3rdterm :--
4th
series
term
series
Fig. 4. The variation of the local Nusseit number as a function of the diitance along the cylinder for r#~= 180’.
1360
D. d. INGHAM and I. POP
Fig. 5. The variation of the local Nusselt number as a function of 4 for various values of z.
is valid at large values of z as obtained in section 3b is presented. It is seen that for z z 0.4 that the asymptotic solution for C#J < 90” has been approached. As the value of z increases so does the approach to the asymptotic solution for larger and larger values of 4. The fact that the governing equations are parabolic in the 4 direction automatically implies that the asymptotic solution is approached quicker for the smaller the value of 4.
I
8
0‘.a
-
0,6-
\_;--
____z
q
0.02
_. ~ -: \.- \
\
0 d-
__.2=0.04 \.!_!;-_i-= 0.1
_-,\\\
2=0.4 and _--*-__ limit solution
0 2-
0 0
I
2
3 Y
Fig. 6. The temperature profiles at various values of the distance along the cylinder as a function of the radial distance: (a) 6 = O”, (b) 4 = 60”, (c) @J= 120”, (d) Q = 180”.
1361
Horkontal flow past a partially heated infinite vertical cylinder
Fig. 6. (Continued)
Usually one is interested in the total heat flux from the cylinder and this is given by (51) This quantity has been obtained from the numerical solution but because of the close agreement between the numerical solution and the four term series solution, which is valid for z e 1, then we may approximate & by 1
z
[0.44375 + 0.29258~~1
(52)
D. B. INGHAM and I. POP
1362
forz<
1. Whenz>
1 weuse
(53)
2 1.
= Gr ‘I2 0.71835 + !?JE! pr O[
(54)
The second term in eqn (54) represents the leading edge correction to the asymptotic solution. Figure 6 show the temperature profiles as a function of y for various values of z along with the asymptotic profiles as given by eqn (35). Again we observe that the smaller the value of 4 the quicker is the asymptotic solution approached. It is also seen that the boundarylayer increases in thickness as # increases and as # - 180” the boundary-layer is of infinite thickness as is indicated by eqn (32). Because of the rapid growth in the boundary-layer thickness near Q, = 180” when z - 0(l) the numerical calculations cannot be continued for all values of Cpuntil the asymptotic solution has been achieved. 5. CONCLUSIONS
The laminar funds-layer which is induced by a horizontal forced flow along a heated vertical cylinder which is main~n~ at a temperature in excess of the su~oun~ng porous media has been investigated. A numerical solution of the governing three-dimensional boundary layer equation has been presented along with the asymptotic solutions which are valid at small and large distances along the pipe. Close to the leading edge the buoyancy force induces a strong free-convection boundary-layer. The importance of the forced convection increases as one moves vertically upwards from the leading edge. It is found that for axial distances greater than about at that the asymptotic solution has been achieved. Thus increasing the Reynolds number or decreasing the Grashof number has the effect of decreasing the distance before the asymptotic solution is reached-this is exactly as one would expect. If one is only interested in the heat transfer from the cylinder it is seen that the series solution which is valid for small values of distance along the cylinder can be used up to z = 1 whilst the asymptotic solution which is valid for large values of z may be used for z > 1. However, if full details of the temperature profiles etc. are required then one must resort to the full numerical solution. NOMENCLATURE radius of cylinder acceleration due to gravity g Grashof number Gr permeability of porous medium thermal conductivity of porous medium f local Nusselt number N”, average Nusselt number RG Prandtl number Pr rate of heat transfer 4 radial co-ordinate Reynolds number Re temperature T u, u, w boundary layer velocities axial co-ordinate a
Greek ~~~bof~ thermal diffusivity thermal expansion coefficient ; constants YTh ratio (: similarity variables 8, l e boundary layer temperature kinematic viscosity azimuthal co-ordinate :
Horizontal Ilow past a partially heated infinite vertical cylinder
1363
Subscripts w surface freestream m Superscripts Dimensions quantities inviscid flow quantities derivation with respect to n Acknowledgements-Part of this paper was completed while the second author (IP) has been in the institute B of The~l~ynami~ (Chairman Prof. D. Vortmeyer) of the Technical University of Munich. The author expresses his heartiest thanks to the Alexander von Humboldt Foundation for supporting his research in Germany and to Prof. Vortmeyer for the very kind hospitality in his Institute. REFERENCES
(I] P. CHENG and M. J. MINKOWYCZ, J. GeopJzys. Res. %I,2040 (1977). [2] [3] (41 [5]
P. CHENG, In?. J. Heat Mass Tram& 21, 1499 (1978). D. B. INGHAM and I. POP, To be published in W&me Stoffibertrug. L. S. YAO, J. Heat Transfe 105, 101 (1983). J. A, D. ACKROYD, Proc. Cum. Phil. Sot. 63,871 (1967). (Received 22 Augusf 1985)