I nternat. J. Math. Mh. Sci.
(1982) 585-594
Vol. 5 No.
585
THE RESPONSE OF SKIN FRICTION,
WALL HEAT TRANSFI AND PRESSURE DROP
TO WALL WAVINESS IN THE PRESENCE OF BUOYANCY
_N.B. RAO and K.S. SASTRI
Department of Mathematics
Indian Instltute of Technology
Kharagpur- 721 302
IDIA
(Received
ABSTRACT.
revLsed form Nomber 5, 1981)
Laminar natural convection flow and heat transfer of a viscous incom-
pressible fluid confined between two long vertical wavy walls has been analysed
taking the fluid properties constant and variable.
In particular, attention is
restricted to estimate the effects of viscous dissipation and wall waviness on the
flow and heat transfer characteristics.
Use has been made of a llnearlzatlon tech-
nique to simplify the governing equations and of Galerkln’s method in the
tlon.
solu-
The solutions obtained for the velocity and the temperature-flelds hold good
for all values of the Grahof number and wave number of the wavy walls.
KEY WORDS AND PHRASES. Fluid Mechanic, Incompressible vco fld, heat transfer, free nvetion, constant fluid properties (CFP), variable fluid pperties (VFP).
1980 MATHEMATICS SUBJECT CLASSIFICATON CODES.
65N30.
76D99, 80A20, 76RI0, 41A60, 41A10,
INTRODUCTION.
Vajravelu and Sastrl [1,2] have reinvestigated Ostrach’s work [3] of laminar
natural convection flow and heat transfer of viscous incompressible fluids confined
between two long vertical walls with a view to estimate the effect of wall-wavlness
on the flow and heat transfer characteristics in two cases, namely,
the walls is wavy and (ll) when both the walls are wavy.
(1) when one of
The results of their
analyses, however, seem to be meaningful for small values of the Grashof number only
C.N.B. RAO AND K.S. SASTRI
586
(the effect of viscous dissipation is likely
to be negligible
then!).
But for
large values of the Grashof number (definitely possible in many free convection
studies), the assumption of constant fluid properties and neglect
heating effects tend to be questionable.
al.
of viscous
Indeed, Hong and Bergles [4] and Hwang et
[5] among several others have pointed out that the assumption of
constant fluid
properties is chiefly responsible for large deviations between analytical predict-
ions and experimental data.
The present work is therefore an attempt to re-examine the works mentioned in
[1,2] when
viscous heating effects are considered and when the fluid properties are
both constant and variable.
Approximate solutions of the governing equations have
been obtained by Galerkin’s method employing orthogonal polynomials which has proved
to give results valid for all values of the wave number of the wavy walls and for
all values of the Grashof number.
With the help of these solutions, the flow and
heat transfer characteristics have been evaluated numerically both when the fluid
properties are constant and when they are variable.
A comparative study of the
results in the two cases when the viscous heating effects are taken into account
5.
has also been made; a detailed description of which has been given in
2.
FORMULATION OF THE PROBLEM.
Consider a channel made up of two long vertical wavy walls parallel to each
Take the X-axis vertically upward
other.
and the Y-axis perpendicular to it in
such a way that the wavy wall at the left may be represented by Y
and the other wall by Y
wavy walls
0(I)).
d
+
Eg
-d
+
g
cos % X
cos % X (E, the ratio of the amplitudes of the
Let an infinite aount of viscous incompressible fluid be con-
fined between the two walls which are at rest and maintained at two constant temperatures
TWL
and
TWR respectively.
Under the assumptions that the flow is laminar, steady and two dimensional and
that the fluid properties are temperature dependent the basic equations that govern
the flow and heat transfer of the problem under consideration are given in Ostrach
[3].
The boundary conditions are the well-known no-slip conditions of the velocity
at both the walls and that the fluid temperature at the wall is equal to that of the wall.
587
RESPONSE OF SKIN FRICTION TO WALL WAVINESS
Making use of the dimensionless variables (refer to [3])
x
, ,
X
Y
y
R
R
V
K.
v
KGr.
p
Ts)d2
8o(TwL-
fx
U
with
U
K.
u
PD
@
0oUR2
T- T s
K.
(2.1)
TWL TS
o
assuming that the fluid properties are temperature dependent (refer to [6]), and
using a linearizatlon technique valid for small values of e (e << I, refer to [i]),
-
we have reduced the governing equations into sets of ordinary differential equations
for the mean parts u
@
o
and the perturbed parts 9,
o
A
(i
(
Bk
+i-
+-A eo)
[iv
(i
+
o
"+
o
212
A
B
(i
8
Bk
+
k
+-- 0o).
-
+ 8o +
o
(’)
o
14 ]
+ ( +
12)
2(1
il Gr
u’.(0"
+
o
+-A Oo
k
+--.
UO
"
k
12)
0) +
(=
u
(Uo’0
(u-’)2"0-o
0
+
0
(2 4)
0$)
(2.5)
@, 0 are related to the perturbed parts
8y
I
Vl
(2.6)
x
RI.{E e ilx (Y)}’
@i
RI.{E
eilX.0(y)},
is the Prandtl number,
Ts)d3
o(TWL
2
Pr
u]
O
u"’.
0]
o
iI PrGr
K
A
(20’. 0’ + O"
o
o
the Grashof number
o
K
12
2u".
0’
o
o
fx
Gr (=
O--
120)
with RI(F) denoting the real part of F, and
C
o p
u"$
(2.3)
0
through the relations
=
Pr (=
(2.2)
C
is taken as zero in the subsequent analysis, a prime denotes differ-
x
01
as
(U’o)2 + K
0o)
u’o" (’’ +
entation with respect to y, the functions
Ul’ Vl’
8’o" u’o
A
2
0’
B
dP o
where C(=
120)
(0"
u
o
+
[2e’_."’ + e"("
o
+
0
A
PrGr.
8 o fxd
C
P
a dimensionless constant,
(2.7)
C.N.B. RAO AND K.S. SASTRI
588
the heat source parameter,
ko (TwL TS
TWR
T
TWL
TS
ao
A (=
2
qd
(= PrGr
s
the wall-temperature ratio,
(TwL- To)
and B
k
(= b
and
(TwL- To))
are dimensionless numbers describing
(It is worth noting that,
the temperature-dependence of the fluid properties
herein, the specific heat C
is taken constant as any changes in it would be much
P
smaller than g).
The relevant boundary conditions on u
o(-I)
0
u
(-I)
0
’
0(-i)
3.
8 (-i)
0
u (+i)
(+i)
0
SOLUTION OF THE MEAN PARTS
8
o
and
o
K,
u’O (-i)
(-I)
O’ (-i)
o
(Uo, 8o)
0
are
(2.8)
8 (+i)
0
mK,
O’ (+i)
Eu’o(+I)
-E0’ (+i)
0(+i)
(2.9)
o
AND THE PERTURBED PARTS
(Ul,
Vl,
el).
In this section we give a brief account of the approximate solutions obtained
for the velocity- and the temperature-fields by the use of Galerkin’s method employ-
ing orthogonal polynomials (refer to [8]).
-
(2.3) are non-linear, we have rphrased them using an
As the equations (2.2)
iteration scheme and taking C equal to zero, as
A
(1 +
(i
+-Bk 0or_l) 0"
or
Uor(-l)
with
and
?
u’or-i
0
or
Z
Uor(Y)
Po (y)
Pl (y)
i=l
K(m-2 i)
i
y
2
u"or + O or + U O’or-i u’or-i
or-i
B
+ k
(3 1)
0
A
(O’or_l)2 + (i +- 0or_l) (Uor_l)2 + K
Uor(+l), Oor(-l)
0’
or-I
Oor-i
Approximating u
where
A
0
K,
Oor(+l)
mK,
r > 0,
0,
(3.2)
(3.3)
0
and O
or
as
air-Pi(Y), 0or(Y)
-Y +
K(m
P2 (y)
+2
i)
y(l
.Y2,
y2 ),
Po(Y)
+
Z
i=l
b
ir. Pi(y),
(3.4)
- -
589
RESPONSE OF SKIN FRICTION TO WALL WAVINESS
P3 (y)
Pi (y)
that
y2
1
(i
y2 )’
P4 (y)
i
y(
y2)
y2)
(i
and so on, (note
satisfy the non-homogeneous and homogeneous parts of the boundary condi-
0 and when i > 0 and further that
tions respectively when i
Pi(y)
O) are
(i
orthogonal to each other over the interval -i -< y < i) and substituting them in
equations (3.1) and (3.2) there result two resldues:
b
lr).
Orthogonalizing r
and r
I
a set of linear equations for a
b
ai,
i
are determined.
2
and
air,
blr
and r 2(y,
Pi(i
bi, solving
air,
y < i gives
which the values of the unknowns
The foregoing method may be termed G(p,q) method whenever a
p-term approximation for u
or
and a q-term approximation for O
Using G(3,3) method, the constants a
and iteratively when r
l(y,
-> I) over the interval -I
with
i
r
# O, till
and b
i
i
or
are employed.
are determined directly when r-0
a convergence criterion is met at r
r (say).
c
(It has been observed that the G(3,3) method gives results of considerable accuracy).
The expressions (3.4) when r
0 and r
r
give the solutions for u and @ in the
o
o
c
absence of dissipation and in the presence of dissipation respectively.
however be noted that the solutions
@o
Uo,
It may
correspond to the CFP-case when
A and
B are both zero and to the VFP-case when they are not zero.
k
In a similar way approximating
by two different orthegonal polynomials
and
and using them in equations (2.4), (2.5) we have obtained solutions for them by
G(3,3) method.
Making use of these
the expressions for the perturbed parts
4.
in the relations (2.6), (2.7) we got
and
Ul, Vl,
@i"
FLOW AND HEAT TRANSFER CHARACTERISTICS.
As in reference [i], here too the shear stress, heat transfer coefficient and
the pressure drop at the two walls are defined and non-dlmensionallzed.
the linearizatlon technique
[i]
we write the shear stress
In view of
(T) and the heat trans-
fer coefficient (Nu) at either wall, in terms of mean and perturbed parts as
1
T + e T (x),
T(x)
Nu(x)
and the nondlmensional pressure drop (PD
pD 1
with
PI(x’Y)
(x)
I dPl I
1
KGr
Pl
I)
+ e
Nul(x),
given as
[Pl(X’)
dx
Nu
Pl
+-y
PI(X’Y) ]yyw’
dy)
perturbed pressure.
C.N.B. RAO AND K.S. SASTRI
590
With the help of the known expressions for
ions for the shear stress
Uo,
0o
and
Ul,
v I,
@i’
the express-
(T), Nusselt number (Nu) and pressure drop (PD I)
at both
the walls have been found, but not presented here for the sake of brevity.
perturbed parts of the shear stress (T
I)
and the Nusselt number (Nu
I)
The
may be expressed
in terms of their amplitudes and phases as
T
L
at the left wall and similarly those at the other wall.
From the definition of the
pressure drop, it may be observed that it indicates the difference of the pressure
at any point
(x,y) in the flow field from that at the center of the channel.
0.72 (air), and given several sets of values to the parameters K,
Taking Pr
m,
=, E,
Gr and
,
the expressions for the shear stresses, Nusselt numbers and
pressure drops at the walls along with those of their amplitudes and phases have
been evaluated numerically in the CFP-case when both
VFP-case when A
T
s
5.
15C and
B
0.0141405
TWL
20C,
k
A5
and B
k
are zero and in the
0.0120485 (fluid properties for air with T
o
refer to [9] and also to [6]).
DISCUSSION OF THE RESULTS.
At the outset it is worth mentioning that the flow and heat transfer results
for the dissipation case only have been presented in the figures i and 2.
Figure 1 shows the mean velocity and mean temperature profiles in both the
CFP- and VFP-cases when m
1 and the parameters K,
take different values.
From
the figure it is clear that the results of the VFP-case are smaller in magnitude
than those in the CFP-case, this behavior being more prominent for large values of
K,
than for their small values.
Figure 2 describes the behaviors of the amplitudes of perturbed shear stresses
and Nusselt numbers.
From the figures 2(a) and 2(b) it is evident that the ampli-
tudes of the former at both the walls increase with the wave number
asymptotically to non-zero constant values.
and tend
This asymptotic nature of the ampli-
tudes holds good in the CFP- and VFP-cases as well as in the presence of viscous
dissipation and in its absence.
On a close look in to the figures 2(a), 2 (b) it
RESPONSE OF SKIN FRICTION TO WALL WAVINESS
591
7
.;""
I
0
0
O. 2
0.4
O. 8
0 6
I. 0
ol
0
"-"
o,
O. 2
O. 4
, ,,-:---4-L’,
---,,...-,
::.,
/ innxio,
,,,,/
0.2
0.4
0.6
<p
!
\,:,,
0.8
I
IO
0
160
(a), (b):
K
16K,
K
3.0,
=o
1.0
.
..---...,
uu
I
_,\
O.S
0.4
0.2
(0) profiles
4=).
(c)
=o
VFP
CFP
1.0
0.0
I
i
1.0
i0.0
II
2
m
0.8
0.8
yO
Mean velocity (uo) and mean
0o
0.6
o0
.,..-_a-_::
yO
Figure i.
..,
yO
yO
I
’’
(d):
K
i0.0
(u
4uo,
1.0
C.N.B. RAO AND K.S. SASTRI
592
becomes evident that the results of the VFP-case are smaller in magnitude than
those of the CFP-case, this behavior becoming significant for large values of
and
Gr.
The amplitude-profiles of the perturbed Nusselt numbers shown in figures 2(c),
2(d) are similar in quality with those of the shear stresses in figures 2(a), 2(b)
but in magnitude, the latter assume larger values than the former in almost all the
cases.
From the figures 2(c), 2(d) it may be further noticed that as in the case
of the wall shear stresses, here too, the results of the VFP-case are, in general,
smaller in magnitude than those of the CFP-case, this behavior becoming prominent
when
and Gr take large values.
From the numerical results it may be pointed out that (I) the effect of viscous
dissipation is to increase the values of the amplitudes considerably and to enhance
the effect of variable fluid properties, (2) the phases of the perturbed shear stress
or perturbed Nusselt number at either wall
are consirably smaller in magnitude
m
than their amplitudes, and that they incrse fr
asymptotically to zero for its large values
a
values of % but decrease
rut holding good
in the CFP- and
VFP-cases, (3) the perturbed parts of shear stress, tummelt number and pressure drop
at either wall are sinusoidal in nature
(see expressions (4.1) for T L
and Nu
),
and they are almost always positive for 0 < %x </2,become negative for
7/2 < %x < 3/2, and take positive values again fo 3]2 <
< 2o
This negative
nature of shear stresses in the trough reglomm of th walls indicates physically
that the perturbed flow tends to get separated there.
because the wavy walls are infinitely long.
Such instances are numerous
However, ths special feature of the
perturbed flow which is of order g, cannot persist in the total flow as this flow
doesn’t dominate the mean one.
From the defnition of the pressure drop and from
its above-noted behavior it may be inferred that th flui pressure at either wall
lags behind (exceeds)that at the center of the cnel henever he pressure drops
are positive (negative) in nature.
Finally i may
computations that in the presence of dIpao
large Grashof numbers (Gr
i00, say) deae
concluded from the numerical
t uls of the VFP-case for
tr ccmmrpats of the CFP-
RESPONSE OF SKIN FRICTION TO WALL WAVINESS
593
,3, III
0
10
20
40
80
60
100
O5
Ic
0
-0.5
"
0
I IO
10
20
40
60
80
02
100
(d)
-Ix10,Ix10__
-02
IV, &,3,111
-1.0
0
10
Figure 2.
20
40
60
80
1
1
AT R
Amplitudes (AT
.0 5 [__.._l___J
0 10 20
100
ANu L
and perturbed Nusselt number
K
i0.0,
m
E
2.0,
s
Gr
0.0
5.0
I
1
0.0
i00.0
II
2
i0.0
i00.0
III
3
VFP
CFP
0.5
1
ANu R
40
60
80
of the perturbed shear stress
100
594
C.N.B. RAO AND K.S. SASTRI
case by about 40 per cent in the case of Nusselt numbers and about 20 per cent in
the case of wall shear stresses.
ACKNOWLEDGEMENTS.
The authors thank the referee for his valuable suggestions for the improvement
of the work.
They express their indebtedness to the RCC, Calcutta for the computer
facilities and the I.I.T., Kharagpur for granting permission to one of the authors
(C.N.B. Rao) to go to Calcutta to carry out the computations needed in this work.
The former author (C.N.B. Rao) gratefully appreciates the financial assistance
received from the CSIR, New Delhi, India.
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Free convective heat transfer of a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall. J. Fluid Mech. 86 (1978) 365-383.
2,
Natural convective heat transfer in vertical wavy channels.
Int. J. Heat Mass Transfer 23 (1980) 408-411.
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OSTRACH, S.
4.
HONG, S.W. and BERGLES, A.E.
5.
HWANG, S.T. and HONG, S.W.
6.
VAJRAVELU, K.
Laminar natural convection flow and heat transfer of fluids with
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NACA Technical note no. 2863 (1952).
Theoretical solutions for combined forced and
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J. of Heat Transfer, ASME. 98 (1976) 1-7.
Effect of variable viscosity on laminar heat transfer in a rectangular duct. Chem. Eng. Pro$r. Symp. Ser. 66 (1970) 100-208.
The effect of variable fluid properties on free convection flow
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31 (1979) 199-211.
7.
New aspects of natural convection heat transfer preliminary study
of effects of frictional heating. Trans. ASME. Paper No. 53-S-43 (January
OSTRACH, S.
1953) 1287-1290.
8.
COLLATZ, L. (Ed.) The numerical treatment of differential equations (4, Chapter i).
Springer-Verlag (1966).
9.
BATCHELOR, G.K. (Ed.) An introduction to Fluid Dynamics (Appendix i), Cambridge
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