IN HEAT AND MASS TRANSFER Vol. 5, pp. 121  130, 1978
PergamxDn Press Printed in Great Britain
VISCOUS DISSIPATION EFFECTS ON HEAT TRANSFER IN AN AXISYM~TRIC STAGNATION FLOW ON A CIRCULAR CYLINDER Rama Subba Reddy Gorla Associate Professor of Mechanical Engineering Cleveland State University Cleveland, Ohio 44115
(C~ilvm~_icated by J.P. Hart~Jett and W.J. Minkc~ycz)
ABSTRACT Boundary layer solutions are presented to study the effects of viscous dissipation on the steady state heat transfer in an axisymmetric stagnation flow on an infinite circular cylinder. The cases of isothermal wall and insulated wall boundary conditions have been treated in this paper. Numerical results for the temperature distribution and the missing wall values of the thermal functions have been given. The range of Prandtl numbers investigated was from 0.01 to 1000 and the Reynolds number was varied from 0,01 to 100,
Nomenclature A
constant used in Equation (5)
a
radius of cylinder
f
velocity profile function
h
heat transfer coefficient
k
thermal conductivity
Nu z local Nusselt number (hz/k) Pr
Prandtl number
P
pressure
qw
heat flux at the wall
Re
Reynolds number (Aa2/2~)
r
coordinate normal to the cylindrical surface
T
temperature
u
velocity component in rdirection
w
velocity component in zdirection
121
122 z
R.S.R. Gorla
Vol. 5, No. 2
coordinate parallel to the wall dimensionless coordinate
@
dimensionless temperature dynamic viscosity kinematic viscosity
p
fluid density Subscripts
w
conditions at the wall conditions far away from the wall Introduction An exact solution of the
boundary layer equations governing the problem
of two dimensional stagnation flow against a flat plato has been given by Hiemenz [i]. Homann [2] later on obtained an exact solution of the NavierStokes equations for the axisymmetric stagnation flow against a plate. Howarth [3] and Davey [4] presented results to unsymmetric cases,
The problem
of axisymmetric stagnation flow on an infinite circular cylinder has been analyzed by Wang [5]o The present work is undertaken in order to investigate the effects of viscous dissipation on the steady state heat transfer in an axisymmetric stagnation flow on a circular cylinder. compressible.
Solutions for the
The flow is assumedto be laminar and in
temperature field are obtained under both
isothermal and insulated wall conditions for a wide range of Prandtl numbers and Reynolds numbers.
Engineering applications of the problem are found in
certain industrial cooling processes, Governing Equations Let us consider a steady, laminar, incompressible flow at an axisymmetric stagnation point on an infinite circular cylinder.
A model of the flow with
the coordinate system is shown in Fig. I,
The flow is axisymmetric about the
zaxis and also symmetric to the z=0 plane,
The stagnation line is at z=0 and
ra.
The temperature of the free stream fluid is taken as T~o The equations expressing conservation of mass, momentum and energy within
boundary layer approximation are given by: Mass:
r ~
*w~
(ru~ = 0
<1)
Momentum: u
U
~r~u +
~z~U =
~w + w ~ ;)r
~z
_
=
~~rl ~P + ~[~2U~r~+ rl ~U~r u2] 
i
3p +
p ~
[32 v
?w 2
+
1
~w
]
(2) (3)
Vol. 5, NO. 2
STA(IqATION F L O W O N A CII~X/LARCYLINDER
J 2a
Figure l,
1.00
Coordinate system and Flow Development
I00
O. 75
f
#
/ /••e"'•e
O. 50
=0. OI
0.25
/
1.0 Figure 2.
2.0
3.0
4.0
5.0
6.0
7.0
Velocity Distribution in the Boundary Layer
123
124
R.S.R. Gorla
Vol. 5, No. 2
Energy:
0Cp ( u
~T + w ~ r ) Dr ~z
= k
[! ~ (r r ~r
~W)] ~r
+
2 = 2~ [ ( ! ~ 2 + u 2 + Qw)] + u[(~w + ~ 2 ] r r ~7 ~F ~Z
where
(4)
The boundary conditions for the velocity field are given by: r=a:
u
=
w
=
0
(s)
r  ~ : u =  A (r  a 2 ) Tw = 2Az For the temperature field we have r=a:
i)
T=Tw for isothermal wall case
ii)
__~T=0 for insulated wall case
(6)
r+~ : T+T ~ r
Coordinate Transformation and Solution In order to proceed with the analysis, we lot 2 u = Aan 1/2" • f ( n ) w = 2 A f ' (n)
It
z
S = Cp(TT~) A2a 2 0 =TT~ can be v~ed that
Upon s u b s t i t u t i n g
(7)
the continuity
the expressions
in
equation (7) i n t o
nf''' + f'' + Re • [i + ff'' p = pop [ A2a2 2
is automatically (2) a n d ( 3 ) ,
satisfied.
we h a v e
(f,)2] _ 0
(8)
f2 + 2vAf' + 2A2z 2] ~
(9)
The primes above denote differentiation with respect to n only.
The boundary
conditions for the velocity field are: f(1) = f'(1) = 0 and f' () = 1 Equations
(8) a n d (10) h a v e b e e n s o l v e d
Kutta method of numerical
integration.
f'
w i t h ~ h a s b e e n shown i n F i g .
It
should be noted that
Re~°
(I0) by using
The v a r i a t i o n
2 for various
corresponds
the fourth
values
order
Runge
of the velocity
function
of the parameter,
to the Hiemenz's problem,
Re.
namely,
VOl. 5, NO. 2
ST~J~%~TION FLOW ON A C l l ~
t h e two d i m e n s i o n a l sults
stagnation
flow against
a flat
CYLI~ER
plate,
125
The n u m e r i c a l
re
for the velocity field are obtained for Re ranging from 0.01 to i00.
The
values of f''(1) for the same range of Reynolds numbers have been tabulated in Table I and can be compared with those given by Wang [5] Isothermal wall with negligible frictional heating: We shall first consider the problem of isothermal wall boundary condition and neglect the viscous dissipation terms in the energy equation.
By using the
expressions in (7), the energy equation may be written as: he'' + [i + (Re • Pr) f]0' = 0
(11)
The boundary conditions are given by 0(i)
= i,
8(~)
I t may b e v e r i f i e d part
order
that
of the total
Equation
(ll)
= 0
(12)
e(n) denotes
energy equation
has been solved numerically
RungeKutta numerical
as parameters.
The v a l u e s
procedure of Prandtl
the Reynolds numbers varied was e m p l o y e d i n a l l
qw
wall heat
flux
(z)
~
=

k
 2K
=
~
on t h e c o m p u t e r b y u s i n g
f r o m O.01 t o 100,
=
o f t h e homogenous
the fourth
number and Reynolds number
n u m b e r s r a n g e d f r o m 0 . 0 1 t o 1000 w h i l e The d o u b l e p r e c i s i o n
In order
to conserve
space,
arithmetic results
for
3,
can be written r
solution
with Prandtl
the computations,
P r = 0 . 7 o n l y a r e shown i n F i g . The l o c a l
the general
(4).
by Fourier's
law a s
a
Dr
(T w  T  ) e ' ( 1 )
(13)
The local heat transfer coefficient is given by h(z) = qw =  2__kke'(1) TwT~ a The local Nusselt number then becomes Nuz = h ( z )
k where
=
(14)
z
(lS)
1/2
2(~
e'(1)
Re z = Az 2 The v a l u e s
of 0'(1)
for
Prandtl
numbers ranging
i n g f r o m 0 . 0 1 t o lO0 h a v e b e e n t a b u l a t e d cations,
it
and the heat
is usually transfer
the surface rates
that
f r o m 0 . 0 1 t o 1000 a n d Re v a r y 
in Table II.
characteristics
I n many p r a c t i c a l
such as the
are of prime importance.
local
wall
The e v a l u a t i o n
applishear of
126
R.S.R. Gorla
Vol. 5, No. 2
Table I  Values of ReI/2f''(1) for various values of Re 1/2 Re f'' (1) Present results
Re
Exact (5)
3.155182 1.946369 1.75770 1.484185 1.316427 1.259642 1.232588
0.01 0.i0 0.20 1.00 i0.00 100.00
1.7577 1.484185 1.31643
1/2 Table II  Values of Re
0'(i) for various values of Pr and Re
Re Pr
0.01
0.01 1.676639 0.1 1.748117 0.7 2.105002 1.0 2.229295 10.0 3.453837 100.0 5.846309 1000.0 10.785380
0.i
0.2
1.0
I0.0
i00.0
0.556623 0.735103 1.105455 1.200518 2.138901 4.085021 8.225758
0.413548 0.602609 0.947625 1.036836 1.928688 3.799283 7.795494
0.2429564 0.4103923 0.7158796 0.7968775 1.622352 3.379548 7.150810
0.139410 0.286157 0.570303 0.646796 1.433432 3,118195 6.739904
0.096469 0.241516 0.520036 0.595211 1.369343 3.028978 6.597576
Table III   Re Pr 0.01 0.i 0.7 1.0 i0.0 100.0 i000.0
0.01
0.07598 0.2195 0.4959 0.5704 1.3389 2.98633 6.52914
Values of M(1) for various values of Pr and Re
0.i
12.034651 8.853412 42.096725 18.096385 109.63155 42.202079 148,96104 50.545100 591.68402 97.816993 982,49951 129.85494 1221.8451 148.51039
0.2
1.0
5.412343 12.677205 24.608735 28.743424 51,362726 66,520185 75.297886
1.934313 3.085311 5.302527 6.069895 10.160273 12.880953 14.157046
i0
0.291120 0.464006 0.643271 0.711059 1.076815 1.325273 1.613712
i00
0.060312 0.072277 0,085531 0.091718 0.126261 0,134236 0.142145
Table IV  Values of N(1) for various values of Pr and Re Re
0.01
0.i
0.2
1.0
i0
100
Pr 0.01 0.1 0.7 1.0 10.0 100.0 1000,0
0.051325 0.369405 1.983285 2.730242 15.828676
67.169154
232.33042
0.091231 0.573904 2.566650 3.309039 14.714691
53.198701
160 22268
0.107314 0.636287 2.678381 3.414705 14.446811
50.177957
145.66621
0.152331 0.773780 2.904313 3.633297 13.910429
0.251342 0.984271 3.141122 3.849079 13.435494
45.002110
41.205362
125.06257 111.48800
0.313143 1.113152 3.252138 3.947072 13.233931
39.779100
105.68213
Vol. 5, No. 2
STA(IWATION FLOW O N A
CII~CYI~ER
127
such quantities requires only the information contained in Tables I and II. Using the transformation f(q)
where
= ReI/2
Rel/2
=
we c a n s e e t h a t
•
. ¢{~) (n

l)
{i~)
equation
(8) r e d u c e s
to the Hiemenz)s problem
@,,,
[email protected]@,,  @,2 + 1 = 0
{17)
with boundary conditions @{0) = ¢ ' ( 0 )
= 0
08)
0'(~) = I Equation {12) can be reduced f o r large values of Re, to the following form as a f i r s t approximation: 0 ' ' + Pr¢O' = 0
(19)
with boundary conditions 0{0) = 1, The e r r o r
in deriving
corresponds flat
E)
= 0
(20)
{17) and (19) i s o f t h e o r d e r
to the heat
transfer
Re  1 / 2 .
a t a two d i m e n s i o n a l
Equation
stagnation
(19)
f l o w on a
plate.
We o b s e r v e
that
the skin
friction
coefficient
Hiemenz problem within 2% difference.
f o r Re * 100 a g r e e s
with the
The heat transfer results of the present
problem for Re = 100 and Pr ~ 0.7 agree with the two dimensional stagnation point results within S% difference. Adiabatic wall includin~ the effect of frictional heating: We shall now consider the boundary condition of adiabatic wall and retain the viscous dissipation terms in the energy equation.
Upon substituting
the ex
pressions in (7) into (4), we have q S"
+ [i + (Re'Pr) f) S' + Pr.[f~2 + 4{f,) 2 _ 2ff' + 4q~ (f")2]= 0 n ~
where
q
(21)
Defining s = MCn) + ~NCn)
we obtain qM'' + [ l + ( R e ' P r ) f ] M' + Pr [f2 + 4 { f , ) 2 _ 2 f f ' ] = 0 qN'' + [ i + {Re.Pr)f] N' + 4 ~r.n { f ' ' ) 2 = 0 boundary conditions are
{22)
n (23)
The a p p r o p r i a t e M'
(I)
= 0
,
M{.)
N'
(1)
= 0
,
N{ ®) = 0
Equations
= 0 (24)
{22) and (23) are solved numerically by means of the fourth order
128
R.S.R. Gorla
Vol. 5, No. 2
1.00 Pr=0,7
0.15 8 O. 50
0.25
I
1.0
2.0
I
I
3.0
I
4.0
!
5.0
i
6.0
7.0
11 Figure 3.
Temperature Distribution in the case of Isothermal Wall Boundary Condition in the Absence of Frictional Heating (Pr=0.7).
0.9 Pr=O. 7 0.8
0.7
0.6
0.5
0.4
0.5
0.2
Figure 4.
M versus q (Pr=O. 7) [A: Mxl0, Re~lO0; B: M , Re10 ; C: MxI0 I, Re=l; D: MxI0 2, Re=0.2; E: Mxl0 2, Re=O.l]
0. I
\
~
~''F
A
B
0
a
i
2
3
4
5
i
I
6
7
Vol. 5, No. 2
ST~]ATICN FLOW O N A
CIfL"UIARCYL]NDER
129
Pr=O. 7
N Re=0. OI
0.1 0.2
I0
~
I
0
I
2
3
4
q
Figure 5.
5
N versus
7
6
n (Pr=0.7)
R u n g e  K u t t a m e t h o d w i t h P r and Re a s p a r a m e t e r s . ed f r o m 0 . 0 1 t o 1000 and 0 . 0 1 t o 100 r e s p e c t i v e l y .
The v a l u e s o f P r and Re r a n g A step size
o f n = 0 . 0 1 was
chosen and the double precision arithmetic was used in all the computations. The distribution of the functions M(~) and N(n) has been illustrated in Figures 4 and S respectively only for Pr = 0.7 for the sake of brevity.
The values
M(1) and N(1) have been tabulated in Tables III and IV for the previously mentioned range of Pr and Re. The g e n e r a l
solution
for a prescribed
and the free stream, namely,
temperature
difference
between the wall
(TwT=), including viscous dissipation effects,
130
R.S.R. Gorla
Vol. 5, No. 2
is thus T(n)  T~ = C • e[n) + A2a2 •S[n)
(25)
Cp where
C = (TwT~)  A2a2 . S(1)
using equation (25), it becomes a straightforward procedure to evaluate the heat transfer rate and Nusselt number. Concludin~ Remarks The governing boundary layer equations are solved for the steady state heat transfer in an axisymmetric stagnation flow on a circular cylinder.
Numerical
results for a wide range of Reynolds and Prandtl numbers have been obtained for two cases, i) ii)
isothermal wall with negligible frictional heating and
adiabatic wall including viscous dissipation effects.
The general solution
corresponding to a prescribed temerpature difference between the wall and the free stream under conditions when the viscous dissipation becomes important, can be obtained from the two cases studied, Acknowledgment A portion of this work has been completed while the author worked at NASA Langley Research Center in Hampton, Virginia, as a summer Research Fellow through the NASAASEE, Old Dominion University program.
The author wishes to
thank Dr. Julius Harris of the Fluid Mechanics Section at Langley Research Center for the facilities provided.
The assistance given by Miss Debra Nagy
in the typing of the manuscript is appreciated also. References i.
Hiemenz, K., Dinglers Journal, Vol. 326, p. 321410, 1911.
2.
Homann, F., Z. Agnew, Math. Me~h., Vol. 16, p. 153164, 1936.
3.
Howarth, L., Phil. Magazine, Vol. 42, p. 14331440, 1951.
4.
Davey, A., Journal of Fluid Mechanics, Vol. 10, p. 593610, 1961.
S.
Wang, C., Quarterly of Applied Mathematics, Vol. 32, p. 207213, 1974.