Viscous dissipation effects on heat transfer in an axisymmetric stagnation flow on a circular cylinder

Viscous dissipation effects on heat transfer in an axisymmetric stagnation flow on a circular cylinder

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 AX...

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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

co-ordinate normal to the cylindrical surface

T

temperature

u

velocity component in r-direction

w

velocity component in z-direction

121

122 z

R.S.R. Gorla

Vol. 5, No. 2

co-ordinate parallel to the wall dimensionless co-ordinate

@

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 co-ordinate system is shown in Fig. I,

The flow is axisymmetric about the

z-axis and also symmetric to the z=0 plane,

The stagnation line is at z=0 and

r-a.

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 ) T-w = 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

Co-ordinate 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(T-T~) A2a 2 0 =T-T~ 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 = po-p [ 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

Runge-Kutta 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) Tw-T~ 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 Re-I/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

= Re-I/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 , Re-10 ; 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

(Tw-T=), 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 = (Tw-T~) - 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 NASA-ASEE, 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. 321-410, 1911.

2.

Homann, F., Z. Agnew, Math. Me~h., Vol. 16, p. 153-164, 1936.

3.

Howarth, L., Phil. Magazine, Vol. 42, p. 1433-1440, 1951.

4.

Davey, A., Journal of Fluid Mechanics, Vol. 10, p. 593-610, 1961.

S.

Wang, C., Quarterly of Applied Mathematics, Vol. 32, p. 207-213, 1974.