Boundary layer flow of a nongray radiating fluid

INT. COMM. HEATleSS TRANSFER 0735-1933/89 $3.00 + .00 Vol. 16, pp. 415-425, 1989 ©PergamunPressplcPrintedinthell%itedStates

BOUNDARY LAYER FLOW OF A NONGRAY RADIATING FLUID

A. Yficel Nuclear Science Center Louisiana State University Baton Rouge, Louisiana 70803 H. Kehtarnavaz Software Engineering Associates, Inc. Carson City, Nevada 89701 Y. Bayazitoglu Department of Mechanical Engineering Rice University Houston, Texas 77251

(O~L,L,.micated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT Radiative heat transfer in boundary layer flow of a nongray, absorbing and emitting fluid is considered. The P-3 spherical harmonics method is used to analyze the radiation part of the problem. The spectral variation of the absorption coefficient of the medium is characterized by the rectangular model. The nonsimilar boundary layer equations are solved by the collocation method. The effects of nongrayness on the temperature profiles and wall heat transfer rates in boundary layer flow over a flat plate and in stagnation flow are discussed.

Introduction Thermal radiation heat transfer is important in many engineering applications involving external flows, e.g. in ablative cooling, atmospheric re-entry and shock wave problems [1,2].

Most studies of radiating boundary layers have

been based on the gray assumption [3-6], with but a few exceptions considering nongray effects [7-9].

The mathematical difficulties involved in handling the

partial differential and integro-differential equations arising from a general treatment of the governing equations have prompted the use of various limiting assumptions and approximations in modeling radiative transfer [1,10].

An ela-

borate scheme that yields high order solutions is the P-N approximation or the 415

416

A. Yuoel, H. Kehtarnavaz and Y. Bayazitoglu

spherical harmonics method.

Despite its effectiveness

solutions to the gray radiative transfer problem,

Vol. 16, No. 3

in generating accurate

little work has been done to

generalize the P-N approximation for nongray problems [11-14]. this study is to develop and

The purpose of

solve the differential relations for nongray ra-

diating boundary layer flows using the higher order accurate P-3 approximation to represent the angular dependence of the radiation intensity.

P-3 approxi-

mation yields satisfactory solutions over the whole range of optical thickness in planar geometry [6,14].

The rectangular model [15] is introduced into the

P-3 formulation in order to account for the spectral variations of the absorption coefficient. curately

Complex variations of the absorption coefficient can be ac-

represented by the

narrower bands or intervals.

rectangular model by

dividing the spectrum into

Multiband cases,

including those with overlap-

ping bands in gas mixtures, are easily handled.

Band absorption coefficients

for infrared radiating gases can be readily obtained from the exponential wide band model using to those media

Edwards' block method [16,17]. for which

The model is also applicable

band absorptance correlations

are not available or

the use of the narrow band approximation is not justifiable. .Analysis The physical model

under consideration

consists of a constant property,

nongray, absorbing and emitting fluid

flowing over a black surface wedge held

at uniform temperature T

The external flow is at a constant tem-

W

(Figure 1).

perature T , and its velocity U~(x) is given by m = al(2r-a)

U®(x) = Cx m, where

a is the wedge angle in radians and C is a constant.

FIG.

1

Schematic Representation of Physical System

Specifically,

the

Vol. 16, No. 3

F I E W O F A NONGRAYRADIATINGFLUID

417

cases m = 0, 1/3, and 1 correspond to flow over a flat plate, over a 90-degree wedge, and stagnation flow respectively.

Steady state laminar flow is assumed

and viscous energy dissipation is neglected.

Under these considerations,

the

momentum and energy equations in the boundary layer are fm + T l+m ff# + m[l-( f, )2] = 0 l+m e # +-~--PrfS'

1 m a0 --~--Prf'f ~ - -

~

(1) , Qr = 0

(2)

subject to the boundary conditions = 0

f = fo = 0

e = 8

~ ~

f' = 1

0 = 1

where the prime d e n o t e s

differentiation

(3a)

W

w i t h r e s p e c t to the

(3b) similarity

vari-

a b l e 7, and f(w) and 8(W,f) r e p r e s e n t the d i m e n s i o n l e s s s t r e a m and t e m p e r a t u r e functions. The

dimensionless

total

radiative heat flux

o b t a i n e d from the s o l u t i o n of the r a d i a t i v e

Qr a p p e a r i n g in Eq. (2) i s

transfer

equation,

which i s g i v e n

f o r an a b s o r b i n g and e m i t t i n g medium in p l a n a r geometry by ( d i v / d z ) + s v i v = ~uebv/~

The radiation field is very well approximated by the

(4)

one-dimensional descrip-

tion in the z-direction [18].

In the P-3 approximation,

of the intensity of radiation

is represented

the angular variation

by a finite series of

Legendre

polynomials: 5 ]p~(~) 4xi (z) = i0p(z) + 3ilv(Z) P~(~) + ~ [3i2u(z) - i0v(z ) + ~7 [i3v(z) - 3ilv(Z) IP~(~)

(5)

where the coefficients are given in terms of the angular moments of intensity. A set of first order

ordinary differential equations

governing the intensity

moments is obtained by multiplying the equation of transfer by the appropriate powers of the

direction cosine

p

and

integrating it

over the solid angle.

Given the rectangular model for the spectral variation of the absorption coefficient, the moment equations can be integrated over the spectral variable for each band.

The integrated moment equations are given in dimensionless form by 35 I~j = ajf [-10 Ilj + --~ 13j]

(6a)

I~j = aj~ (Ebj - I0j)

(6b)

418

A. Yueel, H. Kehtarnavaz and Y. Bayazitoglu

Vol. 16, No. 3

I~j = - a j ~ I l j 1 I~j = a j f [~ Ebj - I 2 j ] with

Ikj = Av. f Ik/ydv

and

(6d)

Ebj = A//fEbvdV,.

J

j = 1,2 . . . . . n

(7)

J

The k th moment of i n t e n s i t y

is d e f i n e d as

Ikv(Z) = f #kIvdn, n Of p a r t i c u l a r

(6c)

interest

k = 0,1,2,3.

is the f i r s t

(8)

moment of i n t e n s i t y which is i d e n t i -

cal to the r a d i a t i v e flux in the z - d i r e c t i o n .

Hence the r a d i a t i v e

f l u x term

in Eq.(2) can be expressed as: n

n

aj (Ebj - I 0 j ) Q~ =j~l I{j = f '= j=l The boundary c o n d i t i o n s are o b t a i n e d by u s i n g the

(9) Marshak approach.

The

i n t e g r a t e d boundary c o n d i t i o n s f o r Eq. (6) are 3 I0j ~ ]6 I l j + 15 12j = 8 Ebj

(10a)

-2 Ioj + 30 I2j ~ 32 I3j = 8 Ebj

(lOb)

where the + sign corresponds to 7 = 0 and the - sign corresponds to 7 ~ @. S o l u t i o n Method Equations (1), tions,

constitute

neously. and

f'

(2), and (6), the system of

t o g e t h e r with the a s s o c i a t e d boundary c o n d i differential

equations to be solved s i m u l t a -

Since the flow f i e l d is independent of temperature,

the values of

r e q u i r e d in the energy e q u a t i o n were o b t a i n e d by s o l v i n g the momentum

equation

s e p a r a t e l y u s i n g the

modified q u a s i l i n e r i z a t i o n

l i n e a r two-point boundary value problems [19]. d i t i o n s were s a t i s f i e d

well b e f o r e 7=20.

algorithm

f o r non-

The maximum value of 7 was set

as 7=20 and a uniform g r i d of 100 d i v i s i o n s was used.

The o u t e r boundary con-

Unlike the v e l o c i t y boundary l a y e r ,

the thermal boundary l a y e r is n o n s i m i l a r , with the temperature f u n c t i o n pending on ential rection

f

7 and f.

The energy e q u a t i o n was reduced from a p a r t i a l

e q u a t i o n to a second order o r d i n a r y d i f f e r e n t i a l by r e p l a c i n g the p a r t i a l

by a two-point f i n i t e t i o n s given by

8 de-

differen-

e q u a t i o n in the

7 di-

d e r i v a t i v e of temperature with r e s p e c t to

d i f f e r e n c e formula.

Eq. (6) were a l s o c a s t

The f i r s t

order s e t of moment equa-

i n t o a second order system in terms of

VOl. 16, NO. 3

FIEWOF

A NCNGRAYRADIATINGFLUID

419

Ioj and I 2 j : 55 Ebj(O) + ~35 I2j - 10 I o j ] = 0 I~j + f.2a2j [---~ I~j + f The r e s u l t i n g

(lla)

2a2 j [Ebj(O) - I o j ] = 0

n o n l i n e a r two-point

(11b)

boundary value problem

the coupled energy and moment equations was solved by the To t h i s end, cubic Hermite basis

c o n s i s t i n g of

c o l l o c a t i o n method.

was used to provide a c c u r a t e r e p r e s e n t a t i o n s

of the temperature g r a d i e n t and the r a d i a t i v e heat f l u x .

Thus s o l u t i o n s were

assumed of the form 1 ¢(7) = ¢(7 i) H?(y) + ¢ ' ( 7 i) H~(7) + ¢(7i+1 ) H?+I(W) + ¢ ' ( 7 i + 1 ) Hi+l(7) for the range and

7 i ~ 7 ~ 7i+ I, i=l,2,...,M,

¢ = {0, Ioj , 12j} , j = 1,2 .... ,n.

(12)

with 0 ~ 71 < 72 < ... < WM+I ~ 7~

The cubic

Hermite polynomials

in the

above equation are given by

H~

= 1-3s2+2s 3

(13a)

1

Hoi+l = 3s2-2s3

(13b)

H~

(13c)

1

= h . ( s - 2 s 2 + s 3) 1

H1 = h i ( s 3 - s 2) i+1

(13d)

where s = (7-7i)/h i and h i = 7i+i-7i. These solutions

were then substituted in the energy and moment equations

and forced to satisfy them at the collocation points

1 1 ( 7 i ) 1 , 2 = 7i + (~ ~ l ~ h i ) This r e s u l t s

in

2M(l+2n)

equations

,

f o r the

i = 1,2 . . . . . M. 2(M+l)(l+2n)

unknowns,

namely

0(7i), 0'(7i), I0j(Ti), I6j(Ti) , I2j(Ti) , I~j(Ti), j = l , 2 . . . . . n, i=1,2 . . . . . M. Boundary c o n d i t i o n s c o n s t i t u t e the remaining of the r e q u i r e d r e l a t i o n s . The procedure used is of the order scheme which is of the order

h4

compared to the

h 2, where

central finite

difference

h is the uniform g r i d s t e p s i z e .

i t s computer implementation is s t r a i g h t f o r w a r d technique of the same o r d e r , for example).

( u n l i k e the

finite

Yet, element

420

A. Yucel, H. Kehtarnavaz and Y. Bayazitoglu

The

Vol. 16, No. 3

solution procedure is initiated by solving the nonradiating

boundary layer equations (~ = 0). increasing values of f. are used as

The system of equations

The converged solutions

initial guesses for the temperature

distributions at each new

~ value.

is then solved for

at the preceding value of and the moments of intensity

Increments as high as A t = 0.05 could be

used as the corresponding converged solutions were found to be within those obtained using A t = 0.005.

f.

8 with respect to

the energy equation reduces to a local similar form

The system of equations

dent of the

can be solved for different values of

"upstream" solutions

tion algorithm [20]

0.5% of

It should be noted that in stagnation flow,

i.e. when m = 1, the term with the partial derivative of vanishes and

thermal

in this case.

was used to solve

in terms of f indepen-

The modified quasilineariza-

the nonlinear system

of equations.

A

performance index P which represents the cumulative error in the equations was used as a convergence criterion.

The algorithm was terminated when P _< 10 -12 .

Rapid convergence was obtained in all cases. Once

the temperature and radiative flux distributions are determined for

a given set of the system parameters,

the total heat flux at the wall

can be

evaluated by #T qw = [-kc ~" + qr]z=O

(14)

In terms of the dimensionless quantities, Eq. (14) is expressed as N 0o Qw = [- ? + Qr]t}=O

(15)

where n n = {04, - 1 - T, [F-~'j(#w),, - uEt-j(1)]}/4 + T, I.. Qr j=z j=l lj

(16)

The total heat flux at the wall is related to the local Nusselt number by

- iN Qw = (1-8 w) NUxRe~l/2

(17)

R e s u l t s and D i s c u s s i o n In order to keep

the number of i n d e p e n d e n t p a r a m e t e r s

to a minimum, the

p r e s e n t c o n s i d e r a t i o n i s l i m i t e d to o n l y those cases with Pr = 1. tion-radiation

parameter

strong radiation effects.

is set at N = 0.1,

which i m p l i e s

The conduc-

relatively

strong

To d e m o n s t r a t e the e f f e c t s of n o n g r a y n e s s , two sim-

p l e models f o r the a b s o r p t i o n c o e f f i c i e n t are employed.

These a r e g i v e n by

421

16, No. 3

vol.

Note

that

to zero

when the in model

2a for

flow

The total gray parent

over

are

optically

the

0 n

for for

0 5 v < uco v 2 vco

plate

as

case,

interval the

models

which

for

B due to the

are

not

are

A or goes

shown

than

contributions

the

For the

approximately

39% of

larger

in model

stagnation

greater

attenuated

medium,

while

flux

2b for are

radiation

a gray

flux

heat

Fig.

c increases.

heat

infinity

wall

A and B

surface

0 5 V 5 3, total

total

(m=Ol and in for

Co

gray.

on the

due to the

limit

Hence,

vco approaches

frequency

regions

in this

model

nv =

in the models,

frequency

for

Model B:

values

is

no such thick

v 1 3.

Osvlv v 2 vco

a flat

This

considered

for for

the medium hecomes

flux

regions

there

n t 0

of nongrayness

heat

case.

nV =

cutoff

B,

The effects

Model A:

heat

cut-off

frequency

contribution

to the

radiative

transSince

value

cco=

radiation

A are

greater heat

the

to the

frequency

model

(m=l). for

tends

blackbody

for

Fig.

medium.

flux

61% comes from the

values

the

flow those

from the

by the

total

in

is

3

from

interval than

flux

those

from the

.v

(a) m = 0

V

Model A

A

Model B

0

-

0.2

0.0

er = 0.1 8 = 0.5 _-W_ _-_-

Gray 0.6

0.4

rI’

0.0

0.6

0.2

Heat

Flux

0

Gray

Model

B

0.6

0.6

<

FIG. Wall

Model A

0.4

<

Total

V A

vs c

Plate

with

(m=O),

N=O.l,

2 Pr=l.O,

bl Stagnation

Yco=3.0: Flow

(m=ll

al

Flow

over

a Flat

422

A. Yuoel, H. Kehtarnavaz and Y. Bayazitoglu

transparent

r e g i o n in model A.

intensified

f o r the case w i t h 0

T ab l e 1

presents

As e x p e c t e d , w

different i.e.,

the e f f e c t s

of the c u t - o f f

is i n c r e a s e d .

CO

model A

f o r the g r a y c a s e ,

t h i c k e r boundary l a y e r s .

f o r a g r a y medium.

Recall that

is

f r e q u e n c y on the t e m p e r a t u r e

p l a t e with 0

the t e m p e r a t u r e g r a d i e n t s

from the r e s u l t s

optically

the o v e r a l l wall h e a t t r a n s f e r

= 0.1.

g r a d i e n t a t the s u r f a c e f o r f l o w over a f l a t It is observed that

Vol. 16, No. 3

= 0 .5 and 0

W

f o r nongray media especially

Thus f o r a g i v e n v a l u e of f ,

= 0.1.

can be q u i t e

for larger ~ values,

The t e m p e r a t u r e g r a d i e n t

model A a p p r o a c h e s

W

is s t e e p e s t

the g r a y l i m i t i n g case i f

the t e m p e r a t u r e g r a d i e n t f o r

i n c r e a s e s from the n o n r a d i a t i n g case r e s u l t

to the g r a y s o l u t i o n .

On

the o t h e r hand, model B becomes t r a n s p a r e n t w i t h i n c r e a s i n g ~ s l o p e of the t e m p e r a t u r e p r o f i l e

: t h e r e f o r e the co These v a r i a t i o n s b e -

d e c r e a s e s in t h i s c a s e .

come more pronounced i f the wall t e m p e r a t u r e i s d e c r e a s e d .

TABLE 1 E f f e c t of C u t - o f f F r e q u e n c y on the Temperature G r a d i e n t a t the F l a t P l a t e S u r f a c e f o r Various Values of f : N=0.1, P r = l . 0 Model

0

W

= 0.5

#

w

=

0.1

co f = . 0 5 f = . 1 0 f = . 2 0 f = . 3 0 f = . 5 0 ~=1.0

f=.05 f=.lO f=.20 f=.30 f=.50 f = l . 0

2 3 5

0.166 0.168 0.172 0.175 0.180 0.187 0.167 0.171 0.181 0.189 0.201 0.213 0.170 0.179 0.199 0.215 0.234 0.252

0.300 0.303 0.311 0.319 0.333 0.354 0.302 0.309 0.328 0.345 0.372 0.408 0.306 0.320 0.354 0.383 0.426 0.475

Gray

0.172 0.185 0.212 0.232 0.254 0.275

0.308 0.328 0.369 0.403 0.449 0.505

2 3 5

0.172 0.184 0.209 0.229 0.252 0.277 0.171 0.181 0.203 0.220 0.244 0.270 0.168 0.173 0.184 0.194 0.209 0.227

0.307 0.324 0.360 0.390 0.431 0.479 0.305 0.318 0.346 0.369 0.402 0.441 0.302 0.307 0.317 0.329 0.343 0.359

A

B

Figure 3 the p a r a m e t e r f case ~

CO

shows the t e m p e r a t u r e p r o f i l e s

as the d i m e n s i o n l e s s d i s t a n c e from the l e a d i n g edge) f o r

= 3 and 0 = 0 . 5 . W

Although the t e m p e r a t u r e s n e a r the wall

in the case of a g r a y medium, the w a l l .

a t the l o c a t i o n ~ = 0 .2 ( v i e w i n g

this situation

the

are higher

i s c o m p l e t e l y r e v e r s e d away from

T h e r e f o r e the thermal boundary l a y e r t h i c k n e s s f o r a nongray medium

c h a r a c t e r i z e d by model A or B

is s m a l l e r than t h a t f o r a g r a y medium.

Varia-

t i o n s in the thermal boundary l a y e r t h i c k n e s s w i t h Vco a r e found to be s i m i l a r to t h o s e o b s e r v e d f o r t e m p e r a t u r e g r a d i e n t a t the w a l l .

Vol. 16, NO. 3

FIEWOFA~RAD~FIEID

423

5 q~

:

V = Model A A = Model B O = Gray

~ [

O

0.0

2'.0

4' .0

FIG.

' 6,0

8.0

3

Temperature P r o f i l e s for Flow over a Flat P l a t e : N=0.], P r = l . , 0w=0.5 , Vco=3.

Nomenclature eb,E b

blackbody emissive power, F.~ = e b / ~ ~

f

dimensionless stream function

i

i n t e n s i t y of r a d i a t i o n

ik, I k

• 4 kth moment of i n t e n s i t y of r a d i a t i o n , I k = lk/4OT~, k = 0,1,2,3

kC

thermal c o n d u c t i v i t y

n

number of bands

N

c o n d u c t i o n - r a d i a t i o n parameter, ~axkc/4O'r~

Nu

local Nusselt number

Pr

Prandtl number

qr,Qr

r a d i a t i v e heat flux, Qr = qr/4~¢l':

qw'%

t o t a l heat flux at the wall, Qw = qw/4°T~

ReX

local Reynolds number

T

temperature

X

424

A. Yuoel, H. Kehtarnavaz and Y. Bayazitoglu

T

e x t e r n a l flow t e m p e r a t u r e

U

e x t e r n a l flow v e l o c i t y

X,Z

spatial

C$

$./~ J

j

Vol. 16, No. 3

coordinates

max

bandwidth nonsimilarity

variable,

1/2 ~maxX/Rex

similarity variable, zRe~/21x o

dimensionless temperature, T / T absorption coefficient max

#

Max {~1,~2 . . . . ~n } direction spectral

c o s i n e in z - d i r e c t i o n variable

(frequency), v = hu/kT

O"

Stefan-Boltzmann constant

fl

solid angle

Subscripts

J

refers

to j t h band

w

refers

to q u a n t i t y a t the wall (z=0)

V

refers

to s p e c t r a l q u a n t i t y References

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d e r V e r f a h r e n s t e c h n i k 22, 51 (1984).

3. R. Y i s k a n t a and R. J . Grosh, I n t . J . Heat Mass T r a n s f e r 5, 129 (1962). 4. R. D. Cess, J . Heat T r a n s f e r ,

86, 469 (1964).

5. N. A. Kacken and J . P. H a r t n e t t , Proceedings o f the 1967 Heat Transfer and Fluid Mechanics I n s t i t u t e , P. A. Libby e t a l , ed, p. 115, S t a n f o r d U n i v e r s i t y P r e s s , S t a n f o r d , C a l i f o r n i a (1967). 6. A. Yficel and Y. B a y a z i t o g l u , AIAA J . 22, 1162 (1984). 7. J . D. Anderson, AIAA J. 6, 758 (1968). 8. A. M. Smith and H. A. Hassan, CR-576, NASA (1966).

Vol. 16, No. 3

~

OF A NCN(~%KY RADIATING FLUID

425

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