LETTERS IN HEAT AND MASS TRANSFER Vol. I, pp. 131 - 138, 1974

Pergamon Press P r i n t e d in the Uni ted S t a t e s

APPLICATION OF THE ENERGY-DISSIPATION MODEL OF TURBULENCE TO THE CALCULATION OF FLOW NEAR A SPINNING DISC

B.E. Launder and B. I . Sharma Department o f Mechanical Engineer ing

Imper ia l Col lege o f Sc ience and Technology, London SW7 2BX

(Communicated by D.B. Spalding and J. Ho Whitelaw)

I n t r o d u c t i o n

In the l a s t few yea r s a number o f models o f t u r b u l e n t hea t and momentum

t r a n s p o r t have been developed in which the e f f e c t i v e t r a n s p o r t c o e f f i c i e n t s

are r e l a t e d t o l oca l va lues o f c e r t a i n t u r b u l e n t c o r r e l a t i o n s ; t h e s e c o r r e l a -

t i o n s a re computed s imu l t aneous ly with the mean f i e l d v a r i a b l e s . Models o f

t h i s kind ach ieve s i g n i f i c a n t l y g r e a t e r b r ead th o f a p p l i c a b i l i t y than do sim-

p l e r approaches based on mean-flow q u a n t i t i e s a lone . One o f the more success -

fu l o f t he s e newer approaches i s the e n e r g y - d i s s i p a t i o n model developed by

Jones and Launder ( 1 , 2 ) . I t s o r i g i n a t o r s app l i ed i t t o the c a l c u l a t i o n o f

numerous boundary l a y e r flows with severe streamwise p r e s s u r e g r a d i e n t or su r -

f ace mass t r a n s f e r . No a p p l i c a t i o n s have been r e p o r t e d , however, o f i t s use

to p r e d i c t s w i r l i n g f lows, an omission t h a t the p r e s e n t no te remedies . The

flow cons ide red ( t ha t gene ra t ed by a r o t a t i n g d i sc in a q u i e s c e n t atmosphere)

produces ve ry high g r a d i e n t s o f swi r l v e l o c i t y in the v i c i n i t y o f the d i s c

which in t u r n b r ings to prominence terms in the k i n e t i c energy and d i s s i p a t i o n

equa t ions t h a t have fo r mer l y been absent or o f on ly small impor tance . This

a p p l i c a t i o n thus p rov ides a t e s t o f the g e n e r a l i t y o f the model f o r an impor-

t a n t c l a s s o f f l u i d f lows.

The Turbulence Model

The t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s UT and r T are ob t a ined from the

fo l lowing system o f d i f f e r e n t i a l and a u x i l i a r y equa t i ons ; exp l ana t i on o f the

131

132 B.]E. L a u n d e r and B . I . S h a r m a Vol. 1, No. 2

o r i g i n o f the equa t ions i s p rov ided in r e f s ( 1 , 2 ) .

Turbu len t v i s c o s i t y : UT = cp pk2/e (1)

where c

Turbu len t thermal or mass d i f f u s i v i t y : F T = PT/0.9p

( i . e . Turbu len t P rand t l /Schmid t number = 0.9) Turbulence k i n e t i c energy equa t ion :

L - ~ j

+ + r - pe - zU

Turbulence energy dissipation equation:

pU ~ + pV 8y r 8y + ~T

+ 2.o o / + r

- 0.09 exp [ - 3 . 4 / ( 1 + RT/50)2]

c 2 ~ 1.92 [1.0 - 0.3 expC-R2T)]

(23

(3)

(4)

(S)

(6)

and R T ~ 0k2/pE, the turbulent Reynolds number.

The other empirical coefficients take the following uniform values:

c I = 1 . 4 4 ; o k = 1 . 0 ; ~E = 1 . 3 .

The independent variables r and y are respectively the radial distance from

the disc axis and the normal distance from the disc surface. The correspond-

ing velocities are U and V; V@ denotes the circumferential velocity. All

other notation is the same as in ref (1,2).

The above system of equations differ from that in (1) and (2) in two

respects:

Vol. I, No. Z SPINNING DISC 133

(i)

(ii)

Extra source terms involving gradients of (Vo/r) appear in the equations

for k and c. Their appearance is due to the conversion of the cartesian-

tensor from of these equations to the present coordinate frame. The 7 are

not ad hoc terms.

The Reynolds number functions c and c 2 and the coefficient c I are

slightly different. This is the result of an overall reoptimisation of

coefficients reported in (3). We repeated the computation of some of the

50

3O

y~

2O

IO

• ERIAN [6] R ¢ = 9 9 3 x l o 5

0 2 O 4 06

V e / T ~

08 iO

50

40

3O

20

-\ \

k %

UITo~ 12

Fig. i Turbulent flow velocity profiles near a spinning disc

flows studied in (1,2) with the present coefficients: there were no sig-

nificant differences in the results from those originally reported.

Solution of Equations

The above system of equations have been solved simultaneously with

that governing the mean flow: i.e. the radial and angular momentum equa-

tions, the continuity equation and (for problems of heat or mass diffusion)

134 B . E . L a u n d e r and B . I . S h a r m a Vol. 1, No. 2

the en tha lpy or chemical spec i e s equa t ion . The mean-flow equa t ions are given

in the Appendix t o g e t h e r with the boundary c o n d i t i o n s . The p r e c i s e form i s

g iven in (4) . The numerica l so lv ing scheme used in b a s i c a l l y the Pa tankar -

Spalding (5) p rocedure modi f ied f o r the i n c l u s i o n o f swi r l as o u t l i n e d in

(4) . Seventy nodes were used to span the boundary l a y e r with a s u b s t a n t i a l

c o n c e n t r a t i o n ve ry near the wall ( t h i s i s about 60% more than are needed to

ob t a in g r i d - i n d e p e n d e n t r e s u l t s when the mix ing - l eng th model i s u sed ) . The

forward s tep used was t y p i c a l l y 15% o f the boundary l a y e r t h i c k n e s s l ead ing

to computer t imes pe r run o f about 50 s on a CDC 6600 computer.

Discuss ion o f Resu l t s

Some p r e d i c t e d c h a r a c t e r i s t i c s o f the r o t a t i n g - d i s c flow a re compared

with exper imenta l da t a in F igs . 1-3. The c a l c u l a t e d r a d i a l and c i r cumfe ren -

t i a l v e l o c i t y p r o f i l e s a re seen, from Fig. 1 to be c l o s e l y in agreement with

E r i a n ' s (6) expe r imen ta l da t a . The mix ing - l eng th r e s u l t s from (4) a re a l so

shown: a l though he re agreement with experiment i s a l so r ea sonab ly s a t i s f a c -

t o r y , the r a d i a l v e l o c i t y f a l l s to zero f a s t e r than exper iment sugges t s due

to the t u r b u l e n t v i s c o s i t y f a l l i n g to zero too qu i ck ly near the o u t e r edge.

CM

O . O I

• T H E O D O R S E N [8] M L H -- X KREITH [7] K-E:

0 0 0 2 I I I I 5 6

IO IO 1"2 ~ / v

Fig. 2

Torque coefficient for spinning disc

Vol. 1 No. 2 SPINNING DISC 135

The cor responding v a r i a t i o n o f to rque c o e f f i c i e n t with Reynolds number

i s shown in Fig . 2. The Reynolds number a t which t r a n s i t i o n occurs must be

p rov ided (when the r a d i a l d i s t a n c e i s so small t h a t the sp in Reynolds number

i s l e s s than Ret ran s the flow i s taken as laminar and only the mean flow

equa t ions are s o l v e d ) . Following our p r a c t i c e in (4) we have adopted the

va lue o f Retran s t h a t seemed to be suggested by any p a r t i c u l a r appa ra tu s ; not

surprisingly there is a significant variation in Retran s from one apparatus

to another due presumably to minor variations in geometry and in surface fin-

ish. The predicted variation of torque coefficient with Reynolds number is

closely in line with the measured behaviour, the line representing predicted

behaviour falling roughly mid-way between the data of Kreith (7) and Theodorsen

and Regier (8).

Heat and mass transfer predictions are shown in Fig. 3. There is

extremely close agreement with Cobb's measured mean Nusselt numbers over the

full range of the experiments; the data of McComas lie about 7% below this.

Agreement with the naphthalene diffusion data of Tien (ii) and Kreith (12)

is not quite as satisfactory, however. While there is close agreement with

experiment for spin Reynolds numbers up to 4 x 105 , beyond this the data

rise progressively faster than the predictions indicate. The same kind of

discrepancies were noted in (4) where the mixing length hypothesis was used.

It does not seem possible to identify the cause of the disagreement in the

absence of measured profiles of species concentration near the disc.

Nu

o$"

,C

i O 2

• McCOMAS [9} ~ , / COBB [IO] ~ / /

0 K I~EITH [12] X U / /

0 " = 2 4

•

~ ER)

o x' 'O'°" M L H - - ° I / o / K -¢

...W-"

, , r ] l l l I i I i I l l l i I0 5 i0 6

T2c~/ ",,;

I i

Fig. 3 Heat and mass transfer from a disc rotating in still air

136 B.E. Launder and B.I. Sharma Vol. i, No. 2

Conclusion

The main conclusion is that the k~E model of turbulence, which had

been devised (1,2) specifically to predict certain low-Reynolds-number phe-

nomena in boundary layers and duct flows, has been found to predict accurately

the flow, heat and mass transfer in the neighbourhood of a rotating disc.

The result is of significance to the problem of predicting convective heat

transfer rates in turbine discs.

Acknowledgement

This r e s e a r c h has been suppor ted by the Sc ience Research Council through

Grant No. B/RG/1863.

References

1. W.P. Jones and B.E. Launder, I n t . J . Heat Mass T r a n s f e r , 15, 301 (1972).

2. W.P. Jones and B.E. Launder, I n t . J . Heat Mass T r a n s f e r , 16, I l l 9 (1973).

3. B.E. Launder, A. Morse, W. Rodi and D.B. Spa ld ing , Langley Free Shear Flows Conference , NASA SP-321 (1972).

4. M.L. Koos in l in , B.E. Launder and B. I . Sharma, J . Heat T r a n s f e r , 96, 204 (1974).

5. S.V. Patankar and D.B. Spa ld ing , Heat and Mass T r a n s f e r in Boundary Layers , I n t e r t e x t Books, London (1970).

6. F.F. Er ian and Y.H. Tong, The Physics o f F lu ids , 12, 2588 (1971).

7. F. K re i t h , Proc. Heat T r a n s f e r and Flu id Mech. I n s t . S t an fo rd (1966).

8. T. Theodorsen and A. Regie r , NACA Rep. 793 (1944).

9. S.T. McComas and J . P . H a r t n e t t , 4th I n t . Conference, Heat T r a n s f e r 3, FC 7.7 (1970).

I0 . C. Cobb and 0.A. Saunders , Proc. Roy. Soc. A236, 343 (1956).

11. C.L. Tien and D.T. Campbell, J . F lu id Mech. 17, 105 (1963).

12. F. Kre i th , J .H. Tay lor and J . P . Chong, J . Heat T r a n s f e r , 9__$5 (1959).

Vol. i, No. 2 SPINNING DISC 137

Appendix

The following is the system of mean-flow conservation equations solved

simultaneously with equations (1-61 describing the turbulence quantities.

Streamwise momentum:

au au pu ~ +pv a-- f =

Angular momentum:

3 rV 0 a rV 0 oU ar + pV ay

I a r ~y (~ + ~T ) + p ~--

1 3

(A1)

ray 3 (n + n T) Ty j (A2)

Enthalpy or spec ies mass f r a c t i o n (¢):

a@ a@ 1 a Ir 3@ 3 (A3) pU ~ + pv a-f = r a y (r + r T) ~f

Boundary conditions are applied at the disc surface (y=O) and beyond

the edge of the boundary layer (y=y~) as follows:

y = O: U = k = ~ = O; V 0 = mr; ¢ = C W

y = y : U = k = e = V 0 = O; ¢ = ¢ .