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I ~ U T T E R W Q R T Hl i l l e I N E M A N N

A v a r ia n t o f th e lo w - R e y n o l d s - n u m b e rt w o - e q u a t i o n t u r b u l e n c e m o d e l a p p l i e d t o

v a r i a b l e p r o p e r t y m i x e d c o n v e c t i o n f l o w sM . A . C o t t o n a n d P . J . K i r w i nScho ol of Engineering, U niversity of Man chester, M anchester, U K

P r e v i o u s w o r k h a s d e m o n s t r a t e d t h a t t h e l o w - R e y n o l d s - n u m b e r m o d e l o f L a u n d e ra n d S h a r m a (1 97 4) o f f e rs s i g n i f i c a n t a d v a n t a g e s o v e r o t h e r t w o - e q u a t i o n t u r b u l e n c em o d e l s i n t h e c o m p u t a t i o n o f h i g h l y n o n - u n i v e r s a l b u o y a n c y - i n f l u e n c e d ( o r " m i x e dc o n v e c t i o n " ) p i p e f l o w s . I t i s k n o w n , h o w e v e r , t h a t t h e L a u n d e r a n d S h a r m a m o d e ld o e s n o t p o s s e s s h i g h q u a n t i t a t i v e a c c u r a c y i n r e g a r d t o s imp le r f o r c e d c o n v e c t i o nf l o w s . A v a r i a n t o f t h e l o w - R e y n o l d s - n u m b e r s c h e m e i s d e v e l o p e d h e r e b y r e f e r encet o d a t a fo r c o n s t a n t p r o p e r t y fo r c e d c o n v e c t i o n f l o w s . T h e r e - o p t im i z e d m o d e l a n d

t h e L a u n d e r an d S h a r m a f o r m u l a t i o n a r e t h e n e x a m i n e d a g a i n s t e x p e r i m e n t a lm e a s u r e m e n t s f o r m i x e d c o n v e c t i o n f l o w s , i n c l u d i n g c a s e s i n w h i c h v a r i a b l e p r o p -e r t y e f f e c t s a r e s i g n i f i c a n t .

Ke ywo r d s : m i x e d c o n v e c t i o n ; v a r i a b l e p r o p e r t i e s ; v e r t i c a l p i p e f l o w ; t u r b u l e n c em o d e l s

Introduction

The relative simplicity of two-equation turbulence models has

resulted in their widespread application to problems of turbulent

fluid mechanics and convective heat transfer. The models consist

of a constitutive equation which relates Reynolds stress to mean

strain rate via a turbulent viscosity and transport equations for theturbulent kinetic energy k and its rate of dissipation e (althoughother parameters have been selected as the second scale-de-termining variable). In thin shear flows, where only one compo-

nent of the Reynolds stress tensor is active in the mean flowequations, model ling at the two-equation level represents the least

complex route by which one might expect to achieve accurate

resolution of some "non-universa l" flow features.A class of thin shear flows which exhibit very marked depar-

tures from universal behaviour (such as adherence to the " law of

the wall") are mixed convection pipe flows. The regime of

mixed convection occurs in vertical pipe flows which are charac-terized by relatively low bulk Reynolds number and relatively

high Grashof number. Thus, the "reference" forced flow (de-

fined in terms of Reynolds and Prandtl numbers) is significantlymodified by the action of buoyancy. In the ascending flow case(to which present attention is restricted), the effectiveness of heat

transfer with respect to forced convection is at first reduced with

increasing buoyancy influence; a further increase promotes apartial recovery, and, at high levels of buoyancy influence , there

is eventual enhancement of heat transfer effectiveness. Reviewsof mixed convec tion research are provided by Petukhov andPolyakov (1988) and Jackson et al. (1989).

Address repr int requests to Dr. M.A. Cotton, D iv. of Mechanica land Nuclear Engineer ing, The Manchester School of Engineer-ing, The U nive rsity of Manchester, Simon Building, Oxford

Road, Manchester, M13 9PL, UK.Received 18 Janu ary 1995; accepted 12 Ma y 1995

Int. J. He at and Fluid Flow 16: 486-49 2, 1995© 1995 by Elsevier Science Inc.655 Avenue of the Americas, New York, NY 10010

Tw o-equat ion mod e ls and the i r app l ica t ion toturbulen t m ixed convec tion

The standard "high- Reynolds-number" two-equation k-c model

is presented by Launder and Spalding (1974) and, in modelling

terms, represents an asymptote towards which models incorporat-ing low-Reynolds-number features revert in fully turbulent flowregions. The modification of the high-Reynolds-number form toinclude functions and additional terms dependent on local flow

features reduces reliance on assumptions of universal behaviour.In particular, low-Reynolds-numberclosures are integrated up to

a solid boundary and therefore do not require the specification of

"wall functions" as used in high-Reynolds-number ormulations

to bridge the near-wall region. The first model to include a

"damping function ," f~(Ret), in the constitutive equation wasdeveloped by Jones and Launder (1972); a re-optimization of that

model due to Launder and Sharma (1974) (hereafter LS) has

since achieved some status as the benchmark low-Reynolds-num-

ber k-e formulation.

The general strategy adopted in order to develop or refine

turbulence closures is to specify model parameters (e.g., the

turbulent Reynolds number) and then to "tune" model constants

and functions by reference to "s imple" flows such as fullydeveloped channel or pipe flow. The level of activity in the

development of alternative two-equation turbulence models isdemonstrated by the review of Patel et al. (1985) in which 10models were examined against data for boundary layers. It w a s

found that on ly four models (including the LS model) performed

at all satisfactorily and that weaknesses were apparent in all themodels tested. Betts and Dafa'Al la (1986)tested 11 two-equation

models (substantially the same models as those considered byPatel et al.) against data for natural convection cavity flow and

found that only 4 models were in reasonable agreement withmeasurements. The LS formulation was the only model deemed

to yield broadly satisfactory results by both Patel et al. and Bettsand Dafa'Alla.

0142-727X/95/$10.00SSD I 0142-727X(95)00040-W

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A k-~ turbu lence mo de / fo r mixe d convect ion: M. A. C ot ton and P. J . K i rwin

C ot ton and J ackson (1990) com pared a cons t an t p rope r t i e s

(B ouss inesq app rox im a t ion ) fo rm ula t i on o f t he LS m ode l aga ins t

t he m ixed convec t i on a i r f l ow m easu rem en t s o f C a r r e t a l . (1973)

and S t e ine r ( 1971) and found gene ra l l y accep t ab l e ag reem en t

w i th t he expe r im e n ta l da t a . Tha t w o rk has r ecen t l y been ex t end ed

by C o t ton e t a l . ( 1995) u s ing t he LS m ode l i n va r i ab l e p rope r ty

f o r m a n d i n c l u d in g c o m p a r i s o n w i t h t h e d a t a o f P o l y a k o v a n d

Sh ind in (1988) . O the r r ecen t com para t i ve s t ud i e s by t he au tho r s

and t he i r co l l eagues (C o t ton and K i rw in 1993 ; M ik i e l ew icz 1994 ;

K i rw in 1995) have dem ons t r a t ed t ha t , o f t he m ode l s t e s t ed , t he

LS m ode l p roduces t he m os t accu ra t e r e su l t s i n t he com pu ta t i on

o f a s c e n d i n g m i x e d c o n v e c t i o n a i r f l o w s .

D e v e l o p m e n t o f a v a ri a n t o f t h e l o w - R e y n o ld s -

n u m b e r k -~ m o d e l

G u i d i n g c o n s i d e r a t io n s

The p r inc ipa l ob j ec t i ve o f t he p r e sen t r e sea rch i s t o p robe fu r t he r

t he r ea sons fo r t he succe ,; s o f t he LS m od e l i n t he ca l cu l a t i on o f

m ixed convec t i on f l ow s . A n e s sen t i a l e l em en t o f t ha t m ode l i s

t ha t a l l depa r tu r e s f rom the h igh -R eyno ld s -num ber fo rm a t a r e

spec i f i ed i n t e rm s o f l oca l f l ow va r i ab l e s . Thus , w e add re s s t he

ques t i on : " I s i t pos s ib l e t o r e fo rm ula t e t he LS m ode l t o ach i eve

be t t e r ag reem en t w i th s im p le f l ow s and subsequen t l y r e t a in a

c o m p a r a b l e l e v e l o f p e r f o rm a n c e i n m i x e d c o n v e c t i o n c o m p u t a -

t i ons ?" In o the r t e rm s , w e seek t o i nves t i ga t e s ens i t i v i ty t o m ode l

de t a i l w h i l e r e t a in ing t he s am e m ode l pa r am e te r s ,

M o d e / e q u a t i o n s

The equa t i ons o f t he LS m ode l and t he p r e sen t va r i an t m ay be

w r i t t en i n t he fo l l ow ing gene r i c fo rm :

[ OU/ OUj ] 2

- o u , u j = i x , / 7 - ~ + ~ l - -~%ok ( 1 )

IX , = C ~ f ~ ( R e t ) pk 2 ( 2 )

D t tx t Ox---~j Ox ~ Ox--~i+ i x +

- p ( ~ + D e ) ( 3 )

- - = - - + - - ~ + - -D(P~)Dt C , 1 - ~ i x t [ ~ x j ~ x i ] O x j O x j f i r :

- c , 2 / 2 - ~ - i x ~ ' [ ° ~ E ( 4 )

T h e h i g h - R e y n o l d s - n u m b e r c o n s t an t s i n b o t h m o d e l s a r e u n -

ad jus t ed f rom those quo t ed by Lau nde r and Sp a ld ing (1974):

C~ = 0 .09 ; % = 1 .0 ; % = 1 .3 ;

C~1 = 1.44 ; C,2 = 1.92 (5 )

N o t a t i o n

B o

c fC p

C ~ I , C ~ 2 ,

C~3C .

D

D ef 2f ~g

G r

H

k

k +

N u

N u oP r

qq +

R e

R e n , R e ,

R e t

- u um +

- u u

UU +

buoyancy pa ram e te r , Equa t i on 12

loca l f r i c t i on coe f f i c i en t , " r w / / ~ 1 pf~2

spec i f i c hea t capac i t y a t cons t an t p r e s su re

cons t an t s i n p roduc t i on and s i nk t e rm s o f e -

equa t i on

cons t an t i n cons t i t u t i ve equa t i on o f k - e m ode l

p ipe i n t e rna l d i am e te r

d i s s i pa t i on t e rm in k - equa t i on ( e = ~ + D , )

func t i on i n s i nk t e rm o f e - equa t i on

func t i on i n cons t i t u t i ve equa t i on o f k - e m ode l

m agn i tude o f acce l e r a t i on due t o g r av i t y

G r a s h o f n u m b e r b a s e d o n w a l l h e a t f l u x,

13bgqD 4 ' hb v 2channe l ha l f -w id th

tu rbu l en t k ine t i c ene rgy

k / V ~N usse l t num ber , q D / h b ( O w - - 0 b)N usse l t num ber fo r f o r ced convec t i on

Prand t l num ber , C p i x b / / ) k b

w al l hea t f l ux

hea t l oad ing pa ram e te r , Equa t i on 13

R e y n o l d s n u m b e r f o r p i p e f l o w , p b U b O / b l , b

R e y n o l d s n u m b e r s f o r c h a n n e l f l o w ,

( 2 p U b H / t x ) , ( p U s H / i x )t u rbu l en t R eyno lds num ber , k 2 / v gR eyno lds s t r e s s

- ~ / u /s t r e a m w i s e v e l o c i t yu / t ; ~

U / , u i m ean , f l uc tua t ing ve loc i t y com p onen t s i n

C ar t e s i an t enso r s

U~ f r ic t ion velo ci ty , ~ r~-w/p

x ax i a l coo rd ina t e

x~ C a r t e s i an coo rd ina t e s

y no rm a l d i s t ance f rom w a l l

y + y U J v

G r e e k

13 coe f f i c i en t o f vo lum e expan s ion

8q K rone cke r de l t a

e r a t e o f d i s s i pa t i on o f k

e + e~ lU ¢

m odi f i ed d i s s i pa t i on va r i ab l e0 t em pera tu r e

k t he rm a l condu c t i v i t y

ix dyna m ic v i s cos i t y

v k inem a t i c v i s cos i t y , i x /p

p dens i t y

t rk , t r~ turbulen t Prand t l num ber for d i f fu s ion of k , e

7 shea r s t r e s s

Subscr ipt sbtW

Supe r s c r i p t

b u l k

tu rbu l en t

w a l l

m e a n

In t . J . Heat and F lu id F low, Vol . 16, No. 6 , December 1995 487

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A k-~ turbulence m ode l for m ixed convect ion: M. A. Cot ton and P. J . K i rwin

T a b l e 1 L o w - R e y n o l d s - n u m b e r m o d e l f e a t u re s

M o d e l f ~ f 2 C ~3

- 3 . 4 1L a u n d e r a n d e x p ( 1 + R ~ - t /5 0 ) 2

S h a r m a 1 9 7 4

P r e s e n t 1 - 0 . 9 7 e x p [ - Ret/160] 1 . 0 0 . 9 5

f o r m u l a t i o n - 0 . 0 0 4 5 R e t e x p [ - ( R e t / 2 0 0 ) 3

1 - - 0 . 3 e x p [ - R e 2 2 . 0

In bo th m ode l s D ~ (w he re ~ = ~ - D ~ ) is de f i ned he re a s fo l l ow s :

D , = 2 v ( a k l / 2 / a x j ) 2 fo r y+ > ~ 2 (6a )

D ~ = 2 v k / x ~ f o r y + < 2 ( 6 b )

I sm ae l ( 1993) has dem ons t r a t ed t ha t ne i t he r Equa t i on 6a no r

6b i s cons i s t en t w i th a quad ra t i c va r i a t ion o f k i n t he im m ed ia t e

nea r -w a l l r eg ion w hen so lu t i on i s ob t a ined us ing a l i nea r d i s -

c r e t i za t i on s chem e ( such a s em ployed he re ) . I t i s f ound , how eve r ,

t h at th e " s w i t c h " a p p l i e d u s i n g E q u a ti o n 6 b i n th e r e g io n y + < 2

p rom o tes conve rgence o f t he d i s c r e t i zed equa t i on s e t. The m o de l sd i f f e r i n t e rm s o f the l ow -R eyno ld s -num ber func t i ons f~ and f2

and t he cons t an t C ~ 3 appea r ing i n t he f i na l t e rm o f Equa t i on 4 .

Tab l e 1 de t a i l s t he se e l em en t s o f c l o su re a s adop t ed i n t he LS

m ode l and t he p r e sen t deve lopm en t .

R e s u l t s o f a f o r c e d c o n v e c t i o n o p t i m i z a t i o n

e x e r c i s e

The c lo su re e l em en t s fv, and C ,3 have been op t im ized by tun ing

the p r e sen t m ode l t o da t a fo r f o r ced conve c t i on channe l and p ipe

f low s . (The func t i on f2 (R e t ) w h ich appea r s i n t he d i s s i pa t i on

equa t i on o f t he LS m ode l depa r t s f rom un i t y on ly a t ve ry l ow

va lues o f t u rbu l en t R eyno lds num ber and has been om i t t ed in t hep re sen t exe rc i s e . ) I t shou ld be no t ed t ha t t he m ode l i s t e s t ed

a g a i n s t m i x e d c o n v e c t i o n f l o w s b e l o w w i t h o u t h a v i n g m a d e a n y

re f e r ence t o such f l ow s i n t he op t im iza t i on p rocedure .

The m ean f l ow equa t i ons o f con t i nu i t y , m om en tum , and

ene rgy a r e so lved i n con junc t i on w i th e i t he r t he p r e sen t o r LS

tu rbu l ence m ode l . Tu rbu l en t P rand t l num ber fo r d i f fu s ion o f hea t

i s s e t t o 0 . 9 t h roughou t t h i s s t udy . The gove rn ing equa t i ons a r e

w r i t t en i n cons t an t p rope r ty t h in shea r fo rm and t he so lu t i on i s

m arched i n t he f l ow d i r ec t i on un t i l a f u l l y deve loped cond i t i on i s

a t t a ined . A f i n i t e vo lum e / f i n i t e d i f f e r ence d i s c r e t i za t i on s chem e

i s em p loyed fo l l ow ing Leschz ine r (1982) . The nea r -w a l l node i s

p o s i t io n e d a t y + - 0 . 5 a n d 1 0 0 c o n t r o l v o l u m e s s pa n th e p ip e

rad ius (o r channe l ha l f -w id th ) . Tw o a lgo r i t hm s a r e u sed t o gene r -

a t e a g r i d d i s t r i bu t i on t ha t expands f rom the w a l l ; 50 con t ro l

vo lum es a r e l oca t ed i n t he r eg ion y + < 60 . A r ange o f s ens i t iv i t y

t e s t s w as pe r fo rm ed t o ensu re t ha t r e su l t s a r e s ens ib ly i ndepen-

den t o f de t a i l s o f t he num er i ca l p rocedure . I n t e rm s o f num er i ca l

s t ab i l i t y and co m pu t ing t im es , no s i gn i f i can t d i f f e r ence w as no t ed

be tw een t he p r e sen t and LS m ode l s .

C om par i son i s m ade be low w i th co r r e l a t i ons fo r f l ow r e s i s -

t ance and hea t t r ans f e r i n p ipe f l ow and w i th D i r ec t N um er i ca l

S im ula t i on (D N S ) da t a fo r channe l f l ow p ro f i l e s .

L o c a l fr i c ti o n c o e f f i c ie n t a n d N u s s e l t n u m b e r f o r p i p e

f l o w

R esu l t s a r e r epo r t ed fo r R eyno lds num ber s o f 5 × 103, 104,

5 X 104 and 105 , and P rand t l num b er s o f 0 . 7 and 6 . 95 . T he

the rm a l bounda ry cond i t i on o f un i fo rm w a l l hea t f l ux i s app l i ed .

C a l cu l a t ed l oca l f r i c t i on coe f f i c i en t i s com pared aga ins t t he

Prand t l equa t i on (Sch l i ch t i ng 1979) :

c f = (4 l og10(2 R e c~ "5) - 1 . 6 ) - 2 (7 )

V a l u e s o f c o m p u t e d N u s s e l t n u m b e r a r e c o m p a r e d a g a i n s t th e

Pe tukhov and K i r i l l ov co r r e l a t i on (K a ys and C raw fo rd 1980) :

( c J 2 ) R e P rN u o = ( 8 )

1 .0 7 + 1 2 . 7 ( c d 2 ) ° 5 ( P r 2/3 - 1 )

w here c f = 0 .25 (1 . 82 l og10 R e - 1 . 64 ) -2 .

In t he ca se P r = 0 . 7 , com par i son i s a l so m ade w i th a fo rm o f

t h e D i t t u s - B o e l t e r e q u at io n p r o p o s e d b y K a y s a n d L e u n g ( 1 9 6 3 ):

N u o = 0 . 0 2 2 R e ° s P r ° 5 ( 9 )

Sch l i ch t i ng (1979) no t e s t ha t Equa t i on 7 i s i n good ag reem en t

w i th da t a ove r a w ide r ange o f R eyno lds num ber . Equa t i on 8

ho lds fo r 1 0 ad R e ~< 5 × 106 and m od e ra t e t o h igh P rand t l

T a b l e 2 L o c a l f r i c t i o n c o e f f i c ie n t a n d N u s s e lt n u m b e rR e

5 × 103 104 5 × 104 105

CfX 1 0 3 :

N u 0 ( P r = 0 . 7 ) :

N u 0 ( P r = 6 . 9 5 ):

E q u a t i o n 7 9 . 3 5 7 . 7 2 5 . 2 2 4 . 5 0

L a u n d e r a n d

S h a r m a 1 9 7 4 m o d e l 8 . 8 6 7 . 3 2 5 . 0 4 4 . 3 8

P r e s e n t m o d e l 9 . 5 6 7 . 7 9 5 . 2 9 4 . 5 8

E q u a t i o n 8 - - 3 0 . 5 9 8 . 2 1 6 6 . 8

E q u a t i o n 9 - - 2 9 . 2 1 0 5 . 7 1 8 4 . 1

L S m o d e l 1 6 . 9 2 9 . 1 1 0 3 . 8 1 8 1 .1

P r e s e n t m o d e l 1 8 . 2 3 1 . 1 1 0 9 . 2 1 8 9 . 5

E q u a t i o n 8 - - 8 6 . 1 3 2 6 . 3 5 8 6 . 8

L S m o d e l 3 5 . 9 6 8 . 6 2 8 8 . 9 5 3 3 . 2

P r e s e n t m o d e l 4 5 . 9 8 4 . 6 3 4 3 . 2 6 2 7 . 5

4 8 8 I n t. J . H e a t a n d F l u i d F l o w , V o l . 1 6 , N o . 6 , D e c e m b e r 1 9 9 5

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A k-~ turbulence mod el for mixed convect ion: M . A. Cotton a nd P. J . / ( i rw in

U +

( a )

20 - - Presentmodel .... .. ... .. . Laundea" nd Sharmamodel _ . - ~

• Data f Kiln -" "" • '~

1510 , - f ~ ~'~'~S g* :"

/ . , ,

t I i I t I t

0 2 5 10 20 50 100 200y÷

_u--q+

k +

5

4

3

0.8

0. 6

0. 4

0.2

00 20 40 60 80 100

y*

Figure 1

2

l

00

f. ol l /(b) 1

• ?. .. .. .-" .. .. . _-_ .. ..

0.8

0. 6

0.4

r - I t : - - P r e s e n t m o d e l J| 1 ( . . . . . . L a u n d e r a n d S h a r m a m o d e l I 0 .

I ~ d , , " • D a t a o f K i m IL , J . / i i t r

( c )

i :: ' - - P r e s e n t m o d e l

- -- ~ - . L a u n d % a n d S h a n n a m o d e l

~ ~ a ~ z ~ l m

I I I I

20 40 60 80 100y*

(d )

, , , ,, , , ,, , , ,, , , . , . o , ~"~ P r e s e n t m o d e l , . . . . . "*""Y " . . .

- - - . L a u n d e r a n d S h a r m a m o d e l

• D a t a o f K i m

= o * i l l l

20 40 60 80 100y+

Ch an ne l fl ow pr of ile s a t R e~ = 395; (a) velocity, (b) Reynolds stress, (c) turbulent kinetic energy, and (d) damping function

n u m b e r s t o w i t h i n a p p r o x i m a t e l y + 1 0 % ( K a y s a n d C r a w f o r d

1 9 8 0 ) . K a y s a n d L e u n g ( 1 9 6 3 ) r e p o r t t h a t E q u a t i o n 9 h a s b e e n

u s e d t o c o r r e l a t e a l a r g e a m o u n t o f d a t a f o r g a s f l o w s a n d

d e m o n s t r a t e g o o d a g r e e r a e n t w i t h d a t a f o r 1 0 4 ~< R e < 1 0 5 a n d

Pr = 0 .7 .L o c a l f r i c t io n c o e f f i c i e n t a n d N u s s e l t n u m b e r c o m p u t e d u s i n g

t h e t w o t u r b u l e n c e m o d e l s a r e c o m p a r e d a g a i n s t E q u a t i o n s 7 - 9

i n T a b l e 2 . O v e r t h e r a n g e 5 × 1 0 3~ < R e < 1 0 5 , v a l u e s o f c /r e t u rn e d b y t h e L S m o d e l l i e b e t w e e n 3 a n d 5 % b e l o w t h e

P r a n d tl e q u a t i o n ; th e p r es e n t m o d e l y i e l d s v a l u e s o f c / b e t w e e n

1 % a n d 2 % a b o v e t h e P r a n d t l e q u a t i o n . I n t h e c a lc u l a t i o n o f h ea t

t r a n s f e r f o r a i r f l o w s ( P r = 0 . 7 ) , t h e L S m o d e l y i e l d s N u s s e l t

n u m b e r d i s c r e p a n c i e s (f o r 1 0 4 6 R e ~< 1 0 5 ) i n t h e r a n g e - 5 % t o

+ 9 % w i t h r e s p e c t t o t h e P e t u k h o v a n d K i r i l l o v c o rr e l a ti o n ;

c o r r e s p o n d i n g f i g u r e s f o r t h e p r e s e n t m o d e l a r e + 2 t o + 1 4 % .

C o m p a r i s o n w i t h t h e K a y s a n d L e u n g ( 1 9 6 3 ) c o r r e l a ti o n o v e r t h e

s a m e R e y n o l d s n u m b e r r a n g e r e v e a l s t h a t t h e m a x i m u m d i s c r e p -

a n c y r e tu r n e d b y t h e L S m o d e l i s - 2 % ; t h e p r e s e n t m o d e l

e x h i b i t s a m a x i m u m d i s c r e p a n c y o f + 7 % . M o r e s t r ik i n g d is c r e p -

a n c i e s a r e e v i d e n t f o r w a t e r f l o w s ( P r = 6 . 9 5 , a n d 1 0 4~ < R e ~<

1 0 5 ) : t h e L S m o d e l c o m p u t e s N u s s e l t n u m b e r s b e t w e e n 9 % a n d

2 0 % b e l o w E q u a t i o n 8 , w h e r e a s t h e p r e s e n t m o d e l r e t u r n s v a l u e s

b e t w e e n 2 % b e l o w a n d 7 ' % a b o v e t h e c o r r e l a t i o n .

V e l o c i ty a n d t u r b u l e n c e p r o f i l e s f o r c h a n n e l f l o w

C o m p a r i s o n i s m a d e w i t h t h e D N S d a t a o f K i m f o r R % = 3 9 5

( c o m m u n i c a t e d t o M i c h e l a s s i e t a l . 1 9 9 1 ) . T h e b u l k R e y n o l d s

n u m b e r o f t h e f l o w i s R e n = 1 2 , 3 0 0 .

F i g u re s l a - l d s h o w t he p re s e nt a n d L S m o d e l s c o m p a r e d

a g a i n s t t h e D N S d a t a o f K d m . V e l o c i t y p r o f i l e s a r e s h o w n i n F i g .

l a t o g e t h e r w i t h t h e v i s c o u s s u b l a y e r p r o f i l e a n d t h e ' l a w o f t h e

w a l l ' w i t h c o n s t a n t s a s s p e c i f i e d b y P a t el a n d H e a d ( 1 9 6 9 ) :

U + = y + ( 1 0 )

U + = 5 .45 + 5 .5 log lo y ~- (1 1)

D i s t r ib u t i o n s o f R e y n o l d s s t re s s ( F i g u r e l b ) s h o w t h a t th e

p r e s e n t m o d e l i s i n b e t t e r a g r e e m e n t w i t h d a t a in t h e " n e a r - w a l l "

( y + < 1 0 ) a nd " l o g a r i t h m i c " r e g i o n s ( y + > 3 0 ) , w h e r e a s t h e L S

m o d e l is c lo s e r to d a t a o v e r t h e i n te r m e d i a t e " b u f f e r " r e g i o n .

T h e p r o f i l e s o f t u r b u l e n t k i n e t i c e n e r g y ( F i g u r e l c ) s h o w t h ep r e s e n t m o d e l t o b e i n c o n s i d e r a b l y b e t t e r a g r e e m e n t w i t h d a t a .

- - +

R e s u l t s s i m i l a r t o t h o s e o b t a i n e d a b o v e f o r U ÷ a n d - u v a re

g e n e r a t e d a t R e , = 1 8 0 ( n o t s h o w n ) ; k + i s , h o w e v e r , i n a d e -

q u a t e l y r e so l v e d b y th e p r e s e n t m o d e l a t v a l u e s o f y + b e y o n d

t h e m a x i m u m i n k ÷ ( K i r w i n 1 9 9 5 ) .

F i g u r e l d s h o w s t h e d i s t ri b u t io n o f t h e d a m p i n g f u n c t i o n f ~

a t R e , = 3 9 5 . T h e p r e s e n t m o d e l i s s i g n i f i c a n t l y c l o s e r t o t h e

D N S d a t a n e a r t h e w a l l ( y + ~ < 2 0 ) , w h e r e a s t h e L S m o d e l i s i n

b e t t e r a g r e e m e n t w i t h d a t a i n t h e r e g i o n y + > t 6 0 . N e i t h e r m o d e l

c o u l d b e s a i d o v e r a l l t o b e i n s a t i s f a c t o r y a g r e e m e n t w i t h t h e

da ta .

Nu/Nu o2

1.8

1.6

1.4

1.2

1

0.8

0. 6

0.4

• P r e s e n t m o d e l A

a L a u n d e r a n d S h a r m a m o d e l

A ~

0 . 2 i i i i

0.0l 0 ,~ l l 1 0 B e

Figure 2 Mixed convection Nusselt number normal ized by the

forced convection value as a funct ion of buoyancy parameter

Int. J. Heat and Fluid Flow, Vol. 16, No. 6, December 1995 489

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8/8/2019 A Variant of the Low-Reynolds-Number

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A k-~ turbulence mode/for mixed convection: M. A. Cotton and P. d./ (i rwin

Nu70

60

50

40

30

P r e s en t m o d e l ( a )

- - - L a u n d e r a n d S l ~ m ~ m o d el" - ~ . . ~ "D a t a o f V i l e m a s e t a l .

" ' " " - " - " . 7 7 . . _ _ - - _ - - _ . _

I I I I I

50 ( b )

40

30

20

1 0 ~ . " ~ r . - . . . . . p _ = ~ _ .0

35

30 ~ t k (C)25

2O

15 ~ em==%

10

5 i i I i i

35:

3 0 ( d )

25

15

10 ~ m ~

5 20 40 60 80 lCl0

x/D

Figure 4 N u s s e lt n u m b e r d e v e l o p m e n t fo r t h e c o n d i t io n s of

t h e V i l e m a s e t a l . ( 19 92 ) e x p e r i m e n t s ( c o n d i t io n s a s g i v e n i nT a b l e 3 )

tained using a test section of over 100 diameters in length. (The

data used here are taken from detailed documentation supplied by

Pogkas which supplements the results presented by Vilemas et al.

A selection of results is presented; Kirwin 1995 reports a compre-

hensive series of comparisons with the data.) Figures 4a-d showdata and model computations for relatively low, approximatelyconstant, values of q+; Be is increased from 0.1 to 7.0 over the

four cases. It is seen from Figure 4a that the present model

returns values of Nusselt number above the data, whereas the LS

model produces lower values. Figure 4b is the clearest example

in the present study of a case where the LS model performs

considerably better then the present formulation. (The buoyancy

parameter of this case represents the maximum impairment re-

gion and it should be noted that heat transfer performance ishighly sensitive to the precise condit ions of an experiment, cf.

Figure 2.) At higher levels of buoyancy influence the experimen-tal data of Figures 4c and d clearly show local recoveries in

Nusselt number. The LS model more accurately captures the

locations of these maxima, however neither model exhibits high

quantitative accuracy.

Concluding remarks

In the work reported here a variant of the Launder and Sharma

(1974) low-Reynolds-number k-e turbulence model is developed

on the basis of an empirical optimizat ion procedure. The present

model is generally seen to have been tuned to be in betteragreement with the reference forced convection cases (which

include DNS data produced some time after the publication of theLS model). Comparison of the two models with experimental

data for mixed convection flows shows that, in some cases, the

performance of the LS model is superior to that of the present

variant, whereas, in other cases, the reverse is true. Thus, no

major improvement of a standard closure could be said to have

been developed. I t is demonstrated, however, that results for

mixed convention c o m p a r a b l e to those obtained using the LSmodel can be generated by an al ternative Re/-damped model.

Acknowledgments

The authors are grateful to their colleagues Professor J.D. Jack-

son and Dr. J.O. Ismael for stimulating discussions in the courseof this work. Dr. Ismael also wrote the channel flow code used in

the optimizat ion procedure. Dr. P.S. Po lkas of the Lithuanian

Academy of Sciences kindly supplied additional data from the

experimental programme of Vilemas et al. (1992). P.J. Kirwin

wishes to acknowledge the support of an EPSRC Research

Studentship. (Authors' names appear in alphabetical order.)

by ~.x=0//hlocal o obtain experimental distributions of Nu basedon local fluid properties.

Figure 3 shows Steiner's (1971) data compared with thepresent and LS models, The level of heat loading is roughly

uniform and the buoyancy parameter increases from approxi-mately 0.1 to 2.0 (Table 3). The case of lowest Be (Figure 3a)shows the present model to be in better agreement with data thanthe LS model. An increase in buoyancy parameter through theregion of maximum impairment and into the recovery region

(Figures 3b-d) reveals both models to be in good agreement with

data.The recent data of Vilemas et al. (1992) shown in Figure 4

represent closely spaced measurements of Nusselt number ob-

References

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