a variant of the low-reynolds-number
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8/8/2019 A Variant of the Low-Reynolds-Number
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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
8/8/2019 A Variant of the Low-Reynolds-Number
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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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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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