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Simulation of air flow over a heated flat plate using anisotropic k-ε model
Journal of Wind Engineering and Industrial Aerodynamics, 46 & 47 (1993) 697-704 Elsevier
S i m u l a t i o n Flat Plate
of Using
Air
Flow over A n i s o t r o p i c
K. S A D A Senior Research Engineer Y. I C H I K A W A P r i n c i p a l Research Engineer, Central Research I n s t i t u t e Industry, 2-11-1 Iwato-kita, Komae-shi, Tokyo, Japan
697
a Heated k-8 Model
of E l e c t r i c Power
Abstract Numerical simulation of air flow over a heated flat plate was performed using the anisotropic k - e model. The wall functions for velocity and temperature were modified using universal functions, and the buoyant term in the dissipation equation was eliminated. The calculated results were compared with those from wind tunnel experiments. The calculated mean velocity, turbulent intensity and mean temperature showed good agreement with profiles measured in the wind tunnel.
i.
I NTRODUCT
I ON
Air f l o w s from the sea to land a r e sometimes o b s e r v e d in c o a s t a l r e g i o n s . Flows are g r a d u a l l y heated, and thermal i n t e r n a l boundary l a y e r s developing leeward and reversed flows are observed above the thermal internal boundary l a y e r . A complete numerical and wind tunnel model to s i m u l a t e a sea b r e e z e should be a b l e to s i m u l a t e (1) the l a r g e - s c a l e flow motions i n c l u d i n g e l e v a t e d p o s i t i o n (2) the s m a l l - s c a l e s h e a r flow motions and t u r b u l e n t t r a n s p o r t formed mainly near the surface, and (3) the i n t e r a c t i o n between these two s c a l e s of motion ~'~. Such sea breezes have been simulated by numerical techniques ~2'~3', but the greater part of these simulations have been f o c u s e d on sea and land b r e e z e c i r c u l a t i o n , and t h e a c c u r a c y of c a l c u l a t e d r e s u l t s f o r s m a l l - s c a l e motions has not been c l a r i f i e d because of flow complications and the lack of f i e l d observation data. A wind tunnel e x p e r i m e n t s i m u l a t i n g such s m a l l - s c a l e motions o f t h e sea b r e e z e was p e r f o r m e d , and t h e e n s e m b l e - a v e r a g e wind f i e l d was s i m u l a t e d by an a n i s o t r o p i c k - e model in this study. Turbulence q u a n t i t i e s were more e a s i l y obtained in the wind tunnel as compared with f i e l d experiments, and a more accurate evaluation of flow prediction was possible.
0167-6105/93/$06.00 © 1993 - Elsevier SciencePublishers B.V. All fights reserved.
698
2.
WIND
TUNNEL
EXPERIMENT
The e x p e r i m e n t r e p o r t e d here was conducted using the wind t u n n e l ( t e s t s e c t i o n dimensions are 20m length, 3m width, 1.5m h e i g h t ) a t CRIEPI. The t e s t s e c t i o n f l o o r was h e a t e d e l e c t r i c a l l y to form an u n s t a b l e thermal i n t e r n a l boundary l a y e r ( s e e F i g u r e 1). An e l e c t r i c h e a t e r , d e s i g n e d to p r o v i d e an e l e v a t e d s t a b l e temperature g r a d i e n t (9 °C/m), was a l s o s e t a t the e n t r a n c e of the wind tunnel t e s t s e c t i o n . Such an e l e v a t e d s t a b l e s t r a t i f i c a t i o n can a l s o be observed in the atmosphere, and r e s t r i c t s secondary flow i n h e r e n t in the device. The f r e e stream v e l o c i t y of U~=l.3m/s was s e l e c t e d to o b t a i n a t h i c k thermal i n t e r n a l boundary l a y e r and to reduce secondary flow. Roughness elements were s e t a t the e n t r a n c e of the wind tunnel t e s t s e c t i o n f l o o r to o b t a i n a t h i c k b o u n d a r y l a y e r and s t i m u l a t e t h e flow n e a r t h e s u r f a c e . Heating of the wind tunnel f l o o r was s t a r t e d 4m leeward of the e n t r a n c e of the t e s t s e c t i o n . The a i r flow in the wind tunnel blew from the " s e a a r e a " (where t h e f l o o r was n o t h e a t e d ) to t h e " l a n d a r e a " (where t h e s u r f a c e t e m p e r a t u r e was i n c r e a s e d almost u n i f o r m l y up to 80 °C). Wind v e l o c i t y and t e m p e r a t u r e were measured by a l a s e r Doppler anemometer and a r e s i s t a n c e thermometer.
Wind
To = 27 °C
U=, = 1, 3m/s
Stable /
Boundary "/
Thermal internal
strati-/ fication
f Sea
~
area
(Unheated zone)
=
Land
X
area
Heated zone; 5m aluminum heating plate)
Figure 1. Schematic diagram of the wind tunnel experiment. The flow f i e l d formed w i t h i n the wind tunnel was compared with p r o t o t y p e flow with s i m i l a r i t y c r i t e r i a s u g g e s t e d by A v i s s a r e t a l . (1990) ~'~. The f o l l o w i n g t h r e e dynamic s i m i l a r i t y c r i t e r i a were a p p l i e d in t h i s s t u d y : (1) bulk Richardson number, (2) s u r f a c e Reynolds number and (3) the r a t i o of s u r f a c e P e c l e t number to Richardson number. Although wind tunnel experiment c o n d i t i o n s , as well as numerical c a l c u l a t i o n c o n d i t i o n s i n d i c a t e d in the next s e c t i o n , were examined u s i n g t h e above s i m i l a r i t y c r i t e r i a , equivalent
699 p r o t o t y p e c o n d i t i o n s could not be determined s o l e l y because of a r b i t r a r y c o m b i n a t i o n s of p r o t o t y p e flow v a r i a b l e s . However t y p i c a l p r o t o t y p e flow c o n d i t i o n s might be s e l e c t e d within the s i m i l a r i t y c r i t e r i a , and these were wind v e l o c i t y 9 m/s, t e m p e r a t u r e d i f f e r e n c e between land and sea 5 °C and length scale r a t i o 500. These v e l o c i t y and temperature d i f f e r e n c e were within the r a n g e of t h o s e o b s e r v e d a t p r o t o t y p e flow. The l e n g t h s c a l e r a t i o obtained here was a l s o within the range of p o s s i b l e wind tunnel experiments. Reynolds number b a s e d on boundary l a y e r t h i c k n e s s was a b o u t 3.104 and t u r b u l e n t shear flow was developed on the wind tunnel floor.
3.
THE
MODEL
Flow c a l c u l a t i o n s using the a n i s o t r o p i c k - e model (4~ were performed. I n s t e a d of the i s o t r o p i c eddy v i s c o s i t y concept, Reynolds s t r e s s and h e a t flux were expressed in the following a n i s o t r o p i c forms in t h i s model.
-u.u, = ~,.e., -§k~., ( Gr _ + ~"
~
8T
Gr
~- ' "-F-~, + ~
8T
~,=Co ~ , e ,
l[SU,
+R'.,-1R'.~., 8T
2 Gr
~ '" g ; ;
a
+Kj. 8 T
OT
~Uj '
R"
8Uj 8U*X axQ a x , /
)
(1)
~ '" ~Z-;~ . ~ ' '
Gr
'-a-~x,÷-a~x,
8U~ axj
Re ~
Gr
• S,
"
*
~
"'
(2)
~ =C '
k"
"' ~
8Ua 8 U , ax~ a x j k ~
k i ~
S
'
.
8U, aU, '"
ax,
ax,
k2 c k~-i z
U, u , k , ~ , T and t are mean v e l o c i t y , v e l o c i t y f l u c t u a t i o n , t u r b u l e n t kinetic energy, its dissipation, mean t e m p e r a t u r e and t e m p e r a t u r e f l u c t u a t i o n , r e s p e c t i v e l y , g l ~ , Re and Gr a r e K r o n e c k e r d e l t a , R e y n o l d s number and Grashof number, r e s p e c t i v e l y . The model constants were C~= O. 0 9 , C~1=0.057, C,2=-0.33, C~8=-0.0067, Ok°p=0.14, CkA=0.032, C~,=0.03, C~2=0. 18 and C~3=0.014. The c o n t i n u i t y e q u a t i o n , momentums in t h r e e d i r e c t i o n s , t u r b u l e n t k i n e t i c energy, i t s d i s s i p a t i o n , mean temperature and i t s variance t r a n s p o r t equations were solved, as indicated below, r e s p e c t i v e l y .
700
aU,
- 0
9x~
__aut ,
(3)
= ---8x,SP~ a_~ { (_ u , u ,+l
__a u x, U,
8 k + OU~k a---t
3
aT+aU, T at ax, -
1 \ Ok ~ - -
a - - ~ , = ~a- - 7 t / ? 7 + ~ ) - a % - ~
a,s+aU,~:
-
~/ v,
-
a---F +
-
a
-
{(vt
a {(_t 8xJ
a ~---Z- = a x~
)e
,
(]rj "~.Re2 ( T - T o ) ~ , 3
8U,
j
+
-
Gr
F " ' " ' $-;7~,-c+R-P - t ,,, ~,~
1'~ az:: 1 _ ,s
/.l
(4)
au,
(5)
_ c = _ ,s Gr
1 a~T l ~ RePrt: 8 x , J -
-
(7)
-
R~P~t
a ~, J
~°-~-7~/-c''
k
(8)
The model constants are C~=l.O, C~=1.3, C~=1.44, C2=1.92, Cx,~o = 0 . 2 2 a n d C~o=1.8. Prt and To are turbulent Prandtl number and r e f e r e n c e temperature, r e s p e c t i v e l y . The treatment of the buoyancy term in Eq.(6) was empirical and was not determined universally. The model constant C3 was c h a n g e d w i t h t h e flow c h a r a c t e r i s t i c s . In this study, the model constant for the buoyancy term in the d i s s i p a t i o n equation was varied. The buoyancy e f f e c t s f o r momentum e q u a t i o n s were t r e a t e d under the assumption of a Boussinesq approximation. The grid number was set at 90.20.21 f o r windward, h o r i z o n t a l and v e r t i c a l d i r e c t i o n s , r e s p e c t i v e l y . These equations were solved in f i n i t e - d i f f e r e n c e forms by the following algorithm of the SMAC method. The i n l e t c o n d i t i o n s for t u r b u l e n t q u a n t i t i e s were set from the values measured in the wind tunnel. The wall function methods were a p p l i e d f o r t u r b u l e n t k i n e t i c energy and i t s d i s s i p a t i o n . The u n i v e r s a l f u n c t i o n methods i n d i c a t e d by f i e l d experiments were a l s o a p p l i e d as wall functions of v e l o c i t y and temperature as follows.
I-~ tl.
= r/ ~ 1-¢h d
Here, ~ and Ch a r e u n i v e r s a l f u n c t i o n s f o r v e l o c i t y and t e m p e r a t u r e , respectively, u . , t . , Tw and v a r e f r i c t i o n velocity, friction temperature, wall temperature and molecular v i s c o s i t y , r e s p e c t i v e l y . The yon
701 Karman constant ~ and constant B a r e 0 . 4 and 9 . 0 2 5 , r e s p e c t i v e l y . The e m p i r i c a l c o n s t a n t s , i n c l u d i n g u n i v e r s a l f u n c t i o n s , were o p t i m i z e d by v e l o c i t y and temperature p r o f i l e s in the wind tunnel. The f o l l o w i n g forms were used for universal functions: ~,
(13)
= (1-40~') - ' / 4
Ch = Prt
(1-25,~) - ' / z
,
Prt=0.84
;
(14)
is the d i s t a n c e from the wall with n o r m a l i z a t i o n by the Monin-Obukhov length L , =
4.
-
(i5)
-
L
RESULTS
AND
DISCUSSION
Calculation r e s u l t s of turbulent k i n e t i c energy, Reynolds s t r e s s and mean temperature with the model constant C, are shown in Figs. 2~4. In the case of a d e v e l o p i n g thermal i n t e r n a l boundary layer, when marine a i r flows over warmer land, m e c h a n i c a l s h e a r and buoyancy f o r c e both a c t to i n c r e a s e t u r b u l e n t k i n e t i c energy. This increase is a t y p i c a l f e a t u r e of a sea breeze and s h o u l d be s i m u l a t e d f o r s m a l l - s c a l e motions. Because the v a l u e of t u r b u l e n t heat flux was p o s i t i v e in the unstable thermal i n t e r n a l boundary layer, d i s s i p a t i o n tended to decrease and turbulent k i n e t i c energy increased when ti~e buoyancy term was eliminated (C3=0). In t h i s c a l c u l a t i o n case (C,=O), maximum t u r b u l e n t k i n e t i c energy and good agreement with those of the wind tunnel experiment were obtained. Although some approximations were applied, t u r b u l e n t length s c a l e was estimated a n a l y t i c a l l y a t almost the same flow f i e l d ~5). The turbulent length scale obtained in this study a t C,=O s h o w e d almost the same values as those from a n a l y t i c a l methods (5~. Reynolds s t r e s s and thermal i n t e r n a l boundary l a y e r t h i c k n e s s a l s o i n c r e a s e d in t h i s case (¢8=0). These i n c r e a s e s of Reynolds s t r e s s and thermal i n t e r n a l boundary layer thickness were due to the increase in turbulent k i n e t i c energy. The c a l c u l a t e d r e s u l t s of t u r b u l e n t k i n e t i c energy, Reynolds s t r e s s and mean temperature with wall functions are also shown in Figs. 2~4. Inclusions of the u n i v e r s a l f u n c t i o n i n d i c a t e d t h a t the Prandtl number was changed to 0.84 with the temperature of the wall f u n c t i o n . The mean temperature and thermal i n t e r n a l boundary layer thickness increased when universal functions Eq.(13) & (14) were used. Because ~h~O at E q . ( l l ) , the temperature near the wall tended to increase. Reynolds s t r e s s and t u r b u l e n t k i n e t i c energy a l s o i n c r e a s e d with the i n c l u s i o n of u n i v e r s a l f u n c t i o n s , Large i n c r e a s e s of turbulent k i n e t i c energy and Reynolds s t r e s s were obtained when the buoyancy
702
term was e l i m i n a t e d . A s m a l l improvement of t u r b u l e n t k i n e t i c e n e r g y was o b t a i n e d when u n i v e r s a l f u n c t i o n s were used. Because good agreement between c a l c u l a t i o n s and e x p e r i m e n t a l r e s u l t s was observed when the buoyancy term of dissipation was e l i m i n a t e d and t h e w a l l f u n c t i o n s i n c l u d i n g u n i v e r s a l f u n c t i o n s were used, the c a l c u l a t e d r e s u l t s f o r t h i s case a r e i n d i c a t e d in the f o l l o w i n g .
50
50
G=0' Eq"(13)&(14) 1 .......
40
40
G=l.44,Eq.(13)&(14)J
.......
x:4m
= 30
lO
10
0
0
k/Uoo
~
O. Ol
Figure 2. T u r b u l e n t k i n e t i c energy. (
1
20
0
~x~.~
f ~1
x:Sm
d; y;,
lo ,,
,~_....J
; ' " ~ ~
0.8
,
Figure 4. Mean temperature,
1 T~-T/T~-To I
•- \
0
flJ.~~~Oo ,
0
-u~ UJ
O. 004
Figure 3. Reynolds stress.
!;,0- C~=0'E(13)&(14) q" 1 I - -c~.+.:+~:, /c,Ll 30
x:4m
o
c4
"20
~20
0
G=l.44,Eq.(13)&(14)J EXP.
-~
~30
G~O'Eq"(13)~(14) 1 - - C3=0,4.=¢ a=l [CAL.
703 Calculated mean v e l o c i t y , and horizontal and v e r t i c a l turbulence i n t e n s i t y are shown in Figs. 5"-7, r e s p e c t i v e l y . Leeward p r o f i l e s of v e l o c i t y became f l a t t e r than those from calculations, and disagreement between c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s was observed in the v i c i n i t y of the f l o o r . In wind tunnel experiments, the temperature d i f f e r e n c e was about 50°C, and i t was presumed that a i r d e n s i t y v a r i a t i o n may not be n e g l i g i b l e in c o n t r a s t to i t s e l i m i n a t i o n in the c a l c u l a t i o n . Because an a n i s o t r o p i c model was used, the h o r i z o n t a l turbulence i n t e n s i t y was larger than the v e r t i c a l i n t e n s i t y . Both turbulence i n t e n s i t i e s decreased in the v i c i n i t y of the wall, and t h i s might be due to the turbulent k i n e t i c energy decrease indicated in Fig. 2. Although some disagreement between c a l c u l a t e d and experimental r e s u l t s was evident, the e x p e r i m e n t a l l y i n d i c a t e d f e a t u r e s of mean v e l o c i t y and t u r b u l e n t i n t e n s i t y were obtained.
50
o
40
x=lm
o
d
2m
o
c
8m
o,
c
4m
c
30 " 20
caL.
C
C
0
o
10
,
,
00
~
,
0
0
0
0,5
1
1.5
Figure 5. Mean velocity.
o/ o/ o,
40
0
0 \~X ' O,,,\ 2,,,
30 o \
/1
o/
,,, 20
~
o\ o
;:;:
I
00
0
0
0
5
10
15
Figure 6. Horizontal turbulence i n t e n s i t y .
20 ,/~-~/ uoo
u/uoo
704
50 o
o o m~
30
e ©
©
2O
CAl.. EXP.
I
O0
0
0
0
5
10
15
'2O
Figure 7. Vertical turbulence i n t e n s i t y .
5.
CONCLUS
I ONS
Wind tunnel e x p e r i m e n t s f o r a h e a t e d f l a t p l a t e were s i m u l a t e d by an a n i s o t r o p i c k - s model. The c a l c u l a t e d r e s u l t s showed good agreement with e x p e r i m e n t a l d a t a . Some d i s a g r e e m e n t was e v i d e n t in the v i c i n i t y of the wall. The treatment of t u r b u l e n t k i n e t i c energy and i t s d i s s i p a t i o n near the wall, i n c l u d i n g the wall f u n c t i o n s , should be modified when more a c c u r a t e c a l c u l a t i o n s are desired.
References 1 A v i s s a r , R., Moran, M.D., Wu, G., Meroney, R.N. and P i e l k e , R.A., Operating ranges of mesoscale numerical models and m e t e o r o l o g i c a l wind tunnels for the simulation of sea and land breezes, Boundary-layer Met., vol. 50(1990), p. 227. 2 Kitada, W., Turbulence s t r u c t u r e of sea breeze f r o n t and i t s i m p l i c a t i o n in a i r p o l l u t i o n t r a n s p o r t , Boundary-Layer Met., vol. 41(1987), p. 217. 3 Ngocly, L., An a p p l i c a t i o n of the E - s t u r b u l e n c e model f o r s t u d y i n g coupled a i r - s e a boundary-layer, Boundary-Layer Met., vol. 54(1991), p. 327. 4 Yoshizawa, A., S t a t i s t i c a l modeling of turbulent thermally buoyant flows, J. Phys. 8oc. J . , vol. 55, No. 9(1986), p. 3066. 5 8ada, K., Wind tunnel experiment of convective p l a n e t a r y boundary layer, Proc. Fluid Engineering Conf. '92, JSME(1992), p. 146.
S i m u l a t i o n Flat Plate
of Using
Air
Flow over A n i s o t r o p i c
K. S A D A Senior Research Engineer Y. I C H I K A W A P r i n c i p a l Research Engineer, Central Research I n s t i t u t e Industry, 2-11-1 Iwato-kita, Komae-shi, Tokyo, Japan
697
a Heated k-8 Model
of E l e c t r i c Power
Abstract Numerical simulation of air flow over a heated flat plate was performed using the anisotropic k - e model. The wall functions for velocity and temperature were modified using universal functions, and the buoyant term in the dissipation equation was eliminated. The calculated results were compared with those from wind tunnel experiments. The calculated mean velocity, turbulent intensity and mean temperature showed good agreement with profiles measured in the wind tunnel.
i.
I NTRODUCT
I ON
Air f l o w s from the sea to land a r e sometimes o b s e r v e d in c o a s t a l r e g i o n s . Flows are g r a d u a l l y heated, and thermal i n t e r n a l boundary l a y e r s developing leeward and reversed flows are observed above the thermal internal boundary l a y e r . A complete numerical and wind tunnel model to s i m u l a t e a sea b r e e z e should be a b l e to s i m u l a t e (1) the l a r g e - s c a l e flow motions i n c l u d i n g e l e v a t e d p o s i t i o n (2) the s m a l l - s c a l e s h e a r flow motions and t u r b u l e n t t r a n s p o r t formed mainly near the surface, and (3) the i n t e r a c t i o n between these two s c a l e s of motion ~'~. Such sea breezes have been simulated by numerical techniques ~2'~3', but the greater part of these simulations have been f o c u s e d on sea and land b r e e z e c i r c u l a t i o n , and t h e a c c u r a c y of c a l c u l a t e d r e s u l t s f o r s m a l l - s c a l e motions has not been c l a r i f i e d because of flow complications and the lack of f i e l d observation data. A wind tunnel e x p e r i m e n t s i m u l a t i n g such s m a l l - s c a l e motions o f t h e sea b r e e z e was p e r f o r m e d , and t h e e n s e m b l e - a v e r a g e wind f i e l d was s i m u l a t e d by an a n i s o t r o p i c k - e model in this study. Turbulence q u a n t i t i e s were more e a s i l y obtained in the wind tunnel as compared with f i e l d experiments, and a more accurate evaluation of flow prediction was possible.
0167-6105/93/$06.00 © 1993 - Elsevier SciencePublishers B.V. All fights reserved.
698
2.
WIND
TUNNEL
EXPERIMENT
The e x p e r i m e n t r e p o r t e d here was conducted using the wind t u n n e l ( t e s t s e c t i o n dimensions are 20m length, 3m width, 1.5m h e i g h t ) a t CRIEPI. The t e s t s e c t i o n f l o o r was h e a t e d e l e c t r i c a l l y to form an u n s t a b l e thermal i n t e r n a l boundary l a y e r ( s e e F i g u r e 1). An e l e c t r i c h e a t e r , d e s i g n e d to p r o v i d e an e l e v a t e d s t a b l e temperature g r a d i e n t (9 °C/m), was a l s o s e t a t the e n t r a n c e of the wind tunnel t e s t s e c t i o n . Such an e l e v a t e d s t a b l e s t r a t i f i c a t i o n can a l s o be observed in the atmosphere, and r e s t r i c t s secondary flow i n h e r e n t in the device. The f r e e stream v e l o c i t y of U~=l.3m/s was s e l e c t e d to o b t a i n a t h i c k thermal i n t e r n a l boundary l a y e r and to reduce secondary flow. Roughness elements were s e t a t the e n t r a n c e of the wind tunnel t e s t s e c t i o n f l o o r to o b t a i n a t h i c k b o u n d a r y l a y e r and s t i m u l a t e t h e flow n e a r t h e s u r f a c e . Heating of the wind tunnel f l o o r was s t a r t e d 4m leeward of the e n t r a n c e of the t e s t s e c t i o n . The a i r flow in the wind tunnel blew from the " s e a a r e a " (where t h e f l o o r was n o t h e a t e d ) to t h e " l a n d a r e a " (where t h e s u r f a c e t e m p e r a t u r e was i n c r e a s e d almost u n i f o r m l y up to 80 °C). Wind v e l o c i t y and t e m p e r a t u r e were measured by a l a s e r Doppler anemometer and a r e s i s t a n c e thermometer.
Wind
To = 27 °C
U=, = 1, 3m/s
Stable /
Boundary "/
Thermal internal
strati-/ fication
f Sea
~
area
(Unheated zone)
=
Land
X
area
Heated zone; 5m aluminum heating plate)
Figure 1. Schematic diagram of the wind tunnel experiment. The flow f i e l d formed w i t h i n the wind tunnel was compared with p r o t o t y p e flow with s i m i l a r i t y c r i t e r i a s u g g e s t e d by A v i s s a r e t a l . (1990) ~'~. The f o l l o w i n g t h r e e dynamic s i m i l a r i t y c r i t e r i a were a p p l i e d in t h i s s t u d y : (1) bulk Richardson number, (2) s u r f a c e Reynolds number and (3) the r a t i o of s u r f a c e P e c l e t number to Richardson number. Although wind tunnel experiment c o n d i t i o n s , as well as numerical c a l c u l a t i o n c o n d i t i o n s i n d i c a t e d in the next s e c t i o n , were examined u s i n g t h e above s i m i l a r i t y c r i t e r i a , equivalent
699 p r o t o t y p e c o n d i t i o n s could not be determined s o l e l y because of a r b i t r a r y c o m b i n a t i o n s of p r o t o t y p e flow v a r i a b l e s . However t y p i c a l p r o t o t y p e flow c o n d i t i o n s might be s e l e c t e d within the s i m i l a r i t y c r i t e r i a , and these were wind v e l o c i t y 9 m/s, t e m p e r a t u r e d i f f e r e n c e between land and sea 5 °C and length scale r a t i o 500. These v e l o c i t y and temperature d i f f e r e n c e were within the r a n g e of t h o s e o b s e r v e d a t p r o t o t y p e flow. The l e n g t h s c a l e r a t i o obtained here was a l s o within the range of p o s s i b l e wind tunnel experiments. Reynolds number b a s e d on boundary l a y e r t h i c k n e s s was a b o u t 3.104 and t u r b u l e n t shear flow was developed on the wind tunnel floor.
3.
THE
MODEL
Flow c a l c u l a t i o n s using the a n i s o t r o p i c k - e model (4~ were performed. I n s t e a d of the i s o t r o p i c eddy v i s c o s i t y concept, Reynolds s t r e s s and h e a t flux were expressed in the following a n i s o t r o p i c forms in t h i s model.
-u.u, = ~,.e., -§k~., ( Gr _ + ~"
~
8T
Gr
~- ' "-F-~, + ~
8T
~,=Co ~ , e ,
l[SU,
+R'.,-1R'.~., 8T
2 Gr
~ '" g ; ;
a
+Kj. 8 T
OT
~Uj '
R"
8Uj 8U*X axQ a x , /
)
(1)
~ '" ~Z-;~ . ~ ' '
Gr
'-a-~x,÷-a~x,
8U~ axj
Re ~
Gr
• S,
"
*
~
"'
(2)
~ =C '
k"
"' ~
8Ua 8 U , ax~ a x j k ~
k i ~
S
'
.
8U, aU, '"
ax,
ax,
k2 c k~-i z
U, u , k , ~ , T and t are mean v e l o c i t y , v e l o c i t y f l u c t u a t i o n , t u r b u l e n t kinetic energy, its dissipation, mean t e m p e r a t u r e and t e m p e r a t u r e f l u c t u a t i o n , r e s p e c t i v e l y , g l ~ , Re and Gr a r e K r o n e c k e r d e l t a , R e y n o l d s number and Grashof number, r e s p e c t i v e l y . The model constants were C~= O. 0 9 , C~1=0.057, C,2=-0.33, C~8=-0.0067, Ok°p=0.14, CkA=0.032, C~,=0.03, C~2=0. 18 and C~3=0.014. The c o n t i n u i t y e q u a t i o n , momentums in t h r e e d i r e c t i o n s , t u r b u l e n t k i n e t i c energy, i t s d i s s i p a t i o n , mean temperature and i t s variance t r a n s p o r t equations were solved, as indicated below, r e s p e c t i v e l y .
700
aU,
- 0
9x~
__aut ,
(3)
= ---8x,SP~ a_~ { (_ u , u ,+l
__a u x, U,
8 k + OU~k a---t
3
aT+aU, T at ax, -
1 \ Ok ~ - -
a - - ~ , = ~a- - 7 t / ? 7 + ~ ) - a % - ~
a,s+aU,~:
-
~/ v,
-
a---F +
-
a
-
{(vt
a {(_t 8xJ
a ~---Z- = a x~
)e
,
(]rj "~.Re2 ( T - T o ) ~ , 3
8U,
j
+
-
Gr
F " ' " ' $-;7~,-c+R-P - t ,,, ~,~
1'~ az:: 1 _ ,s
/.l
(4)
au,
(5)
_ c = _ ,s Gr
1 a~T l ~ RePrt: 8 x , J -
-
(7)
-
R~P~t
a ~, J
~°-~-7~/-c''
k
(8)
The model constants are C~=l.O, C~=1.3, C~=1.44, C2=1.92, Cx,~o = 0 . 2 2 a n d C~o=1.8. Prt and To are turbulent Prandtl number and r e f e r e n c e temperature, r e s p e c t i v e l y . The treatment of the buoyancy term in Eq.(6) was empirical and was not determined universally. The model constant C3 was c h a n g e d w i t h t h e flow c h a r a c t e r i s t i c s . In this study, the model constant for the buoyancy term in the d i s s i p a t i o n equation was varied. The buoyancy e f f e c t s f o r momentum e q u a t i o n s were t r e a t e d under the assumption of a Boussinesq approximation. The grid number was set at 90.20.21 f o r windward, h o r i z o n t a l and v e r t i c a l d i r e c t i o n s , r e s p e c t i v e l y . These equations were solved in f i n i t e - d i f f e r e n c e forms by the following algorithm of the SMAC method. The i n l e t c o n d i t i o n s for t u r b u l e n t q u a n t i t i e s were set from the values measured in the wind tunnel. The wall function methods were a p p l i e d f o r t u r b u l e n t k i n e t i c energy and i t s d i s s i p a t i o n . The u n i v e r s a l f u n c t i o n methods i n d i c a t e d by f i e l d experiments were a l s o a p p l i e d as wall functions of v e l o c i t y and temperature as follows.
I-~ tl.
= r/ ~ 1-¢h d
Here, ~ and Ch a r e u n i v e r s a l f u n c t i o n s f o r v e l o c i t y and t e m p e r a t u r e , respectively, u . , t . , Tw and v a r e f r i c t i o n velocity, friction temperature, wall temperature and molecular v i s c o s i t y , r e s p e c t i v e l y . The yon
701 Karman constant ~ and constant B a r e 0 . 4 and 9 . 0 2 5 , r e s p e c t i v e l y . The e m p i r i c a l c o n s t a n t s , i n c l u d i n g u n i v e r s a l f u n c t i o n s , were o p t i m i z e d by v e l o c i t y and temperature p r o f i l e s in the wind tunnel. The f o l l o w i n g forms were used for universal functions: ~,
(13)
= (1-40~') - ' / 4
Ch = Prt
(1-25,~) - ' / z
,
Prt=0.84
;
(14)
is the d i s t a n c e from the wall with n o r m a l i z a t i o n by the Monin-Obukhov length L , =
4.
-
(i5)
-
L
RESULTS
AND
DISCUSSION
Calculation r e s u l t s of turbulent k i n e t i c energy, Reynolds s t r e s s and mean temperature with the model constant C, are shown in Figs. 2~4. In the case of a d e v e l o p i n g thermal i n t e r n a l boundary layer, when marine a i r flows over warmer land, m e c h a n i c a l s h e a r and buoyancy f o r c e both a c t to i n c r e a s e t u r b u l e n t k i n e t i c energy. This increase is a t y p i c a l f e a t u r e of a sea breeze and s h o u l d be s i m u l a t e d f o r s m a l l - s c a l e motions. Because the v a l u e of t u r b u l e n t heat flux was p o s i t i v e in the unstable thermal i n t e r n a l boundary layer, d i s s i p a t i o n tended to decrease and turbulent k i n e t i c energy increased when ti~e buoyancy term was eliminated (C3=0). In t h i s c a l c u l a t i o n case (C,=O), maximum t u r b u l e n t k i n e t i c energy and good agreement with those of the wind tunnel experiment were obtained. Although some approximations were applied, t u r b u l e n t length s c a l e was estimated a n a l y t i c a l l y a t almost the same flow f i e l d ~5). The turbulent length scale obtained in this study a t C,=O s h o w e d almost the same values as those from a n a l y t i c a l methods (5~. Reynolds s t r e s s and thermal i n t e r n a l boundary l a y e r t h i c k n e s s a l s o i n c r e a s e d in t h i s case (¢8=0). These i n c r e a s e s of Reynolds s t r e s s and thermal i n t e r n a l boundary layer thickness were due to the increase in turbulent k i n e t i c energy. The c a l c u l a t e d r e s u l t s of t u r b u l e n t k i n e t i c energy, Reynolds s t r e s s and mean temperature with wall functions are also shown in Figs. 2~4. Inclusions of the u n i v e r s a l f u n c t i o n i n d i c a t e d t h a t the Prandtl number was changed to 0.84 with the temperature of the wall f u n c t i o n . The mean temperature and thermal i n t e r n a l boundary layer thickness increased when universal functions Eq.(13) & (14) were used. Because ~h~O at E q . ( l l ) , the temperature near the wall tended to increase. Reynolds s t r e s s and t u r b u l e n t k i n e t i c energy a l s o i n c r e a s e d with the i n c l u s i o n of u n i v e r s a l f u n c t i o n s , Large i n c r e a s e s of turbulent k i n e t i c energy and Reynolds s t r e s s were obtained when the buoyancy
702
term was e l i m i n a t e d . A s m a l l improvement of t u r b u l e n t k i n e t i c e n e r g y was o b t a i n e d when u n i v e r s a l f u n c t i o n s were used. Because good agreement between c a l c u l a t i o n s and e x p e r i m e n t a l r e s u l t s was observed when the buoyancy term of dissipation was e l i m i n a t e d and t h e w a l l f u n c t i o n s i n c l u d i n g u n i v e r s a l f u n c t i o n s were used, the c a l c u l a t e d r e s u l t s f o r t h i s case a r e i n d i c a t e d in the f o l l o w i n g .
50
50
G=0' Eq"(13)&(14) 1 .......
40
40
G=l.44,Eq.(13)&(14)J
.......
x:4m
= 30
lO
10
0
0
k/Uoo
~
O. Ol
Figure 2. T u r b u l e n t k i n e t i c energy. (
1
20
0
~x~.~
f ~1
x:Sm
d; y;,
lo ,,
,~_....J
; ' " ~ ~
0.8
,
Figure 4. Mean temperature,
1 T~-T/T~-To I
•- \
0
flJ.~~~Oo ,
0
-u~ UJ
O. 004
Figure 3. Reynolds stress.
!;,0- C~=0'E(13)&(14) q" 1 I - -c~.+.:+~:, /c,Ll 30
x:4m
o
c4
"20
~20
0
G=l.44,Eq.(13)&(14)J EXP.
-~
~30
G~O'Eq"(13)~(14) 1 - - C3=0,4.=¢ a=l [CAL.
703 Calculated mean v e l o c i t y , and horizontal and v e r t i c a l turbulence i n t e n s i t y are shown in Figs. 5"-7, r e s p e c t i v e l y . Leeward p r o f i l e s of v e l o c i t y became f l a t t e r than those from calculations, and disagreement between c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s was observed in the v i c i n i t y of the f l o o r . In wind tunnel experiments, the temperature d i f f e r e n c e was about 50°C, and i t was presumed that a i r d e n s i t y v a r i a t i o n may not be n e g l i g i b l e in c o n t r a s t to i t s e l i m i n a t i o n in the c a l c u l a t i o n . Because an a n i s o t r o p i c model was used, the h o r i z o n t a l turbulence i n t e n s i t y was larger than the v e r t i c a l i n t e n s i t y . Both turbulence i n t e n s i t i e s decreased in the v i c i n i t y of the wall, and t h i s might be due to the turbulent k i n e t i c energy decrease indicated in Fig. 2. Although some disagreement between c a l c u l a t e d and experimental r e s u l t s was evident, the e x p e r i m e n t a l l y i n d i c a t e d f e a t u r e s of mean v e l o c i t y and t u r b u l e n t i n t e n s i t y were obtained.
50
o
40
x=lm
o
d
2m
o
c
8m
o,
c
4m
c
30 " 20
caL.
C
C
0
o
10
,
,
00
~
,
0
0
0
0,5
1
1.5
Figure 5. Mean velocity.
o/ o/ o,
40
0
0 \~X ' O,,,\ 2,,,
30 o \
/1
o/
,,, 20
~
o\ o
;:;:
I
00
0
0
0
5
10
15
Figure 6. Horizontal turbulence i n t e n s i t y .
20 ,/~-~/ uoo
u/uoo
704
50 o
o o m~
30
e ©
©
2O
CAl.. EXP.
I
O0
0
0
0
5
10
15
'2O
Figure 7. Vertical turbulence i n t e n s i t y .
5.
CONCLUS
I ONS
Wind tunnel e x p e r i m e n t s f o r a h e a t e d f l a t p l a t e were s i m u l a t e d by an a n i s o t r o p i c k - s model. The c a l c u l a t e d r e s u l t s showed good agreement with e x p e r i m e n t a l d a t a . Some d i s a g r e e m e n t was e v i d e n t in the v i c i n i t y of the wall. The treatment of t u r b u l e n t k i n e t i c energy and i t s d i s s i p a t i o n near the wall, i n c l u d i n g the wall f u n c t i o n s , should be modified when more a c c u r a t e c a l c u l a t i o n s are desired.
References 1 A v i s s a r , R., Moran, M.D., Wu, G., Meroney, R.N. and P i e l k e , R.A., Operating ranges of mesoscale numerical models and m e t e o r o l o g i c a l wind tunnels for the simulation of sea and land breezes, Boundary-layer Met., vol. 50(1990), p. 227. 2 Kitada, W., Turbulence s t r u c t u r e of sea breeze f r o n t and i t s i m p l i c a t i o n in a i r p o l l u t i o n t r a n s p o r t , Boundary-Layer Met., vol. 41(1987), p. 217. 3 Ngocly, L., An a p p l i c a t i o n of the E - s t u r b u l e n c e model f o r s t u d y i n g coupled a i r - s e a boundary-layer, Boundary-Layer Met., vol. 54(1991), p. 327. 4 Yoshizawa, A., S t a t i s t i c a l modeling of turbulent thermally buoyant flows, J. Phys. 8oc. J . , vol. 55, No. 9(1986), p. 3066. 5 8ada, K., Wind tunnel experiment of convective p l a n e t a r y boundary layer, Proc. Fluid Engineering Conf. '92, JSME(1992), p. 146.