J. of Thermal Science Vol.l, No.3

Journal of Thermal Science Science Press 1992

Turbulent Mixed Convection-Radiation a Cavity Cheng-Xian Lin

I n t e r a c t i o n s in

Ming-Dao Kin

Institute of Engineering Thermophysics, Chongqing University, Chongqing 630044, China

Two-dimensional turbulent mixed convection-surface radiation interaction phenomenon in a back-wall-heated open square cavity is numerically investigated. The flow medium is air, and the turbulence model used is t h e Low-Reynolds-Number k - c Scheme. Calculations have been performed for Grashof numbers Gr up to 101°. And the Richardson number Ri covers a range of 4 × 10-~ - 45. It is shown that within a rather extensive range of Ri, the effects of radiation on the heat transfer and fluid flow in the cavity are significant, should not be neglected and become stronger with the increase of Ri and Gr. Keywords:

turbulent

flow, m i x e d c o n v e c t i o n , surface r a d i a t i o n , o p e n cavity.

INTRODUCTION In the past several decades, the problem of heat transfer to flows past heated open surface cavities has received considerable attention because of its importance in m a n y fields of science and engineering applications, such as ventilation and fire control in buildings, central solar collectors, cooling of electronic equipment, roughness element in heat exchangers, notches in turbine,flow passages and combustion chambers. A great m a n y numerical and experimental studies have been made on this subject, and these studies have contributed much to the undersdanding of the laminar or turbulent, mixed or forced convective transfer processes in cavities. Recent publications include the studies by Amon and Mikic {1], H u m p h r e y and To {2], B h a t t i and Aung [3], Gooray et al[ 4], Glasing et al [5], Showole and Tarasuk [6] , and Richards et al [7], etc. When reviewing the previous literature on this subject, the authors found that though combined convection and radiation exists in m a n y practical situations, almost all the exsiting numerical investigations were conducted under simplified t h e r m a l boundary conditions and ignoring the effects of radiation. To our knowlege, the turbulent mixed or forced convection coupled with radiation in an open cavity h a s not been investigated before. This p a p e r is an effort to bridge, in part, the inforReceived May, 1992.

mation gap existing in the studies on the heat transfer in flow past an open cavity• The interactions between turbulent mixed convection and surface radiation in a two-dimensional open cavity are numerically modeled and analysed. Emphasis is focused on the radiation fraction in the combined complex heat transfer and radiation effects on the convective heat transfer and fluid flow in a cavity.

MATHEMATICAL

MODEL

The two-dimensional open cavity u n d e r question and the c o m p u t a t i o n domain is shown in Fig.l. The cavity consists of a vertical isothermal back wall, two horizontal adiabatic walls (top and b o t t o m walls) and two vertical adiabatic extended walls. All the surfaces are assumed to be diffuse and grey, and of the same emissivity e. The externally imposed flow, with velocity UA and t e m p e r a t u r e 0A, sweeps past the heated cavity. The inertial force competes against buoyancy force for dominating the flow in the cavity. The flow medium air is considered to be transparent, Newtonian, compressible and property-variable (except specific heat}. The Low-Reynolds-Number k - v turbulence model is based on the scheme developed by Jones and Launder Is] , with buoyancy effect taken into account. When' the viscous dissipation and the pressure work in t h e energy equation are neglected, the equations describing the mean flow of the steady turbulent

190

J o u r n a l o f T h e r m a l Science, Vol. 1, No.3, 1992

e

Fi-i # Gr H J k N

Nomenclature emissivity shape factor gravitational acceleration G rashof number,

X ¢

Oo.'U'Sp'~2/ ."2

8o

#

cavity height radiosity turbulent kinetic energy n u m b e r of micro surfaces on the caviW b o u n d a r y average Nueselt n u m b e r coordinate direction perpendicular to a b o u n d a r y pressure Prandtl number heat flux gas constant Reynolds number, [/AHpA/I~A Richardson number, Gr / Re 2 Stanton number, Nu/(RePr) velocity velocity of the externally imposed flow

Nu n P Pr q R Re Ri St U UA

/~ p

o' ~0 * n

rectangular coordinate isotropic dissipation of the turbulent kinetic energy radiation fraction (%) temperature overheat ratio thermal conductivity viscosity density Stefan-Bolt f.mann constant stream function Superscript dimensional quantity iteration n u m b e r Subscrlpts

A c cp E h i,j 1 r t

ambient condition mixed convection part pure mixed convection hydrostatic condition heated back wall spatial indices laminar quantity radiation turbulent quantity

1

a

~ f l I l B [Ill [[}IfffnIIFF ' >J

a

ax~(Pu"u') ~ [ +

..au,

I'"

a~.)l

2

0

aa (puye)= ~a [(~,IPr+#,l(r,)~-~y] e

Oa

L#

ii

7

a x j (pUiK) =

-~ "P ~

(~, + ~ , / , k )

+ "'~-~j + ax, ax~

,akll2,~

3x

(4)

- p,

00

-2#,[-~-f-y ) 4- Gi(1 4- O°O)-zPto'eaX,

~'-S =~=

aP

(5)

aU~ a~ja(pUy.)=~a[(.,+.,/..)aa~]-~c16,YP.ax ' 2

--

Fig. 1 Physical problem mixed convection in Cartesian sionless form can be written

coordinates

as:

p(1 4- 0o0) = M2P 4- p= 0

ax~ (P~')

= o

in dimen-

,2

(1) (2)

a=u~

- C 2 p - ~ 4- 2pd~t ( a x i a x . , ) ' / p

.ok ax, where i,3",m =

1,2.

M 2 = g*H*Oo/(R*O~),l~t

(6) =

C.pk2/s,G, = g*/g*,p= = exp[-g*H'X~/(R*O~)].

Cheng-Xian Lin et al.

Turbulent Mixed Convection-Radiation Interactions in a Cavity

The dimensionless time-averaged variables are defined as

x~. x , = -if:,

o =

o* * , - o~.,

Oo =

o~

o2

The averaged overall Nusselt number based on q = q° + q, for the back wall of the cavity are defined as

N u = Nu~ + N u t _

p = -PA ;-, p

=

P*-

p~ = --;-, PA

P~

p'Ag*H*Oo'

g*H*Oo'

#l = p.AH.(g.H.Oo)t/2,

e'H*

e=

(g*H*Oo)3/2

#t = p ~ H . ( g . H . O o ) l / 2

The transport properties used are taken from Ref.[9]. The empirical coefficients C~,, C1, (72, ak, a , and ors are chosen from the values recommended by Launder and Sharma [1°] . As it is well known that there still isn't a suitable specific value for the coefficient C3, the expression recommended in the recent study by L i n e t al[lll is adopted for this study. The boundary cond~Hons for the problem can be stated as

£

(Ul,k,6)

=0

=0

, U 2 -= - - U A ,

(8)

on free boundaries '2' and '3'

Ui = k = e = O ,

0=1,

(9)

on heated back wall

00 NrCrq~ On + 0 - - - - 7 - = O,

(10)

on adiabatic walls where Cr = 0 for pure mixed convection and Cr = 1 for combined mixed convection and radiation. N r = a*O*A3H*/A *. Taking q~ = • 0A ,4 as a reference parameter, the dimensionless net radiation heat flux qr and radiosity J on one micro-surface si are given by q,*i(si)

q,,(s,)- ~'o7 -

=forK

"---'---' { [ ( 1 + O o O ( . , ) ] ' - J , ( s , ) } 1 - ei

(11)

Ji(si) = J ; ( s i ) [1 + OoO(s,)]' 0r.0--~A4A 4 = ei N + ( i - e,) E g j ( s j ) F , - j j=l

:ox~ q: dX~

~'~ (o;, - o~,)

_~l

+ fo t -~oAq~dX2

(13)

where NrA = a*O~3H*/A*A,K = A*/A~. For pure mixed convection, the convective Nusselt number Nuc = Nucp. The fraction of radiation can be represented by ~ -- N u r / N u x 100%. To represent the flow field, a stream function is defined as follows:

¢(x~, x~) =

/?

p(xl, x?)u~(Xl, x,)dx~

o

+¢(Xo,X2)

(i4)

where ¢(Xo, X2) is assumed to be zero along the vertical walls.

NUMERICAL

(U,,Oo~t,k,d = O, 0 , . = 0 ,

+

(7)

on free boundary 'I'

£

foX~ q: d X ~

~', (o;, - e~,)

U~ = ( 9 . H . 0 o ) ~ 1 ~

k = ~ k*

191

(12)

SCHEME

The formuation of the finite-difference equations is based on the micro-control volume scheme introduced by Patankar and Spalding[12]. An upwind differencing scheme and a central differencing scheme are used for the convective term and the diffusive term, respectively. The resulting finite-difference equations are solved by tri-diagonal matrix algorithm with variables updating according to four llne-by-line sweeps over the celLs in the calculation domain. The SIMPLE itera,' ~ve algorithm of Patankar [131 is used for pressure field. The Gauss elimination m e t h o d is employed to solve Eq.(12) to obtain J and then q, from Eq.(11). To yield well stable results, special linearization techniques are used for source terms in Ui, k and • equations, and the underrelaxation methods are employed for the physical variables with the Rnderrelaxation factors equal to 0.4-0.6. In the near-wall region, a fine-grid is adopted with about six nodes placed within the viscous sublayer region. Numerical explorations showed that for the specially designed nonuniform grid system employed, about 60)<80 - 90 × 130 nodes in the )(1 and X2 directions respectively (with about 32 × 32 - 42 x 42 nodes contained in the cavity) are sufficient to yield grid-independent cavity flow calculations. The greater node numbers are for higher Ra and higher Gr, and the reverse is true for lower Ra and Gr.

192

Journal of Thermal Science, Vol. 1, No.3, 1992

The criteria for convergence of the numerical calculations are that the residuals of mass, momentum, energy, k and e equations are less than 10 -3 respectively, and that the following conditions are simultaneously satisfied:

I¢'"+1

¢"~1 _< lO-4,

-

~bmax

[Nu "+1 - Nu"[ Nu-+X

_

5 x

10 - s

(15)

where ~ represents (Ui~fi) 1/2, 0, k &nd • in the cavity. The basic computer code has been carefuly examined against the previously published numerical and experimental results for turbulent free and mixed convection in open cavities. Some details of the validity studies have been reported by Lin et al.[11] and will not be repeated here. The outcome of this examination is the conclusion that, using the numerical model described above, satisfactory results essentially free of any serious numerical inaccuracy can be abtained. All calculations are performed on the FACOM M340s machine in the computer center of Chongqing University. Typically, about 700-1300 iterations and 4-8h of CPU time are needed to get converged numerical results.

RESULTS

AND DISCUSSION

The following results have been obtained under the condition of 0o -- 1.5. For combined turbulent mixed convection and surface radiation, e -- 0.9.

(i) F l o w C h a r a c t e r i s t i c s Fig.2 shows the streamlines and isotherms at G r =

1.41 x l0 s, Ri = 44.6 with and without radiation taken into consideration. The cavity is occupied by a large recirculating flow zone. As the radiation alters the thermal conditions along the adiabatic walls, the turbulent bouyancy-driving force and then the turbulent mixed convection in the cavity will be affected. Radiation makes the thermal boundary layer along the cavity wall become thicker, and the hot air region inside and outside the cavity larger. So, the bulk temperature in the entire computation domain for the combined convection and radiation is higher than that for pure convection. Fig.2 also shows that when radiation is taken into account, the protuberance phenomenon of the streamlines appears to be more obvious, so the shearing action of the externally imposed flow on the large recirculation flow zone is enhanced. The comparison of the streamlines and isotherms with and without radiation at G r = 1.41 × 101°, Ri = 44.6 is given in Fig.3. Comparing Fig.3 with Fig.2, we see that increasing Gr, the bouyant force increases, the flow in the cavity has more e~;ident nature of free convection. For the conditions of Fig.3, when radiation is considered, the hot air leaving the cavity loses some tendency to be immediately redirected by the external cold air flow, and an obvious free shearing layer is formed outside the cavity. Due to the presence of radiation effect, the number of the recirculating zones or eddies in or about the cavity is reduced, and the vorticity of the fluid flow in the cavity is changed. In distinction to Fig.3(b), Fig.2 and Fig.3(a) show that some hot air that has left the cavity returns there to participate in heat transfer cyclically. Fig.4 gives the streamlines and isotherms with and without radiation at Gr = 1.41 × 10 l°, Ri = 1.41 × 10 -2. As R i is relatively small, the rotating flow in

i___-2

(a) Pure Convection

(b) Convection-Radiation

Fig.2 Streamlines and isotherms for oper cavity with Gr = 1.41 × 10s, Ri = 44.6. 0.03 < o < 1.o, A0 = 0.o6; ~b >_0,,.,~ = 3.2 x lO-S;~b < 0, A~b = 8.8 x 10-s

Cheng-Xian L i n e t al.

Turbulent Mixed Convection-Radiation Interactions in a Cavity

//li k__i

(a) Pure

Convection

193

(b) Convection-Radiation

F l g . $ Streamlines and i s o t h e r m s for o p e n cavity with G r = 1.41 × 1 0 z ° , R i = 44.6. 0.03 < 8 < 1.0, A0 -: 0.06. (a) ~ _ > - 6 x 10 - 4 , A ¢ = 2.85x 10-3;~ <-6x 10 - 4 ' A ~ = 1 . 5 x 10 - 2 . (b) ~b_> 0, A~b = 1 . 8 5 x 10-2;~b < 0, A @ = 1 . 4 x 1 0 - 2 .

e

-

I

°I

(a) Pure Convection

(b) Convection-Radiation

F i g . 4 Streamlines and isotherms for o p e n cavity with G r = 1.41 x 101°, R i = 1.41 x 10 - 2 . 0.03 < 0 < 1.0,

Ae = 0.03.,~ > o,A~ = 3.31 x 10-2;,~ < o, (a) A,p = 0.83; (b) A~ = 1.24. the cavity is dominated by inertial forces, and of the nature of the wan-driven cavity flows. The streamlines about the opening are shrinking inside. The hot air recirculates along the periphery of the cavity from which it escapes mainly through molecular and turbuent diffusion. In the most part of the cavity, the temperature gradient is relatively small. The effect of radiation on the flow field is rather weak. The radiation further keeps the air about the cavity walls hotter, and makes the small cold-aft region near the b o t t o m wall become smaller and even disappear. When Ri is further reduced, the radiation effect on the temperature field also becomes very weak. Numerical computations show that high levels of turbulent kinetic energy k are observed in regions about the heated back wall and in the aperture plane

where shearing action is intense. P a r t of the turbulence in the aperture plane goes into the cavity by convection, the remainder being driven downstream. The effects of radiation are to make the maximum value of k greater, the size of the higher level k zones bigger, and the number of the higher level k zones larger. (ii) H e a t T r a n s f e r The variations of Nusselt numbers and the radiation fraction with Ri are shown in Fig.5 where the values at ( P r R i ) - I = 10 -2 are for the case of free convection, It should be mentioned that for the case of free convection, when Gr = 10s, radiation reduces convective heat transfer; when G r = 101°, radiation enhances free convective heat transfer. Fig.5 shows that for fixed R i , N u is much higher

194

Journal of Thermal Science, Vol. 1, No.3, 1992

than Nuc and NUep. For Gr = 1.41× l0 s, the effect of radiation is to reduce the mixed convective heat transfer, and this effect becomes weaker with the decrease of Ri. For Gr = 1.41× 101°, when Ri > 22.3, the effect of radiation is to enhance the mixed convective heat transfer, and the higher the Ri, the stronger the effect; when Ri < 22.3, the effect of radiation is to weaken the convective heat transfer, and with the decrease of Ri, the effect becomes stronger if 22.3 > Ri > 1.41, and weaker if Ri < 1.41. The changes in convective heat transfer due to the consideration of radiation at Gr = 1.41 x 101° are more intense than at G r = 1.41× 10 s. 90

10 6

higher than 9.98 x 10 - 3 and 1.49 x 10 - 2 respectively, r / > 50%, the dominant mechanism of heat transfer is radiation. And from the tendency of Y7at small Ri, we can estimate t h a t when the Richardson numbers are lower than 5.3 x 10 - 4 and 9.98 x 10 - 4 for lower and higher Gr, respectively, ~7 < 10%, the effects of radiation can be neglected, and the dominant m e c h a h i s m of heat transfer is forced convection. When Ri < 4.46 × 10 - 2 , the influence of forced convection is significant, and the following heat transfer correlations are obtained:

N u = 158.20Ri -°'149, Nuc

8O

6O

Gr

=

1.41

x

lOS°~N~k .

Nuc = 63.39Ri -°'672,

dO 30 20

10 =

'

10-=

*

I

lO-S

10 o

I

10 L

,

Gr

~

I08

!

10 2

10 l0 s

(/,rRq -~

(16)

N u = 559.81Ri -°'221,

50 g-

G r = 1.41 x l ~

29.39Ri -°'3e9,

G r = 1.41 x l 0 s

7O

10 4

=

= 1.41 x 10 l °

(17)

These expressions can be rewritten in terms of Stanton numbers: St = 13.71R/-°'7°3,

St¢ = 4.05 x lO-2Ri-°'? 01, Gr

= 1.41 x 10 s

(I8)

St = 4.507Ri -°'Sss, Fig.5

V a r i a t i o n s of a v e r a g e N u s s e l t n u m b e r s a n d r a d i a t i o n fraction with Richardson number. - - - Nuts, -- Nu¢, ....

Nu,

-.-

,7.

Fig.5 also shows that the variations of Nu, Nuc, Nucp and ,7 with Ri at Gr = 1.41 × 101° is more rapid than at Gr = 1.41 x 10s. For fixed Gr, when Ri is large, minimum Nusselt numbers Nu, Nuc,Nucp, and m a x i m u m radiation fraction ~7 emerge. This phenomenon can be explained as follows. As the freestream velocity initially increases, hot air discharged from the cavity is swept downwards in the free-stream direction and partly re-entrained into the cavity. Consequently, the average t e m p e r a t u r e of stratified air in the cavity is raised, resulting in reduced convective heat transfer from the cavity wall compared with the case of free convection. Further increases in the frees t r e a m velocity induce a recirculating flow inside the cavity which gradually destroys the stable stratification. Eventually, the flow and heat transfer patterns acquire the characteristics of a shear-driven, forced convection flow with large t u r b u h n t fluxes. The Ri at which the above extreme values emerge depends on

Gr. For the case of lower and higher Grashof numbers shown in Fig.5, when the Richardson numbers are

Stc = 1.40 x 1 0 - 4 R i -°'144, Gr = 1.41 x 10 l°

(19)

It should be noticed that within the Ri range investigated in this study, the radiation fraction in combined heat transfer is high. T h e main reason of this is that the emissivity of all the surfaces and the overheat ratio Of the cavity are large, and for not too low Ri, the bouyant force effect is strong and the inertial force effect is weak in the cavity.

C ONCLUSIONS Within the p a r a m e t e r ranges investigated, the following conclusions can be drawn: (1) If the value of Ri is not too small, the radiation effects on the heat transfer and fluid flow 'characteristics in the cavity are significant and should not be neglected within a rather wide range of Ri. The higher the Gr, the stronger the effects of radiation. (2) Radiation can strengthen the turbulent bouyancy-driving effects i n the cavity, and can raise the tendency of hot exiting air not to be redirected by the externally imposed air flow.

Cheng-Xian Lin et al.

Turbulent Mixed Convection-Radiation Interactions in a Cavity

(9) T h e a p p e a r a n c e of different n u m b e r of recirculating zones or eddies with different sizes and intensities is one of the m a i n characteristics of the turbulent mixed convection in an open cavity. T h e r a d i a t i o n can make the n u m b e r , size and intensity of these recirculating zones or eddies change. (4) R a d i a t i o n can m a k e the t h e r m a l b o u n d a r y layer along the cavity walls thicher, and can a u g m e n t the t e n d e n c y of the t h e r m a l stratification to be destroyed with the decrease of R i . (5) W i t h the decrease of Ri, the overall Nusselt n u m b e r , convective Nusselt n u m b e r and radiation fraction vary more rapidly, and t h e y show a stronger t e n d e n c y to gain e x t r e m e values at higher G r t h a n at low G r . (6) T h e inclusion of radiation makes the combined overall heat transfer greatly s t r e n g t h e n e d , and makes the variation of the overall Nusselt n u m b e r s with R i more smooth. (7) Generally speaking, the effect of radiation is to reduce convective heat transfer in the o p e n cavity. For cases of high Gr, radiation can enhance convective heat transfer within certain range of Ri. (8) If R i is not t o o low, the t u r b u l e n t mixed convection is still a relatively weak heat transfer m o d e c o m p a r e d with radiation heat transfer. Only when the value of Ri is very small, for example, R i < 5.3 x 10 - 4 , the effects of radiation can be neglected and the d o m inant m e c h a n i s m of h e a t transfer is forced convection.

REFERENCES [I] Amon, C.H. and Mikic, B.B., "Numerical Prediction of Convective Heat Transfer in Self-Sust~ned Oscillatory Flows," J. Tht.rmophydcm,4, pp.239-246, (1990).

195

[2] Humphrey, J.A.C. and To, W.M., aNumerical Simulation of Buoyant, Turbulent Flow-II. Free and Mixed Convection in a Heated Cavity," I~. J. He~ Mesa ~nafer, 29, pp.B93-610, (1986). [3] Bhatti, A. and Aung, W., "Finite Difference Analysis of Laininar Separated Forced Convection in Cavities," J. He4t Tm~~fer, 106, pp.49-54, (1984). [4] Gooray, A.M. Watkins, C.B., Aung, W., "Numerical Calcuiation of Heat Transfer in Turbulent Recirculating Flow Over an Open Cavity," Proc. ASME/JSME Thermal Engng. Joint Conf., Honolulu, Hawaii, 3, pp.79-86, (1983). [5] Clausing, A.M., LiJter, L.D. and W~dvogel, J~/., aCornbined Convection from Isothermal Cubical Cavities with a Variety of Side-Facing Apertures," Int. J. He¢~Mess ~ . f e r , 82, pp.1561-1566, (1989). [6] Showole, R.A. and Tarasuk, J.D., "An Experimental and Numerical Study of Free and Mixed Convection in a Horizontal and Inclined Open Semi-Cylindrical Cavity," Proc. 9th Int. Heat Transfer Conf., Jerusalem, Israel, 2, pp.453-458, (1990). [7] Richards, R.F., Young, M.F. and Haiad, J.C., "Turbulent Forced Convection Heat Transfer from a Bottom Heated

Open Surface Cavity," 30, pp.2281-2287, (1987). [8] Jones, W.P. and Launder, B.E., "Prediction of Laminari~-ation with a Two-Equation Model of Turbulence," Int. J. Heat M ~ s ~ ] e r , 15, pp.301-314, (1972). [9] Hilsenrath, J., et al, ~Tables of Thermodynamic and Transport Properties of Air, Argon, Carbon Dioxide, Carbon Monoxide, Hydrogen, Nitrogen, Oxygen, Steam~, Pergamon Press, New York, (1960). [10] Launder, B.E. and Sharma, B.I., "Application of the EnergyDissipation Model of Turbulence to the Calculation of Flow near a Spinning Disc," Letters in Heat M u s Transfer, | , pp.131-138, (1974). [11] Lin, C.X., Xin, M.D., Zhang, H.J., Xia, J.L., "A Numeri-. cal Study of the Turbulent Natural Convection in an Open Square Cavity," Proc. 7th Annual Meeting of the Chinese Society of Engng. Thermophysics, Nanjing, 3, pp.S.61-S.66j (1990). [12] Patankar, S.V. and Spalding, D.B., aA Calculation Procedure for Heat, Mass and Momentum Transfer in ThreeDimensional Parabolic Flow," lwt. J. H ~ t M ~ ~ ] ¢ r , 1$, pp.1787-1806j (1972). [13] Patankar, S.V., ~Numerical Heat Transfer and Fluid Flow~:~, Hemisphere Pub. Co., Washington, D.C., (1980).