ACTA MECHANICA SINICA (English Series), Vol.14, No.2, May 1998 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A.

ISSN 0567-7718

N U M E R I C A L S T U D Y O F B U O Y A N C Y - A N D

T H E R M O C A P I L L A R Y - D R I V E N F L O W S I N A C A V I T Y

Lu Xiyun ( ~ ) Zhuang Lixian (~r (Department of Modern Mechanics, University of Science and Technology of China,

Hefei, Anhui 230026, China)

A B S T R A C T : Thermocapillary- and buoyancy-driven convection in open cavities with differentially heated endwalls is investigated by numerical solutions of the two- dimensional Navier-Stokes equations coupled with the energy equation. We studied the thermocapillary and buoyancy convection in the cavities, filled with low-Prandtl- number fluids, with two aspect-ratios A -- 1 and 4, Grashof number up to 10 s and Reynolds number [Re[ <_ 104. Our results show that thermocapillary can have a quite significant effect on the stability of a primarily buoyancy-driven flow, as well as on the flow structures and dynamic behavior for both additive effect (i.e., positive Re) and opposing effect (i.e., negative Re).

K E Y WORDS: thermocapillary-driven flow, buoyancy-driven flow, crystal growth, numerical simulation, flow instability

1 I N T R O D U C T I O N

Thermocapillary- and buoyancy-driven fluid flow under normal gravity or a low-gravity environment has its potential applications in many engineering fields, such as materials sci- ence, space vehicles in orbit or in transit to the moon or other planets, power and energy

systems for space flight, and so on. Combined buoyancy and thermocapillary convection in open cavities, where the flow is driven by a temperature difference between isothermal vertical sidewalls, has its importance for materials processing, such as that of crystal growth where the quality of the crystal growth can be strongly influenced by the fluid motion. In this system with a free liquid-gas interface, the imposed temperature gradient gener- ates simultaneously buoyancy-driven convection due to a density variation in the liquid, and thermocapillary-driven convection due to a surface-tension gradient induced by a tem- perature gradient along the liquid-gas interface. More attention has been focused on the interaction of these two types of convections. Based on previous studies, a fundamental understanding of thermocapillary effect on growing metallic single crystals is obtained, in

particular for low-Prandtl-number fluids, which have their technological importance. But, few experimental velocity and temperature field measurements for the low-Prandtl-number fluids were performed as the experiments are difficult to handle owing to opacity, tempera- ture level, and so on.

Received 20 August 1997

Vo1.14, No.2 Lu & Zhuang: Buoyancy- and Therrnocapillary-Driven Flows in a Cavity 131

Although difficulties exist in experiment, a lot of work based on numerical simulation and theoretical analysis have also been done for buoyancy-driven flow, thermocapillary- driven flow, and combined buoyancy- and thermocapillary-driven flow. For purely buoyancy- driven flows, Winters [1] carried out two-dimensional calculations with bifurcation theory for rectangular cavities, and also determined the critical Grashof number (Grc) for the transition from steady to unsteady flow. A fairly full review for pure thermocapillary flow has been given by Ostrach [21 and Davis [3]. Since then, numerical simulations of thermocapillary- driven flows in open cavities have also been performed by Ben Hadid & Roux [4], Bergman & Keller [5], Zebib et al. [61. Further, coupled buoyancy- and thermocapillary-driven flow in a differentially heated cavity has been investigated. For square cavity A -- 1.0, the computations were taken by Zebib et al.[ 6], Mundrane & Zebib [7], Cuvelier & Driessen Is] and Bradley & HomsyD]; and for A _> 1.0, some work were also performed by Bergman & Keller{5l, Ben Hadid & Roux [l~

In the present study, we consider thermocapillary- and buoyancy-driven flows in open cavities with aspect-ratio, A =(length/height) : 1 and 4, Prandtl number Pr = 0.015, Reynolds number [Re[ < 10 4 and Grashof number values up to Gr = 10 5, which is larger than the critical value for the transition to an unsteady flow and has not been considered in the previous studies. The effect of both negative and positive values of the Reynolds number on the flow structures, dynamic behavior and nonlinear instability character is also investigated.

2 M A T H E M A T I C A L F O R M U L A T I O N

We consider a two-dimensional rectangular cavity with aspect ratio A, where A = L/H , and L is length and H hight as shown in Fig.1. The relevant dimensionless parameters are Prandtl, Grashof and surface-tension Reynolds numbers, which are defined as Pr = u/R,

Gr = g f lH4AT/Lv 2 and Re = (-Oa/OT). H 2 A T / L p v 2, where v is the kinematic vis- cosity, a the surface tension, fl the coeffi- cient of thermal expansion, and p the fluid density. A driving temperature difference ( A T = Th -- To) is imposed in y direction for differentially heated side walls. The surface tension on the upper boundary is assumed to vary linearly with temperature, i.e., a = ao[1-~(T-To)] where ~/= -(1/ao)(Oa/OT) is the temperature coefficient of surface ten-

XjU

H

U=u=O rh

u=v=0 L g~v

Fig.1 Schematic of computing domain. The sidewalls are differentially heated and the upper boundary is a flat free surface

sion and the subscript 0 refers to a reference state. This problem is non-dimensionalized by using H2/v, H and AT~A, as scale quantities

for time, length and temperature, and a characteristic buoyancy velocity, vGr�89 is used as the reference velocity. The dimensionless equations governing the motion of a Newto- nian fluid in the Boussinesq approximation can be written in a vorticity-stream function formulation as

OW 2 OW OW 020) ~2W ~T Grl/2 Ot

(1)

132 ACTA MECHANICA SINICA (English Series) 1998

02r 02r + = (2)

cgT + G r l / 2 ( u O T + v OT~ = 1__( 02T + O2T~ (3) 0-7 ox j j

where w, r and T are the vorticity, stream function and temperature respectively. Two driving forces, buoyancy acting as a body force in the bulk and thermocapillarity acting on the upper surface of the fluid, are involved. The dynamical boundary condition on the upper free-surface relates the velocity gradient to the temperature gradient through the following formula (Birikh[ TM)

Ov Re OT cOx - Grl/2 cOy (4)

and other conditions on the horizontal boundaries are

v (O ,y )=u(O,y )=O; u ( 1 , y ) = 0 (5)

and

for the conducting case; and

T(0, y) = T(0, y) = A - y

OT OT 1 = = 0

for the insulating case. The boundary conditions on the rigid vertical walls are

(6)

(7)

T(x, O) = A; T(x , A) = 0 (8)

u(x, O) = v(x, O) = u(x, A) = v(x, A) = 0 (9)

In order to solve the Poisson equation (2), we set r = 0 along the horizontal boundaries and the vertical walls according to Eqs.(5) and (9). In the present study, we neglected the free-surface deformation. This assumption is valid in crystal growth applications with small capillary numbers and 90 ~ contact angles between the vertical solid wall and the meniscus surface.

3 N U M E R I C A L M E T H O D

The governing equations (1)--~(3) are solved using finite-difference technique with high order formulation. In order to accurately describe gradients in boundary layers, it is neces- sary to use a non-uniform grid along both directions. The spatial derivatives in the equations are discretized by 4th-order compact scheme (Lele [13]) and the time derivatives in Eqs.(1) and (3) are approximated using a second-order scheme. The Poisson equation (2) is solved using multigrid method with V-cycle iteration.

The computation loop to advance the solution from one time level to the next consists of the following three substeps. First, the stream-function Poisson equation is solved using multigrid method based on given vorticity. Then, using the new stream-function, the velocity is gotten and Eq.(3) is advanced to obtain the temperature. Finally, Eq.(1) is solved for new vorticity function.

Vol.14, No.2 Lu & Zhuang: Buoyancy- and Thermocapillary-Driven Flows in a Cavity 133

4 R E S U L T S A N D D I S C U S S I O N

To illustrate the computational procedure, we mainly discuss the results for G r -- 4000

to 105, Re = -104 to 104, P r = 0.015, and A = 4 and 1. The number of mesh points for_

the calculations is 128 x 64 along horizontal and vertical directions for A = 4, and 64 x 64

for A = 1; time-step is 5 x 10 -6. It has been determined that the computed results are

independent of the time steps and the grid sizes. To validate the code, some quantitative

comparisons between the present computation and previous results were carried out for

G r = 6 x 103 and 1.5 x 104, R e = 0 and A = 4. Streamlines and isotherms are shown in Fig.2;

our results are in good agreement with those of Ben Hadid & RouxEtl]. All computations

were done on SGI2-R10K machine; 50 CPU hours were needed for one viscous time unit for

A = 4 with 128 x 64 meshes.

llii J II//ll/ltttl I/Illli/llltl

(c)

iltllJ Il llll/tl///li (d)

Fig.2 Steady solutions in the pure buoyancy-convection case for A = 4: (a) stream- lines and isotherms at Gr = 6 x 103; (b) Gr = 1.5 x 104; as well as the results calculated in Ben Hadid • Roux [lq for (c) Gr = 6 x 103 and (d) 1.5 x 104

4.1 P u r e B u o y a n c y - D r i v e n F lows for A = 4

We first consider the influence of natural convection on the flow when thermocapillary

forces are absent by setting R e = 0. Flow pattern is shown in Fig.3 for G r = 4000; a vortex

progressively forms in the cold region leading to an asymmetric cellular flow. On further

increasing G r , a stationary bifurcation is reached with the appearance of a secondary cell,

as shown in Fig.2 for G r = 1.5 x 104, followed by a transition to unsteady flow, i.e., Hopf

l l l f t / t l l l l (a) (b)

Fig.3 Streamlines and isotherms for A = 4, Gr = 4 x 103

134 ACTA MECHANICA SINICA (English Series) 1998

bifurcation. A periodic flow occurs at Grc ~ 1.9 • 104 for insulating horizontal boundaries

and persists for higher Gr.

In Fig.4, the flow patterns over one period are given at Gr = 4 • 104. The sequence

of streamlines shows a periodic evolution of vortex shape and reveals that two vortices

(primary and secondary) intensify and create two reverse flows (counter-rotating) which

are periodically created and destroyed. A quantitative examination of flow fluctuations

throughout the cavity reveals that the fluctuations increase from the bottom towards the

upper surface and reach their maximum amplitude in the middle part of the cavity (y =

0.5A). The variation of fluctuation behaviour for r with time is illustrated in Fig.5 at

x = 0.5, y = 0.5A. The power spectrum also exhibits several peaks, indicating that this

oscil latory flow varies periodically with time.

(a) (b)

(c) (d)

Fig.4 Flow patterns in one period for A = 4, Gr = 4 • 104 and Re --- 0:

t = (a) 0.62484; (b) 0.63513; (c) 0.64543; (d) 0.65572

:: ,~176 0.6 ..... , ~-

E I0"! 0.4

lo% / 0. 10-3

�9 , , , ~ I ~ i , [

0.0 0.2 0 .4 0 .6 0 .8 1 .0 0 25 50 75 1 0 0

time frequency

Fig.5 Time history of the stream function for A = 4 and Gr = 4 • 10 4

at x = 0.5 and y ---- 0.5A; and its power spectrum

The effect of thermal condit ions along the horizontal boundaries on the flow s t ruc ture

and the corresponding thermal field is also investigated. The streamlines and thermal field

for conduct ing horizontal boundaries are shown in Fig.6 for Gr = 1.5 • 10 4 and Re --- 0.

By compar ing it with the results for insulat ing horizontal boundar ies shown in Fig.2(b), the

liIIIII 'II (a) (b)

Fig.6 Streamlines and isotherms in the pure buoyancy-convection case with

conducting horizontal boundaries at Gr = 1.5 • 104 and A = 4

Vo1.14, No.2 Lu & Zhuang: Buoyancy- and Thermocapillary-Driven Flows in a Cavity 135

isotherms are only slightly distorted indicating that the heat transfer is mainly from con-

duction, while a strong thermal stratification is exhibited in the vor tex regions owing to the

strong convective motion.

4.2 C o m b i n e d B u o y a n c y - a n d T h e r m o c a p i l l a r y - D r i v e n F l o w s for A = 4

With combined buoyancy and thermocapil lary forces two cases will be examined: an

additive (Re > 0) and opposing (Re < 0) effect. The thermocapil lary force can significantly

affect the dynamics of the instability in such a manner tha t oscillatory flow induced by

buoyancy forces can be damped (inverse bifurcation) when [Re I exceeds a certain critical

value.

Here we mainly describe the results of G r = 105. Figure 7 shows the variation of

Cma• with t ime for Gr = 105 and R e = 0 (i.e., pure buoyancy-convection case) and its

power spectrum. Using an instantaneous solution obtained for Gr = 105 and R e = 0 as

initial condition, the calculations for several Reynolds numbers between -105 to 105 were

performed. In Fig.8, the curves of Cma• with t ime are shown for R e = - 8 • 103, - 5 x 103, - 2 • 10 3, 2 • 10 3, 5 X 10 3, and 8 • 10 3. We can identify tha t as the Reynolds n u m b e r

decreases down to R e = - 8 • 10 3, the oscillating solution Cm~x is rapidly damped and

the solution converges to a steady state; and as R e further increases up to R e = 8 x 103,

the solution also goes to a steady state. For the comparison of steady flow structures, the

streamlines are also shown in Fig.9 for R e -- - 8 x 103 and 8 • 103.

.~ 0.75 E 10~ -

0.70 ~ 10-1 ~_

0 .65 ,~

0 .60 =~ 0.551 . ~-10-3~-, , ' ' ' ' ' ' ' ' '

0 .0 0 .2 0 .4 0.6 0 .8 1 .0 0 50 100 150 200

time frequency

Fig.7 Variation of Cm~x with time for A = 4, Gr = 105 and Re = 0;

and its power spectrum

Based on numerous computat ions at carefully selected Gr and R e values, neutral stabil-

ity curve as shown in Fig.10 has been generated in a G r - R e plane for A = 4 and P r = 0.0015.

The Hopf bifurcation of the combined buoyancy and thermocapil lary problem is presented.

I t can be seen that the curve does not intersect the Gr = 0 line at any point. This is

consistent with bo th the steady and unsteady results calculated by Mundrane & Zebib [7], Ben Hadid & Roux [11].

4 . 3 F l o w S t r u c t u r e s f o r A = 1

Pure buoyancy-driven flows as well as combined buoyancy- and thermo-capillary-driven

flows in a square cavity were also investigated. According to our results, the solutions always converge to a steady state for Gr -- 104 to 105 and R e -- - 5 • 10 3 to 5 • 10 3. As an

example, F ig . l l shows the streamlines and the curves of Cmax with t ime for Gr = 4 • 104 and R e = -2000, 0 and 2000. Compare the results of A -- 1 with tha t of A -- 4 for the

same values of Gr and Re, the flow structure has a significant difference.

136 1998 A C T A M E C H A N I C A S I N I C A (English Series)

8 0.34 O r

0.32 E 0~

~ 0.30

~ : 0 . 2 8 ~ . . . . ' . . . . ' 0.0 0.2 0.4 0.6 0.8 1.0

timo (a)

0 . 7 0 E " " , . . . . , . . . . l . . . . , ' " ' ~

0.65 E- -~

o.so ~ , , , , , , , , , , , , , , , v v , , , , , , , , v l ~

0 4 5 ~ . . . . , ~ , . ] . . . . , . . . . I . . . . :I E " O . 0 0.2 0.4 0.6 0.8 1.0

time (c)

0.80~- . , . . , . . . . , . . . . , . . . . , , , , ~ 0 .85 F

. ~ o.78

0.76 ~ 0.80

0.74 0 .72i l l . , , , . . . . I . . . . i . . . . , . . . . :I

0.0 0.2 0.4 0.6 0.8 1.0 ~ 0.7s time (e)

Fig.8

g 0 . 3 4 ~ , , . , . . . . , . . . . , . . . . , ' " ~ 3

o ~

E " 0 Ume (b)

. ~ 0 . 8 0 ~ , , , , , . . . . , . . . . , . . . . , . , , . ~

~ 0.76~- - ~ F

o. |i mmlvmmnmvmmmv,v, " O.0 0.2 0 . 4 0.6 0.8 1.0

tiros (d)

' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 . . . . I ' ' ' ' t

r ,0 0.2 0.4 0.6 0.8 1.0

time (f)

Variation of Cmax with time for A = 4, G r = 105, and R e = (a) - 8 x 103; (b) - 5 x 103; (e) - 2 x 103; (d) 2 x 103; (e) 5 • 103; and (f) 8 x 103

(a ) (b )

Fig.0 Flow patterns for A --- 4, G r = 105: (a) R e = - 8 x 103; (b) R e = 8 x 103

eq

•

120

100

80

60

40

2O

0 ' -10

o

-5 0 5

REX103

i i 10

Fig.10 Neutral stability curve showing for A = 4 and P r = 0.015. Points

marked with o (unsteady) and o (steady) have been calculated

Vol.14, No.2 Lu & Zhuang: Buoyancy- and Thermocapillary-Driven Flows in a Cavity 137

0 O O (a) (b) (c)

0.5

0.4 e -

0.3 E c0

0.2

E

' • . . . . . Re=-2000

0.1 Re=O . . . . . . . . . R e = 2 0 0 0

0.0 , , . i . . . . i . . . . i . . . . ~ . . . . 0.0 0.1 0.2 0.3 0.4 0.5

time

Fig.l l Flow patterns and the curves of Cm~ with time for Gr = 4 x 104, A = 1:

Re = (a) -'2000; (b) 0; and (c) 2000

5 C O N C L U D I N G R E M A R K S

Coupled thermocapil lary and buoyancy convection in open cavities for A = 1 and

4 with differentially heated endwalls for low-Prandtl-number fluids has been numerically

investigated. Our results show strong influence of the parameters Gr, R e and A on the flow

structures and dynamic behavior. The thermocapil lary can have a quite significant effect

on the stability of a primarily buoyancy-driven flow for bo th the additive (Re > 0) and

opposing (Re < 0) cases. According to our and previous results, neutral stability curve has

been given in a G r - R e plane for A = 4 and P r = 0.015. As IRel exceeds a certain critical value, an unsteady flow regime will change to steady flow regime. Pure buoyancy-driven

flows as well as combined buoyancy- and thermocapil lary-driven flows in a square cavity

have also been investigated; solutions always converge-to a steady state for G r = 104 to 105

and R e = - 5 • 103 to 5 x 103. Further work on this problem will investigate the low-Prandtl-

number fluid flow characteristics under combined thermocapil lary and buoyancy convection

for deep cavities (i.e., A < 1) and larger aspect-ratio cavities A > 4.

R E F E R E N C E S

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J Numer Methods Engng, 1988, 25:401~414

2 0 s t r a c h S. Low-gravity fluid flows. Ann Rev Fluid Mech, 1982, 14:313~345

3 Davis SH. Thermocapillary instabilities. Ann Rev Fluid Mech, 1987, 19:403~435

4 Ben Hadid H, Roux B. Thermocapillary convection in long horizontal layers of low-Prandtl- number melts subject to horizontal temperature gradient. J Fluid Mech, 1990, 221:77~103

5 Bergman TL, Keller JR. Combined buoyancy surface-tension flow in liquid metals. Numer Heat Transfer, 1988, 13:49,,,63

138 ACTA MECHANICA SINICA (English Series) 1998

6 Zebib A, Homsy GM, Meiburg E. High Marangoni number convection in a square cavity. Phys Fluids, 1985, 28:3467,,,3476

7 Mundrane M, Zebib A. Two- and three- dimensional buoyant thermocapiUary convection. Phys Fluids, 1993, 4:810,,~818

8 Cuvelier C, Driessen JM. Thermocapillary free boundaries in crystal growth. J Fluid Mech, 1986, 169:1,,~26

9 Bradley MC, Homsy GM. Combined buoyancy-thermocapillary flow in a cavity. J Fluid Mech, 1989, 207:121,-,132

10 Ben Hadid H, Roux B. Buoyancy- and thermocapillary-driven flow in a shallow open cavity: unsteady flow regimes. J Cryst Growth, 1989, 97:217,,,225

11 Ben Hadid H, Roux B. Buoyancy- and thermocapillary-driven flows in differentially heated cavities for low-Prandtl-number fluids. J Fluid Mech, 1992, 235:1,~36

12 Birikh RV. Thermocapillary convection in horizontal layer of liquid. J Appl Mech Tech Phys, 1966, 7:43,~49

13 Lele SK. Compact finite difference schemes with spectral-like resolution. J Comput Phys, 1992, 103:16,-~42