Hysterisis effect on thermosolutal convection with opposed buoyancy forces in inclined enclosures

Pergamon Int. Comm. HeatMass Transfer, Vol. 26, No. 3, pp. 421-430, 1999

Copyright © 1999 Elsevier Science Ltd Printed in the USA. All rights reserved

0735-1933/99/S-see front matter

PII S0735-1933(99)00028-7

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

M. Mamou and P. Vasseur Ecole Polytechnique, University of Montreal

C.P. 6079, Succ. "Centre Ville" Montreal, Quebec, H3C 3A7, Canada.

A B S T R A C T

(Communica t ed by J.P. Hartnet t and W.J. Minkowycz )

The onset of double diffusive convective flows in an inclined fluid layer, when constant fluxes of heat and mass are applied on the two oposing boundar ies of the layer, is investigated. The case of equal and opposing buoyancy forces is considered. A numerical l inear stabil i ty theory is used to determine the critical Rayleigh number for the onset of convection. The existence of a subcri t ical Rayleigh number , for the onset of finite ampli tude convection, is demons t ra ted on the basis of the parallel flow approximation. © 1999 Elsevier Science Ltd

I n t r o d u c t i o n

The growing research on double-diffusive convection is mainly mot ivated by its impor tance in

many na tu ra l and industr ia l problems. For instance, double diffusive convection frequently occurs

in oceans, lakes, a tmosphere as well as in engineering applications such as solar ponds, chemical

processes, crystal growth, etc. A comprehensive review of the available papers on this topic was

presented by Turner [1].

A few studies have been reported in the past for the par t icular configuration of double diffusive

convection in a vertical rectangular cavity with opposing and equal thermal and solutal buoyancy

forces. For this s i tuat ion there exists a trivial steady state solution corresponding to the rest

s ta te where the t empera tu re and concentrat ion profiles are linear. However, in general, this rest

s ta te can be main ta ined only up to a critical Rayleigh number above which a convective flow

421

422 M. Mamou and P. Vasseur Vol. 26, No. 3

occurs. This problem has been investigated numerically in the past by Krishnan [2], Gobin and

Bennacer [3], Ghorayeb and Mojtabi [4] and Mamou [5] for the case of a vertical cavity with imposed

constant temperatures and concentrations along the vertical side-walls. It was demonstrated by

these authors that when the Lewis number is equal to unity the system remains unconditionally

stable. For a square cavity the critical Rayleigh number for the onset of subcritical convection was

obtained, solving the full governing equations by Krishnan [2] and Gobin and Bennacer [3]. On

the other hand, the critical Rayleigh number for the onset of supercritical convection was predicted

by Gorayeb and Mojtabi [4] on the basis of the linear stability theory. Recently, this problem was

reconsidered by Xin et al. [6]. It was demonstrated that, in a vertical slot, the initial instability

of the conductive state is a subcritical circle pitchfork bifurcation. The finite amplitude branch

of steady states is subsequently destabilized by a drift pitchfork bifurcation leading to travelling

waves,

The present investigation considers the onset of double diffusive convection within an inclined

cavity subject to constant fluxes of heat and mass on its long-side walls. A numerical linear stability

analysis of the problem is carried out. For a slender enclosure an analytical finite amplitude solution..

based on the parallel flow approximation, is derived.

M a t h e m a t i c a l Formulat ion

The domain under study is the two dimensional inclined enclosure of aspect ratio A = H ' / W '

shown in Fig. 1. The long-side walls are exposed to uniform fluxes of [teat and solute, q' and j '

respectively, while the other two walls are impermeable and adiabatic. The binary fluid is modeled

as a newtonian Boussinesq-incompressible fluid whose properties are described by its kinematic

viscosity, ~, its thermal and solutal difl]lsivities, (~ and D, and its thermal and solutal volumetric

expansion coefficients, 13r and ~s.

The governing equations are nondimensionalized by scaling length by W', stream function by

a, t ime by c~/W '2 and defining the reduced temperature T = (T' - T ~ ) / A T * and the reduced

concentration S = (S' - S~o)/AS * where the subscript 0 refers to the center of the cavity and

AT* = q ' W ' / k and AS* = j ' W ' / D are the characteristic temperature and concentration differences

respectively while k is the thermal conductivity.

Before introducing the mathematical fornmlation of the present problem, the stremn function,

Vol. 26, No. 3 HYSTERISIS EFFECT ON THERMOSOLUTAL CONVECTION 423

temperature and concentration fields (~,T,S) are decomposed as follows

• = ~ c + ¢ ( t , x , y ) , T = T c + ¢ ( t , x , y ) , S = S c + ~ o ( t , x , y ) (1)

where the static state of the system is described by ~ c = 0 and Tc = Sc = x and the functions

¢(t, x, y), ¢(t, x, y) and ~p(t, x, y) describe the perturbed solution due to convective effects.

olume

FIG. 1 Cavity configuration.

The dimensionless equations governing the present problem then read

0V2~ Ot J (¢ ' V2¢) = Pr V4~ - Pr RaT .7"(¢ -- ~) (2)

0¢ + ¢ ~ - - j ( ¢ , ¢ ) =V2 ¢ (3) Ot

0~o 0--t- + ¢y - J (¢ ' ~) = Le-1 V2~° (4)

where J ( f , g ) = fzgv - fgg~ and P( f ) = sin ~f~ + cos Ofy. The subscripts x and y denote first

derivatives.

The corresponding boundary conditions are homogeneous

. = t l / 2 , ¢=¢x=¢~=¢~=~=0} y = ± A / 2 , ¢ = ¢5 = ¢y = ¢~ = ~y = O

(5)

In the momentum equation (2), the buoyancy forces have been assumed to be opposing and of

equal intensity (/)rAT* = -- f lsAS*). The present problem is thus governed by five dimensionless


parameters, namely the thermal Rayleigh number, RaT = g J3T q' W'4/(ko~u), the Prandt l number,

Pr = u /a , the Lewis number, Le = (~/D, the aspect ratio of the enclosure, A = H ' / W ~, and the

inclination angle, ~.

Linear Stabi l i ty Analys i s

The linear stability of the rest state (~)c = O, T = S = x) is now investigated in terms of

the governing parameters of the problem. Assuming separability the perturbed stream function,

temperature and concentration fields (~b, qS, p) can be expressed as

¢ ( t , x , y ) : ¢o e pt F(x ,y ) ¢( t ,x ,y ) = ¢0 e pt G(x,y) ~p(t,x,y) = ~o e pt G(x ,y) (6)

where F ( x , y ) and G(x,y) are space flmctions describing the fields of "~, ¢ and p at the onset of

convection, p is a complex number and ~0, ~b0 and ~o are constants.

Substi tuting Eqs. (6) into Eqs. (2)-(4) and after neglecting the second order terms, yields

p~9oV2F = ¢oPrV4F - PrRaT(¢O -- qoo)SC(G) ]

peso + ¢oE~ = ¢o V2G

p~oG + ~PoF v = ~OV2G

(7)

A numerical method is used to solve Eqs. (7). The procedure has been described in details

in reference [5] and only the main steps are given here. The calculus domain is discretized into

rectangular Hermite cubic elements. Considering the marginal stability (p = 0) and using the

Bubnov-Galerkin procedure, the resulting space-discretized equations are stated as follows

~/,o[K,,,]{F} = m~,r(¢o - ~o)[B]{G} /

,/,0[L]{F} : ¢01t<1 {C} [ (s)

¢o[L]{F} = P0 ~[K] {C} where [B], [K~0], [K] and [L] are m × m matrices and {F} and {G} unknown vectors of size m.

Here m = 4n, where n is the total nodes mlmber. The corresponding elementary matrices arc

[B] C : ./i~ J : (HjH~dQ, [<.]~ = .f~ V2Yj .V2Hid~ | %

• ' l (9) [~:]~ = j;o var~.vH~d~< [L] ~ = - L ~ H , / < d ~

where N'i(x, y) are the Hermite shape flmetions and ~ is the calculus domain.

Equations (8) can be combined to obtain the following canonical eigenvalue problem

1 [E - ,~ I]{F} = 0 with [E l = [K¢I-I[B][KI-I[L] and )~ -

RAT(1 - Le) (io)


Using double precision subrout ines of the IMSL library we obta in m eigenvalues and the

corresponding fields F ( x , y). If the eigenvalues are rearranged such tha t A] < A2 _< ... _< Am-1 -< Am,

then the supercri t ical the rmal Rayleigh number for the onset of convection is

R sup RaSUp with RaSU p _- 1 when L e > 1 and R a sup 1 when Le < 1 (11) a t e = I - - L e A-~ = A--~

The precision of the value of the critical Rayleigh number predicted by the present numerical

procedure depends on the number of elements used in bo th x and y-directions. Numerical tests

(reference [5]), using various mesh sizes, were done for a given aspect rat io in order to determine

the best compromise between accuracy of the results and comput ing time. From these results, it

was found tha t a mesh size of 10 x 10 was necessary for A -~ 1 and 8 × 40 for A --- 10.

Figure 2 i l lustrates the influence of the inclination angle, ~, on the constant R a sup in the case

of a square cavity (A = 1). Typical flow pa t te rns at the onset of convection, as predicted by the

linear s tabi l i ty analysis, for L e > 1 and • = 45 ° and 90 ° are presented in this figure. The case of

a vertical square cavity (~ = 90 °) will be discussed first. For this si tuation, R a sup = ±19722.7,

the flow pa t t e rns consist of three counterrota t ing cells and two solutions are possible depending

upon whether Le is smaller or greater than unity. For L e > 1, Fig. 2 indicates the existence of a

pr imary circulat ion at the center of the cavity and two secondary ceils one in the upper left corner

and the other one in the bo t tom right corner. For Le < 1, the result ing flow pa t t e rn (not presented

here) is the mirror image of tha t obtained for Le > 1. These two solutions have the same absolute

eigenvalues but different sign (A1 = -Am). As the value of q~ is reduced down to 0 °, when L e < 1,

the s i tuat ion corresponds to a cavity heated (destabilized agent) and salted (stabilising) from below.

The critical Rayleigh number required to destabilize the system is now considerably lower t han for

= 90 ° and the resul t ing flow pa t t e rn (not presented here) is a single square cell occupying the

entire cavity. On the other hand, for L e < 1, Fig. 2 indicates tha t R a ~up increases monotonously

up to infinity as e2 -4 180 ° for which the buoyancy forces induced by the stabil izing agent (heat)

overcomes those due to destabil izing agent (concentration) and vis versa for Le > 1. Figure 2 also

i l lustrates the flow pa t t e rn obtained for L e > 1 at • = 45 ° for which three counter ro ta t ing cells of

approximate ly equal size are observed.

The effect of the aspect ratio, A, on R a s~p is i l lustrated in Fig. 3 for the case of a vertical

cavity. Upon increasing A, R a sup decreases monotonously and the number of the cells wi th in the

enclosure increases. This process is continued as the aspect ratio of the cavity is made larger and


larger, as illustrated by the three cell flow pattern depicted in Fig. 3 for the case A = 4. The

numerical results indicate that, when A is large enough, R sup tends towards a constant value. For

A -~ 0% it was found numerically that Ra s~p = 6511.2 and the flow pattern indicates the formation

of a periodical structure consisting of tilted counter-rotating cells (see Fig. 4d).

.R'- ~ 10 ~

IRaSUPl / ~ J ~ * ~ ~ ~ s (]D=45 Le>l •~. ' (~1=90 Le>l 10 ~ •

&&

1684.6 A&A*,A•~• ; lO

o 20 40 60 80 1 oo 120 140 160 180

FIG. 2 Inclination effect on Ra 's'~p and flow

patterns for A = 1.

25000 [ { A=4, Le>l

-!'i " ! iRa, Up I I

/i "e O e e l e ~ ID • • •

6511 .2 . . . . . • ~=t , [

10

FIG. 3 Effect of A oi1 the constant Ra sup

for ~5 = 90 o.

F i n i t e A m p l i t u d e S o l u t i o n

In the limit of slender enclosures (A > > 1), the steady form of the governing equations (2)-

(4) can be considerably simplified to be solved analytically using the parallel flow approximation

(see reference [5]). Thus, in the central part of the enclosure, we can write that ~b(x, y) ~ ~/J(x)

and ¢(x,y) = Co y + ®o(x), where 0 stands for ¢ and ~p and Co is a constant representing the

temperature or the concentration gradient in the y-direction.

Following the procedure described in details in reference [5] for instance, the present approxi-

mation yields the following profiles for the stream fimction, temperature and concentration

¢(x) = B[cosh(cox) + % cos(co,:)] - a ]

(%(,~) = _ c~B [sinh(~) + ~/,, sin(cox)] + C~a~, (12) CO

e ~ ( x ) -- L e C ~ B [sinh(cox) + 70 sin(cox)] + LeC~Gx CO

where

sinh Wo I co = 4/ (v_RaT~C4~ + L e N C , ~ ) s i n ~ , COo= ~ , "70 -

s i n c o o

R , a T B=W[coshcoo+.,/oCOScoo]. G= ~q- [ ( l+N)s in~+(Ce+NC~)eos~] (13)


and the constants C¢ and C~ can be obtained by performing the energy and mass balances in the

controle volume depicted in Fig. 1. This yields

C¢ + CcF - (1 + C¢G)E = 0 / (14)

C~ + Le2C~oF - Le(1 + LeC~G)E = 0 J where E and F are defined by

E = B [2 (sinhw° + % s i n w ° ) - c ° s h w ° - % c ° s w ° ] B2 (15)

F [w(% 2 + 1) - sinhw + %2 sinw]

V e r t i c a l E n c l o s u r e (~ = 90 °)

For this particular situation, the stream function, temperature and concentration profiles read

(1 % cos/ <

C¢¢0 [sinh(wx) - 70sin(wx)] (16) o¢(z) = ~ ( 1 - 7 o )

O~(x) = C~Le¢o [sinh(wx) - %sin(wx)] J ( i : )

where ¢0 is the stream function value at the center of the enclosure and % -- cosh Wo/cos Wo.

The value of w can be computed numerically by solving tanhwo + tanwo -- 0. From this

equation, it can be demonstrated that there exists multiple solutions. The first one (w0 = 0)

corresponds to the rest state (¢0 = 0) and the others to convective flow modes (wl -- 4.73 and Wn ~-

~r(4n - 1)/2; n=2,3,...; w~ corresponds to a monoeellular flow and w,, correspond to multicellular

flows consisting of (2n + 1) vertical cells. For each value of w there exists two possible convective

solution. One is characterized with a strong flow intensity and the other with a small convective

amplitude. Only the former was found to be stable.

Performing the energy and mass balances in the controle volume (Fig. 1), the constant C¢

and C~ are obtained as follows

4 sinh Wo be0 bLe¢0 1 + Vo 2 and b - (17) C ¢ = l + a ¢ ~ ' C ~ = l + a L e 2 ¢ ~ with a - 2 ( 1 - % ) 2 w ( 1 - % )

that

Upon combining the above expressions with the definition of ¢0, it can be easily demonstrated

a2Le2¢ 4 + a(Le 2 + 1)¢ 2 - 2v/~Ra~(Le 2 - 1)•0 + 1 = 0 (18)


R a T 9. /~= where R a ~ = with R a s~b = w 4 = 1204.93,

According to the above solution, there exists a critical Rayleigh number, called subcri t ical

value, above which a finite ampli tude convective flow bifurcates from the rest state. This value is

given by

R sub R a S U b v ~ [ J aTC = ~-:~-~e2) 2aLe2@c + (Le 2 + 1)¢0c (19)

where ~b0c is expressed by

¢ oe - L e n a Le2 + 1)2 + 12Le2 - (Le2 + 1) (20)

w h e r e s = - i f o r L e > 1 a n d s = l f o r L e < 1.

For large or small Lewis numbers (Le < < 1 or Le > > i), the subcri t ical value reduces to

j~a sub R ~b (21)

a , r c = S l _ Le

Figure 4 shows the effect of the inclination angle of the enclosure, q~, on the constant R a ~*'r'

(Fig. 4a) and the flow s t ructure (Figs. 4c-e) in the case of an infinite fluid layer (A -} oo). Tile

numerical results, predicted by the linear stabil i ty analysis, were obta ined using periodic boundary

condit ions in y directions. In this way, I~,a "sup was computed for various values of A. The min imum

of R a "~p corresponds to the critical Rayleigh number for the onset of convection in an infinite layer,

A c (Fig. 4b) being the wavelength of the s t ructure of the flow pat tern. The results obta ined in this

way are presented by dashed lines in Figs. 4a and b. On the other hand, it can be demonst ra ted

froin Eqs. (14), when C¢ ~ 0 and C~ + 0, tha t the supercrit ical Rayleigh number predicted by

the parallel flow approximat ion is given by

R~ ~p 720 F1 "~'~p - with Ra 's~' - (22)

0'7"(7 1 - Le cos d)

which is valid for 0 < q~ < 90 ° when Le < 1 and [br 90 < q~ < 180 ° when Le > 1.

The above equat ion is depicted in Fig. 4a by solid lines. When Le > 1 (Le < 1), a good

agreement between bo th theories is observed for 111.3 ° <_ • < 180 ° (0 ° < • < 68.7°). For

0 ° < ~ < 111.3 ° (111.3 ° < q5 < 180°), tile mmmrical results predicted by the linear stabil i ty anal-

ysis shows t ha t the flow s t ructure is not parallel bu t nmlticellular (see Figs. 4c-e).


10 s

IRa"Pl

10 4

103

, J

Le> l ~, ,#Le<l

20 40 60 80 100120140160180

(a)

35

3o Ac

25

2o

15

lO

Le>l Le<l /

p

5 "* * d •

o 20 40 6o 80 1oo 12o 14o 18o "18o ¢ (b)

i (c) O = 1 0 5 ° (d) 0 = 9 0 ° (e) 0 - - - 3 0 ° .

FIG. 4 The effect of incl inat ion on R a sup and A c and streamline pa t te rns for A = c~ for Le > 1.

The stabi l i ty diagram for an infinite vertical layer (O = 90 °, R a sup = 6511.19 and A c = 2.48)

as predicted by the linear stabil i ty analysis (linear theory) and the parallel flow approximat ion

(non-l inear theory) is presented in Fig. 5a. In region (I), below the subcri t ical Rayleigh number ,

Eq. (19), the fluid is at rest according to bo th linear and non-linear theories. In region (II), the

l inear theory predicts a stable rest s tate while finite ampl i tude convection is possible according to

the non-l inear theory. In region (III), the system is unstable according to b o t h theories and any

a rb i t ra ry dynamic pe r tu rba t ion can ini t iate a convective flow. Figure 5b i l lustrates the bifurcat ion

d iagram in te rm of the flow intensity ¢0, evaluated at the center of the layer, for Le = 10, A = cx~

and • = 90 °. The solid and dashed lines in this graph are the predict ion of the parallel flow

approximat ion. A numerical solution of the full governing equations (2)-(4) was obtained, using

the technique described in reference [5]. The results, for A = 10, • = 90 °, Le = 10 and P r = 7, are

observed to be in good agreement with the analytical solution. The parallel flow solution indicates a

r~ s u p subcri t ical Rayleigh number ,~aTcl:~ sub =. 122.9 <: £taTC ~- 723.5, at which a finite ampl i tude convection

biflxrcates.

Upon s ta r t ing the numerical code with the rest s ta te solution as init ial conditions, the results

indicate tha t , when R a T increases from 0 to 2000, convective flows are possible only when R a T >

723.5. However, when using a finite ampl i tude flows as initial conditions, a convective solution was


found to be stable for RaT > 122.9. To sammarize, when increasing or decreasing RaT step by step

using the previous solution as initial conditions, the solution follows the path indicated by arrows in

Fig. 5b leading to hysterisis loop. It is worthwhile to mention that, the multicellular flow structure

predicted by the linear stability analysis was found unstable. The numerical results indicate that

the convective solution is in fact unicellular regardless the inclination angle, ~5. Similar results

was reported by Mamou et al, [7] when studying thermosolutal convection in a vertical porous

enclosure.

10 ~ - -

unstable ]~/ . . . . . . e

s..c.,,,.aa_<

10' ( i ) a~b ~"

- 10" 10 10 10 Le

105

10 4 - - Ra T

10 3

i i - -

~' Ra~r b ~ RaCY ~ . . . S

i . nuoer,oa

/ I O0 i ~ - ~ _ ~ .~ . . . . . . . . . . . . . . . 500 1000 1500 2000

RaT

(a) (b)

FIG. 5 (a) Stability diagram and (b) bifllrcation diagrams for Le = 10. A = c,c and q~ = 90 °.

R e f e r e n c e s

1. J S Turner, J. Fluid Mechanics 33, 183 (1968).

2. R Krishnan, ASME Nat. Heat Transfer Conf., Philadelphia, 357 (1989).

3. D Gobin and R Bennacer, Phys. Fluids 6, 59 (1993).

4. K Ghorayeb et A Mojtabi, Phys. Fluids 9, 2339 (1997).

5, M Mamou (1998), Ph.D. Thesis, Ecole Polytechnique de Montr6al, Qu6bec, Canada.

6. S Xin, P Le Qudr6 and L Turckerman, Phys. Fluids 10, 850 (1998).

7. M Mamou, P Vasseur and E Bilgen, J. Fluid Mechanics a68, 263 (1998).

Received November 20, 1998

Hysterisis effect on thermosolutal convection with opposed buoyancy forces in inclined enclosures

Documents

Transcript of Hysterisis effect on thermosolutal convection with opposed buoyancy forces in inclined enclosures