EFFECT OF VIBRATION ON THE ONSET OF DOUBLE-DIFFUSIVE CONVECTION IN POROUS MEDIA

10

EFFECT OF VIBRATION ON THE ONSET OF DOUBLE-DIFFUSIVE CONVECTION IN POROUS MEDIA M. C. CHARRIER MOITABI*, Y. R RAZI^ K. MALIWAN+ and A. MOJTABI+ *Laboratoire d'Energetique (LESETH), EA 810, UFR PCA, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France email: [email protected] +IMFT, UMR CNRS/INP/UPS N°5502, UFR MIG, Universite Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France email: [email protected], mkittinaOme.psu.ac.th and [email protected]

Abstract In this chapter we consider the instability of double-diffusive convection in porous media under the effect of mechanical vibration. The so-called time-averaged formulation has been adopted. This formulation can be effectively applied to study the vibrational induced thermosolutal convection problem. The influence of high-frequency and small-amplitude vibration on the onset of thermo-solutal convection, in a confined porous cavity with various aspect ratios and saturated by a binary mixture has been presented. Linear stability analysis of the mechanical equilibrium or quasi-equilibrium solution is performed. A theoretical examination of the limiting case of the long-wave mode in the case of Soret driven convection under the action of vibration has been carried out. The 2D numerical simulations of the problem are presented which allow us to corroborate the results obtained from the linear stability analysis for both stationary and Hopf bifurcations. Keywords: porous media, vibration, double-diffusive convection, Soret effect, linear stability, long-wave mode, Hopf bifurcation

261

262

10.1

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

INTRODUCTION

In recent years, the effect of mechanical vibration on the stability threshold of thermal systems has been the subject of numerous studies. In thermo-vibrational convection, the energy of mechanical vibration in the presence of a temperature or a concentration gradient can be used to control the onset of convective motion. This type of convective motion, in which the buoyancy force may be thought of as time dependent, has attracted the attention of many researchers. Theoretical studies concerning linear and weakly nonlinear stability analysis of the Rayleigh-Benard convection subjected to a sinusoidal acceleration modulation have been conducted by several researchers, e.g. Gershuni et al. (1970), Gresho and Sani (1970), Biringen and Peltier (1990), and Clever et al (1993). Relative to the classical problem of the Horton-Rogers-Lapwood which has been documented in many books, see, for example, Ingham and Pop (1998, 2002) and Nield and Bejan (1999), only a few works have been devoted to the onset of convection under the action of harmonic vibration. Among existing thermo-vibrational studies in porous media saturated by a pure fluid we mention the works of Zen'kovskaya (1992) and Zen'kovskaya and Rogovenko (1999) in an infinite layer heated from below or above, Khallouf et al (1996) and Sovran et al (2000) in a rectangular cavity heated differentially, Bardan and Mojtabi (2000) in a rectangular cavity heated from below. Also, Jounet and Bardan (2001) consider the thermohaline problem in a rectangular cavity. Finally Sovran et al (2002) consider the effect of vibration on the onset of Soret driven convection in a rectangular cavity. In addition, Rees and Pop (2000,2001, 2003) have recently reported the effect of g-jitter on some boundary-layer problems. It should be emphasized that vibration-induced natural convection may exist even under weightlessness. This phenomenon is in contradiction with the common belief that natural convection cannot exist in space. Further research showed that a spacecraft in orbit is subject to many disturbing influences, see Nelson (1994). These influences result in the production of residual accelerations, which are commonly referred to as 'g-jitter'. It is important to note that these accelerations occurring on microgravity platforms may induce disturbances in space experiments that deal with liquids in the presence of density gradients. The construction of the international space station has increased interest in the influence of g-jitter on convective phenomena. The term g-jitter is used to describe a residual acceleration of 10~^ g fluctuating with a frequency that varies from 10~^ to hundreds of hertz. This acceleration is the result of crew activity as well as machinery on board the space station. One of the aims of the space station is to perform experiments under zero gravity conditions, i.e. without natural convection. It is well known that the g-jitter can produce drastic disturbances in space experiments as, for instance, in solidification processes during which mushy zones modelled as porous media may appear. With reduced gravity, other forces, which are normally masked on earth, may play a dominant role in buoyancy-driven convection. For a better understanding of the g-jitter effects, it was suggested that harmonic oscillations might be used to model this phenomenon, see Alexander (1994).

M. C. CHARRIER MOJTABIET AL.

263

The objective of this chapter is to study the effect of this vibrational mechanism on doublediffusive convection in a porous medium with or without Soret effect. We first present the so-called time averaged formulation applied to the double-convective oscillation in a porous medium in the framework of a Darcy-Boussinesq approximation. This formulation, restricted to the limiting case of high-frequency and small-amplitude vibration, can be effectively applied to investigate thermo-soluto-vibrational convection. We will later show, by using the scale analysis method, what is meant by high frequency and small amplitude. Under the Soret effect, the temperature gradient can produce mass flux in a multicomponent system, see for example De Groot and Mazur (1984). The influence of vibration on the thermo-solutal convective motion with Soret effect has been studied in an infinite horizontal fluid layer, see Gershuni et al. (1997,1999). The governing equations were described by a time-averaged formulation, which can be adopted under the condition of high-frequency and small-amplitude vibration. Their results showed that vibrations could drastically change the stable zones in the stability diagram. Generally, vertical vibrations (parallel to the temperature gradient) increase the stability threshold of the conductive mode. Smorodin et al. (2002) studied the same problem under low frequency vibration. They concluded that the synchronous mode has a stabilizing effect on the onset of convection. In this chapter, we describe the problem of vibrational double-diffusive convection and write down their basic system of equations in the framework of the standard DarcyBoussinesq approximation. The system for the mean field is obtained by applying the averaging technique. In the first part of the chapter, the thermo-convective motion in an infinite horizontal layer and confined cavity saturated by a binary mixture is studied. A linear stability analysis is carried out. The influence of the direction of vibration on the stability threshold is investigated. Stationary and Hopf bifurcations are investigated and the corresponding convective structures under the combined effects of vibration and gravitational accelerations are examined. In the second part, a numerical simulation has been carried out which allow us to corroborate the results obtained from the linear stability analysis.

10.2 MATHEMATICAL FORMULATION The geometry of the problem is a rectangular cavity filled with a porous medium saturated by a binary mixture, as shown in Figure 10.1. The aspect ratio is defined as A = L/H, where H is the height and L is the length of the cavity. The boundaries of the cavity are rigid and impermeable, the horizontal ones can be heated from below or above while the lateral ones are thermally insulated and impermeable. The governing equations are written in a reference frame linked to the cavity. We suppose that the porous medium is homogeneous and isotropic and that Darcy's law is valid. The binary fluid is assumed to be Newtonian and to satisfy the Oberbeck-Boussinesq approximation. The vibration frequency is large and the amplitude of movement is small enough for the averaging method

264

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

H

T = T2, Jm-n = 0,

Vn

=0

6T _ ac dx ~ dx

dx

T = Ti,

J ^ - n = 0,

dx

y.n =0

Figure 10.1 Geometry and coordinate system. to hold. Other standard assumptions, local thermal equilibrium, negligible heating from viscous dissipation, etc., are made. The density p of the binary fluid depends linearly on the local temperature and local concentration of the denser component: p = pref [1 - I3T{T

- Tref) - Pc{C

- Cref)] ,

where the subscript *ref' is taken as the reference state, and the coefficient of volumetric expansion with temperature is represented as PT and for concentration /3c which are assumed constant. It is noted that the expansion coefficient pr is usually positive while 13c can be positive or negative. The entire system, including the cavity and the porous medium oscillates along the axis of vibration e = cos a i -h sin a j , where a = (i, e) is the angle of vibration, following the displacement law b sin cut e, where h is the displacement amplitude and cj is the angular frequency. The gravitational field is then replaced by the sum of the gravitational and the vibrational accelerations in the Darcy equation: g —^ g — huP" smut e. 10.2.1 Direct formulation Using the filtration velocity V, the pressure P , the temperature T and the concentration C as independent variables, the governing equations with Darcy-Boussinesq model can be written as:

v - y = o, PodV W 6*" V " - ' ^ ^ - / ^ o [ / 3 T ( T - T r e f ) - ^ c ( C - C , e f ) ] {g - boj'shiut e) ^V, 8T {pc).-^ + {pc)fV-VT

=

X.^^T,

(10.1)

M. C. CHARRIER MOJTABIET AL.

265

In equation (10.1), (pc)* is the effective heat capacity per unit volume of the medium, {pc)f is the heat capacity of fluid per unit volume, A* is the effective thermal conductivity of porous medium, D^ is the effective mass diffusivity coefficient, a' thermo-diffusion ratio, po density of fluid, and the porosity and permeability of the porous system filling the cavity are e* and X, respectively. The velocity is defined by V = U{t,x,y)i-\-V{t,x,y)j. Then the boundary conditions are given by: V{t,x,y

= 0) = 0,

V{t,x,y = H) = 0,

r(i,x,y = 0 ) = T i ,

T{t,x,y = H)=T2,

dT — {t,x = 0,y) = Uit,x = 0,y) = 0 , dT — {t,x = L,y) = U{t,x = L,y) = 0,

dC dT — + a ' — = 0 ,

^ + "'^=0. dC -^it,x dC —{t,x

(10.2)

= 0,y) = 0, = L,y) = 0.

10.2.2 Time-averaged formulation In order to study the averaged behaviour of the system (10.1), we use the time-averaged method. This method is used under the condition of high-frequency and small-amplitude vibration. The application of the averaging method only allows the mean velocity, concentration and temperature to be solved, see Zen'kovskaya and Simonenko (1966). The basic idea consists of treating the non-stationary flow with an approach similar to that used in the study of turbulent flows. This technique is widely used in different areas of physics and mechanics. According to this method, each field (velocity and temperature) is divided into two parts: the first part varies slowly with time (i.e. the characteristic time is large with respect to the vibration period) and the second part is r periodic and varies rapidly with time, i.e. the characteristic time is of the order magnitude of the vibration period. Under these conditions, it is shown mathematically that two different time scales exist, which make it possible to subdivide the fields into two parts; fast ( V , T', C", P') and slow ( V , r , C , F ) :

y(M, t) = y(M, t) + v'{M, ujt),

r(M, t) = T[M, t) -h T\M,

P{M,t) = P{M,t) -\-P'{M,ujt),

C{M,t) = C{M,t)^

ut),

C(M,oot).

For a given function / ( M , t), the average value is defined by

f{M,t):=-

1 /'*+^/2 ^ Jt-r/2

f{M,s)ds.

On substituting expressions (10.3) into equations (10.1) and boundary conditions (10.2), and averaging the resulting system over the vibrational period, a system for averaged field is obtained. This system is dependent on the average of the product of the oscillating

266

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

parts:

- ^

e dt

= - V P + po [/3r(T - T2) - ^c(C - Cef)] pfe

!ff

{pc).-^

_

_

+ {pc)fV • VT + {pc)fV' • VT' = A.V^T,

f^'n

_

_

e * — + y . VC + ( p c ) / V . VC - iP^CV^T 4- a ' V ^ C ) . On subtracting equation (10.4) from equation (10.1) and using expressions (10.3), we find the oscillating system, which depends on the averaged parts:

v-y' = o, PodV = -WP' + po [0T{T - T2) - /3c{C - Cref)] bu^ sinute e dt + Po WTT' - PcC] [gj + bu^sinute) - pobuj\^TT-l3cC')smujte

dV — (pc)*— + {pc)^V'. VT + {pc)fV'. VT' + {pc)fV • VT' - {pc)fV'' v r =

- ^V

,

Kv^r,

e * ^ 4 - V ' - V C + (pc)/V'-VC' + {pc)fV ' VC" - {pc)fV' 'VC - D,{V'^C' + a'V^T'). (10.5)

It is evident that equations (10.4) and (10.5) are coupled, and the coupling terms are related to po^^^ (i^rT' - ^cC") sin a;te, {pc)fV' • VT' and {pc)fV' • V C which appear following the averaging procedure. The key step in resolving the closure lies in expressing the oscillating (fast) fields ( V , T', C") in terms of the scalar slow fields (T, C). For this reason, an order of magnitude analysis can be carried out on the system of equations having a rapid time evolution. 10.2.3 Scale analysis method In order to obtain the order magnitude of the relevant terms in equation (10.5), we perform a scale analysis method. This method of pedagogical importance has been successfully used in predicting the boundary-layer approximations, optimal geometries and critical parameters, see for example Bejan (1995, 2000, 2003) and Bejan and Nelson (1998). We

M.C.CHARRIERMOJTABIETAL.

267

use the following scales in the oscillatory system of equations: 0(T - Tret) « Ti - T2 = A T ,

0{C - Cref) ^ Ci - C2 = AC ,

o(ffi)..(.,, . dz n HBy introducing these scales into the momentum equation governing the rapid evolution (10.5), and assuming further for the oscillating temperature and concentration fields that T' < AT and C < AC, we obtain: v' ^ e*6a;(^TAr - ^ c A C ) .

(10.6)

Equation (10.6) has been obtained from the consideration that under a high frequency, the buoyancy terms, containing A T and A C are balanced by the inertia term. The domain of validity of this assumption can be obtained from the following inequality (inertia > friction): ^ « l .

(10.7)

By adopting the same procedure in the oscillatory energy and concentration equation, the scales of the oscillating temperature and concentration fields are found: T'«^^^(/3TAT-/3CAC),

(10.8)

C ' « ^ ( ^ T A T - ^ C A C ) .

Equation (10.8) gives the criteria for defining the small-amplitude vibration. The above scales are valid under the following conditions (transient term ^ diffusive terms):

As at the start of our analysis, we assumed that the buoyancy terms containing AT and AC are the dominant convective generating mechanisms, the final step is to validate this assumption. Utilizing expressions (10.8) and with some easy manipulations, we obtain: a;2 > A(e/3^AT - /3cAC). Under condition (10.10), the term po [/3T{T -T2) dominant buoyancy force in the momentum equation.

PC{C

(10.10) - Cref)] buj'^ sinute

is the

10.2.4 Time-averaged system of equations By applying the assumptions (10.7), (10.9) and (10.10) to the oscillatory system of equations, we may simplify the system of equations (10.5). Then by using Helmholtz's decomposition, we eliminate the oscillatory pressure. This allows us to obtain the oscillatory velocity, temperature and concentration, which upon substitution into equation (10.4),

268

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

provides us with the time-averaged system of equations. The dimensionless governing equations for the mean flow, averaged over the vibration period, can be written as follows:

B ^

-\-V = -VP + Ra(T + V^C)j + R v ( W T - h ^ W c ) • V (T-\--CJ dt fir 1 e ^ + V -VC = :^iV^C ot Le V - W T - 0 ,

(cosai +

sinaj),

' -V^T),

V - W C = 0,

where WT and W c are the solenoidal vectors corresponding to the temperature and concentration, respectively. The corresponding boundary conditions are given by: VTT

• n == W c • n = 0,

J ^ - n = 0, J^-n-0,

r-Ti T-r2

dT dC ^ — = ^- = 0

at t / - 0 , at 2/ = l , at

(10.12)

x-0,A.

System (10.12) depends on eight parameters: the thermal Rayleigh number Ra = Kg^/STH/va^, the vibrational Rayleigh number Rv = i?^Ra^, the separation factor ^ — —Ci{l — Ci){Pc/PT)DT/D*, the normalized porosity e (e = e*/a) where a = {pc)^/{pc)f, the Lewis number Le (Le = a* /D* in which a* is the effective thermal diffusivity and D* is the effective mass diffusivity), the coefficient of the unsteady Darcy term in the momentum equation B = Da/aePr* (in porous media B « 10~^ and Da represents the Darcy number Da = K/H^), and finally a the direction of vibration with respect to the heated boundary.

10.3 LINEAR STABILITY ANALYSIS When the direction of vibration is parallel to the temperature gradient, i.e. a = 7r/2, there exists a mechanical equilibrium, for both an infinite horizontal layer and a confined cavity, which is characterized by: Vo^O,

To = 1 - 2 / ,

Co = constant - y ,

WTO = 0,

WCO = 0.

(10.13)

269

M. C. CHARRIER MOJTABIET AL.

However, for other directions of vibration, we may obtain a quasi-equilibrium solution only for the infinite horizontal layer. This is characterized by: Vo = 0,

To = l - y ,

Wroy = 0,

Co = ci-y,

VKTO,

Wco. =C3-ycosa,

=C2-ycosa,

Wcoy = 0 .

(10.14)

It should be noted that, for a confined cavity, upon considering the boundaries conditions we conclude that the solenoidal fields are not equal to zero for a ^ 7r/2. 10.3.1

Infinite horizontal porous layer

The fields are perturbed around the (quasi-) equilibrium state in order to investigate the stability of the conductive solution. Then, after linearization, the disturbances are developed in the form of normal modes. It is assumed that the perturbation quantities are sufficiently small and the second-order terms are neglected (rj = c — 6):

= R a — [(1 + iP)e + VT?] + Rv(W^To. + -\-ipWcoJ

-{WTO.

dxdy [\

^WcoJg-^

eJ

e

H)

+ I1+ - ^ ^

cos a

[(1 + i^)FT + i^Fr,] sina

H)

d^ [{1 + ip) FT+ il^Frj] cos a. dxdy

In equation (10.15), the stream functions are defined as follows: W^Tx

=

dFr dy dF„

^-nx — Q

WTy

,

'W'ny —

sma

cos a

1+

-lyiWro^+Wcojl

. '0

dFr dx OF^ dx

WCx

dFc dy

'^Cy

=

(10.15)

d_Fh dx

- •

_ ^ dx

For the energy equation, we have: dl dt

^ _ y2^ dx

(10.16)

270

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

For the concentration equation, we obtain: dt

dx

(10.17)

Le

For Helmholtz decomposition, we find: V^FT

dy cos adr] V'^Fn = -^cosa dy

de . 7— dx sin a , df]

dx

(10.18)

sin a .

The corresponding boundary conditions are given by: dx dgx,!) dx

' ^ '

dy dx ^ dr]{x,l) ^ dFTJx,!) dy

dx ^ 9F,(x,l)

dx

dx

(10.19) = 0.

The disturbances of the normal mode are introduced as follows: (^(x, y, t),e{x, y, t),r]{x, y, t),FT{x, y, t),Frj{x, y, t)) = (ay), 0{y)My). FT{y),Fr,{y)) exp{Xt + Ikx), (10.20) where P = —1. We may obtain the following amplitude equations. For the momentum equation (D = d/dy): - {XB + 1) {D^ - e) I = IA:Ra [(1 + ^ ) ^ + il^i)] sin a

l + e^V+^r)' / e cos a

-lkD{WTo.^Wco.)

1 +el^V+^^' cos a e

\k{WTo.+WcoJD

;fcMi + ^ lk{l

(1 + V)^T + i^Fr, Sin a +

i, -]D

(1 + IP)FT + rpFr,

)sa > . (10.21)

For the energy equation:

A^ + ifc| = (D^ - k'^)e.

(10.22)

M. C. CHARRIER MOJTABIET AL.

271

For the concentration equation: Le For the Helmholtz decomposition: {D'^ -k'^)FT

= D§ cos a - IkO sin a ,

{D'^ ~ k'^)Fr^ = Dfj cos a - Ikfj sin a .

(10.24)

In the above equations, WTO^ and Wco^ are defined as follows:

The corresponding boundary conditions for equations (10.21)-( 10.24) are given by: |(x,0) = e{x,{)) - Dfi{x,Q) = FT{X,0)

= F^(x,0) = 0,

| ( x , 1) = e^(x, 1) = Dfi{x, 1) = FT{X, 1) = F^(a;, 1) = 0. The systems (10.21)-( 10.24), along with the boundary conditions (10.25) correspond to the spectral amplitude problem with the decay rate, A, as an eigenvalue and with the amplitude as the eigenvector components. The characteristic value of the decay rate depends on all the parameters of the problem namely A = A(Ra, Rv, ^ , a, e, A;, Le) and, generally, the decay rate A is complex, i.e. A = Ar + lA^, because the spectral amplitude problem is not self adjoint. If A^ = 0 then the stability boundary is determined by the condition A = 0, i.e. the stationary bifurcation. If Af ^ 0, then the stability boundary is determined by the condition A^ = 0. In this case \i = Vth which is the frequency of the neutral oscillation. It should be emphasized that in accordance with the timeaveraged formulation this frequency {Vth) is smaller than the frequency of vibration. The stream function perturbation is introduced as 0, the temperature perturbation as 6 and the mass fraction perturbation as c. The stream function perturbations are designated as (j)e and 0c for corresponding solenoidal fields WT and Wc. To facilitate our study, the transformations 7] = c — 0 and ^p^^ = ip^ — ipo are used. For an infinite horizontal porous layer, we introduce disturbances of the normal mode in the following form: N

(f = 22 ^^ sin(i7ry) exp{at -h Ikx), i=l N-l

T] = 2_] ^i cos{i7ry) exp{at 4- Ikx), i=:0 N

(frj = / J 9i sin{i7ry) exp(crt + Ikx), 2=1

N

6=^

bi sm{i7ry) exp{at -\- Ikx),

i=l N

(pe = y j di sm{i7ry) exp{at + Ikx), i=l

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

272

where k is the wave number, a is the decay rate and (a^, bi,Ci,di,gi) are the amplitudes. The corresponding linear stability problem is solved using the Galerkin method. Horizontal vibration Figure 10.2 shows the stability domain for different vibrational parameters in the (Ra, ^p) stability diagram. This diagram is characterized by stationary and oscillatory bifurcations. For ^ > 0 the bifurcation is always of the stationary type, while for ^ < 0, we may obtain oscillatory (Hopf) or stationary bifurcations. The computations are performed for e* = 0.3, B = 10~^ (usual values used in porous media) and Le = 10. From the results we conclude that horizontal vibration has a destabilizing effect on both stationary and Hopf bifurcations. One of the interesting features of Figure 10.2 is that we may obtain long-wave mode instability in the regions which, under static gravity conditions, are infinitelyAinearly stable. The existence of these regions is due to the vibrational mechanism. Vertical vibration The aim of this section is to provide a qualitative picture of the flow and temperature fields to complete the results of our stability analysis. In order to study the effect of vibration on the convective pattern, we set Le — 2, -0 = 0.4, A = 1 and Ra = 30 and changed the R = 0.1^

Rac "^.^

- -A-o---A- -0-

45-

R = 0.5k 35\

Rdics {R = 0.1) Raco {R = 0.1) Racs (i? = 0.1, Arc = 0) Raics{R = 0.5) Raco (i? = 0.5) Racs {R = 0.5, kc = 0)

25-

'A. ••i..

^ 'J

\

15

•©•-9. -0.1

-0.08

-0.06

-0.04

0.02

-0.02

...®

004 ,-•0-

006 .«--e-

-0--9 O08 Ol' .e--9--9

.0*

-15 Figure 10.2 Stability diagram for stationary and oscillatory convection for Le = 10, e = 05 and ^ = 10~^

M. C. CHARRIER MOJTABIET AL.

273

value of the vibrational Rayleigh number Rv. The calculations are performed for e = 0.5 and 0.7 and results are presented in Table 10.1. These values are chosen according to the results of the stability analysis. We conclude from Table 10.1 that, for the selected values of Le, e, i/;, A and Ra, vibration reduces the Nusselt number and we may obtain a conductive solution. In addition we find that for the combination of Rv, V^ and e we have the interesting relation Rvc(l -h i/^/e) = constant. For the case under investigation this constant is 31.5. The linear stability analysis is carried out for different sets of parameters 0 < Rv < 100, 2 < Le < 100, ^ = - 0 . 2 and e = 0.5, 0.7. The results for Le = 2 and e = 0.5 are presented in Table 10.2. As can be observed from this table, two types of bifurcations; namely stationary and Hopf bifurcations, may be distinguished. For the stationary bifurcation, the principle of the exchange of stability is valid, i.e. a G M, and the marginal state is determined (cr = 0). For the Hopf bifurcation (a = ar -\- lujo)'-, the marginal state corresponds to cr^ = 0. In the case of the layer heated from below the Hopf bifurcation is present only for negative separation factors. In this case the Hopf bifurcation is formed before the stationary bifurcation, i.e. Raco < Racs. It can be concluded from Table 10.2 that vertical vibration has a stabilizing effect; it increases the critical value of thermal Rayleigh number for the onset of convection. This conclusion is true for both the stationary and the Hopf bifurcations. In addition, vertical vibration reduces the critical wave number (fcc, kco) and the Hopf frequency (ojo)-

Table 10.1 Effect of mechanical vibration on the Nu number for A = 1, a = 7r/2, Le = 2, V' = 0.4, Ra = 30, and (a) e = 0.5, and (b) e = 0.7. (a) (b) Rv

Nu

Rv

Nu

5 10 15 16 17 17.5 17.53

1.204 1.1483 1.0789 1.0604 1.0359 1.0124 1.0084

5 10 15 16 19 19.7 20.02

1.2057 1.1528 1.0942 1.0604 1.0329 1.0156 1.0034

Table 10.2 Effect of vibrations on the stationary and Hopf bifurcation (Le = 2, ^ -0.2 and e = 0.5). Rv

rC3/cs

0 10 50 100

153.19 157.53 173.63 193.60

r^CS

4.75 4.73 4.65 4.54

iXiOiQQ

95.43 91.1% 107.1 117.8

i^CO

2.59 2.56 2.41 2.26

CJo

10.78 10.75 10.50 10.26

274

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

10.3.2 Limiting case of the long-wave mode The results of the previous section indicate that the long wave mode (k = 0) is the dominant mode of Soret-driven convection under the effect of mechanical vibration in binary liquids. For this reason, we study the special case of long wave mode theoretically. In some related studies, see Gershuni et ai (1997, 1999), this analysis results in a closed form relation for the stability threshold. To obtain such a relation, a regular perturbation method with the wave number as the small parameter is employed: OO

OO

CO

„=0

„=0

„=0

n=0

n=0

n=0

OO

OO

OO

Vxvy.

v//

By substituting expressions (10.26) in the amplitude equations resulting from linear stability analysis, we find for zeroth-order: ^0=0,

^0 = 0,

r]o = constant,

(pQ^ = 0,

(p^^ = 0,

CTQ = 0.

For the first-order: ^1 = ^1=0,

I [Ra -

RYD{WTO.

7/1 == constant,

+ ^pWco^) cos a] V^/yo ,, , 2 ^^ ~ ^^' ipe^ = 0,

iprj^ = 0,

ai = 0.

After invoking the solvability condition, for the second-order we find: (T2

1 eLe

eRa-^ -I- Rvi/;(1 -h ip) cos^ a 12e

It is clear that (72 G R, which means that instability is of a stationary type; consequently for marginal stability (cr2 = 0) the following relation is obtained: ^^^(l + ^)Rvcos^a^^

(10.27)

We may distinguish different physical situations. 10.3.3

Convective instability under static gravity (no vibration)

In order to check the validity of relation (10.27), the classical case of convective instabifity due to static gravity is studied first. For this situation, we set Rv — 0, which results in:

Ra = 4 r - ,

(10.28)

M. C. CHARRIER MOJTABIET AL.

275

which is identical to the published results in the literature; see for example Schopf (1992) and Sovran ^r ^/. (2001). Convective instability under microgravity conditions In this case the instability excitation is due to vibrational mechanism. In order to determine the stable domain under microgravity conditions or weightlessness (g = 0), the following relation may be used: '0(1 -hV^jLecos^a From equation (10.8), we conclude that instability exists in V^ > 0, corresponding to positive Soret effect. By increasing the direction of vibration a from zero to 7r/2, the stable domain increases. In addition in the interval of -0 G [—1,0], the mono-cellular convection is not possible. Also, it can be noted that increasing the Le number decreases the stable domain. In the case of a = 7r/2, the long-wave mode (mono-cellular convection) does not exists under high frequency and small amplitude. Convective instability under the simultaneous action of vibration and gravitation In this case both the instability mechanisms, vibrational and gravitational, are in action. The stability boundary can be determined from the following relationship: ~

2^(1 + 0)Le cos2 a R?

'

^ ^ ^

When the direction of vibration is not parallel to the temperature gradient i.e. a 7^ 7r/2, there is a possibility of mono-cellular convection in all regions of the stability diagram (Ra, 0) except in the region 0; G [—1,0]. However, for the case of vertical vibration in which the direction of vibration is parallel to the temperature gradient, for stability domain we have: Ra== - ^ . (10.31) xpLe

Relation (10.31) demonstrates that under vertical vibration, the critical value of the thermal Rayleigh number for the onset of mono-cellular convection does not depend on the vibrational parameter. But it should be noted that the mono-cellular convection appears from smaller values of separation factors.

10.4

COMPARISON OF THE RESULTS WITH FLUID MEDIA

The objective of the present section is to compare the results of long wave mode in the case of fluid layer with those of porous layer. The results of long wave mode for the onset of mono-cellular thermo-solutal convection in an infinite horizontal layer under variable

276

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION Table 10.3 Comparison of the results of long-wave mode for the onset of convection in the presence or in the absence of vibration.

. ^' .

No vibration

Microgravity conditions

Fluid layer

rv-3, — -77— V'Le

-K/V —

^^^^"^

Ra = - i ^

Rv =

layer

V'Le

, ,. ,—TT-; ^— V(14-'/')Lecos^ a

i2e

Simultaneous action of vibration and gravitation rv,a —

-V>Le±v/(V>Le)2-)-2880i/>(l+V>)Lecos2Qfl2 2V;(H-V')Lecos2aH2

p

- e V > L e ± i / ( e V ; L e ) 2 - f 4 8 e - 0 ( l - f - t / ' ) L e c o s 2 a f l2

_

V(l+V')Lecos2 a

orientation of vibration is reported by Razi ^r al. (2002, 2004). The results are presented in Table 10.3, the comparison of the results reveals that the Darcy model can provide us with a very good approximation for the physical behavior of the fluid system.

10.5 NUMERICAL METHOD The numerical simulations for a confined cavity are performed for vertical and horizontal vibration. The calculations are made for different aspect ratios A — \ and A — 10. The 27 X 27 collocation points are used for A — \, while 63 x 27 collocation points are used for A — 10. In order to solve the system (10.11) with the corresponding boundary conditions (10.12), the projection diffusion algorithm is used, see Azaiez ^r a/. (1994). The linear (viscous) terms are treated implicitly using a second-order Euler backward scheme, while a second-order semi-explicit Adams-Bashforth scheme is employed to estimate the nonlinear (advective) terms. We apply this method to an advection-diffusion equation such as (F is a general coefficient):

g + (n.v/) = rvV,

(10.32)

which can be discretized as follows: (3/2)/"+^ - 2 / " + (1/2)/ n - l

At

py2^„+i _ [2(„ . v / ) " - (u • V / ) " - i ] .

(10.33) Equation (10.33) may be written in the form of the following Helmholtz equation:

(v^-ft)/ n+1

_

(10.34)

A high-accuracy spectral method, namely, the Chebyshev collocation method with the Gauss-Lobatto zeros as collocation points, is used in the spatial discretization of the operators. The successive diagonalization method is applied to the inverse of these operators, see Azaiez et al. (1994).

M. C. CHARRIER MOJTABIET AL.

277

10.5.1 Vertical vibration In order to demonstrate the effect of vibration on the convective structure under different aspect ratios, we begin with the study of convection under gravitational acceleration: A = 10, Le = 2, e = 0.5, ip = 0.4, Racs = 13 and Rv = 0. The streamlines corresponding to this case, which are characterized by six convective rolls, are shown in Figure 10.3. The numerical result for the onset of convection is in good agreement with the result obtained from the linear stability analysis of an infinite horizontal layer heated from below, Racs = 12.95 andfcc= 1.94. It should be added that, based on our numerical simulations, we conclude that .4 = 10 provides a good approximation of an infinite layer. Figure 10.4 represents the effect of vertical vibration. It is clear that vibration changes the convective structure dramatically. The numerical results are in good agreement with the linear stability analysis, Racs = 15.04 andfees= 0.01. The result of the Hopf bifurcation for the temporal evolution of velocity for Le = 2, e = 0.5, Rv = 100 and ip = -0.2 is presented in Figure 10.5. As mentioned earlier, the Hopf bifurcation appears for negative separation factors. The numerical result shows that the critical values corresponding to the Hopf bifurcation are: Raco = 118.5, kco = 2.2 and ujo = 10.08. These are in good agreement with stability analysis results, Raco = 117.83, kco = 2.26 and OUQ — 10.25. 10.5.2 Horizontal vibration In this case, we set A = 1, Ra = 6, Le = 2, e = 0.5, if) = 0.2 and R = 0.3. The value of Ra is set to such a value so that only the vibrational mechanism is in action. Figure 10.6 shows the corresponding fluid flow structure and temperature distribution; the stream functions are characterized by symmetrical four-vortex rolls. This structure is a typical example of an imperfect bifurcation as was observed earlier in convection under microgravity conditions, see Gershuni et al (1982). The sum of stream functions is zero in this case. If we further increase the thermal Rayleigh number to Ra = 13.15, gravitational

Figure 10.3 Onset of stationary convection for A Rv = 0 and Racs = 13.

10, Le = 2, e = 0.5, t/^ = 0.4,

Figure 10.4 Effect of vertical vibration on the onset of convection for A = 10, Le = 2, e = 0.5, ip = 0.4, Rv = 20 and Rac^ = 15.7.

278

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

0.0002

3 7 5 0.0001 [ 3 1 -1

O

S

0

-0.0001

'

0

I

2

1

1

4

6

,|i|||||lli|||||||||||||

[

-0.0002 170

190

Time

210

230

Figure 10.5 Onset of oscillatory convection for A = 10,^ = —0.2,e = 0.5, Rv = 100, Raco = 118.5 andcjo = 10.08.

(b)

Figure 10.6 (a) Stream functions, and (b) isotherms, for A = 1, Le ip = 0.2, Ra = 6 and jR = 0.3.

2, e = 0.5,

acceleration will also be in action, this value is chosen according to linear stability analysis results: for Le = 2, e = 0.5, '0 = 0.5, R — 0.3, the critical Rayleigh number Racs « 14. The intensity of the convective motion will be accordingly increased and the sum of stream functions at all points in the domain is a good criterion for representing the intensity of convective motion. This case is shown in Figure 10.7. As can be seen from the figures, a symmetry breaking structure is obtained. This is explained by the coalescence of the two rolls with the same sign in the diagonal direction and the existence of two separate offdiagonal rolls with weaker intensity. If the thermal Rayleigh number is further increased

M. C. CHARRBER MOJTABIET AL.

279

(b)

Figure 10.7 (a) Stream functions, and (b) isotherms, for A = 1, Le = 2, e = 0.5, il) = 0.2, Ra = 13.15 and R = 0.3.

Figure 10.8 (a) Stream functions, and (b) isotherms, for A = 1, Le = 2, e = 0.5, ip = 0.2, Ra = 15 and i^ = 0.3. to Ra = 15, a single convective roll appears, which means that the gravitational effect is more important than the vibrational effect, see Figure 10.8. For A = 10, the case corresponding to Le = 2, e = 0.5, ip = 0.2 and i? = 0.1 and Ra = 15 is considered. The typical four-vortex structure is presented in Figure 10.9. Further increase in Ra results in the appearance of a multi-cellular convective regime, see Figure 10.10.

280

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION •*'<"

I

0^

7

10

Figure 10.9 Stream functions for A = 10, Le = 2, e = 0.5, tp = 0.2, Ra = 15 and R = 0.1.

Figure 10.10 Stream functions for A = 10, Le = 2, e = 0.5, xp = 0.2, Ra = 17 and R = 0.1.

10.6

THE ONSET OF THERMO-SOLUTAL CONVECTION UNDER THE EVFLUENCE OF VIBRATION WITHOUT SORET EFFECT

In this section, the onset of double-diffusion convection, thermohaline convection, is analyzed. Although the realization of this problem experimentally faces us with practical problems, it has the advantages that we may obtain closed form analytical relations for the onset of convection. Thus, in the present section, in order to gain better understanding of this complex problem, the effect of high-frequency and small-amplitude vibration on the onset of convection in an infinite horizontal porous layer is studied. The direction of vibration is parallel to imposed temperature and concentration gradients. 10.6.1 Linear stability analysis The linear stability analysis procedure is the same as in Section 10.3. For the vertical vibration, the mechanical equilibrium is possible. In order to perform linear stability analysis, the temperature, concentration, velocity and solenoidal field are perturbed around the equilibrium state. Using the linearization procedure, we find the amplitude equation, which admits exact solutions of sinusoidal form. Stationary bifurcation For the onset of stationary double-diffusion convection, the stability domain is obtained from: 1 (P+7r2)2 /^ N\^ e Ra.ct = (10.35) A;2 •^•^•^7j^^fc^ + .^ l-t-ATLe

(Rv = R^Ral).

281

M. C. CHARRIER MOJTABIET AL.

If we set dRast/dfc^ =: 0, the critical Rayleigh and wave numbers can be found from the following simultaneous system of equations: R^stc =

7r2(H-ArLe)fc2

'

(10.36)

7r^(7r^-fcg)(l + iVLe) R' = (fc2+7r2)(27r2-A:2)2(H-iV/e)

Figure 10.11 illustrates the effect of vibration on the stability threshold for Le = 4 and e = 0.5. As can be seen from this figure, vibrations increase the stability domain in the (Rac, N) plane. From a convective pattern formation point of view, vibrations reduce the critical wave number as can be observed in Figure 10.12. Hopf bifurcation For the onset of Hopf bifurcation we obtain the following relations: 1 + eLe (fc2+7r2)2 (e + N)Le k^

-1 =

/ \

iV\ eJ

P k^ -\-7r^

(10.37)

(A:2+7r^)^(l + 6iVLe^)

150

100 Rac

50h

R = 0.12 R = 0.1 R = 0.08 R=0 -0.2

-0.1

0.1

N

0.2

0.3

0,4

0.5

Figure 10.11 The effect of the vibration parameter on the onset of convection for Le = 4 and e = 0.5.

282

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

R=0 R = 0.08 i? = 0.12

..^MR

= 0.16

Figure 10.12 The effect of vibration on the critical wave number for Le = 4 and e = 0.5. Following the same procedure as for stationary bifurcations, we may obtain the following simultaneous system of equation for obtaining the critical parameters (Raco, kco)' '^^co

—

R' =

7r^Le{e + N) en'Le{7r'-kl)

k^

(10.38)

It should be added that the Hopf bifurcation appears for negative values of N and the location of co-dimensional point is given by: A T ^ - -

1 eLe2

•

(10.39)

It is obvious from relation (10.39) that vibrations do not change the location of the codimensional point. Table 10.4 shows the influence of the vibration on the critical values of the Rayleigh number, the wave number and the Hopf frequency. In this study Le = 4, e = 0.5 and A^ = - 0 . 3 . We deduce from Table 10.4 that vibration increases the stability domain of the Hopf bifurcation and it reduces the critical wave number and the Hopf frequency.

M. C. CHARRIER MOJTABIET AL.

283

Table 10.4 The influence of vibration on the onset of oscillatory convection. R

LV^CO

rZco

0 0.01 0.08 0.1

148 148.4 191.2 249.6

3.140 3.136 2.698 2.220

CJo

18.46 18.43 16.04 13.85

Double-diffusion convection under microgravity conditions In this situation, in the interval of N e ]—e, —1/Le[ the onset of stationary convection is possible. The critical wave number and critical vibrational Rayleigh number are found as follows: kc = v27r, 1 277r2 ^ - ^ - - - ( l + iVLe)(l^iVA)-l--

iV6]-e,-l/Le[.

(10.40)

Under microgravity conditions and high-frequency and small-amplitude vibrations, there is no possibility of the Hopf bifurcation.

10.7

CONCLUSIONS

In this chapter two-dimensional thermo-solutal convection both with and without Soret effect, under the influence of a mechanical vibration has been studied analytically and numerically. The time-averaged formulation is used. The influence of the direction of the vibration for different cavity aspect ratios has been studied for the case of Soretdriven convection and the corresponding fluid flow structures explained. For an infinite horizontal layer, linear stability analyses of equilibrium and quasi-equilibrium states have been performed. Our results show that, depending on the orientation of vibration, different effects on the onset of convection may be expected. It is found that, when the direction of vibration is considered parallel to the temperature gradient, vibration has a stabilizing effect on both the stationary and the Hopf bifurcation. The action of vibration reduces the number of convective rolls and the Hopf frequency. However, when the direction of vibration is perpendicular to the temperature gradient, vibration has a destabilizing effect. New instability regions appear in the bifurcation diagram (Ra, V^) in which the preferred pattern is the mono-cellular convective roll. In the limit of long wave mode, the regular perturbation method is used to determine the stability boundary through an analytical relation. A comparison with the long wave mode in a horizontal fluid layer filled with a binary mixture under the action of vibration is given. For a confined rectangular cavity with different aspect ratios, numerical simulations using a spectral method are performed which corroborate the results of the stability analysis. It is

284

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

shown that the vertical vibration can reduce the number of convective rolls. For horizontal vibrations, thefluidflowstructures are sought. For afixedvalue of the vibrational Rayleigh number, we increase the thermal Rayleigh number from a value much less than the critical value corresponding to the onset of convection in an infinite layer. A symmetrical fourvortex structure is observed first, then a diagonal dominant symmetry breaking structure and finally a mono-cellular structure. These results are similar to the results obtained in a cavity filled with a pure fluid under the action of vibration in weightlessness. The interesting result of this study is that, by appropriate use of the direction of residual acceleration in the microgravity environment, significant enhancement in heat and mass transfer rates may be obtained. As the problem depends on several parameters, obtaining closed form relations are of highest importance. The linear stability analysis of an infinite horizontal porous layer under a vertical vibration with imposed temperature and concentration gradients is performed, the Soret effect is neglected. This analysis provides us with closed form relations for the stationary and Hopf bifurcation. It is shown that under microgravity conditions, the onset of the Hopf bifurcation is not possible. This study contributes to the ongoing research in order to define the appropriate microgravity environment.

REFERENCES Alexander, J. I. D. (1994). Residual gravity jitter onfluidprocesses. Micro. ScL Tech. 7, 131^. Azaiez, M., Bemardi, C, and Grundmann, M. (1994). Spectral method applied to porous media. East-West J. Numer. Math. 2, 91-105. Bardan, G. and Mojtabi, A. (2000). On the Horton-Rogers-Lapwood convective instability with vertical vibration. Phys. Fluids 12, 1-9. Bejan, A. (1995). Convection heat transfer (2nd edn). Wiley, New York. Bejan, A. (2000). Shape and structure, from engineering to nature. Cambridge University Press. Bejan, A. (2003). Simple methods in convection in porous media: scale analysis and the intersection of asymptotes. Int. J. Energy Res. 11, 859-74. Bejan, A. and Nelson, R. A. (1998). Constructal optimization of internalflowgeometry in convection. ASMEJ. Heat Transfer 120, 357-64. Biringen, S. and Peltier, L. J. (1990). Numerical simulation of 3D-Benard convection with gravity modulation. Phys. Fluids A 2, 754-64. Clever, R., Schubert, G., and Busse, F. H. (1993). Two-dimensional oscillatory convection in a gravitationally modulatedfluidlayer. J. Fluid Mech. 253, 663-80. De Groot, S. R. and Mazur, P. (1984). Non equilibrium thermodynamics. Dover, New York. Gershuni, G. Z., Kolesnikov, A. K., Legros, J. C, and Myznikova, B. I. (1997). On the vibrational convective instability of a horizontal binary mixture layer with Soret effect. J. Fluid Mech. 330,25169.

M. C. CHARRIER MOJTABIET AL.

285

Gershuni, G. Z., Kolesnikov, A. K., Legros, J. C , and Myznikova, B. I. (1999). On the convective instability of a horizontal binary mixture layer with Soret effect under transversal high frequency vibration. Int. J. Heat Mass Transfer 42, 547-53. Gershuni, G. Z., Zhukhovitskii, E. M., and Yurkov, Yu. S. (1982). Vibrational thermal convection in a rectangular cavity. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza. 4, 94-9. Gershuni, G. Z., Zhukovitskii, E. M., and lurkov, S. (1970). On convective stability in the presence of periodically varying parameter. J. Appl. Math. Mech. 34,470-80. Gresho, P. M. and Sani, R. L. (1970). The effects of gravity modulation on the stability of a heated fluid layer. /. Fluid Mech. 40, 783-806. Ingham, D. B. and Pop, I. (eds) (1998). Transport phenomena in porous media. Pergamon, Oxford. Ingham, D. B. and Pop, I. (eds) (2002). Transport phenomena in porous media. Vol. II. Pergamon, Oxford. Jounet, A. and Bardan, G. (2001). Onset of thermohaline convection in a rectangular porous cavity in the presence of vertical vibration. Phys. Fluids 13, 1-13. Khallouf, H., Gershuni, G. Z., and Mojtabi, A. (1996). Some properties of convective oscillations in porous medium. Numer. Heat Transfer, Part A 30, 605-18. Nelson, E. (1994). An examination of anticipated g-jitter on Space Station and its effects on material processes. NASATM 103775. Nield, D. A. and Bejan, A. (1999). Convection in porous media (2nd edn). Springer-Verlag, New York. Razi, Y. P., Maliwan, K., Charrier Mojtabi, M. C., and Mojtabi, A. (2004). Importance of direction of vibration on the onset of Soret-driven convection under gravity or weightlessness. Eur Phys. J. £15,335-41. Razi, Y. P., Maliwan, K., and Mojtabi, A. (2002). Influence de la direction des vibrations sur la naissance de la convection thermosolutale en presence d'effet Soret. Societe Frangaise Therm. 10, 285-90. Rees, D. A. S. and Pop, I. (2000). The effect of g-jitter on vertical free convection boundary-layer flow in porous media. Int. Comm. Heat Mass Transfer 21, 415-24. Rees, D. A. S. and Pop, I. (2001). The effect of g-jitter on free convection near a stagnation point in a porous medium. Int. J. Heat Mass Transfer^, 877-83. Rees, D. A. S. and Pop, I. (2003). The effect of large amplitude g-jitter vertical free convection boundary layer-flow in porous media. Int. J. Heat Mass Transfer 46, 1097-102. Schopf, W. (1992). Convection onset for a binary mixture in a porous medium and in a narrow cell: a comparison. J. Fluid Mech. 25, 263-78. Smorodin, B. L., Myznikova, B. I., and Keller, I. O. (2002). On the Soret-driven thermosolutal convection in a vibrational field of arbitrary frequency. In Thermal non-equilibrium phenomena in fluid mixtures, Lecture notes in physics, vol. 584 (eds W. Kohler and S. W. Wiegand), pp. 372-88. Springer, Berlin. Sovran, O., Bardan, G., Mojtabi, A., and Charrier Mojtabi, M. C. (2000). Finite frequency external modulation in doubly diffusive convection. Numer Heat Transfer, Part A 37, 877-96. Sovran, O., Charrier Mojtabi, M. C , Azaiez, M., and Mojtabi, A. (2002). Onset of Soret driven convection in porous medium under vertical vibration. In Proceedings of the I2th international heat transfer conference, Grenoble.

286

EFFECT OF VIBRATION ON DOUBLE-DIFFUSIVE CONVECTION

Sovran, O., Charrier Mojtabi, M. C , and Mojtabi, A. (2001). Naissance de la convection thermosolutale en couche poreuse infinie avec effet Soret. CRAS Paris, Serie lib 4, 287-93. Zen'kovskaya, S. M. (1992). Action of high-frequency vibration on filtration convection. Prikl Mekh. Tekh. Fiz. 32, 83-8. Zen'kovskaya, S. M. and Rogovenko, T. N. (1999). Filtration convection in a high-frequency vibration field. 7. Appl MecK Tech. Phys. 40, 379-85. Zen'kovskaya, S. M. and Simonenko, I. B. (1966). Effect of high frequency vibration on convection initiation. Izv. Akad. NaukSSSR, Mekh. Zhidk. Gaza. 1, 51-5.