salt-finger instability in an anisotropic and inhomogeneous porous substrate underlying a fluid...

Saltfinger instability in an anisotropic and inhomogeneous porous substrateunderlying a fluid layerFalin Chen Citation: Journal of Applied Physics 71, 5222 (1992); doi: 10.1063/1.350579 View online: http://dx.doi.org/10.1063/1.350579 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/71/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Equilibrium saltfingering convection Phys. Fluids 7, 706 (1995); 10.1063/1.868596 Influence of viscosity variation on saltfinger instability in a fluid layer, a porous layer, and theirsuperposition J. Appl. Phys. 70, 4121 (1991); 10.1063/1.349134 Onset of thermal convection in an anisotropic and inhomogeneous porous layer underlying a fluid layer J. Appl. Phys. 69, 6289 (1991); 10.1063/1.348827 Saltfinger convection in shear flow Phys. Fluids 27, 804 (1984); 10.1063/1.864708 Thermohaline Instability and Salt Fingers in a Porous Medium Phys. Fluids 15, 748 (1972); 10.1063/1.1693979

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Salt-finger instability in an anisotropic and inhomogeneous porous substrate underlying a fluid layer

Falin Chen Institute of Applied Mechanics, National Taiwan University, Tai$ei, Taiwan 10764, Republic of China

(Received 22 July 199 1; accepted for publication 11 February 1992)

In the directional solidification of a binary liquid, salt-finger and plume convection are the two major types of fluid motion that occur when the lighter species is rejected from the solid. Fingering convection is usually observed prior to the initiation of plume formation. When the nominally planar liquid-solid interface is morphologically unstable, solidification is dendritic. Consequently, the melt is separated from the solid by a mushy zone consisting of solid dendrites and interdendritic liquid. This can be considered to be a porous layer with anisotropic and inhomogeneous permeability and, possibly, thermal and solutal diffusivities. A linear stability analysis is empoyed to study salt-finger instability in a fluid layer bounded above by a rigid walI and below by an anisotropic and inhomogeneous porous medium saturated by the same fluid. The effects of anisotropy and inhomogeneity on the stability characteristics are discussed for a variety of porous media. Results are also presented for a porous medium for which the anisotropy and inhomogeneity of the permeability and thermal and solutal diffusivities have been obtained experimentally during the directional solidification of aqueous ammonium chloride solutions. The nature of the instability modes relevant to the present problem are also discussed. Based on the present results, a relationship is proposed between salt-finger and plume convection during directional solidification. Methods for inhibiting salt-finger convection are also suggested, and the feasibility of their implementation is discussed.

I. INTRODUCTION

Many of the advanced metals currently in use were developed mainly for aircraft gas turbine engines. In de- veloping an advanced metal with the aid of new processing techniques, metallurgists create and exploit irregularities in the metallic crystalline structure to make exceptionally strong alloys that resist heat and corrosion. One of the most important advanced processing techniques is called directional solidification. Using this technique, the result- ant turbine blade consists of several long columnar crystals, whose direction of orientation is roughly the same as that of the enforced centrifugal force. This feature ensures that crystals will not be pulled apart during actual blade operation. The blade thus shows increased creep rupture strain and improved thermal fatigue behavior, two important characteristics for modem gas turbine blades. A general description of the process and mechanical properties that can be achieved in these castings is given by McLean.*

Under some circumstances, however, there are defects in the castings in the form of solute-rich, columnar regions extending longitudinally in the direction of the solidification. These are known as freckles, and their presence causes a deleterious effect on the strength of the castings.2’3 Recently, Sarazin and Hellawel14 used binary Pb-Sn alloys to experimentally study the formation of freckles for various compositions and cooling temperatures. Although the general mechanism for the onset of freckles is known to arise from compositional buoyancy forces resulting from interactions of fluid dynamical and thermodynamical effects in the combined region of the mushy and melt zones, the precise mechanism is not clearly understood at present, nor is there a method by which we can predict the onset of

such occurrences and the aerol distribution of the freckles (see reviews by Fisher,5 and by Glicksman, Coriell, and McFadden6).

During directional solidification of alloys, the planar freezing surface is morphologically unstable’ and solidification is dendritic. As a result, the melt is separated from the completely solid region by a region consisting of solid dendrites and interdendritic liquid, generally referred to as the mushy zone. The mushy zone may be regarded as a porous layer with directionally dependent permeabilities and, possibly, thermal and solute diffusivities.’ After the mushy zone was well established, buoyant liquid jets, ir- regularly spaced, were observed to rise out of the mushy zone. Copley et aL8 were the iirst to attribute the onset of freckles to salt-finger convection, a double-diffusive phe- nomenon much studied by oceanographers.’ Sample and Hellawell’o carried out experiments with the NH,Cl solution as well as the Pb-Sn system. They found that the genesis of such chimney-plume systems is a tinite- amplitude instability originating in the melt and might be induced by the salt-finger convection occurring prior to the initiation of plume. To explicitly pinpoint the conditions for the onset of plume convection during the directional solidification, Chen and Chen’ ’ systematically carried out a series of NH&l-H20 experiments with a constant bottom temperature ranging from - 31.5 to 11.9 “C. They pre- dicted that the critical solutal Rayleigh number across the mushy layer (Rim) for the onset of plume convection lies between 200 and 250. They also reported that the salt- finger convection is always observed in the melt region just above the mush in all experiments, with or without the plume convection.

5222 J. Appl. Phys. 71 (IO), 15 May 1992 0021-8979/92/l 05222-l 5$04.00 0 1992 American Institute of Physics 5222 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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The experimental results of Sample and Hellawell” show that a fmite amplitude disturbance in the liquid near the liquid-mush interface invariably causes the formation of a buoyant plume issuing out the mushy zone. Hence, to predict the onset of plume convection, a study for the stability of the liquid layer overlying a porous layer is necessary. Analysis of the convective motion in solidifying alloys is complicated by the fact that the temperature and com- position in the mushy zone are related by the equilibrium phase diagram, and by the fact that the phase change gives rise to a moving boundary problem. In this paper, we consider the simpler problem of onset of salt-finger convection in a fluid layer overlying a saturating anisotropic and inhomogeneous porous substrate, in which no solidification effect is considered.

Thermal convection in a porous medium with anisotropic permeability was first considered by Castinel and Combarnous.‘2 Epherre’s theoretical work13 extended the analysis to account for thermal anisotropy. WoodingI considered the influences of temperature-dependent viscosity and anisotropic permeability on the size of cells at onset. Kvernvold and Tyvand” analyzed the onset and nonlinear development of thermal convection in more general anisotropic porous media. McKibbini6 conducted an extensive study on the effects of anisotropy on the convective stability of a porous layer. The inhomogeneous effects in terms of permeability and thermal diffusivity on the convective instability in a porous medium were considered by Green and Freehill.r7 For the onset of salt-finger convection in a porous medium, Nield’* analyzed the stability characteristics for various boundary conditions in an isotropic and homogeneous porous medium. In the same porous medium, Taunton and Lightfoot” made an extension of Nield’s analysis to more completely characterize the stability of thermohaline convection. Later Tyvand*’ extended these two analyses to consider an anisotropic and homogeneous porous medium. For the salt-finger convective instability in a fluid layer, Baines and Gill*l have made a rather complete analysis.

The problem of thermal convective instability in superposed fluid and isotropic and homogeneous porous layers has been investigated by several studies. Nield*’ considered surface-tension effects at a deformable upper surface. Som- erton and Cattor? included the viscous effect on the boundaries of the porous layer by including the Brinkman term in the Darcy equation. Chen and Chen24 considered the onset of salt-finger convection in a superposed layers system. Their theoretical results were both qualitatively and quantitatively veritied by their experimental investiga- tion.“5 Taslim and Narusawaz6 later extended the superposed layers configuration to both a porous layer sandwiched between two fluid layers and a lluid layer sandwiched between two porous layers. Recently, Chen, Chen, and Pearlstein and Chen and Hsus8 have included the anisotropic effects and inhomogeneous effects, respectively, of the porous medium into the superposed layers problem.

In the present study, we study the salt-finger convective instability in an anisotropic and inhomogeneous po-

rous medium underlying a fluid layer. We first compare the calculated results with the existing analytical solutions for two special cases and then implement an extensive discussion on the significance of anisotropic and inhomogeneous effects on the stability of salt-finger convection. The real porous medium identified in the experiment of Chen and Chen” is considered to determine the critical solutal Ray- leigh number Rgm of salt-finger convection. The relation between salt-finger and plume convections and the nature of the instability modes are discussed. Finally, we make some suggestions for inhibition of salt-finger convection, hence preventing the formation of plume convection.

II. PROBLEM FORMULATION

We consider a porous layer of thickness d, underlying a fluid layer of thickness d; both layers are of infinite horizontal extent. The top of the fluid layer and the bottom of the porous layer are bounded by rigid walls maintained at different constant temperatures and salinities, both of which are high at the top and low at the bottom. A Car- tesian coordinate system is chosen with the origin at the interface between the porous and fluid layers and the z axis vertically upward. The steady continuity, momentum, energy, and concentration equations for the fluid layer in the Boussinesq approximation are, respectively,

v*u=o, ‘(1)

pou*Vu= -VP+~V2u--~pog[l -a(T- To)

+ P(S - so> lk (2)

U-VT = K=V~T, (3) U ’ vi? = Ksv*s. (4)

The corresponding equations for the porous layer are*’

v*u, = 0, (5)

+ B(S, - So>lkI = 0, (6) u;VT, = V* (K,*VT,), (7)

u;VS, = V* (K,.VS,), (8) in which the Boussinesq approximation is applied to Dar- cy’s law. In these equations, u denotes the velocity vector, P the pressure, T the temperature, and S the salinity. The subscript m denotes the porous medium and 0 denotes condition at the interface. The unit vector in z direction is denoted by k. Moreover, ,U is the dynamic viscosity, p the density, g the gravitational acceleration, (Y the thermal expansion coefficient, p the solute expansion coefficient, K~ the thermal diffusivity of the fluid, and Q the solutal diffusivity of the fluid. The permeability, thermal diffusivity, and solutal diEusivity of the porous medium are assumed to be diagonal tensors,*’

K = Klrll(z)ii + K2rlzM.i + K3773Wfi, (9)

KT= q-m-l(z)ii + Kem(z>jj -t %%(z)kk, (10)

5223 J. Appl. Phys., Vol. 71, No. IO, 15 May 1992 Falin Chen 5223 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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KS = Kslqs~tZ)fi + QzR&%.i + Ks~%dZ)~, t 11) in which i, j, and k are unit vectors in X, y, and z directions, respectively. The vi(Z), q&r), and q,(z), i I- 1-3, are arbitrary functions accounting for inhomogeneities of permeability, thermal diffusivity, and solutal diffusivity, respectively, of the porous medium. It is noted that the thermal diffisivity is obtained by dividing the thermal conductivity by the volumetric heat capacity of the fluid (p&,), where CP is the specific heat of the fluid. The principal axes of permeability and thermal and solutal diffusivities of the porous media are assumed to be coincident. We restrict consideration to horizontal isotropic porous media, namely, K1 = K2r ‘i?(Z) = Q(Z), KTI = KT.z, m-l(z) = rldz),~~l = Kn, and?kl(z) = rl,dz),andso on; this consideration is relevent to the dendritic mushy zone resulted from directional solidification. ‘I

Since the principle of exchange of instabilities holds for salt-finger convection in either a fluid layer21 or a porous layer,18’lg we will assume that it also holds for the present situation. Thus, the time derivatives will be dropped from the disturbance equations, and hence omitted from Eqs. (2)-(4) and (6)-(8). The boundary conditions are those at the upper. boundary T = T,, S = S,, and all velocity components vanish; at the lower boundary T, = T, S, = Sl, and the vertical component of the velocity vanishes.

At the interface (z = 0), the temperature, the salinity, the normal components of heat flux and -solute flux, and the normal stress are continuous. Since in the Darcy equation there is no viscous stress, continuity of shear stress across the interface cannot be enforced. Thus, we use the condition proposed by Beavers and Josephsg in which the slip in the tangential velocity is proportional to the vertical gradient of the tangential velocity in the fluid. In the horizontal directions the boundary conditions are periodic.

The steady basic state is quiescent. After applying the boundary conditions on energy and concentration equations, we obtain the basic temperature and salinity distri- butions in two layers,

T= To+ CT,- T,)z/d, (12)

s = So + (S, - S&/d,

for O<z<d, and (13)

s dz

T,=C, ---+Cc, 77nw

(14)

dz S,=D, -+D2,

77.73 Cz) (15)

for - d,,, < z < 0, in which C1, C2, D1, and D2 are four arbitrary constants depending upon the boundary conditions of T, and S, as q=(z) and vs3(z) are specified. The pressure is hydrostatic and need not be presented here.

We operate on the porous medium momentum equation (6) with VX (K-V) x and then take the vertical component to eliminate the pressure. To render the equations nondimensional, we choose separate length scales for the two layers so that both are of unit depth.” In this manner, the detailed flow fields in both fluid and porous layers can

be clearly discerned for all depth ratios. For the fluid layer, we choose the characteristic length as d, velocity as v/d, temperature as (T, - To>~/~n salinity as (S, ~ &Jv/Ks, where v is the kinematic viscosity. For the

porous layer, the corresponding characteristic quantities are d,, v/d,, (To - T$v/K~, and (Sa - S&V/K~~. We then decompose the velocity, temperature, and salinity fields into basic state and disturbance quantities, and lin- earize the equations for the latter.

The nondimensional disturbance equations, written in the same notation as the dimensional equations (2)-(4) and (6)-(8) are

- V4W = RVfT - R,V$, (16)

W=V”T, (17)

w = v2s, (18)

( 1 a2 zmv:-+&g

1 7; awn2

Ft’“--=

= rld4J%,L - LV;,,&), (19) a a

.4-m-&, +a~ vi-3 a~ ( )I Tnv (20) m m a &lslV:m+~ m

where W is the vertical disturbance velocity, Vz a2/8x2 + a’/ay” is the horizontal Laplacian, and the

pgme denotes the differential operator d/dz,. The ele- ments of the permeability, thermal diffusivity, and solutal diffusivity tensors are scaled according to c = K1/K3, gT = K&K~, and 6s = K&K~~, respectively. With the chosen scaling, the buoyancy terms on the right-hand side of the momentum equations are directly proportional to the Rayleigh numbers. For the fluid layer, the thermal and solutal Rayleigh numbers are

R =ga( T, - To)d3/(vKr), (22)

R,=gP(S,-So)d3/hrs). (23) For the porous layer, the Rayleigh numbers are defined in terms of the vertical components of permeability, thermal diffusivity, and solutal diffusivity at the bottom

R, =ga(T,, - Tl)d&/(v& = RS-4(S~T)2, (24)

R, = gfl(& - Sz)d,,&/(vQ3) = RJ-4(6@2, (25)

where S = &/dm is the Darcy number, eT is the thermal diffusivity ratio K~/K~, and es is the solutal diffusivity ratio K~/K~~.

The dimensionless boundary conditions at the top and bottom walls are

awl) W(1) = T(1) -S(l) Caz’ WA---l)

=T,nt-l)=sm(-l)~o, (26)

5224 J. Appl. Phys., Vol. 71, No. 10, 15 May 1992 Falin Chen 5224 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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and those at the interface are

w = w,?l,

T = t&3 Tm,

(27)

(28)

s = &m2, aT aT, -z-E - az Ta~m 3 as as, -2z-E - az VZ, 9

(29)

(30)

(31)

In addition, the Beavers-Joseph boundary condition, 2

f&-g= A[S-‘[g~l(o)] --‘2 (

aw, g-+

1 , (33)

m is enforced at the interface, where the constant A ranges from 0.1 to 4 as determined experimentally by Beavers and Joseph. 29 The permeabilitv in the Beavers-Joseph condition is chosen to be the horizontal component KIvl(0) [or ~2772uxl-

We apply the normal-mode expansion of the dependent variables in both the fluid and porous layers and use D and D,,, to denote the differential operator d/dz and d/dz,, respectively. We obtain the eigenvalue problem consisting of the following ordinary differential equations (ODEs):

D4W-2a2#W+a4W=a2(RT-RJ), (34)

@T-a2T= W, (35)

02s -a’s= W, (36)

$Dfm-$Dm-~a~)Wm 1

= qlak( - R,T, + %,$,A, (37)

tim-l-$Dm-gT~a~)Tm=-& w,, (38)

(39)

In these final equations, we have nondimensional horizontal wave numbers a and am, which are the separation constants of normal-mode expansion. Since the dimensional horizontal wave numbers must be the same for the fluid and porous layers if matching of solutions in the two layers is to be possible, we must have a/d = am/d, and hence i: = a/a,,,. The boundary conditions on the top and bottom are

W(1) = T(1) =S(l) =DW(l) = W,( - 1)

= TJ - 1) =S,( - 1) =o, (40) and those at the interface are

TABLE I. Parameters considered in Sets. III A-III D.

Parameter

a, (and a) R, (and R,) R, (and R) 7711 73

WI? rl?3 rim rls3

iT 6.7 5 ET ES s

Definition Values

a = a& to be determined ~ Eqs. (23) and (25) to be determined

Eqs. (22) and (24) 1 Eq. (48) - S<A<5 m. (49) - 5<&5 Eq. (50) - 5<c<5

KdK, 10-4<g< 1 KTI~ 1 Ksikn 0.1 G&s< 1

- d/d,,, lo-h<{< 1 Krh 0.7

I. KS/KS3 l/(&T)

$%n 0.003 constant 0.1 porosity 0.39

w=!wm, (41)

T = (&/i-)T,, (42)

s = t4m%z, (43)

DT = eTDmTmr (44)

DS = esD,,&,, (45)

-D3W+3a2DW=t$S-2~-1q;1(0)D,Wm, (46)

@W=&+[<ql(0)] -“2(DW-~2D,Wm). (47)

The eigenvalue problem consists of an eighth-order ODE in the fluid layer and a sixth-order ODE in the porous layer, with 14 boundary conditions. We use a shooting technique based on a hybrid Adams backward difference method to solve this ODE system. An Adams method is used as the family of nonstiff methods, and backward differentiation formulas as the family of stiff methods. The calculation starts at the interface and ends at the top and bottom boundaries. For details of the computation proce- dure, the reader is referred to Chen and co-workers.27

Ill. RESULTS

Since the parameter space of present problem is large, we summarize the parameters needed to be considered in Sets. III A-III D in Table I. The eigenvalues to be sought are R, (and R,) and a, (and a). We select R, = 1 for the present study because it is a reasonable scale to account for the thermal gradient across the dendritic mushy layer as the solute-rich cold plume is initially ejected from the mush.” We also assume the vertical variations of inhomogeneity of the porous medium in both the horizontal and vertical directions are the same,‘7*28 namely,

171bn) = 113hA = 71kn), (48)

qTl(z,) =~73tz,) =~T&,z), (49)

rls*(zm) =r7s3k?J =rlstz,>. (50)


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However, this assumption will be lifted as we consider the case of real dendritic mushy zone in Sec. III E. In the general discussion in Sec. III D of inhomogeneous effects, these inhomogeneity functions are considered to be of exponential form2*

T(Z,) = 6” + =m), rlT(Zm) = &al -I- %A,

q&z,) = eccl + =J, (51)

in which A, B, and C range from zero in homogeneous cases to a nonzero value (5 or. - 5) in inhomogeneous cases. The anisotropies of permeability, thermal diffusivity, and solute diffusivity are represented by c, gn and &, respectively. We considered g ranges from 10 - 4, the lim- iting case for the porous layer with horizontal permeability virtually vanishing, to unity, the isotropic case. The case of c = 10 - 4 is interesting due to the fact that an analytic solution is available for c = 0, which will be discussed in Sec. III A. If no solute diffusion is considered in the solid of the porous medium, the anisotropy of solute diffusivity should be similar to that of permeability; and thus 0.1 < & < 1 is considered. The cases in 10 - 4 < & < 0.1 are skipped over since, iirst, no analytic solution for & = 0 is available and, second, the cases in 0.1 < & < 1 can be representative for the anisotropic effects of present problem. From previous study,2o it was found that the influence due to the variation of f;r on the salt-finger convective stability in a porous medium is much less significant than that due to the variation of either g or & when R, is small. Ac- cordingly, in general discussions in anisotropic effect with R, = 1 in Sec. III C only gT = 1 is considered. The depth ratio g, which plays a decisive role in superposed layers configuration,24 is covered in the range from 10 - 4, a nearly pure porous layer case, to unity, a case relevant to the experiment.” The thermal diffusivity ratio eT = 0.7 is chosen on the basis of considering the averaged thermal diffusivity of the solid of the mush is about the same as that of glass. Considering no solute diffusivity in the solid of the mush, the relation es = l/( &-) is employed.30 In general discussions for both anisotropic and inhomogeneous effects in Sets. III A-III D, the porosity 4 is fixed to be 0.39 since it is one of the representative values for most of the porous media in nature. Thus the Darcy number S = 0.003 is obtained based on the Kozeny-Carman relation31 as well as the consideration of a 3-cm-deep porous medium. When the real mushy zone is considered in Sec. III E, different values of some parameters are selected. A. Horizontally impermeable porous medium, g=O

We consider an anisotropic but homogeneous porous medium, i.e., 7 = VT = Q = 1, for subsequent analyses in this section. As c-0, the porous layer behaves like a conducting solid layer. This is shown in the following. The lack of permeability in the horizontal direction implies that the flow in the horizontal direction vanishes u,(x,,y,,zm) = v,(x y z ) In, no m = 0, where u, = u,i + VJ + W& AS a result, au,/&,,, and Ju,/Jy, both

vanish in the porous medium. From the continuity equation, we obtain Jw,/Jz, = 0. Considering the boundary

condition w,( - I) = 0 and the differentiation Jw,/Jz,,, = 0, we find that w, = 0 for the whole porous layer. Governing equations for the onset of salt-finger convection are therefore Eqs. (34)-( 36) for the fluid layer and

(52)

(53)

for the porous layer. The wave number in the porous layer has been eliminated using relation a, = a/l;. The boundary conditions at the top of the fluid layer are given by Eq. (40). At the interface, which is at the bottom of the fluid layer, we have

W(0) = DW(0) = 0 (54) and

DT(0) =acoth

DS(0) - ’ sa -- ES

(55)

Equation (55) can be obtained by using the solution of Eq. (52) subject to T,( - 1) = 0 to eliminate T,,, from the boundary conditions (42) and (44). Similarly, Eq. (56) is obtained by the same operation using Eqs. (53)) (43), and (45). As for the other boundary conditions, using Eq. (41) and the identity w, = 0, we obtain w = 0, which implies that W(0) = 0. In addition, Eq. (47) suggests that DW(0) = ~‘D,Wm(0) for <=O. From Eq. (46), we have D, W,(O) = 0 for c = 0, which finally follows DW(0) = 0. In summary, the governing equations for the salt-finger convective instability in a horizontal fluid layer bounded by a perfect conducting rigid plate above and by a slab of solid material of finite conductivity and thickness below are given by Eqs. (34)-( 36). The corresponding boundary conditions are

IV(l) =DW(l) = T(1) =S(l) =0 (57) at the top and Eqs. (54)-(56) at the bottom of the fluid layer.

The problem of determining the onset of convective instability in a fluid layer heated from below, where the upper boundary is of finite thickness and thermal conductivity, has been considered by several investigators.32-39 In this section, the salt-finger instability in a horizontal fluid layer bounded above by a rigid boundary of infinite thermal and solute diffusivities and below by a rigid slab of finite anisotropic thermal and solute diffusivities and thickness is considered. This configuration is equivalent to the superposed fluid and porous layers configuration with l-+0. By assuming 6 = 10 - 4, we calculate Rf and a”, which are converted into the corresponding values of R&, and a$ respectively, for various c, CT, and gs and the results are compared with those of g = 0 (Table II). The results are in good agreement while the difference increases

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TABLE II. Comparison between the results of g = 10e4 and those of E=OwhenR, = landgT = 1.

10” :’ ’ * ’ * * * ’ - t I ’ * / ’ ’ * ‘.

5s E =i 10-a l-0

s, R&l a: R&l

1.0 26.4 1693.73 0.6 26.2 1673.49 0.2 25.8 1637.68

1.0 13.0 128.129 0.6 12.9 126.625 0.2 12.7 124.167

1.0 3.21 24.4132 0.6 3.18 23.9615 0.2 3.18 23.3328

1.0 0.6 0.2 1.08

19.3877 19.1877 18.9369

g=o.1 26.6 26.4 26.0

g=o.z 13.1 13.0 12.7

g= 0.5 3.22 3.18 3.18

g = 1.0 0.86 0.96 1.08

1602.95 1584.50 1554.43

120.274 119.624 116.530

22.8379 22.4080 21.8057

18.1604 17.9682 17.7253

with decreasing c. The maximum difference is less than 6% for all the cases considered.

B. Anisotropic effect for pure porous layer: g = 1O-4

Another check is made possible by considering the lim- iting case of 5‘ = 10 - 4, which accounts for the case of a porous layer underlying an extremely thin fluid layer. For the salt-finger instability in an anisotropic porous layer Ty- vand,20 using a linear stability analysis, obtained an exact solution

1 2 J-L Rzm== ’ +m+ (s~,)y~+a [(4J2gs++]. 1

(58) The corresponding critical wave number a”, is determined by the polynomial

(59)

where

Q = #J2/g,

a3 = G%,

1 %ni?~-h 1 ""=g--T .g& =-2jg'

2

(60)

1 ao= -m-

Tyvand proved that there is only one mode for the onset of salt-finger convection since there exists only one positive real root for Q. We calculate the root of Q by the Newton- Raphson iteration scheme and obtain Rim and a; for various 5, gT, &, and R,. These results compare excellently with those obtained by our numerical integration with

K m

(4 b.0 6.2 6.4 6.6 6.8 i.0

10” 0.0 0.2 0.4 0.6 0.8 1.0

c

RG. 1. Effects of anisotropy on the critical conditions in superposed layerswith[=O.lfor& = 1,0.8,0.6,0.4,0.2,andO.l:(a) Rzmand(b) a’,.

i; = 10A4. Note that the definitions of R, and R, in Eqs. (24) and (25)) respectively, differ from those in Tyvand by a negative sign.

C. Anisotropic effects for superposed layers: 0.1 <g<1

For g> 0, we consider four depth ratios, 5 = 0.1, 0.2, 0.5, and 1. For c = 0.1 (Fig.* l), the stability characteristics are significantly influenced by both variations of &J and L& since the porous layer plays a decisive role for the present situation.24 The R:m increases with decreasing &I as well as increasing &. This is due to the fact that smaller ,$ results in higher resistance to the flow and thus leads to a greater stability, and that larger & leads to a more efficient dissipation of destabilizing solutal gradient and hence results in stabilization. Note that for most of the cs considered, both R&, and a', experience a dramatic increase when g is very small. Figure 2 illustrates the onset streamline patterns for four selected 6 corresponding to {.s = 1. For each streamline plot, the width accounts for half the critical wavelength. The dashed line represents the interface between fluid and porous layers. For 5 = 1 [Fig. 2(a)], the isotropic case, the onset of convection occurs in both layers. As g decreases [Figs. 2(b) and 2(c)], both the critical wavelength and the convection in the porous layers decrease due to larger horizontal resistance to the flow. After the dramatic change occurs [Fig. 2(d)], the onset of convection is largely confined to the fluid layer and the porous layer serves only conduction for the system.

For c = 0.2 (Fig. 3), variations in both R&, and a; with either g or cs are similar to those of c = 0.1 except


193.0.65.67 On: Thu, 18 Dec 2014 09:35:05

a

/b

mG. 2. The onset streamline patterns at selected values of 6 for 6~ t: 1 and c~O.1: (a) E= 1; (b) 520.1; (c) {=O.Ol; (d) ~=O.OOl.

that the critical e, at which the dramatic change of a: occurs, is invariably larger than that of c = 0.1 for all & considered. That Rz,,, remaining virtually constant for 0 Q g < &Y means as a”, jumps from a smaller value to a larger value, the stability of motionless state is essentially insen- sitive to the change of either g or & of the porous medium. Since under this circumstance, the onset of convection is largely con&red to the fluid layer which leads to the insen- sitivity of stability characteristics to the anisotropic effects of the porous layer.

For 5 = 0.5 (Fig. 4), decreasing g again Ieads to increasing R&, whereas the variation of a”, with g becomes dependent on the value of L&. For & > 0.4, a: increases with decreasing g invariably. For c(O.2, nonetheless, ai increases with decreasing c first, reaches a maximum, and then decreases. Due to the presence of a larger fluid layer, the motionless state of the present situation is much less stable than that of 6~0.2. For 6 = 1, the variation of R&,

Id (a) 0.0 0.2 0.4 0.6 0.8 1.0 !?

12.0

aLa g." 8.0

3.0 L

-O.O’, . I I, I I I, I. I, I I I, I I. b)

0.0 0.2 0.4 0.6 0.8 1.0

f:

FIG. 3. Effects of anisotropy on the critical conditions in superposed layers with 6 = 0.2 for L& = 1,0.8,0.6,0.4,0.2, and 0.1: (a) Rf, and (b)

c a,.

with 5 is even smaher and the motiomess state is most unstable compared to other f considered [Fig. 5 (a)]. With regards to the variation of a”, with t, the combination of cells in the fluid layer plays a significant role. The onset

(a) 0.0 0.2 0.4 0.6 0.8 1.0 .

5.0

4.0

ac7,z 3.0 2.0

1.0 (b) II.,...,...,.<. .I.,

0.0 0.2 0.4 0.6 i;.6 1.0

F FIG. 4. Effects of anisotropy on the critical conditions in superposed layers with c = 0.5 for &. = 1,0.8,0.6,0.4,0.2, and 0.1: (a) Rzm and (b) a;.


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FIG. 5. Effects of anisotropy on the critical conditions in superposed layers with { = 1 for & = 1,0.8, 0.6, 0.4, 0.2, and 0.1: (a) Rzm and (b) a”,*

streamline patterns for g,s = 0.1, in which a& experiences a dramatic decrease from 3.95 to 1.87 as 6 changes from 0.17 to 0.16 [Fig. 5(b)], are presented in Fig. 6. As one can see, the onset of convection for c = 1 [Fig. 6(a)] occurs mostly in a local region underneath the top boundary and the convection is of multicell. As 6 decreases [Figs. 6(b) and 6(c)] the upper cell becomes increasingly stronger relatively to the other cells. For g = 0.16 [Fig. 6(d)], the critical wavelength increases dramatically and a bicellular convection prevails.

D. Inhomogeneous effects for superposed layers: 0.1 <&I

To examine the inhomogeneous effects, we assume the porous medium to be isotropic and the inhomogeneous variations in the vertical direction to be exponential functions as shown in Eq. (5 1). Accordingly, the inhomogeneous functions are unity at the bottom (z, = - l), increase upwards with z, when the parameters are positive, and decrease when negative. In addition, for positive parameters, the bulk values of permeability and diffusivities are larger than those of the homogeneous case, and vice versa for negative parameters. The inhomogeneous effects can be categorized into local and bulk effects.28 The bulk effect can be detected by observing the variation of R&, with changing parameters while the local effect can be illustrated by the onset streamline patterns.

We first examine the inhomogeneous permeability effect for four depth ratios < = 0.1, 0.2, 0.5, and 1, and the variations of R&, and &, with A are shown in Figs. 7(a) and 7(b), respectively. We also illustrate the onset stream-

..__._.__-....--

C

a

/d FIG. 6. The onset streamliie patterns at selected Values of g for 5s = 0.1 and<= 1: (a)g= 1; (b) 5=0.6; (c) 5=0.17; (d) 5~0.16.

line patterns in Fig. 8 for these four depth ratios with five values of A = - 4, - 2, 0, 2, and 4, where the pictures in the same column are of the same f and in the same array are of the same A. For 5‘ = 0.1, Rzm invariably decreases with increasing A since larger bulk permeability resulted from increasing A leads to destabilization. The corresponding a& decreases with increasing A first and then increases for higher A. It is seen that the onset of convection occurs in both layers when the porous medium is homogeneous [A = 0 for Fig. 8(c)]; as A either increases or decreases, the onset of convection is locally confined to a region in which the permeability is relatively larger than the other regions. Due to the localization of convection, the critical wavelength also changes in accordance with the depth of circu- lation zone.

For j = 0.2, R& remains virtually constant in A < - 3.1 and coalesces with that of 5 = 0.1 for higher A. The slope of the curve of Rfm vs A changes at about A


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(b) -5.0 -3.0 -1.0 1.0 3.0 5.0

A

FIG. 7. Effects of permeability inhomogeneity on the critical conditions in superposed layers with B = C = 0 for c = 0.1, 0.2,0.5, and 1: (a) RTm and (b) &,.

= - 3.1, where af, proceeds a dramatic decrease from 12.57 for A = - 3.1 to 1.83 for A = - 3. The streamline patterns shown in Fig. 8 indicate that, when high a’, mode prevails, the onset of convection is largely conlined to the fluid layer and the inhomogeneous effect of the porous layer virtually vanishes. As A increases, the local effect becomes important and the onset of convection moves to- ward the interface. For c = 0.5, the onset of convection occurs only in the fluid layer when A < 1, in which varying A essentially has no influence on R&. For larger A, however, the onset of convection occurs in both layers due to the relatively larger permeability near the interface and the variation of Rgm is similar to that of 6~0.2. For < = 1, the onset of convection occurs only in the fluid layer for all the A considered and is of multicell type. The strength of the convection of the cell underneath the top boundary is the largest. The number of convection cells in the fluid layer increases with increasing A.

We next examine the inhomogeneous thermal diffusivity effect and the corresponding results are shown in Figs. 9 and 10. It is found that Rzm decreases with increasing B since larger B dissipates more efficiently the stabilizing thermal gradient and thus leads to destabilization. The variation of Rzm vs B is relatively much smaller than that with A [Fig. 7 (a)]. Which means the inhomogeneous thermal diffusivity effect is much less significant than the inhomogeneous permeability effect (as well as the inhomogeneous solutal diffusivity effect, as to be shown later). The local effect due to inhomogeneous thermal diffusivity can be seen from Fig. 10(a), in which B = - 4, the thermal

difhtsivity at the bottom is about 75 times larger than that at the interface. In contrast, for B>2 [Figs. 10(c) and 10(d)], the increasing thermal ditfusivity at the interface makes stronger convection occur near the interface. For 520.2, the onset of convection is essentially confined to the fluid layer. The convection cell moves towards the interface as B increases.

Figure 11 illustrates the variations of Rf, and a”, with C as A = B = 0. It is seen that Rz,,, monotonically increases with C since larger C results in faster dissipation of destabilizing solutal gradient and hence leads to stabilization. For c = 0.1, the onset of convection occurs in both layers and the local effect becomes increasingly important as C increases [Figs. 12(a)-12(d)]. For 6 = 0.2, the slope of the R& curve experiences a discontinuous change at approximately C = 1.2, at which the corresponding a”, jumps from 2.40 to 11.5. As convection is predominated by the high a’, mode [Figs. 12(g) and 12(h)], the onset of convection is essentially confined to the fluid layer and thus the influence of inhomogeneous solutal diffusivity in the porous medium is greatly reduced. For c = 0.5, the onset of convection occurs in the fluid layer only. When C is negative, the relative smallness of C near the interface induces more convection and results in the-downward movement of cell. It is noteworthy that, when C< - 0.5, the stability characteristics in terms of Rf, and a’, for different < are virtually identical. In other words, a larger depth of fluid layer (larger 5) does not provide, in some sense, more degree of freedom for the fluid to convect. Nevertheless, for C > - 0.5, the stability characteristics for different fluid layer depth are quite distinct, because both the occurrence of convection in the porous layer and the formation of an effective sublayer would lead to a greater stability.

E. A dendritic mushy zone underlying a fluid layer

In a series of experiment of directional solidification of aqueous ammonium chloride solution, Chen and Chen,” by using the technique of computed tomography, identified the vertical distribution of porosity in the dendritic mushy zone. Except for the shallow layer close to the interface between the mush and melt, the vertical distribution of porosity of the mush is a linear function,

cj(z,J = 0.082, + 0.67, (61)

where 4 is the porosity of the porous medium and z, is a dimensionless independent variable, which is - 1 at the bottom and 0 at the interface. From the observation of the photos taken in the experiments, they found that the sum of the diameter of primary dendrite arm d2 and the primary arm space dt is approximately 1 mm. The arm space dl increases upwards and we may assume that its vertical variation is also of the form of Eq. ( 6 1) . As a consequence, the averaged dl over the height of the mushy zone is about 0.63 mm, and hence the averaged d2 is 0.37 mm.

To determine the vertical and horizontal permeabilities with the measured porosity (61), d,, and d2, we use Blake- Kozeny models suggested by Poirier.40 For vertical permeability,


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.~ . ..fTlC.. . . J -. ‘\

k!YT=l \

0 ,.F-, ‘; i .\/I ! ---. i a f ..- ~~~

. . . . . . . . . . . . . . ..r......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..-.

k P

b

h r

L--l

’ y&5$5! :

e L---.-Ii FIO. 8. The onset streamline patterns at selected values of A and c (a)-(e) {= 0.1, and A = - 4, - 2, 0, 2, and 4, respectively; (f)-(j) g = 0.2, and A = - 4, - 2, 0, 2, and 4, respectively; (k)-(o) < = 0.5, and A = - 4, - 2, 0, 2, and 4, respectively; (p)-(t) 5 = 1, and A = - 4, - 2, 0, 2, and 4, respectively.

K&z,,) = 1.43x lo-“&z,)d;/[l - 4(zm)] (62)

is chosen, in which K3 and d: are in m2. After applying Eq. (61) and dl = 0.63 mm in Eq. (62)) the averaged value of vertical permeability is about 3.836 X 10 - lo m2. As regards horizontal permeability Kt, we chooseM

K,(z,n) = 1.73x10-3 2 0

1.09

d2$3(z >[I-d(z >I 2 m m 9 2

(63)

in which K,, df, and di are in m2. The averaged horizontal permeability is thus approximately 2.277 X 10 - lo m2. With these, the anisotropy of permeability is obtained as E = 0.581, and we take 0.6 for the subsequent calculations of this section. We assume the solutal diffusivity in the solid of the porous medium is zero, i.e., only the liquid serves to diffuse mass. We thus may infer that & = c = 0.6. With regard to CT, since no existing relation be-

tween either c and gr or & and cr can be applied,20 we leave it as a free parameter for the subsequent analyses.

By substituting Eq. (61) into Eq. (62) and after some arrangements, the vertical inhomogeneous permeability becomes

&(z,) = 3.63~1O-‘~X[(z, + 8.375)3/(4.125 -z,)], (64)

which results in

~~(2,) = (z, + 8.375)3/(5.125 -z,> (65)

5231 J. Appl. Phys., Vol. 71, No. 10, 15 May 1992 Falin Chen 5231

and an averaged Darcy number 6 = /m/dm = 2.159 X 10 - ‘, in which we assume d, = 0.01 m, the

depth of the mushy zone corresponding to the onset of plume convection. * ’ Similarly, we obtain the horizontal permeability

K,(z,) = 1.436X1O-‘2

x [(z, -j- 8.375)3/(4.125 - z,)‘.~~~], (66)

and thus

~~(2,) = (z, + 8.375)3/(4.125 -z~)‘.~~~. (67)

Based on the assumption of no solutal diffusivity in the solid of the mush, we may imply that vl(zm) = qsl(zm) and q3(z,) = ~~(2,). As to the inhomoge-

neous thermal diffusivity function, the exponential function (5 1) is applied for qrl = qT3. According to Chen and Chen,” the R, is small when the plume convection thresh- olds. We thus consider R, = 1 for the present situation. For the other parameters needed to be considered for subsequent analyses, we have3’ =+ = 0.7 and es = l/[+(O)]. In addition, the Beavers-Joseph constant

A = 0.1 is chosen.27J28 We compute Rzm and corresponding af;, for B = 1 and

- 1;ineachcasegT = 0.1,0.5, and 1 are considered. It is found that, for both B considered, R&, decreases with increasing c monotonically except in the range of small 5; in which Rzm is almost constant [see Fig. 13 for B = 11. This

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193.0.65.67 On: Thu, 18 Dec 2014 09:35:05

is due to the fact that increasing g provides more space in the fluid layer for the fluid to convect and hence leads to destabilization. The associated a; experiences a dramatic jump when the slope of the curve of R&, changes. On the basis of previous discussion, we may infer that the onset of convection corresponding to constant R&, as well as low Q”, occurs in both fluid and porous layers and the latter predominates the system by convection. As the high aC, mode convection prevails, the onset of convection is largely confined to the fluid layer.

IV. DISCUSSION

A. Relation between plume and salt-finger ’ convections

Chen and Chen” indicated that the critical Rayleigh number across the mushy layer for the onset of plume convection is within 200-250; meanwhile, the porous layer depth is about 1 cm and the height of the salt-finger convection is larger than the depth of porous layer (the depth ratio 5 is larger than unity). In light of Fig. 13(a), it is found that the R&, for salt-linger convection for c>O.S is generally less than 10, which is much lower than that for the onset of plume convection identified in the experiment. This result is consistent with the experimental finding by Chen and Chen that the salt-finger convection in the fluid layer is always observed prior to the initiation of plume convection.

With regard to the convection in the mushy layer, Chen and Chen,” by employing the dye tracing technique to observe the flow structure in the dendritic mushy zone

RCfm

la’

ld

lo’

18

IO-

lo- w -5.0 -3.0 -1.0 1.0 3.0 5.0

B

14.0 ; ’ ’ ’ ’ q ’ ’ 7 ’ * ’ ’ ’ + ’ ’ * ’ 1..

6.0 4.0 2.0 0.0 b)

---LO -3.0 -1.0 1.0 3.0 5.0

FIG. 9. Effects of thermal diffusivity inhomogeneity on the critical conditions in superposed layers with A = C = 0 for 5 = 0.1, 0.2, 0.5, and 1: (a) R& and (b) (f,.

after the plume convection occurs, found that the convective flow appears to be a bulk flow resulting from the plume convection. With this observation, one may infer two pos- sibilities for the relation between plume and salt-finger convections in the mush: first, the salt-finger convection initially exists in the mush (as well as in the melt as always observed) but is suppressed by the plume convection after its onset; second, the salt-finger convection never occurs in the mush and thus does not exist in the mush as the plume onsets or after the onset. From the observation of corresponding onset streamline patterns, we find that for c)O.3, the onset of salt-finger convection is largely confined to the fluid layer and the convection in the porous layer virtually vanishes. Consequently, we would favor the second possi- bility for the relation between salt-linger and plume convection in the mush. With the discussion above, a clear

h

FIG. 10. The onset streamline patterns at selected values of B and 5: (a) {=O.l,B= -4; (b) c=O.l,B= -2 (c) E=O.l,B=2; (d) c=O.l, B=4; (e) g=O.S, B== - 4; (f) 5 = 0.5, I3 = - 2; (g) g = 0.5, B = 2; (h) p=OS, 8=4.


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picture can be drawn for the flow structure in both the fluid and porous layers as the plume convection initiates: In the melt above the mush, the salt-finger convection is in a highly supercritical stage since at that situation the R, is more than 25 times as large as R&; in the mush, the salt- finger convection virtually vanishes and a bulk convective flow induced by the plume convection predominates. Ac- cordingly, one may imply that the supercritical salt-finger covnection in the melt might play a decisive role for initi- ating the plume convection.

B. Methods of Inhibition of salt-finger convection

To inhibit the salt-finger convection in the directional solidification may be a necessary condition for preventing the occurrence of plume convection. To do so, on the basis of current analysis, one may propose two methods: the passive method and the active method. For the passive method, one may choose a material that can result in a dendritic mushy zone with either smaller g and A, or smaller CT and B, or larger & and C, and so on. Since, in a practical sense, the material used is chosen in advance, this method essentially does not sound feasible. For the active method, one may restrict the fluid layer depth to be small (i.e., small 0 and thus it results in a greater stability for motionless state; or one may increase the the thermal gradient (i.e., increase R,), which stabilizes the motionless state, and hence inhibits the salt-finger convection. In- deed, from the results shown in Fig. 13 (a), one can see that the Rz, for g~O.02 is generally larger than that for @0.4 by more th an three orders of magnitude, 0( 103).

RcS711

lo”

la”

ld

lo”

lo-

lo- (4 -5.0 -3.0 -1.0 1.0 3.0 5.0

c

12.0 10.0 6.0 6.0 4.0 2.0 0.0 (b)

FIG. 11. Effects of solute diffusivity inhomogeneity on the critical condi- FIG. 13. The critical conditions for superposed layers with considering tions in superposed layers with A = B = 0 for 5 = 0.1,0.2,0.5, and 1: (a) the dendritic mushy zone identified by Chen and Chen (Ref. ll), in R& and (b) &. which B= 1 and gr = 0.1, 1, and 5: (a) R& and (b) u$

b f J

d tlh L---.-41

FIG. 12. The onset streamline patterns at selected values of C and 5. (a)-(d) c=O.l, and C= -4, - 2, 2, and 4, respectively; (e)-(h) C; = 0.2, and C = - 4, - 2, 2, and 4, respectively; (i)-(l) 5 E 0.5, and C = - 4, - 2, 2, and 4, respectively.

Nevertheless, to restrict the depth of the melt to be so small is rather difficult practically. By increasing R,, however, is more feasible and one may obtain a greater stability for large c. This is discussed in the following.

We first examine the case with the porous medium described in Sec. III E of R, = 50 for 0.1<5<0.9 and compare the results in terms of Rzm with those of R, = 1 (Ta-

184 ” ” ” q ’ * ” ’ * II ’ 1 ’ * k

10”

Rc 10”

*m lo”

ld

10” (4 0.0 0.2 0.4 0.6 0.6 1.0 c-

lo-l-! a a , a , , , , , , , ~ , , , 1 (b)

0.0 0.2 0.4 0.6 0.6 1.0 il


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TABLE III. Comparison between the R:,,, for R, = 1 and 50 with the porous medium considered in Sec. III E.

5 R,=l R,=SO

0.1 57.64 527.6 0.3 10.67 444.7 0.5 8.957 437.4 0.7 8.652 433.6 0.9 8.096 429.8

ble III). It is found that for R, = 1, as 6 increases from 0.1 to 0.9, Rf, decreases from 57.64 to 8.096, which is about a sevenfold decrease. In contrast, for R, = 50, the decrease of Rf, associated with the same increase of f is less than 20%. We therefore may infer that with a higher stabilizing thermal gradient (R,) the motionless state will not lose its stability as much as it does for smaller R, when the fluid layer depth is increasingly large.

As we consider a porous medium being either anisotropic or inhomogeneous, it is found that the R&, for the’ cases of c>O. 1 are virtually identical when R, = 50 is enforced. To show this, we first calculate the Rim and associated u”, for [ = 0.1 and 0.5 with varying g and &, while gr is fixed to be unity. By comparing the R’& of these two cases, it is found that they are almost identical (see Fig. 14 for c = 0.1). With regards to a”,, the identity between these two cases still holds except in (GO.01 for 0.4 < & a; 1. When we consider the inhomogeneous effects, as shown in Fig. 15, the stability characteristics in terms of Rim are essentially identical for the four depth ratios 4 = 0.1, 0.2,

lO%j s ’ s ’ ’ * * ’ I ’ ’ ’ ’

10”

ld

ld 0.0 0.2 0.4 0.6 0.8 1.0

t

(4

0.0 0.2 0.4 0.6 0.8 1.0

c

FIG. 14. Effects of anisotropy on the critical conditions in superposed layers with R, = 50and c = 0.1 for fs = 1,0.8,0.6,0.4,0.2, andO.l: (a) R;,,, and (b) a$

Kn

-5.0 -3.0 -1.0 1.0 3.0 5.0 B

-5.0 -3.0 -1.0 1.0 3.0 5.0 c

FIG. 15. Effects of inhomogeneity on Rz, in superposed layers with R, = 50 for 6 = 0.1,0.2,0.5, and 1: (a) inhomogeneous permeability effect,

B = C = 0; (b) inhomogeneous thermal diffusivity effect, A = C = 0; (c) inhomogeneous solute diffusivity effect, d = B = 0.

0.5, and 1 considered, except in the range C>4 in which R& is slightly larger for smaller 6.

Without loss of generality, from the above discussion one may infer that for R, = 50 the salt-finger convection is of equal stability for depth ratios 5,O.l. As a result, with imposing larger thermal gradient on the system being so- lidified from below, one may end up with a motionless state with greater stability and thus release the threat from the occurrence of plume convection, which corresponds to the freckle of the casting.

C. Nature of stability modes

It is known that the marginal stability curves (or neutral curves) for layered convecting systems are- multi- branched with different modes broadly corresponding to convection in the different layers, which in turn is condu- cive to the bifurcations, the nonlinear interactions between the modes, and possible resonances when the critical Ray-


193.0.65.67 On: Thu, 18 Dec 2014 09:35:05

FIG. 16. Neutral curves of various A for the case corresponding to Fig. 9 FIG. 17. Neutral curves of various 5 for the case corresponding to Fig. 9 with c = 0.2. with& = 1.

leigh numbers or wave numbers for the different modes are similar. The particular case for the present configuration has been examined by Chen et a1.24P”P28 For thermal convective instability in superposed fluid and isotropic and homogeneous porous layers, Chen and Chen24 found that as { increases from 0.12 to 0.13, the convection shifts from a low a’, mode, in which the onset of convection occurs in both layers [i.e., two-layer flow, (TLF)] and that in the porous layer predominates the system, to a high a; mode, where the convection is largely confined to the fluid layer [i.e., fluid-layer flow (FLF)]. The neutral curves corresponding to this transition are bimodal. Similar results are also obtained by Chen and co-workers27 as well as by Chen and Hsu,~~ in which the variation of permeability anisotropic vectors and permeability inhomogeneous functions, respectively, of the porous medium leads to the same transition (the shift between TLF and FLF) with bimodal instability.

For the present situation, two types of stability mode are identified: One is bimodal and the other is unimodal. For a bimodal stability, the convection shifts between TLF and FLF; which result from a varying 6 as in the cases shown in Fig. 1 [or Figs. 2 (c) and 2 (d)], or from a varying -4 like that in Fig. 7 for g - 0.2 [or Figs. 8(f) and 8(g)], or from a varying C like that in Fig. 11 [or Figs. 12(f) and 12(g)], or from a varying 6 like that in Fig. 13, and so on. We present the neutral curves in Fig. 16 for the case c = 0.2 (Fig. 7) and A = - 5, - 4, - 3.1, - 3, - 2, and - 1, where a5;, experiences a dramatic jump within - 3.1 <.4< - 3. It is seen that the neutral curves for the tran-

sition between TLF and FLF [Figs. 8 (g ) and 8 ( f), respeo

I I ’ II I, I ’ I ’ , I ’ I I

1 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

a"b

tively] are bimodal. For - 3.1 <A< - 3, the R&, for both high and low a”, modes are almost identical.

For an unimodal instability, the onset of convection is largely confined to the fluid layer (FLF) while it is of multicellular convection. The cases of unimodal instability can be found in Fig. 5, which is induced by varying 5 and ends up with a transition between a tricellular convection and a bicellular convection [Figs. 6(c) and 6(d)], or in Fig. 9 induced by varying B. We present in Fig. 17 the neutral curves for the cases corresponding to the transition. As shown in Fig. 17, the neutral curves are unimodal, in which a& decreases from 8.41 for g = 0.004 to 2.86 for g = 0.003.

V. CONCLUDING REMARKS

We have implemented a linear stability analysis for the salt-finger convection in superposed fluid and porous layers, in which the permeability, thermal diffusivity, and solute diffusivity may be anisotropic and inhomogeneous. For anisotropic effects on the stability characteristics, it is con- cluded that, for a larger 5 or {, or a smaller &, the motionless state is less stable. For inhomogeneous effects, both local and bulk effects are the two major factors dictating the stability characteristics. Considering a real porous medium identified experimentally, we found that the salt- finger convection in the melt as the plume convection initiates is in a highly supercritical stage; meanwhile, the salt- tiger convection in the mush does not exist. We also propose both passive and active methods for the inhibition of salt-finger convection and conclude that an increase in R, is the most feasible method practically. Finally, the


193.0.65.67 On: Thu, 18 Dec 2014 09:35:05

nature of the stability mode relevant to the present situation is discussed. It is found that the instability mode may be either unimodal or bimodal, depending on the transition. The transition between TLF and FLF corresponds to a bimodal instability while the transition between two multicellular convections, which occurs only in the fluid layer, is relevant to an unimodal instability.

ACKNOWLEDGMENTS

Financial support for this work from National Science Council Grant No. NSC 80-0401-E-002-26 is gratefully acknowledged.

’ M. McLean, Directional Solidif ied Materials. for High Temperature Setvice (The Material Society, London, 1983).

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salt-finger instability in an anisotropic and inhomogeneous porous substrate underlying a fluid...

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