Open access peer-reviewed chapter

Effect of Depth Length on Mixing in a Double-Diffusive Convection Two-Layer System

Written By

Hideo Kawahara

Submitted: June 17th, 2014 Reviewed: June 1st, 2015 Published: July 29th, 2015

DOI: 10.5772/60954

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1. Introduction

With the rapid development of industrial technologies within recent years, development of multi-function, high-performance materials that can survive harsh conditions is in demand. In regards to the materials manufacturing processes that accompany coagulation, it has been indicated that multilayer convection cell formation and amalgamation occurs due to double diffusive convection, and that this may exert an influence on material quality [1, 2]. When manufacturing this kind of material, solute elimination occurs on the coagulation interface during the multi-component solution coagulation process, and double diffusive convection occurs due to the gradient between temperature and concentration in the droplet phase. It is thought that multilayer cell structure formation and destruction occur due to this phenomenon in the droplet phase, and that this exerts a major influence over crystal growth speeds and material quality. However, many relevant details remain unexplained. As the phenomenon of convection cell amalgamation in particular is complex, the phenomenon of convection between low- and high-concentration layers has come to be considered as a system using experiments and mathematical simulations [3 - 10]. Bergman et al. [5] measured the speed of material passing through the diffusion interface at convection cell amalgamation and between convection cells. Subsequently, Nishimura et al. [7, 8] made broad-ranging changes to control parameters, and proposed a correlational formula for these issues. Hyun et al. [10] considered the amalgamation process in detail using 2-dimensional mathematical calculations, and showed that high-frequency component fluctuation occurs at the average Nunumber.

Convection cell amalgamation is governed by the diffusion interface, where major gradients in concentration are present. As both thermally unstable diffusive-type double diffusive convection and opposing, sheer flows due to thermal convection co-exist in interface areas according to calculation results by Hyun et al. [10], it is anticipated that interface instability can easily occur in these circumstances. Recently, Tanny et al. [9] focused on interface regions and investigated 2-dimensional structures using the Schlieren method. The team also reported that when initial buoyancy is low and thermal Rayleigh number is high, vortex arrays form above and below the interface, and convection cell amalgamation time is shortened. Considered in light of vortex structure, this phenomenon is thought to occur primarily due to sheer current instability. However, while traditional research also leads us to anticipate interface instability due to double diffusive convection, characteristics of diffusive-type double diffusive convection in sheer currents remain unknown.

Figure 1.

Diffusive interface between the convective cell

As shown in Fig. 1, diffusion interface between convection cells is stable in terms of concentration in a gravitational field, but is thermally unstable due to the presence of a vertical temperature differential. Accordingly, it is clear that it takes the aforementioned diffusive state. It has been previously confirmed that non-flowing diffusive interfaces fall into oscillating modality. This is explained via the diffusive interface shown in Fig. 2. As heat diffusion is much faster than material diffusion, high-concentration lower fluid bodies rise due to thermal buoyancy. Afterwards, the fluid body is cooled rapidly while maintaining its concentration, and then proceeds to descend due to reversed buoyancy. The repeated occurrence of this phenomenon at the interface is known as oscillation modality. The diffusive interface between convection cells shown in Fig. 1 contains sheer currents due to thermal convection caused by horizontal temperature gradients above and below the interface. As it is possible for 3-dimensional structures to form due to the presence of sheer currents and the occurrence of oscillation modality at the interface in this sort of situation, a detailed investigation is seen as necessary. In actuality, Chen et al. [11] have reported 3-dimensional structures in cases of finger-type interfaces coexisting with sheer currents.

Figure 2.

Oscillatory modes

In order to investigate 3-dimensional structures in diffusive interfaces in this research, 3-dimensional mathematical calculations were conducted for diffusive interfaces occurring in an aqueous solution with density stratification into a linear state, which is heated and cooled as a system from both sides in a rectangular container. In addition, the computational domain in the direction of depth is changed, and the mixing process of each convection layer was examined.

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2. Numerical analysis

2.1. Mathematical model

The model equations to describe the double diffusive natural convection consist of the continuity, the Navier-Stokes, the energy and the concentration equations in dimensionless forms as follows:

V=0E1
Vτ+(V)V=P+Pr2+PrRa(θNC)κE2
θτ+(V)θ=2θE3
θτ+(V)C=2LeE4

Dimensionless parameters Pr, Ra, Le, N and A are defined as follows:

Pr=νκ,Ragα(ThotTcold)b3κνLe=κD,N=β(CmaxCmin)α(ThotTcold),A=hbE5

Here, the dimensionless variables are defined as follows:

X=xb,Y=yb,Z=zbU=ubκ,V=νbκ,W=wbκτ=tκb2,P=pb2(ρκ2)θ=TTcoldThotTcold,C=CCminCmaxCminE6

2.2. Boundary conditions

The boundary conditions in dimensionless form are illustrated in Fig.3. In the computational domain (see Fig.2), the aspect ratio is 1.25 and length of the direction of depth (Z) examined three kinds (1.0, 1.5 and 2.0). The initial temperature in the system is 0.0 and the initial concentration is linear profile: C=f (Y)

Figure 3.

Boundary conditions

atX=0,U=V=W=0,θ=0.5,CX=0atX=1,U=V=W=0,θ=+0.5,CX=0atY=0,U=V=W=0,θZ,CZ=0atY=1.25,U=V=W=0,θY,CZ=0atZ=0,U=V=W=0,θY,CY=0atZ=1.0,1.5,2.0,U=V=W=0,θY,CY=0

2.3. Computation method

A fine difference method was employed to solve the model equations numerically. The calculation algorithm of pressure terms is the HSMAC method. To solve the mathematical model we developed a home-made code based on the HSMAC method [12,13] that utilizes the finite differences approximation: second order central finite differences for the viscous terms, third order up-wind finite differences (UTOPIA) [14] for the convective terms, and first order forward finite differences for the time derivative.

Details of the numerical algorithm are given next. The discrete form of eq. (2) is

(v˜x)mi,j,kn+1(v˜x)mi,j,knΔτ=(p˜)mi,j,kn+1(p˜)mi+1,j,kn+1Δx˜+(DSFX)i,j,knE7
(v˜y)mi,j,kn+1(v˜y)mi,j,knΔτ=(p˜)mi,j,kn+1(p˜)mi+1,j,kn+1Δy˜+(DSFY)i,j,knE8
(v˜z)mi,j,kn+1(v˜z)mi,j,knΔτ=(p˜)mi,j,kn+1(p˜)mi+1,j,kn+1Δz˜+(DSFZ)i,j,knE9
θi,j,kn+1θi,j,knΔτ=(DSEN)i,j,knE10

Here nis the time step and mis the iteration index. DSFX, DSFY, DSFZand DSENstand for the convection and viscous terms. The grid nodes have the indices (i, j, k) in X, Yand Zdirections, respectively.

The mass balance (1) provides an equation for pressure. Introducing the discrete derivative

Di,j,kn+1=(v˜x)i,j,kn+1(v˜x)i1,j,kn+1Δx˜+(v˜y)i,j,kn+1(v˜y)i,j1,kn+1Δy˜+(v˜z)i,j,kn+1(v˜z)i,j,k1n+1Δz˜E11

The grid pressure P˜i,j,kmay be calculated by Newton method

(p˜)m+1i,j,kn+1=(p˜)mi,j,kn+1Dmi,j,kn+1[(Di,j,kp˜i,j,k)mn+1]1=(p˜)mi,j,kn+1+δ(p˜)i,j,kn+1E12

The velocity is given by

v˜m+1x,(i,j,k)n+1=v˜mx,(i,j,k)n+1+(v˜x,(i,j,k)p˜(i,j,k))mn+1δp˜x,(i,j,k)n+1E13
v˜m+1y,(i,j,k)n+1=v˜my,(i,j,k)n+1+(v˜y,(i,j,k)p˜(i,j,k))mn+1δp˜y,(i,j,k)n+1E14
v˜m+1z,(i,j,k)n+1=v˜mz,(i,j,k)n+1+(v˜z,(i,j,k)p˜(i,j,k))mn+1δp˜z,(i,j,k)n+1E15

The calculation procedure is then:

  1. Use eqs. (7), (8),(9) to compute the velocities {(v˜x)m+1i.j,kn+1,(v˜y)m+1i.j,kn+1,(v˜z)m+1i.j,kn+1}for p˜m+1n+1=p˜mn.

  2. Repeat the calculation until Dm+1n+1εis satisfied, eq. (10)-(15) [in this studyε=103]; report (v˜x)n+1, (v˜y)n+1, (v˜z)n+1, p˜n+1.

  3. Calculate θn+1using eq. (10).

  4. Repeat the cycle 1-3, until the final time.

The third order upwind scheme (Utopia scheme) for the inertial terms of Eqs. (2)-(4) was applied to the calculation. The local Nunumber on the hot wall was calculated from a Taylor series for the temperature field and the average Nunumber was computed by integration over the hot wall. The time step width Δτwas 1×106.

2.4. Computational meshes

Figure 4 indicates computational domain (Z=1.0, 1.5, 2.0). The computational meshes used the uniform grid, and the finite difference grid points in the X-, Y- and Z-directions were 80×100×80,80×100×120,80×100×160.

Figure 4.

Computational domain

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3. Result and discussion

3.1. Overall system behaviour

Figure 5 shows change over time to flow distribution, temperature distribution, and concentration distribution for an X–Y plane at system depth coordinates Z=0.5. As density is stratified linearly under initial concentration conditions in these calculations, density strata interface behavior shows major changes over a short time when compared to standard density stratification. At τ=0.002(Fig. 5 (a)), immediately after calculation starts, currents appear in the previously-still solution, and a large-scale vortex rotating in a counter-clockwise direction is present near the center. Accompanying this, vortices form at opposed upper and lower corners of the system. As heating and cooling is conducted from left and right wall surfaces, influence on temperature distribution gradually becomes apparent in the solution. The initial linear state of concentration distribution strata can be confirmed as decaying from the system’s central region. At τ=0.005(Fig. 5 (b)), the vortex near the center decays, and an inclined interface appears in contrast near the center. This formation is shaped largely in parallel to concentration and flow distributions, and heating and cooling areas from wall areas spread toward the center. Concentration distribution changes in a form similar to temperature distribution, and a concentration plume forms near the interface. Each condition becomes pronounced at τ=0.008(Fig. 5 (c)). Of the multiple plumes formed along concentration distribution lines in Fig. 3 (b), one plume is present in both the upper and lower high- and low-concentration sides. At τ=0.015(Fig. 5 (d)), currents have become extremely complex and 3-dimensional. In terms of temperature distribution, a higher temperature stratum is present at the top, and a low temperature stratum is present at the bottom. Furthermore, the inclined interface becomes vertical in terms of concentration distribution, disorder is present near the interface, and an unstable state can be seen.

Figure 5.

Velocity vector, concentration and temperature distribution for Z=2.0in depth length at Z=0.5.

3.2. Heat transfer characteristics

Figure 6 shows the variation of the average Nunumber in the heated wall. Furthermore, during the figure show Z=1.0, 1.5, and 2.0 results. Overall changes in the Nunumber are different with pure convection as shown in Figure 6, uneven distribution. Minimum Nunumber (position 1) to transition from thermal to thermal convection is less than pure convection. It is in due to the vertical density stratification. Then the Nunumber increased with time and repeat the development layer, Nunumber refers to the maximum value (position 2). However, for changing vertical from horizontal surface as shown in Fig. 5 (d), from heated wall heat transfer of cooling wall will be suppressed. As a result the Nunumber is reduced again. Since then each layer blends Nunumber increases again. And completely mixed, Nunumber indicating steady-state value.

Note changes in the depth direction Z: increases the value of Z, and minimum value of the Nunumber is even smaller. Also the NuMax value has not been observed difference. The Nunumber becomes a steady state value of time becomes shorter with increasing Z.

Figure 6.

AverageNunumber on the hot wall with time for Z=1.0, 1.5 and 2.0 in depth length

3.3. Influence of computational domain (depth length: Z)

Figure 7 shows τ=0.01at depths of Z=1.0, 1.5, and 2.0 for X–Y plane (Z=0.5) and Y–Z plane (X=0.5) and a 3-dimensional contour diagram of concentration. Overall concentration structure near the interface can be confirmed as symmetrical along vertical and horizontal axes. For X–Y plane concentration distribution of 1.0×1.25×1.0 (Fig. 7 (a)), the interface is inclined and interface between high and low concentrations is extended near the center. When Z is increased, it is clear that high and low concentration interface extension increases further near the center as shown in Fig. 7 (b) and (c). On the Y–Z plane, the same patterns form horizontally and vertically at the center as Z rises. In the 3-dimensional contour diagram, at 1.0×1.25×1.0 (Fig. 7 (a)), a concentration distribution with significant curvature appears near the center of the interface. Additionally, concentration distribution is unstable vertically along the Y axis. When Z is increased even further, the curved interface near the center disappears, and the center changes to an extremely smooth interface as shown in Fig. 7 (b) and (c). Disorder near the center decreases in Y axis vertical concentrations as Z increases.

Figure 7.

Concentration profile atτ=0.01

Similarly, Figure 8 shows the result of the concentration distribution in the τ=0.015. Overall concentration distribution, as can be seen in the interface of the X-Y plane than in Figure 5, has the left and right opposite slope. In addition, the interface near it changes to a very complex structure. Further, when Z is large, as can be seen in the contour diagram, toward the center of the interface (Z= 0.5) from the wall direction (X = 0, 1.0) unevenness appeared on the structure of the interface, resulting in three-dimensional structure.

Figure 8.

Concentration profile atτ=0.015

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4. Conclusions

We studied three-dimensional numerical calculations of double-diffusive convection in a rectangular parallelepiped enclosure filled with salt-stratified fluid under a horizontal temperature difference for realistic values of parameters (A=1.25, Ra=2.7×107, N=0.882, Pr=7.15for an NaCl-H2O system). The initial salt concentration linear profile was examined.

Three-dimensional numerical result with a high spatial resolution can capture the fine structures such as salt fingers and traveling plumes. In the linear profile, near the interface, vertical motion is dominant due to salt fingers in each layer. In addition, when the computational domain changes, it has a big influence on the behaviour of plume that appears on the diffusive interface. As a result, the mixing state of the entire faction is greatly different.

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Nomenclature

Aaspect ratio=h/b
bwidth of the system m
Bdimensionless width of the system
Cdimensionless concentration=(ccmin)/(cmaxcmin)
cconcentration kgm-3
cmaxinitial maximum concentration (initial concentration in a linear profile) kgm-3
cmininitial minimum concentration (initial concentration in a linear profile) kgm-3
Ddiffusion coefficient m2s-1
gacceleration due to gravity ms-2
hheight of the system m
Hdimensionless height of the system
LeLewis number = κ/D
Nbuoyancy ratio=β(cmaxcmin)/{α(ThotTcold)}
Nuaveaverage Nusselt number
ppressure Pa
Pdimensionless pressure
PrPrandtl number =ν/k
RaRayleigh number=gα(ThotTcold)b3/(κν)
Ttemperature K
Tcoldtemperature on the cold wall K
Thottemperature on the hot wall K
ttime s
U,V,Wdimensionless velocity
X,Y,Wdimensionless coordinate
Greek symbols
αvolumetric coefficient of thermal expansion K-1
βvolumetric coefficient of expansion with concentration m3kg-1
Δτdimensionless time step
θdimensionless temperature=(TTcold)/(ThotTcold)
κThermal diffusivity m2s-1
νKinematic viscosity m2s-1
ρDensity kgm-3
τdimensionless time

References

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Written By

Hideo Kawahara

Submitted: June 17th, 2014 Reviewed: June 1st, 2015 Published: July 29th, 2015