simulation of solidification with marangoni effects using...

ISTP-16, 2005, PRAGUE 16TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA

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Abstract Numerical simulations of the solidification of liquid bismuth with a free surface are carried out under different gravity conditions. A multiphase approach using the Volume of Fluid (VOF) method is used to solve the flow of the melt, as well as the gas phase. With the VOF method, the temperature distribution at the free surface is solved as part of the solution method. The dynamics of the gas phase affects the position of the solid-liquid interface at the free surface and consequently the shape of the growth front. Higher temperature gradients lead to more deformed interfaces, resulting in poorer quality crystals forming. Under normal gravity, buoyancy and thermocapillarity combine to add to the concavity of the interfaces, further deteriorating the resulting crystal. The effect of a curved and deformable melt free surface is also investigated numerically using an analogue material. Variation in the melt-wall contact angle further affects the degree of crystal interface distortion.

1 Introduction Thermocapillary convection (also known

as thermal Marangoni convection) in crystal growth melts has been the subject of intense research in recent years. Free surface temperature variations cause gradients in surface tension, giving rise to unbalanced tangential shear stresses [1]. The resultant shear stresses cause flow to move from the regions of low to high surface tension. The surface tension in most pure melts decreases with increasing

temperature, inducing the surface flow to move from the hot to cold regions.

Under normal gravitational conditions and large geometries, buoyancy effects dominate and Marangoni effects are masked. However, under microgravity conditions and in small scale geometries, Marangoni convection dominates [2, 3]. Marangoni convection plays an important role in crystal growth [2]. It can affect the quality of the crystal grown with respect to the solid-liquid interface shape or the homogeneity of doped materials and impurities [4, 5]. Many researchers have investigated the influence of Marangoni convection in crystal growth applications, both experimentally and numerically [4, 6, 7]. Most of the numerical work has been conducted assuming either a flat or deformable melt-gas free surface but neglecting the gas phase above the melt. It is known that the gas phase plays a significant role in the heat transfer from the melt free surface [4]. There may also be variations in surface tension coefficients for the melt material, depending on the type of gas that forms the melt-gas interface [8]. These factors have a significant impact on the strength of the Marangoni convection and consequently the shape of the solid-liquid interface. This paper details an investigation where the dynamics of the gas phase above the melt free surface has been completely modelled, in addition to studying solidification within the melt under microgravity and normal gravity

SIMULATION OF SOLIDIFICATION WITH MARANGONI EFFECTS USING A MULTIPHASE APPROACH

L. H. Tan, S. S. Leong, T. J. Barber and E. Leonardi School of Mechanical and Manufacturing Engineering The University of New South Wales, Sydney, Australia

Fax: +61 2 9663 1222 Email: [email protected]

Keywords: CFD, FLUENT, Solidification, Marangoni, VOF

mailto:[email protected]

Le-Han Tan, See-Seng Leong, Tracie Jacqueline Barber, Eddie Leonardi

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conditions. In addition, the effect of the free surface shape and melt-wall contact angles will also be considered.

2 Physical Problem Two different geometries are used in this solidification study, named Model 1 and 2.

2.1 Model 1 Model 1 is used to simulate the solidification of bismuth (Bi) with an argon gas gap (Ar). Pure bismuth has a Prandtl number, kcrP µ= , of 0.019. Here, µ is the viscosity, c is the specific heat and k is the thermal conductivity. The melt and gas are contained in a 2D closed rectangular container. Fig. 1 shows the L = 15 mm (long) by H = 5 mm (high) container. It is initially filled 4 mm high with melt with a 1 mm thick gas gap (pressure of 1 atmosphere) above.

Fig. 1. Model 1

The left wall is at a hot constant temperature, while the opposite right wall is at a cold constant temperature. The melt phase change temperature (544.55 K) lies in between the two wall temperatures. A linear temperature distribution from hot to cold is imposed on the top and bottom walls of the domain.

2.2 Model 2 Model 2 is used to simulate the solidification of an analogue material (Anlg) under zero gravity conditions. A vacuum (vac) is used as the ‘melt

atmosphere’. The analogue material has a Prandtl number (Pr) of 1 or 0.5. The effects of melt-wall contact angles and free surface shape on the crystal interface are studied using this model. Since a non-dimensional study was conducted, the dimensions of the model are given in terms of its length (L). Fig. 2 shows the rectangular geometry of Model 2. It is 0.4L high and is filled with the analogue melt to a height of 4L/15, with the vacuum gap being half of the initial melt height.

Fig. 2. Model 2

Similar to Model 1, the left wall is at a hot constant temperature, while the right wall is at a cold constant temperature. A linear temperature distribution is imposed on the top and bottom walls. The melt phase change temperature also lies in between the two endwall temperatures.

3 Numerical Model A finite volume computational fluid dynamics package, FLUENT V6, is used to simulate the solidification problem. The dynamics of the melt and gas phase are modelled within the Volume of Fluid (VOF) framework [9]. The surface tension forces are incorporated as a source term in the momentum equation via the Continuum Surface Force (CSF) technique [10]. The solidification is modelled using the enthalpy-porosity method [11].


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3.1 Governing Equations The pure bismuth (or analogue material) melt undergoes isothermal solidification at a fixed temperature. The melt and gas phases are assumed to be Newtonian with constant physical properties for each phase. The Boussinesq approximation is used and the flow is assumed to be laminar and incompressible. This is a reasonable assumption since the driving force and fluid velocities are not large. The surface tensions of the melts are assumed to vary linearly with temperature [8], which is typical of most pure liquids.

In the VOF formulation, the density (ρ), thermal conductivity (k), viscosity (µ), specific heat (c) and coefficient of thermal expansion (β) are dependent on the gas volume fraction (F) through the relationship [12]:

( ) meltgas FF φφφ −+= 1 (1)

where φ is either ρ, k, µ, c or β. With the gravity vector g acting in the negative y direction, the 2D governing equations of motion are given by [11, 12, 13]:

Continuity:

( ) 0=⋅∇ uρ (2)

Momentum:

( ) ( ) ( ) Suuuu

+∇−∇⋅∇=⋅∇+∂

∂ pt

µρρ

( )gFs ov T∆−++ βρ 1 (3)

Energy:

( ) ( ) ( ) hSTkTctTc

−∇⋅∇=⋅∇+∂

∂ uρρ (4)

In the preceding equations, u is the velocity, p is the pressure, T is the temperature, and ΔTo is the difference over the reference temperature. The source terms S, Fsv and Sh are defined as:

( )

( ) uSq

A+−−

= 3

21λ

λ (5)

( )( )σκσ

ρρ

ρt

meltgas

sv

F∇+

+

∇= nF ˆ

21

(6)

tL

S fh ∂

∂=

)( λρ (7)

For the Darcy type source term S, λ is the liquid fraction, q is a small number (0.001) to prevent division by zero and A is a large constant to extinguish the velocities in solidified regions. For the surface tension volume force term Fsv (which is nonzero only at the free surface), σ is the surface tension, κ is the interface curvature, n̂ is the unit surface normal and T

T tt ∇∂∂

=∇σσ

is the tangential surface tension gradient, which drives the Marangoni flow. Since σ depends on temperature, higher temperature gradients will give rise to a stronger surface flow. In the energy source term in equation (7), Lf represents the latent heat of fusion.

Interface tracking between the phases is achieved by solving the volume fraction transport equation [9]:

0=∇⋅+∂∂ F

tF u (8)

Reconstruction and advection of the free surface is achieved through use of the Geo-Reconstruct scheme [12, 14], which represents the interface using a piecewise linear approach.

3.2 Boundary Conditions For both Model 1 and 2, the velocity of the melt and gas on the enclosing container walls (x = 0 and x = L, y = 0 and y = H) is u = 0. The temperature boundary conditions are T = Thot at x = 0 (left wall) and T = Tcold at x = L (right wall). A linear temperature distribution from hot


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to cold is imposed on the top and bottom walls of both containers (y = 0 and y = H).

The linear temperature profile always fixes the solidification interface (at its phase change isotherm) at a fixed point on the bottom wall. For Model 1, the fixed point is located at x = 10

mm. For Model 2, the fixed point is at 32

=x L.

The melt is assumed to make a contact angle of θ with the endwalls, measured inside the gas phase from the wall to the tangent at the free surface. The implementation of the wall contact angle is based on Brackbill et al. [10], where the contact angle is used to adjust the free surface normal in cells near the wall. The unit normal vector at the near wall cell is modified as:

θθ sinˆcosˆˆ ww tnn += (9)

In (9), wn̂ refers to the wall unit normal and wt̂ refers to the wall unit tangent. The direction of the normal is then used to adjust the near wall surface curvature [12]. The wall adhesive force is then calculated using the volume force defined in (6). For Model 1, the bismuth melt makes a contact angle of 90° with the coldwall since the material there is always in the solid phase. The hotwall contact angle is also assumed to be 90°. The exact contact angle depends on physical and chemical conditions at the hotwall. Accurate data is not readily available for the problem, and hence the assumption of a 90° angle there. This implies an almost flat free surface, which is a common assumption for metallic melts due to the small dynamic surface deformations [15].

For Model 2, the coldwall contact angle is 90°, since the analogue material present there is solid. The hotwall melt contact angle is varied between 80° and 110° to study the effects of contact angle on the free surface deformation and the crystal interface.

3.3 Initial Conditions The bismuth and argon phases are initially at rest separated by a horizontal free surface. A steady state conduction solution is used to establish the initial temperature field. Hence, the initial crystal interface is planar and vertical, at

x = 10 mm and 32

=x L for Model 1 and 2

respectively.

3.4 Numerical method The unsteady, segregated, 2D solver in FLUENT V6 is used to solve the governing equations for each timestep. The linearized equations are solved sequentially and segregated from each other using a point implicit (Gauss-Seidel) linear equation solver together with an algebraic multigrid method. The convective terms in the governing equations are discretised with a second order upwind scheme while the diffusion terms are central differenced. The grid is generated using GAMBIT, and a co-located scheme is used where the velocity and pressure values are stored at cell centres. The PRESTO! (PREssure STaggering Option) scheme is used for pressure interpolation at the cell faces. This scheme is similar to the schemes used in structured staggered grids. For the transient problem, pressure-velocity coupling is achieved using the PISO algorithm [12].

In the solution of the governing equations, a scaled residual criterion (as defined in [12]) of 1x10-7 was used to judge the convergence of the continuity and momentum equations at each timestep. A residual criterion of 1x10-10 was used for the energy equation.

3.5 Non-dimensional Groups For convenience, non-dimensional groups are used to describe the results obtained. In the definition of the relevant groups that characterise the flow, the physical properties refer to the properties of the melt.


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The strength of the thermocapillary convection is represented by the Marangoni number,

)( µαγ TdMa ∆= , where T∂∂−= σγ is the temperature coefficient of surface tension, d is the initial melt layer height, T∆ is the temperature difference between the hot and cold walls, and α is the thermal diffusivity. Buoyancy strength is represented by the Rayleigh number, υαβ 3TdgRa ∆= where υ is the kinematic viscosity. The other relevant non-dimensional groups include the Prandtl number (Pr), Capillary number ( σγ TCa ∆= ), Bond number ( σρ 2gdBo = ), and the Stefan number ( LTcSte ∆= ).

4 Verification and Validation

4.1 Grid Refinement A non-uniform structured mesh was used for both models with a finer mesh near the free surface. The working mesh for Model 1 had 8680 cells (140 x 62), which was considered to be fine enough to resolve the flow behaviour. To prove this, a mesh sensitivity analysis was carried out using a finer grid (150 x 80 = 12000 cells) and a coarser grid (110 x 60 = 6600 cells). Using 75=−=∆ coldhot TTT K, the results from the three grids of Model 1 showed very little difference in the shape of the melt interface. Table 1 shows the predicted position of the crystal interface at the free surface for the different grids.

Table 1. Mesh refinement for Model 1

Grid (cells) 6600 8680 12000 Crystal interface position at (mm)

11.78 11.85 11.85

The result for the finest grid is identical to the typical working grid. There is only a 0.6 % difference between the results for the finest and the coarsest grid. Hence a grid size of 8680 cells is used for cases with ∆T ≥ 75 K. For cases with ∆T < 75 K, the coarsest grid is used to speed up solution time without loss of accuracy.

The typical working mesh for Model 2 had 6000 cells (100 x 60). A coarser mesh with 4500 cells (90 x 50) and a finer mesh with 9600 cells (120 x 80) were created in order to check grid independence. Table 2 shows the position of the crystal interface at the free surface, using Pr = 1, Ma = 240, Ca = 0.232, Ste = 0.3, Ra = Bo = 0, and θ = 100°.

Table 2. Mesh refinement for Model 2

Grid (cells) 4500 6000 9600 Non dimensional crystal

interface position 2.88 2.88 2.87

The solid-liquid interface shapes for the three grids are almost identical. The results for the coarsest and typical grids are the same. There is only a 0.27 % difference in the free surface interface position between the coarsest/typical and the finest grid. Hence, the typical mesh is used for all the computations.

4.2 Validation of Results Fig. 3 shows the crystal interface shape of the current model and that used by Giangi et al. [4] for single phase solidification with a flat free surface. The value of Ma is 1612, Ra = 5, Ste = 1.8 and Pr = 0.022.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4.4 4.6 4.8 5 5.2 5.4 5.6Non dim X

Non

dim

Y

Current Model, Ma = 1612

Giangi et al., Ma = 1612

Fig. 3. Crystal interface shape for Ma = 1612

The interface shape agreement with the previously reported result is excellent.


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Samsal & Hochstein [16] studied Marangoni convection (without solidification) with a deforming free surface in a cavity using the VOF method. Fig. 4 shows the Nusselt number distribution at the cavity left wall for the current solution and the reported result, for a contact angle θ = 170º, Ma = 100, Ca = 0.1 and Pr = 1.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0Nu

Non

dim

Y

Current Model

Samsal & Hochstein 1994

Fig. 4. Nusselt number distribution at left wall

The agreement between the models is very good, proving that the current model is able to accurately predict large free surface deformations during Marangoni convection.

5 Results

5.1 Model 1 (Bi melt-Ar gap, Microgravity) A typical residual gravitational acceleration of 45 μg [4] was used to simulate microgravity conditions. Four different values of ∆T were used, based on the temperature gradients typical of open boat solidification systems [17]. Table 3 summarises the corresponding non-dimensional numbers. It may be noted that Bo = 1.88x10-4 and Pr = 0.019 for all the cases.

Table 3. Non-dimensional values (μg)

∆T 12 K 24 K 75 K 120 K Ma 244 489 1527 2443 Ra 0.031 0.062 0.19 0.31 Ca 0.0022 0.0044 0.014 0.022 Ste 0.033 0.066 0.21 0.33

It is clear that the contribution from buoyancy is negligible and Marangoni effects dominate the flow. In addition, the small values of Ca indicate that the dynamic free surface deformation due to thermocapillary stresses is negligible [18]. This indicates that the free surface will not deform significantly from its initial flat shape for the wall contact angle of 90°. Examination of the free surface position for the highest Ma shows that the maximum deviation is only 0.25 % of the original layer depth. Fig. 5 shows the steady state melt and gas streamlines for the various values of Ma.

(a)

(b)

(c)

(d)

Fig. 5. Streamlines for (a) Ma = 244, (b) Ma = 489, (c) Ma = 1527 and (d) Ma = 2443

Marangoni convection causes the hot bismuth melt at the left free surface to flow towards the colder right side, where it transfers heat to the crystal interface. This is accompanied by a melt

SOLID

GAS

GAS

SOLID

GAS

SOLID

SOLID

GAS


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return flow in the bulk, away from the interface. This forms a main recirculation in the clockwise direction. For all the values of Ma, a strong primary cell rotating in the clockwise direction forms within the main melt recirculation just before the solidification front, causing the crystal interface to deform. The primary cell and main recirculation induces a secondary cell near the left wall, which rotates in the counter clockwise direction. The size and strengths of the primary and secondary cells rise as Ma increases. The thermocapillary flow also accelerates the argon gas near the free surface towards the right, which is accompanied by a return flow in the bulk to the left. A counter clockwise main gas recirculation is thus formed. For all the cases, there is a concentrated gas vortex embedded within the main recirculation near the solid-liquid interface. At higher Ma (1527 or 2443), the stronger thermocapillary flow increases the gas velocities. The original flow structure then destabilises into a multicellular pattern, consisting of two additional co-rotating cells within the main recirculation, upwind to the concentrated vortex. The crystal interface exhibits increasing concavity as Ma rises. This is due to the higher free surface velocities melting back the crystal and causing the primary cell to become stronger. Fig. 6 shows the shape of the melt interface with increasing values of Ma.

0

1

2

3

4

8 9 10 11 12 13 14x (mm)

y (m

m)

Ma = 244Ma = 489Ma = 1527Ma = 2443

Fig. 6. Crystal interface for various Ma

The influence of Ma on the increased curvature of the interface is evident. The quality of the crystal deteriorates as the concavity increases.

The degree of interface distortion is dependent on the strength of the thermocapillary flow, which is proportional to the free surface temperature gradients. With the VOF method adopted, the temperature distribution at the free surface is solved as part of the solution method, taking the dynamics of the melt and gas phases into account The exact thermal conditions are used and the free surface crystal interface position is located precisely. This avoids the need to assume thermal boundary conditions at the free surface, which may not be as accurate.

5.2 Model 1 (Bi melt-Ar gap, Normal gravity) To study the effect of combined buoyancy and thermocapillarity on the flow, the simulations were repeated using normal gravity conditions (1g). Table 4 summarises the corresponding non-dimensional numbers. The values of Pr, Ste and Ca are the same as before. Bo is now equal to 4.07 for all cases. The free surface shape remains virtually flat.

Table 4. Non-dimensional values (1g)

∆T 12 K 24 K 75 K 120 K Ma 244 489 1527 2443 Ra 672 1344 4200 6720

From the new values of Ra, it is evident that buoyancy now plays an important role in the melt convection. However, Marangoni effects are still very significant due to the small size of the container. Buoyancy drives the free surface melt in the same direction as the thermocapillary stresses (from the hot to the cold wall). In this configuration, buoyancy forces augment the surface tension driven flow. Fig. 7 shows the steady state melt and gas streamlines for the various values of Ma and Ra. The flow patterns have been modified significantly by the presence of gravity.


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(a)

(b)

(c)

(d)

Fig. 7. Streamlines for (a) Ma = 244, Ra = 672, (b) Ma = 489, Ra = 1344, (c) Ma = 1527, Ra = 4200 and (d) Ma = 2443, Ra = 6720

Overall, a main melt recirculation forms in the clockwise direction. Thermocapillary forces are responsible for the presence of a strong co-rotating primary cell embedded within the main recirculation. The cell is located in front of the crystal interface, and increases in strength and size as Ma increases. The main recirculation and primary cell induces a relatively weak secondary cell to form near the bottom wall. This small secondary cell rotates in the counter-clockwise direction, and does not reach the hot wall as before in the microgravity cases. For lower temperature gradients (Ma ≤ 489, Ra ≤ 1344), the shape of the overall main recirculation has now changed compared to Fig. 5 (a) & (b). Near the hot wall, buoyancy now

drives the hot fluid upwards within the clockwise rotating main cell, opposite to the flow direction there for the microgravity cases.

As the temperature gradients increase (Ma ≥ 1527, Ra ≥ 4200), the buoyancy effects become more pronounced. The driving force is now high enough for another “primary” cell to form within the main recirculation, near the hot wall (Fig. 7 (c) & (d)). The flow structure now comprises of two strong co-rotating cells in the clockwise direction, embedded within the main circulation. Due to the augmenting effects of buoyancy on the thermocapillary flow, the argon gas velocities are now higher than before for the microgravity case. The gas flow now comprises of a main counter clockwise recirculation, with multiple co-rotating vortices embedded within it. Even for the lowest temperature gradient (Ma = 244, Ra = 672), the basic gas flow structure in Fig. 5 (a) has destabilised into a two-cell pattern. As the driving forces become stronger (Ma ≥ 1527, Ra ≥ 4200), the gas flow destabilises further into a three-cell pattern within the main recirculation. Fig. 8 shows the combined effect of buoyancy and thermocapillarity on the shape of the crystal interface for Ma = 489 and Ma = 1527 (dashed lines). The original interface shapes for thermocapillarity alone are displayed for reference (solid lines with symbols).

0

1

2

3

4

8 9 10 11 12 13 14x (mm)

y (m

m)

Ma = 489, Ra = 0.062Ma = 489, Ra = 1344Ma = 1527, Ra = 0.19Ma = 1527, Ra = 4200

Fig. 8. Effect of buoyancy on crystal interface

GAS

GAS

GAS

GAS

SOLID

SOLID

SOLID

SOLID


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Buoyancy adds to the thermocapillary effects and causes the crystal interface to acquire greater concavity. Similar results for combined buoyancy-thermocapillarity in melts have been reported by Liang & Lan [6]. They found that the worst case of interface concavity occurred when both types of convection were present. The quality of the crystal grown under conditions of buoyant and surface tension convection is even poorer than before (under just thermocapillary dominated convection alone). From Fig. 8, it can be seen that the angle between the melt interface and bottom wall is now smaller. In crystal growth applications, this might cause parasitic nucleation leading to the formation of polycrystals, when single crystals are ideally desired [6].

5.1 Model 1 (Bi melt-Air gap, Microgravity) In order to further emphasize the importance of taking the flow and the type of gas atmosphere into account, the microgravity simulation with ∆T = 24 K was repeated using air (pressure of 1 atmosphere) as the gas gap instead of argon. Fig. 9 shows the shape of the crystal interface when either air or argon forms the atmosphere above the bismuth melt.

0

1

2

3

4

8 9 10 11 12 13 14x (mm)

y (m

m)

Air Gap, DT = 24 K

Argon Gap, DT = 24 K

Fig. 9. Crystal interface shape for air/argon gas

gaps, ΔT = 24 K The solidification interface is more distorted when the gap is filled with air. The surface tension temperature coefficient of bismuth with

an air atmosphere is –1.3 x 10-4 N/mK, while its coefficient with argon is smaller at –7 x 10-5 N/mK [8]. Due to bismuth-air’s larger surface tension temperature coefficient, the flow is stronger and there is greater concavity of the interface for the air gap case.

5.3 Model 2 (Anlg melt-Vac, Zero gravity) Model 2 is used to study the effects of a deformable curved free surface on the shape of the crystal interface for an analogue material. A vacuum acts as the ‘gas gap’. The non-dimensional study was conducted with Pr = 0.5, Ma = 480, Ca = 0.232 and Ste = 0.3. The values of Ra and Bo are zero, since a zero gravity environment was assumed. With these parameters fixed, only the hotwall melt contact angle (θ) was varied from 80° to 110°. Fig. 10 shows the free surface shape for each value of the specified contact angle. It may be noted that the free surface position and domain lengths have been normalised to the solid material height.

0.50

0.75

1.00

1.25

1.50

0.00 0.75 1.50 2.25 3.00 3.75Non dim X

Non

dim

Y

Theta = 80 degTheta = 90 degTheta = 100 degTheta = 110 deg

Fig. 10. Free surface position for various θ

For θ ≤ 90°, the melt dips at the hot wall, and rises later to conserve the melt volume. The opposite is true for the cases with θ > 90°. As the melt solidifies, the deflections drop and the free surface position approaches Y = 1 (the solid material top surface).


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Fig. 11 shows the melt streamlines, free surface shape and crystal interface for the various contact angles.

(a)

(b)

(c)

(d)

Fig. 11. Steady melt streamlines for various hotwall contact angles (a) θ = 80°, (b) θ = 90°,

(c) θ = 100° and (d) θ = 110°

The melt flow for all the cases consists of a unicellular clockwise recirculation. The centre of the cell is located closer to the crystal interface. Fig. 12 shows the shape of the melt isotherms for the same values of θ. The distortion of the

isotherms due to the clockwise melt recirculation is evident.

(a)

(b)

(c)

(d)

Fig. 12. Melt isotherms for various hotwall contact angles, (a) θ = 80°, (b) θ = 90°,

(c) θ = 100° and (d) θ = 110°

Based on the results of Fig. 11 and Fig. 12, it can be seen that the cases with different wall contact angles exhibit similar flow and isotherm structures. As reported by Rao & Shyy [7], the impact of the curved and deformable free surface is more quantitative rather than qualitative for the range of parameters considered.

SOLID

SOLID

SOLID

SOLID


11

Fig. 13 shows the shape of the solid-liquid interface for the various contact angles and free surface shapes.

0

0.2

0.4

0.6

0.8

1

2 2.4 2.8 3.2Nondim X

Non

dim

Y

Theta = 80 degTheta = 90 degTheta = 100 degTheta = 110 deg

Fig. 13. Crystal interface shape for various θ

As θ changes from 110° to 80°, the crystal is melted back more towards the cold wall, and acquires slightly more interface distortion. This is mainly due to the shape of the melt free surface near the crystal interface. From Fig. 10 and Fig. 12, it can be seen that the melt contact angle at the crystal interface (measured from the horizontal on the liquid melt side) grows as θ changes from 110° to 80°. The melt meniscus causes the local inclination of the temperature gradient and greatly affects the heat transfer to the crystal interface. As a result, the temperature gradient near the interface rises as θ changes from 110° to 80°. Consequently, the free surface velocities there also increase, thus melting the crystal further back and imparting more curvature to it. It may be noted that the melt-crystal interface contact angle is automatically established internally during the solution procedure as mass and volume fraction is conserved. 6 Conclusion Numerical simulation of the solidification of pure bismuth with an argon gas gap above it

was carried out using a VOF approach. The dynamics of the gas phase was shown to be important in establishing the degree of crystal interface deformation. Higher values of Ma gave rise to more distorted interfaces. The combined effects of buoyancy and thermocapillarity increased the concavity of the growth interfaces. Solidification of an analogue material with a deformable and curved free surface was also carried out using the same VOF procedure. The free surface shape did not change the flow patterns qualitatively, but had a quantitative impact on the degree of interface distortion.

Acknowledgements The authors wish to gratefully acknowledge the computing facilities provided by the AC3 Linux Barossa Cluster, Sydney, Australia.

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simulation of solidification with marangoni effects using...

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