- Email: [email protected]
Combined effect of crucible rotation and magnetic field on hydrothermal wave
Journal of Crystal Growth 385 (2014) 72–76
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Combined effect of crucible rotation and magnetic field on hydrothermal wave Y. Takagi a,n, Y. Okano a, H. Minakuchi b, S. Dost c a
Department of Materials Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan Department of Mechanical Systems Engineering, University of the Ryukyus, 1 Senbaru, Nishihara, Okinawa 903-0213, Japan c Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada V8W 3P6 b
art ic l e i nf o
a b s t r a c t
Available online 19 June 2013
The combined effect of crucible rotation and applied magnetic field on hydrothermal wave was investigated through a three-dimensional simulation. The computational domain was a shallow annular pool filled with a silicon melt, subjected to a crucible rotation and a vertical static magnetic field. The thermocapillary flow along the free surface was considered. Governing equations of the system were solved numerically by the finite volume method. The efficiency of controlling hydrothermal wave with crucible rotation and magnetic field was assessed. Results showed that although the hydrothermal wave was completely suppressed under the effect of magnetic field alone, the optimum combination of the body forces induced by the applied crucible rotation and magnetic field provides a better control of the hydrothermal wave. & 2013 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Convection A1. Fluid flow A1. Magnetic fields
1. Introduction
2. Numerical method
Hydrothermal wave (HTW) is the propagation phenomenon of temperature and velocity fluctuations observed along the free surface of a liquid. HTW develops due to the thermocapillary convective flow induced by the surface tension gradient in the liquid along the free surface. The strength of this thermocapillary flow is described by the dimensionless Marangoni number. This flow in the liquid is two-dimensional and stable at low Marangoni number values, however, the flow becomes three-dimensional and unstable as the Marangoni number increases. The critical condition at which a HTW develops was theoretically investigated by Smith and Davis [1,2]. The development of a HTW is clearly seen in a shallow liquid pool during the final stage of Czochralski crystal growth [3]. The HTW analysis is free from the influence of the effect of natural convection. Thus, this makes the HTW analysis very useful in investigating the relative contribution of Marangoni convection. From these points of view, the dynamics and instability of HTW in a shallow annular pool were investigated numerically in Refs. [4–9]. In these studies, the control of HTW was not fully examined, and only the effect of pool (crucible) rotation was taken into account [5]. In the present study, we have examined numerically the combined effect of applied magnetic field and crucible rotation, and introduced an effective method to control HTW.
A schematic view of the computational domain is shown in Fig. 1. The silicon melt is contained in a shallow annular pool of depth d ¼3 mm, inner radius ri ¼ 15 mm and outer radius ro ¼50 mm. The upper boundary is a free surface and other boundaries are solid wall. The pool rotates with an angular velocity ω around the z-axis, and an applied magnetic field in the same direction. It was assumed that the silicon melt is an incompressible, Newtonian fluid, and the Boussinesq approximation holds. Under these assumptions, the governing equations of the liquid phase (silicon melt) take the following forms: Continuity:
n
Corresponding author. Tel./fax: +81 6 6850 5849. E-mail address: [email protected] (Y. Takagi).
0022-0248/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2013.04.062
∇v¼0
ð1Þ
Momentum: ∂v 1 þ ðv ∇Þv ¼ − ∇p þ ν∇2 v þ βgðT−T C Þez þ FM ∂t ρ
ð2Þ
Energy: ∂T þ ðv ∇ÞT ¼ α∇2 T ∂t
ð3Þ
Magnetic field: ∂B 1 2 −∇ ðv BÞ ¼ ∇ B ∂t μ0 s
ð4Þ
where v is the flow velocity vector, t time, ρ the melt density, p pressure, β the thermal expansion coefficient, g the gravitational constant, T temperature, T C temperature of the inner wall, ez the
Y. Takagi et al. / Journal of Crystal Growth 385 (2014) 72–76
73
Table 1 Physical properties of silicon melt [4,12].
Fig. 1. Schematics of the computational domain.
Property
Symbol
Value
Thermal conductivity (W m−1 K−1) Viscosity (kg m−1 s−1) Density (kg m−3) Gravitational acceleration (m s−2) Thermal expansion coefficient (K−1) Surface tension coefficient (Nm−1 K−1) Heat capacity (J kg−1 K−1) Melting temperature (K) Electric conductivity (Sm−1) Magnetic permeability (Hm−1)
λ μ ρ g β γT Cp Tm s μ0
64 7.0 10−4 2530 9.81 1.5 10−4 −7.0 10−5 1000 1683 1.2 106 1.26 10−6
Table 2 Numerical parameters of the HTW simulation.
Fig. 2. Computational grid of the HTW simulation.
unit vector along the z-axis, FM the Lorentz force induced by the magnetic applied field, α the melt thermal diffusivity, B magnetic flux, μ0 magnetic permeability, and s electric conductivity. Eq. (4) is the magnetic field equation derived from the wellknown Maxwell's equations [10]. The Lorentz force in Eq. (2) is calculated by FM ¼ J B ¼
1 ð∇ BÞ B μ0
Parameter
Symbol
Value
Innner wall temperature (K) Temperature diffrence (K) Marangoni number (–) Grashof number (–) Rotation speed (min−1) Rotating Reynolds number (–) Magnetic flux density (mT) Hartmann number (–)
Tc ΔT Ma Gr ω Reω B0 Ha
1683 21 2.91 103 3.01 10−3 0, 1, 2, 5 0, 946, 1892, 4731 0, 26.3, 39.5, 52.6 0, 38, 57, 72
ð5Þ
where J is the induced electric current. The governing equations of the system (Eqs. (1)–(4)) were discretized by the finite volume method, and were solved by the PISO algorithm. Computations were carried out by using the OpenFOAM code. We assume no slip boundary conditions for the flow velocity field on the solid boundary, which determines the circumferential flow velocity on the boundary from the angular velocity of the pool rotation. Along the free surface, the induced flow velocity (by the Marangoni convection) was related to temperature gradient as follows: μ
∂v ¼ γ T ∇T ∂n
ð6Þ
where n is the unit normal to the free surface. The computational grid shown in Fig. 2 was made in a cylindrical coordinate system and the grid points were clustered near the solid wall and the free surface. The grid numbers in the radial – (r), circumferential – (θ), and vertical – (z) directions are 81, 180 and 21, respectively. The maximum values of the dimensionless wall distance yþ (normalized coordinate with respect to kinematic viscosity and friction velocity) on the inner, outer and bottom walls were 3.64, 1.16 and 1.06, respectively. These grid resolutions near the wall were sufficient to capture the boundary layer, therefore, a turbulence model was not adopted. Physical properties of the silicon melt are listed in Table 1. The Prandtl number of the silicon melt is 1:09 10−2 . Other numerical parameters and the dimensionless numbers involved (the Marangoni number Ma ¼ −γ T ΔTðr o −r i Þ=μα, the Grashof number 3 Gr ¼ gβΔTd =ν3 , the rotation Reynolds number Reω ¼ r 2o ω=ν and the Hartmann number Ha ¼ B0 ðr o −r i Þðs=νÞ1=2 ) are presented in Table 2. The relative strength of natural convection with respect to the Marangoni convection, which is estimated with the ratio of Grashof and Marangoni numbers Gr 1=2 =Ma2=3 [11] is small: 2:70 10−4 . Thus, the contribution of the gravitational term
Fig. 3. Surface temperature fluctuation without external force at t¼ 150 s.
βgðT−T C Þez in Eq. (2) was neglected in the present numerical simulation. This condition of neglecting the gravitational effect is valid for very shallow pools.
3. Results and discussion 3.1. Validation of the simulation code Before the actual simulations, we have first validated the simulation code in the absence of external body forces (induced by the applied crucible rotation and magnetic field). The simulation results of the validation are presented in Figs. 3–5.
74
Y. Takagi et al. / Journal of Crystal Growth 385 (2014) 72–76
Fig. 6. Time development of surface temperature at r ¼20 mm with only crucible rotation. Fig. 4. Space-time diagram of surface temperature in the absence of external forces.
Fig. 5. Circumferential temperature (top) and velocity vectors (bottom) in the r¼ 20 mm distance at t¼ 150 s.
Fig. 3 shows the computed surface temperature fluctuation, δT, in the horizontal plane, which is defined by 1 δTðr; θ; z; tÞ ¼ Tðr; θ; z; tÞ− 2π
Z
2π 0
Tðr; θ; z; tÞ dθ
ð7Þ
As seen from the figure, the surface temperature is not axisymmetric and fluctuates in the circumferential direction. This describes the typical spoke pattern of a HTW, which validates the code in terms of spoke pattern. The observed wavenumber (the number of wavy pattern) is 7, which also agreed with that of Li et al. [4]. To examine the temporal motion of HTW, the space-time distribution of the surface temperature was computed, and plotted in Fig. 4. The surface temperature along the circumference of r ¼20 mm was temporally sampled, and plotted (contoured) with respect to the circumferential angle and time. At the fully developed state, after t¼ 110 s, the phase velocity of the HTW (calculated from the slope of the iso-contours) was constant at the value of 0.15 rad/s. Although the sign of phase velocity can be negative (which depends on the initial conditions), its magnitude reaches the same value eventually. In Fig. 5, the computed temperature and velocity vectors in the circumferential plane of r ¼20 mm are presented at t¼ 150 s. The vertical structures were seen in the interval of the spoke patterns (above figures), and the flow velocity (induced by the Marangoni convection due to the circumferential temperature difference) vectors are shown in the figure below. The good agreement between our computed spoke patterns and the motion of the HTW (in the absence of external body
Fig. 7. Time development of surface temperature at r¼ 20 mm with only magnetic field.
forces) with those of Li et al. [4] confirms the validation of the present simulation code.
3.2. HTW control with a single body force The simulations were first carried out under crucible rotation for four rotation speeds (ω ¼ 0, 1, 2, and 5 min−1). Fig. 6 shows the time development of surface temperature at a sampling point of r ¼20 mm. From the figure, we can see that the amplitudes of −1 temperature fluctuations at ω ¼ 1 min and 2 min−1 are in the same order of those without rotation, and the HTW suppression with rotation is not monotonic to the rotation speed. The period of fluctuations at ω ¼ 1 min−1 is very large compared with those at other speeds because the characteristics of HTW changed at this critical condition as discussed in Li et al. [5]. At the higher rotation speed (ω ¼ 5 min−1 ), although the temperature fluctuations became quite smaller, HTW is not still suppressed completely. Next, we have run the simulations under the applied magnetic field without crucible rotation. The simulation results in this case are quite different than those under crucible rotation. Fig. 7 shows the time development of surface temperature at the sampling point of r¼ 20 mm. As the Hartmann number increases, the amplitudes of temperature fluctuation decrease monotonically, and the HTW disappears completely at Ha ¼72. Comparison of the results of these two cases clearly shows that the application of a magnetic field is more effective in suppressing the HTW.
Y. Takagi et al. / Journal of Crystal Growth 385 (2014) 72–76
3.3. HTW control with combined body forces In this case we consider the simultaneous application of magnetic field and crucible rotation to find the most effective combination for suppression. Based on the results of the previous section, we selected Ha¼57 for the magnetic field level, and varied the crucible rotation speed from 0 min−1 to 2 min−1. Fig. 8 shows the time development of surface temperature at the sampling point of r ¼20 mm. As the rotation speed increases, the period and the amplitudes of the temperature fluctuation decreases. At −1 Ha¼ 57 and ω ¼ 2 min , the HTW disappears completely, and the flow becomes two-dimensional. Results indicate that the combination of a magnetic field at Ha¼57 and a crucible rotation at ω ¼ 2 min−1 provides the optimum suppression.
3.4. Effects of body forces In this section we discuss the individual effects of the body forces induced by the applied magnetic field (Lorentz force) and the crucible rotation (centrifugal force). The distributions of the Lorentz and centrifugal forces on the melt surface were computed and plotted in Figs. 9 and 10 at t ¼100 s. As seen, the Lorentz force is stronger around the inner wall and consequently suppresses the radial and circumferential flow velocity components (induced by
75
the Marangoni convection). Figs. 9(a) and (b), at the lower rotation speeds, the HTW spoke patterns develop in the region where the Lorentz force is strong. However, as seen in Fig. 10, the centrifugal force is stronger around the outer wall, in contrast with the previous case, since its magnitude is proportional with the distance from the center. Thus, one can see that these two body forces at their optimum combination provide a complete suppression of the HTW by controlling different regions of the flow in the pool. Finally, we summarize the resultant body force magnitudes for each case in Table 3. Magnitudes of the body force are summed over the pool volume and the results for the cases of crucible rotation and magnetic field are presented. In the case of crucible rotation alone (Ha¼0, ω ¼ 5 min−1 ), although the body force was the largest, the HTW was not suppressed completely (not disappeared). However, in the case of magnetic field alone, the HTW was completely suppressed although the magnetic body force was smaller than the centrifugal force. This is due to the fact that the Lorentz force is stronger in the region where the Marangoni convection occurs, and thus provides an effective suppression. −1 In the combined case of Ha¼57 and ω ¼ 2 min , the total body force is comparable to that of the Lorentz force under the magnetic field alone, and thus the HTW was fully suppressed. Results show that a proper combination of such body forces would provide an effective control of a HTW.
4. Conclusions The combined effect of crucible rotation and applied magnetic field on hydrothermal wave was investigated through a threedimensional simulation study. The following conclusions were drawn from the simulation results:
Fig. 8. Time development of surface temperature at r ¼20 mm with the combination of crucible rotation and magnetic field at Ha¼ 57.
1. Under crucible rotation alone, the effect of hydrothermal wave on temperature fluctuations was not monotonic, and larger rotation speeds are required for significant decreases in temperature fluctuations. 2. The Lorentz force under an applied magnetic field alone, however, acts effectively and provides a complete suppression of the Marangoni convection around the inner wall of the pool. In this case the hydrothermal wave disappears completely. 3. At the optimum combination of crucible rotation and magnetic field, a smaller total body force is required to suppress the
Fig. 9. Surface distribution of the Lorentz force magnitude at t ¼100 s: (a) Ha¼ 57, ω ¼ 0 min−1 ; (b) Ha¼ 57, ω ¼ 1 min−1 ; (c) Ha¼57, ω ¼ 2 min−1 .
76
Y. Takagi et al. / Journal of Crystal Growth 385 (2014) 72–76
Fig. 10. Surface distribution of the centrifugal force magnitude at t¼ 100 s: (a) Ha ¼57, ω ¼ 0 min−1 ; (b) Ha ¼57, ω ¼ 1 min−1 ; (c) Ha¼ 57, ω ¼ 2 min−1 .
Table 3 Resultant body force magnitudes. Hartmann number (–)
Rotation speed (min−1)
Lorentz force (10−4 N)
Centrifugal force (10−4 N)
Total force (10−4 N)
Result flow pattern
0 57 57 57 72
5 0 1 2 0
0.000 1.412 1.416 1.419 2.124
5.297 0.000 0.212 0.848 0.000
5.297 1.412 1.628 2.266 2.124
HTW HTW HTW 2D flow 2D flow
hydrothermal wave completely. This result is particularly important in the development of crystal growth techniques that can be performed under weaker magnetic fields and smaller crucible rotation speeds.
Acknowledgments This work was financially supported by a Grant-in-Aid for Scientific Research (B) (No. 22360343) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] M.K. Smith, S.H. Davis, Journal of Fluid Mechanics 132 (1983) 119–144. [2] M.K. Smith, S.H. Davis, Journal of Fluid Mechanics 132 (1983) 145–162.
[3] T. Azami, S. Nakamura, M. Eguchi, T. Hibiya, Journal of Crystal Growth 233 (2001) 99–107. [4] Y.-R. Li, N. Imaishi, T. Azami, T. Hibiya, Journal of Crystal Growth 260 (2004) 28–42. [5] Y.-R. Li, L. Xiao, S.-Y. Wu, N. Imaishi, International Journal of Heat Mass Transfer 51 (2008) 1810–1817. [6] Y.R. Li, F. Ling, L. Peng, J.W. Tang, Heat Mass Transfer 45 (2009) 1335–1340. [7] Y.R. Li, L. Peng, J.W. Tang, Microgravity Science and Technology 21 (Suppl. 1) (2009) S289–S297. [8] W. Shi, Y.-R. Li, M.K. Ermakov, N. Imaishi, Microgravity Science and Technology 22 (2010) 315–320. [9] W. Shi, X. Liu, G. Li, Y.-R. Li, L. Peng, M.K. Ermakov, N. Imaishi, Journal of Superconductivity and Novel Magnetism 23 (2010) 1169–1172. [10] Y. Takagi, Y. Okano, S. Dost, Journal of Heat Transfer 134 (2012) 012301. [11] Y. Okano, A. Hatano, A. Hirata, Journal of Chemical Engineering of Japan 22 (1989) 385–388. [12] H. Hirata, K. Hoshikawa, Journal of Crystal Growth 125 (1992) 181–207.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Combined effect of crucible rotation and magnetic field on hydrothermal wave Y. Takagi a,n, Y. Okano a, H. Minakuchi b, S. Dost c a
Department of Materials Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan Department of Mechanical Systems Engineering, University of the Ryukyus, 1 Senbaru, Nishihara, Okinawa 903-0213, Japan c Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada V8W 3P6 b
art ic l e i nf o
a b s t r a c t
Available online 19 June 2013
The combined effect of crucible rotation and applied magnetic field on hydrothermal wave was investigated through a three-dimensional simulation. The computational domain was a shallow annular pool filled with a silicon melt, subjected to a crucible rotation and a vertical static magnetic field. The thermocapillary flow along the free surface was considered. Governing equations of the system were solved numerically by the finite volume method. The efficiency of controlling hydrothermal wave with crucible rotation and magnetic field was assessed. Results showed that although the hydrothermal wave was completely suppressed under the effect of magnetic field alone, the optimum combination of the body forces induced by the applied crucible rotation and magnetic field provides a better control of the hydrothermal wave. & 2013 Elsevier B.V. All rights reserved.
Keywords: A1. Computer simulation A1. Convection A1. Fluid flow A1. Magnetic fields
1. Introduction
2. Numerical method
Hydrothermal wave (HTW) is the propagation phenomenon of temperature and velocity fluctuations observed along the free surface of a liquid. HTW develops due to the thermocapillary convective flow induced by the surface tension gradient in the liquid along the free surface. The strength of this thermocapillary flow is described by the dimensionless Marangoni number. This flow in the liquid is two-dimensional and stable at low Marangoni number values, however, the flow becomes three-dimensional and unstable as the Marangoni number increases. The critical condition at which a HTW develops was theoretically investigated by Smith and Davis [1,2]. The development of a HTW is clearly seen in a shallow liquid pool during the final stage of Czochralski crystal growth [3]. The HTW analysis is free from the influence of the effect of natural convection. Thus, this makes the HTW analysis very useful in investigating the relative contribution of Marangoni convection. From these points of view, the dynamics and instability of HTW in a shallow annular pool were investigated numerically in Refs. [4–9]. In these studies, the control of HTW was not fully examined, and only the effect of pool (crucible) rotation was taken into account [5]. In the present study, we have examined numerically the combined effect of applied magnetic field and crucible rotation, and introduced an effective method to control HTW.
A schematic view of the computational domain is shown in Fig. 1. The silicon melt is contained in a shallow annular pool of depth d ¼3 mm, inner radius ri ¼ 15 mm and outer radius ro ¼50 mm. The upper boundary is a free surface and other boundaries are solid wall. The pool rotates with an angular velocity ω around the z-axis, and an applied magnetic field in the same direction. It was assumed that the silicon melt is an incompressible, Newtonian fluid, and the Boussinesq approximation holds. Under these assumptions, the governing equations of the liquid phase (silicon melt) take the following forms: Continuity:
n
Corresponding author. Tel./fax: +81 6 6850 5849. E-mail address: [email protected] (Y. Takagi).
0022-0248/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2013.04.062
∇v¼0
ð1Þ
Momentum: ∂v 1 þ ðv ∇Þv ¼ − ∇p þ ν∇2 v þ βgðT−T C Þez þ FM ∂t ρ
ð2Þ
Energy: ∂T þ ðv ∇ÞT ¼ α∇2 T ∂t
ð3Þ
Magnetic field: ∂B 1 2 −∇ ðv BÞ ¼ ∇ B ∂t μ0 s
ð4Þ
where v is the flow velocity vector, t time, ρ the melt density, p pressure, β the thermal expansion coefficient, g the gravitational constant, T temperature, T C temperature of the inner wall, ez the
Y. Takagi et al. / Journal of Crystal Growth 385 (2014) 72–76
73
Table 1 Physical properties of silicon melt [4,12].
Fig. 1. Schematics of the computational domain.
Property
Symbol
Value
Thermal conductivity (W m−1 K−1) Viscosity (kg m−1 s−1) Density (kg m−3) Gravitational acceleration (m s−2) Thermal expansion coefficient (K−1) Surface tension coefficient (Nm−1 K−1) Heat capacity (J kg−1 K−1) Melting temperature (K) Electric conductivity (Sm−1) Magnetic permeability (Hm−1)
λ μ ρ g β γT Cp Tm s μ0
64 7.0 10−4 2530 9.81 1.5 10−4 −7.0 10−5 1000 1683 1.2 106 1.26 10−6
Table 2 Numerical parameters of the HTW simulation.
Fig. 2. Computational grid of the HTW simulation.
unit vector along the z-axis, FM the Lorentz force induced by the magnetic applied field, α the melt thermal diffusivity, B magnetic flux, μ0 magnetic permeability, and s electric conductivity. Eq. (4) is the magnetic field equation derived from the wellknown Maxwell's equations [10]. The Lorentz force in Eq. (2) is calculated by FM ¼ J B ¼
1 ð∇ BÞ B μ0
Parameter
Symbol
Value
Innner wall temperature (K) Temperature diffrence (K) Marangoni number (–) Grashof number (–) Rotation speed (min−1) Rotating Reynolds number (–) Magnetic flux density (mT) Hartmann number (–)
Tc ΔT Ma Gr ω Reω B0 Ha
1683 21 2.91 103 3.01 10−3 0, 1, 2, 5 0, 946, 1892, 4731 0, 26.3, 39.5, 52.6 0, 38, 57, 72
ð5Þ
where J is the induced electric current. The governing equations of the system (Eqs. (1)–(4)) were discretized by the finite volume method, and were solved by the PISO algorithm. Computations were carried out by using the OpenFOAM code. We assume no slip boundary conditions for the flow velocity field on the solid boundary, which determines the circumferential flow velocity on the boundary from the angular velocity of the pool rotation. Along the free surface, the induced flow velocity (by the Marangoni convection) was related to temperature gradient as follows: μ
∂v ¼ γ T ∇T ∂n
ð6Þ
where n is the unit normal to the free surface. The computational grid shown in Fig. 2 was made in a cylindrical coordinate system and the grid points were clustered near the solid wall and the free surface. The grid numbers in the radial – (r), circumferential – (θ), and vertical – (z) directions are 81, 180 and 21, respectively. The maximum values of the dimensionless wall distance yþ (normalized coordinate with respect to kinematic viscosity and friction velocity) on the inner, outer and bottom walls were 3.64, 1.16 and 1.06, respectively. These grid resolutions near the wall were sufficient to capture the boundary layer, therefore, a turbulence model was not adopted. Physical properties of the silicon melt are listed in Table 1. The Prandtl number of the silicon melt is 1:09 10−2 . Other numerical parameters and the dimensionless numbers involved (the Marangoni number Ma ¼ −γ T ΔTðr o −r i Þ=μα, the Grashof number 3 Gr ¼ gβΔTd =ν3 , the rotation Reynolds number Reω ¼ r 2o ω=ν and the Hartmann number Ha ¼ B0 ðr o −r i Þðs=νÞ1=2 ) are presented in Table 2. The relative strength of natural convection with respect to the Marangoni convection, which is estimated with the ratio of Grashof and Marangoni numbers Gr 1=2 =Ma2=3 [11] is small: 2:70 10−4 . Thus, the contribution of the gravitational term
Fig. 3. Surface temperature fluctuation without external force at t¼ 150 s.
βgðT−T C Þez in Eq. (2) was neglected in the present numerical simulation. This condition of neglecting the gravitational effect is valid for very shallow pools.
3. Results and discussion 3.1. Validation of the simulation code Before the actual simulations, we have first validated the simulation code in the absence of external body forces (induced by the applied crucible rotation and magnetic field). The simulation results of the validation are presented in Figs. 3–5.
74
Y. Takagi et al. / Journal of Crystal Growth 385 (2014) 72–76
Fig. 6. Time development of surface temperature at r ¼20 mm with only crucible rotation. Fig. 4. Space-time diagram of surface temperature in the absence of external forces.
Fig. 5. Circumferential temperature (top) and velocity vectors (bottom) in the r¼ 20 mm distance at t¼ 150 s.
Fig. 3 shows the computed surface temperature fluctuation, δT, in the horizontal plane, which is defined by 1 δTðr; θ; z; tÞ ¼ Tðr; θ; z; tÞ− 2π
Z
2π 0
Tðr; θ; z; tÞ dθ
ð7Þ
As seen from the figure, the surface temperature is not axisymmetric and fluctuates in the circumferential direction. This describes the typical spoke pattern of a HTW, which validates the code in terms of spoke pattern. The observed wavenumber (the number of wavy pattern) is 7, which also agreed with that of Li et al. [4]. To examine the temporal motion of HTW, the space-time distribution of the surface temperature was computed, and plotted in Fig. 4. The surface temperature along the circumference of r ¼20 mm was temporally sampled, and plotted (contoured) with respect to the circumferential angle and time. At the fully developed state, after t¼ 110 s, the phase velocity of the HTW (calculated from the slope of the iso-contours) was constant at the value of 0.15 rad/s. Although the sign of phase velocity can be negative (which depends on the initial conditions), its magnitude reaches the same value eventually. In Fig. 5, the computed temperature and velocity vectors in the circumferential plane of r ¼20 mm are presented at t¼ 150 s. The vertical structures were seen in the interval of the spoke patterns (above figures), and the flow velocity (induced by the Marangoni convection due to the circumferential temperature difference) vectors are shown in the figure below. The good agreement between our computed spoke patterns and the motion of the HTW (in the absence of external body
Fig. 7. Time development of surface temperature at r¼ 20 mm with only magnetic field.
forces) with those of Li et al. [4] confirms the validation of the present simulation code.
3.2. HTW control with a single body force The simulations were first carried out under crucible rotation for four rotation speeds (ω ¼ 0, 1, 2, and 5 min−1). Fig. 6 shows the time development of surface temperature at a sampling point of r ¼20 mm. From the figure, we can see that the amplitudes of −1 temperature fluctuations at ω ¼ 1 min and 2 min−1 are in the same order of those without rotation, and the HTW suppression with rotation is not monotonic to the rotation speed. The period of fluctuations at ω ¼ 1 min−1 is very large compared with those at other speeds because the characteristics of HTW changed at this critical condition as discussed in Li et al. [5]. At the higher rotation speed (ω ¼ 5 min−1 ), although the temperature fluctuations became quite smaller, HTW is not still suppressed completely. Next, we have run the simulations under the applied magnetic field without crucible rotation. The simulation results in this case are quite different than those under crucible rotation. Fig. 7 shows the time development of surface temperature at the sampling point of r¼ 20 mm. As the Hartmann number increases, the amplitudes of temperature fluctuation decrease monotonically, and the HTW disappears completely at Ha ¼72. Comparison of the results of these two cases clearly shows that the application of a magnetic field is more effective in suppressing the HTW.
Y. Takagi et al. / Journal of Crystal Growth 385 (2014) 72–76
3.3. HTW control with combined body forces In this case we consider the simultaneous application of magnetic field and crucible rotation to find the most effective combination for suppression. Based on the results of the previous section, we selected Ha¼57 for the magnetic field level, and varied the crucible rotation speed from 0 min−1 to 2 min−1. Fig. 8 shows the time development of surface temperature at the sampling point of r ¼20 mm. As the rotation speed increases, the period and the amplitudes of the temperature fluctuation decreases. At −1 Ha¼ 57 and ω ¼ 2 min , the HTW disappears completely, and the flow becomes two-dimensional. Results indicate that the combination of a magnetic field at Ha¼57 and a crucible rotation at ω ¼ 2 min−1 provides the optimum suppression.
3.4. Effects of body forces In this section we discuss the individual effects of the body forces induced by the applied magnetic field (Lorentz force) and the crucible rotation (centrifugal force). The distributions of the Lorentz and centrifugal forces on the melt surface were computed and plotted in Figs. 9 and 10 at t ¼100 s. As seen, the Lorentz force is stronger around the inner wall and consequently suppresses the radial and circumferential flow velocity components (induced by
75
the Marangoni convection). Figs. 9(a) and (b), at the lower rotation speeds, the HTW spoke patterns develop in the region where the Lorentz force is strong. However, as seen in Fig. 10, the centrifugal force is stronger around the outer wall, in contrast with the previous case, since its magnitude is proportional with the distance from the center. Thus, one can see that these two body forces at their optimum combination provide a complete suppression of the HTW by controlling different regions of the flow in the pool. Finally, we summarize the resultant body force magnitudes for each case in Table 3. Magnitudes of the body force are summed over the pool volume and the results for the cases of crucible rotation and magnetic field are presented. In the case of crucible rotation alone (Ha¼0, ω ¼ 5 min−1 ), although the body force was the largest, the HTW was not suppressed completely (not disappeared). However, in the case of magnetic field alone, the HTW was completely suppressed although the magnetic body force was smaller than the centrifugal force. This is due to the fact that the Lorentz force is stronger in the region where the Marangoni convection occurs, and thus provides an effective suppression. −1 In the combined case of Ha¼57 and ω ¼ 2 min , the total body force is comparable to that of the Lorentz force under the magnetic field alone, and thus the HTW was fully suppressed. Results show that a proper combination of such body forces would provide an effective control of a HTW.
4. Conclusions The combined effect of crucible rotation and applied magnetic field on hydrothermal wave was investigated through a threedimensional simulation study. The following conclusions were drawn from the simulation results:
Fig. 8. Time development of surface temperature at r ¼20 mm with the combination of crucible rotation and magnetic field at Ha¼ 57.
1. Under crucible rotation alone, the effect of hydrothermal wave on temperature fluctuations was not monotonic, and larger rotation speeds are required for significant decreases in temperature fluctuations. 2. The Lorentz force under an applied magnetic field alone, however, acts effectively and provides a complete suppression of the Marangoni convection around the inner wall of the pool. In this case the hydrothermal wave disappears completely. 3. At the optimum combination of crucible rotation and magnetic field, a smaller total body force is required to suppress the
Fig. 9. Surface distribution of the Lorentz force magnitude at t ¼100 s: (a) Ha¼ 57, ω ¼ 0 min−1 ; (b) Ha¼ 57, ω ¼ 1 min−1 ; (c) Ha¼57, ω ¼ 2 min−1 .
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Fig. 10. Surface distribution of the centrifugal force magnitude at t¼ 100 s: (a) Ha ¼57, ω ¼ 0 min−1 ; (b) Ha ¼57, ω ¼ 1 min−1 ; (c) Ha¼ 57, ω ¼ 2 min−1 .
Table 3 Resultant body force magnitudes. Hartmann number (–)
Rotation speed (min−1)
Lorentz force (10−4 N)
Centrifugal force (10−4 N)
Total force (10−4 N)
Result flow pattern
0 57 57 57 72
5 0 1 2 0
0.000 1.412 1.416 1.419 2.124
5.297 0.000 0.212 0.848 0.000
5.297 1.412 1.628 2.266 2.124
HTW HTW HTW 2D flow 2D flow
hydrothermal wave completely. This result is particularly important in the development of crystal growth techniques that can be performed under weaker magnetic fields and smaller crucible rotation speeds.
Acknowledgments This work was financially supported by a Grant-in-Aid for Scientific Research (B) (No. 22360343) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] M.K. Smith, S.H. Davis, Journal of Fluid Mechanics 132 (1983) 119–144. [2] M.K. Smith, S.H. Davis, Journal of Fluid Mechanics 132 (1983) 145–162.
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