International Journal of Heat and Mass Transfer

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1 International Journal of Heat and Mass Transfer 55 (2012) Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: Large-eddy simulation of melt turbulence in a 300-mm Cz Si crystal growth Lijun Liu, Xin Liu, Yuan Wang Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, School of Energy and Power Engineering, Xi an Jiaotong University, Xi an, Shaanxi , China article info abstract Article history: Received 8 March 2011 Received in revised form 2 August 2011 Available online 16 September 2011 Keywords: Large-eddy simulation SGS model Crystal growth Flow instability We built a curvilinear dynamic Smagorinsky subgrid-scale (SGS) model based on filtering the covariant physical velocity components in the computational space. We implemented our proposed SGS model in large-eddy simulations (LES) of turbulent flows in complex configurations. Our model was validated when compared with direct numerical simulation (DNS) data of the melt turbulent flow in an idealized cylindrical crucible in a Cz Si crystal growth. Then, we carried out LES computations for the melt turbulence in a real ellipsoidal crucible in a 300 mm Cz Si crystal growth. We studied instantaneous behaviors and statistical features of the melt turbulence. Spectral analyses of the temperature fluctuations show that the melt flow is in a soft turbulence state of Rayleigh Bénard convection under the rotating crystal. A cluster of big vortices is formed in the time-averaged bulk flow due to the complex interaction among the thermal buoyancy, surface tension and crucible/crystal rotations. Heat transport in the melt flow is turbulence-dominated with notable fluctuations. The maximal temperature fluctuation in the crystallization zone is close to the crystal edge with a value of 1.8 K. The flow instability mainly attributes to the thermal buoyancy in the melt. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The Cz crystal growth process is the dominant method for producing single-crystalline silicon (Si) that is the mainstay of the microelectronic and photovoltaic industry. The melt flow in the crucible has significant effects on the formation of micro-defects and impurity concentration in the grown crystals. With increasing demand for large crystals in industry, large crucibles are used, in which the melt flow is turbulence-characterized with velocity and temperature fluctuations. Since turbulent fluctuations of the flow could be the dominant mechanism of momentum, energy and mass transport in the melt, a good understanding of the melt turbulence and its instability is very important for further improvement of industrial-scale Cz Si crystal growth [1]. The melt flow and heat transfer in the melt of a Cz Si crystal growth are extremely complex since several affecting factors arise simultaneously in this process and the melt flow is highly unstable when a large crucible is used. Many researches [2,3] have been carried out to investigate the flow instabilities and thermal fluctuations caused by the nonlinear interactions among the centrifugal forces, the thermal buoyancy and the surface tension in the Si melt. Most investigations of melt convection in large crucibles are based on numerical simulation because of its low cost and good visualization capability. Melt turbulence simulations are based either on the direct numerical simulation method [4 6] or the solution Corresponding author. Tel./fax: address: ljliu@mail.xjtu.edu.cn (L. Liu). of the Reynolds-averaged Navier Stokes equations (RANS) [7 10] or the large-eddy simulation method [11 13]. The RANS method was first introduced to account for the turbulence transport in the melt. However, it showed major discrepancies from experiment results and failed to predict the temperature fluctuations. Actually, the RANS method is unable to predict the turbulent fluctuations in the melt flow. On the other hand, the DNS method can predict all scales of turbulent fluctuations in the melt flow. However, it needs huge computational resources and its application is still limited to turbulent flows in some very simple configurations. Therefore, the LES method, as a compromise choice between the efficiency of RANS and the accuracy of DNS, has showed its great potential in many fields as well as in crystal growth simulation. Some researchers used the classical Smagorinsky subgrid scale (SGS) model in combination either with the specialized wall function [11] or with the RANS method for the near-wall regions [12] in their simulations of the melt flow in large Cz crucibles. However, such a SGS model with empirical coefficients cannot guarantee its universality for various flow problems. Hence, the dynamic SGS model based on a dynamic procedure to compute the model coefficients has become popular and has been used in crystal growth modeling [13]. However, its further application to the case of real crucibles of ellipsoidal shape has not been assessed in detail, owing to the computational complexity in body-fitted grid systems. In addition, few works have been published on LES predictions of temperature fluctuations in the melt, which has close correlation with the micro-defect formation and impurity concentration in the grown Si crystals [14] /$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi: /j.ijheatmasstransfer

2 54 L. Liu et al. / International Journal of Heat and Mass Transfer 55 (2012) In this work, we developed and validated a dynamic SGS model for LES in body-fitted grids. Then, we conducted LES computations for the turbulent melt flow in a 300-mm Cz Si crystal growth. We studied the time-averaged bulk flow structure and the flow instability in the melt. 2. Model formulation 2.1. Governing equations of LES for melt turbulence The melt flow is calculated by solving the 3D time-dependent equations of mass, momentum and energy conservation with application of the Boussinesq approximation for an incompressible Newtonian fluid. These equations are filtered implicitly in space by a second-order, finite-volume solution methodology that is equivalent to box filtering [15]. The filtered equations for the resolvable scales of melt turbulence are written as follows in a rotating reference frame: r V ¼ 0; q DV h Dt ¼ rpþr l i eff rv þrv T qgb T ðt T 0 Þ 2qXV; DT Dt ¼ r ðk eff rtþ; where X is the angular velocity of the reference frame, b T is the thermal expansion coefficient of the Si melt, T 0 is the reference temperature and g is the gravitational acceleration. The overbar denotes the implicit grid filter operation. The effective eddy viscosity and effective thermal diffusivity are defined as l eff = l + l SGS and k eff = k + l SGS /r SGS, respectively. The SGS eddy viscosity l SGS is calculated with the dynamic SGS model, and r SGS is the SGS turbulent Prandtl number with a value of 0.9. At solid walls, l SGS was estimated with a generalized three-layer wall function [11]. The physical properties of Si melt are given in Table 1. ð1þ ð2þ ð3þ procedure demands explicit filtering of the resolvable field by a correctly designed test filter. In the traditional method, the test filtering operation is conducted in the physical space. However, Ghosal and Moin [19] proved that the filtering operation and the differential operation do not commute in the non-uniform grids. Moreover, the filtering operation in the physical space is costly to apply due to the non-uniformity of the grid space [20]. Fortunately, we notice the mapping relationship between the physical space and the computational space, as shown in Fig. 1, ifwedoa coordinate transformation and solve the governing equations in the computational space with uniform grids. Since the mesh is uniform in the computational space, the explicit filtering operation can be done very conveniently for the velocity components in covariant form. Therefore, we propose a new evaluation algorithm for C D based on Lily s approach as follows: First, the covariant physical velocity components u nl are obtained by projecting the velocity vector along the tangential direction of the curvilinear coordinates, as shown in Fig. 1(a). The resolvable covariant physical velocity components and other relevant variables are filtered explicitly in the computational space. The 3D discrete filter reads: / i;j;k ¼ F P / i;j;k ¼ XN X N X N l¼ N m¼ N n¼ N a l a m a n / iþl;jþm;kþn ; which is the composition of three one-dimensional (1D) filters applied in each space direction [21]. The second overbar denotes the explicit test filter operation. The coefficients a l, a m and a n are the weighting factors of the 1D filters in each space direction, which is defined as / i ¼ 1 / 6 i 1 þ 4/ i þ / iþ1 in the computational space. Second, the covariant form Leonard tensor yields: ð5þ 2.2. Dynamic SGS model for the body-fitted grids The dynamic Smagorinsky relationship for the SGS eddy viscosity is given as [16] l SGS ¼ 2qC D ðx; tþd 2 jsj; ð4þ where C D is the model coefficient, and the grid-filter scale is taken as D ¼ðVolumeÞ 1=3 in the body-fitted grids. The magnitude of the qffiffiffiffiffiffiffiffiffiffiffiffi resolvable strain rate tensor is calculated as jsj ¼ 2S ij S ij, where S ij ¼ 1. j i Properly evaluating the dynamic model coefficient is critical for successful use of the eddy viscosity relationship for complex configurations. To obtain C D dynamically, we applied the procedure proposed by Germano [17] and modified by Lily [18]. This Table 1 Physical properties of Si melt. Parameter Value Molecular dynamic viscosity, l Pa s Density, q kg/m 3 Specific heat, c p J/(K kg) Prandtl number, Pr Thermal expansion coefficient, b T K 1 Surface tension coefficient, a N/(K m) Melting temperature 1685 K Emissivity 0.3 Fig. 1. Mapping between the physical space and the computational space. (a) Physical space. (b) Computational space.

3 L. Liu et al. / International Journal of Heat and Mass Transfer 55 (2012) L nl n m ¼ u nl u nm u nl u nm : ð6þ The covariant form model stress density, M nl n m, is defined as: M nl n m ¼ 2D 2 C ½ðD C=D C Þ 2 js C js nl n m js C js nl n m Š; ð7þ where D C ¼ Dn is the grid filter width, and D C ¼ 2D C is the test filter width in the computational space. The relevant strain rate tensor is defined as S nl n m ¼ nl m nm, and its magnitude is C j¼ l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S S nl nm n l. nm Finally, the formula for C D is written as: C D ðx; tþ ¼C D ðn; tþ ¼ L n l n m M nl n m M nl n m M nl n m : 3. Results and discussion 3.1. Validation of the curvilinear dynamic SGS model Wagner and Friedrich [5] and Raufeisen et al. [6] investigated the turbulent transport of momentum and heat in the Si melt of an idealized Czochralski crystal growth configuration with the DNS method, which is limited to the turbulent flows in some simple configurations. To validate the proposed curvilinear ð8þ dynamic SGS model for Cz configurations with comparison to these available DNS reference data, we first carried out LES computations of the melt turbulence in the same idealized cylindrical crucible. The idealized geometry of the Cz configuration is presented in Fig. 2(a). The melt domain is cylindrical in shape with a diameter of 340 mm and a height of 85 mm. The crystal diameter is 100 mm. The melt crystal (m c) interface is assumed to be flat with the freezing temperature of Si. The crucible is rotating at a rate of x C ¼ 5 rpm in the counter-clockwise direction, while the crystal is rotating at a rate of x S ¼ 20 rpm in the clockwise direction. A fixed temperature distribution obtained from an experiment is prescribed at the crucible walls [5,6]. We consider the thermal radiation loss at the melt free surface where the surface tension is also taken into account. The validation case leads to the dimensionless numbers of the melt flow of Re = , Gr = , Ma = , and Ra = The non-orthogonal curvilinear grids in a cross plane of the melt domain, shown in Fig. 2(b), were generated for validating the dynamic SGS model in generalized curvilinear coordinates. We used a computational grid with 288,000 control volumes (CVs) and a time step of 0.02 s in the LES computations, corresponding to 2,900,000 CVs and s used in the DNS computations [5]. For the validation case, three sets of meshes were used in the grid independence test. The coarse grid consists of 144,000 CVs. The moderate grid consists of 216,000 CVs, while the fine grid Fig. 2. Cz configuration for LES validation. (a) Idealized geometry of the Cz configuration. (b) Computational grid in a cross section of melt domain for LES.

4 56 L. Liu et al. / International Journal of Heat and Mass Transfer 55 (2012) consists of 288,000 CVs. Though there is no notable difference between the results obtained with the moderate grid and the fine grid, we choose the fine grid in the LES computations in order to improve the resolution in the near-wall region. We used the same initial fields and boundary conditions that were used in the DNS computations [5,6]. We compare the LES results of the first- and second-order statistic fields with the corresponding DNS results in Figs. 3 and 4. Fig. 3(a) shows the time-averaged temperature distribution in a dimensionless form obtained from LES computation, while Fig. 3(b) shows the corresponding results obtained from DNS computation for the purpose of comparison. The dimensionless form of temperature is defined as T =(hti T 0 )/DT, where T 0 =(T max + T min )/2, DT = T max T min, and hti is the time-averaged temperature. The minimal temperature T min and the maximal temperature T max in the melt domain are 1685 K and K, respectively. Therefore, the maximal and minimal values of dimensionless temperature in both figures are 0.5 and 0.5, respectively. The solid contour lines represent positive value and the dashed contour lines represent negative value. The interval of contour lines is As compared in Fig. 3(a) and (b), the time-averaged temperature distribution from LES computation is in very good agreement with the DNS reference data. Fig. 3(c) and (d) shows the melt flow patterns of the time-averaged flow fields obtained respectively from LES computation and DNS computation. We can notice that the characteristic vortices obtained from LES computation are in agreement with the DNS results. Considering the extreme complexity of the melt flow and its affecting mechanisms, the comparison between the LES and DNS results are satisfactory. Some deviations close to the melt free surface and the bottom of the crucible may be caused by different treatments and grid resolutions at the melt boundaries. Fig. 4 presents the comparison of the root-mean-square (RMS) value distribution of temperature fluctuation in a dimensionless form between the LES and DNS computations. The dimensionless form of temperature fluctuation is defined as T rms ¼ T rms=dt, where DT = T max T min and T rms is the RMS value of temperature fluctuation. As can be observed in the figure, both the distribution pattern and the maximal value of temperature fluctuation, as well as its location in the melt, as marked in the figures, are in very good agreement between the LES and DNS computations. The distributions of RMS temperature fluctuation demonstrate the existence of two characteristic regions of temperature fluctuation. One region is right below the melt free surface where the maximal RMS value of dimensionless temperature fluctuation is more than 0.1. Another region is below the crystal. Thus, the proposed curvilinear dynamic SGS model is validated. Compared with the huge computational resource demand in DNS computation, LES computation with this proposed SGS model can predict large-scale flow structures and turbulent fluctuations satisfactorily with much shorter CPU time and much smaller computer memory in modeling of melt turbulence in a Cz Si crystal growth. Our computations also proved that this proposed SGS model requires about 30% less CPU time in LES computations with the explicit filtering scheme in the computational space than that with the explicit filtering scheme in the physical space as used in the traditional LES treatment Investigation of the melt turbulence in a 300-mm Cz Si crystal growth LES computations were conducted for melt turbulence in a 300- mm Cz Si crystal growth for solar cells. The geometry of the computational domain is shown in Fig. 5. We consider a crucible with a convex bottom. The diameter of the crucible is 775 mm. The diameter of the crystal is 306 mm. The melt height is 279 mm. The rotation rates of the crucible and crystal are x C ¼ 6 rpm and Fig. 3. Comparison of the time-averaged fields. (a) Dimensionless temperature distribution from LES. Contour interval is (b) Dimensionless temperature distribution from DNS [5]. Contour interval is (c) Streamline pattern of melt flow from LES. (d) Streamline pattern of melt flow from DNS [6].

5 L. Liu et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 4. Comparison of the dimensionless RMS temperature fluctuation field. Contour interval is (a) LES. (b) DNS [5]. Fig. 6. Turbulence features of the melt flow. (a) Coherent melt flow structure. (b) Instantaneous temperature distribution. Isotherms are plotted every 2 K. (c) Time evolutions of temperatures at the two reading points. Fig. 5. Configuration and computational grids for the melt domain in a 300-mm Cz Si crystal growth. x S ¼ 8 rpm, respectively. A grid number of 182,000 and time step of 0.01 s were used. Due to our special interest in the turbulent fluctuations in the melt, we assumed a flat m c interface. We obtained the temperature distributions at the crucible walls from a 2D global simulation of heat transfer [22]. We applied non-slip conditions for velocities at solid boundaries. At the melt free surface, we took the radiation loss and surface tension into account. The investigated case leads to the dimensionless numbers of Re = , Gr = , Ma = , and Ra = The turbulent nature of the melt convection Fig. 6 shows some turbulence features of the melt flow predicted with LES. Fig. 6(a) shows the coherent flow structure in the turbulent melt flow, corresponding to the instantaneous isosurfaces of the second invariant of the velocity gradient tensor for a value of Q = 0.2. The second invariant of the velocity gradient is defined as [23]: Q ¼ 1 2 where S ij ¼ 1 i ¼ 1 2 ðs ijs ij X ij X ij j i and X ij ¼ 1 i ð9þ are the symmetric and asymmetric part of the resolved velocity gradient tensor, respectively. It is a measure of the magnitude of fluid rotation relative to its strain. When Q is positive, rotation prevails over strain, so that vortices can be identified as positive values of Q. The advantage of Q over the vorticity magnitude as a quantitative vortex identifier is the fact that Q represents the local balance between shear strain

6 58 L. Liu et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 7. Spectral analysis for temperature fluctuations at the two reading points. (a) P1. (b) P2. rate and vorticity magnitude. Hence, Q is not affected by the local shear, which usually prevails in the vicinity of walls. Therefore, it is used for identification of the coherent flow structure in turbulent flows. From Fig. 6(a), we see that the melt turbulence is abundant with small-scale vortices, and the structure is quite complex. Fig. 6(b) shows an instantaneous temperature distribution in the melt. Three-dimensional features of heat transport can be observed in the melt and on the melt free surface due to the nature of the turbulence. Fig. 6(c) shows the time evolutions of temperatures at two reading points P1 and P2, which are located 40 mm below the m c interface at radius R = 100 mm and R = 153 mm, respectively, as shown in Fig. 5. The random behavior of the temperature fluctuation is obvious. The time evolution of melt temperature is evidently chaotic. Information on the state of turbulence is usually derived from spectral analysis of temperature fluctuations [24]. The amplitudes of temperature fluctuations were analyzed by calculating the power spectra of the temporal signal of temperatures at the two points P1 and P2 using FFT. The results are presented in Fig. 7. In the frequency range between 0.1 and 1 Hz, the power spectral density (PSD) decreases almost proportionally to f 4 at the two reading points. This corresponds to the soft turbulence state of Rayleigh Bénard convection, as described in Ref. [25]. For point P1 in Fig. 7(a), the strong peak in the low frequency range means a long-period melt fluctuation, which indicates a flow pattern dominated by the thermal buoyancy in the central region of the melt. For point P2 in Fig. 7(b), the long-period fluctuations are suppressed by the crystal and crucible rotations with increase in radius. A few peaks appear in the higher frequency range Time-averaged fields of velocity and temperature To analyze the statistical behavior of the melt flow, the velocity and temperature fields were averaged in time in the circumferential direction. The statistically averaged flow field can be observed in Fig. 8(a). It consists of three large convection rolls (denoted A, B, and C) extending over the whole melt domain, and two secondary vortices below the melt free surface (denoted D and E). The buoyant vortex A is induced by the axial temperature difference and damped by the effect of crystal rotation. On the other hand, due to high temperature at the crucible side walls, the buoyant upward flow dominates close to the crucible side walls. A vortex denoted C

7 L. Liu et al. / International Journal of Heat and Mass Transfer 55 (2012) radial temperature difference becomes influential. As a result, two vortices (rolls D and E) are formed at both sides of the location in the radial direction with the maximal temperature on the melt free surface. The resultant melt flow pattern is very complex. It consists of a cluster of vortices influenced by complex interaction among the thermal buoyancy, surface tension and crucible/crystal rotations, of which the thermal buoyancy plays a major role. Fig. 8(b) presents the averaged temperature field. The temperature distribution pattern is quite different from the flow structure shown in Fig. 8(a). It shows that the heat transport in the melt is diffusion-dominated. This is due to the enhanced diffusion caused by the notable turbulent fluctuations in the melt Turbulent fluctuation field Fig. 9 shows the RMS contours of temperature fluctuation in the melt. Several characteristic regions of temperature fluctuations can be seen in the melt, which correspond to the bulk melt flow structure, as shown in Fig. 8(a). The melt regions with big fluctuations are located right under the crystal edge and the melt surface, where there is severe interaction between the bulk vortices. This shows that the flow instability is induced by the complex flow structure, in which the thermal buoyancy plays a major role. The maximal temperature fluctuation in the melt is located close to the crucible sidewall with a value of 3.6 K. The characteristic temperature fluctuation in the crystallization zone is close to the crystal edge with a value of 1.8 K. Since fluctuations under the m c interface have close correlation with the impurity concentration and micro-defects in a grown Si crystal, this region is of more interest to a grower. The thermal fluctuations at this region should be deliberately controlled. Obviously, such notable melt flow instability should be controlled by improving the hot zone design of the furnace or by applying any external fields, such as magnetic fields. 4. Conclusions Fig. 8. Averaged fields of the melt flow. (a) Streamlines of melt flow. (b) Temperature distribution. Isotherms are plotted every 2 K. We proposed a dynamic SGS model, based on filtering the covariant physical velocity components explicitly in the computational space, for the melt turbulence in complex configurations with body-fitted grids. We carried out LES computations with this proposed SGS model to study the melt turbulence in a 300-mm Cz Si crystal growth. Some turbulence features of the melt flow were revealed. The LES results demonstrate the typical rotating buoyancy- and surface-tension-driven flow structure in the melt. Heat transport in the melt flow is turbulence-dominated with notable fluctuations. The maximal temperature fluctuation in the crystallization zone is close to the crystal edge with a value of 1.8 K. The power spectra of temperature fluctuation give slopes approximately proportional to f -4 for high frequencies, indicating a soft turbulence state under the rotating crystal. The flow instability mainly attributes to the thermal buoyancy in the melt. Acknowledgments This work was supported by NSFC (No , ), NCET , RFDP (No ) and Fundamental Research Funds for the Central Universities of China. pffiffiffiffiffiffiffiffiffiffiffiffi Fig. 9. Distribution of temperature fluctuation in the melt ht 0 T 0 i. Contours are plotted every 0.2 K. is formed and pushed by the crucible rotation to the crucible side walls. Accounting for the opposing flow directions of vortices A and C, another vortex B is formed between them. For the melt flow close to the melt free surface, the surface tension caused by the References [1] T. Sinno, E. Dornberger, W. von Ammon, R. Brown, F. Dupret, Defect engineering of Czochralski single-crystal silicon, Mater. Sci. Eng. R 28 (5 6) (2000) [2] G. Müller, Convective instabilities in melt growth configurations, J. Cryst. Growth 128 (1 4) (1993) [3] J.R. Ristorcelli, J.L. Lumley, Instabilities, transition and turbulence in the Czochralski crystal melt, J. Cryst. Growth 116 (3 4) (1992)

8 60 L. Liu et al. / International Journal of Heat and Mass Transfer 55 (2012) [4] S. Enger, O. Gräbner, G. Müller, M. Breuer, F. Durst, Comparison of measurements and numerical simulations of melt convection in Czochralski crystal growth of silicon, J. Cryst. Growth 230 (1 2) (2001) [5] C. Wagner, R. Friedrich, Direct numerical simulation of momentum and heat transport in idealized Czochralski crystal growth configurations, Int. J. Heat Fluid Flow 25 (3) (2004) [6] A. Raufeisen, M. Breuer, T. Botsch, A. Delgado, DNS of rotating buoyancy- and surface tension-driven flow, Int. J. Heat Mass Transfer 51 (25 26) (2008) [7] B. Yu, H. Ozoe, Comparison of eight different low-reynolds number K-epsilon models computed for natural convection in a Czochralski configuration, J. Mater. Process. Manuf. Sci. 8 (2) (1999) [8] A. Lipchin, R. Brown, Comparison of three turbulence models for simulation of melt convection in Czochralski crystal growth of silicon, J. Cryst. Growth 205 (1-2) (1999) [9] V. Kalaev, I. Evstratov, Y. Makarov, Gas flow effect on global heat transport and melt convection in Czochralski silicon growth, J. Cryst. Growth 249 (1 2) (2003) [10] G. Müller, J. Friedrich, Challenges in modeling of bulk crystal growth, J. Cryst. Growth 266 (1 2) (2004) [11] I. Evstratov, V. Kalaev, A. Zhmakin, Y. Makarov, A. Abramov, N. Ivanov, E. Smirnov, E. Dornberger, J. Virbulis, E. Tomzig, W. von Ammon, Modeling analysis of unsteady three-dimensional turbulent melt flow during Czochralski growth of Si crystals, J. Cryst. Growth 230 (1 2) (2001) [12] N. Ivanov, A. Korsakov, E. Smirnov, K. Khodosevitch, V. Kalaev, Y. Makarov, E. Dornberger, J. Virbulis, W. von Ammon, Analysis of magnetic field effect on 3D melt flow in Cz Si growth, J. Cryst. Growth 250 (1 2) (2003) [13] A. Raufeisen, M. Breuer, T. Botsch, A. Delgado, LES validation of turbulent rotating buoyancy- and surface tension-driven flow against DNS, Comput. Fluids 38 (8) (2009) [14] K. Kakimoto, T. i Shyo, M. Eguchi, Correlation between temperature and impurity concentration fluctuations in silicon crystals grown by the Czochralski method, J. Cryst. Growth 151 (1 3) (1995) [15] S. Jordan, Investigation of the cylinder separated shear-layer physics by largeeddy simulation, Int. J. Heat Fluid Flow 23 (1) (2003) [16] M. Breuer, Large eddy simulation of the subcritical flow past a circular cylinder: numerical and modeling aspects, Int. J. Numer. Methods Fluids 28 (9) (1998) [17] M. Germano, U. Piomelli, P. Moin, W. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A 3 (7) (1991) [18] D. Lilly, A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A 4 (3) (1992) [19] S. Ghosal, P. Moin, The basic equations for the large eddy simulation of turbulent flows in complex geometry, J. Comput. Phys. 118 (1) (1995) [20] S. Jordan, A large-eddy simulation methodology in generalized curvilinear coordinates, J. Comput. Phys. 148 (2) (1999) [21] P. Sagaut, R. Grohensa, Discrete filters for large eddy simulation, Int. J. Numer. Methods Fluids 31 (8) (1999) [22] L.J. Liu, K. Kakimoto, Partly three-dimensional global modeling of a silicon Czochralski furnace. I. Principles, formulation and implementation of the model, Int. J. Heat Mass Transfer 48 (21 22) (2005) [23] L.F.G. Geersa, M.J. Tummers, K. Hanjali, Particle imaging velocimetry-based identification of coherent structures in normally impinging multiple jets, Phys. Fluids 17 (2005) [24] O. Gräbner, A. Mühe, G. Müller, E. Tomzig, J. Virbulis, W. von Ammon, Analysis of turbulent flow in silicon melts by optical temperature measurement, Mater. Sci. Eng. B 73 (1 3) (2000) [25] S. Togawa, S. Chung, S. Kawanashi, K. Izunome, K. Terashima, S. Kimura, Density anomaly effect upon silicon melt flow during Czochralski crystal growth I. Under the growth interface, J. Cryst. Growth 160 (1 2) (1996)