A modified hypothesis of Reynolds stress tensor modeling for mixed turbulent convection in crystal growth

Three-dimensional modelling of 300 mm Czochralski silicon crystal growth with a transverse magnetic field

“…We adopted the 2D-3D combination method reported by Kalaev, Borisov and Smirnov. 26,27 The external boundary conditions of the 3D local model are provided by the 2D global axisymmetric model. To ensure the rationality of the axisymmetric boundary conditions, the graphite crucible, quartz crucible, melt, crystal and local argon were included in this model, as shown in Fig.…”

“…The interface could be corrected to an isothermal surface or constant growth rate surface. 23,27,33 However, there exist some issues with the methods mentioned above if a non-axisymmetric magnetic field is used. The issues will be introduced at step 2.…”

“…However, these articles do not explicitly state whether the corrected interface is axisymmetric or not, and whether a special method is used in the interface correction process. 26,27,33,34 This lack of information makes it difficult to make comparisons. Fig.…”

Extended Hypothesis for Reynolds Stress Tensor and Turbulent Heat Flux Modeling Within a Novel K‐ε Model for Prediction of Crystal Growth from the Melt

“…In industrial crystal growth modeling, reasonable quality DNS cannot be performed in near future due to extremely high computational requirements. [17] To overcome the limitations of the eddy-viscosity approach, we have developed the Stress Tensor Reconstruction (STR) approach [11] which is aimed at reproduction, the required RST anisotropy by taking into account different factors: i) anisotropy due to influence of the wall and free surface, which appears in relatively more rapid damping of orthogonal velocity fluctuations in the vicinity of the surface, ii) anisotropy due to presence of the mean velocity gradient, and iii) anisotropy due to direct action of mass forces on turbulence, in particular, the buoyancy force which increases the vertical velocity fluctuations. The Generalized Gradient Diffusion Hypothesis (GGDH) in combination with the STR approach successfully reproduces the anisotropy of turbulent fluxes.…”

“…The Generalized Gradient Diffusion Hypothesis (GGDH) in combination with the STR approach successfully reproduces the anisotropy of turbulent fluxes. [11] Based on the STR/GGDH approach, we have developed a RANS STR k model that solves the transport equation only for turbulence kinetic energy while the dissipation rate is calculated using an algebraic expression that uses the characteristic size of energy-containing eddies. This model was applied for the calculation of turbulent heat and mass transport in the silicon melt during growth of 200 mm diameter crystals in an EKZ-3500 furnace.…”