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

Abstract

The major technique of manufacturing silicon monocrystals is Czochralski (Cz) method. To obtain high‐quality crystals, it is necessary to control the melt flow that is turbulent. The most cost‐effective approach for modeling turbulence nowadays is RANS (Reynolds Averaged Navier‐Stokes) that allows for providing steady axisymmetric computations using reasonable grids. However, conventional eddy‐viscosity RANS models usually fail to predict spatial details of turbulence characteristics of the melt flow. To overcome the limitations of the eddy‐viscosity approach the Stress Tensor Reconstruction (STR) approach is developed that is aimed at the reproduction of anisotropy of the Reynolds stresses. The Generalized Gradient Diffusion Hypothesis (GGDH) in combination with the STR approach successfully takes into account the anisotropy of turbulent fluxes in a melt flow. Relying upon the STR/GGDH approach a novel RANS STR k‐ε model is developed. The model is validated using the ILES (Implicit Large Eddy Simulation) data of turbulent melt convection during Cz Si crystal growth, as well, as the experimental data of oxygen concentration in 100 and 200 mm diameter Cz crystals. The results are compared to the results obtained with the conventional low Reynolds number Chien k‐ε model extended by the buoyancy generation term.