LES validation of turbulent rotating buoyancy- and surface tension-driven flow against DNS

The paper is concerned with the validation and error analysis of predictions for the flow and heat transfer in a silicon melt (Pr = 0.013) found in a Czochralski (Cz) apparatus for crystal growth. This system resembles turbulent Rayleigh-Benard-Marangoni convection. Since for practical applications predictions based on direct numerical simulations (DNS) require too many resources to conduct parametric studies or optimizations, nowadays in practice the method of choice is the large-eddy Simulation (LES). The case considered consists of an idealized cylindrical crucible of 170 mm radius with a rotating crystal of 50 mm radius. Boundary conditions from experimental data were applied, which lead to the dimensionless numbers of Re = 4.7 x 10(4). Gr = 2.2 x 10(9). Ma = 2.8 x 10(4), and Ra = 2.8 x 10(7). The filtered Navier-Stokes equations were solved based on a finite-volume scheme for curvilinear block-structured grids and an explicit time discretization. For a comprehensive error analysis, different grid sizes, subgrid-scale models, and discretization schemes were employed. The results were compared to reference DNS data of the same case recently generated by the authors (Int J Heat Mass Transfer, 51 (2008) 6219-6234) for validation. For the finest LES grid (106 control volumes) using a standard Smagorinsky model with van Driest damping or a dynamic model, both with central discretization, the results agree well with the DNS reference while the computational effort could be reduced by a factor of 20. When using an upwind scheme even of formally second-order accuracy, significant deviations Occur. Further stepwise reductions of the grid size decrease the CPU time drastically, but also lead to larger aberrations. When the grid is coarsened by a factor of 32 (resulting in ca. 130,000 CVs), even qualitative differences between the LES and the DNS solution appear. It could be shown in the present work that the LES method is an efficient tool to model the turbulent flow and heat transfer in Rayleigh-Benard-Marangoni configurations. However, care should be taken in the choice of the grid resolution and discretization scheme for the nonlinear convective terms, as too coarse meshes in combination with upwind schemes lead to significant numerical errors. Finally, a quantified relation between the achievable accuracy and the necessary computational effort is presented. (C) 2009 Elsevier Ltd. All rights reserved.