Direct and Large-Eddy Simulation of Turbulent Flows on Composite Multi-Resolution Grids by the Lattice Boltzmann Method
Résumé
In order to properly address the simulation of complex (weakly compressible) turbulent flows, the lattice Boltzmann method, originally designed for uniform structured grids, needs to be extended to composite multi-domain grids displaying various levels of spatial resolution. Therefore, physical conditions must be specified to determine the mapping of statistical information (about the populations of moving particles) at the interface between two domains of different resolutions. It is here argued that these conditions can express quite simply in terms of the probability distributions of the underlying discrete-velocity Boltzmann equation. Namely, the continuity of the mass density and fluid momentum is fulfilled by imposing the continuity of the equilibrium part of these distributions, whereas the discontinuity of the rate-of-strain tensor is ensured by applying a ''spatial transformation'' to the collision term of the discrete-velocity Boltzmann equation. This latter condition allows us to explicitly account for the subgrid-scale modeling in the treatment of resolution changes. Test computations of a turbulent plane-channel flow have been considered. The lattice Boltzmann scheme relies on the standard D3Q19 lattice in a cell-vertex representation, and uses the BGK approximation for the collision term. A shear-improved Smagorinsky viscosity is used for the subgrid-scale modeling. In a quasi-Direct Numerical Simulation at $\mathrm{Re}_\tau=180$ (with two levels of resolution) the results are found in excellent agreement with reference data obtained by a high-resolution pseudo-spectral simulation. In a Large-Eddy Simulation at $\mathrm{Re}_\tau=395$ (with three levels of resolution) the results compare very well with high-resolution reference data. The accuracy is improved in comparison with a large-eddy simulation based on finite-volume discretization with the same subgrid-scale viscosity model and comparable grid resolution. This study demonstrates the good capabilities of the lattice Boltzmann method to handle both Direct and Large-Eddy Simulations of turbulent flows with grid resolutions comparable to those commonly used in simulations based on standard discretization methods, \emph{e.g.} pseudo-spectral or finite-volume methods.
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