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Lattice Boltzmann method with selective viscosity filter. (English) Zbl 1395.76073
Summary: For high-Reynolds number flows, the lattice Boltzmann method suffers from numerical instabilities that can induce local blowup of the computation. The von Neumann stability analysis applied to the LBE-BGK and LBE-MRT models shows that numerical instabilities occur in the high wavenumber range and are due to the interplay between acoustic modes and some other modes. As it is done in the LBE-MRT model, an increase of the bulk viscosity is an efficient way of damping spurious oscillations. However, this stabilization method induces an over-damping of acoustic waves. Some selective spatial filters can be used in order to eliminate the spurious small spatial scales without affecting the large scale physical modes. Three different lattice Boltzmann algorithms based on filtering step are proposed: the fully filtered LBE, the LBE with filtered macroscopic quantities and the LBE with filtered collision operator. The behavior of several explicit filter stencils is studied in the Fourier space. For a given filter stencil, the filtered collision operator approach leads to the highest cut-off wavenumber. In this case, the theoretical wavenumber-dependent viscosity is \(v(k)=c_s^2(\tau /[1-\sigma f(k)]-1/2)\) where \(f(k)\) is the filter shape and \(\sigma \) the filter strength. Under-resolved simulations (high Reynolds number) are performed on the case of the doubly periodic shear layers. The performance of the three filtered LBE is found to be the same as the MRT model for stability control. Propagation of acoustic plane waves is also simulated with the three filtering algorithms. The measured dissipation of acoustic wave compares well with the theoretical results.

MSC:
76M28 Particle methods and lattice-gas methods
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