Semi-Lagrangian lattice Boltzmann method for compressible flows

Abstract

This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method [Dorschner, Bösch, and Karlin, Phys. Rev. Lett. 121, 130602 (2018)], the method operates in a static, nonmoving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent time step size due to the integration along characteristics and for the use of unusual velocity sets, like the D2Q25, which is constructed by the roots of the fifth-order Hermite polynomial. The properties of the present model are shown in diverse example simulations of a two-dimensional Taylor-Green vortex, a Sod shock tube, a two-dimensional Riemann problem, and a shock-vortex interaction. It is shown that the cell-based interpolation and the use of Gauss-Lobatto-Chebyshev support points allow for spatially high-order solutions and minimize the mass loss caused by the interpolation. Transformed grids in the shock-vortex interaction show the general applicability to nonuniform grids.

Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria

“…Simulations are stable for relatively low values of the relaxation time ( τ f = 0.57, Pr = 1), even with the standard BGK operator, while LBMs based on a polynomial discrete equilibrium usually require a large number of discrete velocities (e.g. 81 in 2D [73]), more robust collision models [16,17] or numerical discretizations [86], in order to achieve similar results. Interestingly, the Knudsen number based relaxation times drastically improve the stability of the present model, allowing the simulation of the 1D Riemann problem in the zero-viscosity limit ( τ f = 0.5, Pr = 1), with an accuracy that competes with LBMs coupled with shock capturing techniques [87].…”

Compressibility in lattice Boltzmann on standard stencils: effects of deviation from reference temperature

“…These abscissae are not adapted to the LB method as they do not lead to on-lattice propagation of populations. Such stencils can be used in the context of LB models with off-lattice propagation [16,51]. That is why a third-order quadrature (not correctly recovering the third-order moment) is usually employed instead.…”

Multi-scale analysis of the particles on demand kinetic theory

“…While the LBM has been extensively used in the incompressible flow regime [1], its application in compressible flows is still an open field of study, directing researchers towards developing various models [12][13][14][15][16][17]. Considering that the restrictions in the conventional LB models are mainly due to the fixed velocity sets [1], the idea of shifted lattices was first introduced in [16], which found to be significantly useful in increasing the range of performance of LB simulations of supersonic flows [16].…”