Lattice Boltzmann Relaxation Schemes for High Speed Flows

Abstract

The lattice Boltzmann method (LBM) has emerged as a highly efficient model for the simulation of incompressible flows in the last few decades. Its extension to compressible flows is hindered by the fact that the equilibrium distributions rep- resent truncated Taylor series expansions in Mach number, limiting the strategy to low Mach number flows. Numerous efforts have been undertaken to extend this method to compressible flows recently. Some of these approaches have resulted in complicated or expensive equilibrium distribution functions. A few approaches are limited to subsonic flows while a few others have too many tuning parameters without clear guidelines to x their values. In this context, it is worth exploring newer avenues to develop an efficient lattice Boltzmann method for compressible

fluid flows. In this thesis, we utilize a novel interpretation of the discrete velocity Boltzmann relaxation systems to develop new lattice Boltzmann methods for compressible

flows. In these new lattice Boltzmann relaxation schemes (LBRS), the equilib- rium distributions are free from the low-Mach number expansions. In fact, the equilibrium distributions are simple algebraic combinations of the conserved vari- ables and the fluxes. This novel LB method is tested for the 1-D and 2-D Euler equations using a D1Q3 and a D2Q9 model respectively. Various bench-mark test cases have been considered to demonstrate the robustness of the new scheme. This strategy can be easily extended to other hyperbolic systems of conservation laws representing the shallow-water flows and the ideal magnetohydrodynamics. In this work, the extension of LBRS to the shallow-water equations in both one and two-dimensions and to the 1-D ideal MHD equations is demonstrated through bench-mark test problems. Finally, extension of this new method to parabolic equations is demonstrated by applying it to the viscous Burgers equation. The formulation of LBRS for the viscous case is based on the interpretation of the diffusion term in the resulting relaxation system as a physical diffusion term. Two novel approaches to extend the new scheme to viscous Burgers equation are presented.