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.