Simulation of high-Mach-number inviscid flows using a third-order Runge-Kutta and fifth-order WENO-based finite-difference lattice Boltzmann method

Abstract

A discrete-velocity Boltzmann equation(DVBE) with Bhatnagar-Gross-Krook (BGK) approximation is discretized in time and space using a third-order Runge-Kutta (RK3) and fifth-order weighted essentially nonoscillatory (WENO) finite-difference scheme to simulate benchmark inviscid compressible flows. The implementation of the WENO ensures that solutions behave with minimal or no oscillations, narrowing the gap between the exact and the numerical results. Discrete-velocity sets given by Kataoka and Tsutahara [Phys. Rev. E 69, 056702 (2004)10.1103/PhysRevE.69.056702] are used. The additional dissipation terms as well as artificial viscosity are incorporated in the formulation to solve the compressible flows at high Mach number. Further, the flows which are subjected initially to a high density ratio are effectively simulated. In this article, one-dimensional benchmarks are simulated at initial Mach number up to 30 and density ratio up to 1000, whereas, the benchmarks in two dimensions are simulated with a Mach number up to 10. The algorithm is assessed by simulating numerous benchmarks, namely, (i) one-dimensional Riemann problem for various shock waves combinations [namely (a) shock-shock waves in the case of different Mach numbers, (b) rarefaction-shock waves for various density ratios, (c) sudden contact shock discontinuity, and (d) shock-rarefaction waves for density ratio 5], (ii) isentropic vortex convection test, (iii) regular shock reflection (RR) for Mach numbers 2.9 and 10, (iv) double Mach reflection (DMR) for inflow Mach numbers as 6 and 10, and (v) forward-facing step for inflow Mach numbers 2 to 5. Further, the effect of change in Mach numbers and wedge angles on the flow structures in the case of DMR are detailed. In the case of a forward-facing step, the variations of flow structure (e.g., the Mach stem height, triple points location, and shock standoff distance) are detailed with respect to Mach number, step height, and specific-heat ratios. Finally, the numerical stability of the proposed formulation is carried out to assess the behavior of the free parameters.