Non-Equilibrium Flow Modeling Using High-Order Schemes for the Boltzmann Model Equations

Abstract

We consider application of higher-order schemes to the Boltzmann model equations with a goal to develop a deterministic computational approach that is accurate and efficient for simulating flows involving a wide range of Knudsen numbers. The kinetic equations are solved for two non-equilibrium flow problems, namely, the structure of a normal shock wave and an unsteady two-dimensional shock tube. The numerical method comprises the discrete velocity method in the velocity space and the finite volume discretization in physical space with different numerical flux schemes: the first-order, the second-order minmod flux limiter as well as a third-order WENO scheme. The normal shock wave solutions using BGK and ES collision models are compared to the DSMC simulations. The solution for unsteady shock tube is compared to the Navier-Stokes simulations at low Knudsen numbers and the rarefaction effects in such flow are also discussed. It is observed that a higher-order flux scheme provides a better convergence rate and, hence, reduces the computational effort. The entropy generation rate is shown to be a very sensitive indicator of the onset of non-equilibrium as well as accuracy of a numerical scheme and consistency of boundary conditions in both flow problems.

Numerical Stability with Help from Entropy: Solving a Set of 13 Moment Equations for Shock Tube Problem

“…The nonequilibrium entropy profiles, refined over a few mean free lengths, are peerless amongst moment equations. The maximum exhibited by the specific entropy within the shock wave can only be compared to-and is in agreement with-numerical solutions of the Boltzmann equation; see [8].…”

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An axisymmetric computational model of generalized hydrodynamic theory for rarefied multi-species gas flows

“…To examine the entropy characteristic, the normalized entropy and entropy generation rate of the one-dimensional normal shock are compared with the work by Chigullapalli et al [39]. The test gas is argon.…”