Kinetic algorithms for non-equilibrium gas dynamics
New upwind kinetic-difference schemes have been developed for flows with nonequilibrium thermodynamics and chemistry. These schemes are derived from the Boltzmann equation with the resulting Euler schemes developed as moments of the discretized Boltzmann scheme with a locally Maxwellian velocity distribution. Application of a directionally-split Courant-Isaacson-Rees (CIR) scheme at the Boltzmann level results in a flux-vector splitting scheme at the Euler level and is called Kinetic Flux-Vector Splitting (KFVS). Extension to flows with finite-rate chemistry and vibrational relaxation is accomplished utilizing non-equilibrium kinetic theory. Computational examples are presented comparing KFVS with the schemes of Van-Leer and Roe for quasi-one-dimensional flow through a supersonic diffuser, inviscid flow through two-dimensional inlet, 'viscous flow over a cone at zero angle-of-attack, and shock-induced combustion/detonation in a premixed hydrogen-air mixture. Calculations are also shown for the transonic flow over a bump in a channel and the transonic flow over an NACA 0012 airfoil. The results show that even though the KFVS scheme is a Riemann solver at the kinetic level, its behavior at the Euler level is more similar to the the existing flux-vector splitting algorithms than to the flux-difference splitting scheme of Roe.
A new approach toward the development of a genuinely multi-dimensional Riemann solver is also presented. The scheme is based on the same kinetic theory considerations used in the development of the KF VS scheme. The work has been motivated by the recent progress on multi-dimensional upwind schemes by the groups at the University of Michigan and the Von Karman Institute. These researchers have developed effective upwind schemes for the multi-dimensional linear advection equation using a cell-vertex fluctuation-splitting approach on unstructured grids of triangles or tetrahedra. They have made preliminary applications to the Euler equations using several wave decomposition models of the flux derivative. The issue of the appropriate wave model does not appear to be adequately resolved. The approach taken in the present work is to apply these new multi-dimensional upwind schemes for the scalar advection equation at the Boltzmann level. The resulting Euler schemes are obtained as moments of the fluctuations in the Maxwellian distribution function. The development is significantly more complicated than standard (dimensionally-split) kinetic schemes in that the Boltzmann discretization depends upon the direction of the molecular velocities which must be accounted for in the limits of integration in velocity space. The theoretical issues have been solved through analytic quadrature and Euler schemes have been developed. For this formulation it was not necessary to prescribe any explicit wave decomposition model. Encouraging preliminary results have been obtained for perfect gases on uniform Cartesian meshes with first-order spatial accuracy. Results are presented for a 29° shock reflection, a 45° shear discontinuity, and Mach 3 flow over a step.
Finally, methods for obtaining accurate gas-dynamic simulations in the continuum transition regime are considered. In particular, large departures from translational equilibrium are modeled using algorithms based on the Burnett equations instead of the Navier-Stokes equations. Here, the same continuum formulation of the governing equations is retained, but new constitutive relations based on higher-order Chapman-Enskog theory are introduced. Both a rotational relaxation model and a bulk-viscosity model have been considered for simulating rotational non-equilibrium. Results are presented for hypersonic normal shock calculations in argon and diatomic nitrogen and comparisons are made with Direct Simulation Monte Carlo (DSMC) results. The present work closely follows that of the group at Stanford, however, the use of upwind schemes and the bulk-viscosity model represent new contributions.