TVD flux-difference split methods for high-speed thermochemical nonequilibrium flows with strong shocks

This study is concerned with the numerical solution of high-speed nonequilibrium gaseous flows with strong shocks. The extension of modern total-variation-diminishing (TVD) shock-capturing schemes to include thermochemical nonequilibrium high-temperature effects is of primary interest. Partially decoupled upwind-based TVD flux-difference split schemes for the solution of the conservation laws governing two-dimensional nonequilibrium vibrationally relaxing and chemically reacting flows of thermally-perfect gaseous mixtures are presented. Both time-split semi-implicit and factored implicit flux-limited TVD upwind schemes are described. The semi-implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady-state solutions. As well, a multigrid version of the fully implicit TVD scheme is also proposed for the more efficient computation of time-invariant solutions. The multigrid algorithm is based on the full approximation storage (FAS) and full multigrid (FMG) concepts and employs the partially-decoupled factored implicit scheme as the smoothing operator in conjunction with a four-level V-cycle coarsegrid-correction procedure. In the proposed methods, a novel partially-decoupled flux-difference splitting approach is adopted. The fluid conservation laws and the finite-rate species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially-decoupled gas dynamic and thermodynamic subsystems are then integrated alternately in lagged manner within a two-stage time marching procedure, thereby providing explicit coupling between the two equations sets. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. Moreover, concerns associated with the satisfaction of monotonicity, positivity, and maximum principles are addressed. The proposed partially-decoupled methods are shown to have some computational advantages over chemistry-split and fully coupled techniques in coping with large systems of equations with stiff source terms. The predictive capabilities of the shock-capturing methods are demonstrated and their usefulness appraised, by solving a number of different flows with both complicated shock structure and complex nonlinear wave interactions. The problems considered include nonstationary oblique shock-wave reflections and diffractions and steady high-speed nozzle, compression ramp, and blunt-body flows. The numerical results are compared to available experimental data in many cases.