Exact Jacobians for implicit Navier–Stokes simulations of equilibrium real gas flows

Abstract

This paper documents the extension of several widespread flux schemes, used in finite-volume Navier–Stokes solvers, for the simulation of flows whose fluid properties must be estimated with complex thermophysical models. Exact Jacobian matrices for the convective fluxes are derived with no assumption on the fluid equations of state model for Liou's AUSM+, Toro et al.'s HLLC, and Kurganov and Tadmor's central scheme. The Jacobians of the diffusive fluxes are expressed using the formulation proposed by Pulliam and Steger, resulting in additional terms due to the viscosity and thermal conductivity variations. An efficient look-up table approach is thoroughly studied and proposed as an alternative to the direct solution of the equation of state model for the fluid thermophysical property evaluation. The newly introduced schemes are validated and tested in terms of accuracy and convergence rate on a series of one- and two-dimensional test cases. The results indicate that the Jacobian must be based on the same flux formulation as the one used on the right-hand side of the implicit equation to achieve numerically converged solutions.