A thermodynamically consistent and fully conservative treatment of contact discontinuities for compressible multicomponent flows

A thermodynamically consistent and fully conservative (TCFC) treatment of contact discontinuities (designated as TCFC model in this paper) is proposed for the simulation of compressible multicomponent flows with shock-interface interactions, yielding an accurate capture of contact discontinuities. Starting from the total energy equation of the mixture, a new formulation is developed to define the ratio of specific heats of the mixture, and a governing equation in conservative form for the pressure is subsequently obtained. Another formulation is derived to determine the molecular weight of the mixture from classical thermodynamics. The governing equations in conservative form are further derived to calculate the specific heats ratio and the molecular weight of the mixture with their new formulations and the continuity or mass fraction equations for the individual components. Finally, the governing equations for the ratio of specific heats, the molecular weight and the pressure, combined with the continuity and momentum equations, offer the new hyperbolic conservation laws for the description of multicomponent or multifluid flows with shock-interface interactions. A simple version of the model is also developed without solving the governing equation for the molecular weight of the mixture. A brief analysis of the numerical uncertainties of conventional conservation models and the present TCFC models is presented and the consistency of any additional equation derived within any model with the initial governing equations is investigated. An analysis of the proposed TCFC model and the conventional conservative models is also performed on the pressure evolution for a mixture associated with isolated contact discontinuities. The results show that the present TCFC model produces a uniform pressure field and thus the pressure maintains in equilibrium across the material interfaces. The proposed TCFC model is then implemented into a Godunov-type method based upon a fast, exact Riemann solver. Several multicomponent flow problems with both strong and weak shocks are simulated using this new model and conventional conservation models, and comparisons of the numerical results from the various models and, where possible, exact solutions are performed. It is shown that the proposed TCFC model can handle both strong and weak shocks, and that the numerical solutions are completely oscillation-free through the contact discontinuities. This new approach offers a suitable treatment of contact discontinuities for compressible multicomponent flows and also maintains all the favorable features of fully conservative conservation laws. It belongs to the conservative approaches and is independent of numerical schemes. From the analysis of the numerical uncertainties of several models proposed in the past and the present numerical experiments, it is determined that the conservative or non-conservative form of some additional governing equations is not the ultimate reason that produces oscillating solutions near the material interfaces. The main cause of the oscillating solutions with conventional conservation models is the inappropriate determination of the ratio of specific heats of the mixture. Oscillation-free results are obtained for two test cases with weak shocks even with conventional conservation models using the present Godunov-type method based on a fast exact Riemann solver. This suggests that the problems with weak shocks and strong post-shock contact discontinuities are not suitable for the model validation in multicomponent flows. The capability of handling strong shocks and weak post-shock contact discontinuities is more critical to the demonstration of any model for compressible multicomponent or multifluid flows.