Exact Riemann Solver for Multi-Component 1D Euler Model

Question

I am concerned looking for an exact Riemann solver for compressible 1D multi-component Euler equations, supplied with the ideal gases equation of state and under the assumptions:

• Mechanical equilibrium
• Thermal equilibrium

For example for the form used by Gouasmi et al. 2020 [1]. In a question [2] of this forum, it is very well explained. For me, it seems that he/she is using the Riemann solver for single-component model in the multi-component model. If doing this, I do not see how to obtain the partial densities/mass fractions in the intermediate state. Borisov & Rikov [3] provide the exact Riemann solver for a different equation of state, and Declercq et al. [4] provide it for two-phase model. However, despite used in the literature, I don't find an exact Riemann solver for this model.

If anyone has an idea, I would be very grateful,

Have a nice day,

References: [1] https://www.sciencedirect.com/science/article/pii/S0045782520300955 [2] Exact Riemann solver for perfect gas mixture: problem with Newton's method convergence [3] https://www.researchgate.net/publication/325098653_An_exact_Riemann_solver_in_the_algorithms_for_multicomponent_gas_dynamics [4] https://hal.science/hal-01580049/document

Answer 1

I finally found the solution. In the reference [3] the procedure is very detailed, it was just I did not know that mass fractions (Y) do not change across shocks or rarefactions. They just change across the contact discontinuity, between states 3 and 4 in the figure

Spatio-temporal diagram of the possible wave patterns in the Riemann problems for multi-component Euler model, extracted from [3].

The only complication I found is that if the objective intermediate pressure (the one arising from the solver) is zero, a relaxation coefficient must be applied to the Newton-Raphson loop.

Please notice two possible mistakes in the reference

1. the velociy of the left rarefaction should be $D^{l,*}=U-c^{l,*}$ in case 1 page 14
2. in $P=p_l$ or $P=p_r$, the formulas for calculating the shock velocity blow up (cases 2 and 4 page 14), so it is better to put the case $P=p_l$ and $P=p_r$ in cases 1 and 3 respectively.

For any question do not hesitate to contact me.