Real gas thermodynamics influence on solving Riemann problem under high temperatures

Аuthors

Roshchin A. S.^*, Ananyev A. V.^**, Borisov D. M.^***

Keldysh Institute of Applied Mathematics (Russian Academy of Sciences), 4, Miusskaya sq., Moscow, 125047, Russia

*e-mail: arochtchin@gmail.com
**e-mail: anatoly.v.ananyev@gmail.com
***e-mail: borisovdm62@mail.ru

Abstract

Consideration of temperature dependence of gas thermodynamic properties is necessary to solve the gas dynamics equations at high temperatures.

The effect of the variability air thermodynamic properties such as the heat capacity and the adiabatic coefficient on the Riemann problems solution was studied. The Riemann problem was solved in prediction of constant gas heat capacity in all volume, in prediction of different heat capacity from left and right of the contact discontinuity, and finally considering the real gas thermodynamics. These approaches were compared on the groundwork of three test problems with the solutions containing two shock waves, one shock wave and one depression wave, and two depression waves correspondingly. It was shown that for the Riemann problem with high temperature difference the second approach gives better results as compared with the first one, but sometimes it results some unphysical simple discontinuities appearing at points of contact discontinuities.

The effect of used numerical schemes to the solution of the Riemann problem was also examined. The Euler equation for the supersonic flow around wedge and berm was solved by Godunov scheme and its TVD modification using exact Riemann solvers for gas with constant or variable heat capacity. For all mentioned problems the computed air stagnation temperatures behind shock wave were compared. For the supersonic flow (M=10) around the berm modeled by the 1st order Godunov scheme the stagnation temperature error was less than 0.5%, and the 2nd order TVD-modification scheme resulted the 0.8% and 2% relative errors for supersonic flows with M=6 and M=10 correspondingly.

It was also shown that for weak waves the flow patterns altered insignificantly.

Finally, it can be noted that the use of Riemann problems exact solution combined with high order Godunov-type solvers can be efficient for solution of gas dynamics equations for real gases with variable heat capacities. This approach for flows with strong shock waves allows one to obtain accurately flow patterns, shock wave positions, and gas parameters behind wavefronts.

Keywords:

Riemann problem, real gas thermodynamics, Godunov scheme, TVD

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