Approximate solution to the Riemann problem in non-classical gas dynamics

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Abstract

This study considers an approach to construct an approximate solver for the non-classical Riemann problem. In this regime, the solution of the discontinuity decay problem may contain composite waves, including both classical and non-classical compression and rarefaction waves. The algorithm for finding the exact solution is based on a geometric representation of shock and rarefaction waves on isentropic curves and involves the repeated use of iterative methods to solve local tasks, such as identifying inflection points on isentropes, points of tangency between a straight line and a curve, intersection points, and others. A significant challenge when using iterative methods is the need to specify initial guesses that ensure method convergence. The approach proposed in this work is based on tabulating exact solutions for Riemann problems over a wide range of initial state parameters. These tabulated data are then used to find an approximate solution without requiring iterative methods. The approximate solver was successfully applied to solve two one-dimensional discontinuity decay problems in the non-classical domain.

About the authors

Mariya R. Koroleva

Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences

Author for correspondence.
Email: koroleva@udman.ru
ORCID iD: 0000-0001-5697-9199
https://www.mathnet.ru/rus/person73142

Cand. Phys. & Math. Sci.; Senior Researcher; Institute of Mechanics

Russian Federation, 426067, Izhevsk, T. Baramzina str., 34

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Supplementary files

Supplementary Files
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1. JATS XML
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2. Figure 1. ($p$–$v$) diagram close to the critical point; the inversion zone ($G < 0$) is shaded in gray

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3. Figure 2. Schematic of the Riemann problem solution for the fan–shock case in classical gas dynamics: a) $(p$–$v)$ plane; b) characteristic field; c) pressure distribution; d) specific volume distribution; e) velocity distribution

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4. Figure 3. Schematic of the Riemann problem solution for non-classical gas dynamics: a) $F$-wave pattern from the right initial state for the $f$–$F$ case; b) $Fs$-wave pattern from the left initial state for the $Fs$–$f$ case

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5. Figure 4. DRAKON flowchart of the algorithm for wave solution identification ($v_L < v$)

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6. Figure 5. DRAKON flowchart of the algorithm for wave solution identification ($v_L > v$)

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7. Figure 6. Bilinear interpolation for the solution function

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8. Figure 7. Solution construction

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