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On a supersonic-sonic patch arising from the two-dimensional Riemann problem of the compressible Euler equations

Part of: Equations and systems of special type Partial differential equations Equations of mathematical physics and other areas of application Transonic flows

Published online by Cambridge University Press:  18 September 2024

Yanbo Hu Open the ORCID record for Yanbo Hu [Opens in a new window]  and
Guodong Wang
Yanbo Hu*
Affiliation:
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, PR China (yanbo.hu@hotmail.com)
Guodong Wang
Affiliation:
School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, PR China (yxgdwang@163.com)
*
*Corresponding author
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Abstract

We are interested in the two-dimensional four-constant Riemann problem to the isentropic compressible Euler equations. In terms of the self-similar variables, the governing system is of nonlinear mixed-type and the solution configuration typically contains transonic and small-scale structures. We construct a supersonic-sonic patch along a pseudo-streamline from the supersonic part to a sonic point. This kind of patch appears frequently in the two-dimensional Riemann problem and is a building block for constructing a global solution. To overcome the difficulty caused by the sonic degeneracy, we apply the characteristic decomposition technique to handle the problem in a partial hodograph plane. We establish a regular supersonic solution for the original problem by showing the global one-to-one property of the partial hodograph transformation. The uniform regularity of the solution and the regularity of an associated sonic curve are also discussed.

Keywords

Compressible Euler equationstwo-dimensional Riemann problemsonic curvesupersonic-sonic solutionpartial hodograph transformation

MSC classification

Primary: 35Q31: Euler equations
Secondary: 35Q35: PDEs in connection with fluid mechanics 35M33: Initial-boundary value problems for systems of mixed type 35A09: Classical solutions 76H05: Transonic flows

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 40
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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