Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

doi:10.1007/s42967-021-00185-8

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (3): 1053-1096.doi: 10.1007/s42967-021-00185-8

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Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

Yanbo Hu, Xingxing Li

School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, Zhejiang, China

Received:2021-08-16 Revised:2021-12-19 Online:2023-09-20 Published:2023-08-29
Contact: Yanbo Hu,E-mail:yanbo.hu@hznu.edu.cn E-mail:yanbo.hu@hznu.edu.cn;xingxingli714@163.com
Supported by:
The authors would like to thank the two referees for very helpful comments and suggestions to improve the quality of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang province of China (LY21A010017), and the National Natural Science Foundation of China (12071106, 12171130).

Abstract

Abstract: This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section. Under the conical assumption, the three-dimensional (3-D) steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles. Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane, we construct a classical conical-supersonic solution near the curve under some reasonable assumptions. To overcome the difficulty caused by the parabolic degeneracy, we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane. The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully. We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Key words: Three-dimensional (3-D) full Euler equations, Conical flow, Conical-sonic, Characteristic decomposition, Classical solution

CLC Number:

Yanbo Hu, Xingxing Li. Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations[J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1053-1096.

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Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

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