STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

doi:10.1007/s10473-020-0212-8

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (2): 470-502.doi: 10.1007/s10473-020-0212-8

STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

袁海荣¹, 赵勤²

1. School of Mathematical Sciences, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China;
2. Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

收稿日期:2018-12-04 修回日期:2019-07-23 出版日期:2020-04-25 发布日期:2020-05-26
通讯作者: Qin ZHAO E-mail:qzhao@whut.edu.cn
作者简介:Hairong YUAN,E-mail:hryuan@math.ecnu.edu.cn
基金资助:
This work was supported in part by National Nature Science Foundation of China (11371141 and 11871218), and by Science and Technology Commission of Shanghai Municipality (18dz2271000).

STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

Hairong YUAN¹, Qin ZHAO²

1. School of Mathematical Sciences, and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China;
2. Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

Received:2018-12-04 Revised:2019-07-23 Online:2020-04-25 Published:2020-05-26
Contact: Qin ZHAO E-mail:qzhao@whut.edu.cn
Supported by:
This work was supported in part by National Nature Science Foundation of China (11371141 and 11871218), and by Science and Technology Commission of Shanghai Municipality (18dz2271000).

摘要/Abstract

摘要： Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces. This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.

关键词: Stability, transonic shocks, Fanno flow, three-dimensional, Euler system, frictions, decomposition, nonlocal elliptic problem, Venttsel boundary condition, elliptic-hyperbolic mixed-composite tpe

Abstract: Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces. This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.

Key words: Stability, transonic shocks, Fanno flow, three-dimensional, Euler system, frictions, decomposition, nonlocal elliptic problem, Venttsel boundary condition, elliptic-hyperbolic mixed-composite tpe

中图分类号:

35M32

袁海荣, 赵勤. STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS[J]. 数学物理学报(英文版), 2020, 40(2): 470-502.

Hairong YUAN, Qin ZHAO. STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS[J]. Acta mathematica scientia,Series B, 2020, 40(2): 470-502.

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STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

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