三维矩形喷管中恒定状态稳态跨音速激波在小外力作用下的稳定性

中文摘要 AI 辅助

本文研究了在外部力和出口压力的三维小扰动下，矩形喷管中具有恒定状态的稳态跨音速激波的稳定性。在数学上，该问题可以表述为一个双曲-椭圆耦合偏微分方程系统的非线性自由边界问题。关键困难之一在于背景激波位置的任意性，这可能导致在扰动下对于相同的外力和出口压力存在多个激波解，从而揭示背景激波的不稳定性。我们通过变形和涡量重新表述稳态欧拉方程，以有效分解双曲和椭圆模态。在背景跨音速激波处对重新表述的系统及Rankine-Hugoniot条件进行线性化，我们发现非零水平外力类有助于唯一确定激波位置，其中质量守恒起着关键作用。我们发展了一种精细的迭代方法，以建立跨音速激波解的存在性和结构稳定性。我们的分析表明，平坦和发散喷管中稳态跨音速激波的稳定机制在许多方面存在显著差异。

英文摘要

This paper investigates the stability of steady transonic shocks with constant states in rectangular nozzles under small three dimensional perturbations of the external force and the exit pressure. Mathematically, the problem can be formulated as a nonlinear free boundary problem for a hyperbolic-elliptic coupled PDE system. One of the key difficulties lies in the arbitrary location of the background shock, which may lead to multiple shock solutions for the same external force and exit pressure under perturbations, revealing the instability of the background shock. We reformulate the steady Euler equations in terms of deformation and vorticity to effectively decompose the hyperbolic and elliptic modes. Linearizing the reformulated system and the Rankine-Hugoniot conditions at background transonic shocks, we find that the class of nonzero horizontal external forces helps to uniquely determine the shock location, where the conservation of mass plays a critical role. A delicate iteration is developed to establish the existence and structural stability of transonic shock solutions. Our analysis shows that the stability mechanisms of steady transonic shocks in flat and divergent nozzles are quite different in many aspects.

发表机构

• School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
• School of Mathematics and Statistics, Hubei Key Laboratory of Mathematics & Intelligent Computation and Interdisciplinary Sciences, Wuhan University of Technology（武汉科技大学数学与统计学院）

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