Stability of transonic shocks in steady supersonic flow past multidimensional wedges
Stability of transonic shocks in steady supersonic flow past multidimensional wedges
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经过多维楔块的稳定超音速流中跨音速激波的稳定性
DOI：
10.1016/j.aim.2017.04.019
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发表时间：
2017
影响因子：
1.7
通讯作者：
Fang Beixiang
中科院分区：
文献类型：
--
作者：
Chen Gui-Qiang;Fang Beixiang
研究了定常超音速绕流中多维(M-D)跨声速激波的稳定性。我们的动机之一是，多维情况下的全局稳定性问题比二维情况下的要敏感得多，后者需要更仔细、更严格的数学分析。本文发展了一种非线性方法，并利用它在适当的函数空间中建立了包含跨音速激波阵面的势流相对于楔形边界的M-D扰动的弱激波解的稳定性。为了实现这一点，我们首先将稳定性问题描述为非线性椭圆型方程的自由边界问题。然后，我们引入部分速度图变换，将自由边界问题化为无界区域中二阶非线性椭圆型方程在背景解附近的固定边值问题，该背景解具有完全非线性边界条件。为了解决这个简化的问题，我们将非线性问题线性化到背景激波解上，然后，在求解这个线性化的椭圆问题后，发展了一个非线性迭代格式，证明了它是压缩的。
We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive.