三维弱激波层连续谱的解析研究

中文摘要 AI 辅助

本研究采用精确的与雷诺数无关的基础流和线性稳定性方程形式，对弱可压缩激波层的三维线性稳定性进行分析，该分析通过将问题重新标度到激波自身的固有粘性长度和时间尺度来实现。对于任何仅依赖激波法向坐标且横向速度分量为零的基础流，所得特征值问题在横向波数矢量旋转下被证明是协变的：特征值仅通过横向波数的大小依赖于横向波数，且连续谱中的每个三维本征模都可精确解耦为两个独立分量。二维面内声熵分支承载扰动的全部膨胀、压力和热力学耦合，而一维面外纯螺线管涡分支则由单个标量剪切扩散方程控制。因此，三维激波层的稳定性可精确分解为可压缩分量和涡分量，且通过Grosch-Salwen远场伴生谱以机器精度得到验证。两个解耦算子均带有随总横向波数增大的显式阻尼项，因此三维斜扰动的稳定性绝不低于仅共享其面内波数的二维扰动；涡分支的谱直接由该项控制，其谱随波数平方成比例地远离虚轴。该发现是Squire经典定理的精确、与雷诺数无关的类似物，其中雷诺数的稳定作用在此由横向波数的大小承担。

英文摘要

Three-dimensional linear stability of a weak, compressible shock layer is analyzed using exact Reynolds-number-independent forms of the governing base flow and linear stability equations, obtained by rescaling the problem onto the shock's own natural viscous length and time scales. For any base flow with zero transverse velocity components, depending on the shock-normal coordinate alone, the resulting eigenvalue problem is proved covariant under rotation of the transverse wavenumber vector: the eigenvalue depends on the transverse wavenumbers only through their magnitude, and every three-dimensional eigenmode, throughout the continuous spectrum, decouples exactly into two independent constituents. A two-dimensional, in-plane acoustic--entropy branch carries the disturbance's entire dilatation, pressure and thermodynamic coupling, and a one-dimensional, out-of-plane purely solenoidal vortical branch is governed by a single scalar shear-diffusion equation. Three-dimensional shock-layer stability is thus resolved exactly into its compressible and vortical constituents, confirmed to machine precision by the Grosch--Salwen far-field companion spectra. Both decoupled operators carry an explicit damping term growing with the total transverse wavenumber, so a three-dimensional, oblique disturbance is never less stable than the two-dimensional disturbance sharing its in-plane wavenumber alone; the vortical branch, whose spectrum is controlled directly by this term, is shown to retreat from the imaginary axis in close proportion to the wavenumber squared. This finding is an exact, Reynolds-free analogue of Squire's classical theorem, with the stabilizing role of Reynolds number played here by the transverse wavenumber magnitude.

发表机构

• Faculty of Aerospace Engineering, Technion, Israel Institute of Technology（以色列理工学院航空航天工程学院）

机构由 AI 辅助整理，请以论文原文为准。