The parabolized stability equations (PSE) are a ubiquitous tool for studying the stability and evolution of disturbances in weakly nonparallel, convectively unstable flows. The PSE method was introduced as an alternative to asymptotic approaches to these problems. More recently, PSE has been applied with mixed results to a more diverse set of problems, often involving flows with multiple relevant instability modes. This paper investigates the limits of validity of PSE via a spectral analysis of the PSE operator. We show that PSE is capable of accurately capturing only disturbances with a single wavelength at each frequency and that other disturbances are not necessarily damped away or properly evolved, as often assumed. This limitation is the result of regularization techniques that are required to suppress instabilities arising from the ill-posedness of treating a boundary value problem as an initial value problem. These findings are valid for both incompressible and compressible formulations of PSE and are particularly relevant for applications involving multiple modes with different wavelengths and growth rates, such as problems involving multiple instability mechanisms, transient growth, and acoustics. Our theoretical results are illustrated using a generic problem from acoustics and a dual-stream jet, and the PSE solutions are compared to both global solutions of the linearized Navier–Stokes equations and a recently developed alternative parabolization.

中文翻译：

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抛物化稳定性方程的关键评估

抛物线稳定性方程 (PSE) 是研究弱非平行、对流不稳定流动中扰动的稳定性和演化的普遍工具。PSE 方法被引入作为解决这些问题的渐近方法的替代方法。最近，PSE 已被应用于更多样化的问题集，结果喜忧参半，通常涉及具有多种相关不稳定模式的流。本文通过对 PSE 算子的频谱分析来研究 PSE 的有效性限制。我们表明，PSE 能够准确地仅捕获每个频率上具有单个波长的干扰，并且其他干扰不一定像通常假设的那样被衰减或适当地演化。这种限制是正则化技术的结果，需要抑制因将边界值问题视为初始值问题的不适定性而引起的不稳定性。这些发现对 PSE 的不可压缩和可压缩公式均有效，尤其适用于涉及具有不同波长和增长率的多种模式的应用，例如涉及多种不稳定机制、瞬态增长和声学的问题。我们的理论结果使用来自声学和双流射流的一般问题进行了说明，并且将 PSE 解与线性化纳维-斯托克斯方程的全局解和最近开发的替代抛物线进行了比较。这些发现对 PSE 的不可压缩和可压缩公式均有效，尤其适用于涉及具有不同波长和增长率的多种模式的应用，例如涉及多种不稳定机制、瞬态增长和声学的问题。我们的理论结果使用来自声学和双流射流的一般问题进行了说明，并且将 PSE 解与线性化纳维-斯托克斯方程的全局解和最近开发的替代抛物线进行了比较。这些发现对 PSE 的不可压缩和可压缩公式均有效，尤其适用于涉及具有不同波长和增长率的多种模式的应用，例如涉及多种不稳定机制、瞬态增长和声学的问题。我们的理论结果使用来自声学和双流射流的一般问题进行了说明，并且将 PSE 解与线性化纳维-斯托克斯方程的全局解和最近开发的替代抛物线进行了比较。