Code from Non-normal origin of modal instabilities in rotating plane shear flows. 25 August 2019 12 December 2019

Small variations introduced in shear flows are known to affect stability dramatically. Rotation of the flow system is one example, where the critical Reynolds number for exponential instabilities falls steeply with a small increase in rotation rate. We ask whether there is a fundamental reason for this sensitivity to rotation. We answer in the affirmative, showing that it is the non-normality of the stability operator in the absence of rotation which triggers this sensitivity. We treat the flow in the presence of rotation as a perturbation on the non-rotating case, and show that the rotating case is a special element of the pseudospectrum of the non-rotating case. Thus, while the non-rotating flow is always modally stable to streamwise-independent perturbations, rotating flows with the smallest rotation are unstable at zero streamwise wavenumber, with the spanwise wavenumbers close to that of disturbances with the highest transient growth in the non-rotating case. The instability critical rotation number scales inversely as the square of the Reynolds number, which we demonstrate is the same as the scaling obeyed by the minimum perturbation amplitude in non-rotating shear flow needed for the pseudospectrum to cross the neutral line. Plane Poiseuille flow and plane Couette flow are shown to behave similarly in this context.剪切流中引入的微小扰动会对其稳定性产生显著影响。流动系统的旋转便是典型一例：当旋转速率小幅提升时，对应指数型不稳定性的临界雷诺数（Reynolds number）会急剧下降。我们探究这种对旋转的敏感性是否存在根本性成因，并给出了肯定结论：正是无旋转工况下稳定性算子（stability operator）的非正规性，触发了这一敏感性。我们将旋转存在时的流动视作无旋转情况的扰动项，并证明旋转工况是无旋转工况伪谱（pseudospectrum）中的一类特殊元素。由此可见，尽管无旋转流动对流向无关的扰动始终保持模态稳定，但旋转速率极小的旋转流在零流向波数下会出现不稳定性，且其展向波数与无旋转情况下瞬态增长最高的扰动波数相一致。不稳定性临界旋转数与雷诺数的平方成反比，我们证明该标度关系与无旋转剪切流中，伪谱越过中性线所需的最小扰动振幅所遵循的标度完全一致。在此研究语境下，平面泊肃叶流（Plane Poiseuille flow）与平面库埃特流（plane Couette flow）表现出相似的行为。