Jose Sharath, Govindarajan Rama
Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai, India.
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru, India.
Proc Math Phys Eng Sci. 2020 Jan;476(2233):20190550. doi: 10.1098/rspa.2019.0550. Epub 2020 Jan 29.
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.
已知剪切流中引入的微小变化会显著影响稳定性。流动系统的旋转就是一个例子，在这种情况下，指数不稳定性的临界雷诺数会随着旋转速率的小幅增加而急剧下降。我们要问的是，这种对旋转的敏感性是否有根本原因。我们的回答是肯定的，表明正是在无旋转情况下稳定性算子的非正规性引发了这种敏感性。我们将有旋转情况下的流动视为无旋转情况下的一种扰动，并表明旋转情况是无旋转情况伪谱中的一个特殊元素。因此，虽然无旋转流动对于流向无关的扰动在模态上总是稳定的，但具有最小旋转的旋转流动在零流向波数处是不稳定的，其展向波数接近无旋转情况下具有最高瞬态增长的扰动的展向波数。不稳定临界旋转数与雷诺数的平方成反比，我们证明这与伪谱穿过中性线所需的无旋转剪切流中的最小扰动幅度所遵循的标度相同。在这种情况下，平面泊肃叶流和平面库埃特流表现出类似的行为。
