Predicting the energy stability limit of shear flows using weighted velocity components

Abstract

Predicting the subcritical transition in fluid dynamic systems remains a challenging task, but recent advancements utilizing edge tracking methods, polynomial Lyapunov functions, and various energy norms have shown promise. In this study, we propose a novel approach by defining the general kinetic energy through weighted velocity components. The minimal Reynolds number is determined, where the derivative of this generalized energy with respect to time is zero. The procedure is similar to that of the well-known Reynolds–Orr equation. Unlike traditional methods, our approach does not necessitate the monotonic decay of the classic perturbation kinetic energy, resulting in a larger critical Reynolds number and reduced conservativeness of the Reynolds–Orr equation. However, the energy production of the pressure is not negligible, in contrast to the classical Reynolds–Orr equation. The pressure's implicit dependence on the velocity field complicates the variation process. To address this, a method is presented to handle the problem effectively. Our approach is then applied to analyze parallel flows, specifically the plane Couette and plane Poiseuille flows, wherein the problem can be further simplified using the complex Fourier transformation. The weights of velocity components are optimized to maximize the critical Reynolds number, resulting in a significant increase.