The nonlinear stability of plane parallel shear flows with respect to tilted perturbations

Abstract

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations is studied by energy methods. Tilted perturbation refers to the fact that perturbations form an angle \(\theta \in (0,{\pi \over 2})\) with the direction of the basic flows. By defining an energy functional, it is proven that plane parallel shear flows are unconditionally nonlinearly exponentially stable for tilted streamwise perturbation when the Reynolds number is below a certain critical value and the boundary conditions are either rigid or stress-free. In the case of stress-free boundaries, by taking advantage of the poloidal-toroidal decomposition of a solenoidal field to define energy functionals, it can be even shown that plane parallel shear flows are unconditionally nonlinearly exponentially stable for all Reynolds numbers, where the tilted perturbation can be either spanwise or streamwise.