On the stability and transition for the Navier‐Stokes‐alpha model

Abstract

The main aim of this paper is to investigate the stability and transition of the Navier‐Stokes‐alpha model. By using the continued‐fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non‐dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio‐temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid‐axis of the channel.

Dynamical Transition Theory of Hexagonal Pattern Formations

“…Intuitively, a continuous transition occurs when the system transitions to a nearby local attractor, a catastrophic transition occurs when there are no nearby local attractors after transition and finally random transition occurs when the system either moves to a local attractor or leave the local neighborhood depending on the initial perturbation. For some of the recent applications of this theory, we refer to [39,40,21,15,12].…”