Exploring global dynamics in a low order model of parallel shear flow

Abstract

The study of the onset of turbulence in the flow of fluids, which are governed by the Navier-Stokes equations, has produced some famous instabilities e.g. Kelvin Helmholtz instability. An important, and as of yet mysterious, flow is Plane Couette flow in which turbulence is observable. Another observable phenomenon in Plane Couette flow is a “bursting” process whereby along with the linear laminar profile, non-laminar coherent structures appear and dissipate regularly. Yet, surprisingly V.A. Romanov has proven that the linear laminar profile of Plane Couette flow is linearly stable for all Reynolds number. In order to reconcile these seemingly contradictory facts a search began for numerical periodic solutions to Plane Couette flow. Nagata produced the first such solution and since then many others have been found as well. In order to explain the “bursting” process Fabian Waleffe introduced a low order model of Plane Couette flow that exhibits the phenomenon. We study the bifurcation diagram of a periodic solution of the Waleffe model found by the method of Nagata. Therein we find a codimension 2 bifurcation. The type of bifurcation, namely LPNS, ensures that we can find homoclinic and heteroclinic connections of limit cycles in the parameter space surrounding the it. We use a matrix-free manifold computation method to find a homoclinic orbit of limit cycles. Finally we will discuss a method to continue the homoclinic orbit.