Understanding finite life-times of Newtonian turbulence

Recently, our understanding of the transition to turbulence has significantly changed due to the discovery of exact solutions of the Navier-Stokes equations and the introduction of the self-sustaining process in parallel shear flows. This theory has been very successful in describing the main features of weakly turbulent states, including the metastable nature of turbulence close to the transition and the super-exponential dependence of its lifetime on the Reynolds number.

The main strength of this approach is that it allows for a semi-analytical description of the turbulent dynamics in the form of a rather low-dimensional model. The exact form of such models is typically guided by one’s intuition and DNS. In this talk we present a systematic way of deriving low-dimensional models for plane Couette flow that requires no previous intuition of the system in question or its dynamics. We find that the model exhibits a subcritical transition to turbulent dynamics, contains stable periodic orbits, exact coherent structures and finite turbulent lifetimes. We demonstrate that the super-exponential nature of the lifetimes requires interactions between exact coherent structures of different symmetries and discuss the implications of this discovery for the transition.