A recipe to generate low order models of the transition to turbulence

Recently, our understanding of the transition to Newtonian turbulence has significantly changed due to the discovery of exact solutions of the Navier-Stokes equations and the introduction of the self-sustaining process by which Newtonian turbulence in parallel shear flows is sustained [F. Waleffe, Phys. Fluids, v.9, 883 (1997)]. In this mechanism, a small number of coherent structures (streamwise vortices, streaks and 3D vortices) are able to sustain themselves via a series of non-linear interactions and instabilities. This theory, dubbed the self-sustaining process, has been very successful in describing the main features of weakly turbulent states close to the transition threshold. One of the main predictions of the theory is that close to the transition, turbulence is metastable, and its lifetime depends super-exponentially on the Reynolds number.

The main strength of this approach is that it allows for a semi-analytical description of the turbulent dynamics close to the transition in the form of a rather low-dimensional model. Until now, the exact form of such models was largely guided by the intuition drawn from the self-sustaining process mechanism and direct numerical simulation (DNS). In this talk I present a systematic way of deriving low-dimensional models that requires no previous intuition of the system in question or its dynamics. We test our method by constructing a hierarchy of low-dimensional models for the transition to turbulence in plane-Couette flow. 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 observed in experiments and DNS requires interactions between exact coherent structures of different symmetries and discuss the implications of this discovery for the transition.