Modeling the Direct Transition to Turbulence

After a brief summary of experimental results focussed on plane Couette flow and contrasting the low-Reynolds numbers transition between laminar flow and the turbulent spot regime, with its high-Reynolds counterpart between homogeneous turbulence and turbulent stripes, we introduce a hierarchy of models from conceptual models designed to answer the most basic questions to semi-realistic models more adapted to the study of specific topics. Among conceptual models we distinguish between low-dimensional deterministic dynamical systems apt to analyze problems linked to attractor coexistence and basin boundaries, stochastic models more appropriate to studying the role of noise in the natural transition, and models derived from statistical mechanics focussed on the “macroscopic” growth of turbulent patches in the triggered transition. Semi-realistic models separating cross-stream and in-plane variables are designed in view of concrete analysis of processes at a more “microscopic” scale, related to pattern formation and space-time chaos studies.