On the growth of laminar-turbulent patterns in plane Couette flow

Abstract

The growth of laminar-turbulent band patterns in plane Couette flow is studied in the vicinity of the global stability threshold R_g below which laminar flow ultimately prevails. Appropriately tailored direct numerical simulations are performed to manage systems extended enough to accommodate several bands. The initial state or germ is an oblique turbulent patch of limited extent. The growth is seen to result from several competing processes: (i) nucleation of turbulent patches close to or at the extremities of already formed band segments, with the same obliquity as the germ or the opposite one, and (ii) turbulence collapse similar to gap formation for band decay. Growth into a labyrinthine pattern is observed as soon as spanwise expansion is effective. An ideally aligned pattern is usually obtained at the end of a long and gradual regularisation stage when R is large enough. Stable isolated bands can be observed slightly above R_g. When growth rates are not large enough, the germ decays at the end of a long transient, similar to what was observed in experiments. Local continuous growth/decay microscopic mechanisms are seen to compete with large deviations which are the cause of mesoscopic nucleation events (turbulent patches or laminar gaps) controlling the macroscopic behaviour of the system (pattern). Implications of these findings are discussed in the light of Pomeau's proposals based on directed percolation and first-order phase transitions in statistical physics.