Understanding the sub-critical transition to turbulence in wall flows

In contrast with free shear flows presenting velocity profiles with inflection points which cascade to turbulence in a relatively mild way, wall bounded flows are deprived of (inertial) instability modes at low Reynolds numbers and become turbulent in a much wilder way, most often marked by the coexistence of laminar and turbulent domains at intermediate Reynolds numbers, well below the range where (viscous) instabilities can show up. There can even be no unstable mode at all, as for plane Couette flow (pCf) or for Poiseuille pipe flow (Ppf) that are currently the subject of intense research. Though the mechanisms involved in the transition to turbulence in wall flows are now better understood, statistical properties of the transition itself are yet unsatisfactorily assessed. A widely accepted interpretation rests on non-trivial solutions of the Navier-Stokes equations in the form of unstable travelling waves and on transient chaotic states associated to chaotic repellors. Whether these concepts typical of the theory of temporal chaos are really appropriate is yet unclear owing to the fact that, strictly speaking, they apply when confinement in physical space is effective while the physical systems considered are rather extended in at least one space direction, so that spatiotemporal behaviour cannot be ruled out in the transitional regime. The case of pCf will be examined in this perspective through numerical simulations of a model with reduced cross-stream (y) dependence, focusing on the in-plane (x, z) space dependence of a few velocity amplitudes. In the large aspect-ratio limit, the transition to turbulence takes place via spatiotemporal intermittency and we shall attempt to make a connection with the theory of first-order (thermodynamic) phase transitions, as suggested long ago by Pomeau.