Continuing invariant solutions towards the turbulent flow.

A new mathematical framework is proposed for characterizing the coherent motion of fluctuations around a mean turbulent channel flow. We search for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier-Stokes equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. This is achieved by setting up a continuation procedure of known solutions of the perturbative Navier-Stokes equations, based on the continuous increase of the turbulent eddy viscosity towards its turbulent value. The recovered solutions, being sustained only in the presence of the Reynolds stress tensor, are representative of the statistically coherent motion of turbulent flows. For small friction Reynolds number and/or domain size, the statistically invariant motion is almost identical to the corresponding invariant solution of the Navier-Stokes equations. Whereas, for sufficiently large friction number and/or domain size, it considerably departs from the starting invariant solution of the Navier-Stokes equations, presenting spatial structures, main wavelengths and scaling very close to those characterizing both large- and small-scale motion of turbulent channel flows. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow.

Recent studies have suggested that in some cases transition can be triggered by some purely nonlinear mechanisms. Here we aim at verifying such an hypothesis, looking for a localized perturbation able to lead a boundary-layer flow to a chaotic state, following a nonlinear route. Nonlinear optimal localized perturbations have been computed by means of an energy optimization which includes the nonlinear terms of the Navier-Stokes equations. Such perturbations lie on the turbulent side of the laminar-turbulent boundary, whereas, for the same value of the initial energy, their linear counterparts do not. The evolution of these perturbations toward a turbulent flow involves the presence of streamwise-inclined vortices at short times and of hairpin structures prior to breakdown.