Formation of secondary vortices in turbulent square-duct flow

A linear approach, inspired by hydrodynamic stability theory, is used to describe the formation of large-scale coherent vortices for the turbulent flow that develops in a duct of square cross section. A set of equations for small-amplitude coherent motion is derived and closed with a simple mixing-length strategy. The initial condition that maximizes a chosen functional (related to either the kinetic energy of the coherent motion or the rate of turbulence production) is found through a direct/adjoint numerical approach borrowed from optimal control theory. It is found that different kinds of secondary flows can appear in the duct cross section, sustained by the mean shear. Some of these optimal states display a symmetry about the bisectors and the diagonals of the duct, in agreement with experimental observations and direct numerical simulations.