Travelling-waves consistent with turbulence-driven secondary flow in a square duct

We present numerically determined travelling-wave solutions for pressure-driven flow through a straight duct with a square cross-section. This family of solutions represents typical coherent structures (a staggered array of counter-rotating streamwise vortices and an associated low-speed streak) on each wall. Their streamwise average flow in the cross-sectional plane corresponds to an eight vortex pattern much alike the secondary flow found in the turbulent regime.