Two-dimensional nonlinear plane Poiseuille–Couette flow homotopy revisited

Abstract

A recent paper by Rincon [Phys. Fluids 19, 074105 (2007)] readdressed the question of the existence of two-dimensional steady nonlinear states in plane Couette flow, coming to the conclusion that it is not possible to obtain the nonlinear plane Couette flow solutions reported by Cherhabili and Ehrenstein [Eur. J. Mech. B/Fluids 14, 667 (1995)] using their Poiseuille–Couette homotopy. Exploring the multiparameter space by performing several consecutive and distinct continuations, we show that it is possible to find a complex numerical path from plane Poiseuille streamwise periodic waves to two-dimensional nonlinear steady states for the plane Couette flow limit. The nonlinear Couette flow states are retrieved using three independent solution procedures and the disturbance flow structure is shown to be localized in the streamwise periodic box. Numerical evidence is provided that the width of the nonlinear plane Couette flow disturbance decreases with increasing resolution at fixed boxlength. This singular-type behavior of the solution structure for increasing resolution suggests that the pseudospectral solutions do not converge to smooth two-dimensional physical solutions of the continuous nonlinear system, the existence of which remains an open question.