Nonlinear stability of wall-bounded flows using the One-Way Navier-Stokes (OWNS) Equations

We extend the One-Way Navier Stokes (OWNS) approach to support nonlinear interactions between waves of different frequencies, which will enable nonlinear analysis of instability and transition. In linear OWNS, the linearized Navier-Stokes equations are modified such that upstream propagating modes are removed, so that they can be solved efficiently in the frequency domain as a spatial initial-value (marching) problem. Linear OWNS confers numerous advantages over the parabolized stability equations (PSE) that we seek to extend to nonlinear analysis. In the proposed method, the fully nonlinear Navier-Stokes equations are marched in the downstream direction. At each step of the march, the projection operator from the linear OWNS procedure is applied to (approximately) remove modes with upstream group velocity. We validate the method by examining the nonlinear evolution of two- and three-dimensional disturbances in a low-speed Blasius boundary layer by comparing with PSE and DNS results from the literature.