Stability of wall-bounded flows using one-way spatial integration of Navier-Stokes equations

Abstract

A method for constructing well-posed one-way equations for calculating disturbances of slowly-varying flows was recently introduced (Towne & Colonius, JCP, Vol. 300, 2015). The linearized Navier-Stokes equations are modified such that all upstream propagating modes are removed from the operator. The resulting equations, termed one-way Navier-Stokes equations, are stable and can be solved efficiently in the frequency domain as a spatial initial value problem in which initial perturbations are specified at the domain inlet and propagated downstream by spatial integration. To date, the method has been used to predict large-scale wavepacket structures and their acoustic radiation in turbulent jets. In this paper, the method is extended and applied to wall-bounded flows. Specifically, we examine the spatial stability of two- and three-dimensional boundary layers, corresponding to the Blasius and the Falkner-Skan-Cooke flows, and predict the evolution of unstable Tollmien-Schlichting waves and crossflow vortices, respectively. The method is validated against well-known results from the literature.