Nonlinear Disturbance Evolution in Boundary Layers Using the One-Way Navier-Stokes Equations

Laminar-turbulent transition prediction for boundary-layer flows is a pacing item in engineering design. This work extends the one-way Navier Stokes (OWNS) equations to support nonlinear interactions between waves of different frequencies which enables nonlinear disturbance evolution in spatially-developing shear flows, with the goal of predicting transition for a reduced computational cost relative to direct numerical simulation (DNS). The OWNS approach linearizes the Navier-Stokes equations about a user-specified equilibrium solution, and then evolves disturbances to the equilibrium solution by solving a spatial initial-value problem in the frequency domain. OWNS yields a reduced computational cost compared to global linear stability analysis, while also conferring numerous advantages over the parabolized stability equations (PSE) that we seek to extend to nonlinear OWNS (NOWNS). We validate NOWNS for two- and three-dimensional disturbances to a low-speed Blasius boundary layer by comparing to DNS results from the literature. We further demonstrate that NOWNS can be used to for transition prediction since it accurately predicts the onset of laminar-turbulent transition in low-speed boundary-layer flows, relative to DNS. Subsequently, we extend the approach to high-speed boundary-layer flows, where we apply it to study oblique-wave breakdown of Mack's first and second modes. Finally, we formulate a greedy algorithm for choosing optimal OWNS recursion parameters, which achieves rapid error convergence and a net decrease in computational cost compared to previous approaches to recursion parameter selection.