Mechanical and Civil Engineering: PhD Thesis Defense
Abstract:
We extend the one-way Navier Stokes (OWNS) equations to support nonlinear interactions between waves of different frequencies, which enables nonlinear analysis of instability and transition in spatially-developing shear flows. In OWNS, the linearized Navier-Stokes equations are parabolized and solved in the frequency domain as a spatial initial-value (marching) problem. OWNS yields a reduced computational cost compared to global linear stability analysis, while also conferring numerous advantages over the parabolized stability equations (PSE). We seek to extend these advantages for nonlinear analysis. We formulate and validate the nonlinear OWNS (NOWNS) equations by examining nonlinear evolution of two- and three-dimensional disturbances in a low-speed Blasius boundary layer compared to nonlinear PSE (NPSE) and DNS results from the literature. We demonstrate that NOWNS can be used to simulate flows with blowing/suction strips, is more robust to numerical noise, and converges for stronger nonlinearities, as compared to NPSE. In addition, we show that NOWNS can accurately predict 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 mode. Finally, we formulate a new algorithm for choosing good OWNS recursion parameters to replace previous heuristic approaches. Our greedy algorithm uses the exact eigen-spectrum to choose a sparse set of recursion parameters that achieve rapid convergence in the error. The algorithm can lead to a net decrease in computational cost, even when factoring in the additional cost of finding eigenvalues.