Non-orthogonal plane-marching parabolized stability equations for the secondary instability of crossflow vortices

A detailed derivation, analysis, and verification is given for the non-orthogonal, plane-marching Parabolized Stability Equations (PSE) approach. In applying the approach to a flow distorted by a medium-amplitude crossflow vortex, we determine its linear secondary instability mechanisms. We show that converged solutions can be achieved for a broad frequency range with an existing stabilization method for the line-marching PSE approach. We verify that 1) solutions converge versus grid size in all dimensions, 2) primary disturbance solutions agree with line-marching PSE results, and 3) secondary disturbance solutions match amplitude and growth-rate evolution of reference Direct Numerical Simulation (DNS) results. We show how and why the type-II instability displays a delayed neutral point when modeled with the plane-marching approach versus the considered local stability approaches, whether the streamwise evolution of the distorted base flow is accounted for or not. This may explain why the type-II disturbance is scarcely captured by DNS in the literature.