K.J.Groot, J. Casacuberta, S. Hickel (2026)
Computers & Fluids 306: 106947. doi: 10.1016/j.compfluid.2025.106947
A detailed derivation, analysis, and verification is given for the non-orthogonal, plane-marching Parabolized Stability Equations (PSE) approach. We show that converged solutions can be achieved for a broad frequency range with an existing stabilization method for the line-marching PSE approach. In applying the approach to a flow distorted by a medium-amplitude crossflow vortex, we determine its linear secondary instability mechanisms.
A formulation of the Parabolized Stability Equations (PSE) in a non-orthogonal coordinate system enables simultaneously spanning the direction in which the crossflow vortex is periodic and the direction in which the distorted base flow evolves slowest. Thereby, the effect of the simplifying assumptions for the plane-marching stability approach regarding spatial evolution are minimized. Two local approaches were also considered: in the (proper/consistent) local approach, we drop all out-of-plane derivatives of the distorted base flow, while we keep those terms in what we call the quasi-local approach.
We demonstrated that the stabilization approach of Andersson et al. enables achieving converged plane-marching solutions for a broad frequency range if residual ellipticity of the system would otherwise cause divergence. Therefore, we do not have to drop the pressure gradient term, that can potentially have an important impact on the solution behavior, even if dropping this term yielded practically identical results for the particular conditions considered in this article (a broad frequency range, but a specific crossflow vortex).
We performed several verification studies. First, we demonstrate that solutions converge versus grid size in all dimensions; this was specifically performed for the primary traveling crossflow (TCF) disturbance at 300 Hz, the distorted equivalent of the TCF disturbance (type-III) at 1 kHz, and the proper secondary type-I disturbance at 6 kHz. Second, we matched the aforementioned TCF disturbance against a result computed with a line-marching PSE approach. Third, we matched the amplitude, growth-rate, and disturbance shape evolution against Direct Numerical Simulations (DNS) for the aforementioned type-I and -III disturbances.
For type-III, the plane-marching approach yields both quantitative and qualitative improvement over both local approaches that emphasizes the merits of the plane-marching approach. At the particular frequency, 1 kHz, and particular crossflow vortex considered, we find that the type-III disturbance as modeled by the local approaches develops a secondary structure in its shape function, akin to the shape of the type-II instability, that causes the growth rate to steeply drop. This structure neither emerges in the plane-marching results nor DNS. Therefore, we can conclude that the upstream evolution history of the disturbance is responsible for preventing this type-II-like structure from occurring.
Lastly, we observed that the plane-marching solutions for type-I, -II, and -III disturbances display delayed neutral points as compared to results computed with both local approaches. The delay for the type-II instability is especially pronounced and corresponds to approximately 3 crossflow-vortex wavelengths. The results suggest that this is a consequence of the nonphysical movement of the type-II shape function against the in-plane velocity field near the neutral point, when modeled with a local approach. This may explain why DNS results for the type-II disturbance are scarce in the literature.