A data-driven evaluation of solution strategies for nonlinear parabolized stability equations

Parabolized stability equations (PSE) have been widely employed to understand the development of perturbations in fluid flow and predict laminar-to-turbulent transition since their introduction in the 1990s. The parabolic structure of the PSE framework enables efficient and cost-effective computations. This framework relies on the assumption of “slowly varying shape functions,” implemented through a wavenumber iteration formula typically designed to minimize the streamwise gradient of disturbance kinetic energy for the primary mode. However, it remains unclear whether an iteration strategy effectively mitigates the influence of elliptical terms in the harmonics, thereby affecting both the applicability and accuracy of the equations. To address this concern, we propose a global magnitude analysis combined with a balance model framework utilizing a Gaussian mixture model and sparse principal component analysis. This methodology systematically assesses the relative significance of retained and neglected elliptical terms across the entire flow field and Fourier space, verifying the physical relevance and predictive accuracy of PSE-based analyses.