Stationary Crossflow Vortices' Secondary Instability: Linear Stability Theory and Parabolized Stability Equations

Abstract

Three-dimensional laminar boundary layers over swept wings are susceptible to crossflow instabilities, manifesting as stationary and traveling crossflow vortices. The boundary layer distorted by these vortices is prone to the growth of secondary instabilities. Discrepancies between direct numerical simulation (DNS) and stability methodologies on the development of secondary perturbations of stationary crossflow vortices over swept wings have been reported in the literature. To shed light on the origin of these inconsistencies, a comparison of DNS and linear stability theory is provided here. Secondary disturbances of finite-amplitude stationary crossflow vortices are analyzed for two frequencies: f= 900  Hz (Type III secondary instability) and f= 2200  Hz (Type I secondary instability). Results from linear stability theory (LST-2D) and linear parabolized stability equations (PSE-3D) formulated in a suitable nonorthogonal coordinate system correlate well with DNS data in terms of perturbation shape and location relative to the stationary crossflow vortices. Employing a nonorthogonal coordinate system is crucial for PSE-3D to fulfill slow variation along the streamwise direction and spanwise periodicity, whereas LST-2D, assuming parallel flow, can also use periodic boundary conditions in a vortex-aligned orthogonal coordinate system. However, the LST-2D results underestimate the integrated growth rate, whereas the PSE-3D computations closely match the DNS, highlighting the importance of including streamwise gradients and upstream history in the instability computations.