Floquet analysis of fundamental, subharmonic and detuned secondary instabilities of Görtler vortices

Abstract

Nonlinear parabolized stability equations are employed in this work to investigate the nonlinear development of the Gortler instability up to the saturation stage. The perturbed boundary layer is highly inflectional both in the normalwise and spanwise directions and receptive to the secondary instabilities. The Floquet theory is applied to solve the fundamental, subharmonic and detuned secondary instabilities. With the Gortler-vortices-distorted base flow, two classes of secondary disturbances, i.e. odd modes and even modes, are identified according to the eigenfunctions of the disturbances. These modes may result in different patterns in the late stages of the transition process. Li and Malik [1] have shown the sinuous and varicose types of breakdown originating from the odd and even modes. The current study focuses on the four most amplified modes termed the even modes I & II and odd modes I & II. Odd mode II was missing in the work of Li and Malik [1] probably due to their inviscid simplification. The detuned modes are confirmed to be less amplified than the fundamental (for the odd mode I) and subharmonic modes (for even modes I & II and the odd mode II).