Secondary instabilities of Görtler vortices in high-speed boundary layer flows

Görtler vortices developed in laminar boundary layer experience remarkable changes when the flow is subjected to compressibility effect. In the present study, five Ma numbers, covering incompressible to hypersonic flows, at Ma = 0.015, 1.5, 3.0, 4.5 and 6.0 are specified to illustrate this compressibility effect. Görtler vortices in subsonic and moderate supersonic flows (Ma = 0.015, 1.5, 3.0) are governed by the conventional wall-layer mode (mode W). In hypersonic flows (Ma = 4.5, 6.0), the trapped-layer mode (mode T) becomes dominant. This difference maintains and intensifies downstream leading to different scenarios of secondary instabilities. The linear and nonlinear development of Görtler vortices which are governed by dominant modal disturbances are investigated with direct marching of the nonlinear parabolic equations. The secondary instabilities of Görtler vortices set in when the resulting streaks are adequately developed. They are studied with Floquet theory at multiple streamwise locations. The secondary perturbations become unstable downstream following the sequence of sinuous mode type I, varicose mode and sinuous mode type II indicating an increasing threshold amplitude. Neutral conditions are determined for these modes. The above three modes each can have the largest growth rate under proper conditions. In hypersonic cases, the threshold amplitude A(u) is dramatically reduced showing a great impact of the thermal streaks. To investigate the parametric effect of the spanwise wavenumber, three global wavenumbers (B = 0.5, 1.0 and 2.0 × 10⁻³) are specified. The relationship between the dominant mode (sinuous or varicose) and the spanwise wavenumber of Görtler vortices found in incompressible flows [1] is shown not fully applicable in high-speed cases. The sinuous mode becomes the most dangerous regardless of the spanwise wavelength when Ma > 3.0. The subharmonic type can be the most dangerous mode while the detuned type can be neglected although some of the secondary modes reach their peak growth rates under detuned states.