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

Abstract

Görtler vortices developed in laminar boundary layer experience remarkable changes when the flow is subjected to compressibility effects. 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 these effects. Görtler vortices in subsonic and moderate supersonic flows (Ma = 0.015, 1.5 and 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 is maintained 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. Onset conditions are determined for these modes. The above three modes can each have the largest growth rate under the right conditions. In the hypersonic cases, the threshold amplitude A(u) is dramatically reduced, showing the significant 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 (Li & Malik, J. Fluid Mech., vol. 297, 1995, pp. 77-100) is shown to be 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 sub-dominant secondary modes reach their peak growth rates under detuned states.

Boundary-layer stability of supercritical fluids in the vicinity of the Widom line

“…Note that the current study is up to Ec ∞ = 0.20 (Ma 2) and therefore not influenced by hypersonic effects. In hypersonic flows, temperature perturbation can become the most prominent for ideal-gas flows (for example Görtler instability at Ma = 6 reported in Ren & Fu 2015).…”

The amplification of large-scale motion in a supersonic concave turbulent boundary layer and its impact on the mean and statistical properties

“…The secondary instability and the related breakup of Görtler vortices lead to the final transition of boundary layer (see e.g. Swearingen & Blackwelder 1987;Ren & Fu 2015;Wang et al 2018).…”

The New Mode of Instability in Viscous High-Speed Boundary Layer Flows

“…The 5 × 5 matrices Γ, A, B... are functions of the base flow and dimensionless parameters Re, Ma, Pr. Detailed expressions for these matrices can be found in the authors' previous articles [19,20]. The physical quantities are nondimensionalized with their corresponding free-stream values except pressure p * by ρ * ∞ U * 2 ∞ .…”