Neutral stability curves of compressible Görtler flow generated by low-frequency free-stream vortical disturbances

Abstract

Pre-transitional compressible boundary layers perturbed by low-frequency free-stream vortical disturbances and flowing over plates with streamwise-concave curvature are studied via matched asymptotic expansions and numerically. The Mach number, the Görtler number and the frequency of the free-stream disturbance are varied to obtain the neutral stability curves, i.e. curves in the space of the parameters that distinguish spatially growing from spatially decaying perturbations. The receptivity approach is used to calculate the evolution of Klebanoff modes, highly oblique Tollmien–Schlichting waves influenced by the concave curvature of the wall, and Görtler vortices. The Klebanoff modes always evolve from the leading edge, the Görtler vortices dominate when the influence of the curvature becomes significant and the Tollmien–Schlichting waves may precede the Görtler vortices for moderate Görtler numbers. For relatively high frequencies the triple-deck formalism allows us to confirm the numerical result of the negligible influence of the curvature on the Tollmien–Schlichting waves when the Görtler number is an order-one quantity. Experimental data for compressible Görtler flows are mapped onto our neutral-curve graphs and earlier theoretical results are compared with our predictions.

Görtler Instability and Transition in Compressible Flows

“…Wang et al [145] performed ŕow visualizations of the complete evolution of Görtler vortices at 𝑀 ∞ = 2.95 from the laminar to the turbulent regime. Analogous to the theoretical ŕows studied by Viaro and Ricco [16,100], the unsteadiness of the Görtler vortices was likely caused by the free-stream perturbation, since the plate was free from roughness. Decreasing the concave curvature delayed the boundary-layer transition.…”

“…Their study can be viewed as an extension of the formulation of Wu et al [15] for incompressible boundary layers or as a generalization of the study of Ricco and Wu [80] to include centrifugal effects. As a complementary study to Viaro and Ricco [16], Viaro and Ricco [100] computed the neutral map of the compressible ŕow dominated by Görtler vortices, following the incompressible study of Viaro and Ricco [81]. The neutral map could be uniquely deőned thanks to the receptivity framework, which allowed the precise speciőcation of the oncoming free-stream perturbation ŕow.…”

“…Highly-oblique Tollmien-Schlichting waves, generated by the leading-edge receptivity mechanism discovered by Ricco and Wu [80], were also found to operate when the wall was curved. Viaro and Ricco [16,100] solved the boundary-region equations and also adopted two eigenvalue frameworks, based on the parallel-ŕow and nonparallel-ŕow assumptions and on a high-Görtler-number asymptotic formalism that was revelatory of the different stages of the ŕow evolution. Viaro and Ricco [16,100] investigated the effects of frequency, ratio of free-stream wavelengths, Mach number and Görtler number, focusing particularly on the growth rates, the streamwise length scale, and the location of the velocity and temperature perturbations.…”

“…These streamwise stations were not close to the leading edge and only the receptivity analysis could inform about the locations where the agreement between the two solutions was of good accuracy. Viaro and Ricco [16,100] proved that the inclusion of the correct initial and free-stream forcing was essential to compute the ŕow from the leading edge, especially in supersonic conditions. It was also demonstrated that, when an amplitude was assigned arbitrarily to the eigenvalue solution to start the marching procedure and thus somehow bypass the modelling of the receptivity process from the leading edge, the shapes of the velocity, temperature, and pressure proőles were not accurate because the ŕow could only be computed rigorously by the correct initial conditions, őxed by the receptivity process.…”

“…Viaro and Ricco [16,81,100] considered weak free-stream disturbances, which correspond to a small turbulence Reynolds number (𝑟 𝑡 ≪ 1). Sescu, Afsar, and Hattori [104] extended the work of Ricco and Wu [80] to study the nonlinear Görtler vortices excited by more intense free-stream disturbances, while Es-Sahli et al [108] solved the steady boundary-region equations to study the effects of wall-heat transfer on Görtler vortices excited by wall transpiration.…”