From primary instabilities to secondary instabilities in G\xf6rtler vortex flows

Xi Chen,Xianxu Yuan,Yifeng Zhang,Guohua Tu,Jianqiang Chen

doi:10.1186/s42774-019-0021-8

Xi Chen, Xianxu Yuan + Show 3 more

Open Access

https://doi.org/10.1186/s42774-019-0021-8

Copy DOI

Abstract

We have studied the transformation process from primary instabilities to secondary instabilities with direct numerical simulations and stability theories (Spatial Biglobal and plane-marching parabolized stability equations) in detail. First Mack mode and second Mack mode are shown to be able to evolve into the sinuous mode and the varicose mode of secondary instability, respectively. Although the characteristics of second Mack mode eventually lose in the downstream due to the synchronization with the continuous spectrum, second Mack mode is found to be able to trigger a sequence of mode resonations which in turn give birth to highly unstable secondary instabilities. In contrast, first Mack mode does not involve in any mode synchronization. Nevertheless, it can still “jump" to a sinuous mode of secondary instability with a much larger growth rate than that of the first Mack mode. Therefore, secondary instabilities of Görtler vortices are highly receptive to the primary instabilities in the upstream, so that one should consider the primary instability in the upstream and the secondary instability in the downstream as a whole in order to get an accurate prediction of the boundary layer transition.

Highlights

Görtler instability, manifesting itself as counter-rotating streamwise vortices, is frequently encountered in boundary layer flows over concave walls and other near-wall flows having curved streamlines
The transformation of Görtler vortices to the secondary-instability mode of the second Mack mode in a Mach 6 flaredcone boundary layer studied by [21] belongs to this case. We find another route where the primary instability could translate to the secondary instability through mode synchronization
3.1 From the primary instability to the varicose mode of the secondary instability At first, we examine how the primary instabilities evolve in the Görtler vortex flow with PSE3D

Summary

Introduction

Görtler instability, manifesting itself as counter-rotating streamwise vortices, is frequently encountered in boundary layer flows over concave walls and other near-wall flows having curved streamlines. [16] have utilized blowing and suction to numerically excite Görtler vortices in a Mach 6.5 concave boundary layer. They found (2019) 1:19 that large-wavelength blowing and suction tends to trigger secondary streaks in addition to primary streaks. With DNS and detailed stability analyses, they have found that the boundary layers with secondary streaks are subject to more kinds of secondary-instability modes with generally larger growth rates, and easier to transition

Results

Discussion

Conclusion