Effect of Pressure Gradient on the Development of Görtler Vortices

Leandro Marochio Fernandes,Marcio Teixeira De Mendonça

doi:10.5028/jatm.v11.1069

Leandro Marochio Fernandes, Marcio Teixeira De Mendonça

Open Access

https://doi.org/10.5028/jatm.v11.1069

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Abstract

Boundary layers over concave surfaces may become unstable due to centrifugal instability that manifests itself as stationary streamwise counter rotating vortices. The centrifugal instability mechanism in boundary layers has been extensively studied and there is a large number of publications addressing different aspects of this problem. The results on the effect of pressure gradient show that favorable pressure gradients are stabilizing and adverse pressure gradient enhances the instability. The objective of the present investigation is to complement those works, looking particularly at the effect of pressure gradient on the stability diagram and on the determination of the spanwise wave number corresponding to the fastest growth. This study is based on the classic linear stability theory, where the parallel boundary layer approximation is assumed. Therefore, results are valid for Görtler numbers above 7, the lower limit where local mode linear stability analysis was identified in the literature as valid. For the base flow given by the Falkner-Skan solution, the linear stability equations are solved by a shooting method where the eigenvalues are the Görtler number, the spanwise wavenumber and the growth rate. The results show stabilization due to favorable pressure gradient as the constant amplification rate curves are displaced to higher Görtler numbers, with the opposite effect for adverse pressure gradient. Results previously unavailable in the literature identifying the fastest growing mode spanwise wavelength for a range of Falkner-Skan acceleration parameters are presented.

Highlights

There is a large body of publications addressing centrifugal instability of boundary layers over concave surfaces
The study was conducted based on the linear stability theory, which is valid for Görtler numbers above 7
Stability diagrams for values of the Falkner-Skan similarity solution acceleration parameter βfs ranging from favorable to adverse pressure gradients were compared to the stability diagram of the Blasius boundary layer

Summary

INTRODUCTION

There is a large body of publications addressing centrifugal instability of boundary layers over concave surfaces. In 1994, Finnis and Brown (1994) presented another experimental study of the effect of pressure gradient on the development of Görtler vortices They considered only favorable pressure gradient and showed its stabilizing effect as well as the fact that the spanwise wavenumber does not change due to the flow acceleration. A comparison between the local linear stability theory and a streamwise marching technique was presented by Goulpié et al (1996) They considered base flows given by the Falkner-Skan solution and confirmed previous results, where favorable pressure gradient are stabilizing and adverse pressure gradient are destabilizing. The present investigation tries to clarify the effect of pressure gradient on the fastest growing mode in order to justify the stronger instability of the large spanwise wavenumber vortices in a favorable pressure gradient, confirm the explanation offered by Rogenski et al (2016a) based on the size of the vortices, thickness of the shear layer and velocity gradients.

METHODOLOGY

RESULTS

CONCLUSIONS