Nonlinear Görtler Vortices and Their Secondary Instability in a Hypersonic Boundary Layer

Abstract

Nonlinear development of the Gortler instability over a concave surface gives rise to a highly distorted inflectional flow field in the boundary layer that leads to both wall-normal and spanwise gradients in the flow. Such nonlinear structures are susceptible to strong, high-frequency secondary instabilities that may lead to the onset of laminar-turbulent transition. The present numerical study uses direct numerical simulations and linear secondary instability theory to investigate finite amplitude Gortler vortices and their secondary instability characteristics, respectively, in the hypersonic flow over an axisymmetric cone with a concave aft body. To complement previous studies in the literature wherein the Gortler instability was usually studied for a flat plate and initiated at some upstream location by imposing an eigenfunction as the inflow condition or by blowing and suction at the wall, the present investigation is focused on fully realizable Gortler instability that is excited by an azimuthally periodic array of surface protuberances. Furthermore, while the previous work had mostly focused on the secondary instability of Gortler vortices with cross-plane velocity contours that resembled bell-shaped structures, the present results confirm that fully developed mushroom structures also exist in the hypersonic regime when the Gortler vortex amplitude is sufficiently large. Computations further reveal that the dominant modes of secondary instability in these mushroom-shaped structures correspond to an antisymmetic (i.e., sinuous) “stem” mode that concentrates within the strong, nearly wall-normal internal shear layers surrounding the stem regions underneath the caps of the mushroom structures. Additionally, there exist a multitude of other significantly unstable secondary instability modes of both symmetric and antisymmetric types. Analogous to the secondary instability of crossflow vortices in hypersonic flows, secondary instability modes of both symmetric and antisymmetric types. Analogous to the secondary instability of crossflow vortices in hypersonic flows, secondary instability modes originating from the Mack mode instability play an important role during the nonlinear breakdown process.