Direct numerical simulation of transition to turbulence in Görtler flow

Abstract

Using direct numerical simulation techniques we investigate transition to turbulence in a boundary-layer flow containing two large-scale counter-rotating vortices with axes aligned in the streamwise direction. The vortices are assumed to have been generated by the Görtler instability mechanism operating in boundary-layer flows over concave walls. Full, three-dimensional Navier–Stokes equations in a natural curvilinear coordinate system for a flow over concave wall are solved by a pseudospectral numerical method. The simulations are initialized with the most unstable mode of the linear stability theory for this flow with its amplitude taken from the experimental measurements of Swearingen & Blackwelder (1987). The evolution of the Görtler vortices for two different spanwise wavenumbers has been investigated. In all cases the development of strong inflexional velocity profiles is observed in both spanwise and vertical directions. The instabilities of these velocity profiles are identified as a primary mechanism of the transition process. The results indicate that the spanwise shear plays a more prominent role in the transition to turbulence than the vertical shear, in agreement with the hypothesis originally proposed by Swearingen & Blackwelder (1987). The following features of the transition, consistent with this hypothesis, were observed. Instability oscillations start in the spanwise direction and are followed later by oscillations in the vertical direction. A two-dimensional linear stability analysis predicts that the maximum growth rates of perturbations associated with the spanwise profiles are greater than those associated with the vertical profiles. Regions of high perturbation velocity correlate well with the regions of high spanwise shear and no obvious correlation with the vertical shear regions is observed. Finally, the analysis of the kinetic energy balance equation reveals that most of the perturbation energy production in the initial stages of transition occurs in the region characterized by large spanwise shear created by the action of the vortices moving low-speed fluid away from the wall. Our results are consistent qualitatively and quantitatively with other experimental, theoretical, and numerical investigations of this flow.