Wavenumber selection and irregularity of spatially developing nonlinear Dean and Görtler vortices

Abstract

Spatially developing vortices found in curved channels (Dean vortices) and in a concave boundary layer (Görtler vortices) asre studied numerically using Legendre spectral-element methods. The linear instability of these vortices with respect to spanwise perturbations (Eckhaus instability) is examined using parabolized spatial stability analysis. The nonlinear evolution of this instability is studied by solving the parabolized Navier–Stokes equations. When the energy level of Dean and Görtler vortices in the flow is low, the spatial growth of the vortices is governed by primary instability (Dean or Görtler instability). At this stage, vortices with different wavelengths can develop at the same time and do not interact with each other significantly. When certain vortices reach the nonlinear stage first and become the dominant wavelength, spatial Eckhaus instability sets in. For all cases studied, spatially developing Dean and Görtler vortices are found to be most unstable to spanwise disturbances with wavelength twice or $\frac{3}{2}$ times that of the dominant one. The nonlinear growth of these perturbations generates a small vortex pair in between two pairs of vortices with long wavelength, but forces two pairs of vortices with short wavelength to develop into one pair. For Görtler vortices, this is manifested mostly by irregular and deformed vortex structures. For Dean vortices, this is manifested by vortex splitting and merging, and spatial Eckhaus instability plays an important role in the wavenumber selection process.