WAVE-NUMBER SELECTION AND IRREGULARITY OF SPATIALLY DEVELOPING NONLINEAR DEAN AND GORTLER VORTICES

Spatially developing vortices found in curved channels (Dean vortices) and in a concave boundary layer (Gortler vortices) asre studied numer ically using Legendre spectral-element methods. The linear instability of these vortices with respect to spanwise perturbations (Eckhaus ins tability) is examined using parabolized spatial stability analysis. Th e nonlinear evolution of this instability is studied by solving the pa rabolized Navier-Stokes equations. When the energy level of Dean and G ortler vortices in the flow is low, the spatial growth of the vortices is governed by primary instability (Dean or Gortler instability). At this stage, vortices with different wavelengths can develop at the sam e time and do not interact with each other significantly. When certain vortices reach the nonlinear stage first and become the dominant wave length, spatial Eckhaus instability sets in. For all cases studied, sp atially developing Dean and Gortler vortices are found to be most unst able to spanwise disturbances with wavelength twice or 3/2 times that of the dominant one. The nonlinear growth of these perturbations gener ates a small vortex pair in between two pairs of vortices with long wa velength, but forces two pairs of vortices with short wavelength to de velop into one pair. For Gortler vortices, this is manifested mostly b y irregular and deformed vortex structures. For Dean vortices, this is manifested by vortex splitting and merging, and spatial Eckhaus insta bility plays an important role in the wavenumber selection process.