Splitting, Merging and Wavelength Selection of Vortex Pairs in Curved and/or Rotating Channels

Abstract

In channels that are rotating (about a spanwise axis) or curved, or both curved and rotating, steady two-dimensional vortices develop above a critical Reynolds number Re _c . The stability of these streamwise-oriented roll cells to spanwise-periodic perturbations of different wavelength than the vortices (i.e. Eckhaus stability) is examined numerically using linear stability theory and spectral methods. In curved and/or rotating channels, the Eckhaus stability boundary is found to be a small closed loop. Within the boundary, two-dimensional vortices are stable to spanwise perturbations. Outside the boundary, Eckhaus instability is found to cause the vortex pairs to split apart or merge together in a manner similar to that observed in recent experiments. For all channels examined, two-dimensional vortices are always unstable when Re > 1.7Re _c . Usually the most unstable spanwise perturbations are subharmonic disturbances, which cause two pairs of vortices with small wavenumbers to be split apart by the formation of a new vortex pair, but cause two pairs of vortices with large wavenumber to merge into a single pair. In nonlinear flow simulations presented here and in experiments, most observed wavenumbers are close to those that are least unstable to spanwise perturbations.