SPLITTING, MERGING AND WAVELENGTH SELECTION OF VORTICES IN CURVED AND OR ROTATING CHANNEL FLOW DUE TO ECKHAUS INSTABILITY

In channels with rotation (about their spanwise axis) or curvature or both, steady two-dimensional vortices develop above a critical Reynolds number Re(c), owing to centrifugal or Coriolis effects. The stability of these streamwise oriented roll cells to two-dimensional, spanwise-periodic perturbations (i.e. Eckhaus stability) is examined numerically using linear stability theory and spectral methods. The results are then confirmed by nonlinear flow simulations. In channels with curvature or rotation or both, 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. For all channels examined, two-dimensional vortices are always unstable when Re > 1.7 Re(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. Recent experimental observations of splitting and merging of vortex pairs are discussed. When Re is not too high (Re < 4.0 Re(c)), the wavenumbers of vortices are selected by Eckhaus instability and most experimentally observed wavenumbers are close to the ones that are least unstable to spanwise perturbations.