Onset of wavy vortices in the finite-length Couette–Taylor problem

Abstract

The transition from Taylor vortex flow to wavy vortex flow in the Couette–Taylor problem, for finite annulus lengths, is studied. Using an accurate, fully resolved numerical code, infinite-cylinder nonlinear axisymmetric Taylor vortices are computed and their stability with respect to nonaxisymmetric disturbances corresponding to wavy vortices is determined. The present method computes leading eigenmodes and eigenvalues for arbitrary axial Floquet exponents k and arbitrary azimuthal wave numbers m, so that the full complex dispersion relation ω=ω(k,m) is obtained. The linear coefficients appearing in the Ginzburg–Landau equation are calculated and compared to previous experimental results. It is found that for aspect ratios (length to gap width) less than 30, the Ginzburg–Landau predictions are not reliable; for these aspect ratios the transition is modeled as the superposition of two Floquet modes with k=±π/L, where L is the length of the annulus. This model is verified via a detailed numerical and experimental study at aspect ratios between 8 and 34, for radius ratio 0.87, obtaining good agreement for the critical Reynolds numbers, azimuthal wave numbers and wave speeds over the entire aspect-ratio range.