Front propagation and pattern formation of Taylor vortices growing into unstable circular Couette flow

Abstract

We present numerical solutions of the Navier-Stokes equations for the time evolution of axisymmetric flows between concentric cylinders of three different radius ratios \ensuremath{\eta} in the range 0.5<\ensuremath{\eta}<0.9 after the Reynolds number R is suddenly increased from a subcritical to a supercritical value in the range of \ensuremath{\epsilon}=R/${R}_{c}$-1 between 0.01 and 0.1. Such a procedure generates a propagating interface between the stable Taylor vortex state that grows first near a rigid nonrotating end plate and the unstable homogeneous circular Couette flow which is formed in the bulk of the annulus very shortly after increasing the driving. Step up from stationary flow states at finite subcritical driving and sudden starts from rest are simulated. We determine for both cases the behavior of the Ekman vortex system near an annulus end. The time evolution of the unstable circular Couette flow in the bulk is found to agree perfectly with the analytical solution for an infinite annulus. We discuss the effect of inhomogeneities that break the translational symmetry in axial direction, thereby generate local vortex flow, and thus compete against front propagation in the destruction of the homogeneous flow state. The evolution of the intensity and the structure of the vortex pattern behind the moving interface and various other properties of the fronts are determined in quantitative detail and compared with theoretical and experimental results. The axial intensity profiles of the Taylor vortex fronts are about 20% sharper than those we have derived from the lowest-order amplitude equation. In agreement with the latter, the front extension suitably defined varies as \ensuremath{\sim}${\ensuremath{\epsilon}}^{\mathrm{\ensuremath{-}}1/2}$ and the propagation ve- locity varies as \ensuremath{\sim}${\ensuremath{\epsilon}}^{1/2}$ in the driving range considered here.