Development of spiral and wavy vortices in circular Couette flow

Abstract

Weakly nonlinear calculations are made to describe interactions of a pair of oblique waves inclined at equal but opposite angles with axisymmetric Taylor vortices in the flow between rotating cylinders with a narrow gap. The so-called wavy Taylor vortices observed in a simple case of the outer cylinder at rest and the inner cylinder rotating at a speed slightly above the critical value of the Taylor instability are shown to result from a cooperating interaction between the two oblique waves of the same amplitude and the axisymmetric vortices. In the case of counter-rotating cylinders, on the other hand, spiral vortices appear as the result of an exclusive interaction, in which one of the oblique waves wins their competition in growth and forces the other eventually to disappear, and are found to be stable even to axisymmetric vortices with positive growth rate of linear theory, because their interaction also belongs to the exclusive type. Sometimes both spiral and Taylor vortices form two independent equilibrium solutions of the exclusive interaction in a certain parameter range, and in that case, either of them can be realized in line with the initial amplitudes of the two vortices.