Strongly nonlinear vortices in the Stokes layer on an oscillating cylinder

The Stokes layer on a torsionally oscillating circular cylinder is susceptible to axially periodic vortices which are driven by a centrifugal mechanism which has some similarities to that which is responsible for Gortler vortices in boundary layers on concave surfaces. For small wavelength vortices an analytic description of linear, weakly nonlinear and fully nonlinear perturbations is possible and, as the growth rates of these modes are asymptotically greater than the timescale of the underlying Stokes layer, a quasi-steady stability analysis is attempted in preference to a Floquet-type approach. It is shown that highly nonlinear vortices can be expected to form on the surface of the cylinder; it is noted that these vortices alter the basic mean flow at leading order and occupy a zone which has depth comparable to the Stokes thickness. Depending on the Taylor number of the flow, three distinct types of vortex behaviour are found. In the first of these, corresponding to Taylor numbers just above that at which the Stokes layer becomes unstable over a part of its cycle, the vortices are relatively weak and remain attached to the cylinder throughout their lifetime. At larger Taylor numbers, the vortices initially grow whilst remaining on the cylinder but at some stage they break away and concentrate in a region within the Stokes layer. However, their strength reduces with time and they soon decay away. In the third regime identified, before this vortex activity region dies, the residual flow next to the cylinder becomes unstable once more. In this case a second vortex zone begins to grow so that for a certain time interval two active zones exist. The governing equations for the strongly nonlinear modes take the form of a free boundary value problem subject to ordinary differential constraints. Numerical solutions are presented and, although periodic solutions are by no means guaranteed by use of the quasi-steady approach, the computations show that remarkably periodic vortex configurations tend to result.