Solenoidal spectral formulations for the computation of secondary flows in cylindrical and annular geometries

Abstract

Novel spectral methods are formulated in terms of divergence-free vector fields in order to compute finite amplitude time-dependent solutions of incompressible viscous flows in cylindrical and/or annular geometries. The numerical discretization of the method leads to a simple dynamical system of amplitudes from which the stability properties of the solution can be analyzed easily. In addition, the formulation allows easy implementation of continuation algorithms to track solutions that have bifurcated from a known state, or the search for disconnected solution branches by means of homotopy transformations of the Navier–Stokes equations. The method is succesfully applied to the study of generic double Hopf bifurcations in pressure-driven helicoidal flows and to the search of unstable travelling wave solutions in pipe flow.