Pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem: Taylor vortices in axial flow

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Abstract
A unique pattern selection in the absolutely unstable regime of a driven, nonlinear, open-flow system is analyzed: The spatiotemporal structures of rotationally symmetric vortices that propagate downstream in the annulus of the rotating Taylor-Couette system due to an externally imposed axial through-flow are investigated for two different axial boundary conditions at the inlet and outlet. Detailed quantitative results for the oscillation frequency, the axial profile of the wave number, and the temporal Fourier amplitudes of the propagating vortex patterns obtained by numerical simulations of the Navier-Stokes equations are compared with results of the appropriate Ginzburg-Landau amplitude equation approximation and also with experiments. Unlike the stationary patterns in systems without through-flow the spatiotemporal structures of propagating vortices are independent of parameter history, initial conditions, and system length. They do, however, depend on the axial boundary conditions in addition to the driving rate of the inner cylinder and the through-flow rate. Our analysis of the amplitude equation shows that the pattern selection can be described by a nonlinear eigenvalue problem with the frequency being the eigenvalue. The complex amplitude being the corresponding eigenfunction describes the axial structure of intensity and wave number. Small, but characteristic differences in the structural dynamics between the Navier-Stokes equations and the amplitude equation are mainly due to the different dispersion relations. Approaching the border between absolute and convective instability the eigenvalue problem becomes effectively linear and the selection mechanism approaches that of linear front propagation.
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