Modulated waves in Taylor-Couette flow Part 2. Numerical simulation

Abstract

In part 1 (Coughlin & Marcus 1992a) we described the mathematics of temporally quasi-periodic fluid flows, with application to the flow between concentric cylinders (Taylor-Couette flow). In this paper we present numerical simulations of Taylor-Couette flow, motivated by laboratory experiments which show that periodic rotating waves become unstable to quasi-periodic modulated waves as the Reynolds number is increased. We find that several branches of quasi-periodic solutions exist, and not all of them occur as direct bifurcations from rotating waves. We compute solutions to the Navier-Stokes equations for both rotating waves and two branches of modulated waves; those discussed by Gorman & Swinney (1979, 1982), and those discovered by Zhang & Swinney (1985). A simple physical process is associated with both types of modulation. In the rotating frame the quasi-periodic disturbance field (defined as the total flow minus the time-averaged, steady-state piece) appears as a set of compact, coherent vortices confined almost entirely to the outflow jet region. They are adveeted (with some distortion) in the azimuthal direction along the outflow between adjacent Taylor vortices, with the frequency of modulation directly related to the mean drift speed. The azimuthal wavelength of the disturbance field determines the second wavenumber. We argue that the quasi-periodic flow arises as an instability of the essentially axisymmetric vortex outflow jet. Experimentally, modulated waves appear to bifurcate directly to low-dimensional chaos. Neither the mathematical nor the physical mechanisms of this translation have been well understood. Our numerical work suggests that the observed transition to chaos results from the existence of physically distinct Floquet modes arising from instability of the outflow jet.