Modulated waves in Taylor-Couette flow Part 1. Analysis

Abstract

We present a mathematical analysis of the transition from temporally periodic rotating waves to quasi-periodic modulated waves in rotating flows with circularly symmetric boundary conditions, applied to the flow between concentric, rotating cylinders (Taylor-Couette flow). Quasi-periodic flow (modulated wavy vortex flow) is described by two incommensurate, fundamental, temporal frequencies in an arbitrary frame, but the flow is periodic in the appropriate rotating frame. The azimuthal wavelength of the modulation may be different to that of the underlying rotating wave; hence the flow state is described by two azimuthal wavenumbers as well. One frequency and one wavenumber are determined by the wave state, but no simple physical properties have yet been associated with the parameters of the modulation. The current literature on modulated waves displays both conflicting mathematical representations and qualitatively different kinds of modulation. In this paper we use Floquet theory to derive the unique functional form for all modulated waves and show that the space-time symmetry properties follow directly. The flow can be written as a non-separable function of the two phases (θ −c1t, θ −c2t). We show that different branches of modulated wave solutions in Taylor-Couette flow are distinguished not by symmetry but by the ranges of the numerical values of c1, c2, and the spectral amplitudes of the solution. The azimuthal wavenumber associated with the modulation has a unique physical definition but is not directly expressed in the spatial symmetry of the modulated flow. Because modulated waves should occur generically in systems with rotational symmetry, this analysis has application beyond Taylor-Couette flow.