Short-time Lyapunov exponent analysis and the transition to chaos in Taylor–Couette flow

Abstract

Short-time Lyapunov exponent analysis is a new approach to the study of the stability properties of unsteady flows. At any instant in time the Lyapunov perturbations are the set of infinitesimal perturbations to a system state that will grow the fastest in the long term. Knowledge of these perturbations can allow one to determine the instability mechanisms producing chaos in the system. This new method should prove useful in a wide variety of chaotic flows. Here it is used to elucidate the physical mechanism driving weakly chaotic Taylor–Couette flow.Three-dimensional, direct numerical simulations of axially periodic Taylor–Couette flow are used to study the transition from quasi-periodicity to chaos. A partial Lyapunov exponent spectrum for the flow is computed by simultaneously advancing the full solution and a set of perturbations. The axial wavelength and the particular quasi-periodic state are chosen to correspond to the most complete experimental studies of this transition. The computational results are consistent with available experimental data, both for the flow characteristics in the quasi-periodic regime and for the Reynolds number at which transition to chaos is observed.The dimension of the chaotic attractor near onset is estimated from the Lyapunov exponent spectrum using the Kaplan–Yorke conjecture. This dimension estimate supports the experimental observation of low-dimensional chaos, but the dimension increases more rapidly away from the transition than is observed in experiments. Reasons for this disparity are given. Short-time Lyapunov exponent analysis is used to show that the chaotic state studied here is caused by a Kelvin–Helmholtz-type instability of the outflow boundary jet of the Taylor vortices.