Low-Dimensional Analysis and Numerical Simulation of the Incompressible Flow Between Two Concentric Rotating Cylinders

Abstract

There has been a very large number of experimental and theoretical studies of flow between concentric rotating cylinders in the century, since these pioneering works, the instability of Couette flow and Taylor vortex flow, now known as the “Couette–Taylor problem” is a paradigm of nonlinear problem. In order to explore the transition way of Couette–Taylor flow from laminar to turbulence and the characteristics of chaotic attractors in the turbulent regime of Couette–Taylor flow, dynamical behaviors and numerical simulation of Couette–Taylor flow have been studied by using low-dimensional model analysis method in this paper. Dynamical behaviors of a three-model Lorenz-like system of Couette–Taylor flow have been discussed, such as the stability of equilibrium, presence of limit cycles, occurrence of bifurcation and chaos, as well as the analysis of global stability etc. Moreover, using these results we explain successive transitions of Couette–Taylor flow from Laminar flow to turbulence in the experiment. By means of numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum and Poincare map we analyze complex dynamic behaviors of the system from the bifurcation transition to chaos.