A numerical analysis of the instability in the Taylor Vortex flow by using the visualization of attractors

Abstract

This study clarifies the critical values of bifurcations from a steady Taylor flow into a wavy Taylor flow by a numerical analysis applying chaos theory. In our previous numerical research, we adopt the amplitude fluctuation of the kinetic energy of the axial velocity component as a standard criterion to identify the critical Reynolds number. However, the amplitude fluctuation of the kinetic energy is very small near the critical value, which affects the judgment of Taylor flow or wavy Taylor flow. In this study, in order to improve this problem, we introduce a new method that visualizes attractors in the three-dimensional embedded space composed from the axial velocity component. The targets are the normal 2-vortex, 4-vortex and 6-vortex modes, which stably present in the range of the aspect ratio from 1.0 to 7.2 in a system with stationary end walls of two cylinders with the radius ratio 0.667. The critical values at which the Taylor flow bifurcates into the wavy Taylor flow are determined based on the converged orbit shapes of the attractors. The analytical result shows that, in each flow mode, the critical Reynolds number for the onset of the wavy flow has its peak and presents a quantitative agreement with the experimental result in the lower range than the peak Reynolds number. This study shows that the instability of the Taylor flow is revealed by determining the structure of the attractor applied by the Hopf bifurcation. In addition, it is represented that this numerical approach using attractors can obtain more accurate results than those obtained by a numerical evaluation method using the amplitude of the kinetic energy of the Taylor flow.