A Rational Weakly Non-Linear Theory: Taylor-Couette Instability With a Continuous Spectrum

TL;DR

This paper develops a nonlinear theory for Taylor-Couette flow with a continuous spectrum of unstable waves, revealing non-uniqueness of equilibrium states and narrower stability ranges than linear predictions.

Contribution

It introduces a novel integrodifferential equation for the amplitude-density of continuous spectra, extending classical weakly nonlinear theories and explaining multiple solutions and stability properties.

Findings

Final equilibrium depends on initial disturbance, showing non-uniqueness.

Narrower stable wavenumber range than linear neutral curve.

Unstable flows outside the range excite other waves, consistent with sideband instabilities.

Abstract

Nonlinear evolution of a continuous spectrum of unstable waves near the first bifurcation point in circular Couette flow has been investigated. The disturbance is represented by a Fourier integral over all possible axial wavenumbers, and an integrodifferential equation for the amplitude-density function of a continuous spectrum is derived. The equations describing the evolution of monochromatic waves and slowly-varying wave-packets of classical weakly nonlinear instability theories are shown to be special limiting cases. Numerical integration of the integrodifferential equation shows that the final equilibrium state depends on the initial disturbance, as observed experimentally, and it is not unique. The predicted range of wavenumbers for stable supercritical Taylor vortices is found to be narrower than the span of the neutral curve from linear theory. Taylor-vortex flows with wavenumbers outside this range are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The presence of multiple solutions at a fixed Reynolds number for a given geometry in Taylor-Couette flows has been known since Coles monumental contribution in 1965. It is worthwhile to note that the existence of multiple solutions, found by Coles, differs from current popular bifurcation theories. The multiple solutions in Coles sense have also been found for mixed-convection flows (Yao & Ghosh Moulic 1993, 1994) besides the Taylor-Couette flows. We believe that the nonuniqueness of Coles sense, which complements the bifurcation theories, is a generic property for all fluid flows.