Taylor-Couette Instability With a Continuous Spectrum

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. In all cases, the final equilibrium state consists of a single dominant mode and its superharmonics of smaller-amplitudes. 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 results also show that a linearly stable long wave can excite a short unstable wave through nonlinear wave-interaction. An important implication of the existence of nonunique equilibrium states is that the torque induced by the fluid motion cannot be determined uniquely. The numerical results show that the uncertainty, associated with nonuniqueness, of

Is a direct numerical simulation of chaos possible? A study of a model nonlinearity

“…The essence of their observations has been confirmed by recent direct numerical solutions of the Navier-Stokes equations [2,12]. Similar numerical results are available for the totally different physical example of mixed convection in a vertical annulus [13].…”

“…Otherwise, this discrete system is similar to the discrete system for the amplitude-density functions of a Fourier-eigenfunction spectral method for the Navier-Stokes equations [2,12,13], which suggests that the solution properties studied in this paper are relevant to those of the Navier-Stokes equations.…”

“…The time-step is 0.001 and N = 32. It has been shown that the nonlinear terms represent many wave resonances and are not just limited to resonant trios [12]. Results obtained with the implicit method showing that the energy transfers from A 1 to its harmonics occur simultaneously are displayed in Fig.…”

“…Previous work has shown that such behavior exists for the Navier-Stokes equations [2,12,13]. This suggests that the classical principle of ''dynamic similarity" is strictly valid only if the governing parameter is below its critical value.…”

A resonant wave theory

“…Along the same line of reasoning, Lighthill (1969) proposed that large eddies can be thought of as arising from the primary instabilities of the mean flow profile; he advanced the concept of wave-packet dynamics to describe the energy transfer between turbulent eddies and the mean flow. More recently, taking advantage of modern computing power, instabilities of mixed convection and Couette-Taylor flows have been explicitly determined using a wave expansion method (Yao & Ghosh Moulic 1994, 1995bYao 1995;Ghosh Moulic & Yao 1996). This is the first theoretical verification of multiple solutions for the Couette-Taylor instability, which was systematically studied by Coles (1965).…”