Nonlinear instability of travelling waves with a continuous spectrum

Yao, L. S.; Moulic, S. Ghosh

doi:10.1016/0017-9310(94)00310-r

Cited by 10 publications

(20 citation statements)

References 33 publications

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“…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]. This work establishes the possibility that a instability can lead to multiple solutions.…”

Section: Sensitivity To Initial Conditionssupporting

confidence: 75%

“…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.…”

Section: Discussionmentioning

confidence: 93%

“…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.…”

Section: Introductionmentioning

confidence: 91%

See 2 more Smart Citations

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

Yao¹

2007

International Journal of Heat and Mass Transfer

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Section: Sensitivity To Initial Conditionssupporting

confidence: 75%

Section: Discussionmentioning

confidence: 93%

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

Yao¹

2007

International Journal of Heat and Mass Transfer

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“…The current study indicates that the state of flows on a stable bifurcation branch can involve any wavenumber within a finite band and can not be determined uniquely. 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.…”

mentioning

confidence: 68%

Taylor-Couette Instability With a Continuous Spectrum

Yao

Moulic

1995

Journal of Applied Mechanics

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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

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“…The principal purpose of this paper is to develop a wave theory based on the computation of Yao & Ghosh Moulic (1994, 1995a. This wave theory captures some fundamental properties shared by most solutions to the Navier-Stokes equations.…”

Section: L-s Yaomentioning

confidence: 99%

A resonant wave theory

Yao

1999

J. Fluid Mech.

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Analysis is used to show that a solution of the Navier-Stokes equations can be computed in terms of wave-like series, which are referred to as waves below. The mean flow is a wave of infinitely long wavelength and period; laminar flows contain only one wave, i.e. the mean flow. With a supercritical instability, there are a mean flow, a dominant wave and its harmonics. Under this scenario, the amplitude of the waves is determined by linear and nonlinear terms. The linear case is the target of flow-instability studies. The nonlinear case involves energy transfer among the waves satisfying resonance conditions so that the wavenumbers are discrete, form a denumerable set, and are homeomorphic to Cantor's set of rational numbers. Since an infinite number of these sets can exist over a finite real interval, nonlinear Navier-Stokes equations have multiple solutions and the initial conditions determine which particular set will be excited. Consequently, the influence of initial conditions can persist forever. This phenomenon has been observed for Couette-Taylor instability, turbulent mixing layers, wakes, jets, pipe flows, etc. This is a commonly known property of chaos.

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Nonlinear instability of travelling waves with a continuous spectrum

Cited by 10 publications

References 33 publications

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

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

Taylor-Couette Instability With a Continuous Spectrum

A resonant wave theory

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