Uncertainty of convection

Journal of Applied Mechanics

Moulic

1995

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

Section: Introductionsupporting

confidence: 72%

mentioning

confidence: 67%

Taylor-Couette Instability With a Continuous Spectrum

Journal of Applied Mechanics

Moulic

1995

“…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%

“…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).…”

Section: Introductionmentioning

confidence: 99%

A resonant wave theory

1999

J. Fluid Mech.

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.

“…A sensitivity-to-initial-condition is frequently viewed as a necessary and sufficient condition for the existence of chaos by most readers. However, this property is also noted in the solutions of all nonlinear differential equations when the values of their governing parameters are larger than appropriate critical values (Ghosh Moulic & Yao 1996;Yao & Ghosh Moulic 1994, 1995a, 1995bYao 1999Yao , 2007Yao , 2009).…”

mentioning

confidence: 99%

Computed chaos or numerical errors

2010

NAMC

Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical chaotic solution is currently available. Using the well-known Lorenz equations as an example, it is demonstrated that numerically computed results and their associated statistical properties are time-step dependent. There are two reasons for this behavior. First, chaotic differential equations are unstable so that any small error is amplified exponentially near an unstable manifold.The more serious and lesser-known reason is that stable and unstable manifolds of singular points associated with differential equations can form virtual separatrices. The existence of a virtual separatrix presents the possibility of a computed trajectory actually "jumping" through it due to the finite time-steps of discrete numerical methods. Such behavior violates the uniqueness theory of differential equations and amplifies the numerical errors explosively. These reasons imply that, even if computed results are bounded, their independence on time-step should be established before accepting them as useful numerical approximations to the true solution of the differential equations.However, due to these exponential and explosive amplifications of numerical errors, no computed chaotic solutions of differential equations independent of integration-time step have been found. Thus, reports of computed non-periodic solutions of chaotic differential equations are simply consequences of unstably amplified truncation errors, and are not approximate solutions of the associated differential equations.