Taylor—Couette instability of travelling waves with a continuous spectrum

Abstract

The nonlinear evolution of a continuous spectrum of travelling waves resulting from the growth of unstable disturbances in circular Couette flow has been investigated. Numerical solution of the governing integro-differential equations for different initial conditions shows that the equilibrium states of Taylor-vortex, wavy-vortex or spiralvortex flows are not unique, but depend on the initial disturbance. The presence of multiple solutions at a fixed Reynolds number for a given Taylor–Couette geometry has been known since Coles’ seminal contribution in 1965. The current study indicates that the equilibrium state of flows on a stable bifurcation branch is a natural consequence of nonlinear wave resonance and is dependent on the initial conditions. The resulting wavenumber can take any value within an accessible finite band. Since such multiple solutions have also been found numerically for mixed-convection flows and experimentally for several other flows, there is evidence to support the conclusion that a non-uniqueness in the sense of Coles is a generic property for all fluid flows.

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.…”

“…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 sensitivity-to-initial condition is generally viewed as a necessary and sufficient condition for the existence of chaos. However, this property is also noted in the solutions of all nonlinear differential equations when the value of their governing parameters is larger than their critical value, [2,[12][13][14]. Consequently, it cannot be argued that this sensitivity is a sufficient condition for chaos.…”

Computed chaos or numerical errors

“…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).…”