Nonlinear 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 fully-developed fluid flows is studied. The disturbance is represented in its most general form by a Fourier integral over all possible wavenumbers. The Fourier components of the disturbance quantities are expanded in a series of the linear-stability eigenfunctions, and a set of integro-differential equations for the amplitude density function of a continuous spectrum is derived. No approximations are involved in this reduction; hence, a numerical solution of the integro-differential equations is an exact solution of the Navier-Stokes equations. Numerical integration of the integro-differential equations with different initial conditions shows that the equilibrium state of the flow is not unique after the first bifurcation point, but depends on the waveform of the initial disturbance or, equivalently, on ambient noise which cannot always be controlled in practical situations. Multiple equilibrium states are found to occur at the same value of the dynamic similarity parameters; this implies that any property transported by the fluid can at best be determined within a limit of uncertainty associated with nonuniqueness. A perturbation expansion with multiple time scales is used to show that the equations describing the evolution of monochromatic waves and slowly-varying wavepackets in classical weakly nonlinear theories are special limiting cases of the integro-differential equations near the onset of linear instability. The range of validity of the weakly nonlinear expansions is examined for mixed-convection flow in a heated vertical annulus. The results confirm that weakly nonlinear theories fail to give an adequate description of the physics of the flow even near the onset of linear instability. This is because these theories consider only the most unstable mode, and neglect the contribution from other eigenmodes which can have a large effect on the mean flow distortion. Without considering the leading-order effect of the mean-flow distortion, classical weakly nonlinear instability theories fail to account for proper energy exchanges. The numerical results of the integro-differential equations for the amplitude density function are compared with a direct numerical simulation of the Navier-Stokes equations using a Fourier-Chebyshev spectral method. Complete agreement is found between the two numerical solutions. The solution of the integro-differential equations is simpler than and requires only a small fraction of the computer time necessary for solving the Navier-Stokes equations by a spectral method. The current formulation presents an efficient algorithm to solve the Navier-Stokes equations.