Nonlinear instability theory in fluid dynamics

The nonlinear development of finite-amplitude waves resulting from the growth of unstable disturbances of arbitrary initial waveform in fluid flows is studied. The disturbance is represented by a Fourier integral over all possible wavenumbers. The Fourier components are expanded in a series of the linear stability eigenfunctions. The eigenfunction expansion reduces the Navier-Stokes equations to a system of nonlinearly coupled integrodifferential equations for the amplitude density function of a continuous spectrum. No approximations are involved in this reduction; hence a numerical solution of the integrodifferential equations is an exact solution of the Navier-Stokes equations. This is demonstrated by comparison with a direct numerical simulation of the Navier-Stokes equations using spectral methods. The solution of the integrodifferential equations is simpler than the solution of the Navier-Stokes equations, and requires less computer time. Thus, the current formulation presents a new efficient algorithm to solve the Navier-Stokes equations. The integrodifferential equations have been approximated by a perturbation expansion with multiple time scales. The perturbation expansion demonstrates that monochromatic waves and slowly-varying wavepackets are special limiting cases of the integrodifferential equations in a parameter range close to the onset of linear instability.; The integrodifferential equations have been solved numerically to study wave interactions in Taylor-Couette flow and mixed-convection flow in a heated vertical annulus. The results show that the equilibrium state of the flow is not unique after the first bifurcation point, but depends on the waveform of the initial disturbance, as observed experimentally in Taylor-Couette flow. In all cases, the equilibrium state consists of a single dominant mode and its superharmonics. The range of equilibrium wavenumbers of the dominant mode was found to be narrower than the unstable span of the linear neutral curve. Disturbances with wavenumbers outside this range but within the linearly unstable region are found to decay, but to excite a wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. An important implication of the existence of non-unique equilibrium states is that any physical quantity transported by the fluid can at best be determined within the limit of uncertainty associated with non-uniqueness.