Nonlinear unstable wave disturbances in fluidized beds

Abstract

Instabilities in fluidized beds are interpreted from the two-phase continuum theory of linearized hydrodynamic stability as the result of interactions between wave hierarchies for which the stability condition is violated; that is, in which the lower-order waves propagate at speeds exceeding those of the higher-order waves. For weak nonlinearities a hierarchy of Burgers-like equations is obtained. The nonlinear modifications to the wave speeds point towards the restoration of the stability condition in the linearized sense. A weakly nonlinear hydrodynamic stability analysis yields an amplitude equation that is of second order. It is argued, however, that the major history of the disturbance development may be expressed by a simpler first-order amplitude equation. The Landau-Stuart constant obtained is intimately related to the nonlinear modifications of the wave speeds of the higher- and lower-order wave operators. It is shown that for supercritical disturbances, amplitude and phase velocity equilibration is possible, and that the levels of the equilibration depend on the initial amplification rate, in agreement with observations. The equilibration occurs by cascades of the fundamental wave disturbance into its harmonics.