Subharmonic instabilities of finite-amplitude monochromatic waves

Abstract

Finite-amplitude monochromatic waves in free-shear layers, planar jets, gravity-capillary waves, free-convection boundary layers, and falling films are often first destabilized by disturbances twice or two-thirds their wavelength or period. The transition to the 1/2 mode allows energy transfer to modes with lower wave numbers or lower frequencies and it precedes the onset of wide-spectrum turbulence in the above systems. A center-unstable manifold theory is used to derive the onset criteria for these subharmonic instabilities in terms of a critical amplitude of the fundamental or a critical value of a control parameter. It is shown that, in contrast to classical results, a finite-amplitude wave is always unstable to disturbances with 1/2 its wave number (frequency) if the subharmonic is linearly unstable. This instability is either oscillatory or static depending on the phase difference (frequency mismatch) between the fundamental and the subharmonic and on the amplitude of the fundamental. For a given phase difference, there exists a critical amplitude beyond which the static instability triggers a most efficient energy transfer to the 1/2 mode and causes it to grow monotonically. Inefficient oscillatory interaction occurs below this critical amplitude of the fundamental. In contrast, the fundamental -3/2 interaction is always oscillatory except when there is perfect resonance (no frequency mismatch). The theory presented here contains a rigorous and explicit derivation of amplitude equations for spatially evolving open-flow systems without recourse to Gaster transformation or multiscale expansions. This new formulation is partially demonstrated on the inviscid shear layer. © 1992 American Institute of Physics.