Global instability in amplitude equations produced by infinitesimal roughness #

Jonathan Healey

12:50 Monday in 4Q08.

Part of the Analysis of continuum mechanics session.

Abstract #

Amplitude equations can model dispersion relations for waves in a variety of media. We are interested in unstable waves in fluid flows. Small amplitude disturbances can be represented as a superposition of waves, which usually requires assuming that the flow is homogeneous in the streamwise direction. However, even a nominally ‘parallel’ flow will have small amplitude roughness at solid boundaries. We use amplitude equations with small random spatially inhomogeneous terms to model the effect of rough boundaries. In a parallel flow, ‘absolute instability theory’ describes whether or not wavepackets propagate away from their source. ‘Global instability theory’ is the corresponding theory for a spatially inhomogeneous flow. We use Floquet theory to study the effect of small roughness terms on global stability in amplitude equations. We show there are cases where the homogeneous problem has an O(1) global decay rate, but the inclusion of infinitesimal roughness creates a global instability with O(1) global growth rate. Therefore even small roughness can not be ignored!