Pseudospectral analysis of parallel shear flows using wavepackets

The linearized Navier-Stokes equations for shear flows are typically highly nonnormal, motivating the use of pseudospectral analysis to study linear energy amplification mechanisms. This analysis can be performed via the singular value decomposition of the associated resolvent operator, for specified temporal frequencies and spatial wavenumbers. Here, we investigate an alternative approach to perform this analysis, relying on an assumption that spatial structures associated with large linear energy amplification can be efficiently represented as a sum of wavepackets of a specified form. Thus, this methodology can be recast as an optimization problem for a small number of parameters, and can be assisted by first identifying simplified operators that retain the pseudospectral properties of the original system. We demonstrate that the method can accurately predict mode shape and amplification levels for a variety of mode types, including cases where the modes are attached to one or two boundaries, and/or influenced by multiple critical layers. We additionally show that this methodology can also be successfully applied to the analogous study of suboptimal modes. The method produces closed-form analytic expressions for spatial mode shapes, which can be convenient for ensuing analysis.