On the role of vortex stretching in energy optimal growth of three dimensional perturbations on plane parallel shear flows

TL;DR

This paper investigates how vortex stretching influences the energy growth of three-dimensional perturbations in shear flows, revealing that optimal growth can be understood through two-dimensional shear plane dynamics.

Contribution

It introduces a minimal model explaining the interaction between vorticity and divergence fields, clarifying the role of vortex stretching in energy amplification.

Findings

Optimal perturbations resemble localized plane-waves in high Reynolds number flows.

The phase relationship between vorticity and divergence depends on background shear sign.

A minimal model links 3D growth to 2D shear plane dynamics.

Abstract

The three dimensional optimal energy growth mechanism, in plane parallel shear flows, is reexamined in terms of the role of vortex stretching and the interplay between the span-wise vorticity and the planar divergent components. For high Reynolds numbers the structure of the optimal perturbations in Couette, Poiseuille, and mixing layer shear profiles is robust and resembles localized plane-waves in regions where the background shear is large. The waves are tilted with the shear when the span-wise vorticity and the planar divergence fields are in (out of) phase when the background shear is positive (negative). A minimal model is derived to explain how this configuration enables simultaneous growth of the two fields, and how this mutual amplification reflects on the optimal energy growth. This perspective provides an understanding of the three dimensional growth solely from the two dimensional dynamics on the shear plane.