New relations for energy flow in terms of vorticity

TL;DR

This paper introduces a novel way to analyze energy flow in fluid dynamics by partitioning kinetic energy into pairwise vortex interactions, revealing reciprocity relations and detailed energy transfer mechanisms.

Contribution

It presents a new formulation of energy flow in vortex interactions using vorticity, extending reciprocity relations to three-dimensional flows and streamlines.

Findings

Reciprocity relation for vortex interaction energy in 2D and 3D flows.

Interaction energy flow depends only on specific vortex triples.

Provides a detailed accounting of energy transfer beyond standard conservation laws.

Abstract

Considering the vorticity formulation of the Euler equations, we partition the kinetic energy into its contribution from each pair of interacting vortices. We call this contribution the "interaction energy". We show that each contribution satisfies a reciprocity relation on triples of vortices: $\boldsymbol{A}$'s action on $\boldsymbol{B}$ changes the interaction energy between $\boldsymbol{B}$ and $\boldsymbol{C}$ in an equal and opposite way to the effect of $\boldsymbol{C}$'s action on $\boldsymbol{B}$ on the interaction energy between $\boldsymbol{A}$ and $\boldsymbol{B}$. This result is a curiously detailed accounting of energy flow, as contrasted to standard pointwise conservation laws in fluid dynamics. This result holds for all triples of points $\boldsymbol{A},\boldsymbol{B},\boldsymbol{C}$ in two dimensions; and in 3 dimensions for all points $\boldsymbol{A},\boldsymbol{C}$, and all closed vorticity streamlines $\boldsymbol{B}$. We show this result in 3 dimensions as a consequence of an interaction energy flow around $\boldsymbol{B}$ that is a function only of the triple $(\boldsymbol{A},\boldsymbol{b}\in \boldsymbol{B},\boldsymbol{C})$, a result which may be of independent interest.