Kinetic energy represented in terms of moments of vorticity and applications

TL;DR

This paper develops a framework linking kinetic energy to vorticity moments in 2D vortex sheets with unbounded support, including Kaden spirals, revealing energy dissipation and distribution properties.

Contribution

It introduces a novel formula connecting kinetic energy with vorticity moments for unbounded vortex sheets, including measures with infinite total vorticity.

Findings

Kinetic energy of vortex sheets can be expressed via vorticity moments.

Kaden approximations exhibit energy dissipation near the origin.

Energy is pushed outwards in Kaden spirals, indicating artificial viscosity.

Abstract

We study 2d vortex sheets with unbounded support. First we show a version of the Biot- Savart law related to a class of objects including such vortex sheets. Next, we give a formula associating the kinetic energy of a very general class of ows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only ?-finite measures (in particluar including measures \omega such that \omega(R2) = \infty), but with locally finite kinetic energy. One of such examples are celebrated Kaden approximations. We study them in details. In particular our estimates allow us to show that the kinetic energy of Kaden approximations in the neighbourhood of an origin is dissipated, actually we show that the energy is pushed out of any ball centered in the origin of the Kaden spiral. The latter result can be interpreted as an artificial viscosity in the center of a spiral.