Kinetic Energy Represented in Terms of Moments of Vorticity and Applications

Autor: Marcin Preisner, Krzysztof Oleszkiewicz, Marta Szumańska, Tomasz Cieślak
Rok vydání: 2019
Předmět:
Zdroj: Journal of Mathematical Fluid Mechanics. 21
ISSN: 1422-6952
1422-6928
DOI: 10.1007/s00021-019-0456-z
Popis: 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 flows with certain moments of their vorticities. It allows us to identify a class of vortex sheets of unbounded support being only $$\sigma $$σ-finite measures (in particular including measures $$\omega $$ω such that $$\omega (\mathbb {R}^2)=\infty $$ω(R2)=∞), 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.
Databáze: OpenAIRE
Externí odkaz: