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Seminar

Euler/Navier Stokes (Part 1): Vortex layers of small thickness May 06, 2021 (08:00 AM PDT - 09:00 AM PDT)
Parent Program:
Location: MSRI: Online/Virtual
Speaker(s) Marco Sammartino (Università di Palermo)
Description

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

 

 

Video

Vortex Layers of Small Thickness

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

We consider a 2D vorticity configuration consisting of highly concentrated vorticity around a curve and exponentially decaying away from it: the intensity is O(1/ε) close to the curve while it decays at an O(ε) distance from the curve itself. 

Besides their intrinsic interest, vortex layers are relevant because they are regularizations of vortex sheets. 

We prove that, if the initial datum is analytic, Euler solutions preserve the vortex-layer structure for a time that does not depend on ε. 

Moreover, for a short time, Birkhoff-Rott equation well approximates the motion of the center of the layer. 

We shall also show the results of numerical simulations for Navier-Stokes equations at high Reynolds number. Investigating the typical roll-up process, we shall see that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterize the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics.

We reveal complex singularities in the solutions of Navier-Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff-Rott singularity suggests that vortex layers, in the limit Re → ∞, behave differently from vortex sheets.

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Vortex Layers of Small Thickness