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Viscous vortex crystals

Michele Dolce, Martin Donati

Abstract

We study the solution to the two-dimensional incompressible Navier-Stokes equations arising from a sum of Dirac masses in a particular co-rotating configuration. This configuration consists of a polygonal vortex crystal with or without a central vortex. By exploiting the symmetries and stability properties of the system, we describe and control the solution up to sub-diffusive time scales, prior to the expected onset of vortex merging.

Viscous vortex crystals

Abstract

We study the solution to the two-dimensional incompressible Navier-Stokes equations arising from a sum of Dirac masses in a particular co-rotating configuration. This configuration consists of a polygonal vortex crystal with or without a central vortex. By exploiting the symmetries and stability properties of the system, we describe and control the solution up to sub-diffusive time scales, prior to the expected onset of vortex merging.
Paper Structure (43 sections, 46 theorems, 403 equations, 9 figures)

This paper contains 43 sections, 46 theorems, 403 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Mathematical setting
  3. Main result
  4. Non radial viscous corrections
  5. Analogies and differences with previous related results
  6. Plan of the paper
  7. Reformulation of the problem
  8. Notations and conventions
  9. Symmetries and self-similar variables
  10. Expression of the system in self-similar variables and rotating frame
  11. Choice of position and angular speed
  12. Equations in reduced form
  13. Function spaces, operators and their asymptotic expansion
  14. Function spaces
  15. Diffusion and advection operators
  16. ...and 28 more sections

Key Result

Proposition 1.1

Let $K > 0$ be a fixed constant. Then there exist constants $\delta_0 \in(0,1)$ and $C_0>0$ such that for every $\delta \le \delta_0$, the unique solution $\omega$ of equations eq:NS with initial datum eq:omegain satisfies where $\delta,\varepsilon(t)$ and $T_{\rm adv}$ are defined in def:adpar.

Figures (9)

  • Figure 1: Two examples of polygonal vortex configuration as described at \ref{['eq:zin']}, with $N=6$ and $\gamma = 1$ (left), and $N=5$ and $\gamma =0$ (right).
  • Figure 2: Initially radially symmetric concentrated vortices (top) evolve in time (bottom), becoming more elliptical with different orientations in the case $\gamma = 10$ (left) and in the case $\gamma = -10$ (middle). In the case $\gamma= \gamma_6^* = \frac{5}{12}$ (right), the vortices do not display any clear 2-fold symmetric deformation. The 3-fold deformation happening in that case, as well as the 6-fold deformation of the central vortices in each case, are too subtle to be seen here. The bottom pictures are taken in all three cases after the whole configuration has completed a rotation of angle $2\pi/3$.
  • Figure 3: Values of the distance parameter $d$ in \ref{['def:d']} and $\alpha_4$ in \ref{['def:alha4']} computed for $N=3,\dots,10$ (left) and plot of $\alpha_4$ as a function of $N$ (right). For $N=5$, we have $\gamma_5^*=0$ and note the behavior $\alpha_4=\mathcal{O}(N^{-1})$ for large values of $N$.
  • Figure 4: The pair of initially Gaussian vortices rotates and deforms.
  • Figure 5: Three identical vortices rotate and deform.
  • ...and 4 more figures

Theorems & Definitions (101)

  • Proposition 1.1: Corollary of Gallay2011
  • Theorem 1.2
  • Remark 1.3: Bounds on the time scale
  • Remark 1.4: On the circular vortex sheet limit
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 91 more