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Fast relaxation of a viscous vortex in an external flow

Martin Donati, Thierry Gallay

TL;DR

This work rigorously analyzes the evolution of a concentrated vortex in a smooth external flow within the 2D Navier–Stokes framework. By introducing self-similar variables and constructing a high-order approximate solution, the authors capture both the Lamb–Oseen core and its deformation under external shear, with a detailed perturbative expansion in the diffusion length $igl( u tigr)^{1/2}$. They develop a robust energy framework in weighted spaces and prove precise relaxation results: for point-vortex initial data the solution stays close to the approximation with the vortex center following a viscous-corrected ODE, while for ill-prepared Gaussian data the solution rapidly relaxes to the well-prepared state thanks to enhanced dissipation in the vortex core. The latter relies on intricate estimates for the linearized operators around the Lamb–Oseen vortex, combined with multi-region energy controls. Overall, the paper provides a rigorous, quantitative description of vortex deformation and relaxation in external flows with high Reynolds numbers, validating the asymptotic picture and highlighting the role of enhanced dissipation in shaping vortex dynamics.

Abstract

We study the evolution of a concentrated vortex advected by a smooth, divergence-free velocity field in two space dimensions. In the idealized situation where the initial vorticity is a Dirac mass, we compute an approximation of the solution which accurately describes, in the regime of high Reynolds numbers, the motion of the vortex center and the deformation of the streamlines under the shear stress of the external flow. For ill-prepared initial data, corresponding to a sharply peaked Gaussian vortex, we prove relaxation to the previous solution on a time scale that is much shorter than the diffusive time, due to enhanced dissipation inside the vortex core.

Fast relaxation of a viscous vortex in an external flow

TL;DR

This work rigorously analyzes the evolution of a concentrated vortex in a smooth external flow within the 2D Navier–Stokes framework. By introducing self-similar variables and constructing a high-order approximate solution, the authors capture both the Lamb–Oseen core and its deformation under external shear, with a detailed perturbative expansion in the diffusion length . They develop a robust energy framework in weighted spaces and prove precise relaxation results: for point-vortex initial data the solution stays close to the approximation with the vortex center following a viscous-corrected ODE, while for ill-prepared Gaussian data the solution rapidly relaxes to the well-prepared state thanks to enhanced dissipation in the vortex core. The latter relies on intricate estimates for the linearized operators around the Lamb–Oseen vortex, combined with multi-region energy controls. Overall, the paper provides a rigorous, quantitative description of vortex deformation and relaxation in external flows with high Reynolds numbers, validating the asymptotic picture and highlighting the role of enhanced dissipation in shaping vortex dynamics.

Abstract

We study the evolution of a concentrated vortex advected by a smooth, divergence-free velocity field in two space dimensions. In the idealized situation where the initial vorticity is a Dirac mass, we compute an approximation of the solution which accurately describes, in the regime of high Reynolds numbers, the motion of the vortex center and the deformation of the streamlines under the shear stress of the external flow. For ill-prepared initial data, corresponding to a sharply peaked Gaussian vortex, we prove relaxation to the previous solution on a time scale that is much shorter than the diffusive time, due to enhanced dissipation inside the vortex core.

Paper Structure

This paper contains 27 sections, 27 theorems, 237 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Self-similar variables and approximate solution
  3. Expansion of the external velocity
  4. Functional framework
  5. Construction of the approximate solution
  6. Second order vorticity profile
  7. Third order vorticity profile
  8. Fourth order vorticity profile
  9. The solution starting from a point vortex
  10. The short time estimate
  11. Construction of the energy functional
  12. The large time estimate
  13. Control of the diffusion terms
  14. Control of the advection terms
  15. End of the proof of Proposition \ref{['prop:largetime']} and Theorem \ref{['thm1']}
  16. ...and 12 more sections

Key Result

Proposition 1.2

Fix $\Gamma > 0$ and $z_0 \in \mathbb{R}^2$. There exist positive constants $K_0,\delta_0$ such that, if $0 < \nu/\Gamma < \delta_0$, the unique solution of eq:NSf, eq:BS satisfying eq:omunique has the following property: where $d = \sqrt{\Gamma T_0}$ and $\hat{z}(t)$ is the unique solution of the differential equation

Figures (3)

  • Figure 1: Numerical simulation of a vortex in an external field with Gaussian initial data. The vorticity distribution (left) and the deviation from the Lamb-Oseen vortex (right) are represented at nine different times, using standard color codes for the vorticity levels. The final state at $t = 7$ is close to the approximate solution defined in \ref{['def:omapp']}. This simulation is made with the free software http://basilisk.fr/, and the external field is chosen as in Section \ref{['ssecA1']}.
  • Figure 2: The function $w_2$, which enters the definition of the approximate solution \ref{['def:omapp']} and describes to leading order the deviation of the vorticity distribution from the Gaussian profile, is represented as a function of the radius $r = |\xi|$.
  • Figure 3: Level lines of the perturbation $\Omega_2: \xi \mapsto w_2(|\xi|)\sin(2\theta)$ on the square $[-6,6]^2$ (right), and of the approximate solution $\Omega_0 + 0.04*\Omega_2$ on the smaller square $[-1.4,1.4]^2$ (left).

Theorems & Definitions (54)

  • Remark 1.1
  • Proposition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 44 more