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The long way of a viscous vortex dipole

Michele Dolce, Thierry Gallay

TL;DR

The paper analyzes the viscous evolution of a two-dimensional vortex dipole formed by counter-rotating point vortices. It constructs a two-parameter asymptotic expansion in the dipole’s aspect ratio and inverse Reynolds number, and develops a rigorous framework based on self-similar variables and Arnold’s variational principle to control the Navier–Stokes solution over a long time scale. The authors obtain a precise leading correction to the dipole’s translation speed and prove long-time proximity between the exact solution and the approximate viscous dipole, with detailed error bounds. The work provides a robust methodology for understanding finite-size effects in vortex interactions and lays the groundwork for extending to more complex vortex configurations.

Abstract

We consider the evolution of a viscous vortex dipole in $R^2$ originating from a pair of point vortices with opposite circulations. At high Reynolds number $Re >> 1$, the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier-Stokes equations remains close to the approximation on a time interval of length $O(Re^σ)$, where $σ< 1$ is arbitrary. This improves upon previous results which were essentially restricted to $σ= 0$. As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.

The long way of a viscous vortex dipole

TL;DR

The paper analyzes the viscous evolution of a two-dimensional vortex dipole formed by counter-rotating point vortices. It constructs a two-parameter asymptotic expansion in the dipole’s aspect ratio and inverse Reynolds number, and develops a rigorous framework based on self-similar variables and Arnold’s variational principle to control the Navier–Stokes solution over a long time scale. The authors obtain a precise leading correction to the dipole’s translation speed and prove long-time proximity between the exact solution and the approximate viscous dipole, with detailed error bounds. The work provides a robust methodology for understanding finite-size effects in vortex interactions and lays the groundwork for extending to more complex vortex configurations.

Abstract

We consider the evolution of a viscous vortex dipole in originating from a pair of point vortices with opposite circulations. At high Reynolds number , the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier-Stokes equations remains close to the approximation on a time interval of length , where is arbitrary. This improves upon previous results which were essentially restricted to . As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.
Paper Structure (25 sections, 30 theorems, 304 equations, 2 figures)

This paper contains 25 sections, 30 theorems, 304 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The rescaled vorticity equation
  3. Self-similar variables
  4. Formula for the vertical speed
  5. The approximate solution
  6. Function spaces and operators
  7. The second order approximation
  8. The induction step
  9. Leading order correction to the vertical speed
  10. Functional relationship in the inviscid case
  11. Correction to the approximate solution
  12. Main difficulties and Arnold's variational principle
  13. The weighted space and its properties
  14. Bound on the vertical speed
  15. The energy functional
  16. ...and 10 more sections

Key Result

Theorem 1.1

Gallay2011 Assume that the point vortex system eq:PV with circulations $\Gamma_i$ and initial positions $x_i$ is well posed on the time interval $[0,T]$. There exists a constant $C_0 > 0$ such that the unique solution $\omega^\nu$ of the vorticity equation eq:NS with initial data def:initvort satisf where $z_1(t)\,\dots,z_N(t)$ is the solution of eq:PV such that $z_i(0) = x_i$ for $i = 1,\dots,N$.

Figures (2)

  • Figure 1: The level lines of the function $\Phi_\mathrm{app}^E$ defined in \ref{['def:PhiappE']}, which correspond to the stream lines of the inviscid approximate solution $\Omega_\mathrm{app}^E$ in the co-moving frame, are represented for $M = 2$ and $\varepsilon = 1/50$ (left) or $\varepsilon=1/8$ (right). Large positive values of $\Phi_\mathrm{app}^E$ are depicted in red, and large negative values in blue. The flow has two elliptic stagnation points located at $\xi = 0$ and $\xi = (-1/\varepsilon,0)$ in our coordinates, as well as two hyperbolic points on the black line which separates the vortex dipole from the exterior flow. Near the vortex centers, the stream lines are nearly elliptical with a major axis in the $\xi_2$-direction, which reflects the fact that $\mathsf{w}_2 > 0$ in \ref{['Om2exp']}.
  • Figure 2: A schematic representation of the graph of the weight function $W_\varepsilon$ defined in \ref{['def:Weps']}. In the inner region $\mathrm{I}_\varepsilon$, the weight is close for $\varepsilon > 0$ small to the radially symmetric function $W_0(\xi) = 4|\xi|^{-2} \bigl({\rm e}^{|\xi|^2/4}-1\bigr)$. It then takes constant values in the intermediate region $\mathrm{II}_\varepsilon$, and grows like $\exp(|\xi|^{2\gamma}/4)$ in the outer region $\mathrm{III}_\varepsilon$. The dashed lines illustrate the bounds \ref{['bd:Weps']}, where the constants $C_1, C_2$ are independent of $\varepsilon$.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 58 more