[Paper Review] Desingularization of vortex rings in 3 dimensional Euler flows: with swirl

Daomin Cao, Jie Wan+1 more|arXiv (Cornell University)|Sep 1, 2019

Navier-Stokes equation solutions27 references4 citations

TL;DR

This paper constructs a two-parameter family of steady vortex rings with swirl in 3D axisymmetric Euler flows via a variational method, achieving desingularization of the classical circular vortex filament. The key result shows that as the regularization parameter β → 0⁺, the vorticity support shrinks to a circle of radius r* = 1/(4πW), with energy and Lagrange multiplier scaling logarithmically in 1/β, and the limiting vorticity distribution is radially nonincreasing in the weak L² topology.

ABSTRACT

We study desingularization of steady vortex rings in three-dimensional axisymmetric incompressible Euler fluids with swirl. Using the variational method, we construct a two-parameter family of steady vortex rings, which constitute a desingularization of the classical circular vortex filament, in several kinds of domains. The precise localization of the asymptotic singular vortex filament is shown to depend on the circulation and the velocity at far fields of the vortex ring and the geometry of the domains. We also discuss other qualitative and asymptotic properties of these vortices.

Motivation & Objective

• To construct desingularized steady vortex ring solutions for 3D axisymmetric incompressible Euler flows with swirl.
• To overcome compactness issues in exterior domains by developing a new variational approach distinct from Turkington's method.
• To precisely characterize the asymptotic location of the singular vortex filament in terms of circulation, far-field velocity, and domain geometry.
• To establish the weak limit of the vorticity family as a radially nonincreasing function in L².

Proposed method

• Formulates the problem using the modified azimuthal vorticity ζ = ω^θ / r and derives a nonlinear elliptic equation (Bragg-Hawthorne/Long-Squire equation) for the stream function ψ.
• Introduces a regularized energy functional Eβ(ζ) involving kinetic energy, swirl-induced potential energy, and a penalization term for vorticity concentration.
• Employs a constrained variational framework over symmetric, compactly supported functions in L∞(D) with fixed total circulation.
• Uses Steiner symmetrization to ensure radial symmetry of the minimizer and proves existence of a maximizer ζβ via compactness and lower semicontinuity.
• Applies Lagrange multiplier theory to derive the Euler-Lagrange equation linking ζβ to the stream function ψβ and the multiplier μβ.
• Analyzes asymptotic behavior as β → 0⁺ by deriving logarithmic scaling of energy and multiplier in terms of W and r* = 1/(4πW).

Experimental results

Research questions

• RQ1 How can a two-parameter family of desingularized vortex rings with swirl be constructed in 3D Euler flows with non-zero circulation and far-field velocity?
• RQ2 What determines the asymptotic location of the singular vortex filament in the limit β → 0⁺?
• RQ3 How does the energy and Lagrange multiplier scale with the regularization parameter β in the desingularization process?
• RQ4 What is the weak limit of the vorticity family {gβ} as β → 0⁺, and what symmetry properties does it possess?

Key findings

• As β → 0⁺, the energy Eβ(ζβ) scales as (r*/(4π) - W r*²/2) log(1/β) + O(1), where r* = 1/(4πW).
• The Lagrange multiplier μβ scales as (r*/(2π) - W r*²/2) log(1/β) + O(1), with r* = 1/(4πW).
• The diameter of the vorticity support satisfies diam(supp(ζβ)) ≤ R₁β for some R₁ > 1 and all sufficiently small β.
• The asymptotic distance from the support of ζβ to the circle Cr* tends to zero as β → 0⁺.
• Every weak limit point of {gβ} in L² is a radially nonincreasing function.
• The vorticity profile ζβ is symmetric under Steiner symmetrization and converges to a singular limit concentrated on a circle of radius r*.