Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Point Vortex Dynamics on Closed Surfaces

Marcel Padilla

TL;DR

The work provides a comprehensive guide to point vortex dynamics on closed genus-zero surfaces, unifying plane, sphere, and arbitrary genus-zero geometries under a zero total vorticity condition. It develops a coherent framework based on Green's functions, stream functions, and a Hamiltonian/symplectic view, augmented by Boatto–Koiller conformal-metric theory to handle general surfaces via a conformal map to the sphere. The approach combines rigorous derivations for planar, spherical, and closed-surface cases with a detailed implementation path, leveraging discrete conformal mappings (Kazhdan et al.) and discrete exterior calculus (DEC) to enable real-time vortex simulations on meshes. The practical impact is a scalable, geometry-agnostic vortex simulation pipeline with demonstrable behaviors such as geodesic motion, leapfrogging, and Taylor-vortex analogues, useful for graphics, visualization, and mathematical experimentation.

Abstract

The theory of point vortex dynamics has existed since Kirchhoff's proposal in 1891 and is still under development with connections to many fields in mathematics. As a strong simplification of the concept of vorticity it excels in computational speed for vorticity based fluid simulations at the cost of accuracy. Recent finding by Stefanella Boatto and Jair Koiller allowed the extension of this theory on to closed surfaces. A comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity is presented here. Additionally fundamental knowledge of fluid dynamics and surfaces are explained in a way to unify the theory of point vortex dynamics of the plane, the sphere and closed surfaces together with implementation details and supplement material.

Point Vortex Dynamics on Closed Surfaces

TL;DR

The work provides a comprehensive guide to point vortex dynamics on closed genus-zero surfaces, unifying plane, sphere, and arbitrary genus-zero geometries under a zero total vorticity condition. It develops a coherent framework based on Green's functions, stream functions, and a Hamiltonian/symplectic view, augmented by Boatto–Koiller conformal-metric theory to handle general surfaces via a conformal map to the sphere. The approach combines rigorous derivations for planar, spherical, and closed-surface cases with a detailed implementation path, leveraging discrete conformal mappings (Kazhdan et al.) and discrete exterior calculus (DEC) to enable real-time vortex simulations on meshes. The practical impact is a scalable, geometry-agnostic vortex simulation pipeline with demonstrable behaviors such as geodesic motion, leapfrogging, and Taylor-vortex analogues, useful for graphics, visualization, and mathematical experimentation.

Abstract

The theory of point vortex dynamics has existed since Kirchhoff's proposal in 1891 and is still under development with connections to many fields in mathematics. As a strong simplification of the concept of vorticity it excels in computational speed for vorticity based fluid simulations at the cost of accuracy. Recent finding by Stefanella Boatto and Jair Koiller allowed the extension of this theory on to closed surfaces. A comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity is presented here. Additionally fundamental knowledge of fluid dynamics and surfaces are explained in a way to unify the theory of point vortex dynamics of the plane, the sphere and closed surfaces together with implementation details and supplement material.
Paper Structure (49 sections, 31 theorems, 134 equations, 56 figures, 1 algorithm)

This paper contains 49 sections, 31 theorems, 134 equations, 56 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. History of Fluid Dynamics
  3. Modern Fluid Dynamics
  4. History of Vortex Dynamics
  5. Supplement Material Software
  6. Fluid Dynamics and the Vorticity Equation
  7. The Navier-Stokes Equations
  8. The Euler Equations
  9. Notes on Common Fluid Methods
  10. Vorticity of Fluids
  11. The Vorticity Equation
  12. Surface Fluids
  13. Surface Fluids
  14. Green's Functions on Surfaces
  15. Stream Functions
  16. ...and 34 more sections

Key Result

corollary 1

A vector field $\vec{u}$ of a fluid flow satisfying the incompressibility condition $\frac{D}{D t}\rho=\frac{\partial }{\partial t}\rho + \vec{u}\cdot\nabla\rho = 0$ for it's density $\rho$ consequently has to fulfill the divergence free condition: If so, $\vec{u}$ is called divergence free.

Figures (56)

  • Figure 1: The great wave of Kanagawa by Katsushika Hokusai. Around 1829 in Japan. Public domain.
  • Figure 2: Me trying to understand fluids by looking at the Red Sea.
  • Figure 3: A simplification to illustrate the complexity of the demands made by fluid dynamics applications.
  • Figure 4: The effects of a smoke grenade next to a wall in the game Rainbow Six Siege seen from two perspectives. The smoke clearly penetrates through the surface.
  • Figure 5: Images created using the supplement material.
  • ...and 51 more figures

Theorems & Definitions (56)

  • definition 1: Incompressible Navier-Stokes equations
  • definition 2: Incompressible flow
  • corollary 1: Divergence free condition
  • definition 3: Incompressible Euler Equations
  • definition 4: Cross Product
  • definition 5: Curl
  • lemma 1
  • definition 6: 2D Curl
  • lemma 2
  • definition 7: Closed Surface in $\mathbb{R}^3$
  • ...and 46 more