11 - Vortex Methods: An Introduction and Survey of Selected Research Topics

Summary

Introduction

Vortex methods are a type of numerical method for approximating the solution of the incompressible Euler or Navier-Stokes equations. In general, vortex methods are characterized by the following three features.

1. The underlying discretization is of the vorticity field, rather than the velocity field. Usually this discretization is Lagrangian in nature and frequently it consists of a collection of particles which carry concentrations of vorticity.

2. An approximate velocity field is recovered from the discretized vorticity field via a formula analogous to the Biot-Savart law in electromagnetism.

3. The vorticity field is then evolved in time according to this velocity field.

In the past two decades a number of different numerical methods for computing the motion of an incompressible fluid have been proposed that have the above features. In this article we consider a class of such methods which are based on the work of Chorin [1973, 1978, 1980, and 1982]. Members of this class are related by the manner in which a vorticity field in an inviscid, incompressible flow is discretized and subsequently evolved. It is common practice to use the term vortex method or the vortex method to refer to a member of this class when it is used to model the incompressible Euler equations. One can modify the vortex method and use it to model the incompressible Navier-Stokes equations by adding a random walk. This is known as the random vortex method. One can also replace the random walk by some non-random technique for solving the diffusion equation. Such methods are generally referred to as deterministic vortex methods.