The discrete vortex method

The motion of M discrete vortices in two space dimensions is given by the system of complex ordinary differential equations,

$$\frac{dz_\mu(t)}{dt} = i \sum^M_{\nu=1 \atop \nu \ne \mu} \frac{Q_\nu} {\overline{z_\mu(t)-z_\nu(t)}}.$$ (1)

The complex number $z_\mu(t)$ represents the location of the discrete vortex $\mu$ at time t. The value $Q_\nu$ represents the rotation of the vortex $\nu$. Note that the denominator is a complex conjugate. If time is discretized by a simple forward difference, we get the numerical method,

$$z^{n+1}_\mu = z^n_\mu + \Delta t \cdot \left( i \sum_{\nu=1} \frac{Q_\nu} {\overline{z^n_\mu-z^n_\nu}} \right).$$ (2)

In this forward Euler formula $z^n_\mu$ approximates $z_\mu(t)$ at $t=t^n=n\Delta t$.

The discrete vortex method simulates the motions of discrete vortices but it can also be seen as a method for simulating more general incompressible flows. In some of the examples below we line up a large number of discrete vortices after each other and use this configuration to simulate the motion of a vortex sheet. The discrete vortices can also be spread out to represent a continuous vorticity distribution in the simulation of incompressible and inviscid flow.

In [3] it was proved that a point vortex method approximation of flow with continuous vorticity converges as the number of numerical vortices grows to infinity.