Vortex tracking (2) model

The vortex tracking (2) model is a simplified variant of the vortex tracking (1) model. It shares many common features with model 1, and differs in the following ways:

• it uses a constant time step, equal to the simulation outer time step
• it does not use sheet detachment to separate the vortices coming from one side of the disc into separate sheets
• it does not use entrainment, nor re-discretisation to keep the vortices at equally spaced arc lengths along the sheet
• it uses a coalescing algorithm to control the number of vortices

Coalescing

To keep down the number of vortices being tracked, model 2 tries, at each time step, to coalesce pairs of vortices that have come very close to each other.

At each time step the model finds, for each vortex, the nearest neighbouring vortex. If two vortices are each other's nearest neighbours then they are called mutually nearest neighbours, and such a pair are considered for possible coalescing into one.

A pair of mutually nearest neighbours are coalesced if their separation is less than $s\urm{c}$, where $s\urm{c}$ is a coalescing separation that depends on the distance $d$ from the two vortices to the disc surface. $s\urm{c}$ is given by \begin{equation*} s\urm{c} = \begin{cases} \lambda r \quad&\text{if }d\leq 2r \\ \lambda r \left(\dfrac{d}{2r}\right)^2 \quad&\text{if }d\gt 2r \end{cases} \end{equation*}

where $r$ is the disc radius and $\lambda$ one of the user-defined coalescing thresholds. Which threshold is used depends on whether the two vortices are of the same sign or opposite signs.

These formulae for $s\urm{c}$ mean that the data define, in disc radius units, the coalescing separation for vortices that are within 2 disc radii of the disc surface; for vortices further away, the coalescing separation increases according to the square of the distance from the disc surface. The aim of this is to restrict coalescing near to the disc but encourage it once the vortices have convected significantly away from the disc.

If the two vortices are coalesced then they are replaced by one vortex whose strength is the sum of their signed strengths and which is placed at their centroid of absolute vorticity.