The vortex tracking (1) model is our implementation and development of the vortex tracking model originally developed by Sarpkaya and Shoaff. It shares many common features with the vortex tracking (2) model, and differs in the following ways:
- It uses a variable time step, $\Delta t = 0.2\, r/v$ where $r$ is the radius of the line and $v$ the fluid velocity relative to the node. Assuming a Strouhal number of 0.2, this value is 1/50th of the (instantaneous) Strouhal period.
- Vortices shed from the same side of the disc are grouped to represent the vortex sheets coming off that side. The model tries to detect a suitable point in the cycle at which to break the attached sheet away from the disc and start a new attached sheet.
- At each time step each sheet is re-discretised in a way that keeps the vortices at equally spaced arc lengths along the sheet.
- At each time step the model searches for vortices from detached sheets that have moved in between the two attached sheets. Any such vortices are entrained into the attached sheet of opposite sign to its vorticity.
Sheet detachment and coalescing
At any given time there are usually two vortex sheets being fed from the disc – one from each side. These are called the attached sheets.
As the flow progresses, an attached sheet can become detached and a new attached sheet then starts forming on that side. Typically this happens first on one side of the disc, then on the other, etc., and this alternating behaviour tends to be synchronised with the oscillatory nature of the lift force.
In reality the vortex sheets form, become detached and flow downstream ad infinitum, with their effect decreasing as they move further away from the disc. OrcaFlex, of course, has to limit the number of vortices it keeps track of. In model 1 this is done as follows:
- The two attached sheets and the one most recently detached sheet are modelled in detail, i.e. as linked sequences of discrete vortex points.
- Older detached sheets (providing they are not still close to the disc) are simplified by 'coalescing' them to single vortex points. In other words whenever a sheet becomes detached the previous detached sheet is usually replaced by a single vortex point whose strength is the total vorticity in the sheet and whose position is the centroid of the vorticity in that sheet. The sheet is then referred to as a coalesced sheet.
- When a coalesced sheet gets beyond a certain distance from the disc, its effect is assumed to have decayed to the point where it is no longer significant and so it can be removed from the model. OrcaFlex does this by transferring its vorticity to the nearest coalesced sheet of the opposite sign. Doing the removal in this way has the advantage that the total vorticity present is preserved, and so vorticity is only being moved a small distance, rather than being destroyed.
To summarise, the wake is represented by
- detailed modelling of two attached vortex sheets and (usually) one detached vortex sheet, plus single-vortex modelling of a number of coalesced sheets – illustrated below, where the two attached (one green, one red) sheets and the most recent detached sheet (red) can be seen as vortex lines
- single-vortex point modelling of the earlier detached sheets, now coalesced – shown in the drawing below 1(and OrcaFlex 3D view) as circles
Wake line and entrainment
In practice the tail end of the detached sheet trails into the spiral part of the preceding attached sheet. This can lead to very contorted situations and to modelling problems if vortex points come very close to each other. This is handled in model 1 as follows.
We calculates the tangent line between the two attached sheets. This line, called the wake line, is the dashed line in the drawing above.
The region bounded by this wake line, the two attached sheets and the disc itself is called the wake region. Vortex points in the detached sheet that trail into the wake region can then be absorbed into the attached sheet of the opposite sign. We refer to this process as entrainment.
