Manual

Here everything will be listed, the routines for computing

Vortex filament structure on GPU

In the standard implementations of the Vortex Filament Method (VFM), filaments are constructed by a Fortran 'type' array. Unfortunately, though structure types are native to Julia, structures defined and adapted using the Adapt package are required to be immutable. The implementation of the VFM requires structures to mutate according to the number of vortex points, the value will fluctuate to maintain discretisation between filaments.

Boundary Conditions

As of v1.0.2, only periodic boundary conditions and open boundary conditions are supported.

#Initialises all open boundary conditions 
boundary_x = OpenBoundary(1) #x direction
boundary_y = OpenBoundary(2) #y direction 
boundary_z = OpenBoundary(3) #z direction

#Initialises all periodic boundary conditions 
boundary_x = PeriodicBoundary(1) #x direction
boundary_y = PeriodicBoundary(2) #y direction
boundary_z = PeriodicBoundary(3) #z direction

Periodic and open boundary conditions can be mixed. For example, if we define a helical or straight line vortex, we require periodic boundary conditions in the $z$ direction, however we may also set open boundary conditions in the $x$ and $y$ directions.

#Set a straight line vortex - this condition  requires  periodic boundary in z
IC = SingleLine() 

#Set open conditions in x,y and periodic in z 
boundary_x = OpenBoundary(1)#x direction
boundary_y = OpenBoundary(2)#y direction
boundary_z = PeriodicBoundary(3)#z direction 

Vortex Configurations

Here is a list of all of the vortex configurations that can be used for the vortex filament method. Specific parameters relating to the initialisation of the vortex line are stored in the corresponding structure blocks.

List of implemented initial conditions

• Single helix
• Single vortex ring
• Straight line

Derivatives

The Vortex Filament requires the computation of spatial derivatives. The following methods compute these derivatives using finite difference adaptive techniques.

First order derivative using a second order adaptive-mesh finite difference scheme:

\[\frac{d\mathbf{s}_i}{d\xi} = \frac{\ell_{i-1}\mathbf{s}_{i+1} + (\ell_{i+1} - \ell_{i-1})\mathbf{s}_i + \ell_{i+1}\mathbf{s}_{i-1}}{2\ell_{i+1}\ell_{i-1}} + {\cal O}(\ell^2) \]

Second order derivative using a second order adaptive-mesh finite difference scheme:

\[\frac{d^2 \mathbf{s}_i}{d \xi^2}=\frac{2\mathbf{s}_{i+1}}{\ell_{i+1}(\ell_{i+1}+\ell_{i- 1})}-\frac{2\mathbf{s}_i}{\ell_{i+1}\ell_{i-1}}+\frac{2\mathbf{s}_{i-1}}{\ell_{i-1}(\ell_{i+ 1}+\ell_{i-1})}+{\cal O}(\ell^2)\]