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Leapfrogging vortices simulation

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Published on Oct 17, 2009

Four leapfrogging vortices in two-dimensions. Breadth ratio a=0.3. 100,000 passive particles to visualise flow field around vortices, coloured by velocity magnitude (spectrum palette, red high purple low). Solution method: standard Runge-Kutta 4th order with adaptive step sizing. Truncation error estimated by step halving. Spatial tolerance per time step was 1E-8.

Model - two dimensional point vortex singularities (incompressible potential flow)

dx(i)/dt = - Sum(j=0..N, j/=i) (k(j) * rij,y / rij^2 )
dy(i)/dt = Sum(j=0..N, j/=i) (k(j) * rij,x / rij^2 )

where:
rij,x = (x(i) - x(j)) ; rij,y = (y(i) - y(j)) ; rij^2 = rij,x^2 + rij,y^2

Initial conditions: all initial velocities zero
r(0) = (0,1) ; r(1) = (0,-1) ; r(2) = (0,a) ; r(3) = (0,-a)
k(i)=(-1)^i

Coded in Visual Basic .NET (yes, stuff like this can be done in basic, despite its name :-D)

Enjoy!

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