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Boundary conditions for vortex particle passing through a square domain

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Old   August 18, 2017, 08:47
Default Boundary conditions for vortex particle passing through a square domain
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Case

I am trying to simulate a single vortex particle which convects along with the freestream velocity through a free-surface square domain. Initially the particle is outside of the domain, so I simulate the particle approaching, entering, travelling through, exiting and distancing from the domain.

Approach

For this case the analytical solution for the velocity and pressure are known in closed-form. So I can compute the velocity and pressure at any given point in space and instance in time. I would like to use the analytical solution to impose boundary conditions.

Results and problem

Previously, I had good results for a case where the particle passes the domain from the outside (so it never enters or exits the domain). In this case I used the analytical solution to impose a non-uniform Dirichlet b.c. for the velocity on all four sides. This worked well, but for the case where the particle passes through the domain I get wiggles along the side at which the particle exits the domain.

Question

What type of boundary condition(s) could I use for this case?

Notes

Analytical solution:

The velocity equals the freestream velocity plus the induction by the particle. The pressure is calculated via Bernoulli's principle.

u = uInf + Gamma/(2*pi*r^2)*(1 - exp(-0.5*(r/e)^2))*ry
v = vInf - Gamma/(2*pi*r^2)*(1 - exp(-0.5*(r/e)^2))*rx
p = pInf + 0.5*[(uInf*uInf + vInf*vInf) - (u*u + v*v)]

uInf = freestream velocity in x direction
vInf = freestream velocity in y direction
pInf = freestream pressure
Gamma = particle strength
e = particle core radius
r = Euclidean distance between particle and point of evaluation
rx = x component of r
ry = y component of r

ps: The analytical solution is NaN at r = 0. I have ensured that r never becomes 0 by positioning the particle such that it never coincides with a cell center or face center.
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boundary conditions, non-uniform bc, potential flow, unsteady b.c.

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