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Boundary conditions at infinity for the velocity potential

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Old   July 11, 2012, 09:36
Default Boundary conditions at infinity for the velocity potential
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Hi guys, I have been struggling lately with a potential flow code around a 2D airfoil and I found one problem.

I compute the potential flow around the airfoil on a body-fitted grid that I generated solving a Poisson equation (elliptic grid). Later on some circulation is explicitly added around the airfoil in order to simulate the effect of the boundary layer (the circulation \Gamma will be such that the Kutta condition is enforced, i.e., the flow separates at the trailing edge).
The outer domain is a circle with a radius much longer than the chord of the airfoil.

The question is: to which value do I set the velocity potential \Phi at the outer domain?

Thank you for your help.

Cheers,
Michujo.
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Old   July 11, 2012, 09:57
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Ok, I could figure out that the contribution of the velocity far from the airfoil is \Phi_{\infty}=\vec{V_{\infty}\cdot\vec{x}}, where the infinity symbol denotes at the outer boundary, far from the airfoil.

This is the potential due to the free stream. However, as circulation is being generated around the airfoil, what is the effect of this circulation on the velocity potential?

Thanks again.
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Old   July 11, 2012, 10:05
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Far from the airfoil you can add velocity vector of magnitude = Circulation/(2 Pi r) with direction tangetial to the redius-vector r to the free stream velocity vector

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