Boundary conditions at infinity for the velocity potential

michujo · July 11, 2012, 08:36

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.

michujo · July 11, 2012, 08:57

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.

truffaldino · July 11, 2012, 09:05

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

Truffaldino

July 11, 2012, 08:36	Boundary conditions at infinity for the velocity potential	#1
michujo Senior Member Join Date: Dec 2011 Location: Madrid, Spain Posts: 134 Rep Power: 15	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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July 11, 2012, 08:57		#2
michujo Senior Member Join Date: Dec 2011 Location: Madrid, Spain Posts: 134 Rep Power: 15	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.

July 11, 2012, 09:05		#3
truffaldino Senior Member Join Date: Jan 2011 Posts: 249 Blog Entries: 5 Rep Power: 17	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 Truffaldino