Incompressible potential flows in 2D are governed by two Laplace equations for velocities.

I discretize these equations by FEM on triangular grids, and solve the resulting linear system by Gauss-Seidel iteration, for a flow around a Karman-Trefftz airfoil with the exact solution on the outer boundary, but a tangency condition on the airfoil surface.

The tangency condition: I simply ignore the normal component of the velocity at the boundary nodes produced by Gasss-Seidel iterations. (i.e. update the velocity at a boundary node, subtract the normal component, go update to the next boundary node.)

QUESTIONS:

(1)Are there any other ways to impose tangency on the airfoil?

(2)How fine the grid should be to obtain a good solution (say, 1% pointwise error ON the airfoil), especially for a lifting problem?

According to my results, FEM is very inefficient for lifting problems as it requies extremely fine grids (more than 1000 nodes on the boundaries) to get such good solutions while such solutions are more cheaply obtained by panel methods. Do you think that there is a bug in my code?

Thank you for reading my message,

Hiro

Hi,

You can apply the tangency condition simply by changing velocity unknows (u,v) by (u_n,u_s) on the airfoil, where u_n is the normal velocity and u_s the tangential component. I have not the references in mind but you can see my paper on

http://www.math.u-psud.fr/~moussaou/Papers/unify.ps.gz

Farid++