1. Dirichlet formulation with constant potential Ana Laverón Simavilla Mª Victoria Lapuerta González

2. Dirichlet Formulation  The integral equation must be solved making the point P tend toward the body surface Doublet potential:  d Source distribution Doublet distribution Doublet potential:  d

3. Dirichlet Formulation Making the inner potential zero: Doublet distribution 

4. Constant potential method (2D) The profile’s contour is modeled with N panels, that will be straight segments. Panels are determined by N+1 nodes Collocation points are the midpoints of each panel: Node of trailing edge node panel j node j node j+1 Path followed around the profile

5. Constant potential method (2D) The following conditions are imposed: Constant potential on each panel The previous equation is rewritten for the collocation points k  N -  1

6. Constant potential method (2D) To solve the integrals on each panel it’s suitable to choose axes bound to the panel ( u,v ) v u C.P. k j j+1 panel j k j j+1

7. Constant potential method (2D) For the panel, k=j For the layer surface, a semi-infinite panel is used Finally:

8. Constant potential method (2D) Matrix form: with and

9. Constant potential method (2D)

10. Flow around a Karman-Trefftz profile  x y R x0x0 y0y0 0 0 a  kaka We will compare the numerical and the analytic results for a Kármán-Trefftz profile.

11. 1.For n panels calculate the pressure coefficient on the profile’s lower and upper,. For this, calculate the speed on the nodes using the expression: where d i is the distance between the collocation points i and i+1. 2.Calculate the global circulation around the airfoil: 3.Compare the results with those obtained exactly using the Kármán-Trefftz transformation. Required calculations

12. Comments for the resolution Function that gives the nodes’ coordinates: function [ ξ,η ]= function_profile ( n, t 0,k, R ) Function that gives the analytic value of : function [ ξ p,lower, C p, lower, ξ p,upper, C p, upper, η p,lower, η p, upper,  ] = function_karman( t 0, k, , n _kam, R, )

13. low Results Profile for t=-0.3+i0.2, k=1.5, R=1, n=100,  num  6.1939 upp