1. Panel method with doublets of constant intensity. Neumann resolution Mª Victoria Lapuerta González Ana Laverón Simavilla

2. Neumann Formulation The equation for the velocity potential, obtained using the Dirichlet formulation for constant potential, is: angle reference discontinuity panel

3. Neumann Formulation The velocity potential due to an uniform distribution of doublets of Ф j intensity over the panel is equivalent to the potential due to a couple of vortices placed in the edges of the panel with Ф j and – Ф j intensities.

4. Neumann Formulation The velocity potential due to a uniform distribution of doublets with Ф j intensity over the panel is equivalent to the potential due to a couple of vortices located at the extremes of the panel with Ф j and – Ф j intensities.  After deriving and applying boundary conditions: we obtain: with which can be rewritten as: with

5. Neumann formulation 1.Model: We replace the outline of the airfoil with n panels, assuming constant distributions of doublets. This is equivalent to locating vortices at the edges of each panel. We make the strength of the trailing edge vortex zero (so there are N-1 unknown variables). 2.Boundary condition: We impose the condition that the normal component of velocity is zero on the collocation points of N-1 panels. The collocation points are located in the middle point of each panel.

6. Airflow around a Karman-Trefft profile  x y R x0x0 y0y0 0 0 a  kaka Compare the numerical results with the analytical ones for a Kármán-Trefftz profile:

7. 1.For n panels, calculate the in both the upper and lower surfaces (intrados and extrados). To do that, the velocity at the nodes is calculated as: where d k is the distance between the collocation points k and k+1. The circulation is calculated with: 2.Compare the results with the ones obtained exactly with Kármán-Trefftz transformation. Required Calculations

8. Notes On Solving The Problem Function that gives the nodes : function [ ξ,η ]= function_profile ( n, t 0,k, R ) Function that gives the analytical : 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, )

9. Results Airfoil with t=-0.3+i0.2, k=1.5, R=1, n=100,  num  6.2717

10. Dirichlet-Neumann comparison Airfoil with t=-0.3+i0.2, k=1.5, R=1, Neumann Dirichlet

11. Differences between the formulations DirichletNeumann Bodies with non-zero thickness   Slender airfoils   Precision for N panels  