## Linear-Strength Vortex Unsteady Panel Method (2D)

Hi everyone,

I am working on a project for the development of a 2D Unsteady Panel Method for Airfoils. As often already suggested in this forum I have been using the book "Low Speed Aerodynamics" which has helped me a lot to produce the Steady State solver (I am writing the code in Matlab at the moment), which works very well.

However I am having great problems in validating my unsteady code.
I have followed the book, mainly the steps are:

1) compute the steady solution at first time step
2) move the airfoil via a prescribed motion path (in my case pitching A*sin(wt))
3) position a concentrated vortex at a certain distance from the T.E., which has unknown intensity GAMMA_W and which is therefore part of the solution
4) use the Kelvin theorem for completing the coefficients matrix A which will be N+2 times N+2 (N+1 gammas on the nodes of the panels) + 1 (latter) unknown shed concentrated vortex
5) compute the RHS (right-hand-side) term considering free stream velocity and wake induced velocity
6) solve the system

The problem seems that the Cp variations are not big enough. I am using a NACA report on wind tunnel experimental data (pitching motion 5 + 5sin(Wt) motion) to check the code. But also the comparison with results contained in the Katz and Plotkin (plot of CL-ALPHA of a 0012) isn't satisfactory.

I compute the Cp at each collocation point via the formula:

Cp_i = 1 - Vel_i^2

where Vel_i is the velocity at the i-th collocation point obtained from the different contributes: free stream (similar to steady state) + kinematic movement (sinusoidal motion) + induced velocities (by panels and by wake).
I consider the free stream V to be equal to 1.Honestly I am not quite sure that all of this is right (talking about Cp calculation).

Moreover I have great doubt about the Kutta condition: in the steady state case the last line of matrix A is set equal to: [1 0 0 0 . . . 0 1] so that vorticity at T.E. is 0:

gamma_1 + gamma_(N+1) = 0

I did not change this in the unsteady code only because I couldn't find any indication on how to modify it (the book says that for small oscillations the steady case condition should still work fine).

I would be really thankful if any of you could help me out, as it is more than a month that I am trying to solve the problem. I have also read many articles like Hancock and Mook's one, or similar, but they don't really say explicitly how to do things.

Thank you very much.

PS: do ask questions if you don't understand some of the procedures I have used, I will be happy to explain more thoroughly.
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 It looks to me like the problem is how you are actually calculating the Cp. You are using the result of the steady Bernoulli equation, but your flow is obviously unsteady. You should have a term that contains the time derivative of the velocity potential in you Cp calculation. Take a look through the book Low Speed Aero and you will find it.
 Well the formula would be: Cp = 1 - (V/U)^2 - (2/U^2)*dPHI/dt but I do not know exactly how to calculate my potential PHI from my Gammas. Do you have any idea? I have tried in a few ways but the results are not correct.

 Tags aerodynamics, airfoil, panel method, vortex