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3D Panel Method: Implementing Kutta Condition

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Old   May 8, 2017, 22:14
Default 3D Panel Method: Implementing Kutta Condition
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I'm attempting to develop a 3D, unsteady, inviscid panel method using constant-strength source and doublet panels. My problem: I have validated the method for the steady case with an quasi-infinite length wake panel, but, when I attempt to allow the model to shed vorticity into the wake (even in steady flow, no body kinematics), I get a nonzero pressure difference across the panels at the upper and lower surface of the wing's trailing edge (pressure distribution shown in attachment). For context, I'm using Plotkin and Katz's [U]Low Speed Aerodynamics{/U] as my guide, although this paper might be a more at-hand guide for the theory:

I currently enforce the Kutta condition by ensuring that vorticity at the trailing edge is equal to zero (described in more detail on the fourth page, numbered as pg 379, of the posted link). However, as noted, this just doesn't seem to hold water for the unsteady case, even if I let the wake convect for over 6 chords downstream of the wing (for reference, this is done by computing the induced velocities at all wake panel corners + the freestream velocity and then using a 3rd order explicit Runge-Kutta scheme to march forward in time). The solution converges to this mess that I previously mentioned. I compute the pressure coefficient via the unsteady Bernoulli equation, or Cp_local = 1 - v_local^2/v_inf^2 + (2*4*pi/v_inf)*(dphi/dt), where the doublet strength is equivalently the local velocity potential.

At the same time, I'm still observing wake rollup and deformation as I think I should, at least for the limited model (2 spanwise panels, 19 chordwise panels on a NACA0012) that I'm currently using as a test case. So that's good. But I have no idea what's causing the Kutta condition to bug out on me.

Any help or suggestions would be greatly appreciated...I'm running out of ideas of how to debug this very quickly.
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Old   May 12, 2017, 17:41
james nathman
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Imagine an airfoil slowly increasing it's angle of attack. The circulation of the airfoil increases with alpha. To do this, the airfoil sheds vorticity into the wake.
In fact, the unsteady Kutta condition should be used to determine the shedding rate, not to impose zero vorticity. The increase in circulation of the wake panel attached to the trailing edge is then vorticity_rate*delta_time.

For thin airfoils at low angle of attack, you can assume the wake convects downstream at freestream velocity.

(The discussion about the imaginary flow inside the airfoil in both of your references is nonsense. See Luigi Morino's papers from the 1970s for rational explanations)
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Old   May 12, 2017, 19:06
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Hi blackjack -- ironically, I just figured out my issue prior to reading your message. I'm sticking with the Dirichlet BC formulation (as developed by Plotkin and Katz, Hess and Smith, and others), but, as you pointed out, the use of the Kutta condition I mentioned is nonsense for the unsteady case. Following an example from David Willis (, I devised a nonlinear, explicit equal-pressure condition for the trailing that is quite promising as it has already delivered some excellent results. Thanks, though!
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inviscid flow, panel method, panel methods, potential flow

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