Calculating Velocities on Airfoil Surfaces with Vortex Sheet

[aeHiFlyer210]

First post here on pf!

I wanted to ask if it's feasible to determine velocities at the top / bottom surface of an airfoil due to the vortex sheet placed on the camber line (due to thin airfoil theory).

I'm attempting to do this because the big picture is to determine the Cp distribution on the top and bottom surface of the airfoil.

Thanks!

 

[boneh3ad]

I would have to go back and look through my literature to confirm, but I believe that is possible. Only one way to find out: try it! It seems you would be better off using (creating) a panel method code with a trailing edge vortex to enforce the Kutta condition though.

 

[aeHiFlyer210]

Thanks for responding!

I've read up on the panel method code and would probably do a complete turn around on my code if not for the time constraints. I'll probably go ahead and do it later on.

If I wanted to find the induced velocity at the top and bottom surfaces due to the vortex sheet on the camber line, my thinking would be:

1. Find the coordinates which make up the top and bottom surface.
2. Break up the vortex sheet into individual vorticies (say 100 vorticies) that lie on the camber line.
3. Determine the contribution for each individual vortex.
4. Due to the fact that the direction of the velocity is perpendicular to the distance vector, I would have to decompose my velocity into x and y components. I would sum the x and y components of each contribution.
5. Find the magnitude of the total induced velocity and use this to find your coefficient of pressure.

I've been toying with this idea and haven't been able to get it to work. I'd really appreciate knowing if there was an error in my thinking.

Thanks again in advance for your help!

 

[RandomGuy88]

There is no reason that shouldn't work. Have you properly determined the strength of each of your discrete vortices?

Is the flow normal to the surface at each control point zero?
Is the Kutta condition satisfied?

Assuming you have done this correctly you can determine the pressure distribution by determining the total induced velocity at each of your control points.

 

[aeHiFlyer210]

Would you be able to clarify if my equations are correct?

I determine the strength of the vortex sheet using the thin airfoil theory equation:

γ = 2U_∞ * (A₀((1 + cos(θ)) / sin(θ)) + ƩA_nsin(nθ))

To find my induced velocity for each vortex, I use the equation:

W(z) = γ / (2πr) <---- This is the equation I'm not sure about.

 

[RandomGuy88]

Ok, I see what you are doing wrong. I believe I misunderstood what you were trying to do. You can't use the results of thin airfoil theory to determine the velocities on the surface of the airfoil away from vortex sheet.

In thin airfoil theory you generally place the vortex sheet on the x-axis and solve for the distribution of vorticity that satisfies the Kutta condtion and the boundary condition which is "No flow normal to the x-axis". This can be extended by placing the vortex sheet on the camber line and then the boundary condition becomes "No flow normal to the camber line".

The problem with what you are doing is that the thin airfoil results you are using result from the boundary condition of no flow normal to the camber line. That means that in your model, on the airfoil surface there is both flow tangential and normal to the surface. And this is not physically correct.

What you need to do is determine where you want your discrete vortices to be. This can be on the x-axis, the camber line, or what is normally done in this case the airfoil surface. So this requires you to write a panel method.

The typical panel method involves first dividing your surface up into N number of panels. On each panel you then place a vortex of unknown strength at the quarter chord of the panel. Then you place a control point at the 3/4 chord location of the panel. Then you set up your system of equations which states that the total velocity normal to each control point is zero. Then you add the Kutta condition which gives you an overdetermined system so you must remove the no flow normal condition at one control point. Now you have N unknowns vortex strengths, N-1 control points and 1 Kutta condition giving you N equations.

Finally once you know the strength of each vortex you can determine the total velocity at each control point, which is only tangential to the surface and you can use these velocities to determine your Cp distribution.

 

[aeHiFlyer210]

Well that clears up a lot of questions.

Thanks much for responding so quickly and I'll be sure to keep working on the vortex panel method.