CL-alpha curve nonlinear effects

[Gustavo Luiz Halila]

Hi,

While running some CL-alpha curve cases with OpenVSP (v.3.25 and also v.3.23) I noticed that, beyond a certain angle of attack, some nonlinear effects pop up. Are those related to some numerical (side) effect or is there any viscosity correction under the hood causing this?

Thank you,

Gustavo

[Rob McDonald]

Depending on your case, it is likely a real inviscid phenomena.

For example, a wing with a swept, non-planar trailing edge will show substantial curvature of the CL vs. alpha and CM curves.

Rob

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[Gustavo Luiz Halila]

Hi Rob,

Thank you for getting back. I ran a simple test case - a rectangular, planar trailing edge, zero-sweep wing at subsonic conditions.

My runs considered the VLM approach only. I was expecting a straight CL-alpha curve for this case. Any comment on this?

Best,

Gustavo

[Rob McDonald]

How high of alpha?

Even thin airfoil theory says the lift is cl=2*pi*sin(alpha).  We apply a small-angle approximation to get a linear cl-alpha curve.

Rob

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[Gustavo Luiz Halila]

Hey Rob,

Thanks for following up.

I see some nonlinear effects for angles of attack beyond 15 degrees (see attached file).

The wing is based on the NACA 2412 airfoil and the flow is subsonic (M=0.1, Re=1E06).

Best,

Gustavo

[Cibin Joseph]

Gustavo,

This might be because of the expression used for lift computation being slightly different from the popular form of the Kutta-Joukowski formula (rho V gamma).

If you look up textbooks like Low Speed Aerodynamics by Katz, J. and Plotkin, A., you'll notice that in panel methods the panel pressure difference is first computed and then the panel lift vector. 

For a 2D flat plate/panel of chord dx, inclined at alpha with freestream velocity V, the panel pressure difference is typically computed using the local tangential velocity (V cos(alpha)), giving,

panel pressure difference, dP = rho V cos(alpha) Gamma / dx

The panel normal force then turns out to be

panel normal force, dN = dP dx

and panel lift, dL = dN cos(alpha) = rho V Gamma cos^2(alpha)

This approach results in an additional cos^2(alpha) term along with the linear part. It may be intuitive in some sense to think about the limiting case when alpha=90 deg and the linear theory still produces 'unnatural' lift. On the other hand, having the cos^2(alpha) term will cause the lift to fall to zero. It is a consequence of computing lift from the normal force (from unsteady Bernoulli theorem to be precise). 

If you divide your 'nonlinear' curve by cos^2(alpha) and you get back a straight line, it's almost definitely due to this reason. The attached figure shows the discrepancy for a 2d flat plate. You will see differences depending on your wing's aspect ratio and local angle of attack.

Disclaimer: I'm not sure if VSPAero does exactly this, since the force calculation was recently switched over to computing using Kutta Joukowski. Maybe there's something related that causes the cos^2 term to come up. However, I've seen the cos^2(alpha) term show up in a few other inviscid solvers too.

Cibin

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[Gustavo Luiz Halila]

Hi Cibin,

Thank you for sharing these thoughts. The idea makes sense. I divided the CL values by cos**2(alpha) and got a straight line.

Best,

Gustavo