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Is there an equation relating AoA to lift coefficient?

I have been searching for a while: there are plenty of discussions about the relation between AoA and Lift, but few of them give an equation relating them.

I know that for small AoA, the relation is linear, but is there an equation that can model the relation accurately for large AoA as well? (so that we can see at what AoA stall occurs)

I am not looking for a very complicated equation. Can anyone just give me a simple model that is easy to understand?

(Of course, if it has to be complicated, then please give me a complicated equation)

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  • $\begingroup$ Are you asking about a 2D airfoil or a full 3D wing? $\endgroup$ – MikeY May 19 at 1:42
  • $\begingroup$ @MikeY 3D is good, but 2D is OK $\endgroup$ – Holding Arthur May 19 at 3:56
  • $\begingroup$ For a 3D wing, you can tailor the chord distribution, sweep, dihedral, twist, wing airfoil selection, and other parameters to get any number of different behaviors of lift versus angle of attack. So your question is just too general. $\endgroup$ – MikeY May 19 at 13:30
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No, there's no simple equation for the relationship.

Here's an example lift coefficient graph:

A lift coefficient graph for the NACA 0015 airfoil

(Image taken from http://www.aerospaceweb.org/question/airfoils/q0150b.shtml.)

This is actually three graphs overlaid on top of each other, for three different Reynolds numbers. I'll describe the graph for a Reynolds number of 360,000.

We see that the coefficient is 0 for an angle of attack of 0, then increases to about 1.05 at about 13 degrees (the stall angle of attack). From here, it quickly decreases to about 0.62 at about 16 degrees. Then it decreases slowly to 0.6 at 20 degrees, then increases slowly to 1.04 at 45 degrees, then all the way down to -0.97 at 140, then...

Well, in short, the behavior is pretty complex. The most accurate and easy-to-understand model is the graph itself.

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  • $\begingroup$ "there's no simple equation". Could you give me a complicated equation to model it? $\endgroup$ – Holding Arthur May 19 at 4:53
  • $\begingroup$ You could take the graph and do an interpolating fit to use in your code. $\endgroup$ – MikeY May 19 at 13:28
  • $\begingroup$ @Holding Arthur, the relationship of AOA and Coefficient of Lift is generally linear up to stall. So for an air craft wing you are using the range of 0 to about 13 degrees (the stall angle of attack) for normal flight. There is an interesting second maxima at 45 degrees, but here drag is off the charts. This is why coefficient of lift and drag graphs are frequently published together, $\endgroup$ – Robert DiGiovanni May 19 at 14:33
  • $\begingroup$ @HoldingArthur Perhaps. What are you planning to use the equation for? I don't want to give you an equation that turns out to be useless for what you're planning to use it for. $\endgroup$ – Terran Swett May 20 at 0:55
  • $\begingroup$ Great graph and source (there's also a graph for drag), but the conclusion seems wrong. Based on those graphs, lift is clearly proportional to sin(2α) in the post-stall region. $\endgroup$ – Rainer P. May 21 at 20:27
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In the post-stall regime, airflow around the wing can be modelled as an elastic collision with the wing's lower surface, like a tennis ball striking a flat plate at an angle. Lift and drag are thus:

$$c_L = sin(2\alpha)$$ $$c_D = 1-cos(2\alpha)$$

I superimposed those (blue line) with measured data for a symmetric NACA-0015 airfoil and it matches fairly well. I don't know how well it works for cambered airfoils.

cl-vs-angle cd-vs-angle

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The lift coefficient is linear under the potential flow assumptions. So just a linear equation can be used where potential flow is reasonable.

Potential flow solvers like XFoil can be used to calculate it for a given 2D section. Or for 3D wings, lifting-line, vortex-lattice or vortex panel methods can be used (e.g. using XFLR5).

When the potential flow assumptions are not valid, more capable solvers are required.

XFoil has a very good boundary layer solver, which you can use to fit your "simple" model to (e.g. a spline approximation). And I believe XFLR5 has a non-linear lifting line solver based on XFoil results.

For 3D wings, you'll need to figure out which methods apply to your flow conditions. Possible candidates are: experimental data, non-linear lifting line, vortex panel methods with boundary layer solver, steady/unsteady RANS solvers, ...

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