The Sleekest Airfoil

What does a minimum force airfoil look like? I basically want to make a fairing that minimizes the total force over the range of 0-5 degrees. This mostly means reducing the lift generated at non-zero angles of attack, while keeping the drag manageable. Minimizing lift on an airfoil feels oddly perverse. I suppose that’s why I’ve never seen it before?

We need a couple more parameters: the thickness and the Reynolds number. Just for fun, use a thickness of 15%. To keep with the previous post, we’ll use a Reynolds number of 3 million.

Again, I’ll use XFOIL to estimate the forces. It shouldn’t be used for separated flow (very thick airfoils, sharp corners, etc) or transonic flow, but otherwise it’s very accurate for two-dimensional airfoils. It combines a boundary layer model with an inviscid panel method solver, allowing it to give drag as well as lift.

Given a list of the airfoil’s surface points, XFOIL will return the lift and drag coefficients for a given Reynolds number and angle of attack. How do we generate the airfoil’s geometry? Previously, I used a modified version of the NACA 4-digit airfoil equation. I’ll use that equation, as well as a new parameterization based on Bézier curves.

Since we want a rounded leading edge, the first point on the curve should be fixed at the leading edge. The second point should be directly above that, and of controllable height. This gives the desired blunt leading edge. The final point should be fixed at the trailing edge, giving a sharp trailing edge.

How to optimize this? It depends on how many parameters there are. With one parameter, a linear search is the right option. With more, I’ve found that a version of random search works quite well. So, let’s throw the optimizer at the problem!

Modified NACA 0015

How many parameters are needed? The following plot shows the average force magnitude as a function of the number of parameters of the modified NACA function:

The average forces can be reduced to 20% of the baseline values! That is a very successful fairing. The best parameters were [0.5532, 0.1164, -0.3277, -0.3994]. Have a look at it, and the resulting forces and moments:

As you see, the max thickness is much farther back: at 52% of the chord. This keeps a favorable pressure gradient along the foil, reducing the tendency to flow separation. The thinner leading edge reduces the intensity of the lift peak, and reduces the overall lift.

Bezier Airfoil

How does this compare to the spline-based optimization? With three parameters, this results in a max thickness that is even farther back (59% of the chord), reducing the forces to 12% of the baseline value! The image at the top of this post shows the new (weird) airfoil below the NACA 0015.

This one is over-specialized, of course. For a more useful airfoil, I should also include a range of Reynolds numbers in the optimization, but this is good for now. Hooray for weird airfoils!

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