# What is the drag optimized wing form for given lift and bending moment?

I am trying to build a passive hydrofoil board.

Something like this:

I want to use wood and i am coming to the conclusion that i want to optimize drag for a given lift and bending moment.

Doing my research i found out that an elliptical wing isnt the best approach. But a linear tapered wing should work better. Is there a rough formula for this optimization?

I already came across two articles: "Lift Distributions for Minimum Induced Drag with Generalized Bending Moment Constraints" and "THE SPANWISE DISTRIBUTION OF LIFT FOR MINIMUM INDUCED DRAG OF WINGS HAVING A GIVEN LIFT AND A GIVEN BENDING MOMENT"

I only need a rough formula/shape

You have already found the most relevant articles. In 1933 Ludwig Prandtl published "Über Tragflügel kleinsten induzierten Widerstandes" in the Zeitschrift für Flugtechnik und Motorluftschiffahrt. 17 years later R.T. Jones came to the same conclusions in NACA-TN-2249. The 2013 article "Lift Distributions for Minimum Induced Drag with Generalized Bending Moment Constraints" by David J. Pate and Brian J. German goes one step further and optimises drag at a lower and structural mass at a higher lift coefficient.

But that is not all that relevant to your case. You want a wing which supports your weight plus the small surfboard which sticks out of the water. At those small loads, structural mass is almost negligible. Scaling laws mean that structure becomes more important as aircraft grow in size and mass. You know that elephants have much more massive legs relative to their body size than antelopes (or even ants, for an even more drastic comparison), since body mass scales with the cube of linear dimension while structural strength scales only with the square of linear dimension. This means that wing spar weight will be proportionally higher for larger aircraft.

As a consequence, insects have more elliptic wings than albatrosses, and model aircraft have optimum wings which are much more elliptic than the optimum wing of an airliner. The optimum shifts from an elliptic load distribution at very small scales to an almost triangular distribution at large scales.

What is more relevant is the angle of attack range in which this wing is meant to operate. The pumping motion seems to lead to quite large angle of attack variations, so you cannot optimise that wing for just one flow condition. Instead, it must be robust enough to produce tolerably low drag over the whole operating range.

Drag has two components: Pressure drag and friction drag. Pressure drag has one part which stems from lift creation, which in turn depends on the mass of fluid affected by the wing. That part drops with speed and wing span, so you want to be fast with super wide wings in order to minimise that kind of drag. However, friction drag increases with speed, so there is an optimum which varies with wing planform.

But there is another issue. Wide wings are very sensitive to angle of attack variations while more narrow, stubby wings will tolerate a wider angle of attack range. Also, the increased chord length of more stubby wings produces less friction drag per area than the narrow but wide wing.

Next to consider is the relative thickness of the wing's cross section (called an airfoil) and its camber. A thicker airfoil will create more static lift (buoyancy) but will create less dynamic lift. Make the airfoil too thin and its useable angle of attack range will become very small. I would select a thickness between 12% and 15% of chord length. Camber should reflect the operating lift coefficient: For highly loaded wings, use more camber. The tail surface, however, should be uncambered.

Another important factor is the incidence of wing and tail. The tail should have less for positive static stability, but you need to include the wing's camber into that calculation.

All those questions can only be answered when we know the flow speed, the mass on the board and the size of both surfaces. But you have to start somewhere. For that, it might be best to start with a proven design, collect the operating parameters and then go back to the drawing board with a sound basis for optimization.

For a first shot, start with a trapezoidal wing planform in which the tips have 70% of the chord of the root. Aspect ratio should be 5 or 6. You may add highly swept tips and 3D-shaping as you desire, but don't overdo it.

• Thank you so much. I already do have a first outline. I am trying to optimise it for 3m/s. The wood i am using now can handle the bending moment xflr calculated for my foil. I feel like i lost the overview though. To reach my goal i have to optimise poweroutput. i want to be able to start from a deck. exactly like in this video.
– x3oo
Aug 27 '19 at 12:37