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Virtually all airplane wings are convex on both the top and at least part of the bottom. But thus convexity on the bottom creates suction which reduces lift. I can only think of two reasons to have it: negative AoA performance (important only for aerobatics), and the need to accommodate a spar. Suppose these two were possible to neglect. Would a concave bottom give a better L/D?

So, is the L/D a cruising aircraft best when bottom is concave?

When you look at applications where the structural and AoA concerns are absent, it does seem that the answer is yes.

enter image description here

The blackbird is small, hence there are less structural concerns. It has a concave bottom.

Turbofan blades are also relatively small. Also, I think the forces are dominated by centrifugal force, not bending force. Once again, the bottom.is concave.

The same is true of the turbine blade.

Curiously, the propeller, which also has similar conditions, has a convex bottom.

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    $\begingroup$ There is no ideal airfoil for all situations - but optimized for the specific szenario it is designed to operate in - and in most cases, there will be quite a lot of compromises... $\endgroup$
    – tsg
    Mar 4 at 7:49
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    $\begingroup$ Get the book The Miracle of Flight by Stephen Dalton. It's written for laypeople. He goes though aerodynamics and lift generation, and mechanics, starting with insects. then birds, and moving up to aircraft. At very low Reynolds Numbers, heavily cambered single surface wings work best. As RN goes up, wings get thicker until the thickness becomes a drag problem at high speed, then they get thinner again. The ideal supersonic airfoil is a flat thin uncambered sheet. Efficiency of undercamber depends on the operating angle of attack. $\endgroup$
    – John K
    Mar 4 at 19:10

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So, is the L/D [of] a cruising aircraft best when bottom is concave?

If we use gliders as a reference, the bottom of the best airfoils is only very slightly concave, almost straight, in order to allow for a long laminar run while still achieving high lift. Up to the Nineties of the last century, the best designs used rear loading over the last 15% - 20% while the most recent ones do away with even that to squeeze a few percent more of laminarity from the lower side. This has to do with flying technique as well as better understanding of laminar flow: Instead of optimizing the design for two polar points (thermalling and fast flight), now the ideal is to also look at intermediate speeds like when flying under lined-up clouds.

While birds have wings optimized for low speed because they can retract and fold their wings when going fast, airplanes have to design their wings for high speed and add flaps and slats for low speed flight. When you look at an airfoil of an airliner in landing configuration, it is not so different from a bird's wing.

Typical landing configuration of an airliner wing

Typical landing configuration of an airliner wing, from an article by A. M. O. Smith, McDonnell-Douglas, in Journal of Aircraft, Vol 12 No 6, 1975.

Also, birds do not have to deal with trans- and supersonic effects which dictate the airfoil shape of almost all jet-powered airplanes.

Curiously, the propeller, which also has similar conditions, has a convex bottom.

Propeller and especially rotor airfoils should have a low pitching moment change over angle of attack to avoid twisting the propeller rsp. rotor blade. This is achieved by reduced camber, and the result is a convex bottom surface: A symmetrical airfoil has no pitching moment variation within the linear range of angles of attack. In order to be able to work over a range of angles of attack, they must have a certain thickness, which of course also benefits the structure.

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  • $\begingroup$ Is the parasitic drag less for the cambered "low speed" airfoil? $\endgroup$ Mar 5 at 2:56
  • $\begingroup$ @AbdullahisnotanAmalekite Depends on angle of attack. Generally, thinner airfoils have less parasitic drag over a small angle of attack range while thicker airfoils have a bit more parasitic drag over a wider angle of attack range. $\endgroup$ Mar 5 at 8:45
  • $\begingroup$ What about the lowest achievable values? $\endgroup$ Mar 5 at 9:57
  • $\begingroup$ @AbdullahisnotanAmalekite Flat plate at zero angle of attack, obviously. Equals twice the friction drag at the respective Reynolds number. Another interesting concept uses a slotted flap for pressure recovery to enable laminar flow over the first 80% of both sides. See here for more. $\endgroup$ Mar 5 at 18:05
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    $\begingroup$ @AbdullahisnotanAmalekite That would be a thin, birdlike airfoil like a cambered plate similar to the Göttingen 417. $\endgroup$ Mar 6 at 8:35
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Back to the roots. An airfoil can be basically described by 3 parameters:

  • the shape/curvature of its camber line;
  • the radius of its leading edge (the trailing edge is almost always pointy);
  • the distribution of the thickness upon the camber line.

Those 3 parameters are enough to define any airfoil of practical use (source Wikipedia):

 Airfoil nomenclature

Without any claim to thoroughness (that would cover an entire book) the following can be said:

  • a rounder leading edge and higher thickness give a smooth stall with a progressive loss of lift; pitching moment also changes smoothly approaching the stall region; thick airfoils have also a big volume that can be used for fuel and structure $\rightarrow$ good for the root of a jetliner's wing or of a propeller.
  • a thinner leading edge and thickness give a more nervous stall with a higher pitching moment variation; anyway at high subsonic speeds a thinner airfoil possesses a lower drag than a thick one since it retards the formation of shock waves (which give rise to the boundary layer's detachment and stall) $\rightarrow$ due to structural and aerodynamical reasons, lift on a jetliner's wing and on a propeller should diminish toward the tip: twisting the wing/propeller so that the AoA at its tip is lower than that at the root allows the use of these thinner airfoils with a benefit in terms of drag reduction.
  • a pointy leading edge is necessary at supersonic speeds in order to avoid the formation of a detached (aka bow) shock which increases drag and temperature $\rightarrow$ virtually all the wings of supersonic fighters are built around a very thin, pointy airfoil; the loss in internal volume for fuel and structure is compensated for by the use of delta planform for the wing.
  • camber is mainly used to control/delay stall and limit pitching moment; a very pronounced camber (one that makes the airfoil look like an arc) is used when lift needs to be generated at very high AoA i.e. low speeds; this is typically the shape that a bird's wing possesses or that a jetliner's wing assumes when its high-lift devices are deployed; in those cases a round, thick leading edge helps too, as already said; anyway a lot of camber doesn't work at high speeds since the boundary layer tends to detach and make the airfoil stall; the only exception where high camber and high speed work together is within an axial compressor/turbine: here the adjacent blades "channel" the airflow and make it stay attached also at the high speeds seen by the blades. As a side effect, an airfoil with high camber generates also a high pitching moment which needs a lot of structure to be carried: also for this reason the turbine blade is quite tick; to partially offset this, the camber can be built in such a way to go upward toward the trailing edge: so the first, say, ¾ of the chord is arched like the turbofan blade in your picture while the last ¼ is arched the other way around, in a sort of "S" shaped pattern; in this way the good aerodynamic characteristics at high AoA are retained but without the increased torsional moment $\rightarrow$ this is normally seen in helicopter blades (CH-47, source) enter image description here

To get to your original question:

Is the ideal airfoil concave on the underside?

A highly cambered airfoil (such that also the belly is arched) works only when lift is needed at high AoA and low speeds and the high torsional (aka pitching) moment doesn't play an important role. For airfoils in cascade (like in an axial compressor/turbine) the limitation on the speed can be dropped due to the channeling effect. For all the other cases a more subtle camber or even an S-shaped camber is needed.

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is the L/D for a cruising aircraft best when the bottom is concave?

It depends on the Reynolds number. Full scale aircraft have a high enough value that they can create most of their required lift with the top of the wing. Even though the bottom of the wing is convex, it still contributes to lift due to its positive angle of attack.

In cruise, AoA will be small, and the convex wing bottom best serves by being a very low drag shape and adding to wing structural strength.

"Top lift" provides the best L/D up to the airspeed where Mach effects increase drag of the upper wing.

The effects of Reynolds number on L/D can be seen at airfoil tools website.

Cambering the underside does provide a higher coefficient of lift, but at the expense of higher drag. Where efficiency is not important, such as landing, adding camber (with flaps) can be very useful in lowering landing speed.

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