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My understanding of the reason why a plane fly is summarized by the following figure: because the upper part of the wing is curved, the air above the wing has to flow faster than the wing below. This induces a force that lift the wing.

enter image description here

My question is:

  1. How is the exact shape of the wing determined?

  2. What is the "optimal" curve for a wing? By "optimal", I mean the best lift.

  3. How would the answer (shape of the wing) differ if we were looking for optimization of other parameters like stability.

PS: I am definitely not a physicist. If the question is too naive, please help to improve it.

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    $\begingroup$ What induces a lifting force on the wing is that air behind the wing moves downwards. A flat plate at the right angle of attack flies just fine, it just has an atrociously bad lift to drag ratio which makes it kind of useless past the RC model scale. There is no "optimal" wing shape. What a "good" wing looks like depends on the performance requirements. A fighter jet will have different wings than a passenger jet, which will differ from a space shuttle, which will differ from a glider. $\endgroup$ – CuriousOne Jun 1 '16 at 9:20
  • $\begingroup$ Thanks. That was my intuition that the notion of "best" wing was the naive part of the question. Could you expand on the typical performance requirements, so I can factor it into the question. I guess lift to draft ratio, stability of the wing,... what else? $\endgroup$ – Taladris Jun 1 '16 at 9:53
  • $\begingroup$ Also, if you have any good introductory reference about the problems of wings, I would be glad to hear about it. My background in physics is really low, but I am quite strong in maths. $\endgroup$ – Taladris Jun 1 '16 at 9:56
  • $\begingroup$ The exact profile of the wing, as well as the size and shape, obviously vary from aircraft to aircraft, depending on their intended purpose, as CuriousOne says. Here is Wikipedia on the various types: en.m.wikipedia.org/wiki/Wing_configuration $\endgroup$ – user108787 Jun 1 '16 at 10:02
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    $\begingroup$ It is a common mistake that the top shape of the wing has to be more curved than the bottom shape. After all, aeroplanes can fly upside down. $\endgroup$ – Feyre Jun 1 '16 at 10:15
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A modern airfoil is designed on the basis of the desired pressure distribution over the chord length of both sides. In some cases, only a single angle of attack is relevant while in others the airfoil must be a compromise of the pressure distributions over a range of angles and flap deflections.

Two parameters can be used to tailor the desired pressure distributions:

  • Relative thickness: Any body needs to displace air when it travels through it. This displacement causes supervelocities on both sides and extra drag (called wave drag) at supersonic speed. For better structural efficiency, a thicker airfoil is better, while a supersonic wing needs to be as thin as practical. Typical cases range from 22% at the root of WWII bombers over 13% for a modern airliner root all the way down to the 4% of a supersonic delta wing.

  • Local camber: Positive local camber spreads the pressures on the lower and the upper side of an airfoil apart, such that is shows higher pressure on the lower and lower pressure on the upper side. This pressure difference is the source of lift.

Now it might seem that we ideally create a lot of suction on the upper surface while maximizing the pressure on the bottom surface, and that is indeed what an airliner wing with flaps extended for landing tries to do, but a few limitations exist.

  • Pressure recovery: Once the flow leaves the airfoil at the trailing edge, its pressure should have returned to its ambient value. A big suction area on the rear-facing part of the airfoil will cause drag, so the area of low pressure should extend only over the forward and mid part of the chord length. The pressure gradient cannot be too steep in order to avoid flow separation. Separation makes further pressure recovery impossible and is very drag-intensive. The maximum pressure gradient at the desired maximum operating angle of attack therefore limits the minimum pressure over the forward part of the airfoil.

  • Maximum local Mach number: Lower pressure means higher speed (the total energy of the flow is constant, and pressure is equivalent to potential energy), and since the flow behaves differently once it crosses into the supersonic realm, the minimum pressure can also be limited by the maximum local Mach number where recompression is still possible without causing a drag-intensive shock.

  • Positive wing thickness: Sometimes reaching the theoretical optimum is prevented by the fact that the upper side of the airfoil must be above the lower side contour.

Please follow the links if you want to know more - here I could only scratch the surface.

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There are lots of answers on this site regarding "how is lift produced" etc. but this post asks about the determination of a typical wing profile, so it might be appropriate to follow through on that aspect.

  1. Determine the chord (length from wing leading edge to the trailing edge).

  2. Decide on your camber line, for a flat bottom surface wing, this is the shape of the curved upper surface. As in the comments above this parameter is determined by the use to which the aircraft is to be put and may involve camber on the lower wing surface. In general, the faster the speed of your aircraft, the less camber thickness you need.

The first picture shows a Sopwith Camel c. 1915 with thick cambered wings and the bottom image is of an almost camberless winged F105 60 years later.

Sopwith Camel c. 1915 with thick camber

F105 Wing Profile with almost no camber

From here on physics is replaced by engineering, and a good source for basic principles is NASA Aircraft Design, from which my few comments are taken.

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    $\begingroup$ No offense, but that's the state of airfoil design around approx. 1930 (or not even?) and even then it would already have been engineering. I think we should better send this one off to another SE where they know how it's being done today, rather than 85 years ago... $\endgroup$ – CuriousOne Jun 1 '16 at 11:09
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    $\begingroup$ I'll have you know good men died testing these sections, Lilienthal and others later, but yeah I take your point :). I looked for a side on image of a Sopwith Camel versus an F105, I will update it. Thanks $\endgroup$ – user108787 Jun 1 '16 at 11:15
  • $\begingroup$ I just looked it up... so the Goettingen airfoils seem to go back to Prandtl in 1917! Isn't that amazing how fast they went from guessing to making it a strict engineering discipline? Maybe war is the mother of all things, after all... That a lot of courageous people died so we can fly safely today is absolutely true. The pioneers in every field leave a mark... and a hundred years later a staff of a hundred bored out of their minds aerodynamics designers at Boeing has to run hundreds of thousands of CFD simulations for every new plane they are designing... $\endgroup$ – CuriousOne Jun 1 '16 at 11:19
  • $\begingroup$ Thanks for sharing... your Dad could have replied that it was all safe as long as none of the 1.5 million 5/32 inch rivets broke... $\endgroup$ – CuriousOne Jun 1 '16 at 11:47

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