# (Lifting of aircraft wings) Source of centripetal force on a curved streamline?

I am trying to understand how wings lifting work in airplanes. One explanation is found here

Basically a fluid portion on curved streamline will experience a centripetal force. This centripetal force comes from the pressure difference ( as all other forces are ignored by the author). I would like to understand what is the nature of the centripetal force the author is talking about( in case of the curved streamline).

My question is, fundamentally what causes this pressure difference? In case of a stone attached to a string, the centripetal force is provided by the tension (electromagnetic in nature) in the string, in case of planets moving around the sun, this centripetal force is Gravitational force. Similarly in case of curved streams (I agree that there is a pressure difference) what exact forces are involved?

Note: The corresponding youtube video does not show the slides thanks to glitches in camera. So it is to be watched along with separate slides

The author explains everything in the article here

I have seen some answers on wings lifting, but those are either very complex or they have not explained in terms of centripetal forces in this particular case.

The centripetal force is caused by the local wing surface having a greater inclination than the local streamline. In order to follow the wing contour, the streamline needs to curve around it. This inclination can either be caused by camber or by angle of attack. It is important, however, that it changes gradually in order to avoid flow separation. That is the reason why wing airfoils have a rounded forward edge.

• Only when there is atmospheric pressure, air speed and angle of attack can centripetal force be generated. – enbin zheng Sep 30 '19 at 21:51
• @enbinzheng: Right. But you need some pressure to have a gas, so pressure and motion are preconditions for fluid dynamics. – Peter Kämpf Dec 27 '19 at 7:23
• @PeterKämpf: note sure.. the radial pressure gradient would exist in any type of fluid that's under pressure (in a curved flow), on the other hand you could maybe design a system that's under negligible pressure .. assume a blob of water floating in space: a hydrofoil in there would probably split the water blob and not produce very curved streamlines – Carl Berger Dec 27 '19 at 11:08
• @CarlBerger: That thought experiment is worthy of its own question. I would expect some force from the collision between the hydrofoil and the water molecules, but would that be lift? And during the collision we would have some non-negligible pressure gradient in that blob of water from inertial forces, right? So only as long as there is pressure, there could be lift. – Peter Kämpf Dec 28 '19 at 7:26

You ask about the “source of the force”. But lift, at least in the stationary case, is more subtle than that. The shape of the wing induces the flow pattern, which affects the pressure distribution, which affects the flow pattern, and eventually it all reaches a consistent stable “stationary” flow.

If at first the pressure is high here and low there, the flow changes to correct that.

For example, initially horizontal flow over a curve or angle of attack would leave a vacuum below. That pressure difference bends the flow down, but moves air into the low pressure region raising it a little. This continues until equilibrium is reached.

(Its a bit like how stable currents arise in a DC circuit: if it’s too much here and too little there, electrical forces arise that try to push things back)

When a parcel of air, along it's streamline in a flow, curves due to the presence of an obstacle, it must develop an acceleration toward the center of the curvature, because $$F=ma$$. It's that simple. Not because of gravity. There is pressure around the the parcel. A change in flow speed requires a pressure difference, and same thing about a change in flow direction. The only thing that can change a velocity vector is a difference of pressure. Air has mass. So if a change of direction occurs, there must be a force to balance it. Which, in this case is an acceleration toward the center of curvature. Lift is a mechanical phenomena, it requires the deviation of a flow to occur. When a flow turns to follow a curved surface, the pressure field adjusts to provide the force needed to accelerate the parcel toward the center of curvature. the force exerted is normal as in every fluid. Deviation, changes in flow speed and pressure cannot exist without one another, and they support and cause each other in a mutual, circular and reciprocal way. How does a small ball generate centripetal force when it moves on a curved surface? The reason is gravity. When the small ball has a velocity along the red arrow, the small ball tends to leave the curved surface, so the force of the small ball on the curved surface is reduced, thus the centripetal force of the small ball moving along the curve is obtained.

We change the small balls on the surface into air. When the air does not move, assume that the force of air on the curved surface is F. When the air moves in the direction of the red arrow, the force of air on the curved surface is f. Because the air moves along the curved surface, it tends to leave the surface, so F > f. So the air has a centripetal force, which makes the air move along the surface. The force exerted by air on the curved surface is air pressure. The decrease in air pressure represents a decrease in the force exerted by air on the curved surface.

The curved surface here is similar to the wing.

I think the lecturer that you link to did himself a disservice by mentioning the concept of a centripetal force.

That's unpractical in the following sense: while it is true that whenever there is a centripetal force the resulting motion is curvilinear, there is logically no reason to reverse the causal relation and posit: Here we notice a curvilinear motion, let's attribute that to a centripetal force.

I recommend that you drop the idea of attributing the air flow to some centripetal force.

The particular cross section of a wing is good for efficiency, but simpler shapes can produce lift too.

In most cases optimizing for fuel efficiency is the way to go.
There is one class of aircraft where the wings are not optimized for fuel efficiency, but for versatility: aircrafts that are built for aerobatics.

For aerobatics performance you want wings that can readily produce lift both when they are right side up and when they are upside down.

Aerobatics aircraft wings are very flat; if the top of the wings would be curved like normal wings then flying upside down would be more difficult. The wings are almost like flat boards. Those wings will certainly leave a lot of turbulance in their wake, which of course is wasted energy. Normal aircraft wings are designed to minimize energy waste, that is the reason for the shape of the cross section

This demonstrates that the primary mechanism of wings producing lift is the angle of attack. Every wing, be it a normal wing or an aerobatics wing must have an angle of attack in order to produce lift at all.

Of course, understanding angle of attack is very straightforward. For instance, when you are in car traveling at highway speed stick your hand out the window. You feel the lift.

The whole issue of wing lift should always be discussed in terms of the following two separate issues:
- What is it that produces lift: angle of attack
- How do you optimize lift/drag ratio: that requires thorough understanding of aerodynamics

Let me elaborate on the implications of angle of attack. Using the example of a hand sticking out of car window: you hold your hand at an angle such that you try to optimize lift/drag ratio. At that angle of attack two things are happening: compression of the air moving underneath, rarefaction of the air moving over the hand. The compression and the rarefaction are counterparts. While the compression is very palpable the rarefaction is something you will not readily notice. However, the rarefaction is as important, if not more important. As long as the air flow over the hand doesn't become detached it will be deflected downwards, just as the compressed air moving underneath is deflected downwards. If the angle of attack is too large the air flow over the wing does become detached, it becomes turbulent, resulting in catastrophic loss of lift.

Later edit:
A webpage with a description of the design of the Mudry CAP 232 aerobatics aircraft

The Cap wing [... ] uses a symmetrical airfoil cross section [...]. The CAP wing has the same curvature on the top and the bottom surfaces. [...] this CAP symmetrical wing works equally well whether the angle of attack is [...] upright or inverted. And the stall speed is the same both ways. This makes axial rolling more precise.

• Compression and rarefaction at fully subsonic speeds? You are explaining subsonic lift with supersonic physics! – Peter Kämpf Sep 15 '19 at 23:06
• Angle of attack is not necessary to produce lift. Wings with camber have lift without it. – D. Halsey Sep 15 '19 at 23:44
• @Cleonis, Thanks for your detailed answer. Are there other examples, apart from wing lifting case, in nature where we have a curvilinear motion but centripetal forces are absent? – gpuguy Sep 16 '19 at 2:49
• @gpuguy Let me make a somewhat whimsical comparison: explanations and taxi rides have the following in common; they should get you to your destination in the most efficient way. If a taxi ride takes a detour something's wrong. In the case of the air flow around an aircraft wing: there are pressure gradients at play, so if you want to express things in terms of force the natural choice is pressure gradient force: how much of it everywhere along the wing. Reexpressing that in terms of a centripetal force is not inherently wrong, but in this particular case it is a form of taking a detour. – Cleonis Sep 16 '19 at 21:45
• @D.Halsey: well.. any wing has a zero-lift angle of attack... for a cambered one the value is just a few degrees less. If you had an inverse camber, the zero-lift angle would indeed be positive. Plus (ok, being a bit pedantic) - an angle of zero degrees doesn't mean "no angle". – Carl Berger Dec 27 '19 at 11:13