What is the actual air speed over and under a wing due to Bernoulli's Principle?

Question

Bernoulli's Principle states that as a fluid speed increases, its pressure decreases, and vice versa. Air flowing over the wing of an aircraft flows faster than its neighboring air flowing under the wing.

But how much faster? Using the cruise conditions listed for a Boeing 747-400 – airspeed Mach 0.85 (567 mph, 493 knots, 912 km/h) flying at 35,000 ft – what are the corresponding maximum and minimum speeds of the airflow over the wing?

For the purposes of the calculation, assume standard atmospheric conditions for an EAS of 273 knots.

In general:

1. How much does this depend on the shape of the airfoil (or: does this translate to general aviation as well)?
2. Does changing the angle of attack (without stalling) affect the airspeed/pressure differential? If so, by how much?
3. Typically, where on the wing (inboard to outboard) is the most Bernoulli-derived lift generated (i.e. due to airspeed/pressure differential)? Or does this vary greatly with wing design?
4. Where on the airfoil (leading edge to trailing edge) is the region of highest and lowest pressures (and thus, the slowest- and fastest-moving air)?

Answer 1

If you know the pressure coefficient of the flow, the rest is easy. The equation for the pressure coefficient $c_p$ is: $$c_p = \frac{p - p_{\infty}}{q_{\infty}} = 1 - \left(\frac{v}{v_{\infty}}\right)^2$$ $q_{\infty}$ is the dynamic pressure and contains air density $\rho$ and the flight speed $v_{\infty}$: $$q_{\infty} = \frac{\rho}{2}\cdot v_{\infty}^2$$ The equation for the local speed, relative to flight speed, is:$$\frac{v}{v_{\infty}} = \sqrt{1 - \frac{p - p_{\infty}}{q_{\infty}}} = \sqrt{1 - c_p}$$ Now you need to know the pressure coefficient. At Mach 0.85 the local Mach number of the 747 wing, which has a wing sweep of 37.5°, is 0.674. Since I do not have the airfoil of the 747-400, I used one from the same family (BACJ), found here. If you are interested, many more can be found in Michael Selig's immense airfoil database.

The plot is from a TSFOIL calculation of the 2D flow around BACJ at Mach 0.8, so it needs to be taken with a grain of salt. Note that it shows a stagnation point, and, therefore, Mach = 0 at the leading edge. The leading edge of the 747 wing is swept by 45°, so it will only slow the orthogonal speed component to zero. Consequently, the 747 wing has a stagnation line where the lowest speed will still be Mach 0.6.

On the suction side, the airfoil reaches Mach = 1.2 quickly and even Mach 1.3 near the trailing edge. Modern, so called supercritical airfoils can tolerate mildly supersonic speeds on their suction side, and the Boeing 747-400 makes use of them. However, since the wing sweep reduces Mach effects, the top speed will be Mach 1.2 or slightly lower, and the TSFOIL calculation shown here is not exactly correct for flight at Mach 0.85, but gives a general idea what happens on the wing.

To answer the many questions you posted directly:

1. The shape of the airfoil is immensely important. Thicker airfoils need to displace more air, and higher flow speeds are created on both sides. Camber will increase the pressure (and speed) difference between both sides.
2. Changing the angle of attack will add a roughly triangular difference, which is highest at the leading edge, to the difference between both sides. Lower side pressure is increased slightly, and upper side suction goes up by a lot at the nose.
3. Good wing designs have a triangular lift distribution over wing span, with the highest lift at the root. Since the displacement effect of the fuselage is added to the flow effects, the highest speeds will be encountered at the wing root.
4. Just look at the pretty pictures ...

Answer 2

Wishing I had my copy of Basic Aerodynamics handy to provide some more detailed references and numerical examples (may come back later to add more detail), but here goes the highly abbreviated version.

1. Airfoil shape is absolutely critical in determing the distribution of lift (and therefore air pressure differential between the two sides of the wing) and this holds true for literally anything that makes use of them. The exact specifics as to how much this varies differ from one airfoil to the other, but you should be able to just look up the airfoils you're interested in and compare their coefficients of lift, one of the variables in the generation of aerodynamic force (i.e. horizontal lift and induced drag), at various alphas (a.k.a. angle of attack, angle of incidence).

2. The coefficient of lift is indeed determined by the angle of attack. The greater the angle of attack, the greater the coefficient of lift, right up until the point where airflow separates from the wing and you enter a stall. The end result is that, for a given airfoil and constant airspeed, the higher the angle of attack, the more lift your wing will generate, and the greater the air pressure differential.

3. A very interesting question and the answer is, it depends. For a constant camber airfoil (think your basic Hershey-bar style wing), you should be able to get more lift being generated closer to the wing roots. This happens due to the fact the airflow tends to "slip" around the wing edges due to the fact that we do not have infinite span wings and the higher pressure air underneath the wing tries to fill the lower-pressure areas on top and has an outlet to do so where the wing ends. The end result of this phenomenon is known as a wingtip vortice and one of the effects is that the aerodynamic force vector usually has a somewhat different direction at the wing edges vs. the tip, usually leading to less vertical lift being generated their as opposed to the wing root. Other things to keep in mind, many modern manufacturers will use variable-camber designs today to build wings that stall at the wing root first in order to ensure aileron control as far into the stall as possible. Recommended further reading on this topic: wingtip vortices, wake turbulence, winglets, Spitfire wing

4. For most subsonic airfoils, the center of pressure will occur quite close to the leading edge. Again, this depends on the wing design itself, which is dependent on aircraft role and requirements. Aircraft designed to fly slowly will typically have relatively "fat" wings with a pronounced camber close to the leading edge. Aircraft design to fly faster will seek to reduce frontal area and will have their thickest camber set farther back. Supersonic airfoils generally have a vaguely diamond-shaped profile owing to the need to keep the leading edge very sharp and the frontal area minimized.

If I might suggest, if this is something you're curious about, look into picking up an introductory text on aerodynamics. The topic is truly fascinating and the answers to your questions can run very deep indeed.

In the meantime, hope this helps.

Follow-up

@erich @FreeMan I've been doing some thinking about the 747 question and I think it's worth delving into some of the underpinnings to clarify a few things.

Disclaimer: My expertise here is as a pilot rather than an aero engineer (big, big difference in the depth of study of aerodynamics), so if any mistakes follow, that's why.

One thing to remember about an aircraft in flight is that it's not so much air "flowing" over the wing, as the aircraft "flowing" through the air. We often illustrate air moving over the wing because this is how it works in wind tunnels and, as an added benefit, it simplifies things when introducing this topic to new students. We often use the term "Relative Wind" to refer to this phenomena.

As such, it is important to remember that when the aircraft moves through the air, it displaces air and pushes it above and below the wing. To further simplify our discussion, let us assume a positively-cambered wing at 0 alpha. What will happen in this situation is that the surface area on the top of the wing, being greater than the surface area below the wing, will have fewer air particles per unit of area leading to lower pressure on the top of the wing than the bottom, as air density is relatively constant around the aircraft. This pressure differential results in a force, whose vertical component counteracts the aircraft's weight, resulting in level flight. Extrapolate from here for climbs, descents, banking turns, etc.

Digression: you may point out that if this is the case, how come airplanes don't just sort of float away when they're on the ground, seeing how the wing area differential still holds. The intuitive reason for that is that unless the aircraft is moving and "disturbing" the air around it, air particles will be free to float around the wing and fill any low pressure areas until the pressures equalize.

Going back to our 744, what you would really be interested then is to actually calculate pressure differentials and, working from there, calculate "speed" differentials (keeping in mind that you would actually be calculating more of an "average speed", as different parts of the wing will generate different amounts of force). Someone else will have to do that math however, as I can't seem to find the necessary technical data. Also, in case you're wondering, yes, the fuselage also generates a certain amount of lift, while the tailplane horizontal stabilizers generate a aerodynamic force downwards (think "negative lift"). Have fun digging :)