This Bernoulli Stuff

We’ve all heard that an airfoil develops lift by means of this esoteric thing called Bernoulli’s Principle. We have all been taught in private pilot ground school that the air going over the more highly-curved upper surface of a wing’s airfoil must go faster to meet up with those molecules of air that pass under the flatter underside of the wing. And unlike cars on the ‘freeway’ during rush hour, when a fluid flow is constricted the individual air molecules speed up! What’s more — unlike anything else you might try to cram into a smaller space (like intake air during the compression stroke of an internal combustion engine), when a fluid flow is constricted, its pressure goes down… not up! I’m sure many of you have wondered just how it is that, as the air is moving faster, being squished (that’s a technical term) it does so not at a higher pressure, but in fact a lower one! Why?

For one thing, the statement that we’ve all read in ground school references, encyclopedias and physics texts, saying that air parcels traveling over and under a wing meet up again at the trailing edge is… just plain wrong! Actually, the truth is that with a high camber airfoil at near the maximum angle of attack, the air that travels over a wing can be accelerated to nearly twice the free-stream velocity! (Consider that the next time you wonder why bizjet manufacturers make such a big deal about top speeds that approach Mach 0.9.)

Example: If you inject pulses of dye in a wind tunnel, synchronized in a vertical line in front of an airfoil (see image below), you will see that the columns of dyed pulses traveling above the wing arrive at the trailing edge much sooner than their companion columns underneath.

It looks something like this:

That said, it is also true that a wing doesn’t need to have any camber at all to develop lift. So, there’s another factor involved in generating lift and this is Newton’s Third Law of motion: Every action is accompanied by an equal and opposite reaction. The proverbial flying barn door will produce ‘lift’ by virtue of angle of incidence — and enough power to tug it through the air — as the molecules of air hitting it from below would cause ‘impact pressure’ and an upward ‘lift vector.’ After all, a rubber band powered balsa wood airplane flies, doesn’t it? The truth on that one, dear reader, is mostly that it’s basically Bernoulli, and not Newton. Here’s a picture of a flying barn door airfoil:

Air is a continuous fluid, not a fusillade of molecule sized bullets hitting the wing. Although there IS impact pressure, it is not the main reason for our barn door slipping the surly bonds.

Why IS it that a fluid which is obviously squished, rather than having a higher pressure (which is what you would intuitively expect) instead has a lower pressure?! Part of it is that air — or any fluid — has something we know as potential energy, and another thing we know as kinetic energy.

Let’s start out with pressure. It’s a force per unit area (equation 1). If we multiply both numerator and denominator (the ‘F’ divided by the ‘A’) by a length dimension (call it ‘D’, for distance, in equation 2), we get work (or energy) divided by volume.


  P = F / A   (1)    P = (F . D) / (A . D)  (2)  P = Work / Volume


It turns out that 1/2 times mass times velocity squared is the kinetic energy of a physical object in motion. It looks like this:

  KE = 1/2 . m . v2  (3)

Well since density (usually represented by a lower case Greek letter, rho) equals mass per unit volume (4), if we solve for mass we get density times volume (5), or going one step further, density times area times length (6).

  ρ = m / V   (4)    m = ρ . V   (5)  m = ρ . (A . D)  (6)

If we consider everything we now have, instead of one half x mass x velocity squared, we get 1/2 times density times area times length times velocity squared (7).

  KE = 1/2 . ρ . (A . D) . v2 (7)  KE = 1/2 . ρ  . v2  (8)

And per unit of volume, it’s just 1/2 x density x V-squared (8). There’s the kinetic part. The sum total — let’s call it ‘E’ — must never change. So if the sum of ‘potential’ (pressure) energy and motion-induced kinetic energy is constant (9), when one goes up, the other must go down. (Yes, even this is a bit over-simplified.)

  E = P + (1/2 . ρ . v2)  (9)

Well, think of it this way: When you turn on a faucet and a sink (or tub) starts filling up, what happens? Well a lot of things, but one is that water near the stream flowing in gets ‘entrained’ (OK sucked in) due to the viscosity — cohesive forces — of water. The same thing happens with air zooming over a wing. More air gets sucked in when a mass of high-pressure air moves towards an area of lower pressure. Another even simpler way to look at it, is that, since the air molecules above the wing have to travel a greater distance than the ones below it (in less time, let alone the same amount of time), they must be “stretched out”, kind of like a spring, and therefore the density is less there (and from that, the pressure, as well).

So, the next time you wonder why wings fly… well, that’s why.