Why Do Planes Get Lift?

When someone asks: why does a plane fly, I'd say: "because the shape of the wings, the plane gets pulled up, it's called the Bernoulli force ..." But is this force really enough to overcome the force of gravity for a Boeing 747? The aerodynamics involved are actually extremely complicated, but you can get a "back of the envelope" figure by considering a wing, as drawn below.



Assume that the air flow is non-turbulent, (basically, this means that two air molecules next to each other at the front of the wing will meet at the back of the wing if one takes the upper path and the other takes the lower path. If this is not the case, air would be accumulating above or below the wing and at the back, you'd get vortex effects - i.e. turbulence.)

Note that the wing is slightly longer on top then on the bottom. The bottom is of length L, and the top is of length L + dL, where dL is small compared to L. To emphasize this, lets say:

d = dL/L ... (1)

Since the flow above the wing has to travel a farther distance than the that of the bottom path in the same amount of time (non-turbulent flow), the air on the top is going faster than the air on the bottom.

Since velocity = distance/time, we have:

velocity on bottom = vb = L/t ... (2)

and on top = vt = (L+d)/t = vb + d/t = vb + L/t*d/L = vb(1+d) ... (3)

where in that last step, I used (1).

Now it's time for Bernoulli's equation: it says that, assuming non-turbulent flow, the pressure P, velocity v, and height h, of a fluid with density ρ at points htop and hbot are related by:

Ptop + 1/2 vtop^2 + ρ g htop = Pbot + 1/2 ρ vbot^2 + ρ g hbot ... (4)

The first thing to note is that htop~ hbot, so the lift due to these terms (buoyancy) can be neglected:

ρ g htop ~= ρ g hbot ... (5)

To be convinced of this, the density of air is around 1 kg/m^3 and a wing is about 0.1m thick. Since g is about 10 m/s^2, the differential pressure is on the order of 1 Pascal. For a 10 m^2 wing, this could lift about 1kg, which is much less than the wing itself would weigh.

Next note that the quantity that we're interested in is Force due to the differential pressure. Since Force = Pressure x Area, this is, for wings of area A:

Flift = (Pbot - Ptop)*A ... (6)

Combining (4) through (6), we get:

Flift = 1/2 ρ (vtop^2 - vbot^2)*A ... (7)

We can now use (2) and (3) to get:

(vtop^2 - vbot^2) = vbot^2 (2d -d^2) ~ 2dvbot^2 ... (8)

In the last step, we used the fact that d << 1bot = v, the velocity of the plane, we get:

Flift =ρ A d v^2 ... (9)

... the force of lift due to the wings.

Let's say that the top path of the wing is 5% longer (d=0.05) and that each wing is 10 m^2. Using ρair = 1 kg/m^3, we get:

Flift = v^2, with Flift in Newtons and v in meters/second. In order for the plane to fly upwards, the force of lift must be greater than the force of gravity Fg = m g.

A typical jet can fly at around 200 to 250 m/s, so the force of lift is then about: 62,500 N (now we see why the ρ g h terms could be ignored.) This can lift a mass of 6,250 kg, but a jet weighs in at a few 100,000 kg, so why does it fly?

First off, this is an oversimplified picture: this would work well for a glider or bird (a bird with 10cm x 40cm = .04 m^2 wings and d = 10% would have to go about 30 m/s to glide if it weighed a pound - any slower and it'd have to flap its wings.) A jet on the other hand, has propulsion which pulls it upward, in addition it gets an upward lift from the normal force of the wind. This is the same force that you feel when you stick your hand out of the window while driving down the highway. For a large jet such as a Boeing 747 this actually contributes to most of the lift force.

The point of this was to do a back of the envelope calculation of the "popular" idea of lift, and show that this simplified picture alone doesn't really explain why a jumbo jet flies way up in the sky. For a more details, the complexity of the problem goes up exponentially and you soon have to resort to numerical simulations. However, a really nice (slightly less simplified) discussion is found here!

Twinkling Streetlights

If you ever look out over a city on a clear night, like on one of those 'make-out points' from cheesy 80's movies, you notice that distant streetlights seem to flicker and dance around a little, whereas closer ones don't. If your not making out with Winnie Cooper at the time, you inevitably start to wonder why that is the case. Notice that the same thing is seen in the sky: stars twinkle, but planets don't (the ones that wee can see, that is.)

The reason for the "twinkling of the stars" is what is called "atmospheric seeing": the atmosphere has layers of turbulent air of varying density and temperature, which leads to a time-varying index of refraction (The speed of light in a vacuum (outer space, say), divided by the speed of light in the material (air here.)) To see why this would make a star twinkle, recall that a lens is just a material of certain index of refraction, shaped in a certain way so as to focus (or diverge) light. The pockets of air blowing around the atmosphere, means that light from a star would rapidly become focused and unfocused as the air above blows around. This is kind of like the pattern you see on the sand, underneath shallow water: the light jumps around like crazy from being randomly bent at the surface. For water the effect is way more pronounced, since the index of refraction of water is about 1.33, whereas for air it's 1.0003 - just barely different than for a vacuum for which, by definition, it's 1. You can check how bad the current "seeing" is here.

So that's why stars twinkle, but then what about planets? The same "seeing effects" would be present for both planets and stars, but we only notice it for planets (actually, if you look through a telescope at Saturn or the Moon, you really start to notice the effects of seeing on a good night vs. a bad night.) The reason is that to us, stars come from a point in the sky - we can't resolve with our eyes what shape they are or what they look like. All our eyes know is that light is coming from one specific direction. When we see an object, say Winnie Cooper, her left ear is focused to one point on our retina, and her right ear is focused to another. That is, we can resolve the shape of the image. For stars, this is not the case: they're so far away that the left part of the star is totally smeared over with the right part of the star. Because light is wave, it can't be focused to an infinitely small point, the focus is limited (by diffraction) to a certain size, and for stars, this size is much bigger, that the size that they would be focused to on our retina. For planets however, we can make out the shape of them and although each point of the planet is experiencing this seeing effect, all of the combined effects average out into a steady (slightly blurred, if you look through a telescope) image. To verify this, we can to a rough calculation.

The minimum resolvable angular distance between two objects which are sent though a lens with diameter D is: sin(θ) = λ/D times some constant which is close to 1 and depends on the type of wave and the exact shape of the lens. For a plane wave through a perfectly circular lens, it is 1.22. For us, D is the diameter of our pupil, say 5 mm, at night. The wavelength of visible light is on the order of 600nm, so λ/D is about 10^(-4) which means that the minimum angle is about θ = 0.0001 (since sin(θ) is pretty much equal to θ when θ is so small.) Now by definition, sin(θ) = Dia/dist, where Dia is the diameter of the object and dist is the distance to it. We now have a test to see if an object is resolvable or not: if Dia/dist is bigger than 0.0001, it should be, if Dia/dist is much less that 0.0001, it's not (keeping in mind the roughness of the calculation.)

Several cases:

Mars: Dia = 6800 km, dist = 55000000 km, Dia/Dist = 0.00012
i.e It should just be resolvable.

Jupiter: Dia = 140000 km, dist = 700000000 km, Dia/Dist = 0.0002
i.e. Again, in the window of resolvability

α-Centauri (nearest star): Dia: 10^6 km, dist = 2*10^13 km, Dia/Dist = .0000005
=> much less that that of the planets.

Now, bringing this back to the original discussion, say that a street lamp has a diameter of about 10 cm. The distance at which it takes up about the same angle as a planet is: dist = Dia/θ = 1 km. Much farther than this, and the lights become point sources, like the stars. From where I live, the lights at the train-station 1 km from my place don't twinkle, but the lights on the ski-hill, 5 km away do. However, the giant billboards by the hill do not - they have a bigger Dia. and therefor are resolvable. These calculations were pretty rough, but they sketch out the main point - that the "twinkling" of stars and distant lights, stems from them being "point-sources" of light to our eyes.