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!

The Jog Mapper

Sometimes, when I've watched to much "The biggest Loser", I decide to go for a run. I put on my shorts and runners, start my watch and go. Roughly 30 minutes later, I stop my watch and keel over. I look at my time and realize that I don't know if it's good or not, because I have no idea just how far I've run. I decided to go on to mapquest to trace out my root, but this was awkward since I don't usually run along roads. I then just grabbed the image of the map and found out how many 'pixels' I ran, and then converted it from there, but that took a long time. Finally, while running today, I thought "why not just make a basic program to do this: You give it a map and tell it the scale and then draw your path, and it'll tell you how far your went?" - I didn't actually say this out loud. I was conserving every precious bit of oxygen I could.

Here's a screenshot of the program in action. The calibration is done at the beginning where you type in the scale and then drag across the scale-bar to define a conversion fact of meters/pixel. Then you click out your path in line-segments and the program keeps track of the total distance.


Right now, the program is very bare-bones. First off, you have to acquire the map image first, put it into bmp format, and make sure the scale is visible. What would be nice would be to dynamically load the map from mapquest. Second, the user interface is usable, but not friendly. However it does what it does well enough. Thirdly, it would be nice to be have segments that aren't lines, but as long as you don't mind a lot of clicking, it's not a problem.

The guts of the program are as follows: The program starts out in calibration mode. You type in the physical reading from the map's scale, and then click the mouse at the start and end of the scale to get the conversion factor of physical length per pixel. then the program is in the main mode. A display on the top left corner gives you the mileage (kilometerage actually.) When you click at your starting point the program is in draw mode, and a line follows the mouse cursor. Click again and the line segment formed is added to the Path() class. Each click thereafter adds a line segment and the distance continuously updates.

As of now it's a quick and dirty program. If I were to make some changes, it would be (in this order:)

1. Make the program, resettable.
2. Design the UI properly, including a file dialog box for the map.
3. Make the window scrollable to allow for bigger maps.
4. Get the maps real-time off the information super-highway. (I have no idea how to do this)

For now, the code and zipped visual c# project is accessible here!