We’ve got some nice data for our experimental wing, but this data is all for a very specific geometry at a specific speed. It’s useful, but there’s only so much we can do with it.
Let’s think about how we can take a measurement of forces on a wing and turn it into a value we can apply to any wing size at any condition. We’ve going to nondimensionalize, or remove the units from, our test results.
To start, we know that the single value of lift we measured is actually the sum of all of the lift acting across the surface of the wing.
Say the wing generated 30 lbs of lift in our testing, and the wing itself is 6 feet across (wingspan) and 1 foot wide. Our wing’s total surface area is therefore 6 square feet.
If we want to apply this data to any size of wing with the same airfoil as what we tested, we need to find the amount of lift generated per chunk of wing surface area. Dividing 30 lbs by 6 square feet gets us 5 lbs of lift per square foot.
Take note of the fact that our result is in units of pressure: pounds per square foot. This is important.
If we want to be able to figure out the lift generated by this wing shape at any airspeed, too, then we need to take our testing speed into account. But what really matters when you’re talking about aerodynamics is the relative pressure of all those air molecules pushing on your wing.
Because airplanes fly at different altitudes, and the density of air decreases as you go up in altitude, we need to take both the speed of the air molecules, and how many there are, i.e. the air density, into account. You might be flying at the same speed whether you’re at 100 ft or 5,000 ft, but the air will be less dense at 5,000 ft so you will actually be experiencing less pressure across your wing.
We call this quantity dynamic pressure—the pressure exerted by a flow due to the energy of movement within the flow.
We can calculate it for our testing: pretend we did the test at sea level, and that we drove at 40 miles per hour. This gets us a dynamic pressure of 4.1 pounds per square foot, or psf. (The exact equation isn’t as important—just know it uses density and speed.)
Now for the magic trick: we want to find the amount of lift generated per section of wing, per unit of pressure applied by our airflow. Divide 5 psf by 4.1 psf, and we get 1.2 as our answer.
Yes, just 1.2. There are no dimensions attached. This is our coefficient of lift, for our particular wing. Neat!
…but what do we do with it?