2. How Airplanes Produce Lift

Problem:

How do airplanes produce lift?

Vocabulary:

Airfoil – The two dimensional image of an airplane’s wing. Imagine cutting a thin slice of the wing while looking from the side of the airplane.

Background:

The problem of how airplanes produce lift has been tackled many times. Most people know the basic theory. The shape of the foil causes air to move faster over the top of the wing than it moves over the bottom. Bernoulli’s Equation implies that pressure and speed are inversely related. Applying Bernoulli’s Equation, it can be seen that the pressure on upper portion of the wing must be lower than the pressure on the bottom of the wing. This pressure differential results in a net force that causes the airplane to move upwards. This explanation is correct at its core, but the model simplification leads to a few issues:

1) How can airplanes fly upside down? In the above model, the pressure differential will cause the airplane to plummet towards the earth.

2) How do airplanes or gliders with flat wings fly?

The answer is that the curvature is not the only factor that causes a pressure differential to occur over the wing. I hope to present an explanation that is rigorous in its physical derivation, but simple enough for the layman to understand.

Solution:

Before looking at flow over an airfoil, it is important to first look at the Law of Conservation of Matter. The Law of Conservation of Matter states that matter can be neither created nor destroyed, but it can change form. For a fluid like air, the law of conservation of matter can be expressed mathematically as:

d1 * v1 = d2 *v2

Where d represents the distance between the walls at a location, and v represents the speed at that location. The flow was assumed to be incompressible and one dimensional (namely the flow goes from right to left). In this context, incompressible means that you can’t “squeeze” a given volume of fluid down to a smaller size. This equation is called the continuity equation.

Now imagine a pipe of varying wall spacing, and apply the continuity equation. As the pipe wall spacing changes, the fluid speed must change as well. In the example below, setting d1=5, v1=3, and d2=3, we can derive v2=5.

Thus, the equation implies that as the wall spacing decreases, the fluid speed increases. This is a very significant result, and it is the basis of our solution.

Now the example can be extended to an airfoil. For generality, the airfoil will not be level: it will be at an angle of attack with respect to the freestream flow. This will help to demonstrate why flat wings can produce lift. The sketch below is an approximation of what streamlines would look like over the upper surface of an airfoil at an angle of attack.

Pretend that the regions labeled as a dotted lines are a walls. This is a valid assumption since there will be no flow into or out of these lines (they are horizontal streamlines). Now the continuity equation can be used. Focusing in on the top region, it can be seen that the flow must speed up while traveling over the top of the airfoil. The wall spacing is smaller over the top surface of the airfoil; therefore, the air must move faster. For the same reason, the flow must speed up as it goes along the bottom surface. Placing the airfoil at any angle such that air flow is faster over the top surface than over the bottom will produce lift.

Model Limitations:

This analysis assumed an idealized (1D) steady incompressible (irrotational) flow. In plainer words, it assumes that the airplane is flying below Mach 0.3 speed and that there are neither frictional nor “3D” effects. In general, none of these things are true in real aviation. While this invalidates the model in terms of rigorous mathematical application, I believe the model still serves as a useful tool to model trends for all types of flows. This is supported by the fact that airplanes fly.