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Circulation
Category Miscellaneous
Posted by Taylor Grayson Submitted on 01/01/2008 12:00 AM
How a wing generates lift remains one of the most argument-inducing subjects in all aerodynamics, at least among pilots. What isn't commonly known, though, is that scientists have well understood the process for close to 100 years, and there is no debate within the scientific community. When evaluating any potential explanation for lift, there is one important concept to keep in mind which will help you cut through the smoke: nature only has two ways for a fluid to transmit a force to an object:
  1. pressure
  2. friction
Pressure is merely the sum total of the random motion of the atoms of a fluid. The more atoms that bounce off an object, and the faster they move, the greater the pressure felt by the object. Pressure acts perpendicular to the surface of the object.

Friction is due to the viscosity of the fluid. Yes, air has viscosity, even though it's very low compared to fluids such as molasses, or even water. Friction acts parallel to the surface over which the air flows.

So no matter what you hear about the "true" cause of lift, in the end, unless those causes can create a pressure or friction changes around the airfoil, they can have no effect on lift at all. Now, even though technically friction can contribute to lift, it should be apparent that its contribution is small, as indicated in this diagram:

Since friction acts parallel to the surface, there are only two small areas in which friction acts in the vertical direction. Not enough to matter, so it's ignored. So the only possible source for the forces to produce lift are the pressure differences around the airfoil. What we need to achieve looks something like this:

Now, one way of demonstrating the concept is by placing a disk that fits tightly in a cylinder and then evacuating the air above the disk, like so:

This creates a pressure difference between the top and bottom that produces a net force that lifts the weight. However, for generating lift on a wing, we could not evacuate the air fast enough to keep the surrounding air from rushing in, at least with this method.

The Bernoulli Equation

The Bernoulli equation offers another potential for lowering the pressure on the top of the wing, if we can figure out how to use it:

The Bernoulli Equation says that the sum of static pressure and dynamic pressure is always the same. If you increase the dynamic pressure, then static pressure must go down; if dynamic pressure goes down, then static pressure goes up. If we assume that the density of the air remains the same, the only way to change dynamic pressure is via changing the velocity.

One way to think of the Bernoulli Equation is that it says that the total energy of an airstream is a constant, since energy can never be created or destroyed. The static pressure is potential energy per unit volume of air and the dynamic pressure is the kinetic energy per unit volume. You can convert one to the other, but the total energy is a constant.

The Bernoulli Equation acts like a roller coaster, as in the picture below. Car A is at the top of a hill; it has converted its velocity into altitude and arrives at the peak with high potential energy, but low kinetic energy. Momentarily, it will convert the potential energy (altitude) back into kinetic energy as it moves to the position of car B.

Now, the classic way to use the static pressure to speed up the air is a Venturi, as shown below:

Can we use the Venturi Effect to make the air go faster over the top of the airfoil? Let's put an airfoil into the airstream and find out:

Putting an airfoil into the airflow does indeed cause the Venturi effect to kick in. The airfoil impedes the flow of air, acting as one side of the Venturi, and the atmosphere above the airfoil cannot "bump up" in order to accommodate the air trying to make a detour. It speeds up because it must. The problem is that the airflow speeded up on both sides of the airfoil, creating the exact same pressure drop. They cancel either other out and there is no pressure difference between the top and bottom of the airfoil that we need for lift.

The Magnus Effect

One hint as to the proper direction is indicated in something called the "Magnus Effect." Consider a cylinder placed in an airflow:

As you can see, it behaves pretty much like the airfoil above; the air speeds up on both sides of the cylinder, so no net lift is generated. But look what happens we start rotating the cylinder clockwise:

The rotating cylinder drags the air above along with it, speeding it up, but below the cylinder, the rotation fights against the airflow, slowing it down. The net result is a positive amount of lift.

Although the Magnus Effect isn't a practical means of generating lift on a wing, it does suggest that imparting some overall rotation to the airflow in addition to the freestream flow could produce lift. Our goal, then, is to somehow produce this situation:

Circulation

It turns out than when we place an airfoil in an airflow at an angle of attack, we get exactly the rotational effect that we're looking for, and it's called circulation.

No Viscosity

If air had no viscosity, the flow of air around an airfoil would look like this:

As you can see, there is a normal stagnation point on the leading edge, but the rear stagnation point is on the top. In order for the rear stagnation point to be there, the airflow must be like this:

This can't happen, except at very low airspeeds. The airflow simply cannot follow the rapid change in direction that would be required to negotiate the sharp trailing edge, and the airflow separates from the surface into a tight vortex:

This vortex, called the starting vortex is carried away down stream, but it leaves behind another vortex that has the same quantity of rotation, but in the opposite direction. The follow graphic uses the analogy of cogs to show how the starting vortex creates its opposite around the airfoil:

So what we're left with is a permanent vortex, called the bound vortex that speeds up the air on the top of the wing, and slows it down in the bottom, like this:

The air never flows backwards along the bottom of the airfoil as the bound vortex seems to show. Instead, it adds to the freestream; since the freestream is faster, the net result is movement in the freestream direction, just slower.

Is It Real?

Photographs of the airflow over a wing made visible by such techniques as smoke show that the initiation of lift creates the starting vortex, just as the theory shows. The bound vortex is evident as it spills over the wing tips into the familiar wingtip vortices. Each change in angle of attack is accompanied by a new starting vortex. So, yes, real world evidence supports that circulation theory is more than just a mathematical tool that works.

Anderson, John D., Jr., Fundamentals of Aerodynamics. McGraw-HIll, New York, 2001.

Carpenter, Chris. Flightwise: Principles of Aircraft Flight. Airlife Publishing Ltd, Shrewsbury, UK. Copyright 1996.

Kundu, Pijush K. and Ira M. Cohen. Fluid Mechanics. Academic Press, San Diego, CA. Copyright 2002.

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