Bernoulli’s Principle
and the Equal Transit Myth
Like most
kids that watched Top Gun, I thought airplanes were pretty cool and wanted to be
a pilot. When I tried to look into the
encyclopedia to understand how airplanes fly, I was given an explanation that
depends on Bernoulli’s principle. But
what is Bernoulli’s principle?
Bernoulli discovered in 1738 that
as a fluid’s speed increases, its pressure decreases. Intuitively, this makes some sense. Imagine you are a particle of air approaching
a high-pressure region. As you approach
it, you start crashing into more and more other particles since they are packed
more tightly together there. Crashing
into particles is a drag and slows you down.
If, on the other hand, you are approaching a lower pressure region, then
there are fewer particles in front of you to counter the force of those
crashing into your posterior. As a
result, you accelerate into the low-pressure region. In practice this phenomenon works in both
directions. If you cause an increase in
pressure you will cause a decrease in speed.
If you cause a degree in speed you will cause an increase in pressure.
That’s
fine, but what does it have to do with airplanes? Well, the conventional explanation asks us to
look at a cross-section of an airplane wing.
Since the wing is shaped so that there is a longer curved surface on top
and a shorter, straighter surface on the bottom the air across the top has to
move faster in order to reach the trailing edge at the same time. Thanks to Bernoulli we know that
if the air on top of the wing is moving faster than the air on the bottom, it
will have a lower pressure. Since
there’s more pressure below the wing than above it, the result will be to push
the wing, and whatever’s attached to it, into the air. This explanation is sometimes called the
“equal transit myth” and it is plain wrong.
Even as a
child, I could never understand what principle compelled the air on top of the
wing to reach the trailing edge at the same time as the air on the bottom. In fact, there is no such principle. For one thing, consider that unlike an
airplane wing, a sail is nearly two-dimensional: the air flowing over the
windward surface travels almost exactly the same distance as the air flowing
over the leeward surface. If this
conventional explanation for lift were correct, sails couldn’t produce lift,
but we all know that isn’t true.
Interestingly, experimental evidence has shown that not only do particles
going over the two sides of a wing get to the trailing edge at different times,
but in fact the particles that go over the top (or leeward side on a sail)
reach the trailing edge long before
their siblings on the other side.
Bernoulli’s principle is still true, but the equal transit theory isn’t
the reason that the particles on the leeward side are going faster.
Air Circulation and
Starting Vortices
So let’s forget about that equal
transit and start from the beginning.
Imagine a sail that’s luffing in the wind. The air pressure is equal on the top and
bottom of the sail. Now start sheeting
the sail in slowly. All of a sudden, the
sail is catching air on the windward side and deflecting it. This causes a higher pressure and hence lower
speed region on the windward side.
However, this isn’t lift. It’s
drag. Right now all that pressure is in
the same direction as the wind, which is of no use in going upwind. As those particles reach the trailing edge of
the sail, the pressure difference between the windward side and the leeward
side will be so great that the air coming off that edge will swirl into a tiny
tornado, called the starting
vortex. This vortex is absolutely
essential to setting up the conditions for real lift to obtain.
Thanks to
the Baron
Kelvin, we know that angular momentum in a bounded fluid is conserved. In other words, when the starting vortex is
created by our sail, there must be another vortex that spins in the opposite
direction but with the same strength.
This is called Kelvin’s
circulation theorem. By some miracle of physics, the
equal-and-opposite vortex is centered on the sail itself, and induces the air
around the sail to go even faster over the leeward side and even slower over
the windward side. This further enhances
the pressure difference between the two sides of the sail and begins to
generate lift proper. Now that it isn’t
just the force of the wind hitting the sail but the force of the wind being
bent around the sail that can push a boat in directions other than straight
downwind.
The insight
that there must be a circulation field to help drive the flow of air around the
sail is called the Kutta-Joukowski
theorem. Its major limitation is
that it only explains the flow of a two-dimensional fluid. A lot of strange phenomena
happen at the edges of airfoils that have to contend with three dimensions, but
I’ll have to move on to those later.
Science in your Home
Most of this
information is a rehashing of the explanations put forth by Arvel Gentry. He describes a simple experiment that you can
perform in your bathtub in order to see the circulation fields around a
sail-like airfoil. I really recommend
trying this out at home.
Fill your
bathtub with just a few inches of water.
Not too much, just three inches or so should do. Then, sprinkle some kind of dark powder over
the surface of the water. Gentry
recommends pepper. I used ground nutmeg
with success. Now, fashion an
airfoil. A simple one can be made from a
small piece of waxed cardboard like a milk carton. Start with a flat piece and bend it very slightly into a gentle curve. It shouldn’t curve as fully as a laser’s sail; Because water is so much thicker than air it
doesn’t take as much curve in the airfoil to generate the same lift. Make sure the piece is tall enough to reach
completely to the bottom of the tub and still be sticking up above the surface
of the water. Place the cardboard
airfoil into the water near one edge of the tub (although don’t let it touch
the edge). Place it so the length of it
is pointing to the other edge of the tub and parallel to the long axis. Now, move it in a steady line across the length
of the tub and lift it out of the water when it’s still a decent distance from
the other edge. Thanks to the spices in
the water you should be able to see a vortex left behind where you first
started the motion of the airfoil. That’s
the starting vortex. When you lift it
from the water you should see another swirl, opposite in direction and also
larger and slower than the starting vortex.
This is the circulation vortex that was following the airfoil along and
is left behind when it is lifted from the water.
The whole
operation should look a bit like this. Note the starting vortex that trail behind
each time the wing is accelerated, and the effect of circulation around the
particles just in front of the wing. Further
experimentation can be had right in your browser thanks to some folks at
NASA. I definitely recommend playing
with that for a few minutes.
Let’s Be Honest, This
Doesn’t Explain Everything
The biggest
problem in this model of lift is that there is no explanation of why the
circulation field that gets kicked off by the starting vortex should follow the
airfoil as it moves through the airstream.
It’s just as much a bit of voodoo as the old equal transit explanation’s
assertion that two particles just had to get to the back of the wing at the
same time. According to some new research,
this circulation field doesn’t even exist, although after doing the bathtub
experiment, I’m inclined to believe my eyes.
As mentioned above, this description of things is inherently two-dimensional. The effects of a third dimension are harder
to understand but also important to getting the most out of a sail, a topic
that we’ll cover in future installments.
Until then, stay cool.
