Bernoullis
and the Physics of Lift
Bernoullis discovered the theoretical basis
for lift produced
by a rigid wing.
Alexander McKee begins his fascinating book,
Great Mysteries of Aviation, with
the observation that the most
puzzling mystery in the history of aviation is why it
took so long for humankind to
learn to fly. With so much
intellectual and physical energy devoted to a single problem
for so long, one might have
expected someone to stumble on the
secret, if only by accident, long ago.
What was the obstacle?
The problem is that the physical principles that lie
at the foundation of
flight are counterintuitive; indeed, the
mechanics of flight were ultimately revealed after
some fancy manipulation of
the physics and mathematics created by Sir Isaac Newton in
the late 1600s. Not only were the
theories of Aristotle, Bacon, Leonardo, and the
rest all wrong, but the
true principles of flight, including
how birds stay aloft, were simply unguessable
and unobservable.
It took several remarkable scientists,
including members of a celebrated
family of scientific giants, to
piece together the puzzle. For all the triumphs of
Newtonian physics—from
explaining the tides to predicting comets—Newton had little
success in applying his methods to
fluids and fluid dynamics.
Along came the Bernoullis, a Swiss family among
whom were some of the most
important contributors to the
development of mathematics and science in the seventeenth
and eighteenth centuries. The two key figures in
this family were Johann (1667—1
748), who made the
University of Basel in Switzerland the centre of European
science in its day, and his
son Daniel (1700—1782). In 1725,
Daniel accepted an appointment in St. Petersburg,
Russia, where he stayed for eight years and did some
of his most important work. He
managed to take a friend
with him: the great mathematician Leonhard Euler, who
had been a student of
Johann Bernoulli back in Basel.
In 1734, Daniel completed his famous work
Hydrodynamica, which
was not published until 1738. In
addition to coining the word “hydrodynamics,” Daniel
laid out the basic
principles of the new science, applying
Newton’s basic laws to simplified cases of
fluid dynamics. Out of this work came Bernoulli’s
Principle (or Law), which Euler
helped express as a mathematical
equation known as Bernoulli’s
Equation. What Bernoulli found
boiled down to this: when a
fluid is moving—through a pipe or conduit, or simply
over any surface—it exerts
pressure in all directions:
against anything that is in the way of its flow, as well as
against any surface it
touches. For example, as water
flows through a garden hose, you can feel the pressure of
the water against the inner
wall of the hose if you try to
squeeze the hose. Now, if the fluid is
noncompressible
(meaning it can’t he squeezed into a smaller volume,
which is true of water, in
most ordinary circumstances), and
if there is no change in the amount of fluid flowing
(meaning nothing is leaking
out or coming in), then the faster
the fluid is flowing, the lower its pressure
against the surface it’s
flowing over will be.
That means that
when you pinch the garden hose
slightly in the middle and the
water keeps coming out of the end at the same
rate, then the water must
be travelling through the pinched
portion a little faster (since the same amount of
water is passing through
that section of the hose as
before). Our intuition is that faster water exerts
greater pressure (and it
does, but only in the direction of the
flow), but the pressure of the faster
water against the wall of
the hose (which is perpendicular to the direction of the
flow) is less—a total
surprise. Euler gave Bernoulli’s work
mathematical form (with the help of the work of
French mathematician Jean le Rond
d’Alembert), and Johann, Daniel’s
father, made it intuitively
palatable in his 1743 work,
Hydraulica (which he tried to pass off as having
been written in 1728).
Now, as to flying: if a sleek, symmetrical
wing is in an air flow so that air
is passing over it and under it,
the flow can be considered noncompressible
and a closed system—a few feet
back (if the wing is sleek enough
and the wind is not too strong),
one wouldn’t even know the air
took a little detour around the
wing. As the air flows over the
wing’s surface, it too exerts pressure in
two directions—in the
direction of its flow (that’s the force of
the wind) and perpendicular to its
flow against the surface of the wing. But since the
air has to travel a greater
distance to flow around the wing,
it speeds up, and by Bernoulli’s
Principle it exerts less pressure on the surface
of the wing.
Since the wing is symmetrical (a teardrop
shape in crosssection), the reduced
pressure is the same both
above and below. Now what happens
if we slice the wing in half, so
that the lower surface is
straight (and the air flows across
it in a straight line), hut the
upper surface is curved (and the
air speeds up only when flowing
over that surface)? The pressure of
the air on the upper
surface drops, making the pressure
of the air on the underside
greater. The
difference
between the pressure upward on the underside of
the wing and the force downward on
the top surface is called
“lift”; the curve of the top surface of a wing over
its under surface is called
its “camber.”
After centuries of believing the
very reasonable notion that, like ships
floating on the ocean,
birds flew in a sea of air, and that
a wing (of a bird or of a successful aircraft)
would have a crosssection
that, like a boat, would be curved on the
bottom and flat on top, the
exact opposite turned out to he
the case. Flight is made possible by the lift created by
the pressure difference
resulting from air flowing over a
wing with camber, and that’s the secret of
flight.
