Lift is the force that holds an aircraft
in the air. How is lift generated? There are many explanations for
the generation of lift found in encyclopedias, in basic physics
textbooks, and on Web sites. Unfortunately, many of the explanations
are misleading and incorrect. Theories on the generation of lift have
become a source of great controversy and a topic for heated
arguments for many years.
The proponents of the arguments usually fall into two camps: (1)
those who support the "Bernoulli" position that lift is generated by
a pressure difference across the wing, and (2) those who support the
"Newton" position that lift is the reaction force on a body caused by
deflecting a flow of gas. Notice that we place the names in
quotation marks because neither Newton nor Bernoulli ever
attempted to explain the
aerodynamic lift of an object.
The names of these scientists are just labels for two camps.
Looking at the lives of Bernoulli and Newton we find more similarities
than differences.
On the figure at the top of this page we show portraits of Daniel Bernoulli,
on the left, and Sir Isaac Newton, on the right.
Newton worked in many areas of mathematics and physics.
He developed the theories of gravitation
in 1666, when he was only 23 years old. Some twenty years later, in 1686, he
presented his
three laws of motion
in the Principia Mathematica Philosophiae Naturalis .
He and Gottfried Leibnitz are also credited with the development of the
mathematics of Calculus.
Bernoulli also worked in many areas of mathematics and physics and
had a degree in medicine. In 1724, at age 24, he had published a mathematical
work in which he investigated a problem begun by Newton concerning
the flow of water from a container and several other problems involving
differential equations. In 1738, his
work Hydrodynamica was published. In this work, he
applied the conservation of energy to fluid mechanics problems.
Which camp is correct? How is lift generated?
When a gas flows over an object, or when an object moves through a gas,
the molecules of the gas are free to move about the object; they are not
closely bound to one another as in a solid. Because the molecules move,
there is a velocity associated with the gas. Within the gas, the
velocity can have very different values at different places near the object.
Bernoulli's equation, which was named for
Daniel Bernoulli, relates the pressure in a
gas to the local velocity; so as the velocity changes around the
object, the pressure changes as well. Adding up (integrating) the
pressure variation
times the area around the entire body determines the aerodynamic
force on the body. The
lift
is the
component
of the aerodynamic force
which is perpendicular to the original flow direction of the gas.
The
drag
is the component of the aerodynamic force
which is parallel to the original flow direction of the gas.
Now adding up the velocity variation around the object instead
of the pressure variation also determines the aerodynamic force.
The integrated velocity variation around the object produces a net
turning
of the gas flow. From
Newton's third law
of motion, a turning action of the flow will result in a reaction (aerodynamic
force) on the object.
So both "Bernoulli" and "Newton" are correct. Integrating the effects
of either the pressure or the velocity determines the aerodynamic force on
an object. We
can use equations developed by each of them to determine the
magnitude and direction of the aerodynamic force.
What is the argument?
Arguments arise because people misapply Bernoulli and Newton's equations
and because they oversimplify the description of the problem of aerodynamic
lift.
The most popular incorrect theory of lift arises from a misapplication of
Bernoulli's equation. The theory is known as the
"equal transit time" or "longer path"
theory which states that wings are designed with the upper surface longer
than the lower surface, to generate higher velocities on the upper surface
because the molecules of gas on the upper surface
have to reach the trailing edge at the same time as the molecules on the lower surface.
The theory then invokes Bernoulli's equation to explain lower pressure on the
upper surface and higher pressure on the lower surface resulting in a lift
force. The error in this theory involves the specification of the velocity
on the upper surface. In reality, the velocity on the upper surface of a lifting
wing is much higher than the velocity which produces an equal transit time.
If we know the correct velocity distribution, we can use Bernoulli's equation
to get the pressure, then use the pressure to determine the force. But the equal
transit velocity is not the correct velocity.
Another incorrect theory uses a
Venturi flow
to try to determine the velocity. But this also gives the wrong answer
since a wing section isn't really half a Venturi nozzle.
There is also an incorrect theory which uses Newton's third law
applied to the bottom surface of a wing. This theory equates
aerodynamic lift to a stone
skipping
across the water. It neglects the physical reality that both the lower
and upper surface of a wing contribute to the turning of a flow of gas.
The real details of how an object generates lift are very complex and do
not lend themselves to simplification. For a gas, we have to simultaneously
conserve the
mass,
momentum, and
energy
in the flow. Newton's laws of motion are statements concerning the conservation
of momentum. Bernoulli's equation is derived by considering conservation of
energy. So both of these equations are satisfied in the generation of lift; both
are correct. The conservation of mass introduces a lot of complexity into the
analysis and understanding of aerodynamic problems.
For example, from the conservation of mass, a change in the velocity of a gas
in one direction results in a change in the velocity of the gas in a direction
perpendicular to the original change. This is very different from the motion of
solids, on which we base most of our experiences in physics.
The simultaneous conservation of mass, momentum,
and energy of a fluid (while neglecting the effects of
air viscosity)
are called the
Euler Equations
after Leonard Euler. Euler was a student of Johann Bernoulli, Daniel's
father, and for a time had worked with Daniel Bernoulli in St. Petersburg.
If we include the effects of viscosity, we have the
NavierStokes Equations
which are named after two independent
researchers in France and in England. To truly understand the details of
the generation of lift, one has to have a good working knowledge of the
Euler Equations.
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