Lift is created by deflecting a flow of
air, and drag is generated on a body in a
wide variety of ways. From Newton's second law
of motion, the
aerodynamic force F
on the body
is directly related to the change in momentum of the fluid
with time t.
The fluid momentum is equal to the mass m times the
velocity V of the fluid.
F = d (m * V) / dt
F = constant * V * m / t
Since the air moves, defining the mass is tricky. If the mass of
fluid were brought to a halt, it would occupy some volume in space.
We can define the density (r) of the fluid to be the
mass divided by the volume v.
r = m / v
Since the fluid is moving, we must
determine the mass in terms of the mass flow rate.
The mass flow rate is the amount of mass passing a given point during
some time interval t and its units are mass/time.
We can relate the mass flow rate to the density mathematically.
The mass flow rate mdot is equal to the density
times the velocity times the area A through which the mass passes.
mdot = m / t = r * V * A
With knowledge of the mass flow rate, we can express the aerodynamic
force as equal to the mass flow rate times the velocity.
F = constant * V * r * V * A
A quick units check:
mass * length / time^2 = constant * length/time * mass/length^3 * length/time * length^2
mass * length / time^2 = mass * length/time^2
Combining the velocity dependence and absorbing
the area into the constant, we find:
F = constant * r * V^2
The aerodynamic force equals a
constant times the density times the velocity squared. The
of a moving flow is equal to one half of the density times the velocity squared.
The aerodynamic force is directly proportional to the
dynamic pressure of the flow.
Effect of Velocity on Aerodynamic Forces
The velocity used in the aerodynamic equation is the relative
velocity between an object and the flow. The aerodynamic force
depends on the square of the velocity.
Doubling the velocity quadruples the force.
The dependence of lift and drag on the square of the velocity has been
known for more than a hundred years. The Wright brothers used this information
in the design of their first aircraft.
Effect of Air Density on Aerodynamic Forces
The aerodynamic force depends linearly on the density
of the air. Halving the density halves the force. As altitude
increases, the air density decreases. This explains why airplanes
have a flight ceiling, an altitude above which it cannot fly.
As an airplane ascends, a point is reached where there is not enough
air mass to generate enough lift to overcome the airplane's weight.
The relation between altitude and density
is a fairly complex exponential.
You can investigate the effect of momentum on lift by using the
FoilSim II Java Applet. Set a small angle of
attack using the slider, then vary the "Speed" and "Altitude." Try
doubling the speed and notice the effect on lift. Change the altitude
until the air density is half of its previous value. What happened to
the lift? You can use the browser "Back" button to return to this
page. If your browser does not support Java, or you just want your
own copy of FoilSim to play with, you can download
it at no charge.
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