Aerodynamic forces
are generated whenever an object moves through a liquid or gas.
From Newton's second law
of motion, the aerodynamic forces on the body
are directly related to the change in
momentum of the fluid with time. The fluid momentum is equal to
the mass times the velocity of the fluid. Since the fluid is moving,
defining the mass gets a little tricky. If the mass of fluid were
brought to a halt, it would occupy some
volume
in space; and we could
define its density to be the mass divided
by the volume.
With a little math
we can show that the aerodynamic forces are
directly proportional to the density of the fluid that flows by the
rocket.
Beginning with Newton's second law:
F = d (m * V) / dt
where F is the force, m is the mass, t is time,
and V is the velocity. If we integrate this equation, we obtain:
F = constant * V * m / t
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 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 r
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
dynamic pressure
of a moving flow is equal to one half of the density times the velocity squared.
Therefore, the aerodynamic force is directly proportional to the
dynamic pressure qof the flow.
F = constant * q
where the value of this constant is different than the previous constant.
During the launch of the Space Shuttle you may hear the commentator call out,
"MaxQ on the vehicle." This indicates the area of maximum aerodynamic forces
on the spacecraft.
Lift and drag depend linearly on the density of the fluid. Halving
the density halves the lift, halving the density halves the drag.
The fluid density depends on the type of fluid and the depth of the fluid.
In the atmosphere, air density decreases as altitude increases.
The relation
between altitude and density is a fairly complex exponential
that has been determined by measurements in the atmosphere.
A similar model has been developed for the
Martian atmosphere based on satellite
measurments.
You can investigate the effect of density and the other
factors on the flight of a rocket by using the
RocketModeler II Java Applet.
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