When two solid objects interact in a mechanical process,
forces
are transmitted, or applied, at the point of contact. But when a
solid object interacts with a fluid, things are more difficult to
describe because the fluid can change its shape. For a solid body
immersed in a fluid, the "point of contact" is every point on the
surface of the body. The fluid can flow around the body and maintain
physical contact at all points. The transmission, or application, of
mechanical forces between a solid body and a fluid occurs at every
point on the surface of the body. And the transmission occurs through
the fluid pressure.
Variation in Pressure
The magnitude of the force acting over a small section of an
object immersed in a fluid equals the pressure p times the area A
of the section. A quick units check shows that:
p * A = (force/area) * area = force
As discussed on the fluid pressure slide, pressure is a
scalar quantity related to the momentum of the molecules of a fluid.
Since a force is a
vector quantity,
having both magnitude and direction, we must determine the direction of the
force. Pressure acts perpendicular (or normal) to the solid surface
of an object. So the direction of the force on the small section of
the object is along the normal to the surface. We denote this
direction by the letter n.
The normal direction
changes from the front of the rocket to the rear and from the top to
the bottom. To obtain the net mechanical force over the entire solid
object, we must sum the contributions from all the small sections.
Mathematically, the summation is indicated by the Greek letter
sigma ()
The net aerodynamic force F is equal to the sum of the product of
the pressure p times the area A in the normal direction.
F = p * A * n
In the limit of infinitely small sections, this gives
the integral of the pressure times the area around the closed
surface. Using the symbol S dA for integration, we have:
F = S (p * n) dA
where the integral is taken all around the body. On the figure, that is why the
integral sign has a circle through it.
If the pressure on a closed surface is a constant, there is
no net force produced because the summation of the directions of the
normal adds up to zero. For every small section there is another
small section whose normal points in exactly the opposite
direction.
F = S (p * n) dA = p * S n dA = 0
For a fluid in motion, the velocity has different values at different
locations around the body.
The local pressure is
related
to the local velocity, so
the pressure also varies around the closed
surface and a net force is produced.
Summing
the pressure perpendicular to the surface times the area
around the body produces a net force.
F = S (p * n) dA
Definitions of Lift and Drag
Since the fluid is in motion, we can define
a flow direction along the motion. The
component
of the net force
perpendicular (or normal) to the flow direction is called the
lift; the component of the net force along
the flow direction is called the drag. These
are definitions. In reality, there is a single, net, integrated force
caused by the pressure variations along a body. This aerodynamic force
acts through the average location of the pressure variation which is called the
center of pressure.
Velocity Distribution
For an ideal fluid with no
boundary layers,
the surface of an object is a
streamline.
If the velocity is low, and no energy is
added to the flow, we can use Bernoulli's
equation along a streamline to
determine the pressure distribution for a known velocity
distribution. If boundary layers are present, things are a little
more confusing, since the external flow responds to the edge of the
boundary layer and the pressure on the surface is imposed from the
edge of the boundary layer. If the boundary layer separates from the
surface, it gets even more confusing. How do we determine the
velocity distribution around a body?
To correctly determine the velocity distribution, we have to solve
equations
expressing a conservation of
mass,
momentum, and
energy for the fluid passing the object.
In some cases, we can solve
simplified versions
of the equations to determine the velocity.
For coasting rockets, this is particularly difficult because the flow
normally separates at the nozzle end of the rocket.
Summary
To summarize, for any object immersed in a fluid, the
mechanical forces are transmitted at every point on the surface of
the body. The forces are transmitted through the pressure, which acts
perpendicular to the surface. The net force can be found by
integrating (or summing) the pressure times the area around the
entire surface. For a moving flow, the pressure will vary from point
to point because the velocity varies from point to point. For some
simple flow problems, we can determine the pressure distribution (and
the net force) if we know the velocity distribution by using
Bernoulli's equation.
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