In the 1700's, Daniel Bernoulli investigated the forces present in
a moving fluid. This slide shows one of many forms of Bernoulli's
equation. The equation appears in many physics textbooks, as well
as fluid mechanics and airplane textbooks.
The equation states that the static pressure (p) in the flow plus one half
of the density (r) times the velocity (V) squared is equal to a constant throughout
the flow, which we call the total pressure (pt) of the flow.
pt = p + .5 * r * V^2
Again, this is only one
form of the equation and the restrictions for this form are
that the flow is inviscid, incompressible, steady, without heat addition,
and with negligible change in height.
If we consider other properties of the fluid, we can
derive other forms of the equation.
The equation was originally derived by considering the
conservation of mechanical energies within the fluid. The molecules
within a fluid are in constant random motion and collide with each
other and with the walls of an object in the fluid. The motion of the
molecules gives the molecules a linear momentum and the fluid pressure
is a measure of this momentum. If a gas is at "rest", the motion
of all of the molecules is random and the pressure that we detect is the
total pressure of the gas. If the gas is set in motion or
flows, some of the random components of velocity are changed in favor
of the directed motion through the collisions of the molecules.
We call the directed motion "ordered," as
opposed to the disordered random motion.
We can associate a "pressure" with the momentum of the ordered
motion of the gas. We call this pressure the dynamic pressure.
The form of the dynamic pressure is the density times the
square of the velocity divided by two. This form is similar to
a kinetic energy term (1/2 mass time velocity squared).
The remaining random motion of the molecules still produces a
pressure called the static pressure. At the molecular level,
there is no distinction between random and ordered motion. Each
molecule has a velocity in some direction until it collides with
another molecule or the walls of a container and the velocity is then changed.
But when you sum up all
the velocities of all the molecules you will detect the ordered
motion. From a conservation of energy and momentum, the static
pressure plus the dynamic pressure is equal to the original total
pressure in a flow (assuming we do not add or subtract energy in the
flow).
It is important when applying any equation that you are aware of
the restrictions on its use; the restrictions usually arise in the
derivation of the equation when certain simplifying assumptions about
the nature of the problem are made. If you ignore the restrictions,
you may often get an incorrect "answer" from the equation. For
instance, this form of the equation was derived while assuming that
the flow was incompressible, which means
that the speed of the flow is much less than the speed of sound. If
you use this form for a supersonic flow, the answer will be
wrong.
In the derivation of the equation, the dissipation of energy by
viscous forces was also neglected (inviscid
flow), and there was no introduction of energy allowed (no heat
addition). In the integration of the general equations, an assumption
was also introduced that restricts this equation to be true only
along a single streamline. If you have a
flow with many streamlines, the total pressure is a constant along
any streamline, but may vary from streamline to streamline. (If we
make an additional assumption that the flow is irrotational, then the
constant does not vary from streamline to streamline as long as the
height difference is small. If the height distance between
streamlines becomes large, you have to add a buoyancy term {density x
gravity x height} to the left side. Irrotational flows are
flows that conserve angular momentum, which seems fairly restrictive,
but, in fact, occurs quite often in aerodynamics. An ideal source,
sink, uniform flow, and point vortex are all irrotational flows.
The fluids problem shown on this slide is low speed flow through a
tube with changing crosssectional area. For a streamline along the
center of the tube, the velocity will slow from station one to two.
Bernoulli's equation would describe the relation between velocity,
density, and pressure for this flow problem. Along a low speed
airfoil, the flow is incompressible and the density remains a
constant. Bernoulli's equation then reduces to a simple relation
between velocity and static pressure. Since the velocity can vary
along the streamline, this equation can be used to compute the change
in pressure. The static pressure integrated along the entire surface
of the airfoil will give the total aerodynamic
force on a body. This force can be broken down into the
lift and drag of the airfoil.
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