I posed some questions in hopes of getting people to think about what
"pressure" means in a moving fluid.
Here are some of the answers:
1) Terminology: when people talk about "the pressure" of a fluid, unless
context demands otherwise, that means "static pressure" not any other kind
of pressure. This is important for reasons explained below.
2) Suppose we have two big airmasses sliding past each other. We have
various observers, including some on the ground and some in balloons
comoving with the airmasses.
If you ask various observers to evaluate the pressure (i.e. the static
pressure) of a given parcel of fluid, they will all give the same
answer. Static pressure is independent of the observer's frame of reference.
Why is that?
That's because the static pressure is DEFINED in the rest frame of the fluid!
It's like asking for the rest mass of an electron; it doesn't matter what
the observer's frame of reference is. I don't know whether this is a deep
principle of physics or a tautology (or both :-).
-- It's a tautology in so far as the pressure is DEFINED in the rest
frame of the fluid.
-- It becomes an interesting principle of physics when you add the idea
that two airmasses moving past each other will be out of equilibrium unless
their pressure (i.e. static pressure) is the same. This means that
pressure is not just a defined quantity, but an important defined quantity.
(Nitpickers note: Yes, I am aware that special relativity
introduces corrections to the foregoing. We don't want to go there.)
Remember: static pressure is what you would observe on a pressure gauge
carried by a balloon comoving with the fluid.
3) In contrast, kinetic energy is not independent of frame of
reference. Just as the kinetic energy of a baseball depends on the
velocity of the observer, the kinetic energy of an air parcel depends on
the velocity of the observer.
4) The pressure P (i.e. static pressure) tells us the potential energy per
unit volume of the fluid.
The dynamic pressure Q = 1/2 rho V^2 tells us the kinetic energy per unit
volume of the fluid.
The total mechanical energy per unit volume of the fluid is P+Q. Potential
plus kinetic. Hardly surprising.
5) Suppose you have some fluid which is stationary in the lab
frame. According to a lab-frame observer it has
Plab = whatever
Qlab = 0
and to another observer walking past with some nonzero velocity it has
Pwalk = Plab
Qwalk > Qlab
6) More terminology: The combination P+Q has dimensions of pressure, and
has been given the name "stagnation pressure" because it tells us something
about the stagnant air in a Pitot tube.
Warning: This terminology is confusing. In everyday speech, "stagnant
fluid" and "static fluid" mean more-or-less the same thing, but the
technical terms "static pressure" and "stagnation pressure" mean quite
different things -- and people tend to have trouble remembering which is which.
When talking to non-experts, my personal preference is to avoid the term
"stagnation pressure". One can use other terms e.g. "Pitot pressure" or
"head" or "mechanical energy density" instead.
7) Bernoulli's principle tells us that moving air should have less
pressure, Other Things Being Equal. But less than what? And what Other
Things are Being Equal?
Answer: a parcel of moving air has less pressure than a parcel of
nonmoving air _having the same energy_! (Specifically, having the same
mechanical energy density P+Q).
The simplest sufficient condition for applicability of Bernoulli's formula
is to consider the _same_ air parcel as it moves along a streamline, in
conditions where
--) we can neglect energy dissipation (such as friction and/or thermal
conduction), and
--) we can neglect PdV work done on neighboring parcels.
These conditions often hold to high accuracy, making Bernoulli's formula
extremely useful even in its simplest form.
There are other sufficient conditions that hold more generally. If you
have air parcels on different streamlines, you can (with a little extra
effort) check to see that they have the same P+Q, and then apply
Bernoulli's formula in its simplest form. Or you could generalize the
formula to account for the variation in P+Q.
Even more generally you can account for "PdV" if you need to (when the
speed is not small compared to the speed of sound.
To say the same thing in a negative way: If you have two different air
parcels that started out with different energy (for instance if they reside
in different airmasses) you will get utter nonsense if you blindly plug
numbers into Bernoulli's formula. In general it is a bad idea to turn the
crank without knowing what the symbols mean.
Charlatans often take a curved piece of paper and blow over the top and
pretend that this exemplifies Bernoulli's principle. Utter nonsense! The
mouth-air flowing over the paper has a different P+Q, not comparable to the
ambient-air underneath the paper. For the experimentally inclined: Try
blowing on an _uncurved_ piece of paper or card-stock, at any angle you
like, with any velocity you like. Observe that the high-velocity mouth-air
does _not_ have a lower pressure than the low-velocity ambient-air.
8) Here's how altimeters work: The air outside the window of my airplane
has energy density P+Q, where Q depends on my airspeed. The faster I go,
the greater the Q.
If I take this air and bring it to rest (relative to the airplane) in a
Pitot tube, i.e. without losing any of the energy, the local pressure p in
the Pitot tube will be p=P+Q. (Little p refers to the local pressure; big
P refers to the pressure in the free stream.)
On the other hand, if I don't bring the air to rest, and just let it rush
past the side of the airplane at 100% of my true airspeed, the local
pressure will be less than P+Q. According to Bernoulli's formula, the
local pressure will be 1 Q less than that. So the moving air exerts a
pressure on the side of the airplane equal to P+Q-Q. I can capture some
air at this pressure and feed it to my altimeter. So my altimeter measures
P+Q-Q = P which is the static pressure, the same thing as would be measured
by a balloonist comoving with the air at my altitude.
The place where the altimeter samples the air is called the "static port"
because it measures the static pressure P -- but that does not mean that
the air there is "static" in the vernacular sense -- far from it! It is
crucial that the air be rushing past that point at 100% of the true airspeed.
===================
Bottom line: This isn't very tricky. It's just basic physics.
-- you need to think about the potential energy
-- you need to think about the kinetic energy
-- you need to think about what changes and what doesn't when you
change from one reference frame to another.