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Re: [Phys-L] Bernoulli's equation



On 12/15/2014 08:27 PM, I wrote:
Every aircraft airspeed indicator on earth is basically
a pressure gauge.

Let me explain why that is important. This winds up being
a nice exercise involving Bernoulli's equation plus scaling
laws.

Suppose one day you take off from Phoenix / Deer Valley
http://www.airnav.com/airport/KDVT
fly around for a couple of hours, and then land back at
Deer Valley.

Suppose the next day you take off from Deer Valley, fly
around for a couple of hours, and land at Springerville
http://www.airnav.com/airport/KJTC

That's no big deal, except that the elevation is more
than 5500 feet higher, 7055 versus 1478. The air is
noticeably thinner. The pressure distribution above
and below the wing can be understood in terms of
Bernoulli's principle, ½ ρ v^2, so when ρ is smaller
you need more v to support the weight of the airplane.

This is a big deal, because if you are too fast or too
slow on final approach, bad things are going to happen.

As mentioned above, the airspeed indicator is a pressure
gauge. It can be understood in terms of Bernoulli's
principle, ½ ρ v^2 ... so it is affected by the density
in exactly the same way as the wing. Therefore you fly
the approach using the exact same indicated airspeed as
the day before. The air temperature doesn't matter,
the humidity doesn't matter, the pressure doesn't
matter ... the only thing that matters is ½ ρ v^2,
and that affects the wing and the airspeed indicator
in the same way. This is pretty nifty. (*)

There is of course a difference between indicated
airspeed and true airspeed. At the high-elevation
airport, your true airspeed will be higher, and you
will therefore need a longer runway. However, on
final approach all you care about is the indicated
airspeed, not the true airspeed.

So far we have implicitly been assuming that the weight
of the airplane was the same for both landings. This
is not necessarily the case. Depending on how much
fuel and how much payload you have, the weight of the
airplane could change by a factor of 2 from one day
to the next.

So the question arises, how does approach speed scale
with the mass of the airplane? Give me a scaling law.
If you know how to land the airplane at one weight,
what do you have to change to land at another weight?

For today only you cam make the assumption that
the Pitot-static system is ideal. This is *not*
a safe assumption in the real world. You seriously
need to convert IAS to CAS, apply the scaling law,
and convert back. But let's not worry about that
right now. Assume that IAS=CAS, for today only.

You can also assume that the shape of the airflow
around the airplane is controlled by the geometry
of the situation. That is, if you fly slower at
the same angle of attack, the velocity field is
the same, just slower, everywhere in the same
proportion. This is actually true to an excellent
approximation. (*)

===========

This is a 100% real-world topic. It is definitely
not plug-and-chug. To make progress you need to
figure out the relationships between pressure, area,
force, weight, density, and speed. A lot of that
stuff drops out from the final answer, but still
it is part of the logical reasoning process.

Interesting aside: The Pilot's Operating Handbook
is the official instruction book that comes with
the airplane. Typically it says *nothing* about
correcting the stalling speed, approach speed, or
anything else when the weight changes. A
fundamentalist who insists on doing things strictly
according to The Book is going to be in serious
trouble. The guy who has a clue about Bernoulli
is going to do a lot better. Don't laugh, I've had
people argue with me that they /have/ to use the
approach speed in The Book, because it's in The Book.
At that point I say "Why it is that you are using
twice as much runway length as The Book says it
should take? Either The Book is wrong about the
approach speed or it's wrong about the runway
requirements; take your pick. You'd better mend
your ways, or one of these days you're going to run
off the end of the runway and wind up in the ditch."

---------------

(*) The general rule in fluid dynamics is that everything
is nastier and more complicated than you expected it to
be. However, in this situation we have two things that
are simpler than you might have expected: The invariance
of the lift/IAS relationship, and the simple similarity
of the flow field (at any given angle of attack).