This caused a more in depth discussion of lift on another board. Here is one response that discusses Bernoulli in depth, but pretty well says it's much better to keep it simple for most pilots <g>
Let me know if you'd like to see the rest of the post.
Best,
Dave
es, I can hear the groans out there. First, please, if you reply to this
(and it's OK not to), remember to trim it down before posting.
I come to praise
Bernoulli, not to bury him, his dad or his uncle. In the
periodic
Bernoulli lift debates on rec.aviation.<pick your favorite> there
is always a great deal of heat generated. Personally, I believe that most
schoolkids (and pilots) are best served by a pure Newtonian description of
air being forced downward (and lots of it) instead of a discussion of
differential pressures and
Bernoulli, and I'd like to briefly describe why
in a qualitative and historical way. In short, it's because most of you guys
aren't ready for it and I think it hinders, not helps, the understanding of
what is going on. And, I should mention, I am also not competent to apply
Bernoulli's equations to *any* real system today, either. It is not a tool I
use, and I also make no claim to understand the nuances of lift of a real
wing in real air.
When you first study physics as a schoolkid you do very simple problems.
Basic time, speed and distance problems like you might get in fifth grade
math word problems, or the FAA knowledge exam route planning questions.
Motion in one dimension. In a traditional high school physics class you
might get bombarded by coordinate systems and transformations that expect a
budding knowledge of trigonometry, and you learn some basic Newtonian
physics. More detail of motion in one dimension. Two dimensions, even.
Conservation of energy, conservation of linear and rotational momentum. Neat
stuff, often relying on fictions such as massless pulleys and ropes,
frictionless bearings and other flights of fancy, and there is a great deal
of insight that can be gained with some elementary one variable differential
Calculus (Calculus, not coincidentally, is also something Newton figured out
first).
Go on to college to study more science and you get more of the above your
freshman year. Including more massless and frictionless pulleys. Digging out
my old Halliday & Resnick "Physics", there were about 650 pages devoted to
25 chapters that I think were covered in the first year. "
Bernoulli's
equation" is presented in chapter 18 but only in a simplified form that
assumes steady and incompressible flow. Chapter 18 (22 pages) is an
introduction to fluid dynamics, but it's mostly very basic stuff devoid of
calculus, in part because none of the frosh involved had yet taken any
significant multivariable calculus, or how to actually make any sense of
difficult differential equations (or diffyQ's as we'd call them),
multivariable or not. The fluid used in the modeling at this level is what
Feynman described as "dry water", every bit as real as those frictionless
and massless pulleys.
Freshman (2nd semester, maybe), Sophomore and Junior years you take more
math like multivariable calculus, linear analysis and applied analysis.
Methods to solve fiendishly complex multivariable differential equations,
partial differential equations. I reached my own level of incompetence in
math with a class in Complex Analysis where math really does diverge from
Reality. Powerful stuff, nearly indistinguishable from magic.
Late in the game, someone actually majoring in physics would take a class in
Mechanics and another in Statistical and Thermal Physics. If engineering,
Fluid Dynamics, maybe a course studying Compressible Flow afterwards. In
these classes (I took the physics route) you start dealing seriously with
different ways of doing that same old Newtonian physics but this time in
much more general terms and with mathematical tools which allow a different
sort of more powerful physical modeling. Let me quote from the introduction
of the text "Mechanics" by Symon:
"A great many of the applications of classical mechanics may be based
directly on Newton's laws of motion. ... There are, however, a number of
other ways of formulating the principles of classical mechanics. The
equations of Lagrange and Hamilton are examples. THEY ARE NOT NEW PHYSICAL
THEORIES [my caps], for they may be derived from Newton's laws, but they are
different ways of expressing the same physical theory. They use more
advanced mathematical concepts, they are in some respects more elegant than
Newton's formulation, and they are in some cases more powerful in that they
allow the solutions of some problems whose solution based directly on
Newton's laws would be very difficult."
Newton's Principia was first published in 1686. In 1687, James
Bernoulli,
father of Daniel (the eventual author, in the 1730's, of Hydrodynamica and
the
Bernoulli who is most often remembered) was appointed chair of a math
department at a university in Switzerland. James was a noted mathematician
who was a very early proponent of the new calculus. Skip forward, and the
great mathematician and physicist Leonhard Euler, a friend of Daniel's, was
the son of one of James's students, and studied mathematics under James'
brother younger brother John
Bernoulli. The forefront of physics and math
were concentrated in a few centers in the 18th century, and these guys were
at the center of it as were the aforementioned Lagrange and Hamilton. Rather
than writing equations of motion directly in the style of Newton, the
Hamilton,
Bernoulli and Lagrange equations were derived from the calculus
and Newton's principles of conservation of energy and momentum. Equations of
motions of fluid particles by their positions were first developed by Euler
and are the so-called "Lagrangian equations" of fluid mechanics. The
equations of motion of a moving fluid in terms of density and velocity at
each location are sometimes referred to as the "Eulerian equations"; the
derivation of
Bernoulli's equations involve the energy of a moving fluid
using these. Very interesting, powerful stuff, but the basic physics is
exactly the physics of Newton. No new principles, just new and potentially
more powerful tools to use for further study and application.
Let me restate the above: there is no
Bernoulli Theory that generates lift
in a way counter to Newton's F=mA. What
Bernoulli did in his day (along with
many others) is essentially to take Newton's axioms of physics and the
calculus and apply them in very beautiful, insightful and innovative ways.
Bernoulli saw far because he stood on the shoulders of giants, as did
Newton.
However, none of these tools are useful for kids and pilots who are
interested in the principles that keep airplanes up in the air. Yes,
Bernoulli's insights can model the pressures above and below a wing and the
lift. It still boils down to a Newtonian "the wing generates lift up because
a lot of air is moved down". Why not keep it simple? Proper, efficient
airfoils and barn doors will generate lift at speed if at a proper and
positive angle of attack to move the air down. Airfoils have much better
low speed handling characteristics and less drag at cruise than barn doors,
but angle of attack remains the critical concept that pilots must be aware
of.
Langewische was right. The wing of an airplane in flight bats the air down.
Using this requires at most a very basic knowlege of 17th century physics
and perhaps 11th century algebra. Leave the 18th century physics and the
newer stuff to the specialists.
-Greg
PS A close relation (he was a CFI and ATP in a past life, before he started
flying satellites for a living) has a real aeronautical engineering degree,
and I did have enough of a clue to run the above by him before posting
today. If you want more, keep reading. He added,
"That's a good emphasis on the point that
Bernoulli's equation is simply an
energy equation and is ultimately based on Newtonian formulations in the
first place. And the fact that
Bernoulli's equation is used with the
assumption that the flow is 'steady and incompressible'. To go a little
further, it also assumes inviscid and irrotational flow - of which, air is
neither. Which leads to the question - why mention
Bernoulli in the first
place?
And the answer is, because you use other boundary conditions and flow models
to overlay (superpose) real fluid conditions on the flow so that you can
then pull out the
Bernoulli equation to solve for the pressure distribution.
And in so doing, you mask the reality that - as you said - the wing is
turning a mass of air downward to create the resultant lift force on the
wing. At which point, you remember that F=MA is a VECTOR equation with a
little arrow over the F and the A.
So in essence I agree with you, although I think you gave them too much
history. They don't deserve it.
The standard method prior to computer solution of the Navier-Stokes
equations went like this:
1. Assume an inviscid, irrotational flow (Air is neither, but we'll work
that out..)
2. Assume a potential flow, centered at the aerodynamic center of the wing,
that creates a 'vortex flow' about the wing in a clockwise direction.
3. Impose a boundary condition at the trailing edge that requires the
trailing edge upper and lower flow directions to be parallel, and of the
same magnitude.
4. Assert that a 'starting vortex' of opposite sense to the main vortex is
shed by the wing to establish Condition 3. This starting vortex never dies
off in the inviscid formulation. It remains where it originally was in
space, enforcing the irrotationality of the flow which is disrupted by the
assertion of Condition 3 (commonly known as the 'Kutta condition').
5. Now superpose flows 1. and 2., impose Condition 3, include assumption 4,
and calculate the velocity potential wherever you want. Use this velocity
with
Bernoulli's to calculate the pressure.
Simple, huh? And real physically intuitive. [he also is more sarcastic than
I am]. It's just a bunch of ROT designed to model a flow in a way that an
ordinary human can calculate it.
The reason a 'vortex' is assumed is to IMPOSE the downward flow seen behind
the wing - to visualize this, add a circulatory flow to a linear flow and,
at arbitrary points ...
