Index
October 29, 2002: from Andrew
October 29, 2002: from
Andrew #2
October 30, 2002: from Andrew
October 30, 2002: from Andrew #2
November 1, 2002, from Andrew
November 5, 2002: Run Ads
November 8, 2002: from Andrew
November 12, 2002: from Andrew
November 14, 2002: from Andrew
November 14, 2002: to
Professor Weltner
November 22, 2002, From Andrew
October 29, 2002:
from Andrew
Index
I reviewed the web site, and I'm afraid that your friend's
description of current theories of lift is a bit oversimplistic to the
point of inaccuracy. I would not recommend that he attempt publication
without a more rigorous derivation.
I think it would also be good if he did some more background research
on compressible flow and the kinetic theory of gasses  his basic
concepts are not entirely wrong, but he is clearly missing quite a bit
of background work that has been done. It is incorrect to
suggest that the drop in pressure from an incompressible radially
expanding fluid is due a density change. Water will not change density
sufficiently to account for lift in hydrofoils for example. In general,
water can be treated as a totally incompressible fluid (as can be seen
in the PV diagram for water), and you can generate lift quite well with
a hydrofoil wing  there is no radial expansion there ...
I would like to go into more detail on all of the fluid flow concepts
he is missing (the difference between static pressure and total pressure
jumps out immediately), but I don't really have the time or energy to
pursue this. I would strongly suggest that he take some courses in fluid
dynamics before attempting to publish anything. I'm afraid he won't be
taken seriously until he addresses the actual state of knowledge in a
much more rigorous way.
I don't want to be a downer, but proposing a new theory of lift is
not at all trivial (I don't mean to imply that what he has done is
trivial  it's just that it's clear that his understanding of the
existing theories is not accurate). In reality, there is more than one
mathematically correct way to explain many phenomena. An explanation of
lift can be made with Bernoulli or from a pure conservation of momentum
argument (entirely ignoring pressure). I'm sure his mathematical
explanations could work for the particular scenarios he has laid out,
but knowing when to apply what formulas (and for what reason) is what
really matters, and for that he needs some more background in the field
of fluid dynamics. I am not an expert in this field, but it is clear
from his writings that I am somewhat more familiar with it than he is. I
would encourage him to study the field in greater depth before
attempting to publish a theory about this field. Sorry to be a
whistleblower.  Andrew 
Response by Ed
1. Andrew states that "It is incorrect to suggest that the drop
in pressure from an incompressible radially expanding fluid is due a
density change."
Note that from PV = NRT, P = N/V * RT. Since N/V is density, Pressure
is indeed proportional to density.
Current theories of lift rest on the notion of incompressible flow.
Radial momentum theory rests on the notion that fluids do indeed
compress. Indeed, compression and decompression are at the very heart of
the theory. As fluid expands radially, it occupies greater volume and
the density decreases, the fluid decompresses and the pressure drops.
Actually, socalled incompressible fluids such as water show much more
lift than compressible fluids such as air. This is because the fluid
"resists" compression (decompression in the case of lift) and
the effect of radial momentum to separate the molecules results in a
much greater pressure drop. Notice that it takes much more
pressure to compress water than it takes to compress air.
2. In the case of the hydrofoil wing, to the extent that lift occurs
as a result of the angle of attack, Andrew is correct in that radial
momentum does not apply. Nor does Bernoulli's Principle apply. This case
simply resolves by applying linear momentum models such as when pool
table balls strike each other.
To the extent that the hydrofoil curves, such as in the classic
diagrams of air flowing around airplane wings, then there is some
lift where the air, in trying to navigate the curve, separates from the
wing in a radial expansion pattern.
Radial Momentum explains the TubeandCone
experiment and the Levitator
Table Experiment. Theories resting on noncompressible flow fail to
explain these experiments.
3. I appreciate Andrew's encouragement to survey the current
literature, and I have done so. I have posted several examples
from standard texts on the subject. By nature, a new theory that
challenges an existing theory is going to conflict with standard
practice. To evaluate a new theory, one must temporarily suspend the
notion that the old theory is correct, and proceed from deeper and more
fundamental notions. 
October
29, 2002 From: Andrew
Index
Responses from Ed Seykota, to this email are in red.
ok, I'll bite this one time, but I really don't have time to teach
fluid dynamics over email, so after this one, I am going to have to
leave it up to Ed to do what he thinks is right. You both know that I
think he should learn more about the existing subject before strongly
publicly advocating his own theory  it makes him seem less
credible. In any case, I have inserted my response to what he
wrote:  Andrew
Response 1. Andrew states that "It is incorrect to
suggest that the drop in pressure from an incompressible radially
expanding fluid is due a density change." Note that
from PV = NRT, P = N/V * RT. Since N/V is density, Pressure is indeed
proportional to density.  That is the ideal gas law. It does NOT
apply for incompressible fluids such as water  I challenge you to show
experimentally that it does if you don't believe me  does the volume
of water in a rigid container change when you pressurize the
container??? No! (at least not appreciably, and definitely not in
accordance with the ideal gas law). In an incompressible fluid flow the
density is constant (hence incompressible).
If you perform the spool and levitator
experiments using water, you get much more lift. Indeed, a cavitation
ring appears just around the center of the disk, just outside the
entrance, just beyond the place where the fluid enters the interplate
gap, as in Textbook example #1.
Since water is much less compressible than air, it does takes much more
pressure to compress it. Similarly, the attempt to decompress water,
gives rise to substantially greater lift, precisely since the water is
incompressible. My empirical studies show radial momentum induces more
lift with incompressible fluids.
Air is a compressible fluid (and generally does behave like an ideal
gas), but it will behave as an incompressible fluid for low Mach number
flow (generally Mach numbers below 0.3 or so). When the speed of a
compressible flow approaches or is above the speed of sound in the flow,
the density of the fluid will change appreciably. The speed of sound is
given by a = sqrt(gamma*R*T)  this can be derived from kinetic theory
and is about 340 m/s in air at room temperature. Therefore for absolute
upstream to downstream pressure ratios much less than the critical
pressure ratio for the choking condition (which for air is around 2),
the flow of air can be approximated as incompressible.
This is also the case for aircraft that fly at airspeeds below Mach
0.3 (below roughly 100 m/s). If your vehicle travels faster than that
through the air, then you will see significant compressible effects and
you could have local density changes, but since there are many airplanes
that fly at speeds lower than 100 m/s, it is not necessary to have a
density change in order to have lift ...
I agree that the Reynolds Number and behavior
of air changes as it approaches fractional Mach numbers. My own flow
studies confirm this. This, however is only marginally applicable to
lift since almost all lift has to do with the angle of attack of the
wing, and not the curvature of the wing. The curvature of the wing
mostly entrains the air into laminar flow, and the resultant decrease in
drag can appear as net forward thrust, and this increases lift. If wing
curvature and if curvatureinduced lift induction were significant
factors, then upsidedown air flight would be impossible and helicopter
blades with symmetrical rotor cross sections would not rise.
2. In the case of the hydrofoil wing, to the extent that lift occurs
as a result of the angle of attack, Andrew is correct in that
radial momentum does not apply. Nor does Bernoulli's Principle
apply. This case simply resolves by applying linear momentum
models such as when pool table balls strike each other.
You can explain wing lift via a momentum argument, that is valid, but
it is also valid to explain wing lift via Bernoulli for flow speeds much
lower than Mach 1. They are two equally valid descriptions of the same
physical phenomenon. As I said before, there is often more than one way
to correctly look at a physical phenomenon, but implying that a density
change is necessary for lift is flat out wrong. To the
extent that the hydrofoil curves, such as in the classic diagrams
of air flowing around airplane wings, then there is some lift
where the air, in trying to navigate the curve, separates from the
wing in a radial expansion pattern.  No, actually, in a
properly designed wing, flow separation is avoided  if the flow does
separate from the surface, a condition known as "stall" is
encountered  this is very bad for lift  it generally occurs at
angles of attack greater than 10 or 15 degrees depending on airfoil
geometry. It is also possible to have essentially "flat plate"
airfoils that produce lift (without geometric curvature) at small angles
of attack  just stick your hand out of a car window and experiment...
you can feel it for yourself  and I guarantee you that you can feel
this even if you are driving at substantially subsonic speeds ;)
The angle of attack is the main factor in lift
at most speeds. There is a very small amount of lift at curvatures,
where the flow, due to momentum, tends to continue in one direction
while the wing continues in a different direction. At these places, the
air decompresses due to radial momentum and there is a pressure drop.
This only offsets the negative lift that accompanies the initial
diversion of the air flow up and over the wing. None of these curvature
effects, however are significant relative to the angle of attack of the
entire wing. For very large angles of attack and for very sharp
curvatures, the air separates and becomes turbulent and dissipates
power.
Radial Momentum explains the TubeandCone experiment and the
Levitator Table Experiment. Theories resting on noncompressible flow
fail to explain these experiments.  I don't doubt that you
can come up with some formulas that describe what you see, but the text
book figures you have compared to offer very simplistic explanations
that are meant to convey a concept  not be taken as absolute
statements of the current state of knowledge of this subject, or even as
real analyses of what is depicted. You've got to dig deeper.
I have dug deeply into the math and have
demonstrated fallacies in the math in Textbook examples #1,
#2 and #3.
I have also constructed a model based on radial momentum that explains
the cavitation ring and that also provides reasonably accurate qualitative
and quantitative validation.
3. I appreciate Andrew's encouragement to survey the current
literature, and I have done so. I have posted several examples
from standard texts on the subject. By nature, a new theory that
challenges an existing theory is going to conflict with standard
practice. To evaluate a new theory, one must temporarily suspend
the notion that the old theory is correct, and proceed from deeper
and more fundamental notions.  I still think you should look a
little harder at the existing literature. In order to garner interest
from the scientific community, a new theory should be able to explain a
phenomenon that is of interest AND is as yet unexplained.
The levitator
and the tube and cone
experiments nicely isolate the lift phenomena and allow differential
diagnosis of the Bernoulli Principle (velocity induces lift, based on
incompressible fluid) and the Radial Momentum Theory (based on radial
expansion induction of reducing density).
Right now, we have very good models to explain lift  we have been
building aircraft based on these models for almost 100 years, and they
work very well.
The design of aircraft relies heavily on
empirical testing, such as with wind tunnels, not only on formulas,
since many of the formulas, such as NavierStokes, are fundamentally
intractable. In particular, the practical formulas for airplane wing
lift mostly rely on angle of attack and do not rely on Bernoulli's
Principle to compute lift. Attempts to employ Bernoulli's Principle to
explaining lift leads to complicated patches such as
"circulation". Again, curvature is not a big contributor to
lift, except in that it entrains laminar flow and reduces drag.
I
don't have the time to do the detailed analysis of your experiments that
would be necessary. I suggest you learn how to do them based on
Computational Fluid Dynamics and see if that will explain what you see.
I'd be willing to bet that it would. Good luck!  Andrew
I would be willing to bet that radial momentum
explains the levitator much better than the Bernoulli Principle does. If
you are certain that I am wrong, I wonder how much you would be willing
to bet, and at what odds, to take the other side. For example, would you
be willing to bet $100.00 at 100:1 in your favor?
I have actually studied the literature. Indeed
some of the examples I use for incorrect analysis come directly from
Potter and Foss.
See the model
of the levitator, that shows a cross section of density, pressure,
Reynolds number and lift along the contour of the levitator interplate
gap. Conventional approaches do not provide such analytics. 
October 30, 2002 from
Andrew
Index
Responses from Ed Seykota, to this email are in red.
Send #1:
Inspection of this link: http://www.seykota.com/rm/model/model.htm
suggests that Ed is using fundamental fluid dynamics (not new theories
of lift) in his model of the levitator  I guess he's just calling it
"radial momentum," in which case the difference is purely
semantic. It would appear that he is correct in stating that the
Bernoulli principle is inaccurate to completely describe "the
levitator" and therefore perhaps it should not be used as an
example of Bernoulli in the text books, but in his model it appears he
has basically implemented at simple CFD routine based on the fundamental
equations of fluid dynamics  there is nothing "new" about
the equations in the model that I could see, so the fact that it
accurately describes the measured lift is not surprising. What then is
different about the theory of radial momentum? I suggest that your
numerical models ARE selfderived CFD, which is very impressive, but not
akin to a "new theory of lift." I would agree that the use of
the levitator example should be removed from discussions of the
Bernoulli effect without some type of qualification. Perhaps what you
should propose is a correction to the text book examples citing your CFD
analysis. I do not think that this constitutes a new theory of lift,
however. I wish I had seen the modelling page earlier  you
clearly base your model on the fundamentals of fluid dynamics, which is
good, but again not really "new." Good work on spotting
the text book error! I still don't see what the new theory of lift is
though. Andrew
The Bernoulli Principle has been used for a
long time as a universal explanation of flowinduced lift, in the
literature, on websites and throughout academia. The classic thinking is
that, based on Bernoulli's Principle, fast fluid == lift, assuming
incompressibility.
What is new is that many of these phenomena can
more accurately map onto a model where radial expansion == lift,
assuming compressibility. Applying Bernoulli's Principle to lift
situations results in delusions.
Application of Radial Momentum is new and results in more accurate
modeling of the physical universe.
The course of scientific advance seems to
follow several steps.
1. The community thinks you are crazy.
2. The community gets irritated since the new
way is difficult to disprove, and they attack the person, not the idea.
3. The community accepts the idea as obvious
and not really very new or insightful.
4. Other members of the community claim they
thought of it first.
Send # 2:
Please add this to the correspondence page: I looked at
the textbook problems again, and I think the root of the discrepancy is
that the texts use these "levitator" configurations as
examples of Bernoulli's Principle, when in a real, physical levitator,
the actual flow cannot accurately be described by Bernoulli. Ed
is absolutely correct in pointing that out. I don't want to make
excuses for the authors of the textbooks because I don't believe that
they should use nonphysical examples to relate concepts, but I would
bet that if confronted they would say that they were just using the
problem to illustrate a concept of incompressible fluid flow  and that
it should not be taken as an actual technique to estimate the forces. I
would disagree with their use of that example, but I would not disagree
with the general correctness of Bernoulli's principle for isotropic,
incompressible flow.
There is almost universal employment of the
Bernoulli Principle to explain flowinduced lift. Proper employment to
situations of isotropic, incompressible flow are certainly appropriate.
However, these situations do not include lift.
I don't know if this has been at all helpful, but I hope that
Ed has found this discourse of some value. I was definitely too
quick to discredit his work early on in my mind because of the
bold statements he made ("new theory of lift" etc.) and
a few mistaken concepts that he has, but it turns out his analysis
looks like it is based on real fluid dynamics principles. So even
if they are not "new" it is possible that they are
correct, and he is correct that the levitator example given in the
texts is not an accurate physical description of the actual
levitator phenomenon. Although I have not spent the time to double
check all of the math, I would not be surprised if Ed's model of
the levitator flow is at least prettyclose to what he estimates.
There are still some concepts he is missing, but his analysis of
the levitator at least starts off well.
I find the discourse very useful, both to
challenge my understanding and also to help to focus my presentation
along the lines of currently conventional methods.
I would be interested in an enumeration of my
missing concepts. I have taken the liberty to employ some variables not
generally in the lexicon of conventional fluid dynamics, in order to
name auxiliary variables useful in the Euler stepwise integration
model, such as "pushacceleration." I gave names to such
auxiliaries as best as I could, knowing that they were not in general
use.
 In order to garner credibility in academic circles it's
always good to start from the conservation equations (mass, momentum,
and energy)  that will get people to listen to the next steps...
starting with "here is a new theory of lift" is a sure
way to get people to think you are crazy. If you start with the
old and derive the new you will be in a much better position than
if you just state your concept at the beginning. Academic progress
is all about building on previous work. I would encourage you to
show how you have done that and give credit to the people who have
done the work that you base yours on  that is how to get
published. Andrew
I shall review the presentation with an intent
to reduce the level of contentiousness and raise the level of
credibility. I do not have much experience in how to confront a thesis
with an antithesis without some degree of contentiousness. I realize
that striking the correct balance is likely quite important.
Send #3:
The tube and cone experiment can also be explained with existing
fluid mechanics and structural mechanics  one doesn't need a
"new" theory of lift.  Andrew
I would be very interested in observing
these analytics. I feel you are underestimating the complexity of this
offer. I doubt there is a way to explain the phenomenon without using
radial momentum. If you can produce such analytics, for the levitator,
or even for the cone, it would argue very effectively against my thesis
and, indeed, put it, once and for all, to rest. The cone might be more
complicated since the control volume varies inversely with the square of
r, whereas the levitator control volume varies inversely with r.
Send #4:
Ok, sorry to keep bugging you but this problem got really stuck in my
head... as it turns out, you CAN show why the parallel plate doesn't get
sucked up to the levitator using only Bernoulli and conservation of mass
 so the text books
are not wrong after all  I was wrong (as is Ed). Here is the
reasoning: the parallel channels on the plate in the levitator
produce a flowpath with constant crosssectional area.
The levitator
plates are circular and the fluid enters from a plenum at the center of
the disks. Therefore, the crosssectional area = the gap height times
the circumference of the control volume. So Cross_Section = Height * 2 *
pi * r. The Cross section is not constant; it varies with the
radius. The speed of the molecules is not highest at the center of the
disk. The velocity actually increases toward lower pressure ahead. The
molecules also proceed, by momentum, to fan out into larger and larger
volumes. This occasions a decrease in density, a decrease in
temperature, and a decrease in pressure. As the fanning continues,
friction and backpressure eventually act to reduce velocity until the
velocity reaches a steadystate contour along a gradual pressure
gradient to the edge of the disk. The action zone, that occasions
the lift, occurs close to the center of the disk, in which zone the
pressure drops very quickly, at approximately 1/r and then rises in a
hydraulic jump. To trace all these interactions, see the model
behavior.
Since the mass flow and density are constant for the flow along the
path (assuming steady state and low Mach # or incompressible liquid) the
bulk fluid velocity is also constant along the path since v = mdot/(rho*A)
where mdot is the mass flow (kg/s) A is the local crosssectional area
of the channel (m^2) rho is the fluid density (kg/m^3) v is the local
bulk fluid velocity (m/s) Since the velocity is constant,
Bernoulli's Principle states that the static pressure must be constant,
and since the static pressure is fixed at the exit plane by the
atmosphere boundary condition, the static pressure along the entire
channel is equal to normal atmospheric pressure.
The velocity is not constant. See the model.
In response to this argument, I constructed a businesscard
levitator that constrained the flow to a parallel channel. The
result, as predicted by the model, was to eliminate the lift.
The pressures on both sides of the disk are therefore equal and
opposite and the disk is thereby free to fall under the influence of
gravity  in reality it will also be pushed down by the dynamic
pressure of the air/water source acting on the area defined by the
radius of the inner hole. When there is an expanding conduit, the
crosssectional flow area gets bigger as radius increases, in order to
maintain conservation of mass, the
flow velocity must be much higher near the center. For the flat disk
the flow velocity will go like 1/r Bernoulli's principle states
that P + 1/2*rho*v^2 + rho*g*h = constant since the height change is
insignificant we can ignore that term P + 1/2*rho*v^2 = const
Therefore the faster fluid velocity in the center thereby necessitates a
lower static pressure.
See the model
to notice the dynamic interaction on velocity and pressure. They do not
correlate in a linear fashion. In the active region, you might even make
an empirical, although flawed, case for hysteresis. I take no issue with
Bernoulli's Principle, since it follows from energy balance. My issue is
that universal application of the Principle to lift situations is
inappropriate, since it violates the fact of compressibility, that you
need to get lift.
Again the boundary condition is atmospheric pressure out at the edge,
so the static pressure will drop below atmospheric in towards the
center. If we then integrate the pressure over the entire surface area
of the disk we will wind up with a net positive force pushing upwards
(we then need to subtract the weight of the disk and the dynamic
pressure force acting on the innermost radius to see if the disk will
actually be suspended), but it is definitely possible to have a
suspended disk IF the flow area expands. This explains your
experimental results using only Bernoulli and conservation of mass. It
also explains why you see your cavitation ring in near the center during
you water flow tests (local pressures much lower than atmospheric).
Sorry it took so long for me to come up with this  I should have
started here with my own analysis instead of critiquing yours. No hard
feelings.
I would be interested in tracing your math
on this. I do not think you can get there without using radial momentum
and without using Euler's method for solving simultaneous integral
equations. I would be very interested in your derivation of the
velocity, density, pressure and massflow contours, as I have done using
the model. Note: I have
simplified the model so it does not include temperature. Inclusion of
temperature would likely produce more accurate results, as some of the
motive energy results in refrigeration of the flow stream, and so less
might appear as lift.
Andrew 
October 30, 2002:
from Andrew #2
Index
Responses from Ed Seykota, to this email are in red.
To Ed: my previous email illustrated the results of your
experiments using only Bernoulli (P+1/2*rho*v^2 = constant) and
conservation of mass (mdot = rho*A*v = constant) and without the need
for a compressible fluid, I explained first why the business card
levitator does not work and then why the normal levitator (or the
levitators with expanding passageways) can be made to work. One only
needs to come up with an expression for the flow passage crosssectional
area as a function of radius.
In the business card levitator, the passage area is constant in
crosssection irrespective of radius. By conservation of mass, the flow
velocity must be constant throughout the passage, therefore, by
Bernoulli, since the velocity is constant, the static pressure is
constant in the entire flow passage. And since the static pressure must
be equal to atmospheric pressure at the exit (because there is a direct
fluidfluid interface), the pressure all along the passage is
atmospheric. Since there is no pressure that is lower than atmospheric
pressure on the top of the business card, the pressure forces balance
and the card is free to fall under the influence of gravity. Bernoulli
and conservation of mass show that the business card levitator will NOT
work.
Actually, in practice the business card
levitator can work, sort of. This is in the situation in which one side
of the card slopes slightly downward, establishing an angle between the
bottom of the levitator and the card, allowing radial expansion of the
air. I call this effect external
coning.
In the conventional levitator the crosssectional area of the flow
passage is just the height of the gap between the spool and the disk (h)
times the circumference at a given radius (2*pi*r). A = 2*pi*r*h if we
then algebraically rearrange the conservation of mass equation to solve
for velocity and substitute in the above function for area we get v =
mdot/(rho*2*pi*r*h) Everything in that expression is constant except for
r. so, v is proportional to 1/r. This means that the flow is going much
faster near the center than out at the edge. If we now use Bernoulli, we
can substitute our expression for v in and get: P+1/2*rho*(mdot/(rho*2*pi*r*h))^2
= const.
We can now solve for the static pressure as a function of radius [1]
P = const  1/2*rho*(mdot/(rho*2*pi*r*h))^2 Again, everything is
constant except for r, so [2] P is
proportional to 1/r^2. We can solve for the constant by plugging in
whatever mass flow density and height are in the experiment and set r
equal to the radius of the spool and the static pressure to atmospheric
pressure. This is as simple function that can be easily plotted. It can
be seen that the pressure at r < r_spool is LOWER than atmospheric
pressure resulting in LIFT  and all that with only Bernoulli and
conservation of mass  no compressible fluid is necessary. Hope
that clears things up. Let me know if you have any problems following
the math this time.  Andrew
I do not exactly follow the math above, and
feel there might be a sign inversion. For instance, from
[1] P = const  1/2*rho*(mdot/(rho*2*pi*r*h))^2
this does not seem to lead to
[2] P is proportional to 1/r^2
and seems to lead, instead to
[3] P = k  c/r^2,
where k = const and c = 1/2*rho*(mdot/(rho*2*pi*h))^2
This minus sign in [3] seems vital,
since it leads counterindications for employing Bernoulli's Principle
to explain lift.
For each value of k and c in [3], there must
exist some r = sqrt (c/k), such that the absolute pressure, P = 0. There
then also exist many smaller values of r such that P < 0, even P
<< 0, indicating negative absolute pressure.
Also, the pressure at the plenum must be
greater than atmospheric pressure or the device would not function in a
forward direction; for very small values of r, even with P
0, [3] then indicates high pressure at the plenum directly facing
the lowest pressure in the system. Thus, [3] indicates a substantial
discontinuity in pressure along the flow path.
Moreover, [3] also indicates fluid flowing from
the center of the disk outward toward the perimeter, against a gradient
of monotonically increasing pressure.
For more on this derivation, which I have also
performed, see my discussion
of Potter and Foss problem # 2.32. Similar discussions appear at
Textbook examples #1, #2
and #3.
Thus, I feel that employing Bernoulli's
Principle indicates:
1. Negative absolute pressure near the center
of the disk.
2. A pressure discontinuity at the exit from
the plenum.
3. Fluid flowing uphill against a pressure
gradient.
I feel these indications confirm that
Bernoulli's Principle does not apply to such lift situations. The
literature typically interprets Bernoulli's Principle as [high fluid
velocity == low fluid pressure]. This is incorrect for compressible
fluids. I feel that levitator lift results from density reduction so the
incompressibility requirement does not apply.
Using the Radial
Momentum model does produce reasonable contours for all variables.
Daniel
Bernoulli himself stated "It would be better for the true
physics if there were no mathematicians on earth." Perhaps, in some
way he predicted the mathematical misapplication of his own principle.
If there is some way to employ Bernoulli's
Principle to these lift analytics, in such a way that produces
reasonable contours for the key variables, then I would be more inclined
to accept the derivation. 
November
1, 2002, from Andrew
Index
Responses from Ed Seykota, to this email are in
red.
Another one to pass along:
1. The derivation indicates negative absolute pressures near the
center of the disk. You are correct that there cannot be a
negative pressure.
In reality what happens is when the static pressure drops below the
vapor pressure of water (which at 283K is around 10 mmHg), the water
does vaporize  this is the cavitation bubble that you see in your
experiments. The vapor pressure is effectively the minimum static
pressure in the system  so you are correct that Bernoulli does not
explain this nonideality, but Bernoulli never claims to take into
account nonideal fluids. Bernoulli can still be used to estimate the
onset of cavitation.
There are other real effects that Bernoulli does not take into
account either  such as viscous effects  indeed it is not a perfect
model for a real world system, but it is still a good model where it
applies.
That the Bernoulli derivation solves to
negative pressure, indicates a flaw in the application. The choice
between ideal and nonideal fluids in an actual apparatus would have no
bearing on the flaw in the math. Not only does the derivation indicate
negative pressure, it indicates arbitrarily enormous negative pressure.
For a more detailed discussion of the exact nature of the conceptual and
derivational flaws in employing Bernoulli to lift situations, see
Textbook examples #1, #2
and #3.
2. The derivation indicates a discontinuity, an instant drop in
pressure at the exit from the plenum, with no apparent
cause. There is a discontinuity in the flow path/direction which
gives a discontinuity in this model  of course if we were to do a real
CFD analysis of the flow around the corner you would see a separated
flow region and a vena contracta below it  near the cavitation bubble
 that CFD analysis would show very steep gradients in the static
pressure in that region, not actual discontinuities.
The discontinuity in the model around the
corner arises from the derivation to pressure = a  b/r^2. This is
incorrect and it places the lowest pressure at the center of the disc,
adjacent to the flow motivation. Explaining this discontinuity in terms
of flow separation and vena contracta seems like little more than a
guess to try to patch up an equation that does not work. The Radial
Momentum model places the maximum velocity at a small distance from
the center, since the lower pressure ahead accelerates the fluid. The
minimum pressure occurs even further out. The model contours make more
intuitive sense and also confirm the actual observations. It seems that
the radial momentum model explains all the facts, whereas the Bernoulli
model has several shortcomings.
3. The derivation indicates fluid flowing uphill against a
pressure gradient. The flow DOES go against the static
pressure gradient, but there are many flows that go against the static
pressure gradient (diffusers of pumps, compressors etc.  this flow is
very much like a pump diffuser)  the key is that the total pressure is
constant (because we are assuming an inviscid Bernoulli flow).
The factor that accounts for fluid flowing
against a pressure gradient in various types of diffusers, including the
levitator is, I feel, momentum. The geometry of the device sets the
fluid molecules radiating outward, radially. Radial momentum carries the
particles into larger and larger control volumes. As the fluid thereby
becomes less dense, lift appears. Eventually, the pressure gradient
absorbs the momentum and the molecules slow into a hydraulic jump.
In actuality, the total pressure would be dropping slightly due to
viscous effects, but the static pressure would be rising. FYI, the total
pressure (Ptot)= the static pressure(P) + the dynamic pressure(1/2*rho*v^2)
= the constant in the Bernoulli equation.
Yes, Bernoulli's equation certainly holds, in
some circumstances, since it is basically an energy balance. Applying it
to indicate a relationship between high fluid velocity and low fluid
pressure requires careful assessment of the situational geometry and
observation of the requirement for constant fluid density. In this case,
where density variation is the very property that generates the lift
effect, Bernoulli's Principle fails to provide a satisfactory
explanation.
Flows against a static pressure gradient are often unstable and are
characterized by an increased likelihood of flow separation and
nonisentropic conditions.
That is correct, and the instability occurs
where the radial momentum no longer supports laminar flow. Note that in
the Radial Momentum model where
the radial momentum dissipates, there is a resolution into a hydraulic
jump. This jump, that marks the end of the cavitation ring, also does
not appear in the Bernoulli derivation. The Radial Momentum model
correctly indicates a "U" shape for the pressure contour,
characteristic of a cavitation ring. The Bernoulli model incorrectly
indicates a cavitation region increasing hyperbolically toward the
center.
4. Attempts to reconcile these artifacts result bizarre
conclusions. I don't think the above explanations are bizarre, but
perhaps you do. The conclusions that I take away are just that the
models are good, but they have only limited regimes of applicability.
It is critical to know what those limitations are. Just as your model
cannot explain lift from a hydrofoil wing in water, Bernoulli cannot
explain the effects of real world viscosity and cavitation. That does
not mean that it is entirely irrelevant or incorrect  it just means
that it is important to know when and how to apply it correctly.
For our previous examples, we cannot assume that Bernoulli is a
perfect explanation because in the real world there are viscous effects
and there is a finite vapor pressure for real liquids, but that does NOT
mean that we cannot use Bernoulli as an incompressible flow model to get
a good idea of what is going on in the levitator (after all the
cavitation ring is relatively small and the viscous effects can probably
be shown to be small relative to the bulk flow by calculating the
Reynolds number of the device in a given configuration).
I do not think that the Bernoulli Model applies
at all. First, my analysis of textbook examples #1,
#2 and #3
show that the application does not follow from the physics of the
configuration. Second, the Bernoulli derivation indicates substantial
discontinuities and negative pressures, even for ideal gasses.
Third, the Bernoulli derivation does not produce realistic contours for
velocity, pressure, and density.
The earlier send, from October 30, send #4,
states that the
fluid must flow faster at the center. This seems to miss the physics
(faster than what?) and lead to strange implications. If the meaning is
that fluid at the center flows faster than the fluid further out, then
since neither flow has any particular relationship to the flow
(nonexistent) below the disk, none of this has any relationship to
lift. If, instead, the meaning is that the flow at the center flows
faster than the air below the disk (that does not flow at all), then, if
that causes lift, the parallel channel with high fluid velocity would
also lift.
Indeed, from that logic, one might also argue
that a sealed vial of air, in a race car traveling at high velocity,
would show low pressure. I have dealt with these formulations in
addressing the textbook examples. The literature is replete with such
examples of incorrect, nonrigorous and lazy applications of Bernoulli's
Principle to the physics of lift. Perhaps they appear since there is no
other handy theory, until now. I have found no careful and rigorous
applications of The Bernoulli Principle that can produce reasonable
contours for the key variables in the levitator.
I hope I've been of some help. I've really been spending way too much
time on this over the past few days, and I should really start doing my
work. It's been fun though. If you want to continue this discourse, I do
occasionally do consulting work ... I'm having fun with this ... if you
are interested  after all, this is only bordering on my primary field
of expertise. If not, best of luck with promoting your theory. I am
encouraged that you seem open to critique. I would guess that paying for
an actual class on fluid dynamics would probably be [a good idea].
Anyway, good luck!  Andrew
Yes, this has been very helpful and I
appreciate your taking the time and effort to help this project along by
taking it for a road test.
In particular, I appreciate the help in:
1. focusing my attention on areas of my
presentation that are unclear.
2. showing me how a credentialed physicist
views the physics.
3. confirming that there is no convincing
factual basis disputing my claim that the radial momentum model explains
the levitator (and other lift phenomena) whereas applications of
Bernoulli's Principle contain inherent conceptual and computational
flaws and do not produce reasonable contours for the key variables.
4. reminding me that being overly
confrontational is likely counterproductive to acceptance.
At this point, I would like to increase the
effectiveness of my argument, including clarifying the concepts, and
finding a good way to present it.
To the extent you might like to continue with
the project, I am open for effective support in publishing the Theory of
Radial Momentum. I am considering inviting a tenured professor with a
publication history (who might be able afford to take some risks on a
new approach) to champion the project. I am also considering inviting a
technical writer to put the work into a more standard form, as you
suggest. 
November 6, 2002: Run
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Index
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November 8, 2002: from Andrew
Index
Thanks. I feel like the lessons he took away based on his web site were
not exactly what I was trying to get across, but oh well  he's very
determined to avoid seeing the correct application of existing theories
... I can't really convince him otherwise. In any case, it was a fun
exercise for me.
 Andrew

Ed
Replies:
I
would be very happy to see a correct application of existing theory. In
particular, I would like to see:
(1)
A derivation of lift for the levitator that follows the geometry of the
device. The standard derivation that follows Potter and Foss, in Example
#3, uses the Bernoulli equation. This is incorrect since:
(a) the flow is motivated;
(b) the flow is not part of a closed loop;
(c) the flow is arguably compressible.
(2)
A sketch of the contours for pressure, velocity and density in Example
#3 that follow laboratory observation and common sense. Sketches of
the Potter and Foss derivation reveal flaws:
(a) there is a huge discontinuity in pressure at the plenum exit;
(b) the equations indicate arbitrarily large negative pressures.
(3)
Numerically reasonable results. Example
#2, showing another standard derivation indicates that one can
levitate a piece of glass about the size and shape of a small book
simply by breathing briskly over it.
I
feel that these flaws are sufficient to throw substantial doubt on the
applicability of the Bernoulli approach. Meanwhile, the Radial Momentum
model succeeds in all these areas.
Note:
Ockham's razor (also Occam's razor and the Law of Parsimony) is a rule
in science and philosophy stating that the simplest of two or more
competing theories is preferable and that an explanation for unknown
phenomena should first be attempted in terms of what is already known.
While the Bernoulli Principle seems simpler and is already known,
attempts to use it to explain the levitator seem to fail.
Andrew
seems as yet undecided as to whether or not he agrees with the Radial
Momentum theory:
Ed
is absolutely correct ... seems to agree
the
text books are not wrong after all ... seems
to disagree
It
would seem that if the Bernoulli application has merit, Andrew would
have produced convincing answers to questions 1, 2 and 3 above.
(Especially 1b, 2a and 2b). That he has not done so, seems to argue
in favor of Radial Momentum.
In
particular, I feel proponents of applying the Bernoulli principle cannot
do the following by following the Bernoulli model:
1.
Show the levitator flow is part of a closed loop (a necessary condition
for applying Bernoulli's Principle.
2.
Show there is no discontinuity in the pressure profile.
3.
Show there are no negative pressures (a necessary condition for a
reasonable solution).
In
any event, I feel Andrew, by volunteering to argue the Bernoulli case,
has had a very beneficial impact by helping to motivate focus onto a few
key issues. 
November 12, 2002:
from Andrew
Index
Tue, 12 Nov 2002 15:58:44
A gift is really not necessary. I was happy to help out. The address
is correct except for the apt #  it's apt # 2 if he is really
insistent. I would be happy to talk about the possibility of doing some
consulting work for Ed, but I do have significant time constraints now
because of school and other projects.
I would need to know what kind of stuff he would want me to do
before committing to anything. Does he just want to continue the
correspondence, or does he want me to develop some type of model,
or build an experiment, or what?
Offhand I can tell you that I certainly don't have time for
the experiment, but I could definitely continue the correspondence
if he thinks that it is worth paying for.
If he wants me to develop a model based on my understanding of
existing theories of lift, that is a possibility, although it
would take some time and it would depend on what he wants included
in the model. Also, it should be made clear up front that if he
does want me to do consulting work for him, there could be no reference
to [my university] in any type of publication (web or otherwise)
with regard to this work. That could get us both in trouble. I'll
take a look at the web page again. Andrew
Date: Tue, 12 Nov 2002 16:37:55 0500 From: Andrew
Re: Andrew seems as yet undecided as to whether
or not he agrees with the Radial Momentum theory: Ed is
absolutely correct ... seems to agree the text books are not wrong
after all ... seems to disagree
Just for the record, I don't doubt that it is possible to derive
valid explanations for fluid flow that you could call a new theory of
lift if you want, but I do think that it's possible to use existing
fluid dynamics to accurately describe the levitator.
It would seem that if the Bernoulli application has
merit, Andrew would have produced convincing answers to questions
1, 2 and 3 above. (Especially 1b, 2a and 2b). That he has not
done so, seems to argue in favor of Radial Momentum.
You will have to do more than be verbally provocative in order to
garner a serious response from me from this point on. You know that I
have some background in this area. Make me an offer if you think I can
add value to this discourse, if you don't think it's worthwhile, please
don't try to be publicly provocative to incite a response from me; I
don't appreciate it.
FYI: Your logic is flawed: Failure to correctly apply the Bernoulli
theory does not inherently imply failure of the theory, in addition,
failure of Bernoulli (if it should fail in this case) would not imply an
argument in favor of radial momentum.  Andrew
Response from Ed:
My thesis is that explanations for the levitator (and other
flowinduction lift devices) based on an interpretation of Bernoulli's
Principle are fundamentally flawed. Lift in the levitator is a function
of density reduction, and this follows from momentum. Attempts to solve
the levitator based on Bernoulli's relationship of velocity inverse to
pressure, miss the physics, and generate numerically unreasonable
results.
The Radial Momentum model follows the physics and predicts the
behavior of actual laboratory devices.
The
challenge for the defenders of the Bernoulli principle is as follows:
1.
Show the levitator flow is part of a closed, unmotivated loop. This is
a necessary condition for applying Bernoulli's Principle.
2.
From the Bernoulli derivation, sketch the contours for pressure,
velocity, mass, mass flow and density along the radius extending from
the center of the device to the periphery. From these, (a) defend the
assumption that density is constant and (b) defend the pressure
discontinuity, from the maximum to the minimum pressure, all occurring
at the plenum exit, by mathematical analysis of control volumes at and
around the discontinuous zone.
3.
Define "negative absolute pressure" and defend the occurrence
of arbitrarily large negative pressures that appear in the Bernoulli
derivation.
I
have standing offers with numerous physicists to develop answers to the
above challenges. I have offered prize money and I have offered
consultancy arrangements, for a clear solution to the levitator in terms
of the textbook method of applying Bernoulli's Principle, that velocity
is inverse to pressure, and have received none.
If
no convincing counterargument to my thesis appears, then I intend to
find a champion to help get these ideas into wider circulation, for
discussion, review and extension within the scientific community. 
November 14, 2002:
from Andrew
Index
Hello, It does not appear that Ed has changed the web page as
you suggested he would. I would appreciate it if he would do so. Since
he has not clearly specified an offer, I will suggest a rate (X$/hr)
which is lower than what I normally ask since this is not my primary
area of expertise. At the end of this email I will tell him how long it
took me to address his questions and I will ask for payment by check to
be sent to me at the address that you have by first class mail. I will
allow up to two weeks for the check to arrive and clear. If I have not
received payment by that time, I will cease consultation and recommend
to anyone who may ask me that they not pursue any such arrangement with
you. Those are my terms. If you don't agree, please tell me directly and
I will not expect payment. Otherwise I will expect to see a return on my
time starting now.
I will now address his specific requests:
1. Show the levitator flow is part of a closed, unmotivated loop.
This is a necessary condition for applying Bernoulli's Principle.
The levitator is clearly not part of a closed loop system, but I don't
know why you think that is necessary to make use of Bernoulli; the
necessary conditions for application of Bernoulli (as far as I know) are
a) Idealized incompressible flow assumption b) Idealized invicid
(isentropic) flow These conditions are admittedly not perfectly
met by an actual levitator, but it is possible to model the flow by
using Bernoulli with corrections for the viscosity and the real vapor
pressure of the fluid for water, or with compressible flow relations for
air.
2. From the Bernoulli derivation, sketch the contours for
pressure, velocity, mass, mass flow and density along the
radius extending from the center of the device to the
periphery. From these, (a) defend the assumption that density is
constant and (b) defend the pressure discontinuity, from the
maximum to the minimum pressure, all occurring at the plenum exit,
by mathematical analysis of control volumes at and around
the discontinuous zone. This should be doable as long as you
permit the application of other known fluid mechanics (other than pure
idealized Bernoulli).
3. Define "negative absolute pressure" and defend the
occurrence of arbitrarily large negative pressures that appear in
the Bernoulli derivation. There obviously are not real negative
pressures; as stated above, the Bernoulli model is not perfect for every
situation, but that does not mean that the levitator cannot be explained
by existing fluid mechanics. So in response to your request I would say
that the Bernoulli model does not apply when the static pressure drops
below the vapor pressure of the liquid (as in the cavitation region). In
this region, the flow is no longer incompressible thereby negating the
Bernoulli model.
This response took me 32 minutes. I will round down and charge $X.
Please let me know whether to expect a check. If you do not agree,
please tell me ASAP and I will remove this charge and stop the
correspondence.  Andrew
Response by Ed:
Your check is in the mail.
1. Unless the system is a closed loop, or equivalent, then the total
energy at the beginning and end of the flow path cannot be shown to be
equal, so the Bernoulli Principle would not apply. Bernoulli's Principle
is an energy balance, where, by conservation of energy, the sum of
thermal (pressure), kinetic (velocity) and gravitational (height)
energies must stay constant in an incompressible flow path. This is not
true where the beginning and end of the flow path do not have the same
total energy. The total energy before the pump is ambient energy, as is
the total energy after the fluid exits the device. The energy at the
plenum, downstream, from the pump, however, is necessarily higher, as it
motivates the flow through the device. So using an energy balance
equation from the plenum exit to the edge of the device, does not make
sense.
2. The challenge here is to demonstrate that the Bernoulli Principle
explains the levitator. If you need to apply "other" methods
to derive the variable contours, that indicates the Bernoulli Principle
is, indeed, insufficient. Furthermore, unless the other methods
acknowledge variable density, they, too, will not produce accurate
contours.
3. The region in which you admit (1) compressibility and (2) that the
Bernoulli Application does not apply is the the very region that
accounts for the lift.
If you wish to consult further, I would like you to either (1)
successfully defend these three points or (2) concede there is no
defense and that Bernoulli does not apply. 
November 22,
2002, From Andrew
Hi Ed,
I have some interesting information for you. First of all, I cannot
claim that Bernoulli accounts for cavitation, so in that sense, it does
not apply to the cavitated region of the levitator  if you wish to
treat that as a consession, you may, but I would caution you from
arguing that it cannot apply at all.
Of note is the essentially constant density of water at all pressures
from the vapor pressure all the way up to high pressures (above
atmospheric). This would tend to suggest that Bernoulli is applicable in
all the flow regions that are not cavitated. If that is the case, lift
will be generated from noncavitated regions as well as the cavitated
region.
This invariance of the density is also damaging to the case for radial
momentum (if you believe the best models of the National Institute of
Standards and Technology, which I do) because it suggests that the
density does not change in any noncavitated flow of water  but it is
clearly possible to get lift from noncavitated flows.
I have also spent some time thinking about possible experiments that
could show how well a Bernoulli model (appropriately modified for the
cavitation bubble) would actually match reality. I think that putting
static pressure taps in the levitated disk at various radii would
illuminate whether a Bernoulli model or a Radial Momentum model is
closer to reality. It would seem that these modifications could be done
to your apparatus with relatively little effort (you would need to
purchase pressure transducers that would span the range of predicted
pressures for both models  the minimum pressure in the modified
Bernoulli model is simply the vapor pressure of water at the apparatus
temperature).
Ed Responds
... "essentially constant density" is not the same as
incompressible ... in the case of water, very small density changes
accompany large pressure changes ... indeed, this is essential to the
Theory of Radial Momentum ...
I feel we are communicating rather randomly ... almost seems like two
different languages ...
All the best to you in your studies ...
Ed 
