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


I reviewed the web site, and I'm afraid that your friend's description of current theories of lift is a bit over-simplistic 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 P-V 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 whistle-blower.  - 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 de-compresses and the pressure drops.

Actually, so-called incompressible fluids such as water show much more lift than compressible fluids such as air. This is because the fluid "resists" compression (de-compression 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 Tube-and-Cone experiment and the Levitator Table Experiment. Theories resting on non-compressible 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


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 inter-plate 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 de-compress 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 curvature-induced lift induction were significant factors, then upside-down 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 Tube-and-Cone experiment and the  Levitator Table Experiment. Theories resting on non-compressible 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 Navier-Stokes, 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 inter-plate gap. Conventional approaches do not provide such analytics.



October 30, 2002 from Andrew


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 self-derived 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 flow-induced 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 non-physical 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 flow-induced 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 step-wise integration model, such as "push-acceleration."  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 cross-sectional area.

The levitator plates are circular and the fluid enters from a plenum at the center of the disks. Therefore, the cross-sectional 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 back-pressure eventually act to reduce velocity until the velocity reaches a steady-state 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 cross-sectional 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 business-card 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 cross-sectional 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 mass-flow 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.




October 30, 2002: from Andrew #2


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 cross-sectional area as a function of radius. 

In the business card levitator, the passage area is constant in cross-section 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 fluid-fluid 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 cross-sectional 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 counter-indications 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 mis-application 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


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 non-ideality, but Bernoulli never claims to take into account non-ideal 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 non-ideal 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 non-isentropic 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 (non-existent) 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, non-rigorous 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.

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November 8, 2002: from Andrew

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 1-b, 2-a and 2-b). 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


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?

Off-hand 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 1-b, 2-a and 2-b). 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 flow-induction 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, un-motivated 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 counter-argument 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


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, un-motivated 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 14, 2002: to Professor Weltner



Dear Professor Weltner,

I like your Misinterpretations of Bernoulli's Law at http://www.rz.uni-frankfurt.de/~weltner/Mis6/mis6.html

I, too feel Bernoulli's Principle does not apply to lift.  I have a web-site at www.radialmomentum.com that demonstrates an alternative theory, with experiments and a model.

I would like to publish this material and do not know how to proceed. I think your help might be invaluable. If you would be willing to help me, please let me know on what basis we might proceed.

Yours truly,

Ed Seykota



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 non-cavitated 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 non-cavitated flow of water -- but it is clearly possible to get lift from non-cavitated 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 ...