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Physics Start‑class Mid‑importance

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What about the Superfluidity experiments on He? While "no drag" has not been observed, most other non-viscous elements have been observed, including no vortexes and no temperature hotspots.

This article is more or less incomprehensible to the layman, unfortunately. I think what's needed is an opening section that explains in non-mathematical terms what the paradox is, and what it means in practice, and what its consequences for the field of fluid dynamics were. While what's there is no doubt of value for the more enthusiastic or technical reader, at present it is not terribly useful in a general purpose encyclopedia. Graham 02:20, 18 August 2005 (UTC)

Yes, I agree. I have no idea what the bits of mathematical logic have to do with d'Alembert's paradox. I thought it had to do with zero drag in inviscid flow.

I agree too on the need for an opening section on what the paradox is before getting into an in-depth discussion over how it's been resolved, whether it's been resolved, and whether there's a controversy over whether it's been resolved or not.

I went to Misplaced Pages after reading that D'Alembert's paradox had "proven" that airplanes were impossible. Is that true? What does it mean? Whether or not the "parodox" still seems paradoxical, and no matter to what extent it's been satisfactorily resolved, the history of the effect it had in the years following its discovery seems the most important thing this page should describe.

Also, the article isn't neutral: apparently some people think there's no remaining controversy and the current author thinks there is. Somehow there has to be a neutral way to describe the situation rather than one guy laying out long technical arguments and documentation to bolster one position.

Also, if someone has resolved the paradox, that person should not be writing the article about it, it's a Misplaced Pages rule as I understand it. SteveWitham (talk) 23:40, 16 May 2008 (UTC)

You have correctly stated the crux of the paradox. Inviscid fluids do exhibit drag but the bits of mathematical logic in the peice show that this set of very useful equations predict that they will not. Yet the equations are properly derived and make useful predictions about real fluids behavior.

I found this page very very helpful - In fact I believe I may have derived a solution to the paradox. If I am correct it will be an important acheivement. I have written up my thoughts in a small article and I will be seeking to have this work published in an appropriate journal. In all events I will soon publish my thoughts here as well. Many thanks to the author of this article.

Sincerly,

Tony Gallistel A. Gallistel Innovation tgallist@aol.com

Sorry to break your bubble, but the paradox has already been "solved". It is even a matter of opinion whether this is a paradox at all. It is thought to be one because we expect to have drag when an object is in a moving fluid from our everyday life experience, yet the theory predicts that there is none. However, we can't blame a set of equations which are based on big assumptions (inviscid) for not representing the exact physical model and call the result paradoxal! it's like assuming that if gravity is neglected, a ball which is thrown accross a room will travel in a straight line, then carrying the experiment and finding out that it does not (although part of the result is still correct: the x velocity will be the same in both cases. Just like part of the result of assuming inviscid can still be correct)

New version

I am sorry to having implemented such a major change of the page without discussing it here first, but it appears that there has been some debate over the content of this paradox-page and I would like to help to make things more clear. I have made a complete rewrite (apart from the introduction) that hopefully will be helpful for anyone interested. I realize there are some mathematical terms that are not fully explained here or elsewhere in Misplaced Pages, but I intend to add material when neccessary. I also plan to add some pictures illustrating the article. The content in this article now also conforms with the German version of Misplaced Pages. Visitor22 09:37, 16 August 2007 (UTC)

The new version seems to be very biased towards the resolution of Hoffman and Johnson. The paradox is well known in fluid dynamics and I thought that Prandtl's resolution of the paradox is universally excepted. Apparently, Hoffman and Johnson put out a preprint last year in which they proposed a different resolution. This preprint has apparently been rejected by three journals . I doubt that Hoffman and Johnson's theory can be included in Misplaced Pages in the light of the verifiability and no-original-research polices. Even if it is to be included, the "undue weight" provision implies that it should be treated very briefly, in one sentence or so. -- Jitse Niesen (talk) 05:04, 18 August 2007 (UTC)

I read the policies and I think you are right; it seems that the idea is that the amount of text should somehow reflect the acceptance of the different views. I will edit and rearrange things the coming days to better follow this policy. As for the verifiability, the reference in the new version of the article is to the book (not the preprint), in which the underlying research (several articles referenced in the book) is published in well established journals in the field (Journal of Fluid Mechanics, Computational Mechanics etc.). So the underlying arguments (turbulent Euler solutions, computational method, etc.) are published and thus accepted by the scientific community, even though the consequences for the d'Alembert paradox is still under debate (as is clearly indicated in the new version of the article). Visitor22 07:52, 20 August 2007 (UTC)

I should add that apart from that, it is an immense improvement. What article on the German Misplaced Pages are you referring to? -- Jitse Niesen (talk) 05:13, 18 August 2007 (UTC)

Thanks! The german link (which I have nothing to do with) is: http://de.wikipedia.org/D%27Alembertsches_Paradoxon Visitor22 07:52, 20 August 2007 (UTC)

Also, I should add that I am the first author of the book referenced in the article (Hoffman), so I may of course be considered biased as a person. But my Misplaced Pages-article is based on (published) research that I have encoutered in my work, and I would be very happy to discuss the content of the present article with you or anyone else interested on this discussion page (or elsewhere). My main motives for updating this page is not to market my own research, but to carry out my duty as a researcher to communicate the present state of research in areas that I am familiar with, including my own findings, to the public. Visitor22 08:11, 20 August 2007 (UTC)

I have now minimized the material on the new resolution. Hopefully this gives a better balance to the article. Visitor22 10:02, 20 August 2007 (UTC)

First, despite the name, I don't really consider this a paradox at all. The "no drag with steady inviscid irrotational flow" is an provable mathematical result. The reason this doesn't agree with experiments is that the real flow isn't everywhere steady inviscid and irrotational.

It looks to me that this article is still lacking a full account of the accepted resolution: namely how even at high Reynolds numbers there's still a thin viscous boundary layer and this can lead to separation, with a low pressure region behind the body. It would also be good to include details of the agreement between boundary-layer theory and experiments in the case of slender bodies (where separation is not an issue). There must be lot of citable work out there to support the boundary-layer separation view -- lets see some more of that referenced. And some images would be really good too.

I haven't seen what's in the referenced book, but if it's anything like the preprint it leaves a lot to be desired in terms of good scientific argument. In the pre-print:

• The numerical scheme is not fully described, nor is the effect of numerical diffusion considered. Therefore no weight can be given to the numerical results.
• There is no satisfactory explanation for how vorticity is generated computationally in the inviscid flow -- something that is not permitted by the equations that are claimed to be being solved. There's some suggestion of unbounded velocity gradients, but how you expect to capture these numerically I'm not sure.
• There's no evidence offered to suggest that the traditional explanation is unsatisfactory at explaining any experimental results, and hence no reason for a new explanation to be needed.
• Since the Reynolds number doesn't appear in the system solved, the results must be independent of Re, if (as it is argued) viscous boundary layers are unimportant for ${\displaystyle Re\gg 1}$. However, experiments show a clear transition in behaviour between ${\displaystyle 10^{5}}$ and ${\displaystyle 10^{6}}$, which the results cannot be used to explain.

In conclusion, the article still seems very biased in favour of the Hoffman ideas, considering that they have yet to be accepted for publication in a peer-reviewed journal.

-- Rjw62 13:35, 23 August 2007 (UTC)

What would seem to be needed to support the commonly accepted resolution attributed to Prandtl, is original scientific work claiming to resolve the paradox, since Prandtl does not do so in his 1904 article. Otherwise, Prandtl´s resolution would not be suitable to present on Misplaced Pages because of its verifiability and no-original-research policies. The review article of Stewartson seems to indicate that no such original work is available up to 1981, and so far I have been unable to find any such work later either.

Regarding your concerns about the book; it is available for download for anyone to inspect. The points your are listing (numerical diffusion, drag crisis, vorticity generation etc.) are all clearly presented in the book as well as published by leading peer-reviewed journals of the field. Visitor22 07:18, 24 August 2007 (UTC)

About references to boundary layer theory: I have added an internal link to "Boundary_layer", and among the references in the current article are Stewartson and Schlichting. Visitor22 07:22, 24 August 2007 (UTC)

I'm not sure what you consider the actual 'paradox' that needs to be resolved to be, but in my mind it's the discrepancy between the mathematically rigorous steady Euler solution and what is observed in reality at large Reynolds numbers. The theory for a resolution is provided by Prandtl in his 1904 paper -- to claim otherwise is simply incorrect. He develops boundary layer theory, showing boundary layers can exist for arbitrarily large Ra, and that adverse pressure gradients can cause these layers to separate. As a result the limit as ${\displaystyle Ra\to \infty }$ does not have to correspond with the Euler solution. Separation can break fore-aft symmetry and a major way and allow a new pressure force on the body to be responsible for the observed drag. Even if Prandtl didn't feel in necessary to spell out all the details, there are numerous respected text books that do.

I've had a look through the book chapter and it appears quite similar to the pre-print. I stand by what I said above, and see nothing there (or in the referenced material) to address those issues (execpt a description of the numerical scheme). I'm also rather concerned about the claims in the book that steady potential solutions can not generate lift, and that fact that you've missed the whole class of lift-generating solutions for the model problem you consider in 10.6.

I'm also rather confused by your analysis of Stewarton's paper. I can't access the full text, but the abstract clearly states This three-pronged attack has achieved considerable success, especially during the last ten years, so that now the paradox may be regarded as largely resolved. I find it hard to believe that he would contradict this inside the paper. Moreover, I believe what he is referring to here is not just the essential theory of boundary layers and separation, but also a full understanding of the details of real flows. And even that he finds largely resolved.

Unless or until there is a peer-reviewed paper directly advancing your claims, I feel any suggestion in the[REDACTED] article that the drag can be accounted for without viscous boundary layers should be flagged as 'unproven' and confined to a couple of sentences at the most.

-- Rjw62 10:09, 24 August 2007 (UTC)

It seems that we agree on the definition of the paradox. I agree that Prandtl in his 1904 paper introduces boundary layer theory, for 2d steady laminar flow. Prandtl also presents a scenario for separation (p.6): with an adverse pressure gradient the flow may separate due to loss of kinetic energy in the boundary layers. This appears reasonable for laminar boundary layers up to Reynolds number Re of about 10 for e.g. a circular cylinder. Although for higher Re the boundary layers undergo transition to turbulence, which results in delayed separation and reduced drag, so called drag crisis. To get to Re=infinity one has to pass through turbulent boundary layers and drag crisis, so it appears that the boundary layer theory of Prandtl's 1904 paper (2d steady flow) is an over simplification for Re beyond drag crisis. And this theory also seems unable to explain the subsequent rise in drag for Re beyond drag crisis, reported in experiments.

On the other hand, in the book (Section 35.5-35.8) it is shown that it is possible to simulate drag crisis, including the rise in drag for very high Re, by parameterizing the boundary layer by simply a friction coefficient corresponding to the skin friction. This skin friction is then reduced towards zero corresponding to increasing Re, resulting in delayed separation (drag crisis) and the development of streamwise vorticity in the separation points (with increasing drag).

Discontinuous potential solutions is discussed in Section 3.3.

I have access to the full Stewartson paper, and it is clear to me that although he is satisfied with recent work within the area, he admits that it is still a long way to go to fully characterize boundary layer flow. In particular since unsteady flow and 3d flow is poorly understood. His statement that: "...the paradox may be regarded as largely resolved.", does not sound very convincing to me. Either the paradox is resolved or not. I also quote from his summary: "...Much remains to be done; in particular, the development of a rational theory for three-dimensional, and possibly also unsteady two-dimensional flows may be in its infancy....".

The book is published with an established scientific publisher, and the results on the computational method, simulation of drag crisis etc. are published as peer-review papers in scientific journals, which is evident from the list of references in the book. Visitor22 14:01, 24 August 2007 (UTC)

Removed section: Boundary condition: slip or no-slip?

This section is original research and contradicts measurements, due to confusing skin friction coefficient with skin friction. See for instance: . The skin friction (or wall shear stress) τ is related to the far-field velocity U (or another characteristic velocity for the problem at hand), by:

${\displaystyle \tau =C_{f}{\frac {1}{2}}\rho U^{2},}$

with C_f the skin friction coefficient and ρ the mass density of the fluid. From experiments, the skin friction coefficient C_f decays with the Reynolds number as:

${\displaystyle C_{f}\sim R_{\text{e}}^{-0.2},}$   with   ${\displaystyle R_{\text{e}}={\frac {U\,L}{\nu }}}$

the Reynolds number depending on a characteristic length scale L and kinematic viscosity ν.

As a result, for a given object and fluid, i.e. L, ρ and ν constant, with increasing velocity U :

• the Reynolds number R_e will increase:  ${\displaystyle R_{\text{e}}\sim U,\,}$
• the skin friction coefficient C_f will decrease:  ${\displaystyle C_{f}\sim U^{-0.2}\,}$ and
• the skin friction τ will increase:  ${\displaystyle \tau \sim U^{1.8}\,}$,

as expected. — Crowsnest (talk) 10:06, 21 May 2008 (UTC)----

I do not see that this section contradicts measurements: it should be clear that what is referred to is the skin friction coefficient, if the "coefficient" was missing that was a typo. What is relevant in terms of the d'Alembert paradox is the relative importance of the drag force from the pressure drop over the body compared to the drag force from skin friction. Thus you have 2 options: (i) either compare the skin friction coefficient C_f with the normalized drag coefficient C_d connected to the pressure drop, where C_d is about 0.5-1 and ${\displaystyle C_{f}\sim U^{-0.2}\,}$, or (ii) compare the skin friction "τ" with the actual drag force from the pressure drop that will increase as ${\displaystyle \sim U^{2}\,}$.

That is, I do not see that the section on slip vs no slip boundary conditions is either confusing nor misleading.

Visitor22 (talk) 15:08, 11 June 2008 (UTC)

This is the removed section "Boundary condition: slip or no-slip?", with the lines numbered by me:

1. Experiments show that the skin friction from a turbulent boundary layer decreases towards zero as Re as the Reynolds number Re increases .
2. This indicates that for large Reynolds numbers (small viscosity) a slip boundary condition (or more generally a friction boundary condition with small friction), is a better model than a no-slip boundary condition.
3. Computational simulation of drag crisis supports this approach,
4. which opens entirely new possibilities for simulation of high Reynolds number flow without resolving very thin boundary layers.
5. This is contrary to Prandtl's claim that even for very high Reynolds numbers thin boundary layers need to be resolved (which is impossible) to get a correct drag.
6. This indicates that a correct resolution of d'Alembert's paradox can have important consequences also in applications.

References:

1. Hermann Schlichting, Boundary layer theory, McGraw Hill, 1979
2. Johan Hoffman and Claes Johnson, Computational Turbulent Incompressible Flow, Springer, 2007

On which I have the following comments, refering to the above line numbers:

1. This has been addressed. If it should read "skin friction coefficient" instead of "skin friction", the form drag divided by the skin friction increases as R_e. This means that for very large to huge Reynolds numbers R_e, say in the range 10–10, skin friction is of the order of 1% of the form drag. Which is small, but may at all not be negligible with respect to the current claims regarding the d'Alembert paradox.
2. The conclusion that a slip (zero friction) or partial-slip (friction proportional to velocity to some power at the wall) boundary condition is better, is speculative. Especially with respect to the claims regarding the d'Alembert paradox.
3. I do not see how a complex computational model, which may contain bugs, numerical diffusion, round-off errors, etc., can support these claims with regard to mathematical-physical modelling.
4. Only in case of a slip boundary condition there is no boundary layer. Partial slip conditions also have a boundary layer to be resolved.
5. Prandtl did not claim that thin boundary layers need to be resolved (in a numerical model), only that "In the thin transition layer, the great velocity differences will then produce noticeable effects in spite of the small viscosity." How these effects are brought into account, will depend on the solution method used (so also if there are problems with "resolving" the boundary layer).
6. This is also an unverifiable claim.

So in my opinion there is much more wrong with this removed section than the omission of the word coefficient in the first sentence. -- Crowsnest (talk) 17:18, 5 July 2008 (UTC)

Good, this is a very important discussion. Let me address your points:

1. We agree that skin friction in problems of practical interest may be of the order 1%. For many applications this may be considered to be negligible, for example it is typically less than the experimental margin of error of the total drag (e.g. for the well known circular cylinder with experimental accuracy of a few percent). The most important cause for the dominating form drag is the separation of the flow, which determines the size of the wake, which gives the pressure drop and drag. A key question is then if it is possible to capture correct separation with slip boundary conditions. This is shown to be possible in the work cited in the article, with also in particular the following article to be published in Journal of Mathematical Fluid Mechanics this year: http://www.csc.kth.se/~jhoffman/archive/papers/dal-jmfm.pdf
2. That slip/friction bc is better is in the sense of computational efficiency: with no slip bc and full resolution of the boundary layer, computation of high Re flow is simply not possible whereas with slip/friction bc it is, as is shown in the references.
3. This is fully described in the references: basically it comes down to well-posedness of mean value output such as drag, using mathematical techniques of weak solutions and duality. That is, drag can be computed with reliable a posteriori error control (that is the result can be verified after having been computed from the computational model).
4. Yes, but this boundary layer does not attach to the wall with the same velocity as the wall (typically zero for a stationary wall), but instead just a few percent for high Re. This means that the corresponding boundary layer is not thin, and thus cheaply computationally resolvable.
5. But still Prandtl claims that there are "noticeable effects". Computing accurate drag using slip bc (without any boundary layer effects), in particular showing non-zero drag for inviscid flow past a cylinder without any boundary layer effects, shows that Prandtl's claim is wrong.
6. It is obvious that if you can use slip/friction bc to compute high Re flow in advanced applications, compared to the inability to solve high Re flow with resolved boundary layers in no slip bc, this has dramatic practical consequences.

Visitor222 (talk) 08:52, 10 July 2008 (UTC)

I am not going to comment on computational efficiency since this article is on d'Alembert's paradox: inviscid potential flow solutions to the Euler equations predict zero drag, which contradicts measurements finding substantial (form) drag — not reducing with increasing Reynolds number — while, in non-dimensional formulations, the Navier-Stokes equations converge asymptotically to the Euler equations with increasing Reynolds number.

Using the same numbering as above:

1. Although a skin friction of the order of one percent of the total drag is often not interesting from a practical point of view, it is important with respect to the solution of d'Alembert's paradox: the energy dissipation in the boundary layer attributes to the start of separation of the flow, which – after the separation points move upstream – leads to the much larger form drag. The article, cited by you, uses a numerical approach not solving the Euler equations but a "regularized" form including a heuristic non-isotropic dissipation. How such a numerical approach may lead to a solution of d'Alembert's paradox is unclear to me:
   • The viscous dissipation of the Navier-Stokes equations (whose formulation is based on measurements, physical and symmetry considerations) has been replaced by some other, heuristic (non-symmetrical tensor) form.
   • As is well known, the addition of dissipation in non-linear systems may have a destabilizing effect and lead to chaotic behaviour.
   • Perhaps the Navier-Stokes equations with slip boundaries will give similar numerical results. But this is all speculative and ignores the experimental fact of the occurrence of viscous boundary layers.
2. Better in the sense of computational efficiency is no argument with respect to the resolution of d'Alembert's paradox. There is no proof that a slip boundary condition is "better".
3. This only says that the code converges to some — reasonable looking or not — solution, not that it proofs the solution of d'Alembert's paradox, nor that a slip boundary condition is better than a no-slip.
4. Yes.
5. The numerical solutions are not inviscid, since explicitly some dissipation has been added. An option is, that any form of dissipation may trigger separation and lead to a reasonable prediction of form drag. Prandtl gave a theory, backed up by experimental evidence, that there is a viscous and turbulent, dissipative boundary layer at high Reynolds numbers. This becomes thinner with increasing Reynolds numbers, but does not vanish.
6. I agree. But how does this relate to the claim that "...a correct resolution of d'Alembert's paradox can have important consequences also in applications...". All kinds of CFD codes exist giving in many instances reasonable predictions of the flow characteristics.

Crowsnest (talk) 15:03, 29 July 2008 (UTC)

Garrett Birkhoff

Why the variant spelling of his name? Richard Pinch (talk) 11:06, 13 July 2008 (UTC)

I can see no valid reason to depart from the spelling used in the Wiki article of the same name - Garrett Birkhoff. I have rectified the error. Dolphin51 (talk) 11:57, 13 July 2008 (UTC)

Vague reference?

Jitse Niesen removes a reference to a closely related published article claiming that the article is vague. The criticism of the article should be made more explicit to motivate the removal. —Preceding unsigned comment added by Egbertus (talk • contribs) 12:34, 5 August 2008 (UTC)

Well, there is already some discussion, see above. Further, a relation between the solution of d'Alemberts paradox and the solution of the Clay millenium prize problem on the existence and smoothness of solutions to the Navier-Stokes equations seems to be speculative, at best. So I support the removal of this off-topic reference, and removed it again. -- Crowsnest (talk) 13:20, 5 August 2008 (UTC)

Speculative? In what sense? In fact, showing blowup of vanishing viscosity solutions of the Euler equations is precisely what solves d'Alembert's paradox: It is shown that initially inviscid incompressible irrotational (smooth potential laminar) flow (with zero drag) over time breaks down into a non-smooth turbulent Euler solution (with substantial drag). So there is a very strong connection between the two problems, which motivates cross-reference, —Preceding unsigned comment added by Egbertus (talk • contribs) 13:55, 5 August 2008 (UTC)

What are "...vanishing viscosity solutions of the Euler equations..."?. The Euler equations, according to all renowned textbooks on the subject, do not have viscosity by definition.

Further, why do you stay re-reverting this edit, which was reverted by two editors; instead of discussing this on the talk page. That is where talk pages are for. Also be aware that it is not done to stay reverting edits, see WP:3RR. -- Crowsnest (talk) 14:24, 5 August 2008 (UTC)

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