Talk:D'Alembert's paradox

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What about the Superfluidity experiments on 4He? 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 Wikipedia 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 Wikipedia 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)

[edit] 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 Wikipedia, 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 Wikipedia. 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 [1]. I doubt that Hoffman and Johnson's theory can be included in Wikipedia 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 Wikipedia 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/wiki/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 Wikipedia-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 Re \gg 1. However, experiments show a clear transition in behaviour between 105 and 106, 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 Wikipedia 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 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 wikipedia 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 105 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 [4] 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)

[edit] 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: [2]. 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:

\tau = C_f \frac{1}{2} \rho U^2,

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

C_f \sim R_\text{e}^{-0.2},   with   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 Re will increase:  R_\text{e} \sim U,\,
  • the skin friction coefficient Cf will decrease:  C_f \sim U^{-0.2}\, and
  • the skin friction τ will increase:  \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 Cf with the normalized drag coefficient Cd connected to the pressure drop, where Cd is about 0.5-1 and 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 \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)