Paradoxes of Fluid Mechanics

The book Hydrodynamics A study in logic, fact and similitude (1950) by Garrett Birkhoff gives a long list of paradoxes of fluid mechanics including the following concerning incompressible flow:

1. D'Alembert's paradox (zero drag of (potential) inviscid flow). 
2. Reversibility paradox (reversion of flow direction does not reverse flow).
3. Fatness paradox of Kutta-Joukowsky theory (lift decreases with thickness of wing).
4. Magnus effect (lift of backspin tennis ball opposite to that of table tennis ball).
5. Eiffel paradox ("drag crisis" as sudden drop of drag). 
6. Dubaut paradox (smaller drag of stationary pole in streaming fluid than pole moving through stationary fluid).  

Birkhoff describes the role of paradoxes:

• These paradoxes have been the subject of many witticisms. Thus, it has recently been said  that in the nineteenth century, " fluid dynamicists were divided into hydraulic engineers who observed what could not be ex­plained, and mathematicians who explained things that could not be observed." (It is my impression that many survivors of both species are still with us.)—Again, Sydney Goldstein has observed that one can read all of Lamb without realizing that water is wet!  
• I think we should welcome the discovery of hydrodynamical paradoxes—recognizing frankly the inadequacy of existing mathematics (and logic) to analyze the complex wonders of Nature. Experience shows that man's imagination is far more limited than Nature's resources: as Pascal wrote, "l'imagination se lassera plutot de concevoir que la nature de fournir.

Solving paradoxes thus may open roads to progress. The mother of all paradoxes of incompressible flow is d'Alembert's paradox. Prandtl was crowned Father of Modern Fluid Mechanics because he saved the face of fluid mechanics confronted by the paradox, by coming to rescue in a short 1904 article selling the idea that drag somehow is an effect of a boundary layer caused by an imagined necessity of a no-slip boundary condition forcing a fluid to "stick" to a solid wall. However, Prandtl's boundary layer theory came with a serious side effect as the impossibility to computational resolve thin boundary which has paralysed Computational Fluid Dynamics CFD throughout the 20th century.

  

The paralysis was lifted only in 2008 with a new resolution of d'Alembert's paradox (check video) showing that inviscid flow modeled by the Euler equations can be described as potential flow modified by 3d rotational slip separation into turbulent flow, which can be resolved computationally by Direct Finite Element Simulation DFS. This is described in Computational Turbulent Incompressible Flow with subsequent elaborations, resolving all the paradoxes listed by Birkhoff, and more as shown in the previous post with references including revealing The Secret of Flight (check video).

One Paradox Enough to Kill a Theory

A scientific paradox is a contradiction between the prediction of a scientific theory and observation.

The detection of a paradox of a scientific theory means the end of the theory, unless the contradiction between theory and observation can somehow be resolved, either by changing the theory or the observation.

As an illuminating illustration let us consider D'Alembert's paradox in fluid mechanics, which compares the zero drag of potential flow as stationary incompressible inviscid flow around a body of any shape, with the observation of substantial drag in slightly viscous flow of aero/hydro-mechanics. Theory says drag = 0, while observation shows drag = 1, a glaring contradiction.

D'Alembert's paradox was formulated in 1752 but was not solved until 2008 after 256 years of brooding by the most able minds of mathematics and mechanics.

How was the paradox then handled during its 256 years of existence as a potentially lethal poison to the science of fluid mechanics? Since observation of drag =1 could not be argued away, it was necessary to blame one of the assumptions of potential flow for the unrealistic prediction of zero drag, and then the assumption of zero viscosity of inviscid flow was the first choice:

It was argued that even if the viscosity of air or water is very small, it is not zero and drag = 1 thus results from an arbitrarily small positive viscosity. This led from the zero viscosity Euler equations, allowing unphysical potential solutions which could be computed analytically, to the positive viscosity Navier-Stokes equations with solutions which could not be computed at all.

The paradox was thus wiped away by a gesture without any real scientific substance. If solutions to the Navier-Stokes equations could not be computed then no predictions could be made assuming positive viscosity, which thus was a useless theory. Practical engineering was thus thrown back into d'Alembert's paradox of zero drag of inviscid flow incompatible with the observed lift of a wing in many experiments in the late 19th century preparing for the powered sustained flight demonstrated by the Wright brothers in 1903.

The observations required a change of theory and since positive viscosity had led to a dead end,  a new card had to be pulled and this was the circulation theory of Kutta and Zhukovsky changing potential flow by large scale circulation around the wing section. From circulation followed lift (but still no drag) and the problem now was to explain the generation of circulation by a wing.

To this end, viscosity was again brought in, which of course led to the same dead end as before, but with a bit of mathematics of analytical functions, this was covered up and kept the paradox under control for over 100 years, although many scientists had their doubts.

A non-correct resolution of a paradox thus showed to be better than no resolution at all (which as indicated was unacceptable). One way of maintaining an incorrect theory involving a paradox, thus amounts to presenting one incorrect resolution after the other of the paradox as a way to delay the final verdict that the theory is incorrect.

If you follow this blog, you know about the correct resolution of d'Alembert's paradox presented in 2008 and you may well understand the far-reaching consequences of the the resolution.

Mathematical Secret of Flight 6: Wikipedia Cover Up

To see that our new theory of flight fills a need, it is instructive to study how Wikipedia covers up the lack of a convincing theory in the literature:

• There are several ways to explain how an airfoil generates lift.
• Some are more complicated or more mathematically rigorous than others; some have been shown to be incorrect.
• For example, there are explanations based directly on Newton’s laws of motion and explanations based on Bernoulli’s principle.
• Both principles can be used to explain lift, but each appeals to a different audience.
• This article will start with the simplest explanation; more complicated and alternative explanations will follow.
• Explaining lift while considering all of the principles involved is a complex task and is not easily simplified.
• In attempting to explain why the air flows the way it does (e.g. why the flow follows the upper surface of the airfoil and why the streamtubes change size), the situation gets considerably more complex.
• It is here that many simplifications are made in presenting lift to various audiences.

We see that one part of Wikipedia struggles to hide that there is no theory of flight, while another part tells the truth by citing John D. Anderson, Curator of Aerodynamics at the National Air and Space Museum:

• It is amazing that today, almost 100 years after the first flight of the Wright Flyer, groups of engineers, scientists, pilots, and others can gather together and have a spirited debate on how an airplane wing generates lift. Various explanations are put forth, and the debate centers on which explanation is the most fundamental.

As a last line of defense Wikipedia presents the classical theory by Kutta-Zhukovsky (which we have shown to be incorrect).

• The effects of viscosity are contained within a thin layer of fluid called the boundary layer, close to the body. As flow over the airfoil commences, the flow along the lower surface turns at the sharp trailing edge and flows along the upper surface towards the upper stagnation point. The flow in the vicinity of the sharp trailing edge is very fast and the resulting viscous forces cause the boundary layer to accumulate into a vortex on the upper side of the airfoil between the trailing edge and the upper stagnation point.[26] This is called the starting vortex. The starting vortex and the bound vortex around the surface of the wing are two halves of a closed loop. As the starting vortex increases in strength the bound vortex also strengthens, causing the flow over the upper surface of the airfoil to accelerate and drive the upper stagnation point towards the sharp trailing edge. As this happens, the starting vortex is shed into the wake, and is a necessary condition to produce lift on an airfoil. If the flow were stopped, there would be a corresponding "stopping vortex". Despite being an idealization of the real world, the “vortex system” set up around a wing is both real and observable; the trailing vortex sheet most noticeably rolls up into wing-tip vortices.

In both politics and science, cover up is a most essential part of the game, because admitting that there are no answers to questions which should have answers, destroys credibility and authority, the core values of both politics and science. But pretending to have answers when no answers are available has a high cost, as demonstrated in Dr Faustus of Modern Physics.

The above connects to my old controversy with Wikipedia about d'Alembert's paradox discussed in posts on d'Alembertgate and the 2009 knol Wikipedia Inquisition, leading to a banning of my voice on Wikipedia. This makes it impossible to give any form of link to the new theory of flight on Wikipedia, as if understanding what keeps an airplane in the air would be dangerous knowledge which must be kept hidden to the people.

Standard (Mal) Practice of Science

In the UK MPs Science and Technology Commitee Inquiry into Climategate the key scientist Phil Jones claimed that it was "standard practice" to not release data and computer models so other scientists could check and challenge research. See a summary of the British reaction to the  tragi-comics here. Then compare with the stunning NYT message: Peers Say IPCC Conclusions Remain Sound! Peers.

What Jones says may be true: The standard practice of science may not be what it is supposed to be, as expressed by the Institute of Physics in its submission to the inquiry, namely:

• The principle that scientists should be willing to expose their results to independent testing and replication by others, which requires the open exchange of data, procedures and materials, is vital.
  

The "standard practice" seems to be that computer codes and data can be kept as secret belongings to scientists, and that the peer-review process is closed to inspection and biased (corrupted). 

Much of science is of little interest to society, politics and people, and  it does not matter much exactly what standards are used in the internal fights between different cohorts of scientists. 

But some science is of critical importance, and then the standards matter. Climate science is

an example where highest standards are required, at least if it as now is supposed to determine global politics. 

A good thing with Climategate is that it exposes scientific malpractice, and can lead to a renaissance of the holy principle of (nonclassified) science: data, codes and methods open to scrutiny by anybody. 

We are still waiting for the Royal Swedish Academy to express its support of this principle, in the foot-steps of the Institute of Physics and the Royal Society of Chemistry and the Royal Society of Statistics, in a withdrawal of its statement of support of the "standard practice" of IPCC and thereby indirectly of the "standard practice" of Phil Jones. When will it come?

In science, credibility is the most important asset of any scientist or scientific organization. 

To hand out Nobel Prizes requires scientific credibility. 

A recent poll gives the credibility of IPCC a 81% F, the worst grade. What would a poll for the Academy give? When will Swedish MPs inquire Swedish climate scientists? Compare Climategate hits Westminster: MPs spring a surprise:

• MP Graham Stringer had done his homework, and through patience and dogged persistence, he began to swing the chairman behind him. Mirroring the collapse in public sympathy for climate science since the scandal broke, the stalwarts so vocal at 3pm were silent by the close.
  

It took 2 hours to go from alarmism to scepticism...

The Norwegian Nobel Peace Prize Committee giving the 2007 Prize to Gore/Pachauri is now 

in trouble as demands of withdrawing the Prize are being made.

Which Committees, Academies and Societies will sink together with Jones/Gore/Pachauri? So far the Royal Society has said nothing indicating any departure from the sinking ship, nor has the Royal Academy...

But the Royal Society was shaking already on January 23: Royal Society capitulates on climate debate in worst week for global warmers since Climategate and thus can be expected to give in and follow the Royal Societies of Chemistry and Statistics and Institute of Physics any moment...and then the Royal Academy will have to follow as well...Eller hur Gunnar Öquist?

Now NYT! reports: 

• Ralph J. Cicerone, president of the National Academy of Sciences, the most prestigious scientific body in the United States, said that there was a danger that the distrust of climate science could mushroom into doubts about scientific inquiry more broadly. He said that scientists must do a better job of policing themselves and trying to be heard over the loudest voices on cable news, talk radio and the Internet.

This also applies to the Royal Academy in particular. Let's hear!  

According to NYT at least Pachauri speaks out in an e-mail message:

• Scientists must continually earn the public’s trust or we risk descending into a new Dark Age where ideology trumps reason.
  

PS1 Listen to a BBC reporter revealing that he is not allowed to say that he does not believe in global warming... Compare with Biased BBC.

PS2 To get some perspective take a look at historic reports on climate change.

d'Alembertgate

Another example of the (mal)practice of peer-review in science revealed in Climategate, is my own experience with the reception of my resolution of d'Alembert's paradox with Johan Hoffman, first submitted to the leading journal in the field Journal of Fluid Mechanics and rejected with the following referee's reports:

Referee 1:

• I strongly recommend that you reject this paper.
• 1. The authors show no understanding of fluid mechanics, of how small viscosity leads to experimentally verified thin boundary layers, and of how experimentally verified these boundary layers detach when the external flow decelerates.
• 2. The authors show no understanding of mathematics, of how in the Euler equations streamlines leaving a surface will be vortex sheets which divide the domain into regions in which there is separately potential flow.
• 3. The authors show no understanding of numerics, of how finite element methods introduce an artificial viscosity (or hyper-viscosity in high-order methods) through the truncation error, and so their solutions are not solutions of the Euler equation but solutions of a Navier-Stokes equation.
• I am alarmed that these authors have another paper accepted by JFM: I recommend that someone checks it for logic.
  

Referee 2:

• THE AUTHORS PROPOSE RESOLVE D'ALEMBERT'S PARADOX BY SHOWING THAT THE ZERO DRAG POTENTIAL SOLUTION OF EULER'S EQUATIONS IS UNSTABLE AND INSTEAD A TURBULENT (APPROXIMATE) SOLUTION DEVELOPS WITH A NON ZERO DRAG, EVEN WITHOUT BOUNDARY LAYER EFFECTS.
• THEY BASE THEIR RESULT ON NUMERICAL CALCULATION MOSTLY DONE IN 2006 WITH THEIR OWN NUMERICAL PACKAGE.
• THE EQUATIONS THEY STUDY (2.1) HAVE NO PARAMETERS OTHER THAN THE VELOCITY SO THAT THEIR CONCLUSIONS MUST APPLY TO ALL REYNOLDS NUMBERS.
• SINCE TURBULENCE DOES NOT DEVELOP AT LOW RE THEIR RESULT GOES TOO FAR. I DO NOT KNOW HOW THEY GET VORTICITY WHEN THE INTIAL FLOW IS IRROTATIONAL, BAROTROPIC AND WITHOUT SHEAR LAYERS.
• I DO NOT NOT UNDERSTAND THEIR USE OF SEPARATION WHEN THERE ARE NO BOUNDARY LAYERS TO SEPARATE. I HAVE ATTACHED A CITATION FROM LIGHTHILL IN WHICH HE ADVOCATES REPLACING d'ALEMBERTS PARADOX WITH d'ALEMBERTS THEOREM.
• THIS PAPER SHOULD BE REJECTED.
• I DID NOT READ THEIR NUMERICAL PAPERS WHICH ARE VERY NEW AND APPARENTLY NOT VALIDATED AGAINST EXACT SOLUTIONS AND OTHER TURBULENT CODES.
  

It is clear that these referee's are out kill, and they do it! The paper was then submitted to Journal of Mathematical Fluid Mechanics and was accepted and published in Dec 2008.

The story is told in my previous post The Sciencegate of Fluid Mechanics: It shows how a small group of scientists can control a whole scientific discipline by suppressing new information and new discoveries showing classical "truths" to be empty/false. See my interview with the Editors of Journal of Fluid Mechanics JFM.

It shows how this small group of scientists also controls Wikipedia, and blocks any reference

to our published work there. More examples are given at the webpage of our book Computational Turbulent Incompressible Flow.

Climategate has shown that some of climate science has been a dirty business, and I have experienced similar practice in fluid mechanics. The Climategate drama is now unfolding and its consequences must be far-reaching...carbon trading is losing momentum...Maybe the emails behind dAlembertgate will also be made public...maybe according to FOI...

If you look into the details of dAlembertgate you will discover that the new resolution of d'Alembert's paradox fundamentally changes the mathematical basis of fluid dynamics and that the attitude of JFM is untenable...

For more insight into the essentials of peer-review see an example from 1945.

Time for Fight, in Climate Science

The basic scientific questions in climate science are

• (i) Is there today unequivocal unprecedented global warming?
• (ii) Is there scientific evidence of major AGW?

In a BBC interview Phil Jones as central scientific advocate of AGW, answers No to (i) and Maybe Not to (ii), as evidenced in:

Question A - Do you agree that according to the global temperature record used by the IPCC, the rates of global warming from 1860-1880, 1910-1940 and 1975-1998 were identical?

Phil Jones: An initial point to make is that in the responses to these questions I've assumed that when you talk about the global temperature record, you mean the record that combines the estimates from land regions with those from the marine regions of the world. CRU produces the land component, with the Met Office Hadley Centre producing the marine component.Temperature data for the period 1860-1880 are more uncertain, because of sparser coverage, than for later periods in the 20th Century. The 1860-1880 period is also only 21 years in length. As for the two periods 1910-40 and 1975-1998 the warming rates are not statistically significantly different (see numbers below).I have also included the trend over the period 1975 to 2009, which has a very similar trend to the period 1975-1998.So, in answer to the question, the warming rates for all 4 periods are similar and not statistically significantly

different from each other.

Question D - Do you agree that natural influences could have contributed significantly to the global warming observed from 1975-1998, and, if so, please could you specify each natural influence and express its radiative forcing over the period in Watts per square metre.

Phil Jones: This area is slightly outside my area of expertise. When considering changes over this period we need to consider all possible factors (so human and natural influences as well as natural internal variability of the climate system). Natural influences (from volcanoes and the Sun) over this period could have contributed to the change over this period. Volcanic influences from the two large eruptions (El Chichon in 1982 and Pinatubo in 1991) would exert a negative influence. Solar influence was about flat over this period. Combining only these two natural influences, therefore, we might have expected some cooling over this period.

Question E - How confident are you that warming has taken place and that humans are mainly responsible?

Phil Jones: I'm 100% confident that the climate has warmed. As to the second question, I would go along with IPCC Chapter 9 - there's evidence that most of the warming since the 1950s is due to human activity.

Question V - If you have confidence in your science why didn't you come out fighting like the UK government's drugs adviser David Nutt when he was criticised?

Phil Jones: I don't feel this question merits an answer.

Both question (i) and (ii) are scientific questions which can be answered today, or not. It is up to the scientific community including the Royal Society and Royal Swedish Academy of Sciences to present answers or if this is not yet possible, to make this clear.

It is time for Phil Jones and the Royals to come out in the open and fight. This is how science is performed, unlike business and politics taking place in closed boards behind closed doors, and how merits in science are gained.

Or rather, this is how science is supposed to work, but the practice is different, as now being

brought into light in Climategate and Climategate Analysis by John Costella

• So what reception do they get? Instead of embracing this diversity of knowledge—  thanking them for their experience (no one knows everything about everything)  and using that knowledge to improve their own calculations—these power-brokers  of climate science instead ignore, fob off, ridicule, threaten, and ultimately black-  ball those who dare to question the methods that they—the power-brokers, the  leaders—have used. 
• And do not be confused: I am here talking about those  scientists within their own camps, not the “skeptics” which they dismiss out of  hand.    
• This is not “climate science”, it is climate ideology; it is the Church of Climatology.    It is this betrayal of the principles of science—in what is arguably the most  important public application of science in our lifetime—that most distresses  scientists. 
  

A parallel story, on a much smaller scale but with the same ingredients, is my own experience with my resolution together with Johan Hoffman of the 250 year old d'Alembert's paradox as a central problem of fluid mechanics.

Sciencegate of Fluid Dynamics

The origins of the collapse of climate science now unfolding in Climategate can be traced back to the basic scientific discipline of fluid dynamics, because the global climate is created by the fluid dynamics of the coupled system ocean and atmosphere driven by the Sun and the rotation of the Earth.

The truth is that, which is hard to believe at first sight,  fluid dynamics collapsed as a science in the beginning of the century along with the collapse of classical mechanics as observed in The Age of Unreason and Reason, when Ludwig Prandtl presented a resolution of d'Alembert's paradox from 1752 of zero drag and lift in slightly viscous flow. 

d'Alembert's had shown that theoretical fluid dynamics predicted that an object would move through air without any force acting on the body from the fluid. Thus wings could not generate lift to carry a bird or airplane and there would be no resistance to motion. This made fluid dynamics into a mystery since start. 

This mystery lasted until the beginning of the 20th century, because a rational resolution of the paradox seemed impossible, but then the came the collpase of the Age of Reason which openend to irrational resolutions: Ludwig Prandtl quickly cooked up an explanation for drag and the mathematicans Kutta and Zhukovsky for lift, which lasted uncontested through the 20th century. But both explanations were physically incorrect, which was understood but kept secret in order to maintain scientific credibility. 

This was precisely what the Pythagoreans did with the irrationality of the squareroot of two, which threatened their school based on natural numbers. But the secret was revealed by a whistle-blower and the geometric school of Euclid took control, until Descartes resurrected numbers and initiated the scientific revolution leading into our digital world.

But the truth implicit in presentations by NASA is that:

• The fluid dynamics of generation of lift by a wing is unknown!
• The fluid dynamics of generation of drag is unknown!  
  

NASA presents three incorrect theories of lift, but no correct theory! Read and check! NASA does not seem to know more than Leonardo da Vinci, but is kept as a secret.

But there is a new resolution of d'Aembert's paradox which resolves these issues as shown in Why It Is Possible to Fly. The new resolution is published in a refereed journal of high class Journal of Mathematical Fluid Mechanics. It has massive mathematical, computational and experimental support. But it is suppressed by the scientific journal controlling the minds of fluid dynamicists in the World: Journal of Fluid Mechanics JFM. 

Unfortunatley science is extremely authoritarian: What the chief editor JFM says dictates the beliefs of all fluid dynamicists on the globe, more effectively than the Pope controls all catholics.

This breathtaking story is told in my previous blog posts under theory of flight including

• The Mathematical Secret of Flight
• New Theory of Flight is Taking Off
• A Paradoxical Paradox Resolution

and interviews with key actors such as NASA and JFM. I hope you will read and get amazed, by scientific fraud and real science, just as in Climategate! It is a good story! 

In short, since fluid dynamics is a mess, no wonder that climate modeling is a mess. And the moment when this will be acknowledged in "I was wrong" from Al Gore or Michael Mann or Jim Hanson seems to be approaching...Or that The Royal Society and The Royal Swedish Academy say "We were wrong".

A key question is the heat transport from ocean through the troposhere by turbulent convection to the stratosphere and out by radiation. It should be possible to simulate this process with the new fluid simulators now available, as well as ocean circulation. We are gearing up to this challenge ...hopefully filling some space left free in the aftermath of Climategate...