New Theory of Flight Presented at 2nd AIAA High Lift Prediction Workshop

The Direct Fem Simulation (DFS) of the flow of air around an aircraft underlying the New Theory of Flight, which was rejected 2012 by AIAA Journal and is now in review by Journal of Mathematical Fluid Mechanics, was presented by Johan Jansson at the 2nd AIAA High Lift Prediction Workshop in San Diego June 22-23, 2013.

As shown in the Agenda and Summary our contribution stands out as the most advanced simulation technique based on ab initio solution of the incompressible Navier-Stokes equations, without any turbulence model and with a slip boundary condition as an accurate model of small skin friction of high Reynolds number flow, using a stabilized adaptive finite element method.  

Our results of the workshop show that ab initio simulation (DFS) of the complex aerodynamics of complete aircraft during start and landing is possible, which reveals the true secret of flight and opens the possibility of constructing fully realistic flight simulators.

Separation in Slightly Viscous Flow

The New Theory of Flight presented on The Secret of Flight offers a new analysis of the central problem of separation in slightly viscous flow modeled by the Navier-Stokes equations with a slip boundary condition. This analysis is fundamentally different from the classical analysis by Prandtl of boundary layer separation with a no-slip boundary condition, which has dominated modern fluid mechanics.

Separation with slip show to take one of the following forms illustrated in the oil film visualizations above:

1. Rotational separation with opposing flows (top)
2. Parellel flow separation (bottom)

The analysis is presented in the upcoming book The Secret of Flight in Chapters 24 - 30. Why not take a look!

New Theory vs Old Peer Review System

Our article New Theory of Flight, first rejected by AIAA Journal and now under review by Journal of Mathematical Fluid Mechanics (JMFM), raises an important question as concerns the peer review system with anonymous referees used in scientific publishing.

The negative referee reports from AIAA show that the fluid dynamics community will do whatever is needed to suppress the New Theory in defense of the Old Theory.

If JMFM were to choose referees from this community the New Theory would have to be rejected by JMFM. On the other hand, JMFM has already published our article Resolution of d'Alembert's Paradox underlying the New Theory by relying on expertise outside the fluid dynamics community, with basis in mathematics and computation.

But Wikipedia being controled by the fluid dynamics community has blocked every reference to this published article in the Wikipedia article on d'Alembert's paradox, with details recorded on Wikipedia Inquisition, and so the publication in JMFM does not count. JMFM is outside while Journal of Fluid Mechanics (JFM) and AIAA Journal are inside.

It is thinkable that JMFM could likewise publish the New Theory, by finding referees endorsing the New Theory. But these referees could not come from inside the fluid dynamics community and thus would lack scientific credibility. In short:

• Referees from inside the fluid dynamics community will reject the New Theory, even if it is correct, because it challenges the Old Theory. 
• Referees outside the fluid dynamics community cannot endorse the New Theory, even if it is correct, because they lack scientific credibility.   

This shows that the conventional referee system does not seem to be functional in this case of a New Theory challenging an established Old Theory: Either the New Theory will be rejected without good reason, or it will be published without good reason.

The only possibility in this case is to publish the article without the usual refereeing process and invite to an open discussion of the New Theory vs the Old Theory.

What seems to be needed is thus

• A: open refereeing process after publishing with non-anonymous referees,

instead of

• B: closed refereeing process before publishing with anonymous referees.

If there once in the era of analog printing on paper was a reason for preferring B before A, because what was once printed on paper remained printed on paper, this reason no longer holds in the digital era of electronic publishing where rejection and erasing can be done at any time.

Functional science like functional society can only be maintained through open discussion by people with names and faces. Science and politics behind closed doors is against the basic principle of science and democracy of open critical inquiry.

In this light the censorship by KTH of my book Mathematical Simulation Technology recorded as KTH-gate, is troublesome for KTH and a nuisance for me.  

New Theory of Flight and Practice

The New Theory of Flight revealing the Secret of Flight is based on 3d rotational separation at the trailing edge which allows the flow to leave the wing without the high pressure of potential flow destroying the lift created at the leading edge, as illustrated in the above left picture, in the case of a circular cylinder.

The right picture show vortex generators on an upper wing surface with the objective of delaying stall according to the new version P180 3xFJ33 of P180 Avanti. We see a close connection between theory and practice.

Liberation from the Prandtl Spell 2

Continuing the argument in the preceding post, we consider a (laminar) shear layer L between to regions of fluid with different velocity. If epsilon is the viscosity, the width of the shear layer then scales like square-root-of-epsilon = epsilon^0.5 and the velocity gradient like 1/epsilon^0.5, which gives a total dissipation

• D = integral-over-L epsilon x grad u^2 ds ~ A x epsilon^0.5 

where A is the area of L. If the area is bounded, which is the case for a boundary layer of a bluff body, then D tends to zero with epsilon, which indicates a vanishing effect with vanishing viscosity, contrary to Prandtl's basic postulate.

On the other hand, the length of the rolls of 3d rotational separation stretching into the flow behind the body may increase with decreasing viscosity thus keeping D positive under vanishing viscosity. The effect of the rolls on the boundary of a bluff body and total dissipation, which determine drag and lift, would thus stay the same under vanishing viscosity, while the rolls after the body would get longer.

In the case of a turbulent shear layer, D ~ A x epsilon^0.2 according to experimental data, but the conclusion would be similar.

See also The Secret of Flight.

Liberation from the Prandtl Spell 1

The New Theory of Flight explaining for the first time the mathematics and physics of the flight of birds and airplanes is based on solving the Navier-Stokes equations with slip boundary conditions as a model for slightly viscous flow.

The Navier-Stokes equations with slip boundary conditions and vanishingly small viscosity are scale invariant in the sense that a change of the scale in space leaves the equations invariant (assuming a vanishingly small viscosity has no scale). We see this effect as large-scale features of bluff body flow reflecting the geomety of the body, which remain the same under mesh refinement (modulo gradient sharpening effects) such as the 3d rotational separation pattern of the flow around a circular cylinder shown above, which is also seen at the trailing edge of a wing and what lies behind the secret of flight.

It is the scale invariance of solutions to the Navier-Stokes equations with slip boundary condition and vanishing viscosity, which make solutions computable on meshes resolving only the large scale features of the flow in what can be referred to as Large Eddy Simulation (LES), without the need of user-defined turbulence models.

This makes computational solution of slightly viscous flow possible by removing the Prandtl spell of having to resolve thin boundary layers from no-slip boundary conditions, which has parlalyzed fluid mechanics for so long time. Follow the story in the upcoming book The Secret of Flight.

In fact, a no-slip boundary condition is mathematically incompatible with vanishing viscosity and Prandtl's insistence on using no-slip can be seen as a consequence of Prandtl's limitation as mathematician witnessed by e.g. von Karman.

Incorrect Explanation of Magnus Effect

Above is the the standard explanation of the Magnus effect, giving a backspin tennis ball an upward lift force, as potential flow (left) augmented with large scale circulation (middle) to give an unsymmetric flow with lift (right). The same type of argument is used in the circulation theory of Kutta-Zhukovsky supposedly explaining the lift of a wing.

However, the above standard explanation of the Magnus effect is incorrect. The correct explanation is by unsymmetric separation as seen in the following picture of real flow:

and not by large scale circulation as in the above standard explanation. We see that in reality the incoming flow is not affected by the rotation of the ball, while in the standard explanation the incoming flow is changed from horizontal direction without circulation to a slanted upward direction with circulation.

We thus see that the standard circulation theory for the Magnus effect is unphysical, and so is Kutta-Zhukovsky circulation theory for the lift of a wing. For a correct theory see The Secret of Flight.

New Theory of Flight Submitted to JMFM

Our new article New Theory of Flight has now been submitted to Journal of Mathematical Fluid Mechanics, after having been rejected by AIAA Journal.

New Theory of Flight is a spin-off of our earlier article Resolution of d'Alembert's Paradox published in Journal of Mathematical Fluid Mechanics in 2008, which has been one of the 5 most downloaded articles and therefore now can be downloaded for free.

Take the chance to read this article, which finally after 256 years solves a problem preventing a fruitful interplay between theoretical and practical fluid mechanics, and then continue with the New Theory of Flight.  

The JMFM referee reports for Resolution of d'Alembert's Paradox, can be studied here.

Lift from Unsymmetric Attachment-Separation

The lift of a wing in the form of a tilted flat plate with a certain angle of attack can be explained as an effect of

• unsymmetric attachment-separation in real flow (right picture),

which reflects different stability aspects of attachment and separation of symmetric potential flow (left picture) with the flow diverging in attachment and converging in separation. The result is downwash with redirection in real unsymmetric flow, which gives lift as a reaction, while in symmetric potential flow there is no downwash and the lift is zero. Since the downwash is proportional to the angle of attack, lift can be expected to be proportional to the angle of attack, which is also observed:

 

This is all explained in more detail in The Secret of Flight. 

Lifting Line Theory Illposed

Circulation theory in the form of the Prandtl/Lanchester Lifting Line Theory relies on the following theorems by Helmholtz concerning "vortex filaments" or lines of concentrated vorticity:

• Theorem 1: The strength of a vortex filament is constant along its length.
• Theorem 2: A vortex filament cannot end in the fluid; it must extend to the boundaries of the fluid or form a closed path.
• Theorem 3: In the absence of of rotational forces, a fluid that is initially irrotational remains irrotational. (Kelvin's theorem).

Here, Theorem is 3 is not wellposed and thus does not describe physics: An infinitesimal perturbation can give rise to substantial vorticity without rotational forcing, as shown in the Basic Principle of New Theory of Flight. 

Nevertheless, Lifting Line Theory is based on Theorem 3 with a closed lifting line of vorticity formed by (i) transversal vorticity along the wing, (ii) streamwise vorticity extending from the wing tips (iii) transversal starting vortex, formally adding to zero as illustrated in the above left picture. The right picture shows Prandtl's conception of streamwise vorticity from the trailing edge compensating variation in transversal vorticity again from an idea of closed circuits of vorticity summing to zero

The streamwise vorticity emerging from the airfoil compensating variation of transversal vorticity in Lifting Line Theory superficially connects to the counter-rotating vortex lines of 3d rotational slip separation of the New Theory attaching to the trailing edge, but the physics is entirely different:

In the New Theory "vortex filaments" are really created on the surface from opposing flow instability, while in the Lifting Line Theory they appear formally from variation of transversal vorticity as a consequence to Helmholtz's (illposed) Theorem 3.

Note that the "vortex filaments" of streamwise vorticity attaching to the trailing edge in the New Theory conforms to Helmholtz's Theorem 2 with "vortex filaments" allowed to start from the boundary, as a correct statement. Helmholtz Theorem 1 is questionable, Theorem 2 is ok and Theorem 3 is illposed. Lifting Line Theory is based on Theorem 3.

One may say that the New Theory is based on unstable physics, while Lifting Line Theory is based on illposed formal mathematics. Unstable physics is real, while unstable formal mathematics is unreal.

Clarification by Dover

Here is the reaction from Dover upon my request to see the review of The Secret of Flight.

Dear Claes,

In response to your recent messages, a few comments:

In Dover's 70-year history, the company has published perhaps 20 or 25 books on aerodynamics.  This is a minor yet important category for us and we have no contacts in the aerospace industry. As far as we know, no one in that industry is aware of the projects we are considering, including yours, nor are they pressuring us in any way concerning them.  How anyone would try to do that is, frankly, beyond our comprehension.

I've discussed your request for our review with my department head and regret that we must decline.  All of the reviews we obtain concerning projects under consideration are requested by us under a strict belief that anonymity must be preserved, not only the identities of the reviewers, but the content of what they say. We are not engaged here in scientific debate or discussion, we are making publishing decisions. We ask qualified people to tell us what they think secure in the knowledge that they will not later find their comments posted on the Internet without their permission.  This system works well for us-- your first Dover book was reviewed the same way--and we are unwilling to make any exceptions to it.

While we must decline to go ahead, I wish you every success in finding a publisher for your manuscript.

Best regards,

Rochelle Kronzek
Engineering, Science and Mathematics Editor
Dover Publications, Inc. 


OK, so Dover is not engaged in "scientific debate or discussion", only publishes, or not publishes, scientific books based on "publishing decisions". This gives material to the current discussion concerning the role and functioning of publishing companies as a service to science. The system with secret reviews "works well for Dover" but is against the basic principle of science of open discussion.  

Dover Pushed to Not Publish The Secret of Flight

I took a risk by putting up a post announcing that Dover had approached me through Rochelle Kronzek with a proposal to publish my upcoming book The Secret of Flight.  I was well aware of the risk that Dover could be put under pressure from the aerodynamics community to suppress the book, since the book questions the established thinking of this community, but I did not expect Dover to be overly sensitive. After all it was a "delightful book" that could give Dover good money, in addition to the good money from my finite element book, which I gave away to Dover.

But I was wrong, as evidenced by today's message from Rochelle:

• Unfortunately, the reviewer feedback that we've received about your project was very negative and my publisher has decided to back away from the project. I'm sorry that "The Secret of Flight" is not a good fit for our list. I wish you every success in finding a suitable publisher for your book manuscript.

We see Rochelle changing from 

• What an absolutely delightful book, Claes!!  We at Dover are very interested in publishing this work for you and your co-authors when you feel ready to do so. 

to

• The Secret of Flight is not a good fit for our list.

I have asked Rochelle to get to see the reviewer feedback and will report. Stay tuned...Scientific publication can be tricky business...What pressure was put on Dover to not publish the book?
What would Dover risk by publishing the book? What was stronger than good money and good service to science? 

Otto Lilienthal: The Flying Man

Otto Lilienthal, German aviation pioneer named the Glider King and The Father of Flight, reported in Birdflight as the Basis of Aviation measurements during 1866 - 89 of The Carrying Capacity of Arched Surfaces in Sailing Flight collected in a set of diagrams giving the lift L for different angles of attack.

The above legendary Table IV shows a lift coefficient C_L= 0.1 x alpha, with alpha = angle of attack (in degrees), of an arched wing with camber 1/12, up to stall at 15-20 degrees (compared to a plane wing with much smaller C_L as the dotted curve).  This is close to the generic formula found to describe many wings, also captured by the basic lift formula of circulation theory.

Lilienthal computed from the diagram that a wing of area of A = 15 m2 should be able to carry himself in sustained flight at a speed V = 10 m/s powered by 2/3 hp. It took 100 years for Lilientahl's dream to come true in the form of the Gossamer Condor on 60 m2 wings at a speed of 5 m/s powered by human 1/3 hp.

Making of the Prandtl Myth by John D Anderson

We cite from the book Fundamentals of Aerodynamics by John D Anderson addressed in a previous post, concerning Ludwig Prandtl, the father of modern aerodynamics:

• The first practical theory for predicting the aerodynamic properties of a finite wing was developed by Ludwig Prandtl and his colleagues at Gottingen, Germany, during the period 1911-1918, spanning World War I. The utility of Prandtl's theory is so great that it is still in use today for preliminary calculations of finite-wing characteristics.
• The modern science of aerodynamics rests on a strong fundamental foundation, a large percentage of which was established in one place by one man-at the University of Gottingen by Ludwig Prandtl. Prandtl never received a Noble Prize, although his contributions to aerodynamics and fluid mechanics are felt by many to be of that caliber.
• By the 1930s, Prandtl was recognized worldwide as the "elder statesman" of fluid dynamics. Although he continued to do research in various areas, including structural mechanics and meteorology, his "Nobel Prize-level" contributions to fluid dynamics had all been made. 
• Prandtl remained at Gottingen throughout the turmoil of World War II, engrossed in his work and seemingly insulated from the intense political and physical disruptions brought about by Nazi Germany. 
• In fact, the German Air Ministry provided Prandtl's laboratory with new equipment and financial support. 
• Prandtl was considered a tedious lecturer because he could hardly make a statement without qualifying it. However, he attracted excellent students... 
• Prandtl died in 1953. He was clearly the father of modern aerodynamics-a monumental figure in fluid dynamics. His impact will be felt for centuries to come.

Anderson captures the essential quality of Prandtl's theory as a practical method for preliminary calculation of wing characteristics (lift),  and not a physical theory describing the true aerodynamics of the generation of lift of a wing. Anderson further emphasizes the strong connection between Prandtl's work and German war efforts, and thus gives the reason why Prandtl was not near to get a Nobel Prize.

Paradoxical Circulation Theory of Lift

The New Theory of Flight presented on The Secret of Flight shows that the classical circulation theory of lift by Kutta-Zhukovsky is unphysical and thus incorrect, which is also supported by the fact that it builds on a paradox:

• Kutta-Zhukovsky lifting flow around a 2d airfoil is obtained by augmenting non-lifting potential flow by large scale circulation around the airfoil.
• The sharp trailing edge of an airfoil is supposed to be necessary for generation of lift.  
• The lift is proportional to the circulation, and the circulation is determined so as eliminate the singularity of potential flow arising from the sharp trailing edge.
• Lift is thus paradoxically obtained by elimination of what has been introduced, that is by eliminating the singularity introduced by the sharp trailing edge. This is like obtaining an effect by first introducing something and then removing it, as if (+ 1 - 1) could be greeter than zero. 
• The paradox is supposedly resolved by claiming that the singularity is eliminated by effects of viscosity. 

The New Theory shows that lift is generated, not by a sharp trailing edge, but by wings with smoothly rounded trailing edges with diameter up to 10% of the chord length.

The Enigma of the Aerofoil 3

                                                         Sir Leonard Bairstow (1880-1963)

The Enigma of the Aerofoil by David Bloor describes the dispute in England after the 1st World War between Leonard Bairstow representing Cambridge mathematics and Hermann Glauert influenced by German aerodynamics engineering led by Ludwig Prandtl, concerning the fundamental problem of aerodynamics of flight with the following highlite:

• The International Air Congress for the year 1923 was held in London. It provided a further occasion for assessing the advances that had been made in aeronautics during the war years and for addressing unresolved problems. It was a highly visible platform on which the supporters and opponents of the circulatory theory could express their opinions and, in some cases, air their grievances. 
• In the morning session of Wednesday, June 27, there were three speakers: Leonard Bairstow, Hermann Glauert, and Archibald Low. The first to speak was Bairstow, whose talk was titled “The Fundamentals of Fluid Motion in Relation to Aeronautics.” 
• Bairstow was explicit: his aim was nothing less than the mathematical deduction of all the main facts about a wing from Stokes’ equations and the known boundary conditions. 
• The work of Stanton and Pannell had shown that eddying motion did not compromise the no-slip condition and had established kinematic viscosity as the only important variable. “These experiments appear to me,” said Bairstow, “to remove all doubt as to the correctness of the equations of motion of a viscous fluid as propounded by Stokes and the essential boundary conditions which give a definite solution to the differential equations. 
• The range of these equations covers all those problems in which viscosity and compressibility are taken into account, and from them should follow all the consequences which we know as lift, drag etc. by mathematical argument and without recourse to experiment. Such a theory is fundamental”.
• The boundary conditions were empirical matters, but thereafter everything should follow deductively: lift, drag, changes in center of pressure, the onset of turbulent flow and stalling characteristics, along with a host of other results. 

In short, Bairstow stated that theoretical aerodynamics could be reduced to solving the Navier-Stokes equations for slightly viscous flow using mathematics. Bairstow thus reduced aerodynamics to mathematics, in the same way as celestial mechanics was by Newton reduced to solving the equations of motion for a set of n bodies subject to gravitational attraction, referred to as the n-body problem. 

Newton showed that the equations of motion could be solved analytically for a sun with one planet (n=2) and by followers using perturbation techniques for a planetary system like ours (n=10), which made Newton immensely famous. 

But Bairstow (and nobody else at his time) could solve the Navier-Stokes equations in any case of interest of aerodynamics,  which made his grand plan useless. Bairstow thus had nothing to match Glauert's The Elements of the Aerofoil and Air Screw Theory published in 1926 based on circulation theory for inviscid flow with a fix-up in the form of the Kutta condition at the trailing edge, which gave some results and set the text book standard into our time. 

Today it is possible to solve the Navier-Stokes equations using computers and Bairstow's grand plan can finally be realized.  This shifts the weight back to fundamentals without fix-up boosted by computation, and circulation theory with fix-up has no longer any role to play, as explained in detail on The Secret of Flight.

Letter to AIAA Journal Chief Editor into Waste Basket

The following letter was sent to AIAA Journal Chief Editor Peretz P. Friedman on Sept 17.
The letter was thrown into the waste basket by AIAA without answer.

Concerning New Theory of Flight submitted to AIAA Journal

In March 2012 we submitted the article New Theory of Flight to AIAA
Journal. The article presents a new mathematical description of the
generation of lift and drag from the flow of air around a wing, which is
fundamentally different from the current textbook theory based on the work
by Kutta-Zhukovsky-Prandtl. Our new theory is based on a mathematical
analysis of computed solutions of the 3d Navier-Stokes equations with lift
and drag within experimental error tolerance for all angles of attack
including stall and beyond.

The assigned AIAA Journal editor Gregory Blaisdell acknowledges that our
article "is unusual in that it challenges our existing understanding of
aerodynamics". However, the challenge is not really evaluated by the
reviewers and the editor, who instead respond by a lengthy defense of
exactly the existing theory we question using exactly the same arguments
we question, without actually considering our arguments and evidence.

One of the reviewers states: "The authors provide no documented scientific
evidence to discredit the current state of the art”, but acknowledges
that: “At most, perhaps they present a numerical model of the governing
equations which avoids the need to discretize the boundary layer“.

Our objective is not to "discredit" classical theory. But we do present a
new theory made possible through the recent developments in computational
modeling of turbulent flow based on the Navier-Stokes equations, which
allows for quantitative prediction of the full 3d flow field including
aerodynamic forces, also for separated flow beyond stall. Our evidence
consists of accurate solutions of the 3d Navier-Stokes equations in close
correspondence with observation, and an analysis of these solutions.

We therefore request a new review process by AIAA of our article, without
being simply discarding in a defense of the classical theory.

Sincerely, JH, JJ and CJ

Lift (Not) Explained at National Air and Space Museum

                                  A museum full of objects but empty of understanding.

Here is the response from John D. Anderson, Curator of Aerodynamics, Smithsonian National Air and Space Musuem, to my question how lift of a wing is explained by state-of-the-art expertise of today:

• There are several explanations that have been offered for how aerodynamic lift is produced on an airfoil. 
• Some of these explanations are given with almost religious fervor, which sometimes generates heated intellectual debate over which is correct. 
• Such discussions are a matter of philosophy as to which explanation is the most fundamental. 
• The following is my philosophical conviction as to the most fundamental mechanisms used by nature to generate lift: 
• The lift is mainly due to the pressure distribution on the surface; shear stress has only a small effect on lift. In general, for an airfoil that is generating lift, the average pressure exerted on the top surface (pressing down on the airfoil) is smaller than the average pressure exerted on the bottom surface (pressing up on the airfoil). With lower pressure on the top and higher pressure on the bottom, presto -- lift on the airfoil. 
• Therefore, let us concentrate on how nature creates this situation, i.e., how nature produces the pressure distribution over the surface of the airfoil. 
• First, the flow velocity is faster over the top surface and slower over the bottom surface. Why? Because mass must be conserved. 
• Now let us invoke the famous “Bernoulli principle”:
• Because the flow speeds up as it flows over the top surface of an airfoil, as explained above, then the pressure over the top surface must go down. 
• The flow velocity over the bottom surface is lower than over the top, hence the pressure over the bottom surface is higher than on the top. This imbalance of pressure (higher on the bottom and lower on the top) creates an upward force on the airfoil – the lift. 
• Please note: In nature, the velocity increase does not come first, and the pressure decrease does not come second, or vice versa. 
• Nature does all this simultaneously, applying the fundamental laws all at the same time. Nature sets up a velocity field and a pressure field simultaneously, fields that obey the fundamental principles. 
• So there you have it. The absolute fundamental principles of mass conservation and Newton’s 2nd law are the reasons why lift is produced, and we have just explained how nature invokes these two principles to create lift on an airfoil. 
• This is what nature does. 
• I can not advance a philosophy on the generation of lift more fundamental than this, although it takes a full page to explain it.

To sum up, lift results from Newton's 2nd law and mass conservation, according to the highest authority of state-of-the-art science represented by John D. Anderson, author of the book Fundamentals of Aerodynamics, professor em of aerodynamics at Univ. of Maryland and Curator of the National Air and Space Museum.

But this is science reduced to empty triviality. Anderson sends me the following excuse:

• My explanation is based on the fundamentals of physics that apply to any flow, incompressible to hypersonic. This explanation is different from yours, but neither is necessarily “wrong.” 

No, it is not "wrong" because it is completely empty, and an empty statement cannot be judged as "wrong", only void of any meaningful content.

But a Museum of Science which is empty has no role to play, nor does a scientific theory which is a triviality.

PS1 Note that Anderson does not mention circulation theory, the only non-trivial theory of lift presented in text books,  presumably because Anderson share our insight that it is an unphysical and thus incorrect theory.

PS2 The aerodynamics experts at KTH (Art Rizzi and Ulf Ringertz) are smarter than Anderson; they simply refuse to answer any question posed by me about state-of-the-art. But to say nothing is less than repeating a triviality.

PS3 Anderson does not say that our New Theory of Flight is "necessarily "wrong"", which could mean that Anderson believes that it could well be correct. But Anderson does not say that he has studied the New Theory to find out if is correct or not. It may be that Anderson would rather see our theory published by AIAA as new input to a discussion without conclusion since 100 years, than a rejection aimed at stifling discussion. I have asked him about his opinion of the New Theory and will report the answer.

PS4 The generation of lift of a wing is described by Anderson as follows on the museum's web site How Things Fly educating the next generation of US scientists:

• A wing is shaped and tilted so the air moving over it moves faster than the air moving under it. As air speeds up, its pressure goes down. So the faster-moving air above exerts less pressure on the wing than the slower-moving air below. The result is an upward push on the wing—lift!

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.

Feyerabend on the New Theory of Flight

Paul Feyerabend expresses the central message of Against Method as follows:

• My intention is not to replace one set of general rules by another such set: my intention is, rather, to convince the reader that all methodologies, even the most obvious ones, have their limits. The best way to show this is to demonstrate the limits and even the irrationality of some rules which she, or he, is likely to regard as basic. 

Let us speculate about what Feyerabend might have said about the New Theory of Flight explaining the generation of large lift at the price of small drag of the wing of an airplane or bird, that is the secret of flight. Feyerabend could have posed the following questions:

1. When and how was the secret of flight revealed in the form of the New Theory of Flight?
2. Why did it take so long time (100 years after the first sustained powered flight)?

We could have answered Feyerabend as follows:

The New Theory of Flight came out as a natural consequence of the resolution of D'Alembert's paradox by Hoffman and Johnson in 2008 describing slightly viscous bluff body flow as potential flow modified by 3d rotational separation. This description was rationalized from a stability analysis of potential flow after inspection of computed turbulent solutions of the Navier-Stokes equations. 

The "method" used was to solve the Navier-Stokes equations computationally, understanding that slightly viscous flow should be well described by the Navier-Stokes with a slip boundary condition, and then look for features of the turbulent solutions which could be grasped by analytical mathematics. This "method" worked in this case because the main feature of 3d rotational separation could be connected to the basic instability mechanism of potential flow at separation.

We could thus "understand" slightly viscous bluff body flow, such as the flow around a wing, in analytical mathematical terms as potential flow modified by 3d rotational separation resulting from instability of potential flow separation.

We then used the general principle of science that "understanding" of a physical phenomenon means capturing the main features of a mathematical model of the phenomenon in "understandable" analytical terms.

We thus used a "method"consisting of

• solving the Navier-Stokes equations computationally, 
• comparing the solution with the potential solution,
• analyzing the stability of the potential solution.  

We could have asked Feyerabend if this can be described as "anything goes" in the spirit of Against Method, or if it represents an approach which is less ad hoc.

Why did it take so long time to reach this relatively simple insight? Because computational solution of turbulent solutions of the Navier-Stokes equations became possible only 100 years after the first powered sustained flight.