A ToE for Fluid Mechanics

Einsteins ideal as a Theory of Everything ToE is a mathematical model of physics without any parameters. 

The standard model of particle physics contains 18 parameters. It is a very complicated model. To determine the parameters experimentally is impossible.

The standard model of isotropic linear elasticity contains 2 parameters. This is a very simple model but for a non- isotropic body the number of parameters includes 18 parameters. 

To be a useful model the values of its parameters must be supplied as input determined from experiments or more basic model, which in general is very difficult. The 2 parameters of isotropic linear elasticity can be determined from simple tests, but the 18 parameters for non-isotropic linear elasticity are difficult to determine, not to speak of non-linear elasticity and all the parameters of the standard model. 

Are there any parameter-free models of physics? A basic example is a circle described as the set of points in a plane with a certain distance to a given mid-point from which the value of Pi can be computed as the quotient between circumference and diameter. That is a very simple model. Is there any model of more complex physics which is parameter-free? 

Yes, there is one, and maybe this is the only one: Euler's equations for incompressible fluid flow are expressed in terms of velocity and pressure without any parameter: Input is geometry, in/out-flow conditions and external forces, but no parameter, since viscosity is set to zero.   

The remarkable thing is now that the drag and lift of a body moving through a slightly viscous fluid like air and water can accurately be predicted by computing turbulent solutions to the Euler equations with only geometry of the body as input. This is like computing the ratio of circumference/diameter of a circle (that is computing Pi), but just more astounding. Drag and lift coefficients (scaling with $speed^2$) of a body only depend on the geometry of the body! No parameter input needed! See Computational Turbulent Incompressible Flow and Breakthrough of predictive simulation.

The Euler equations for incompressible flow is a ToE for slightly viscous incompressible flow like air (subsonic) and water.  This is remarkable. Is this is the only ToE in physics.

Well, Newton's law of gravitation contains the gravitational constant G connecting gravitational force to mass as parameter, but may be viewed as a ToE in the sense of correctly predicting that all bodies independent of composition move the same way subject to gravitation. 

PS Von Neuman famously claimed that he (in principle) could model an elephant with 4 parameters, and make it wiggle its trunk with a 5th, but in practice how would he determine the parameters?  Elephant experiments are costly and cumbersome.

Euler Was Right, Prandtl Was Wrong I

Euler vs Prandtl

In 1755 the great mathematician Euler formulated the Euler equations for slightly viscous nearly incompressible flow (of air and water) with the following prophetic declaration:

• My two equations contain all what is contained in the theory of fluid mechanics. It is not the principles of mechanics we lack to pursue this analysis but only Analysis (computation), which is not sufficiently developed for this purpose.  

Euler's equations are formulated in terms of fluid velocity and fluid pressure depending on space and time as an expression of force balance (Newton's 2nd Law) and incompressibility complemented by a slip boundary condition with only pressure forces from a solid wall meeting the fluid, that is, with zero skin friction allowing the tangential flow velocity to be non-zero restricting only the normal flow velocity to be zero on a wall.  Euler's equations are parameter-free (formally zero viscosity), thus meeting Einstein's ideal of a mathematical model. The only force acting on fluid particles is pressure and shear forces are assumed to be negligible.  Euler made the assumption about zero skin friction from experiments showing very small skin friction in slightly viscous flow with massive evidence in modern times. 

Eulers adversary d'Alembert quickly crushed Euler's grand plan by showing that Euler's equations admitted certain solutions (potential solutions) showing zero net forces (drag, lift) of a body moving through air or water, in direct contradiction to observation. This was coined d'Alembert's Paradox which from start, as expressed by Chemistry Nobel Laureate Hinshelwood:

• separated practical fluid mechanics (hydraulics) describing phenomena (drag, lift), which cannot be explained, from theoretical fluid mechanics explaining phenomena (zero drag, lift), which cannot be observed.       

Zero lift is incompatible with flight and so d'Alembert's Paradox had to be resolved, in particular after powered human flight was shown to be possible by the Wright brothers in 1903, and so the young fluid mechanician Prandtl presented a resolution in a sketchy 8-page conference contribution in 1904, where he discriminated potential flow with zero skin friction claiming that a real fluid always meets a solid wall with zero tangential velocity named no-slip.  Prandtl thus "resolved" d'Alembert's Paradox by declaring that Euler's equations with slip had to be replaced by the Navier-Stokes equations including small viscosity and no-slip. But no-slip was an ad hoc assumption which Prandtl could not justify since the exact nature of the microscopic contact between fluid and wall was unknown to him and so has remained into our days. 

Prandtl in 1904 with his self-built fluid test channel resolving d'Alembert's Paradox.

Anyway, the scientific community was by Prandtl relieved from a main headache making theory of fluid mechanics into a joke and accordingly Prandtl was named Father of Modern Fluid Mechanics based on the Navier-Stokes equations with no-slip and not Euler's equations with slip. 

But there was one main caveat: The Navier-Stokes equations with no-slip have solutions with boundary layers so thin that computational resolution is impossible with any forseeable computational power.  Prandtl's resolution thus came with the cost of making Computational Fluid Dynamics CFD into an impossibility asking for resolution of atomistic scales in a macroscopic setting.

In 2010, Hoffman and Johnson published in Journal of Mathematical Fluid Mechanics a different resolution of d'Alembert's paradox showing that the reason zero-drag/lift of potential flow cannot observed, is that potential flow (in fact any laminar flow) is unstable and thus turns into turbulent flow. This was shown by computing turbulent solutions to Eulers equations with slip with drag and lift in close correspondence to observations supported by stability analysis, as exposed in detail in the book Computational Turbulent Incompressible Flow. As a spin off a New Theory of Flight was developed revealing the true Secret of Flight in physical terms, very different from the unphysical lifting line theory advocated by Prandtl. 

Since then massive evidence has been accumulated by Johan Jansson showing that computing turbulent solutions of Euler's equations with slip opens basically all of slightly viscous nearly incompressible flow to predictive simulation without parameter input and need to resolve thin no-slip boundary layers, thus with readily available computing power, all along Euler's prophecy. More evidence: HighLift Workshop.

Euler was thus right, and he understood that he just had to wait for computing power to see his prophecy become true. It took 250 years, but now it is here.

It means that Prandtl was wrong claiming drag and lift to be effects of thin no-slip boundary layers thereby making CFD into an impossibility. 

Question

How will the fluid dynamics community react to replacing Prandtl by Euler as Father of Modern Fluid Mechanics thus changing CFD from impossible to possible? 

Further Important Facts

Turbulent solutions to Euler's equations are computed as best possible approximate solutions in the sense of having residuals which are small in a weak sense and not too large in a strong sense, in a situation when all solutions with small residual in a strong sense (laminar solutions) are unstable and do not persist over time. We thus face a new situation where only turbulent flow is computable and laminar not, as an expression of the fluctuating nature of turbulence, as seen in a waving flag showing the only motion which can persist. The control of the residual in strong sense introduces a viscous effect as a form of turbulent viscosity set by computation alone without need to model or measure turbulent viscosity beyond human comprehension.  

Euler was a mathematician while Prandtl as Father of Modern Fluid Mechanics was more of an engineer. Replacing Prandtl by Euler means freeing the full power of mathematics with computation in a rare example of parameter-free mathematical model with very rich applicability.

Standard CFD under a Planck dictate of no-slip has developed complicated wall models as well as turbulence models including many parameters, and an agreement has been made to adjust parameters to give  50% or more of total drag to skin friction. Turbulent Euler computations with zero skin friction show correct drag in a large variety of situations, which is incompatible with the 50% skin friction from standard CFD.

Total drag consists of pressure drag and skin friction drag. Turbulent Euler computations show that pressure drag dominates skin friction by a factor of at least 10, and so standard CFD claiming 50% skin friction must underestimate pressure drag by a factor 2. The CFD community is now wrestling under this contradiction. The investments in standard CFD are huge and will loose their value if Euler is allowed to take over from Prandtl...Compare with posts on Prandtl Medal.

Incompressible flow is well captured by the Euler equations  for Reynolds numbers (scaling with 1/viscosity) larger than about 500.000 associated with the so called drag crisis when drag of a bluff body drastically decreases with a factor 2-3 as the boundary condition effectively turns into slip from limited velocity strains, with late separation and small wake of low pressure, in particular with lift/drag around 15 for a wing allowing flight at affordable power.

Euler vs Navier-Stokes: What is viscosity?

The Navier-Stokes equations connect fluid velocity strains (derivatives in space) with shear forces through a positive coefficient of viscosity $\nu$ as a parameter to be supplied as input, assumed to be constant independent of fluid velocity in the basic case, but in general with a very complex unknown non-linear dependence on local flow velocities. Formally $\nu =0$ in the parameter-free Euler's equations.

In slightly viscous flow the coefficient of viscosity is small with a Reynolds number $Re = \frac{UL}{\nu}$ beyond drag crisis (bigger than 100.000- 500.000) with $U$ typical flow speed and $L$ typical spatial scale L. 

The Navier-Stokes equations can be complemented by a (skin) friction boundary condition with a friction parameter $\beta$ connecting (tangential) shear stress to tangential flow velocity, with slip corresponding to $\beta =0$ and effective no-slip for $\beta >1$, thus covering a range from slip to no-slip with important effects on flow separation and drag (as exposed in Computational Turbulent Incompressible Flow). 

To determine the viscosity as input to the Navier-Stokes equation experimentally or theoretically has shown to be virtually impossible in the case of slightly viscous flow, which is always partially turbulent with a very complex expression of viscosity. Using Navier-Stokes equations for true prediction of slightly viscous flow has not been shown to be possible. With parameter fitting in viscosity models standard CFD can match measured drag, but generally fail in blind tests without prior knowledge of the correct value to match.

Computing turbulent solution to the Euler equations includes automatic modeling of viscosity

through weighted strong residual control as a dissipative effect with a complex flow dependence beyond viscous shear stress.  It appears as a solution to the open problem of turbulence modeling. In particular, size of the strong residual measures the turbulent dissipation as a mesh independent quantity meeting Kolmogorov's conjecture. 

The Navier-Stokes equation model (1823) with constant positive viscosity is generally viewed to be a better/more complete model then the Euler equations (1755) with formally zero viscosity. This was picked up by Prandtl in 1904 using in particular no-slip from the presence of positive viscosity as a way to discriminate potential flow and get around d'Alembert's paradox. But the more complete model showed to be boundary layer uncomputable and asking for parameter input and so non-predictive, while the basic Euler model showed to be more useful by being both computable (no boundary layers) and predictive as parameter free. 

The ultimate quest for a physicist is to find a Theory of Everything ToE as a parameter free model explaining all of basic physics. Computing turbulent solutions to the Euler equations is a ToE for fluid mechanics. 

Towards Resolution of Gray's Paradox

Gray's paradox concerns the contradiction between standard fluid mechanics predictions of the resistance to motion (drag) of a dolphin and the observed speed of a dolphin. Gray estimated that the required muscle power would be seven times bigger than that available. A real paradox!

The search to resolve the paradox has tried different routes: One is to claim that the muscle power of dolphin in fact is much bigger (seven times) than what can be envisioned. Another is to claim that the skin of a dolphin has a magical composition somehow decreasing drag by a factor of seven.

None of the attempts has been successful.

Let us see if the new approach to computational fluid dynamics presented in Computational Turbulent Incompressible Flow and The Secret of Flight, offers a resolution.

We term the new approach DFS Direct Finite Element Simulation (of turbulent flow) based on computing turbulent solutions to the Euler equations for incompressible flow with slip/small skin friction boundary conditions. We have found that DFS predicts the drag of a wing, full airplane and car in close correspondence with observation, with slip as zero skin friction.

Our conclusion is that skin friction gives a minor contribution to total drag as pressure drag plus skin friction in many applications of aero- and hydromechanics, including the locomotion of a dolphin.

This is against the common view of the fluid dynamics community that skin friction is 50-90% of total drag. DFS thus gives design fundamentally new conditions to work from.

DFS in particular seems to offer a resolution of Gray's paradox, by showing that the drag of a dolphin is severely overestimated by conventional techniques as being based on a formula for flat plate drag.

The resolution is a spin-off of the resolution of D'Alembert's paradox (check video) as the mother of the paradoxes of fluid mechanics, a resolution which is intimately connected to DFS.

That skin friction drag predicted from flat plate experiments gives an overestimate of the drag of a streamlined body, like a dolphin, is supported by the article TURBULENT SKIN-FRICTION DRAG ON A SLENDER BODY OF REVOLUTION AND GRAY’S PARADOX, by Nesteruk and Cartwright (13th European Turbulence Conference (ETC13), Journal of Physics: Conference Series 318 (2011) 022042):

• The presented analysis shows that turbulent frictional drag on a slender rotationally symmetric body is much smaller than the flat-plate concept gives and the flow can remain laminar at larger Reynolds numbers. Both facts are valid for an unseparated flow pattern and enable us to revise the turbulent drag estimation of a dolphin, presented by Gray 74 years ago, and to resolve his paradox, since experimental data testify that dolphins can achieve flow without separation. The small values of turbulent skin-friction drag on slender bodies of revolution have additional interest for further experimental investigations and for applications of shapes without boundary-layer separation to diminish the total drag and noise of air- and hydrodynamic hulls.

We will now compute the drag of a dolphin by DFS and report the results shortly. Reducing prediction of skin friction from 70% to 10% may correspond to Gray's factor seven...

PS  From Passive and Active Flow Control by Swimming Fishes and Mammals by F.E. Fish and G.V. Lauder:

• Dolphins have the muscular capacity to swim at high speeds for short durations while maintaining a fully attached turbulent boundary layer. The turbulent flow conditions would delay separation of the boundary layer (Figure 1; Rohr et al. 1998). When the boundary layer separates from the skin surface and interacts with outer flow, this results in a broader wake and increased drag, so delaying separation is beneficial to the dolphin. Separation is more likely to occur with a laminar boundary flow, producing a greater drag penalty compared to turbulent boundary conditions. Thus, the turbulent boundary layer remains attached longer because it has more energy than the laminar boundary layer. The increased drag of a turbulent boundary layer is small compared to the increase in drag due to separation, which is more prone to occur with a laminar boundary layer.

This conforms with the theory and practice presented in Computational Turbulent Incompressible Flow showing in particular that flow with a slip boundary condition stays attached with small drag, while flow with a no-slip laminar boundary layer separates early with large drag. The observed small drag of a dolphin thus can be explained by the theory behind DFS, but not by any commonly accepted theory seeking the origin of drag in thin boundary layers following the legacy of Prandtl.       

New Theory of Flight Accepted for Publication in Journal of Mathematical Fluid Mechanics

The ground-breaking article New Theory of Flight is now accepted for publication in Journal of Mathematical Fluid Mechanics. The paralyzing spells of Prandtl, father of modern fluid mechanics, and Kutta and Zhukovsky, fathers of modern aerodynamics, are now finally broken after more than 100 years of misleading unphysical mathematics. A post-modern era of (computational mathematical) fluid mechanics and aerodynamics is now approaching...

The Ludwig Prandtl Medal

Today I received the following message:

Dear Professor Johnson, It is our great pleasure to inform you that you are the winner of the 2014 edition of the ECCOMAS Ludwig Prandtl Medal. The decision has been taken by the ECCOMAS Award Committee in a two-round voting procedure. Please receive our warmest congratulations. 

The Ludwig Prandtl Medal will be delivered at the Opening Session of the WCCM-ECCM-ECFD 2014 Conference in Barcelona, July 21, 2014 (8:30-10:30). We would very much appreciate if you can confirm your participation.

With our best regards and congratulations,

Ekkehard Ramm
ECCOMAS PresidentJosef Eberhardsteiner
ECCOMAS Secretary

Here is my answer:

Dear Profs Ramm and Eberhardsteiner

Thank you for this great honor, which I will be very happy to receive in person at the conference opening. 

The award has an interesting aspect from scientific point of view in that my work (with Johan Hoffman), shows that Prandtl's main idea of the fundamental role of the boundary layer, for both separation and drag and lift, crowning him as the Father of Moden Fluid Mechanics, is incorrect. We show that separation, drag and lift originate from instability of slightly viscous flow and not from a boundary layer. The evidence comes from solving the Navier-Stokes equations with slip boundary condition, which does not give rise to any boundary layer, and we obtain results in full agreement with observations. We conclude that separation, drag and lift in slightly viscous flow do not originate from a boundary layer and thus that Prandtl's main idea is not in agreement with observations.

I would appreciate if this will be made clear to the public at the conference and I would certainly be willing to shortly expose the reasons why Prandtl was wrong. 

The fluid dynamics community will not applaud the award, since 20th century fluid mechanics has followed the Father in search of the origin of separation, drag and lift in the boundary layer. This has had a catastrophic impact on computational fluid mechanics leading to the strong belief that correct results require resolution of boundary layers, which however is impossible even in thinkable future since quadrillions of mesh-points would be required. The result is a dead-lock of rational science. We show that drag and lift of an airplane can today be accurately computed over the entire range of angles of attack including stall, by solving the Navier-Stokes equations with slip using a couple of millions of mesh points. 

The award thus brings a major scientific question to the podium and I hope it can be accompanied by a scientific discussion.
Sincerely, Claes Johnson

Parameter-Free Fluid Models: How to Make Einstein Happy

In recent work (here and here) we have shown that mean-value outputs of computed turbulent solutions of the incompressible Navier-Stokes equations with small viscosity, vary little with the absolute value of the viscosity. This makes this mathematical model to an example of Einstein's ideal as a physics model without parameters or coefficients requiring experimental determination.

For example, we show in New Theory of Flight that the lift and drag of an airplane can be accurately computed using this model, thus ab initio without input of any experimental measurement. This is very remarkable and would have made Einstein very happy.

Augmenting this model to include self-gravitation as in the blog post Equivalence of Inertial and Heavy Mass, gives a parameter-free cosmological model, by choosing the unit of mass so that

• $\Delta\phi (x,t) = \rho (x,t)$, 

where $\phi (x,t)$ is the gravitational potential and $\rho (x,t)$ mass density for a given unit of length specified by the $x$ coordinate, and the unit of time $t$ so that Newton's 2nd law takes the form

• $\ddot x = - \nabla\phi (x,t)$,

connecting particle accelleration $\ddot x(t)$ to the gradient $\nabla\phi (x,t)$, where $x(t)$ is the trajectory of a particle of unit mass.

Such a  model can describe galactic scales after galaxy and star formation from interstellar dust under compression, as an incompressible fluid of small viscosity under self-gravitation, without any parameter to determine experimentally. This could have made Einstein even happier.

Quantum Contradictions 23: The Truth

Daniel Greenberger states in the Preface to Fundamental Problems in Quantum Theory: A Conference Held in Honor of Professor John A. Wheeler:

• Quantum theory represents one of the great and most beautiful structures in all of physics.  
• Nonetheless, despite its uncontrovertible experimental successes, the theory has a very shaky philosophical foundation. 
• The standard Copenhagen interpretation(whatever that is) requires us to accept so many assumptions that defy common sense that ever since the theory was first developed it has led to enormous debates concerning its interpretation. 
• Most modern physicists accept it without qualification and, indeed, one can develop a creative intuition for using it. 
• The fact that many of its founding fathers turned against the standard interpretation, whereas their followers have tended to accept it without second thoughts can only partly be ascribed to the circumstance that anything tends to grow more familiar with repeated use. 
• Part of the explanation must be related to the fact that those very founders were much moreculturally well rounded than most modern physicists. 
• They were philosophically trained and philosophically inclined and did not like what they saw.
• In spite of their doubts, the subject grew rapidly and it became fashionable to avoid questions concerning the foundations. 
• This attitude only started to change after Bell’s famous theorem in 1964. He showed that one could pose some of one’s intuitive doubts experimentally. 
• Since then, a number of alternate interpretations have grown and new experimental tests devised.
• Today, we know that the strange predictions of the theory hold up experimentally (even though the foundations remain shaky). 
• We will never go back to classical physics - we must learn to accept and live with the world as it actually is.
• What makes quantum mechanics so much fun is that its results run so counter to one’s classical intuitions, yet they are always predictable, even if unanticipated. 
• That is why I like to say that quantum mechanics is magic, but it is not black magic. 

This may well be the truth about quantum mechanics as one of the two pillars of modern physics:  

• A magic perfect theory counter to classical intuition, which we all have to accept without understanding its foundations and without asking the questions the founding fathers posed without ever giving any answers. 
• A magic perfect theory which (self-proclaimed) physics experts (like Lubos Motl of the Reference Frame) pretend to understand perfectly well, but refuse to answer any question with the excuse that all questions were answered by the founding fathers.  

This is the truth also of the other pillar of relativity theory: a magic theory which we all have to accept without understanding, a magic theory which the experts claim to understand but are not willing to explain with the excuse that all questions were answered by its founding father Albert Einstein (who explained very little),  

The same phenomenon has come to dominate climate science with a magic counter-intuitive  "greenhouse effect" which we all have to accept without understanding, a magic theory which the experts claim to understand but are not willing to explain with the excuse that all questions were answered by the founding fathers (Tyndall and Arrhenius, who explained very little).

The same phenomenon has come to dominate modern fluid mechanics with a magic counter-intuitive boundary layer theory which we all have to accept without understanding, a magic theory which the experts claim to understand but are not willing to explain with the excuse that all questions were answered by the founding father Ludwig Prandtl (who explained very little).  

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.

The Secret of Flow Separation Uncovered

Real flow as potential flow modified by vortical slip separation with point stagnation.

The flow around a body moving through a slightly viscous incompressible fluid like water or air at subsonic speeds, attaches at the front as fluid particles approach the body and separates in the back as fluid particles part from the body.

In slightly viscous flow fluid particles glide along the surface with small skin friction which can be approximated as a slip boundary condition.

Observation shows laminar attachment and turbulent separation, with the flow being close to potential flow (inviscid irrotational flow with slip) prior to separation. In potential flow the separation is simply reverse attachment and thus real flow differs from potential flow at separation.

Ever since the basic mathematics of fluid mechanics was formulated by Euler and d'Alembert in the mid 18th century, a prime goal has been to describe slightly viscous flow around a body as potential flow subject to some modification at separation:

• Prandtl as the father of modern fluid mechanics described the modification as a boundary layer effect from a no-slip boundary condition.
• Kutta and Zhukovsky as fathers of modern flight mechanics described the modification as large scale circulation around a wing section.

In the new article Analysis of Separation in Turbulent Incompressible Flow together with Johan Hoffman (submitted), we show that the answer is neither boundary layer nor circulation, but instead

• vortical slip separation with point stagnation.

We give evidence in the form of

• mathematical analysis of basic instability of potential flow at separation
• computational solution of Navier-Stokes equations
• experimental observation.

We thus present strong evidence that the dream of Euler and d'Alembert can be fulfilled by describing slightly viscous flow around a body as

• potential flow modified by vortical slip separation with point stagnation.

The article gives the details.

As an application we uncover The Secret of Flight (upcoming book) and The Mathematical Secret of Flight (article and talk).

Large Boundary Layer Collider: Why Prandtl Was Wrong 3

Part of the Large Boundary Layer Collider at the European Spallation Source in Lund, Sweden.

According to Ludwig Prandtl, named the father of modern fluid mechanics, both drag and lift of a body moving through air or water originate for a thin boundary layer.

This is the fundamental postulate of modern fluid mechanics formulated in 1904, but it is now being questioned. Is modern fluid mechanics based on a postulate which is does not correspond to physical reality?

The answer may be given by the European Spallation Source (ESS) in Lund, Sweden: The world's biggest proton accelerator (see picture).

The idea is to eliminate the boundary layer by bombarding it with high energy protons, and once the boundary layer has been removed completely this way, drag and lift will be measured. If drag and lift remain the same under removal of the boundary layer, then drag and lift do not originate from any boundary layer, and modern fluid mechanics is based on incorrect physics.

But ESS will not be ready to use before 2020, and thus it is natural to ask if there is some other quicker and cheaper way of eliminating a boundary layer? Yes, there is. But what is it?

Follow the thrilling uncovering of one of modern physics most well kept secrets...

PS An alternative to ESS would be to use liquid helium with next to zero viscosity, but to reach a sufficiently large Reynolds number, the dimension of the experiment needs to be 10 times bigger than that of the Large Hadron Collider and thus is out of reach, for the moment at least.

But as UN global warming alarmism is now fading away maybe this experiment could become the next big initiative by the UN backed by EU. DS

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...

Hurrah! Scientists Back in Spotlight!

100 years ago scientists were leading the debate by forming the new wonderful world of modern physics. Einstein was preparing his entree to the general public to become the most famous person all times. Then came the wars, science was discredited as something useful for evil purposes and the interest decreased,  among students and in the general public.

But with the debate on global warming scientists again are back on track: Today nothing is more important than mathematical climate models and scientific measurements and their accuracy and relevance. Are we approaching a hot hell or not? Only scientists can tell. Politicians listen. The general public waits...

But scientists do not agree, and so there is an intense debate going, in particular as a preparation for the Copenhagen meeting on climate in December, which can be used in a case study of scientific method and practice.  The web is filled with conflicting information, but let us start with something which just appeared:

• The Yamal Implosion about cherry-picking data showing global warming.
• Cooling Down the Cassandras about the plateau in temperature the last decade. 
• Flawed Climate Data about hockey stick mathematics.

This is bad news for climate alarmism, but the debate is not over. Global climate is a complex dynamical system and a real challenge to science. As long as the debate is raging money will flow into climate research and give a boost to classical physics and fluid mechanics...and above all to computational mathematics...and there are new computational tools ready to do their service to humanity...as we will report on later...

An overview is given in the 4-slide presentation The Actual State of Climate Science by Roger Pielke SrRead, also presented on a video featuring 2 alarmists and 2 critics allowing a study of typical arguments of the debate.

Consensus in Science and Sports: An Inconvenient Truth

The idea of scientific consensus is used by the alarmists of global warming and by the wikipedians controling the information on Wikipedia. However, scientific truth cannot be determined by majority voting, only by scientific facts and arguments. One fact or argument by one scientist can outweigh the consensus of billions of people. Scientific consensus can lead astray, since it can give the false impression of scientific truth, when it is only the superstition of many.

Political truth in democratic societies is determined by consensus of the majority,  but scientific truth should not be determined by consensus, in particular not by third-party majority consensus, but by real combat between active living scientists. 

It is the same in sports: The Wimbledon matches between Borg and McEnroe were not determined by consensus of the spectators, but by Borg and McEnroe alone. 

Borg and McEnroe represent the active living scientists carrying the scientific knowledge at any given time, who by playing matches of science or disputations in seminars and journals decide the current truths. 

In sports, you win by walkover if your opponent does not show up to the match, and it is the same in science. You cannot defend your position by saying nothing, neither can the songbird  defend its territory by singing nothing.

An illustration is given by the match about how to resolve  d'Alembert's paradox in fluid mechanics which has been going on for 255 years. This long match has now come to an end by the publication of my resolution together with Johan Hoffman in the leading Journal of Mathematical Fluid Mechanics. The victory is declared by Google putting our resolution in top position in a search on "resolution of d'Alembert's paradox". It is a walkover victory because the entire editorial board of the Journal of Fluid Mechanics says nothing.

d'Alembert's paradox of zero drag in inviscid flow is important since much of modern fluid mechanics is related to the paradox in one way or the other, as explained in my knols on fluid mechanics.

Michel Crichton expresses the essence very clearly: 

• Let's be clear: the work of science has nothing whatever to do with consensus. Consensus is the business of politics.  Science, on the contrary, requires only one investigator who happens to be right, which means that he or she has results that are verifiable by reference to the real world.  In science, consensus is irrelevant. What is relevant is reproducible results. The greatest scientists in history are great precisely because they broke with the consensus.
• The work of science has nothing whatsoever to do with consensus. There is no such thing as consensus science. If it is consensus, it isn't science. If it's science, it is not consensus. Period."

Conservation of Momentum or Newton's 2nd Law?

                                          Incremental or Conservation Party?

Continuing the discussion from the previous post, let us note that the Navier-Stokes equation

expressing conservation of momentum, alternatively can be expressed as Newton's 2nd law 

                                                             F = ma = m dv/dt

with F the force acting on an element of fluid of mass m and acceleration a = dv/dt. We can view these formulations to be equivalent from analytical mathematical point of view, but we may ask if they also are equivalent physically, or computationally? 

Of course they are equivalent, you may say, because mathematics rules the game, but it is not so simple and clear if we recall that the Navier-Stokes equations cannot be solved exactly analytically, only approximately digitally by computers. The equivalence is then not so clear anymore.

So which formulation is most suitable to computation?  Newton's 2nd law because it can be solved by time-stepping moving forward in time with small increments of time: The force F gives the acceleration dv/dt = F/m which tells the change of velocity which tells the change in position, from one time level to the next. 

On the other hand, conservation of momentum is not directly ready for time-stepping, since it just expresses that something is conserved, namely momentum.

We are thus led to prefer an interpretation of a law of nature, which is most accessible to computation.  We may prefer such an interpretation also from physical point of view, if we view real physics as some form of analog computation, as discussed in the knol Is the World a Computation?

Light refraction is a result of the wavelike nature of light as propagating electromagnetic waves. Light refraction can alternatively be described as shortest time of travel of light rays. Wave propagation can be time-stepped, while shortest time of travel is a global minimization problem, for which computational solution is less direct. We are led to view light as waves from physical and computational point of view, rather than as rays of particles. 

An equilibrium states may be described as a state of balance of forces without any net force driving change. To find an equilibrium state of a system, we may time-step the system starting from some out-of-balance non-equilibrium state, with the hope that the system by itself approaches equilibrium. A physical law could then express the dynamics of a system computable by time-stepping, rather than a balance of forces at equilibrium, since this balance may not be directly computable. 

A minimization principle in physics, like minimal time of travel of light, would then not qualify as a physical law unless augmented by e.g. time-stepping into computable form.

 

Climate and Turbulence Modeling

Global climate models are based on turbulence models, since the slightly viscous flow of air in the atmosphere and water in the oceans is turbulent. Turbulence modeling, in the form of  analytical mathematical models, is a main unsolved problem of fluid mechanics.

In our book Computational Turbulent Incompressible Flow, with prel. version for download, Johan Hoffman and I present a new approach to turbulence modeling based on ab initio numerical computation with turbulence automatically modeled by the stabilization of the numerics, thus without any explicit analytical turbulence model.  

We show that mean-value quantities of turbulent flow such as drag and mean temperature can be accurately computed without analytical turbulence model, thus circumventing the main unsolved problem of fluid mechanics. We plan to test this approach on climate modeling with hopefully a connection between turbulence and the main unsolved problem of climate modeling: cloud formation. 

We will report as soon as we have something to report on...hopefully before the Copenhagen meeting in December...since the outcome of this meeting critically depends on computational modeling of turbulence and cloud formation...and dark clouds over the meeting are already forming... 

Coin Tossing: Cold or Warm?

Newscientist reports:

• Forecasts of climate change are about to go seriously out of kilter. One of the world's top climate modellers said Thursday we could be about to enter "one or even two decades during which temperatures cool."People will say this is global warming disappearing," he told more than 1500 of the world's top climate scientists gathering in Geneva at the UN's World Climate Conference.
• "I am not one of the sceptics," insisted Mojib Latif of the Leibniz Institute of Marine Sciences at Kiel University, Germany. "However, we have to ask the nasty questions ourselves or other people will do it.
• "Few climate scientists go as far as Latif, an author for the Intergovernmental Panel on Climate Change. But more and more agree that the short-term prognosis for climate change is much less certain than once thought.
•  "In many ways we know more about what will happen in the 2050s than next year," said Vicky Pope from the UK Met Office.

The message is that global climate models cannot predict year or decade meanvalues, but can predict centennial meanvalues. How can this be? What is the mathematics behind such a belief? 

The first idea that come to mind is the law of large numbers of statistics offering prediction of the meanvalue 0.5 of many cointosses between 0 and 1, but no prediction of the meanvalue of a few tosses. But is climate modeling the same as coin tossing between cold and warm? 

Newscientist concludes:

• The world may badly want reliable forecasts of future climate. But such predictions are proving as elusive as the perfect weather forecast.
  

The future of mankind thus seems to lie in the hands of mathematicians running the climate models...but coin tossing statistics does not seem to be enough...what can be done or said? 

Well, let us recall that the 0.5 probability of heads in coin tossing is computed mathematically using the fact that a rotating coin has head up half of the time, that is using a short-time-accurate mathematical model, see the discussion in Chapter 13 Turbulence and Chaos in Computational Turbulent Incompressible Flow. Without a short-time-accurate model, nothing can be be predicted about long-time...Compare with the UK Met Office assurement:

• There have been major advances in the development and use of models over the last 20 years and the current models give us a reliable guide to the direction of future climate change.
• Computer models cannot predict the future exactly...
• Current models enable us to attribute the causes of past climate change, and predict the main features of the future climate, with a high degree of confidence.

What are "the advances in the development and use of models"? What is meant by "direction of future climate change"? Colder or warmer? Does "direction" indicate that the size of the change cannot be predicted? What is the meaning of "computer models cannot predict the future exactly"? That computer models can predict the future almost exactly? Who is the inventor of this form of newspeak? Note the clever construction of the following key statement by Met Office:

• As well as producing CO2, burning fossil fuels also produces small particles called aerosols which cool the climate by reflecting sunlight back into space. These have increased steadily in concentration over the 20th century, which has probably offset some of the warming we have seen.
  

Note the clever use of "probably" and "some of the warming". Very clever doublespeak: Clearly suggesting something, without saying anything! This is not the language of science. 

Can really these semantic tricks help save the World?

Scoping the 5th Assessment Report of IPCC

                                                       The core of a climate model.

The upcoming 5th Assessment Report of IPCC AR5 was scoped in IPCCs 30th Session in Antalya, April 21-23, 2009, as a Proposal for an IPCC Expert Meeting on Assessing and Combining Multi Model Climate Projections:

• Climate model results provide the basis for IPCC projections of future climate change. Previous  assessment reports included model evaluations but avoided weighting or ranking models.  
• Defining a set of minimum criteria for a model to be 'credible' or  agreeing on a metric of performance is therefore difficult and the criteria are likely to depend on the  variable and timescale of interest. Combined with an estimated data volume exceeding 1000  Terabytes, the AR5 faces immense obstacles in trying to make sense of the deluge of model runs and  data that it will produce. 
• Recent studies have started to address these issues by proposing ways to weight or rank models, based  on process evaluation, agreement with present day observations, past climate or observed trends.  While there is agreement that 'the end of model democracy' may be near, there is no consensus on how  such a model selection or weighting process could be agreed upon. 

Reading these reservations, we get a warning that we should not expect any significant improvement of climate model reliability from AR4 to AR5. World leaders preparing far-reaching reductions in CO2 emissions thus have to look to the stars for guidance, or simply rely on the predictions of IPCC based on climate models of unknown but most likely low reliability.

 

Feedback, Sensitivity, Cancellation and Duality

The sensitivity of a mathematical model is a measure of the effect on a certain model output from variation of certain model input data. The sensitivity to errors in data, modeling and computation directly connects to the accuracy of a model. 

Climate sensitivity is primarily concerned with the effect on the global mean temperature from increasing the CO2 concentration in the atmosphere. 

Concerning the climate sensitivity of current climate models, IPCC states:

• Spread in model climate  sensitivity is a major factor contributing to the range in  projections of future climate changes. 
• Consequently, differences in climate  sensitivity between models have received close scrutiny in all  four IPCC reports.
• Climate sensitivity is largely determined  by internal feedback processes that amplify or dampen  the influence of radiative forcing on climate. 
• (A) To assess the  reliability of model estimates of climate sensitivity, the ability  of climate models to reproduce different climate changes  induced by specific forcings may be evaluated. 
• (B) An alternative approach, which is followed here,  is to assess the reliability of key climate feedback processes  known to play a critical role in the models’ estimate of climate  sensitivity. 

Here (A) is a reasonable way of testing climate sensitivity, and gives a large spread shown in Fig 10.2, while (B) boils down to 

• To assess the reliability of model estimates of climate sensitivity, we assess the reliability of key climate feedback processes  known to play a critical role in the models’ estimate of climate sensitivity. 

In other words, assessment of climate model sensitivities, is replaced by assessment of the feedback processes built into the model. But this is an internal check which appears to be circular: You build in a certain feedback process into the model and you then test model sensitivity by testing the validity of the feedback process you have put in. But in most cases you cannot isolate and experimentally test the validity of the feedback process you have put in: If you could directly observe climate sensitivity experimentally, then climate models would serve no purpose. 

But some sensitivities can be observed experimentally, and thus can serve as reliability tests of climate models. This is done in a recent article by Richard Lindzen showing that the radiation sensitivity of current climate models with respect to surface temperature, does not fit with observations, as shown in the above figure with ERBE radiation measurements: Climate models show too small radiation. Something is apparently wrong with the climate models, and there are many things that could be wrong...

In our exploration of the secret of turbulence by computation, we have studied output sensitivity by duality techniques based on solving associated dual linearized problems, and we have found that local exponential turbulent perturbation growth is controled by effects of cancellation. 

In turbulent flow, an important charcteristic of climate atmosphere/ocean circulation,  cancellation means that the worst combination of effects does not occur: Increase in space-time is balanced by decrease in space-time so that the net effect is smaller than worst case. Duality techniques should be able to offer important information on sensitivity also in climate models, but current models lack this capability and there seems to be room for improvement...will cancellation and duality help save humanity?

Fear of Flying?

Are you afraid of flying? If so, you should not read the following, except the very last paragraph. If you are not afraid of flying, and want to know the whole thruth, you should read the following and see if there is any need to change your mind.

Summing up my blog-poll with aerodynamics experts on why it is possible to fly and sail, or how a wing generates lift at small drag, I find that no convincing theory is presented. Either the expert flatly refuses to say anything whatsoever about the basic question of the expert's area of expertize, which is common but a bit disappointing as an expression of expertize, or the expert including NASA presents nonsense-theory such as

• lift comes from the pressure distribution around the wing
• lift comes from downwash
• lift comes from Newton's law 
• lift comes form Bernuoilli's law

or refers to a "mathematical" circulation theory stating that

• lift comes from large scale circulation around the wing
• lift comes from wing-tip vortices
• lift comes from a Prandtl lifting line including a starting vortex.

The circulation theory for lift of a wing was suggested by Kutta-Zhukovsky in the beginning of the 20th century as an analog to an earlier circulation theory for the Magnus effect causing a top-spin tennis ball to curve down. 

However, according to state-of-the-art fluid dynamics, the Magnus effect results from unsymmetric separation and not from large scale circulation. Likewise, circulation is not the origin of lift of a wing. The failing belief in circulation theory is expressed by the fact that in state-of-the-art it is described as a "mathematical" theory, which means that it is an un-physical theory, which does not correctly describe physics; it is only a "mathematical trick" without physical meaning, see The Spell of Kutta-Zhukovsky's Circulation Theory.

The net result is that according to experts there is no physical theory of flying. Surprised? How do you feel now? No fear of flying even if experts don't know what keeps an airplane in the air?

Should I then cancel my next flight, you ask? No, you don't have to. Take a look at Why It Is Possible to Fly and you will discover an understandable new correct physical theory of lift and drag of a wing. You can take it as medication against fear of flying!