Light-Matter Interaction: Nobel Prize in Physics 2012

The 2012 Nobel Prize in Physics awarded Serge Haroche and David Wineland, concerns experimental investigations of light-matter or radiation-matter interaction for small (quantum) systems: ions in a harmonic trap and photons in a cavity.

There is a connection to the study of blackbody radiation on Computational Blackbody Radiation based on a mathematical model of the form W + R = E of radiation-matter interaction, including the following components:

• wave equation for matter (W)
• small radiative damping: outgoing radiation from matter (R)
• excitation from incoming radiation (E).

The study brings out a basic aspect of near-resonance resulting in matter waves out-of-phase with excitation waves leading to a balance between incoming and outgoing radiation in the form of Planck's radiation law.

Of particular interest is the following information suggesting a connection to near-resonance:

• In most experiments performed by Haroche’s group, the atom and field have slightly different frequencies. 

The basic idea of Computational Blackbody Radiation is to replace statistical quantum mechanics by deterministic wave mechanics with finite precision high-frequency cut-off.  Change of spin may then be thought of as change of phase-shift.

It remains to be seen if this idea is productive. Statistical quantum mechanics is troubled with unsolved riddles and it is not clear (to me at least) that deterministic finite-precision wave mechanics cannot model small systems. 

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.

New Theory for Wind Turbines

                                                 Transport of 50 meter wind turbine blade.


Wind turbines commonly use stall control to limit the power at higher wind speeds, as described in the 2001 thesis Flow Separation on Wind Turbine Blades by Gustave Paul Corten. The thesis starts with the following description of state-of-the-art:

• Presently it is impossible to calculate the lift and drag characteristics of an airfoil accurately. Especially beyond the stall angle, the calculations can be off by tens of percents. For that reason airfoil characteristics still have to be determined in wind tunnels.

We show on The Secret of Flight based on New Theory of Flight that today it is possible to compute lift and drag characteristics of airfoils in the full range of angles of attack including stall and beyond to a tolerance of 5%, by solving the Navier-Stokes equations with slip boundary condition. This opens new possibilities in wind turbine design and control.

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.

Stefan Löfven: Matematik-Syndrom?

Trots flera påstötningar reagerar inte S på mitt förslag om uppföljning av mitt tidigare möte med Socialdemokraterna i Riksdagens Utbildningsutskott om av IT-Matematik för ett år sedan.

Kristina Persdotter, politisk S-sekreterare i Utbildningsutskottet, lovar att svara om S är intresserat av IT-Matematik eller inte, men svarar inget alls.

Nu är det ju så att IT-Matematik precis passar Stefan Löfvens satsning på modern utbildning och innovation, som är kärnan i den nya moderna S-politik som nu skall säljas.

Man frågar då fråga varför Löfven inte har begripit detta?  Kan det vara så att Löfven, i likhet med så många med erfarenhet från den traditionella skolan, lider av en mer eller mindre allvarlig form av det matematik-syndrom som gör att tanken lamslås när matematik och matematik-prov kommer på tal, så att säga?  Och om tanken är lamslagen kan man ju inte säga vare sig ja eller nej till en diskussion om IT-Matematik. Kan det vara så?

Björklunds Mattelyft verkar också lamslaget: Trots oändliga resurser för utveckling av material för omskolning av mattelärare har endast en liten 15 min video producerats hittills, som visar hur man skall lära elever efter 8 år av skolmatematik,  hur man utan miniräknare kan räkna ut vad 6 delat med 0.1 blir.

Efter att lagt upp denna post kommer så prompt ett svar från Kristina Persdotter:

Hej Claes,

Vi bjöd in dig och Trevor till ett möte som vi uppskattade mycket och vi tar med oss de lärdomarna i vårt fortsatta jobb. Socialdemokraterna är mycket intresserade av att undervisningen utvecklas och kvaliteten höjs, inte minst i matematik. Vi ser att det är lärare, skolledare och forskare som ska utveckla undervisningen, inte politiker och politiska partier, och därför har vi bland annat föreslagit betydligt större resurser till lärares kompetensutveckling och till att nytt institut för lärande som vi tror är rätt väg att gå för att höja kvaliteten på undervisningen och resultaten i skolan, se t ex Forskningsinstitut för lärande

Med vänlig hälsning,
Kristina Persdotter

OK, vad säger man då om detta? Uppenbarligen backar S från uppgiften att ansvara för att skolan är anpassad till det moderna IT-samhället. Björklund sätter ner foten med ett diktat att återgå till 1950-talets katederundervisning, medan S vill satsa 25 miljoner på ett Forskningsinstitut för Lärande, som förmodligen skall befolkas med samma lärare, skolledare och forskare som inkompetensförklarats genom samma S-förslag, som ju innebär att universitetsforskningen kringgås.

Varför är det så svårt att få till stånd en diskussion om matematikundervisningen i våra skolor, där inte alla är lamslagna av frågans oerhörda obegriplighet? Varför vågar Löfven inte tänka själv? Är det inte matematik-syndromet som spökar?  Och i så fall, är det bra för S? 

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.