## fredag 21 maj 2021

### Svar 3 av Utbildningsdepartementet

Här är svar från Utbildningsdepartementet på mitt tidigare brev till Anna Ekström om skolmatematikens kris, som Anna Ekström sagt sig vara lyckligen omedveten om. Den senaste turbulensen kring Sveriges resultat i PISA18 ger perspektiv på den försköning av Sveriges prestationer som utövats av både Skolverk och Utbildningsminister. Soppan ifrågasätts av Oppositionen i Utbildningsutskottets utfrågningar av Anna Ekström och Peter Fredriksson, medan S försvarar med att svensk skola står stark trots PISA, vilket framgår av svaret. Allt är lugnt!

Hej

Tack för din fråga på ett tidigare svar. För regeringen är det viktigt att kunskapsresultaten stärks inom matematik såväl som i andra ämnen i skolan. Som du själv nämner i ditt brev pågick under 2012-2016 satsningen Matematiklyftet som var en fortbildning genom kollegialt lärande för alla lärare i Sverige som undervisar i matematik. Satsningen utvärderades av Umeå universitet. Du kan ta del av utvärderingen via Skolverkets hemsida: Utvärdering av Matematiklyftets resultat – Slutrapport - Skolverket

Därtill arbetar regeringen med att utveckla ett professionsprogram som syftar till att stärka professionens kompetens och därmed höja kunskapsresultaten. Arbetet har pågått en längre tid och bereds för närvarande i Regeringskansliet. Regeringen har fört dialog med olika nyckelaktörer både enskilt och inom ramen för Samling för fler lärare. Av den sakpolitiska överenskommelsen mellan Socialdemokraterna, Centerpartiet, Liberalerna och Miljöpartiet de gröna framgår att programmen ska inrättas med grund i betänkandet Med undervisningsskicklighet i centrum – ett ramverk för lärares och skolledares professionella utveckling (SOU 2018:17). Du kan ta del av utredningen här: Med undervisningsskicklighet i centrum – ett ramverk för lärares och rektorers professionella utveckling - Regeringen.se

Det mesta av det faktiska innehållet i programmen behöver tas fram och beslutas i nära dialog med professionerna.

Med vänlig hälsning

Alexander Widergren

Departementssekreterare

Utbildningsdepartementet

## torsdag 20 maj 2021

### 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.

## tisdag 18 maj 2021

### 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.

## måndag 10 maj 2021

### Ämneslärarutbildningar i Matematik: Bristande Kvalitet enligt UKÄ

Universitetskanslersämbetet UKÄ har avslutat sin granskning av ämneslärarutbildningar inledd 2018. Följande utbildningar bedöms ha bristande kvalitet:

Ämneslärare matematik gymnasium:

• Göteborgs Universitet
• Högskolan i Kristianstad
• Luleå Tekniska Universitet
• Lunds Universitet
• Uppsala Universitet.

Ämneslärare matematik åk 7-9:

• Göteborgs Universitet
• Högskolan i Kristianstad.
Dystert, särskilt vad gäller gymnasielärare. Ytterligare uttryck för skolmatematikens kris.

Lösningen på krisen består av fortbildning för lärare i matematik för det digital samhället: DigiMat. Läs och begrunda!

## tisdag 4 maj 2021

### Pisa, Anna Ekström och Peter Fredriksson

Anna Ekström har blixt-inkallats till Utbildningsutskottet med anledning av två promemorior av tjänstemän på Utbildningsdepartmentet om felaktigheter i PISA 2018, som Ekström mörkat i sin presentation inför svenska folket av en påtaglig resultatförbättring jämfört med PISA 2015. Speciellt kan det vara så att den lilla uppgången från 2015 för matematikämnet i själva verket innebär fortsatt nedgång och fördjupning av skolmatematikens kris.

Både Anna Ekström och Skolverkets generaldirektör Peter Fredriksson har som framgår av tidigare poster visat sig vara totalt ointresserade av möjlig lösning av skolmatematikens kris. Ekström säger sig inte känna till någon kris alls och Fredriksson säger sig "inte ha möjlighet" att betrakta frågan överhuvudtaget.

Det är inte så ansvaret för skolan skall bäras. Frågan är om Ekström och Fredriksson kan sitta kvar. Tänk om det tillsattes ny ledning med vilja att lyfta svensk skolmatematik. Så bra det vore för elever, föräldrar, lärare och samhälle.