Update of DigiMat: Basic + BodyandSoul

The DigiMat Project has been updated with new material:

• Basic: New version of Preschool App
• BodyandSoul: Model Work Shop 

Take a look.

Frågor till Skolverket 2

Eftersom Lotta Ramqvist inte svarat på mina frågor, vilket Lotta sagt mig skola göra, har jag ställt följande frågor till Skolverket inskickade till Registrator:

Till Skolverket
Generaldirektör Peter Fredriksson

Undertecknad prof em i tillämpad matematik vid KTH önskar svar på följande frågor:

1. Vilka befattningshavare vid Skolverket har huvudansvar för matematikämnet?

2. Hur väl uppfylls den nya läroplanen med programmering som del av matematikämnet i den svenska skolan idag?

3. Vilken fortbildning har Skolverket levererat (planerar att leverera) i programmering som del av matematikämnet i egen regi eller i samarbete?

Vänliga hälsningar
Claes Johnson

Nu får vi se om det blir några svar. Det är inte säkert...

DigiMat vs Skolverket

DigiMat  presenterades för Skolverket representerad av undervisningsråden Lotta Ramqvist, Mats Hansson, Mats Hansson och Daniel Siksjö, vid ett web-möte 28/4, med avsikt att få Skolverket att medverka till spridning av DigiMat som fortbildning för lärare  i matematik/teknik + programmering.

Lotta Ramqvist svarar 8/5 enligt följande:

Hej Claes och Johan!

Efter att ha granskat och diskuterat materialet bedömer vi att det inte är aktuellt att Skolverket sprider materialet. Det grundar sig bland annat på att det inte ligger tillräckligt i linje med kurs- och ämnesplaner och att målgruppen är för smal. Skolverket tipsar mycket sällan om enskilda resurser för lärande/undervisning.

Vi inser ändå att materialet är intressant och användbart för en del lärare. Ett möjligt sätt för er att nå ut med det till många lärare är genom diskussionsforum, så som gruppen Matematikundervisning på Facebook (som troligtvis är Sveriges största forum för matematiklärare).

Vi vill tacka er för att vi har fått tagit del av ert mycket omfattande material.

Och vi önskar er en trevlig helg!
Med vänliga hälsningar
Lotta Ramqvist

Jag svarar 10/5:

Hej Lotta

Ditt svar ställer följande frågor:

1. Vem vid Skolverket är högst ansvarig för beslutet att Skolverket inte på något sätt kan “sprida” DigiMat?

2. Vilket ansvar har Skolverket vad gäller uppfyllande av den nya läroplanen med programmering som del av matematikämnet?

3. Vilket ansvar har Skolverket för fortbildning av lärare när ny läroplan med programmering införs?

4. Anser Skolverket att den nya läroplanen med programmering uppfylls i landets skolor?

5. Vilken fortbildning i matematik/teknik + programmering erbjuder/sprider Skolverket?

6. Vår målgrupp är samtliga lärare i matematik/teknik med programmering som del av undervisningen. Hur kan denna målgrupp anses vara "för smal"?

7. DigiMat uppfyller med råge den nya läroplanen med programmering. Hur kan då Skolverket anse att DigiMat inte ligger “i linje med kurs/ämnesplaner"?

8. Skolverket anser att DigiMat är intressant och användbart för en del lärare. Hur stor del? Varför kan Skolverket inte medverka till att DigiMat når dessa lärare?

9. Vem vid Skolverket har fattat beslutet att Skolverket “mycket sällan skall tipsa om enskilda resurser för lärande/undervisning”? Är DigiMat en “enskild resurs” och då i vilken mening?

10. Det det digitala samhället ställer nya krav och villkor för skolutbildningen i matematik/teknik + programmering. DigiMat är utvecklat av högsta akademiska kompetens och sätter en ny standard för den nya läroplanen. Vill verkligen inte Skolverket medverka till en standardhöjning i svensk skola?

Detta är viktiga frågor som Skolverket har att svara på.

Vänliga Hälsningar
Claes 

Lotta säger sig komma med svar, som kommer att presenteras nedan.

Remembering Fred Singer 1924-2020

I had the great pleasure to meet Fred Singer in a historic seminar. A great scientist and person!

Inbjudan Fortbildning Lärare Matematik Teknik

I dessa Coronatider inbjuder jag tillsammans med Johan Jansson till

• Fortbildning för Lärare i Matematik Teknik Naturvetenskap:
• Matematik + Programmering för den Digitala Världen.

på nätet med stöd för de lärare som så efterfrågar, efter anmälan.  Utbildningen kommer att annonseras av NCM och Skolverket inom kort.

Take a look and see that this may be something for you!

DigiMat Web Education

The Corona crisis asks for school mathematics education on the web.

DigiMat Basic is web based

• New School Mathematics Education for the Digital World

which will soon be available on edX as MOOC as an expansion of DigiMat Pro now running as

•  High Performance Finite Element Modelling Part 1 and 2.

DigiMat is constructive mathematics opening to interactive learning in both individual and group form,  without need of traditional class room teacher instruction, 

Stay tuned! The World is not what it used to be.

Drag Crisis and Slip at Reynolds Number 1 million

This is a continuation of the previous post identifying three types of contact between a fluid and a fixed smooth solid wall:

1. laminar slip/small skin friction
2. laminar no-slip 
3. turbulent no-slip

where DFS Direct Finite Element Simulation uses 1 while standard CFD uses 2 and 3. 

No-slip forms a thin boundary layer connecting fluid with zero velocity on the wall with free flow velocity away from the wall. Slip allows fluid particles to glide along a smooth solid wall without boundary layer at small skin friction.

Standard CFD uses no-slip with thin boundary layers beyond direct computational resolution thus requiring wall models for turbulent flow, which have shown to be elusive. Standard CFD therefore is not truly predictive and thus not very useful. 

DFS uses slip/small friction as an effective boundary condition, which does not form a boundary layer. This makes DFS computable, with true predictive capability demonstrated. 

The appearance of slip/small friction connects to the so called drag crisis observed to occur in slightly viscous bluff body flow with drag drastically dropping at a Reynolds number $Re\equiv\frac{UL}{\nu}$ of around 1 million (or 500.000), where $U$ is typical flow speed, $L$ typical length scale and $\nu$ kinematic viscosity. With $U=1$ and $L=1$, the drag crisis thus connects to $\nu\approx 10^{-6}$ or $Re =10^6$. 

For Reynolds numbers below drag crisis the effective boundary condition can be viewed to be no-slip, which forces early separation into a large turbulent wake and large drag.  For Reynolds numbers above drag crisis separation is delayed to form a narrow wake with small drag,  which the analysis of DFS shows to connect to the appearance of an effective slip/small friction boundary condition. 

Let us seek to follow this transition, thus starting before drag crisis with a laminar no-slip layer of width $d=\sqrt{\nu}$ and shear $\frac{1}{\sqrt{\nu}}$ with free stream velocity $U=1$, and corresponding Reynolds number based on $L=d$ of size $\frac{1}{\sqrt{\nu}}$.

A laminar no-slip layer is an example of shear flow, which shows to develop into a turbulent no-slip layer for Reynolds numbers of size $10^3$ as described in detail in the book Computational Turbulent Incompressible Flow. This connects to a drag crisis at $\nu =10^{-6}$ with $\sqrt{\nu}=10^{-3}$.

In a first step a laminar no-slip low shear layer thus develops into a turbulent no-slip high shear layer which in a second step can develop into an effective slip/small friction condition as an effect of plastic yield in high shear turbulent flow, with a corresponding maximal shear force of size $\sqrt{\nu}=10^{-3}$ appearing as small skin friction of size 0.001. 

The transition from laminar no-slip to turbulent no-slip to slip can be followed in the flow over a convex surface which as laminar no-slip flow separates, because the pressure gradient normal to the boundary is small in a laminar shear layer,  and so develops into a turbulent no-slip layer which can reattach by effectively forming a slip layer with pressure gradient preventing separation.  

Summary: Drag crisis connected to slip occurring at a macroscopic Reynolds number of about $10^6$ with a shear of $1000$ and corresponding skin friction $0.001$, can thus be connected to 

• transition from laminar no-slip at $Re =10^6$ to turbulent no-slip with shear exceeding $10^3$,
• transition from turbulent high shear with layer to effective slip skin friction $0.001$ as an effect of visco-plastic flow.  

  

Laminar Slip Layer vs Turbulent No-Slip Layer: Change of Paradigm

A turbulent no-slip  boundary layer is uncomputable and lacks mathematical model. A troublesome concept. Modern fluid dynamics has been obsessed with the problem of tackling this problem, without success. The result is CFD which is not predictive  and thus not very useful.

DFS Direct Finite Element Simulation as a new paradigm in Computational Fluid Dynamics CFD exhibits a new basic phenomenon of

• laminar slip boundary layer 

to be compared with the basic elements identified by Prandtl as the Father of modern fluid mechanics of:

• laminar no-slip boundary layer, 
• turbulent no-slip layer.

The appearance of a laminar slip boundary is connected to the so called drag crisis occurring in bluff body slightly viscous flow such as air and water at a Reynolds number $Re\approx 500.000$ with the drag of a bluff body drastically dropping beyond $500.000$. 

The reduction is the result of delayed separation with reduced wake as an effect of a shift from a laminar no-slip boundary layer, which trips the flow to early separation,  to effectively a laminar slip boundary layer, which allows a different form of separation as 3d rotational slip separation without tripping.

The appearance of a turbulent no-slip layer is typically artificially induced in experiments through a transversal ribbon/strip attached to the body thus effectively changing the shape of the body, which trips the flow into separation and turbulent wake. The idea is that this way force the experiment to fit with a preconceived notion by Prandtl of a turbulent no-slip boundary layer, but this is against the most basic principle of science to fit theory to observation and not the other way around.    

The result of using an effective laminar slip boundary condition without any artificial tripping, is that fluid flow beyond the drag crisis is computable by DFS because impossible computational resolution of thin turbulent boundary layers required in Prandtl CFD,  is no longer needed. A non-computable turbulent no-slip boundary is thus replaced by a computable laminar slip layer. 

DFS shows to accurately predict fluid flow beyond the drag crisis by computing best possible turbulent solutions of Euler's equations as first principle physics without parameters with slip as wall model and a turbulence model as emergent from computation. This makes CFD computable from being uncomputable to all Prandtl followers, and thus represents a veritable change of paradigm.

A key to the breakthrough is the concept of laminar slip boundary layer of a fluid which is viscous-plastic with fluid particles sliding along a smooth wall with skin friction coefficient of size 0.001 at drag crisis and decreasing beyond. 

DFS shows that slightly viscous flow is not Newtonian with a constant (small) viscosity since the emergent turbulence model in DFS does not reflect a constant viscosity, nor does the viscosity-plastic slip boundary condition. 

This gives perspective on the Clay Navier-Stokes problem which concerns a Newtonian fluid seemingly without relevance for slightly viscous flow as the main challenge of fluid mechanics.           

Banach Documentary: Digital Math: Body and Soul

Together with Per Enflo and Johan Jansson I participate in a documentary about the great Polish mathematican Stefan Banach, to be shown in Polish TV in March. The film was shown to an invited audience at Fokus in Östervåla Fokus 22/2 followed by a discussion about Banach and our connections to his work. The historical event is recorded at the Per Enflo web site and featured on Icarus Digital Math.

The film has appeared in festivals of documentary film and will be shown at KTH in the Spring and maybe also on Swedish State Television. Stay tuned.

The title of the film is

• Banach: Between Spirit and Matter

with a connection the inscription on the grave stone of Steinhaus, who discovered Banach's talent and became his teacher:

• Mathematics connects Soul to Matter

which is basically the same as the leading theme of the series of books:

• Applied Mathematics: Body and Soul Vol 1-4

as made clear in the film.

DFS: Change of Paradigm in CFD

DFS Direct Finite Element Simulation is change of paradigm of Computational Fluid Dynamics CFD by correctly predicting the forces acting on a body moving through a slightly viscous fluid such as air or water with the shape of the body as only input, through computation of best possible solutions to Euler's equations expressing first principle physics without parameters.

DFS takes CFD out of the conundrum of finding turbulence and wall models, which despite efforts over more than 100 years has not led to true predictive capability. Standard CFD is typically fitted to match observation but does not deliver correct prediction without prior (wind tunnel) observation and so is not very useful for design.

DFS combines the Euler equations in the fluid domain with a slip boundary condition on the smooth wall of the body modeling vanishing viscous skin friction. DFS shows to correctly predict drag as form/pressure drag within experimental precision and thus shows that the contribution from skin friction is negligible. This is in direct contradiction to standard CFD which attributes $50\%$ or more of drag to skin friction for slender bodies.

As an example we consider the case of drag and lift coefficients $C_D$ and $C_L$ for the basic test case of a long Naca0012 wing, as function of angle of attack $\alpha$. DFS delivers the following results for $0\le \alpha\le 15$ well below stall:

• $C_L(\alpha ) \approx = 0.1\times\alpha$, 
• $C_D(\alpha ) \approx = 0.004 + 0.001\times\alpha$.        

This fits wind tunnel experiments (without artificial tripping) by Ladson within experimental precision. 

The Ladson value $C_D=0.005$ for $\alpha =0$ instead of $0.004$ with DFS, stands out as a limit case for which extrapolation from $\alpha\ge 2$ as in DFS may well be more relevant than direct measurement with tripping as an issue ($C_D=0.008$ with tripping).   

We see a linear variation of both $C_L$ and $C_D$ with the angle of attack $\alpha$ as an expected effect of changing geometry.  For lift it connects to effective downwash scaling with $\alpha$ and for drag with an effective frontal area also scaling with $\alpha$    

The efficiency of the wing is measured by the lift $L$ to drag $D$ quotient $\frac{L}{D}=\frac{C_L}{C_D}$ ranging from 33 for $\alpha =2$ over 60 for $\alpha =6$ to 75 for $\alpha =15$, thus with steadily increasing $\frac{L}{D}$ before stall. 

The common view is that for a short wing $C_D$ has a contribution scaling with $C_L^2$ thus quadratically in $\alpha$  due to a wing tip effect, which suggests that for a long wing $C_D$ is constant as being dominated by skin friction, however without support in observation.  

Summary: 

• DFS shows that for slightly viscous flow beyond the drag crisis for Reynolds number around $500.000$, total drag is mainly form/pressure drag with a very small (at most $10\%$) contribution from skin friction. 
• Standard CFD attributes instead $50\%$ or more to skin friction for an airplane or ship.  

The consequence for design is a change of paradigm from an old standard bogged down by unsuccessful attempts to decrease skin friction, to a new standard focussing on form, where possibilities for improvements are many.  

The dogma of $50\%$ skin friction is upheld by tripped experiments where e.g. a ribbon is fastened on the body transversal to the flow to generate turbulence increasing drag which is then attributed to skin friction, while it effectively instead corresponds to a change of form. This way observation is fitted to theory prescribing massive skin friction, while in correct science theory is fitted to observation.