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.

Prandtl's Tripped Science vs Boeing Max

Prandtl making tripped experiments

Danger of tripping 

Ludwig Prandtl is viewed as the Father of Modern Fluid Mechanics because he offered a resolution of the pressing problems of fluid mechanics in the beginning of the 20th century including d'Alembert's paradox through his discovery of the laminar and turbulent boundary layer in wall bounded fluid flow.

The legacy of Prandtl is described in Prandtl-Essentials of Fluid Mechanics edited by Herbert Oertel, Springer 2004, with the following introduction

• The development of modern fluid mechanics is closely connected to the name of its founder, Ludwig Prandtl. 
• In 1904 it was his famous article on fluid motion with very small friction that introduced boundary-layer theory. 
• His article on airfoil theory, published the following decade, formed the basis for the calculation of friction drag, heat transfer, and flow separation.
• Prandtl was particularly successful in bringing together theory and experiment, with the experiments serving to verify his theoretical ideas. 
• It was this that gave Prandtl’s experiments their importance and precision. His famous experiment with the tripwire, through which he discovered the turbulent boundary layer and the effect of turbulence on flow separation, is one example. 
• The tripwire was not merely inspiration, but rather was the result of consideration of discrepancies in Eiffel’s drag measurements on spheres. 
• Two experiments with different tripwire positions were enough to establish the generation of turbulence and its effect on the flow separation. For his experiments Prandtl developed wind tunnels and measuring apparatus, such as the Göttingen wind tunnel and the Prandtl stagnation tube. 
• His scientific results often seem to be intuitive, with the mathematical derivation present only to provide service to the physical understanding, although it then does indeed deliver the decisive result and the simplified physical model. 
• According to a comment by Werner Heisenberg, Prandtl was able to “see” the solutions of differential equations without calculating them.

To give the highlighted parts perspective recall that when I was awarded the Prandtl Medal in 2014 by ECCOMAS, I stated that I would receive the medal under the condition that it would be expressed that the New Theory/Computation of Flight developed together with Johan Hoffman and Johan Jansson showed that Prandtl had misled modern fluid mechanics into a fruitless search for the origin lift and drag of an airplane wing in a boundary layer so thin that it could never be resolved in computation. This was not allowed to be expressed and the result was that I did not accept to receive the medal. The story can be read here.

The New Theory of Flight supported by refined computations since 2014 shows that contrary to Prandtl wall bounded slightly viscous flow can be modeled by a slip boundary condition without any boundary layer, which makes the flow computable as time variable turbulent flow. There is thus now massive evidence that Prandtl was wrong, seriously wrong. 

Signs that there is something fishy with Prandtl's boundary layers as the origin of drag and lift can be seen in the above highlights: 

1. Prandtl use a tripwire to change the flow to fit what he could "see" without mathematics and computation. 
2. His results were intuitive.     

The effect of artificially tripping the flow in experiments has led to the misconception that skin friction drag is a major part of total drag with form/pressure drag a minor part, viewed to be relevant  also for an airplane wing without tripping device. The New Theory gives hard evidence that this is seriously misleading by computing drag and lift with slip in close accordance to observations.

The lesson is that if you rely on intuition rather than correct mathematics and are ready to trip experiments to fit, then you can end up with something with little connection to reality. Evidently Prandtl did so. The consequences are severe with the Boeing Max debacle a result of misconceived engineering computation following Prandtl.

PS The following question/answer appears on FAQ at Secret ion Flight:

Q30: Why is the flow tripped by a wire, strip or ribbon in wind tunnel measurements of drag of wing, when a real wing does not have any tripping device and the tripping thus appears to be artficial?

A30: The rationale presented is that the tripping will force the development of a turbulent boundary layer with substantial skin friction,  which according to Prandtl should be present. The tripping is thus done to artificially fit reality to theory, which is opposite to the basic principle of science to fit theory to reality. In the New Theory, which fits with untripped experiments, the flow of air meets the wing with a slip boundary condition modeling vanishing skin friction.

Fundamentals of Aerodynamics by John D Anderson as Old Theory of Flight

The book Fundamentals of Aerodynamics by John D Anderson describes the standard theory of flight as Old Theory of Flight with basic ingredients expressed in Chapter 4:

• The purpose of this chapter is to present theoretical methods for the calculation of airfoil properties.
•  In most of this chapter we will deal with inviscid flow, which does not lead to predictions of airfoil drag; indeed, d’Alembert’s paradox says that the drag on an airfoil is zero—clearly not a realistic answer.
• However, if we lived in a perfectly inviscid world, an airfoil could not produce lift. 
• Indeed, the presence of friction is the very reason why we have lift. These sound like strange, even contradictory statements to our discussion in the preceding paragraph. 
• What is going on here? The answer is that in real life, the way that nature insures that the flow will leave smoothly at the trailing edge, that is, the mechanism that nature uses to choose the flow shown in Figure 4.18c, is that the viscous boundary layer remains attached to the surface all the way to the trailing edge. 
• Nature enforces the Kutta condition by means of friction. If there were no boundary layer (i.e., no friction), there would be no physical mechanism in the real world to achieve the Kutta condition. 
• So we are led to the most ironic situation that lift, which is created by the surface pressure distribution—an inviscid phenomenon, would not exist in a frictionless (inviscid) world. In this regard, we can say that without friction we could not have lift. 

We read that the Old Theory is strange, contradictory, not realistic and ironic. The New Theory of Flight presented on Secret of Flight shows that this characterisation is correct. It is now time to allow the Old Theory to retire since it is physically incorrect and no longer is needed as a facade when there is a physically correct theory.     

DFS vs standard CFD: Form vs Skin Friction Drag

Direct Finite Element Simulation DFS, as a new revolutionary methodology/software for Computational Fluid Dynamics CFD, computes best possible turbulent solutions to Euler's equations as first principle physics without parameters.

DFS gives results in close agreement with observations as true prediction without adjustment of parameters to match each computation with observation in non-predictive mode. In particular, DFS uses a slip boundary condition on a smooth solid wall as a model of vanishingly small skin friction.

DFS thus computes the drag of a bluff body as form/pressure drag with vanishingly small contribution from skin friction, in close agreement with observation.

For a Naca0012 airfoil at zero angle of attack DFS delivers a drag coefficient $C_D =0.006$ as form/pressure drag, which fits well with untripped measurements (red) by Abbott and von Doenhoff:

The figure also shows measurement (blue) by Ladson with artificial tripping not present for a real wing with larger $C_D\approx 0.008$, thus not applicable to a real wing which does not carry any tripping device.

We now compare DFS with standard CFD where we find the following account in the standard reference Fundamentals of Aerodynamics 5th ed by John D Anderson p. 381 with reference in particular to Lombardi, G., Salvetti, M. V. and Pinelli, D.: Numerical Evaluation of Airfoil Friction Drag, J. Aircraft, vol. 37, no. 2, March–April, 2000, pp. 354–356:

• total drag 0.00623 
• skin friction drag 0.00534 ($85\%$ of total)

into (see also this post): 

• DFS: form/pressure drag $95-100\%$ of total drag.
• Standard CFD: form/pressure drag $15\%$ of total drag.

We see a vast difference of form/pressure drag with a factor 6! There is no way both DFS and standard can be correct.

We have massive evidence that DFS without parameters gives correct drag. The conclusion can only be that standard CFD does not capture anything like the truth.

Standard CFD includes turbulence and wall models with many parameters, with the wall model delivering large skin friction ($85\%$) of total drag. The parameters are then adjusted to give total drag in accordance with observations, which means that form/pressure drag comes out as a small portion ($15\%$) of total drag. 

The conclusion can only be that standard CFD is not useful, acknowledged by many users, since by the necessity of parameter fitting is not predictive and does not capture true physics.

The reason standard CFD does not capture physics is rooted in the wall model used, which prescribes a separation pattern which is not physical. In DFS the separation is not prescribed by a model and instead follows the physics.  

It is clear that aerodynamic design will be very different if based on predictive DFS with form/pressure drag dominating skin friction drag, instead of as now non-predictive standard CFD postulating dominating skin friction in contradiction with physics.

It is the same story in ship hydromechanics with a current consensus of $70\%$ skin friction drag and  correspondingly small form/pressure drag, misleading design into (resultless) efforts to reduce skin friction.

PS More precisely John D Anderson reports the following results from Lombardi at al for NACA0012 at zero angle of attack at $Re =3\times 10^6$:

  We see standard CFD delivering skin friction drag even larger than observed total drag.        

Good Example: NRK Denies Denial

Klimarealsitene reports that Norwegian State Radio NRK has declared a new Strategy on Climate Alarmism:

• Dekningen skal i hovedsak handle om hvordan, og ikke om, det skal handles for å tilpasse seg eller dempe den globale oppvarmingen.
• De stadfester at menneskeskapte klimaendringer er reelt, og at NRK skal legge dette til grunn for journalistikken.
• NRK skal være oppmerksomme på den «falske balansen». Hvis de slipper til klimafornektere, skal de stille de rette motspørsmålene.

In English:

• NRK is to report only about how to stop global warming, not if it is needed or meaningful.
• NRK declares that humans control the climate.
• Climate denialism will only be allowed to be voiced if directly countered with complete denial by NRK. 

We now await Swedish State Radio SR to follow with a similar declaration to meet the mission of public radio, although effectively such a policy is already implemented.  

Strong State: Scaring or Comforting?

Swedish Social Democracy 2020: Stronger Safer

A Strong State pretending to have a mission, can use one of two basic strategies:

1. Fear-Mongering: Imminent Threat! Alarm! The State protects you.
2. Comforting: No Threat! No Alarm! The State takes care of you.

There are many examples of 1. through history collapsing to state terror. Swedish Social Democratic Society peaking in the 1950s is an example of 2.

Today we see a Swedish Social Democratic Society, which has switched mode from 2 to 1, in the name of climate alarmism set on a road supported by a new climate law to be the first fossil free welfare state by 2050 in another great leap forward, as an example of a new brave world to be followed by the entire world. 

Once that new brave world is reached, Swedish Social Democratic Society will return to mode 2. Only 30 years of great leap state terror with the end as usual justifying the means.

But is climate alarmism starting to crack already today? Yes, there are signs like this one.

Theoretical Pole Vault Limit for Duplantis?

                Mondo Duplantis breaks own pole vault world record with 6.18 clearance. 

Duplantis says that $6.30$ is not impossible. What does theory say? Inertial energy is $m\times \frac{v^2}{2}$ with $m$ mass in $kg$ and $v$ speed in $\frac{m}{s}$, and potential energy is $m\times g\times H$, where $g =9.81\,\frac{m}{s^2}$ is gravitational constant and $H$ is height in $m$.  

Duplantis horisontal speed is 10 which gives an inertial energy of $m\times 10^2/2$, which if completely converted into potential energy corresponds to $H=5$. This requires the pole to be fully elastic as well as the ground support for the pole. Duplantis uses a very stiff pole and so may reach the height 5.00 from the pole alone. The remaining height must come from the arms. Since the limit of high jump using the legs is about 2 $m$, it is apparently possible to reach 1.18 $m$ with the arms alone.

Another way is to recall that a human being can develop 1 hp momentarily and thus lift 75 $kg$ 1 $m$ during 1 $s$. So 2 $m$ may be impossible, and thus 7.00 $m$ may be the theoretical limit? 

Letter to Scientific American on New Theory of Flight

To: Scientific American

Att: Article by Ed Regis: No One Can Explain Why Planes Stay In The Air

Yes, it is true, that there is no scientific explanation to be found within the established aerodynamics community in academics and industry, why it is possible to fly.

But there is an explanation as a New Theory of Flight developed outside the established aerodynamics community based on mathematics and computation, which allows accurate prediction of full flight characteristics of an aeroplane and which is well documented in articles, books and open source software.

The New Theory is presented in full on the web site  Secret of Flight and summarized in compact form in this flyer.

Basic scientific documentation consists of

• New Theory of Flight (J of Mathematical Fluid Mechanics 2015)
• Resolution of d'Alembert's Paradox (J of Mathematical Fluid Mechanics 2011)
• Time-resolved adaptive Direct FEM Simulation of High-Lift configurations (Springer 2017)
• Computational Turbulent Incompressible Flow (Springer 2008)

We invite Ed Regis to go through this material and consider a follow up article in Scientific American, which we offer to contribute or write together.

The fact that the aerodynamics community despite massive efforts over 100 years has not been capable of explaining why planes stay in the air is an unbelievable story, yet true. The debacle of the Boeing 737 Max is an expression of the limitations of conventional theory and computation, in particular concerning the dangerous phenomenon of stall.

The New Theory of Flight explains the physics of generation of lift and drag of a wing including stall supported by solid mathematical analysis and computation, and more generally allows the computational simulation of the full flight characteristics of an airplane.

But New Theory is not welcomed by the aerodynamics community since it threatens making established routines obsolete. An article in Scientific American could be instrumental in breaking the silence and allow progress.

The New Theory of Flight offers an exciting story of historic dimensions which could also capture the interest of a broad public.

We hope to get in direct contact with Ed Regis to have a discussion of the subject and how to proceed.

Sincerely

Johan Jansson assoc prof scientific computing KTH Stockholm
Claes Johnson prof em applied mathematics KTH Stockholm

Ridgway Scott prof em mathematics and computer science UChicago

Scientific American: No One Can Explain Why Planes Stay in the Air. We can.

Scientific American sends the shocking message:

• No One Can Explain Why Planes Stay In The Air.

Yes, it is true that there is no scientific explanation to be found within the established aerodynamics community in academics and industry why it is possible to fly. Without proper understanding the design of airplanes has to rely on trial and error and the Boeing 737 Max debacle gives harsh evidence of the danger of this approach.    

But there is an explanation as a New Theory of Flight developed outside the established aerodynamics community based on mathematics and computation which allows accurate prediction of full flight characteristics of an aeroplane and which is well documented in articles, books and open source software, and which the reveals the Secret of Flight for scientists and engineers and is also understandable to the general public.  Take a look an get enlightened! 

I will now take contact with Scientific American and suggest a follow up article presenting the New Theory of Flight. Attention is needed.

To connect to the Scientific American article check out Old Theory, a travesty. 

Euler's Dream, Einstein's Ideal and Leibniz' Best Possible of Worlds

DFS Direct Finite Element Simulation offers a veritable breakthrough in Computational Fluid Dynamics CFD in the form of best possible solution of Euler's equations expressing first principle physics in the form of Newton's 2nd Law and incompressibility.

DFS is a realisation of Euler's dream formulated in 1755:

• All of aero/hydrodynamics is captured in the equations I have formulated, Euler's equations.

DFS is a realisation of Einstein's ideal:

• Mathematical model of physics without parameters.

DFS closely connects to Leibniz's grand vision:

• The real world as the best of all possible worlds as the most perfect world being richest in phenomena from simplest laws. 

DFS in particular offers a solution to the problem of computing/simulating/modeling turbulent flow with turbulence as an expression of instability of slightly viscous flow preventing exact conservation of momentum and mass in both physics and computation as the essence of turbulence.

DFS generates turbulent flow as complex phenomena from simple laws and as such is best possible.

Standard CFD in the form of RANS generates simple non-turbulent phenomena from complex laws and as such is the opposite.

A short presentation of the DFS breakthrough is given here with further material on Secret of Flight.

Skin Friction is Small

DFS with slip (zero skin friction) correctly predicts lift and drag within 2-5 percent to observations for Reynolds numbers Re beyond the drag crisis that is Re bigger than about 500.000 of relevance for flight. This shows that the contribution from skin friction to drag at high Re is at most 2-5 percent. In DFS drag is pressure/form drag with zero skin friction, in close agreement with observation.

This is in direct contradiction to a widespread belief in the fluid dynamics community that skin friction is 50 percent or more of total drag for streamlined bodies. The support comes from flat plate experiments where the flow by an attached transversal ribbon trips the flow and thus creates drag, which is translated to arbitrary streamlined body without ribbon claiming that the ribbon drag on the flat plate becomes skin friction on the body. Compare with this post.

The logic is missing and the 50% skin friction drag is an artefact from tripped experiments combined with the fact that standard CFD has been fitted to a preconceived artefact of 50% skin friction thus lacking true prediction.

To see how deeply rooted the belief in 50% skin friction is at e g Airbus, take a look at this picture:

(also note that predictive CFD is claimed to require 800.000 years of computing) as presented by Philipp Schlatter KTH:

When will Boeing and Airbus open to DFS and turn into a new era of predictive CFD?

Are Greenhouse Gasses Warming or Cooling?

Infrared atmospheric window to the right of the peak.
The dip under the peak is the effect of CO2 which changes very little upon doubling of CO2.
Compare with Will Happer's diagram in PS below.

This is a comment to recent posts:

The basic dogma of CO2 alarmism is that a bit more CO2 as a "greenhouse gas" in the atmosphere will have such a big warming effect that the Earth will pass a "tipping point" into a "run-away greenhouse effect" into extinction of human civilisation.

Are then the greenhouses gasses in the atmosphere (mainly water vapour and a bit CO2) warming or cooling? The answer is: they are both cooling and warming.

They are cooling in the sense of infrared radiation into outer space as emission from the "top of the  atmosphere" at an effective emission level of 5 km and temperature -18 C, balancing the radiative input from the Sun.

They are warming in the sense of allowing the Earth surface temperature, by a gravito-thermodynamic  effect with lapse rate of 6.5 C/km, to be 33 C warmer than that at the top of the atmosphere. This is then a combined total effect of thermodynamics with gravitation and radiation,
not radiation only.

The "radiation only" effect can be estimated to be 1/3 of the total effect by seeing that the radiative heat exchange between an Earth surface and top of the atmosphere is 2/3 thermodynamics and 1/3 radiation. This gives a greenhouse effect from radiation of about 9 C with thus a surface temperature  of + 6 C, which is the grey body temperature of an Earth without greenhouse gasses.

The greenhouse gasses thus have the double role of radiating into outer space as a cooling effect and together with thermodynamics+gravitation of keeping the Earth surface at a higher temperature than the top of the atmosphere as a warming effect.

To compare with an Earth without greenhouse gasses is not so relevant, but to compare with the warming effect of a fully opaque is enlightening. Recalling that the step from the the semi-opaque atmosphere of the Earth to a fully opaque atmosphere corresponds to completely closing the "infrared atmospheric window" from 5/6 shut, leaving a total effect of 9/5 thus less than 2 C. Since doubling of CO2 is far from closing the window, its effect as climate sensitivity can not be bigger than 1 C, probably much smaller.

PS Below is Will Happer's picture (from COP25 Madrid There is No Climate Emergence) of the spectrum of outgoing infrared radiation showing that doubling of CO2 as represented by the difference between the black and red curves visible only in the CO2 ditch below the peak, is at most 3 W/m2, which gives less than 1 C warming by Stephan-Boltzmann.