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.

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.