Boeing Halts Production of 737 Max vs DFS

Boeing will halt production of troubled 737 Max airplane. It’s unclear how long the suspension will last.

Compare with CFD State-of-the-Art/NASA 2030 Vision vs DFS recalling that Boeing and its competitors are very conservative companies and penetration of CFD is gradual.

Yes, Boeing indeed took a very conservative approach when launching the new 737 Max by equipping a design from 1967 by new larger supposedly more fuel efficient engines, which had to mounted farther forward and higher to clear ground. 

The result showed to be an airplane with a tendency to stall in low-speed climb and turn, which must have come as a surprise, because the standard software for CFD Computational Fluid Dynamics used by Boeing does not have the capability to predict the complex flow dynamics of stall. 

But the design was kept by Boeing, in line with its conservative company strategy, and to fix the instability the MCAS software was installed with the objective to automatically pitch the nose down on input from an angle of attack sensor. But the system malfunctioned with catastrophic consequences. 

The idea has then, since the the 737 Max fleet was grounded in March 2019, been to improve the software to make it safe, but reauthorisation by FAA is dragging and may never come. So now the production is halted and may never be resumed. 

DFS Direct Finite Element Simulation from Icarus Digital Math  is new software for CFD with the capability if predicting full flight characteristics of an airplane including stability and tendency to stall, as a realisation already today of NASA CFD Vision 2030.  DFS comes with new mathematical theory explaining for the first time The Secret of Flight.             

DFS as new computation/theory is now being presented to Boeing towards evaluation of the new predictive capabilities of DFS and possible incorporate into the designs process. Big values are at stake.

The catch for Boeing is that if the Max requires stabilising software, then it will be very hard to demonstrate that the software always will operate as intended and thus for FAA to re-authorise. The other possibility is that in fact the software is not needed, but that requires predictive computational simulation capability at Boeing trusted by FAA, since real flight testing of extreme situations is hazardous. In both cases, both Boeing and FAA have a problem, for which the only real solution may well be to put an end to the whole story of 737 Max.

How long time would it take for Boeing to make a whole new design (for the new engine or better) meeting todays expectations, using a tool like DFS and then start production?  Two years?      

Prescription vs Prediction in CFD

Recent posts compare the standard methods of CFD based on turbulence and wall models (RANS, LES and DES as a combination), with DFS Direct Finite Element Simulation without turbulence and wall models as best possible solution of Euler's equations.

DFS has shown to accurately predict complex aerodynamics such as the stall of an airplane by capturing both turbulence and flow separation from first principle physics. DFS thus predicts the full flight characteristics of an airplane with the only input being the shape of the airplane. DFS not only predicts flow separation but also makes it understandable as 3d rotational slip separation with point or line stagnation.

This is a stunning example of the ideal according to Einstein of a mathematical model capable of predicting true physics without input of physical parameters. It is like predicting the circumference of a circle with radius 1 to be $2\pi$, just much more complicated and surprising.

With the standard methods of RANS-LES prediction is replaced by prescription mediated through the turbulence and wall models containing many parameters.  As a result RANS-LES cannot truly predict flow separation since that has to be built into the wall model, by either prescribing the flow to stay attached to a smooth solid wall and separate at a corner, or separate under influence of an "adverse pressure gradient".

The main novelty of DFS is thus the possibility of true prediction, which is not possible with RANS-LES which include prescription.  This connects to Bohr's comment to Einstein's claim that God does not trow dice, in the form:

• Einstein, stop telling God what to do!

RANS-LES tells physics what to do. DFS predicts what physics does.

DFS prediction of stall of a jumbojet with flow separation on top of the
inner part of the wing, in close agreement with observation.

The Difference Between DFS and RANS-LES, DNS and DES

The main methods in CFD Computational Fluid Mechanics are:

• DFS Direct Finite Element Simulation. 
• RANS-LES Reynolds Averaged Navier-Stokes-Large Eddy Simulation.
• DNS Direct Numerical Simulation.

The characteristics are:

• DFS: Best possible solution of Euler's equations with force boundary condition as slip/small friction without turbulence and wall model.
• RANS-LES:  Turbulence model and wall model specifying velocity profile into no-slip on wall.
• DNS: Navier-Stokes equations without turbulence/wall model with no-slip on wall.  

The capabilities/limitations are:

• DFS: Captures high Reynolds number flows (beyond drag crisis around $10^6$) with slip in large generality including separated flow, and through drag crisis with small friction.
• RANS-LES: Large difficulties of turbulence/wall modeling and flow separation. 
• DNS: Restricted to low Reynolds numbers.  

For a review of the state-of-the-art of RANS-LES and DNS (2016), see 

• On the role and challenges of CFD in the aerospace industry 

by P. R. Spalart and V. Venkatakrishnan, Boeing Commercial Airplanes Seattle.

DFS prescribes a force boundary condition on a solid wall as slip/small friction, while RANS-LES and DNS both prescribe velocities to be zero on wall as no-slip.

A force boundary condition is a so called natural or weak boundary condition, which mathematically can be imposed in variational form and as such represents a physical boundary condition, which can be controlled as slip/small friction.

On the other hand, a no-slip boundary condition on velocity is mathematically referred to as an unnatural or strong boundary condition, which is unphysical in the sense of being possible to impose in reality, only by paper and pen in a mathematical model or computer code.  

DFS captures flow separation by using a force boundary condition allowing the simulation to "follow the physics".

RANS-LES does not capture flow separation by artificially prescribing the velocity close to the wall which does not "follow the physics".

DNS for high Reynolds number flow requires computational power estimated to be reached only in 2080, as predicted by Spalart in 2000 and repeated in the above review. 

In short, DFS is the only CFD method which today can deliver simulations of high Reynolds number capturing the essential aspects of turbulence and flow separation. Compare with  Spalart's bleak perspective for RANS-LES and DNS:

• Our expectations for a breakthrough in turbulence, whether within traditional modelling or LES, are low and as a result off-design flow physics including separation will continue to pose a substantial challenge, as will laminar-turbulent transition.

As a key example, DFS allows accurate simulation/prediction of the full flight of an airplane including flow separation as stall, and thereby reveals The Secret of Flight, for the first time in the history of science, from first principle physics without turbulence and wall modeling.

RANS-LES handles separation by ad hoc prescription of velocities close to the wall, and not in true computation simulation. But ad hoc prescription is not prediction.

The difference between unnatural unphysical paper and pen velocity boundary condition comes to expression in the famous Kutta condition, where the velocity in a (potential) flow computation is artificially ad hoc prescribed to be zero (stagnation) at a sharp trailing edge of an airfoil. The separation is thus prescribed to take place at the trailing edge, which corresponds to artificially introducing a massive force to this effect, for which however the physics is lacking. The fake explanation is that the singularity of a sharp trailing edge "prevents" the flow from earlier separation with loss if lift.  With the Kutta trick lift is generated, but the physics is missing.

To come to grips with the unphysical flow separation in RANS-LES by ad hoc prescription of velocities close to the wall, remedies such as Detached Eddy Simulation DES have be tried but again relying on velocity prescription without true predictive capability.

A solid wall can force the normal fluid velocity to vanish as a non-penetration condition (ultimately realised by a force), but the tangential velocity cannot be prescribed e g as a no-slip condition; only tangential forces can be prescribed, such as zero skin friction or slip.

The change from early separation on the crest of the flow around a sphere with no-slip for Reynolds numbers below the drag crisis with massive wake, to later 3d rotational slip separation for Reynolds numbers through and beyond the drag crisis with smaller wake diameter and corresponding drastic drop of drag, can be followed in these pictures:

      

Models of Flow Separation

The holy grail of CFD as computational fluid mechanics is:

• Turbulence modeling.
• Flow separation.

DFS as Direct Finite Element Simulation offers answers to these problems:

• Turbulence captured as best possible computational solution to the Euler equations.
• Flow separation described as 3d rotational or parallel slip separation.  

We here give the elements of separation in bluff body flow as illustrated in the pictures above of an airplane landing gear (with further details in New Theory of Flight and Secret of Flight):

1. 3d rotational slip with point stagnation (back and side of wheels).
2. 3d parallel slip with 2d line stagnation (top of wheel support). 

We start from the following basic observations:

• Separation in 2d potential flow can only take place a stagnation with zero flow velocity.
• Accelerating flow is stable in velocity and unstable in vorticity.
• Decelerating flow is unstable in velocity and stable in vorticity.
• Rotational flow is neutrally stable. 

We consider 2d potential flow in a $(x_1,x_2,x_3)$ coordinate system around a long cylinder with axis in the $x_3$-direction and flow in the $x_1$-direction, in the back modeled by the velocity    

• $u(x)=(x_1,-x_2,0)$ in the half-plane $\{x_1>0\}$                                              (1)

We observe the critical element of separation away from the plane $\{x_1=0\}$ representing the back surface of the body, through the positive velocity $u_1=x_1$, which is balanced to maintain incompressibility by the opposing flow $u_2=-x_2$, with 2d stagnation with $x_1=x_2=0$ along the $x_3$-axis. We recall that opposing flow is unstable in 3d and thus $u_2=-x_2$ generates rotational flow from a perturbation oscillating in the $x_3$-direction:   

• $u(x)=(0,x_3,-x_2)$ in the half-plane $\{x_1>0\}$

as counter-rotating tubes of stream-wise vorticity in the $x_1$-direction attaching to the plane $\{x_1=0\}$. This leads to a combined quasi-stable separation pattern of the form  

• $u(x)=(2\epsilon x_1,x_3-\epsilon x_2,-x_2-\epsilon x_3)$ in the half-plane $\{x_1>0\}$ (2)

with some $\epsilon \gt 0$, which is characterised as rotational flow with 3d point stagnation as seen in the oil film visualisation above, in the rotational flow in a bath-tub drain and in the rotational rising (separating) flow of a tornado. Instability of potential flow with separation from 2d line stagnation is thus turned in 3d quasi-stable rotational separation from 3d point stagnation. The flow accelerating in the $x_1$-direction is stable in velocity, but unstable in stream-wise vorticity which intensifies the swirling motion into turbulence (vortex stretching).

The oil film picture also shows parallel separation from lines of converging flow lines with transversal stagnation superimposed on a main flow, which we model by the velocity   

• $u(x)=(1,x_2,-x_3)$ in the half-plane $\{x_2>0\}$,                                (3)

with flow separating from the surface $\{x_2>0\}$ with velocity $u_2=x_2$, balanced by the opposing flow $u_3=-x_3$. In this case the instability of opposing flow potentially generating vorticity in the $x_2$-direction, is "swept" away by the main flow $u_1=1$.  

The vortical flow in (3) with the $x_2=0$-plane as the upper surface of a wing can be seen to be generated by vortex stretching in accelerating flow on the leading edge. In this case the stabilisation from the main flow may be insufficient, which may lead to 3d rotational slip separation into the half space $\{x_2>0\}$ and then connects to stall. This phenomenon is also seen on the inner side of the wheels above.

We can thus summarise quasi-stable patterns of flow separation with slip as: 

• 3d rotational with point stagnation modeled by (2). (back of wheel)
• Parallel with 2d line stagnation modeled by(3). (top of wheel support)
• Parallel 3d rotational modeled by (3) + properly modified form of (2). (inner side of wheel)  

In the standard boundary layer theory with no-slip by Prandtl, flow separation is connected to stagnation from adverse pressure gradient. With this theory flow separation has remained a mystery. As a consequence CFD with no-slip following Prandtl does not truely capture flow separation.

In short: DFS offers a resolution to the two main open problems of CFD: turbulence and flow separation. DFS also opens to theoretical understanding for the first time of the complex phenomenon of partly turbulent bluff body flow, which is captured in the following mantra:

• bluff body flow = potential flow modified by 3d rotational or parallel slip separation. 

We understand that flow separation in potential flow is unstable, while flow attachment is more stable because the opposing flow is not present. This is what makes bluff body flow largely stay potential until separation, as seen on the outside of the wheels. We see that flow separation is a large scale phenomenon and that turbulence arises in the vortical swirling flow after separation.   

By-Pass from Laminar No-Slip Boundary Layer to Slip without Layer

Artificial vibrating ribbon in flat plate experiments with objective to generate Tollmien-Schlichting waves.

When theory does not fit experiment, one approach is to change the experiment. This is an established technique in fluid mechanics since the discovery of d'Alembert's paradox in 1755 separating from start fluid mechanics into theory explaining what cannot be observed in reality, and real observation which cannot be explained theoretically.

There are thus basic experiments in fluid mechanics which are manipulated in the form of artificial forcing containing:

1. Artificial generation of Tollmien-Schlichting waves by a heavily vibrating ribbon in experiments on transition from laminar to turbulent flow in a shear layer. 
2. Artificial tripping of the flow over a wing by a fixed rib or wire to generate a turbulent boundary layer with substantial skin friction to fit Prandtl's boundary layer theory.

Computational Turbulent Incompressible Flow presents a different non-artificial real scenario for transition to turbulence in a shear later such as a laminar boundray layer. The scenario is that

weak streamwise vorticity always present from small perturbations, acting over long time by non-modal linear growth restructures the flow in a laminar shear layer into high and low speed streamwise streaks (increasing transversal velocity gradients) which when big enough triggers transition to turbulence. This effect is damped in streamwise accelerating flow, but not so in constant or decelerating flow. 

The result is that a laminar shear layer over a flat plate (without acceleration) turns turbulent if the Reynolds number is big enough and the plate long enough. 

On the other hand, in the accelerating flow on the upper part of the rounded leading edge of a wing,
the transition does not take place. Instead the laminar no-slip boundary layer present at the stagnation on the leading edge stays laminar (as well as on the lower pressure side of the wing) and if the Reynold's number is big enough effectively acts and can be modeled as a slip boundary condition without boundary layer.  

The change from laminar no-slip boundary layer to effectively slip without boundary layer, thus without transition to a turbulent boundary layer, can be connected to a Reynolds number of size     

$10^6$ with thus a laminar boundary layer of thickness 0.001 with free stream velocity and size normalized to 1. 

Slip would then result when the thickness of the boundary layer is about 0.1% of the gross dimension. For a wing with chord 1 m this would be 1 mm. 

We thus add theoretical evidence that the slip condition used in DFS as well as the New Theory of Flight has a sound rationale. 

In particular DFS shows that total drag is more than 90% form/pressure drag and skin friction drag less than 10%, while standard theory and computation says that skin friction dominates form/pressure drag.  

Connecting to 2. above, the direct passage from laminar no-slip boundary to slip without boundary layer, thus in real cases "bypasses" the generation of a turbulent boundary from artificial forcing. 

Likewise, without the artificial vibrating rib transition to turbulence is not by Tollmien-Schlichting waves, but instead through the scenario presented after 2. 

In short, reality does not do what standard theory says reality should do. Reality "bypasses" standard theory, but standard theory is nevertheless claimed to be correct because it fits experiments with artificial forcing! This is state of the art. Something to think about.

How Big is Skin Friction?

Tripping along leading edge of wing creating thick turbulent boundary layer causing drag. 

The drag of a body moving through air (airplane) or water (ship) consists of

• form/pressure drag + skin friction drag. 

It is generally believed from experiments dragging a plate through water, that for an airplane and ship skin friction may be 50-70% of total drag. Experiments are performed with (i) untripped/free transition and (ii) tripped/forced transition creating a turbulent boundary layer, with (ii) showing a bit bigger drag than (i).

Tripping us done e g by mounting a rib along the upper part of the leading edge of a wing. The effect of creating a thick turbulent boundary layer is illustrated in the above image.

Computations with DFS Direct Finite Element Simulation with zero skin friction (slip boundary condition on wall) shows drag in close accordance with drag experiments with free transition.

The DFS results thus show total drag as pure form/pressure drag with zero skin friction, in accordance with free transition experiments. This gives evidence that drag with free transition has very little contribution from skin friction, and further that the measured (small) difference between tripped and untripped drag can be used to assess the skin friction, which is forced by tripping and is thus absent without tripping.

Now, a real airplane is not equipped with tripping devices on wings or fuselage since that would increase drag for no use, and DFS with slip shows close correspondence to experiments with free transition.

Altogether, there is strong evidence that skin friction drag for an airplane or ship is an order of magnitude smaller than that commonly used based on experiments from tripping. The results indicate that what is believed to be a thick turbulent boundary layer forced by tripping with substantial skin friction, in fact is absent i reality without tripping and thus that the interaction between fluid and solid acts as slip/small friction (without boundary layer to resolve computationally).

Obviously, if skin friction in reality is less than 10% of total drag, instead of an unreal tripped imagination of 50-70%, the design of airplane or ship will work from different premises.

DFS with slip makes CFD computable, whereas std CFD with no-slip tripped boundary layers is uncomputable.

Why is then tripping used in experiments if in reality not? This is to make experiments fit with the boundary layer theory of Prandtl as the Father of Modern Fluid Mechanics tracing drag to the presence of a thick turbulent boundary layer. But to fit unreal experiments to theory is opposite to the idea of real science to fit theory to real experiments.

Drag coefficients for NACA0012 by Ladson with free and tripped transition. Note the small dependence on Reynolds number for free transition and that difference between tripped and untripped drag is about 0.001 as about 10% of total tripped drag as an estimation of skin friction drag.

      

The Mystery of Skin Friction from Tripping Resolved

This is a continuation of the previous post on DFS as the first predictive CFD methodology based on first principle physics without need of turbulence or wall models. In particular, DFS uses a slip boundary condition on a solid wall as expressing physics of the observed very small skin friction of a slightly viscous fluid.

DFS is a new approach to CFD which for over a century has been dominated by a dictate by Prandtl as the Father of Modern Fluid Dynamics, that thin boundary layers will have to be computationally resolved, which however is projected to be possible only in 2080. 

DFS shows that a slip boundary condition circumvents the Prandtl dictate and makes CFD computable already today meeting in particular the NASA 2030 vision.

The total drag of a body has contribution from (i) form or pressure drag and (ii) skin friction drag.

It is commonly believed that skin friction drag can be 50% of total drag. This is based on flat plate experiments where the force from the fluid over a flat surface is measured to a give a skin friction coefficient. Typically the flow is tripped by a flow transversal device like a rib with the objective to create a turbulent boundary layer. Experiments show that the skin friction with tripping is bigger than without tripping, in which case the boundary layer is less turbulent than with tripping.

To estimate the skin friction of a bluff body like an airplane or wing the tripped flat plate skin friction cofficient (multiplying the area of the body) is used although the flow around the bluff body is not tripped. This may give a skin friction up to 50% of total drag for a slender body, but there is a caveat: The skin friction coefficient is the result of tripping, while the bluff body flow has no tripping. If the un-tripped skin friction coefficient was used a much smaller skin friction for the body would result.

There is thus a lack of logic in conventional CFD: The skin friction coefficient is determined with tripping, while real flow is without tripping. The result is large skin friction drag, up to 50% of total drag.

In DFS with slip, skin friction drag is zero, yet DFS gives correct total drag for an airplane and wing without tripping.

The conclusion is that conventional CFD attributes too much to skin friction by using a skin friction coefficient determined from tripped flat plate experiments, which comes out to be too large when applied to a non-tripped real case.

DFS with slip thus resolves a basic open problem of fluid mechanics. DFS makes CFD computable.

A slip boundary conditions models physics, while the conventional no-slip condition does not.

More precisely, the boundary layer of a real smooth body is neither fully turbulent (too much drag), nor fully laminar (no-slip condition), but instead acts with slip as if non-existent. This is major news.
        

How to Make CFD Truely Predictive: DFS

The global market for CFD Computational Fluid Dynamics software reaches soon $2B per year with Ansys dominating, but still struggles with basic difficulties of computational simulation including turbulence and flow separation from solid walls, despite major efforts over many years.

The effect is that CFD is not predictive, which means that design still needs time consuming and expensive experimental testing in wind tunnels or ship tanks. At best CFD can be used to support already known facts from experiment or accumulated experience, by suitable fitting of parameters in turbulence and wall models.

DFS Direct Finite Element Simulation represents a breakthrough meeting the NASA 2030 Vision by offering for the first time predictive computational simulation of wall bounded turbulent fluid flow. DFS is predictive because it is based on first principle physics without use of turbulence or wall modeling.

The first principle physics of DFS consists of best possible computational solution of equations expressing incompressibility and Newton's 2nd law combined with a slip boundary condition at solid walls reflecting the observed very small skin friction for Reynolds numbers larger than $10^6$of relevance for airplanes, ships and cars.

In particular DFS has been shown to correctly capture the physics of flow separation as 3d rotational slip separation with point stagnation, and more generally bluff body flow as potential flow modified by 3d rotational slip separation. DFS gets around the obstacle of computational resolution of thin boundary layers, which has so long prevented predictive CFD simulation.

In short, DFS is the first truely predictive CFD code.

DFS is presented to the market by Icarus Digital Math in basic open source form with add-ons for different complex applications including F1 racing and flight simulation.

Ludwig Prandtl was given the role of Father of Modern Fluid Mechanics because he presented a resolution in 1904 of d'Alembert’s paradox formulated in 1755 and so gave new promise to a fluid mechanics haunted by a fundamental contradiction for 150 years. But Prandtl’s resolution came with the severe side effect of making predictive CFD impossible by asking for computational resolution of thin boundary layers.

DFS frees CFD for the first time from the spell of Prandtl.

Understanding that Prandt’s resolution was physically incorrect and giving a different physically correct resolution, represented key first steps towards the predictive CFD now being realised in fully developed form as DFS with key scientific references:

• Resolution of d'Alembert's Paradox
• Computational Turbulent Incompressible Flow
• New Theory of Flight
• 3rd High Lift Workshop

DFS as New Design Tool: As an example from the 3rd High Lift Workshop, standard CFD computes the drag of an airplane as 50% form and 50% skin friction drag with the total drag matching experiments, while DFS with zero skin friction computes correct drag then as 100% form. This means that standard codes miss form drag by 50% by missing physically correct flow separation,  which is captured by DFS from first principle physics! In other words, standard codes appear to give a completely wrong picture of the contribution to total drag from form and skin friction, thus misleading design. The fact that standard CFD despite missing form drag with 50% gets total drag right, indicates that standard CFD is fitted to observation and thus does not deliver true prediction.

DFS reveals New Theory of Flight: DFS comes with mathematical theory offering a true explanation of the miracle of flight for the first time.   

The Future in CFD is Already Here

The following two documents describe the vision of state-of-the-art of Computational Fluid Dynamics (CFD):

• CFD Vision Study: A Path to Revolutionary Computational Aerosciences
• Future Directions in Computational Fluid Dynamics 

The breakthrough of the G2/Unicorn/FEniCS fluid solver at the 3rd AIAA  HighLift Prediction Workshop (HighLiftPW-3) as documented here (and commented on at The Secret of Flight), shows that the future is already here! 

More specifically G2/Unicorn/FEniCS today delivers what is described as Grand Challenge Problem 1: 

• LES of a powered aircraft configuration across the full flight envelope. 
• This case focuses on the ability of CFD to simulate the flow about a complete aircraft geometry at the critical corners of the flight envelope including low-speed approach and takeoff conditions, transonic buffet, and possibly undergoing dynamic maneuvers, where aerodynamic performance is highly dependent on the prediction of turbulent flow phenomena such as smooth body separation and shock-boundary layer interaction.

This is made possible by a combination of the following elements:

1. Euler/Navier-Stokes as parameter free model for high Reynolds number flow.
2. Slip boundary condition as basic simple parameter-free wall model.
3. Time resolved computational solution by G2 as residual-stabilised finite element solver.
4. Automatic turbulence model from computational residual stabilisation.
5. Automatic residual-based tetrahedral mesh adaptivity.
6. Automatic duality-based output error control.

LES by G2/Unicorn/FEniCS of a powered aircraft configuration across the full flight envelope. 

Talk at NSCM26: Bluff Body Drag

The slides to my talk at NSCM26 at Simula Oslo Oct 23 - 25 to the honor of my friends and collegues Juhani Pitkäranta (65) and Rolf Stenberg (60), is now available as

• Direct Fem-Simulation of Bluff Body Flow.

Abstract Nordic Seminar in Computational Mechanics Oslo Oct 23-25: Breaking the Spell of Prandtl

                                                          Movie is here

The abstract for my upcoming talk at NCSM26 in Oslo Oct 23-25 is now available as

• Direct Fem-Simulation of Turbulent Bluff Body Flow. 

The talk presents in particular the first ab initio direct computational simulation of the flow of air around an airplane at large angle of attack and low velocity at landing, in close accordance to measured pressure distributions. This represents a major breakthrough in Computational Fluid Dynamics blocked for hundred years by a spell of Prandtl requiring impossible computational resolution of thin boundary layers demanding trillions of mesh points. 

We use a slip boundary condition modeling the small skin friction of slightly viscous air flow which does not generate and thus does not require resolution of boundary layers. We obtain pressure distributions using three millions of mesh points matching observations, and conclude that boundary layers have little impact (for slightly viscous flow) and thus do not have to be resolved. The spell of Prandtl is thereby broken.

The Mathematical Secret of Flight 4

The secret of flight is hidden in the above picture showing the flow separation at the trailing edge of a wing, as explained in detail in the article The Mathematical Secret of Flight and the upcoming book The Secret of Flight.

Mathematical analysis shows that the swirling flow separation shown in the picture results

from an instability of opposing flows meeting behind the trailing edge, and the swirling motion

allows the flow to separate with little retardation requiring high pressure. Instead low pressure

develops inside the swirling flow which does not like high pressure destroy the high lift/suction established on the crest of the wing, while causing only small drag because of the small diameter of the trailing edge.

This is the miracle of flight revealed by mathematical analysis, whether you like it or not in the words of Richard Feynman.

The swirling motion is similar to that used by noble men when backwards leaving the king after an audience; an elegant form of separation without high pressure destruction of what was gained during the meeting.

I have asked Antony Jameson about a comment but not received any response so far.

Compare with 50 ways to leave your lover: The most elegant and thus best way is with a swirling motion avoiding build up of high pressure.

The Mathematical Secret of Flight 3

After my talk The Mathematical Secret of Flight at Svenska Mekanikdagar 2011 an remark was made by Laszlo Fuchs recalling early attempts to compute the lift and drag of a wing by solving the Euler equations by Antony Jamseon (left) and Art Rizzi at KTH in the 1980s.

In essence what Jameson and Rizzi did was to solve the Navier-Stokes equations with a slip boundary condition at very high Reynolds number, essentially the same as we are doing, and observing the appearance of unsteady fluctuating solutions.

However, these solutions were regarded with suspicion by the fluid dynamics community as some kind of ghost solutions with unclear physical significance. In particular the slip boundary condition without any boundary layer was against the dictate of Prandtl that no-slip is the only physically correct boundary condition and that boundary layers have to be resolved because the truth is to be found there.

The result was that the (incompressible) Euler solvers of Jameson and Rizzi became marginalized, just as Birkhoff had been in the 1950s when asking if there were any steady solutions at all.

What we have shown is that Birkhoff, Jameson and Rizzi were on the right track and that the effective suppression of their ideas has delayed the advancement of computational fluid mechanics by several decades. The solutions computed by Jameson and Rizzi were not ghost solutions but true turbulent solutions carrying important information of real physics.

The suppression of correct science is often more harmful than the promotion of wrong science.

Mathematical Secret of Flight 2

Computed turbulent flow velocity around a NACA0012 wing at 15 degrees angle of attack in beginning stall with separation on top of the wing.

An updated version of my talk on June 15 at Svenska Mekanikdagar 2011, is now available for preview as

• The Mathematical Secret of Flight

describing joint work with Johan Hoffman and Johan Jansson.

This work shows that computation of mean values such as lift and drag of an airplane, car or boat can be accurately computed without resolving thin boundary layers. Based on the computations a new mathematical theory for flight is presented which is fundamentally different from that by Kutta-Zhukovsky-Prandtl filling text books. The new flight theory was first published in Normat 57:4 (2009) as The Mathematical Secret of Flight.

The spell of Prandtl as the father of modern fluid mechanics of attributing both lift and drag as effects of thin boundary layers requiring unreachable quadrillions of mesh points for computational resolution, can thus be broken. This opens a wealth of applications of computational fluid dynamics suddenly reachable with millions of mesh points.

The new theory of flight is presented in Mathematical Simulation Technology, a book which was officially banned by KTH in November 2010, as described in posts on KTH-gate. The reader can act as referee and decide if the ban is motivated from a scientific point of view. The last time a math book was banned was in 1632.

Mathematical Secret of Flight 1

Computed Lift and Drag of a 3d NACA0012 wing for different angles of attack by Unicorn (blue) compared with different experiments.

My talk on June 15 at Svenska Mekanikdagar 2011, is now available for preview as

• The Mathematical Secret of Flight

describing joint work with Johan Hoffman and Johan Jansson.

Based on accurate solution of the incompressible Navier-Stokes equations we identify the true mechanism for the generation of large lift L at small drag D of a wing with lift to drag quotient L/D of size 10 - 50, which is not described in the literature.

We combine the Navier-Stokes equations with a slip boundary condition on the wing motivated by the experimental fact that the skin friction is small for a slightly viscous fluid such as air or water, and we exhibit the role the slip condition in two crucial aspects:

• prevention of separation at the crest of the wing generating large lift
• 3d slip-separation at the trailing edge not destroying large lift and causing small drag.

Text books claim following Prandtl, named the father of modern fluid mechanics, that both lift and drag result from a boundary layer arising from a no-slip condition.

We obtain lift and drag in full accordance with experiments by solving the Navier-Stokes equations with a slip condition, which does not generate any boundary layer, and we thus present strong evidence that lift and drag do not originate from any boundary layer.

In short, we show that solutions to the Navier-Stokes equations with slip are computable and

correctly capture the physics of (subsonic) flight. See also

• The Mathematical Secret of Flight (Normat 57, 2009)
• Why It Is Possible to Fly (Google Knol) .

Compare with state-of-the-art as expressed by Moin and Kim in Tackling Turbulence with Supercomputers:

• To solve the Navier-Stokes equations for, say, the flow over an airplane requires a finely spaced computational grid to resolve the smallest eddies.
• Consider a transport airplane with a 50-meter-long fuselage and wings with a chord length (the distance from the leading to the trailing edge) of about five meters. If the craft is cruising at 250 meters per second at an altitude of 10,000 meters, about 10 quadrillion (10^16) grid points are required to simulate the turbulence near the surface with reasonable detail.

Kim and Moin express the necessity dictated by Prandtl to resolve thin boundary layers to correctly compute lift and drag of a wing or an entire airplane, which would require 50 years of Moore's law to increase the computing power with a factor 10^10 to reach the dictated 10^16 points.

We show that this is possible already today using 10^6 points by using slip without boundary layers to resolve.

Climate and Turbulence Modeling

Global climate models are based on turbulence models, since the slightly viscous flow of air in the atmosphere and water in the oceans is turbulent. Turbulence modeling, in the form of  analytical mathematical models, is a main unsolved problem of fluid mechanics.

In our book Computational Turbulent Incompressible Flow, with prel. version for download, Johan Hoffman and I present a new approach to turbulence modeling based on ab initio numerical computation with turbulence automatically modeled by the stabilization of the numerics, thus without any explicit analytical turbulence model.  

We show that mean-value quantities of turbulent flow such as drag and mean temperature can be accurately computed without analytical turbulence model, thus circumventing the main unsolved problem of fluid mechanics. We plan to test this approach on climate modeling with hopefully a connection between turbulence and the main unsolved problem of climate modeling: cloud formation. 

We will report as soon as we have something to report on...hopefully before the Copenhagen meeting in December...since the outcome of this meeting critically depends on computational modeling of turbulence and cloud formation...and dark clouds over the meeting are already forming... 

Coin Tossing: Cold or Warm?

Newscientist reports:

• Forecasts of climate change are about to go seriously out of kilter. One of the world's top climate modellers said Thursday we could be about to enter "one or even two decades during which temperatures cool."People will say this is global warming disappearing," he told more than 1500 of the world's top climate scientists gathering in Geneva at the UN's World Climate Conference.
• "I am not one of the sceptics," insisted Mojib Latif of the Leibniz Institute of Marine Sciences at Kiel University, Germany. "However, we have to ask the nasty questions ourselves or other people will do it.
• "Few climate scientists go as far as Latif, an author for the Intergovernmental Panel on Climate Change. But more and more agree that the short-term prognosis for climate change is much less certain than once thought.
•  "In many ways we know more about what will happen in the 2050s than next year," said Vicky Pope from the UK Met Office.

The message is that global climate models cannot predict year or decade meanvalues, but can predict centennial meanvalues. How can this be? What is the mathematics behind such a belief? 

The first idea that come to mind is the law of large numbers of statistics offering prediction of the meanvalue 0.5 of many cointosses between 0 and 1, but no prediction of the meanvalue of a few tosses. But is climate modeling the same as coin tossing between cold and warm? 

Newscientist concludes:

• The world may badly want reliable forecasts of future climate. But such predictions are proving as elusive as the perfect weather forecast.
  

The future of mankind thus seems to lie in the hands of mathematicians running the climate models...but coin tossing statistics does not seem to be enough...what can be done or said? 

Well, let us recall that the 0.5 probability of heads in coin tossing is computed mathematically using the fact that a rotating coin has head up half of the time, that is using a short-time-accurate mathematical model, see the discussion in Chapter 13 Turbulence and Chaos in Computational Turbulent Incompressible Flow. Without a short-time-accurate model, nothing can be be predicted about long-time...Compare with the UK Met Office assurement:

• There have been major advances in the development and use of models over the last 20 years and the current models give us a reliable guide to the direction of future climate change.
• Computer models cannot predict the future exactly...
• Current models enable us to attribute the causes of past climate change, and predict the main features of the future climate, with a high degree of confidence.

What are "the advances in the development and use of models"? What is meant by "direction of future climate change"? Colder or warmer? Does "direction" indicate that the size of the change cannot be predicted? What is the meaning of "computer models cannot predict the future exactly"? That computer models can predict the future almost exactly? Who is the inventor of this form of newspeak? Note the clever construction of the following key statement by Met Office:

• As well as producing CO2, burning fossil fuels also produces small particles called aerosols which cool the climate by reflecting sunlight back into space. These have increased steadily in concentration over the 20th century, which has probably offset some of the warming we have seen.
  

Note the clever use of "probably" and "some of the warming". Very clever doublespeak: Clearly suggesting something, without saying anything! This is not the language of science. 

Can really these semantic tricks help save the World?

The Threat of Computed Climate

Suppose a team of mathematicians today announces that the Earth will be hit by a large meteor in 5 years and 20 days, according to their computations. How should we react?Ask about the reliability of their computations, I guess. If we then get the answer that the computations are so complicated that there is no way we can check that they are correct. How are we then to react? Should be panick and launch a massive exodus to Mars? Well, this is all fiction and has not yet happened, so we don't have to worry.

Suppose a couple of teams of mathematicians today are telling us that according to their computations, in a few decades the sea level will rise by 5 meters and 20 centimeters and billions of people will have to move to avoid getting drowned, in a great exodus. Again we ask about how the computations are performed and what their reliability may be? But again we get the answer that the computations are so complicated that nobody outside the teams of mathematicians involved, can understand them. How are we then to react?

And this is not fiction: Global warming from burning of fossil fuels is today predicted by a couple of computer codes simulating the global climate. One of these codes is the National Center for Atmospheric Research Community Atmosphere Model CAM 3.0 also referred to as CCSM  presented as follows:

• CCSM belongs to an elite category of computer-based simulation models known as general-circulation models. Such models use mathematical formulas to recreate the chemical and physical processes that drive Earth's climate. What emerges from trillions of computer calculations is a picture of the world's climate in all its complexity. 

The future of mankind seems to be governed by this code. What is then the reliability/accuracy of CCSM? Is there anyway we can tell or is CCSM so complex that its beyond scrutiny by people outside the coding team?

Well, the first thing is to take a look at the documentation to learn that the numerics involves 

the following fixes of an otherwise unstable or overly stable useless code: 

• time filter
  
• horisontal diffusion correction
• initial divergence damping
• mass fixers
• energy fixer
• monotonicity fixer.

Further, the codes uses mathematical formulas to model the following physics:

• deep convection
• moist convection
• precipitation
• cloud fraction
• short wave radiation
• long wave radiation
• surface exchange
• vertical diffusion
• boundary layers
• sulfur chemistry
• prognostic of greenhouse gases

plus a number of other fixes and models. The most advanced version CCSM3 uses a horisontal computational grid of mesh size of about 150 km x 150 km and about 30 vertical levels, which must be regarded as a coarse grid requiring a lot of difficult mathematical modeling. The numerics seems to be a concoction of rather old-fashioned techniques.

Looking at this information, as an expert of computational fluid dynamics, my scientific self protests. It does not seem right to base far-reaching limitations on energy consumption on such a code. Or is the accuracy of the code irrelevant, as long as it serves to limit the use of energy?

What if a code tomorrow shows that CO2 can increase without negative climate effects? What if this code is wrong, but we don't understand that?

Of course we should be careful how we use resources on our only Earth, but believing that we will be hit by a meteor in 5 years and 20 days may not be the best basis for rational decisions, if the meteor is only fiction predicted by an incorrect computer simulation.

The numerics could be upgraded using modern adaptive finite element methods with flexible meshing and a posteriori error control. This may not do wonders, but the effort can not be spared...For the first time in history the future of mankind is in the hands of computational mathematicians... 

Sheep Herd Accuracy?

The next UN Climate Conference will take place in Copenhagen in December under Swedish chairmanship of EU. UN Climate chief Yvo de Boer hopes the conference will in particular reach agreements to limit the growth of emissions in developing countries, required to be necessary by predictions of catastrophical global warming.

The most advanced computational methods for long-time predictions of the global climate, the AOGCM coupled Atmospheric Ocean General Circulation Models, use computational grids of size 

• surface: 2.5 x 3.75 degrees, height: 19 levels HadAM3
  
• surface: 1.4 x 1.4 degrees, height: 26-40 levels CCSM3

to simulate turbulent convective radiative reactive phenomena. This means a surface grid of 30 x 30  to 60 x 60 for a quarter hemisphere including complex geometry of land and ocean.

From my experience from computing turbulent flow like the flow around a car or wing, the resolution of HadAM3 is definitely not impressive, and the grid of CCSM3 also seems too coarse. For such grids turbulence modeling is required, which however is an open problem.

What about the accuracy of these models? According to IPCC TAR underlying the Kyoto Protocol

• The model mean exhibits good agreement with observations.
• The individual models often exhibit worse agreement with observations.
  

The logic here is to take the mean of many incorrect models, to get a correct model! As a scientific principle this is wonderful, because you can generate so much knowledge out of ignorance: 

• The mean value of many incorrect values, is the true value!
  

But of course you say: What about the sheep herd effect? Just because all sheep rush in certain direction, does it mean it is the right direction? Not even in politics, and definitely not in science.

Mathematics of Global Warming?

One of the few dissidents in the climate debate is Lars Bern coauthor with Maggie Thauersköld of the Chill-Out - The Truth about the Climate Bubble, which questions the accepted truth that global warming is such a threat to humanity that in particular poor people should be prevented from increasing their energy consumption.  

Decisions aimed at controling global warming are based on predictions obtained by computational solution of basic equations of physics expressing conservation of mass, momentum and energy. Correct decisions require correct predictions and the question of the accuracy and reliability of the predictions must be addressed.

The Intergovernmental Panel of Climate Change IPCC  states in its Fourth Assessment Report (AR4) released in 2007 about Climate Models and their Evaluation:

• One source of confidence in models comes from the fact that model fundamentals are based on established physical laws, such as conservation of mass, energy and momentum,  along with a wealth of observations.  
• A second source of confidence comes from the ability of  models to simulate important aspects of the current climate. 
• A third source of confidence comes from the ability of models to reproduce features of past climates and climate changes. 
• Nevertheless, models still show significant errors. 
• This is partly due to limitations in computing power, but also  results from limitations in scientific understanding or in the  availability of detailed observations of some physical processes.  Significant uncertainties, in particular, are associated with the  representation of clouds, and in the resulting cloud responses  to climate change. Consequently, models continue to display a  substantial range of global temperature change in response to  specified greenhouse gas forcingof model development, they have consistently provided a robust  and unambiguous picture of significant climate warming in response to increasing greenhouse gases.
• In summary, confidence in models comes from their physical basis, and their skill in representing observed climate and past  climate changes. Models have proven to be extremely important  tools for simulating and understanding climate, and there is  considerable confidence that they are able to provide credible quantitative estimates of future climate change, particularly at larger scales. 
• Models continue to have significant limitations,  such as in their representation of clouds, which lead to uncertainties in the magnitude and timing, as well as regional details,  of predicted climate change. 
• What does the accuracy of a climate model's simulation of past or contemporary climate say about the accuracy of its projections of climate change? This question is just beginning to be addressed...
  
• Nevertheless, over several decades  of model development, they have consistently provided a robust  and unambiguous picture of significant climate warming in response to increasing greenhouse gases.
• Over several decades of development, models have consistently provided a robust and unambiguous picture of significant climate warming in response to increasing greenhouse gases.

We read that IPCC states that

• the models still show significant errors
• models continue to have significant limitations,

which expresses that the accuracy of the predictions is very questionable, according to the highest authority of IPCC. Compare with a Global Climate Modeling: Fate of Humanity?

The statement that the confidence comes from the fact that the models have physical basis, shows an alarming innocence as concerns computational mathematical modeling. Physical basis? What else? Religion or parapsychology?

This statement is aimed at impressing the uninformed, but is a triviality to the informed.  Is the message that the poor should feel the pressure of established physical laws and not ask for more?

The statement cited last is the key: models show response to increasing greenhouse gases.

Yes, but this is because they are so designed. Models are not reality. 

It may be that there is antropogenic warming derived from human activities, but statements that this is proved by simulations using Global Climate Models can be questioned.

 

If mathematical predictions are to serve as the basis of decisions influencing the lives of billions of people, the accuracy has to be improved. Suddenly nothing seems more important to humanity than computational mathematics, fluid dynamics and simulation technology ...because this is the only way to get information about the future we know of ...

Compare Climate Debate Daily and Resisting Climate Hysteria, Global Warming - Sensibilities and Science and Global Warming - A Peculiar Issue by Richard Lindzen stating:

• What was done by IPCC, was to take a large number of models that could not reasonably simulate known patterns of natural behavior (such as ENSO, the Pacific Decadal Oscillation, the Atlantic Multidecadal Oscillation), claim that such models nonetheless accurately depicted natural internal climate variability, and use the fact that these models could not replicate the warming episode from the mid seventies through the mid nineties, to argue that forcing was necessary and that the forcing must have been due to man. 
• The argument makes arguments in support of intelligent design sound rigorous by comparison.  It constitutes a rejection of scientific logic, while widely put forward as being ‘demanded’ by science.
  

If we reject scientifc logic, what shall we then rely on?