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.

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.   

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.   

Banach and DFS and Clay Navier-Stokes Problem

This is an exercise in preparation for participation in a film about the Polish mathematician Stefan Banach who advanced functional analysis as mathematics describing relations between functions or analogies between analogies. My punch line is that the finite element method, as the subject of my work, is (nothing but) computational functional analysis following the spirit of Banach.

The crown of my work, together with Johan Hoffman and Johan Jansson, is Direct Finite Element Simulation DFS as solution of the Navier-Stokes-Euler equations without turbulence model or complicated wall model from a principle of best possible solution, in a situation where there is no exact solution. DFS brings revolutionary new capacity to Computational Fluid Dynamics CFD, which we (as a show case) claim resolves the Clay Navier-Stokes Problem by computation.

Functional analysis was formed by the mathematician Hilbert at the switch to modernity around 1900, with contributions from the Swedish mathematician Fredholm, and was further developed by Banach starting in 1920.  A prime objective was to justify mathematical models in the form of partial differential equations of solid and fluid mechanics and electromagnetics formulated during the 19th century by Laplace, Fourier, Navier, Stokes and Maxwell, by answering basic questions concerning existence and uniqueness of solutions, as well a construction of solutions by computation.

The basic element of functional analysis is a collection of functions named Hilbert space or Banach space equipped with a structure or geometry generalising that of ordinary three dimensional space. The solution of a given partial differential equation is then an element of a suitably chosen Hilbert or Banach space in basic cases determined by a principle of energy minimisation. The differential equation, which is impossible to solve directly by symbolic computation with pen and paper,  is thus reformulated into a minimisation problem over a function space, which allows construction of solutions as a limits of functions with decreasing energy computed according to the Banach Contraction Mapping Theorem.

Starting in the 1950s this form of computational functional analysis has been developed under the name of the finite element method into a universal method for computing solutions of the differential equations of science and engineering bringing revolutionary new capacities.  This success story was darkened only by Navier-Stokes-Euler equations of fluid mechanics, which were believed to demand computational power beyond anything which could be envisioned, the reason being the phenomena of turbulence and thin boundary layers involving small scales too costly to resolve computationally, the impossibilities presented in NASA CFD Vision 2030.

We show that with DFS the NASA CFD Vision 2030 is realised already today. By computational functional analysis in the spirit of Banach.

DFS and functional analysis gives a new perspective on differential equations representing ideal physics, however with uncomputable or non-existing exact solutions as in the case of Navier-Stokes-Euler,  and reformulations in terms of functional analysis with computable approximate solutions representing real physics.