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.           

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.

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.        

Skin Friction is Small

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

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

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

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

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

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

On Real Simulation

The painter, the painting and the nude physical origin.

                                           
Icarus Digital Math will be launched under the banner:

• automated real simulation. 

Here "real" signifies that the simulation can serve as true prediction of reality and "automated" that the key elements of the simulation

1. mathematical modeling in terms of differential equations
2. discretisation in terms of algebraic equations
3. computational solution of algebraic equations, 

are carried out in the form of computational software, such as FEniCS. 

The term "real simulation" may seem like an oxymoron with an apparent contradiction between reality as ”what is” and simulation as ”what is computed/imagined”. But the relation between reality as god-given physical origin and simulation as man-made mathematical model or picture, is in fact very complex and as such an important aspect of both science and arts. 

The postmodernist philosopher Baudrillard coined the concept of ”hyperreality” with the image filling the empty place of a non-existing origin, as true reality with Disneyland as example. 

The computer scientist Dijkstra similarly elevated the model as the "abstract machine" more true than the "physical machine": 

• Originally I viewed it as the function of the abstract machine to provide a truthful picture of the physical reality. Later, however, I learned to consider the abstract machine as the true one, because that is the only one we can think ; it is the physical machine's purpose to supply a working model, a (hopefully) sufficiently accurate physical simulation of the true, abstract machine.

It also comes to full expression in the Copenhagen Interpretation of Quantum Mechanics with the observer intimately coupled with the observed reality, in particular deciding the fate of Schrödinger's cat in the process of observation. 

In arts, the relation between the painter, the picture and the model is also filled with deep meaning, connecting to Plato's allegory of the cave with shadows forming the reality. 

Boeing Flight Simulator?

The Guardian reports:

• Boeing now recommends 737 Max flight simulator training for pilots.
• Decision is a reversal of company’s long-held position that computer-based training alone was adequate.

In order for a flight simulator to correctly prepare pilots to handle real flight, the simulator must represent reality, which requires CFD (computational fluid dynamics) software capable of predicting the real reaction of the airplane upon control input from the pilot, including the critical phenomenon of stall which must be avoided. Standard CFD does not have this capability since it is based on prescriptive modeling.

Without CFD the action of an airplane, beyond simply guessing, will have to be discovered from extensive experience in real flight, where extreme situations such as stall are hazardous to test and thus must be avoided.     

DFS Direct Finite Element Simulation is new CFD software based on first principle physics without prescriptive modeling, which has shown to be truly predictive of complex flight dynamics including stall, beyond the capability of standard CFD.    

The design debacle behind the two 737 Max crashes can be connected to the use of standard CFD without stall prediction.

The question is now if Boeing is going to use DFS to design the intended upgrade of the automatic stall prevention system MCAS and the flight simulator preparing pilots to handle 737 Max with MCAS. Or if Boeing will continue to rely on standard software without stall prediction.

See also: Boeing employees’ frightening internal messages released in 737 Max investigation:

• Would you put your family on a Max simulator trained aircraft? I wouldn’t.
• This airplane is designed by clowns who in turn are supervised by monkeys.

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.

Flow Separation in RANS-LES and DNS vs DFS

The standard methods for CFD Computational Fluid Dynamics are RANS-LES with, and DNS without turbulence and wall models. Both RANS-LES and DNS use a no-slip boundary condition prescribing zero relative fluid velocity on a solid wall, as the corner-stone of Prandtl's boundary layer theory dominating modern fluid dynamics.

DNS is restricted to Reynolds number well below drag crisis at around $5\times 10^5$, because computational resolution of thin boundary layers is required.

RANS-LES uses a wall model prescribing the transition from zero relative velocity on a wall to free stream velocity.

Reynolds numbers for vehicle fluid dynamics of cars, airplanes and boats lie in the range $10^6 -10^9$ beyond the drag crisis.

DFS is a new method for flows beyond the drag crisis based on best possible solution of Euler's equations with a slip boundary condition as a force boundary condition expressing vanishing skin friction without boundary layer.

The drag crisis appears to represent a switch from a no-slip to effectively a slip boundary condition. In CFD with Reynolds numbers in the range $10^6-10^9$ of relevance for vehicles, it thus appears to be possible use a slip boundary condition which does not generate a boundary layer. The evidence is DFS with slip for a wide range of vehicle fluid dynamics in close agreement with observations.

DFS can be viewed as a form of DNS which works for high Reynolds numbers beyond the drag crisis, works because then the fluid effectively satisfies a slip boundary condition.

In particular DFS has shown to correctly predict the critical element of flow separation from a solid wall as 3d rotational slip separation. 

On the other hand, in RANS-LES the flow velocity is prescribed close to the wall and thus also flow separation (or non-separation) is prescribed and prescription is not prediction.

DNS with no-slip as being restricted to low Reynolds numbers, cannot predict flow separation beyond the drag crisis and and so separates on the crest of a wing and not at the trailing edge required for generation of lift (before stall).

In short, DFS represents a major advancement in CFD by allowing prediction of flow separation through the use of a force boundary condition expressing observed vanishingly small skin friction
allowing the simulation to "follow the physics", in contrast to RANS-LES where instead the simulation "prescribes/dictates the physics". The difference is huge.

In fluid dynamics according to Prandtl, flow separation is connected to the presence of an "adverse pressure gradient" retarding 2d flow to stagnation followed by separation as a 2d phenomenon. Accordingly flow separation in RANS-LES is prescribed by "adverse pressure gradients", which however not physics.  True flow separation is a 3d phenomenon which is captured in DFS.

     

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:

      

The (Unphysical) Kutta Condition

Kutta and Zhukovsky named Fathers of Modern Aerodynamics saved fluid dynamics from collapse after the Wright brothers with their Flyer in 1903 had shown that powered human flight was possible, in blatant contradiction to the theoretical prediction that it is not possible. The great idea of Kutta and Zhukovsky was to add large scale circulation to potential flow according to this generic picture:

We see potential flow (left) with zero lift and drag from cancelling low (L) and high (H) pressure and flow separation before the trailing edge,  modified by large scale circulation (middle) around the wing section into flow with lift and separation at the trailing edge (right). Kutta a Zhukovsky claimed that the large circulation was generated by a sharp trailing edge preventing the flow from turning around the edge, as in potential flow creating high pressure on top of the wing destroying lift.

The argument was that it was the singularity of the sharp trailing edge that was powerful enough to generate the large scale circulation around the section, with the effect of creating lift. A wing thus had to have a sharp trailing and so the concept of airfoil was born as a wing with a sharp trailing edge and to help the design of airplanes, a database with 1600 wing sections was created, all airfoils with sharp trailing edge.

But there was one caveat: It was early on observed that wings with rounded trailing edges worked just as fine as wings with sharp trailing edge. Rounded edge of diameter up to 2% of the chord gave the same lift and drag as with sharp edge, and the same lift but a bit larger drag for up to 10%, that is of the same diameter as the leading edge, see Trailing Edge Geometry.The conclusion could only be that lift was not an effect of a sharp trailing edge.

But that did not prevent the KZ circulation theory to serve as the salvation from collapse all through the modern era of aviation. After all, the data base only listed airfoils with sharp trailing edges and so the conclusion was that it had some effect, albeit somewhat mysterious.

But the KZ theory is an example of Aristotle's logical fallacy of confirming the consequent of an assumption. The argument started with the correct implication that if there is circulation, then there is lift, and concluded from observing lift (the consequent) that there must be circulation (the assumption). This type of argument is common in science as a technique to affirm an assumption, but the logic is missing and science with incorrect logic is non-science, that is nonsense.

The singularity of the sharp trailing edge was thus used to explain lift and it also came to be a crucial element of computational fluid mechanics CFD: The presence of the singularity allowed prescribing the velocity in a potential flow CFD code at the trailing edge and thus moving the separation back to the trailing edge from its position in potential flow. The argument appeared to be that from a singularity anything can happen. The effect was to artificially introduce strong suction (or blowing) on top of the wing thus causing circulation around the wing as in KZ theory and lift. The trick to prescribe the velocity in a CFD code at the trailing edge (easy to do) was celebrated as the Kutta condition. 

Standard CFD codes such as RANS or LES thus implement the Kutta condition and so they are able to give reasonable predictions to lift for airfoils with sharp trailing edge, but not to drag because circulation does not change the zero drag of potential flow.

On the other hand, DFS Direct Finite Element Simulation computes lift and drag of wings with rounded trailing edges without any trick of artificially specifying the velocity at the trailing edge, all in close accordance with observation.

A relevant question is then what standard CFD would give for wings with rounded trailing edges? Results are sparse because airfoils are supposed to have sharp trailing edges and so standard CFD comes with the Kutta condition. Or the other way around, without the Kutta condition standard CFD gives completely wrong lift.

However the arcticle Numerical Study Comparing RANS and LES Approaches on a Circulation Control Airfoil by Rumsey and Nishino offers information. The study concerns the flow around a wing subject to a mechanism of blowing on the leading edge which creates circulation and thus enhances lift. The interesting thing is that the trailing edge is rounded allowing us to study the performance of RANS and LES without the singularity of a sharp trailing edge. The reason it is rounded is to not prevent circulation like a sharp trailing edge.

The figures below show the wing section with blowing mechanism at leading edge (right) and rounded trailing edge (left). We see a pressure distribution with unphysical (not observed) high pressure at the trailing edge connecting to a (not observed) separation pattern. We thus see that RANS and LES without sharp trailing edge and Kutta condition gives incorrect pressure distribution.

On the other hand, DFS shows the observed pressure distribution of separation without pressure rise.

Altogether, standard CFD comes with the Kutta condition, which artificially creates circulation and lift, which means that standard CFD is unphysical.

DFS does not use any Kutta condition and is physical because it is based on first principle physics.

Connecting to the discussion on no-slip vs slip, recall that standard no-slip CFD flow without the Kutta condition, will separate on the crest of the wing and then give little lift.

Role of Shear Layer: No-Slip vs Slip

The book Computational Turbulent Incompressible Flow (Chap 36) describes in theory and computation the transition to turbulence in parallel shear flow such as Couette flow between two parallel plates and in a laminar boundary layer. The basic mechanism is the action of streamwise vorticity, generated from perturbations in incoming flow, which slowly redistributes the shear flow transversally into high and low speed streamwise flow streaks with increasing transversal velocity gradients, which trigger turbulence when big enough.

The transition is a threshold phenomenon based on the product of perturbation growth (scaling with Reynolds number and shear strength) and perturbation level, which if large enough triggers transition to turbulence through the above mechanism acting in a shear layer. See this picture from the book:

In particular, without shear the transition to turbulence does not get triggered. This closely connects to the discussion in recent posts on a no-slip vs a slip boundary condition on a solid wall: With no-slip there is a boundary shear layer, while with slip there is no shear layer. In other words:

• A no-slip laminar shear boundary layer may turn into a no-slip turbulent boundary layer. 
• Laminar flow with slip does not develop a turbulent boundray layer.    

This makes a difference for skin friction, where no-slip connects to large skin friction of a turbulent boundary layer, while slip is seen as a bypass limit of a laminar boundary layer with small skin friction. 

Standard CFD is calibrated to large skin friction from tripped flat plate experiments forcing transition to a turbulent boundary layer, which then attributes most of drag to skin friction for a streamlined body like an airplane wing.

DFS with slip computes drag of all bodies including streamlined bodies (for Reynolds numbers bigger than $10^6$ beyond the drag crisis) in accordance with observations, thus as form/pressure drag with no skin friction.  This gives strong evidence that flow beyond drag crisis acts as effectively satisfying a slip boundary with small skin friction, and thus that calibration to tripped flat plate experiments has led CFD in a wrong direction. 

The real catch: With slip there are no thin laminar or turbulent boundary layers to resolve computationally, and this makes DFS computable while standard CFD with boundray layers is not.

DFS supports the following conceptual understanding:

• bluff body flow = potential flow with 3d rotational slip separation into a turbulent wake.  

In particular, turbulence is not generated by tripping the flow by no-slip in boundary layers, but instead from 3d rotational slip separation in the back (with small damped contribution from flow attachment in the front). This is a radical step away from Prandtl's scenario which has paralysed CFD by asking for computational resolution of very thin boundary layers beyond any forseeable computer power. 

     

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.

      

Breaking The Prandtl Spell: Do Not Trip!

A body moving through a fluid like air or water is subject to a resistance force referred to as drag. In 1755 the French mathematician d'Alembert showed that the drag of potential flow, which is a mathematically possible flow according to Euler's equations, is zero. Since zero drag was in direct contradiction to observation of substantial drag even in slightly viscous fluids such as air and water, this was coined  d'Alembert's paradox. It sent fluid mechanics from its promising start with Euler's equations into scientific collapse with theory in blatant contradiction to observation. 

D'Alembert's paradox remained without resolution until 1904 when the young German fluid mechanician Ludwig Prandtl (later named Father of Modern Fluid Mechanics) suggested that substantial drag could result from the presence of a thin boundary layer connecting free flow velocity to zero relative velocity on the boundary of the body as if the fluid somehow was sticking to the boundary with a no-slip condition thereby causing positive skin friction. This discriminated potential flow because of zero friction or slip. 

Prandtl's suggested resolution of d'Alembert's became the lead star of the modern fluid dynamics of the 20th century, but it made Computational Fluid Dynamics CFD into an impossibility by asking for impossible computational resolution of very thin boundary layers to correctly compute drag. 

In 2005 I gave together with Johan Hoffman a new resolution of d'Alembert's paradox showing that potential flow is unstable and develops into a quasi-stable flow with 3d rotational slip separation creating a low pressure wake with substantial drag. We thus showed that the main drag of a body comes from form/pressure drag and not from skin friction drag. This gave new life to CFD in the form of Direct Finite Element Simulation DFS allowing computation of the drag of any body at affordable computational cost by not requiring resolution of thin boundray layers; with slip there are no boundary layers! DFS computes best possible turbulent solutions to Euler's equations.

DFS correctly computes the drag of a body as form/pressure drag thus giving evidence of very small contribution from skin friction drag ($1-10\%$), whereas conventional Prandtl CFD predicts at least $50\%$ skin friction drag. 

So how big is then skin friction? Experiments should give answers. And yes, there are tables and data banks of skin friction for various surfaces presented in the form of skin friction coefficients $c_f$

usually in the range $0.003$ which can give $50\%$ skin friction for long slender bodies. The experiments typically use flat plates dragged through water. 

But the experiments always use some form of tripping by a rib or wire fastened to the flat plate with the effect of forcing the development of a heavily turbulent boundary layer with up to a factor 10 larger skin friction than without tripping. 

This is illustrated in the plot blow from Vinuesa et al:Turbulent boundary layers around wing sections up to Rec = 1, 000, 000, where we see the friction force over the span of a wing from leading edge left to trailing edge right with the blue curve with tripping and the black curve without tripping. Here the friction force in the middle of the span from 0.2 to 0.7 is the relevant part, with special irrelevant effects at leading and trailing edge. We see the effect of tripping at 0.1 giving skin friction a kick which remains over the span (blue curve), to be compared with the very small skin friction without tripping (black curve) as smaller than say 0.0006, a factor 5 from 0.003, from $50\%$ to $10\%$ or smaller.

                 

Prandtl CFD thus uses tripping in experiments to inflate skin friction coefficients, which are then used to support a std scenario with $50\%$ skin friction asking for modeling or computational resolution of very thin boundary layers as the dictate Father Prandtl, however impossible to follow.

But a real wing does not have a rib fastened at the leading edge to force the development of a heavily turbulent boundary layer, because that would decrease lift and increase drag, and so the experiments with tripping are not relevant to real cases. 

Instead, untripped experiments are relevant and they show much smaller skin friction. This gives experimental support to DFS with slip. DFS computes both lift and drag of a wing or whole airplane within experimental accuracy of untripped experiments. DFS has no parameters to fit and thus computes lift and drag from form only. Amazing! 

From the perspective of DFS, putting a rib on a wing would correspond to changing the form of the wing and thus would be computable from form only, and would then show increased drag. But real wings don't have such ribs, since it would not serve any real cause. 

The ribs are used only in order to make experiments fit theory. Removing the rib, theory can be brought into contact with reality and Prandtl's spell can be broken.

DFS with slip shows that the connection between fluid and solid wall can be viewed to be effectuated as a "thin film" the action of which can be modeled by slip/small friction without creation of any thin boundary layer to resolve. The thin film then does not act like a laminar no-slip boundary layer, nor as a fully turbulent tripped no-slip boundary layer, but as a new connection between fluid and wall ready to model as slip/small friction.

Here are more results from Philipp Schlatter et al: Progress on High-Order Simulations of Turbulence Around Wings showing that skin friction can be small (red curve):       

To see the tripping used in so called Direct Numerical Simulation DNS over a wing, take look at this video:

and this presentation:

Follow also the heavy tripping in this monster DNS for a NACA4412 wing with 5 degrees angle of attack at Re = 350.000, with 1 billion mesh points taking 1500 hours on 1024 processors, showing unphysical  separation before trailing edge:

You see a DNS with no-slip which does not capture the real flow around a wing despite the pretention  of DNS as true physics. But DFS with slip does, as true physics!

Paradoxes of Fluid Mechanics

The book Hydrodynamics A study in logic, fact and similitude (1950) by Garrett Birkhoff gives a long list of paradoxes of fluid mechanics including the following concerning incompressible flow:

1. D'Alembert's paradox (zero drag of (potential) inviscid flow). 
2. Reversibility paradox (reversion of flow direction does not reverse flow).
3. Fatness paradox of Kutta-Joukowsky theory (lift decreases with thickness of wing).
4. Magnus effect (lift of backspin tennis ball opposite to that of table tennis ball).
5. Eiffel paradox ("drag crisis" as sudden drop of drag). 
6. Dubaut paradox (smaller drag of stationary pole in streaming fluid than pole moving through stationary fluid).  

Birkhoff describes the role of paradoxes:

• These paradoxes have been the subject of many witticisms. Thus, it has recently been said  that in the nineteenth century, " fluid dynamicists were divided into hydraulic engineers who observed what could not be ex­plained, and mathematicians who explained things that could not be observed." (It is my impression that many survivors of both species are still with us.)—Again, Sydney Goldstein has observed that one can read all of Lamb without realizing that water is wet!  
• I think we should welcome the discovery of hydrodynamical paradoxes—recognizing frankly the inadequacy of existing mathematics (and logic) to analyze the complex wonders of Nature. Experience shows that man's imagination is far more limited than Nature's resources: as Pascal wrote, "l'imagination se lassera plutot de concevoir que la nature de fournir.

Solving paradoxes thus may open roads to progress. The mother of all paradoxes of incompressible flow is d'Alembert's paradox. Prandtl was crowned Father of Modern Fluid Mechanics because he saved the face of fluid mechanics confronted by the paradox, by coming to rescue in a short 1904 article selling the idea that drag somehow is an effect of a boundary layer caused by an imagined necessity of a no-slip boundary condition forcing a fluid to "stick" to a solid wall. However, Prandtl's boundary layer theory came with a serious side effect as the impossibility to computational resolve thin boundary which has paralysed Computational Fluid Dynamics CFD throughout the 20th century.

  

The paralysis was lifted only in 2008 with a new resolution of d'Alembert's paradox (check video) showing that inviscid flow modeled by the Euler equations can be described as potential flow modified by 3d rotational slip separation into turbulent flow, which can be resolved computationally by Direct Finite Element Simulation DFS. This is described in Computational Turbulent Incompressible Flow with subsequent elaborations, resolving all the paradoxes listed by Birkhoff, and more as shown in the previous post with references including revealing The Secret of Flight (check video).

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.