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The Secret of FlightApplied Mathematics Body&Soul Vol 6

Johan Hoffman, Johan Jansson and Claes Johnson

December 23, 2008



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Contents

I In the Rear Mirror 2

1 Classical Theory of Flight 3

1.1 Early Pioneers . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Cayley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Lilienthal and Wright . . . . . . . . . . . . . . . . . . . . . 81.4 Status Quo . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 New Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 A Glimpse of The Secret . . . . . . . . . . . . . . . . . . . . 131.7 The Secret of Lift According to NASA . . . . . . . . . . . . 14

2 Confusion: Lack of Theory of Lift 17

3 Aristotele 21

3.1 Liberation from Aristotle . . . . . . . . . . . . . . . . . . . 22

4 Medieval Islamic Physics of Motion 23

4.1 Avicenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Abu’l-Barakat . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Biruni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Ibn al-Haytham . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.5 Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26



iv Contents

5 Leonardo da Vinci 275.1 The Polymath . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 The Notebooks . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 The Scientist . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 The Mathematician . . . . . . . . . . . . . . . . . . . . . . . 305.5 The Engineer . . . . . . . . . . . . . . . . . . . . . . . . . . 315.6 The Philosopher . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Newton’s Incorrect Theory 33

7 D’Alembert and his Paradox 357.1 d’Alembert and Euler and Potential Flow . . . . . . . . . . 357.2 The Euler Equations . . . . . . . . . . . . . . . . . . . . . . 377.3 Potential Flow around a Circular Cylinder . . . . . . . . . . 387.4 Non-Separation of Potential Flow . . . . . . . . . . . . . . . 41

8 Robins and the Magnus Effect 43

9 Lilienthal and Bird Flight 45

10 Wilbur and Orwille Wright 49

11 Lift by Circulation 5511.1 Lanchester . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.2 Kutta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.3 Zhukovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

12 Prandtl and Boundary Layers 5912.1 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2 B oundary Layers . . . . . . . . . . . . . . . . . . . . . . . . 6012.3 Prandtl’s Resolution of d’Alembert’s Paradox . . . . . . . . 63

II Preparing for Takeoff 65

13 Navier-Stokes Equations 6713.1 Small or Very Small Viscosity . . . . . . . . . . . . . . . . . 6713.2 The Squeeze . . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.3 The Way Out . . . . . . . . . . . . . . . . . . . . . . . . . . 6913.4 Navier-Stokes with Force BC . . . . . . . . . . . . . . . . . 6913.5 Exponential Instability . . . . . . . . . . . . . . . . . . . . . 7013.6 Energy Estimate with Turbulent Dissipation . . . . . . . . . 7113.7 G2 Computational Solution . . . . . . . . . . . . . . . . . . 7313.8 Wellposedness . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.9 Wellposedness of Mean-Value Outputs . . . . . . . . . . . . 7413.10Stability of the Dual Linearized Problem . . . . . . . . . . . 75



Contents v

13.11Turbulent Flow around a Car . . . . . . . . . . . . . . . . . 75

14 Resolution of d’Alembert’s Paradox 7714.1 Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . 7714.2 Stability of Corner Flow . . . . . . . . . . . . . . . . . . . . 7814.3 Potential Flow as Navier-Stokes Solution . . . . . . . . . . . 7814.4 Turbulent Flow around a Circular Cylinder . . . . . . . . . 79

15 Flow Separation 8315.1 Lift and Drag from Separation . . . . . . . . . . . . . . . . 8315.2 Separation in Pictures . . . . . . . . . . . . . . . . . . . . . 8415.3 Critique by Lancaster and Birkhoff . . . . . . . . . . . . . . 8515.4 Can You Prove that Prandtl Was Incorrect? . . . . . . . . . 8715.5 Separation vs Normal Pressure Gradient . . . . . . . . . . . 8815.6 Laminar Separation with No-Slip . . . . . . . . . . . . . . . 8915.7 Turbulent Separation with Slip . . . . . . . . . . . . . . . . 9015.8 Potential Flow and Non-Separation . . . . . . . . . . . . . . 9015.9 Mechanics of Separation . . . . . . . . . . . . . . . . . . . . 9115.10Flow around a Cylinder and Sphere . . . . . . . . . . . . . 9215.11Turbulent Separation and Drag Crisis . . . . . . . . . . . . 93

15.12Separation vs Normal Pressure Gradient . . . . . . . . . . . 9415.13Scenario for Separation without Stagnation . . . . . . . . . 9415.14Separation Experiments on Youtube . . . . . . . . . . . . . 96

16 Effects of Non-Symmetric Separation 9716.1 Magnus Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.2 The Reverse Magnus Effect . . . . . . . . . . . . . . . . . . 9816.3 T he Coanda Effect . . . . . . . . . . . . . . . . . . . . . . . 9816.4 More NASA Confusion . . . . . . . . . . . . . . . . . . . . . 98

17 Boundary Layer Turbulence from Separation 101

III Flying 10218 Gliding Flight 103

18.1 Mechanisms of Lift and Drag . . . . . . . . . . . . . . . . . 10318.2 Phase 1: 0 ≤ α ≤ 4 − 6 . . . . . . . . . . . . . . . . . . . . . 10418.3 Phase 2: 4 − 6 ≤ α ≤ 16 . . . . . . . . . . . . . . . . . . . . 10618.4 Phase 3: 16 ≤ α ≤ 20 . . . . . . . . . . . . . . . . . . . . . . 10618.5 Lift and Drag Distribution Curves . . . . . . . . . . . . . . 10618.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11118.7 Comparing Computation with Experiment . . . . . . . . . . 11118.8 Kutta-Zhukovsky’s Lift Theory is Non-Physical . . . . . . . 112

19 Flapping Flight 113



vi Contents

20 At the Horizon 11520.1 Complete Airplane . . . . . . . . . . . . . . . . . . . . . . . 11520.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11520.3 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11520.4 Flight Simulator . . . . . . . . . . . . . . . . . . . . . . . . 115

References 117



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Foreword

To those who fear flying, it is probably disconcerting that physicistsand aeronautical engineers still passionately debate the fundamentalissue underlying this endeavor: what keeps planes in the air? (Ken-neth Chang, New York Times, Dec 9, 2003)

In this book we present a mathematical theory of subsonic flight basedon a combination of analysis and computation. By computing and ana-lyzing turbulent solutions of the incompressible Navier-Stokes equationsfor slightly viscous flow subject to force boundary conditions, we uncovera mechanism for the generation of substantial lift at the expense of smalldrag of a wing, which is fundamentally different from that envisioned in theclassical theories by Kutta-Zhukovsky for lift and by Prandtl for drag. Weshow that lifting flow results from an instability at rear separation gener-

ating counter-rotating low-pressure rolls of streamwise vorticity inititatedas surface vorticity resulting from meeting opposing flows. This mecha-nism is entirely different from the mechanism based on global circulationof Kutta-Zhukovsky theory. We show that the new theory allows accuratecomputation of lift, drag and twisting moments of an entire airplane us-ing a few millions of mesh-points, instead of the impossible quadrillionsof mesh-points required according to state-of-the-art following Prandtl’sdictate of resolution of very thin boundary layers connected with no-slipvelocity boundary conditions. The new theory thus offers a way out of thepresent deadlock of computational aerodynamics of slightly viscous turbu-lent flow.

Stockholm Christmas Eve 2008, Johan Hoffman and Claes Johnson



Part I

In the Rear Mirror

2




1Classical Theory of Flight

I feel perfectly confident, however, that this noble art will soon bebrought home to man’s general convenience, and that we shall beable to transport ourselves and families, and their goods and chattels,more securely by air than by water, and with a velocity of from 20to 100 miles per hour. (George Cayley 1809)

I can state flatly that heavier than air flying machines are impossible.(Lord Kelvin 1895)

You can’t make unseen, what a man has seen. (Bertold Brecht inLife of Galileo)

1.1 Early Pioneers

The flight of birds has always challenged human curiosity with the dream of human flight described already in the Greek myth about the inventor andmaster craftsman Deadalus , who built wings for himself and his son Icarus to escape from imprisonment in the Labyrinth of Knossos on the islandof Crete. The leading scholar Abbas Ibn Firnas of the Islamic culture inCordoba in Spain studied the mechanics of flight and in 875 AD survivedone successful flight on a pair of wings made of feathers on a wooden frame.Some hundred years later the great Turkish scholar Al-Djawhari tied twopieces of wood to his arms and climbed the roof of a tall mosque in Nisabur,Arabia, and announced to a large crowd:



4 1. Classical Theory of Flight

• O People! No one has made this discovery before. Now I will fly before your very eyes. The most important thing on Earth is to fly to the skies. That I will do now.

Unfortunately, he did not, but fell straight to the ground and was killed.It would take 900 years before the dream of Al-Djawhari became true,after many unsuccessful attempts. One of the more succesful was made byHezarfen Ahmet Celebi , who in 1638 inspired by work of Leonardo da Vinci,after nine experimental attempts and careful studies of eagles in flight, tookoff from the 183 foot tall Galata Tower near the Bosphorus in Istanbul andsuccessfully landed on the other side of the Bosphorus. The word Hezarfenmeans expert in 1000 sciences and a reward of 1000 gold pieces was givento Hezarfen for his achievement.

FIGURE 1.1. Abbas Ibn Firnas flying from the Mosque of Cordoba in 875 AD.

The understanding of why it is possible to fly has haunted scientists sincethe birth of mathematical mechanics in the 17th century. To fly, an upwardforce on the wing, referred to as lift L, has to be generated from the flowof air around the wing, while the air resistance to motion or drag D, is nottoo big. The mystery is how a sufficiently large ratio L

D can be created. Inthe gliding flight of birds and airplanes with fixed wings at subsonic speeds,LD is typically between 10 and 20, which means that a good glider can glideup to 20 meters upon loosing 1 meter in altitude, or that Charles Lindberg could cross the Atlantic in 1927 at a speed of 50 m/s in his 2000 kg Spirit of



1.1 Early Pioneers 5

St Louis at an effective engine thrust of 150 kp (with LD = 2000/150 ≈ 13)

from 100 horse powers (because 1 hp = 75 kpm/s).By elementary Newtonian mechanics, lift must be accompanied by down-

wash with the wing redirecting air downwards. The enigma of flight is themechanism generating substantial downwash under small drag, which isalso the enigma of sailing against the wind with both sail and keel actinglike wings creating lift.

Classical mathematical mechanics could not give an answer. Newton computed the lift of a tilted flat plate (of unit area) redirecting a ho-risontal stream of fluid particles of speed U and density ρ, but obtaineda disappointingly small value approximately proportional to the square of the tilting angle or angle of attack α (in radians with one radian = π

180degrees):

L = sin2(α)ρU 2, (1.1)

since sin(α) ≈ α (for small α). The French mathematician Jean le Rond d’Alembert (1717-1783) followed up in 1752 with a computation based onpotential flow (inviscid incompressible irrotational stationary flow), show-ing that both the drag and lift of a body of any shape (in particular a wing)is zero, referred to as d’Alembert’s paradox , since it contradicts observations

and thus belongs to fiction. To explain flight d’Alembert’s paradox had tobe resolved.

But the dream was not given up and experiments could not be stoppedonly because a convincing theory was lacking; undeniably it was possiblefor birds to fly without any theory, so maybe it could somehow be possiblefor humans as well.

The first published paper on aviation was Sketch of a Machine for Flying in the Air by the Swedish polymath Emanuel Swedenborg , published in1716, describing a flying machine consisting of a light frame covered withstrong canvas and provided with two large oars or wings moving on ahorizontal axis, arranged so that the upstroke met with no resistance whilethe downstroke provided lifting power. Swedenborg understood that themachine would not fly, but suggested it as a start and was confident thatthe problem would be solved:

• It seems easier to talk of such a machine than to put it into actual-ity, for it requires greater force and less weight than exists in a hu-man body. The science of mechanics might perhaps suggest a means,namely, a strong spiral spring. If these advantages and requisites are observed, perhaps in time to come some one might know how better toutilize our sketch and cause some addition to be made so as to accom-plish that which we can only suggest. Yet there are sufficient proofs and examples from nature that such flights can take place without danger, although when the first trials are made you may have to pay

for the experience, and not mind an arm or leg.




Swedenborg would prove prescient in his observation that powering theaircraft through the air was the crux of flying.

1.2 Cayley

The British engineer Sir George Cayley (1773-1857), known as the father of aerodynamics , was the first person to identify the lift and drag forcesof flight, discovered that a curved lifting surface would generate more liftthan a flat surface of equal area. and designed different gliders as shownin Fig.1.2. In 1804 Cayley designed and built a model monoplane gliderof strikingly modern appearance. with a cruciform tail, a kite-shaped wingmounted at a high angle of incidence and a moveable weight to alter thecenter of gravity.

In 1810 Cayley published his now-classic three-part treatise On Aerial Navigation , the first to state that lift, propulsion and control were thethree requisite elements to successful flight, and that the propulsion systemshould generate thrust while the wings should be shaped so as to create

lift. Cayley observed that birds soared long distances by simply twistingtheir arched wing surfaces and deduced that fixed-wing machines would flyif the wings were cambered. Thus, one hundred years before the Wrightbrothers flew their glider, Cayley had established the basic principles andconfiguration of the modern airplane, complete with fixed wings, fuselage,and a tail unit with elevators and rudder, and had constructed a seriesof models to demonstrate his ideas. In 1849 Cayley built a large glidingmachine, along the lines of his 1799 design, and tested the device with a10-year old boy aboard. The gliding machine carried the boy aloft on atleast one short flight.

Cayley recognized and searched for solutions to the basic problems of flight including the ratio of lift to wing area, determination of the center of wing pressure, the importance of streamlined shapes, the recognition that atail assembly was essential to stability and control, the concept of a bracedbiplane structure for strength, the concept of a wheeled undercarriage, andthe need for a lightweight source of power. Cayley correctly predicted thatsustained flight would not occur until a lightweight engine was developedto provide adequate thrust and lift.

Cayley’s efforts were continued by William Henson who designed a largepassenger-carrying steam-powered monoplane, with a wing span of 150 feet,named The Henson Aerial Steam Carriage for which he received a patent in1843, but it could not fly. The Aerial Transit Company’s publicist, FrederickMarriott, had a number of prints made in 1843 depicting the Aerial SteamCarriage over the pyramids in Egypt, in India, over London, England, andother places, which drew considerable interest from the press.



1.2 Cayley 7

FIGURE 1.2. Different gliders designed by Cayley.




In 1856, the French aviator Jean-Marie Le Bris made the first flighthigher than his point of departure, by having his glider L’Albatros arti-

ficiel pulled by a horse on a beach. He reportedly achieved a height of 100 meters, over a distance of 200 meters. In 1874, Felix du Temple builtthe Monoplane , a large plane made of aluminium in Brest, France, with awingspan of 13 meters and a weight of only 80 kilograms (without pilot).Several trials were made with the plane, and it is generally recognized that

it achieved lift off under its own power after a ski-jump run, glided for ashort time and returned safely to the ground, making it the first successfulpowered flight in history, although the flight was only a short distance anda short time.

The British marine engineer Francis Wenham (1824-1908) discovered,while unsuccessfully attempting to build a series of unmanned gliders, thatthe most of the lift from a bird-like wing was generated at the leadingedge, and concluded that long, thin wings would be better than the bat-like ones suggested by many, because they would have more leading edgefor their weight. He presented a paper on his work to the newly formedRoyal Aeronautical Society of Great Britain in 1866, and decided to proveit by building the world’s first wind tunnel in 1871. Members of the Societyused the tunnel and learned that cambered wings generated considerably

more lift than expected by Cayley’s Newtonian reasoning, with lift-to-dragratios of about 5:1 at 15 degrees. This clearly demonstrated the ability tobuild practical heavier-than-air flying machines; what remained was theproblem of controlling the flight and powering them.

In 1866 the Polish illiterate peasant Jan Wnek built and flew a control-lable glider launching himself from a special ramp on top of the Odporyszowchurch tower 95 m high above the valley below, especially during religiousfestivals, carnivals and New Year celebrations.

1.3 Lilienthal and Wright

The German engineer Otto Lilienthal (1848-1896) expanded Wenham’swork, made careful studies of the gliding flight of birds recorded in Bird-

flight as the Basis of Aviation [25] and designed a series of ever-better hanggliders allowing him to make 2000 successful heavier-than-air gliding flightsstarting from a little artificial hill, before in 1896 he broke his neck fallingto the ground after having stalled at 15 meters altitude. Lilienthal rigor-ously documented his work, including photographs, and for this reason isone of the best known of the early pioneers.

The first sustained powered heavier-than-air flights were performed bythe two brothers Orwille and Wilbur Wright , who on the windy dunes of Kill Devils Hills at Kitty Hawk, North Carolina, on December 17 in 1903,



1.4 Status Quo 9

FIGURE 1.3. Tautological explanation of the flight of The Flyer by NASA.

managed to get their 400 kg airplane Flyer off ground into sustained flightusing a 12 horse power engine. The modern era of aviation had started.

1.4 Status Quo

It is natural to expect that today the mechanics of gliding flight is wellunderstood. However, in the presentation [61] directed to math and scienceteachers, the authority NASA first dismisses the following three populartheories for lift of a wing as being incorrect:

• longer path above: low pressure above the wing because of higher

velocity because of equal transit time above and below,

• Venturi nozzle: higher velocity and lower pressure above becauseof wing asymmetry,

• skipping stone: force from fluid particles hitting the wing from be-low,

with the first two being based on Bernouilli’s principle combining lowpressure with high velocity and vice versa, and the third on Newton’s 3rdlaw. NASA then hints at a trivial tautological fourth theory similar toskipping stone, see the end of this chapter for details:

• lift from flow turning,




and ends with a disappointing seemingly out of reach:

• To truly understand the details of the generation of lift, one has tohave a good working knowledge of the Euler Equations.

The Plane&Pilot Magazine [66] has the same message. The ambitiousteacher thus does not get much help from expertize to explain the mysteryof flight to curious students, not because NASA wants to hide the secret,

but simply because there is no convincing scientific explanation of why itpossible to fly in the existing literature, however incredible this may seem.

But how can this be? Can you design airplanes without understandingwhat makes it possible to fly? Yes, you can. Just look at the birds, whoare able to fly without scientific understanding. You can replace theory bytrial and error practice. This is what Lilienthal and the Wright brothersdid. On the other hand, nothing is more practical than a good theory,but such a theory is missing for flight. In short, state-of-the-art literature[2, 54, 60, 68, 71] presents a

• theory for drag without lift in viscous laminar flow

by Ludwig Prandtl [67], called the father of modern fluid mechanics [69],

and a

• theory for lift without drag in inviscid potential flow

at small angles of attack by the mathematicians Martin Kutta and Nikolai Zhukovsky , called the father of Russian aviation . Kutta and Zhukovsky au-gumented inviscid zero-lift potential flow by a large scale circulation of airaround the wing section causing the velocity to increase above and decreasebelow the wing as illustrated in Fig.6.1, thus generating lift proportionalto the angle of attack [60, 71, 47]:

L = 2παρU 2. (1.2)

Kutta-Zhukovsky thus showed that if there is circulation then there is lift,

which by a scientific community in desperate search for a theory of lift afterthe flights by the Wright brothers in 1903, was interpreted as an equivalence([71], p.94):

• If the airfoil experiences lift, a circulation must exist .

State-of-the-art is described in [2] as:

• The circulation theory of lift is still alive... still evolving today, 90 years after its introduction .

However, there is no

• theory for lift and drag in turbulent incompressible flow



1.5 New Theory 11

such as the flow of air around a wing of a jumbojet at the critical phaseof take-off at large angle of attack (12 degrees) and subsonic speed (270km/hour). The first start of the new jumbojet Airbus 380 must have beena thrilling experience for the design engineers, since full scale tests cannotbe made.

FIGURE 1.4. High (H) and low (L) pressure distributions of potential flow (left)past a wing section with zero lift/drag modified by circulation around the section(middle) to give Kutta-Zhukovsky flow (right) leaving the trailing edge smoothlywith downwash/lift and a so-called starting vortex behind.

FIGURE 1.5. Lift theory by circulation as typically presented at typical websources such as http://galileo.phys.virginia.edu and http://www.densmore.org.

1.5 New Theory

In this book we present a theory for lift and drag in turbulent incom-pressible flow based on computing turbulent solutions of the incompressible Navier-Stokes equations for slightly viscous flow , which reveals mechanisms




of gliding flight fundamentally different from those envisioned by Kutta-Zhukovsky and Prandtl.

In particular, we show that lift comes along with drag, in contradictionto a common belief supported by zero-drag Kutta-Zhukovsky theory, that[3]:

• A truly inviscid fluid would exert no drag.

The new theory of flight comes out of a new resolution of d’Alembert’sparadox [39, 40, 37], showing that zero-lift/drag potential flow is unstable and in both computation and reality is replaced by turbulent flow withboth lift and drag.

The new resolution is fundamentally different from the classical officialresolution attributed to Prandtl [68, 70, 60], which disqualifies potentialflow because it satisfies a slip boundary condition allowing fluid particlesto glide along the boundary without friction force, and does not satisfya no-slip boundary condition requiring the fluid particles to stick to theboundary with zero relative velocity and connect to the free-stream flowthrough a thin boundary layer , as demanded by Prandtl.

In contrast to Prandtl, we complement in the new theory Navier-Stokesequations with a friction force boundary condition for tangential forces on

the boundary with a small friction coefficient as a model of the small skin friction resulting from a turbulent boundary layer of slightly viscous

flow . In the limit of zero boundary friction this becomes a slip boundarycondition, which means that potential flow can be seen as a solution of theNavier-Stokes equations subject to a small perturbation from small viscousstresses. In the new theory we then disqualify potential flow because it isunstable, that is on physical grounds, and not as Prandtl on formal groundsbecause it does not satisfy no-slip boundary conditions.

As an important practical consequence of the new theory, we show thatlift and drag of an airplane at subsonic speeds can be accurately predictedby computing turbulent solutions of the Navier-Stokes equations using mil-lions of mesh-points without resolving thin no-slip boundary layers. Instate-of-the-art dictated by Prandtl this is impossible, because resolvingthin no-slip boundary layers for slightly viscous flow requires impossiblequadrillions of mesh points [59]. State-of-the-art is decribed in the sequenceof AIAA Drag Prediction Work Shops [11], focussing on the simpler prob-lem of transonic compressible flow at small angles of attack (2 degrees) of relevance for crusing at high speed, leaving out the more demanding prob-lem of subsonic incompressible flow at low speed and large angles of attack at take-off and landing, presumably because a workshop on this topic wouldnot draw any participants.

The new theory is supported by solving the Navier-Stokes equationswith friction boundary condition using an adaptive stabilized finite element method with duality-based a posteriori error control referred to as Gen-eral Galerkin or G2 presented in detail in [39] and available in executable



1.6 A Glimpse of The Secret 13

open source form from [50]. The stabilization in G2 acts as an automatic computational turbulence model , and the only input is the geometry of thewing. We thus find by computation that lift is not connected to circula-tion in contradiction to Kutta-Zhukovsky’s theory and that the curse of Prandtl’s laminar boundary layer theory (also questioned in [73, 48, 72])can be circumvented. Altogether, we show in this book that ab initio com-putational fluid mechanics opens new possibilities of flight simulation ready

to be explored and utilized.

1.6 A Glimpse of The Secret

The new resolution of d’Alembert’s paradox [39, 40, 37] identifies a basicinstability mechanism of potential flow arising from retardation and ac-celleration at separation, which generates counter-rotating rolls or tubesof streamwise vorticity forming a low-pressure wake effectively generat-ing drag. For a wing this is also an essential mechanism for generatinglift by depleting the high pressure before rear separation of potential flowand thereby allowing downwash. This mechanism is illustrated in Fig.1.6showing a perturbation (middle) consisting of counter-rotating rolls of low-pressure streamwise vorticity developing at the separation of potential flow(left), which changes potential flow into turbulent flow (right) with a dif-ferent pressure distribution at the trailing edge generating lift. The rolls of counter-rotating streamwise vorticity appear along the entire trailing edgeand have a different origin than the wing tip vortex , which adds drag butnot lift, which is of minor importance for a long wing [72].

FIGURE 1.6. Stable physical 3d turbulent flow (right) with lift/drag, generatedfrom potential flow (left) by a perturbation at separation consisting of counter-ro-tating tubes of streamwise vorticity (middle), which changes the pressure at thetrailing edge generating downwash/lift and drag.

We see that the difference between Kutta-Zhukovsky and the new ex-plantion is the nature of the modification/perturbation of zero-lift poten-tial flow: Kutta and Zhukovsky claim that it consists of a global largescale two-dimensional circulation around the wing section, that is transver-sal vorticity orthogonal to the wing section (combined with a transversalstarting vortex), while we find that it is a three-dimensional local turbulentphenomenon of counter-rotating rolls of streamwise vorticity at separation,




without starting vortex. Kutta-Zhukovsky thus claim that lift comes fromglobal transversal vorticity without drag, while we give evidence that in-stead lift is generated by local turbulent streamwise vorticity with drag.

We observe that the real turbulent flow like potential flow adheres to theupper surface beyond the crest and thereby gets redirected, because thereal flow is close to potential before separation, and potential flow can onlyseparate at a point of stagnation with opposing flows meeting in the rear, as

shown in [40, 37]. On the other hand, a flow with a viscous no-slip boundarylayer will (correctly according to Prandtl) separate on the crest, becausein a viscous boundary layer the pressure gradient normal to the boundaryvanishes and thus cannot contribute the normal acceleration required tokeep fluid particles following the curvature of the boundary after the crest[41]. It is thus the slip boundary condition modeling a turbulent boundarylayer in slightly viscous flow, which forces the flow to suck to the uppersurface and create downwash, as analyzed in detail in [41], and not anyCoanda effect [47].

This explains why gliding flight is possible for airplanes and larger birds,because the boundary layer is turbulent and acts like slip preventing earlyseparation, but not for insects because the boundary layer is laminar andacts like no-slip allowing early separation. The Reynolds number of a jum-

bojet at take-off is about 108 with turbulent skin friction coefficient < 0.005contributing less than 5% to drag, while for an insect with a Reynolds num-ber of 102 viscous laminar effects dominate.

1.7 The Secret of Lift According to NASA

NASA presents on [61] three incorrect theories of lift plus a fourth tau-tological theory named lift by flow turning claimed to be (more) correct,see Fig.1.7-1.10. To present incorrect theories at length can be risky peda-gogics, since the student can get confused about what is correct and not,but signifies the confusion and misconceptions still surrounding the mecha-

nisms of flight. If a correct theory was available, there would be no reason topresent incorrect theories, but the absence of a correct theory is now seem-ingly covered up by presenting a multitude of incorrect theories. There canbe only one correct theory, but there are infinitely many incorrect ones.



1.7 The Secret of Lift According to NASA 15

FIGURE 1.7. Incorrect theory of lift according to NASA.






FIGURE 1.10. Trivial tautological theory of lift presented as correct by NASA.




2Confusion: Lack of Theory of Lift

Many sources give witness of the lack of convincing scientific answer of how a wing can generate lift with small drag. We give here a small samplestarting with [6]:

• To those who fear flying, it is probably disconcerting that physicists and aeronautical engineers still passionately debate the fundamental issue underlying this endeavor: what keeps planes in the air?

• “Here we are, 100 years after the Wright brothers, and there are peo-ple who give different answers to that question, said Dr. John D.Anderson Jr., the curator for aerodynamics at the Smithsonian Na-tional Air and Space Museum in Washington. “Some of them get tobe religious fervor.

• The answer, the debaters agree, is physics, and not a long rope hang-ing down from space. But they differ sharply over the physics, espe-cially when explaining it to nonscientists. “There is no simple one-liner answer to this, Dr. Anderson said.

• The simple Newtonian explanation also glosses over some of the physics,like how does a wing divert air downward? The obvious answer – air molecules bounce off the bottom of the wing – is only partly correct.

• If air has to follow the wing surface, that raises one last question.If there were no attractive forces between molecules, would there be no flight? Would a wing passing through a superfluid like ultracold helium, a bizarre fluid that can flow literally without friction, produce



18 2. Confusion: Lack of Theory of Lift

no lift at all? That has stumped many flight experts. “I’ve asked that question to several people that understand superfluidity, Dr. Ander-son, the retired physicist, said. “Alas! They don’t understand flight.

We cite from [87]:

• You’d think that after a century of powered flight we’d have this lift thing figured out. Unfortunately, it’s not as clear as we’d like. A lot of

half-baked theories attempt to explain why airplanes fly. All try to take the mysterious world of aerodynamics and distill it into something comprehensible to the lay audience–not an easy task. Nearly all of the common ”theories” are misleading at best, and usually flat-out wrong.

• How can aviation be grounded in such a muddy understanding of the underlying physics? As with many other scientific phenomena, it’s not always necessary to understand why something works to make use of it. We engineers are happy if we’ve got enough practical knowledge to build flying aircraft. The rest we chalk up to magic.

We cite from [52]:

•It is important to realize that, unlike in the two popular explanations

described earlier (longer path and skipping stone), lift depends on significant contributions from both the top and bottom wing surfaces.While neither of these explanations is perfect, they both hold some nuggets of validity. Other explanations hold that the unequal pressure distributions cause the flow deflection, and still others state that the exact opposite is true. In either case, it is clear that this is not a subject that can be explained easily using simplified theories. Likewise,predicting the amount of lift created by wings has been an equally challenging task for engineers and designers in the past. In fact, for years, we have relied heavily on experimental data collected 70 to 80 years ago to aid in our initial designs of wings.

We cite from [51]:

• Few physical principles have ever been explained as poorly as the mechanism of lift.

• It’s all one interconnected system. Unless the overall result of that system is for air to end up lower than it was before the plane flew by,there will be no lift. Wings move air downward, and react by being pushed upward. That’s what makes lift. All the rest is just interesting details.

We cite from [49], which contains many additional references and links:

• How do airplane wings really work? Amazingly enough, this question is still argued in many places, from elementary school classrooms all



2. Confusion: Lack of Theory of Lift 19

the way up to major pilot schools, and even in the engineering de-partments of major aircraft companies. This is unexpected, since we would assume that aircraft physics was completely explored early this century. Obviously the answers must be spelled out in detail in nu-merous old dusty aerodynamics texts. However, this is not quite the case. Those old texts contain the details of the math, but it’s the in-terpretation of the math that causes the controversy. There is an

ongoing Religious War over both the way we should understand the functioning of wings, and over the way we should explain them in children’s textbooks.

From [45]:

• Lift is a lot trickier. In fact it is very controversial and often poorly explained and, in many textbooks, flat wrong. I know, because some readers informed me that the original version of this story was inaccu-rate. I’ve attempted to correct it after researching conflicting ”expert”views on all this....If you’re about fed up, rest assured that even engi-neers still argue over the details of how all this works and what terms to use.

You can watch the various incorrect theories and mystifications beingpresented on Youtube:

• http : //www.youtube.com/watch?v = uUMlnIwo2Qo

• http : //www.youtube.com/watch?v = ooQ1F 2 jb10A

• http : //www.youtube.com/watch?v = kXBXtaf 2T T g

• http : //www.youtube.com/watch?v = 5wI q 75BzOQ

• http : //www.youtube.com/watch?v = khca2FvGR − w



20 2. Confusion: Lack of Theory of Lift




3Aristotele

Probable impossibilities are to be preferred to improbable possibili-ties. (Aristotle)

In all things of nature there is something of the marvelous. (Aristotle)

We have seen that the secret of flying is how to generate large lift by themotion of a wing through air at the expense of small drag. Approachingthis problem we face the general problem of motion , seriously addressedalready by the Greek philosopher Zeno in his famous paradox about thearrow, which in every single moment along its path seems to be frozen intoimmobility, but yet effectively is moving. Today we are familiar with manyforms of motion and most people would probably say that Zeno’s paradoxmust have been resolved since long, although they would not be able toaccount for the details of the resolution.

However, the true nature of e.g. the motion of light through vacuumstill seems to be hidden to us, while the motion through a gas/fluid likeair or water can be approached following an idea presented already byAristotle in the 4th century BC known as antiperistasis . Aristotle states inhis Physics that that a body in motion through air is pushed from behindby the stream of air around the body contracting in the rear after havingbeen expanded in the front. This is like the peristaltic muscle contractionsthat propels foodstuffs distally through the esophagus and intestines, whichis like the squeezing of an object through a lubricated elastic tube by thecombined action of the object expanding the tube in the front and the tubecontracting in the rear of the object, as expressed in the words of Aristotle:



22 3. Aristotele

• Thirdly, in point of fact things that are thrown move though that which gave them their impulse is not touching them, either by reason of mutual replacement, as some maintain, or because the air that has been pushed pushes them with a movement quicker than the natural locomotion of the projectile wherewith it moves to its proper place. But in a void none of these things can take place; the only way anything can move is by riding on something else.

• Fourthly, no one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than here? So that a thing will either be at rest or must be moved ad infinitum, unless something stronger than it impedes it.

• Fifthly, things are now thought to move into the void because it yields;but in a void this quality is present equally everywhere, so that things should move in all directions.

Of course, we say today that according to Newton’s 2nd law, a bodywill continue in rectilinear motion at constant speed unless acted uponby some force, while to Aristotle sustained motion would require a forcepushing from behind. Nevertheless, we will find that there is something inAristotle’s antiperistasis which correctly describes an important aspect of motion through air, if not through vacuum, but you cannot fly in vaccum...

3.1 Liberation from Aristotle

The renewal of learning in Europe, that began with 12th century Scholasti-cism, came to an end about the time of the Black Death, but the NorthernRenaissance (in contrast to th Italian) showed a decisive shift in focus fromAristoteleian natural philosophy to chemistry and the biological sciences.Thus modern science in Europe was resumed in a period of great upheaval:the Protestant Reformation and Catholic Counter-Reformation; the discov-

ery of the Americas by Christopher Columbus; the Fall of Constantinople;but also the re-discovery of Aristotle during the Scholastic period presagedlarge social and political changes. Thus, a suitable environment was cre-ated in which it became possible to question scientific doctrine, in muchthe same way that Martin Luther and John Calvin questioned religiousdoctrine. The works of Ptolemy (astronomy) and Galen (medicine) werefound not always to match everyday observations.

The willingness to question previously held truths and search for newanswers opened the Scientific Revolution by Copernicus’ De Revolutionibus in 1543 stating that the Earth moved around the Sun, followed by Newton’sPrincipia Mathematica in 1687.




4Medieval Islamic Physics of Motion

When hearing something unusual, do not preemptively reject it, forthat would be folly. Indeed, horrible things may be true, and familiarand praised things may prove to be lies. Truth is truth unto itself,not because [many] people say it is. (Ibn Al-Nafis, 1213-1288 A.D.)

We start by observing reality we try to select solid (unchanging) ob-servations that are not affected by how we perceive (measure) them.We then proceed by increasing our research and measurement, sub- jecting premises to criticism, and being cautious in drawing conclu-sions In all we do, our purpose should be balanced not arbitrary,the search for truth, not support of opinions...Hopefully, by follow-ing this method, this road to the truth that we can be confident in,we shall arrive to our objective, where we feel certain that we have,by criticism and caution, removed discord and suspicion...Yet we arebut human, subject to human frailties, against which we must fightwith all our human might. God help us in all our endeavors. (IbnAl-Haytham)

The knowledge of anything, since all things have causes, is not ac-quired or complete unless it is known by its causes. (Avicenna)

Medieval Islamic developments in mechanics prepared for the liberationof science from Christian Scholasticism following Aristotle’s legacy, throughthe new mechanics of Galileo and Newton leading into the Enlightmentand modern Europe. We recall some early Islamic scientists questioningAristotle, and preparing for human flight...



24 4. Medieval Islamic Physics of Motion

4.1 Avicenna

Avicenna (980-1037) a foremost Persian polymath developed an elaboratetheory of motion, in which he made a distinction between the inclinationand force of a projectile, and concluded that motion was a result of aninclination (mayl) transferred to the projectile by the thrower, and thatprojectile motion in a vacuum would not cease. He viewed inclination as

a permanent force whose effect is dissipated by external forces such asair resistance. He also developed the concept of momentum, referring toimpetus as being proportional to weight times velocity. His theory of motionwas also consistent with the concept of inertia in classical mechanics, andlater formed the basis of Jean Buridan’s theory of impetus and exerted aninfluence on the work of Galileo Galilei.

4.2 Abu’l-Barakat

Hibat Allah Abu’l-Barakat al-Baghdaadi (1080-1165) wrote a critique of Aristotelian physics where he was the first to negate Aristotle’s idea that a

constant force produces uniform motion, as he realized that a force appliedcontinuously produces acceleration as an early foreshadowing of Newton’ssecond law of motion. He described acceleration as the rate of change of velocity and modified Avicenna’s view on projectile motion stating that themover imparts a violent inclination on the moved and that this diminishesas the moving object distances itself from the mover. Abu’l-Barakat alsosuggested that motion is relative.

4.3 Biruni

Another prominent Persian polymath, Abu Rayan Biruni , engaged in awritten debate with Avicenna, with Biruni criticizing the Peripatetic schoolfor its adherence to Aristotelian physics and natural philosophy preservedin a book entitled al-As’ila wal-Ajwiba (Questions and Answers). al-Biruniattacks Aristotle’s theories on physics and cosmology, and questions al-most all of the fundamental Aristotelian physical axioms. He rejects thenotion that heavenly bodies have an inherent nature and asserts that their“motion could very well be compulsory” and maintains that “there is no ob-servable evidence that rules out the possibility of vacuum” ; and states thatthere is no inherent reason why planetary orbits must be circular and can-not be elliptical. He also argues that “the metaphysical axioms on which philosophers build their physical theories do not constitute valid evidence

for the mathematical astronomer”, which marks the first real distinctionbetween the vocations of the philosopher-metaphysician (like Aristotle and



4.4 Ibn al-Haytham 25

Avicenna) and that of the mathematician-scientist (al-Biruni himsel). Incontrast to the philosophers, the only evidence that al-Biruni consideredreliable were either mathematical or empirical evidence, and his systematicapplication of rigorous mathematical reasoning later led to the mathema-tization of Islamic astronomy and the mathematization of nature.

Biruni began the debate by asking Avicenna eighteen questions, ten of which were criticisms of Aristotle’s On the Heavens , with his first question

criticizing the Aristotelian theory of gravity for denying the existence of lev-ity or gravity in the celestial spheres, and the Aristotelian notion of circularmotion being an innate property of the heavenly bodies. Biruni’s secondquestion criticizes Aristotle’s over-reliance on more ancient views concern-ing the heavens, while the third criticizes the Aristotelian view that spacehas only six directions. The fourth question deals with the continuity anddiscontinuity of physical bodies, while the fifth criticizes the Peripatetic de-nial of the possibility of there existing another world completely differentfrom the world known to them. In his sixth question, Biruni rejects Aristo-tle’s view on the celestial spheres having circular orbits rather than ellipticorbits. In his seventh question, he rejects Aristotle’s notion that the motionof the heavens begins from the right side and from the east, while his eighthquestion concerns Aristotle’s view on the fire element being spherical. The

ninth question concerns the movement of heat, and the tenth question con-cerns the transformation of elements. The eleventh question concerns theburning of bodies by radiation reflecting off a flask filled with water, andthe twelfth concerns the natural tendency of the classical elements in theirupward and downward movements. The thirteenth question deals with vi-sion, while the fourteenth concerns habitation on different parts of Earth.His fifteenth question asks how two opposite squares in a square dividedinto four can be tangential, while the sixteenth question concerns vacuum.His seventeenth question asks “if things expand upon heating and contract upon cooling, why does a flask filled with water break when water freezes in it?” His eighteenth and final question concerns the observable phenomenonof ice floating on water. After Avicenna responded to the questions, Biruniwas unsatisfied with some of the answers and wrote back commenting onthem.

4.4 Ibn al-Haytham

Ibn al-Haytham (965-1039) discussed the theory of attraction betweenmasses, and it seems that he was aware of the magnitude of accelerationdue to gravity and he stated that the heavenly bodies “were accountable to the laws of physics”. Ibn al-Haytham also enunciated the law of inertia,later known as Newton’s first law of motion, when he stated that a bodymoves perpetually unless an external force stops it or changes its direction



26 4. Medieval Islamic Physics of Motion

of motion. He also developed the concept of momentum, though he did notquantify this concept mathematically. Nobel Prize winning physicist AbdusSalam wrote the following on Ibn al-Haytham: Ibn-al-Haitham was one of the greatest physicists of all time. He made experimental contributions of the highest order in optics. He enunciated that a ray of light, in passingthrough a medium, takes the path which is the easier and “quicker”. In thishe was anticipating Fermat’s Principle of Least Time by many centuries.

He enunciated the law of inertia, later to become Newton’s first law of mo-tion. Part V of Roger Bacon’s Opus Majus is practically an annotation toIbn al Haitham’s Optics.

4.5 Others

Ibn Bajjah (d. 1138) argued that there is always a reaction force for everyforce exerted, connecting to Newton’s 3rd law, though he did not refer to thereaction force as being equal to the exerted force, which had an importantinfluence on later physicists like Galileo. Averroes (1126-1198) defined andmeasured force as “the rate at which work is done in changing the kinetic condition of a material body’ and correctly argued “that the effect and measure of force is change in the kinetic condition of a materially resistant mass.” In the 13th century, Nasir al-Din al-Tusi stated an early versionof the law of conservation of mass, noting that a body of matter is able tochange, but is not able to disappear. In the early 16th century, al-Birjandideveloped a hypothesis similar to Galileo’s notion of “circular inertia .

At night I would return home, set out a lamp before me, and de-vote myself to reading and writing. Whenever sleep overcame meor I became conscious of weakening, I would turn aside to drink acup of wine, so that my strength would return to me. Then I wouldreturn to reading. And whenever sleep seized me I would see thosevery problems in my dream; and many questions became clear to mein my sleep. I continued in this until all of the sciences were deeply

rooted within me and I understood them as is humanly possible. Ev-erything which I knew at the time is just as I know it now; I have notadded anything to it to this day. Thus I mastered the logical, natu-ral, and mathematical sciences, and I had now reached the science.(Avicenna)




5Leonardo da Vinci

The noblest pleasure is the joy of understanding. (da Vinci)

Although nature commences with reason and ends in experience itis necessary for us to do the opposite, that is to commence withexperience and from this to proceed to investigate the reason...Hewho loves practice without theory is like the sailor who boards shipwithout a rudder and compass and never knows where he may cast.(da Vinci)

For once you have tasted flight you will walk the earth with youreyes turned skywards, for there you have been and there you willlong to return. (da Vinci)

Life is pretty simple: You do some stuff. Most fails. Some works. You

do more of what works. If it works big, others quickly copy it. Thenyou do something else. The trick is the doing something else. (daVinci)

Nothing strengthens authority so much as silence...You do ill if youpraise, but worse if you censure, what you do not understand...Thereare three classes of people: those who see, those who see when theyare shown, those who do not see...And many have made a trade of delusions and false miracles, deceiving the stupid multitude...Bewareof the teaching of these speculators, because their reasoning is notconfirmed by experience. (da Vinci)



28 5. Leonardo da Vinci

5.1 The Polymath

Leonardo da Vinci (1452-1519) is the greatest polymath, universal genious,homo universale or renaissance man all times, with remarkable achieve-ments as a scientist, mathematician, engineer, inventor, anatomist, painter,sculptor, architect, botanist, musician and writer.

Born as the illegitimate son of a notary, Piero da Vinci, and a peasant

woman, Caterina, at Vinci in the region of Florence, Leonardo was educatedin the studio of the renowned Florentine painter, Verrocchio. Much of hisearlier working life was spent in the service of Ludovico il Moro in Milan.He later worked in Rome, Bologna and Venice and spent his last years inFrance, at the home awarded him by King Francois I.

As an artist Leonardo created the most famous, most reproduced andmost parodied portrait and religious painting of all time: Mona Lisa andThe Last Supper. As a scientist, he greatly advanced the state of knowledgein the fields of anatomy, civil engineering, optics, and hydrodynamics. As anengineer he conceptualised a helicopter, a tank, concentrated solar power,a calculator, the double hull and outlined a rudimentary theory of platetectonics.

5.2 The Notebooks

Da Vinci recorded his discoveries in journals or Notebooks mostly written inmirror-image cursive, probably for practical expediency because Leonardowas left-handed, rather than for reasons of secrecy because it appears theywere intended for publication. Although his language was clear and ex-pressive, Leonardo preferred illustration to the written word stating, in thespirit of modern pedagogics:

• The more detail you write concerning it the more you will confuse the reader .

It is believed that there were at least 50 notebooks left in the hands of da Vinci’s pupil Francesco Melzi at the master’s death in 1519, of which28 remains, but they were virtually unknown during his life-time and re-mained hidden for over two centuries. His wonderful ideas were forgotten;his inventions were not tested and built for hundreds of years. Dan Brown’sbest-seller The Da Vinci Code have stimulated renewed interest in da Vinciand his complex and inquiring intelligence. Today we can recoognize as anearly precursor of an entire lineage oc scientists and philosophers whosecentral focus was the nature of organic form [4].

Leonardo planned to treat four major themes: the science of painting,architecture, the elements of mechanics, and a general work on humananatomy. To these themes were eventually added notes on his studies of



5.3 The Scientist 29

botany, geology, flight, and hydrology. His intention was to combine all hisinvestigations with a unified world view:

• Plan of Book 15: First write of all water, in each of its motions; then describe all its bottoms and their various materials, always referring to the propositions concerning the said waters; and let the order be good, for otherwise the work will be confused. Describe all the forms

taken by water from its greatest to its smallest wave, and their causes.Da Vinci made impressive and comprehensive investigations into aero-

dynamics collected into his Codex on the Flight of Birds from 1505, anddesigned a large variety of ornithopters for muscle-powered human flightusing flapping wings. After extensive testing da Vinci concluded that evenif both arms and legs got involved through elaborate mechanics, humanpower was insufficient for flapping flight, but during his last years in Flo-rence he began to experiment with designs of flying machines that had fixedwings, not unlike modern hang-gliders.

5.3 The Scientist

Da Vinci observes the dynamics of the physical world with mountains,rivers, plants and the human body in ceaseless movement and transforma-tion, according to a basic principle of science:

• Necessity is the theme and inventor of nature, the curb and the rule.

Da Vinci recognized the two basic forces of fluid mechanics to be iner-tial and viscous forces, realized that water is incompressible and thoughit assumes an infinite number of shapes, its mass and volume is alwaysconserved. Below we will return to the following deep insights expressed byda Vinci:

• In order to give the true science of the movements of the birds in the

air, it is necessary to first give the science of the winds.• As much force is exerted by the object against the air as by the air

against the object.

• The spiral or rotary movement of every liquid is so much swifter as it is nearer to the center of revolution. What we are here proposing is a fact worthy of admiration, since the circular movement of a wheel is so much slower as it is nearer to the center of the rotating object.

• I have found among the excessive and impossible delsusions of men,the search for continuous motion, which is called by some the perpet-ual wheel.




FIGURE 5.1. Da Vinci studies of bird wings and flight

5.4 The Mathematician

Da Vinci had a great admiration for mathematics:

• A bird is an instrument working according to mathematical law, which is within the capacity of man to reproduce.

• There is no certainty, where one cannot apply any of the mathemat-ical sciences, nor those which are connected with the mathematical sciences .

• Mechanics are the Paradise of mathematical science, because here we come to the fruits of mathematics.

• Let no man who is not a mathematician read my principles .

Although da Vinci had little technical training in mathematics, he under-stood basic principles such as the law of free fall motion long before Galileoand conservation of mass:

• The natural motion of heavy things, at each degree of its descent ac-quires a degree of velocity.

• If the water does not increase, nor diminish, in a river which may be of varying turtuoisities, breadths and dephts, the water will pass in



5.5 The Engineer 31

equal quantities in equal times through every degree of the length of the river

Da Vinci adopted Aristotle’s principle of motion:

• Of everything that moves, the space which it acquires is as great as that which it leaves .

He also formulated basic priniciples of differential geometry:

• The line is made with the movement of the point.

• The surface is made by the tramsversal movement of the line.

• The body is made by the movement of the extension of the surface.

FIGURE 5.2. Da Vinci design of a glider

5.5 The Engineer

Between 1480 and 1505 da Vinci made a series of studies of birds and batsand developed sketches of flying machines, including gliders and more orless impossible devices including a flying machine like a boat. The pilotwas intended to lie stretched out and to pull at oars which would propelthe craft through air rather than water. Although this does not work forlarger devices, this is essentially the mechanism for flight of small insectsexperiencing a substantial viscosity of air.

The modern helicopter invented by the Ukrainian-American engineerIgor Sikorsky in the 1930s, was probably inspired by a design by da Vinci




with a helical screw instead of rotor blades (which of course did not work).The most inventive of da Vinci’s flying machines was the glider in Fig.5.2with the following control technique:

• this [man] will move on the right side if he bends the right arm and extends the left arm; and he will then move from right to left by changing the position of the arms.

5.6 The Philosopher

Da Vinci expressed a view on the interaction of body and soul connectingthat of Descartes leading into modern conceptions of mind-brain interac-tion:

• It could be said that such an instrument designed by man is lacking only the soul of the bird, which must be counterfeited with the soul of man...However, the soul of the bird will certainly respond better to the needs of its limbs than would the soul of the man, separated

from them and especially from their almost imperceptible balancing

movements

• Spiritual movement flowing through the limbs of sentient animals,broadens their muscles. Thus boadened, these muscles become short-ened and draw back the tendons that are connected to them. This is the origin of force in human limbs...Material movement arises from the immaterial.




6Newton’s Incorrect Theory

Dubito ergo cogito; cogito ergo sum. (Descartes)

The foolish dog barks at the flying bird. (Bob Marley)

The man who has no imagination has no wings. (Muhammad Ali)

When a distinguished but elderly scientist states that something ispossible, he is almost certainly right. When he states that somethingis impossible, he is very probably wrong. (Arthur C. Clarke)

We know that a surfing board or water skis can carry the weight of aperson, but only in sufficiently rapid motion depending on the weight of the person and the area exposed to the water surface. The vertical force orlift is a reaction to a constant downward push of water as the board meets

new water in its horsiontal motion. Without horisontal motion the boardwith the person will sink into the water. This is illustrated in Fig.??.Newton was the first scientist to seek to develop a theory of lift and

drag, and suggested that they should both be proportional to the densityof the fluid and the square of the speed, which turns out to be more orless correct. Using a surfing board (or skipping stone) argument, whichaccording to NASA we now know is wrong, Newton derived the aboveformula

L = sin2(α)ρU 2, (6.1)

for the lift L of a tilted flat plate of unit area with a quadratic dependenceon the angle of attack α. This formula follows from the fact that the massρU sin(α) hits the plate from below per unit time and gets redirected with a



34 6. Newton’s Incorrect Theory

downward velocity U sin(α) corresponding to a change of momentum equalto sin2(α)ρU 2, which equals the lift force L by Newton’s 2nd law.

Newton’s formula explains the force acting on a surf board on water. Theratio of density of water to that of air is about 1000 and thus surfing onair requires about 30 times as large speed as surfing on water, because liftscales with the speed squared. Water skiing is possible at a speed of about20 knots, which would require a speed about 1000 km/h, close to the speed

of sound, for surfing on air. We understand that Newton’s formula grosslyunder-estimates the lift, at least for subsonic speeds. We understand thatflying in the air is not at all like surfing on water. Newton could thus provethat subsonic flight is impossible in theory, and so must have viewed thethe flight of birds with surprise. Apparently birds were not willing to abideby the laws of Newtonian mechanics, but how could they take this liberty?

It is possible that Newton contributed to delaying human flight by mak-ing it seem impossible. Only after powered human flight had been demon-strated to be possible by the Wright brothers in 1903, did mathematiciansreplace Newton’s erronous lift formula with a formula compatible withflight, although the derivation of the new formula again turned out to beincorrect, as we will discover below...

FIGURE 6.1. Incorrect explanation by Newton of lift by surfing.




7D’Alembert and his Paradox

To those who ask what the infinitely small quantity in mathematicsis, we answer that it is actually zero. Hence there are not so manymysteries hidden in this concept as they are usually believed to be.(Leonhard Euler)

High office, is like a pyramid; only two kinds of animals reach thesummit– reptiles and eagles. (d’Alembert)

Just go on . . . and faith will soon return. (d’Alembert to a friendhesitant with respect to infinitesimals)

If one looks at all closely at the middle of our own century, the eventsthat occupy us, our customs, our achievements and even our topics of conversation, it is difficult not to see that a very remarkable change

in several respects has come into our ideas; a change which, by itsrapidity, seems to us to foreshadow another still greater. Time alonewill tell the aim, the nature and limits of this revolution, whoseinconveniences and advantages our posterity will recognize betterthan we can. (d’Alembert on the Enlightment)

7.1 d’Alembert and Euler and Potential Flow

Working on a 1749 Prize Problem of the Berlin Academy on flow drag,d’Alembert was led to the following contradiction referred to as d’Alembert’s paradox [75, 76, 77, 78, 83] between observation and theoretical prediction:



36 7. D’Alembert and his Paradox

• It seems to me that the theory (potential flow), developed in all pos-sible rigor, gives, at least in several cases, a strictly vanishing resis-tance, a singular paradox which I leave to future Geometers to eluci-date .

The great mathmatician Leonard Euler (1707-1783) had come to same con-clusion of zero drag of potential flow in his work on gunnery [79] from 1745

based on the observation that in potential flow the high pressure formingin front of the body is balanced by an equally high pressure in the back, inthe case of a boat moving through water expressed as

• ...the boat would be slowed down at the prow as much as it would be pushed at the poop...

This is the idea of Aristotle adopted by da Vinci, which we met above inthe form of peristaltic motion.

More precisely, d’Alembert’s paradox concerns the contradiction betweenobservations of substantial drag/lift of a body moving through a slightlyviscous fluid such as air and water, with the mathematical prediction of zero drag/lift of potential flow defined by the following properties:

(i) incompressible ,(ii) irrotational ,

(iii) inviscid ,

(iv) stationary .

Evidently, flying is incompatible with potential flow, and in order to explainflight d’Alembert’s paradox had to be resolved. But d’Alembert couldn’tdo it and all the great mathematical brains of the 18th and 19th centurystumbled on it: Nobody could see that any of the assumptions (i)-(iv) werewrong and the paradox remained unsolved. We shall resolve the paradoxbelow and find the true reason that potential flow with zero drag/lift isnever observed. And the true reason is not (iii).

We recall that a flow is irrotational if the flow velocity u has zero vortic-ity , that is if ∇×u = 0, in which case (for a simply connected domain) thevelocity u is given as the gradient of a potential function: u = ∇ϕ where ϕis the potential. If u is also incompressible, then

∆ϕ = ∇ · ∇ϕ = ∇ · u = 0

and thus the potential ϕ is a harmonic function satisfying Laplace’s equa-tion :

∆ϕ = 0. (7.1)

This promised to open fluid mechanics for take-over by harmonic functionsin the hands of mathematicians, supported by Kelvin’s theorem stating that



7.2 The Euler Equations 37

FIGURE 7.1. D’Alembert formulating his paradox.

without external forcing, an incompressible inviscid flow will stay irrota-tional if initiated as irrotational. Mathematicians thus expected to find anabundance of potential flows governed by harmonic potentials in the fluidmechanics of slightly viscous flow, but such flows did not seem to appearin reality, and nobody could understand why.

In the words of Chemistry Nobel Laureate Sir Cyril Hinshelwood [63]:

• (Because of d’Alembert’s paradox) fluid mecahnics was from start split into the field of hydraulics, observing phenomena which could not be explained, and mathematical or theoretical fluid mechanics explaining phenomena which could not be observed.

We shall se that this unfortunate split has remained into our days. A reso-

lution of d’Alembert’s paradoxis is necessary to uncover the secret of flight,but the paradox has remained unsolved until very recently.

7.2 The Euler Equations

The basic equations in fluid mechanics expressing conservation of momen-tum or Newton’s 2nd law connecting force to accelleration combined withconservation of mass in the form of incompressibility, were formulated byEuler in 1755 as the Euler equations for an incompressible inviscid fluid(of unit density) enclosed in a volume Ω in R3 with boundary Γ: Find the




velocity u = (u1, u2, u3) and pressure p such that

u + (u · ∇)u + ∇ p = f in Ω × I,∇ · u = 0 in Ω × I,u · n = g on Γ × I,

u(·, 0) = u0 in Ω,

(7.2)

where the dot signifies differentiation with respect to time, n denotes theoutward unit normal to Γ, f is a given volume force, g is a given in-flow/outflow velocity, u0 is a given initial condition and I = [0, T ] a giventime interval. We notice the slip boundary condition u · n = 0 modeling anon-penetrable boundary with zero friction.

The momentum equation can alternatively be formulated as

u + ∇(1

2|u|2 + p) + u × ω = f (7.3)

whereω = ∇× u

is the vorticity of the velocity u, which follows from the following calculus

identity: 1

2∇|u|2 = (u · ∇)u + u × (∇× u).

For a stationary irrotational velocity u with u = 0 and ω = ∇× u = 0, wefind that if f = 0, then

1

2|u|2 + p = C (7.4)

where C is a constant, which is nothing but Bernouilli’s principle couplingsmall velocity to large pressure and vice versa.

We conclude that a potential flow velocity u = ∇ϕ solves the Eulerequations with the pressure p given by Bernouilli’s law.

Kelvin’s theorem states that if the initial velocity u0 is irrotational and

∇×f = 0 and g = 0, then a smooth Euler solution velocity will remain

irrotational for positive time. Below we will question the validity of Kelvin’stheorem on the ground that solutions of the Euler equations in general arenot smooth, even if data are.

7.3 Potential Flow around a Circular Cylinder

To understand how a wing generates lift and drag it is instructive to firstconsider the corresponding problem for a circular cylinder, which we canview as a wing with circular cross-section. Of course you cannot fly withsuch a wing, but a wing is similar to a half cylinder which can be analyzedstarting with a full cylinder.



7.3 Potential Flow around a Circular Cylinder 39

FIGURE 7.2. First page of Euler’s General Principles concerning the Motion of

Fluids from 1757 [80].




We start considering potential flow around a circular cylinder of unitradius with axis along the x3-axis in three-dimensional space with coordi-nates (x1, x2, x3), assuming the flow velocity is (1, 0, 0) at infinity, see Fig.14.1 showing a section of through the cylinder with the flow horisontal fromleft to right. We can equally well think of the cylinder moving transversallythrough a fluid at rest. Potential flow around the cylinder is constant in

FIGURE 7.3. Potential flow past a circular cylinder: streamlines and fluid speed(left) and pressure (right) in a (x1, x2)-plane with horisontal x1-axis in the flowdirection.

the x3-direction and is symmetric in x1 and x2 with zero drag/lift with theflow velocity given as the gradient of the potential

ϕ(r, θ) = (r +1

r) cos(θ),

in polar coordinates (r, θ) in the (x1, x2)-plane. The corresponding pressure(vanishing at infinity) is determined by Bernouilli’s law as:

p =−

1

2r4+

1

r2cos(2θ).

In its simplicity potential flow is truely remarkable: It is a solution of the Euler equations for inviscid flow with slip boundary condition, whichseparates at the back of cylinder at the line (1, 0, x3), with equally highpressure in the front and the back and low pressure on top and bottom(with the low pressure three times as big as the high pressure), resulting inzero drag/lift. This is d’Alembert’s paradox: All experience indicates thata circular cylinder subject to air flow has substantial drag, but potentialflow has zero drag.

We understand that the high pressure in the back, balancing the highpressure up front, can be seen as pushing the body through the fluid ac-cording to the principle of of motion of Aristotle and da Vinci. We shall



7.4 Non-Separation of Potential Flow 41

discover that there is something which is correct in this view. But the netpush from behind in real flow must be smaller than in potential flow, andso real flow must be different form potential flow in the rear, but how andwhy? We shall see that the correct answer to these questions hide the secretof flight.

7.4 Non-Separation of Potential Flow

Direct computation shows that on the cylinder boundary

∂p

∂n=

U 2

R, (7.5)

where n is the outward unit normal to the boundary, ∂p∂n is the gradient

of the pressure in the unit normal direction or normal pressure gradient into the fluid, U is the flow speed and R = 1 the radius of curvature of the boundary (positive for a concave fluid domain thus positive for thecylinder). The relation (7.5) is Newton’s law expressing that fluid particlesgliding along the boundary must be accellerated in the normal direction by

the normal pressure gradient force in order to follow the curvature of theboundary. More generally, (7.5) is the criterion for non-separation : Fluidparticles will stay close to the boundary as long as (7.5) is satisfied, whileif

∂p

∂n<

U 2

R, (7.6)

then fluid particles will separate away from the boundary tangentially. Inparticular, as we will see below, laminar flow separates on the crest ortop/bottom of the cylinder, since the normal pressure gradient is small ina laminar boundary layer with no-slip boundary condition [70, 41].

We sum up so far: The Euler equations express conservation of mass andmomentum for an inviscid incompressible fluid. Potential flow is smoothand satisfies the Euler equations. D’Alembert’s paradox compares invis-cid potential flow having zero drag/lift with slightly viscous flow havingsubstantial drag/lift. Potential flow has a positive normal pressure gradi-ent preventing separation allowing the pressure to build up on the back topush the cylinder through the fluid without drag. We shall see that this isa bit too optimistic, but only a bit; there is some push also in real (slightlyviscous) flow...







8Robins and the Magnus Effect

The greates part of military projectiles will at the time of their dis-charge acquire a whirling motion by rubbing against the inside of their respective pieces; and this whirling motion will cause them tostrike the air very differently, from what the would do, had they noother but a progressive motion. By this means it will happen, thatthe resistance of the air will not always be directly opposed to theirflight; but will frequently act in a line oblique to their course, andwill thereby force them to deviate from the regular track, they wouldotherwise describe. (Robins in [65])

The English engineer Benjamin Robins (1707-1751), called the father of ballistics, introduced the concept of rifling the bore of guns to improve theaccuracy of projectiles by spinning. In experiments with a whirling armdevice he discovered that a spinning projectile experiences a transverse liftforce, which he recorded in [65] in 1742. Euler translated Robins’ book toGerman, but added a critical remark stating that on mathematical symme-try grounds the lift must be zero, and thus the measured lift must have beenan effect of a non-symmetric projectile resulting from manufacturing irreg-ularities. Recognized as the dominant hydrodynamicist of the eighteenthcentury, Euler far overshadowed Robins, and thus Robins’ finding was nottaken seriously for another century. In 1853 Gustav Magnus (1802-1870) in[62] suggested that the lift of a spinning ball, the so-called Magnus effect ,was real and resulted from a whirlpool of rotating air around a ball creatinga non-symmetric flow pattern with lift, an idea which was later taken upby Kutta and Zhukovsky as the decisive feature of their lift theory basedon circulation.



44 8. Robins and the Magnus Effect

Magnus suggested that, because of the whirlpool, for a topspin ball the airvelocity would be larger below than above and thus result in a downward liftforce because the pressure would be smaller below than above by Bernouilli.Magnus thus claimed to explain why a topspin tennis ball curves down, anda backspin curves up.

We shall see below that this explanation is incorrect: There is no whirlpoolof rotating air around a spinning ball, nor is it any circulation around a

wing. In both cases the lift has a different origin. We shall see that findingthe real cause of the Magnus effect will lead us to an explanation of alsothe lift of a wing.

In 1749 Robins left the center stage of England, when he was appointedthe engineer-general of the East India Company to improve the fortifi-cations at St. David, Madras, where he died of fever at an early age of forty-four.

FIGURE 8.1. Built in 1930 (USA), the 921-V is reported to have been flownat least once - ending it’s short carreer with a crash landing. Three cylinderswith disks performing as winglets driven by a separate engine. Probably the onlyaircraft equipped with cylinder wings which made it into the air...




9Lilienthal and Bird Flight

To invent an airplane is nothing. To build one is something. But tofly is everything. (Lilienthal)

Sacrifices must be made! (Lilienthal near death after breaking hisspine in an airplane crash in a glider of his design).

No one can realize how substantial the air is, until he feels its support-ing power beneath him. It inspires confidence at once. (Lilienthal)

We returned home, after these experiments, with the conviction thatsailing flight was not the exclusive prerogative of birds. (Lilienthal,1874)

Otto Lilienthal gives in Bird Flight as a Basis of Aviation [25] the fol-lowing description of bird flight:

• From all the foregoing results it appears obvious that in order to dis-cover the principles which facilitate flight, and to eventually enable man to fly, we must take the bird for our model. A specially suitable species of birds to act as our model is the sea-gull.

• How does the gull fly? At the very first glance we notice that the slen-der, slightly curved wings execute a peculiar motion, in so far as only the wing-tips move appreciably up and down, whilst the broader arm-portions near the body take little part in this movement, a condition of things which is illustrated in Fig. 76.

• May we not assume that the comparatively motionless parts of the wings enable the gull to sail along, whilst the tips, consisting of easily



46 9. Lilienthal and Bird Flight

FIGURE 9.1. Lilienthal’s analysis of a stork in flight.

rotating feathers, serve to compensate for the loss of forward velocity ? It is unmistakable that the wide Portion of the wing close to the body, which does little work and has little movement, is intended for sustaining, whilst the narrower tips, with their much greater ampli-tude of movement, have to furnish the tractive power necessary tocompensate for the resistance of the bird’s body and for any possible restraining component.

• This being conceded, we are forced to consider the flying apparatus of the bird as a most ingenious and perfect mechanism, which has its fulcrum in the shoulder joint, which moves up and down, and by virtue of its articulation permits of increased lift or fall as well as of rotation of the light tips.

• The arm portion of the wing is heavy, containing bones, muscles ,and tendons, and therefore opposes considerable inertia to any rapid movement. But it is well fitted for supporting, because being close to the body, the air pressure upon it acts on a short lever arm, and the bending strain is therefore less severe on the wing. The tip is very light, consisting of feathers only, and can be lifted and depressed in rapid succession. If the air pressure produced by it increased in proportion to the greater amplitude of movement, it would require a large amount of work; and would also unduly strain the wings; we therefore conclude that the real function of the wing-tips is not somuch the generation of a great lifting effect, but rather the production of a smaller, but tractive effect directed forward.



9. Lilienthal and Bird Flight 47

• In fact, actual observation leaves no doubt on this point. It is only necessary to watch the gull during sunshine, and from the light effects we tan distinctly perceive the changing inclination of the wing-tips, as shown in Figs. 77 and 78, which refer to the upstroke and downstroke of the wings respectively . The gull , flying away from us, presents at the upstroke, Fig. 77, the upper side of its wings strongly illuminated by the sun, whilst during the downstroke (Fig. 78) we have tlie shaded

camber presented to us from the back. The tip evidently ascends with the leading edge raised, and descends with the leading edge depressed,both phases resulting in a tractive effect.

Da Vinci had made similar observations in Codex on Bird Flight :

• Those feathers which are farthest from their fastening will be the most flexible; then the tops of the feathers of the wings will be always higher than their origins, so that we may with reason say, that the bones of the wings will be lower in the lowering of the wings than any other part of the wings, and in the raising these bones of the wing will always be higher than any other part of such a wing. Because the heaviest part always makes itself the guide of the movement .

• The kite and other birds which beat their wings little, go seeking the course of the wind, and when the wind prevails on high then they will be seen at a great height, and if it prevails low they will hold themselves low.

• When the wind does not prevail in the air, then the kite beats its wings several times in its flight in such a way that it raises itself high and acquires a start, with which start, descending afterwards a little, it goes a long way without beating its wings, and when it is descended it does the same thing over again, and so it does successively, and this descent without flapping the wings serves it as a means of resting itself in the air after the aforesaid beating of the wings.

• When a bird which is in equilibrium throws the centre of resistance of the wings behind the centre of gravity, then such a bird will descend with its head down. This bird which finds itself in equilibrium shall have the centre of resistance of the wings more forward than the bird’s centre of gravity, then such a bird will fall with its tail turned to the earth.

• When the bird is in the position and wishes to rise it will raise its shoulders and the air will press between its sides and the point of the wings so that it will be condensed and will give the bird the movement toward the ascent and will produce a momentum in the air, which momentum of the air will by its condensation push the bird up.



48 9. Lilienthal and Bird Flight

The observations of the wingbeat cycle by da Vinci and Lilienthal daVinci can be summarized as follows:

• a forward downstroke with the wing increasingly twisted towards thetip with the leading edge down,

• a backward upstroke with the wing twisted the other way with theleading edge up.

Below we shall analyze the lift and drag generated at different momentsof the wingbeat cycle, and thus give a scientific explanation of the secretof bird flight. We compare with with the lack of a scientific theory of birdflight according to state-of-the-art [17]:

• Always there have been several different versions of the flapping flight theory. They all exist in parallel and their specifications are widely distributed. Calculating the balance of forces even of a straight and merely slowly flapping wing remained difficult to the present day. In general, it is only possible in a simplified way. Furthermore, the known drives mechanism and especially wing designs leave a lot to be desired.In every respect ornithopters are still standing at the beginning of their

design development .

FIGURE 9.2. Lilienthal getting ready to simulate a stork in flight.




10Wilbur and Orwille Wright

It is possible to fly without motors, but not without knowledge andskill. (Wilbur Wright)

The desire to fly is an idea handed down to us by our ancestorswho...looked enviously on the birds soaring freely through space...onthe infinite highway of the air. (Wilbur Wright)

The natural function of the wing is to soar upwards and carry thatwhich is heavy up to the place where dwells the race of gods. Morethan any other thing that pertains to the body it partakes of thenature of the divine. (Plato in Phaedrus )

Sometimes, flying feels too godlike to be attained by man. Some-times, the world from above seems too beautiful, too wonderful, toodistant for human eyes to see . . . (Charles Lindbergh in The Spirit

of St. Louis )

More than anything else the sensation is one of perfect peace mingledwith an excitement that strains every nerve to the utmost, if you canconceive of such a combination. (Wilbur Wright)

The exhilaration of flying is too keen, the pleasure too great, for itto be neglected as a sport. (Orwille Wright)

The first successful powered piloted controled flight was performed by thebrothers Orwille and Wilbur Wright on December 17 1903 on the windyfields of Kitty Hawk, North Carolina, with Orwille winning the bet to bethe pilot of the Flyer and Wilbur watching on ground, see Fig 10.1. In thewords of the Wright brothers from Century Magazine, September 1908:



50 10. Wilbur and Orwille Wright

• The flight lasted only twelve seconds, a flight very modest compared with that of birds, but it was, nevertheless, the first in the history of the world in which a machine carrying a man had raised itself by its own power into the air in free flight, had sailed forward on a level course without reduction of speed, and had finally landed without being wrecked. The second and third flights were a little longer, and the fourth lasted fifty-nine seconds, covering a distance of 852 feet

over the ground against a twenty-mile wind.

FIGURE 10.1. Orwille Wright (1871-1948) and Wilbur Wright (1867-1912) andthe lift-off at Kitty Hawk, North Carolina, the 17th December 1903.

The work preceeding the success was described by Wilbur Wright in anaddress to the Western Society of Engineers in 1901 entitled Some Aero-nautical Experiments :

•The difficulties which obstruct the pathway to success in flying-machine construction are of three general classes: (1) Those which relate tothe construction of the sustaining wings; (2) those which relate to the generation and application of the power required to drive the machine through the air; (3) those relating to the balancing and steering of the machine after it is actually in flight. Of these difficulties two are already to a certain extent solved. Men already know how to construct wings or aeroplanes which, when driven through the air at sufficient speed, will not only sustain the weight of the wings themselves, but also that of the engine and of the engineer as well. Men also know how to build engines and screws of sufficient lightness and power todrive these planes at sustaining speed. As long ago as 1884 a ma-chine weighing 8,000 pounds demonstrated its power both to lift itself



10. Wilbur and Orwille Wright 51

from the ground and to maintain a speed of from 30 to 40 miles per hour, but failed of success owing to the inability to balance and steer it properly. This inability to balance and steer still confronts students of the flying problem, although nearly eight years have passed. When this one feature has been worked out, the age of flying machines will have arrived, for all other difficulties are of minor importance.

•The person who merely watches the flight of a bird gathers the im-pression that the bird has nothing to think of but the flapping of its wings. As a matter of fact this is a very small part of its mental labor.To even mention all the things the bird must constantly keep in mind in order to fly securely through the air would take a considerable part of the evening. If I take this piece of paper, and after placing it paral-lel with the ground, quickly let it fall, it will not settle steadily down as a staid, sensible piece of paper ought to do, but it insists on con-travening every recognized rule of decorum, turning over and darting hither and thither in the most erratic manner, much after the style of an untrained horse. Yet this is the style of steed that men must learn to manage before flying can become an everyday sport. The bird has learned this art of equilibrium, and learned it so thoroughly that

its skill is not apparent to our sight. We only learn to appreciate it when we try to imitate it. Now, there are two ways of learning to ride a fractious horse: One is to get on him and learn by actual practice how each motion and trick may be best met; the other is to sit on a

fence and watch the beast a while, and then retire to the house and at leisure figure out the best way of overcoming his jumps and kicks. The latter system is the safest, but the former, on the whole, turns out the larger proportion of good riders. It is very much the same in learning to ride a flying machine; if you are looking for perfect safety, you will do well to sit on a fence and watch the birds; but if you really wish to learn, you must mount a machine and become acquainted with its tricks by actual trial.

•Herr Otto Lilienthal seems to have been the first man who really com-prehended that balancing was the first instead of the last of the great problems in connection with human flight. He began where others left off, and thus saved the many thousands of dollars that it had thereto-

fore been customary to spend in building and fitting expensive en-gines to machines which were uncontrollable when tried. He built a pair of wings of a size suitable to sustain his own weight, and made use of gravity as his motor. This motor not only cost him nothing to begin with, but it required no expensive fuel while in operation,and never had to be sent to the shop for repairs. It had one serious drawback, however, in that it always insisted on fixing the conditions under which it would work. These were, that the man should first be-take himself and machine to the top of a hill and fly with a downward




as well as a forward motion. Unless these conditions were complied with, gravity served no better than a balky horse – it would not work at all. Although Lilienthal must have thought the conditions were rather hard, he nevertheless accepted them till something better should turn up; and in this manner he made some two thousand flights, in a few cases landing at a point more than 1,000 feet distant from his place of starting. Other men, no doubt, long before had thought of trying

such a plan. Lilienthal not only thought, but acted; and in so doing probably made the greatest contribution to the solution of the flying problem that has ever been made by any one man. He demonstrated the feasibility of actual practice in the air, without which success is impossible. Herr Lilienthal was followed by Mr. Pilcher, a young En-glish engineer, and by Mr. Chanute, a distinguished member of the society I now address. A few others have built gliding machines, but nearly all that is of real value is due to the experiments conducted under the direction of the three men just mentioned.

The Wrights built The Flyer in 1903 using spruce and ash covered withmuslin, with wings designed with a 1-in-20 camber. Since they could notfind a suitable automobile engine for the task, they commissioned their

employee Charlie Taylor to build a new design from scratch. A sprocketchain drive, borrowing from bicycle technology, powered the twin pro-pellers, which were also made by hand. The Flyer was a canard biplaneconfiguration. As with the gliders, the pilot flew lying on his stomach onthe lower wing with his head toward the front of the craft in an effort to re-duce drag, and steered by moving a cradle attached to his hips. The cradlepulled wires which warped the wings and turned the rudder simultaneouslyfor lateral control, while a forward horisontal stabilizer (forward canard)was controled by the left hand.

To sum up, the Wright brothers were the first to solve the combinedproblem of (1) generation of lift by (sufficiently large) wings, (2) generationof thrust by a propeller powerd by a (sufficiently light) combustion engineand (3) horisontal control of balance under different speeds and angles of

attack as well as lateral control. The data of the Flyer were:

• wingspan: 12.3 m

• wing area: 47 m2

• length: 6.4 m

• height: 2.8 m

• weight (empty): 274 kg

• engine: gasoline 12 hp



10. Wilbur and Orwille Wright 53

At a lift/drag ratio of 10 the drag would be about 35 kp to carry a totalweight of 350 kp, which at a speed of 10 m/s would require about 5 effectivehp.

The Flyer had a forward canard for horisontal control, like the modernSwedish jet fighter JAS Gripen , which is an unstable configuration requiringcareful control to fly, but allowing quick turns. The Wrights later replacedthe canard with the conventional aft tail to improve stability. The stability

of an airplane is similar to that of a boat, with the important design featurebeing the relative position of the center of gravity and the center of theforces from the fluid (center of buoyancy for a boat), with the center of gravity ahead (below) giving stability.







11Lift by Circulation

Man must rise above the Earthto the top of the atmosphere andbeyondfor only thus will he fully understand the world in which helives. (Socrates)

A single lifetime, even though entirely devoted to the sky, would notbe enough for the study of so vast a subject. A time will come whenour descendants will be amazed that we did not know things thatare so plain to them. (Seneca)

All the perplexities, confusion and distress in America arise, notfrom defects in their Constitution or Confederation, not from wantof honor or virtue, so much as from the downright ignorance of thenature of coin, credit and circulation. (John Adams)

If you would be a real seeker after truth, it is necessary that at leastonce in your life you doubt, as far as possible, all things. (Descartes)

11.1 Lanchester

Frederick Lanchester , (1868-1946) was an English polymath and engineerwho made important contributions to automotive engineering, aerodynam-ics and co-invented the field of operations research. He was also a pioneerBritish motor car builder, a hobby he eventually turned into a successfulcar company, and is considered one of the big three English car engineers,the others being Harry Ricardo and Henry Royce.

Lanchester began to study aeronautics seriously in 1892, eleven yearsbefore the first successful powered flight. Whilst crossing the Atlantic on



56 11. Lift by Circulation

a trip to the United States, Lanchester studied the flight of herring gulls,seeing how they were able to use motionless wings to catch up-currentsof air. He took measurements of various birds to see how the centre of gravity compared with the centre of support. As a result of his deliberations,Lanchester, eventually formulated his circulation theory, which still servesas the basis of modern lift theory. In 1894 he tested his theory on a numberof models. In 1897 he presented a paper entitled The soaring of birds and

the possibilities of mechanical flight to the Physical Society, but it wasrejected, being too advanced for its time. Lanchester realised that poweredflight required an engine with a far higher power to weight ratio than anyexisting engine. He proposed to design and build such an engine, but wasadvised that no one would take him seriously [1].

Stimulated by Lilienthal’s successful flights and his widely spread bookBird Flight as the Basis of Aviation from 1899, the mathematician Martin Kutta (1867-1944) in his thesis presented in 1902 modified the erronousclassical potential flow solution by including a new term corresponding toa rotating flow around the wing with the strength of the vortex determinedso that the combined flow velocity became zero at the trailing edge of the wing. This Kutta condition reflected the observation of Lilienthal thatthe flow should come off the wing smoothly, at least for small angles of

attack. The strength of the vortex was equal to the circulation around thewing of the velocity, which was also equal to the lift. Kutta could this waypredict the lift of various wings with a precision of practical interest. Butthe calculation assumed the flow to be fully two-dimensional and the wingsto be very long and became inaccurate for shorter wings and large anglesof attack.

11.2 Kutta

Stimulated by Lilienthal’s successful flights and his widely spread bookBird Flight as the Basis of Aviation from 1899, the mathematician Martin Kutta (1867-1944) in his thesis presented in 1902 modified the erronousclassical potential flow solution by including a new term corresponding toa rotating flow around the wing with the strength of the vortex determinedso that the combined flow velocity became zero at the trailing edge of the wing. This Kutta condition reflected the observation of Lilienthal thatthe flow should come off the wing smoothly, at least for small angles of attack. The strength of the vortex was equal to the circulation around thewing of the velocity, which was also equal to the lift. Kutta could this waypredict the lift of various wings with a precision of practical interest. Butthe calculation assumed the flow to be fully two-dimensional and the wingsto be very long and became inaccurate for shorter wings and large anglesof attack.



11.3 Zhukovsky 57

FIGURE 11.1. Generation of lift according to Kutta-Zhukovsky theory, as ex-tended by Prandtl by connecting the circulation around the wing to the startingvortex by so-called trailing vortices from the wing tips. The circulation aroundthe wing and starting vortices are unphysical, while the trailing vortices from thewing tips are real and often can be observed by condensation in damp weather.

11.3 Zhukovsky

The mathematician Nikolai Zhukovsky (1847-1921), called the father of Russian aviation , in 1906 independently derived the same mathematics forcomputing lift as Kutta, after having observed several of Lilienthal’s flights,which he presented before the Society of Friends of the Natural Sciences inMoscow as:

• The most important invention of recent years in the area of aviation is the flying machine of the German engineer Otto Lilienthal .

Zhukovsky also purchased one of the eight gliders which Lilienthal sold tomembers of the public.

Kutta and Zhukovsky thus could modify the mathemathical potentialtheory of lift of a wing to give reasonable results, but of course could notgive anything but a very heuristic justification of their Kutta-Zhukovskycondition for the velocity at the trailing edge of the wing, and could nottreat realistic wings in three dimensions. Further, their modified potentialsolutions are not turbulent, and as we will see below, their calculationswere merely happy coincidences (knowing ahead the correct answer to ob-tain) without connection to the physics of real turbulent flow: There is nocirculation around a wing, and connecting lift to circulation is unphysical.

It is remarkable that 400 years passed between Leonardo da Vinci’s inves-tigations and the largely similar ones by Lilienthal. Why did it take so longtime from almost success to success? What was the role of the misleading



58 11. Lift by Circulation

mathematics of Newton and d’Alembert, still influencing the judgement of e.g. Lord Kelvin in the late 19th century?

FIGURE 11.2. Hurricane with physical circulation.




12Prandtl and Boundary Layers

On an increase of pressure, while the free fluid transforms part of its kinetic energy into potential energy, the transition layers instead,having lost a part of their kinetic energy (due to friction), have nolonger a sufficient quantity to enable them to enter a field of higherpressure, and therefore turn aside from it. (Prandtl)

The modern world of aerodynamics and fluid dynamics is still dom-inated by Prandtls idea. By every right, his boundary-layer conceptwas worthy of the Nobel Prize. He never received it, however; somesay the Nobel Com- mittee was reluctant to award the prize foraccomplish- ments in classical physics...(John D. Anderson in [2])

No flying machine will ever fly from New York to Paris. (OrvilleWright)

12.1 Separation

The generation of lift and drag of a wing is closely connected to problemof separation in fluid mechanics: As a body moves through a slightly vis-cous fluid initially at rest, like a car or airplane moving through still air,or equivalently as a fluid flows around a body at rest, fluid particles aredeviated by the body in a contracting flow switching to an expanding flow at a crest and eventually separate away from the body somewhere in therear, at or after the crest. In the front there is typically a stagnation point ,where the fluid velocity vanishes allowing laminar attachment at stagnation to the boundary. On the other hand the fluid mechanics of the turbulent



60 12. Prandtl and Boundary Layers

separation occuring in the rear in slightly viscous flow, which creates drag and lift forces, appears to be largely unknown, despite its crucial impor-tance in many applications, including flying and sailing. The basic studyconcerns separation from a convex body like a sphere, circular cylinder,wing, car or boat hull.

FIGURE 12.1. Prandtl’s idea of laminar viscous separation with no-slip causedby an adverse pressure gradient, which is does not describe the turbulent slightlyviscous separation with slip in the flow of air around a wing.

12.2 Boundary Layers

In 1904 the young German physicist Ludwig Prandtl (1875-1953) suggestedin a 10 page sketchy presentation entitled Motion of Fluids with Very Lit-tle Viscosity [67] at the Third International Congress of Mathematics inHeidelberg, that the substantial drag of a bluff body moving through afluid with very small viscosity (such as air or water), possibly could arisefrom the presence of of a thin laminar boundary layer , where the fluid ve-locity radpidly changes from its free-stream value to zero on the boundarycorresponding to a no-slip boundary condition , causing the flow to sepa-

rate from the boundary brought to stagnation under an adverse pressure gradient (negative pressure gradient in the flow direction), to form a low-pressure wake behind the body. But the acceptance of Prandtl’s ideas wasslow [2]:

• Prandtls idea (about the boundary layer) went virtually unnoticed by anybody outside of G¨ ottingen... The fifth and sixth editions of Lambs classic text Hydrodynamics published in 1924, devoted only one para-graph to the boundary-layer concept.

However, Prandtl had two forceful students, Theodore von Karman (whoemigrated to the US in 1930) and Hermann Schlichting (who stayed inGermany), who crowned Prandtl as the father of modern fluid mechan-






• The form drag which does not exist in frictionless subsonic flow, is due to the fact that the presence of the boundary layer modifies the pressure distribution on the body as compared with ideal flow, but its computation is very difficult.

• The origin of pressure drag lies in the fact that the boundary layer

exerts a displacement action on the external stream. This modifies somewhat the pressure distribution on the body surface. In contrast with potential flow (d’Alembert’s paradox), the resultant of this pres-sure distribution modified by friction no longer vanishes but produces a preessure drag which must be added to skin friction. The two to-gether give form drag.

• In the case of the most important fluids, namely water and air, the viscosity is very small and, consequently, the forces due to viscous

friction are, generally speaking, very small compared with the remain-ing forces (gravity and pressure forces). For this resaon it was very difficult to comprehend that very small frictional forces omitted in classical (inviscid) theory influenced the motion of a fluid to so large extent .

Prandtl described the difficulties himself in Applied Hydro- and Aerome-chanics from 1934:

• Only in the case where the “boundary layer” formed under the influ-ence of the viscosity remains in contact with the body, can an approx-imation of the actual fluid motion by means of a theory in terms of the ideal frictionsless fluid be attempted, whereas in all cases where the boundary leaves the body, a theoretical treatment leads to results which do not coincide at all with experiment. And it had to be con-

fessed that the latter case occurs most frequently.

In a nutshell, these quotes present much of the essence of modern fluid me-chanics propagated in standard books and courses in fluid mechanics: Dragand lift in slightly viscous flow are claimed to arise from separation in a thinviscous laminar boundary layer brought to stagnation with reversed flowdue to an adverse pressure gradient. On the other hand, both Prandtl andSchlichting admit that this standard scenario does not describe turbulentflow, always arising in slightly viscous flow, but persists that “it is never-theless useful to consider laminar flow because it is much more amenable tomathematical treatment”. However, turbulent and laminar flow have differ-ent properties, and drawing conclusions about turbulent flow from studiesof laminar flow can be grossly misleading.






FIGURE 12.2. Horisontal boundary layers with streaks of streamwise vorticity(top view above), and turbulent boundary layer (side view below).



Part II

Preparing for Takeoff

65



66




13Navier-Stokes Equations

Waves follow our boat as we meander across the lake, and turbu-lent air currents follow our flight in a modern jet. Mathematiciansand physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an under-standing of solutions to the Navier-Stokes equations. Although theseequations were written down in the 19th Century, our understand-ing of them remains minimal. The challenge is to make substantialprogress toward a mathematical theory which will unlock the secretshidden in the Navier-Stokes equations. (Clay Mathematics InstituteMillennium Problem [5])

Because things are the way they are, things will not stay the waythey are. (Bertold Brecht)

Everything is self-evident. (Descartes)

13.1 Small or Very Small Viscosity

To uncover the secret of flight we first have to resolve d’Alembert’s paradox.State-of-the art concerning its resolution is expressed on [53] as follows:

• The general view in the fluid mechanics community is that, from a practical point of view, the paradox is solved along the lines suggested by Prandtl. A formal mathematical proof is missing, and difficult toprovide, as in so many other fluid-flow problems modelled through the NavierStokes equations...The viscous effects in the thin boundary lay-ers remain also at very high Reynolds numbers they result in friction



68 13. Navier-Stokes Equations

drag for streamlined objects, and for bluff bodies the additional result is flow separation and a low-pressure wake behind the object, leading to form drag.

The suggestion is apparently that substantial drag results from the pres-ence of a thin boundary layer even for arbitarily small viscosity, that is asubstantial effect from a vanishingly small cause [88]:

• ...great efforts have been made during the last hundred or so years to propose alternate theories and to explain how a vanishingly small

frictional force in the fluid can nevertheless have a significant effect on the flow properties.

But to claim that something substantial can result from virtually nothing,is very cumbersome from a scientific point of view, since it requires ac-cess to an infinitely precise theory for justification, which is not available.Moreover, d’Alemberts paradox concerns a contradiction between math-ematical prediction and practical observation and can only be solved byunderstanding the mathematics and the reason mathematics leads astray.It is precisely a “mathematical proof” which is needed, which the fluid me-chanics community apparently acknowledges “is missing”. The trouble is

that mathematics predicts zero drag, not that observation shows substan-tial drag.

If it is impossible to verify Prandtl’s theory, it can well be possible todisprove it: It suffices to remove the infinitely small cause (the boundarylayer) and still observe the effect (substantial drag). This is what we willdo. But we will not remove the viscosity in the interior of the flow whichwill create turbulent dissipation manifested in drag.

To resolve d’Alembert’s paradox we have to expand the scope from theincompressible Euler equations (7.2) for an ideal fluid with zero viscosityto the incompressible Navier-Stokes equations for a real fluid with smallor very small viscosity. For air the kinematic viscosity (normalized to unitdensity) is about 10−5 and for water about 10−6. Normalizing also withrespect to velocity and length scale, the viscosity is represented by theinverse of the Reynolds number , which in subsonic flight ranges from 105 formedium-size birds over 107 for a smaller airplane up to 109 for a jumbojet.We are thus considering normalized viscosities in the range from 10−5 to10−9 to be compared with density, velocity and length scale of unit size.We understand that 10−5 is small compared to 1, and that 10−9 comparedto 1 is very small .

The Navier-Stokes equations represent a more realistic model of physicsthan the Euler equations, and also are more meaningful from mathematicalpoint of view. By inherent instability, solutions of the Euler equations, suchas potential flow, show blowup into turbulence [37] and then cease to exist,while solutions of the Navier-Stokes equations with positive viscosity, seemto exist for all time, even if solutions are turbulent and very complex.



13.2 The Squeeze 69

13.2 The Squeeze

Massive evidence indicates that the Navier-Stokes equations constitute anaccurate mathematical model of aerodynamics, but according to state-of-the-art there is a serious crux: Solutions are turbulent and can only bedetermined by computation, but resolving thin boundary layers seem torequire so many mesh-points that no thinkable computer would suffice.

According to state-of-the-art confessing to Prandtl [59], 1016 mesh-pointswould be required to simulate the air flow around a jumbojet by ab ini-tio simulation solving the Navier-Stokes equations without any turbulencemodel. This seems to put Computational Fluid Dynamic CFD into a hope-less squeeze: Either invent turbulence models (which has shown to be im-possible) or resolve very thin boundary layers (which is impossible).

13.3 The Way Out

But there is a way out of the squeeze: We shall see that we can, using mil-lions of mesh points instead of impossible quadrillions, compute turbulent

solutions of the Navier-Stokes equations which carry correct informationof mean-values such as lift, drag and twisting moment of a wing or entireairplane, without resolving thin boundary layers. This is made possiblebecause

(i) we use skin friction force boundary conditions for tangential stressesinstead of no-slip boundary conditions for tangential velocities,

(ii) the skin friction is small from a turbulent boundary layer of a fluidwith very small viscosity,

(iii) it is not necessary to resolve the turbulent features in the interior of the flow to physical scales.

We shall below give evidence that (i)-(iii) can take CFD out of its presentdeadlock, through ab initio computational simulation of turbulent slightlyviscous flow, without resort to turbulence models.

13.4 Navier-Stokes with Force BC

The Navier-Stokes equations for an incompressible fluid of unit density withsmall viscosity ν > 0 and small skin friction β ≥ 0 filling a volume Ω in R3

surrounding a solid body with boundary Γ over a time interval I = [0, T ],read as follows: Find the velocity u = (u1, u2, u3) and pressure p depending




on (x, t) ∈ Ω ∪ Γ × I , such that

u + (u · ∇)u + ∇ p −∇ · σ = f in Ω × I,∇ · u = 0 in Ω × I,

un = g on Γ × I,σs = βus on Γ × I,

u(

·, 0) = u0 in Ω,

(13.1)

where un is the fluid velocity normal to Γ, us is the tangential velocity, σ =2νǫ(u) is the viscous (shear) stress with ǫ(u) the usual velocity strain, σs isthe tangential stress, f is a given volume force, g is a given inflow/outflowvelocity with g = 0 on a non-penetrable boundary, and u0 is a given initialcondition. We notice the skin friction boundary condition coupling thetangential stress σs to the tangential velocity us with the friction coefficientβ with β = 0 for slip, and β >> 1 for no-slip. We note that β is related tothe standard skin friction coeffieient cf = 2τ

U 2 with τ the tangential stress

per unit area is, by the relation β = U 2 cf . In particular, β tends to zero

with cf (if U stays bounded).Prandtl insisted on using a no-slip velocity boundary condition with

us = 0 on Γ, because his resolution of d’Alembert’s paradox hinged ondiscriminating potential flow by this condition. On the oher hand, withour new resolution of d’Alembert’s paradox, relying instead on instabilityof potential flow, we are free to choose instead a friction force boundarycondition, if data is available. Now, experiments show that the skin frictioncoefficient decreases with increasing Reynolds number Re as cf ≈ 0.07 ∼Re−0.2, so that cf ≈ 0.0005 for Re = 1010 and cf ≈ 0.007 for Re = 105.Accordingly we model a turbulent boundary layer by friction boundarycondition with a friction parameter β ≈ 0.03U Re−0.2.

We have initiated benchmark computations for tabulating values of β (or σs) for different values of Re by solving the Navier-Stokes equationswith no-slip for simple geometries such as a flat plate, and more generallyfor different values of ν , U and length scale, since the dependence seems

to be more complex than simply through the Reynolds number. Early re-sults are reported in [39] with σs ≈ 0.005 for ν ≈ 10−4 and U = 1, withcorresponding velocity strain in the boundary layer 104σs ≈ 50 indicatingthat the smallest radius of curvature without separation in this case couldbe expected to be about 0.02.

13.5 Exponential Instability

Subtracting the NS equations with β = 0 for two solutions (u,p,σ) and(u, ¯ p, σ) with corresponding (slightly) different data, we obtain the following



13.6 Energy Estimate with Turbulent Dissipation 71

linearized equation for the difference (v , q , τ ) ≡ (u − u, p− ¯ p, σ − σ) with :

v + (u · ∇)v + (v · ∇)u + ∇q −∇ · τ = f − f in Ω × I,∇ · v = 0 in Ω × I,v · n = g − g on Γ × I,

τ s = 0 on Γ × I,v(

·, 0) = u0

−u0 in Ω,

(13.2)Formally, with u and u given, this is a linear convection-reaction-diffusionproblem for (v , q , τ ) with the reaction term given by the 3 × 3 matrix ∇ubeing the main term of concern for stability. By the incompressiblity, thetrace of ∇u is zero, which shows that in general ∇u has eigenvalues withreal value of both signs, of the size of |∇u| (with | · | som matrix norm),thus with at least one exponentially unstable eigenvalue.

Accordingly, we expect local exponential perturbation growth of sizeexp(|∇u|t) of a solution (u,p,σ), in particular we expect a potential so-lution to be illposed. This is seen in G2 solutions with slip initiated aspotential flow, which subject to residual perturbations of mesh size h, inlog(1/h) time develop into turbulent solutions. We give computational evi-dence that these turbulent solutions are wellposed, which we rationalize bycancellation effects in the linearized problem, which has rapidly oscillatingcoefficients when linearized at a turbulent solution.

Formally applying the curl operator ∇× to the momentum equation of (13.1), with ν = β = 0 for simplicity, we obtain the vorticity equation

ω + (u · ∇)ω − (ω · ∇)u = ∇× f in Ω, (13.3)

which is a convection-reaction equation in the vorticity ω = ∇×u with coef-ficients depending on u, of the same form as the linearized equation (15.5),with similar properties of exponential perturbation growth exp(|∇u|t) re-ferred to as vortex stretching . Kelvin’s theorem formally follows from thisequation assuming the initial vorticity is zero and ∇× f = 0 (and g = 0),

but exponential perturbation growth makes this conclusion physically in-correct: We will see below that large vorticity can grow out of vanishingviscosity, and that this effect is part of the secret of flight...

13.6 Energy Estimate with Turbulent Dissipation

The standard energy estimate for (13.1) is obtained by multiplying themomentum equation

u + (u · ∇)u + ∇ p −∇ · σ − f = 0,




with u and integrating in space and time, to get in the case f = 0 andg = 0, t

0

Ω

Rν (u, p) · u dxdt = Dν (u; t) + Bβ(u; t) (13.4)

whereRν (u, p) = u + (u · ∇)u + ∇ p

is the Euler residual for a given solution (u, p) with ν > 0,

Dν (u; t) =

t0

Ω

ν |ǫ(u(t, x))|2dxdt

is the internal turbulent viscous dissipation , and

Bβ(u; t) =

t0

Γ

β |us(t, x)|2dxdt

is the boundary turbulent viscous dissipation , from which follows by stan-dard manipulations of the left hand side of (13.4),

K ν (u; t) + Dν (u; t) + Bβ(u; t) = K (u0), t > 0, (13.5)

where

K ν (u; t) =1

2

Ω

|u(t, x)|2dx.

This estimate shows a balance of the kinetic energy K (u; t) and the turbu-lent viscous dissipation Dν (u; t) + Bβ(u; t), with any loss in kinetic energyappearing as viscous dissipation, and vice versa. In particular,

Dν (u; t) + Bβ(u; t) ≤ K (0)

and thus the viscous dissipation is bounded (if f = 0 and g = 0).Turbulent solutions of (13.1) are characterized by substantial internal

turbulent dissipation , that is (for t bounded away from zero),

D(t) ≡ limν →0 D(uν ; t) >> 0, (13.6)

which is Kolmogorov’s conjecture [84]. On the other hand, the boundarydissipation decreases with decreasing friction

limβ→0

Bβ(u; t) = 0, (13.7)

since β ∼ ν 0.2 tends to zero with the viscosity ν and the tangential velocityus approaches the (bounded) free-stream velocity. Kolmogorov’s conjecture(13.6) is consistent with

∇u0 ∼ 1√ ν

, Rν (u, p)0 ∼ 1√ ν

, (13.8)



13.7 G2 Computational Solution 73

where · 0 denotes the L2(Q)-norm with Q = Ω × I . On the other hand,it follows by standard arguments from (13.5) that

Rν (u, p)−1 ≤√

ν, (13.9)

where · −1 is the norm in L2(I ; H −1(Ω)). Kolmogorov thus conjecturesthat the Euler residual Rν (u, p) for small ν is strongly (in L2) large, whilebeing small weakly (in H −1).

Altogether, we understand that the resolution of d’Alembert’s paradoxof explaining substantial drag from vanishing viscosity, consists of realizingthat the internal turbulent dissipation D can be positive under vanishingviscosity, while the boundary dissipation B will vanish. In contradictionto Prandtl, we conclude that drag does not result from boundary layereffects, but from internal turbulent dissipation, originating from instabilityat separation.

13.7 G2 Computational Solution

We show in [39, 37, 40] that the Navier-Stokes equations (13.1) can besolved by a least squares stabilized finite element referred to as G2 as anacronym for General Galerkin . G2 produces turbulent solutions character-ized by substantial turbulent dissipation from the least squares stabilizationacting as an automatic turbulence model, reflecting that the Euler residualcannot be made small in turbulent regions. G2 has a posteriori error controlbased on duality and shows output uniqueness in mean-values such as liftand drag [39, 34, 33, 38, 31, 32]

We find that G2 with slip is capable of modeling slightly viscous turbu-lent flow with Re > 106 of relevance in many applications in aero/hydrodynamics, including flying, sailing, boating and car racing, with hundredthousands of mesh points in simple geometry and millions in complex geom-etry, while according to state-of-the-art quadrillions is required [43]. Thisis because a friction-force/slip boundary condition can model a turbulent

blundary layer, and interior turbulence does not have to be resolved tophysical scales to capture mean-value outputs [39].The idea of circumventing boundary layer resolution by relaxing no-slip

boundary conditions introduced in [31, 39], was used in [26] in the form of weak satisfaction of no-slip, which however misses the main point of usinga force condition instead of a velocity condition.

An G2 solution (U, P ) on a mesh with local mesh size h(x, t) according to[39], satisfies the following energy estimate (with f = 0, g = 0 and β = 0):

K (U (t)) + Dh(U ; t) = K (u0), (13.10)

where

Dh(U ; t) =

t0

Ω

h|Rh(U, P )|2 dxdt, (13.11)




is an analog of Dν (u; t) with h ∼ ν , where Rh(U, P ) is the Euler residualof (U, P ) We see that the G2 turbulent viscosity Dh(U ; t) arises from pe-nalization of a non-zero Euler residual Rh(U, P ) with the penalty directlyconnecting to the violation (according the theory of criminology). A tur-bulent solution is characterized by substantial dissipation Dh(U ; t) withRh(U, P )0 ∼ h−1/2, and

Rh(U, P )−1 ≤√

h (13.12)in accordance with (13.8) and (13.9).

13.8 Wellposedness

Since Hadamard [82] it is well understood that solving differential equa-tions, such as the Euler equations, perturbations of data have to be takeninto account. If a vanishingly small perturbation can have a major effect ona solution, then the solution is illposed , and in this case the solution may notcarry any meaningful information and thus may be meaningless from bothmathematical and applications points of view. According to Hadamard,

only a wellposed solution, for which small perturbations have small effectson certain solution outputs , can be meaningful.We show that a potential solution is illposed with respect to all outputs,

including drag/lift, and thus explain why the zero-drag prediction of po-tential flow carries no information. We show that computed turbulent G2solutions are wellposed with respect to drag/lift and thus can give valuableinformation.

Although wellposedness in the form of hydrodynamic stability is a keyissue in fluid dynamics literature, an stability analysis of potential solutionsseems to be lacking.

13.9 Wellposedness of Mean-Value Outputs

Let M (v) = Q

vψdxdt be a mean-value output of a velocity v defined

by a smooth weight-function ψ(x, t), and let (u, p) and (U, P ) be two G2-solutions on two meshes with maximal mesh size h. Let (ϕ, θ) be the solu-tion to the dual linearized problem

−ϕ − (u · ∇)ϕ + ∇U ⊤ϕ + ∇θ = ψ in Ω × I,∇ · ϕ = 0 in Ω × I,ϕ · n = g on Γ × I,

ϕ(·, T ) = 0 in Ω,

(13.13)

where ⊤ denotes transpose. Multiplying the first equation by u − U andintegrating by parts, we obtain the following output error representation



13.10 Stability of the Dual Linearized Problem 75

[39, ?]:

M (u)− M (U ) =

Q

(Rh(u, p)− Rh(U, P )) · ϕdxdt (13.14)

where for simplicity the dissipative terms are here omitted, from whichfollows the a posteriori error estimate:

|M (u)− M (U )| ≤ S (Rh(u, p)−1 + Rh(U, P )−1), (13.15)

where the stability factor

S = S (u,U,M ) = S (u, U ) = ϕH 1(Q). (13.16)

In [39] we present a variety of evidence, obtained by computational so-lution of the dual problem, that for global mean-value outputs such asdrag and lift, S << 1/

√ h, while R−1 ∼

√ h, allowing computation of of

drag/lift with a posteriori error control of the output within a tolerance of a few percent.

13.10 Stability of the Dual Linearized ProblemA crude analytical stability analysis of the dual linearized problem (13.13)using Gronwall type estimates, indicates that the dual problem is point-wise exponentially unstable because the reaction coefficient ∇U is locallyvery large This is consistent with massive observation that point-values of turbulent flow are non-unique or unstable.

On the other hand we observe computationally that S is of moderatesize for mean-value outputs of turbulent solutions. We explain in [39] thisremarkable fact as an effect of cancellation from the following two sources:

(i) rapidly oscillating reaction coefficients of turbulent solutions,

(ii) smooth data in the dual problem for mean-value outputs.

For a laminar potential solution there is no cancellation, and therefore noteven mean-values are wellposed.

13.11 Turbulent Flow around a Car

In Fig. 13.1 we show computed turbulent G2 flow around a car with sub-stantial drag in accordance with wind-tunnel experiments. We see a patternof streamwise vorticity forming in the rear wake. We also see surface vor-ticity forming on the hood transversal to the main flow direction. We willdiscover similar features in the flow of air around a wing...




FIGURE 13.1. Velocity of turbulent flow around a car




14Resolution of d’Alembert’s Paradox

How wonderful that we have met with a paradox. Now we have somehope of making progress. (Niels Bohr)

...do steady flow ever occur in Nature, or have we been pursuingfantasy all along? If steady flows do occur, which ones occur? Arethey stable, or will a small perturbation of the flow cause it to driftto another steady solution, or even an unsteady one? The answer tonone of these questions is known. (Marvin Shinbrot in Lectures onFluid Mechanics, 1970)

14.1 Ingredients

We will now present a resolution of d’Alembert’s Paradox [75, 76, 77, 78, 83]comparing observations of substantial drag/lift of a body moving througha slightly viscous fluid such as air and water, with the mathematical pre-diction of zero drag/lift of potential (inviscid) flow.

We present analytical and computational evidence that (i) potential flowcannot be observed because it is illposed or unstable to perturbations, (ii)computed solutions of the Navier-Stokes equations (13.1) with slip bound-ary conditions initiated as potential flow develop into turbulent solutions ,which are wellposed with respect to drag/lift and show substantial drag/lift.

We will in the next chapter identify the basic mechanism of instabil-ity of potential flow as a combined effect of retardation and accelerationat rear separation which generates rolls of streamwise vorticity effectivelyproducing drag, and also lift for a wing. Without any presence of viscous



78 14. Resolution of d’Alembert’s Paradox

boundary layers, we thus obtain substantial drag/lift in accordance withobservations and the conjecture of Birkhoff.

We motivate the slip boundary condition by bench-mark computationsand observations indicating that the skin friction decreases to zero withthe viscosity [?]. These facts are in direct contradiction to Prandtl’s claimthat effects of skin friction on drag/lift remain substantial under vanishingviscosity.

14.2 Stability of Corner Flow

We analyze the stability at rear separation of potential flow around a circu-lar cylinder in the simplest possible setting of potential flow in a halfplanex1 > 0 tangent to the cylinder at the line of separation (0, 0, x3), with ve-locity u(x, t) = (2x1,−2x2, 0) and corresponding pressure p = −2(x2

1 + x22).We check by direct computation that (u, p) solves the Euler equations.

The stability of this flow is governed by the linearized equation (13.2)with u = u, with the crucial coefficient matrix ∇u of the reaction term(v · ∇)u, equal to a diagonal matrix with diagonal (2,−2, 0) (which is thereaction term for potential flow at the line at separation), thus with onepositive (stable) and one negative (unstable) eigenvalue). We thus get astrong indication that potential flow is exponentially unstable at separation,and we are thus prepared to discover some effects of the instability.

14.3 Potential Flow as Navier-Stokes Solution

Potential flow (u, p) is a solution to the Euler equations with zero forcingf = 0 and slip boundary conditions, and can also be seen as a solution of the Navier-Stokes equations for slightly viscous flow with a slip boundarycondition, if we consider

•the viscous term

∇ ·(2νǫ(u)) as a perturbation of the volume force

f = 0,

• the tangential boundary stress σs = 2νǫ(u)s as a perturbation of zerofriction with β = 0.

with both perturbations being small because ν is small and a potential flowvelocity u is smooth. Potential flow can thus be seen as a solution of theNavier-Stokes equations with small force perturbations proportional to theviscosity.

We thus find that the Navier-Stokes equations admit potential solutionswith zero drag/lift, but we cannot observe potential flow in real slightlyviscous flow, which is d’Alembert’s paradox. Something is seriously wrong,and the question is what?



14.4 Turbulent Flow around a Circular Cylinder 79

After a moments thought we understand that we cannot put the blameon the Navier-Stokes equations, since they express Newtons 2nd law andmass conservation, which we have no reason to doubt, and so there must besomething wrong the potential solutions as solutions to the Navier-Stokesequations, and it is not the force perturbations, but instead instability.We have discovered exponential grwoth of perturbations in a model case,and this is what we also see in G2 computations: A G2 solution initiated

as potential flow with zero drag/lift develops over time into a turbulentsolution with substantial drag.

The potential solution is like an inverted pendulum, which cannot beobserved in reality because it is unstable and under infinitesimal pertur-bations turns into a swinging motion. A stationary inverted pendulum isa fictious mathematical solution without physical correspondence becauseit is unstable. You can only observe phenomena which in some sense arestable, and an inverted pendelum or potential flow is not stable in anysense.

We shall find that point-values of turbulent solutions are not stable, butmean-values such as drag and lift turn out be, which make them physicallyobservable and meaningful.

14.4 Turbulent Flow around a Circular Cylinder

We consider the bench-mark problem of flow around a circular cylinder.We initiate a G2 solution as potential flow and find that it develops intoa time-dependent flow with a turbulent wake with streaks of low-pressurestreamwise vorticity generating substantial drag as displayed in Fig. 14.1.The drag coefficient cD is about 1.0 in accordance with experiments in-dicating that for large Reynolds numbers beyond the so-called drag crisisoccuring around Re ≈ 106 with a drop of drag from 1.0 to about 0.5, thedrag climbs back to about 1.0.

We see in Figs.14.3 and ?? rolls of strong streamwise vorticity with low

inside pressure creating drag. In the next chapter we analyze in detailthe main mechanism of instability generating these rolls, with a similarmechanism generating both drag and lift of a wing.




FIGURE 14.1. Potential (above) and turbulent G2 flow past a circular cylinder:streamlines and fluid speed (left) and pressure (right) in a (x1, x2)-plane withhorisontal x1-axis in the flow direction.



14.4 Turbulent Flow around a Circular Cylinder 81

FIGURE 14.2. Computational solution of the Euler equations for flow past acircular cylinder; colormap of the pressure (left) and streamlines together with acolormap of the magnitude of the velocity (right) (t = 11.0)

FIGURE 14.3. Levels surfaces of strong streamwise vorticity in EG2 solution atdifferent times seen from above (in the x1x3-plane).







15Flow Separation

The fear of making permanent commitments can change the mutuallove of husband and wife into two loves of self - two loves existingside by side, until they end in separation. (Pope John Paul II)

15.1 Lift and Drag from Separation

We shall now discover that the generation of both lift and drag of a bodymoving through air, such as a wing, is closely related to the dynamics of

flow separation . We know that the flow around the body attaches some-where in the front, typically around a point of stagnation , where the flowvelocity is zero, and separates somewhere somehow in the rear. In manycases attachment is governed by smooth (laminar) potential flow, whileseparation effectively is a generator of turbulence. We shall thus find thatdrag can be seen as a “cost of separation”, which for a wing also pays forgenerating lift.

We will present a scenario for separation in slightly viscous turbulentflow, which is fundamentally different from the scenario for viscous laminarflow by Prandtl based on adverse pressure gradients retarding the flowto stagnation at separation. We make a distinction betweeen separationfrom a laminar boundary layer with no-slip boundary condition and froma turbulent boundary layer with slip. We thus make a distinction between

• laminar separation with no-slip in (very) viscous flow

considered by Prandtl of relevance for viscous flow, and



84 15. Flow Separation

• turbulent separation with slip in slightly viscous flow

of relevance in aerodynamics.

We noted above that separation occurs if ∂p∂n < U 2

R , where ∂p∂n is the

pressure gradient normal to the boundary into the fluid, U is a flow speedclose to the boundary and R the curvature of the boundary, positive fora convex body. We note that in a laminar boundary layer ∂p

∂n > 0 only in

contracting flow, which causes separation as soon as the flow expands afterthe crest of the body. We observe that in a turbulent boundary layer withslip, ∂p

∂n > 0 is possible also in expanding flow which can delay separation.We present a basic mechanism for tangential separation with slip basedon instability at rear points of stagnation generating low-pressure rolls of streamwise vorticity reducing ∂p

∂n . We present new explanations of the drag-reducing effect of the dimples of a golf ball, the Magnus effect, the reverseMagnus effect and the Coanda effect, all related to delayed separation froma turbulent boundary layer with slip.

We give evidence that Prandtl’s boundary layer theory for laminar sepa-ration has fallen into this trap, with the unfortunate result is that much re-search and effort has gone into preventing laminar separation in flows whicheffectively are turbulent with turbulent separation . We present a scenario

for turbulent separation without stagnation supported by analysis, compu-tation and experiments, which is radically different from Prandtl’s scenariofor laminar separation at stagnation. The fundamental question concernsthe fluid dynamics of separation without stagnation , since in slightly vis-cous flow the friction is too small to bring fluid particles to rest. We shallfind an answer which connects to the familiar experience of the rotatingflow through a bathtub drain, which in reality replaces the theoreticallypossible but unstable fully radial flow.

We show that laminar separation with no-slip occurs at the crest of aflow, while turbulent separation can be delayed. We show that drag can beseen as cost of separation, which for a wing also generates lift as shown in[38]. We show that the difference between laminar and turbulent separationcan give rise to non-symmetric separation, which underlies both the Magnus effect and the reverse Magnus effect generating lift by rotation. We alsoshow that the Coanda effect arises from delayed turbulent separation withslip. We start recalling some critcism of Prandtls boundary layer theory.

15.2 Separation in Pictures

The scenario for separation can briefly be described as follows: Velocityinstability in retardation as opposing flows meet in the rear of the cylin-der, generates a zig-zag pattern of surface vorticity from which by vorticityinstability in accelleration, a pattern of rolls of low-pressure vorticity de-



15.3 Critique by Lancaster and Birkhoff 85

velops. We depict this scenario is depicted in Fig.15.1 and give into a moredetailed analysis below.

FIGURE 15.1. Turbulent separation without stagnation in principle and simula-tion.

15.3 Critique by Lancaster and Birkhoff

Prandtl’s contribution to fluid mechanics was to explain separation, dragand lift as effects of a very small (vanishingly small) viscosity. This viewhas been seriously questioned, however with little effect since no alternativeto Prandtl’s theory has been in sight. Lancaster states already in 1907 inhis in Aerodynamics [44]:

• According to the mathematical theory of Euler and Lagrange, all bod-ies are of streamline form (with zero dragh and lift). This conclusion,which would otherwise constitute a reductio ad absurdum, is usually explained on the gorund the fluid of theory is inviscid, whereas real possess viscosity. It is questionable of this expanlanation alone is ad-equate.




FIGURE 15.2. Separation in flow over a smooth hill by generation of surfacevorticity.



15.4 Can You Prove that Prandtl Was Incorrect? 87

Birkhoff follows up in his Hydromechanics from 1950 [73]:

• The art of knowing “how to apply” hydrodynamical theories can be learned even more effectively, in my opinion, by studying the para-doxes I will describe (e.g d’Alemberts paradox). Moreover, I think that to attribute them all to the neglect of viscosity is an oversim-plification. The root lies deeper, in lack of precisely that deductive

rigor whose importance is so commonly minimized by physicists and engineers.

However, critique of Prandtl was not well received, as shown in the reviewof Birkhoff’s book by James. J. Stoker [86]. The result is that Prandtl stilldominates fluid mechanics today, although the belief in Prandtl’s boundarylayer theory (BLT) seems to be fading as expressed by Cowley [48]:

• But is BLT a 20th century paradox? One may argue, yes, since for quantitative agreement with experiment BLT will be outgunned by computational fluid dynmaics in the 21st century.

The 21st century is now here, and yes, computational fluid mechanics re-veals a different scenario than Prandtl’s.

But Prandtl’s influence is still strong, as evidenced by the common belief that accurate computational simulation requires very thin boundary layersto be resolved. Thus Kim and Moin [43] claim that to correctly predictlift and drag of an aircraft at the relevant Reynolds number of size 108,requires computation on meshes with more than 1016 mesh points, whichis way out of reach for any foreseeable computer. This puts CFD intoa deadlock: Either compute at irrelevant too small Reynolds numbers orinvent turbulence models, which has shown to be very difficult.

Techniques for preventing laminar separation based on suction and blow-ing have been suggested. In the recent study [46] computational simulationsare presented of synthetic jet control for a NACA 0015 wing at Reynoldsnumber 896.000 (based on the chord length) for different angles of attack.As indicated, the relevant Reynolds number is two orders of magnitude

larger, and the relevance of the study can be questioned. The effects of thesynthetic jet control may simply be overshadowed by turbulent boundarylayers.

15.4 Can You Prove that Prandtl Was Incorrect?

Lancaster and Birkhoff did not accept Prandtl’s explanation of the gener-ation of drag and lift as an effect of a vanishingly thin boundary layer. Wehave said that it is difficult to directly prove that an infinitely small causecannot have a large effect, without access to an infinitely precise mathe-matical model or laboratory, which are not available. So Prandtl can be




pretty safe to direct attacks, but not to indirect: Suppose you eliminatethat vanishingly small cause from the consideration altogether, and yet ob-tain good correspondence between theory and experiment, that is, supposeyou observe the effect without the infinitely small cause. Then you can saythat the small cause has little to do with the effect.

This is what we do: We compute turbulent solutions of the incompressibleNavier-Stokes equations with slip boundary conditions, requiring only the

normal velocity to vanish letting the tangential velocity be free, and weobtain drag and lift which fit with experiments. We thus obtain the effect(drag and lift) without Prandtl’s cause consisting of a viscous boundarylayer with no-slip boundary condition requiring also the tangential velocityto vanish. We conclude that the origin of drag and lift in slightly viscousflow, is not viscous boundary layers with no-slip boundary conditions.

We have motivated the use of slip boundary condition by the fact thatthe skin friction of a turbulent boundary layer (the tangential force froma no-slip boundary condition), tends to zero with the viscosity, whichis supported by both experiment and computation, also indicating thatboundary layers in general are turbulent. More generally, we use a friction-force boundary condition as a model of the skin friction effect of a tur-bulent boundary layer, with a (small) friction coefficient determined by

the Reynolds number Re = ULν , where U is a representative velocity, La length scale and ν the viscosity. The limit case of zero friction withslip then corresponds to vanishing viscosity/very large Reynolds number,while large friction models no-slip of relevance for small to moderately largeReynolds numbers. In mathematical terms we combine the Navier-Stokesequations with a natural (Neumann/Robin type) boundary condition forthe tangential stress, instead of an essential (Dirichlet type) condition forthe tangential velocity as Prandtl did.

15.5 Separation vs Normal Pressure Gradient

Fluid particles with non-zero tangential velocity can only separate froma smooth boundary tangentially, because the normal velocity vanishes onthe boundary. By elementary Newtonian mechanics it follows that fluidparticles follow the curvature of the boundary without separation if

∂p

∂n=

U 2

R(15.1)

and separate tangentially if

∂p

∂n<

U 2

R, (15.2)

where p is the pressure, n denotes the unit normal pointing into the fluid,U is the tangential fluid speed and R is the radius of curvature of the



15.6 Laminar Separation with No-Slip 89

boundary counted positive if the body is convex. This is because a certainpressure gradient normal to the boundary is required to accelerate fluidparticles to follow the curvature of the boundary.

We understand that flow separation is directly related to the pressuregradient normal to the boundary accelerating fluid particles in the normaldirection, while Prandtl instead makes a connection to an adverse pressuregradient retarding the flow in a tangentially to the boundary. We exhibit

the difference in several examples below.

15.6 Laminar Separation with No-Slip

The classical (stationary) boundary layer equations for laminar viscous flow proposed by Prandtl in 1904 [67], take the following form assuming thatthe fluid occupies the half plane x2 ≥ 0 with main flow in the positivex1-direction with u3 = 0: Find (u1, u2, p) such that

u1∂u1

∂x1+ u2

∂u1

∂x2+

∂p

∂x1= ν

∂ 2u1

∂x22

,

∂u1∂x1

+ ∂u2∂x2

= 0,

∂p

∂x2= 0,

(15.3)

combined with the no-slip boundary condition u1 = u2 = 0, where ν > 0denotes the viscosity. These equations are formally derived form the Navier-Stokes equations assuming ν to be small, that the flow is constant in thex3-direction and does not vary quickly in the x1-direction. An importantfeature of the boundary layer equations is that the pressure is constantin the x2-direction as expressed by the ∂p

∂x2= 0, resulting from inertial

momentum balance in the x2-direction:

∂p

∂x2≈ −u1

∂u2

∂x1− u2

∂u2

∂x2, (15.4)

where in particular u1∂u2∂x1

is small because u1 = 0 on the boundary by theno-slip condition.

In the converging flow around a convex body before the crest a positivenormal pressure gradient satisfying (15.1) can be balanced by a negativenormal gradient of momentum, but not in the diverging flow after the crest.Assuming the plane x2 = 0 is tangent to the body at the crest at x1 = 0with the flow in the (positive) x1-direction, we have there u2 = 0 and inthe case of no-slip also u1 = 0, which forces ∂p

∂x2= 0 thus violating (15.1)

and causing separation. A flow with no-slip thus separates on the crest, butnot before.





15.9 Mechanics of Separation 91

15.9 Mechanics of Separation

We recall the linearized Navier-Stokes equations (with β = ν = 0 forsimplicity):

v + (u · ∇)v + (v · ∇)u + ∇q = f − f in Ω × I,

∇ ·v = 0 in Ω

×I,

v · n = g − g on Γ × I,v(·, 0) = u0 − u0 in Ω,

(15.5)

where (u, p) and (u, ¯ p) are two Euler solutions with slightly different data,and (v, q ) ≡ (u − u, p − ¯ p). Formally, with u and u given, this is a linearconvection-reaction problem for (v, q ) with growth properties governed bythe reaction term given by the 3×3 matrix ∇u. By the incompressiblity, thetrace of ∇u is zero, which shows that in general ∇u has eigenvalues withreal values of both signs, of the size of |∇u| (with | · | som matrix norm),thus with at least one exponentially unstable eigenvalue. In particular thereis exponential perturbation growth in regions where the flow is retardingin the streamwise direction.

Alternatively, applying the curl operator ∇× to the momentum equationwe obtain the vorticity equation

ω + (u · ∇)ω − (ω · ∇)u = ∇× f in Ω, (15.6)

which is also a convection-reaction equation in the vorticity ω = ∇×u withcoefficients depending on u, of the same form as the linearized equation(15.5), with a sign change of the reaction term. Also the vorticity is thuslocally subject to exponential growth with exponent |∇u|.

The linearized equations (15.5) and (15.3) indicate exponential growth of velocity perturbations in retarding flow and of streamwise vorticity in ac-cellerating flow. We identified in [37] a corresponding basic instablity mech-anism generating counter-rotating low-pressure streaks of strong stream-wise vorticity attaching to the rear of the body allowing separation with-out stagnation, as well as the associated cost for separation in terms of increased drag.

Note that in classical analysis it is often argued that from the vorticityequation (15.3), it follows that vorticity cannot be generated starting frompotential flow with zero vorticity and f = 0, which is Kelvin’s theorem . Butthis is an incorrect conclusion, since perturbations of f of f with ∇× f = 0must be taken into account, even if f = 0. What you effectively see incomputations is local exponential growth of vorticity on the body surfacein rear retardation and by vortex stretching in accelleration, even if f = 0,which is a main route of instability to turbulence as well as separation.




15.10 Flow around a Cylinder and Sphere

We recall potential flow in R3 with coordinates x = (x1, x2, x3) around a

circular cylinder of unit radius with axis along the x3-axis, assuming theflow velocity is (1, 0, 0) at infinity, see Fig. 14.1. Potential flow is constantin the x3-direction and fully symmetric in x1 and x2, with zero drag/liftand separates at the plane of stagnation x2 = 0 in the rear. It is given (in

polar coordinates (r, θ) in a (x1, x2)-plane) by the potential function

ϕ(r, θ) = (r +1

r)cos(θ)

with corresponding velocity components

ur ≡ ∂ϕ

∂r= (1 − 1

r2) cos(θ), us ≡ 1

r

∂ϕ

∂θ= −(1 +

1

r2)sin(θ)

with streamlines given as the level lines of the conjugate potential function

ψ ≡ (r − 1

r)sin(θ).

By Bernouilli’s principle the pressure is given by

p = − 1

2r4+

1

r2cos(2θ)

when normalized to vanish at infinity. We compute

∂p

∂θ= − 2

r2sin(2θ)),

∂p

∂r=

2

r3(

1

r2− cos(2θ)),

and discover an adverse pressure gradient in the back, while the normalpressure gradient

∂p

∂r = 4 sin2

(θ) ≥ 0

is precisely the force required to accelerate fluid particles with speed 2| sin(θ)|to follow the circular boundary without separation, satisfying the condition(15.1). We note, coupling to the above discussion relating to (15.4), that∂us∂r = 2

r3 sin(θ) = 2 at the crest. We further compute

∂ψ

∂r=

1

r2sin(θ)

which shows that fluid particles decrease their distance to the boundary infront of the cylinder and increase their distance in the rear, but the flowonly separates at rear stagnation.



15.11 Turbulent Separation and Drag Crisis 93

15.11 Turbulent Separation and Drag Crisis

We find that G2 solutions with slip initialized as potential flow developinto time-dependent flow with a turbulent wake with counter-rotating low-pressure rolls of streamwise vorticity generating substantial drag, as dis-played in Fig. 15.3.

FIGURE 15.3. Levels surfaces of strong vorticity in EG2 solution: streamwise |ω1|(left) and transversal |ω2| (middle) and |ω3| (right), at three times t1 < t2 < t3(upper, middle, lower), in the x1x3-plane.

We compare with G2 computations reported in [?, ?] with variable fric-tion coefficient β . If β > 0.02 the effect is no-slip with laminar separationat the crest according to Fig. 15.4 below with a drag coefficient cD

≈0.7.

If β < 0.002, then the effect is slip with cD ≈ 0.4. Varying the frictionparameter we can thus simulate the drag crisis with a drastic reductionof drag due to a switch from laminar separation at the crest to delayedturbulent separation with increasing large Reynolds numbers (in the range105 − 106.)

We display similar results for the flow around a sphere from [7] in Fig15.4 with β = 2, 10−2, 10−2, 5, 10−3, with a corresponding drop of dragfrom 0.5 to 0.2, showing for small friction a pattern of four low-pressureco-rotating streaks of streamwise vorticity which are analogous to the pat-tern of streamwise streaks behind the cylinder. This indicates that thedrag-reducing effect of the dimples of a golf ball is by triggering turbulentseparation.




FIGURE 15.4. Flow around a sphere under decreasing viscosity. Notice the sep-aration at the crest in the case of substantial viscosity, and delayed separationwith small viscosity.

15.12 Separation vs Normal Pressure Gradient

We show in Fig. 15.2 the normal pressure gradient ∂p∂n on the boundary

for different patterns of separation varying with the friction, and noticeas expected that tangential separation coincides with small ∂p

∂n (but is notrelated to an adverse pressure gradient).

G2 with variable friction thus opens to computational simulation of highReynolds number flow without resolving thin boundary layers, with po-tentiall very many applications, considered impossible in state-of-the-art[43].

15.13 Scenario for Separation without Stagnation

We now present a scenario for transition of potential flow around a cir-cular cylinder into turbulent flow, based on identifying perturbations of strong growth in the linearized equations (15.5) and (15.3), which is alsoa scenario for separation without stagnation. We will find that the pertur-bations consist of low pressure streamwise streaks attaching to the rear of the cylinder with small normal pressure gradient allowing tangential sep-

aration. We shall see that the scenario captures essential features of flowseparation and can be used to explain how both drag and lift arises in tur-bulent incompressible flow. In particular, it suggests a new explanation of why gliding flight is possible [?], indicating that the classical explanationsare inadequate.

As a model of potential flow at rear separation, we consider the potentialflow u(x) = (x1,−x2, 0) in the half-plane x1 > 0. Assuming x1 and x2

are small, we approximate the v2-equation of (15.5) by

v2 − v2 = f 2,

where f 2 = f 2(x3) is an oscillating mesh residual perturbation depending onx3 (including also a pressure-gradient), for example f 2(x3) = h sin(x3/δ ),



15.13 Scenario for Separation without Stagnation 95

with δ > 0. It is natural to assume that the amplitude of f 2 decreases withδ . We conclude, assuming v2(0, x) = 0, that

v2(t, x3) = t exp(t)f 2(x3),

and for the discussion, we assume v3 = 0. Next we approximate the ω1-vorticity equation for x2 small and x1 ≥ x1 > 0 with x1 small, by

ω1 + x1

∂ω1

∂x1 − ω1 = 0,with the “inflow boundary condition”

ω1(x1, x2, x3) =∂v2∂x3

= t exp(t)∂f 2∂x3

.

The equation for ω1 thus exhibits exponential growth, which is combinedwith exponential growth of the “inflow condition”. We can see these fea-tures in Fig. ?? showing how opposing flows on the back generate a patternof co-rotating surface vortices which act as initial conditions for vorticitystretching into the fluid generating rolls of low-pressure streamwise vortic-ity, as displayed in Figs.15.1 and 15.3.

Altogether we expect exp(t) perturbation growth of residual perturba-tions of size h, resulting in a global change of the flow after time T

∼log(1/h), which can be traced in the computations.We thus understand that the formation of streamwise streaks as the re-

sult of a force perturbation oscillating in the x3 direction, which in theretardation of the flow in the x2-direction creates exponentially increasingvorticity in the x1-direction, which acts as inflow to the ω1-vorticity equa-tion with exponential growth by vortex stretching. Thus, we find exponen-tial growth at rear separation in both the retardation in the x2-directionand the accelleration in the x1 direction. This scenario is illustrated inprinciple and computation in Fig. 15.1. Note that since the perturbation isconvected with the base flow, the absolute size of the growth is related tothe length of time the perturbation stays in a zone of exponential growth.Since the combined exponential growth is independent of δ , it follows thatlarge-scale perturbations with large amplitude have largest growth, whichis also seen in computations with δ the distance between streamwise rollssas seen in Fig. 15.3 which does not seem to decrease with decreasing h.

Notice that at forward separation the retardation does not come fromopposing flows, and the zone of exponential growth of ω2 is short, resultingin much smaller perturbation growth than at rear separation.

We can view the occurence of the rear surface vorticities as a mechanismof separation with non-zero tangential speed, by diminishing the normalpressure gradient of potential flow, which allows separation only at stag-nation. The surface vorticities thus allow separation without stagnationbut the price is generation of a system of low-pressure tubes of stream-wise vorticity creating drag in a form of “separation trauma” or “cost of divorce”.




15.14 Separation Experiments on Youtube

Experiments showing fluid separation can be watched on

• http : //www.youtube.com/watch?v = jiWa4uzOynk

• http : //www.youtube.com/watch?v = I 6NQhBY 5L80




16Effects of Non-Symmetric Separation

Although symmetry breaking can be applied to various things, likedetermining the shape of a snowflake based on its composition, theteam comprised of the 3 researchers focused on the creation of theuniverse. Basically, their theories indicate that, during the universe’sevolution, matter somehow gained the upper hand in its struggleagainst antimatter, leading to the formation of today’s celestial bod-ies. (Sofpedia on the 2008 Nobel Prize in Physics)

Common sense is the most fairly distributed thing in the world, foreach one thinks he is so well-endowed with it that even those who arehardest to satisfy in all other matters are not in the habit of desiringmore of it than they already have. (Descartes)

16.1 Magnus Effect

Observations show that a top-spin tennis ball curves down, and a back-spin curves up, as a result of the Magnus effect creating a lift force per-pendicular to the flow. For top-spin this can be explained as an effect of non-symmetrical separation occurring because the friction on top of theball is larger than below, because the relative velocity is larger, and thusthe separation occurs later below with a corresponding increase of tangen-tial velocity and pressure drop, resulting in a downward force. Similarly, aback-spin ball curves up because of a delayed separation on top.

In G2 with force boundary conditions with the tangential stress propor-tional to the the difference of the ball surface velocity and the free-stream



98 16. Effects of Non-Symmetric Separation

fluid velocity, the rotation effectively causes non-symmetric friction whichcauses non-symmetric separation with lift, see [35]. In classical fluid me-chanics the Magnus effect is described as an effect of large scale rotationof air around a spinning ball, while today the true reason is believed to benon-symmetric separation, which is confirmed by G2 computation.

16.2 The Reverse Magnus Effect

Observations show that a ping-pong ball with strong backspin can curvedown, seemingly subject to a reverse Magnus effect . This can be again beunderstood as a result of an non-symmetric separation, but this time as aneffect of laminar separation at lower Reynolds number on top because of lower relative speed, and delayed turbulent separation below because of ahigher relative speed and higher effective Reynolds number. G2 computa-tions with varying friction coefficients confirm this scenario [?].

16.3 The Coanda Effect

Holding a spoon vertically under a water faucet, shows the Coanda effect of a stream of fluid staying attached to a convex surface. The principlewas named after Romanian discoverer Henri Coanda, who was the firstto understand the practical importance of the phenomenon in aircraft de-velopment, and patented several devices such as the Coanda saucer . It iscommonly believed that the Coanda effect arises from surface tension orVan der Waals forces, but in the new scenario it is instead seen to be adirect consequence of the tendency of turbulent incompressible Euler flowwith slip to stick to a solid boundary.

16.4 More NASA Confusion

We cite from [64] from 2008:

• State-of-the-art CFD is by no means fully adequate for predicting separated flows and buffet onset for aircraft configurations

• If state-of-the-art CFD is not fully adequate for subsonic fixed wing aircraft operating in cruise through buffet onset conditions, then pre-cisely how does it fall short? We have seen that, in isolated cases, it can perform reasonably well even for seprated flow conditions....Does this mean that these models ( k − ω turbulence models) are known tobe adequate for aerodynamics flows in general? The answer to the last question is clearly no....It is not known whether any definitive



16.4 More NASA Confusion 99

failings that have been identified through unit problems, necessarily have a significant effect for some of the characteristics of interest for

full-aircraft configurations in cruise-through buffet conditions.

• With circulation control, a tangential wall jet is used primarily for the purpose of enhancing lift over an aerodynamic surface. A wall jet emanating from the plenoum inside an airfoil or wing sticks to the

rounded trailing edge surface due to the Coanda effect, causing delayed separation and thus increasing circulation and producing higher lift...One of the main conclusions to come out of the 2004 Circulation Control Workshop was the inconsistency in the CFD for capturing the Coand jet flow physics .

We understand that state-of-the-art CFD is incapable of predicting startand landing of a jumbojet with instationary separated flow referred to asbuffect onset. Further state-of-the-art is that a tangential wall jet sticksto the trailing edge because of the Coanda effect (which we have seen iswrong), and that delaying separation increases circulation (which we haveseen is wrong) and that lift requires circulation (which we have seen iswrong).



100 16. Effects of Non-Symmetric Separation




17Boundary Layer Turbulence fromSeparation

The experience reported above suggests the following scenario for separa-tion into a turbulent boundary layer over a flat plate or more generally asmooth boundary with large radius of curvature: (i) Rolls of streamwisevorticity are formed by non-modal linear perturbation growth referred toas the Taylor G¨ ortler mechanism in [39]. (ii) The rolls create opposingtransversal flows (as in the back of cylinder), which generate surface vor-ticity which is stretched into the fluid, while being bent into the free streamflow, as evidenced in e.g. [29, 30].

We note that by energy balance it follows that the total turbulent dissipa-tion in a turbulent boundary layer of width δ b equals σsus which indicatesthat δ b ∼ ν 0.2U 0.8, assuming that cf ∼ Re−0.2.



Part III

Flying

102




18Gliding Flight

More than anything else the sensation is one of perfect peace mingledwith an excitement that strains every nerve to the utmost, if you canconceive of such a combination. (Wilbur Wright)

He who would learn to fly one day must first learn to stand and walkand run and climb and dance; one cannot fly into flying. (FriedrichNietzsche)

The Wright Brothers created the single greatest cultural force sincethe invention of writing. The airplane became the first World WideWeb, bringing people, languages, ideas, and values together. (BillGates)

The desire to fly is an idea handed down to us by our ancestorswho, in their grueling travels across trackless lands in prehistoric

times, looked enviously on the birds soaring freely through space,at full speed, above all obstacles, on the infinite highway of the air.(Wilbur Wright)

18.1 Mechanisms of Lift and Drag

We are now ready to uncover the secret of flight using our insights gainedfrom the new resolution of d’Alembert’s paradox including the new sce-nario for flow separation. In this central chapter of the book we shall thusdiscover a mechanism for the generation of lift of a wing, which is fun-damentally different from that by Kutta-Zhukovsky coupling lift to large



104 18. Gliding Flight

scale circulation around the wing. We shall find that lift results from amodification of zero-lift potential flow consisting of counter-rotating rollsof low-pressure streamwise vorticity generated by instability at separation,which reduce the high pressure on top of the wing before the trailing edgeof potential flow and thus generate lift, but which also generate drag. At acloser examination of the quantitative distributions of lift and drag forcesaround the wing, we discover large lift at the expense of small drag result-

ing from leading edge suction, and we can thus give an answer the openingquestion of this book of how a wing can generate a lift/drag ratio largerthan 10.

The secret of flight is in concise form uncovered in Fig. 18.1 showing G2computed lift and and drag coefficients of a Naca 0012 3d wing as functionsof the angle of attack α, as well as the circulation around the wing. Wesee that the lift and drag increase roughly linearly up to 16 degrees, with alift/drag ratio of about 13 for α > 3 degrees, and that lift peaks at stall atα = 20 after a quick increase of drag. We find that the circulation remainssmall for α less than 10 degrees without connection to lift, and concludethat the theory of lift of by Kutta-Zhukovsky is fictional without physicalcorrespondence: There is lift but no circulation. Lift does not originate fromcirculation.

Inspecting Figs. 18.2-18.4 showing velocity, pressure and vorticity andFig. 18.5 showing lift and drag distributions over the upper and lowersurfaces of the wing (allowing also pitching moment to be computed), wecan now, with experience from the above preparatory analysis, identify thebasic mechanisms for the generation of lift and drag in incompressible highReynolds number flow around a wing at different angles of attack α: Wefind two regimes before stall at α = 20 with different, more or less lineargrowth in α of both lift and drag, a main phase 0 ≤ α < 16 with the slopeof the lift (coefficient) curve equal to 0.09 and of the drag curve equal to0.08 with L/D ≈ 14, and a final phase 16 ≤ α < 20 with increased slopeof both lift and drag. The main phase can be divided into an initial phase0 ≤ α < 4 − 6 and an intermediate phase 4 − 6 ≤ α < 16, with somewhatsmaller slope of drag in the initial phase. We now present details of thisgeneral picture.

18.2 Phase 1: 0 ≤ α ≤ 4 − 6

At zero angle of attack with zero lift there is high pressure at the leadingedge and equal low pressures on the upper and lower crests of the wingbecause the flow is essentially potential and thus satisfies Bernouilli’s lawof high/low pressure where velocity is low/high. The drag is about 0.01and results from rolls of low-pressure streamwise vorticity attaching to thetrailing edge. As α increases the low pressure below gets depleted as the



18.2 Phase 1: 0 ≤ α ≤ 4 − 6 105

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50 60 70

2

4

6

8

10

12

14

FIGURE 18.1. G2 lift coefficient and circulation as functions of the angle of attack (top), drag coefficient (middle) and lift/drag ratio (bottom) as functionsof the angle of attack.




incoming flow becomes parallel to the lower surface at the trailing edgefor α = 6, while the low pressure above intenisfies and moves towards theleading edge. The streamwise vortices at the trailing edge essentially stayconstant in strength but gradually shift attachement towards the uppersurface. The high pressure at the leading edge moves somewhat down, butcontributes little to lift. Drag increases only slowly because of negative dragat the leading edge.

18.3 Phase 2: 4 − 6 ≤ α ≤ 16

The low pressure on top of the leading edge intensifies to create a normalgradient preventing separation, and thus creates lift by suction peaking ontop of the leading edge. The slip boundary condition prevents separationand downwash is created with the help of the low-pressure wake of stream-wise vorticity at rear separation. The high pressure at the leading edgemoves further down and the pressure below increases slowly, contributingto the main lift coming from suction above. The net drag from the up-per surface is close to zero because of the negative drag at the leading

edge, known as leading edge suction , while the drag from the lower surfaceincreases (linearly) with the angle of the incoming flow, with somewhatincreased but still small drag slope. This explains why the line to a flyingkite can be almost vertical even in strong wind, and that a thick wing canhave less drag than a thin.

18.4 Phase 3: 16 ≤ α ≤ 20

This is the phase creating maximal lift just before stall in which the wingpartly acts as a bluff body with a turbulent low-pressure wake attachingat the rear upper surface, which contributes extra drag and lift, doublingthe slope of the lift curve to give maximal lift

≈2.5 at α = 20 with rapid

loss of lift after stall.

18.5 Lift and Drag Distribution Curves

The distributions of lift and drag forces over the wing resulting from pro- jecting the pressure acting perpendicular to the wing surface onto relevantdirections, are plotted in Fig.18.5. The total lift and drag results fromintegrating these distributions around the wing. In potential flow compu-tations (with circulation according to Kutta-Zhukovsky), only the pressuredistribution or c p-distribution is considered to carry releveant information,



18.5 Lift and Drag Distribution Curves 107

FIGURE 18.2. G2 computation of velocity magnitude (upper), pressure (middle),and non-transversal vorticity (lower), for angles of attack 2, 4, and 8 (from leftto right). Notice in particular the rolls of streamwise vorticity at separation.




FIGURE 18.3. G2 computation of velocity magnitude (upper), pressure (middle),and non-transversal vorticity (lower), for angles of attack 10, 14, and 18 (fromleft to right). Notice in particular the rolls of streamwise vorticity at separation.



18.5 Lift and Drag Distribution Curves 109

FIGURE 18.4. G2 computation of velocity magnitude (upper), pressure (middle),and non-transversal vorticity (lower), for angles of attack 20, 22, and 24 (fromleft to right).




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−2

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−1

0

1

2

3

4

5

FIGURE 18.5. G2 computation of normalized local lift force (upper) and dragforce (lower) contributions acting along the lower and upper parts of the wing,for angles of attack 0, 2 ,4 ,10 and 18, each curve translated 0.2 to the right and1.0 up, with the zero force level indicated for each curve.



18.6 Summary 111

because a potential solution by construction has zero drag. In the perspec-tive of Kutta-Zhukovsky, it is thus remarkable that the projected c p-curvescarry correct information for both lift and drag.

18.6 Summary

We understand that the above scenario of the action of a wing for differentangles of attack, is fundamentally different from that of Kutta-Zhukovsky,although for lift there is a superficial similarity because both scenariosinvolve modified potential flow. The slope of the lift curve according toKutta-Zhukovsky is 2π2/180 ≈ 0.10 as compared to the computed 0.09.

The lift generation in Phase 1 and 3 can rather easily be envisioned,while both the lift and drag in Phase 2 results from a (fortunate) intricateinterplay of stability and instability of potential flow: The main lift comesfrom upper surface suction arising from a turbulent boundary layer withsmall skin friction combined with rear separation instability generating low-

pressure streamwise vorticity, while the drag is kept small by negative dragfrom the leading edge. We conclude that preventing transition to turbulenceat the leading edge can lead to both decreased lift and increased drag.

18.7 Comparing Computation with Experiment

Comparing G2 computations with about 150 000 mesh points with ex-periments [55, 58], we find good agreement with the main difference thatthe boost of the lift coefficient in phase 3 is lacking in experiments. Thisis probably an effect of smaller Reynolds numbers in experiments, with aseparation bubble forming on the leading edge reducing lift at high anglesof attack. The oil-film pictures in [55] show surface vorticity generatingstreamwise vorticity at separation as observed also in [37, 41].

A jumbojet can only be tested in a wind tunnel as a smaller scale model,and upscaling test results is cumbersome because boundary layers do notscale. This means that computations can be closer to reality than windtunnel experiments. Of particular importance is the maximal lift coefficient,which cannot be predicted by Kutta-Zhukovsky nor in model experiments,which for Boeing 737 is reported to be 2.73 in landing in correspondencewith the computation. In take-off the maximal lift is reported to be 1.75,reflected by the rapidly increasing drag beyond α = 16 in computation.




18.8 Kutta-Zhukovsky’s Lift Theory isNon-Physical

Fig.18.1 shows that the circulation is small without any increase up toα = 10, which gives evidence that Kutta-Zhukovsky’s circulation theorycoupling lift to circulation does not describe real flow. Apparently Kutta-Zhukovsky manage to capture some physics using fully incorrect physics,

which is not science.Kutta-Zhukovsky’s explanation of lift is analogous to an outdated ex-

planation of the Robin-Magnus effect causing a top-spin tennis ball tocurve down as an effect of circulation, which in modern fluid mechanicsis instead understood as an effect of non-symmetric different separation inlaminar and turbulent boundary layers [41]. Our results show that Kutta-Zhukovsky’s lift theory for a wing also needs to be replaced.




19Flapping Flight

We recall the observations of the wingbeat cycle by da Vinci and Lilienthalalso described on [17]:

• a forward downstroke with the wing increasingly twisted towards thetip with the leading edge down,

• a backward upstroke with the wing twisted the other way with theleading edge up.

• The downstroke, which requires muscle power, gives positive lift andforward thrust from propeller action of the twisted wing.

• Large twist gives large thrust but requires quick downstroke to givepositive lift.

• The upstroke with large twist gives positive lift and negative thrustby turbin action, with the lift replacing muscle power.

• A quick upstroke with small twist can give forward thrust and nega-tive lift with propeller action.

• The twist/motion of the part of the wing close to the body is relativelysmall and gives consistent lift through the complete wingbeat cycle,while the twisted wing gives forward thrust at least in the downstroke.



114 19. Flapping Flight




20At the Horizon

20.1 Complete Airplane

20.2 Dynamics

20.3 Control

20.4 Flight Simulator



116 20. At the Horizon




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