Why It Is Possible to Fly

· flying, theory of flight

  • 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?(Kenneth Chang, New York Times, Dec 9, 2003)

The Wright Flyer 1903: the first sustained powered heavier-than-air flight.

The material of this knol is developed in further detail in the new book The Secret of Flight.

The Mystery of Gliding Flight

The problem of explaining why it is possible to fly in the air using wings has haunted scientists since the birth of mathematical sciences. To fly, an upward force on the wing, referred to as lift L, has to be generated from the flow of air around the wing, while the air resistance to motion or drag D, is small. The mystery is how a large ratio L/D can be created (see video of model airplane).

In the gliding flight of birds and airplanes with fixed wings, L/D is typically between 10 and 20, which means that a good glider can glide up to 20 meters upon loosing 1 meter in altitude, or that a 400 ton jumbojet can cruise at an engine thrust of 20 tons, while about 400 tons is needed in take-off.

By elementary Newtonian mechanics, upward lift must be accompanied by downwash with the wing redirecting air downwards. The enigma of flight is how a wing generates substantial downwash; with downwash there is lift.

Classical mathematics and mechanics could not give an answer: Newton computed the lift of a tilted flat plate bombarded by a horisontal stream of fluid particles from below and obtained a disappointingly small lift, proportional to the square of the tilting angle or angle of attack.

French mathematician d’Alembert followed up in 1752 with a computation based on potential flow (inviscid incompressible irrotational steady flow), showing that both the drag and lift of a wing is zero, referred to as d’Alembert’s paradox , since it contradicts observations and thus belongs to pure fiction.

Explanation of Lift by Kutta-Zhukowsky

It took 150 years before someone dared to challenge the pessimistic mathematical predictions by Newton and d’Alembert, expressed by Lord Kelvin as:

  • I can state flatly that heavier than air flying machines are impossible.

In the 1890s the German engineer Otto Lilienthal made careful studies of the gliding flight of birds, and designed wings allowing him to make 2000 successful heavier-than-air gliding flights starting from a little artificial hill, before in 1896 he broke his neck falling to the ground after having stalled at 15 meters altitude. The first powered heavier than-air flights were performed by the two brothers Wilbur and Orwille Wright , who on the windy fields of Kitty Hawk, North Carolina, on December 17 in 1903, managed to get their airplane Flyer off ground using a 12 horse power engine.

The mathematicians Kutta and  Zhukovsky (called the father of Russian aviation) then quickly modified potential flow around the section of a wing with zero lift/drag by introducing a large scale circulation or rotation of air around a two-dimensional wing section as illustrated in the following figure showing the zero lift/drag potential solution supplemented by large scale circulation into the Kutta-Zhukovsky flow pattern with lift (but no drag):

     Kutta-Zhukovsky explanation of the generation of lift by adding large scale circulation to potential flow

We see how the zones of high (H) and low (L) pressure of potential flow with zero net lift, by the circulation are changed to produce net lift by low pressure on top and high pressure from below. Kutta-Zhukovsky suggested that the circulation around the wing section was balanced by a counter-rotating so-called starting vortex behind the wing as shown in the figure, giving zero total circulation according to Kelvin’s theorem.

Kutta-Zhukovsky’s formula for lift (proportional to the angle of attack) agreed reasonably well with observations for long wings and small angles of attack, but not for short wings and large angles of attack, and the drag was still zero. Despite these shortcomings, the explanation of lift by Kutta-Zhukovsky, is the only one available in the literature [1][2] See further The Spell of Kutta-Zhukovsky’s Circulation Theory.

NASA Confusion

To get an idea of the confusion surrounding the generation of lift,  take a look at the presentation by  National Aeronautics and Space Adminstration dismissing three popular explanations as being incorrect: (i) longer path/equal-transit theory, (ii) skipping-stone  theory and (iii) Venturi-Bernouilli theory, but offering no theory claimed to be correct. Can it really be that generation of lift is a mystery to NASA? You find the same  confusion in the book [3] with the contradictory title Understanding Flight. 

You find more confusion on video1, video2, video3, video4, video5, video6,video7 and (among many):

Non-Physical Fiction of Kutta-Zhukovsky

The problem with Kutta-Zhukovsky’s theory is that it is purely fictional mathematical theory, which does not describe physics: In reality there is 

  • no large scale circulation around the section of the wing
  • no starting vortex behind the wing.

Thus the matematical theory of lift by Kutta-Zhukovsky based on modified potential flow, is non-physical, and does not explain the origin of lift and why it is possible to fly.  It rests on the following incorrect logic: Circulation around a wing (=A)  implies lift (=B) and since there is lift (B is true) there must be circulation (A is true). But from A implies B, you can only conclude that B is true if A is true, not that A is true if B is true, since this corresponds to the reverse implication, that B implies A. The incorrect logic is like saying that since eating cakes makes you gain weight, and you have gained weight, you must have eaten a lot of cakes. But you can get fat by eating pasta as well.  This is shown below: lift has another origin than circulation.

Pilots Misconceptions

But there is no other theory in the aerodynamics literature with a better explanation. In fact, state-of-the-art  claims that mathematical computation of the lift and drag of an airplane is impossible. This could make you a bit nervous as you lean back for take off in a new airplane, like the Airbus 380 or Boeing 787. In particular, don’t read Plane and Pilot Magazine: Misconceptions abound about one the most important forces in flying, where it is made clear that pilots are not supposed to understand what keeps an airplane in the air. 

New Mathematical Theory of Lift and Drag

But there is hope: The new resolution of d’Alembert’s paradox [4] and the book [5] offers an explanation of how lift is generated by a wing in incompressible flow, which is fundamentally different from the accepted explanation by Kutta-Zhukowsky. The new theory is developed in detail in The Mathematical Secret of Flight [6] and in the new book The Secret of Flight [7].

Watching the   movies of pressure  and velocity of turbulent computational solutions of the  incompressible Navier-Stokes equations with slip/small friction boundary conditions, around a three-dimensional[8] Naca0012 wing under increasing angle of attack,  you can yourself uncover the secret of flight. What you see can be described in pictures as follows:

We see a potential flow with high pressure at the separation on top of the wing before the trailing edge, being modified by
  • low-pressure counter-rotating rolls of streamwise vorticity generated at separation
  • depleting the high pressure moving it to the trailing edge
  • thereby generating lift. 

You can follow the development of this scenario on the following snapshots for angles of attack 2,4,8,10,14,16,18,22, of


The mechanism for generation of streamwise vorticity at separation is described in the Knol on  d’Alembert’s paradox d’Alembert’s paradox. You can watch the streaks of streamwise vorticity (and the absense of large scale circulation) generated at the trailing edge on McDonnel Douglas MD-11, a Boeing 757 767  767 (very instructive) 777 and a 747 about to land on the eFluids Gallery:
Watching the movies we see that for small angles of attack, as in the figure above, the main lift comes from
the low pressure on top of the leading edge, while for larger angles also the upper rear part contributes lift.
We find that the same turbulent mechanism which gives drag, also gives lift to a wing: The low pressure turbulent wake at separation forces the flow to separate at the trailing edge resulting in downwash and thereby lift. The rolls of streamwise vorticity appear along the entire trailing edge and have a different origin than the wing tip vortex, which is of minor importance for a long wing.
You can watch the rolls of streamwise vorticity generated at separation on top of a delta wing in a computational movie with the following snap shot to the left (see also eFluids ):

with an experiment showing the same feature in the middle and a Concorde  in landing configration to the right.  

We see that the difference between Kutta-Zhukovsky and the new explanation of lift is the nature of the modification/perturbation of potential flow: Kutta-Zhukovsky claim that it consists of a large scale circulation around the wing section, while we show that it is a three-dimensional instability phenomena generating turbulent flow.

In the related Knol on why a topspin tennis ball curves down, it is explained why the flow does not separate on the crest of the wing based on the fact that the boundary layer is turbulent and therefore acts like very small boundary friction on the free stream flow, which makes the (incompressible) flow follow the boundary. It is shown that a flow with laminar boundary layer would separate on the crest of the wing and give poor lift. Gliding flight without turbulence is thus impossible, which gives further evidence that the non-turbulent circulation theory is incorrect. 

The new resolution opens the possibility of computing lift and drag of a complete airplane, considered impossible in state-of-the-art [9][10] of obvious use in the design and control of airplanes. 

The new theory is presented in more detail in the upcoming book (draft) The Secret of Flight.

The Secret of Lift and Drag

The following plots of lift, circulation and drag as functions of the angle of attack (aoa), reveals the secret of flight:


Figure 1.  Lift and circulation (x2)  of the Naca0012 airfoil as functions of the angle of attack. 

                                         Figure 2.     Drag as function of angle of attack.

                                                 Figure 3. Lift/drag ratio as function of angle of attack.


We see  in Fig.1 that the lift under small drag increases linearly for aoa < 16 degrees with the slope of the lift (coefficient) curve equal to 0.09,  whereafter the drag increases quickly, and reaches a maxium before stall at 20 degrees. Figs. 2 and 3 show that drag increases roughly linearly up to 16 degrees (with a somewhat increasing slope) with the slope of the drag curve equal to 0.08, with a L/D of about 13 for  3 < aoa < 16.

Evidence of Lift without Circulation

We see in Fig. 1 that lift increases linearly up to 16 degrees, while the circulation stays practically zero. These computations are thus not compatible with Kutta-Zhukovsky’s theory of lift, where correct lift is captured in accordance with experiments, without circulation.

Details of the Secret

We can understand details of the generation of lift and drag by carefylly studying the following plots of the distribution of  the lift force (upper) and drag force (lower) along the lower and upper sides of the wing, for angles of attack 0, 2 ,4 ,10 and 18 degrees, each curve translated 0.2 to the right and 1.0 up, with the zero force level indicated for each curve.

   Fig. 4 Lift and drag distributions on upper and lower sides of a Naca0012 for
   angles of attack 0,2,4,10 and 18, each curve translated 0.2 to the right and 1.0 up.

The main phase can be divided into an initial phase 0< aoa <4-6 and an intermediate phase 4-6 < aoa <16 with the following details:

Phase 1: 0 < aoa < 4-6

At zero angle of attack with zero lift there is high pressure at the leading edge and equal low pressures on the upper and lower crests of the wing because the flow is essentially potential and thus satisfies Bernouilli’s law of high/low pressure where velocity is low/high. The drag is about 0.01 and results from rolls of low-pressure streamwise vorticity attachingto the trailing edge. As  aoa increases the low pressure below gets depleted as the incoming flow becomes parallel to the lower surface at the trailing edge for aoa = 6, while the  low pressure above intenisfies and moves towards the leading edge. The streamwise vortices at the trailing edge essentially stay constant in strength but gradually shift attachement towards the upper surface. The high pressure at the leading edge moves somewhat down, but contributes little to lift. Drag  increases only slowly because of negative drag at the leading edge.

Phase 2: 4-6< aoa <16

The low pressure on top of the leading edge intensifies to create a normal gradient preventing separation, and thus creates lift by suction peaking on top of the leading edge. The slip boundary condition prevents separation and downwash is created with the help of the low-pressure wake of streamwise vorticity at rear separation.  The high pressure at the leading edge moves further down and the pressure below increases slowly, contributing to the main lift coming from suction above. The net drag from the upper 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 surface increases (linearly) with the angle of the incoming flow, with somewhat increased but still small drag slope. This explains why the line to a flying kite can be almost vertical even in strong wind, and that a thick wing can have less drag than a thin.  

Phase 3: 16 < aoa <20

This is the phase creating maximal lift just before stall in which the wing partly acts as a bluff body with a turbulent low-pressure wake attaching  at the rear upper surface, which contributes extra drag and lift, doubling the slope of the lift curve to give maximal lift  ~  2.5 at aoa  = 20 with rapid loss of lift after stall.

Comparison with Experiments/Reality

Comparing the above computational results (with about 100.000 mesh points) with experiments in Nasa Technical Memorandum 100019 and Aeronautical Research Council Reports and Memoranda and with representative values, we find good agreement (see further discussion below) with the main difference that the  boost of the lift coefficient in phase 3 is lacking in experiments. This is probably an effect of smaller Reynolds numbers in experiments,  with a separation bubble forming on the leading edge reducing lift at high angle of attack. The oil-film pictures in the second reference show surface vorticity generating streamwise vorticity at separation as observed also in  d’Alembert’s paradox. 

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 not scale. This means that  computations can be closer to reality than wind tunnel 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 correspondence with the computation. In take-off the maximal lift is reported to be 1.75 (reflected by the rapidly increasing drag beyond aoa =16 in computation)  and for a Cessna 172, L=1.4 for aoa =10.
The following plot from The Evolution of Modern Aircraft, NASA Scientific and Technical Information Branch shows the development of max L/D over time:

chart illustrating trends in maximum lift-drag ratio from 1920 to 1980
Figure 7.9 – Trends in maximum lift-drag ratio of propeller-driven aircraft.


The following plot from US Centennial Flight Commission shows L/D as a function of the Mach number with Mach = 1 for the speed of sound and M = 0 for incompressible flow: 

Lift-to-drag ratio vs Mach number for different wingsThis figure shows (L/D)max, a measure of aerodynamic efficiency, plotted against Mach number for an optimum straight-wing and swept-wing airplane.Credits – NASAThe following plot from Allstar Aeronautics Learning Laboratory shows consistent results:

The maximum L/D of a Boeing 767-200 is estimated to~ 18, possibly a bit optimistic. In the Gimli Glider incident a Boeing 767 running of out fuel managed to glide to a safe landing at L/D = 12.
Low L/D ~ 3 – 6 is of interest in unpowered landing of a space shuttle.
The L/D of a kite (which acts like a wing) can be directy read from the angle of the line to the kite
with common values 5-10. For the extreme Globalflyer, the first airplane to fly around the globe without refueling, L/D is estimated to a possibly optimistic 37: A rough calculation assuming a mean weight of
5.000 kg with corresponding mean thrust of 250 kp at L/D = 20 balances about 10 tons of fuel consumption at efficiency of 50%. Thus L/D = 20 may be more realistic.
Sir Hiram Maxim (1840-1916) made a fortune with his famous Maxim machine gun, but his main goal was powered, manned flight. Maxim tested airfoils in a wind tunnel, 12 feet long with a test section 3 feet square, with twin coaxial fans driven by a steam engine blowing air into the test section at 50 miles per hour, and obtained L/D=14 for a cambered airfoil at aoa=4. 

The Wright Brothers made careful wind tunnel tests in 1901 and obtained L/D~10 for a cambered airfoil with aspect ratio 6, see Fig.9 in An Engineering Analysis of the Wright Brothers 1902 Glider. 

Ice skate sailing inside wings shows L/D = 6 simply because it shows to be possible to go 6 times as fast as the wind speed (assuming zero friction with the ice). The fact that L/D < 10 can be an effect of the short length of the wing:

Compare with the movie of a windjet (with more substantial ground friction) with top speed of 187 km/h about 5 times the wind speed (+ windjet movies + greenbird movies):


water the speed record is 90 km/h set on a kiteboard and with a winged boat 86 km/h.

Correct Drag and Lift with Better Resolution

With better resolution of the separartion at the trailing edge correct lift and drag are obtained, as shown
in the following plot comparing Unicorn computations with different experiments for a long NACA0012 
with L/D up to 70, see also [13]:

L/D = 100 Is Not Possible in Reality

In Nasa Technical Memorandum 100019 results from wind tunnel tests of a Naca0012 (without wingtips as in the above computation) are reported indicating maximal L/D ~ 100, seemingly allowing ice skating at 1000 m/s, three times the speed of sound! Similar results can be read from lift data and drag data. and on Aerospaceweb. These results are not in accordance with the above plots representing real wings with instead max L/D = 5 – 20, but they are supported by standard software such as XFOIL.
The mismatch of these windtunnel results with reality may be an effect of incorrect scaling of boundary layers (and not absence of wingtips), as supported by Fig. 8 in Aeronautical Research Council Reports and Memoranda showing L/D ~ 80  for a very clean smooth surface of airfoil, but L/D < 30 with 0.3 mm ballotini on the surface (at aoa = 10).  
Evidently, a somewhat rough surface triggering a turbulent boundary layer brings the windtunnel test closer to the reality. The slip condition in the computation also models a turbulent boundary layer, which can explain why the computations do not give the same large values of L/D as do windtunnel tests with a clean surface. The difference between laminar and turbulent boundary layers is discussed in more detail on The Spell of Prandtl’s Laminar Boundary Layer. 
More precisely, the large values L/D ~ 100 in the wind tunnel tests are most pronounced for small aoa, in accordance with a common belief that D increases slower than linearly in aoa (e.g. quadratically in L with 
L being linear in aoa),  in the case of a laminar boundary layer. However, in Figure 2 we rather see a linear 
growth of D in aoa, as a result of using slip as a model of a turbulent boundary layer. 
Thus there seems to be evidence that the wind tunnel flow with clean surface for small aoa has a non-separating laminar boundary layer with small drag increasing very slowly with aoa seemingly giving L/D = 100, while in reality the boundary layer is turbulent with drag increasing linearly with aoa giving instead L/D = 10-30. As always, discrepancy between model and reality must be blamed on the model, and the wind tunnel result L/D = 100 thus seems to lack significance. 

L/D of Gliders

Max L/D for gliders range from 17 for a classic  Grunau Baby over 34 for a standard Libelle up to 50 for an extreme ASG29. The polar curve connects different forward speeds with sink speeds and the tangent to this curve through the origin gives the best glide angle or max L/D. With non-separating laminar boundary layer max L/D would occur for small aoa and high speeds. In reality, max L/D is achieved at low speeds because the boundary layer becomes turbulent for higher speeds which reduces L/D according to the above discussion. See Arizona Glider Stuff and compare with the polar curve for Sparrowhawk showing stall speed at 50 km/h and max L/D = 37 at 70 km/h and that L/D drops with a factor 2-3 for higher speeds.

The Secret of Sailing

Both the sail and keel of a sailing boat under tacking against the wind, act like wings generating lift and drag, but the action, form and aoa of the sail and the keel are different.  The boat is pulled ahead by the sail by a forward force component from lift but also from a component from negative drag on the leeward side of the sail at the leading edge (close to the mast) compensating for the positive drag from the rear leeward side of the sail, while there is little positive drag from the windward side of the sail (as opposed to a wing profile).

The result is a forward pull from the sail combined with a side force from lift which tilts the boat and needs to be balanced by lift from the the keel in the opposite direction. The shape of a sail is different from that of a wing which gives smaller drag from the windward side and thus improves forward pull, while the keel has the shape of a wing and acts like a wing.

The aoa of a sail is typically around 20 degrees to give maximal pull forward from maximal lift with contribution also from the rear part of the sail, like for a wing just before stall, while the drag is smaller than for a wing at aoa of 20 degrees, as just motivated. The aoa of a keel is about 10 degrees with an efficient lift/drag ratio about 13. This explains why the keel in modern designs often is much more narrow than the sail.

Inspection of the movies shows that for small angles of attack the lift is concentrated to the forward upper part of the wing, while for maximal lift at about 22 degrees, the whole upper wing is engaged.  Compare the computational movies with  WindTunnel Movies and notice in particular the separation on the leeward side of the main sail.

The material in this section is developed in more detail on the Knol Why It Is Possible to Sail.
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