*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):

- How Airplanes Work
- How-do-airplanes-fly-really?
- Airfoil-Lifting-Force-Misconceptions
- NY Times: What does keep them up there?
- NPR Radio: Why Planes Fly.
- See How It Flies (
*The wing produces lift “because” of circulation)*. - Bernoullli Or Newton: Who’s Right About Lift? (
*There’s more than one correct way to explain lift*) - Aircraft Owners and Pilots Association:
*Air flowing over the longer distance of**the curved upper wing surface, must travel faster than the air flowing the shorter distance under the flatter bottom surface of the wing…produces lower pressure above…* *Recreational Aviation Australia: There is a long-held and still-continuing argument, particularly in newsgroups and other internet venues, about the pros and cons of the various lift generation theories. None of the arguments put forward (often ill-informed) affect in any way how an aircraft flies, how it should be safely and economically operated, or how it should be built; so it is best to ignore them…*- Hyperphysics:
*Which is best for describing how aircraft get the needed lift to fly? Bernoulli’s equation or Newton’s laws and conservation of momentum? This has been an extremely active debate among those who love flying and are involved in the field….this treatment will not settle it. But perhaps it can at least indicate the lines of the discussion.*

## 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:

**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

velocity:

**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 ):

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.

**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 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.

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.**

#### Phase 1: 0 < aoa < 4-6

#### Phase 2: 4-6< aoa <16

#### Phase 3: 16 < aoa <20

## Comparison with Experiments/Reality

**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**.

**L/D**over time:

*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:

This figure shows (L/D)max, a measure of aerodynamic efficiency, plotted against Mach number for an optimum straight-wing and swept-wing airplane.* Credits – NASA*The following plot from Allstar Aeronautics Learning Laboratory shows consistent results:

**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**.

**L/D**of a kite (which acts like a wing) can be directy read from the angle of the line to the kite

**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.

**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):

### Correct Drag and Lift with Better Resolution

## L/D = 100 Is Not Possible in Reality

**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.

**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**).

**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.

**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

**D**in

**aoa**, as a result of using slip as a model of a turbulent boundary layer.

**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.