Normat 57:4, 145–169 (2009) 145

The Mathematical Secret of Flight

Johan Hoffman and Claes Johnson

Computational Technology Laboratory

School of Computer Science and Communication

KTH

jhoffman@csc.kth.se, cgjoh@kth.se

1 The Mystery of Flight

When you lean back for take-off in a jumbojet, maybe the following question ﬂashes

through your mind: How is it possible that the 400 squaremeter wings can carry 400 tons

at a wingload of 1 ton per squaremeter in sustained ﬂight in the air? Or maybe you are

satisﬁed with some of the explanations offered in popular science, like higher velocity and

lower pressure on the upper surface of the wing because it is curved and air there has a

longer path to travel than below? Or maybe you are an aeroplane engineer or pilot and

know very well why an airplane can ﬂy?

In either case, you should get a bit worried by reading that the authority NASA on its

website [43] dismissses all popular science theories for lift, including your favorite one,

as being incorrect, but then refrains from presenting any theory claimed to be correct!

NASA surprisingly ends with an empty out of reach: To truly understand the details of the

generation of lift, one has to have a good working knowledge of the Euler Equations. The

Plane&Pilot Magazine [44] has the same message and New York Times [8] informs us:

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?

2 Overview

In this arcticle we present a new mathematical and physical explanation of the generation

of lift L and drag D of a wing based on new discoveries of the dynamics of turbulent air-

ﬂow around a wing, obtained by computational solution of the basic mathematical model

of ﬂuid dynamics: the Navier-Stokes/Euler equations. When ﬂying in the air, the down-

ward gravitational force is balanced by upward wing lift L, while backward wing drag D

is balanced by forward thrust from engine, and wing-beat for birds, or descent in gliding

ﬂight without forward thrust.

The ﬁrst author would like to acknowledge the ﬁnancial support from the Swedish Foundation for Strategic

Research, and the Starting Grant of the European Research Council.

146 Johan Hoffman and Claes Johnson Normat 4/2009

We show that a wing creates lift as a reaction force from redirecting air downwards,

referred to as downwash, with less than 1/3 coming from the lower wing surface pushing

air down and the major remaining part from the upper surface sucking air down, with a

resulting lift/drag quotient

of size 10 − 20.

The enigma of ﬂight is why the air ﬂow separates from the upper wing surface at the

trailing edge, and not before, with the ﬂow after separation being redirected downwards

according to the tilting of the wing or angle of attack. We will reveal the secret to be

an effect of a fortunate combination of features of slightly viscous incompressible ﬂow in-

cluding a crucial instability mechanism at separation analogous to that seen in the swirling

ﬂow down a bathtub drain, generating both suction on the upper wing surface and drag.

We show that this mechanism of lift and drag is operational for angles of attack smaller

than a critical value of about 16 − 20 degrees depending on the shape of the wing, for

which the ﬂow separates from the upper wing surface well before the trailing edge with a

sudden increase of drag and decrease of lift referred to as stall.

It is absolutely crucial that

is large, of size 10 or bigger, since otherwise the muscle

power of a bird would not sufﬁce, and the fuel consumption of an airplane would be pro-

hibitive. Flying on a tilted barn door at 45 degrees angle of attack with

≈ 1, is not an

option.

An outline of the article is as follows: We ﬁrst recall classical theories for lift and drag

and then in pictures describe the new theory. We support the new theory by computational

solutions of the Navier-Stokes equations, also showing that the classical theories are in-

correct. We then present basic aspects of the mathematics of turbulent solutions of the

Navier-Stokes equations underlying the new theory.

3 Newton, d’Alembert and Kutta-Zhukovsky

The problem of explaining why it is possible to ﬂy in the air using wings has haunted

scientists since the birth of mathematical sciences. The mystery is how a sufﬁciently large

ratio

can be created.

In the gliding ﬂight of birds and airplanes with ﬁxed wings at subsonic speeds,

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 Charles Lindberg could cross the Atlantic in 1927

at a speed of 50 m/s in his 2000 kg Spirit of St Louis at an effective engine thrust of 150

kp (with

= 2000/150 ≈ 13) from 100 horse powers.

By Newton’s 3rd law, lift must be accompanied by downwash with the wing redirecting

air downwards. The enigma of ﬂight is the mechanism of a wing generating substantial

downwash at small drag, which is also the enigma of sailing against the wind with both

sail and keel acting like wings creating substantial lift [30].

Classical mathematical mechanics could not give an answer to the mystery of gliding

ﬂight: Newton computed by elementary mechanics the lift of a tilted ﬂat plate redirecting a

horisontal stream of ﬂuid particles, but obtained a disappointingly small value proportional

to the square of the angle of attack. To Newton the ﬂight of birds was inexplicable, and

human ﬂight certainly impossible.

D’Alembert followed up in 1752 by formulating his paradox about zero lift/drag of

inviscid incompressible irrotational steady ﬂow referred to as potential ﬂow, which seemed

to describe the airﬂow around a wing since the viscosity of air is very small so that it can

Normat 4/2009 Johan Hoffman and Claes Johnson 147

be viewed as being inviscid (with zero viscosity). Mathematically, potential ﬂow is given

as the gradient of a harmonic funtion satisfying Laplace’s equation.

At speeds less than say 300 km/h air ﬂow is almost incompressible, and since a wing

moves into still air the ﬂow could be expected to be irrotational without swirling rotat-

ing vortices. D’Alembert’s mathematical potential ﬂow thus seemed to capture physics,

but nevertheless had neither lift nor drag, against all physical experience. The wonderful

mathematics of potential ﬂow and harmonic functions thus showed to be without physi-

cal relevance: This is D’Alembert’s paradox which came to discredit mathematical ﬂuid

mechanics from start [26, 48, 7].

To explain ﬂight d’Alembert’s paradox had to be resolved, but nobody could ﬁgure out

how and it was still an open problem when Orwille and Wilbur Wright in 1903 showed

that heavier-than-air human ﬂight in fact was possible in practice, even if mathematically

it was impossible.

Mathematical ﬂuid mechanics was then saved from complete collapse by the young

mathematicians Kutta and Zhukovsky, called the father of Russian aviation, who explained

lift as a result of perturbing potential ﬂow by a large-scale circulating ﬂow or circulation

around the two-dimensional section of a wing, and by the young physicist Prandtl, called

the father of modern ﬂuid dynamics, who explained drag as a result of a viscous boundary

layer [45, 46, 47, 9].

This is the basis of state-of-the-art [16, 37, 14, 2, 19, 49, 50], which essentially is a

simplistic theory for lift without drag at small angles of attack in inviscid ﬂow and for

drag without lift in viscous ﬂow. However, state-of-the-art does not supply a theory for

lift-and-drag covering the real case of 3d slightly viscous turbulent ﬂow of air around a 3d

wing of a jumbojet at the critical phase of take-off at large angle of attack (12 degrees)

and subsonic speed (270 km/hour), as evidenced in e.g. [1, 3, 4, 6, 8, 10, 34, 36, 41].

The simplistic theory allows an aeroplane engineer to roughly compute the lift of a wing

at cruising speed at a small angle of attack, but not the drag, and not lift-and-drag at the

critical phase of take-off [42, 13]. The lack of mathematics has to be compensated by

experiment and experience. The ﬁrst take off of the new Airbus 380 must have been a

thrilling experience for the design engineers.

4 From Old to New Theory of Flight

A couple of years ago we stumbled upon a resolution of d’Alembert’s paradox [25, 26],

when computing turbulent solutions of the basic mathematical model of ﬂuid mechanics,

the Navier-Stokes equations. The resolution naturally led us to a new theory of ﬂight,

which we will explain below. You will ﬁnd that it is quite easy to grasp, because it can be

explained using different levels of mathematics. We start out easy with the basic principle

in concept form and then indicate some of the mathematics with references to more details.

Supporting information is given in the Google knols [32] and [33].

Before proceeding to work we recall both folklore and state-of-the-art mathematics ex-

plantions of ﬂight as being either correct but trivial, or nontrivial but incorrect, as follows:

Downwash generates lift: trivial without explanation of reason for downwash from

suction on upper wing surface.

Low pressure on upper surface: trivial without explanation why.

148 Johan Hoffman and Claes Johnson Normat 4/2009

Low pressure on curved upper surface because of higher velocity (by Bernouilli’s

law), because of longer distance: incorrect.

Coanda effect: The ﬂow sticks to the upper surface by viscosity: incorrect.

Kutta-Zhukovsky: Lift comes from circulation: incorrect.

Prandtl: Drag comes mainly from viscous boundary layer: incorrect.

5 The Principle of Flying

We will ﬁnd that the secret of ﬂight is revealed in Fig. 1: To the left we see potential ﬂow

around a portion of a long wing with zones of high (H) and low (L) pressure giving no

net lift, because the pressure is high on top of the wing at the trailing edge and low below.

This makes the ﬂow leave the wing in the same direction as it approaches, thus without

downwash and lift.

Potential ﬂow is a mathematical solution without lift/downwash of the Navier-Stokes

equations (with vanishing viscosity), which however is fundamenatlly different from the

ﬂow observed in reality with lift/downwash. Potential ﬂow is a ﬁctional mathematical

solution without physical relevance, and the reason hides the secret of both d’Alembert’s

paradox and ﬂight: Potential ﬂow is very sensitive to a speciﬁc form of perturbation and

thus is unstable and non-physical.

Potential ﬂow is similar to an inverted pendulum in upright equilibrium or a pen balanc-

ing on its tip, which is a mathematical solution of the equations of motion, but an unstable

non-physical solution which under a small perturbation away from the fully upright po-

sition will change into a different swinging motion. Potential ﬂow without lift/downwash

changes under a speciﬁc form of perturbation into a different more stable physical ﬂow

with lift/downwash, with a turbulent ﬂuctuating layer including the perturbation attaching

to the trailing edge, as we will see in computational simulations below with movies on

[31].

Figure 1: Correct explanation of lift by perturbation of potential ﬂow (left) at separation from

physical low-pressure turbulent counter-rotating rolls (middle) changing the pressure and ve-

locity at the trailing edge into a ﬂow with downwash and lift (right).

The speciﬁc form of perturbation is illustrated in the middle picture of Fig.1 showing

a layer of counter-rotating rolls of swirling ﬂow attaching to the trailing edge, with each

roll similar to the swirling ﬂow in a bathtub drain. The layer of rolls is distributed all along

the trailing edge and is not related to the wing tip vortex, which often is seen at landing

in moist air, since we assume the wing to be long. The perturbation switches the pressure

Normat 4/2009 Johan Hoffman and Claes Johnson 149

distribution of potential ﬂow at the trailing edge since the pressure inside the rolls is low,

into the ﬂow depicted to the right which has both lift, downwash and drag.

The speciﬁc perturbation thus hides the secret of ﬂight as a ﬂow with both lift, down-

wash and drag. By understanding mathematically the origin and nature of the instability

mechanism generating the counter-rotating rolls at the separation of potential ﬂow, which

we do in more detail below, we will be able to reveal the mathematical secret of ﬂight. In

short, the counterrotating rolls develop when the opposing ﬂows from above and below

meet on top of the wing before separation and ﬁrst are retarded and then accellerated and

stretched in the ﬂow direction, as shown in detail in [25, 26, 24, 23]. We understand that

inside the rolls of swirling ﬂow the pressure must be low to keep the roll together, and it

is this low pressure that annihilates the high pressure on top to allow the ﬂow to leave the

wing in the direction of the upper surface tangent with substantial downwash as illustrated

in the ﬁgure.

We see that the fundamental instability mechanism changes the ﬂow at the trailing edge

to give lift, but does not change the ﬂow at the leading edge where the ﬂow gives positive

lift. Real ﬂow thus shares a very important property with potential ﬂow, namely to not

separate at the crest of the ﬂow above the leading edge. If it did, downwash and lift would

be lost: This is what happens when a wing stalls at a too large angle of attack.

Summing up we have that lift comes from the instability mechanism at separation con-

sisting of counter-rotating low-pressure rolls of swirling ﬂow, which also creates drag by

suction from the low pressure. Thus lift comes along with drag: No lift without drag. Lift

without drag is an illusion, although still a common dream.

6 Comparison with Kutta-Zhukovsky

We compare with the classical explanation presented by Kutta-Zhukovsky illustrated in

Fig.2, which you ﬁnd in most books claiming to explain ﬂight: We see again potential ﬂow,

now around a section of the wing, but combined with a different perturbation consisting of

large scale circulating ﬂow around the wing. This perturbation also changes the pressure

distribution to give lift/downwash as illustrated in the picture to the right. However, as

we will see below, the circulating ﬂow around the wing does not arise in reality: Kutta-

Zhukovsky’s circulating ﬂow is purely ﬁctional and generates lift/downwash by a non-

physical mechanism which does not occur in reality.

Nevertheless, with no alternative in sight, Kutta-Zhukovsky’s trick to generate down-

wash (lift) is generally viewed as a mathematically sophisticated way of explaining ﬂight,

beyond comprehension for most people. We shall ﬁnd that the true reason it cannot be un-

derstood, is that it does not make sense, simply because there is no physical mechanism to

generate the large scale circulation around the wing, nor the associated so-called starting

vortex behind the wing supposedly balancing the circulation indicated in the right picture

of Fig.2.

We observe that Kutta-Zhulovsky ﬂow is two-dimensional, since both potential ﬂow

and circulation is constant in the wing direction and thus can be depicted in a plane ﬁgure,

while the true ﬂow is fully three-dimensional with the speciﬁc perturbation bringing in a

variation in the wing direction. Kutta-Zhukovsky ﬂow is like potential ﬂow a non-physical

two-dimensional stationary ﬂow, while the real ﬂow around a wing is a three-dimensional

partially ﬂuctuating turbulent ﬂow.

150 Johan Hoffman and Claes Johnson Normat 4/2009

Figure 2: Incorrect Kutta-Zhukovsky explanation of lift by perturbation of potential ﬂow (left)

by unphysical circulation around the section (middle) resulting in ﬂow with downwash/lift and

starting vortex (right).

7 Effects of Small Viscosity

We conclude that ﬂying is possible because of a fortunate combination of the following

properties of real slightly viscous incompressible ﬂow:

non-separation at the crest of a wing because the ﬂow is there similar to potential

ﬂow,

the instability mechanism of potential ﬂow at separation changes the pressure distri-

bution at the trailing edge to give lift, and drag.

Slightly viscous ﬂow has small skin friction along the boundary, which makes it similar

to potential ﬂow with zero skin friction satisfying a slip boundary condition at a solid

boundary modeling that ﬂuid particles can slide along the boundary without friction. Small

skin friction can thus be modeled by zero skin friction requiring the normal velocity to

vanish at the boundray, but imposing no restriction on the tangential velocity.

For a more viscous ﬂuid like syrup with larger skin friction, instead a no-slip boundary

condition is used requiring that both normal and tangential ﬂow velocities vanish on the

boundary modeling that ﬂuid particles close to the boundary have small speed and connect

to the interior ﬂow by a boundary layer where the ﬂow speed changes from zero to the free

stream speed. The effect of a no-slip boundary condition causing a boundary layer, is that

the ﬂow separates at the crest with loss of lift as compared to slightly viscous ﬂow. This

is because in a viscous boundary layer the pressure gradient normal to the boundary van-

ishes and thus cannot contribute to the normal acceleration required to keep ﬂuid particles

following the curvature of the boundary after the crest, as shown in detail in [27]. It is thus

the slip boundary condition modeling a turbulent boundary layer in slightly viscous ﬂow,

which forces the ﬂow to suck to the upper surface and create downwash. Gliding ﬂight in

viscous ﬂow is thus not possible, which explains why small insects do not practice gliding

ﬂight because to them air appears to be viscous.

8 Wellposedness vs Clay Millennium Problem

In order to judge the physical relevance of a mathematical solution, stability must be as-

sessed. Only wellposed solutions which are suitably stable in the sense that small pertur-

bations have small effects when properly measured, have physical signiﬁcance as observ-

able pheonomena, as made clear by in particular the mathematician J. Hadamard in 1902

Normat 4/2009 Johan Hoffman and Claes Johnson 151

[15]. However, the completely crucial and fundamental question whether solutions of the

Navier-Stokes equations are wellposed, has not been studied because lack of mathematical

techniques for quantitative analysis, as evidenced in the formulation of the Clay Millen-

nium Prize Problem on the Navier-Stokes equations excluding wellposedness [28, 24]. G.

Birkhoff was heavily criticized for posing this question in [7], and refrained from further

studies. The ﬁrst step towards resolution of d’Alembert’s paradox and the mathematical se-

cret of ﬂight is thus to pose the question if potential ﬂow is wellposed, and then to realize

that it is not. It took 256 years to take these steps.

9 Computed Lift and Drag

We now a take a closer look at solutions of the Navier-Stokes equations, computed by the

General Galerkin ﬁnite element method G2 [25]. These solutions should tell us the truth

because the Navier-Stokes equations express the basic laws of physics of conservation of

mass and momentum (Newton’s 2nd law), which cannot be doubted. We focus on the case

of slightly viscous incompressible ﬂow of relevance for airplanes at subsonic speeds and

larger birds. The fact that the ﬂuid has small viscosity is of crucial importance both for

physics and computation: First, the ﬂow is then turbulent with a turbulent boundary layer

allowing the ﬂow to suck to the upper surface of the wing and cause downwash and lift.

Second, a turbulent boundary layer can be modeled by a slip or small friction boundary

condition which makes it possible to simulate the ﬂow without computationally resolving

thin boundary layers, which is impossible with any forseeable computer [42].

We have indicated that the basic mechanism for the generation of lift of a wing consists

of counter-rotating rolls of low-pressure streamwise vorticity (swirling ﬂow) generated by

instability at separation, which reduce the high pressure on top of the wing before the

trailing edge of potential ﬂow and thus allow downwash, but which also generate drag. At

closer examination of the quantitative distributions of lift and drag forces around the wing,

we discover large lift at the expense of small drag resulting from leading edge suction,

which answers the opening question of of how a wing can generate a lift/drag ratio larger

than 10.

The secret of ﬂight is in concise form uncovered in Fig. 3 showing G2 computed lift

and and drag coefﬁcients of a Naca 0012 3d wing as functions of the angle of attack α,

as well as the circulation around the wing. We see that the lift and drag increase roughly

linearly up to 16 degrees, with a lift/drag ratio of about 13 for α > 3 degrees, and that lift

peaks at stall at α = 20 after a quick increase of drag, and ﬂow separation at the leading

edge.

We see that the circulation remains small for α less than 10 degrees without connection

to lift, and conclude that the theory of lift by Kutta-Zhukovsky is ﬁctional without physical

correspondence: There is lift but no circulation. Lift does not originate from circulation.

The incorrect explanation by Kutta-Zhukovsky is illustrated in Fig. 2 which is found in

books on ﬂight aerodynamics.

Inspecting Figs. 4-6 showing velocity, pressure and vorticity and Fig. 7 showing lift and

drag distributions over the upper and lower surfaces of the wing (allowing also pitching

moment to be computed), we can now, with experience from the above preparatory anal-

ysis, identify the basic mechanisms for the generation of lift and drag in incompressible

slightly viscous ﬂow around a wing at different angles of attack α: We ﬁnd two regimes

152 Johan Hoffman and Claes Johnson Normat 4/2009

0 5

5 30

0.5

1.5

2.5

3.5

0 5

5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 30

0 50 60

Figure 3: G2 lift coefﬁcient and circulation as functions of the angle of attack (top), drag

coefﬁcient (middle) and lift/drag ratio (bottom) as functions of the angle of attack.

Normat 4/2009 Johan Hoffman and Claes Johnson 153

before stall at α = 20 with different, more or less linear growth in α of both lift and drag,

a main phase 0 ≤ α < 16 with the slope of the lift (coefﬁcient) curve equal to 0.09 and

of the drag curve equal to 0.008 with L/D ≈ 14, and a ﬁnal phase 16 ≤ α < 20 with

increased slope of both lift and drag. The main phase can be divided into an initial phase

0 ≤ α < 4 − 6 and an intermediate phase 4 − 6 ≤ α < 16, with somewhat smaller slope

of drag in the initial phase. We now present details of this general picture.

10 Phase 1: 0 ≤ α ≤ 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 attaching

to the trailing edge. As α increases the low pressure below gets depleted as the incoming

ﬂow becomes parallel to the lower surface at the trailing edge for α = 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.

11 Phase 2: 4 − 6 ≤ α ≤ 16

The low pressure on top of the leading edge intensiﬁes to create a normal gradient pre-

venting 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.

12 Phase 3: 16 ≤ α ≤ 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 α = 20 with rapid loss of lift after stall.

154 Johan Hoffman and Claes Johnson Normat 4/2009

Figure 4: G2 computation of velocity magnitude (upper), pressure (middle), and non-

transversal vorticity (lower), for angles of attack 2, 4, and 8

◦

(from left to right). Notice

in particular the rolls of streamwise vorticity at separation.

Normat 4/2009 Johan Hoffman and Claes Johnson 155

Figure 5: G2 computation of velocity magnitude (upper), pressure (middle), and topview of

non-transversal vorticity (lower), for angles of attack 10, 14, and 18

◦

(from left to right).

Notice in particular the rolls of streamwise vorticity at separation.

156 Johan Hoffman and Claes Johnson Normat 4/2009

Figure 6: G2 computation of velocity magnitude (upper), pressure (middle), and non-

transversal vorticity (lower), for angles of attack 20, 22, and 24

◦

(from left to right).

Normat 4/2009 Johan Hoffman and Claes Johnson 157

0 0

6 0

1 1.2 1.4 1.

−2

0 0

6 0

1 1.2 1.4 1.

−1

Figure 7: G2 computation of normalized local lift force (upper) and drag force (lower) contri-

butions acting along the lower and upper parts of the wing, for angles of attack 0, 2 ,4 ,10 and

◦

, each curve translated 0.2 to the right and 1.0 up, with the zero force level indicated for

each curve.

158 Johan Hoffman and Claes Johnson Normat 4/2009

13 Lift and Drag Distribution Curves

The distributions of lift and drag forces over the wing resulting from projecting the pres-

sure acting perpendicular to the wing surface onto relevant directions, are plotted in Fig.7.

The total lift and drag results from integrating these distributions around the wing. In po-

tential ﬂow computations (with circulation according to Kutta-Zhukovsky), only the pres-

sure distribution or c

-distribution is considered to carry releveant information, because a

potential solution by construction has zero drag. In the perspective of Kutta-Zhukovsky, it

is thus remarkable that the projected c

-curves carry correct information for both lift and

drag.

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) intricate interplay of stability and instability

of potential ﬂow: The main lift comes from upper surface suction arising from a turbulent

boundary layer with small skin friction combined with rear separation instability generat-

ing low-pressure streamwise vorticity, while the drag is kept small by negative drag from

the leading edge.

14 Comparing Computation with Experiment

Comparing G2 computations with about 150 000 mesh points with experiments [20, 40],

we ﬁnd good agreement 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

angles of attack. The oil-ﬁlm pictures in [20] show surface vorticity generating streamwise

vorticity in accordance with [24, 27, 23].

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 compu-

tations 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, corresponding to the

maximal lift of 2.5 in computation for a long wing and not a full aircraft. In take-off the

maximal lift is reported to be 1.75 with 1.5 in computation at a similar angle of attack.

We compute turbulent solutions of the Navier-Stokes equations using a stabilized ﬁnite

element method with a posteriori error control of lift and drag, referred to as General

Galerkin or G2, available in executable open source from [17]. The stabilization in G2

acts as an automatic turbulence model, and thus offers a model for ab initio computational

simulation of the turbulent ﬂow around a wing with the only input being the geometry of

the wing. The computations performed on a single workstation show good agreement with

experiments. We are now performing computations on super-computers allowing more

precise comparisons and paramater studies.

15 Navier-Stokes with Force Boundary Conditions

For the reader interested in the mathematics we now present the Navier-Stokes equations

along with a stability analysis exhibiting the basic instability mechanism at separation

which is crucial for the generation of lift, at the expense of some drag.

Normat 4/2009 Johan Hoffman and Claes Johnson 159

The Navier-Stokes equations for an incompressible ﬂuid of unit density with small

viscosity ν > 0 and small skin friction β ≥ 0 ﬁlling a volume Ω in R

surrounding a solid

body with boundary Γ over a time interval I = [0, T ], read as follows: Find the velocity

u = (u

, u

) 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,

= g on Γ × I,

= βu

on Γ × I,

u(·, 0) = u

in Ω,

(1)

where u

is the ﬂuid velocity normal to Γ, u

is the tangential velocity, σ = 2ν(u) is the

viscous (shear) stress with (u) the usual velocity strain, σ

is the tangential stress, f is a

given volume force, g is a given inﬂow/outﬂow velocity with g = 0 on a non-penetrable

boundary, and u

is a given initial condition. We notice the skin friction boundary condi-

tion coupling the tangential stress σ

to the tangential velocity u

with the friction coef-

ﬁcient β with β = 0 for slip, and β >> 1 for no-slip. We note that β is related to the

standard skin friction coefﬁeient c

2τ

with τ the tangential stress per unit area, by the

relation β =

. In particular, β tends to zero with c

(if U stays bounded).

Prandtl insisted on using a no-slip velocity boundary condition with u

= 0 on Γ,

because his resolution of d’Alembert’s paradox hinged on discriminating potential ﬂow

by this condition. On the oher hand, with the new resolution of d’Alembert’s paradox,

relying instead on instability of potential ﬂow, we are free to choose instead a friction

force boundary condition, if data is available. Now, experiments show [47, 11] that the

skin friction coefﬁcient decreases with increasing Reynolds number Re as c

∼ Re

−0.2

so that c

≈ 0.0005 for Re = 10

and c

≈ 0.007 for Re = 10

. Accordingly we model

a turbulent boundary layer by a friction boundary condition with a friction parameter β ≈

0.03URe

−0.2

. For very large Reynolds numbers, we can effectively use β = 0 in G2

computation corresponding to slip boundary conditions.

As developed in more detail in [27], we make a distinction between laminar (boundary

layer) separation modeled by no-slip and turbulent (boundary layer) separation modeled

by slip/small friction. Note that laminar separation cannot be modeled by slip, since a lam-

inar boundary layer needs to be resolved with no-slip to get correct (early) separation. On

the other hand, as will be seen below, in turbulent (but not in laminar) ﬂow the interior

turbulence dominates the skin friction turbulence indicating that the effect of a turbulent

boundary layer can be modeled by slip/small friction, which can be justiﬁed by an poste-

riori sensitivity analysis as shown in [27].

We thus assume that the boundary layer is turbulent and is modeled by slip/small fric-

tion, which effectively includes the case of laminar separation followed by reattachment

into a turbulent boundary layer.

16 Potential Flow

Potential ﬂow (u, p) with velocity u = ∇ϕ, where ϕ is harmonic in Ω and satisﬁes a

homogeneous Neumann condition on Γ and suitable conditions at inﬁnity, can be seen

as a solution of the Navier-Stokes equations for slightly viscous ﬂow with slip boundary

condition, subject to

160 Johan Hoffman and Claes Johnson Normat 4/2009

perturbation of the volume force f = 0 in the form of σ = ∇ · (2ν(u)),

perturbation of zero friction in the form of σ

= 2ν(u)

with both perturbations being small because ν is small and a potential ﬂow velocity u

is smooth. Potential ﬂow can thus be seen as a solution of the Navier-Stokes equations

with small force perturbations tending to zero with the viscosity. We can thus express

d’Alembert’s paradox as the zero lift/drag of a Navier-Stokes solution in the form of a

potential solution, and resolve the paradox by realizing that potential ﬂow is unstable and

thus cannot be observed as a physical ﬂow.

Potential ﬂow is like an inverted pendulum, which cannot be observed in reality be-

cause it is unstable and under inﬁnitesimal perturbations turns into a swinging motion. A

stationary inverted pendulum is a ﬁctious mathematical solution without physical corre-

spondence because it is unstable. You can only observe phenomena which in some sense

are stable, and an inverted pendelum or potential ﬂow is not stable in any sense.

Potential ﬂow has the following crucial property which partly will be inherited by real

turbulent ﬂow, and which explains why a ﬂow over a wing subject to small skin friction

can avoid separating at the crest and thus generate downwash, unlike viscous ﬂow with no-

slip, which separates at the crest without downwash. We will conclude that gliding ﬂight

is possible only in slightly viscous incompressible ﬂow. For simplicity we consider two-

dimensional potential ﬂow around a cylindrical body such as a long wing (or cylinder).

Theorem. Let ϕ be harmonic in the domain Ω in the plane and satisfy a homogeneous Neu-

mann condition on the smooth boundary Γ of Ω. Then the streamlines of the corresponding

velocity u = ∇ϕ can only separate from Γ at a point of stagnation with u = ∇ϕ = 0.

Proof. Let ψ be a harmonic conjugate to ϕ with the pair (ϕ, ψ) satisfying the Cauchy-

Riemann equations (locally) in Ω. Then the level lines of ψ are the streamlines of ϕ and

vice versa. This means that as long as ∇ϕ 6= 0, the boundary curve Γ will be a streamline

of u and thus ﬂuid particles cannot separate from Γ in bounded time.

17 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 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,

= 0 on Γ ×I,

v(·, 0) = u

− ¯u

in Ω,

(2)

Formally, with u and ¯u given, this is a linear convection-reaction-diffusion problem for

(v, q, τ) with the reaction term given by the 3 × 3 matrix ∇¯u being the main term of

concern for stability. By the incompressiblity, the trace of ∇¯u is zero, which shows that

in general ∇¯u has eigenvalues with real value of both signs, of the size of |∇¯u| (with | · |

some matrix norm), thus with at least one exponentially unstable eigenvalue.

Normat 4/2009 Johan Hoffman and Claes Johnson 161

Accordingly, we expect local exponential perturbation growth of size exp(|∇u|t) of a

solution (u, p, σ), in particular we expect a potential solution to be illposed. This is seen

in G2 solutions with slip initiated as potential ﬂow, which subject to residual perturbations

of mesh size h, in log(1/h) time develop into turbulent solutions. We give computational

evidence that these turbulent solutions are wellposed, which we rationalize by cancellation

effects in the linearized problem, which has rapidly oscillating coefﬁcients when linearized

at a turbulent solution.

Formally applying the curl operator ∇× to the momentum equation of (1), with ν =

β = 0 for simplicity, we obtain the vorticity equation

˙ω + (u · ∇)ω −(ω · ∇)u = ∇ × f in Ω, (3)

which is a convection-reaction equation in the vorticity ω = ∇ × u with coefﬁcients

depending on u, of the same form as the linearized equation (2), with similar properties

of exponential perturbation growth exp(|∇u|t) referred to as vortex stretching. Kelvin’s

theorem formally follows from this equation assuming the initial vorticity is zero and

∇ × f = 0 (and g = 0), but exponential perturbation growth makes this conclusion

physically incorrect: We will see below that large vorticity can develop from irrotational

potential ﬂow even with slip boundary conditions.

18 Energy Estimate with Turbulent Dissipation

The standard energy estimate for (1) is obtained by multiplying the momentum equation

˙u + (u · ∇)u + ∇p − ∇ · σ − f = 0,

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

Ω

(u, p) · u dxdt = D

(u; t) + B

(u; t) (4)

where

(u, p) = ˙u + (u · ∇)u + ∇p

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

(u; t) =

Ω

ν|(u(

t, x))|

dxd

is the internal turbulent viscous dissipation, and

(u; t) =

β|u

(

t, x)|

dxd

is the boundary turbulent viscous dissipation, from which follows by standard manipula-

tions of the left hand side of (4),

(u; t) + D

(u; t) + B

(u; t) = K(u

), t > 0, (5)

162 Johan Hoffman and Claes Johnson Normat 4/2009

where

(u; t) =

Ω

|u(t, x)|

dx.

This estimate shows a balance of the kinetic energy K(u; t) and the turbulent viscous

dissipation D

(u; t) + B

(u; t), with any loss in kinetic energy appearing as viscous dis-

sipation, and vice versa. In particular,

(u; t) + B

(u; t) ≤ K(u

and thus the viscous dissipation is bounded (if f = 0 and g = 0).

Turbulent solutions of (1) are characterized by substantial internal turbulent dissipa-

tion, that is (for t bounded away from zero),

D(t) ≡ lim

ν→0

D(u

; t) >> 0, (6)

which is Kolmogorov’s conjecture [18]. On the other hand, the skin friction dissipation

decreases with decreasing friction

lim

ν→0

(u; t) = 0, (7)

since β ∼ ν

0.2

tends to zero with the viscosity ν and the tangential velocity u

approaches

the (bounded) free-stream velocity. We thus ﬁnd evidence that the interior turbulent dissi-

pation dominates the skin friction dissipation, which supports the use of slip as a model of

a turbulent boundray layer, but which is not in accordance with Prandtl’s (unproven) con-

jecture that substantial drag and turbulent dissipation originates from the boundary layer.

Kolmogorov’s conjecture (6) is consistent with

k∇uk

∼

√

, kR

(u, p)k

∼

√

, (8)

where k · k

denotes the L

(Q)-norm with Q = Ω × I. On the other hand, it follows by

standard arguments from (5) that

(u, p)k

−1

≤

√

ν, (9)

where k · k

−1

is the norm in L

(I; H

−1

(Ω)). Kolmogorov thus conjectures that the Euler

residual R

(u, p) for small ν is strongly (in L

) large, while being small weakly (in H

−1

Altogether, we understand that the resolution of d’Alembert’s paradox of explaining

substantial drag from vanishing viscosity, consists of realizing that the internal turbulent

dissipation D can be positive under vanishing viscosity, while the skin friction dissipation

B will vanish. In contradiction to Prandtl, we conclude that drag does not result from

boundary layer effects, but from internal turbulent dissipation, originating from instability

at separation.

Normat 4/2009 Johan Hoffman and Claes Johnson 163

19 G2 Computational Solution

We show in [25, 24, 26] that the Navier-Stokes equations (1) can be solved by G2 pro-

ducing turbulent solutions characterized by substantial turbulent dissipation from the least

squares stabilization acting as an automatic turbulence model, reﬂecting that the Euler

residual cannot be made pointwise small in turbulent regions. G2 has a posteriori error

control based on duality and shows output uniqueness in mean-values such as lift and drag

[25, 21, 22]

We ﬁnd that G2 with slip is capable of modeling slightly viscous turbulent ﬂow with

Re > 10

of relevance in many applications in aero/hydro dynamics, including ﬂying,

sailing, boating and car racing, with hundred thousands of mesh points in simple geom-

etry and millions in complex geometry, while according to state-of-the-art quadrillions is

required [42]. This is because a friction-force/slip boundary condition can model a turbu-

lent boundary layer, and interior turbulence does not have to be resolved to physical scales

to capture mean-value outputs [25].

The idea of circumventing boundary layer resolution by relaxing no-slip boundary con-

ditions introduced in [21, 25], was used in [39, 5] in the form of weak satisfaction of no-

slip, which however misses the main point of using a force condition instead of a velocity

condition in a model of a turbulent boundary layer.

A G2 solution (U, P) on a mesh with local mesh size h(x, t) according to [25], satisﬁes

the following energy estimate (with f = 0, g = 0 and β = 0):

K(U(t)) + D

(U; t) = K(u

), (10)

where

(U; t) =

Ω

h|R

(U, P )|

dxdt, (11)

is an analog of D

(u; t) with h ∼ ν, where R

(U, P ) is the Euler residual of (U, P ).

We see that the G2 turbulent viscosity D

(U; t) arises from penalization of a non-zero

Euler residual R

(U, P ) with the penalty directly connecting to the violation (according

the theory of criminology). A turbulent solution is characterized by substantial dissipation

(U; t) with kR

(U, P )k

∼ h

−1/2

, and

(U, P )k

−1

≤

√

h (12)

in accordance with (8) and (9).

20 Wellposedness of Mean-Value Outputs

Let M(v) =

vψ dxdt 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 solution 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)

164 Johan Hoffman and Claes Johnson Normat 4/2009

where > denotes transpose. Multiplying the ﬁrst equation by u − U and integrating by

parts, we obtain the following output error representation [25]:

M(u) − M(U) =

(u, p) − R

(U, P )) · ϕ dxdt (14)

where for simplicity the dissipative terms are here omitted, from which follows the a pos-

teriori error estimate:

|M(u) − M(U)| ≤ S(kR

(u, p)k

−1

+ kR

(U, P )k

−1

), (15)

where the stability factor

S = S(u, U, M) = S(u, U) = kϕk

(Q)

. (16)

In [25] we present a variety of evidence, obtained by computational solution of the dual

problem, that for global mean-value outputs such as drag and lift, S << 1/

√

h, while

−1

∼

√

h, allowing computation of of drag/lift with a posteriori error control of the

output within a tolerance of a few percent. In short, mean-value outputs such as lift and

drag are wellposed and thus physically meaningful.

We explain in [25] the crucial fact that S << 1/

√

h, heuristically as an effect of can-

cellation of rapidly oscillating reaction coefﬁcients of turbulent solutions combined with

smooth data in the dual problem for mean-value outputs. In smooth potential ﬂow there is

no cancellation, which explains why zero lift/drag cannot be observed in physical ﬂows.

As an example, we show in Fig.8 turbulent G2 ﬂow around a car with substantial drag

in qualitative accordance with wind-tunnel experiments. We see a pattern of streamwise

vorticity forming in the rear wake. We also see surface vorticity forming on the hood

transversal to the main ﬂow direction. We see similar features in the ﬂow of air around a

wing.

Figure 8: Velocity of turbulent G2 ﬂow with slip around a car with courtesy of geometry Volvo

Cars and computations by Murtazo Nazarov.

Normat 4/2009 Johan Hoffman and Claes Johnson 165

21 Scenario for Separation without Stagnation

We now present a scenario for transition of potential ﬂow into turbulent ﬂow, based on

identifying perturbations of strong growth in the linearized equations (2) and (3) at sepa-

ration generating rolls of low pressure streamwise vorticity changing the pressure distri-

bution to give both lift and drag of a wing.

As a model of potential ﬂow at rear separation, we consider the potential ﬂow u(x) =

, −x

, 0) in the half-plane {x

> 0}. Assuming x

and x

are small, we approximate

the v

-equation of (2) by

˙v

− v

= f

where f

= f

) is an oscillating mesh residual perturbation depending on x

(including

also a pressure-gradient), for example f

) = h sin(x

/δ), with δ > 0. It is natural to

assume that the amplitude of f

decreases with δ. We conclude, assuming v

(0, x) = 0,

that

(t, x

) = t exp(t)f

and for the discussion, we assume v

= 0. Next we approximate the ω

-vorticity equation

for x

small and x

≥ ¯x

> 0 with ¯x

small, by

˙ω

+ x

∂ω

∂x

− ω

= 0,

with the “inﬂow boundary condition”

(¯x

, x

) =

∂v

∂x

= t exp(t)

∂f

∂x

The equation for ω

thus exhibits exponential growth, which is combined with exponential

growth of the “inﬂow condition”. We can see these features in Fig. 9 showing how oppos-

ing ﬂows on the back generate a pattern of co-rotating surface vortices which act as initial

conditions for vortex stretching into the ﬂuid generating rolls of low-pressure streamwise

vorticity, in the case of a wing attaching to the trailing edge.

Altogether we expect exp(t) perturbation growth of residual perturbations 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 result of a force

perturbation oscillating in the x

direction, which in the retardation of the ﬂow in the

-direction creates exponentially increasing vorticity in the x

-direction, which acts as

inﬂow to the ω

-vorticity equation with exponential growth by vortex stretching. Thus, we

ﬁnd exponential growth at rear separation in both the retardation in the x

-direction and

the accelleration in the x

direction. This scenario is illustrated in principle and computa-

tion in Fig.9. Note that since the perturbation is convected with the base ﬂow, the absolute

size of the growth is related to the length of time the perturbation stays in a zone of ex-

ponential growth. Since the combined exponential growth is independent of δ, it follows

that large-scale perturbations with large amplitude have largest growth, which is also seen

in computations with δ the distance between streamwise rolls as seen in Fig.3 which does

not seem to decrease with decreasing h.

166 Johan Hoffman and Claes Johnson Normat 4/2009

Notice that at forward attachment of the ﬂow the retardation does not come from op-

posing ﬂows, and the zone of exponential growth of ω

is short, resulting in much smaller

perturbation growth than at rear separation.

We can view the occurence of the rear surface vorticities as a mechanism of separation

with non-zero tangential speed, by diminishing the normal pressure gradient of potential

ﬂow, which allows separation only at stagnation. The surface vorticities thus allow sepa-

ration without stagnation but the price is generation of a system of low-pressure tubes of

streamwise vorticity creating drag in a form of “separation trauma” or “cost of divorce”.

The scenario for separation can summarized as follows: Velocity instability in retar-

dation as opposing ﬂows meet in the rear of the cylinder, generates a zig-zag pattern of

surface vorticity shown in Fig.9, allowing separation into counter-rotating low-pressure

rolls, attaching to the trailing edge in the case of a wing, as shown in Fig. 1.

Figure 9: Turbulent separation by surface vorticity forming counter-rotating low-pressure rolls

in ﬂow around a circular cylinder, illustrating separation at the trailing edge of a wing [23].

22 Stability of the Streamwise Vorticity Perturbed Flow

The rolls of streamwise vorticity swirling ﬂow appearing at separation because of the

instability of potential ﬂow represent a more stable ﬂow pattern. An indication of stability

is given by an analysis of the stability of the rotating ﬂow ﬁeld u = (0, x

, −x

) with a

linearized problem of the form

˙v

+ v

= 0, ˙v

− v

= 0 (17)

which does not have any exponentially unstable solutions. The swirling ﬂow at separation

is similar to the vortex seen at the drain of a bathtub.

23 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, geometrical shape and angle of attack of the sail

and the keel are different. The effective angle of attack of a sail is typically 15-20 degrees

and that of a keel 5-10 degrees, for reasons which we now give.

Normat 4/2009 Johan Hoffman and Claes Johnson 167

The boat is pulled forward by the sail, assuming for simplicity that the beam is parallel

to the direction of the boat at a minimal tacking angle, by the component L sin(15) of

the lift L, as above assumed to be perpendicular to the effective wind direction, but also

by the following contributions from the drag assumed to be parallel to the effective wind

direction: The negative drag on the leeeward side at the leading edge close to the mast gives

a positive pull which largely compensates for the positive drag from the rear leeward side,

while there is less positive drag from the windward side of the sail as compared to a wing

proﬁle, because of the difference in shape. The result is a forward pull ≈ sin(15)L ≈ 0.2L

combined with a side (heeling) force ≈ L cos(15) ≈ L, which tilts the boat and needs to

be balanced by lift from the the keel in the opposite direction. Assuming the lift/drag ratio

for the keel is 13, the forward pull is then reduced to ≈ (0.2 −1/13)L ≈ 0.1L, which can

be used to overcome the drag from the hull minus the keel.

The shape of a sail is different from that of a wing which gives smaller drag from

the windward side and thus improved forward pull, while the keel has the shape of a

symmetrical wing and acts like a wing. A sail with aoa 15−20 degrees gives maximal pull

forward at maximal heeling/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 15-20 degrees aoa (for

which the lift/drag ratio is 4-3), with the motivation given above. The lift/drag curve for a

sail is thus different from that of wing with lift/drag ratio at aoa 15-20 much larger for a

sail. On the other hand, a keel with aoa 5-10 degrees has a lift/drag ratio about 13. A sail at

aoa 15-20 thus gives maximal pull at strong heeling force and small drag, which together

with a keel at aoa 5-10 with strong lift and small drag, makes an efﬁcient combination.

This explains why modern designs combine a deep narrow keel acting efﬁciently for small

aoa, with a broader sail acting efﬁciently at a larger aoa.

Using a symmetrical wing as a sail would be inefﬁcient, since the lift/drag ratio is poor

at maximal lift at aoa 15-20. On the other hand, using a sail as a wing can only be efﬁcient

at a large angle of attack, and thus is not suitable for cruising. This material is developed

in more detail in [30].

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