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New Theory of Flight

Johan Hoffman∗

, Johan Jansson†and Claes Johnson‡

Abstract

We present a new mathematical theory explaining the ﬂuid mechanics of sub-

sonic ﬂight, which is fundamentally different from the existing boundary layer-

circulation theory by Prandtl-Kutta-Zhukovsky formed 100 year ago. The new the-

ory is based on our new resolution of d’Alembert’s paradox showing that slightly

viscous bluff body ﬂow can be viewed as zero-drag/lift potential ﬂow modiﬁed by

3d rotational slip separation arising from a speciﬁc separation instability of po-

tential ﬂow, into turbulent ﬂow with nonzero drag/lift. For a wing this separation

mechanism maintains the large lift of potential ﬂow generated at the leading edge

at the price of small drag, resulting in a lift to drag quotient of size 15 −20 for

a small propeller plane at cruising speed with Reynolds number Re ≈107and a

jumbojet at take-off and landing with Re ≈108, which allows ﬂight at affordable

power. The new mathematical theory is supported by computed turbulent solu-

tions of the Navier-Stokes equations with a slip boundary condition as a model

of observed small skin friction of a turbulent boundary layer always arising for

Re > 106, in close accordance with experimental observations over the entire

range of angle of attacks including stall using a few millions of mesh points for a

full wing-body conﬁguration.

1 Overview of new theory of ﬂight

We present a new theory of ﬂight of airplanes based on computational solution and

mathematical analysis of the incompressible Navier-Stokes equations with Reynolds

numbers in the range 106−108of relevance for small to large airplanes including

gliders. The new theory of ﬂight explains how an airplane wing can generate large lift

Lat small drag Dwith a lift-to-drag ratio L

Dranging from 15 −20 for a standard wing

up to 70 for the long thin wing of an extreme glider, allowing ﬂight at affordable power,

which can be viewed as a form of a miracle of real physics.

We show that the ﬂight of an airplane is computable with millions of mesh points,

without any need of resolving turbulent boundary layers because their effect of small

skin friction shows to be small on main ﬂow characterstics such as lift and drag. By

∗School of Computer Science and Communication, KTH, SE-10044 Stockholm, Sweden, and Basque

Center of Applied Mathematics (BCAM), Bilbao, Spain.

†Basque Center of Applied Mathematics (BCAM), Bilbao, Spain, and School of Computer Science and

Communication, KTH, SE-10044 Stockholm, Sweden.

‡School of Computer Science and Communication, KTH, SE-10044 Stockholm, Sweden.

1

a mathematical stability analysis of computed Navier-Stokes solutions we then show

that ﬂight is also understandable.

First outlined in [1, 2] the new theory of ﬂight comes out of our new resolution of

d’Alembert’s Paradox [3], explaining that zero lift/drag potential ﬂow around a bluff

body cannot be observed as a real physical ﬂow, because it is unstable with a speciﬁc

basic mode of instability arising from opposing ﬂow retardation at stagnation before

separation, with line stagnation along the trailing edge of a wing associated with a

zone of high pressure. This basic mode of instability develops into a quasi-stable sep-

aration pattern followed by a turbulent wake, which we describe as 3d rotational slip

separation with point stagnation. Here the high pressure zone of potential ﬂow is re-

placed by an oscillating pressure with a net suction effect thereby generating drag, and

for a wing also lift. We thus ﬁnd that the ﬂow of air around a wing can be described as

potential ﬂow modiﬁed by 3d rotational slip separation with point stagnation, a large-

scale ﬂow which is computationally resolvable except in a turbulent wake where the

under-resolution is of little effect for drag and lift.

The new theory shows that the miracle of ﬂight is possible because the ﬂow around

a wing: (i) is incompressible, and (ii) satisﬁes a slip boundary condition as a model of

the small skin friction of the turbulent boundary layer always arising for high Reynolds

number. It follows from (i) and (ii) that the ﬂow around a long smooth wing is 2d poten-

tial (incompressible, irrotational, stationary and constant in the axial direction) before

separation into a turbulent wake, and as such can only separate at stagnation with zero

ﬂow speed, because (as shown below) the wing section boundary is a streamline. 2d

potential ﬂow thus requires a normal pressure gradient to accelerate the ﬂow to follow

the upper wing surface directed downwards for a positive angle of attack (assuming

a symmetric airfoil), see Fig. 1, into what is referred to as downwash, which requires

low pressure or suction on the upper surface creating about 2/3 of the total lift with 1/3

from high pressure on the lower wing surface. In particular, the ﬂow does not separate

on the crest of the wing because there the ﬂow speed is maximal, far from stagnation,

and there maximal lift is generated.

Accordingly there is no miracle of ﬂight in moderate or low Reynolds number

viscous ﬂow, with a laminar boundary layer satisfying a no-slip boundary condition

with zero normal pressure gradient, which separates on the crest without much lift

from the upper surface. A small ﬂy cannot glide on ﬁxed wings like a big albatross

because the Reynolds number is too small, and has to compensate by very rapid wing

ﬂapping.

The new theory makes ﬂight conceptually readily understandable by ﬁrst recalling

that for (horisontal) potential ﬂow around a circular cylinder the pressure drop on top

(and bottom) is 3 times the pressure rise in the front (and back) with the ﬂow speed on

top twice the incoming speed in accordance with Bernoulli’s law. Viewing then a wing

as being formed by two circular cylinders of different radii stretching a tube of fabric

as shown in Fig. 2, suggests a pressure drop on top about three times the pressure rise

up front, which, combined with 3d rotational slip separation without mean pressure

rise or drop and with a stretching factor of 3, would give L

D>9. Flight thus shows

to be understandable as potential ﬂow modiﬁed by 3d rotational slip separation with

point stagnation, as illustrated in Fig. 1 showing the generation of large lift on top

of the leading edge of potential ﬂow and the separation without mean pressure rise

2

maintaining large lift at the price of small drag, which can be viewed as a form of

elegant separation. Large lift thus results from strong suction on the upper part of the

leading edge of non-separating potential ﬂow redirecting incoming ﬂow downwards

followed by elegant separation without mean pressure rise or drop.

Figure 1: New theory showing potential ﬂow (upper left) being modiﬁed at separation

by the main instability mode consisting of counter-rotating rolls of streamwise vortic-

ity attaching to the trailing edge (upper middle) to avoid the high pressure buildup at

separation which results in a ﬂow with downwash and lift and also drag (upper right),

where high (H) and low (L) surface pressure is indicated in the ﬁgure. Snapshots of

instantaneous ﬂow past a NACA 0012 airfoil at angles of attack 4◦(lower left) and

10◦(lower right), simulated by computational solution of the Navier-Stokes equations,

visualised as magnitude of velocity and limiting streamlines, showing counter-rotating

rolls of streamwise vorticity at separation.

In this paper we ﬁrst highlight differences between the new theory and the classical

textbook theory in Section 2, we then present our basic model of subsonic ﬂight in

Section 3, which we validate against experimental data in Section 4. In Section 5 we

present the basic elements of the new theory, which we summarise in the concluding

Section 6 and Section 7.

3

Figure 2: A model of a wing can be constructed by stretching a tube of fabric around

two circular cylinders.

2 New theory vs classical textbook theory

The new theory of drag and lift of a wing is fundamentally different from the textbook

theory [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] of Prandtl for drag (without lift) based on re-

duced Navier-Stokes equations in the form of boundary layer equations with no-slip

boundary conditions, and Kutta-Zhukovsky theory for lift (without drag) of potential

ﬂow with slip boundary conditions augumented by a certain amount of large scale cir-

culation around the wing section determined by a Kutta condition of smooth separation

at the trailing edge. This is the Prandtl-Kutta-Zhukovsky theory which was developed

shortly after powered ﬂight was shown to be possible by the Wright brothers in 1903,

breaking the perceived mathematical impossibility of powered ﬂight based on New-

ton’s incorrect theory of lift from air hitting the wing from below. We ﬁnd that the

textbook theory of ﬂight attributed to Prandtl-Kutta-Zhukovsky is incorrect, since it is

based on phenomena which are not carried by solutions of the Navier-Stokes equations.

Speciﬁcally we ﬁnd that Prandtl’s boundary layer theory with no-slip boundary

condition for drag is incorrect, since we combine Navier-Stokes equations with a slip

boundary condition modeling the observed small skin friction of high Reynolds num-

ber ﬂow, which does not give rise to boundary layers, and yet compute drag in close

agreement with experimental observation. We conclude that the main part of the ob-

served drag of a wing does not originate from boundary layers with no-slip boundary

conditions.

Further, we ﬁnd that the Kutta-Zhukovsky theory for lift, based on circulation gen-

erated by a sharp trailing edge is incorrect, since we combine Navier-Stokes equations

with a rounded trailing edge and yet get lift in close agreement with experimental ob-

servation. We conclude that the observed lift of a wing does not originate from a sharp

trailing edge as postulated in the Kutta-Zhukovsky theory. On the other hand, we re-

cover the Kutta condition demanding smooth (or elegant) separation at the trailing edge

as effectively being realized as 3d rotational slip separation, but without connection to

large scale circulation as suggested by Kutta-Zhukovsky.

Although circulation can theoretically generate lift, the logic that since lift is ob-

4

served there must be circulation, is an example of Aristotle’s logical fallacy of conﬁrm-

ing the assumption by observing the consequence. In his book Hydrodynamics [14]

from 1950, the mathematician Garret Birkhoff also questioned [15] Prandtl’s boundary

layer resolution of d’Alembert’s paradox [16], but was criticized in a review by James J.

Stoker [17] in strong support of the Prandtl-Kutta-Zhukovsky theory. In fact, as noted

e.g. in [18, 19, 20], the generation of circulation has never been given a convincing

explanation, and the new theory shows that there is none.

The new theory shows that ﬂight is both computable and understandable and chal-

lenges the legacies of Prandtl and Kutta-Zhukovsky, and thereby asks for a complete

revision of fundamental parts of textbooks. The state-of-the-art is that computational

simulation the ﬂow of air around an airplane to capture lift and drag, requires bound-

ary layer resolution with more than 1016 mesh points [21], which would require more

than 100 years of Moore’s law to be reached. In short, state-of-the-art is that ﬂight is

not computable and therefore not understandable: Prandtl tried to explain drag without

lift from a very thin no-slip viscous laminar boundary layer, and Kutta-Zhukovsky lift

without drag from a slip boundary condition without a boundary layer, but no explana-

tion for lift and drag of real slightly viscous ﬂow with a turbulent boundary layer was

given. We show that ﬂight today is computable using millions of mesh points, and un-

derstandable as potential ﬂow modiﬁed by 3d rotational slip separation, as documented

in particular at the AIAA Highlift Workshop 2013 [22].

We hope this paper will result in a constructive discussion including representatives

of state-of-the-art ﬂight theory about the central question voiced in the New York Times

article What Does Keep Them Up There? by Chang from 2003 [23]:

”To those who fear ﬂying, it is probably disconcerting that physicists and aeronau-

tical engineers still passionately debate the fundamental issue underlying this endeav-

our: what keeps planes in the air?”

3 Mathematical model of subsonic ﬂight

3.1 Navier-Stokes equations

We simulate subsonic ﬂight of an airplane by computational solution of the incom-

pressible Navier-Stokes equations in the air volume surrounding the airplane with a

combined velocity-stress Dirichlet-Neumann boundary condition on the surface of the

airplane prescribing the normal ﬂuid velocity and tangential stress to be zero, with the

airplane ﬁxed and the air moving with respect to the coordinate system. This boundary

condition is referred to as a slip condition valid for an inpenetrable surface with zero

friction as an approximation of the observed small skin friction of a turbulent bound-

ary layer at high Reynolds number Re, with Re ≈106for a glider, Re ≈107for a

propeller plane at cruising speed and Re ≈108for a jumbojet at take-off and landing,

recalling that Re =U L

ν, where Uis a charateristic velocity (m/s), La characteristic

length (m) and ν≈10−5is the kinematic viscosity of air (m2/s).

The Navier-Stokes equations for exterior incompressible ﬂow of (small positive)

constant kinematic viscosity νin a (bounded) volume Ωin R3around a ﬁxed rigid

body (wing or full wing-body of an airplane) with smooth frictionless boundary Γ(slip

5

boundary condition) over a time interval I= [0, T ], take the following form: Find the

velocity u= (u1, u2, u3)and pressure pdepending on (x, t)∈Ω∪Γ×I, such that

˙u+ (u· ∇)u+∇p−∇·τ= 0 in Ω×I,

∇ · u= 0 in Ω×I,

un= 0 on Γ×I,

τs= 0 on Γ×I,

u(·,0) = u0in Ω,

(1)

where ˙u=∂u

∂t ,un=u·nis the ﬂuid velocity normal to Γwith na unit outward

normal vector, τ=τ(u)=2ν(u)is the viscous stress with (u)the standard velocity

strain, τsis the tangential stress, u0is a given initial condition and we assume suitable

far-ﬁeld inﬂow/outﬂow boundary conditions. We focus on high Reynolds number in-

compressible ﬂow with the slip boundary condition un= 0 and τs= 0 on Γmodeling

observed small skin friction, which is instrumental for the generation of L

Dlarger than

10 required for powered ﬂight. We assume the boundary Γto be smooth thus elimi-

nating the possibility of attributing lift to a sharp trailing edge, in accordance with the

observed fact that lift does not disappear by rounding the trailing edge [24, 25].

We identify a potential solution as an incompressible, irrotational and stationary

velocity u=∇φwhere φis a harmonic function in Ωsatisfying a homogeneous Neu-

mann boundary condition on the boundary Γcorresponding to un= 0, which is an

approximate solution of the Navier-Stokes equations (1), together with a correspond-

ing pressure, with a residual scaling with the viscosity ν(∼Re−1) and thus being

vanishingly small for vanishing viscosity (increasing Reynolds number). We shall ﬁnd

that potential solutions have fundamental importance because Navier-Stokes solutions

at high Re can be viewed as potential solutions modiﬁed by 3d rotational slip separa-

tion.

3.2 DFS: Weighted residual stabilized ﬁnite element method

We compute approximate solutions of the Navier-Stokes equations (1) by a weighted

residual stabilized ﬁnite element method referred to as Direct Finite Element Simula-

tion DFS, described and analyzed in detail under the acronym G2 as General Galerkin

[1]. DFS takes the principal form: Find ˆu= (u, p)∈Vhsuch that for all ˆv= (v, q )∈

Vh

[R(u; ˆu),ˆv]+[hR(u; ˆu), R(u; ˆv)] = 0,for all ˆv∈Vh,(2)

where Vhis a space-time ﬁnite element space with velocities vsatisfying v·n= 0 on Γ,

[·,·]is an L2(Ω×I)inner product, R(U; ˆu) = ( ˙u+U· ∇u+∇p, ∇ ·u)is the residual,

and his the local mesh size. The ﬁrst term in (2) establishes ˆuas a weak solution of

(1) and the second term introduces kinetic energy dissipation [hR(u; ˆu), R(u; ˆu)] =

kh0.5R(u; ˆu)k2

L2bounded by data with k · kL2an L2(Ω ×I)norm. Basic analysis

shows that ˆuis an approximate weak solution with residual R(u; ˆu)in H−1(Ω ×I)

scaling like h0.5[1].

Notice that here νis set to zero with instead the weighted residual stabilization

introducing a dissipative effect as an automatic turbulence model. This is analogous to

the dissipative weak solutions introduced by Duchon and Robert [26], with dissipation

6

caused by a lack of smoothness in the velocity ﬁeld, unrelated to viscous dissipation.

The dissipative effect of a turbulent DFS solution ˆudoes not disappear when the mesh

is reﬁned, instead a Law of ﬁnite dissipation is observed where the local dissipation

converges to a ﬁnite value under mesh reﬁnement [27, 28]. On the other hand, for

a smooth solution ˆu, corresponding to a small residual in L2(Ω ×I), the dissipative

effect of the weighted residual stabilization vanishes.

3.3 Stability: output error representation by duality

We choose a target output M(ˆu) = [ˆu, ˆ

ψ]+[pn, ψΓ]Γ, where ˆ

ψ= (ψ, χ)is a given

weight function and ψΓis boundary data for the dual velocity, with [·,·]Γan L2(Γ ×I)

inner product. The difference in output M(ˆu)−M(ˆ

U) = [ˆu, ˆ

ψ]+[pn, ψΓ]Γ−[ˆ

U, ˆ

ψ]−

[P n, ψΓ]Γof two DFS solutions ˆuand ˆ

Uon different meshes with maximal mesh size

h, can be represented as

M(ˆu)−M(ˆ

U)=[R(u; ˆu)−R(U;ˆ

U),ˆϕ](3)

where ˆϕ= (ϕ, θ)is a solution of the dual linearized problem

−˙ϕ−(u· ∇)ϕ+∇UTϕ+∇θ=ψin Ω×I,

∇ · ϕ=χin Ω×I,

ϕ·n=ψΓon Γ×I,

ϕ(·, T )=0 in Ω,

(4)

where (∇UTϕ)j=P3

i=1 ∂Ui/∂xjϕi. For example, to choose the lift and drag of an

airplane as the target output the dual data is set to ˆ

ψ= 0 and ψΓ=vD·n+vL·n,

with vDand vLunit vectors in directions opposite and normal to the ﬂight direction,

respectively. Basic analysis shows that (assuming the time step to be bounded by h)

|M(ˆu)−M(ˆ

U)| ≤ Ckh0.5ˆϕkH1,(5)

with a constant Cdepending on data and bounds on the magnitude of the discrete

velocities uand U, and thus establishes stability in output if kh0.5ˆϕkH1is small [1].

To construct an adaptive algorithm we compute a DFS solution ˆ

Ufor which we

use the error representation (3) to estimate the output error |M(ˆu)−M(ˆ

U)|, with ˆu

any DFS solution computed on a ﬁner mesh than ˆ

U. If the estimated error is too large

the mesh is locally reﬁned to increase the precision in M(ˆ

U). The adaptive mesh

reﬁnement algorithm is based on an estimate of the local contribution to the global

error, which is obtained by splitting the error representation (3) into a sum of error

indicators En,K over the discrete space mesh Thand time intervals In= (tn−1, tn),

n= 1, ..., N , that is

M(ˆu)−M(ˆ

U) =

N

X

n=1

X

K∈T h

En,K =

N

X

n=1

X

K∈T h

[R(u; ˆu)−R(U;ˆ

U),ˆϕ]n,K (6)

with [·,·]n,K an L2(K×In)inner product. The error representation (6) can be the basis

for a number of different adaptive algorithms based on optimization of the mesh and

7

the ﬁnite element space Vh. One basic adaptive algorithm is to keep the mesh constant

in time and then iteratively reﬁne the spatial mesh based on (6) until convergence in

M(ˆ

U)is observed. If the estimated global error in (6) is within the tolerance, the

adaptive algorithm is terminated.

There are several technical aspects with respect to how to apply the error represen-

tation formula (6), in particular with regards to the approximation of ˆϕ, the solution

to the dual problem. The dual problem is linear, but runs backward in time which is a

challenge since the DFS solutions ˆuand ˆ

Uact as coefﬁcients in the dual problem and

thus have to be stored as data. In practise, the dual problem is solved using a similar

ﬁnite element method as for the primal DFS problem, and the coefﬁcients ˆuand ˆ

Ucan

be interpolated in time to minimize data storage. The function ˆucan be taken as a DFS

solution on a different mesh in the adaptively generated hierarchy of meshes, or be

an ideal function representing a ﬁnest possible resolution with M(ˆu)the target value

for the output approximation M(ˆ

U), where we then use the approximation ˆu≈ˆ

Uin

the dual problem. Also, several different approaches are possible for how to apply the

error representation formula (3) in practise, to retain sharpness and achieve maximal

robustness, see e.g. [29].

For simple geometries high Reynolds number bluff body ﬂow has been investigated

in detail [1, 30, 31, 32, 33, 28], where it is found that a computed DFS solution shows

to have small residual (of order h0.5) in the weak norm of H−1(Ω ×I), and to be

turbulent in the wake identiﬁed by locally large residual (order h−0.5) in the stronger

L2(Ω ×I)-norm [27]. Combined with observed boundedness in H1(Ω ×I)of the

dual solution (weighted by h−0.5), we ﬁnd computational evidence that errors in lift

and drag on the level of a few percent can be reached with a few hundred thousands

mesh points for simple geometries. The observed boundedness in the weighted H1(Ω)

norm of the dual solution, with distributed input data for lift and drag output, can be

understood as an effect of cancellation from a highly oscillating reaction coefﬁcient

∇uin the wake.

3.4 DFS of subsonic ﬂight

We compute DFS solutions of (1) for a long wing and a wing-body conﬁguration,

choosing lift and drag as mean value output quantities with corresponding data in the

associated dual problem, and with adaptive mesh reﬁnement geared by the error rep-

resentation (3). The ﬂow shows to be computationally resolvable except in a wake

behind the wing, as potential ﬂow modiﬁed by 3d rotational slip separation into a tur-

bulent wake stretching out into the farﬁeld behind the airplane, see Fig. 4. Computed

solutions show the pressure distribution on the body surface with lift Land drag D

obtained by integration of the surface pressure. The computed results are in close

agreement with experimental observations for a long NACA 0012 wing with maximal

L

D≈50, and for a complete wing-body conﬁguration with L

D≈20 over the entire

range of angle attack from zero to beyond stall [34, 22].

A computed partly turbulent solution displays a large scale resolvable quasi-stable

ﬂow except in a not fully resolved turbulent wake. The computational resolution of

the turbulent wake shows to have little inﬂuence on the pressure distribution on the

body and thus on lift and drag, which is supported by the nature of the dual solution

8

Figure 3: Overview of the computational domain Ωfor the DLR-F11 aircraft model

(upper) and detail of wing pressure side (lower). On the detail snapshot of the wing

pressure side, slat tracks and ﬂap fairings are seen.

Figure 4: Snapshot of a DFS solution of the ﬂow around of a wing-body conﬁguration

with vorticity visualised by a Q-criterion (left), with an adaptively reﬁned computa-

tional mesh optimised for lift and drag approximation [22].

9

displayed in Fig. 5 with little weight in the wake. A computed solution can thus be

viewed as a representative resolved Navier-Stokes solution with small residual in the

weak global norm of H−1(Ω ×I)in accordance with [35], and locally small residual

in the strong norm of L2(Ω ×I), except in a turbulent wake with large residual in

L2(Ω×I), with wake under-resolution of little impact on surface pressure distribution,

lift and drag.

Figure 5: Snapshot of the solution of the dual linearised Navier-Stokes equations with

data corresponding to the output quantities lift and drag on the wing-body conﬁgura-

tion. We note the magnitude of the dual solution which is small in the downstream

wake [22].

DFS/G2 is extensively documented and validated with experiments [36, 1, 30, 31,

32, 33, 34, 37, 38, 39, 40, 22] and can for high Reynolds number ﬂow be viewed as a

parameter-free model since the weight in the residual stabilization is set equal to the

local mesh size of a ﬁnite element mesh (without small angles). The effect of mesh

reﬁnement is resolution of the near-ﬁeld around the wing-body, see Fig. 4, with further

resolution concentrating on the far-ﬁeld wake with little impact on lift and drag. Lift

and drag thus show to be computable within a few percent using a couple of hundred

thousands of mesh points in simple geometry and millions in complex geometry.

G2/DFS is the ﬁrst methodology based on the full time-dependent Navier-Stokes

equations capable of simulating high Reynolds number bluff body ﬂow in complex

geometry, without model or parameter input from the user. We note that G2/DFS can

be used with non-zero skin friction, and also with boundary layer resolution using no-

slip boundary conditions although at a much higher cost which effectively prevents

simulation of high Reynolds number ﬂow. In [41, 42, 34, 37, 39, 40, 22] we give

computational evidence that for Re > 107the skin friction is so small that it can be

put to zero as a slip boundary condition without noticeable effect on lift and drag.

3.5 The Clay Navier-Stokes Millennium Problem

The fact that a computed solution is partly turbulent with locally large residual in

L2(Ω×I), can be seen as evidence of non-existence of a well-posed smooth solution to

the Navier-Stokes equations with smooth data, in the case of small viscosity bluff body

ﬂow, and thus as a negative answer to the Clay Navier-Stokes Millennium Problem

[43], which unfortunately does not contain the fundamental aspect of well-posedness

identiﬁed by Hadamard [44]: For bluff body ﬂow, physically meaningful well-posed

10

solutions are non-smooth even if data is smooth, and globally smooth solutions are not

well-posed.

4 Validation of the computational model

In the previous section we introduced the Navier-Stokes equations with slip bound-

ary condition as the basic mathematical model of subsonic ﬂight, with G2/DFS as a

methodology to compute solutions to the mathematical model. The purpose of this

section is to present a validation of the model (1) by comparing DFS computational

results with experimental data in two examples related to ﬂight. The ﬁrst example is

the NACA 0012 wing [34] which is a standard test case, and the second example is a

full wing-body conﬁguration used as benchmark in the AIAA HighLift workshop [22].

4.1 Long NACA 0012 wing

We model the ﬂow around a wing in a virtual wind tunnel. The computational setup

is a wind tunnel with square cross-section 2.7×2.7meters (m) and of length 5.32

m, in which we place a NACA 0012 wing with a chord length of 0.76 m. On the

tunnel walls we use free slip boundary conditions, at the inlet we set a constant inﬂow

U= 1ms−1, and at the outlet we use zero stress outﬂow boundary conditions [1]. At

the wing surface we use zero skin friction boundary conditions, corresponding to a free

slip boundary condition, modelling the small skin friction of high Reynolds number

ﬂow.

For a set of angles of attack α, DFS simulations are performed over the time interval

I= [0,10], where for each angle of attack the computational mesh is adaptively reﬁned

based on a posteriori estimation of the error in mean total force on the wing (both

drag and lift) until convergence, over the time interval Im= [5,10] (where the ﬂow

is assumed to be fully developed). Results from DFS reported in [34] show lift and

drag in close agreement with experimental measurements [46, 45], see Fig. 6. The

corresponding surface pressure distributions for a pre-stalled angle of attack is shown

in Fig. 7, also close to experimental data.

The experimental measurements are performed at high Reynolds numbers, but still

signiﬁcantly lower than in the case of a real aircraft. One effect of a lower Reynolds

number is the development of laminar separation bubbles at the leading edge of the

wing at high angles of attack, which causes a decrease in lift [46]:

”...it might be that an intermediate range of Reynolds numbers exists over which

the section exhibits a combined leading-edge and trailing-edge type stall. This implies

that above some Reynolds number within the range of the tests, the aerofoil starts to

stall with turbulent-boundary-layer separation moving forward from the trailing edge,

but the ﬂow breakdown is completed by an existing laminar separation of the ﬂow in

the leading-edge region failing to re-attach. At some higher Reynolds number, how-

ever, transition from laminar to turbulent ﬂow would be expected to precede laminar

separation so that the stall would then be entirely controlled by the forward movement

of turbulent separation.”

11

Figure 6: Lift and drag coefﬁcients from DFS for a NACA 0012 wing plotted against

the angle of attack, together with experimental results from Ladson [45].

Figure 7: G2 computation of pressure coefﬁcient for a NACA 0012 wing, averaged in

time and in the spanwise direction, along the lower and upper parts of the wing for an

angle of attack 10◦. Note in particular the low pressures near the trailing edge of the

wing corresponding to suction from the attached ﬂow over the rounded trailing edge.

12

In [46] it is reported that a laminar separation bubble is present from roughly angle

of attack 9◦, which coincides with an observed ﬂattening of the lift curve [46, 45].

Since the DFS simulation is an idealisation of an inﬁnite Reynolds number no laminar

separation bubbles develop, and thus lift is higher than in the experiments for high

angles of attack, and likely also closer to the real case of a an aircraft at higher Reynolds

numbers where turbulent boundary layers are fully developed.

4.2 Complete wing-body

Next we report results from the 2013 AIAA CFD High Lift Prediction Workshop [47],

where DFS was used to simulate the time-dependent ﬂow past a full wing-body conﬁg-

uration [22]. The DFS solver was the only fully time-dependent Navier-Stokes solver

in the workshop, and no other methodology (such as RANS [48]) showed better agree-

ment with experimental observation, even at much higher computational cost (in terms

of the number of degrees of freedom). Free slip boundary conditions (zero skin fric-

tion) was used in the DFS model to simulate the low skin friction of high Reynolds

number ﬂow, and the mesh was adaptively reﬁned based on a posteriori estimation of

the error in lift and drag.

0 5 10 15 20 25

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

α [degrees]

CL

exp

iter0

iter1

iter2

iter3

iter4

0 5 10 15 20 25

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

α [degrees]

CD

exp

iter0

iter1

iter2

iter3

iter4

Figure 8: Lift and drag coefﬁcient for angles of attack 12◦and 22.4◦, under adaptive

mesh reﬁnement with the computational approximations approaching the experimental

reference values.

Simulations were performed for a number of angles of attack, from attached ﬂow

through partly separated ﬂow to stall. We ﬁnd that for angles of attack corresponding to

a pre-stalled ﬂow, DFS simulation captures the lift and drag in experiments within a few

percent, see Fig. 8. For a stalled aircraft the computed lift is also close to experiments,

while the error in drag is somewhat larger. The local surface pressure over the wing

is well approximated in the DFS computation, see Fig. 9-10. Under adaptive mesh

reﬁnement the lift and drag approach the experimental reference values [22], and with

additional mesh reﬁnement the precision is expected to increase further.

13

0 0.5 1

−4

−2

0

2

x/c

cp

PS01 y = 0.20967m

sim

exp

0 0.5 1

−6

−4

−2

0

2

x/c

cp

PS05 y = 0.7603m

0 0.5 1

−6

−4

−2

0

2

x/c

cp

PS07 y = 1m

0 0.5 1

−6

−4

−2

0

2

x/c

cp

PS11 y = 1.3488m

Figure 9: Mean velocity contours and measurement plane locations (upper), and pres-

sure coefﬁcient, CP, vs. normalized local chord, x/c, (lower) for the angle of attack

α= 12o, for a complete wing-body conﬁguration.

0 0.5 1

−6

−4

−2

0

2

x/c

cp

PS01 y = 0.20967m

sim

exp

0 0.5 1

−20

−10

0

10

x/c

cp

PS05 y = 0.7603m

0 0.5 1

−20

−10

0

10

x/c

cp

PS07 y = 1m

0 0.5 1

−6

−4

−2

0

2

x/c

cp

PS11 y = 1.3488m

Figure 10: Mean velocity contours and measurement plane locations (upper), and pres-

sure coefﬁcient, CP, vs. normalized local chord, x/c, (lower) for the angle of attack

α= 22.4o, for a complete wing-body conﬁguration.

14

5 Elements of the new theory of ﬂight

We now present the key elements of the new theory of ﬂight, in the form of the drag

and lift generated by a wing. Contrary to the classical theory which is based on 2d

potential ﬂow, effects from a vanishingly thin boundary layer, and an inﬁnitely sharp

trailing edge, we construct the new theory directly from the fundamental model (1) of

3d ﬂow past a smooth wing. To model the ﬂow around a wing we start from the circular

cylinder, for which we analyse the stability of ﬂow attachment and separation.

5.1 From circular cylinder to wing

Experimental studies of the ﬂow past a circular cylinder under increasing Reynolds

number shows a sudden drop in drag at Re ≈105−106referred to as drag crisis,

which is a consequence of delayed separation due to transition to turbulence in the

boundary layer, into the development of a pattern of streamwise vortices for Re > 106

that we refer to as 3d rotational slip separation [49, 50, 51, 52, 53].

G2/DFS simulations exhibit this scenario under decreasing skin friction towards

free slip at the boundary, as a model of a turbulent boundary layer under increasing

Reynolds number, with close agreement in drag compared with experiments [42]. As

we reﬁne the mesh we ﬁnd that that the large scale features of the ﬂow around the

cylinder including separation pattern, surface pressure and drag, change little under

decreasing mesh size. The effect of mesh reﬁnement downstream the cylinder is an

extension of the length of the turbulent wake, as shown in Fig. 11 .

The ﬂow around a circular cylinder can be used as a model of both leading edge

attachment and rounded trailing edge separation for a long wing viewed as two circular

cylinders joined by two sheets, see Fig. 2. We now analyse the stability of potential

ﬂow as an approximate solution of (1), where we ﬁnd that 3d rotational slip separation

is realized as a quasi-stable ﬂow with minimal retardation by point stagnation and with

oscillating high and low pressure, to be compared with unstable 2d irrotational potential

ﬂow separation from line stagnation with high pressure. We start with the following

observation:

Theorem 1:Two-dimensional potential ﬂow in a domain Ωexterior to a wing sec-

tion with smooth boundary Γ, can only separate at a stagnation point with zero ﬂow

velocity.

Proof: Let the potential ﬂow velocity u=∇φwith φa harmonic function and let ψ

be the corresponding conjugate function. The level lines of φand ψare orthogonal,

with the boundary Γa level line of ψwherever u=∇φis bounded away from zero.

Fluid particles close to Γfollow streamlines as level lines of ψwhich are parallel to

the boundary Γwherever u=∇φis bounded away from zero, because level lines of φ

are orthogonal to Γ. Fluid particles thus cannot separate from the boundary as long as

u=∇φis bounded away from zero.

In particular, it follows that potential ﬂow around a long wing cannot separate on

the crest of the wing, because there the ﬂow velocity cannot vanish. The slip condition

satisﬁed by potential ﬂow is thus instrumental for the generation of lift. We note that 2d

potential ﬂow past a wing section with smooth boundary must have a stagnation point

15

Figure 11: Snapshots of velocity magnitude (upper) and streamlines (lower) illustrating

the quasi-static large scale ﬂow features which are stable under mesh reﬁnement. The

ﬁgure displays the solution after 13, 15, 17, 19, 21 and 23 adaptive mesh reﬁnements

with respect to the error in drag [42].

16

at the trailing edge from meeting opposing ﬂow coming from above and below the

section. Potential ﬂow past a long wing thus necessarily separates with line stagnation

and thus the basic instability necessarily appears, as the origin of stable lift and drag

making controlled ﬂight possible.

5.2 Potential ﬂow

Potential ﬂow around a long circular cylinder, of unit radius with axis along the x3-

axis in R3with coordinates x= (x1, x2, x3), assuming the ﬂow velocity is (1,0,0)

at inﬁnity in each x2x3-plane, is given by the following potential in polar coordinates

(r, θ)in a plane orthogonal to the cylinder axis, see Fig. 12:

ϕ(r, θ)=(r+1

r) cos(θ)

with corresponding velocity components

ur≡∂ϕ

∂r = (1 −1

r2) cos(θ), us≡1

r

∂ϕ

∂θ =−(1 + 1

r2) sin(θ)

and with streamlines being level lines of the conjugate potential function

ψ≡(r−1

r) sin(θ).

Potential ﬂow is constant in the direction of the cylinder axis with velocity (ur, us) =

(1,0) for rlarge, is fully symmetric with zero drag/lift, attaches and separates at the

lines of stagnation (r, θ) = (1, π)in the front and (r, θ) = (1,0) in the back. By

Bernouilli’s principle the pressure is given by

p=−1

2r4+1

r2cos(2θ)

when normalized to vanish at inﬁnity. We compute

∂p

∂θ =−2

r2sin(2θ),∂p

∂r =2

r3(1

r2−cos(2θ)),

and discover an adverse pressure gradient in the back. We shall now see that the poten-

tial ﬂow around a circular cylinder shows exponentially unstable 2d irrotational sepa-

ration and quasi-stable 2d attachment, and is a useful model for both attachment at the

leading edge of a wing and separation at a rounded trailing edge.

5.3 Linearized Navier-Stokes equations with vanishing viscosity

We analyze the stability of a potential solution as a smooth solution of the Navier-

Stokes equations (1) with vanishing viscosity and skin friction through the linearized

Euler equations

˙v+ (u· ∇)v+ (v· ∇)¯u+∇q=f−¯

fin Ω×I,

∇ · v= 0 in Ω×I,

v·n= 0 on Γ×I,

v(·,0) = u0−¯u0in Ω,

(7)

17

Figure 12: Potential ﬂow around circular cylinder: colormap of pressure, velocity ﬁeld

(left), velocity potential and stream function (right).

where (u, p)and (¯u, ¯p)are two Euler solutions with slightly different initial data (u0

and ¯u0) and volume forcing (fand ¯

f), and we denote the difference by (v, q)≡(u−

¯u, p−¯p). Formally, with uand ¯ugiven, this is a linear convection-reaction problem for

(v, q)with growth properties governed by the reaction term given by the 3×3matrix

∇¯u. By the incompressiblity, the trace of ∇¯uis zero, which shows that in general ∇¯u

has eigenvalues with real values of both signs, of the size of |∇u|(with |·| some matrix

norm), thus with at least one exponentially unstable eigenvalue, except in the neutrally

stable case with purely imaginary eigenvalues, or in the non-normal case of degenerate

eigenvalues representing parallel shear ﬂow [1].

The linearized equations in velocity-pressure indicate that, as an effect of the reac-

tion term (v· ∇)¯u:

•streamwise retardation is exponentially unstable in velocity,

•transversal acceleration is neutrally stable,

where transversal signiﬁes a direction orthogonal to the ﬂow direction.

5.4 The vorticity equation

Additional stability information is obtained by applying the curl operator ∇× to the

momentum equation ˙u+ (u· ∇)u+∇p=fwith forcing fviewed as a perturbation

of the zero forcing in (1), to give the vorticity equation

˙ω+ (u· ∇)ω−(ω· ∇)u=∇ × fin Ω,(8)

which is also a convection-reaction equation in the vorticity ω=∇×uwith coefﬁcients

depending on u, of the same form as the linearized equations (7), with a sign change of

18

the reaction term. The vorticity is thus also locally subject to exponential growth with

exponent |∇u|:

•streamwise acceleration is exponentially unstable in streamwise vorticity (vortex

stretching).

It thus seems possible that vorticity may grow large from a small force perturbation f

with ∇ × fless small, and this is what we see in computer simulations. This means

that Kelvin’s theorem, often cited in ﬂuid dynamics textbooks, stating that vorticity

without forcing will remain zero if it is zero initially and at inﬂow, as a consequence

of the vorticity equation (8) with ∇ × f= 0 and un= 0 on Γ, is a result which is

not stable under small perturbations and as such is not well-posed in Hadamard’s sense

and thus is not a result about physical ﬂow. Viewing Kelvin’s theorem as a result about

real ﬂow, as is often done, is thus based on a misconception. In particular, Kutta and

Zhukovsky made the world believe that a wing must have a sharp trailing edge, since

the circulation they needed to get lift could not result from anything but a singularity

defying Kelvin’s theorem.

We shall now exhibit in more detail how 3d rotational slip separation, as the basic

mode of instability in 2d irrotational potential ﬂow separation, can develop from the

exponential instability in streamwise retardation identiﬁed in (7) followed by vortex

stretching in acceleration identiﬁed in (8). We thus give evidence that the possibilty of

instability, so obvious by the presence of the reaction terms with coefﬁcient ∇uin (7)

and (8), can be made into a reality of turbulence, recalling that it is turbulence which

allows a wing to generate stable lift and drag.

5.5 Exponential instability of 2d irrotational separation

We identify the basic instability mechanism at separation of 2d potential ﬂow around a

circular cylinder resulting from opposing retarding ﬂow with line stagnation connected

to high pressure, which creates a quasi-stable large-scale ﬂow with minimal retardation

and oscillating surface pressure in a pattern of point stagnation, which we describe as

3d rotational slip separation. We ﬁnd that the ﬂow has large scale resolvable features

except in a turbulent wake where under-resolution shows to have little impact on lift

and drag.

To analyze the stability of 2d irrotational separation we use potential ﬂow in a half-

plane as a model of trailing edge separation: u(x) = (x1,−x2,0) in the half-plane

{x1>0}with stagnation along the line (0,0, x3)and

∂u1

∂x1

= 1 and ∂u2

∂x2

=−1,(9)

expressing that the ﬂuid is squeezed by retardation in the x2-direction and acceleration

in the x1-direction. We ﬁrst focus on the retardation with the main stability feature of

(7) captured in the following simpliﬁed version of the v2-equation of (7), assuming x1

and x2are small,

˙v2−v2=f2,

19

where we assume f2=f2(x3)to be an oscillating perturbation depending on x3of a

certain wave length δand amplitude h, for example f2(x3) = hsin(2πx3/δ), expect-

ing the amplitude to decrease with the wave length. We ﬁnd, assuming v2(0, x) = 0,

that

v2(t, x3) = (exp(t)−1)f2(x3).

We next turn to the accelleration and then focus on the ω1-vorticity equation, for x2

small and x1≥¯x1>0with ¯x1small, approximated by

˙ω1+x1

∂ω1

∂x1

−ω1= 0,

with the “inﬂow boundary condition”

ω1(¯x1, x2, x3) = ∂v2

∂x3

= (exp(t)−1) ∂f2

∂x3

.

The equation for ω1thus exhibits exponential growth, which is combined with expo-

nential growth of the “inﬂow condition”. We can see these features in principle and in

computational simulation in Fig. 13 showing how opposing ﬂows at separation gener-

ate a pattern of alternating surface vortices from pushes of ﬂuid up/down, which act as

initial conditions for vortex stretching into the ﬂuid generating counter-rotating low-

pressure tubes of streamwise vorticity.

The above model study can be extended to the full linearized equations linearized

at u(x)=(x1,−x2,0):

Dv1+v1=−∂q

∂x1,

Dv2−v2=−∂q

∂x2+f2(x3),

Dv3=−∂q

∂x3,

∇ · v= 0,

(10)

where Dv = ˙v+u·∇vis the convective derivative with velocity uand f2(x3)as before.

We here need to show that the force perturbation f2(x3)will not get cancelled by the

pressure term −∂q

∂x2in which case the exponential growth of v2would get cancelled.

The force perturbation f2(x3)will induce a variation of v2in the x3direction, but this

variation does not upset the incompressibility since it does not involve the variation in

x2. Thus, there is no reason for the pressure qto compensate for the force perturbation

f2and thus exponential growth of v2is secured.

5.6 3d rotational slip separation for circular cylinder

We thus discover streamwise vorticity generated by a force perturbation oscillating

in the x3direction, which in the retardation of the ﬂow in the x2-direction creates

exponentially increasing vorticity in the x1-direction, which acts as inﬂow to the ω1-

vorticity equation with exponential growth by vortex stretching. That is, we ﬁnd ex-

ponential growth at rear separation in both the retardation in the x2-direction and the

acceleration in the x1-direction, as a result of the squeezing expressed by (9).

20

The corresponding pressure perturbation changes the high pressure at separation of

potential ﬂow into a zig-zag alternating quasi-stable pattern of moderate pressure vari-

ations with elevated pressure zones deviating opposing ﬂow into non-opposing streaks

which are captured by low pressure zones to form rolls of streamwise vorticity allow-

ing the ﬂow to spiral away from the body. This scenario is similar to the vortex formed

in a bathtub drain.

The larger the scale of the vortices, the more stable is the rotational separation,

since unstable streamwise retardation can be minimized. The most stable conﬁguration

of the zig-zag pattern of vortices is thus the one corresponding to the maximal size of

the vortices allowed by the geometry with minimal retardation. At start-up from zero

velocity, streamwise vorticity in computer simulations is triggered from perturbations

on the same scale as the mesh size, which is rapidly modiﬁed into mesh independent

large scale streamwise vortices of the same scale as the geometry. We shall see that

the tubes of low-pressure streamwise vorticity change the normal pressure gradient to

allow separation without unstable retardation, but the price is generation of drag by

elimination of the high pressure of zero drag potential ﬂow as a “cost of separation”.

We have now identiﬁed 3d rotational slip separation as one of two crucial parts of the

miracle of ﬂight, the other being potential ﬂow with slip boundary condition.

Figure 13: Illustration of quasi-stable separation pattern with point stagnation for a cir-

cular cylinder (upper left), and observed in experiment using silk threads attached at the

upstream attachment line [50] (upper right), with corresponding computer simulation

[42] showing surface pressures (lower left) and velocity streamlines (lower right).

21

5.7 Quasi-stable 3d rotational slip separation

As a model of 3d rotational slip separation at x= (0,0,0) into the halfplane x1>0,

we consider u(x) = (2x1, x3−x2,−x2−x3)with the corresponding linearized

equations

˙v1=−2v1,˙v2+v3=v2,˙v3−v2=v3,(11)

where ≥0. With = 0, the ﬂow u(x)represents rotating ﬂow with neutrally stable

transversal acceleration. With > 0the ﬂow u(x)is retarding in x2(and x3), but in

this case a perturbation f2(x3)will be affected by the rotation preventing the instability

of 2d irrotational separation. We thus ﬁnd analytical evidence the computationally

observed quasi-stability of 3d rotational slip separation, connecting to the observed

stability of the rotating ﬂow through a bathtube drain emerging from the instability of

non-rotating radially symmetric and retarding drain ﬂow.

Further, the surface pattern of 3d rotational slip separation given in Fig. 13 shows

high pressure stagnation points as 2d saddle points in the plane x1= 0 of the form

(0,−x2, x3), to be compared with the unstable ﬂow (x1,−x2,0) analyzed above (as

well as stable stagnation points inside low pressure rotational ﬂow). The analog to

the unstable perturbation f2(x3)for (x1,−x2,0), would be a perturbation f2(x1)po-

tentially generating transversal vorticty in the x3direction, which however would be

blocked by the surface (similar to that in attachment) and also be subject to destruction

by the main ﬂow in the x1direction away from the surface.

The observed quasi-stability of the surface ﬂow of 3d rotational slip separation

including 2d saddle points depicted in Fig. 13, thus can be understood from a lin-

earized stability analysis, an essential aspect being the different stability properties of

(x1,−x2,0) and (0,−x2, x3)with the main ﬂow in the x1-direction, which we record

as a basic fact in

Theorem 2:The saddle point ﬂow u(x) = (x1,−x2,0) is stable in the plane x3= 0,

but unstable in the half space x1>0.

Proof: We have already proved instability in the half-space x1>0as the origin of

3d rotational slip sepration, and we thus consider the 2d case in the plane x3= 0.

Fluid particles follow trajectories (x1(t), x2(t)) as solutions of ˙x1=x1,˙x2=−x2,

while the linearized equations in velocity perturbations (v1, v2)take the form ˙v1=

−v1,˙v2=v2. Exponential growth/decay along a ﬂuid particle trajectory is thus bal-

anced by exponential decay/growth in perturbation, with the effect that the velocity per-

turbation v2which is subject to exponential growth will be initialized as a perturbation

of an exponentially small particle velocity u2, without the possibility of a substantial

oscillating perturbation f2(x3)as in the case x1>0, with the effect that exponential

growth and decay cancel under restriction to x3= 0, and stability follows.

5.8 Quasi-stable potential ﬂow attachment

The above analysis also shows that potential ﬂow attachment, even though it involves

streamwise retardation, is quasi-stable. This is because the initial perturbation f2in

the above analysis is forced to be zero by the slip boundary condition requiring the

normal velocity to vanish. In short, potential ﬂow attachment is stable because the ﬂow

is retarded by the solid body and not by opposing ﬂows as in separation.

22

α=4◦α=4◦

α=10◦α=10◦

α=17◦α=17◦

α=20◦α=20◦

Figure 14: Flow around NACA 0012 wing under increasing angle of attack, illustrated

by plots of the magnitude of the velocity (left) and surface pressure (right).

23

5.9 Practical consequence: rounded trailing edge

In the classical Kutta-Zhukovsky circulation theory there is no lift without a sharp

trailing edge. The impact of this theory on practice is seen in the UIUC Airfoil Co-

ordinate Database [54] which lists 1550 airfoils, all with sharp trailing edge, despite

early experiments [24, 25] showing that a rounded edge of diameter less than 1% of

the chord length gives essentially the same lift and drag as a maximally sharp edge,

while a moderate increase of drag was noted for 2%.

The new theory assumes that the trailing edge is more or less smoothly rounded,

which opens to both mathematical analysis and computation, and also ﬁts with practice:

in general real wings do not have knife-sharp trailing edges, with up to 10% rounding

of modern wind turbine wings with improved stall characteristics.

5.10 Lift and drag invariance from scale invariance

The Navier-Stokes equations with slip and vanishingly small viscosity are scale in-

variant in the sense that a change of the scale in space leaves the equations invariant.

We therefore expect the 3d rotational separation pattern at the trailing edge including

the pressure variation on the trailing edge surface to remain the same with total effect

tending to zero with the radius of the trailing edge. We may thus expect lift and drag

to vary little for small radii of the trailing edge, which is also observed experimentally

[24, 25].

As the radius tends to zero the wake ﬂow can be described as a complex ”vor-

tex sheet” of counter-rotating rolls of streamwise vorticity with the length increasing

with decreasing viscosity (increasing Reynolds number), analogous to what we see in

simulations when the computational mesh is reﬁned, see Fig. 11.

6 Descriptive scenario for different angles of attack

The elements of the new theory are now collected into a description of the ﬂow around

a long wing, e.g. formed by stretching a fabric between two cylinders, as potential

ﬂow modiﬁed by 3d rotational slip separation at a smoothly rounded trailing edge as

illustrated in Fig. 1-2.

By Theorem 1 the ﬂow being 2d potential cannot separate from the upper part of the

wing before stall, and therefore is redirected downwards, which requires low pressure

on the upper surface creating lift which is kept by 3d rotational separation without the

mean pressure rise of 2d irrotational potential ﬂow separation cancelling lift.

We now give a detailed description of the ﬂow pattern for different angles of attack

α, exempliﬁed by the NACA 0012 wing, see Fig. 6. Altogether, we see lift increas-

ing linearly with the angle of attack as a consequence of redirection into downwash

until stall for α > 17, drag staying nearly constant for α < 14 as a consequence of

unchanging separation pattern, with only weak linear increase due to an increasing

effective thickness of the wing (projection in the ﬂow direction) with the separation

moving up on the trailing edge onto the upper surface, and ﬁnally quickly increasing

24

drag under beginning stall for 14 < α < 17 with separation on the upper surface, after

which the wing essentially behaves as a bluff body, see Fig. 14.

7 Summary of New Theory of Flight

The new theory shows that the miracle of ﬂight is made possible by the combined ef-

fects of (i) incompressibility, (ii) slip boundary condition and (iii) 3d rotational slip

separation, creating a ﬂow around a wing which can be described as (iv) potential ﬂow

modiﬁed by 3d rotational separation. The basic novelty of the theory is expressed in

(iii) as a fundamental 3d ﬂow phenomenon only recently discovered by advanced com-

putation and analyzed mathematically, and thus is not present in the classical theory.

Finally, (iv) can be viewed as a realization in our computer age of Euler’s original

dream to in his equations capture an uniﬁed theory of ﬂuid ﬂow.

The crucial conditions of (ii) a slip boundary condition and (iii) 3d rotational slip

separation show to be safely satisﬁed by incompressible ﬂow if the Reynolds number

is larger than 106. For lower Reynolds numbers the new theory suggests analysis and

design with focus on maintaining (ii) and (iii). In forthcoming working work we will

in more detail study the mechanism and computational prediction of stall.

Acknowledgements

The authors would like to thank our collaborators Niclas Jansson and Rodrigo Vilela

De Abreu who generously contributed images from [42, 34, 22]. The authors also ac-

knowledge the ﬁnancial support from EU-FET grant EUNISON 308874, the European

Research Council, the Swedish Foundation for Strategic Research, the Swedish Re-

search Council, the Basque Excellence Research Center (BERC 2014-2017) program

by the Basque Government, the Spanish Ministry of Economy and Competitiveness

MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323 and the Project of the

Spanish Ministry of Economy and Competitiveness with reference MTM2013-40824.

We acknowledge PRACE for awarding us access to the supercomputer resources

Hermit and SuperMUC based in Germany at The High Performance Computing Cen-

ter Stuttgart (HLRS) and Leibniz Supercomputing Center (LRZ), from the Swedish

National Infrastructure for Computing (SNIC) at PDC – Center for High-Performance

Computing and resources provided by the “Red Espa˜

nola de Supercomputaci´

on” and

the “Barcelona Supercomputing Center - Centro Nacional de Supercomputaci´

on”.

The initial volume mesh was generated with ANSA from Beta-CAE Systems S. A.,

who generously provided an academic license for this project.

References

[1] J. Hoffman and C. Johnson, Computational Turbulent Incompressible Flow,

Springer 2007.

25

[2] J. Hoffman and C. Johnson, Mathematical Secret of Flight, Normat, Vol.57,

pp.145-169, 2009.

[3] J. Hoffman and C. Johnson, Resolution of d’Alembert’s paradox, J. Math. Fluid

Mech., Vol.12(3), pp.321-334, 2010.

[4] T. von Karman and W. R. Sears, Airfoil theory for non-uniform motion, AIAA

Journal. Vol. 41, no. 7A, pp. 5-16. 2003.

[5] D. A. Peters, S. Karunamoorthy and W.-M. Cao, Finite state induced models part

1: two-dimensional thin airfoil, AIAA Journal of Aircraft, Vol. 32(2), 1995.

[6] Prandtl, Essentials of Fluid Mechanics, Herbert Oertel (Ed.), 2004.

[7] Introduction to the Aerodynamics of Flight, Theodore A. Talay, Langley Reserach

Center, 1975.

[8] Aerodynamics of the Airpoplane, Hermann Schlichting and Erich Trucken-

brodt,Mac Graw Hill, 1979.

[9] Fundamentals of Aerodynamics, John D Anderson, 2010.

[10] Aerodynamics, Aeronautics and Flight Mechanics, McCormick, 1995.

[11] Aerodynamics, Krasnov, 1978.

[12] Aerodynamics, von Karmann, 2004.

[13] Theory of Flight, Richard von Mises, 1959.

[14] G. Birkhoff, Hydrodynamics: a study in logic, fact, and similitude, 1950.

[15] Garret Birkhoff in Hydrodynamics, 1950: I think that to attribute dAlemberts

paradox to the neglect of viscosity is an unwarranted oversimpliﬁcation. The root

lies deeper, in lack of precisely that deductive rigour whose importance is so com-

monly minimised by physicists and engineers.

[16] L. Prandtl, On Motion of Fluids with Very Little Viscosity, Third International

Congress of Mathematics, Heidelberg, 1904.

[17] J. Stoker in Review of [14] in Bull. Amer. Math. Soc. 57 (6): 497499:On the other

hand, the uninitiated would be very likely to get wrong ideas about some of the

important and useful achievements in hydrodynamics from reading this chapter.In

the case of air foil theory, for example, the author treats only the negative aspects

of the theory. It has always seemed to the reviewer that the Kutta-Joukowsky theory

of airfoils is one of the most beauti- ful and striking accomplishments in applied

mathematics. The fact that the introduction of a sharp trailing edge makes possible

a physi- cal argument, based on consideration of the effect of viscosity, that leads to

a purely mathematical assumption regarding the behavior of an analytic function

which in its turn makes the solution to the ﬂow problem unique and also at the same

time furnishes a value for the lift force, represents a real triumph of mathematical

ingenuity.

26

[18] David Bloor, The Enigma of the Aerofoil: Rival Theories in Aerodynamics, 1909-

1930, The University of Chicago Press, 2011.

[19] J. Hoffren, Quest for an improved explanation of lift, AIAA 2001-0872, 39th

AIAA Aerospace Sciences Meeting and Exhibit, Reno, 2001.

[20] J. Hoffren, Quest for an Improved Explanation of Lift, AIAA Journal 2001:The

classical explanations of lift involving potential ﬂow, circulation and Kutta condi-

tions are criticized as abstract, non-physical and difﬁcult to comprehend. The basic

physical principles tend to be buried and replaced by mystical jargon...Classical

explanations for the generation of lift do not make the essence of the subject clear,

relying heavily on cryptical terminology and theorems from mathematics. Many

classical texts even appear to have a fundamental error in their underlying as-

sumptions...Although the subject of lift is old, it is felt that a satisfactory general

but easily understandable explanation for the phenomenon (of lift), is still lacking,

and consequently there is a genuine need for one.

[21] J. Kim and P. Moin, Tackling Turbulence with Supercomputer, Scientiﬁc Ameri-

can.

[22] J. Hoffman, J. Jansson, N. Jansson and R. Vilela De Abreu, Time-resolved adap-

tive FEM simulation of the DLR-F11 aircraft model at high Reynolds number,

Proc. 52nd Aerospace Sciences Meeting, AIAA, 2014.

[23] Kenneth Chang; Staying Aloft: What keeps them up there?, New York Times,

Dec 9, 2003.

[24] J. Stack, W. F. Lindsay, NACA Technical Report 665, 1938.

[25] L. Joseph Herrig, James C. Emery and John R. Erwin, Effects of Section Thick-

ness and Trailing-Edge Radius on TBE Performance of NAAC 65smms Compres-

sor Blades in Cascade, NACA Research Memorandum L51J16, 1956.

[26] J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incom-

pressible Euler and Navier-Stokes equations, Nonlinearity, Vol.13, pp.249-255,

2000.

[27] J. Hoffman and C. Johnson, Blowup of Euler solutions, BIT Numerical Mathe-

matics, Vol 48, No 2, 285-307, 2008.

[28] J. Hoffman, J. Jansson and R. Vilela De Abreu, Adaptive modeling of turbulent

ﬂow with residual based turbulent kinetic energy dissipation, Comput. Meth. Appl.

Mech. Engrg., Vol.200(37-40), pp.2758-2767, 2011.

[29] W. Bangerth and R. Rannacher, Adaptive ﬁnite element methods for differential

equations, Springer, 2003.

[30] J. Hoffman, C. Johnson, A new approach to computational turbulence modeling,

Comput. Methods Appl. Mech. Engrg. 195 (2006) 2865–2880.

27

[31] J. Hoffman, Adaptive simulation of the turbulent ﬂow past a sphere, J. Fluid

Mech., 568 (2006), pp. 77–88.

[32] J. Hoffman, Computation of mean drag for bluff body problems using adaptive

dns/les, SIAM J. Sci. Comput., 27(1) (2005), pp. 184–207.

[33] J. Hoffman, Efﬁcient computation of mean drag for the subcritical ﬂow past a

circular cylinder using general galerkin g2, Int. J. Numer. Meth. Fluids, 59(11)

(2009), pp. 1241–1258.

[34] J.Jansson, J.Hoffman and N.Jansson, Simulation of 3D ﬂow past a NACA 0012

wing section, in review (available as CTL technical report kth-ctl-4023).

[35] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta

Mathematica, Vol.63, pp.193-248, 1934.

[36] J. HO FFM AN, J. JAN SSO N, R. VIL EL A DE ABR EU , C. DEGIRMENCI, N. JANS-

SO N, K. M ¨

UL LE R, M. NAZ AROV AN D J. HI ROMI SP¨

UH LE R,Unicorn: paral-

lel adaptive ﬁnite element simulation of turbulent ﬂow and ﬂuid-structure interac-

tion for deforming domains and complex geometry, Computers and Fluids, Vol.80,

pp.310-319, 2013.

[37] R. Vilela De Abreu, N. Jansson and J. Hoffman, Adaptive computation of aeroa-

coustic sources for rudimentary landing gear, in proceedings for Workshop on

Benchmark problems for Airframe Noise Computations I, Stockholm, 2010.

[38] N. Jansson, J. Hoffman, M. Nazarov, Adaptive Simulation of Turbulent Flow Past

a Full Car Model, in: Proceedings of the 2011 ACM/IEEE International Confer-

ence for High Performance Computing, Networking, Storage and Analysis, SC

’11, 2011.

[39] R. Vilela de Abreu, N. Jansson and J. Hoffman, Computation of Aeroacoustic

Sources for a Complex Landing Gear Geometry Using Adaptive FEM, Proceed-

ings for the Second Workshop on Benchmark Problems for Airframe Noise Com-

putations (BANC-II), Colorado Springs, 2012.

[40] R.Vilela de Abreu, N.Jansson and J.Hoffman, Adaptive computation of aeroa-

coustic sources for rudimentary landing gear, Int. J. Numer. Meth. Fluids, Vol.,

Vol.74(6), pp.406-421, 2014.

[41] J.Hoffman, Simulation of turbulent ﬂow past bluff bodies on coarse meshes using

General Galerkin methods: drag crisis and turbulent Euler solutions, Comp. Mech.

38 pp.390-402, 2006.

[42] J. Hoffman and N. Jansson, A computational study of turbulent ﬂow separation

for a circular cylinder using skin friction boundary conditions, in proceedings for

Quality and Reliability of Large-Eddy Simulations II, Pisa, Italy, 2009.

[43] C. L. Fefferman, Ofﬁcial clay prize problem description: Existence and smooth-

ness of the navier-stokes equation, 2000.

28

[44] J. Hadamard, Sur les probl ´

emes aux d´

eriv´

ees partielles et leur signiﬁcation

physique. Princeton University Bulletin, pp. 4952, 1902.

[45] C. Ladson, U. S. N. Aeronautics, S. A. Scientiﬁc, T. I. Division, Effects of inde-

pendent variation of Mach and Reynolds numbers on the low-speed aerodynamic

characteristics of the NACA 0012 airfoil section, NASA technical memorandum,

National Aeronautics and Space Administration, Scientiﬁc and Technical Informa-

tion Division, 1988.

[46] N. Gregory, C. C.L. O’Reilly, Low-speed aerodynamic characteristics of NACA

0012 aerofoil section, including the effects of upper-surface roughness simulating

hoar frost, Aeronautical Research Council (Great Britain), Reports and memo-

randa, H.M.S.O., 1973.

[47] Rumsey, C., 2nd AIAA CFD High Lift Prediction Workshop HiLiftPW-2,

(http://hiliftpw.larc.nasa.gov/), 2013.

[48] P. Sagaut, Large Eddy Simulation for Incompressible Flows (3rd Ed.), Springer-

Verlag, Berlin, Heidelberg, New York, 2005.

[49] M. M. Zdravkovich, Flow around circular cylinders: a comprehensive guide

through ﬂow phenomena, experiments, applications, mathematical models, and

simulations. Vol.1 [Fundamentals], Oxford Univ. Press, Oxford, 1997.

[50] A. I. Korotkin, The three dimensionality of the ﬂow transverse to a circular cylin-

der, Fluid Mechanics - Soviet Research, Vol.5, 1976.

[51] J. S. Humphreys, On a circular cylinder in a steady wind at transition Reynolds

numbers, Journal of Fluid Mechanics, Vol.9(4), pp.603-612, 1960.

[52] G. Schewe, Reynolds-number effects in ﬂow around more or less bluff bodies,

4 Intern. Colloquium Bluff Body Aerodynamics and Applications, also in Journ.

Wind Eng. Ind. Aerodyn. 89 (2001).

[53] B. G ¨

olling, Experimentelle Untersuchungen des laminar-turbulenten Ueber-

ganges der Zylindergrenzschichtstr¨

omung, DLR, 2001.

[54] UIUC Airfoil Data Site, http://www.ae.illinois.edu/m-selig/ads.html

29

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