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We present a new mathematical theory explaining the fluid mechanics of subsonic flight, which is fundamentally different from the existing boundary layer-circulation theory by Prandtl–Kutta–Zhukovsky formed 100 year ago. The new theory 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 potential 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 \({Re\approx 10^{7}}\) and a jumbojet at take-off and landing with \({Re\approx 10^{8}}\) , which allows flight at affordable power. The new mathematical theory is supported by computed turbulent solutions 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 \({Re > 10^{6}}\) , 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.
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New Theory of Flight
Johan Hoffman
, Johan Janssonand 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 106108of 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 ν105is 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 in 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)Γ×I, such that
˙u+ (u· ∇)u+p−∇·τ= 0 in ×I,
∇ · u= 0 in ×I,
un= 0 on Γ×I,
τs= 0 on Γ×I,
u(·,0) = u0in ,
(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 satisfying 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(Re1) 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 ˆvVh,(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 H1(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 Mu) = [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×I)
inner product. The difference in output Mu)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
Mu)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 ×I,
∇ · ϕ=χin ×I,
ϕ·n=ψΓon Γ×I,
ϕ(·, T )=0 in ,
(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)
|Mu)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 |Mu)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= (tn1, tn),
n= 1, ..., N , that is
Mu)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 Mu)the target value
for the output approximation M(ˆ
U), where we then use the approximation ˆuˆ
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 H1(Ω ×I), and to be
turbulent in the wake identified by locally large residual (order h0.5) in the stronger
L2(Ω ×I)-norm [27]. Combined with observed boundedness in H1(Ω ×I)of the
dual solution (weighted by h0.5), 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
D50, and for a complete wing-body configuration with L
D20 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 for 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 H1(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= 1ms1, 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 12and 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 105106referred 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 exterior 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), us1
r
∂ϕ
∂θ =(1 + 1
r2) sin(θ)
and with streamlines being level lines of the conjugate potential function
ψ(r1
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
r2cos(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 ×I,
∇ · v= 0 in ×I,
v·n= 0 on Γ×I,
v(·,0) = u0¯u0in ,
(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 ,(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,
˙v2v2=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), 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”
ω1x1, 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,
Dv2v2=∂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) = (2x1, x3x2,x2x3)with the corresponding linearized
equations
˙v1=2v1,˙v2+v3=v2,˙v3v2=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=4α=4
α=10α=10
α=17α=17
α=20α=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.
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25
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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
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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
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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-
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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,
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29
... CMI has been at the forefront of the coordination of the worldwide search for the general solution(s) (or proof of no solution(s)) to these relations. A check on the status of the problem on its website (accessed on December 9, 2021) still reveals it to be unsolved despite attempts by various researchers [2,8,[8][9][10][11][12][13][14][15][16][17][18][19][20][21]. However, with limiting but suitable assumptions and approximations, exact and approximate analytical solutions of these relations have been obtained for some special flows. ...
... This has proven successful in over a century of applications, and it has been generally considered as the resolution of the d'Alembert's Paradox. Nonetheless, there are conflicting discussions on the validity of the no-slip boundary condition in the literature [25,26] as exemplified by the work of Hoffman et al. [12,27]. They computed drag with a slip boundary condition from the computational solution of the unsteady Euler equations and obtained results comparable to actual experiments [2]. ...
... When the fluid is assumed to have no viscosity, the only wall condition imposed on Eqn. 12 is the impermeability of the normal velocity [2,7]. However, Hoffman et al. [12] employed the slip condition at the wall to avoid the difficulty of numerically resolving boundary layers in high Re flows. They report values for aircraft lift and drag comparable to actual experiments. ...
Preprint
Full-text available
The three main approaches in fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics.Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author refined CPT. He introduced refined potential theory (RPT) that provides a viscous potential/stream function as a physically consistent solution to the NSE problem. This function captures observable unsteady flow features including separation, wake, vortex shedding, compressibility, turbulence, and Reynolds-number-dependence. It appropriately combines the properties of a three-dimensional potential function that satisfy the inertia terms of NSE and the features of a stream function that satisfy the continuity equation, the viscous vorticity equation, and the viscous terms of NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere, and spheroid.
... CMI has been at the forefront of the coordination of the worldwide search for the general solution(s) (or proof of no solution(s)) to these relations. A check on the status of the problem on its website (accessed on November 28, 2021) still reveals it to be unsolved despite attempts by various researchers [2,8,[8][9][10][11][12][13][14][15][16][17][18][19][20][21]. However, with limiting but suitable assumptions and approximations, exact and approximate analytical solutions of these relations have been obtained for some special flows. ...
... This has proven successful in over a century of applications, and it has been generally considered as the resolution of the d'Alembert's Paradox. Nonetheless, there are conflicting discussions on the validity of the no-slip boundary condition in the literature [25,26] as exemplified by the work of Hoffman et al. [12,27]. They computed drag with a slip boundary condition from the computational solution of the unsteady Euler equations and obtained results comparable to actual experiments [2]. ...
... When the fluid is assumed to have no viscosity, the only wall condition imposed on Eqn. 12 is the impermeability of the normal velocity [2,7]. However, Hoffman et al. [12] employed the slip condition at the wall to avoid the difficulty of numerically resolving boundary layers in high Re flows. They report values for aircraft lift and drag comparable to actual experiments. ...
Preprint
Full-text available
The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics. Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author embarked on a doctoral research on the refinement of CPT. He introduced the Refined Potential Theory (RPT) that provides the Kwasu function as a physically consistent solution to the NSE problem. The Kwasu function is a viscous scalar potential function that captures known and observable unsteady features of experimentally observed wall bounded flows including flow separation, wake formation, vortex shedding, compressibility effects, turbulence and Reynolds-number-dependence. It is appropriately defined to combine the properties of a three-dimensional potential function to satisfy the inertia terms of the NSE and the features of a stream function to satisfy the continuity equation, the viscous vorticity equation and the viscous terms of the NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere and spheroid. It is concluded that the Kwasu function is a physically consistent and closed-form analytical solution to the NSE problem.
... Interestingly, if one wishes to consider a smooth trailing edge, however small the trailing-edge radius (i.e., however close to a sharp trailing edge), the classical aerodynamic theory collapses; there are no theoretical models that can predict lift on a body with no sharp edges (only few ad-hoc methods with no theoretical basis). In fact, some authors even consider the sharp edge as a lifting mechanism; i.e., an airfoil must have a sharp trailing edge to generate lift (see Ref. [8]). ...
... Whether the sharp edge is a lifting mechanism or not, the accepted wisdom by the fluid mechanics community asserts that the lift development is a viscous process (lift is due to vorticity in the boundary layer); and the Kutta condition is a manifestation of viscous effects-it implicitly accounts for viscous effects in a potential flow formulation. In fact, this view has been challenged by several authors [8,9], but without providing a mathematical proof or theoretical basis; Hoffren mentioned "it is readily admitted that there is no rigorous proof for the claim" [9], describing his criticizing claim of the accepted wisdom. It may be the right time to recall Chang's 2003 New York Times article: What Does Keep Them Up There? ...
... The fact that the minimization principle (13) reduces to the Kutta condition in the special case of a sharp-edged airfoil, wedded to the fact that this principle is an inviscid principle (equivalent to Euler's momentum equation), imply that the classical Kutta-Zhukovky lift over an airfoil with a sharp edge is both computed and explained from inviscid considerations; the Kutta condition is not a manifestation of viscous effects, rather of inviscid momentum effects. This result explains the several inviscid computations that converged to the Kutta-Zhukovky lift without viscosity [8,11,12,14]. Moreover, while their authors legitimately questioned the underpinning lift mechanism in the absence of viscosity in their computations, attributing it to compressibility [14], artificial viscosity (or numerical dissipation) [13], or 3D rotational slip separation [8]; the current result provides a convincing argument about the mechanism behind circulation development over the airfoil. ...
Preprint
Full-text available
We exploit a special, less-common, variational principle in analytical mechanics (the Hertz' principle of least curvature) to develop a variational analogue of Euler's equations for the dynamics of an ideal fluid. We apply this variational formulation to the classical problem of the flow over an airfoil. The developed variational principle reduces to the Kutta-Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom that lift generation is a viscous phenomenon wherein the Kutta condition is a manifestation of viscous effects. Rather, it is found that lift arises from enforcing a necessary condition of momentum preservation of the inviscid flow field. Moreover, the developed variational principle provides a closure condition for smooth shapes without sharp edges where the Kutta condition is not applicable. The presented theory is validated against Reynolds-Averaged Navier-Stokes simulations. Finally, in the light of the developed theory, we provide a simple explanation for lift generation.
... CMI has been at the forefront of the coordination of the worldwide search for the general solution(s) (or proof of no solution(s)) to these relations. A check on the status of the problem on its website (accessed on November 16, 2020) still reveals it to be unsolved despite some recent attempts by various researchers [23][24][25][26][27][28][29][30][31][32]. [34,35] as exemplified by the recent work of Hoffman et al. [27,36] who were able to compute drag with a slip boundary condition from the computational solution of the unsteady Euler equations with values comparable to those obtained in actual experiments. ...
... A check on the status of the problem on its website (accessed on November 16, 2020) still reveals it to be unsolved despite some recent attempts by various researchers [23][24][25][26][27][28][29][30][31][32]. [34,35] as exemplified by the recent work of Hoffman et al. [27,36] who were able to compute drag with a slip boundary condition from the computational solution of the unsteady Euler equations with values comparable to those obtained in actual experiments. ...
... When the fluid is idealized as having no viscosity, it is generally the case that the only condition imposed on Eqn. 12 at the surface is the impermeability of the normal velocity [21]. properties of the flow are pressure-based, rather than viscous driven [27]. ...
Thesis
Full-text available
The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. In classical potential theory, the steady inviscid incompressible flow over a body can be obtained by the superposition of elementary flows with known analytical solutions. Analytical solutions can offer huge advantages over numerical and experimental solutions in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, the classical potential theory falls short of reconciling the actions of viscosity in an experimentally observed flow with the theoretical analysis of such a flow. As such, it is unable to resolve the boundary layer and predict the especially important flow separation phenomenon that results in the pressure drag experienced by a body in the flow. This has relegated potential theory to idealized flows of little practical importance. Therefore, an attempt is made in this thesis to refine the classical potential theory of the flow over a circular cylinder to bridge the gap between the theory and experimentally observed flows. This is to enhance the ability to predict and/or control the flows' aerodynamic quantities and the evolution of the wake for design purposes. The refinement is achieved by introducing a viscous sink-source-vortex sheet on the surface of the cylinder to model the boundary layer. These vortices, sources and sinks introduced at the cylinder surface are modeled as concentric at every location. The vortices are modeled as Burgers' vortices, and analytic expressions for their strengths and those of the sinks/sources are obtained from the classical theory. These are employed to obtain a viscous and time-dependent stream function that captures critical qualitative features of the flow including flow separation, reattachment, wake formation, and vortex shedding. After that, a viscous potential function, the Kwasu function, with which the pressure field is obtained from the Navier-Stokes equation, is derived from the stream function. It is obtained by defining the viscous stream function on a principal axis of the flow about which the vorticity vector is identically zero. Strategies have also been developed to account for the finite extent of the cylinder and dynamic unsteadiness of the flow, and to predict the points of separation/reattachment/transition and the boundary layer thickness. Additionally, the strategies are used to obtain forces and apply the solution to arbitrary geometries focusing on spheres and spheroids. These strategies include the gravity analogy that considers a fluid element-cylinder scenario to be like a two-body problem in orbital mechanics. This analogy introduces the perifocal frame of fluid motion and exploits it to resolve the d'Alembert's Paradox. The perifocal frame is also used to predict flow separation/reattachment/transition and explain the observation of sign changes in the shear stress distribution at the rear of a circular cylinder in a crossflow. The refined potential theory is verified against experimental and numerical data on the cylinder in an incompressible crossflow at Re∞=3,900. Its drag prediction is within the error bound of measured data and tHRLES (transitional Hybrid Reynolds-averaged Navier-Stokes Large Eddy Simulation) prediction. The predictions of the pressure distribution, separation point and Strouhal number are also within acceptable ranges. Its prediction of the force coefficients over the range 25≤Re∞<300,000 is validated against experimental and theoretical data on the cylinder in crossflow. There is a good agreement in the magnitude and trend for Re∞>100. For Re∞<100, there is a disparity in magnitude that is unsafe for design purposes. Similarly, it under-predicts the coefficient of drag in some of the explored axial flow configurations. However, at Re∞=96,000 and an aspect ratio of 2, the RPT drag prediction falls within 1.2% of validated computational result. The energy spectra of the wake velocity display the Kolmogorov's Five-Thirds law of homogeneous isotropic turbulence. This verifies and validates the unsteadiness in refined potential theory as turbulent in nature. The drag coefficient of a sphere for 25≤Re∞<300,000 is explored to demonstrate the application of refined potential theory. Additionally, the flow over a sphere at Re∞=100,000 is explored in detail. A generally good agreement is observed in the prediction of the experimental trend for Re∞≥2,000. The transitional incompressible flows over a 6∶1 prolate spheroid at an angle of attack β=45° for Re∞=3,000 and Re∞=4,000 are also explored. The present theoretical pressure distribution has a close agreement with the DNS (direct numerical simulation) result in the starboard rear of the spheroid. However, the magnitude of the predicted force coefficients are generally less than five times the corresponding DNS results. The asymmetry of the DNS pressure distribution in the meridian plane is not captured. Therefore, further analyses of the spheroid flow including the separation locations are recommended for further studies. It is concluded that the refined potential theory can be used to resolve, explore and/or control the aerodynamic quantities of the flows around canonical bluff bodies as well as the evolution of their wakes.
... We remark that the slip BC is also applied to the ground wall. Although the slip BC is only well defined for inviscid fluids, it is a good approximation in high Reynolds number (Re) scenarios [88]. Bear in mind that imposing a noslip condition in the ground would have required the use of an overkill boundary layer mesh to properly capture the wall viscous effects. ...
Article
Full-text available
This work presents a Fluid-Structure Interaction framework for the robust and efficient simulation of strongly coupled problems involving arbitrary large displacements and rotations. We focus on the application of the proposed tool to lightweight membrane-like structures. Nonetheless, all the techniques we present in this work can be applied to both volumetric and volumeless bodies. To achieve this, we rely on the use of embedded mesh methods in the fluid solver to conveniently handle the extremely large deflections and eventual topology changes of the structure. The coupling between the embedded fluid and mechanical solvers is based on an interface residual black-box strategy. We validate our proposal by solving reference benchmarking examples that consider both volumetric and volumeless geometries. Whenever it is possible, we also compare the embedded solution with the one obtained with our reference body fitted solver. Finally we present a real-life application of the presented embedded Fluid-Structure Interaction solver.
... A possible explanation of force generation in fluid dynamics was presented recently by Hoffman, Jansson and Johnson, mathematicians from KTH Stockholm [6]. They supposed fully 3D turbulent flow around the body. ...
Article
Full-text available
The aim of this paper is to study the physics related to lift generation on an airfoil. A new hypothesis [1] of physical mechanism of flight relies on existence of streamwise vortical structures above the wing and inside the wake. The vortices origin as a consequence of flow instability inside the boundary layer developed under adverse pressure gradient. These structures are highly dynamical in nature, they change position and size very rapidly. A simple airfoil in the form of a flat plate with moderate angle of attack is considered in the presented research that generates suitable flow at rather low Reynolds number. Stereo PIV time resolved measurement technique is used to capture high-dynamic data in several planes which are located in the wake and are perpendicular to freestream or parallel to the airfoil. The overall image of the flow field dynamics will be created using POD decomposition. Distinct flow patterns with associated kinetic energy are to be described as well as their role in the studied case. Existence of streamwise vorticity is proved, topology and other parameters are estimated in the paper; however related pressures and forces are not evaluated.
... Over the past two decades, a substantial amount of evidence has been published to suggest that residual-based stabilized formulations for incompressible flow are effective as implicit largeeddy simulation (LES) models, both in fundamental turbulence studies [1][2][3] and industrial-scale applications [4][5][6]. The connection between residual-based stabilization and LES was illuminated by variational multiscale (VMS) theory [7][8][9][10], in which unresolved fine-scale solution components are driven by the residual from introducing the resolved coarse-scale solution into the partial differential equation (PDE) system. ...
Preprint
We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier-Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.
Article
In this paper we revive a special, less-common, variational principle in analytical mechanics (Hertz’ principle of least curvature) to develop a novel variational analogue of Euler's equations for the dynamics of an ideal fluid. The new variational formulation is fundamentally different from those formulations based on Hamilton's principle of least action. Using this new variational formulation, we generalize the century-old problem of the flow over a two-dimensional body; we developed a variational closure condition that is, unlike the Kutta condition, derived from first principles. The developed variational principle reduces to the classical Kutta–Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects. Rather, we found that it represents conservation of momentum. Moreover, the developed variational principle provides, for the first time, a theoretical model for lift over smooth shapes without sharp edges where the Kutta condition is not applicable. We discuss how this fundamental divergence from current theory can explain discrepancies in computational studies and experiments with superfluids.
Conference Paper
View Video Presentation: https://doi.org/10.2514/6.2022-1665.vid Not exactly!
Article
We introduce a residual-based stabilized formulation for incompressible Navier–Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf–sup stable spaces with H1-conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier–Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.
Article
Full-text available
This work presents a direct comparison of unsteady, turbulent flow simulations with measurements performed using a Gulfstream G550 nose landing gear model. The experimental campaign, which was carried out by researchers from the NASA Langley Research Center, provided a series of detailed, well documented wind-tunnel measurements for comparison and validation of computational fluid dynamics (CFD) and computational aeroacoustics (CAA) methodologies. Several computational efforts were collected and presented at the Benchmark for Airframe Noise Computation workshops, BANC-I and II. For our simulations, we used a General Galerkin finite element method (G2), where no explicit subgrid model is used, and where the computational mesh is adaptively refined with respect to a posteriori estimates of the error in a quantity of interest, here the source term in Lighthill's equation. The mesh is fully unstructured and the solution is time-resolved, which are key ingredients for solving problems of industrial relevance in the field of aeroacoustics. Moreover, we choose to model the boundary layers on the landing gear geometry with a free-slip condition for the velocity, which we previously observed to produce good results for external flows at high Reynolds numbers, and which considerably reduces the amount of cells required in the mesh. The comparisons presented here are an attempt to quantify the accuracy of our models, methods and assumptions; to that end, several results containing both time-averaged and unsteady flow quantities, always side by side with corresponding experimental values, are reported. The main finding is that we are able to simulate a complex, unsteady flow problem using a parameter-free methodology developed for high Reynolds numbers, external aerodynamics and aeroacoustics applications.
Article
The book is unique in that it is the only one of its kind devoted entirely to the subject of (large-eddy simulation). It presents a comprehensive account of the available knowledge in this field, and also the first unified view of the various existing approaches. Large-eddy simulation is the only efficient technique for approaching high Reynolds numbers when simulating industrial, natural or experimental configurations. The author concentrates on incompressible fluids. The book gives a complete account of this young but very rich discipline. The topics are well chosen and both the mathematical ideas and the applicatons are presented with care. The book addresses researchers as well as graduate students and engineers.
Article
This report contains a comprehensive data base on the low-speed aerodynamic characteristics of the NACA 0012 airfoil section. The Langley Low-Turbulence Pressure Tunnel was used to obtain the data. Included in the report are the effects of Mach number, Reynolds number, and transition fixing on the aerodynamic characteristics. Also presented are comparisons of some of the results with previously published data and with theoretical estimates. The Mach number varied from 0.05 to 0.36. The Reynolds number based on model chord varied from about 2 to 12 × 106.
Book
This is Volume 4 of the book series of the Body & Soul mathematics education reform program, and presents a unified new approach to computational simulation of turbulent flow starting from the general basis of calculus and linear algebra of Vol 1-3. The book puts the Body & Soul computational finite element methodology in the form of General Galerkin (G2), up against the challenge of computing turbulent solutions of the inviscid Euler equations and the Navier-Stokes equations with small viscosity. The book shows that direct application of G2 without any turbulence or wall modeling, allows reliable computation on a PC of mean value quantities of turbulent flow such as drag and lift. The power of G2 is demonstrated by resolving several classical scientific paradoxes of fluid flow and by uncovering secrets of flying, sailing, racing and ball sports. The book presents new aspects on both mathematics and computation of turbulent flow, and challenges established approaches. The book is directed to a wide audience of computational mathematicians fluid dynamicists and scientists. The G2 solver is available as part the free software project FEniCS at www.fenics.org. The book has a dedicated dynamic web page, including movies from a wide variety of simulations, at www.bodysoulmath.org. The book is focussed on incompressible flow, but opens to compressible flow continued in Vol 5 on thermodynamics. The authors are experts on computational mathematics and technology. © Springer-Verlag Berlin Heidelberg 2007. All rights are reserved.