THE KUTTA-JOUKOWSKY CONDITIONS IN THREE-DIMENSIONAL FLOW

NASA TECHNICAL TRANSLATION:

NASA TT P-l4,9l8

THE KUTTA-JOUKOWSKY CONDITION IN
THREE-DIMENSIONAL FLOW

Robert Legendre

(NASA-TT-F-14918) THE KUTTA-JOUKOWSKY
CONDITIONS IN THREE-DIMENSIONAL FLOW
(Linguistic Systems, Inc. , Cambridge,
Hass.) 24 p HC $3.25

N73-24319

Unclas
H3/,12 0A3J0 J

Translation of: "La condition de
Joukowskl en ecoulement trldi^menslonnel, "
La Recherche Aerospatiale ^ 1972, no. 5
(Sept. -Oct. ), pp. 241-247.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D.C. 205^6 MAY 1973

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NASA TT F- l4,9l8

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4. Title and Subtitle

THE KUTTA-JOUKOWSKY CONDITION IN
THREE-DIMENSIONAL FLOW

a. Report Date

MAY 1973

6. Performing Organization Code

7. Author(s)

Robert Legendre

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LINGUISTIC systems; inc.

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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D.C. 20546

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15. Supplementary Notes

Translation of: "La condition de Joukowskl en ecoulement
trldlmenslonnel," La Recherche Aerospatiale, 1972, no. 5
(Sept. -Oct. ), pp. 241-247.

•16. Abstract

The separation line along which a Vortex sheet is attached
on a wing Is not always limited to the conventional trailing
edge. It may extend to wing tips or even to parts of the
leading edges.

Prom observations of the flow over marine propeller models
and delta wing models, a discussion is started, aiming at
Improving the description of the flow over any wing, and
giving a better basis for an accurate calculation of the
perfect fluid flow used as a reference.

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NASA TT F-:;4,9l8

THE KUTTA-JOUKOWSKY CONDITION IN
THREE-DIMENSIONAL PLOW+

tt
Robert Legendre

1. Introduction

The supporting surface theory of Ludwig Prandtl and part leu- / 242 *
larly his diagramming of the so-called supporting line have enabled
predictions to be made on the lift of elongated wings and even dis-
tribution of this lift along the wingspan, with sufficient accuracy
for sizing the structure.

With the increase in aircraft speed, wings have been decreased
in length and swept further back. The supporting surface theory
has been substituted for the supporting line concept.

Calculation of the exact shape of the vortex sheet could be X'
passed over, since modifications to this shape have only a minor
influence on the distribution of pressure on the wing. However,
control experiments have revealed anomalies, even for relatively
small incidences, particularly at the wing tips.

It is too easily accepted that experimental checking cannot
fully take into account the predictions established for a flow pat-
tern in ideal fluid, and that it is thus futile to keep on perfect-
ing such a pattern.

In quite a number of cases, certain patterns of behavior in
real fluid, apparently abnormal and attributed in too facile a man-
ner to viscosity effects, have been explained by the ideal fluic
calculation.

"^Communication presented at the 13th Congress of I.U.T.A.M., Moscow,
August, 1972.

'^''"Member of the Academy of Sciences, Senior Scientific Adviser to

O.N.E.R.A.

*Numbers in righthand margin indicate pagination of foreign text.

1

Since the advent of powerful computers , It has become possible
to determine the shape of the vortex sheet over several mean chords
of the wing and establish curvature of the sheet's edges as a
fully-predictable phenomenon. Two-dimensional calculations In
sections distant from the wake took this curvature Into account
only very Incompletely. We still have to determine exactly the
length of the line of the wing from which the sheet has originated.

The present article Is devoted to discussion of this problem.

2. Marine Propellers

The blades of marine propellers for fast ships are short sup-
porting surfaces, which were very rapidly introduced into naval
engineering.

The appearance of cavitation must be delayed, or its develop-
ment tempered, by giving a large surface to the blades, but their
diameter is limited since the propellers must not extend too far
beyond the ship's beam. j

In 1932, while a young engineer at the Carenes, Paris Test
Tank, the author was taken, to participate in propeller studies for
the liner "Nor mandie". I first familiarized myself with the "J
theory, far ahead of its time, developed by my friend Roger Brard,
now Chairman of the Paris Academy of Sciences, then interpreted
the observations made in the viewing channel on flow around the
propeller models.

At the time, flow was made visible in the water by suspending
fine particles of aluminum by a process largely developed by
Camichel at Toulouse. Moreover, certain propeller models were ar-
ranged for distribution of air bubbles at various points suitably
chosen on the surface of the blades. Stereoscopic photographs
taken with adjustable exposure times enabled the three-dimensional
nature of the flow to be restored fairly correctly and, in particu-
lar, the curvature of the vortex sheets, as shown by the helio-
coidal trajectories of the air bubbles, to be clearly observjed.

It appeared to the author that the origin of curvature was
not the point where the blade was tangential to the maximum cylin-
der of the propeller- but at a pointy whj.ch is usuaily_. considered as
the leading edge, located far upstream, where the angle of the tan-
gent to this pseudo-leading edge with the radius Is about 45° (Fig.
1).

pig. 1. Formation of a vortex sheet on a marine
propeller blade.

These observations were discussed at great length between the
author and his supervisor, Emlle Barrlllon. -Tests were conducted
at very low Reynolds numbers, and viscosity could be held respon-
sible. However, this too-simple hypothesis could not be retained,
and already the Idea that the Joukowsky condition could apply to a
substantial, fraction of the leading edge with very low viscosity
was formulated.

It was out of the question to work out this Idea at the time.
The limited computing facilities available were already overflow-
ing by processing Roger Brard's theory, and it was impossible to
begin a truly three-dimensional flow calculation. However, it was
important for the future to have cast doubt on the validity of a

3

perfectly clear boundary between the leading edge and trailing
edge, and to have suggested for the first time that Joukowsky's
condition could apply to a leading edge under certain conditions.

It is possible to make the idea intelligible in the vernacu-
lar by stating that the fluid cannot "remember" the orientation of
its infinite speed upstream to reconstitute in addition, in the
case of a propeller, the transition velocity triangle from the
fixed reference point to the moving reference point. Any surface
line to which the fluid is moving, on both sides of this line., is
a trailing edge independently of any geometrical consideration that
is too elementary.

Unable to make calculations, Emile Barrillon asked several of
his colleagues in the Academy whether the problem of determining
the flow of the ideal fluid would be properly formulated if the
origin of the vortex sheet was not defined from the beginning. To
the knowledge of the author he received no reply and the studies
he made himself gave no satisfaction. Henri Vlllat did take the
trouble to come and discuss, with his colleague and the present
author, the observations made and the theory that might be devel-
oped to take account of them.

3. Delta Wings / 243

The studies undertaken in 1950 at O.N.E.R.A. directed by
Maurice Roy [1] to study the behavior of delta or swallowtail
wings made the above considerations once more topical.

The experiments in water with air bubbles and in the air with
wires attached to the nodes of a grating, recommended by the au-
thor, and the experiments done in water by H. Werle with colored
meshes, converged towards the diagram worked out by Maurice Roy
of "cone-shaped vortex sheets" (Fig. 2) which he would later in-
terpret by formulating the mathematical properties of the limiting
case of the delta wing, plane, and undefined.

Fig. 2. Cone-shaped sheets on a delta wing. [

For this diagram, corresponding to evanescent viscosity, the
vortex sheet separates from the whole contour of the wing, leaving
no distinction between trailing edge and leading edge.

It remained to be verified whether such a diagram fitted the
properly stated- mathematical problem, but the opportunities for
discussion and calculation were still slim.

The author was satisfied, as were all his successors In the
next two decades, with the thln-bodles theory and the conical-flow
theory [2, 12]. There was no difficulty In extrapolating the law
of pressure continuity through a- sheet and the law of drag of the
sheet by the fluid, already classical for Prandtl's sheets. The
only novelties were extending the origin of the sheet to the two
leading edges and writing In a Joukowsky condition on these lead-
ing edges. It remained to be verified, at the end of the calcula-
tion, that the parietal flow moved towards the leading edge as well
as towards the trailing edge on both the upper and lower surfaces
of the wing.

To write the correct conditions Is not, however, to resolve
the problem, and the author, anxious to find a digital approxima-
tion quickly, set up a simplified diagram, concentrating the en-
tire vortex Intensity on two lines, without even trying to con-
struct a physically acceptable picture [2]. The pressure balance,

5

across the sheet was represented by the uniformity of the potential
derivatives of the simplified field and the evolution of the sheet,
reduced, for the angular plane wing, to the effect of a change of
scale of transversal sections with increasing abscissae, had for
a homolog the speed Induced by the entire field in the center of
each vortex. Because of a confusion in determining a logarithm,
which was corrected later, the pressure continuity across the sheet
was not satisfied in the first calculation.

Later, Brown and Michael in the United States [4], Mangier
and Smith in Great Britain [9], and Gorston in Germany [10] took
the work of O.N.E.R.A. and simplified less grossly than had the J
author the conical sheets described and interpreted by Maurice Roy.
The results of Mangier and Smith are particularly accurate and
give a good determination of the shape of the cone sheet.

Today, it is possible to study not only the permanent flow
but the nonstatlonary flow around a moving wing or a wing subject
to vibration. It is not necessary to assume that the leading edges
are rectilinear and that the wing is flat, provided the sweepback
angle at each point is sufficient for the calculation to supply a
flow moving towards the leading edge, on both the upper and lower
surfaces of the wing. The use of the thin-body theory Is not es-
sential, and the calculation may be done in the framework of three-
dimensional flows or Incompressible fluids, or, those comparable to
such fluids by the approximation of Prandtl-Glauert . It is not
yet possible to calculate transsonlc on supersonic flows, but it
is almost certain that the major effects of vortex sheets have
been worked out by the flow calculation on the incompressible fluid,
and that the effects of compressibility in shocks with mechanisms
Independent of those of vortex sheets must be sought for. More-
over, from the point of view adopted in the present article, the
laws governing the appearance of vortex sheets remain valid at
transsonlc or supersonic speeds, even if it is not yet possible to
make exact and accurate calculation of the entire flow.
6

The calculations above demand very powerful computers, and
the number of applications made, especially by Rehbach at O.N.E.R.A.
[17, 18], remains small. It is no less important to discuss care-
fully the laws of formation of sheets to avoid engaging in calcula-
tions that are both costly and futile.

Rehbach formulated computer programs inspired by the work of
Belotserkovskii, Butter, and Hancock [ISj I6].

4. Thin Wings

It is evident that the extension to a leading edge of a
Joukowsky condition, of finite speed, brings up difficulties for
a wing of practical interest whose leading edge must be neither
angular nor tapered. From this viewpoint, the work at O.N.E.R.A.
has been wrongly interpreted by those who recommend beginning on
vortex sheets at the leading edges. The directives given by
Maurice Roy to his co-workers were quite clear:

- find wing shapes "adapted" to cruising speed, i.e., such
that the lower and upper wing surface flows would separate on a
line close to the leading edge, which strictly excludes separation
of vortex sheets.

- in addition, exploit the remarkable stability of the conical
sheets which appear at higher incidences so that performances
evolve continuously and faithfully in a broad domain.

There was no question of confusing these two points of view
by making the formation of sheets a pre-condition, even a cruising
speed, since it is well known that creation of a vorticity is not
gratuitous.

To properly separate the two distinct objectives without pre-
maturely complicating them by thickness effects, it is convenient
to begin by studying infinitely thin wings considering them to be
thin but having a leading edge around which the flow might pass.

Development of computer techniques gave a practical answer / 2hh
to the problem discussed between the author and Emlle Barrlllon: J
the boundary between the leading edge region, effectively separa-,.
'tlng_the flow, and the region where a_vortex sheet_appeared_ls not_J
established by computation; It Is very largely arbitrary and the
problem Is difficult to state In mathematical terms. Other cri-
teria must be Introduced, as empirical as Joukowsky's condition
Itself, to define the solution to be reconciled with the experi-
mental results.

The most striking Illustration of the above statement Is sup-
plied by actual calculation of several types of flow around a
hlghly-sweptback wing, as done by Rehbach at O.N.E.R.A. [I?].

An Initial solution Is obtained for a vortex sheet coming only
from along the conventional trailing edge (Pig. 3a). In this case,
the flow passes around the leading edge and the wing tips. In
theory at Infinite speed: this gives no cause for concern, since
It Is manifested only If the leading edge Is rounded and If the
tip fairings had a large enough radius of curvature.

Pig. 3a. Vortex sheet coming from a trailing edge,

A second solution is obtained for a vortex sheet arising si- •
multaneously on_the conventional trailing edge and along the wing]
tips (Pig. 3t)). The flow still passes around the leading edge.

Pig. 3b. Vortex sheet coming from one trailing edge
and the wing tips.

Although the computation has not yet been made it is evident
that there is no fundamental difference between the wing with a
very sharp sweepback angle and the delta wing, especially in the
vicinity of the apex. It is thus possible to find a solution such
that the vortex sheet is emitted over the entire contour of the
wing (Pig. 3c).

Pinally, we can look at the possibility of selecting the ori-
gin of the sheet at an arbitrary point on the leading edge, even
deciding that the sheet will pass around certain portions of the
leading edge while other portions will give rise to isolated
sheets. This last form of computation could provide an interpre-
tation for the shredded sheets described by Maurice Roy from ex-
perimental results on medium swept wings [7].

The digital computations discussed above make it futile to
seek to demonstrate a theorem of existence and uniqueness, at least
as long as precise criteria have not been worked out.

Fig. 3c. Vortex sheet coming from the whole contour
of the wing.

The above comments relate to flow around a given wing. For
the converse problem of designing a wing for flow to satisfy given
conditions J and in particular for the first problem of adaptation
raised by Maurice Roy, it is possible to have camber at the lead-
ing edge to obtain the appearance of a sheet as desired. The
adaptation must be made such that a sheet would not form, and the
flow would separate at the leading edge Instead of converging to-
wards the leading edge. The configuration is then two-directional.
However, as the shape of the wing has been designed to satisfy
these conditions, the slightest change in incidence causes the al-
ternative of sheet separation to reappear.

To the aerodynamicist , who does not have to be concerned with
properly stated problems, the freedom of interpretation he is of-
fered is an advantage, since it allows him to adapt the ideal
fluid flow picture to experimental findings and thus to search
for criteria defining the solution.

10

5. Sharp Edges

There Is little point In discussing the contouring of sharp
edges since, for supporting surfaces of practical Interest, such
edges are placed only along the actual trailing edges onto which
converge two boundary layers of viscid fluid, tending to become / 2^5
mixed together In a vortex sheet with evanescent viscosity.

Mangier and Smith [l4] studied the behavior of the sheet
coming from a sharp edge, and their findings are summarized below.

The basic hypothesis is that two alternatives only can arise:

- either the flow passes around the edge and the speed be-
comes theoretically infinite in incompressible non-cavitatlng fluid,

- or a vortex sheet is detached from the trailing edge, thus
preventing infinite speed from being reached.

We must not attempt to substantiate this hypothesis mathema-
tically, as It is built into the concept of ideal flow for evanes-
cent viscosity. Only a computation Including viscosity and turbu-
lence could furnish such a substantiation, and this will be beyond
our reach for some time to come.

To study the behavior of a trailing edge sheet, it is suffi-
cient to examine the consequences of limitation of speed to a
finite value.

At a point M on the sharp edge (Pig. 4) the two determinations
of speed, V and V. to either side of the sheet, upper and lower
wing surfaces, are in the plane tangential to the sheet even if,
being confused in direction, they no longer make such a tangential
plane.

Moreover, the speed V is contained in the plane tangent to
the upper surface at M, and V. is contained In the plane tangen-
tial to the lower surface at the same point.

Thus, one of the two speeds, V or V., is tangential to the
edge and the vortex sheet is tangential to the wing surface

11

corresponding to the other speed, except perhaps in the degenerate
case where both speeds are tangential to the edge or in the even
more degenerate case when they are zero.

Pig. 4. Speeds at one point of an edge.

The line of reasoning assumed that the two speeds are finite
but introduced no hypothesis as to their intensities. It thus re-
mains valid if the wing is moving or subject to vibration.

If the flow is permanent, the pressure continuity across the
sheet requires that the intensities of both speeds be equal, but
this adds nothing essential to the conclusions reached.

However, we will examine the degenerate case in which both
speeds are tangential to the edge only when the flow is permanent,
since it does not appear urgent to perfect the calculation by in-
troducing refinements encumbering the programs.

If the two speeds V and V. are equal and tangential to the
edge, two alternatives may be considered:

- either the two speeds are the same and circulation is sta-
tionary; for any arbitrary wing the points where this circumstance
arises are isolated and, for a wing of practical interest, attacked

12

without skidding, only the point on the longitudinal plane of sym-
metry satisfies the condition. It would obviously be possible to
design a wing for circulation to stay constant over a finite ex-
panse of the trailing edge, but In this case no element of the
vortex sheet would have its origin on this expanse and the sheet
would be torn;

- or the speeds V and V. are opposed, in which case it is
more difficult to show that this circumstance can arise only at
Isolated points. Here, it is sufficient to indicate that, if the
phenomenon took place on a finite expanse of a sharp trailing edge,
the parietal streamlines would travel toward two distinct nodes;
even if the nodes defined at a regular point of the surface degen-
erated when they came to an edge, they would cause the streamlines
to converge on both sides. This is because a node on the wing sur-
face corresponds to a dip on the vortex sheet.

Actually, to extend the calculations discussed in the preced-
ing paragraph to thick wings with a sharp tailing edge, it is suf-
ficient to state that the product of the speed components V^ and
V. normal to the tangent of the edge is zero. At the end of the
calculation, it must also be checked whether the normal component
which is not zero is oriented such that the fluid leaves the wing
and enters the vortex sheet [l4].

Provided the origin of the sheet has been defined elsewhere,
the problem seems to be properly stated, at least for digital compu-
tation. The individual isolated points where the two speeds are
tangential to the trailing edge are supplied by computation.

6. Wing Tips

Engineers performing aircraft model tests in wind tunnels
have discovered experimentally the Importance of wing tip fairings,
giving better continuity to the variable incidence and skid charac-
teristics, both in steady state and in a changing pattern. Indeed,
cone-shaped vortex sheets are not only advantageous: a rectilinear

13

wing tip In the longitudinal plane In normal flight can give rise
to very different sheets under skid conditions according to whether
the sign of the skid Is positive or negative. As a result undesir-
able wing dropping behavior arises whose disadvantages are even
greater at high speeds when Interaction takes place between the
fields Induced by the fields and shocks.

Designing wing tip fairings with adequate mean radii can al-
low the flow to pass around the wing tips without separations gen-
erating bumps or short bubbles, or detachment of vortex sheets.

The difficulty of generalizing the Kutta-Joukowsky condition,
already In planar flow when the trailing edge Is rounded, gives us
to understand that It Is not possible to define precisely the limit
flow for evanescent viscosity around a wing tip with a fairing.
If the wing sweepback Is sufficiently moderate for the sheet not
to appear as far forward as the leading edge, the best design is
that of Fig. 3a, if the mean radius of the fairing is large, and
that of Fig. 3b if this radius Is negligible. Between the two
extreme cases the development pattern of the three-dimensional
boundary layers must be examined to decide whether the flow be-
comes detached or not (Fig. 5). In the case of detachment, we must / 246
examine what happens to the separation line when viscosity decreases
to see whether a boundary has any chance of existing. This diagram
would be valid for zero viscosity.

"1

Fig. 5. Plow without and with detachment at the wing tip.

The author has always declared himself to be opposed to
straight wing tips in longitudinal planes, precisely because of

14

the ambiguity In defining an ideal reference fluid flow. He re-
commends tapering the wing tips leeward of the leading edge, even
for substantial skid, so that they always behave just as trailing
edges (Pig. 6). There is now no disadvantage to thinning them into
sharp edges.

Pig. 6.

Plow on the upper and lower surfaces of a win§
with trailing edge tapered leeward of the
trailing edge.

It is possible to avoid flow passing around the wing tips at
cruising incidence when performing the adaptation calculation, but
this leads to reducing local lift without reducing the friction
resistance. Manufacturers will thus choose a compromise solution.

7. Leading Edge

The difficulties experienced in defining flow around the wing
tips foreshadow those raised in that around the leading edge, with

15

its failures that can lead even to complete separation, beginning
formation of a cone-shaped vortex sheet.

This is why all basic research both in Prance and abroad has
been focussed on wing models with sharp leading edges, giving
edges so sharp that they behave like trailing edges. Further
studies on Concorde models, whose leading edge radii were small
to be compatible with supersonic speeds, confirmed the validity
of such basic research, at least at high incidences. The joining
of the upper and lower wing surface boundary layers, which takes
place a little beyond the minimum radius of curvature, can be dia-
grammed by a vortex sheet in an extrapolation towards zero viscos-
ity.

However, it would be desirable to make the separation line
more precise as soon as separation is manifested so that a refer-
ence flow diagram in ideal fluid can be better defined. This would
certainly be of use in the future.

Unfortunately, the research program defined by the author in
collaboration with H. Werle has not thus far given significant re-
sults for two reasons: on the one hand, to a number of varying
parameters, numerous models had to be built over which other work
had priority, and the second, that for the only model properly
tested, it appears that the Reynolds number of the viewing channel
designed by Maurice Roy, although appropriate for working out the
basic flow characteristics, is inadequate for detailed analysis
of the origin of vortex sheets. The comments below are thus more
in the nature of a priori viewpoints. than solidly established ex-
perimental findings. They ask questions rather than answer them.

First of all, the radius of curvature of the leading edge and
its change along the wingspan must be basic factors in separation.
For the Concorde model, the local sweepback variation is also im-
portant, but seems to be less determinant than the above parameters

16

since separation does not appear near the fuselage, although the
local sweepback is considerable, because the radius of curvature
is large.

The carefully-tested model is delta-shaped and, designedly,
the evolution in^the leading edge radius of curvature is very dif-
ferent from that of the Concorde. The transversal sections are
elliptical and the minor axis of the ellipse is defined by a lens-
shaped median longitudinal section. Thus the apex is tangential
to an elliptical cone.

Unfortunately, the desire to observe the phenomena at the
leading edge with a good scale led to the choice of rather a large
thickness. As a result, with a very low testing Reynolds number,
separation goes back to the trailing edge when incidence increases
at the same time as interesting phenomena developi-.at the leading
edge (Fig. 7).

The mechanism of vortex sheet generation is better shown on
a heavily-tilted ogive which can represent a large fraction of the
leading edge (Fig. 8). Colored threads show convergence of the
parietal streamlines into a focus, after which they extend into
the heart of the fluid with a single streamline shown up by a very
fine thread. The configuration is exactly that predicted by an
examination of the individual points in solutions of differential
equations [5].

The phenomenon apparently has no relationship to the genera-
tion of a vortex sheet since none of the parietal streamlines
either converge towards the focus nor have a particular property
permitting them to be viewed as the attachment line of a sheet.
The entire boundary layer of the ogive surface is evacuated in a
turbulent vortex around the single thread leaving the surface.

One must imagine how the configuration described above evolves / 2^7
when viscosity tends towards zero. The author does not believe

17

■■i iiw i i i . I iM i j ; 'ff. '|i - ' gi J *| ' '.^ ' ■". ' ■• ' "■ '".■J.t » '..-"t,t* ! ^ !" J » l

Fig. 7. Formation of foci gen-
erating vortices at the upper
surface of a delta wing.

Fig. 8. Formation of §. focus
generating a vortex at the upper
surface of a sharply tilted ogive i

that a definite answer can be deduced by logic alone. He is of the
opinion that this must be given mechanically, improving the obser-
vations for smaller and smaller viscosities, and building up dia-
grams representing the phenomena in the best way. Just as in the
case of thin wings, there is probably no single solution in an
ideal fluid, but rather an infinity of solutions, all individually
coherent, from which selection must be made by empirical criteria
yet to be established. For example, it is possible to define a
flow of ideal fluid around a delta wing with rounded leading edges
by causing the vortex sheet to start from the trailing edge alone;
but adding two isolated vortices normally beginning from two sym-
metrical points on the leading edges, which would be fairly

18.

arbitrary. The diagram derived Is certainly coherent and may repre-
sent flow observed with low viscosity fairly correctly.

In addition, the comment on the quasl-lsotropy of a focus Is
based on the hypothesis of regular behavior, which Is faulty as
soon as a vortex sheet appears. A diagram of Ideal fluid flow
with a vortex sheet based on one of these parietal streamlines
ending up at the focus can thus be coherent and define a valid
limit for viscosity tending towards zero.

An acceptable configuration of parietal streamlines at the
upper surface of a delta wing can be reconstructed qualitatively
(Pig. 9). We may Imagine that the center of the focus is the ori-
gin of an Isolated vortex or that the streamline leaving the throat
and ending at the focus is already the start of a vortex sheet.

8.

Pig. 9. Focus near the leading edge of a delta wing.

Conclusions

The supporting surface theory must be added to by a more pre-
cise interpretation of the anomalies appearing on wing tips, and
■often on large 'sections of the leading, edge . 1

19

For this purpose, we must accept that Joukowsky's condition
of finite speed on a separation line extending to the heart of
the fluid by a vortex sheet at the ideal limit for the zero vis-
cosity fluid applies not only to the traditional trailing edge de-
fined geometrically. It can also apply to the wing tips, and even
to more or less extended portions of the leading edge.

Determination of the size of the origin of sheets raises a
difficult problem , the study_ of.._which..is _merely_ in its infan_cy,^ andj
can require the use of wind tunnels with a high Reynolds number,
well-equipped with viewing facilities.

In the first stage, taking advantage of the development in / 248
methods of calculating three-dimensional boundary layers, it seems
possible to proceed with fairly detailed experimental research and
undertake empirical studies to locate the origin of sheet formation,
i.e., (as long as viscosity is not completely zero) evacuation of
parietal boundary layers.

Profound understanding of the laws of separation seems to be
further in the future, but Its basis will certainly be closely
associated with the study of nodes, throats, foci, and solutions
of differential equations. The difficulty will be to predict their
degeneration when viscosity tends towards zero, a difficulty mirror-
ing those already involved in studies of degeneration when the above
above special points come on single points or single points of the
surface.

For the study of wings adapted to supersonic flight such as
the Concorde, radii of curvature at the leading edge are rather
small so that there should not be too great an uncertainty as to
the location of separation lines.

It is rather on wings for low-speed subsonic flight that anal-
ysis of these phenomena is behindhand in governing the choice of
suitable configurations serving as a basis for ideal fluid refer-
ence flow computation. ~
20 Manuscript submitted 8/18/72.

References

i. Roy, M. , Flow characteristics around a highly swept-back
wing, C. R. Ac. Sc.,vol. 254, 1952, p. 2501.

2. Legendre, R. , Plow In the vicinity of the forward point of

a highly swept-back wing with average Incidences, Rech. Aeron.,
no. 30 (Nov. -Dec. 1952), p. 3-8; Rech. Aeron, no^ 31 (Jan.-
Peb. 1953), p. 3-6, andRech. Aeron, no. 35 (Sept .-Oct .. 1953>J
p. 7-8.

3. Adams, M. C, Leading-edge separation from delta wings of
supersonic speeds, J.A.S., vol. 20, June 1953, P- ^30.

4. Brown, C. E. and W. H. Michael, Effect of leading-edge separa-
tion on the lift of a delta wing, J.A.S., vol. 21, October

■ . 1954, p. 690.

5. Legendre, R. , Separation of three-dimensional laminar flow,
Rech. Aeron., no. 54, Nov. -Dec. 1956, p. 3-8.

6. Roy, M. , Delta wing cone sheet and apex vortex, C. R. Ac. Sc,
vol. 249, no. 9, 1957, p. 1105.

7. Roy, M. , Delta wing theory, Rech. Aeron, no. 56, Jan.-Peb.
1957, p. 3-12.

8. Roy, M. , Notes on flow around swept-back wings, Zeit. f. ang.
Math, und Phys., vol. 9 b, no. 5-6, 1958, p. 554.

9. Mangier, K. W. and J. B. Smith, A theory of the flow past a
slender delta wing with leading-edge separation, Proc. Roy.
Soc, vol. 251 A, no. 1265, May 1959, p. 200.

10. Gersten, K. , Nonlinear supporting surface theory, expecially
for airfoils with small side ratio, Ing. Arch., vol. 30,

no. 6, 1961, p. 431.

11. Legendre, R. , Delta wing conical vortices, C. R. Ac. Sc,
vol. 257, 1963, p. 3814 and vol. 258, 1964, p. 429.

12. Legendre, R. , Vortex sheets from the leading edges of delta
wings, Progr. in Aeron. Sc, vol. 7, 1965.

13. Belotserkovskli, S. M. , Calculation of the flow about wings
of arbitrary planform at a wide range of angle of attack,
Mekhanica Zhidkostl 1 Gaza,1968.

21

14. Mangier, K. W. and J. H. B. Smith, Behaviour of the vortex
sheet at the tralllng-edge of a lifting wing., R. A. E. report
63.049, March 1969.

15. Legendre, R. , Effect of wing tip shapes on curvature of vor-
tex sheets, Rech. Aerosp., no. 1971j 4, p. 227-228.

16. Butter, D. J. and G. J. Hancook, A numerical method for-' cal-
culating the trailing vortex system behind a swept wing at
low speed, Aeron. Journal, vol. 75j August 1971.

17. Rehbach, C, Numerical study on the Influence of the shape of
a wing tip on curvature of the vortex sheet, Rech. Aerosp.,
no. 1971-6, p. 367-368.

18. Rehbach, C, Calculation of flow around thin wings with evolu-
tional vortex sheet (to appear).

22