RIJKSUNIVERSITEIT TE GRONINGEN
ON SAILING TO WINDWARD
Proefschrift
ter verkrijging van bet doctoraat in de Wiskunde en
Natuur-wetenschappen aan de Rijksuniversiteit te Groningen op gezag
van de Rector Magnificus Dr. J. Borgman in bet openbaar te
verdedigen op vrijdag 12 oktober 1979 des namiddags te
4.00 uur
door
ADRIAAN KLAAS WIERSMA
geboren te Leeuwarden
Krips Repro Meppel
1979PROMOTOR : Prof.dr. J.A. Sparenberg
STELLINGEN
De methode die Kalman e.a. gebruiken ter berekening van de gelnduceerde weer-stand in de Vortex Lattice Method kan nauwkeuriger gemaakt worden door de circulatieverdeling in spanwijdterichting met een kubische spline te benaderen.
T.P. Kalman, J.P.Giesing, W.P. Rodden: "Spanwise distribution of induced drag in subsonic flow by the Vortex Lattice Method", Journal of Aircraft, Vol. 7, no. 6 (1970) 574-576.
2
Beschouw een tweedimensionaal profiel, gegeven door y = a f(x) (x E [0,1],
f E c2[O,l], 0 < a « 1), geplaatst in een uniforme stroming evenwijdig aan
de X-as. De exacte uitdrukking voor de lift, ondervonden door dit profiel, bezit een asymptotische ontwikkeling naar machten van a, waarin de even machten ontbreken.
3
In de mathematisch georienteerde inleidingen tot de stromingsleer zou bij de
afleiding van de bewegingsvergelijkingen meer gewezen kunnen worden op een
gevoig van de Hodge-decompositie stalling [', p.76]. Dit orn de invoering van een funktie, die i.h.a. de druk wordt genoemd, mathematisch te recht-vaardigen.
[*] J. Marsden: "Applications of global analysis in mathematical physics" Publish or Perish, Inc.; Boston Mass.
4
Met richtend moment van een zeiljacht kan vergroot worden door gebruik te maken van een uitlegger, aan hat eind waarvan een klein dragend vlak is be-vestigd dat zich met de juiste invalshoek onder het wateropperviak bevindt.
5
De verkiaring, die Marguerie e.a. geven voor de latente periode die optreedt
in de door thrombine geinduceerde viokking van fibrine is onnodig
gecompli-ceerd.
G. Marguerie, Y. Benabid, M. Suscillon: "The binding of calcium to fibrinogen: influence on the clotting process.
Biochimica et Biophysica Acte, 579 (1979) 134-141.
6
Beschouw twee rechte vliegtuigvleugels, gemonteerd in V-stalling zodat de
hook tussen elke vleugel en het horizontale viak 5 is. Bu een voorgeschreven
lift L hoort dan een minimale geinduceerde weerstand R. ter grootte
2
L
1-k
4 2 12 cos J(t -p) 2 - t) 2 dt
p
waarin p de dichtheid van hot medium U de uniforme aanstroomsnelheid en 1 de lengte vari de vieugels is,
terwiji S k 2 1-k p = -1+k 1+k = 1-k 2 7
Bet is niet te verwachten dat sen platte, brede giek de optimale stuwkracht
van een jacht, zeilend can de wind, merkbaar zal vergroten.
8
"Things ain't what they used to be" (M. Ellington), daarom moet de Volvo Amazone weer in produktie genomen worden.
ACKNOWLEDGEMENTS
CHAPTER 1 DEMARCATION OF THE PRESENT STUDY
Introduction
Some remarks on the study of dynamics and motion of a sailing yacht
3 The meaning of sailing to windward
4 The mathematical model
5 Formulation of the propulsion problem and the related
optimization problem
6 Summary of the work presented in this thesis and
suggestions for further research References
CHAPTER II J.A. Sparenberg and A.K. Wiersma: "ON THE MAXIMUM THRUST
OF SAILS BY SAILING CLOSE TO WIND" (1976) - Journal of
Ship Research, Vol. 20, No. 2
pp.
98 - 106Introduction
Formulation of the problem The variational' problem
The integral equations belonging to and
Asymptotic expansion of maximum thrust for narrow gap Numerical results
Discussion of the results
TABLE OF CONTENTS
References 27
CHAPTER III A.K. Wiersna: "ON THE MAXIMUM THRUST OF A YACHT BY SAILING CLOSE TO WIND" (1977) - Journal of Engineering Mathematics,
Vol.
11, No.
2, Pp.
145 - 160
29Introduction 29
Formulation of the problem 3D
The optimization problem 33
The two coupled lifting lines 36
Optimization of T for given angle of heel 37
Approximate dependence of T on the righting moment 39
A numerical example 4' References 44 3 6 8 10 15 17 19 19 19 21 22 23 26 27
CHAPTER IV A.K. Wiersma: "ON THE PROFIT OF OPTIMIZING THE FIN-KEEL OF A YACHT SAILiNG CLOSE TO WIND" (1978) - Journal of
Engineering Mathematics, Vol. 12, No. 4, pp. 357 - 364 45
Introduction
Statement of problem Some necessary formulae The representation of the hull Thrust production with and without Discussion of the results
Acknowledgement References 45 46 47 48 optimized underwatership 50 52 52 52 SAMENVATTING 95
CHAPTER V A.K. Wiersma: "ON THE OPTIMIZATION OF THE THRUST OF A
YACHT SAILING TO WINDWARD" (1979) Journal of Engineering
Mathematics, Vol. 13, No. 4
, pp. 289 - 316 53
Introduction 53
On the O(a2) contribution to the thrust of the forces
generated by the sails 56
Statement of the problem for the sails 57
The optimization problem for the sails 61
The formulae for the underwatership and the coupling
of the two systems 66
Numerical results 68
The asymptotic case of sailing close to wind 77
Concluding remarks 79
References 80
CHAPTER VI A.R. Wiersma: "NOTE ON THE INTERACTION OF TWO OVERLAPPING
RIGID SAILS'. Part I: TWODIMENSIONAL SAILS (1979) International Shipbuilding Progress, Vol. 26, No. 293,
pp 9 - 20 81
Introduction 81
The exact integral equations 81
The forces on the sails 82
Numerical approach of the integral equations 83
Some numerical, results 84
The linearized problem 89
References 92
Acknowledgements
I wish to express my gratitude to
prof.dr. J.A. Sparenberg. He suggested the problem. In many discussions he took great pains to make me "forget" sailing practice when doing
theo-retical research on yachts in order to ensure an open-minded approach
to the problem. I am sure the course of the study would have been quite different had it been done at an institute with a tradition in sailing
yacht
research.
prof.ir. J. Gerrítsma. Already in an early stage he stimulated the study by showing his interest and making helpful suggestions. He also brought me
into contact with the experimental side of yacht
research.
prof. J. Goodrich. Thanks to his hospitality I could spend fiveweeksof the
summer of 1977 at his institute for Ship Science of Southampton
Univer-sity.
There
I learned a lot about the interaction of experimental andtheoretical sailing yacht
research.
Ian Campbell, John Flewitt and Ian Howiett from Southampton University, Lars Larsson from Statens Skeppsprovningsanstalt Gteborg, Gerbrarìd Moeyes from the Technische Hogeschool Delft for their many discussions
on sailing yacht
research.
ir. B. Bennekers from the Nationaal Lucht- en Rujmtevaart Laboratorium of
The Netherlands. He nada me familiar with a numerical lifting surface
program, which was kindly put at my disposal by the NLR.
Eugen Botta from Groningen University. His great knowledge of numerical methods for solving integral equations was of great help to me.
Matty Folkertsma, Trudy Klosse-van den
Heuvel
and Maria de Werker. Theyall typed part of the manuscript.
mr. D. Huisman. He made the drawings for chapters II and III.
my family and friends. They knew that I was doing "something on sailing
yachts",seemingly an outstanding research topic to be elucidated in
every-day language. Yet they respected my
reluctance
to talk about it and'All vessels are handled in the same way as far as theory goes, just as you may deal with all men on
broad andrigid principles.( ) Both men and
ships live in ari unstable element, are subject to
subtle and powerful influences, and want to have their merits understood rather than their faults found out.
After all, the art of handling ships is finer, perhaps, than the art of handling men".
1. Introduction
CHAPTER I
DEMARCATION OF THE PRESENT STUDY
Ships became an official subject of experimental and theoretical
research
in a time when mercantile sailing vessels already lost their importance. So
from the beginning ship research was chiefly focused on ships which obtain
their thrust from screw propellers not from sails. Furthermore the shapes of
hull and sails of the traditional sailingvesselsdiffer substantially from
those of modern sailing yachts. Consequently the methodology and the various
concepts in ship research are not primarily developed for sailing yachts.
Yet the propulsion mechanism of sailing yachts involves typical problems
which require special attention. Originally sailing yacht research was a
mixture of the hydrodynamics of propeller driven boats and the aerodynamics
of aeroplane wings. Some new methods and concepts were introduced and often
they are modifications of or additions to the existing ones. An example is
the concept of 'resistance due to heel'. A sailing yacht can move stationary
with an appreciable angle of heel. The resistance in this
position is
con-sidered to be the sum of the resistance in upright position and the resistance
due to heel. In itself this latter concept has no physical meaning, but it
becomes a reasonable approach if one remembors that so many results are
al-ready avàilable for hulls in upright position. As an example of a result,
which is inaccurately adopted from aerodynamics, we mention the widespread
belief that an elliptical liftdistribution of sails (or a keel) is favourable,
because then the induced resistance is minimal. For aeroplane wings, which
move in an unbounded medium, this is true because they only have to produce
a prescribed lift force. Yacht sails however move close to a boundary (the
watersurface) and apart from their lift also their heeling moment must be
taken into account. Therefore the optimum induced resistance is generally
not obtained by an elliptical spanwise circulation-distribution.
From the beginning of this century an increasing number of books and
articles has been devoted to the subject of sailing and sailing yacht
been published before 1972, is presented by Thieme [261. We mention only a
few. In 1936 Davidson [1] already appreciated the typical form of yacht
hulls as compared with those of power ships. In the discussion following the presentation of his paper it was thought useful for yacht designers to make
a systematic investigation of yacht hull forms. Only 40 years later this
investigation was initiated by Gerritsma [2]. In the meantime numerous
ex-periments have been carried out to study the properties of specific hulls.
However in these cases it is doubtful whether the results are general enough
to be usable for other hull forms. The first experiments,
giving insight in fin-hull interaction, are reported by De Saix [31. Other studies followed,
e.g. byMacLaverty [4], Kerwin and Herreshoff [5], Beukelman and Keuning [6].
Here we can see that in general the test results will depend on the specific
hull-fin combination used in the experiments. For Mac Laverty and Beukelman
& Keuning come to different conclusions regarding the optimum sweep angle
of the fin. Letcher [71 considers a yacht's hull plus fin as a slender body
of revolution with two wings attachted to it (i.e. the fin and its reflection
in the undisturbed watersurface), following Newman and Wu [8]. He claims a
good theoretical estimate for the lift curve slope and the drag due to
lift. With regard to the aerodynamics of sails much pioneering work has been
done by Curry [e.g. 9]. With hardly any theoretical information he has drawn
a lot of original conclusions and he made many interesting suggestions. Abt
of experimental and some theoretical work has been done by the Advisory
Com-mittee for Yacht Research of Southampton University, embodied in various
reports. Many of these results are incorporated in Marchaj's "Sailing Theory
and Practice" [IO] . We also mention the research activities of the
Institut
fUr Schiffbau of Hamburg University, which seem to be not so well-known. For
instance in "Windkanaluntersuchungen einer Segeljacht", Wagner and Boese [11]
supply very complete information on sailforces for all courses with
re-spect to the true wind. A first step towards the analytical design
and
optimization of yacht sails was made by Milgram [12]. In his work he also
took into account the heeling moment exerted by the sails and he tried to
reduce it by a proper design of the sails. The present study canbe considered
as an extension of parts of the theoretical work initiated by Milgram.
The results have to be interpreted with the necessary reserve. We do not
claim them to be reliable in a quantitative sense. For that the used
mathe-matical model is too crude. Rather they can be used to obtain some insight in
2. Some remarks on the study of dynamics and motion of a sailing yacht.
Nobody ever gave a full description of the origin of all forces and
moments acting on a sailing yacht for all its possible courses with respect
to the true wind and neither we will venture upon it. Norwill we discuss
all the motions of a yacht under the influence of the various forces and
moments. The principal reason is that the calculations in the following
chapters only have a bearing on yachts sailing to windward and evenfor these
situations we do not consider all dynamics and motions of the yacht (The meaning of the concept "sailing to windward" will be given later on).
Apart from the aforementioned reason for not attempting to give a com-plete description of "what happens when a yacht is sailing", there is an-other equally important one, namely it would be a rather precarious and complicated enterprise. Precarious because one easily overlooks or even isn't aware of some details and complicated because generally there is a
strong interaction between the various forces, moments and motions of a
yacht. We could for example try to list all factors which fix the ultimate displacement of a yacht sailing stationary to windward. First there is the yacht's weight. Then, because the waterparticles underneath the hull obtain
another velocity the pressure there will be different from the normal hydro-static presure. If the yacht heels there will be a vertical force component generated by the sails and the keel. In general there will be incidentwaves
and the yacht itself will generate a wave pattern which depends on the form
and the motion of the yacht. This wave pattern which is not known before-hand will in its turn determine the submerged part of the hull. Furthermore
one has to take into account the boundary layer on the submerged part of the
hull because it determines among other things a major part of the yacht's
resistance and hence of its velocity. And this velocity again influences the generated wave pattern, heeling angle etc. We will stop here but the com-plexity can be augmented at will. An example of the abundance of parameters,
even if one only considers part of the whole problem of the dynamics and
motions of a yacht, can be found in "Mechanik des Segelantriebes" by Thieme
[13].
The usual approach to the mentioned difficulties is first to introduce
an artificial subdivision of the whole problem in a number of subproblems,
proposed to separate the resistance of a ship in frictional resistance and "residuary" resistance, which then consists chiefly of wave resistance and
sone resistance due to the formation of eddies [ 14] It has proved to be a
very fruitful idea, although it has its shortcomings of which Froude himself was aware. Wehausen gives in "The wave resistance of ships" [ 15] a clear
insight, physically as well as mathematically, in how frictional and wave resistance of a ship are interwoven. A general introduction to the methods and problems in ship research can be found in Newman's "Marine Hydrodynamics"
16] . A discourse on the factors governing the subject of sailing yacht de-sign is given by 1-laminitt in his book "Technical yacht design" [27].
A typical example of the separation of effects in sailing yacht research is the division of resistance in upright and heeled resistance, which we mentioned in the introduction (see also fig. 2.1). We emphasize here that even when a simplification is too crude to yield reliable quantitative re-sults, it still can be used to gain insight or to predict some trends. For instance the flow around two overlapping sails is not described by a chord--wise twodimerisional potential theory. This theory however can provide us with useful information on the distribution of liftforces over fore- and mainsail when the slot between the sails becomes smaller.
Even a subproblem can be too complicated for a complete treatment. In
that case one often distinguishes between influences and effects of different
order of magnitude and then only quantities of lowest order are taken into account. For example the first order sideforce of a sailing yacht's hull is due to the small leeway angle. On the other hand in the study of wave resistance this leeway angle usually is assumed to generate a higher order effect. The extra drag due to leeway is attributed to the induced resistance which is inherent in the sideforce production. Things can be elucidated with the aid of fig. 2.1. It is partially copied from Marchaj [IO] and it may be considered as a typical speed-resistance diagram. The extra resistance under
heel is attributed to the aforementioned induced resistance. The
side-force from which it originates, is connected with the heeling side-force which is necessary to keep the yacht in its heeled position.
In this context we can draw attention to the care which has to be taken when
comparing the theoretical orders of magnitude of the various subproblems.
Peters and Stoker [17] for instance use a small parameter, say t, in their
asymptotic theory of the motion of a ship in a seaway. This parameter C is
boat speed (knots)
Fig. 2.1 Typical speed-resistance diagram.
that the wave resistance is of order 2. Now for a sailing yacht it seems
reasonable to use this saine
t
as a measure for the angles of attack of thesails when sailing to windward. Then we find a driving force of order
t
andusing this force inthe problem of wave resistance would lead to the con-clusion that a sailing yacht is always accelerated on windward courses. Hence
lifting surface theory and the theory of Peters & Stoker cannot be combined in this way into one asymptotically valid theory for predicting the yacht's stationary motion. We can think of various reasons for this discrepance. It can be that one of the other kinds of resistance has the required order of magnitude (0(t)). Since the induced resistance is by the used asymptotic
theory of order e2, the remaining candidate is the frictional resistance. But
from fig. 2. 1 we see that for the relevant boat speeds it is of the same
order of magnitude as the other types of resistance. So the cause must be sought in the mathematical approach. We try to combine two mathematical theories, each of which is asymptotically valid,imto one asymptotically valid theory. We do not know however whether the same small parameter may be used for both asymptotic theories. This has to be checked by experiments. Another possibility is that the nwnercal value of the total resistance is comparable with the nwnerzcal value of the first order driving force.
In places we have already partially schematized the sailing yacht problem. Now let us confine our attention to the specific schematization of this problem.
There is a general concensus that ít can be divided into the fÏoating body extra resistance due to heel mainly wave-res is tance upright
-
friction1
3problem and the propulsion problem, each with its own simplifications. The floating body problem is generally tackled with the methods also used for other ships, however one needs extra concepts owing to the special propul-sion system. We mentioned already resistance due to heel. Typical topics of study in the floating body problem are wave- and eddy resistance, lateral stability, the yacht's response to incident waves, etc.. In the propulsion problem one considers the action of sails, keel, rudder and hull, in so far as it is producing lift. Furthermore there is the rather isolated determina-tion of form resistance of the above-watership and the fricdetermina-tional resistance of above- and underwatership. A combination of the experimental and theore-tical data then helps to predict the yacht's velocity, heeling- and leeway angle, directional stability etc.
Our study concerns the propulsion problem when sailing to windward. Before
discussing the mathematical model we will first give a meaning to theconcept
of sailing to windward.
3. The meaning of sailing to windward
The air is assumed to be incompressible and nonviscous (as in the water).
Then we can define a yacht to be sailing to windward when
its sails are acting 85 lifting surfaces. A picturesque description bf the
action of a lifting surface, due to Lanchester, is given with Giacomelli
[is]. A mathematical treatment can be found e.g. in Ashley and Landahl [19]. The essential condition is that the local angles of attack of the sails are 'small' and the special property of a lifting surface is that it experiences a force perpendicular to the undisturbed incoming flow (the lift). Now our
definition includes more than what is generally understood by sailing to
windward. It certainly comprises those courses which make an angle with the true wind which is less than 900 (fig. 3.Ia). This angle nay even be some-what larger, because the apparent wind always points more backwards. But
also an ice-yacht, which is tacking downwind in order to get a velocity greater than the true wind speed, is sailing to windward (fig. 3.lb).
Theore-tically even the situation drawn in fig. 3.lc can occur. Here the apparent wind comes 'from behind', but the sheets are eased so far that the angle of
incidence of the sail still is small. It is interesting to note that this yacht needs a negative leeway angle and that the induced drag yields a positive
contribution to the driving force.
(C)
apparent wind, -.-.- true wind, ----+ yacht's velocity
Fig. 3. 1 Possibilities of sailing to windward. L lift, R. = induced resistance
We do not know whether situation 3.lc ever occurs. Presumably here the sail will not produce enough driving force to give the yacht a reasonable veloci-ty. Or anyway the sgil will produce more driving force by acting as a drag
device. By this we mean that the flow pattern around the sail is so
irregu-lar that we cannot speak of a laminar flow anymore, so that lifting surface
theory is not applicable. Then the forces on the sail are mainly due to friction. We observe that it is not easy to draw a strict dividing line, theoretically or experimentally, between the action of a sail as a lifting surface or as a drag device. When the angle of incidence of a lifting sail
increases, the laminar flow outside the thin boundary layer (which mostly
is turbulent) will separate at the leeward side and eddies will be formed. With increasing angle of incidence the point of separation will move towards the leading edge and a smaller and smaller part of the sail will produce lift until at last the sail becomes a drag device. In our investigations we assume that the angle of incidence is small enough to enable the sail to act completely as a lifting surface.
Tacking downwind as in fig. 3.Ib is not possible for a sailing yacht because the necessary yacht speed would involve too much resistance of the immersed part of the hull. (Catamarans do have this possibility, but they are not included in our study). At the other hand the frictional resistance of the skates of an ice-yacht is almost nil. Consequently the yacht's velocity here can become very large. Thieme [13] mentions velocities of 185% of the
true wind speed. By this great speed of the yacht its velocity component in the
direction of the true wind can become greater than the true wind speed itself. When a yacht sails in water, which itself has no velocity, we can say, in analogy with the sails, that under normal conditions it is always sailing to waterward. We will not extend the analogy (for instance one can talk about
to the production of driving force there is no fundamental difference be-tween air and water.
4. The mathematical model
First we want to give the mathematical model and subsequently discuss some aspects of it.
The sailing yacht, considered as a propulsion unit, consists of two systems of lifting surfaces without thickness, each one protruding into one of the media air or water. The interface between air and water is rigid and flat. The system of lifting surfaces protruding into the air consists of two sails, a fore- and a mainsail. The gap, which can exist between sails and deck, is represented by a gap of the same width between sails and
water-surface. The system protruding into the water consists of the rudder and the projection of fin and hull on the longitudinal centreplane of the yacht, together called the underwatership. There is no gap between underwatership and watersurface. Both systems are coupled to one propulsion unit, which translates with constant velocity. For the rest it has solely the freedom
to rotate around the line which forms the intersection of the underwatership
in upright position with the undisturbed watersurface. Air and water are incompressible and inviscid. The water has a uniform velocity with respect to the yacht and the air has a 'small' nonuniformity in its velocity, only depending upon the height above the watersurface.
Now let us discuss the model in some detail.
i) It is generally accepted that in the calculation of the propulsion
forces one can neglect viscosity and compressibility of air and water. Only when the slot between fore- and mainsail becomes small phenomena connected with viscosity could become important and hence be taken into account (see the chapter on sail interaction). If the water shows only
wave motion, generated by the wind, the assumption of a uniform velocity
is not too crude, and if we assume a rigid and flat water-surface
(zero Froude-number) it is exact. Justification for a 'small' non-uniformity of the wind can be found in the chapter on the wind in
Marchaj's book [101 and with Wieghardt[20J . The latter derives an
ex-pression for the wind strength as function of the height (U(z))above the water-surface, valid for wind strengths from 5 Beaufort. It reads
0(z) = U(1O).()h/b0, where z is the height in metres, and thus shows indeed in the region of interest a small variation with the height. All
these assumptions allow us to use the common theory for lifting surfaces, leaving behind a trailing vortex sheet.
The influence of the rigging on the airflow around the sails is
negli-gible. Warner and Ober [21] investigated already in 1923 the influence
of the mast upon the flow around the mainsail, and demonstrated that it can be substantial. Since then many attempts have been made to reduce
this harmful effect, see e.g. Marchaj [10] pp. 91-100. The hull causes some upward flow which tends to nullify the small velocity gradient of the wind. The crudest approximation concerning the sails is maybe the omission
of the hull and the location of the gap between sails and deck at the watersurface. It certainly is too low and maybe too small. Calculations
in chapter II however show that the important question is whether there is a gap or not. Moreover its exact magnitude becomes of less importance below a certain value (perhaps 5 cm). For below that value viscous or
non-linear effects cannot be neglected anymore, so that potential theory, for which our calchlations hold true, loses its validity. An interesting remark with regard to the gap between sails and deck is made by Spens
in the discussion following the presentation of Herreshoff's paper 'Hydrodynamics and Aerodynamics of the Sailing Yacht' [25]. We quote: 'In connection with sail testing it is perhaps useful to remark that in some cases the image effect is quite important. This is illustrated by the fact that introducing a hull model below a Dragon Rig increased both lift and drag force by some 10 percent, presumably as a result
of closing the gap below the genoa jib. The image effect is less signifi-cant in the case of a mainsail alone'. Otherwise we remark that sailing practicenowadays is to keep the genoa as close to the deck as possible. Letcher [71 analysed the full-scale towing tank tests of the 5.5 meter yacht Antiope. Some of his conclusions are of interest for us:
drag due to wave-making and due to lift are substantially additive and independent
zero Froude-number calculations (i.e. rigid and flat watersurface) for the sideforce of the underwatership agree well with experiments.
Conclusion b) has also been reported by Gerritsma [22]. He makes the
restriction that the Froude-number should not exceed the value of 0.4, because otherwise the waves generated by the yacht cannot be neglected
anymore. We mention another restriction (which was implicitly imposed by Gerritsma) that the heeling angle should not become too large (< 300?),
in order that the keel remainssufficiently immersed in the water. Conclusion a) gives evidence that we may indeed tackle separately the problems of wavemaking and the production of sideforce, with which is connected the induced resistance. Further justification, that we can
- with respect to the sideforce production - consider the underwatership
as a lifting surface, is obtained from experimental observations,
re-ported by various authors (e.g. Gerritsma [2]) : for fixed velocity
the sideforce depends linearly upon the leeway angle and for fixed leeway angle the sideforce varies as (velocity)2.
iv) For the calculation of the heeling moment by sails and underwatership the position of the axis of rotation is immaterial when the heeling forces of sails and underwatership are of equal magnitude.
y) We do not consider the weather turning moment, which is the moment
around some axis perpendicular to the watersurface. This moment certainly can be calculated within the proposed mathematical model. It depends upon the distribution of forces over foresail, mainsail, keel, hull and rudder, and hence upon the angles of attack of the sails, the
leeway angle and the rudder angle (i.e. the angle between the rudder and the yacht's centreplane). It can however not be taken into account in the optimization problem which will be formulated in the next section. There we will return to this matter.
vi) All sorts of motions of the yacht are neglected in the hope that they are of minor importance in the propulsion problem. There are however circumstances where the assumption of a steady translational motion must become inadinissable. For instance a yacht sailing in a heavy sea
can oscillate with appreciable amplitudes.
5. Formulation of the propulsion problem and the related optimization problem
We assume that the geometry of underwatership and sails is given. Then the parameters in the propulsion problem are (fig. 5.1)
U true wind velocity
true wind angle
V : yacht's velocity
A : leeway angle
f' m' r
: 'angles of attack' of foresail, mainsail and rudder,
respectively
the direction of the driving force.
We observe that the symbols used here do not all correspond with those commonly used in sailing litterature. Possibly the most annoying differences
occur for the heeling- and leeway angle, which usually are denoted by P and
instead of and A respectively.
In fig. 5.1 we have drawn the parameters, forces and moments which play a role in the propulsion problem (see next page).
The parameters X, S, 5 and 5r are of the same sort. They determine the
angles of attack of the lifting surface. Now the parameters are not all mutually independent. In practice U is given, the yacht's course is fixed
by a and when for example also and m are chosen the other parameters
follow from the demand that an equilibrium of forces and moments must establish itself. The equilibrium conditions of interest for the propulsion problem are:
The driving force T and the resistance R must cancel each other. The resulting sideforce perpendicular to the driving force must be zero.
The heeling moment generated by the sails and the underwatership must be balanced by the righting moment of the yacht, which we assume to be known.
From the total resistance experienced by the yacht, only the induced resistance R. of sails and underwatership can be calculated within our mathematical model. The remaining part, which we could call the external
resistance Re consists of frictional and form resistance of the whole yacht
and the wave resistance. There are experimental and theoretical methods to determine the magnitude and direction of this external resistance for every
combination of the parameters (U, V, a, , A). On the other hand the
mathe-matical model for the propulsion problem enables us to calculate for every
choice of (U, V,cxPfS ) the driving force T, the induced resistance R. and
3, A
'
, S), under the constraint that b) and c) are satisfied
(we do not know whether the Outcome of this calculation, which we call the
apparent win
projection lift sails
weather turning momeo
water driving force
-
---pF-
-sideforce sailstotal resistance above-watership tal resistance of yacht
--CL
total resistance underwatership
projection
lift2
underwatership
sideforce underwatership
heeling moment
heeling force under wateTship...I7
(c)
Fig. 5.1 Illustration of parameters, forces and moments in the propulsion
problem parameters
forces and momenta projected on the watersurface
forces and moments projected on a plane perpendicular to the yacht's centreline
heeling force sails
wate rs urf ace
results mustbe combined with R in an iterative procedure to find the
corn-e
bination of parameters which also satisfies a). And since more than one
combination can satisfy a), b) and c) one can also look for the best, i.e.
the one which involves the largest yacht speed V. More detailed descriptions of similar iterative procedures are given for instance by Wagner [23] and Kerwin and Newman [24].
The optimization problem, which is tackled in our study refers to the 'best' equilibrium situation mentioned above. The sails and underwatership generate a certain lift and heeling moment and coupled with these an induced resistance. This induced resistance, which is present as extra kinetic energy
in the wake, is represented by the free trailing vortex sheets shed by the lifting surfaces.Linearized lifting surface theory permits minimization of the induced resistance under the constraint that a prescribed lift and bee-11mg moment are generated. The induced resistance of a lifting surface is completely determined by its free vortex distribution, which in its turn depends only on the spanwise circulation distribution of the lifting surface. Therefore we can represent without loss of generality the lifting surface by a lifting line when we are interested in minimizing its induced resistance. The optimization problem then is formulated as follows:
Represent each sail and the underwatership by a lifting line.
Minimize for an 'equilibrium combination (U, V, a, X, )
the induced resistance under the constraint that the lifting forces and heeling moments belonging to the equilibrium position are generated. This formulation will be made more precise in the next chapters, where it also will be leading a life of ita own, i.e. in the numerical examples we will use fictive equilibrium positions.
The outcome of this optimization process consists of the optimum span-wise circulation-distributions of each sail and the underwatership, which
in general will be different from the initial ones. The next step is then to investigate whether the minimum induced resistance differs substantially from the induced resistance belonging to the original sails and
underwater-ship. If this is so sails and underwatership have to be redesigned in order
chat they generate the optimum circulation-distribution. Some remarks have to be made:
i) The whole underwatership is represented by one lifting line. The reason
for this is that the vortex sheets from keel and hull and a separate rudder cannot be distinguished.withina linearized theory, due to the
smallness of the leeway angle.
The result of the optimízation is the spanwise circulation- (or lift-) distribution of the various lifting surfaces. Hence in the design of these surfaces we can choose freely with respect to the theory we use the pressure distribution in chordwise direction within certain limits allowing e.g. for flow separation. This is particularly favourable for the distribution of lift over the hull-keel combination and a separate rudder. As mentioned before the whole underwatership has to be
repre-sented by one lifting line, so only the total lift of the
underwater-ship is of importance in the optimization problem. This enables us to control the weather turning moment after the optimization problem has been calculated.
The optimization process can itself form part of the aforementioned iterative procedure since the driving force of the eventually redesigned sails and underwatership will not be fully compensated by the resistance anymore.
The optimization problem can also be considered irrespective of a
prescribed equilibrium position. For example one can investigate the influence of a reduction of the heeling moment with a fixed lift.
y) An ice-yacht forms an almost ideal subject for the study proposed here
with respect to the sails. It moves very smoothly over the ice, which
indeed is rigid and flat. It has a large initial righting stability
so that generally the heeling can be neglected. The carriage isvery
slender so that it could, wíth even more right than the hull of a
sailing yacht be considered as part of a lifting surface system. By
its large speed the apparent wind angle will be small. All this makes
it likely that the asymptotic theory for sailing close to wind,
des-cribed in chapter II, will be rather reliahl. A fruitful use can be
made, then, of the result that zero gap between sails and bounding
surface gives the largest driving force. Furthermore one can probably
make a fairly good estimate of the resistance as function of the
velo-city so that possibly a rather good prediction of the final velocity
can be made.
vi) From the speed-resistance diagram (fig. 2.!) we see that generally we
are optimizing the driving force in a steep part of the resistance curve. For instance a yacht sailing close to wind in wind force 5
4 rn/s 6 knots. This means that we need a rather large increase of the
driving force in order to obtain a perceptible increase of the yacht's speed.
vii) The proposed mathematical model and the related formulation of the propulsion problem have no preference for either of the systems of lifting surfaces (sails or underwatership). In chapter V we try to
elucidate why generally the sails contribute more to the driving force
than the underwatership. In essence this is due to the fact that the gravitational force (which does not occur in the propulsion problem)
is neutralized by the
hydrostatic
forces on the yacht's hull. Hence dueto the motion of the hull through the water the direction of the yacht's total resistance is closer to the relative water velocity than to the
relative velocity of the air. If however the force, necessary to
neu-tralize the gravitational force, is supplied by a body, floating in the
air, this does not hold anymore. Now we could for instance give the
underwatership the role which is generally played by the sails, and
sail under very small angles (maybe 50) with respect to the apparent
wind.
6. Summary of the work presented in this thesis and suggestions for
further research
The next three chapters refer to the case of sailing close to wind. By this we mean that the apparent wind angle is small of the same order of magnitude as the leeway angle (We mean here small in a mathematical sense;
in practice the leeway angle does not exceed 50, whereas the apparent wind
angle generally is larger than 200). The mathematical assumption is then that
the apparent wind angle is of order r, where e is the small linearization parameter. Consequently the sails have to be represented by one lifting line, just as the underwatership. Another consequence is that the driving force is of the same order of magnitude (O(e2)) as the induced resistance. This
erablesus to determine the maximum resultant forward force and the related
sideforce and heeling moment. Chapter II considers this optimization problem for the sails alone. Investigated is among other things the influence of the gap between sails and deck on the maximum forward force. In chapter III we consider the optimization problem for the whole yacht. In chapter IV the
performance of a yacht with optimum sails, however with an underwatership which is not optimized, is compared with its performance when also its
underwatership would be optimal.
In chapter V the case of sailing to windward is considered, when the
apparent wind angle is of order r0. Special attention is paid here to the fact that now in the optimization problen we can take full account of the distribution of the sideforce over fore- and mainsail separately. We also
investigate the mathematical validity of the theory of sailing close to wind. This theory we described in the previous paragraph.
The natural extension of the research is now the investigation of the performance of a yacht of which not only the underwatership but also the sails are non-optimal. Then we run up against the well-recognized problem
(e.g. Milgram [12]) of lifting surface calculations for interacting and overlapping sails. To this end a preliminary study is made of the potential flow around two overlapping twodimensional sails. It is presented in chapter
VI. Exact results are compared with those, obtained rom a twodimensional
analogue of the threedimensiOnal Vortex Lattice Method. In this chapter also the numerical approach of the occuring integral equations is described. The boundary value problems in the previous chapters lead to integral equations of a similar type. The singularities are equal to those of the equations in chapter VI, only the boundary conditions of the '3nknown functions are dif-ferent. Hence both sets of integral equations can be cracked with the same numerical methods.
Unfortunately time failed to check whether lifting surface calculations of threeditnensional sails, based on the recommendations obtained from the
twodimerisional analogue, have any success. We hope to do this in the near
future and, if the calculations are successful, to compare current sails with their 'optimized equivalent'. Then, if the differences in performance are sub-stantial, a method has to be developed to design sails which generate the desired spanwise circulation-distributions. Milgram [12] reports progress towards it. A related problem, which can also be studied without the inten-tion of optimizing circulainten-tion-distribuinten-tions, is how the interacinten-tion of fore-and mainsail can be used to counteract flow separation fore-and turbulence at the mainsail.
In order to make the optimization problem not too complex we have kept fixed the relevant dimensions of a yacht. A by road of the optimization problem would be the study of the influence on the optimum driving force of
alterations in quantities such as the length of the mast, the ratio of mast length and depth of the fin. We stress however that our theory gives
no information on the changes in wave-, form- and frictional resistance
due to the suggested alterations.
The optimization theory provides us with circulation distributions
which are different for every different set of parameters. Hence in the
design of optimum sails and underwatership we can try to find those which
realize an optimum which is as 'flat' as possible. By this we mean that
also for off--design conditions the circulation-distributions are as near
to the optimum ones as possible.
References
K S.M. Davidson: "Some experimental studies of the sailing yacht (1936). Transactions SNAME, pp. 288-334.
J. Gerritsma, G. Moeyes, R. Onnink: "Test results of a systematic yacht hull series" (1977). Proc. of the 5th HISWA-symposium,
pp. 150-196. Interdijk B.V., Amstelveen, Holland.
P. de Saix: "Fin-hull interaction of a sailing yacht model' (1962).
Stevens Institute of Technology, T.M. No. 129.
K. Mac Laverty: "Tests of a 5.5 meter yacht form with various fin
sweep-back angles" (1966). University of Southampton,
SUYR rep. No. 17.
J.E. Kerwin, H.C. Herreshoff: "Sailing yacht keels" (1973). Proc. of the 3th HISWA-symposium, pp. 157 - 186. Interdijk B.V., Amstelveen, Holland
W. Beukelman, J.A. Keuning: The influence of fin keel sweep-back
on the performance of sailing yachts' (1975).
Proc. of the 4th HISWA-symposium, pp. 7-51. Interdijk B.V., Amstelveen, Holland.
J.S. Letcher, Jr. : "Sailing hull hydrodynamics, with reanalysis of
the Antiope data" (1975). Transactions SNANE, pp. 22-40. J.N. Newman, T.Y. Wu: "A generalized slender-body theory for
fish-like forms" (1973). Journal of Fluid Mechanics, Vol. 57 pp. 673-693.
M. Curry: "Regatta-Segeln. Die Aerodynamik der Segel" (1960).
Schweitzer Druck- und Verlagshaus AG., Zurich.
C.A. Marchaj: "Sailing theory and practice" (1964). Dodd, Mead & Company, New York.
B. Wagner, P. Boese: "Windkanaluntersuchungen einer Segeljacht"(1968). Schiff und Hafen, Heft 9, pp. 619-624.
J.H. Milgram: "The analytical design of yacht sails" (1968). Transactions SNAME, pp. 118-160.
H. Thieme: "Mechanik des Segelantribes" (1955). Institut fuir Schiff-bau der Universituit Hamburg, Ifs-Bericht No. 23.
W.Froude"0bservations and suggestions on the subject of determining
by experiment the resistance of ships" (1868). In: The papers
of William Froude; Inst. of Naval Architects, London.
J.V. Wehausen: "The wave resistance of ships" (1973). Advances in
applied mechanics, Vol. 13. Academic Press. New York.
J.N. Newman; "Marine hydrodynamics' (1977). M.I.T. Press,
Cambridge, Massachusetts.
J.J. Stoker: "Water waves" (1957). Interscience Publishers Inc., New York.
R. Giacomelli: "Historical Sketch". In: Aerodynamic Theory,Vol. I ed. W.F. Durand, Peter Smith, Gloucester, Mass. (1976). H. Ashley, M. Landahi: "Aerodynamics of wingsand bodie.s"(1965).
Addison-Wesley Publishing Comp., Reading, Mass.
K. Wieghardt: "Zum Windprofil iiber See" (1972). Schiffstechnik Bd. 19, Helft 96, pp. 35-37.
E.P. Warner, S. Ober: "The aerodynamics of yacht sails" (1925). Transactions SNAME, pp. 207-232.
J. Gerritsma: "Course keeping qualities and motions in waves of a
sailing yacht" (1968). Lab. voor Scheepsbouwkunde, Technische
Hogeschool Deift, Rapport No. 200.
B. Wagner: "Fahrtgeschwindigkeitsberechnung fuir Segelschiffe". Jahrbuch der Schiffbautechnischen Gesellschaft, 61. Band 1967. J.E. Kerwin, J.N. Newman: "A summary of the H. Irving Pratt ocean
race handicapping project". Paper, presented at the Chesapeake
Sailing Yacht Symposium, Annapolis, Maryland, January 20, 1979. H.C. Herreshoff: "Hydrodynamics and aerodynamics of the sailing
yacht" (1964). Transactions SNANE, pp. 445-492.
H. Thieme: "Literaturverzeichnis zum O.N.R. -Manuskript: Increased
performance of sailing ships" (1972). Institut fuir Schiffbau
der Universitt Hamburg, Schrift Nr. 2197.
A.G. Hammitt: "Technical yacht design" (1975). Adlard Coles Limited, Granada Publishing, London.
Journal of Ship Research, Vol. 20. No, 2. Juse 1976. pp. 98-106
On the Maximum Thrust of Sails by Sailing Close to Wind
J. A. Sparenberg1 and A. K. Wiersma'The maemuni thrust of sails when sailing close to wind is discussed. The influence of the gap between sails and hell and the influence of a reduction of the heeling force and the heeling moment on the maoimum thrust are considered Numerical results are giaen.
Introduction
FROSt 'titE POINT OFVttSV cftheracingyacht,the primary
objective is to achieve a large dris'ing force and simultaneously to
reduce to a minimum the harmful heeling force. This creed. which is to be realized by designing the sails of a yacht in a suitable way, comes from thebookSailing Theory and Practice by Marchaj
111.0 Here we use the concept of heeling force: however, it is clear
that perhaps even more dangerous is the heeling moment, which
in essence is independent of the heeling force. We will try to
elucidate some aspects of this principle by discussing a special case. We consider the sails only and not the properties of the hull, keel and rudder, which are of course to the same entent importattt for the performance of a yacht. We assume that the sails deliver their driving force ht "static" action and do not discuss dynamic effects caused by the motion of the yacht in a seaway or by the more ar-tificial action called puntping.
The specific situation we describe is sailing close to the apparent wind. The angle between the direction in which the driving force is defined and the direction from wi'ch the apparent wind comes is assumed to be a small angle of O(c), where r is a small parameter with respect to which the theory will be linearized. Hence we consider a limited case in ss'hich the range of applicability has to be checked by experiments.
For tise driving force created by the relative motion of sails and air, it is important that the resistance forces be small. These may be listed as induced drag, friction drag, form drag, and additional
resistance of rigging. Etere we will consider only the induced drag, which, as discussed for instance by Milgram 2,31, isa substantial part of the total drag of the rig.
The dependence of the maximum thrust, e'2), on a number of parameters is the' subject 0f this paper. Important geometrical parameters are the heeling angle and lite gap between sails and
hull.
When it becomes too large, the heeling angle is detrimental to the gond performance' of the yacht: hence it will be of interest to investigate what happens Io the maxinicleil titrust when we put a constraint on the heeling moment and o the heeling force. This hind of constraint will also appear when a yacht is considered as i system willi Iv, o types of wings moving at the boundary of two inedia with a relative motion. The sails, which are one type of ss'ing. are prcctrudling ill the air: the other type, the keel, is in the water. 'rhen of course for a stationary situation there has to he neither a resulting lateral f orce nor a resulting heeling moment on lie yacht; hettce there is a relation between these forces of the nails
.1ttdtite'keel. Titis will lii' considered in a following paper. The
,sfluenceof the gap xs'ill he discussed by simulating in a simple
tttanfler several values for its width.
University nf Cronicsgc'cc, Grorsingc'n. Tic'Nelhe'rlamls. '2 Ntinctx'rs ici hrackets designate References at cru I nf paper
Ntactniseript received alSNAMEt,eacle1riarte'rsjsm' 23,I 1175, revised manuscript received December 2, 975
Some ideas have beecu used in practice to diminish the induced drag of sails. The only way to do this is by spreading out the shed-free vorticity in an appropriate way. We mention the tall
hollow masts by which sails with a long trailing edge can he used, and a wide boom which has to prevent the flow around the boom caused by the pressure differences across the sail il.
We do clot take icuto account the dependence of the velocity of
the wind ore tite' height above tise water surface. When these
variations of the wind velocity are small O(e), they only have in our case of sailing close to xvind an influence of O(es) on the thrust and hence can be neglected. When they are finite, hence nf 0(r°), then
the concept of trailing vortices is lost and a more complicated
theory is needed 141. These variations will cause a still stronger decline of tite maximttm thrust by heeling than follows from our numerical results.
Finally, we point out that this paper does not claim tobe ap.
plicable directly to the analytical design of sails; instead, it gives a consistent linearized theory for sailing close to wind acid intends
to enrich the background already available for more practical
considerations. Possibly the most ideal realization for zero heeling angle occurs in the case of an ice yacht sailing close to wind.
Formulation of the problem
Figure 1 represents a coordinate system IX, Y. Z) which is in rest
with respect to the ship. The air is in the half-space region Z >
O. The relative velocity of the scind makes a small angle ii with
the X-axis and has the magnitude t T. Tise angle ii' is assumed to be of O(i) where isa small parameter se ith respect to svhich the theory is linearized. Because of this small angle the vortex sheets of foresail and mainsail are separated icy a distacccn' nf O(e) and hence cad he' considered to coincide, in a linearized optimization theory, lifting surfaces can be replaced by lifting lines, which can be chosen at an arbitrary place at the vortex sheets. Hence also
the lifting lineen can be asscime'd to coincide. So we' re'piai,'e the' sails
of the ship by one lifting line' (.f. - R) ticrough the origin (land in the lY. ZI-placie wïth a heeling angle , O t3 tr/2. The free
vortes sheet H stretches doss'cistream from (A - B). Because the unchsttcrbed mainstream has ocsly a component of O(e) perpen-dii.'ular to the' X -axis. H can he chosett svititin the accuracy of tice linearized theory parallel to the X-axis.
In order to simulate' the' boundary between air acid svate'r. sse consider also. as usual, the image (A - B) of (A - B) and the image H of H; both reflertions tire with respect to the piatte Z =O. We define the + (-1 side of H acid H to be oriented in the positive
(negative) Y direction acid the unit normal vector ri on ¡J artel on
¡lirons the + side toseard the - side. Oct H and JI we introduce a cartesian coordinate system s) where inc Il lice cocurdinate s 5 tile distance from a picicit to the X-axis. svhile lun H this coordi-nate is minus this distance'. The hoitctciaries of H are denoted by
= a acid s=b, with O a <b, acid of JI by s=a and s=
b.
Reprinted from JOURNAL OF SHIP RESEARCH, ViiI. 20, June 976, pp.98-bR
The width of Ji and Ñ is denoted by / = (b - a). We assume the whole space to be filled with air of which the incoming velocity
for Z < O is the same as for Z > O, The air is assumed te, be in-compressible and inviscid and of density p
The driving force or thrust T )Ø(2)) of the sails is defined as the component in the X-direction of the total force acting on the bound
vorticity of the lifting line)A - B);it is reckoned positive in the
negative X-direction. It is our intention to determine within the realm of a linear theory the maximum thrust T,,,.
It can be made plausible that an upper bound exists for T. Suppose we have at (A - B) and )A - B) some bound vorticity
distribution rI'(s) (z' is some factor O z') reckoned positive with
a right-hand screw in the positives-direction. Thus by symmetry
('(s) = ('(-s) (1)
The thrust T becomes
= p ul'(s) (lJa ens/i + z'th,,)s)( ds (2)
where z')s) is the component of the induced velocity in the di-rection ñ at (A - B). When z' increases starting from z' = O, first
T also increases; however, the induced velocity z'tZ',,(s) also be-comes stronger and will more and more counteract the component
(1cv cop! until for some value O < z' the thrust T is again zero.
Hence because T depends quadratically on z', one maximum value of T exists for the I')s) under consideration. When we assume that arz optimum F)s) exists the thrust will have for this distribution an optimum value denoted by T,,,,
We define next the heeling force F acting on )A - B) as the force perpendicular to (A - B) and the incoming flow, reckoned positive
in the negative ñ direction The constraint that F has to have a prescribed value Fi, which in general will be chosen smaller than the value which belongs to T,,, has the form
F = -pU I'(s)ds Fi, (3)
Analogously the constraint that the heeling moment has to have a prescribed value Mi,, which is reckoned positive wizen it is
con-nected with a right-hand screw to the negative X-direction, is
M (e + s)F(s)ds = Mi, (4)
where r is some constant depending on the line with respect to
which the moment is calculated.
We now change the formulation in such a way that we obtain
a problem of energy extraction out of a slightly disturbed fluid.
Consider the cartesian coordinate system (r, y, z) which is related to the former system by
xXt't, yY, zZ
(5) JUNE 1976 u . o C BFig. i Lifting line and free sortes sheet
X
The air is in rest with respect to (x, y, z) except for a small
homo-geneous flow of magnitude Un (O(s)) in the y-direction and for a negligible flow of O)v) in the x-direction The lifting lines (A - B) and (A - B) then move in the negative x-direction with
ve-locity U, along the strips H and H, on which from now on we use the coordinates (x, s).
The only way in which(A -B) and(A - B)in a linearized theory sense the homogeneous floss in the y-direction is by its component (Jo ens/i in the ñ-direction, The most direct way to induce this normal velocity field is by placing at H and H a
suitable vortex layer parallel to the x-axis of strength -y(.$). Ve
reckon -yo(s) positive when it is connected with a right-hand screw
to the positive x-direction. We have
= y)'-s)
(6) In order to define this vortex layer uniquely we assume that its tcstal circulation is zero and hence has a finite kinetic energy around itper unit of length in the x-direction.
We can then consider the following problem. We have a fluid
at rest with respect to a coordinate system )x, y, z). At the two
strips H and H, now considered to be two-sided infinite, we place vorticity 'yo(s) of O(/i, which creates the desired normal velocity
(Jo ens/i. Next the lifting lines (A - B) and )A - B), corning
from r +", move' along H and H in the negative x-direction and have to extract, under the -onstraints (3) and (4), as much kinetic
energy as possible out of the fluid Then clearly the thrust will be
a maximum.
When there are no constraints )3) and (4), the solution to our
problem is very simple. The bound vorticity l')s) has to be such that its trailing free vorticity annihilates the free vorticity yo(o).
Then all the kinetic energy is taken out of the fluid The meaning
of this latter result for the original problem is the following. The
bound vorticity of the sail has to leave behind trailing vorticity such
that at the Trefftz' plane (far downstream) it induces a normal velocity at H which is opposite to the normal component of the
original flow; the surface of the water has to be considered as a rigid
boundary. Of course this condition is tightly connected to the condition for the minimum induced resistance of a wing
The optimization method just described has been chosen in favor Pf a more direct method based on (2), because it gives a good in-sight into more general linearized problems of energy extiaction.
Consider a fluid with a time independent velocity field ii'(x. yz)
of O),) caused by a vorticity distribution ' (r, y, z). In this fluid
a smoothly curved strip ¡lis present, along which a flexil,Ie wing
W cari move The question is what part of the kinetic energy can
be extracted out of the fluid by W The procedure is then
anal-ogous to the one followed in the foregoing Consider the strip H
in a fluid al rest. Place at H vorticity 5'o which induces a velocity
field i15)x, y,z) with thc same normal component at Has was
in-duced by ', (r, y. z) and which has zero total circulation. Then
the kinetic energy belonging to y and not more can be extracted
B a a s xb sO-b
out of the fluid. The wing W has to have hound vorticity which leaves behind free vorticity of strength 'ij.
Thevariational problem
Consider the incompressible and inviscid fluid with respect to
the cartesian coordinate system lx. q, z), n which we hase the
two-sided infinite sheets H and H (Fig. 2). On these strips we have
vorticity 'o(s) = 'yO)s) svith total strength zero
yo(n)ds= 0 (7)
which induces a velocity field L (y. z) with a normal component
lin cosa. From (7) it follows that outside H and H, i5 (y, z) can
be described by a uniquely defined potential which has a jump across the strips. We write
= (0, vi, wt) = isV cosi'3 gradçi (y, z), (8)
where (y, z) is a potential function with lirnl (y, z) = O for y2
+ z2 - and with 2t/dn = I atH and H. Along the strips
move the lifting lines (A B) and (A B) which have to extract
as much kinetic energy as possible out of the fluid, taking into
account (3) and (4) When the lifting lines have passed and are
"far away" at z = , the kinetic energy per unit of length in the x-directiou has changed by the trailing vorticity y(s) of (A B)
and (A B) left behind at H and H. Outside these strips the
ve-locity field i (y, z) induced by the trailing vorticity can be de-scribed by a potential function ç (y. z)
= (0v, w) = grad(y,z( (9)
which also has a (limp across H andÑ. The kinetic energy per
unit of length in the x-direction becomes
Ebe(=p
s s.:
(_xnUcosß.(+t)2
+
(oUcos-'+)2dydz (10)
which is a functional of the unknown potential .
From (2) it is easily seen that the circulation for a certain value of s around the sail in the case of optimum driving force and no
constraints is proportional to n[!, The heeling force and heeling
moment in that case are proportional to nfl2. We now denote by
pM and pM2 the heeling force and moment when u = t and ti
= 1. Then the actual heeling force and moment, again inthe case
of no constraint, are xsfJ2pM and uvfJ2pM 2. Using the potential
the constraints(S) and (4) can be written as
«X H
Fig 2 The two-sided tree vortex sheets ¿-land H
H
K(sc)def.(2HJIxUMI
+ Js;
dsj 0,j
1,2 (11) where)(s)] = r"(s) - (s( = I'(a( (12)
TIse indices - (+) denote the values of the potential at the )+)
side of the H and H; p is tite fraction of nfJ2pM, which is
toler-ated; and
f(s(ee 1.J(s) = r + IsI (13)
The interval a <s <a gives no contribution to the integral in
(11) because there (17 vanishes.
Introducing Lagrange multiplierscvlJpcosd !1'"/X1 where the factor ixVp cosd l) is only for convenience, we have to minimize
the functional
G)) = E() + sUp cosß l°_11)sjKj(çe) (14) In order tobe acceptable in our problem, the potential sr must be symmetric with respect to the z-coordinate and one-valued in the whole (y, z(-plane, especially when H or H is encircled. The one-valuiedness of follows directly from the fact that the trailing
vorticity of the lifting lines each have total strength zero.
Consider a perturbation h of such that t also satisfies the
conditions just mentioned for . hi order that G will be extreme,
we have the necessary condition
r+ r /
S¿lGp I
- J--' \ ay /13/ ay+
(au cosfi'+ ) -1)-} dyd:
+ aUpcosa
jij 5
ds 0 (15)By partial integration of the two terms in the first integrai, using
the fact that 'u and are potential functions and combining some
terms, we can reduce (15) to
j'
+haU cos/ (s) +
xxV cosi 1(i'JtXf(s)j ¡s)) da = 0 (16)
where because of the symmetry
)()I = )As)J
(17)Now we take s) zero everywhere except in the neighborhood
of s=SO and s=s0 with a <so < b. That these boundary values
of bç on H and H can be extended in the (y, z (-plane by a one-valued potential function ¿ip(y,z),and hence are adnstssable as a variation of , is seen as follows. Consider for instance at H two disturbance vortices of opposite strength, parallel to the x-axis, each at a side of s Soand close to it. The same is done at H for s = SOwhere the vortices have opposite strength of the
corre-sponding vortices at H Then the potential A(y. z) outside H and H belonging to these four vortices is symmetric and one-valued in the whole (y, z (-plane.
Using this we find from (16) as a necessary condition for
optimality
(18) at H andÑ,where we used the boundarycondition for r and (13),
We now introduce the potential function (y. z) which satisfies the boundary condition
= IsP (19)
at H andÑand lini =O for (y2 -i- z)'/ - . Then it follows
from (18)
y,z)=oU cosd {(i +x, + x2 ,(y. z)
+
T
A2C2(y z) (20)Using (18) we find for the total velocity component w in the
di-rection of ñ at the lifting line (A - B), because then the trailing vorticity is "only half infinitely' long
w)s)=oU cosd + (s)
2 en
= aU cosa {_l + A, + 2(r + si)]
Hence the thrust Thas the valus'
T= ,
fw(s)kr(s));
dui +A"
+ s)l-..po2U2cos2lj
5b[
+A, J
+x,
+ x2) ),)s)J
+The Lagrange multipliers Xi and X2 follow from (Il)
iMi + cosa
5"
(i +
x,+ x2) )Ci(s))a
-+-)CS(a))*jds=0
2M2 + cosiJ
(r -t. s) A2 -j(i + Xi + X2) )i(s) +T
)(s)),
ds=O where M,=cosa
fb
ds M2=5h
(r + s)(,(sfl ds (26)In order to obtain dimensionless formulas we introduce the potential functions .(y, z) and z) by
JUNE 1976
'lj(y. z) !i,c1)Iy. lz), j= 1.2 (27)
These functions besides on y and z depend only on the dimen-sionless parameters ß and a/I. The equations (23) and (24) for A, and A2 become by using (25), (26) and (27):
f:+1
{(i
-
+ A +x5)
+ A2(s)] ds =0 (28)
£r
(1+s){(lns+Xi+X21))i(s));
r+ X2is(s)(}ds=0 (29)
From(20)and (27) itfollowstha the optirnilrsi boundvorticity has the representation
l')s)= oU cosI
j(i + X, +
x2)[1
(;)]
+
X2{tiS()]
j,asb (30)
ThethrustT,the heeling force F,,,and the heeling moment M,, can he written as T pco2 U2 cos2ß 12 (i + X, + A2
( +
j(i
+,
+)
,)sfl +X2)a(s)]j ds (31) F,,=,peiU2cos/312J '
)tfi)s) ds (32) ri/iM,,=2poU2 cosl3 I
5/+i
G + ),(s)J ds (33) We now discuss briefly equations (28) and (29). First suppose= ,' =1. Then we have no constraints, whieh is in agreement
with the fact that A,=A2 O satisfies these equations. When there is one constraint, for insiance, on the heeling force, then we have only the multiplier X,. We then put X2 = 0, neglect equation (29) and calculate X, from (28). Then the heeling moment M,, has to he calculated by means of (4) where r(s)from (30), with X2 0, has to be used. When there is only a constraint on the heeling moment, we neglect equation (28). put A,=O, arid X2 follows from
(29). The heeling force F,, then follows from (.3) where 1'(s)
fol-lows from (30) with A,=0. This latter case is dealt with explicitly in the next section.
The integral equations belonging to and i'2
We consider in this section the case of only a constraint on the heeling moment. Furthermore, we takt' the heeling moment with respect to the origin, r=0. Under these assumptions we have to neglect (28). and (29) becomes
s;' s)(1 -
y2)Ii',(s)( + X2(2(s))) ds OThe equations (30), (31), and (33) specialize to F(s)=aUcosti!
{[1 G)].
+ X2(»]* j
(35)(34)
T= pa2U2 cos2ß 1
