The tactical implications of the polar curve of yacht performance

16 NOV. 1976

ARCHIEF

I

w

UNIVERSITY

OF

SOUTHAMPTON

department of

aeronautics

and astronautics

S.TJ.Y.R. Report No. 20

THE TACTICAL IMPLICATIONS OF THE

POLAR CURVE OF YACHT PERFOR}IANCE

P.V. NacKinnon

ADVISORY COMMITTEE FOR YACHT RESEARCH

Lab. y. Scheepsbouwkunde

Technische HogeschooI

Deift

S.U.Y.R Report No. 20.

The Tactical Impliòations of the Polar Curve of Yacht Performance

by

P.V. MacKinnon.

January, 1962 Reissued, 1967.

Summary

S.U.Y.R. Report No. 9, which coritaiis

the

_{papers presented} at the Conference on Yacht Design and Research held at

SouLhamptn University in 1962, has been out of print for some time. _{Most of the material in this reprrt is now available}

in other pub ications, but this does rot appear to be true of the paper presented by P.V. acKmnnon, entitled 'A Scientific Approach to Yacht Racínci'.

The present report is a reprint of this paper. It discusses the tactical implications, for sailing in winds _{of varying strength} and direction and in tidal streams, of the knowledge of yacht

rerforifiance which is repi;unted by the polar curve relating the yacht's speed (V) to the true wind angla (ì).

A recent addendum comments on the application of the saíne principles to modern catamarans.

A second addendum gives estimated polar _{curves for a 12-metre} yacht.

A Scientific Approach to

_{Yacht Racinq}

I shall not attempt _{to discuss the}

tactical problems which arise from

_the

_{presence on the course of}

other co:petitors. My approach is that, for _{a given course in given}

conditions of wind and tide, the problem _{is to discover the}

quickest way to sail from each mark

to

_{the next and so to finish}

in tbe

shortest possible time. _{For consideration}

of this problem it. Is necessary first to know the maxirTum

_performance

of the

yacht,

_{and this «111 dIffer} for different wind _velocities.

Polar Curv

Let

_{us therefore suppose}

that _{know our yacht's maximum} speed, sailing at _{any given angle to the}

true wind direction, _and over a range of true wind _speeds.

This information 1 most

conveniently shown by means of a series of polar

_curves.

Each curve will _{apply for one wind}

speed only and will depict graphically the performance

of the yacht when _{sailed to best} advantage. _{The distance of}

the curve from

the _{r1gin in any} given direction represents the maximum

speed of which the yacht _is capable In that _direction.

There will, of _{course, be}

gap in the curve directly _{upwind, and}

without attempting

to be precise

about

exact speeds, the polar _{curve for a}

normal

yacht in a moderate wind wil.

is the origin and the curvo is so

_drawn

_{that any radius vector V} from origin to

curve

_{represents the yacht's maximum speed}

in that direction

?

to the given

_wind.

If the curve has either _{a hollow or a gap, thon an external} tangent is

drawn

and the limit of performance _{to windward is the} tangent. _{It will, however, be}

necessary to sail on

two

tacks to achieve lt. In Fig. 2 thé

_{polar curve}

_{is the saíne as in Fig. 1,}

but

the windward tangent has

_been

_{drawn and the yacht's performance}

Is limited by it.

_That

_{is to say, the}

_{best speed}

dead to'windward Is represented by the vectors OB. _{But to sail In any} given direction in the

_windward

_{section, lt is necessary to make}

two

or more tacks, each of which _{is in aithe}

the direction or the direction OB.

The polar

_curve

_{is the simplest}

way of setting out full information on the performance _{of the yacht.}

A complete set of measurements has, so far _{as I know, nover been made,}

it It is nevertheless possible to give _{a demonstration of how these curvos} can be used (and how they will _{doubtless be used when}

they have

_been

measured). _{For a normal keel}

yacht they are likely to _{have the} following

features-Except possibly in the _{downwind quadrant, the curve} is convex everywhere.

The velocity Is _{greatest in the réaching direciion} (approximately across wind).

The

two

points where maximum

_speed

_{is made good to} windward are at the highest _{points of curves which}

are

smooth and continuous.

Even if nothing more is known about the curves than this, some interesting and valuable

deductions

can be made.

I propose to show how a number of special cases In the prcblem of sailing in variable winds _{can be dealt with. It is a common} experience that a change of wind direction results _{in a yacht}

previously out of the running being elevated _{to first place. Such} a happening Is not mere luck if the helmsman (a) foresaw _{that as a} matter of geometry he would be favoured if the change of wind

direction took

place, (b) had some reason,

_{whether from meteorological} knowledge or just

a "hunch", to think that lt might take place.

My object Is to draw attention _{to the methods which can be used,} given the yacht's polar curve, to examine the effect of possible future changes in wind direction _{or strength, and}

_{so to}

_{assess what} will happen if these changes take place. _{The geometry Is not} always simple and the results not entirely as

one might

_{expect, so} that what can be learnt from a few examples is helpful towards an understanding of general principles.

Sail jnq to Windward

When faced with the problem of _{sailing a windward leg in a} variable wind, the

_helmsman

_{cannot of course know what is best to do} unless he can foresee all the _{future behaviour of the wind für the} period of time Involved. _{It wotìd then be possible in theory to} select the optimum path. _{In practice all that can be done is to}

rk to a few general principles,

and an examination of .certain special cases throws

light

_{on these principies.}

-3-Case i

In an

otherwise steady wind, there Is a

change

of direction for a short period, after which the wind reverts to its original direction. In Fig. 3, 0 represents the point reached when the

wind

changes direction; M is the windward mark. The figure is drawn for the case where the

temporary

change Is a

ciockwi.se.one,

and we draw from O

as origin a scaled down copy of the

polar curve which

indicates the

limit

of distance attainable in any chosen direction during the period.for which the

wind

remains changed. On this limiting curve it is obvious that

P is the

point from which, in the normal wind, M can be reached in the shortest'

possible time.

If the polar curve conforms to feature (jij) of those quoted above as applying generally, P does not coincide with A.

Thus we sec that during the

wind shift,

ndt only is It Important

to be on one particuiar

tack, but it actually pays to sail on that tack wide of the normal angle used in

windward work.

There is also the obvious limitation that the result

only

applies when the

course from P

to v1

is within the

windward quadrant and so cannot be

sailed with advantage

on one tack.

Case 2

Shortly before reaching the windward mark, after a long windward leg, the wind changes direction and the change

persists

for

a ionge± time than is needed to rach the

mark.

What is

the most advantageous position to be in wt-in the change takes place? In

in the given time. This line is of course limited in length but in the nature of the problem the limits do not

enter Into the

result.

A polar diagram reversed is

drawn

wIth the windward mark t

as origin, and

for

the new wind

direction.

This

represents the locus of poInts from which M can be reached in equal times, arid if the scale is such that it touches XY, then the point of contact is th solution to the problem.

This time the solution sho that after the change of wind

direction it

pays

to sail the final leg to the mark on one

particular

tack, and

wide of the normal

angle for windward work.

Case

A windward

leg is to be sailed from the starting

position M1

in Fig. 5

to the

w1ndrd mark M2

and after the lapse of approxirndtely

half the t.me

required

the wind changes direction and the change is permanent. In Flg 5 the polar curve H1 A1 B1 is drawn with M1 as origin to such a scale that it represents the locus of points which can he

reached from

within

the time which will elapse before the wind changes. Another polar

curve reversed is drawn from M2 as

origin, and for the new wind direction.

This represents the locus

of points from which M2 can be reached

in equal times, arid if

the scalo of

the second

curve is such that It touches the first, then the point of contact is the solution to the problem.

Fig. 5 shows that in sorne cases, depending

on the relative

positions

of M1

and M2, and on-the amount of, and time

_{of, the change}

the normal

angle for

_{windward work.}

-1aving exarnnod three particular cases, I will ' tö introduce some generality. Let us consider a

windward leg to be sailed in a

wind which chances direction frequently _{and at random,}

_{but having a}

mean direction such that the leg is always to be _{considered a}

windward one.

From the starting point, for the wind direction prevailing

_{at the}

start, a small polar curvo

can be drawn for

points accessible within the interval of time for which the wind is expeted to remain the s arno.

Let us now

_{forecast (or sornetrow have knowledge of)}

the next wind direction. From each of the points _{on the firt curve a}

polar can be drawn for points

_{accessible within the next}

_{interval of}

time.

These polars will have

_an

_envelope

which is the locus

of

points accessible within the _{second interval of}

_time.

So, if we cn forecast the wind changes, ? step-by-step process is possible which will in the end _{colve the problem of how quickly,}

und by

what route,

the

_{windward end of}

_{the leg will}

be reached. The process just. described

is exactly anlogous to that in current use for

_{optimising ocean}

_{steamship routes.}

Forecasts are made of wind and sea for the duration of the voyage, and _{from these}

forecasts are deduced the shlp's

_possible

_{speed on each day of}

_the

voyage and at each point

_in

_{the ocean.}

The process is, of course, of such

ctrnplication that it

can never be carried _{it on board a}

yacht;

hut a

race is won '

the halrnsrnan who

_{makes the}

nearest

approach

o t.he correct

solution. It Is therefore _{necessary for}

and such

principles cari he best evolved by the methods used to illustrate Cases 1, 2 and 3 rather than by the method of successive envelopes. It is, for examples a ;.ìrnplc extension of Case 3

to

deduce that if

the

wind is fluctuating in

drectn about a

fixed average, then whenever the wind has shifted clockviso of

the average

the yacit should be on the starboard tack,

and moreover should

be

sailing wide of the

normal windward angle to the wind as lt is at

the moment.

It will be noticed that Cases 1, 2 and 3

are all

examples of thu use cf two polar curves, one drawn with the startthq point as origin

and the other drawn reversed with the

goal

as origin.

right hand half of it is the line C1 B1 K1. For the adverse In Case 1

the proportions are such that only the straight portion of the secorid polar is

relevant.

In Case 2 only the straight

portion of

the

first polar is relevant. It was not

until

Case 3 was reached that the idea was mentioned of using two polar curves which touch,

xit in

fact

Casos i

and 2 use the same process.

One further examples of sailing to windward is instructive as showing that the polar curve method can be applied to charges in conditions resulting from differential. current movements.

Case 4 is shown in FIg. 6 where M is the windward mark, moored in an adverse tidal stream, and to be reached in a wind steady both

in

strength

and direction. There is,

however,

a strip of water

unaffected by

the tide

and

obviously

the yacht should tack up in this until the mark can

be reached

in a single tack. ini 11g. 5 the polar

diagram

for still water Is shown with origin 01 and the

current section the polar diagram is altered for two _reasons.

F.. L the adverse current reduces the vîind relative to the

sea surtace, and the polar diagram relative _{to the sea is therefore}

tu

be drawn br a reduced wind

_{speed, as}

_{shown by the line H2 A9 C2} with origin O.. NOW _{take a new origin 02 such that}

02

0

represents the

current, and

_{the polar diagrdm relative to the mark}

will then be the . lue H2 A9 C2 with origin O:,

The problem is to sail from a given starting point, say

_01,

to M in the shortest çosible _{iine. if M}

is parallel to At,, 0a course such as 0 _{Q1 M is possible (the yachts}

course throuqh th water on the lE-q _{M bein9 parallel to 01 A2). There is,}

a faster course such as

01 T2 Q2 M, on Which the yacht is uled in the normai manner to Q, and then 'from _{to M wide of the}

normal. windward courses The vector ovei' Lhe ground _{on this leg}

will be

₀₂

_{P and through the water}

1.

The proof of the assertion that. such a faster course exists will

nnt

he given here, but requires _{on .y the ssumptJon (already}

mentioned -as feature (iii) of _{those generally obtaining) that the} curve A, P l-{, _{is tn'entiai to, and does not}

start off at an angle ¿

with, -the straight line C2'A2. _{it is then posibìn to show that}

the couxse to Q1

and thence direct 'Lo M 'is actua?ly _{slower than one} which can be found caiLing th _{final 1-eq slightly free. Given the}

actual polar curves and the current velncity the problem could be

crnnp.Letely

solved.

I

I

Sailing Down Wind

The expression "tacking to leeward" is

11 known, but the

practice of this manoeuvre is not as frequent as it cught to bc.

Let us first consider the case of a yacht with

a very high reachtn.q

speed. The ice yacht Is the supreme example, but the sama considerations can

arise w:ith a

catamaran or a sailing canoe, arid

for other light

displacement

craft.

The po1r curve of

performance may have a hollow in the duwrì wind

direction as in

Fig. 7, and if this is the case it is obvious that In a steady wind any down wind leg of the course within the sector FCX _{should be}

sailed

tacas parallel to OF and

,

and the proportion of time

spent on thase two directions adjusted to give the

required

resultante

Hovvor, the yachts to which this applies are rare.

But a

change of

wind direction may make

tacking to leeward proftahte in

any yacht, even tf the polar curve is like that showi :n FIg. 1.

The argument is similar to that already applIed In Case 3.

Case

is illustrated in Fig. 8.

The leg to be sailEd Is

from M1 to M.,

with wind dlreçtlort initially straight down the page,

But befoe It Is possible to reach M9 in this wind, a clockwise change

of direction takts place.

The polar curve is drawn with

as

origin to the scale which represents the locus of pints attainabte

within th

time before the wind

changes

direction.

From M, arid

for the naw wind direction a reversed polar is then drawn to such a

sca].e thti it just touches the first.

The point cf contact, P,

io

-to be sailed arc M1 P and

This example shows that a

change of wind direction may make "tacking to leeward" advantageous in certain circumstances, even when tho yacht's

polar may

not

suggest

this 9fl first inspection.

( in Fig. shows that it is possible for a straight course

to be advantageous even

when the wind changes direction,

but this

I

will be comparatively rare, as it occurs by chance only for certain relative positions of M1 and M2.

To

conclude this series of examples, Case

7 is one

where

the

most advantageous course is

not a straight one because of a change

in nrid treng th. Fhis can arise because the

polar

curve for the same yacht is usually of quito a different shape in a light wind to that in a strong wind. Fig. 10 shows the form the polar curves commonly take. The one

for light wind

is of a different shape, and in Fig. 11 the

result is shown when the log to he sailed is a broad

reach

and the wind falls from strong to light at

half time.

The pr.coss and the

lettering is exactly the same as for Case 5, Fig. 8. The result, we 1 known to

yachtsmen, is

tht ji pays to git to leeward 'f the direct course while the wind is strona.

Certerol Discuscion

lt cviy be

asked how

these polîr curves hivc been measured, and r must confess tho'l.

they

have not. rheir shape is based on the eccum'jLtcd experience of yachtsmen, and to sorne extent ust1fied by

th sucessfu.1 application of the

results to which

they 1ad. The purpose of this

paper is

to show that knowledge of tha complete

I

out of the best way to get round a course in predicted wìnd and tide conditions. The complications are formidable even when the polar curves are available, but all know so far as yachtsmen is that a

few generai

principles bring success - for example,

that it is

better to go

quickly somewhero

than slowly on the direct course. This maxim is illustrated by several of the examples already given.

I hope that i may focus

attention on the

need for

measurcrnents

of the polar curves of a number of representative yachts, arid for a more detailed study along the lines described, but using po]ars a.

actually measured. Only in

this way

can it be ascertained with certainty

how much wide of

the normal close-hauled course one

should

sail in given conditions, or how far to leeward of the direct course one should sail on a broad reach when expecting the wind to die away.

The work which is taking place at Southampton University will no doubt provide comprehensive information of the kind required, arid a

study of the polir curves and their behaviour when used in ossihie cases of changes of wind direction and strength is the only way to reduce to a scientific

basis the

art of sailing in i

variable wind.

The same methods require extension to the cases where

the wind

direction varies, not with timo but with position, and the problem js

to

pick the most advantageous course through such variations.

The complications are no more formidable thon

have

been

deì.t

with in other fields -

for

example, studies of the art of gliding for maximum duration, or

for maximum

range in a head wind, Yacht;men as

- 12

-a whole h-ave le-arnt the hard way a great deal which could have bean predicted ori the basis of such polar curves had they been available., and if ty can be measured, or computed from otherperformance measuremento, the nethods evolved by yachtsmen are capable of further refinonent.

i

i

Ad dendurn

Five years have elapsed since this paper was written aiìd there is

little to

add. However the development o

the C class

Catamaran has enormously increased the need for full

understanding of "tacking ta leeward".

These cro lt have deve IOL:U the una rig o the most effective overall, but in iigìt or moderate winds the L:st. downwir. performance is on a course at approximately

450 ta

th& true wind direction. The polar curve i.s similar tn

rig. 7 but

the angle k0G is about 900 and the hollow between E and G is very marked.

The result is Lhat coisiderations similar to those descrìhe in cases, 1, 2 and 3 apply also to dQwn wino sailing, and if the wind is fluctuating in direction abut a fixed average, then whenever

tfl

wind has shifted clockwise of the averagc the yacht should be on the port qybe, and marcver sriouid be sailing slightly higher than the noimal angle tor Leeward saiLing in a steady wind.

The well known working rule tar windward sailinu; "Tack whenever the 'und heads you" how has a corresponding comparison

for leeward sailing which maj be expressed as "Gybe whenever

the wind comes more att". This rule must he

urierstad as Ieterriûg

to the true wind direction; and this must be carefully

distinguished

from that of the apparent wind. Generally speaking the difference in direction between the two i; with this te of yacht, nuch greater ir downwind than in upward sailing.

PJLAR CURVES FOR A J 2-4ETRE YACI{T

AtflENDU IT

S.U.Y.R. REFORT No. 20.

In order to aive a more precise idea of the form of the polar curves

for real

ychts, Figure 12 ha been prepreo. It shows estirnited polar

curves for a modern

12 mntre yacht silirg in calm wter in trua wind

strenqths at'

7, 12 and 20 knots,

carrying genoa or spinoake.r as 3propriate.

The curves are in reasonable agreement

with observations ot 12 metre performance as Lve11 as with the

performance eected

rm tank tests 01 hulls ¿md wind tunnel tests of the seiLs.

However as both fufljze

ohservatiors arid predictions frocs model tests are subject to error these curves must

. .

only

be

regarded as

approximate.

Furthermore iii open water the stronger winds would

ciuse a cons!ierble sea.

This would lead to a marked reduction in the speed made

qod to windward, and would also affect, to some extent, the speeds on

other headings.

Thofitr--

marked alongside the polar curves indicate the apparent

wind angle,

i.. the angle between the yocht's centreline and the direction of

the wind

_elative

le.

the yacht.

This is the angle that would be indicated by a wiridvane on the yacht,

could he located at a positiun where the air flow was not affected

by the

rsdmity ot the sailc.

It is ir1t?restinq to see

much the forward speed ot th

yr;ht causes the apparent wind to draw ahead, especially it the lower wirìdspeeds.

The

of the yacht also affects the velocity of the

apparent

wind, which is reduced

rD

some three knots when running in a 7 knot true wind.

This results iii a holl

ir

th' donwi.rd

part of the polar dia.ram which becomes less marked as the wind velocity

boccncs laroer relative to the yacht's speed.

The form of the polar cuve

ts sonewhat dependent on the

fact that the

12-metre is a yacht cf rather narrow beam, rather large

displacement/length ratio

ur her size arid hig

ballast, ratio.

Speeds up to about 7- knots

(V,4rL

1.1)

.in thus be achie'ed relatively easily, but thereafter the resistance rises increasingly

ridly and vory lar;e toxces are required to drive the hull above

about lO knots

1.5).

Hulls of lower dtsplacernnt/length ratio, more easily driven at high

speds, would Lravel laster in broad reaching conditions and

would show a hollow In

the runninQ part of Ihe polar curve at higher windspeeds than the

12-metre.

The

Cataniaran referred

us earlier is an extreme exampie of this

behaviour.

For other displacement yachts the polar curVe5 would be of broadly similar

form

to those for .he 12-metre.

For smaller yachts having hull and

rig

characteristics silar to those of the 12.-metre, polar curves

giving the

approximate perto:ance might be obtained from those for the

12-metre by reducing

'J

and V. values iii proportion to the square root of the waterline

lengtho

WIND

FIG.1

POLAR CL!V FOR GIVEN

WIWO SPEED

WIND

FIG.2

sìj.

¡VII/LE

P

HT

SLACK

WATER

WIND DIRECTtOH AND

STRENGTH

4 WiND FROM Mi

TO P

C-, I I

-x

I

o

I I I I W IN

WIND____

DIRECTION

WIND FROM

M1 TO P

C.)

/

WP4D

DIRECTOH

lo

ç) e ' d

Vs (Kn)

400 50

/

120° 90° ¶10

Figl2.Poar diagram

_{for 12-mQtrQ.}

True

wind

d rec t iOn

Irue wind

Speed V T