The induced drag of a yachts hull

ADVISORY COMMITTEE FOR YACHT RESEARCH

Lab. v. Scheepsbouviktit

Technische Hogeschool

Delft

NMI

UNIVERSITY

OF

SOUTHAMPT_

ON

_

department of

aeronautics

and astronautics

S.U.Y.R. Report No. 24

THE INDUCED DRAG OF A YACHT'S HULL

by

A. Millward

1 6 NOV. 1976

ARCHIE

Department of Ac.ronautios and Astronautics

Advisory Committee for Yacht Research

S.U.Y.R. Report No. 24

THE INDUCED DRAG OF A YACHT'S HULL

by

A. Millward

March 1968

-PIDEX

Paw,

Index

Notation

Forces on a Vertical Hydrofoil

St., merged Hydrofoil

Surface piercing Hydrofoil

Forces on a Yacht Hull ...

Effect of Reynolds Number

Synbol

2

d2

At Aspect Ratio (6 /s or

/s)

CD Total hydrodynamic drag

coefficient (parallel to Vs). (D/1.v2s) 2$,

Di

CD, Hydrodynamic induced drag coefficient (

/102s)

CL Total hydrodynamic side force coefficient

(normal to Vs) (//1,.v2s)

zw s

C Local hydrodynamic side force coefficient (CL Ctds)

Hull hydrodynamic drag (parallel to Vs).

Vs/

Froude number ( igt ).

sAtCDT Induced drag factor ( 2 )

CL Reynolds number (Vs/iv).

Submerged profile area (mean chord x s or d)

Yacht's velocity along its track

Draught below static waterline

Depth of immersion of upper

tip

of hydrofoil

9. A characteristic length

of the body (usually the chord or water line

iength)

Vertical span of fully submerged hydrofoil.

Distance along vertical axis (positive downwards). NOTATION ii h

I

D F K. R

Symbol

A Kull displacement

Heel angle

A Yaw or leeway angle

Kinematic viscosity of water NOTATION (Continued)

1.

1N1ROWeCIION

The to:dtody,amic drag of a yacht's hull when sailing to windwarA, and cherefere making leeway, can be considered in four separate, though

noz necessarily independent, parts, namely

Skin friction Wave drag

Form (or pressure) drag Induced drag

The first three increments are familiar to yachtsmen, partieularly skin friction which can be noticeably altered by the smoothness of the

huli surface. Recent trends in yacht design have also arlected skin

friction by reducing the wetted surface area of the keel,

although

la :svme

cases this has resulted in a reduetion of directional stability which has

made the boat more

dif:icult

to steer.

On the other hand little is known about the induced drag of the hull. This is defined as the added resistanee caused by the orodeetien of side force aad it may not be generally appreciated that the induced drag van amount to a quarter of the total drag of a displacement yacht

at high speeds, as illustrated in Fig. 1. Thus it is important to km,,,4

the relationship between the side force and the induced drag so that their

effect en

the tntal hell drag is minimised. At preseA there is no hvhrr,-dynamic thcry which might he used to predict these farces hut

it has

ieP.eested that aerodynamic theories for wings could be used it crit

eurface was at,sumed to act as if it were a plain se)id

this afumptio-, Tanner (1959) found however that the foeces 77easured

In

towing tank tests on hulls did not agree with existine eeradynerle

ue?

At was fwied titat. these had bt,:u 4c:ivcd to:

slender

uh-ertas -,,,arhY, resembles a thick wtng-bk

_comhtnatioc

In

the first

patt et the

present work Lhereioie

t.itnriittr6 thnii'

med to tind the effect

of the free surface and henLe to

teSt

whethe, it dces MA. as a platy solid boundary. The 'hull'

Is

chosen

ti

be a laige aspect riii, rectangu'Lar foil

lot

_{which both experinenzai and}

_the:ret1J-11

ai.rodynimic data is readily available. _{It is in effect a carefully}

stream-lived dagger-plate, such as might be used on the centre-in'

or

a ,,atamaian,

or a, a 3,1.1.roia

einehy rudder

sti:h leading edge, le is appre.iatr-d

that the canoe body above the keel may have a significan, effect and

this

d:zcw:sed leter.

Although there is ac aerodynamic theory which wou,d be apolreaht, to a A,apt iwlin a icht hull it would still be usciW

ti. rteLinitl

Witt t the wat.:,' surfv.:c acvs if it

tif"ii

a sin,e it

this were it would Enable useful tests of a 'doehle medel' to oe made

in lhe wind tunnel. Several advantages would be obtained from this, e.p,

it

Ruining time in the wind runnel is unlimited whereas

ih

the rowing Lank

:it

is determined by th: length the tank.

b) _{The force measuring balance in the wind}

_Lc.,tsel

likely to be more accurate since

it

is sta:iarary.

z) Larger Reynolds numbers can be attained which more

nearly approachthe corre,st Reynolds number for the full-eize

d) It is often easier to use flow visualisation teLtiniciues with a continuous running time and the observcr

see the model.

2

-readilt

-..f.1 Submerged Hydrofoil

It is cite:: simpler to obtain the us

or A

wind tunnel than a towing rank which is cTlinped ".:() test yacht models properly.

IHL e0KGES ON A VERTICAL HIDROFOIL

discussion which follows is based on a theoretical and exnerimentai :ns.tsf.i.gbtien of the side ferec :41:: induced drag of a cmoletely

rtLadydrowil at an angle of yaw.

The work is givcn in detail

hv

MiIluard ;1967) with the more important results described below since theN

are relevant to the present

consideration of yacht hulls.

In order to obtain the theoretical results a number of assumptions were made,

the

most important of which were that the foil should be of

r,-ie th.it the rioude numlier should ho nr.,11 !7 <

Ihn theorctizai results shcv that the indnced drag et:eft:Tient cnn be ex-7rts,c,d in tera

ne

Aide fr.rce coefficient as

2

KC,

/ITS

e AI is tae aspect ratio of the foil and K is called tic ihdutte'' 'ran

Fer ly4c speed aeroplane wings K is normally a constant with

vL,*ce slightly greater than ur.i;.y. The theoretical varlanDn cif re

irr-ced

_drag

factor for be sithmergod hydrofoil with e.1,

numher is givvn in FiE. 2. It will be seen thot th.e

Lr:14-t!

dog fao-or d creases slowly as the hydrofoil approach che varer Jur:are

'"u thc _{ariation with Froude number is small and is neglibl:, at a der.th} n

)

2.

ratio related to the span of the

foil

greater than a third whereas the

value of L/ shown in Fig. 3, increases near the water surface but decreases as the Froude number becomes larger.

The experimental measurements showed that the variation of side force

coefficient and induced drag factor with Froude number wae greater

near-to the surface than the theoretical results indicated but was again negligible for a depth ratio greater than a third.

Although close to the surface the experimental variations of aide force coefficient and induced drag factor with Froude number are greater

than the theoretical values they are still comparatively small. It can therefore be concluded that, at least for small Froude numbers (F < 1) the variations of side force coefficient and induced drag factor with

Froude number are small and diminish rapidly with depth of'immersion of

the foil, becoming negligible at depths greater than a third

of

the span.

Frcm a comparison of the experimental with

the

theoretical results

it is clear that the measured value of side force coefficient is slightly lower than the theoretical figure but the measured induced drag factor

Is

murh higher. In particular as the depth increases towards infinity the

measured valve of K tends to 1.58 whilst the theoretical value tends to 1.025. Since the theory applies only to a non-viscous fluid it is

thought that this large discrepancy can be attributed to

the

noundary layer on the foil in a real fluid, particularly since at the low Reynads Nunher

of 105 used in this experiment the effects of viscosity

woo;d be exaggerated,

An estimate was made of the effect of the boundary layer thickness on the induced drag factor using the work of Gardner and

Weir

(196b) and this

provided a correction which agreed with the apparent discrepancy between

the theoretical and experimental values of the induced drag fa:i.z*r at a large depth of immersion. It is therefore concluded that the Reynolds

cal. nave A _{large effect uu the induced dr.}

fa,:tor.

..-24tLer

_{effect on the side force coefficient,}

This concluion

_ih

used to

_account

_{for a numh,r of}

_a003Vvne

discrenancte2 CA later ..,-tiors

sh:vld

he carefully assessed in tank

testing pro(edures.

Surface Piercing Full

The experimental inyetige _{luns were extende4}

t.. i

_{SIMI-a:AI, ;vserr;n!,}

ioii hot as yet it has not been found possible _{to extend tett,eorl} of the difficulties in _{defininz the boundary}

conditions

at

the ,4arer

geriacz where the foil passes through.

Measurr-ments of

the forces

_on

_{the high}

_aspeCt

ratio foil shot.,ed

imiiar trends to the fully _{submerged foil, i.e. the}

ide

a,.!crased and the induced drag factor increased _{As te rr:AlOp nurh-r} tear-e l:!rer, hut the variations

_were

rather bigger for the surface piercing foil.

An additional experiment _{was made to measure the}

Di'esure

and heai_e

_{the distribution of side}

force on a surface piercing _{foil, ..ut}

thtse measuremers were restricted to a single

_{Froude number h}

_the

aj!eenninment. The

_{distribution of side force}

along the f:oi

_Is

shown in Fig. 4. _{The curve for the}

surface niercine foi) has nct -,en extended very close to the water surface since the _{exact shai:t, oi}

curve could not

_{be determined, particularly as the}

wave Conlati:31 1,r41

appr;_iCiabTe. Alsc, in Fig. 4 the

distribution

of side force is

tao,,,,same-f74 in

Zwind tunnel

and with the effect of a r)iain hr,undz;r, renresented by _{an image model.}

The

results clearly

.at near ti) the free surface the _{local side}

force

is increased hur

_{is7 ec.}

rapid _{dimir.:Acs as the distanrp}

below

the surface t.Neot;nes

and

3. THE FORCES ON A YACHT HULL

The aerodynamic and hydrodynamic forces acting _{on a sailing yacht are}

illustrated in Fig. 5 from which it _{can be seen that the hull will move} at an angle

to

_{the direction of its centre-line in order}

to develnp the side force necessary to balance the lateral force of the sails.

Accompanying this side force there will be _{an increment of resistance,} usually called the induced drag, so that the total water resistance of a

yacht hull in steady motion in calm water is made

_up

_{of four parts, namely}

skin friction, form drag, wave drag, and the induced _{drag. It should he} noted that the wave drag is due to the bound waves, that is, the wave

pattern travelling with the hull. _{The extra drag caused by rough water} is a further prnblem which is not discussed in _{the present work, As a} first approximation the four increments of drag are 'sometimes

assumed to

he

independent of each other in order to gain an understanrOng of

their

relationship to hull shape, but it is found in the

present work that this

approximation may lead to significant errors. _{For example a}

_study

_of

the wave pattern araand a yacht hull shows that there _{are wave nattcrni} 1101;7e erag,

awn;_ate.i

It

_{Lnth forn at}

_"

that there will be an intereaction between the two wave formations_

Davidsnn (1936) developed a technique for

_{testing mo,2e14}

n'

vacnt

huas in a towing tank and suggested that the underwater

nentnn

yacht's hull right be

considered as half a body which reser:bled _an

plane.

This

suggestion implies that the water surface arts

as a refect:.c,r,

plane (or solid boundary), in which case the relationshin

_{t'et.een the aLje}

force and induced drag coefficients would be CDT _{KC-1.RA uhere 15}

6

-

an

dna:.

Lle

is the aspect ratio

of

the hull and its rmaye

erbr.

Jwin: 1avidson's paper von Kerman pointed out 111,;,.

'

c:--nctant Alessure houndafy ape the

42 he ?renter than if it were 'a refletien

r-red the results'for.a yneht hull, ,-hich tlas An asnerA

fkr--If atout i. the ime?e system is included, with the LanchesteT-It'

is oily

applicable,

to ings of high aspect ratio

Iritra..1

6ince the only theer) then existing. Tanner (105g) h

cnated

with a more recent theory, for a low asnect ratio slender

as tfis resembies the shape of a yacht hull more closet', than

Coes ,J

aspe,:.t ratio Wing.

P.as found however that the measered side fo-rce

:!o-effiflev. are induced drag factor were larger than the theorY supgesten.

1- Tier measured the forces on various centre-boards

rz4

Ccnoe.

These have

an aspect retie of s.73

t'

the

model is

,ssume

The forces

were compared with the Lancbester-rranet1

fl,

in

the --aaurcd -.aloes of side force and inducee dric.

ir

eement

with

the theoretical values.

n.Jmparisom howeter ,ere all hetwen an exnerirental result

for a 0r,li-atcd

teldi ardrh. ry devi:ed for

enba.;c gu.s,

g.

an unLn hi:h Asoer

rat'o or a stend:r aspeet ratio

'Tere ':acT.averty (1(4-) bas tested a daubl cdc nf ,

ya0.t

in c!,e wind runnel ane by co

arin hs results

rtrr

h1t ie !tie .(5ns, tank it is possible to determine .tfore A-rect

nre caused bv the use n

over si-rpliiied equiva:er.t wing theory on hether tbc efeet of the tree

surface i- significant in the case of r: yacht's hull.

-he 1,odea

p.te

nonion cf the hull at an angle of l

el

the

7-ay., lieu

-Janry

results

4 .aldition

a

represented, as shown in Fig. 6. The tests were designed to investigate

the merits of several fins by varying the leading edge sweep hack angle of the fin while the hull shape and fin area were kept constant. Tt

was assumed that the effect of varying Froude number

an

the fin would he small and the order of merit of fins with different sweepback angles would be the same for the double model in the wind tunnel and for towing tank model even though the magnitudes of the differences night not he the same

in each case. This assumption is sunported in the present wirk by the tests on the vertical hydrofoil which showed that the effect of varying froude nu-nber was small and diminished rapidly with distance below the water surface.

aA3itioa a series of tests was recently made in the towing tank

at Southampton University by Koekebakker using a normal node/ of the same 5.5-metre hull with the same series of fin shapes. Koekehakker's results showed the same order of merit as MacLaverty's tests, namely that as the

sweepback angle of the fin was increased the drag for a given side force was reduced, but the magnitude of the change was a little different.

The

variation of siae force coefficient with yaw angle for the hull with one of the fins is shown in Fig. 7 for both the wind tunnel and tank models. Although there is a small increase of side force coefficient with Froude number in the tank tests, the results from the wind tunnel are generally

in close agreement with them. Similarly the variation of drae coefficient

with

side farce coefficient is River for the two tent nethods in

rift.

g.

This shows that there is no significant variation of induced drlp eoefficient with Froude number since all the curves have nearly the same slope, and

that the tank and wind tunnel results are in general agreement. There is

number of the wind Lunnel tests wns 3 x

106

whereas in

-"Onnv

tests it

1..as fi A 105.

An estimate has been made of the eft

of

charn,,e

in Rtynnlds nicaber on the side force and induced drag cnnfficitn.;

and iT is found that this could account, for the d4fference between

rile 1.60d

results..

MacLaverty, in his orininal analysis of the wind tunnel result:,

corharnn nith a

:null:

test

ar

the 5.5-retre model, 7th one

fin, made

en:

British

_{Kovercraft Corporation. Yn this ca,:e it tnnInn}

that the

coefficient

_{at a given yaw angle was greater in}

_ele

wind znanni than in

the towing tank but

the induced drag

factnr was the

san.:. in ench _{The model used by the Rritish novercraft Cornntatinn}

was larger in

tat

proportion 6/5 so that the Reynolds ninl,nr is a

blizer.

If a correction is made for this then the dIfference in side force

coefficient between rank and wind tunnel becomes very srall but

_{the inJuned}

dran are now no longer the same. It is possible that- there

-.insliht

differences in model shape which might have affected the results

:n additinn i

shonld be noted that

the tank results 'ere nbtainnd

frrm

a conven*ional tank test. This is adequate for the cnmmercial rnonirements

of a customer but covers too restricted a range of condttons to detFrmine

accnrate]n-* the curve of side force coefficient ae,ainst ynt., angle ,n.-2

particularly the induced

drag

factor.

Thus the difference between the two

set of tnnk results is

not considerrd to

bP unreasonably tarse.

Further ,nformetion on the variation of side force -coefficinnt and.

indv_ed drag factor with Fronde nnmber is given bynTtarla

(1,V11 Hp

testnd fonr

different

models, all related to another 5.5-metre hut],

_anu

fnund that

th:,

_{side force coefficient remained constant with increasInn}

rrnude nunbnr, or increased sliehtly depending an tl-e nnttienlar nnnel

9.

tanic

by

slight

but

A

ind hr.:A angle, while the induced drag factor decreased a little. It is noticelble that .gekebakker's results, gi.vi ;ri Fig. 7 and 8, showed that

the

side force coefficient increased with a larger Froude number hut the induced drag factor remained collet-ant.

These differing results suggest that the variations of side force coefficient and induced drag factor with Froude number are small hut are

closely related

to

hull shape. It is concluded therefore that the douhle

(or siragef) yodel is not completely valid hut since the effect of Froude number is small it seems likely that it can-he used to investigate alterations

such as changes of fin shape provided they are well below the water surface. As .7entioned previously, Tanner has shown quite large differences when

comparing the results of tests on several centreboards of an Tnternational

10 sq.m. Canoe with theoretical results for a high aspect ratio wing. In the comparison it was assumed that the effect of the hull and the water surface would resemble a reflection plane so that the aspect ratio of the centre-board and its image would be 5.73. If this were so then from high aspect ratio wing theory (AR*4) the side force coefficient would he given by

AR 210, ( )

AZ+ 7

where AKis the aspect ratio and X is the angle of

yaw

in radians,

i.e. Cc 4.66 A in this case, and the induced drag factor would he 1.0

approximately. The experimental measurements however gave CI . 4.n5 and an induced drag factor of 2.45 respectively.

Most of the results quoted by Tanner were however for centre-hoards

which had sharp leading edges and it is possible that, at the law Reynolds numbers of the tests (R 5 x 105), there was leading edge separation of

the flow. One of the centre-boards did however have a rounded ading edge

10 4 4 -A

and if the results, shown in Figs. 9 and 10, are taken for this hoard alone then an experimental value of side force coefficient is 5.0 A and the

Induced drag factor is 1.0 approximately. These

results, which were ehtained

on a full-size hull being towed at realistic speeds, particularly erphasise

the advantages of a centre-hoard with a rounded leading edge. Tr A correction is made _{for the effect of Reynolds number on the houndary layer thickness}

then the value of side force coefficient corresponding to an infinite

Reynolds number, is slightly increased and the induced drag factor is decreased. When compared with high aspect ratio theory therefore, which is also for

an infinite Reynolds number, then the measured side force coefficient for this one centre board is apparently too high and the induced drag factor too low.

The initial assumption in comparing the experimental results on the

centre-board with high aspect ratio wing theory was that the hull and water surface acted as a reflection plane. A possible explanation for the

disagreemert between the experimental and theoretical results is that the

yawed hull has a noticeable effect on the flow and this is confirmed i)y De Sa:y. (1962) who measured the forces developed by a 5.5-retrP fin wben

attached to the hull and when 'Isolated' - that is, mounted under a large board. He found that there was a large cross flow under the hull which increased the measured value of side force coefficient by about 507 as

shown in Fig. 11.

The

_{measurements of drag coefficient are also shown in Fie. 11 ana it} can be see, that the values are a little higher than for the 'isolao.P fin. _{However because of the large increase in side force the induced drap}

factor is decreased by the presence of the hull. _{Thus the 5.5-metre}

measurements agree qualitatively with the International ranoe rezolts that

11

-4

-the

side force is greater than would be expected from simnle aerodynamic

theory and the induced drag factor is smaller. It seems that this is caused by a large crossf low under the yawed hull and suggests that

com-parisons with such aerodynamic theory are unlikely to be satisfactory unless

this cross flow is taken into consideration.

4. THE EFFECT OF REYNOLDS NUMBER

Several apparent differences have been found when comparing results from wind tunnel and tank tests or when comparing the theoretical and

experimental work. It is suggested that in a

number:

of cases these are due

to the effect of the boundary layer thickness which changes with Reynolds

num/3or and this affects both the side force and induced drag. The theoretical

estimates were based on the work of Gardner and Weir (1q66) which was for Reynolds numbers greater than 106. To use this in the present work it is

necessary to extend the range of Reynolds numbers as low as 105, and it also necessary to assume that the results can be applied to a lkyw aspetr ratio wing-body combination resembling a yacht hull. Although considerable extension of the available data will be necessary before corrections to results for a yacht hull can be made with certainty it is shown that Jerre

changes in Reynolds number can have a significant effect, especially on the induced drag factor and are likely to be particularly important when naking comparisons between theoretical work and experimental results obtained on

soall models.

In the case of towing tank tests however the effect of different Reynolds numbers on the side force and induced drag appears to he less imnortant since

the variation in Reynolds number between model and full-size is ,Isually oniv

about a factor of ten and in addition the induced drag of a conventional

12

-yacht is usually less than a quarter of the total drag. _{A recent correlation} of towing tank tests of both the full size 5.5 _{metre yacht "Antiope" and its} model confirms this conclusion since there is little discernable _difference

between the model and full size induced drag factors. _{This suggests that}

present methods of scaling up results of tests on model hulls _{are adequately}

reliable, at least for conventional yachts.

5. _CONCLUSIONS

The results suggest that the side force coefficient and induced _drag factor of a yacht vary slightly with Froude number _{so that the 'image' model}

idea is not entirely valid. _{The results also indicate that the variation}

of the forces with Froude number is related to hull shape.

Although the results show that the image model is not entirely valid the discrepancy is not as large as has been suggested in _{the peat. This is}

attributed to previous comparisons being made using _{theories appropriate}

to simple wing shapes whereas a yacht hull should be regarded _{as a wing-body} combination for which no theory is at _{present available.}

Tests of various fins on a model 5.5 metre hull in the towing tank

show the same order of merit for the fins as when tested on a double model in the wind tunnel. _{This result, together with both theoretical}

and experimen-tal data on submerged and surface piercing foils _{suggests that the effect of} the free surface becomes very small _{as the distance below the surface} in-creases so that it should be possible to investigate small alterations _of hull shape well below the water surface by tests on a double model in the wind-tunnel. Such tests can be run at higher Reynolds _{numbers than tank} tests, approaching those of the full-size yacht, _{and other advantages can}

be

galy!., 'tom Lhe

greater accuracy of force measurement together with ease of flat; visualisation.

The regults of tests on the hydrofoils and on yacht hulls show that the side force and induced drag coefficients are dependent on Reynolds and

this should be taken into account when

comparing

experimental and theoretical

results where the effect of changing Reynolds number can be expected to be large.

The effect of Reynolds number on the side force and induced drag

coefficients when scaling the results of towing tank tests from model to full size is likely to be less significant for a conventional yacht, This

is confirmed by the only available model - full size correlation for a yacht and suggests that existing towing tank procedures are likely to 'Ne arleguate.

REFERENCES

Barkla H.M. (1962) Tests of Four Related Yacht Forms

Stevens Institute of Technology, TM 132.

Davidson K.S.M. (1936)

Cardn - D. and Weir J. (1966)

De Sal,: P. (1962)

Tanner T. (1159)

15

Some Experimental Studies of the Sailing Yacht.

Trans. S.N.A.M.E., vol. 44

The Drag due to Lift of Plane T:q...frs at

Subsonic Speeds. J.R.Ae.S., May 1966.

MacLaverty K.J. (1966) Tests of a 5.5 metre Yacht Form s?ith Various Fin Sweepback Angles.

S.U.Y.R. 17

Millward A. (1967) The Induced Drag of A Vertical Hydrofoil, University of Southampton, Ph.D Thesis,

Fin-Hull Interaction of

a Saiiine Yacht

Model.

Stevens Institute of Technoogy, TM t20

A Preliminary Report on the

Cornbetween

Theory and Experiment in Rela6.)1

to the

Effects of Aspect Patin on the

Side Force and Induced Drag of Keel

S.H.Y.R. 1.

300

250

200

Hull

drag

D lb

150 100

50

Hull drag

for

zero

sideforce

10°

156 4 5 6

Vs

Boat speed - knots

Fig. 1

Heeled

resistance of a displacement yacht.

5

Induced

drag

56

106

Drag with zero

side force

-1 1 1 .0

_0.9

0.8

0.7

Increasing

Froude

no.

I I

0.01

0.1

1 .0 5 10

Fig. 2

The

variation

of

induced drag factor

with

depth

of immersion and

Froude

no. for a

submerged

foil

( A4 F<1)

F:0

4.5

4.4

_4.3

CL

I>'

x

rad.'

4.2

4.1

4.0

/(--//

Increasing

Froude no.

i 1 5 10

Fig. 3

The

variation

of

side force

coefficient

with depth

of

immersion and Froude

no.

for a submerged foil

( AR=4 :F<1)

-

-I 1 i

0.01

1.0

0.1

vs

6-0

4.0

CL/x

rad"'

2-0

Wind tunnel model

_{(with image)}

Static water surface

0.2

0.4

_z/d

0-6

AR=2

I"'

t

z

Fig. 4

The spanwise

variation

of

local

side

force

coefficient

for a surface

piercing

foil (F= 0-47)

LA

Fig.5

_{The aerodynamic and}

_hydrodynamic

forces

on a sailing yacht.

!IND

Incidence

control wire

Attachments

to wind

/c

tunnel balance

Turbulence trip wire

Line of fin attachment

to hull

Fig. 6. Double model

of

5. 5 metre

hull

in

the wind tunnel

.

-0-14 0-12 0-10

0.08

0-06

0.04

0-02

Legend :

o

0-23

x

-28

4

0.32

o

0 . 38

v Wind tunnel

7t.

Fig. 7. The variation

of side force coefficient with leeway

for a 5.5 metre yacht

( cp, = 100)

Legend:

(PI"'

i

0-020

6

./

I D)( . 0-01554

0

.0 :

Wind tunnel model,

4-1

0.01

C2

Fig.8

The variation Of

dragi coefficient with side force

coefficient for a

5.5 metre yacht

(

7.-

id),

10-02 o

0-23

x

0-28

A

0.32

io

0-38

0,030

0-025

Co

.

,11.q& I MO

453 lb

kri.

a = 411lb

.0

2

06

3 4

lA -*Displacement.

2

4 A Ii ..a

,

01

2

0.2

0.3

CL P.

Fig.9,

The variation of side force coefficient

with

leeway

10 sq. mu. canoe

Fig. 10. The variation of induced

drag coefficient

with side side force coefficient -

10 sq: m.

,canoe

111 5!4 4

0.4

0.2

6

-CL

0.08

0-06

0-04

002

Fin on hull

../

\

..---_--ç

0----...0--r----)2.--

'\

Isolated fin

Fin on hull

A°

Fig. 11

.

The variation of side

force and drag coefficients

with leeway for a 5.5

metre

fin

.

/<

/Isolated

I in

0-03

CD

0.02

0-01 0.14 0-12 0-10

3