Predicted and measured performance of a day sailing catamaran

ABSTRACT

A simplified theory is given for predicting the performance of light displacement craft sailing at small heel angles The theory predicts and experiment confirms that the ratio of boat speed to true wind speed is virtually independent of true wind speed for all headings

to tha true wind for this type of boat Only two additional dimension-less parameters and the aerodynamic drag angle are required to completely specify the performance within the approximations set forth. Performance tests of a fourteen feet catamaran substantiate the theory to within the experimental uncertainty of the investigationS A simple, direct method used for experimental performance evaluation is described

PPEDI CTED AND MEASURED PERFORMANCE OF A

DAY SAl LING CATAMARAN

by

WS0 Bradfield*

*Professor of Engineering, State University of New York at Stony Brook

NOTATION

aspect ratio ((height)2/a.rea)

a/2

thrust index, defined under equation (18)

hydrodynamic drag product, defined under equation (18)

e

efficiency factor

H

hydrodynamic sideforce

k

hydrodynamic drag ratio

_rR

_{+ R}

+ L

measured course length

LWL

load waterline

R

bydrodynamic resistance

S

area

s

mean surface roughness

T

total force (aerodynamic

or hydrodynaniic)

(t2-t1)

measured time in time trap

W

weight

displacement., weight

incremental valu

of measured quantity

-1

DA

_-1

_B

drag angie

= tan

= tan

-angte between Lru w nd Ve'tor and sh p VP3J)Cj ty vec or

defined under equation (18)

SUBS OP FPTB

A

ar'odduamic

o crew f friction i induced o zero yaw r residuary S "ship" T true

W water (or wetted)

w wave

w wetted

INTODU OTION

Full scale performance is the obvious criterion of worth of

any sailing yacht

Performance testing is expected to be a part of the

;rcess of des±n of every boat that comes off the drawing board.

Never-theless, sailing yacht performance data are extremely difficult to come by

The present ine3tigation was partially dedicated to the production and

publication of a modest store of such information

Being able to predict performances on the other hand, is as

important as being able to sail it since each effort contributes to the

other and to the better understanding of sailing mechanics

Obviously the

mechanics of sailboat performance involves aerodrnmic as well as

hydro-dynamic considerations

Both sails and hull serve as "lifting" surfaces

in the aerodynamic sense

The sail utilises the reaction with the apparent

wind to provide thrust and the hull operates at an angle of yaw to develop

the side force jhich counteracts leeway

The similarity to the operation

of aeronautical lifting surfaces is obvious

As a result, one might expect

that aerodynamic data on wings accumulated from decades of experience in

aeronautical engineering would combine with hydrodynathc data on hulls to

permit the prediction of the performance of a specified sailing yacht.

So far, attempts to apply such information have not met with

notable success

One reason is that the aerodynamics of the traditional

saiLing yacht is

uit

complicated when compared to that of the average

glider or subsonic airplane

The complication is j.art1y due to the

aero-dynamic interaction among such elements as multiple softsails, exposed

stand-ing and rurnistand-ing riggstand-ing, hulls anc crew, and surface waves.

The underwater

for operation at or near the air-water interface In addition, the motion of a yact is es en.ially ui.. tendy motion, The disturbance velocities due tc gustiness and wave motion are frequently not small relative to a

characteristio velocity of the yacht Thus, even though steady state analyses are useful for airplane performance predictions, there is Some question, in the practical sense, concerning the applicability

of such

analyses to yacht performance Another difficulty in the

analysis is due

to the couuling which occurs among aerodynamic and hydrodynani c variables. For example, a sudden gust may increase the sail force

which increases heel

and lewa, d.écreases thrust, and increases the hydrodynamic d.rag,

and so

on Finally, the s:arc*ty of published results of performance

investi-gations and of aerodyna.mc investigations of sails and hulls makes it difficult to evaluate simplifying assumptions which would make the problem

tractable

Tanner1

(1962) identified the variables in the theoretical problem, discussed the coupling among variables, and described

graphical

solutions of general utility for somewhat restricted operating conditions and for piasibi.e physIcal restrictions, Herreshoff

(i961

discussed

the application of the digital computer to the solution of

the steady

state equations of motion. Taken together with intensive model testing

and performance testing of a given yacht,

this procedure would be expected

to yield a maximum of precise information useful for optimizing a specific design as well as for predicting its performance

However, these facilities

and techniques are ot generally avai1able

Myers

(l96)

pub1ised a steady state theory for catamaran

per-1 References are listed

alphabetically by authors name at the end of the paper

:rmance

. snecial effort was made to account for all major factors

influencing sailing performance. A comparison of the predicted perfor-mance of individual boats with measured results indicated "that they do not arpar to disagree with the theory within the limits of the measure-ment accuracies". However, as Myers pointed out, the measurements tended

obe. "an order-of-magnitude less accurate than desired for a careful cmarison of theory and experiment". This state of affairs appears to be typical of performance measurements up to the present time (Tanner

(1968)). Thus, although the groundwork for systematic performance

analysis has been laid for some time (see, for example, Davidson (1956)), a comparison between predicted and measured performance together with an evaluation of the methods does not appear to be generally avai1ab1e

In this paper an attempt is made to generalize the performance prediction and measurement of a limited class of sailing vehicles. This

;as undertaken in two steps: (1) the formulation of a dimensionless

theory of performance with the aid of a series of simplifying assumptions; and, (2) selection and testing of a vehicle which fits into the framework of physical restrictions imposed by the theory. The predicted and

measured performance are then compared and evaluated relative to each other in the light of known theoretical and experimental uicrtainties.

THEORY

The type of craft to be considered here is the light displace-ment daysailing or racing sailboat Generally speaking, the analysis s limited to cases where the neglect of unsteady forces, heel angle, vertical components of resultant forces, trim angle, and changes in attitude due to

changes in pitching and heeling moments is justified0 These approximations seem especially appropriate for catamarans and for many trapeze equipped daysailing monohulls0 For normal operation, heel angles for iuch craft

are restricted to ten degrees or lesa The cat"aran attitude is compara-tively sensitive to changes in pitching moment because of the fine bows0

However, for normal operation the pitch angle and heel angle are of the same order of magnitude0 Furthermore, for daysailing and raci:1F catamarans the crew weignt is usually of the same order as the weight of the

boat0

Thus, changes in heel and pitch can be compensated by shifts in crew position. On these grounds, the pitch angle is assumed negligible and the neglect of changes in attitude with changes in magnitude of pitching and heeling moments is likewise justified0

Suarizing, th

physical pic-ture can be simplified at the outset by the following restrictions

(i) the motion is regarded as steady and rectilinear and the water surface as "flat"1;

the rudder normal force is assumed zero;

heel and pitch angles are assumed negligible; and,

(14) sail planform, camber and twist are assumed independent

of wind force0

1. "Flat" is defined by the condition that a characteristic wave amplitude is small compared to the waterline length of the

hull0

:liit in the third assiption

is the neglect of vertical components of

aero- and hydrodynaiaic resultant forces. The last assumption is based partly on the common use of fully battened or other "hard" mainsails by catamarans. Variations in the shape of foresails are simply ignored in the absence of data.

The foregoing restrictions reduce the problem to a simple matter of longitudinal and lateral equilibrium; i.e., thrust equals drag and aerodynamic sideforce equals hydrodynainic sideforce. In terms of the appropriate vector quantities as the he1man sees them (Figure i), these

component equations are:

TA sin - CA) = R (i)

and

TA cos _- CA) = H

(2)

The symbols are defined on Figure 1 and in the table of nomenclature. The hydrodynainic drag is expressed as the sum of its components by

R=

R +RJ +

R.

-r

f 1

when the sum of the residuary and friction drags is regarded as uncoupled from the drag due to yaw. With this assumption, the two quantities can be independently evaluated.

Firstly the sum of the residuary and friction drags is assumed directly proportional to the friction drag; i.e., for zero yaw or leeway (R. = 0),

= = kR.

The implication is that the zero yaw hull drag is parabolic with velocity

at constant Cf which is, of oourse. not true for all speeds even for light displacement hulls. However, a reasonably good fit to existing data can be obtained by using two or more parabolas to cover he opera-tional range of a given craft. For operational catamarans, the speed-length ratio range may well vary from zero to five. Therefore, two ranges were selected for fitting parabolas:

and

V

2<

<5

(6)

/LWL

The overlap is intended to emphasize the tentative nature of the results in the range 2

vs

< 3,

Ych (1965) published data on high speed displacement forms which was useful for establishing empirical relations for k over the selected

speed ranges. It was found that average values of residuary drag co-efficient for hulls of typical catamaran geometry could be approximated by a sole linear dependence on displacement coefficient over the speed-length ratio ranges specified. Hull interference effects were assumed. negligible.

Specifcally, two ewion1oi average values

0.05

vs

_Ci)

r (0.01 LWL)3 x

L2<

c

0.032L

vs

(ooi

LwL)3 x 10 '

/ThJ

< 5 (8)

6.

were obtained, where LWL is the design waterline length. These equations permtthe evaluation of the zero yaw drag parameter

V

L-

--

<3J

(5)

Thus, Cf Rf 2 pwvs 2

w

In order to express performance in terms of boat speed and wind speed, the aerodynamic coefficient

TA

CT _{= 2}

A

_PAVA 2 SA

and the hydrodynaxnic induced drag coefficient R. 3. CR ₂ 1

_pV

S. 2 1

are defined. Here, S is the projected effective sideforce producing

area of the hull and boards

Combining equations (1), (2), (3), (1), (9), (10), (ii) and (12), a first expression for the apparent boat speed ratio

A CTA SA

CR

1

-

-

_.-

[sin( _{- CA) - -a- cos(A -} CA)J_j (13)

is obtained. This can be made re useful by expressing Cf and in terms of geometric parazneters

For smooth surfaces and small values of (Re) the dependence of

friction drag on velocity is well known.. However, it is equally well C r R SW (10) (12) k 1 +

_rf

(9)

in terms of the loading and the speed range. In equation (9), both coefficients C and are defined on the friction (or wetted")

f r

area S

established that for rough surfaces

and for larger values of

the

velocity

dependence

decreases

In fact, in certain regions of the

fri-LWL

tion versus Reynolds number

( )

curve the function may be

regarded

as piecewise

constant

to within the approximations already made (see, for example, Schiicting (1968),

p6ii)

Under these circurstances, the

friction coefficient has been shown to

depend only on

parameters

of surface

roughness. For most of the speed range

involved in the present

investi-gation this latter situation

exists0

Therefore, the hull friction

coefficient was assumed to be represented by

Cf =

LL89 + l62 log10

(LWL/20S

where

"S

is the

mean

surface roughness0 Typical values of s

are

given in Marchaj (196)4), p02)4l0

Note that an average value of

friction coefficient is required

in order to include

board and rudder values0

For the present investigation

Cf-

i-

[s

_Cf+SbCft+S

Cf _{_I}

null

rud

rud

was used.

An equation relating friction area

and

displacement

can

be established from the lines of a given

boat,

For the catamaran

of the present investigation, numerical integration of

the lines yielded the

linear empirical equation

S

=O052 W

+0cj)4c

W

+ s

+ 10

w S c b rud

over the range

of displacement from 400 to 900

pounds which encompassed

the

measured

operating range0

Letcher (1965)

assumed

aerodynamic lifting line theory

to

be

applicable to hull geometries,

This

hypothesis

was

accepted in the

('5)

present case to the extent of ass11mng that CR 1

-ç

C,

CR f C C

--

Lb

as is commonly assumed

in the aircraft industry.

The empirical factor

eb

provides a

means

for evaluating various board shapes

and boa.rd.-hu.l1 junctions

relative to the elliptic shape (see, for example, Wood (1955) Chapter . Combining equation (2) in coefficient

form with equation (16)

I

e

ids

A SA

CT _A

-C p S. e

TfiR

F.

Wi

b

irhich makes

clear

the variation of hydrodynaan.ic

induced drag ratio with

heading to the apparent wind.

Miliward

(1967) has shown by theory and

experiment that there also exists a dependence of this ratio on Froude number. However, this is assumed negligible within the approximations of the present treatment

The final expression for apparent boat speed ratio is obtained by combining equations (13)

and (17).

The result is

rv

-CT

[sin(A -

)

/sin2(A

1b

2 it A (18)

-where

_-

C

-AA

TA C A' -specifically, VA V S 2

(16)

(17)

retresents the ratio of available aerodynamic force to the zero yaw hull drag, (ror brevity tiis will be referred to as the "thrust index" in what follows). The quantity

kCS

lb 4

fw

le ASS.

Lb

1

represents the product of the zero yaw hull drag and. the hull induced drag (this will, hereafter, be termed "hydrodynamic drag product"). The quantity

(BA - CA(BA)) =

is the difference between the heading to the apparent wind and the aero-dynamic drag angle; note that (90 - -r) is the angle between C and

A (Figure i).

In equation (18) a choice of the negative root implies that decreasing the drag results in a decrease of apparent boat speed ratio which is physically unrealistic. Therefore, the positive root was selected.

A limit on the solution is imposed by the fact that the quantity under the radical is negative for a certain range of _{(BA -} regardless

of the maiitude of Physically speaking, the appearance of the imaginary root implies that the fundamental hypothesis is no longer appli-cable, Le0, static equilibrium is no longer possible. To the heIrn.mari,

this situation corresponds to pinching to the point where the yacht speed is no longer great enough to develop the sideforce required to balance the leeway component of the aerodynamic force and the yacht "falls off", i.e,, acceleration occurs0 This condition can be expressed in terms of the ratio of aerodynamic thrust to hydrodynamic sideforce; viz, when

'A sin (B - 1kc S

fw

cot(BA - cA)

H

71%S.

static equilibrium in the balanced condition (which was postulated) can no longer be maintained. This expression also shows how minimal aerodynamic

drag interacts with minimal zero yaw and induced hydrodynamic drags to produce high pointing.

For regions outside the pinching limit, the three perforniance parameters listed below equation

(18)

characterise the performance potential of any specific yacht. In the thrust index of equation (18), the density ratio and sail area are fixed factors for a specific

con-figuration. The friction coefficient depends mainly on surface smoothness The wetted area and the drag factor "k" depend strongly on the sailing weight.

The hydrodynamic drag product similarly depends on weight. The other factors entering this parameter reflect the magnitude of the hydro-dynamic drag due to the aerohydro-dynamic sideforce. The board efficiency factor mainly expresses the hydrodynathc cleanliness of the board-hull juncture and the effect of planform. Its maiitude ranges between about one half arid nine tenths. The smoother and tighter the seal, the higher the value. The aspect ratio should be as high as practicable to minimize the hydrodynamic drag product. The ratio of total wetted area to

side-force producing area should be minimized. This ratio is bounded since S. includes a proportion of hull projected area which provides

s.J.

The

maximum board area for a well designed boat has presumably been established to give minimum induced hydrodynamic drag to sideforce ratio sailing

closehauled. This establishes a maximum for S..₁

-

) 900 this factor in equation

(18)

approaches zero and the

:oars ar

o'ly up

Calculating the optimum board setting (hence,

s.; for each heng in between

is an obvious extension of the present analysis since equations (.17) and (18) can be combined to do so0 However, it is beyond the intended scope of the present paper0

All of the factoro just discussed can be rcgardcod as parameters (except possi'oly for boa:d ara which should be continuously varied with

ein.).

Tab±e I contain the value e in pro di ctirig the performance

of the catar1euof th

present inlrestigation.

Thre rcmans the evaluation of the aerodynamic functions

A

it is possible to design and operate a rig so

that

0T

and _CA are independent of heading to the apparent wind A

are cntaat. in valu.

The c

ile'rerod warig is one exie

(3radfieid

(1968)).

The faster catamaran unarigs should also be approxi-mately in this category when tacked downwind0 Under these circumstances

and s should be relatively simple to evaluate0 The rig of the

present investigation, however, was not aerodynamically simple and the boat was relatively hea and slow so that tacking downwind was of mar-ginal benefit, Thus, the evaluation of the aerodynamic functions was both difficult and uncertain,

!o data on the sloop rig of the boat of the present investigation were available at the time of writing. The Draor. rig data of Mercaj

and Tanner (196I) were available and include an evaluation cf th effects of hull, stamling

rlgglflg,

and main-foresail interaction for different slot confiurations These results were adapted. The empirically deduced aerodynamic functions are shown as Figure 2. A rig operation schematic

In constructing

CT (BA)

EA(BA)

the five regimes shown

A

on Figure 3 were considered separately.

The rig is closehauled from the

pinching limit

(BA =

22°) to

BA 1400

(Note that corresponding values

of the heading to the true wind

(y

= 30

and

y =

65°)

are given for

the present boat across the top of Figure

2.)

The pinching limit is

defined as that heading to the apparent wind

beyond which minimum the boat

will no longer maintain a steady, rectilinear course.

The upper limit

of the close hauled regime is taken as the

heading off the apparent wind

at which it becomes necessary to increase

the boom angle arid to start

opening the slot in order to maintain optimum flow

conditions.,

This also

marks the beginning of the close

reaching regime which extends (by

definition) to the apparent beam reach

(BA = 90 + i0)

Broach reaching

is defined as that range of apparent

headings between the beam reach and

"wing-arid-wing" operation (BA = 150, in this case),

For 150

B

180

the boat was considered to be

running.

The initial values of

CT

arid

CA were

calculated from per-.

A

formance data at the pinching

limit.

Beyond this point, the

CT and.

A

£A

functions were constructed entirely from

wind tunnel data.

Between

22 < 8A < 140, the incidence

increases at constant slot configuration to

CT .

However, the drag angle

CA

also increases (Marchaj and Tanner

A max

(19614) Fig. 25).

Beyond

BA =

140, the slot begins to open to avoid stall

but with the prsent rig it was

possible to maintain both main and jib

shape for boom angles up to

30°.

Thus, for 140. < BA

70,

the

_ICT

A

decreases because of the decreasing

jib-main interaction.

However, as the

c1oe r

ch.ag region corresponds to the peak performance of the boat,

tb car:e:ponding values of the true heading are 65 < < iiO eyord = TO, the boom angle increases beyond the limit of the rasheet track and the boom starts to rise introducing twist, At the same time, the jIb has reached the limit of its sheeting control and begiis to backind the main0 Once the boom limiting angle (established

br

the shrouds) is reached, stalled operation begins and continues to develop for TO

<

180,

There is no data available for this type of operation.

However, for _A = 180, the drag angle is = 90 by definition. Also, for wing and wing operation the drag angle will vary linearly with

Thus a seent of

EA(A) for

150

<

<

180

was established. Para-chute and flat plate data (for example, Hoerner

(1958))

and data of

metallic sail models (for example, Marchaj

(l961)

have shown that CT = A

1.2 is a reasonable average value for surfaces at right angles to the air flov. This value was assumed in the present case. In the absence of better

information, a linear interpolation between the established points was assumed in order to fill in the beam reaching and broad reaching regimes The interpolated segments are shown dashed on Figure 2

Using values at' _{CT (eA)} and cA(8A) from Figure 2 and tie

A

paramete.s a/2 and 4b/Tr from Table I, the apparent boat speed ratio

ias calculated from equation

(18)

for all headings to the apparent wind, These re.ults ere o::pressed in terms of earth fixed axes using the trans-formation relation

lL,

= tan'

- (vs/VA)

etveen the appent- wiiid reading id the true wind heading, and

1 VT

- /ivAIvs)2 + - ii

-exrresses the true boat speed ratio in terms of the apparent boat speed ratio and the true wind heading. The transformation relations are obtained trigonometrically from the diagrams of Figure 1.

The resulting predicted performance for the fourteen foot

cata-maran of the present investigation is shown as Figure 14 in polar co-ordinates. The maximum true speed ratio lies between the headings = 90° and y = 100° and it is predicted that V

_{= O73VT}

The heading for best speed made good to windward is y = The magnitude cf best VMG is 0.36 VT as

can be seen from the tangent to the polar at y = 143, A tangent to the polar at y =

i6o°

shows negligible advantage to be expected from tacking downwind for this boat Within the approximations of this theory, the true speed ratio on any selected heading is predicted to be independent of the true wind speed itself except for the parametric dependence of "k" on

speed-length ratio (equations

_(T)

and (8)).

EXFEPOfl'ENT

It is apparent from a comparison of equations (18) and (20) that there are two equally valid approaches to sailing vehicle perfor-mance measurement according to theory0 One approach is to measure

VA, and from the test boat0 VT and y are then calculated with

help from equation (19) thus producing data for comparison with the pre-dicted performance of Figure . The other method is to measure

VT and y over a measured course directly from a moored instrument boat. The method proposed for the experiment was the latter method;

namely, to make times. runs over a course of known length at different headings relative to the true wind0 The results of these runs were reduced in terms of the dimensionless ratio of boat speed to true wind

speed. They were then compared with the predicted performance.

The method was chosen becaue of the inherently smaller experimental uncertainty which it offers Determining performance from measurements aboard the test vehicle has several fundamental dis-advantages. This is based partly on the difficulty in measuring VA,

and V with sensors which are located within the influence of the flow field of the test vehic1e Tidal current velocity measurements are also especially difficult to obtain from the test vehicle, The unsteadiness of the vehicle motion coupled with natural unsteadiness of the air flow introduces further complication, and the necessity of

calculating true boat speed

and

true wind speed

and

direction from measurements aboard tends to increase the experimental uncertainty.

inally, it was felt that the fewer the distractions for helmsman and crew the greater the likelihood that data could be repeated. For these reasons and because of practical problems of stowage space and added weight aboard the test vehicle chosen, instrument boats were made available and a suit-able area for fixed course testing was chosen.

The limitations of the theory require that the sea should be "flat" and heel and. trim angles negligible. Accordingly, an upper limit of 15 knots true wind speed was set0 The length of run was deter-mined by the essentially unsteady nature of the wind vector, The theory

assumes the existence of meaningful quasi-steady values of wind speed and direction. In order to obtain such values in practice, a time

average was indicated0 In the absence of other information, an averaging period was sought long enough to encompass several fluctuations of the wind vector but short enough to avoid significant "permanent" shifts of speed and/or direction. A time trap length of 200 yards was finally selected giving average run times of the order of one minute.

The course was set in Port Jefferson Harbor, New York0 Shelter from Long Island Sound is provided by a low sand bar which completely encloses the harbor on the north except for a narrow entrance channel This bar ensured flat water over a sufficiently wide range of wind speeds without disrupting the natural air flow. A sketch of a typical course setting is shown as Figure 5

The instrument boat was a 26 foot work boat with an instrument mast on the foredeck. In testing, this boat was moored fore and aft at midcourse and to leeward A l foot outboard was used for setting marks and other miscellaneous tasks and errands. The test vehicle was a

standard DC-l)4P catamaran A sketch of the configuration is shown as Figure

6.

Specifications are given in Table I. Figure 7 is a photo-graph shoving the end of a timed run in a light breeze. The mark shown is a time trap buoy. The sand bar previously mentioned can be seen in the background.

The run shown, although typical of all runs, is of a single handed unarig configuration. The standard racing rig of main and. jib with helmsman and crew (Table I and Figure

6)

was tested first. The research crew consisted of two men on the instrument boat in addition to the two men on the test vehicle. While the instrument boat crew were setting the time trap marks, the test vehicle crew set inrun and outrun markers (Figure 5). Normally, it was found unnecessary to set the

course at re than three different headings to the true wind. Typical headings selected were y = 90, y = 65 and y = 1Q. The usual shifts

in wind direction were sufficient to provide coverage of the complete range of headings between those set.

Each course was run in both directions. Thus, the single course setting shown in Figure 5 would provide points close hauled and

broad reaching. On the inrun, all four marks were lined up and marks were kept aligned thereafter. MarkB were taken as close aboard as possible. Final ad.justments to running rigging were made between marks

(2) end (3) (or (5) and (14)) and steady state was established. As the time trap marks were passed, the crew signalled the instrument boat.

Runs could be aborted from either vehicle. Typical reasons were sudden and extreme fluctuations of the wind vector, failure to establish equili-brium before entering the time trap, and extreme accelerations while in

the time trap. Vehicle control settings (except rudder) were locked during the timed run. Information from the test vehicle was relayed

to the instrument boat by walkie talkie radio. All data were recorded aboard the instrument boat

Performance prediction according to the equations (18),

(19)

and (20) involved approximately twenty measureable quantities. For-tunately, most of these data take the form of geometric configuration parameters (see Table I) which are measured prior to performance testing Additionally, a group of control parameters, although not explicitly involved in the performance equations, require monitoring during testing in order to ensure repeatability of the data. These included (in the present case) hydrodynamic yaw (leeway angle), rudder angle, heel angle, trim angle, boom angle, slot configuration, and sail camber and twist. The three test variables are boat speed, wind speed and wind direction. Sea state and tidal current appear in the investigation as environmental parameters. The test variables were measured from the moored instrument boat. The control parameters were monitored aboard the test vehicle,

the information being relayed to the instrument boat after each run. Wind speed and direction were measured on the instrument boat at rig center of pressure height.

The instrumentation involved in this first investigation was relatively unsophisticated. Included were a marine compass, a water speed meter, a cup anemometer, a masthead wind direction indicator, a thermometer, and an aneroid barometer, A list of instrumentation is given in Table II together with pertinent specifications0 With this information

=

The biat speed was determined from the equation

-minus (tidal current component)

The predicted experimental uncertainty (see, for example, Schenck

(196)4),

p52) in boat speed measurement by this method is obtained from

AL

Lt

A(tc:c-)

- t1)V (t2 - t1)Zvs

-where quantities are determined from steady state calibration

results

applied at midrange values of the experiment The resulting predicted probable errorV

(0:637

x std deviation) in boat speed was

-4- )4 percent

Of this, one percent was attributable to the

uncertainty in course length

measurement (line length and sag plus buoy scope)

Three percent was

attributable to uncertainty in tidal current component measurement

The

contribution to uncertainty due to time interval measurement was

negligible

The cup anemometer used for wind velocity measurement was

calibrated by

the manufacturer The least count of the instrument was

accepted as its

precision-. Eased on an average wind speed of ten miles per

hour, the

predicted uncertainty was + 3 percent As a result of these

too individual

uncertainties, an experimental probable error of +

)4 percent in the ratio

(vs/VT)

was predicted. The heading of the test vehicle to the

true wind

is the algebraic sum of the ccurse heading, the instrument

vehicle heading

and the wind vane deflection Hence, the error in

heading is the sum of

individual errors. Accordingly, the experimental uncertainty was

predicted

to be

+ 5

degrees

20

(21)

The measured. performance results are shown on Figure 8 This plot shows the ratio of measured boat speed to true wind speed versus the angle between the true wind vector and the boat velocity vector

(Figure i) The solid line is the mean line fitted by computer to the data points by the method of least squares. The dashed lines represent

the band of neasured probable error

(0.637

x (standard deviation)). The average measured probable error is accordingly + 7 percent. a result of the statistical analysis of the data, two

points were reved by

application of Chauvenet's Criterion (see, for example, Schenck (19614) p.136). These points were:

VS/VT =0.14147 at y = 29°; and, VS/VT -0.500 at y = 115°,

The results presented represent twenty four separate runs over a period of sixteen days. The different symbols represent different runs

and ranges of wind speed0 This information is shown in Table 3. When it is considered that this data represents results involving numerous launch-ings and delaunchlaunch-ings, riggings and deriggings, and course settings as well as a reasonable spectrum of weather conditions, the repeatability is

considered good. It tends to confirm the use of the ratio (VS/VT) as a "universal" performance parameter, at least for the vehicles under dis-cussion; specificafly, for light displacement boats operated under

the

conditions of the theory previously enumerated0

It will be noted that the measured probable error is almost twice that predicted. This discrepancy is attributable in part to

imprecision in repeating control parameters on the test vehicle from run to run and from week to week. This is especially true of sail camber,

twat, leading edge configuration, and slot corifiguration However, the largest s:ngie uncertainty factor is undoubtedly the essentially unsteady character of the wind This uncertainty is felt at the instru-ment boat in the process of assigning an average value to the wind

velocity and direction0 It is also felt on the test vehicle where the effect of unsteady changes of sail incidence on drive and sideforce are not yet fully understood,

In the present case, the latter effect was obviously ignored in the experisient since the ri controls were locked before each ruxi This process has the advantage of simplicity and also provides a known geometry from which a mean value of aerodynamic incidence can be calculated when average values of true wind speed and direction are obtained These single run averages were taken arithmetically in this first approach experiment "Instantaneous" values and resulting mean values from selected runs of the present investigation are shown on Table 4 Included are extreme cases and a typical case Note that run times vary with boat speed which influenced the number of samples obtained per run. In a typical run of less than a minute, the velocity varied 20 percent and the wind direction varied 13 degrees. In the unsteady run chosen, the wind speed varied 0 percent and the wind, direction changed by 30 degrees during 90 seconds

It is difficult to believe that such large and rapid changes are without effect on the boat performance0 However, the scatter of the present experi-mcntal data is sufficient to prevent an answer to the question as to whether a quasi-steady state actually does exist and, if so, in what terms to

define it0 It will require a more precisely controlled experiment to shed

additional lIght in

that area.

The present experimental results are

considered good. in view

of the state of the art of sailing

yacht

performance investigations. At

the same time,

it is

obvious from the foregoing discussion

that

several

things can be done to reduce the experimental uncertainty

of such

investi-gatio- -. The course length should be shortened

to 100

yards to reduce

still further the probability of large

changes in wind speed and

direc-tion. The precision of the tidal

current measurement should and can

easily be improved by a factor of 10 which will

bring

the

uncertainty in

average boat speed down to the order of one

percent The limit of

pre-cision of the wind speed and direction sensors should be reduced by a factor of ten and

they

should be continuously recorded

throughout the

runs. Operational amplifiers

should be used to average the fluctuating

quantities. Hydrodynrric yaw and

rudder angle should be tape recorded

throughout every run. All of this can be done with

available

instrumen-tation. With these improvements and using a

rig with precise controls,

COMPARISON OF TIOEY AND EXPEFIMENT

The predicted and measured performance are compared in polar form on Figure 9 The shaded band represents the data of Figure 8 together with the experimental uncertainty. The solid line is the pre-dicted performance of Figure

4.

Substantial agreement is shown0

Correctly predicted are the best speed ratio made good to windward, the heading for best speed made good; maximum speed ratio; heading for maximum speed ratio; and, speed ratio running The predicted and measured perfcrmance disagree only in the broach reaching region between

125 degrees and 160 degrees to the true wind This is the range of heading corresponding to stalled performance of the rig (Figures 2 and

3) where aerodnrrc data are not, at present, available0 Even in this region the maximum excursion of the theory from the data is only four percent Therefore, the measurements are considered to verify the theory within the experimental uncertainty of the data,

According to the theory, performance in the range 60 < < 180 is predominantly sensitive to changes in the thrust index (defined under equation (l8)) It is less sensitive to changes in aerodynamic drag angle and insensitive to changes in the hydrodynathc drag product Over this range of headings, the apparent performance can be approximated by

= L

CT(2 sin(A -

LA)):

(23)

At the pinching limit, on the other hand,

BA]pinch = (tan

21)

_{+ CA}

and the pinching (or pointing) limit is sensitive to both the drag pro-. duct and the aerodynamic drag angle. it is entirely indifferent to magnitudes of the thrust index and the aerodynamic resultant force co-efficient, assuming only that they exist.

Thus, in planning performance experiments, the theory indicates certain sailing regimes as favorable for performing specific investi-gations. For example, investigations of rig changes and. rig control parameter changes should be carried out reaching because performance is highly sensitive to thrust changes in this regime and insensitive to the changes in sideforce which accompany thrust changes. An assessment of

the effect of rig changes on aerodynamic d.ra.g angle can be obtained at the pinching limit (equation 23). Geometric configuration parameter

changes can be evaluated accordingly.

In the closehauled regime, between the pinching limit and the close reach, all performance parameters are important. Performance is equally sensitive to changes in _{CT CA} and "a" and is importantly

A

(though somewhat less) sensitive to changes in "b". Optimisation of all these factors is required to bring out the best "speed made good" in a given boat; i.e., _CT and "a" have to be maximized and _CA and

A

"b" have to be minimized by design and tuning. CT and

Am

usually do not occur at the same sail angle of incidence. The wing sail is the current rig which comes closest to accomplishing this ideal

value is obtained0 However, increasing the thrust index by decreasing the zero yaw hull drag is doubly beneficial since the drag product is

thereby autoticafly reduced

The existence of this double benefit places strong emphaai.s on light weight; especially, as it enters the geometric parameters very nearly as W This is because changing weight affects both wetted

area arid residuary drag (equations (7),

(8)

and (15)). For example, the theory predicts that if sail area is decreased by 30 percent, the effect on performance can be compensated by a weight reduction of only 20

per-cent0 Such an experiment was run as part of the present investigation

and the results are shown as Figure l0 As in Figure

8,

the dashed.

lines mark the limits of probable error of the measurements0 The solid line is the predicted performance of the configuration The agreement is within the experimental uncertainty of the measurements0

The results on Figure 10 were obtained from the unarig con-figuration shown in Figure 7. It will be noted that only three points

(badly scattered) were obtained for headings higher than the beam reach

(y < 8) This was true despite the fact that the theory indicates a pinching limit of roughly 30 degrees The reason so few points were obtained was the virtual impossibility of sailing any closer to the wind with that configuration0 This poor performance was attributed to the

extreme shape of the sail which was as shown on Figure 11- The sail camber on Figure 11 was 12 percent- In order to obtain this camber the mast was rotated to such an extreme that the spreaders were nearly paral-lel to the boom (Figure 1). Hence, the leading edge of the configuration was nearly perpendicular to the apparent wind vector It is probable

that leading edge stall resulted under these conditions and contributed to the difficulty.

A series of runs on a true beam reach showed the sensitivity of this sail configuration to aerodynamic incidence (Figure 12). The results are normalised with respect to the maximum performance obtained on that heading. Note that a + 10 degree change of incidence "off peak" resulted in a roughly 15 percent decrease of performance. When this information is compared with the data of Table I, it is apparent that the test vehicle is continuously "integrating" a series of thrust

cycles. It is concluded that, entering the close hauled regime, the

aerodynamic drag angle was abnormally large due to leading edge stall This, combined with the unsteadiness of the flow, is considered respon-sible for the balkiness of the vehicle for headings to the true wind

less than 75 degrees

A major difficulty in applying any performance theory to the evaluation of configuration changes is the frustrating unavailability of

data on the aero and hydrodynamics of sailing yachts Most of the data which exist have not yet been published in the open literature and. many of the problems simply have not been adequately investigated at present. A case in point is spinnaker performance. The boat of the present investigation was equipped with a standard Blue Jay spinnaker of 115 square feet measured area. The results of four different day's runs with main and spinnaker are shown as Figure 13. These represent

V a range of wind speeds 5 < VT < 11 miles per hour (1.1 <(

S

< 32).

:: deees.

The corresponding apparent wind angle is 60 degrees an apparent close reach, At this heading, the vehicle exceeded wind speed. Heading higher resulted in intermittent collapse of the spinnaker luff and loss of performance as shown by the measurement at y = 117 degrees. Since this represents an unsteady condition, the point

can be eliminated from consideration in fairing a curve through the data0 However, there is considerable scatter in the remaining points and it would be helpful in evaluating the data to have a well-established set of main-spinnaker aerodynamic coefficients from close reach to run Unfortunately, none exist. However, existing main-genoa data (Marchaj

(l96))

together with parachute data with varying vent area (Hoerner

(1958),

page 3-21.) would lead one to believe that 0,8 _{< CT} < 2.2

A

represent reasonable bounds on the aerodynamic resultant force coefficient. With these bounds in mind, equation (18) of the present theory was used

to calculate values of _CT from the measured main-spinnaker performance A

Values of drag angle were assumed from Figure 2. The results of this "educated guess" at aerodynamic coefficients is shown as Figure 11, Three f the nine points yielded values of aerodynamic

coefficient beyond the bounds established and were, thereby, regarded as wild points. The fairing shown was performed by connecting the remaining six points by straight lines on the rectangular co-ordinates of Figure 13.

The results which are of greatest interest to the racing sailor are summarized in the polar diagram of Figure 15 where the performance

of the boat with colete

racing rig is presented0 In addition to the

of Figure l3

previous conclusions based on Figure

9,

it is clear that running with spinnaker is no advantage. This fact was borne out in competition.

However, the results indicate that substantial advantage is possible with the spinnaker by tacking downwind at 35 to Lo degrees. Furthermore, for broad reaching the main-spinnaker configuration improves performance by

50 percent over the range 120 <

150.

The tactical implications are

obvious.

The foregoing discussion points up several uses of the theory in predicting performance and in analyzing performance data. Thus, the theory is useful for: (1) predicting gross performance characteristics; (2) planning performance experiments; (3) predicting effects of config-uration changes on performance; and,

(4)

assisting the analysis of performance data.

The validity of every physical theory must be established by measurements fraught with a degree of experimental uncertainty. In the present case, the experimental uncertainty was established as 7 percent. The present measurements are as good as or better than other existing measurements known to the author. Nevertheless, the, degree of precision to which absolute performance predictions by the present theory can be relied upon must be evaluated in terms of the experimental uncertainty. Relative performance predictions by the theory should be more certain since their validity depends only on the inclusion of essential physical parameters and not on absolute values. The gross agreement between theory and experiment confirms this physical validity of the theory.

CONCluSIONS

Information of general engineering value as well as tactical and tuning importance for the racing sailor was obtained from the present investigation. In the former category are the development of the theory and the performance testing technique; in the latter con-nection, are specific tactics and tuning for the particular boat tested as discussed in the previous section.

The simple dimensionless theory was obtained mainly as the result of three assumptions with respect to hydrodynamic drag:

(1) that the dependence of zero yaw hull drag on velocity can be approxi-mated by a parabola or series of parabolas; (2) that residuary drag for

light displacement hull forms can be approximated by a linear function of displacement coefficient (if two ranges of speed-length ratio are selected); and, (3) that lifting line theory can be applied to the evaluation of hydrodynamic induced drag The theory was successful in predicting the gross performance of a day sailing catamaran, The

dimensionless speed ratios _(Vs/VT and _Vs/VA) are shown to be "universal" performance variables for negligible heel. It is probable that this useage

can be extended to apply for parameters of heel for light displacement nohull boats.

The construction of the aerodynamic functions CT (8A) and A

£A(A)

depends on wind tunnel measurements since presently existing

aerodynamic theories do not apply and full scale measurements are practi-cally non-existent. Existing wind tunnel data, although far from

plentiful, are quite useful in the closehauled and close reaching regimes.

However, data in the stalled aerodynamic regimes are especially badly needed as is information on the unsteady effects which are predominant under sailing conditions.

The experimental investigation was performed within the physical restrictions placed upon the theory (to the best ability of

the research crew). These restrictions are to: (1) steady motion; (2) zero helm;

(3)

negligible heel and trim; and,

(4)

sail shape invar-iant with wind force The difference between the predicted probable error (+ 1 percent) and the measured probable error T percent) tends

to reflect the lack of control

of these factors.

The measurement

errors in the present case can easily be red.uce by a. factor of two by improvements of instrumentation and technique. A corresponding overall reduction in

the

scatter of results wifl depend partly on whether it is possible to exert tighter control on factors (2), (3) and (n). However, the area of greatest ignorance at the present time is with

respect to the effects of factor (1) ... the unsteady flow. These effects remain to be studied.

The present investigation confirms the conviction that per-formance measurements should be made by a skilled crew over a fixed

course equipped with

"fixed" instrumentation at least

until the proved

difficulties

with performance measurement are well understood. The

present investigation has

shown that the technique can be applied to

producing performance polars, evaluating gross geometrical configuration changes, and, to a lesser degree of certainty, gross aerodynamic charac-teristics. More refined applications depend on reducing the experimental

uncertainty of the technique0

It would be an easy matter to increase the sophistication of the theory by substituting less gross assumptions and thereby freeing the physical restrictions. However, this process would inevitably bring with it greater complication and less generality of the theory0

Furthermore, it is doubtful whether, in view of the experimental un-certainty of existing performance measurements, additional theoretical sophistication is justified at present. It would appear more to the point to concentrate further on refining techniques for full scale

performance measurement and, especially, on gaining better understanding the effects of unsteady flows on sailing performance.

ACOWLEDGE}T

The writer wishes to express his appreciation for the assistance of Messrs. George Hoschel, Neal Townsend and Gilbert Frele who were members of the research crew

during

the experi-mentation at the State University of New York at Stony Brook. They were partially supported by a National Science Foundation Undergraduate Research Participation Grant which help is gratefully

acknowledged.

The analysis and

write-up were

assisted by many

stii1ating

discussions

with members of the

sailing

yacht research groups at M.I.T.

and at

Southampton during the writer's sabbatical year. The writer wishes to express his appreciation to those men

R'ERENCES

Hra±field, W.S., (1968).

A Simple Performance Theory for Light

Displacement Multihull Sailboats, State University of New York at Stony Brook, Department of Mechanics, Report No lO3

Davidson, KSOM,

₍₁₉₅₆₎

Surveys in Mechanics, pi. Cambridge University Press.

Herreshoff, H.00

(l96),

Hydrodynamics and Aerodynamics of the Sailing Yacht0 Transactions of the Society of Naval Architects and Marine Engineers.

Hoerner, S.F.

_(1965),

_{Fluid-Dynamic Drag0 Published by the Author.} Letcher, J,S., Jr0

_(1965).

Balance of Helm and Static Directional

Stability of Yachts Sailing C1ose-Hau1ed J. Roy Aero0

Soc., 69, pp.2l2-248.

Marchaj, C.A0

(l961)

Sailing Theory and Practice0 Dodd, Mead and Company, New York

Marchaj, C.A0 and Tanner, T0

(l9614)

Wind Tunnel Tests of a

-sca1e Dragon Rig.. University of Southampton, Deparbment of Aeronautics, SUYR Paper No0 i1

Miliward, A0

(l967)

The Induced Drag of a Vertical Hydrofoil0 University of Southampton, Department of Aeronautics, PhOD. Thesis0

Myers, H.A0

_(l961.)

Theory of Sailing - With Application to Modern Catanarans, Marine Technology, October

pp0lO-280

Farham, Ma.jor General HF., Farrar, A. and Macalpine-.Downie, J.R.

(196T)

Class 'C' Racing Catamarans. Transactions Royal Institution of Naval Architects.

Schenck, H. (l96b)0 Engineering Experimentation McGraw-Hill Co., New York0

Schlicting, H

_(1968)..

Boundary Layer Theory0 McGraw-Hill

Co0,

New York.

Tanner, T. (l962) A Survey of Yacht Research at Southampton University. J, Roy. Aero. Soc., Vol.

66, pp. 62 - 6i8.

Tanner, T.

₍₁₉₆₈₎

The Measurement of Windward Performance. Yachting World, May0

?eerce (continued)

Wood, K.D. (1955). Technical Aerodynamics. Third Edition. Published by Author, Boulder, Colorado.

Yeh, E,Y.H. (1965). Series 6i Resistance Experiments on High Speed Displacement Forms; Marine Technolor, Vol. 2,

LIST OF TABLES

Specifications and Performance Parameters for Catamaran of Present Investigation.

Instrumentation List0

3,

Symbol Identification for Data of Figure 8

4. Time Dependence of Wind. Speed and. Direction.

[z/'LJ

=1.6

S ave

Speed-1engh ratio data:

[v /VLWLJ

. = 0.6; [v5//LWLJ = 3 3;

S max

(measured. operating range

over

186 runs).

obtained

from hull plus

image

reflected

about sailing waterline. Table 1. SPECIFICATIONS AIID PERF0RMAICE PARA1ThRS

a/2 = _0.215; ws = 430 lb (racing

wtj;

if = 0.042; w C = 340 ib; Sb

s

= =

45 It2, (total wetted);

54

It6;

= 770 ib; tons / - 78.5 per hull); (0.01 LWL)3

ft

S. = 3.0 ft2 1 S.. jib = 40ft2; SA = 1407 ft2/ton;

S.

_main = 100 ft2 k = 2145;

Sd

= 14 ft (total wetted); LWL = 12.5 ft.; S spin = 115 It2; (LWL/s)

L5 x

1o14;

SA = 140 ft2 (main + jib); S =

00O2 in-;

=

5*;

pw

Table 2, INSTRUMENTATION DATA I Measureable Instrument(s) Manufacturer Range Count4

Resonse

Time Method . Calibration Limit 7' Precision V ship speed VT true wind.speed true wind direction

200 yard line stop watch water speed mtr cup anemomenter marine comp0 masthead wind vane

-Cenco RiteKnot Simerl Kenwind

-0 to 15 mm 0 to

30 k

0 to

15 k

0 to

30

mphl 0 to 15 mph 0 to

360

0 to

360

-000l sec

O1 k

02 k

mph mph 2 degr 2 degr

-0.1 sec44

N.A444

1/10 sec 1/10 sec

NA

1/10 sec Steel tape lab0 stan- dard towing on timed runs mfg mfg swung on std

hdg0

bread board set up + 1 in L.c

+02 k

+O.25 k L, c.. + 2

degr.

standard deviation of

replicated calibration data;

*

defined as

smallest scale division;

*4

estimated time lag due to

signalling and "snapping" button;

*4*

NA0

T1e 3.

SYMBOL IDENTIFICATION FOR TEE DATA OF FIGURE 8

Symbol Run Series Date Wind

Number Speed Range

knots YRP-5

Aug23

YRP-6 Aug 21 _{7 < VT} 15 YRP-7

Aug21

G<VT<12

YRP-8 Aug 30 5 < VT < 13 YRP-1O Atg 31 6 < VT < 11 YRP-11 Sep 8 li < VT < 11

T&1e 1 TI) Dkk'DENCE OF WIND SPEED AND DIRECTION

Test Vehicle

Steady Run Unsteady Run Typical Run

start idcotrse finish Ave Wind Direct Degrees 115 112 Wind Speed VT knots

8

9

8

9 9

96

Run

Time

t2-t1 sec Wind Direct Degrees 15 15 25 40 55 Wind Speed VT knots 10

14

9

10

8

Run

Time

t2-t1 sec Wind Wind Direct Speed

VT

Degrees knots

96

13

99

111

109

16

104

16

Run

Time

t2t: see 115 120 115 115

873

42

102

914

102

14,8

4o8

/

/

/

/

20

1 .6

W 12

U

0

IL -J

4

0.8

0

w

4

I

0.4

I-U

0

close

hu led

- HEADING TO THE

TRUE WIND

a p parent

close reach

I I

20

40

60

80

100

120

140

160

IA_ HEADING TO THE APPARENT

WIND

180

130

142

162

180

w -J

z

4

0

4

apparent

broad reach

100 '1

O

beam reach

U

80

4

z

60

0

40

4

20 ()

I I I

65

83

100

110

30

45

i2

main

a) CLOSE HAULED

d) BROAc REACH

'

c) BEAM REACH

b) CLOSE REACH

e) RUN

4.

Predicted

Performance of

DC-14P Catamaran

Instrument boat

Inrun & outrun buoys

(TJ()

Time trap buoys

Fig 6

131/2

_ft

1 O

9

.8

.7

5

.4

.3

2

I

BEsr FT TO DATA

BY METHOD OF LEAST SQUA1ES.

BAND OF EROBABLE ERROR

4 KNOTS <VT <15 KNOTS

0

20

40

60

80

100

120

140

160

I 80

20

30

0

7VT6

80

70----90

₁₀₀

10

120

50

b 0

iil

Vs/VW

400

4

VS/il

130

VT.

1

40

150

160

170

180

(0

20

0

0

I I

40

60

80

100

120

140

160

-

HEADING TO THE TRUE WIND

10.

Predicted and Me :sured Performance

(Mainsail Only)

180

C

10

(helmsman

,

(

mainsail only)

least squa es fit)

Predicted Performance

only

ig.11

a

..

:25°

LENGTH

INCHES

/

a

a

as

b

28Y2

32/2

C

1O/4

12

07

v/v1J

0.6

Vs/ VT]

MAX

1.0

09

0.8

12

Effect of Unarig

Sail

Incidence on Beam Reaching Performance.

-T

10

20

30

40

50

60

70

80

90

IJ

I-i2

10

_{Fatrinq (aided}

_{by theory)}

Wiidward limit

of spinnaker performance

0

20

40

60

80

100

120

140

160

180

'- HEADING TO TRUE WIND

0

.8

4

.4

8.0

7O

6.0

5.0

U

4 4.0

z

3

0

L&i

4

3.0

I-z

-J

2O

U

1.0

0 wild

Owild

- - - - Aerodynamic coefficients calculated

by equation 23 from data of Figure 13

0 high

-

-N

N

N

N

N

'0

J

I.I

80

100

120

140

160

180

PAm MEASURED HEADING TO THE APPARENT WIND

14.

Spinnaker plus Main Aerodynamic

Coefficients

10

Main pius jib data

( Figure

)

Main plus SpinnKer data (Figure 13)

80

100

70

,11O

20

60

120

130

1.0