Method for the calculation of the resistance and side force of sailing yachts

THOD FOR THE CALCULATION OF THE RESISTANCE AND SIDE FORCE OF SAILING YACHTS

TECBNISCIIE UNWERSflT

Scheepahydroner!h11 i ca

Archief

Dr. Peter van Oossanen

Mekelweg 2, 2628

Delft

Netherlands Ship Model Basin, Wageningen, The. Netherlands.

Tel: 015-2786873/Fax:2781836

SUTYIMARY

In this paper a new method for the calculation of the hydrodynamic resistance and side force of sailing craft with an arbitrary huil (canoe body), keel, trim tab, skeg and rudder :configu-ration is presented. The method is basd on theoretical resistance and side force formula-tions, in which the coefficients have been tuned to fit the results of a significant amount of experiments. The results of the method compare satisfactorily with, those of fu.U-scale test's on the

5.5

Metre Class Yacht "Antiope". The required calculations can be performed with relative ease on a pocket calculator or (for repeated use) on

a small computer.

-1. IHTRODUCION

At some stage during the design of a sailing craft the hydrodynamic resistance and side force properties have to be determined for various boat speeds. Usually, a method for the calcu-latïoñ of the resistance and side force of the underwater hull configuration constitutes one of the most important tools of the naval archi-.. tect-designer. In some cases, speed-related as-pects of the hull become dominating factors in the design, often requiring a continuous series of such calculations, culminating in a final check of the design by means of model tests. This paper presénts a new method for the'calcu-lation of the resistance and side force of sai-ling craft with an arbitrary hull (canoe body), keel, trim tab, skeg and rudder configuration. The method is based on mainly theoretical for-mulations for resistance and side force, of which the (empirical) coefficients have, been tuned to fit the results of a significant. amount of experiments. The method was ,deve].oped.to pré-diet the resistance and side .force perforance of 12 Metre Class Yachts for the Americas Cup on,' which subject

_a

_{paper was 'presented by the au-"} thor in 1979 [i]*. A compariso±iof the .result-of this method with those' .result-of other calculation procedures and full-scale tests, for the case of the

5.5

Metre Class Yacht "Antiope"., was'- carried ,out by Larsson _{121, who demonstrated that the}

method adopted to calculate the resistance was exceptionally good. The published comparison 'for the side force pròperties of "Antiope" showed that the authort s method underestimated the side

Numbers in brackets. refer to the list of ref-erences given 'at th end of the paper.

force by 30 percent. However, an error was made by Larsson in applying the procedure given in,. Ref. 1. When accounted for, anunderestimationof the side force of "Antiope" by 11 percent is ob-tained instead. Since then, the procedure for the calculation of the side force has been

slightly modified (relative, to that given in Ref. 1).to take into account the increment in side force on the hull and keel, due to hull-keel interaction, in accordance with results of aerodynamic wing-body studies.

A numerical example of how the method is 'adopted (for "Antiope") is included in the paper.

2. CALCI.ILATION OF THE HYDRODYNAMIC SIDE FORCE 2. 1 The Side. Force on Keel and Trim Tab 2.1.1 The Basic Equation for the Keel

The side force or lift produced by the keel of a yacht, for small yaw angles

_8,

can be considered to be ali-near function of $, analogous to the lift of a wing as a function of angle-of-attack. Thebasic 'expression forthe side force of akeel canbe'written as:

-2DCLk

= PVB

8AcosO

. (i)

lift (side force) of keel density of sea water boat speed

drift angle

lift-curve slope, i.e.' the'slopé of the lift coefficient 'curve öf'. the' keel (CLk) against.tliè:azig3.é=of-attack

₍₈₎

- -. ,

lateral area

of-keel''..-angle of heel of the yaòht. For not too large angles-of-attack, the lift slope of the keel can be considered as a con-stant for a' given keel geometry. The' drift anglet ß is the angle-of-attack at -which the hull of the yacht passes through the water'.' It is mea- -sured by the angle b,etweeñ t'he course and the centre line of the' yacht. The angle-of-attack range, or drift-angle range, for which the

lift-curve slope is constant, depends primarily on the aspect ratio'of the keel. Figure 1, taken' from "Fluid-Dynamic Lift" by Hoerner .and Borst

[31,

shows the value of the lift coefficient _CL, as a function of angle-of-attack

_a,

_{as measured} on wings of various aspect ratios AR. The lift coefficient of the keél is defined as:

where Lk = p = V = = acLk aB

+ NACA, t/C:121. IN 2- DIM L TUNNEL * NACA TAPERED AR:6 AND :12

ÑACA AND ARC,0012, AR6

A NACA.CLÀRK Y.'RIOUS A RATIOS

e AVA. SHARP-EDGED RECTANG WINGS

V BRUNSWICK, AR:3

INVESTIGATION-A ARC.RECTANGULAR,AR:0.5 AND 2

* SCHOLZ. RECTANGULAR. AR:0.5 D NACA.RECTANGULAR PLATE

WftITER STRIP WITH AR: 1/30 ,2Tc3'18o a 16 _/PAR:12 CL t2

f

/1

,1

_/

f,,

/// /

.

-LINEAR COMP

ìy//

/4

o-:r

/1/

3 LINEAR COMP 0F

LIFT FOR ARO.5

40

Fig. i Lift coefficient of profiled, sharp-edg-ed rectangular (and of some other) wings, asa function of angle of attack (adjusted for zero lift where necessary) for various aspect ratios

(from Ref. 3).

.2 . i . 3 The Three-Dimensional Lift-Curve Slope It follows from Fig . i that for one value of the aspect ratio the lift slope is consta.nt,for all practical purposes,up to angles-of-attack vary-ing from about 5 degrees for an aspect ratio of

0.5,

to about 10 degrees for an aspect ratio of

I.O,and to about 15 degrees for aspect ratios higher than

6.

As the aspect ratio becomes smaller , the non-linear component of lift be-comes more important. The non-linear compo-nent of lift for an aspect ratio equal to 0.5 becomes discernable at

an

angle-of-attack of about 5 degrees. Analyses of the

perfonance of sailingyachts have shown that the drift angle B usually attains maxi-mum values of about 7 degrees. In some

cases values of up to 10 degrees are found. It follows that in the present

context the assumption of a constant lift slope value is valid for values of the (effeàtive)

aspect ratioin excess of about 1.0, or for geo-metric aspect ratio values in excess of about 0.5. Por smaller aspect ratios the concept of a constant lift-curve slope could lead to an under-estimation of the side force of the keel.

Various formultions have been derived which ex-press the value of the 3-dimensional lift-curve slope as a function of the (effective) aspect. ratio. A widely-used formulation is that derived by Whicker and Péhiner [14]. The relation obtain-ed by then is valid for the lift-curve slope, at zero angle-of-attack, of control surfaces (rud-ders, keels, etc.) with a taper ratio A equal to 0.145*. This value of the taper ratio nearly leads to elliptical spanwise loading for a quar-ter-chord, sweep-angle of zero. Also, the

Whicker and Pehiner relation is val-id for square tip shapes. For rounded pl-anforms and rounded lateral edges the lift-curve slope Is markedly reduced, particularly at low aspect ratios while, according to Hoerner and Borst'.[3j, the lift-curvé slope for differing taper ratios is hardly affected. To account for the efféct of rounded planforms and rounded lateral - edges on the lift-curve slope, the concept of an "-effec-tive span" can be adopted. This concept leads to the possibility of deriving the lift-curve slope of keels and rudders with rounded plan-forms or rounded lateral edges (or both) from the Whicker -and Fehlner lift-curve slope equa-tion,- valid for tapered control surfaces with

square -tips. The Whicker and Fehlner relation, in the présent nomenclature, is as follows:

acL

- - -

-.

21raAB

where aok=

' ß square.

-2 14

- 2aok+cosAkv' AP/cos A.+14

lift-curve slope factor of the 2-dimen-sional section shape composing the keel (a value equal to i.Ö corresponds to the theoretical lift-curve slope of 2v),

sweep-angle of quarter-chord line of keel.

(6)

*The _{taper ratio A is defined as the ratio of} the chord length at the tip to that at the root of the rudder or keel.

(2)

pvBAkcose

while the geometric aspect ratio bf the keel is defined as:

geom (3)

where b = span (height) of the keel

= average chord length of the keel When. the yacht heels at an angle

e,

the area of the projection of the keel on a vertical plane is reduced by cas

e,

which results in a reduc-tion of the side force.

2.1.2 The Effective Aspect Ratio

It can be shown theoretically that the lift on a wing, protruding from an infinitely long and wide. va].l, can be derived by neglecting the in-fluence of the wall and by assuming that the effective aspect ratio is doubled through re-flection in the wall. In the case of a yacht's keel-this also holds since no loss of lift oc-curs at the keel-hullintersection at mòderate angles of heel of the yacht, because the pres-sure differencé between the two sides of the keel is maintained. Hence the efféctive aspect

ratio ARk can be considered to be double the geometric aspect ratio, viz.:

ARk

2bk 2b

(4)

The effect of heel on the effective aspect ratio can be approximately accounted for by reducing the spañ of the keel by multiplying by cos O, where O is the angle of heel, viz.:

2b

TABLE I Typical values for the effective span ratio tb/b

planform lateral edges- L1'o/b (1+b/b)2

I-.

DESIGN

For round planforms and for round lateral edges:

= (1+b/b) round

Where ¿b is the effective reduction of the geo-metric span b. Hoerner and Borst [3] list values

for ¿b/b, which are presented here in Table I. The marked reduction in ¿b/b for rounded plan-forms and rounded lateral edges listed in Table I is due to the inward movement of the trailing (or free) vortices near the tip, leading to a reduction in the "vortex span" (see e.g. Fig. lo, page 3-7 of Eoerner and Borst [3J).

The symbols used here and elsewhere in this pa-per, relative to the keel of a yacht, are de-fined in Fig. 2.

2.1.14 The Two-Dimensional Lift-Curve Slope For individual design calculations, after the section profile to be used for the keel has been selected, it is appropriate to determine the 2-dimensional lift-curve slope from the results of tests, as tabulated e.g. by Riegels

151 and

Abbott and Doenboff [6]. For parametric design studies, however, it is often more practical to use a relation between a0 and the parameters on which it is dependent. Besides the thickiess-chord ratio, Hoerner and Borst [3] conclude that the _{trailing-edge "wedge" angle of the section} shape is also important. Sections with cusped contours near the trailing edge (leading to small trailing-edge angles) such as the NACA 63

WATERLINE

Croot Ak

-..OUARTER-CHORD LINE

PROJECTED (UPRIGHT) LATERAL AREA = Ak

Fig _{2 Definition of symbols used to describe} the geometry of a keel.

and 64 series, have high lift-curve slopes. Sections such as the NACA 4-digit series, with relative large trailing-edge angles, display a significantly lower lift-curve slope. On the basis of these facts the following relation was obtained for the 2-dimensional lift-curve slope

factor a.ok. dCL -1+0.82 (t/c)k_tanTT( + + 3.2(t/c)k+3.9(t/c)) (8) where: dC

2-dimensional lift-curve slope at zero angle-of-attack

(t/c)k thickness-chord ratio of section shape of keel

= half trailing-edge angle of section shape of keel.

Figure 3 shows how the angle TTk and the ratio (t/c)k are defined. Sometimes the angle T is not given. In that case the value of TTk canThe found from the slope at the trailing edge (,j) by use of the following relation:

tan

TT = CT . (t/c)k

k k

Riegels _{[51 has listed values for for most} known section shapes.

Comparisons between the values following from equation 8 and experimental values - for the NACA 00, 23, 63, 64, 65, 66 and some DVLprofiles are given in Fig. 14. The a0 values for the NACA 63, 64, 65 and 66 profiles vere taken from Abbott and Doenhoff [6], who (in Fig.. 57) give a set of figures showing dCL/da (a in degrees) as a function of thickness-chord' ratio for a Reynolds number of 6x106. They provide a faired curve for each type of section for both smooth and rough surface conditions. .Thè experimental values for the NACA 00 and 23 series were taken from Rieels [51 for ReynOlds number of 8.2 10 to 8.4 x 100. At a Reynolds number of 6x10°, the lift-curve slopes of these sections are locally about _{5 to 10 percent higher. The} experimental values for the DVL profiles were also taken from Riegels, and are valid fo ReYnoldsfnumbers varying between 2.5 X 10 to

3.2 x 100.

T

Tk

Fig. 3 Definition of symbols used to describe the average (mean) section of a keel.

- Rectangular or tapered sharp 0 1.0 - Elliptical or rounded sharp -0.014 _0.92 - Rectangular or tapered round -0.09 0.83 - Rounded round -0.12 _0.78

As follows from Fig. 14, the calculated and expe-rimentaJ. values for a0 agree satisfactorily for all practical purposes. Equation 7 can be used for most types of sections for Reynolds numbers in excess of 2x106 and for camber-chord ratiós up to about 0.014.

2.1.5 The Effect of Hull-Keel Interaction on the ifydrodynamic Side Force

The presence of the bull influences the flow along the keel and the presence of the keel chances the flow along the hull. As described by Schlichting and Truckenbrodt [7], additional velocities are induced along the hull by the keel which are directed to windward in front of the keel and to leeiard behind the keel. The hull is therefore in a curved flOw which influences (increases) the side force on the hull. The ef-fect of this cross flow along the hull on the flow about the keel is to induce additive upwash velocities in the vicinity of the keel which effectively increases the angle-of-attack to a-bout 2 just where the keel intersects the hull. It follows that the presence of the hull in-creases the side force on the keel. Both effects will be acccunted for in this section because the adopted method uses the basic liftof the keel, as follows from equation 1, to calculate the increment in side force on the hull and on

-1.3 -0.9 tic =0.3 0.27 0.24 021 0.18 0.15 0.12 0.09 0.06 0.8 a0 -0.7 -0.6

tant1 (Ey tic)

0.05 0.10 0.15 0.20 t/c =0.09 =0.12 =0.15 ,. =0.18 s

0.0

.35

Fig. 14 Cömparison of calculated and measured two-dimensional lift-curve slope values. The test data for the NACA 63, 614 65 and 66 series are averaged values accorging to Ref. 6, for a Reynolds number of 6 x 10 and camber to chord ratios of up to 0.014. The test data for the NACA 00 (4 digit) and 23 series, and for the DVL sections, are those given in Ref. 5, for Reynolds nuber values ranging from 2.5 X 10

to 8.5

X lOu.

the keel.

In aerodynamics, the effect of interactions be-tween a wing and the fuselage of ari airplane, on the total lift, poses a similar problem to that of the keel and the hull of a sailing craft.. In a review on this topic , Ashley and Rodden [8]

conclude that the net effect of centering an elongated fuselage of circular cross-section in a wing is to slightly increase the total lift of ftselage (hull) and wing (keel) over that of the wing alone, for small ratios of fuselage diame-ter to wingspan. For large fuselage diamediame-ter to wingspan ratios the total lift decreases with respect to that of the wing alone. This can be seen in Fig. 5, taken from berner and Borst [3], in which the increments in the lift-curve slope of wing-fuselage combinations due to interaction1 effects, for effective aspect ratios greater than 3, are shown as a function of the fuselage diameter-wingspan ratio, a = d/b. From this fi-gure it follows that the total side force pro-duced by keel and hull, for hull draught -keel depth ratios up to about 0.14 or 0.5 can be

calculated with reasonable accuracy by assuming that the keel extends to the waterline and set-ting the total hull-keel lift equal to the side force of this "equivalent" keel for which a = O.

Indeed, this procedure was first suggested by Gerritsma [9], and has been used with success by Beukelman and Keuning [1O],and others. One

drawback of this "equivalent" keel method, how-ever, is that it does not provide any insight into the relative significance of the side force. of the hull itself or the side force increments on hull and keel due to interaction effects. Al-so, for hull draught to keel depth ratios in ex-cess of about 0.5 and for geometric aspect

:a-OF WING WITHOUT FUSELAGE

O ü2 0.4 0.6 0.8 1.0

G=d/b

MEASURED DATA SHOWN COMPRISES WEBER EXPERIMEN'

TAL DATA (AT AR:3 AND 10) JUNKERS RECTANGULAR LOW WING (AT AR:6) AND NACA RECTANGULAR MID -WING (AT AR =6) AND WING COMPONENT DATA

Fig. 5 The lift-curve slope of wing-fuselage combinations as a function of the diameter ratio d/b (from Ref. 3).

o a a o s £ DVL

- EQUATION

8 SERIES 00 23 63 64 65 66 PROFILES

tios smaller than about 1.5, the "equivaJ.ent" keel method will overestimate the side force. For these reasons the results derived by Flax

[ii] are here preferred, who- d.erived approximate formulae for the separate interaction effects mentioned above, approximately valid for small to moderate aspect ratios, typical of yacht keels. When applied to the hull-keel case under consideration, these formulae are as follows:

= aLk (io)

= a(1+oLk where:

= side force increment induced on keel due to cross flow on hull

= side force increment induced on hull by the keel

Lk = side, force of exposed part of keel alone (without keel-hull interaction) as follows from equation 1

a = hull draught to keel depth ratio _{(=Th ITk;}

see Fig. 2). k

The total side force produced by the keel, com-prising the total lift on the keel and the lift induced on the hull by the keel,. then becomes:. LTk Lk + k + h = (i +

a)Lk

or (12)

2 CLk Thk 2

k PVB . . (i +-j) A,cos 0 (13) k

Figure 6 shows a comparison between the results of Viaden [12], valid for geometric aspect ratios greater than about 1.5, the results of Flax [11} as used in equations 10 and 11, and slender wing theory as presented by Schlichting and Truckenbrodt [îj, for the ratio of the lift of a wing alone(without body influence) to the sum of the lift of the wing-body configuration.

FLAX (11.AS USED IN EQN.i

(AN APPROXIMATION FOR ALL ASPECT RATIOS) O 1.0 -0.8 -0.6

-0.4

-0.2 LK CI-G) (1 -p2)2 (1.G)(I-G)% -

i

a G: Thk/Tk 0.2 0.4

SLENDER WING THEORY

(FOR SMALL ASPECT RATIOS

0.6 0.8

Fig. 6 Ratio of side force on the. keel alone (with-out interaction) to the sumof the total side force

on th&keel (with interaction) and the keel-induced side force on the hull according to 3 different approximations.

2.1.6 The Effect of a Trim Tab on the Side Force of the Keel

2.1.6.1 The Basic Equation

The effect of a trim tab on the side force of the keel is analogous to the effect of a move-able flap in a wing of an airplane. A flap de-flection domward causes an increase in the ef-fective camber of the wing and hence an increase in lift. The curves of lift coefficient against angle-of-attack, for several flap deflection angles, are therefore parallel to each other. If the angle of deflection of the trim tab is

the dependence of the side force of a keel on the drift angle ß and the trim tab angle can be written as: 3CLktt aCLk

C-+ cStt tt or (i1) cIrt; ß

(8

aCLk + (15)

where the subscript ktt implies that the respec-tive value of the combined keel and trim tab configuration is to be adopted. The total side force of a keel-trim tab configuration then be-comes (see equation 13):

LT

=2

ktt B CLktt Thktt2

(8+---T

)Akcos0

°tt

ktt (16) The linear relations 1 and 15 are approximately valid for trim tab angles from -10 to +10 degrees. The coefficient

_3ß/

_{is commonly referred to} as the flap effectiveness ratio. Figure 7 shows

a typical result of the effect of a flap on the. lift of a wing,. while Fig. 8 shows the depen-dency of the flap effectiveness ratio on the

t/c:O13

20 30 40

FLAP ANGLE §

'

_---.

-UFT PRODUCED IN SPITE

OF FLOW SERATION

-,.C?SEPARATION FROM FLAP

':

-/7.7

_/:.5

/

AR:3.5

-Rn :6x105 C1 :0.4 c

Fig. _{7 Lifting characteristics of an isolated} rectangular horizontal tail surface with flap tested in an open wind tunnel (from Ref. 3).

dC1jdö0.Q37 0.8

'

.& 06

_/

-0.4

/ f

/

_;

t2 CL

to

-I.

flap-wing chord ratio. Both figures

are taken

from Hoerner and Borst

_[3].

When the chordlength

of the trim tab is not constant over the height

of the keel, as is sometimes the case, it is

more appropriate to use the ratio

of the trim

tab area to the (total) keel area Att/Aktt, in-stead of the ratio of the trim tab chord to the (total) chord Ctt/Cktt. A

typical

trim tab con-figuration is

shown

in Fig.

_9.

2.1.6.2 The Flap Effectiveness Ratio

For a flapped wing (keel) of infinite aspect ra-tio

and

small flap (trim tab) angles, Glauert [13] derived a theoretical expression for the flap effectiveness ratio _{/35. The dérived,} expression (valid for zero wing

and

zero flap thickness), in terms of the keel-trim tab nomen-clature is:

th

*

+ arcsin /Att/Aktt (17)

The curve denoted as "théory" in Fig. 8 is that

according to equation

17.

From Fig.

8 it

follows

that due to viscous effects, experimental values

for the flap effectiveness rátio

are

always

smal-ler than the theoretical values. Since the

dif-ferences

can

be of the order of 10 to 20 percent

or more for frequently used section shapes

and values of Att/Aktt, it is appropriate to correct equation 17 to yield more realistic values. The observation that the effectiveness ratio is re-duced by section thickness

and particularly

by

the trailing-edge "wedge" angle

TTtt (see

Hoer-and Borst

_[3],

page

9-3)

led,

through

ap-plication of a trial and error procedure, to the

following equation forthe flap effectiveness ratio, corrected for the effects of viscosity.:

(..L)

_{(_)th(0.?5+0.25(_)01}

₊ - 1.0 db 0.8 0.6 0.4 0.2

'ç d'AS TESTED ON PLAIN

Z/ Y

ANDCLEAN CONFIGURATIONS

//,/ A) SECTIONS. WITH PLAIN

-

t49

- Fl.APS:

-:f

r'

00

_02.

_{0.4 0.6}

0B

FLAP/CHORD RATIO 2.i

_-C

OR

±i

A

Fig. 8 Flap effectiveness ratio of various types of trailing-edge control flaps (elevators, rudders,.a.ilerons) as a function of their chord ratio (from Ref. 3).

1.0

-

7.35

(1-()01)tan

_TT

Aktt

tt

In equation 18, TT

.

is half the trailing-edge "wedge" angle of te trim tab (see also equation: 8). Values for the effectiveness ratio according to equation 18 are shown in Fig. 10 together with some measured values. Values för the ratio of the corrected to the theoretical flap effeô-tiveness value is given as a function of

tan for various trim tab-keel area ratios.

2.1.6.3 Effects

of Finite Aspect Ratio

and

Sweep Anple

According to Hoerner and Barst

_[3],

the flap ef-fectiveness ratio only slightly varies down to aspect ratios equal to about 2. For smaller

as-pect ratios, experimental

and lifting surface theory results indicate that as the aspect ratio decreases, the flap effectiveness increases. The flap effectiveness approaches unity as the as-pect ratio approaches zero.

From results compiled by Hoerner and Barst

_[31.

shown in Fig. 11, it is apparent that the effect3

of aspect ratio -is noticeable for-flap effecti-veness values, as calculated from equation 18, larger than about 0.17 ARktt, where _{ktt is the}

effective aspect ratio of the keel-trim

tab

áon-figuration, as follows

from

equation

_5.

An

ap-proximation of this influence of the aspect ratio was found by assuming, that when

0.17 ARktt, the effectiveness tends to unit3 as does a 1-(sinx)°25 function. The actual equation obtained -is:

1 (1

(._a_) )(

O.l7lTARktt

_)0.25

(i

)

-

--

o

S1fl2(/)

9 when

> 0. 17

_ARktt

tto

-According to various authorities, the flap

effec-Fig.

9 Definition of symbols used to describe

the geometry of a keel

and

trim tab con-

-figuration.

(18),

X NACA 000619 NACA-vARIous £ ARC -VARIOUS OTHER SOURCES

WITH OVERHANGINGNOSE

BALANCE

NACA HORIZONTAL TAIL -OTHER SOURCES

ON ISOLATED ICRIZONTAL. TAILS:

e GOETHERT SYSTEMATIC

tiveness ratio varies with the cosine of the in which:

weepback

angle

of the hinge line of the flap (0.117

AHtt [3].

Hence equation 18 should be multiplied Oktt_1+0.S2(t/ktt_tanTTtt(t/C)k

by cos Aif when the hinge line of the trim tab

has sweepck, as follows: 2

+ 3.2(t/c)ktt+

3.9(t/c)ktt)

(--)

_th

(o.75+o.25(__)0.1 +

_and

Aktt

O.lTlrARktt

_0.25

)°)tanTTt)cosAH

tt (20) (sin

_2(B/36)0

(19)

-2.1.7 The Final Equations for the Side Force of in which

-Keel and Trim Tab

_I_)

(o.75o.2s(_)0.1

+

On the basis of the results presented on the tt th preceding pages, the total side force of a.

keel-(20)

trim tab configuration, including the side force -

7.35(1-on the hull induced by the keel, can be calcu- Aktt tt

lated as follows: - where

LT 2 3CLktt

(+

tt)(1+Ttt)2tc0s9

h

*(1Att/A,tt(1_Att/Aktt)

+ 36 1t

aß

_tt ktt tt ktt

(16)

where:

21raOk(1+b/b)2

-0.9

aß

-0.8

-07

0.2 0.1

òtt

-06

EQUATION 18

'bbtt'ttl

NACA 0009 + NACA 0015 x NACA 23012 NACA

66-009

NACA 66(2x15)-216(a-:0.6) NACA 745 A 317 D NACA 653 - 418 Att/Akt z 1.0 0.8 0.6 0.4 0.-3 i/AR ¿

/

2aok+cosAtt

ktt

cosA,+14

(6)+(7) 0.05 010 0.15 0.20 0.28

tan tT(t1 t/c)

Fig. 10 Comparison of calculated

and

measured

flap effectiveness ratio values. The test data shown

are

from Refs. 3 and 6, for Reynolds number values

_{ranging from 1.14}

x 10 to 8 x

io6,

for plain flaps with a.led gaps. Only the infinite aspect ratio case is

considered.

It should be

noted that when

then

aß!

The effective aspect

ratio of the keel-trim tab configuration i,s:

2btt

2(Tkt. Thkt

cose

ARktti

cosO=

cosê

ktt

Aktt

Aktt

(5)

When a trim tab is not fitted to the keel, or when t=0 the total side force of the keel be-comes:

23CLk

kB

(1.+;---) -AkcosO

with

3CLk

2Wa.cARk( 1+b/b)2

313 + aresin /Att/Aktt

2a0+cosA/ AR/cosA,+14

(i')

Fig. il Influence

of the aspect ratio

upon the

flap effectiveness ratio (from Ref. 3).

+

where a0 =

1+0.82(t/c)k_tanTk(°7

_{+ 3.2(t/c)k+} (t/c)k + 3.9 (t/c)) (8) where

2(Tk_T2cosO

ARk =

2.2. The Side Force on Skeg and Rudder

2.2. 1 Side Force on, a Combined Skeg and Rudder Configuration

The total side force on a combined skeg and rudder configuration, such as'depictedin Fig.

12a, can be determined along the saine lines as presented in section 2.1 for a keel-trim tab

configuration when the entities related to the skeg and those related to the rudder take the plaöe of those of the keel and the trim tab, respectively. Two additional corrections to the formulas must be made, however. The first of these correction is for the speed of the flow relative to the rudder. Since thé rudder is lo-cated in the wake (dovnvash) of the keel, it is appropriate to assume that its effective speed through the water is not equal o _VB. Also, the thickness of the boundary layer along the hull at the location of the rudder is relatively greater than at the location of the keel. To correct for these, effects, it is assumed. that the effective speed of the rudder through the water is equal to approximately _0.SVB. Secòndly, the rudder is often situated close to the trans-om or stern of the yacht, in which case the hull-rudder intersection is often above the water

sur-face. In this case the effective aspect ratio is no longer double the geometric aspect ratio, but appreciably less. It is assumed that then the effective aspect ratio approximately equals the geometric aspéct ratio. The equations for the total side force of a skeg and rudder confi-guration, such as shown in Fig. 12 a, then be-come as follows:. . .

0CL Th

1 2 sr

sr2

(+

)(i+ i-) ASrCOSO (21). L AR( 1+Lcb/b)2 where. 3CL sr in which a= 1+O.82(t/ô)'..taflTT (t/'C) + 3.9(t/c)2) .. ₍₂₃₎ d - . -,. O 77TAR )O.25 - 1_(1_(6r)oXS1fl2(8/3r)o Ak

2a0+cosAv/AR2/cos4A'+14

(5). (22) (2') where

(...)

=a

_{)th(0.75+0.25(A}

_r¡A_sr

)O.1735(1

+

-(A ¡A

_)°1).tanTT

)cosA r sr

r r

and

(--)

= (v'A ÏA '(1-A ¡A ) +' arcsin,/A ¡A

3rth

ir r sr r sr r sr

(26)

Again, when (3/

r

) < 0.17 AR , then

o sr

The effective aspect ratio follows from the following relations: 2(T

-T

)2cose AR sr hsr

forT

>0

sr A h and T

cos®

AR

forT

<0

sr c

h-sr sr

The case of O occurs when the hull-rudder intersection is above the water surface. Then the value of _(1+Th n1'Tsr)2 in equation 21be-comes equal to unity. In equation 21 through 28 the subscript sr implies that the respective value of the combined skeg and rudder configu-ration is to be adopted. For examplè, Asr is the lateral area of the combined skeg and rud-.

(25)

AVERAGE (MEAN)

SECTION OF.

SKEG-RUDDER CONFIGURATION

HINGE lINE OF RUDDER

ATR-LINE

PROFILE

QUARTER -CHORD

LINE OF SKEG AND

RUDDER

CONFIGU-RATION SKEG; COMBINED

AREA 'OF SKEG

AND RUDDERAsr SECTION AA LINE PROFILE 12a 12 b

Fig. 12 Definitiön of symbols used to describe the geometry of a skeg and rudder configuration (Fig.

12a), andofarudderalone (Fig. 12b'), forwhich the caseT<0 is drawn.

der configuration. Similarly, the ubscript r implies that the respective value of the rudder is to be taken only. For example _r is the rud-der angle and TT is the half. trailing-edge angle of the ruder section. The quantities used in these equations are defined in Fig. 12a. 2.2.2 Side Force on a Rudder Alone

The side force on a spatype rudder, as de-pictd in Fig. 12b, at zero rudder angle, can be determined from an equation similar to equa-tion 13 valid for a keel without a trim tab. In general, the side force on a rudder at any angle

r becomes as follows:

2aCL

Th ₂ LT,, 0.8vB)

rr)(1) ArcosO

(29) where, as before: 21ra. AR ( l+b/b)2 aß 2a.0+cosA,/AR2/cos4A+4 in which 0.117 a0 1+0.82(t/c)r_tanTT((t/) +3.2(t/c) +3.9(t/c)2) where 2(T -T. ) cos O ' 2

AR-

-r _Ar when Th > O r when Thr < O

Since the canoe body has a round "planform" and (30) _{rounded "lateral edges" a value of -0. 12 can be}

adopted for Lxb/b (see Table i). Hence the follow-ing expression is obtained:

acL

+ CLh= B 0.T8(ARh-I i.8ß)ß (38)

and

Here also an effective speed óf the rudder through the water of about 0.8 VB is á,ssumed. Again, the case that Tb. < O corresponds to the

situation when the top of the rudder is above the water surface at the angle of hel 6. Also, the effective value of (i + T./Tr) i-s equal to unity in that case since no side force.is. induced on the hull by the keel and no extra side force is induced on the rudder by the hull.

The definition of the various eñtities in equa-tions 29 through 33 is shown in Fig. 12b. 2.3 The Side Force on the Hull (Canoe Body) The hull also experiences a side force at

non-zero drift angles. Similarly to the procedures followed above, this side force can be written

as:

h

aCLh

cosO

:(3

Since the effective aspect ratio of the hull is small (usually 0.30 or less), the theory of low aspect ratio wings (see for example Weinig [114]) can be applied, which leads to the result that the lift-curve slope of such a wing is equal to _{ARh. On applying this theory to the hull} of a yac'ht it is assumed that the hull is mod-elled by a thin flat plate with the same (late-ral) area and profile as the hull. Hoerner and Borst [3] show that this result is not correct due to the occurrence of a non-linear lift

com-ponent equal to the force developed in the lift direction by drag. They give the following re-lation for the non-linear lift component for bodies with an aspect ratio of about 0.2:

CL = 1.8 sin2ßcosß h

(35)

which, for drift angles less than about 10 de-grees, leads to:

CL = 1.8ß2 h

(36)

It follows then that the lift-curve slope of the canoe body can be written as:

___k=(.Pth+ 1.8ß)(l + Lb/b)2 (37)

LT pV . _{0.78(irTcosO + l.OAhß)ßcose (39)}

where it is assumed that the effective aspect ratio of the hull (canoe body) ,2at a heel angle e, is approximately equal to 2Thcosß/Ah, where' Th is the maximum draught of the canoe body and Ah the lateral area.

2.14 Side Force Calculations for 5.5 Metre Yacht "Antiope" and Comparisons with Results of Mea-surements

Measurements of the resistance and side force on the actual "Antiope" hull were carried out at the David Taylor Model Basin (now the David TaylorNaval Ship Research and Development Center) for, the Technical and Research Panel H-13

(Sailing Yachts) of the Society of Naval Archi-tects and Marine Engineers. The results of these measurements were presented by Herreshoff and Newman [15]. Resistance and side force measure-ments were carried

_out

_{for various combinations} of yaw angle _{B, rudder angle 6r' heel angle} and speed VB. A lines drawing of "Antiope",. taken from Letcher [16], is shown in Fig. 13. For the calculation of the side force on the keel and rudder (which- is - attached to the, keel), the rud- -der cnbé. consi-dered as a trim tab so that-equa-tion 16. applièá. From the data given in referen-ces 15 -and 16 the- following data, required for the calculations, were derived:

-= 0.58 m -. = 0.716 radians (tic), = 0.07 TTt+ = 1.13x0.O7=0.079 A.ttlAhtt = 0.12 = -0.514 ad.].ans Aktt - 1.90 n2 Ah

3.05m

The aspect ratio is: and

T cos 8 AR = r

r c

2(1.Lvl_0.58)2cos8

= 0.725 cose

ktt

1.90

The -theoretical flap effectiveness ratio is:

+ arcsïn/0. 12)=0.1132

and

(38 )_= 0.1132(0.75+0.25(0.12)0.1 + -7.35(1-(0.12)0Htan(0.Oî9flcos(-0.511) = 0.312. Thus:

1_(1_O.312)(sin(17:)co5O))O.2S

= 1-0.688(sin(Q.62lcosO))°25 -The two-dimensional lift-curve slope is:

a0 =

1+0.82(0.07)-tan(0.079)(7+3.2(Q.07)+

ktt

:---+

3.9(0.07)2)=Q.906

The keel has rounded lateral edges, so that

tb/b=-0.09. Hence:

2ir(0.906)0.]'25cos8(1-0.09)2

38 2(-0.906)+cos(Q.716)/ (0.725cosO) cos4(0.716) 3.418 cosO i .812+0.7-51/i.622cos2O+11

The side force of keel and rudder (trim tab) -is:

3.418cosO(8+( i-0.688)

LTktt pV

-1.812 +

-..

'p 9

-,

4 3 1t?

LI

LP IFT.

Fig. 13--Lines of "Antiope" showing the test waterline.

(sin(o.62icoO))°25)6tt)(1*---)21.9cose

+ O.751W,i 622cos-O4

The side force of the canoe

body is:

0. 78) ( ,r(0. 58)

cos9+1.

8( 3. 05)8)8cos0

=pV(O. 8248cos20+4 .28282cos0)

The results of the measurements, as presented by Letcher [16], are given as lift coefficients,, using the upright pro.jécted lateral area

(4.95 m2) as the reference area. On following this example the following expression for the

total side förce coefficient is obtained:

CL 2.61-3 +

p4A.

1.812 +

+( 1-0.688(sin(0.62lcosO) )°25)tt)cos20

-+ 0.7-54/1 .622cos20+11

+O.1665ßcos28 + 0.86-5iß2cosO

Comparisons between calculated values according

to this equation and Letcher's reduced data are

made in Table

2. Only the measured data for

which the standard

deviation

is less than 0.1

has been considered.

On not considering the comparisons for data points 5, 7, 13, 114 and 15, for which the

-mea-sured values are probably sübject to relatively

large errors, the average &bsolute difference

between measured and calcúlated values i-s about

11.9%-which is sufficiently small for

all

prac-tical purposes. + -1 -2 -3 67 I

1...=

10

iiviiiii

IMAUI

.. -4

TABLE II Comparison between calculated and mea-sured side force values for "Antiope"

3. CALCULATION OF THE HYDRODYNAi.0 RESISTA10E 3.1 The Viscous Resistance

3. i.. i _{The Basic Equation for the Viscous}

Resistance

The total hydrodynamic resistance. of a flat plate which is deeply submerged, at zero angle-of-attack, is equal tothe frictional resistance Hp0 which-cai be written as:

Hp0

CpOV2S

.

. . . -(4o)

where C _{= specific frictional resistance} coef-ficient in a two-dimensional flow,. V _{= velocity of the plate through the}

fluid,

S _{= wetted area (of both sides) of flat} plate.

This relation, for the. frictional resistance can also be adopted for the calculation of the fric-tioñal resistance of the hul]., keel or rudder of a. sailing yacht. In that case, however, HF is not, the total resistance, nor is it the toal viscous resistance because, at the water surface,

wavema.king occurs. For surface ships it -is cus-tomary to divide the total resistance into a non-viscous part and a viscous parta The viscous resistance Rv is considered equivalent to the sum of the frictional resistance R of the three-dimensional hull (as distinct from RF for a flat plate) and a pressure resistance component of viscous origin R7. The frictional resistance is associated with the force required to overcome the tangential stresses developed between the hull and the fluid, while the viscous pressure resistance is due to a pressure difference be-tween the forebody and the a.ftbod.y of the hull. The growth . of the boundary layer along the hull

causes the pressure on the aftbôdy to be smaller than on the .forebody leading to a resultant pressure force on the hull, of viscous origin, accordingly termed the viscous pressure resis-tance. Since Rpv is usually small, the viscous resistance is á±teri written as:

=

CrPV2S

(41)

where

CV = (42)

in which C = specific total viscous resistance coèfficient ...

k = three-dimensional., form factor on .flat plate friction,

'S = wetted, surface of hull.

The form factor k:thus accounts for the effects of the .three-dimensional form on' the 'value of CF 'and for the viscous pressure resistance R. FrSm detailed boundary layer calculations

carried out by Larsson for the 5.5 Metre Yacht "Antiope" [2], it is clear that' for yacht-like hull forms, the effects of the form of the hull on the value of CF0 is very small, and that the form factor maih1,r accounts for the viscous pressure resistance. Larssòn found fr a yacht speed of 3.05 ni/sec that C=2.6Ox1O _{. According} to equation 1#6.. (see. section'3.1.2), the'flat plate value CF =2.62x103' when'basing the, calcu-lation of the Lyriolds number on an effective length of 0.8 Lt, _{(to account for the. shorter} length of the keel). It is _{ssumed, therefère,} that CFCp and that the form factor k approxi-mately accounts' for the effect of'the viscous pressure resistanòe only.

In the calculation _{f the Reynolds number, an} average hull length equal' tó 0.7 or 0.8 of the' waterline lenh..is usually adopted, following

the' practice: of Davidson some. 40' years 'ago: [.17]. In the last decade', however, -the';development in

-the design 0±' keels 'and: rudders. of sailing.' yachts has' been such 'that .they can 'now be

'con-sidered 'as appendages "rather than _{as an intégral} part. of the hull. Keels and rudders 'a.e now pro-portioned in accorda.nce.to their main function,

as control surfaces. Since control surfaces be-come more effective as their aspect ratio in-creases, yacht keels and rudders 'are nowadayS appreciably shorter. Also, rudders are now no longer placed iminediatély behind the. keel, _but at the aftermost part of the hull, where they are most effective. It follows that one particu-lar value of the frictional resistance coeffi-Dàta heel drift rudder mea- calculated

point angle angle angle sured side force side force 00 tt° C1,c1O0 CLTX100 o o 0

.014

0 2 1.0 2.70 0 14.56 11.140 3 3.0 2.714 0 .. 4.71 14.45 14 5.0 .2.79 0 14.8o 4.52 5 20.0 3.15 0 3.99 4.61 6 21.5 3.19 Ö 4.914 4.58 7 22.8 3.22 0 5.23 4.55 8 21.4 7.01 0 10.72 10.7)4 9 23.3 7.05 o . 10.27 _10.55 10 _25.9 7.11 0 10.30 10.26 11 13.0 6.82 0.

ii.4o

11.31 12 _{19.6 6.97} 0 11.70 _10.90 13 9.0 0.22 0 0.01 0.34 1)4 _{9.3 0.23} 0 _0.50 . 0.35 15 _9.5 0.23 0 0.79 0.35 16 10.1 2.92 3.0 6.34 6.12 17 11.0 2.95 3.0 6.4)4 6.14 18 1)4.6 3.03 .3.0 6.66 6.11 19 10.4 _2.94 6.o _7.8? 7.62 20 16.1 3.07 6.0 7.98 7.53 21 10.9 2.94 1.6 _{5.91 5.43} 22 12.3 2.98 1.6 _6.09 5.145 23 14.2 3.02 1.6 6.20 5.144 .2)4 10.2 _2.92 0 _{4.78 4.63} 25 13.8 3.01 0 5.12 _4.67 26 11.3 2.95 3.0 5.89 6.13 27 14.7 _{3.03 3.0} . 6.27 6.11 28 6.9 _6.67 0 11.52 11.41 29 10.4 6.76 0 12.00 11.39

dent, based on an average vaJ.ue of the Reynolds number, cannot rightly reflect the considerable variation in length with draught of the under-water hull, keel and rudder of modern yachts. Obviously then, the best method to determine the viscous resistance of a sailing yacht is to per-form the calculation of equation i2 separately for hull, keel and rudder, viz:

Rv=pv(cF( 1+kb)Sh+CF ( 1+kk)Sk+(O.8)2Cj(1+kl)s)

R3)

in which the subscripts h, k and r denote hull (canoe body)2 keel and rudder, respectively. The factor (0.8) apprôximately accounts for the reduced speed of the flow relative to the rudder (see section 2.2.1).

3.1.2 The Frictional Resistance Coefficient There is considerable evidence that the boundary layer along parts of the hull, keel or rudder of a sailing yacht can be laminar. Tanner [18] tank tested a full-scale International 10 Square Metre Class Canoe, fitted with six different

centre-boards, from which he found that the boundary layer over the centre-boards was domi-nantly laminar. The Reynolds number of the centre boards (based on the average chord lenth in the tests extended up to 9x105. Crago L19J noted that important extents of laminar flow exist

along the hulls of sailing yachts "up to the size of a Dragon". It has been found that even larger yachts can "ghost" at i or 2 knots in absolutely

calm conditions L2o1. In a very comprehensive

study

on the significance of scale effects in resistance tests with sailing yacht models, Kirkman and Pedrick [21] conclude that laminar flow effects are very obious up to Reynolds numbers of.at least 1x10

To account for the effects of laminar flow along the hull, keel or rudder, it is possible. to adopt a formula for CF such as was devised by Prandtl and Schlichting [22], viz

CF =. CF

_T

turb n

(44)

where CF = skin friótion coefficient for a turb turbulent boundary layer along a

flat plate,

c = a constant,

Rn = Reynolds number.

From the results obtained by Gebers.. and Blasius

t

23] for the skin friction of flat plates with sharp leading edges, Prandtl concluded that la.-minar to turbulent transition began for R=5x105 and that the skin friction coefficient assumed values accordingto fully turbulent flow at ap-proximately 5x100. Accordingly, he adoptedthe following formula:

c_0T4

1700 ...

F _(Rn)O2

- Rn . 5

wbichreflects the influence of laminar flow n CF

i9

the Reynolds number range betveén 5x107 and 5x100. On adopting the 1957 ITTC friction formu-lation for a turbulent boundary layer, the same

influence of laminar flow is obtained when the following equation is used:

flfl7Ç

CF

(log10R_2)2

Values according to equation 45 are shown in Fig! 14. For flat pltes with rounded leading edges, Wieselsberger j24} and others have found that the value of .the skin friction coefficient reflects the eisterxe of a urbulent flow down to a Reynolds number of 1x10 . Subsequent studies have revealed that this is caused by early. traxis.. ition, due to the existence of an adverse

pres-sure gradient just downstream of the rounded leading edge. Sudh adverse pressure gradients do not occur, however, So close to the leading ed-ges of keels and rudders normally applied, at angles-of-attack less than about 7 degrees. Ac-cordingly, equation 146 would seem to be appli-cable for the calculation of realistic Cpvalues for the hull, keel and rudder of sailing craft, when hydrodynamically smooth for Reynolds num-ber values in excess of 5x107. The Reynolds nun-bèr to be used in combination with equation 146 is defined as: R (47) 5 R

VBk

0.8V . R

Br

n U r Io and. (48) (49) where the factor 0.8 in equation 49 approximate-ly accounts for the decreased velocity of the flow to the rudder (see section 2.2.1). The kine-matic visgosty y of sea water equals

1.191x10 m /sec at 15°C. For some other water temperatures, the corresponding values are given in Table 3 for both seawater and fresh water.

5 s-0 40

vi

,J5 I0 R VL

Jill!

I I

II

-t{EASUREMEÑTS a WIESELSBERGER s GEBERS A FROUDE £ KEMPF 2 4 I

Fig. 114 Theoretical andmeasuredvalues of the fictional resistance coefficient of smooth flat plates CFO. Theoretical curves are: 1, laminar. (Blasius); 2, turbulent (Prandtl); 3, turbulent (Pra.ndtl-Schiichting; 3a, transition laminar-turbulent (Prandtl-Schlichting); 14, laminar-turbulent (Schuitz-Grunov). Equation 14/5 corresponds to curves 3 and 3a (from Ref.

7).

R (46)

TABLE III Value of the kinematic viscosity

y

at

various temperatures for sea water

and

fresh water

Kinematic vis- Kinematic viscosi-Temp. (°c)

cosity

fr sea

ty for fresh water

water (m ¡sec) (m2/sec)

3.1.3 The

Form

Factor of Hull, Keel and Rudder The form factor, as defined in equation _142,

ac-counts for the increase

in viscous resistance of

a

three-dimensional form (bu.].].,

keel or

rudder)

over that of a flat plate, with thè same wetted area. In the method outlined in reference 1, the

author adopted. the

form

factor formüla derived

by Holtrop [25] for the canoe body. Larsson [2] found that this formula

resulted in a value

which was too high for "Antiope" (the formula

gives a value of 1.23, while Larsson's results indicate a value of about 1.07).' Since more ac-curate formulae for the form factor are not a-vailable it is more appropriate not to _{dopt the} form factor concept for the canoe, body and to use the flat plate approximation to the viscous resistance. The effect of not incorporating a form factor for the canoe body will not influ-ence the result foi'the total resistance because

in the calculation of the, wave

resistance (seé

section' 3.2.1) an approach is' used based on the residual resistance BR, as obtained from towing tank measurements by subtraction of' the equiva-lent flat plate frictional', resistance R _from' the total resistance HT, i.e.

RR=Rr.RF .

It,foJ.-lows 'that the viscous pressure

resistace-being the difference between _R

_and

_{RF, Ìs,} ïnclu-ded in this RB-value, as used in section 3.2.' The effect of thickness on the drag of typical keel and rudder sections has béen studied by Hoerner [26]. He derived formulas for NACA .14-digit-type profiles (typical of rùdder sections)

and

for 1VACA 63, 614 and 65 profiles (typical _of keel sections) which, as an approximation, can -:

be adopted as expressions for the- form factor of the rudder and keel, respectively. _These for-mulas are as follows: ' ' ' ' ' '

-kk

1.2(t/c)k+

70(t/c) '

(50)

kr

2(t/+ 60(t/c)14

.

','

(51)-where t/c is the effective (average) thick ess-to-chord ratio. - '

3.1.14 An Approximate Equation for

_{the Wetted'}

Surface of the Hull

When

the wetted surface of the canoe body is not

knom, it can

be approximately deduced from a

formula given by Holtrop [25], vii:

Sh=I(?Th+B)CGS(o.1453o+0.141425CB_o.2862C

-

0.0031467L/Th+0.3E96C)

. (52)

where L.length of the design waterline, Tb = maximum draught of canoe body,

B,, beam

of the design waterline,

C = maximum section coefficient .(for yacht. hulls not necessarily the

midship

section coefficient) =

A

T'

CBh= block coefficient of canoe body, C='wáterplane coefficient =

=

area of maximum section

and

A

area of 'design waterplane.

.To demonstrate the

ccuracy of- equation 52 the

calculated value for ".Antiope" can be compared

with the actual value. According to Letcher [16], the total wetted surface of ttAntiope" (in-cluding the keel) is 114.80 m2. On subtracting

the wetted surface of the keel,' which is approxi-mately equa], to 3.8 m2 (approximately 2 'times. the lateral area), the wetted- surface of the canoe body is found to be approximately 11.0 m2.

With L= 7.41 m, Th=O.58 m, B=1.75 m,'-C= -0.567, C=O.3O6- and Cp=O691 (where _{C&, CB} and

C were btáined from the lines drawing proide'd by- Letcher in reference 16 and reproduced here

in Fig. 13), the wetted surface 'according to e-quation 52 is calculated to be 10.83 m2 which is

very close to t1e actual

value of approximately

11.0

m2. - - ' . .

3.1.5 The Final Equations for the Viscous

Resis-tance , ' - .

The fina], equations for the viscous resistance of 'a hull-keel-rudder configuratido become _as follows:

RjFBhF( ik,)

PVBSk+CF.(

1+k)(O

.

8v/S

(53)

where

CF' = 0.075, ' 2

.1800

-' '' (146) h,k,r

(log1R

- -2) - n k -h, ,r-in which _Rflh

k follow from equations 47, 148

and

149, respctively. -If required,- the wetted

-surface 'of the canoe body Sb'

can

_{be determined}

-approximately from equation 52. The form

factors

and kr follow from equations 50 and 51, -res-pectively, while the wetted areas S, and

_Sr

are approximately.: Sj ₌

Sr =

2(Tr_Th) and (514) (55) + 5 1.565x1Ó6 1 .519x106 10 _1.356x106

i .308x106

15

_l.191x106

_112xlO6

20

1.056x106

1 .007x106

25

0.9458x106

_0.8965x106

30

O.8528x106

0.80L5x'106

in which Th

and Th

are the averagehuJ.l draughts

at the locaions ofrthe keel and rudder, and Ck

and Cr are the average chord lengths of keel and

rudder. In calculations for a keel-trim tab

aridi

or a skeg-rudder configuration,the entities with

a k and/or r subscript are replaced with the

sane entities for the combined keel and trim tab

and/or the combined skeg and rudder

configura-tion, with the ktt and/or sr subscript,

respec-tively. For example, the wetted surface Sk is

replaced by the wetted surface of the combined

keel-trim tab configuration Sktt.

3.2. The Wave Resistance

On subtracting the viscous resistance (or the

approximate-flat plate-equivalent thereof) from

the total measured resistance the so-called

re-sidual resistance RR is obtained, which is

main-ly the resistance associated with wavemaking R.

The residual resistance is often expressed as a

fraction of the displacement weight; i.e. as

RR/. Many compilations of RR/-values have been

published for different types of hull forths.

U-sually these residual resistance values have been

derived from model tests on systematic series of

hull forms. Sometimes these RR/-values have been

correlated with the hull form parameters which

constituted the associated hull series, either

graphically or numerically. An ovérview of

avail-able data for small, high-speed displacement

vessels,

usefu.l for making resistance

predic-tions, is presented in reference 27.

A particular successful correlation of RR/-values

with hull form parameters, for 970 data points of

93 differeit hull forms of small craft, has been

derived by Van Oortmerssen [28]. Van Oortmerssen

used: amultiple regression tehxiique to obtain

an analytical expression for the residual

resis-tance. The basic expression adopted for the

resi-dual resistance was derived from Havelock's work

on the theory of the wavemaking resistance of a

twO-dimensional pressure disturbance at forward

speed, having peaks at the equivalent stem arid

stern positions of a fictitious ship, and a

pressure minimum in between [29]. The range of

parameters for which the coefficients of the

ba-sic expression are valid, are as follows:

- waterline length L

between 8 and 80 metres;

- displacement volume V between 5 and 3000 cubic

metres;

-- length--beam ratio L/B between.3 and 6.2;

- beam-draught ratio BIT between 1.9 and 14.0;

- prismatic coefficient C

between 0.50 and 0.73;

- midshipsection coefficient CM between 0.70 and

0.97;

- longitudinal centre of buoyancy LCB between

-7% L and

+2.8% L forward of Q.5L;

- half angle of entrance of design waterline iE

between 100 and 146°.

Because of the theoretical nature of the basic

expression, some extrapolation beyond the ranges

of values given above is permissible

particular-ly for the lesser significant terms in the poparticular-ly-

poly-nomials constituting the coefficients, such as

for LCB, -BIT and CM. The speed range covered by

the 970 data points is between Froude number

values (based on the waterline length) F

equal

to O and 0.5 (equivalent to a range in V&

be-TABLE IV Values of the coefficients in equation

58 for C1, C2, C3 ad C14

between O and 1.70). The basic expression is as

follows:

BR

.,. .,-2.

T

= C1e

-2

2

+ C e'' .cosF2

in which m =

o.11435C2.19

and the coefficient C1,C2,C3 and C14

mials as follows:

C1

+eC+a8C+a9B/T+a10(B/T)2+a1

1C

(58)

in which the coefficients a0through a11, for C1,

C2, C

and C14 are given in Table IV.

tn this

e-quatin Cw71pL/B, where

is the half angle of

entrance.

ofthe design waterline in degrees.

The values of L

B, T are all based on:the design

waterline for the upright condition (at zero

heel), while C

is the prismatic coéf-ficient and

the ma.ximumrsection coefficiént, both based

on the canoe body only. LCB is the longitudinal

centre of buoyancy of the canoe body, in percent

of I, forward of the midship section.

The calculation of the wave resistance R,,

as-.sumed equa], to the residual resistance BR for the

upright condition, then follows from:

Bw

:2

..-:

:.

cos ø.

where RR/

follows fron equations 6, 57 and 58.

t Is the displacement weight in the same.units

in which theresistance force. is to be calculated..

In the SI (Système International d'Unités)

sys-tem of units, force and weight are expressed in

Newton (N). If the volume of displacement V is

expressed in m3, then

pgV100614V Newton. In

equation 59 it is assumed that the wave resis

tance at a heel angle 8, increases with 1/cos 8,

following the practise of Myers [30].

C3e''. sinÇ2

+

are

polyno-aj

C1x103

C2x103

C3x103

Cx

a

79.321

67114.9

-908.144

3012.1

a1

_-0.09287

19.830

2.5270

2.711414 a2

_-0.00209

_2.6700

_O.357914

_0.25521

a3

-2146.146

_-19662

_755.19

-9198.8

a4

187.114 114100

-148.940

6886.6

a

-1.14289

137.314

9.8687

-159.93

a

0.11898

-13.369

-0.77652

16.236

a

0.15727

-14.14985

3.7902

-0.820114

-0.000614

0.02100

-0.01879

0.00225

a

-2.5286

216.145

-9.21440

236.38

a0

0.50619

-35.076

1.2857

_414.178 a11

1.6285

-128.73

250.65

207.26

3.3 The Induced Resistance

3.3.1 The Basic Equation for the Induced Resis-tance

Lift is generated by deflecting a flow over an angle a. downward (or sideways) from its undis-turbed .irection. The force generated by the body which induces (deflects) this flow is directed at approximately right angles to the direction of the deflected flow, as shown in Fig. 15. It follows that the component of this force, F sin a, then acts against the direction of motion.

This

force is called the induced drag force B1, because it is associated with the induced flow

fiel& From Fig. 15 it follows that R1=F sin = = L tan a. Hence CR1= CT tan a It can be shown that the induced flow angle a1 is related to the lift coefficient and the aspect ratio an-cording to:

CL

vAR (60)

therefore, CR - (61)

since for small a-angles, _CRTCTC*i.

Use of

iu-tion 2 (for a keel) together *itE

RIkpVCRIAcôse, leads to:

2 2

R1pV

hARk k PVAkcose.rAB,

-AcosG

k (62)

where tfollows from equation 13;

The induced resistance of a sailing yacht can then be determined on adding the induced resis-tance of the hull, keel (and trim tab) and rudder (and skeg), as follows:

R ---( + +

'T pV hrAOcosO.ABh hrAkcose.ARk LT2

+ (0.8)2vArcose.AR ) (63)

Fig. 15 Induced drag: a component of the lift force.

where t _, Liand I follow from equations 39, 13 and , respectivly.

3.3.2 Effects of Planform and Sweep Angle of Keel and Rudder on Induced Resistance

Equation 60 is strictly only valid for an ellip-tical lift distribution over the span of the lif-ting surface. The planforms of present-day keels and rudders rarely lead to an elliptical span-wise loading, however. Appreciable -increments in

induced drag are found in planforms that are either extremely tapered or close to a rectan-gular shape. For taper ratios between 0.3 and

an

elliptical spanwise loading is nearly obtained. In that case the additional induced drag is very small (about i or 2%). For other values of the taper ratio the following

expres-sion can be used,.which fits the results given by Hoerner [26]; the expression--itself is va],id

for the keel: V

RÇ

_{71Akcose.ARk.(}

STAk V

+ 0.095A- o.o'oÀ)AR,)

(61) where = root with

= chord length of the keel at the tip V

(bottom), and V

Ct = chord length of the keel at the hull-keel intersection.

It should be noted that whereas eliiptièal or rounded planforms might be advantageous in mini-mizing induced drag, they also lead to a reduc-tion in the total lift. As discussed in secreduc-tion 2.1.3, the effective span of rounded planforms is less than that of rectangular planforms. A consequence of this fact is that rectangular plan-forms often lead to;the highest lift to drag, ratios.

The effect of sveep is toincrease the loading. near the ti of the lifting surface. According to Hoerner [26], a sweep-bank angle of 30° re-quires a taper ratio of about 0.15 to obtáin near-elliptical loading (instead of about 0.35 for zero sweep-back). Since such taper ratios are rarely .Vractiea]. (exdept in delta configurátions) it follows- that the spanwise loading of svept-: back lifting surfaces is not often

near-ellipti-cal, leading to somewhat higher induced dxag values Also, the lift force of eachchrdwise segeent Vof the lifting surface' apohi thèV V

tip is tilted further "backward" becausèofan V

increasing deflection of- the induced flow, lead-ing to larger flow angles a1 (see F1g. 15). It follows that because of this the component of the lift in the direction of the undisturbed flow. becomes greater with increasing sweep 'back.

Ac-S

cording to Hoerner [26] the induced drag :increas-es proportionally with sweep angle according to

1/cosA, where A is the sweep back (or forward) ' angle of the quarter-chord line of thel lifting -.

-surface. Such an increase in induced drag, how-ever, is never found in experimental sailing yacht studies. It would appear that the increase

in induced

_drag

due to sweep is

nearly

comple-tely compensated for by the favourable influ-ence of sweep on the wave resistance

ioJ.

The

final

expression for the induced resistance (for the keel) is therefore

assuméd

to be given by equation 61k.

3.3.3 The Final Equations for the Induced Resistance

For a keel-trim tab and skeg-rudder

configura-tion, the total induced resistance can be

writ-ten as:

R =R1

+.R1

+R1

'T

ktt sr h

there

Akttcose.Att)(1012

+ R1

sr

- 0.057A

+O.O95À2-O.010A3 )AR

sr

)

(67)

sr

sr

-

0. O57À.+O .095Att_0 9.tttJ

2

sr

P(o.8vB)2

cos8.AR

)(1(o.o12

+

sr

sr

(65)

In the induced resistance calculation for the

hu.).l (i.e. the

canoebody),

the effects of taper

ae neglected. The values for

-.

"T _r

and LTh follow from equation 16, 21

an

39. When a trim täbisnot fitted

tò

thekeel,

or when 6tt=0, thè indüced resistance of the

keel becomes as given by équation

6l.

Tor the

case of a rudder

alone

(without a skeg),the

induced resistance of thé rudder is:

r2

2

r

)(.1+(o.o12

+

I

- .. 2

irAcose.AB

-r

_(o.BvB)

r

r

-

0.057A+0.095À2-O.0140A3)AR)

(69)

3.b Resistance Calculations for 5.5 Metre Ycht

"Antioe" and Comparisons with Results of

Mea-surements

-

-From the datagiven in referénôes 15

and16th..

following values, required for the calculations,

were derived

!

T Sh =

_{114.80 - 3.80}

= 11.0 n2 A

_ktt

=

1.90 n2

-= 3.oO

Tkt

-= 1141

n...-

V tt =

(3.03+1.61)/2

= 2.32 m

(t/c)ktt= 0. 16/2.32

=

_0.07

tt

1.61/3.034= 0.531

V =

1.003x10

m2/sec

The valu

of the Froude and Reynold.s numbers,

based

on the waterline length of 7.141 n, for speeds of 2, 3,

4, 5,

6,

and 6,5 knots

(at which the

measùreménts

vere carried out) have been

cal-culated by Letcher [16]. Usinj these Rn-values, the frictionaj. resistan....coeffièients of the -hull

can

be.. èalculated by

means

of equation 146.

The frictional resistance coefficients of the keel, baséd on an avéiage length of the keel

of.

2.32 n, cañ be calculat

Th. the ..sane

manner The

form

factor of the )èel thid rudder configu-ration (the

rudder

is

again

considered to be a trim tab) is:

V

(66)

kktt= 2(0.07) + 60(0.07) = 0.14114 so that

-1 2 V

R.=PVB

(CF xli + CF xl.11414x3.8).

-h ktt

The

results of

thecalculations for

_{CFh, CFt}

RF re

éive 1n

Tble.V The..ca.lculation o

he

wave resistance can

best béstä.rted by first de-.

ternining the values of C1, C2, C

aridC

in

equation

56.

This is carried out ty

multiplying

the values of. theaóefficients a0

thróugh a1

i

(68)

(given in Table IV). by the values of LCB, LCB2,

Cpa,

etc., as prescribéd by equation

_58.

The

results of these muj.tip.lications arg given in

-Table VI. Also,

inQ.11435(0.51+)219

=

_0.5560.

V

"

-ô.Ö6vr7F2

Hence, -

= 0.003709e

i...n +

,

2

-+

O.55O524e55n

+

e-2.

-

0.182252sin(F2)eSSFn

+

O.037332cos(F2)e55'n

- --n..-.-

..-.-.-..--...

For the Froude number values corresponding to

the. boat-speeds-already -used--.for tbe- frictional

resistance calculations (see Table V) -the

corres-ponding values of RR/h have been

calculated

and

are given2iTäble

VII ,

together with the values

ofRR/pVB. Note that:

RH ER

23127

ER 145.126 = pv :

Ç1)

VB2

To calculate the inded resistance it is

neces-sary

to first deter]nine the side forco?i the

canoe

body and on the kèel-trim taoigurat ion. ]V

Table II only the combind sid-'force is

given. In section 2.14 the side 'force-of-the keel

_--

ç;-,.-____----V

an

trim tab: conftgura-tion..was..-calculated .to

-be:

.2

12.936.cbs O (B

+

1ktt

B 1.812

+(1..O.688(sin(0.621cosO))°25))

+0.754/1.

622

cos2O

+

14

Ch

=

0.306

Vh

=2.30m....

=

_{2.3x1025x9.81N}

23127 N

C

0.567

-Cph

=0.514

Cp

= 0.691: LCB =

.2.2

-. 1E =

200; Cjj

20

X

14.23 =814.6

Ah

3.05m

and

R1=

QV2B Ahcos.O.ABh

and the side force of the canoe body vas found to

be:

L=pV(O.

82148cos28+14.28282cos6)

With AR, t=0.725c0s0 (see section 2.14), it fol-lows tha%: 2

R1=

(1.9o.T25cose10012_00303+

+ 0.0268 - 0.0060)0.725cós0) or 1 2 1+0.0018 cosO

RIktt2

ktt 14.3280os20 or PVB 2 Rlktt _{ktt 1+0.00l8cosO} 1 2 - 1 2 2 14.328 cos2O PVB (pvB)

Likewise for the canoe body, with

h3.05

= 0.2206 cose

v(3.b.5

cosO')o.2206 cosO or RIh

pV

- (pV)2

2.1137 cos2O In Table VIII the calculated values

f and LThp\1, and finally

RIh/2PVB

RIT/PV, are

given according to the equations g3.ven above. The calculations have been carried oùt for the various heel angles,drift angles and rudder angles pre-viously considered in Table II.

The results of the measurements, as presented by Letcher [16], are given as resistance coefficients, using the total upright projected lateral area of 14.95 m2 as the reference area. On following this procedure, the total. resistance coefficient

becomes': RT .RF+RW+RIT

pVA

pV _14.95 where

i =

2 (see equation 59) PVB PVB O

For the data points already used inTable II, the. results for the total resistance are given in

Täble IX. , ' '

The comparison of calculated and measured coeffi-cients of total resistance is in most cases good. For some cases, notably for some of the larger drift and rudder angles, the resistance is over-predicted, which would indicate that at those

drift- and rudder angles the induced resistance is too great. To check this, some additional com-parisons were made at zero drift- and rudder angls, as shown in Table X. By comparing the

calculated and measured values of the total re-sistance coefficient shown in this table, it can

be concluded that no large systematic errors oc-cur in the calculation of the total resistance coefficient for zero heel angles, zero drift an-gles and zero ruddér anan-gles. This demonstrates that indeed the over-prediction of the total

re-sistance for relative large drift and/or rudder angles (particularly for the data points No. 8, 9, 10 and 11 in Table IX) is due to the over-prediction of the induced resistance in these cases.. Since the side force is predicted quite accurately for the data points concerned (see Table II), it follows 'that the errormust be in

the formu.lation of 'the basic induced resistance equation itself (equation 61).

The accuracy of the total resistance prediction is nevertheless quite göod. Not including data points No. 13 and 30 ('for which the measured va-lues are probably subject to relatively large errors), the average (absolute) difference be-tween the calculated and measured total resis-tance coefficient of the remaining 32 different data points coveredin Tables IX and Xis 6.7%, distributed' as follows: 14.7% for zero or very-small drift and ruddér angles; 14.0% for drift angles of about 30 and zero rudder angle; 4.9% for drift angles of about 30 and rudder angles between O and 30; 12.1% for drift angles of about 7 and rudder angles of O and 6°. No discernable correlation in the distribution of the errors can be found with heel angle.

14. CONCLUSIONS AID FINAL REMARKS

The method for the calculation of the resistance and side force properties of sailing yachts,

pre-sented in detail in this paper, yiélds results which compare well with full-scale test data on the

_5.5

Metre Class Yacht "Antiope". The side, force is predicted with an absolute difference with measured values of about 4.9%, while the total resistance is predicted with an average absolute difference with measured'yalues of about 6.7%. Only in t'he case of relative high drift

and rudder angles can a consistent (over-predic-tion) error be discerned. Evidencè can in that

case be found which would indicate that for rela-tively la.rge'drift and/or rudder angles the

cal-culation of the induced resistance leads'to va-lues which are about 10 to 15 percent too high. Nevertheless,'the predictions for some 34 consi-,

dered combinations of boat speed, heel, drift 'and rudder angles for "Antiope" can all be

con-sidered, accurate for most practical purposes.' Even though the accuracy of the' method is only demonstrated for "Antiôpe",' it has been found to be applicable to other yacht forms as well. Satis-facto±7 comparisons' of calculations with model test data for 12 Metre Class Yaòhts andwith the

test data published by Gerrïtsina et al L3111 for the so-called Deift Series of Yacht Forms, have been obtained. This is probably due to the

rela-tive wide applicability of the basic equations adopted and,, particularly, to the successful tuning of the coefficients of these equations to fit a vide range of appropiiate experimental data.