On the maximum thrust of sail by sailing close to wind

ARCH1EF

On the maximum thrust of sails by sailing close to wind.

by J.A. Sparenberg and A.K. Wiersma.

Summary.

The maximum thrust of sails when sailing close to wind is discussed. The in-fluence of the gap between sails and hull and the inin-fluence of a reduction of the

heeling force and the heeling moment, on the maximum thrust is considered. Numerical results are given.

1. Introduction.

From the point of view of the racing yacht the primary objective is to achieve

a large driving force and simultaneously to reduce to a minimum the harmfull

heel-ing force. This creed which is to be realized by designheel-ing the sails of a yacht in a suitable way, comes from the book "Sailing Theory and Practice" by Marchaj [i]. Here is used the concept heeling force, however it is clear that perhaps even more dangerous is the heeling moment which in essence is independent of the heeling force. We will try to elucitate some aspects of this principle by discussing a

special case.

We consider the sails only and not the properties of the hull, keel and rudder, which are of course to the same extent important for the performance of a yacht. We assume that the sails deliver their driving force by "static" action and do not discuss dynamic effects caused by the motion of the yacht in a see way or by

the more artificial action called pumping.

The specific situation ve will describe is sailing close to the apparent wind. The angle between the direction in which the driving force is defined and the

direc-tion from which the apparent wind comes is assumed to be a small angle of 0(c) where c is a small parameter with respect to which the theory will be linearized.

Hence we consider a limit case of which the range of applicability has to be

checked by experiments.

Important for the driving force created by the relative motion of sails and air is that the resistance forces are small. These can be listed as induced drag, friction drag, form drag and additional resistance of rigging. Here we will con-sider only the induced drag, which as is discussed for instance by Miigram [2),

[3] is an important part of the total drag of the rig.

The dependence of the maximum thrust, which is 0(c2), on a number of para-meters is the subject of this paper. Important geometrical parameters are the heeling angle and the gap between sails and hull.

Technische Hogeschool

The heeling angle, when it becomes too large is detrimental for the good

perfor-mance of the yacht, hence it will be of interest to investigate what happens to

the maximum thrust when we put a constraint on the heeling moment and on the heel-ing force. This kind of constraints will also appear when a yacht is considered as a system with two types of wings moving at the boundary of two media with a relative motion. One type of wings, the sails, are protruding in the air, the other type, the keel, in the water. Then of course for a stationary situation

there has to be neither a resulting lateral force nor a resulting heeling moment on the yacht, hence there is a relation between these forces of the sails and the keel This will be considered in a following paper. The influence of the gap will be dis-cussed by simulating in a single manner several values for its width.

Some ideas have been used in practice to diminish the induced drag of sails.

The only way to do this is by spreading out the shed free vorticity in an

appro-priate way. We mention the tall hollow masts by which a higher aspect ratio of the sails can be obtained and a wide boom which has to prevent the flow around the boom caused by the pressure differences across the sail [i].

We do not take into account the dependence of the velocity of the wind on the height above the water surface. When these variations of the wind velocity are small 0(c), they only have in our case of sailing close to wind an influence of 0(c3) on

the thrust and hence can be neglected. When they are finite hence of 0(c0) then

the concept of trailing vortices is lost and a more complicated theory ís needed [4). These variations will cause a still stronger decline of the maximum thrust by heeling, than follows from our numerical results.

At last we remark that this article has not the claim to be applicable

directly to the analytical design of sails, it gives a rigorous linearized theory for sailing close to wind and intends to enrich the background already available for more practical considerations.

2. Formulation of the problem.

We consider a coordinate system (X, Y, Z) which is in rest with respect to the ship. The air is in the half space region Z > 0. The relative velocity

of the wind makes a small angle ct with the X axis and has the magnitude U. The angle is assumed to be of 0(c) where

Û,

"T

Fig. 2.. Lifting line and free vortex sheet.

e is a small parameter with respect to which the theory is linearized. Because

of this small angle the vortex sheets of fore sail and main sail are separated by a distance of 0(c) and hence can be considered to coincide. In a linearized opti miEation theory, lifting surfaces can be replaced by lifting lines, which can

be chosen at an arbitrary place at the vortex sheets. Hence also the lifting lines can be assumed to coincide. So we replace the sails of the ship by one lifting line (A - B) through the origin O and in the (Y,Z) plane with a

heeling angle , O < ß < ¶12. The free vortex sheet H stretches downstream from

(A - B). Because the undisturbed main stream has only a component of O(e) per-pendicular to the X axis, H can be chosen within the accuracy of the linearized

theory parallel to the X axis.

In order to simulate the boundary between air and water, we consider as usual, also the image (' - ) of (A - B) and the image of H, both reflections

are with respect to the plane Z = 0. We define the + ( - ) side of H and to

be oriented in the positive (negative) Y direction and the unit normal vector on H and on ' from the + side towards the - side. On H and Ii' we introduce a Cartesian coordinate system (X,$) where on H the coordinate s is the distance

of a point to the X axis, while on this coordinate is minus this distance.

The boundaries of H are denoted by s = a and s = b, with O < a < b and of by s = -a and s = -b, The width of H and will be denoted by . = (b-a). We assume the whole space to be filled with air of which the incoming velocity for Z < O is the same as for Z > O. The air is assumed to be

in-compressible and inviscid.

The driving force or thrust T

(O(E2))

of the sails is defined as the component in the X direction of the total force acting on the bound vorticity of the lifting line (A - B), it is reckoned positive in the negative X direc-tion. It is the intention to determine within the realm of a linear theory

the maximum thrust Tm.

It can be made plausible that an upper bound exists for T. Suppose we have

at (A - B) and ( - ) some bound vorticity distribution vr(s) Cv is some

factor O < y) reckoned positive with a right hand screw in the positive s direc-tion. We remark that by symmetry

F(s) = F(s). (2.1)

The thrust T becomes

b

T =

p5

' F(s) { - U c cos + y w (s)} ds.

(2. 2)

where y w(s) is the component of the induced velocity in the direction

at (A - B). When y increases starting from ' = O first also T increases, how-ever the induced velocity yw(s) also becomes stronger and will more and more

counteract the component -U cos until for some value O < y the thrust T is again zero. Hence it can be expected that a maximum value of T exists for the

F(s) under consideration. When we assume that an optimum F(s) exists the thrust will have for this distribution ari optimum value denoted by Tm.

We define next the heeling force F acting on (A - B) as the force

perpen-dicular to (A - B) and the incoming flow, reckoned positive in the negative direction. The constraint that F has to have a prescribed value Fh which in general will be chosen smaller than the value which belongs to Tm has the form

Analogously the constraint that the heeling moment has to have a prescribed value Mh, which is reckoned positive when it is connected with a right hand screw

to the negative X direction, is

a

M = - p U .1 (r+s) r' (s) ds (2.4)

b

where r is some constant depending on the line with respect to which the moment

is calculated.

We now change the formulation in such a way that we obtain a problem of energy extraction out of a slightly disturbed fluid. Consider the Cartesian

coordinate system (x, y, z) which is related to the former system by

x = X - Ut , y = Y , z = Z

. (2.5)

The air is in rest with respect to (x, y, z) except for a small homogeneous flow

of magnitude

U (0(c))

in the y direction and for a negligible flow of 0(c2) in

the x direction. The lifting lines (A - B) and (X

- '')

then move in the negative

x direction with velocity U, along the strips H and ,on which from now on we use the coordinates (x, s).

The only way in which (A - B) and (X

-

) in a linearized theory sense the homogeneous flow in the y direction is by its component - U cos in the di-rection. The most direct way to induce this normal velocity field is by placing

at H and 11' a suitable vortex layer parallel to the x axis of strength y0(s). We

reckon y(s) positive when it is connected with a right hand screw to the positive

x direction. We have

y (s) = - y (-s)

o o

(2.6)

In order to define this vortex layer uniquely we assume that its total circulation is zero and hence has a finite kinetic energy around it per unit of length in the

x direction.

Then we can consider the following problem. We have a fluid at rest with re-spect to a coordinate system (x, y, z). At the two strips H and i, now considered

to be two sided infinite, we place vorticity y(s) of 0(c), which creates the

de-sired normal velocity - U a cos . Next the lifting lines (A - B) and (X

-coming from x = + move along H and in the negative x direction and have to

extract, under the constraints (2.3) and (2.4) as much kinetic energy as possible

When there are no constraints (2.3) and (2.4), the solution to our problem s very simple. The bound vorticity r(s) has to be such that its trailing free

rorticity annihilates the free vorticity y(s). Then all the kinetic energy is :aken out of the fluid. The meaning of this latter result for the original

pro-dem is the following. The bound vorticity of the sail has to leave behind

trai-Ling vorticity such that at the Trefftz'plane (far down stream) it induces a iormal velocity at H which is opposite to the normal component of the original Elow, the surface of the water has to be considered as a rigid boundary. Of :ourse this condition is tightly connected to the condition for the minimum induced resistance of a wing.

At last we remark that the above described optimisation method has been :hosen in favour of a more direct method based on (2.2), because it gives a good insight into more general problems of energy extraction. Consider a fluid with a Lime independent velocity field (x, y, z) of O() caused by a vorticity

distribu-tion y (x, y, z). In this fluid is present a smoothly curved closed or periodic

strip H along which a flexible wing W can move. The question is what part of the

kcínetic energy can be extracted out of the fluid by W. The procedure is then

analogous to the one followed above. Consider the strip H in a fluid at rest. Place at H vorticity which induces a velocity field (x, y, z) with the same normal

component at H as was induced by Ç (x, y, z) and which has zero total circulation. then the kinetic energy belonging to Ç and not more can be extracted out of the fluid. The wing W has to have bound vorticity which leaves behind free vorti.city

of strength

-3. The variational problem

Consider the incompressible and inviscid fluid with respect to the Cartesian coordinate system (x, y, z), in which we have the two sided

A

Fig. 3.1. The two sided infinite vortex sheets H and H.

infinite sheets H and 11. On these strips we have vorticity y(s) = - y(-s) with total strength zero

b

y(s) ds = O , (3.1)

which induces a velocity field (y, z) with a normal component - U COS From (3.1) it follows that outside H and 1(y, z) can be described by a uniquely defined potential which has a jump across the strips. We write

= (O, y1, w1) = - c U cos grad p1 (y, z) (3.2) where (y, z) is a potential function with hm (pj (y, z) = O for y2 + z2 -,

and with (p / n = i at H and î!. Along the strips move the lifting lines (A - B)

and ( - î!) which have to extract as much kinetic energy as possible out of the

fluid, taking into account (2.3) and (2.4). When the lifting lines have passed and are "far away" at x = - the kinetic energy per unit of length in the x direction has changed by the trailing vorticity y(s) of (A - B) and ( - î!) left behind at

H and H. Outside these strips the velocity field (y, z) induced by the trailing vorticity can be described by a potential function (p (y, z)

= (O, y, w) = grad (p (y, z) (3.3)

which also has a jump across H and

î!.

The kinetic energy per unit of length in

>x

the x direction becomes

E ((p) IP

f

f

{ (-a U 6 P1

)2

+ (-a u cos 6

p1 +p

)2 } dy dz, (3.4)

- -

y

y

z z

which is a functional of the unknown potential (p

From (2.2) it is easily seen that the circulation for a certain value of s around the sail in the case of optimum driving force and no constraints is pro-portional to a U. The heeling force and heeling moment in that case are propro-portional

to a u2. We now denote by p M1 and p M2 the heeling force and moment when a =

and U = 1. Then the actual heeling force and moment, again in the case of no

con-straint, are a U2 p M1 and a U2 p M2. Using the potential (p the constraints (2.3) and (2.4) can be written as

b

K. ((p) 2 p. a U M. +

f

f. (s) I p (s) ds } = O, j = 1, 2,

-b3

where

Ip(s) ]

= p

(s) - q (s) , (3.6)

the indices - (+) denote the values of the potential at the - (+) side of the H and p. is the fraction of a U2 p M. which is tolerated and

f1 (s) 1 , f2 (s) = r + (3.7)

The interval - a < s < a gives no contribution to the integral in (3.5) because

there [ (p J vanishes.

Introducing Lagrange multipliers a U p cos B 2, where the factor

J

a U p cos ß 2.' 31 is only for convenience, we have to minimize the functional

2

G ((p) = E ((p) + a U p cos B E 2,(1 - K. ((p)

j=1

(3.5)

(3.8)

In order to be acceptable in our problem the potential (p must be symmetric with respect to the z coordinate and one valued in the whole (y, z) plane,

especi-ally when H or is encircled. The one valuedness of (p follows directly from the

fact that the trailing vorticity of the lifting lines each have total strength zero.

Consider a perturbation ó (p of . such that also c (p satisfies the conditions

[6 (p

(5)

]₊ = [6 (

(-s)

]

Now we take 5 (p j(s) zero everywhere except in the neighbourhood of s

= s and s = - s with a < s < b. That these boundary values of 6 (p on H and IT can be

ex-tended in the (y, z) plane by a one valued potential function 5 (p (y, z) and hence are admissable as a variation of (p is seen as follows. Consider for instance at H

two "disturbance vortices" of opposite strength, parallel to the x axis, each at a

side of s s and close to it. The same is done at IT for s = - s where the

vor-o o

tices have opposite strength of the corresponding vortices at H. Then the

poten-tial S (p (y, z) outside H and IT belonging to these four vortices, is symmetric and one valued in the whole (y, z) plane.

Using this S (p we find from (3.10) as a necessary condition for optimality

(3.11) (3.12) (3. 13) r X2 (p (y , z) = a U cos { (1 + A1 + A2

-) (p (y, z) + -i-- .p (y, z)} . (3.14)

necessary condition

+cO +

SG

= p

_{f ¡}

{

(-

au cos (p1 + P)---ó p+ (- aU cos p1

.?__5p}dy dz +

co

y

_az

2 .

+ U p cos $ E (1 -

3x.

j f.(s) [ 5 p (s) ] ds = O . (3.9)

-b

By partial integration of the two terms in the first integral, using the fact that

.p and are potential functions and combining some terms we can reduce (3.9) to

f

{ -

a U cos p(s) + ap(s) - a U cos E Z - A. f.(s)} [ S p (s) Çds = o

-b n Bn j=1

(3. 10)

where because of the symmetry

Bn

B (p(s) = o. U cos (1

+ + 2 + A2

r

kL)

at H and IT, where we used the boundary condition for and (3.7). We now

intro-duce the potential function (y, z) which satisfies the boundary condition

Bp2

- si ,

Using (3.12) we find for the total velocity component w in the direction of

at the lifting line (A - B), because then the trailing vorticity is "only half

infinitely " long,

w(s) = - . U cos ß + p (s) = a U cos (- I

+ + A2 r + isi) ) 2.

hence the thrust T has the value

b

₂₂

b 2 T p

_f

Wn (s) [ .p (s) ds p a U cos

f (-

I + X1 + A2 (r + s) ). a a (3.16) { (1 + A1 + A2 r )[p (s) ]

+22

[2(s) ] } ds. £ 2.

The Lagrange multipliers A1 and A2 follow from (3.5),

M1 + cos

_J

(1 + A1 + A2 .) (s) J +

.-2 (s) ] } ds a 2.. 2. b 2 M2 + cos

J

(r + s) { (1 + X1 + X2r)[(p (s) j X2 2 (s)J

+

-2. _£ where b M1 = - cos

_f

(s) ] ds b M2 = - cos $

_f

(r + s) (s) ] ds

In order to obtain dimensionless formulae we introduce the potential functions

(y, z) and

2 (y, z) by

(y, z) 2. (P (Ly, £Z) j = 1.2, (3.21)

These functions besides on y and z depend only on the dimensionless

parameters and a _I 2.. The equations (3.17) and (3.18) for A1 and A2 become by

using (3.19), (3.20) and (3.21)

f

{ (I- p1+A1+X2 r) [i

(s)1

+ X 2 (s)J )ds = 0, (3.22) £ (3. 15) = O, } ds = 0, (3.17) (3.18) (3.19) (3.20)

a

*1

2.

f

(r + s) { (1 - 12

_{1 «2}

r)[i4 (s)] 2 (s)] } ds = O. a 2. 2. 2.

From (3.14) and (3.21) it follows that the optimum bound vorticity has the representation F (s) = o. U cos ß 2.{ (1 + A1 + A2

_-fl

+ x2 (s) ]), a < s < b. £

r

2. = - p2 p a 3 j (r + s) (s) ] ds. a/2.

Ï

We now discuss shortly equations (3.22) and (3.23). First suppose p1

= p2 =

then we have no constraints, which is in agreement with the fact that A1 = A2 = O satisfy these equations. When there is one constraint for instance on the heeling force, then we have only the multiplier A1, we then put X2 = O , neglect equation

(3.23) and calculate A1 from (3.22). Then the heeling moment _Mh has to be calcu-lated by means of ( 2.4) where

r

(s) from (3.24) with A = O, has to be used. When there is only a constraint on the heeling moment, we neglect equation (3.22) put A1 = O and A2 follows from (3.23). The heeling force _Fh then follows from (2.3) where F (s) follows from (3.24). with A1 O. This latter case will be dealt with

explicitly in the next section.

(3.23)

(3.24)

The thrust T,the heeling force Fh and the heeling moment can be written as

+ 1

(3.25)

T = P o.2 U2 cos2 ₁ (-1 +A1+A2(r+ s)){(1 + A1 + A2 r)

kb1(s)]

+A2[2(s)]}ds.

a/2. 2. 2. (3.27) Fh - P1 p a u2 cos 2 j

[i

_(s) ] ds , (3.26) a

4. The integral equations belonging to and 12. = -'2 p a u2 cos

f

s [p1 (s) J ds. a

+1

2. (4.4) (4.7)

The heeling force Fh follows from (2.3) and (4.2)

Fh - a p U2 cos (s)] + X [*2 (s)J } ds. (4.5)

We now introduce the functions a

+1

r

m I. (a, )

= J

s [i4. (s)] ds,

j

= 1,2 ; rn = 0,1. (4.6) a 2.

Then we can rewrite (4.1), (4.3), (4.4) and (4.5) as

We consider in this section the case of only a constraint on the heeling moment. Furthermore we take the heeling moment with respect to the origin, r = O. Under these assumptions we have to neglect (3.22) and (3.23) becomes

f

s (1 - u2) [(s)]

2 [2(s) ]

} ds = O (4.1)

a

2.

The equations (3.24), (3.25) and (3.27) specialize to

r (s) = U 2. { [ (s)] + A2 (4.2) 2. 2. T = p 2 U2 cos2 2 (- + s) { [ (s)] + A2 2 (s)] }ds, (4.3)

22

2 2

pa U

cog 8 - _{U cUs p2} - - U2 2 _{{110121 -} - P2) 111120 121 J-

_i

(s) ds = O, a L 2 j = 1,2.

which the solution of (4.12) becomes unique. Integration of the vorticity yields

a

+1

9.

[ip. _(s)]

= +

X

'r

(s) ds

e the function I. ( a, ) (4.6) follows by an other integration.

j 9, (4.9) (4.10) (4.13) i = 1,2 , (4.14)

The problem is now reduced to finding i (s) and

2

(s), then in a simple way

relevant quantities are defined by the I.. This is done for (s) by

calcu-ing a vortex distribution (s) at H and which induces a velocity with nor-component "1" in the direction and for

2 (s) a vortex distribution '2 ch induces a normal component s

' in both cases a < s < a ₊ . Because of the L £ imetry we have (s) = - y.(_s) ', i = 1,2. (4.11)

vorticity is reckoned positive when it is connected with a right hand screw the positive x direction. Using Biot and Savart's law we find the singular egral equations

a

+1

y. (s){ I - (a + s cos 2 ) } ds = g. (a), a <a< a + I , (4.12)

(a-s) 2 2 - 9. - - 9.

9. (s + 2 s a cos 2 + a )

re g1 (o) = I and g2 (a) = a. Because of the one valuedness of (y, z) and

(y, z), we know that the total circulation of each strip H or i1 has to be zero,

5. Asymptotic expansion of maximum thrust for narrow gap.

We consider the case of a mast perpendicular to the water surface ₍ = O) with

no constraint. It can be simply argued that in the case of a narrow gap O < a « 1,

I

we have in the neighbourhood of this gap a very complicated trailing vortex system

in the optimum case, which for too small values of a will not be computable by numerical means. In order to obtain insight for whin small values of a

_I

numerical results are still reliable we give in this section some asymptotic

considerations.

-f3:')

A A

?ig. 5.1. Two dimensional flow problem for the potential

.

We have to determine the function (s), where s stretches along the z axis.

ience we have to solve the two dimensional flow problem around two strips as

irawn in figure 5.1. This can be done most easily by conformal mapping as is

indi-:ated in [5] page 244. The resulting complex flow potential w(z + iy) (z and y are

real variables and i is the imaginary unit), becomes

(z + i y) 2 2

w(z+iy)-i J

(q -T)dT

O

{((a2

_{- T2)((} )2 2)} L (5.1)

le square-root in the integrand is defined such that for small real values of r it

s real and positive. The quantity q2 is determined so that the total circulation

tround one strip is zero, this yields a

+ I

L 2

2)dT

=0,

J

(q

-r

a 2

a2

a - () )(( + 1)2_ T2)}2 (5.2)

L)

'tS

T T

where the integration is along the real axis. Equation (5.2) defines q2 as a function of a

_I

9J Introducing the functions

L1 ()

=

J

T dT a (a)2)((a _{+ 1)2} -and a

=1

dT , (5.4) \ 2 {(T2 (a)2)((a )2 2 1

i

-

T)}

which are closely related to Legendre integrals {6], we find L

2 '

q2 (a / L) =L1 (a /

i) I

L (a / i)

From (5.1) and the definition of the square-root we find s

2 2

(e)]; = Re (w (s - iO) - w (s + iO))= - 2

J (q

- T )

dT (5 6)

a 2

{( (a)2)((a

)2 2)

}

Hence we find by (4.3) with À2 = O and a change of order of integration, for the maximum thrust (8 0) T = 2 2 2 (( + 1)-

T)

(q2 - T2) d.T a i {(T2 (a 2 a 2 2 -

r

) )((_+ 1)

T )

i

The numerator of this integrand is a polynomial in -r of third degree, hence

the integral can be split in four parts. Two of these according to (5.2) cancel

each other while the remaining ones have very simple results [ 6 ]. Using the

re-presentation (5.5) for q2 we obtain from (5.7) the following expression for T

222

2

a

T = T _{p a U i} {(I + 2 a + 2 a ₎ - 2 L _(-a) / L _(.) } . (5.8)

I p

Expression (5.8) is still exact. However we can find by using expansions given in [7] the following asymptotic expressions

(5.3)

(5.5)

Fig. 5.2. Asymptotic and numerical vaiues

222

POE U no constraints.

in figure 5.2 we have plotted numerically calculated values and asymptotically values of the maximum thrust. In the interval 10 < a

/

_L < 1O, both methods

give practically the same results within a relative error of about 10_2%. It follows from figure 5.2 that the maximum thrust increases very sharply in the neighbourhood of a

/

0. For a narrow gap with a

/.

10, still about 20 %

of the maximum thrust T = ir p

2

u2 L2 which is possible for a

/

p, = 0, is lacking.

In table (5.1) we give further information about this extraordinary behaviour.

of

2T

';

L1 ()1

+

-

ln + (ln2 - + in ) (5.9)

t2 ()

-

in + 2 in 2 + in + (1-2 in 2)- 5 2ln

2(7

+ 5 in 2)+

O(3ln)

4 4 2 (5. 10)

10

222

log a / i

2T/pa U

i - 10 1,4421 - Il 1,4532 - 12 1,4625 - 13 i ,4705 - 1,5708

Table 5.1. The asymptotic behaviour of the driving force for a- 0, = 0,

6. Numerical results.

In this section we give a number of graphs which show numerical results of the previous theory. In figure (6.la) is drawn the maximum thrust for the heeling angle = O, as a function of the width a

I

Q of the gap between "sail and deck"

and of the heeling moment constraint coefficient p2. On the horizontal axis is

0, o,

,

ç

-00,,

5 -/

3,

Fig. 6.1. The dependence of the maximum thrust on a

I

and on p2.

a) = 00 , b) =

450

plotted a / on a logarithmic scale which is appropriate because of the sensiti-vity of the maximum thrust for narrow gaps. At the right hand side of the lines are given the values of p2. At the T axis are denoted the values for a / = O. In figure (6Jb) are given the same results but now for 450 In fact the value a / is to a certain extent an ambigous measure for the gap because for

large heeling angles it becomes large even when the distance of the lowest part of the sail to the water surface remains the same. However seen as a distance of the sail to the deck it is more correct. Anyhow it is good ro realize that a / 9 for $ = 00 and for

450

is not quite the same. It is seen that due to reflection properties of the water surface, the values of the functions drawn in figure (6Jb) are greater than the corresponding ones of figure (6.Ia). This increase however is entirely counteracted by the factor cos2 which is

In figures (6.2) and (6.3) are given analogously the values of the heeling LL f, s-0,8 o,6

o,ç

0,11 0,2. I_7' I i i -.---., o 'L -S---L

>

/

L i L

-

--/

3

-

-i

o -.S

-'/

-3

-2- -/ 0

'o'l

Fig. 6.2. The dependence of the heeling force on a

I

Q and on

a)

=0°

, b)

3=45°.

force and the heeling moment. The heeling force is in this linearized theory the force acting on the sail and is up to a component of O (c2) (the thrust)

perpendicular to the plane determined by the lifting line and the x axis. In figure (6.3) the lines for 1 follow directly from the line for

'2

= by multiplication by p2.

It turned out from our numerical calculations that the values given in figures (6.1), (6.2) and (6.3) for = O differ only by at most 6 % from the corresponding values for = 300. This confirms a statement of Miigram [3] that mostly it will be sufficient to consider the case 00.

In fig. (6.4) we give the optimum circulation distributions

r

(s) around

the lifting line as a function of the dimensionless coordinate s, a < s < a + 1, along the mast. Two cases are considered a / = 0 and a / =0,01 both fr

= 00. The numbers inside the graphs denote the values of p2. Again the very strong dependency of optimum quantities on a narrow gap is evident.

In figure (6.5) are given the distributions which multiplied by an appro-priate factor have to be subtracted in the case 1, from the

r

distribution

-1/ I &

1f

-

-3

-2. _:,

;

_7'

'°4a1 -S -1 -3 -2. -I O

Fig. 6.3. The dependence of the heeling moment on a

I

£ and on p2.

a) = 00 , b) = 45°.

s

o) $

o,6

04

-II

o,8 0, , f I 0,9 o,8

-02

Fig. 6.4. The dependence of the optimum circulation around the sail as a function

of s and p2, = 0. a) a / i 0, b) a / i = 0,01.

o,

References

Marchaj, C.A. Sailing theory and practice, Dodd, Mead and Company, New York, 1964.

Milgram, J.H. The aerodynamic of sails,

7th Symponium on Naval Hydrodynamics, 1968.

Miigram, J.H. The analytic design of yacht sails,

Annual meeting, New York, N.Y. The Society of Naval architects and Marine engineers, 1968.

Von Karman, T and H.S. Tsien, Lifting-line theory for a wing in a non

uniform flow,

Quarterly of applied mathematics, Vol 3, no 1, 1945.

Betz, A. Konf orme Abbildung, Springer Verlag, Berlin, 1948.

Gr6bner, W and N. Hofreiter, Integraltafel I and II, Springer Verlag, Wien, 1973.

. Jahnke, E. and F. Emde, Tables of functionswith formulae and curves,