A method of calculating the lift on submerged hydrofoils

NORWEGIAN SHIP MODEL EXPERIMENT TANK

THE TECHNICAL UNIVERSITY u,- NORWAY

A METHOD OF CALCULA TING THE LIFT

ON SUBMERGED HYDROFOiLS

by

Harald Aa. Waiaerhau.g

NORWEGIAN SHIP MODEL EXPERIMENT TANK PUBLICA TI()N N 71

NOVEMBER 1963

LIST OF CONTENTS

THE 3 DIMENSIONAL HYDROFOIL

ABSTRACT

Page

i 2

NOTATI ON te OS0eøO0S 0@0000 Q t 0QQ Ott Cotto

THE 2 DIMENSIONAL HYDROFOIL FUNDANENTALEQUATIONS

THE POINT V'ORTEX . . . 6

¶t'IiE DIPOLE , O O O Q t O O C O t O O QOOtOt 9

THE HYDROFOIL

,,,,,,,,,,,,,

... 10

CIRCULATION OF THE HYDROFOIL . . . ... . e.,. 18

SUBSTITUTION BYA VORTEX 20

SUBSTITUTION BY A VORTEX AND A DIPOLE ,. 22

THE HYDROFOIL AT HIGH SPEEDS . 23

THE CIRCULATION REDUCTION FACTOR ...., 28

COMPARISON BETWEEN THE SYSTEMS

OF SUBSTITUTION o o o t . . o o t e o t e e s o , o o o o o _37' CHORDWISE DISTRIBUTION 0F CIRCULATION .. 1i8

ANALYTICAL MODEL OF THE FOIL ... . . . 59 MOMENT OF DIPOLE DISTRIBUTION _{000 63}

THE DOWNWASH VELOCITY 66

DESIGN OF A HYDROFOIL 73

ANALYSIS OF A HYDROFOIL ...O

8)-i-EXPERIMENTAL PART

THE DYNAMOMETRE . . .. .,. ....,,, .. 90

EXPERIMENTALSETUP

₉₅

T:: - :::::NsIONAL IZDROFOIL 95

THE) DIMENSIONAL HYDROFOIL lii

DISCUSSION

0t.

00

0 tO O O 00000 116

12)-l.

ABSTRACT

Applying Kotchin's

[1]*com1ex

velocity potential for a vortex in the vicinity of a free surface and the corresponding potential for a dipole developed in an analogous way, the complex velocity potential for a 2 dimensional hydrofoil has been found by substitution of a vortex at the approximate centroid of

circulation, alternatively by substitution of a vortex and a dipole

at the centre of the hydrofoil. The two ways of approximating the

hydrofoil have been discussed. The strength of the vortex and the moment of the dipole have been determined by satisfying the Kutta

condition at

3/k

e. Expressions for the circulation reduction due to the free surface at finite and infinite speed have been

found and streamlines have been drawn by substituting the hydrofoil

with a) a vortex, b) a vortex and a dipole, both cases with and

without gravity terms. Further the hydrofoil has been approximated by a vortex sheet and the circulation distribution and lift on the vortex sheet have been calculated. Some values have been computed

on a digital computer.

An analytical model has been proposed for the

3 dimensional hydrofoil applying the results obtained for the

2 dimensional foil. Further an example of lift calculation and analysis is given for a ₃ dimensional hydrofoil.

A dynamometre suitable for hydrofoil testing has been designed, and a series of experiments have been conducted to cheek

the validity of the theoretical results.

*

See list of references.

NOTATI ON

a radius of Joukowski. circle0

b span of ₃ dimensional hydrofoil0

e chord of hydrofoil.

g acceleration due to gravity.

h distance from undisturbed water surface to centre of hydrofoil.

k circulation reduction factor for 2 dimensional flow,

koo

circulation reduction factor at infinite speed.

i factor in equation of Joukowski transform0

q resultant of free stream. and induced velocities.

r radius0

s distance from transformed circle to centre of Joukowski profile. length (span) of ₃ dimens ona1 foil measured along foil

centreline.

t thickness of hydrofoil. time.

V total induced velocity vector0

u x-component of induced velocity.

y

y-

I? It ft

w z- ¡I II Vt

complex velocity potential in z-plane0

WD " of dipole0

WV 't It Vt vortex.

A a/i.

Fh depth Froude number,

u/Va

K0 g/U2.

U velocity of undisturbed flow along x-axis, W complex velocity potential in '-plane,

incidence relative to chord line.

.4 It It II line of zero lift, corrected for

3 dimensional effects, i0e.

=4g

i4g incidence relative to line of zero lift.

dihedral angle0

' (x) circulation along foil chord.

6 sweep back angle.

*

J

displacement thickness of boundary layer.

E downwash angle due to ₃ dimensional effects0 b 9 angle defining coordinate y through y = - cos

3

kinematic visc4tty.

9 mass density.

T1 angle between foil chord and tangent to foil mean line at

point Zje

angle defining coordinate x through x = -(l - cos

r circulation.

in unbounded fluid. at infinite speed. velocity potential.

THE 2 DIMENSIONAL HYDROFOIL

FUNDAMENTAL E QUATI OMS

We cDnsiier an ideal, incompressible and homogeneous fluid with a free surface which is horizontal when undisturbed. Recti-linear, rectangular systems of coordinates are selected with the x-axis in the horizontal plane of the undisturbed, free surface and

with the y.-axis directed vertically upwards.

A hydrofoil is assumed to be moving under the free surface on a straight, horizontal course and with constant speed U directed

along the positive x-axis

The axis of coordinates x , y are assumed to be moving with

the hydrofoil, and the axis x' , y' to be fixed in space.

Hence

XI = X + Ut ,

y'=y

when x' y' are coinciding with x , y at the time t = O.

When an irrotational motion is assumed, we may define a

potential for the absolute velocity:

4,'-

'(x',y',t)

4,

(x'l./t,y')

,

and hence:

òx

ò24'

-

¿j2 O2e ?t2

òx

For waves of small height and slope the dynamical condition at the

free surface is:

(3) = ò4' / ai? ec

.,_,1ò?p

_(x,O),

"òx

d

y}/ hi/ /,(C fè(c.

(4)

_at

ò'

_ày

-

ò4

ày

when we define

¿,y_

From (2),

₍₃₎

and (4) we obtain:

Ò77 ¿jZ _ò24'

9

àx

=-,

or: òy

(5)

with 5

and the kinematical condition

_L_oz4

_-o

k

òx2

òy

K-B o '.7

jj'a

I ,4 _c7d«

1jjhe/

r1$c

wheny= O

The boundary condition on the hydrofoil surface may be written:

- _{¿I cos}

_(nx).

The velocity components must remain finite when

x2#y2co,

and at infiniti in front of the hydrofoil there must be no waves, i.e.:

1/fl,

)2

(

)2}

-*--1' 00

We now introduce the complex variable

z = X ¡y

and the complex velocity potential

where is the stream function, and ¿1=- _ay ¿1=

y ax

-Since

di,'

ò+.òIi

dz - / ÔX

=4;ì&_

òx

we obtain d2w ò24

_{- ¡}

dz2

òxX òyòx

'

and hence the condition (5) may be written

mn{} im(

}o

when y = o ,

or

(8)

'n

fiì;'

=o

_{when y = O.} THE POINT VORTEX

We consider a vortex of strength r/2crr at the point

under a free surface. For this case the complex velocity function is holomorphic in the entire region yO except at the point

z = , and may be regarded as built up by the complex velocity

of a vortex in an unbounded fluid together with a certain perturbation velocity due to the free surface, thus:

a'k#'y,C_

/ +

z-where g'(z) is holornorphic for yO. Referring to

₍₈₎

we introduce the function:

E(z) = ¡

_dzZ

dw,

dz

which for yLO may be written:

f

f

- 2Ti (z-0)2

# ,9(Z)

where is holomorphic for

yL

O. From

(8)

we find that f(z) has real values on the x-axis, hence by Schwartz's reflection principle it may be continued analytically into the region y> O,

with:

In the region y.O we have:

(9)

1(z)

f-

(z."z0)2

j;

_Z

and in the whole plane:

1(z)=¡

¿2,

g dw

dz2

dz

r

/

-2

.r_.

i

¡A."_r

211 (z0-0)2

2-7e

2iT (z-z0)2 ° 2iT

z-z0

with the condition:

¿in, ft'z)O.

Z-oo

The homogeneous equation

dwv

/ _dz2

_0dz

has the solution

WV C/

c)t

't"

and making use of Lagrange?s method (variations of the constants) for the solution of the inhomogeneous equation, we write:

(io) We take: dC,

dC

-/Jc'z (II) _{dz dx} e

=0,

hence by derivation of (10):

dw

d,

dC

_{k,z_;j(cC_i4z}

dz a'z

= - ''

C

e4'02

and:

--

e'

dC

e1'('Z

-K02_(2e

-

/

_{° dz}

Substituting in

₍₉₎

we have: -M;' z

r

i

+ (zz)]

r,

t

i

(i4)

1e0c,/2 _e

L(z0)2

02'_i_

_1.z-

z-z0J

from which we obtain by partial integration:

z

(15)

C2eZ_

L._t_"

7' /

\i.i_Ce_'C'oZ Çei('ot

K0

2Çf2-,

_z-go)

IT

j

t-0

¿t

+00

when

òm C,"tim

2

Z--?co

2___._

d_

LI_

t

J

d

K0 2

_{[(Z_Z0)2(2_'Z0)2]}

ZZ0

hence:

r

/ f .

r

K0 2Ti' z-i0

-0)

2iT

ZZ0

and

from (io),

(15) and (17):

z

j.

,

r

e0Z

(

;JY

dt.

''2TÍ og

2-2e

'rr

o

From (12) and (15):

dw.

r

₍

_{1_}

_/

2e'ot

S

t-z0

19)

d _{-, 2i \z-z}

_z-z0

THE DIPOLE

Defining the complex velocity potential of a dipole of

moment M in

an

unbounded fluid

D

2ì(z-z0)

we may write

in analogy with (9):

We have:

C

e'0Z

(20)

/ dz'

'° dz

i, aw,

'

1e e

and taking: we find: (21) and hence:

(2)

_{VD _ Zr} IO

ti

e'9

.,L/

1t'.// e9ek'.z

2 e

z-0

2í z-z0

t-z0

00 z

(2k)

If

e

fi

c-9

_¡

N

e

,

%2,?

_{e'g -}

¡/(0 Ç 1k:

dt.

-

2'ÎT

(zi)2

W(zz,)z

'T z-z0

ii

j

tz0

THE HYDROFOIL

It may be shown (see f.ex.. ) that at sufficiently large distance from a Joukowski hydrofoil, the flow is that due to a vortex and a dipole both at the centre of the hydrofoil,

i

alternatively a single vortex, the substitutional vortex, placed at the centrold of circulation.

We shall consider these two systems of substitution

more closely. Referring to Fig. 1, the circle V'-ika is

transfonned into a Joukowski profile by:

dC

dC+

e"=O.

dz

dz

N

I

e (22) _Ç#

eZ

/ 2rh

_Lz

z ___

dt

-zo (25) lo -1..d_..

1e'

-io

-i ₁

io

e

e

-/

2'Th.L(z-0)

i_z)2j

_2'iT1z-0

.

z-zò

J

e'0

1f

e'

_;

''"

)2 _{(z -Z0)2] T}

_{z -}

_i0 ejr

or:

z1*V_42

2

With this reversal the usual expression for the complex velocity

potential in the -plane of a circle with circulation:

L/ = (le '°' ¡

2T1 -s)

or, since s for thin and comparatively straight profiles is

very small: ¿/e4Ca2 (27) 2q-r ¿O9.( (26) may be written: li

-FIG. i

-. r 12 -, -/ 2 Ue a ,

(/ec

(zv/z2 4L/Z,)#

2'#Vz'2_4c12

2

t

r

_¿

_2]

2'Tt

For a hydrofoil moving under a free surface, it is more convenient to use the axes x, y of Fige 1. We then have:

Z'- Ze'

and:

e'

_(ze°'# /_2')#

2¼a2

.

iogze1#i2(

2

ze'vÇ'

Z2'fT

211a2 r r .' V- ae12 #1 j z ' / 2 # Ic' 2 2 /1 ¿'e

where the constant K may be omitteth Equation (29) may also be

written:

2V_Ì2i2

/1O9

(30)

w=k#

z

"e) yt2_2/2oc)

z z

/

2 .

f-)

I liC 2ee z z

Writing: (32)

¿o-2

21T we obtain: (31)

20Z

{i_

and when I.

'?ií./(a2-1'e

¡2.0

r'

,

(t

/_(2e14')ì

is small, we may write

,2Ç1() (at_/2ei2oC)

r

13 t', ¡ t_il

.-

. r

/ (2/eYi

i

z

V'vP';

¡

log ', {e 2 ,2G'

4' (2/e')Z

2 ext-i

rz1f#/

(2/«.

₎

-f_(21e)2

;2u(a/zei2

L 2

r2

For a flat plate in an unbounded fluid, we have

r=

'-i-r ¿IL sin oC e'

and consequently

20.leb0C _,

When z is large, we obtain the usual, approximate expression for

(33)

L./1Jz#i1og

VOR7TX

Writing

(3k)

k'=

we find a virtuel centroid of circulation for the system vortex +

dipole: (35)

_{; - z}

{ I-exp -;241¿('02_12e ¡2 J

rz

VORTEX *D(POIJ

which may be compared with:

1k

-which is at the quarter point of the hydrofoil, and which has been used as a centroid of circulation by several authors.

The exact equation:

/

121e" )2

¿1(a_I2eo)

1'l/t-

z

(30)

w=lJz#i

_2fl'

2 4/2121C 2

(XACT

may now be compared with the two approximations:

/_2e/2C)

z VORTEX # DlPOLi r ,2TrV(o2 12.c)] 12

r

z 20

-

_{2 [}

I

f'.k4' (21iøG)zexp(i4(rr1/(a2_Zze ¡2.G)

2

_rz

EXACT z

¡2J(a2_4')

_r

VORTEX

15

-o

16

-As a practical example we choose

i/ /C ¡21e aMd /e

and consider the foils defined by:

i = 0.025 a = 0.0275 oc 20 =

i = 0025

a = 0.02625 0

0c=

2 »3 = i = 0.025 a VT 20 /3 = k) 1 = 0,025 a t? I O

= Lf3

T? 5) 1 = 0.025 a 20 /3

The 5 foils are shown in Fig. 2.

Further we define the circulation

P=zj 4'uia tJsìn,4

where k = i in an unbounded fluid, and where k#1 in the

neighbourhood of a free surface. As an example we choose k = 057

TABLE 1:

Centroid of circulation

E

= Exact equation

VD =

Vorte> # dipole

a

I

,

'

Substi-1

_tutfon

_{zil e'}

Observation

_{z=i21 e}

point

z

_zai4ie'

lQ

E

o.ôo5O.008

o.òc6*zO.o/5

Ò0C+

O022

V*D

_{O. oo3 -t-}

. ooi

o. cog ,

_0.025

ö.cc8'+ ¿

ô.o

V ô. O/i -t-

i. 0.032

. °1/ # ¿ 0.032 C.c// -t-

¿ 0.032

57

_{_VD}

E

_{O.00/ -}

o. o .'.

¿ o.oi

_{¿ 0.023}

0.007 1- ¿ o.02g

_{ô0o5 * ¿ 0.035}

0.0/0 -0.01/ -e'-

_{¿ 0044}

0.039

V O. ô/9 -i'- ¿ 0.c.57 O, 0/9 -t- ¿ 0.057

0.019 i- ¿ Ò.57

29

E

Ô. cI -+ ¿

O. 024 O. ocG i

¿ 0.O3

0.0/2 -

¿ O,o5

VD

-c ô, + ¿ 0.025

0. ö

4 ¿0.04e

O.b// -

¿ OO9

V

0.037 + ¿

0.1/2

.0.037 -

0.1/2

0.037y-/ 0.1/2

7.0

E --

¿. ûoo/

o.009 -t- ¿ OooS c.Ó1ô-t. ¿

c.oic

0.ôo-t- ¿ 0.0/3

O.00-t- ¿ 0.O/5

0.0/o + ¿ cJ.O/5 V 0.01/ ,'-

¿ ô.017

o.ô/f

¿ oöí7

o.oi/

. ¿

.0/7

2E

.57

ò.o(i

-t- ¿ Q.00 0.0/2 .,'-

¿ 0.0/5

o.ö,5

- ¿ 0.021 V+b o. ôo,

¿ 0.0/9

0.0/f -i'-

¿ 0024

0.0/4 -t- ¿ 0.025 y

_{ô.öi.9 -t- ¿ 0.029}

o-dg

-t-

¿ 0.029

0.0/9 -t- ¿ 0.029

'29

E -t-

¿ O02o

C0/4 -t- ¿ 0.032

0.022 .'. ¿ 0.042

V*D

O. oo2 # ¿ &025

0.01/ -t. ¿ Ô.O39 0.021

0.047

V C. 3g -t-

¿ ôo5g'

0038 -t-

o.o58 ô.03e

¿ 0058

3

1.0

E

0.0/2 - ¿ 0.014

00/2 - ¿ 0..oío

0.0/2 - ¿ C.005

Vt-D

0.0/i

-t- ¿O.003

0.0/2

-t- ¿C.ôo2

0.0/2+ C 0.00I

V

0.0/2 -f

¿ O,öoo

0.0/2 + ¿ Oôøö

0.0/2

¿ Ö.ôa.o

57

I E

0.02o - L O.O/

0.020 - ¿ 0,o

0.o2o ¿ 0.003

EVD

o.o/g + ¿ O.ôo9

d.02o # ¿ O.005

0.02o -t- ¿ 0.002

0.020 -t- C coot

V

0.020 + ¿ OooI

0.o2ö + ¿.C.ööJ

29

E

0.ö35 -

0003

0.037 -t- ¿0.004'

o.04o ¿ 0.002

VD

0.ô2G -I- ¿ 0.027

0. o3, -t ¿0.017

0.039 -- ¿ o.006' V

0.o4o + ¿ Ooo/

O.04o -t. ¿ 0.00/ ô.c40 1- ¿ Oôö /

4

7'Q

- E

ó023- ¿ O.00

0.024 - ¿ 0.004

0.025 - ¿ 0.003

V+D 0. 02o -t-

¿ 0.0/3

0.023 * ¿ 0,009

0024 .t- ¿ 0004

V.

0.025

_{¿ O.ôo2}

0.025 * ¿ O,002

Ö.025-f--i 0.002

'57

-E

0.034 -# ¿ OÖoê

0.039 * ¿ Oco6

0.042+1 0.00G

VtO

0.023 + ¿ 0.030

0.037 ¿ 0.02/

0.o4/ +

V

0.044 -

¿ Cô3

ao44 -t-

¿ c.003 0.o44 -e.- ¿ 0.003

'2.9

E o.03c -i'- ¿ oò4

0.055 * i O.o44

0.o73+C 0032

Vi-D

-0.o,4 -

¿ o.c

c.c44

- ¿ 0.059

0.072 -i-iÖ.o4o

V e'.o96 + ¿ Ô.ÖO C.og +C O.oÔG C.Og -t-COOoG

5

1.0

E

O.c24 - ¿OOo7

0.0.2

-00.005

0.o25-'..0.002

VD

0.. 02/ + ¿ 0.0/2

0.024 -

e O.007

0.024 + ¿0.004

V

0.025 + ¿ OôôJ

O,.Ö25* 00o/

0025.t- ¿

'57

E O. o3 -t- C 0,. oô9 0. o4c - - o. o42 c o. co5 . Vt-O 0.02-4 -t- ¿ O.03i 0.038' -t- ¿ O.oiQ

0.042 ¿0-0/1

V

O.e44

O.co2

0.044 ¿ 0.062

oo'i4 + ¿0.002

E

0.032 -i- ¿0.049

0.059 ,'-

0,043

Ö..ö7 * ¿ C.03o

29

V+D

- o.o/o -t ¿ oo4g

0.047 -i- C 0059

0.075+ ¿ 0.037

18

-With the given values, Table 1, of z0 has been computed.

Some information may be obtained from this table:

In an unbounded fluid (k = i) , substitution by a vortex and

a dipole at the foil centre as well as substitution by a vortex

at the approximate centroid of circulation, both give a good

approximat1on to the complex potential at a distance from the

foil at least equal to the foil chord.

As we approach the hydrofoil,substitution by a vortex seems to give the best approximation to the potential for infinitely thin hydrofoils, whereas substitution by a vortex and a dipole gives the best approximation for hydrofoils of some thickness.

The quarter point may be regarded as ceritroid of circulation

only for infinitely thin and straight foils.

In the nei:hbourhood of a free surface k1 , substitution

by a vortex and a dipole at the foil centre seems to give the best approximation to the complex potential at all distances

from foils of finite thickness. For infinitely thin, straight

or curved foils, there seems to be little difference between

the two systems. Substitution by a vortex at the quarter point is in no case permissible.

The effective centroid of circulation of a hydrofoil In the vicinity of a free surface is not the same as that given by z0 in (31), (32) or (35), since the image system will influence

the velocity distribution. This will be shown later by substituting

the foil with a vortex sheet

CIRCULATION OF THE HYDROFOIL

In an unbounded fluid the circulation around a

Joukowski

profìle may be evaluated in the -plane of the

Joukowski

circle

by making a stagnation point the point which transforms into the

trailing edge of the airfoil, I.e

the point -le"

Strandhagen and Seikel ¡2J applied a related method for evaluating the

circulation around a flat plate hydrofoil under a free surface They substituted the flat plate by a vortex at the i/k point,

le"

, and satisfied the Kutta condition at -le"'

- 19

In this paper we consider the flow around a Joukowski hydrofoil of finite thickness and in the vicinity of a free surface as approximately equivalent to the flow around a dipole and a

vortex of suitable strength at the centre of the foil together with

the images due to the presence of the free surface. In analogy

with the case of unbounded flow, we now regard as "undisturbed" flow the free water stream together with the disturbances due to

the image system. The circulation is found by making the point

on the Joukowski circle a stagnation point, we then consider the perturbation velocities at -le1" in the zplane instead

of in the plane. This may be done since the perturbation

velocities at leC re approximately the same in the z-plane and

the ' -plane. By derivation of (25) we find the velocities in

the z-p1ane

dl'

dz dl'

dz

dw

Hence the flow at some distance from the dipole and the vortex,

i.e. for large

'//

_, will be approximately the same in the

z-plane and in the r-plane. We shall discuss this at the end of

the paper.

The considerations above will of course also be valid when the hydrofoil is substituted by a vortex.

For evaluating the circulation we apply the formula

(36)

P=4ia,

Sin

where q is the velocity due to the free stream and the image

system, ,4 is the absolute incidence and the correction to ,3 due to free surface effects, see Fig. 3 The velocities u and y on Fig. _{3 are the components of the velocity induced at}

-le' by the image system,and since they are small compared with the free stream velocity U , we may use the approximations

or:

41Ta(U-u)(/3*

)=krra«

From the last equation we obtain:

4

(-4ç;

iì )

SUBSTITUTION BY A VORTEX

Approximating the complex velocity potential of a hydrofoil in an unbounded fluid by:

r=br',

,

20

-'- ¿Iu

I,,

¿1-For thin hydrofoils the angle ,4# is small and

1cin

(/3#

/34.

With these approximations (:36) may be written:

y

Denoting the circulation of the hydrofoil in an unbounded fluid

by

Ç

, we define

and putting

- ¡2'rru(

a-Le

2 I2OC₎ 1/, z0=

r

-

-i2'TWe'a2- l2e12' ) , 20

_r

we obtain with reference to (19) the complex velocity due to the

image system: (40) 2' dz 2T( z ,.,, 21 eìA'e

dt.

¡h

(42)

" = i

_-

_'

_-

_e'

b 4a ¿j'

(-le 'A)

e

dt

For thin profiles, we may use the approximation

and

20 -;2r(1J

(I - éc)

= ¿Si»ol

e

k

or:

(ki) _z0., e'°'# iii

With this 1.st approximation we obtain for the complex velocity

at -1e'-ih due to the image system:

-le hh1c/h

¿11 J2T(V(a2- ¿2e-/2.

r

*14

and

VJrr7

we may evaluate the circulation reduction factor k of

(38)

using

(21.2).

SUBSTITUTION BY A VORTEX AND A DIPOLE

If we approximate the complex velocity potential of a hydrofoil in an unbounded fluid by (see (34) ):

(43)

_w/z#i-2-1ogz

¿/(a2_12e121C)

z

and put

('ti

we obtain with reference to (19) and (2k) the complex velocity

due to the image system:

z

dw,

_{=.;_[_}

/

dz

_2i

_z-1h

_t-;A

dt

£

_L._ e'8

_./k'

N

e

,'2.JL. i9e_'0Z

Ç ,'4Y

2í (z-1h)2

O IT

_z-1h

o

\ tii

di',

or, at the point

-1e'-ih

- ¿'e ' ,A

dw

_{- ¡}

421/a srnS

dz

- le

°- ¡ZA

k, k 41/a 8ífl/

iÁ'

(-le ''-1it) Ç

_g

_dt

f-liz

cO -le '/A ík,t

dt

¿Ja2

¡K

21/a2

K22

¿Ja

zj

14 (-le 'i. ¡h)

('.1e'"-l21')

-le

i'd_124

k

.-e

Ht

-

-23-../ '-í1i

ic' 2012e -20C

I

e

IkOt

_dt.

(-te ''-2h)2

-'

°

_{-c '-i2A}

t -liz

00

This is a more convenient expression than when evaluating the circulation reduction the hydrofoil by a vortex and a dipole at

foil.

THE HYDROFOIL AT HIGH SPEEDS

From (18) and (23) we find the complex velocity potential

for the system vortex + dipole at the centre of the hydrofoil, i.e. at = -1h

z z ¡h

r

- f14, z C

(46)

wUz-/f.tog

co

N

e'9

N

i'9

#/1_ejo2

C e°t

29T

z*/h

2T1 z-1h

O9

\ t-i

dt.

At high speeds when

the complex potential approaches:

(47)

M

e'8

ÌY

e'8

'2'íT

Z.'ih

2T z-,h

j-2w

(42),

and we shall therefore, factor k , substitute

hydro-With the approximations: -/e''-/2A--- 121z and denoting

I

/

fl

f. =

-we obtain from

₍₄₉₎

'A

(50) a

¿í"kM

-4

P s2oc' - i

c

82

1'

i T ços2ac - T

- 7.42

-Á voo= 2k -or:

t,= ¿'z .

¡b 2 ¿la gi,/

¿09

(z ¡h) ¡k 2

¿Ja sin/3 log (z -

¡h)

lía2 ¿Ja2 tí2eI)G

jj/2

e

z#ih

Z-lA

zìA

.

An expression for the complex velocity at the point

-le°- ¿k

due to the image systems is found from

(45)

dw...; A2Ya ¿J&

25

-The circulation reduction factor is from (39)

j=i

.a_,

V

¿1 _¿1,4

and using (50) we find the following expression valid at high

speeds:

(52'

A00--

z(

_{n-),' 4}

_{coi2oc(ßt t,f)'}

LA

i' o

8

2(Á

_i)

has been computed for several values of A , h/c , o and

4 and the result Is given in

Fig0

.

From (11.8) and (52) an expression for the streamfunction

is easily found by introducing z = x + iy :

(6.3)

Vyik2Ua s/n,4 ¿n

y/2*(y,.h)2.A

2Va sin,, ¿n

Vx2(yh)2'

=/

¿la2- ¿l/2o2

_(yi'h)

*

¿la2-E1/2,2

_{(y h)}

5

= X X2 L (y'h)2

¿/'21/fl

2 2 X

26

-FIG.4

1.0

5

o .3 10

-

W--.-

_-=_-=-=:--kUi

_r

A=1'l

/1'

/

t/c''11

-[3r43

/I'\\

o 11f .5

10

_h/c

15

ILI_________

A.

1/f t 5

U.

A =1.

1-ro 1f ft

/c

ii

__________________

Zero lift

Û 4

lo

_h'

_{___}

1.5

¡C

- 27

the ₅ parts of which may be written implicite like:

(54) _y (.55) x2#

(y#h)2

-

(ep4 )2 )C:Z _{,' (} 42)2

-(eXpA _{21/a s/n/s /} ' ( U2sin2oC 2 _{#12 #} (ai4_2a2/2os.2ot) 2L) , (59)

*

[

/z

-

-

(a2_I2as2oc)}=

(a#/-2a2/2c,s2oc,).

When the circulation reduction factor for the foil at high speed is found from (52), it may be more convenient to substitute the hydrofoil by a vortex at the appropriate centroid of circulation

expressed by:

(5Q')

/ (a

2-í'e

i2)

h

-

A2as/n/3

At high speeds we obtain the complex velocity potential:

,ia*__¿2 /2oC)

(60)

_{-Uz*iÁ2(Jai/n/J1og}

{

42a

.

#-ik,211a

6/fl/ ¿

[

and the stream function:

we find

28

=(Iy

,'Aoe2tJa sfn,,4 ¿a

y"('x ¿'im2oc 2

/

a2-c'2co.c2a

Áoe2a

5th)

IY

2a

# k 2 ¿la g/n,/ i

/(

¿25in 2 c 2

/

a212co52

Áoe2 a ( Y 2a _GInA

h )

2 =.

or

W' ¿1 '

(

¿2.9in2o '\2

a2-tcos2oc

\2

/

Áoe2a /n,,4/1 Aoe2a

(/?,) -Çe

(6k)

l25jn2oC \2

(

,

*

a2-Ico2c '2

/

2

Aoe 2a 2 a = (e

oe2aUin)

THE CIRCULATION BEDUCTION FACTOR

Substituting the hydrofoil by a vortex and a dipole, we shall find a general expression for the circulation reduction factor

k , and consider first the integral: z

(65)

T=e0hS eY

dt

Introducing

¿ =k('h)

)2

J

(66)

7=e_2K0

oo

e'

_da

D,r #iZ/-4)

or, with

C X

y=-Á

(67)

r=_e21co

e"

-K0 -;21t,Á

The integral (67) may be divided in

3

parts like (see Fig.

_5):

29

-/00 -24'0h

(68)

T = -

Ç e

du

_S Q/U

au #

S e¿L

du

-i2K0h

-/2Kh

e2Koh

The last integral3 III, is equal to zero9 and the remaining 2 may

be developed as follows:

(69

i

If#14

'urther:

-0-2N,A

I, = -

du

u

du

-

-/2K04) * Ûc'o -i2kA)

=

';(ì'f "

Ci(í24Ç i»

and: -4

-2K0/z

;=-/Ç

_'1'

du

-/20h

-12123 -2K0h Ç _{03 ¿}

_{du .}

_{/ Ç} sÑ u

_du

u

-

-i24 h

-

h

-2k;,fr

-e

31

--S;(-K0 -,2X0A)

- /3'!

(i

;'4 L

In integral II we substitute

V-/L(

hence

r &'

-2AÇIz

- E«JK0h

)#/ iT,

and consequently:

T_ie.21'0'1 {C/(/(,f*/2A'0h»C/(/241z)

[C,(*k/# i2Koh) Ci(/21c',h)

32

-With the approximations

_1e"_/2#4sf -/2,4

and

I/fl

S#/4

we find from

_(k5)

the induced velocity at the point ('.- - 1h)

due to the image system:

.- -/4

(7k)

42/Ja_-/21 D

Ç eAt

¿t

)

t-1h

00

--i,

¿la2

A',21a.L

_#

2(a2e16"f et

(-

-,2h)2 /

/2k

t -1h

dt

Co ¿/Icoo2oc

/ 2cJtc,

K,2//2o2oc

(_J2%,)2

(_/21z)2

/ KD22O1eCOs 2oCe''

'

C

--/2h

dt

CO

-i-iA

#

K2/2oe'0 (-J-1h)

Ç

e

jt-,½

dz'.

With the abbreviations:

FREE SURFACE

Line

of zero

lift

FREE 5'L/RFACE

zero

!1.

/3 = 4.3° const.

h

lo

Line

of

t/c

O

-O.O6

-uO'l1

4.3

3.15

2

-

--.4

U9uIPuuI!IPuII____

_i

_.____

Ii

3

4

5

8

7

8

9

lo

F

15

-

35

-E ,4s/n X - co.s

gin X 2C

CO.9 X

T4Z

cos

A o2c

C06

)2'c

.5/fl A

6 =sin À *,,4co A

N =,. .in ).. -

CO.5 A

,,_ì-i

AL

L

and the approximation

¿fl%d,

the circulation reduction factor derived from (51) and (Vi.) may be written down in the following form:

(76)

A

_32",4BLAAß2

_{[C,FeTmI/JmT]}

#

#A

[L Co62 c,3j

#2A2ß2

[(Çt*r),peTt (,F)Jm TJJ

With K0 = 0, (76) reduces to (52). (76) bas been evaluated for

several values of a/i , h/c , U , oC and , , and the result is shown in Figs. 6 and 7.

The calculations were performed as shown in Table 2, and for the smaller values of K0h, the following approximations

-

36

-TABLE 2

CALCULATION OF CIRCULATiON REDUCTION FACTOR k

Ex.: A=11

c=2°

®

.020

©

si/C

©

A01/

.0(67

-2.58

((7

ii J

(2K0h)

-2.60

[C(c2I<6h)]

-2.64

® rí-

_{' ]} 1.57

®

R.[S(1\.+26h))

02

©Irn[

J

o

®

-5.18

© D-

2.54

©

®++T

473

®

©-©

© exp(-2K0h)

®

-2.4i

©

LmTx

®

/c)2+o.o63

₂₃

®

(h/c2 0063

297

®

01G7

©

coX

y

®

A2 x

0202

®

co2o xJ

01G7

®

2o<. '

Oo/!7

®

f?

'00/25

®

A2

1210

©

.99g

j::i

2o

-070

©

«-'®

075

®

.0902

®

®-®

.0735

.999

@

'2/I

I®

+®

-09/7

4

-®

-0. 9988

e -®

.o0o

,®

r»©

.022

®

297

®

®0.125

®

®

.98°

®

©+

(.191

©

_®

2.8/O

©

-®

2.73C

®

©x©

-2.9c5

®

9.3,o

®

-/2.265

©

®x®

-0.22'

©

3/'2o

©

©-©

-3.644

®

.179

®

co2o

'268

(J

.002

©

co2&

.0

r'-o

®

® ©+++

©

A2,(®

¿324

© k-©

017

©

xA2x©x@x®

C17

®

2®ÇJ®

'002

®

©+©+c

f02

©

ô.9

®

248

3 Ax®x®

.- oo48

'29e

0.34

-37-Si (4-K,h/#

i2JÇ,1z)

'

124',h)

0.5772 '.

{(/

Á)2#(2Xh)2

2/61z 1 iaia,? 6'

o 'C

These approximations are readily obtained from the general expressions for Si(x + iy) and Ci(x + iy).

COMPARISON BETWEEN TEE SYSTEMS OF SUBSTITUTION

We shall make a comparison between the systems:

Vortex

Vortex + dipole

Vortex + gravity terms

Vortex + dipole + gravity terms

The systems a) and b) may further be divided into:

a)1 and b)1

r=k ç

a)2 and b)2

r=kr

For systems a)1 and b)1 the effect of gravity is disregarded altogether, whereas for systems a)2 and b)2 the effect of gravity upon the circulation is taken into account although the gravity

terms in the complex velocity potential are disregarded

We shall compare these 6 systems and choose 2 particular

h/c=

oii

U = 6 rn/see = 2° /3 = k

= O57

k00=

0.63 cases: i) A = 1.1 This gives:

C =

O1 m

_{= 9.6}

2) As 1) but with U = 1.897 rn/sec This gives: Fh = k = 0,29

k= 0.63

Hydrofoil case i substituted by a vortex with r = k fl, In this case we make use of (62), (63), and (6k), and the

streamlines shown in Fig. 8 are readily drawn.

Hydrofoil case i substituted by a dipole and a vortex with r=kj'e, The streamlines for this case are drawn making use of (5k) through

(58), and he result is shown in Fig. 9.

Hydrofoil case i substituted by a vortex with r =k r

In this case we also make use of (62), (63) and (6k) but replaue

k,

in these equations with k from (76). The result is shown

.

-_Lt

Hydrofoil case 1 substituted by a dipole and a vortex with r = k

Ç,

We again make use of (5k) through (58) replacing k,,, in these

equations with k from (76). The result is shown in Fig. 9.

Hydrofoil case i substituted by a vortex with gravity terms. The complex velocity potential may be written:

(77)

_{w=Uz#ik2rillain,41o9}

£ìb

2

59

-FIG. ê

CASE1

U=6m/sec.

.O5

15

2O

-'25

VORTEX

r=kr

_ ---

o

Q5

-VORTEX

=k1,

VORTEX AND

GRAVITY TERMS

I

r=

k

-

ko

FIG. 9

f10

VORTEXDIPOLE

r=

VORTEXDIPOLE

r- kr

-

₀₅

VORTEXDIPOLE 1

GRA VI T Y TERM S J'

r- k

LJ

where

a2-t2e

12CC _¡1

Zo=/A2a$jn/3

=Xo _'Yo ¿5ii72oC

Ia2_/2c62ac

k2a5/n

'

j k2a5m

h]

zo=xo

-/yo

With the substitution

a = -,h', (t

₂₀₎

the integral in the last term of

(77)

may be transformed like:

z -;Áz 'ik'0z0 e 6¡A;z0

e

du

t-z0

oc Oc z0

This is a generalization of the integral in (67), and writing

the last integral of

₍₇₉₎

7 =

du

we shall evaluate the integral for different values of 'X and Y. We distinguish between 4 different cases

-In 1. quadrant we have:

x,y

(81)

e"

du

x#,y

X

Ç

e

_da

* S

e"

X

=z,' iz7

In integral we substitute a = -iv and obtain

-y/X

IV

dv

ix

-y#,X oc X

_yI/x

C _O5V = V

dv #

# i

s/nv V dv oo#1X

_¡X

_¡X

'- y ¡X) - C/(1X)

i .5, (- Y' ì')

Further

e"

_du

X-ioo

42

X

=Ti(-X).

Hence, with X + IY in 1. quadrant, we obtain

1,

Ci(-Y*i X) -C1iix) ¡

S (-Yx) #L7 (-x)

e _a'u

= Cì( Y/X) -cjX) -i

s,(» ¡ Y) *Ei(-X) # i ri'.

The second case is

e

_du

X,00

=C;(YiX) C/(/X) , 5/(Y*iX) #(j(X)

-In

_3.

quadrant: -X-iX (81#)

Z =Çetdu

_J

jU

X--

11.3

--C,(Y/X) - c(ìX) + ¡ 5i(Y íX)#Ei(X),

and in 4 quadrant:

X-i)'

(85)

X-ioo

We may now write (77) in the form:

from which the stream function is readily formed:

2aJin,i3.tn

4'a ¿ls/ríA

o'Yo CO'I':

xa)

.7

4ai,A.e1Y'Y0in/4(x) .7ml

The stream function has been evaluated for different x and y

and the streamlines are shown in

FigS

8.

Hydrofoil case i substituted by dipole and vortex with gravity

terms The complex velocity potential of (24.6) may be written

as:

w=LJz/k2lJsÌn./og

4 (y Á) - /4Ç

¡k - Ç

(86)

\JUz

¡k2alIein/. log

(?'-x0 í('yy0

4(y*y iK0/x.-x)

145

-*

a2_2eM0C) /J2_I2eiA'OC)

S

e

_a'u

-X0h -,00

from which we deduce the streamfunction

z*ih

2-112

The streamlines are shown in Figs lo and iL

(89)

=Uy142a1Lci17./n

x2,' (h)2

-x # (¼2 U/2cos2oc) (y#12)

x2*(y#h)2

¿'125m2c.X

(y-h)

fr4.

Srìf3CO.5Ax"Za.2(.OSk, x X,2 /2CO5(Ç #2oc)] ¿Je 'e»

ee I

#/Ika sIn,,in

/{o#Ko2a6in X0x-K 212s1n(Kx#2o4Je"0")Jpn I.

The resulting streamlines are shown in Fig 9

FIG. lo

CASE 2

_h/c0.04

U=1897m/sec.

'O2

-

----o---VORTEX

AQ

04

06

VORTEX rkr

02

p04

06

VORTEX AND

R.

GRAVITY TERMSJ

r=

FIG. 11

CASE 2

U=1897m/sec.

04

-VORTEXDIPOLE

r

Içr

VORTEXDIPOLE

=kfi,

O2

VORTEXDIPOLE

GRA VI T Y TERMS

r

k

H

or, with and C

(91)

- ¿i */v

k

/ y(x)dx

*

/

Ç

¡

Y'x) dx

j

xi-x

j-i-i2A

IT o o

-CHORDWISE DISTRIBUTION OF CIRCULATION

In the preceeding chapters a method is given for

calculating the total lift on a submerged hydrofoil, and in the following we shall find the chordwise distribution of vorticity when the hydrofoil is regarded as built up by a distribution of vortices along the hydrofoil mean line together with the images of

these vort1ces

We consider a 2 dimensional vortex sheet built up by

a row of vortices and their images, see Fig. l2. Across the vortex sheet there must be no flow of water, and at the trailing edge the

velocities must be finite,

hence the vortex strength at that point must be zero The induced velocity at

the point z due to the

I

vortex element at and

i _I its image at z, may be

-.

f dund by applying (l9).

Z _{Integrating this equation}

we find the complete

induced velocities at the

FIG. 12

point z. (90)

-=-avv

o C C

z.

r /Y(x)d.'c

C ¡r()d

+ _a't

2'rr(z -i)

j 2

( -z) Y (x ) dx

z1 Ç e

1K0 t o 0 0 00 >4-o x-1i5

y () dx

''T

di'.

From this equation we obtain the following expression for the vertical component of the induced velocity at the point x - 1h

k9-C C C

Çe't

(92) _{= 2/ri}

L.Çr&a'x

₂

/

j

(jx)2 4hz

.Çr(x)a'x .Jm [ex(-i(x.-i4)J

dt

o o O cG

or, with

X0(e-x-/Á)A,

c c C / Ç Yfx)dx .j. Cr(x)(x.-xg'x

()

v1

_-

X

2TJ ()2

V(x)dx .JmfrxLiK.óv4Jt4-dÀJ

O O O oo-th,h IntroducIng

(9k)

x =

f(/co

)

jai--(/CO59j)

cy

c--

c'

we may write

₍₉₃₎

in the following form:

Ir (Ir

(95)

v,1

_2'rrj

Çy(p)s/npd_{co p-coi ç1 2fT}

_L(

_(cos-cos»

ç)(co-cosçjsinp 4f

_#.(4ih)

o o

e P dp

Jm

f (ces ip - ces

1erT

o

-Regarding the circulation distribution, we shall make the f ollowir

assumption which is wellknown from airfoil theory:

/

/-cA1p

_{4LJA ./n ,np}

*

satisfying the condition

where

Hence:

(98)

-

50

Since there shall be no transport of water across the vortex sheet, we must satisfy the condition (see f.ex [i4J)

(97) ,

çiuk

e-angle between foil chord and tangent to vortex

sheet at point z

j

= angle between U and foil chord (incidence)

IT IT

i-o(-=4S

/-osq

d'f-co -cos

Applying the integral formula

IT COSn'f COStqCosLq1 efl.

(I-

os

r

côs IT j (eost CO5 p7) # (4' .4'-) 00

z.A

sinnwsi>npdq

n.t ¿7 COS q -cas IT

*

_sFr)

(co -co»2

(42L)2 o

ir

&2

_{j'/-ca5tP)a'LrJm}

_{[exflL_/Ahncoo6ije__a'Al}

J

Ir

4hJ(CO5p-COSf)-i2k0h

;LAÇh -2X0h

,4sin

n q'

5/n p 4 mFxp{IItchß(cw -°

dÀ à 5/fl q'.,. J Cf - C4 _.

:0

-I, _o

jL

't

-together with the relation

sin n 'p s/n ip

=f{cos(n-/)p-

ca

the two first integrals in

(98)

are readily integrated We find:

(99)

¿

I Ç 4srnn If $/ %9 = -2 I i,, Ca.5 Pl _9j.

(loo)

_i-r

\

cos-cos1

o

It is convenient to write

(98) in

a more compressed form by use

of the following parametre:

A0 C ₌

_-A0

Co.54-ca5f1 o

AhMJ -/24,h

i-r k

e2

Jm(exL-/4',h

JÇe/

_dAj

Qa-ìIt;#4 o

Introducing the notation: iT IT o/mp a'p. where (102) = cos i - cas and hence: IT

(lo))

-Z,1-0c

(105)

4;- 2 co n

+

2jj

si» n in a4

we find:

- (106) - -o = A0 ¿ .4,.

The coefficients A0 , A1 , A2 will determine a circulation

distribution given by

(96)

satisfying the conditions

1(0)

= O and

no flow of water across the vortex sheet at the point

Zj

However, (106) must be valid all along the vortex sheet, and we must therefore determine a set of coefficients which are solutions

of (106) with

j= 0,12,

---In practice we shall find a set of 10 coefficients, A0, A1 ....A9

making (106) valid at the 10 points:

(109)

D

= rc COS

5;i

,

= O, 1,2-.

-9.

We find the solutions: (108) _A

_n-D

_{- I 2c1LJ} where 470

,46/

,..-.

....

-

52 , .. 499 (107) xi = C

and

where

is

obtained by substituting -T,,-(

-t-c

in the

(n#/)5t

column of

D. For example

-

53

-"fl'

.b00

-

-

c

,

. . . b,9 t

b9,

-7;,

-

oC "92

The lift on the vortex

sheet is:

i =9ír

=p1J5(9dx

.pli 21A0

pdp.

o o

By making use

of the formula

rn n w 5m (p = ;f (cos ( -1)

L - cos

# ,) pl we find

(iii)

LfpO2C2'iT('4o#4)

2'1TM,#4).

t L: )

51.

-The coefficients A0 A9 have been caiculatedjon a digital computer for the mean line of foil no. i (or foil no. 3) shown in

Fig. 2, for the conditions:

oC

=2°.

The coefficients are given in Table

_3.

We could also solve (98) by making use of tables of sine and cosine Integrai functions for complex arguments. Referring to

(loi)

we find: (112)

S

EÀ

_{dÀ -it4}

; 5iA/. 244)

-0o and hence:

(,(-2kh-L/(2k,4)

2 a> .

4jÁ

f

e21<°1Jm

{exIiki4h4JF[C4'Á4A4j -J2Kh)

.

¡5/(K0hñJ -/2AÇ4')

,.

y

-2Ah)-4i(2.th) P

2

.iJ'

A'Qh = 0.25

¿.01 0

H/C =

0.3 ¿'.6 1.6

TABLE 3

9 - -o- -

h/c

15

3

-5

15

-3

-6

1-5 A0

Jo4bo

8.27x/O3/828/02 2.283x2 2.14o*102

2o9x iò2 2.733x

1O23109xb02 34/1k

1O A1

3.125x102 3.292x,o2

3433

2.%fO2

3. O/2x/O

33/02 222x/O2

ìo2 309xío

6. 93gm io

i.

/O

f. O7 /0

I. 57

io

3, 9S5 /O

i, 7 g ¡o

A2

7.3/2

/2

2. o8 Jo

2.595 io

A3

2.43 fO

3.392

1. 439x/O5 9 33x fÒ

2.24x /o

I. O5 /O .

72x

-4

/.39x I0

5. 929x Io A4

-97/o

Io

- ó5f°

-4.7oc

3.563f0

2/x/0

_/. 25/O

7ô5x107

A5

-2./72x /O

-7. ¡27 ¡c

-g.573 1o8 -642x

1S

-.o7x70

6.4x fO

2.492x(O

.55x /O

5.f2ôx /0 A6

-/.8o

4.fÌ

o78fo

4.2x/ô6

6,378/O

7.g27A/O

/.ô2x/O

5g<f7

/.o7o1O

A7

4.993/o

2.313 to-3.9o4/o

2./4ôx107 //44/7

-iíi /ô

9/27/c

754;1x/O'32KIO

/.&2fO

4.573/ô

/.R45/ô

/.3o41ô

-8.4f?/o

2.944/o

A9

2.33x/O

3.158 fO

2.2(

Io

4/8310

_/34%f0g

4.07xfô

/.53fO

-/.347/0

A9

257/0 -7.742f0-(.3210

7251O

CL

-272

25

330

-311

323

.33

330

388

200

4.47

25

05

-

56

-97

/1(/-CO?) tj

(ilk)

6=

jZ _{#(4 _»Z}

O

e2f(/

co

)1e5/(4hhj"i244)Jm Û(A1,2Áh)

(j7 _h o

-

co (JcÇh f,i )

d,

"i K0h _{c 21(h\(/..c05ç0)LJ?fl}

11h

o EÍ-24'04) -E,/2K04) Js/n(Á'Dk

_-;)dLp

2

i-r c 24k

*

e -cod )E-,&

5i(-K

+iÁh)

Im C/(-Jh 24,J «,!KD 4)

tj

_frrrj

K0h c

_-

24h

--e

p))Jìn

t9j

-2Ah)-(2kh)]51(_,

2

and

i-r » W S/I?

(115)

4»-2eo n

4

O

±

_'ir

_h n EEeC$/aÁÂ #;2Ç4)2m C/(Á11

,z

*)

57

-,h

_{- ern n ip4srn}

wLJm Si(k,4fr1 #iZ4),'k'eC/(K,%2j #,'4 A)

o (;(-2KÇ)-L7(2k'04) }smoc'04

frcj)d

2

ir

2t'4

e26Aj'irn

n 'ç -c/n Eieesi(-#t hA ; #/

%4)#17m',(Kh f

/2k, h)

-fr!1

côs(#t,h f1)d,

'Tr 2K,#Ç

f

e2t'1Çrn

_{n q5fn}

/2kh)#&C/%j1,I244)

"J

-2

where we have made use of the Identities:

(116)

Sì(-x /y)* Siór#iy)

C'x-iy) - C/tx vyi

C//-xtiy) -

''iy)' ¡IT

The circulation distribution has been calculated by means

of

₍₉₆₎

with the values given in Table ₃ and the result is shown

-

58

-FIG. 13

-I

.3

U

.1

o

0

3

1

2

3

4

5

6

7

89 10

2

= 4.

4, 7

h/;±..

i

-

---0

/

0

3

1

2

3

4

5

6

7

8.9

10

2

f=2

0

O 1

2

34 56

7

8.9

10

-

59

-It may be observed from this figure that the resultant centroid of circulation on a hydrofoil in the vicinity of a free surface is shifted backwards as the foil approaches the free surface

or as the depth Froude number decreases. This is in agreement with

the results of pressure measurements carried out by Ausman [20] and

also with the observation that hydrofoils exhibit better cavitation

characteristics nearer the surface 1211

THE ₃ DIMENSIONAL HYDROFOIL

ANALYTICAL MODEL OF THE FOIL

In [7] Lunde has given the velocity potential of a moving source at the point (0,0,-h):

41/2

L

sec2 Odû '

°

_{¶;» 8)ex4K(zh)1dK}

(117)

_A'-1sec

o O

- m K

6/fl(1CX ec8)cos(Ayxc2&s,»O)ex'1á (z -h) ec28J d&.

o

Here the axis x, y, z form a right handed system with the x-axis in the direction of motion, and the z-axis pointing vertically

upwards. The velocity potential of a dipole may be found from that

of the source by applying the operator

(t'-m

ò

-and substituting the moment of the dipole for the strength of the

source. Here J. , ni and n are the direction cosines of the

dipole axis. For a dipole distribution along the line z = -h

x = O , between y = ±b/2 , and with the axis of the dipole elements being perpendicularto the y-axis and making the angle X with

(1l8)

.6/2

xco.#í'z,.4)th

I

J

1T/2 Sec 60 -XCO5zWz-h)5m

z

I

N7,)& 4r îT b/Z

sìKxca8)caEK(y-)sth G3e.L K/i-I2 )1 co

KdJ

K-Çsec29

Cú6(KxCt5 9)c'o

LK/y-)s/i1ex,vIK(z-h)L;ìz

z1C'dK

6/2 tTr/2

'kif

Çft()d

sec

cs ( x cc )cak (y-ec

20 ,

-6/2

.6/2 1V2

K02 1ft/&d Çc.si

(K, xseco)csL x0 (y-) 5ec20

_{8]exp[&-h) ec9J5'}

dO.

-.6/2

The velocity potential of a vortex distribution along the line z = -h , x = O , between y = * b/2, including the trailing vortex system, has been given by Wu in 181

(119)

_(yzh)2

1

[x2#(y?)2#(z#)2t/2]d?

- 61

6/2

zh

Ç

r

(y-7)2(z-h)2

[/#

[X2#(Y)2(2h)t]/d7

- -i'-

_cÎrj r( )th2

d

Jec2ed9 coo(Á'.cs &

Li'(y-r),,' 9]

_{.(- ,}

_sec2cx,

[k(z-h)lic'dK

-b/2 o p b/2

4

r()dpScos[#t'e'y)] exp[4ë(2-h)} d1t' -b/2 O b/2 91/2

-

çs(kxe

-4V2

Substituting a finite hydrofoil by a dipole distribution and a bound vortex distribution along the centreline of the foil

together with the appropriate trailing vortex sheet, the velocity potential of such a foil without dihedral or sweep back will be

given by

(120)

Applìcation of (120) will make the practical design of a hydrofoil very cumbersome, and especially so if dihedral and sweep back Is

introduced. As shown earlier the 2 dimensional hydrofoil may be represented by a vortex at the centroid of circulation, and we should expect that the finite hydrofoil might be represented by a vortex line along a line of centroids of circulation. Provided

that this line of centroids of circulatïon was straight and with

a dihedral angle ' we could make use of the velocity potential

as given by Nishlyama _[9] for a submerged finite vortex line with

dihedral:

(121)

-

ß2

j

(r)deee8d8eok(_116mt)4to]dK

-b/2 JT o -b/2 'TT

A'?7Y ¡r(

912 _7'

$

iT

89T2J

()

dÇse

84' aSexp

{-íc(z

#h - 1J ia r) /Kp}dK

b/2

62

-OC

I ep&K(h- I 1Ism Y-z)#/1'pJ

kAec

$/i?V_ç

00

±

(IT

r()a'p expL/'(z 'h

- i l 3/frVY)JS/>? K

c r)dK

o -4/2 6/2

S/a '

o

where

-4'2

p x

4/2 00 6/2 lT 00

dK

jCD5 y4 1r()a'ec 38a'81

X-% Se( 28

8

4972 -WZ tiT _o 4/2 00

(05V 1(

_rr

)dpexfl Et'(z h -1

3,>? y)] os

frt'(y--A'2 o

-Is/n Y-Z)}J

K(y-ces Y)d

4,/2

±

r()déan 8d8expEk(zi'h -

Ile/>z r)# i/(pJdK

J

.9/121'

8'1r2 s

Thin and deeply submerged hydrofoils with dihedral, might

be substituted by (121) when 'Y and h in (121) were replaced by the hydrofoil dihedral angle and submergence at the centre of the

hydrofoil. Referring to the previous discussion of 2 dimensional hydrofoils, such a substitution is doubtful for hydrofoils of some

thickness and with a small submergence, and moreover the practical design or analysis of a hydrofoil would be very cumbersome by

application of (121). We shall therefore make use of a simpler analytical model of the foil based on the results obtained earlier for the infinite foil and which lends itself for practical

applica-ti on.

A usual assumption in ordinary airfoil work is to

substitute the finite wing by 2 dimensional strips and with the wing followed by a trailing vortex sheet. The lift force on such a 2 dimensional strip is found from the 2 dimensional lift

characteristics of the section when the induced velocity components from the other strips and the trailing vortex sheet have been

added to the free stream velocity.

The velocity potential as given by Wu for a submerged 3 dimensional vortex without dihedral or sweep back, has been discussed by Kaplan, Breslin and Jacobs [lO'J . They were able to

show that at high speed the potential approaches that of a vortex and its biplane image when x is small (near the foil) and that

of a vortex and its wall image at large distance downstream.

Since we are interested in finding the liftcharacteristics of the foil, or the down-wash velocity at the foil, we shall make the assumption that the foil is built up by a series of

2 dimensional strips whose lift characteristics are found by means of the results derived earlier for the 2 dimensional hydrofoil. Further we assume that the hydrofoil is followed by a trailing

vortex sheet and its biplane image. For the application of the results obtained for the 2 dimensional hydrofoil, we shall find the corresponding moment of the ₃ dimensional dipole distribution.

MOMENT OF DIPOLE DISTRIBUTION

The velocity potential of an element of a ₃ dimensional dipole distribution along the y-axis is (see Fig. lu):

(122)

d

//JCOÔZdJ Wr2 -

63

-4 ç- _{J c} ('.'S t

I / _ò/j.

a3

,

=

H.smrdy

The component of dv along the x-axis is:

4,.5mz

with a negative direction. The radial velocity component is:

ò '4'3

= /fcoZdy

2íT3

The component of dv along the x-axis is:

CO5't

hence the total velocity along the x-axis due to the dipole

element M3dy is:

-

64

and

65

-= ,% Z a'y

ra

2L-3

rrr3

We further observe that

2

1 = ay

_i

-C062t hence:

(2c

v-srn'rcor)dr

Integrating we find the total velocity along the x-axis at the

point (R, O):

vTv

(2co3

z-5m2Z

m/z

When is considered constant, we find:

,32TT/?2

MJ

The corresponding velocity due to a 2 dimensional dipole of moment M2 is

y2 !'2 2íi,c2

- 66

-A circle in a 2 dimensional flow may be represented by a dipole

of moment M2 , and we conclude that an infinitely long circular cylinder with the axis normal to the flow direction and with the same diameter as the circle, may be represented by a constant

dipole distribution of the same moment per unit length as the moment

of the 2 dimensional dipole.

In conformity with the strip method of representing the

3 dimensional foil, we assume that the moment of the dipole

distribution for a strip of the foil of unit length, is equal to the moment of the dipole representing the 2 dimensional hydrofoil of the same cross section, the same incidence and the same inflow

velocity.

THE DOWNWASH VELOCITY

The 2 dimensional circulation reduction factor given in

Fig. 6 has been used to obtain the necessary incidence to give a

certain lift coefficient in 2 dimensional flow. The result is

given in Fig. 15 where the parameter _CL h/C is used instead of CL in order to make the diagram easier to read. To the incidence

found by means of Fig. 15 we must add the effect of the trailing

vortex sheet and its biplane image. The two vortex sheets induce the downwash velocity w at the hydrofoil, and when the foil has sweep back, we must add to w also the downwash velocity due to

the bound vortices.

For the calculation of the downwash velocity of the foil

with dihedral and sweep back, we apply the formula of Biot-Savart:

(125)

r(s)_{9T r3}

where vectors are indicated by a bar, and where x indicates a vectorial multiplication.

Let the circulation along the hydrofoil be

_r(5)

and

projected onto the y-axis _rl,» so that

s/2

Sr(s)d5=r(?)dF.

-5/2 -6/2

.. ::::::::::::::::::..

ù..J...:.:.:

U...p.Ub.

..

... 1 _ :

:':L9::.uj

.. .. F .. ..

: ::

L: .

.:

JLJj i4

h3:F .:

Wh1h

IJ'$

FJ!P. .... ¶!. ...:il::::::u::2#I ...ifl :r.'-.... :!!u:q::::r.'-....:r.'-...::::: :::r:::::q::::r ::h. g.

r

g:

rir" r:::

: L

:::

L 1 : r aua

..:..n..u'Ji:::q:::..

....fl. ... s. .._ . :L...:. ...: 18U R. rn #::: :

:ll"

il L. Lrn: L..L r ::p

r9dJj.:

.. s . .;.h;.uI;;;;

r

Y g z

1fr. ::1j

r...

.ri.

:!::j9

k!!: Ig%5! 9hz

:P f ::

:

r

::9g :gg'.

...:zzmr

:::u:::::::::w:mu;:.

...

m!... ii!!R..

-r-R. !

:L

! :VØ !!.!ç . .ijj.. ! .:::zzu:: u:: ; ir::::.. '

z::::ir

h:::::: .. z:::

68

-Applying (125) we may write the contribution to the induced velocity

at the point (X, y, z) on the port wing from bound and trailing

vortex elements as (see Fige 16):

i) From element of starboard wing:

j,A

(126)

4

₌

'rr f[

S 24y- J2#jy#

rJ2j3/2

From element of starboard image wing:

I

,j,

(l27)dv='

r()

rc

Q/76

/

'

.

(-y-')tj, (y-e) ,-2h1'(y-

')irn

From element of trailing vortex behind port wing:

ò

d

{[-ytanî-J2*_{Ly_12*L(y_)t712}3/2}

ò)

.

'2)3/2 (129 )a'i ₌ ')wnj

_j

,

/ , (-y

-,)Qft5,

_{&-'),&')a#rï}

L

/

7

/

,c,

O

(-ytm-- )

_{,(yi),'yiXm;'}

k)

From element of trailing vortex behind starboard wing:

7'

,o

,

a

(-ytp$.), ( -),(y#&anT

-

69

70

-5) From element of trailing vortex behind port image wing:

ÒP(V

(l30)d

è., c17

4i1 {f-yi*nJ- Ly_12#[-2y#Fr?l312J312

6)

From element of trailing vortex behind starboard image wing:

t1)

a42

(131 ) d3=' _4cr

j

,j,

T

7' 0

(-yt4P S-

_{),(yi),24 (yq)vT}

/

,j,

4

/ ,0,

O

(-y4nS-V,(y-)j2h -)6vi

From (126) through (131) we deduce the following

expression for the downwash velocity at the point (x, y, z) of

the port wing:

o Z _'1) o

(132)

_2T

Ç

j ty_p);]2#Lyij2(y)tì2J3'2

-A'2 o

C5fcO tôìiSy

_I

2Tr

J

a(_Y)

JJ2fty_,J2,.[_2,4 (y_)t,nJ2JJi

4'12 -o. Ç('Y

,)orb)d

-à_{'z ) ([-jtQoã_}32#Ty,12#L(y _,)tu32jsI2} o n

--

- )

4

[[y.J

)2 ₎ 3/2

6/2

-k (y-)

òr)

d

_]2 [-2 h ¿y 3/2 o

-?'

-oc

/

-

_a -4/2 -

71

-When y+

, integration of the last term of the k double

integrals will give:

o cas

J

)2#t,f '1y - ,)2,L

(y#r)2,,>JJ/2

-4/2 o

Ct.CO5ttP1J.0

\ 2 îJ-J

[(

a12d;(yi)2#L_2A#(y?)?,7l2}J/4 -,2 6/2 ' / ç

Òr),,1

Yf#toé'ta#7) i#nj

t' _1'

(1)3)

w-6/2

-h5&

4/2

(l2)

_L.

a?

/çjft

_TJ t,

72

-a r (-ag)

/

J

)»)

{t(rp2t4J

y-ï

y

e teìn J

-fr('-

1

#(7)2

2hy)fr7ça)1

--Wi

When the foil has no sweep back, i.e.

¿

O

, we

have:

1

-

T J ,,' ( o o

I

ri

_/

)T

_y1

-liz

)

I,!t24 )21

o

If

t

j

-b/2

òr(

and when the angle of sweep back as well as the angle of dihedral

are zero, i.e, S

_r

= O , we have: 6/2

/

(135)

w-The last expression is the usual one for the downwash velocity of biplane wings, and a similar expression may be found f.ex in

[1k]

DESIGN OF A HYDROFOIL

By means of the foregoing expressions, calculation of the downwash velocity is straight forward when the circulation

distribution is given. Analysing a given wing under given flow conditions is more difficult, but may be performed by successive

approximations.

For the experimental check of the strip method of

representing the ₃ dimensional hydrofoil, we shall design a

hydro-foil for the following conditions:

speed of advance 6 rn/sec

sweep back 00

o

dihedral angle 15

total lift 12 kg

circulation distribution clliptic

draught at centre

oo6

¡n

foil span O.32 m

foil chord at centre 0-10 m

chord distribution elliptic

thickness/corde ratio 0.11 = constant

75

-We now divide the foil in 8 parts, and for the numerical computation of the downwash, we substitute (134) by the system (see Fig. 17):

(136)

where L = total lift

hence: a

/

'¼

rr2v

_'b/2

f*(

6/2

/

:5-Q /(/YFt2

)2 _y

/

_>°

I

¿IL 49Tp(1 _-6/2

/(Qf) y-.

The lift at the centre of the foil is

9-r-b

L0 =

47.8

kg/rn

1/2

¿iL

¿iL

and the lift distribution is given by:

(3)2

e