On the chordwise pressure distribution on submerged hydrofoils

(1)

THE NOR Vv'EGIA N SHIP MODEL EXPERIMEN T TA NK

THE TECHNICAL UNIVERSITY OF NORWAY

ON THE CHORD WIS E PRESSURE DISTRIBUTION

ON SUBMERGED HYDROFOILS

by

Harald Aa. Wa/derhaug

NOR WEG IA N SHIP MODEL EXPERIMENT TANK PUBL ¡CA TION N 75

(2)

page

INTRODUCTION i

NOTATION 2

THE SUBMERGED VORTEX SHEET

TIOTICAL

PRESSUHE DISTRIBUTION ₇

EXPERIMENTAL INVESTIGATIONS 21

SUMMARY 26

(3)

INTRODUCTION.

In aerodynamic theory, the thin and. slightly cambered airfoil has successfully been substituted by a continuous distribution of vortices along the mean line of the profile. The imderlying theoretical considerations have been described in several text books, see for example reference [2] . By prescribing the two conditions: continuous flow at the

trailing edge and zero flow across the mean line, the vortex distribution has been determined for varying amount of camber and incidence.

A related procedure may be applied to the case of a thin hydrofoil under a free surface. Substituting the hydrofoil by a vortex distribution along the mean line together with the images of the vortices, lt is possible to calculate the pressure distribution on the mean line.

(4)

-1-NOTATION.

c chord of hydrofoil.

g acceleration due to gravity.

h distance from undisturbed water surface to hydrofoil or vortex centre.

t thickness of hydrofoil.

u

x-component of induced velocity.

at point

X.

ujj x- ti

"

due to image syseni.

V

y-

ii ii Ii

Vj

y-

n at point

Xj.

w complex velocity potential. depth Froude niunber, ïj/ /jii K0 g/U2.

L lift.

Reynolds' number.

U

velocity of undisturbed flow.

X

incidence relative to chord line.

circulation along foil chord.

ti ti ti ti

r

total circulation of foil. mass density.

angle between foil chord and. tangent to foil mean line at point

Z.

(5)

-3-THE SUBMERGED VORT SHEET.

We substitute a 2 dimensional, thin and slightly curved hydrofoil by a continuous row of vortices and their images in the free, undisturbed water surface as shown in Fig.la, where the x-axis coincide with the undisturbed water surface. The induced velocity at z3 due to the vortex element at and its image at z, may be found by applying the theory of a single, submerged vortex, as given in 153 :

I4

_(L

1)

Integrating and putting

Zj

Xj - ih z = x -s- ih we find: C C C _r

(rC,)a'<

1('(c)dx

±

r(

e

uJJ

o _o

where we approximate by integrating along the chord instead of along the actual mean line. For pract1tal, slightly curved mean lines, this approximation is very close. Introducing

k-x-h

A

cos)

(6)

h

FIG. la

X_m

L

/1

2h

j=O

1 2 3 4 5 6 7 8 9 lO

FIG. lb

(7)

I ii:

i

p)(c-cos1)q,c

,

(.q.)Q

-5-h(c-co)-ch

We represent the circulation distribution by the series: oO

i- CÒ3(Q

'(p) UA0 ' +

4UEAsv-nj7

5(i2

n1

satisfying the condition:

o.

The resultant velocity perpendicular to the vortex sheet

must

be zero, hence:

Us,n(r1c)-t-i.

or, with good approximation:

= - u(

)

The coefficients A0 A1 _{will determine a circulation}

dis-tribution satisfying the conditions: continuous flow at the trailing edge

and

_{zero flow across the vortex sheet. In practice we shall} satisfy the latter condition at 10 points along the vortex sheet:

J

lo

(7)

CMCCoS

5-j

j=oI1.

₉

c -2K,h

r

e

tKhos-s)]

e

(8)

From (3), (4)

and

(5) we now find:

r

(8) -'r-c'=- 1-COSSU

Qcosq49

o 9

&

(cs 6os))

dsp + 2r(03,

-

;i;1 A Sir?flS°

c#(Ç)

ssq)Z+(4 )Z

¶)

(cs

s))Z

(4

₎ o _o

ir

I(5)thP

p1

(cosp-f

_{da J}

C21

_1i-p

_srnd

e

j

d

o where j = 0, 1, 2 9.

In this formula h is the distance from the vortex to the free, un-disturbed surface. We have considered this distance as constant, i.e. the mean line camber and the surface deformation due to the wave system have been disregarded. The first approximation may be considered as satisfactory for slightly curved mean lines. The latter approximation is more uncertain. As a matter of fact the complex

velocity potential for the discrete vortex has been found by

applying Schwartz's reflection principle, where in this case the un-disturbed free surface

_and

_{the x-axis coincide.}

Obviously the formation of a wave above the vortex will disturbe the reflection condition, but it was shown in [5] that a satisfactory approximation may be reached by adding the wave height above the vortex to the widisturbed depth h.

(9)

-

-7-on the submerged vortex sheet are: _U

More or less this result might have been anticipated, e3;e7

when the wave is long and flat, which is mostly the case since the hydrofoil usually travels at high speed.

It is now possible to evaluate the coefficients A0 A9 and the circulation distribution (x) by using a digital computer. This has been done for a parabolic mean line, NACA 1.75 5, with the following characteristics taken from [6] :

TABLE 1.

In unboimded flow, the ideal angle of attack and ideal lift coeffi-cient of this mean line are 00 and 0.219 respectively. The computed circulation distribution is given in Table 2 and in Figs. 2,3 and 4

= 0 .035 .070

THEORETICAL PRESSURE DISTRIBUTION.

When the velocity values on the upper and lower sides of the mean line are known, it is possible to calculate the pressure distri-bution by means of Bernoulli's equation. The velocity values on the two sides due to the free stream velocity U and the vorticity

j 0 1 2 3 4 5 6 7 8 9 10

y/c 0 .0063 .0112 .0147 .0168 .0175 .0168 .0147 .0112 .0063 0

r..O7O

.056 .042 .028 .014 0 _{-.014 -.028 -.042 -.056 -.070}

for the conditions:

K0h = 1.5 0.5 0.05 Fh = .817 1.414 4.47

(10)

¡<OH ALFA COMPUTED TABLE 2a CIRCULATION DISTRIBUTION NACA 1.75 5 U X.5o H/C

1.0 0.043,78

_{3. 500,}

_-2

_7.000,-2 1.187,-3 4. 380, -a _{9. 946,}

-3

7.830,-2 _9.824,-2 _{X. 182,-!}

I.034,I

I. _327,1

t .6i 9,1

I

173,-I _'.547,-' _1.923,-x Z.235,-I _1.697,-X

_3.157,-I

I.

238,-I

I

181,-i 1.796,-I X.86o,-i

3.354,!

2537,-?

I.

o 58,

-I

I. _900,1

_2.742,-I 8. 437,3 1.946,-I

_{3 048,1}

4.531,3

Z XI

3,1

3.773,-I = '.50 H/C =

0.043,78

7.102,-3 8.110,-2

I.077,1

1.228,-I

1.304,-I

I.

3X8,-! I 274,-I

1.164,-I

9.683,-3 6.195,-2

2.0

3.500,-z 5.492,-2 Z. 384,-I

1.630,-I

I. 8o,-x

1.933,-T 2.028,-I

2.105,-I

3.203,-I

2.476,-I 7.

000,

3 1.169,-Z

I. 23I,1

i

692,-I

2.033,1

2.307,1

2.548,-I 2.782,-I 3.047,-I 3.438,-I 4.33!,-! KOH = ALFA r.so H/C = = 0.043,-78

=.817

7.

0002

0.

3 3.500,-2 O

1.339,2

_{4. 088,} ₃ _{9 516,}

_-2

I 8.049,-2 1.362,-I

I. _920,1

2 9.599,-3

I.

730,-X _2.500,-r 3 9.767,-2 1.863,-1 _3.749,-I 4 9.291,-3 1.865,-I 5 8.519,-2

I. 795,1

2.738,-Z 6

_7.594,3

_1.698,-, _3.638,-I 7 6.463,-3 1.614,-1 _3.583,-I 8 4.834,-3 _1.589,-I _3.695,-I 9

I.783,3

I. 752,1

_{3. 327,} ₁

J=0

I

2 3 4 5 6 7 8 9 ¡<OH = AL FA ₌

J=

O t 2 3 4 s 6 7 8 9

(11)

TABLE 2b

COMPUTED CIRCULATION DISTRIBUTION

NACA 1.75 5

= 1.414

C'

-KOH =

0.50 H/C =

KOH =

o.o

H/C

2.0

ALFA =

0.043,-78

3.500,-2

_{7. 000, -2}

O

1.549,-3

3. 904,3

9.356,-2

I

7.880,3

9.781,2

r. ¡68,-x

2

1.041,-I

I. 324, -I

_{I. 607 ,}

3

T.179,-1

1.547,_1

I 914,1

4

1.242,-I

1.697,-z

2.152,1

5

I.241,I

I. 795,I

2.349,-I

6

1.177,-1

I

852,-I

2.537,1

7

X.041,I

I. 88o,-

2.7I8,7

8

8.046,-a

1.899,-I

_{2. 994,}

_I 9

3.707,2

2.009,1

3.647,-1

1.0

ALFA =

0.043,-78

3.500,-3

7. 000, -2

j=0

y a

2.466,2

7.471,2

9. 770, -3

2.094,-2

9.617,2

I. 287,-I

6.652,-a

I. 776,-I

i. 8,-r

3

X. 093,1

I. 470,-I

I.846,1

4 I

X 35,-I

I. 581,-I

3. 027,

1 5

1.112,-I

I.633,1

2.155,-r

6

I.025,I

7.638,-I

2 251 -7

7

8.609,-2

I 600,-I

2.340,1

8

5.845,-3

I. 537,-I

2.469,-I

9

6.693,-3

1.453,-I

3.838,-z

KOH = ALFA

i = o

I 2 3 4 5 6 7 8 9 =

0.50 H/c =

0.043,-78

3. 252,-a

3. 307,

2.696,-2

X.569,2

5.573,_3

-I. 15X,3

4.984,3

9. 045,3

2.007,3

-5.584,2

0.3 3.500,3

3.326,-3

7.386,-I

I. 570,-I

1.494,-X

I 313,1

1.105,-I

9. x6ç,-z

7. 633, -3

6.370,-2

5.134,2

7.000,2

3.978,-3

2. 442, -X

3

2.837,-1

2.567,-I

2.221,1

1.883,-I

i 6z',,-i

1.474,-I

X. 585,1

(12)

TABLE 2c

COMPUTED CIRCULATION DISTRIBUTION

NACA 1.75

5

/7; = 4.47

U KOH ALFA =

0.05 H/c =

0.043,-78

0.3 3.500,-2

7.000,-3

j-_o

-5.566,-3

4.566,2

9.688,2

I

7.230,-z

I.o86,-i

1.448,-I

2

8.875,-3

1.375,-I

1.862,-I

3

9.359,-2

1.488,-I

3.039,-I

4

9.282,-3

!.507,I

2.086,-X

5

8.923,-2

1.482,-!

2.072,-I

6

8.394,-3

I.449,I

2.058,-!

7

7.660,-2

1.442,-I

2.117,-I

8

6.503,-2

1.507,-7

_2.364,-I

9

4.279,-2

1.791,-I

3.154,-I

KOH =

o.og H/C

ALFA

0.043,78

3.500,2

7.000,-3

C)

7.264,3

5.028,2

1.078,-I

X

7.815,-2

9.630,2

I.145,1

2

!.o36,-1

1.306,-I

1.576,-I

3

I.t78,X

I.531,1

1.883,-X

4

I.249,-I

r.688,-x

3.128,-1

5

t.36o,-t

i.8or,-i

2.34!,-!

6

I.315,1

1.882,-I

_2.548,-I

7

I.to8,-t

I.948,I

2.788,-I

8

9.180,-3

2.033,I

_3.146,-X

9

5.793,2

3.276,-I

3.973,-I

KOH =

0.05 H/C =

2.0

ALFA =

0.043,78

_{3. 500, -a}

_7.000,-3

O

4.410,3

5.953,-3

1.335,-I

I

8.153,-a

I 021,I

_{I. 237,}

₁

3

1.084,-I

1. 393,-I

_{I. 703,-I}

3

1.338,-I

1.644,-I

2.051,!

4

1.318,-I

1.827,-I

2.335,-I

5

1.338,-1

1.963,-I

2. 588, I

6

1.300,-I

_{2. o68 -1}

_2.837,-I

7

1.199,-I

2.161 -I

_{3. 124,}

_I 8

1.017,-I

2.283,-I

3. 549, -I

(13)

-n-FIG. 2

COMPUTED CiRCULA TION DISTRIBUTiON

K0h= 1.50

0.817 NA CA

1.755 .4

o h/c=20

-'i- =

--

=

1.0

0.3

I

-d'i

.

'2

c=O

o

.4

::

-

__at=2°

.4

_A

O 1

2

3

4

5

6.7

8

9

₁₀

J

(14)

FIG. 4

COMPUTED CIRCULATION DISTRIBUTION

Kh=O.O5

_Fh=4.47

NA CA

1.75 5

A

o h/c2.O

:::

!

::

_t

z;

=4o

+

i

O 1

2

3

4

5 6'78

9 iO

(15)

(9)

-14-To be exact, the velocity U should have been modified due to angle of attack as well as camber of the mean line, but these effects are negligible. To the velocity above we must add the tangential,

or approximately the horizontal, velocity component induced by the image system:

C C. i -kj-ih)

I

e

_dt

Jx_x_ih

o

For moderate or large h/c values this velocity is small compared with '(9). Moreover it is approximately constant along the sub-merged vortex sheet, and since computing Uil according to (9) is very complicated, and since its influence on the pressure distri-bution is small compared with the effect of the y(q) ) distridistri-bution of the vortex sheet itself, we shall make use of an approximate value of u.1. At larger h/c values we take u to be equal to the velocity induced at -1h by the vortex image system _{at 1h with} cir-culation, r . Referring to [5] this velocity may be written:

(lo)

_L

r

4'ch

Idi Introducing we find: r (11) s - -iK0(t - 1h) 'OhP

e'°' E(h).

rch Lh

r

1e

J -fr-(h

co

(16)

(12)

C

L= ?Ufz

o I- 9

Uf2UA0

-

4 U A o o

which, making use of the identity

=

LcO5(n

-

1)tp - cos(n1)cç2] is equal to:

L

Uc(A0-41).

The velocity U is of course influenced by the image system, so that the lift expressed by (13) is only approximately correct. A more exact expression would have been:

'r

L=

d9

however,

Uji

and Vj are small compared with U so that (14) has only theoretical interest in the present connection.

Since

L

we may now write (li):

L

_}çL

-fl'0h E(.c0h) (16) ll

4Uh

rUh

e L

r

_-

4h

_()h)]

- 4Uh

L

U(A0tA1) r

-(0h

4 h/c

L

-

4h e

(17)

At smaller h/c values we shall make use of a closer approximation to uji. We divide the image system into 5 discrete systems as

shown in Fig. lb. The horizontal velocity induced at the point - 1h by such a discrete system at xm 1- 1h is:

(17) where (18) =

-h(m)

-(h -(

e ° e

a(m)

Ç

du

°

The induced velocity at

Xj

may then be written:

I

(19) 1A. Jt-where -

h-i(1Xm)

r

(20)

T

e du., .1 IA

With reference to [5] this _{integral may be written:}

-16-x-h

+()fe

J

h 0 00

rn-111-

s.

Ju

-IÇh-co

m) 5x h m)

-kh

kh

[Som)

Tmt']

(21)

_T

-

+

(18)

when Xj > x

,

Slid

'

=

S [

1)t2Kh]+ E [wij -

it'

when Xj < Xm.

Applying these expressions, we may write for the velocity

on the back of the mean line, the suction side:

tL

and on the face, the pressure side:

1J

u--(tp)-u

Applying Bernoulli's equation we readily find the pressure

on the back of the vortex sheet:

iSP

J)

_Fpj12+

+

U

LUJ

U U

and on the face:

Pjpo

-(j"i

_p)

(C1'

u

By means of the theory developed above, the

_{pressure distribution}

has been calculated for NACA

_1.75

5

mean line under the conditions

described earlier, At h/c

= 0.3 we made use of (19) and at h/c

and 2 we made use of (16) for calculating Ujj. The calculated

pressure distributions are given in Table 3 and in Pigs. 6

through 14.

ft

(19)

= 2.0 C o 1 2 3 4 5 6 7 8 9

-18--TABLE 3 a

COMPUTED PRESSURE DISTRIBUTION (F.

-

P0)/u2.

NACA 1.75 5

O=

00 -.016 -.003 .011 -.099 -.084 -.072 -.126 -.110 -.100

-.143

-.125 -.117 -.150 -.131 -.125 -.152 -.131 -.127 -.148 -.125 -.123

-.137

-.110 -.113 -.117 -.090 -.0 3 -.081 -.041 -.ObO =

0.3

J 'J

Face

Back

Fh

.817

1.414

4.47

.817

1

h

1.414

4.47 O 1

-.092

,00

.003

,0b2

.035 .10 -.092 -.159

.030

-.004

-.03.035 2 ,020 .054

.12

-.175

₀

_-.04 3 .028 .041 .134 -.171 .009 -.050 4 ,0'31

.028

.135

-.i6o

.015

_-.048 .029 .018 .027 .010 .022 .002 .012 -.013 .131 .126 .117 .10 -.145 -.130 -.111 -.086 .020 .020 .020 .027 -.044 -.039

-.033

-.025 9 -.011 -.055 .07 -.049 .057 -.009 =

1.0

o 028 -.004 .01 -.028 - . 004 019 i .049 .

069 .09

-.108

_-.080 _-.060 2 _.072 091 .120 -

.13

- .104

-

_.087 3 .086 .102

.133

- .14

-

.115

-.102

4 .091 .106 .140 -.155e

-.120

-.110

02 .10

.141

-.15o

-.118

-.111

0o7

_.09

_.137

_{- .14}

_-.109

_{- . 106}

9

.075

.080

.05 .053 .01 .003 .12 .10 .076 - .13 -.114 -.074

-.02

-.0o3 -.011 -.095 -

.075

-.040

-.016

-.003 .011 .063 .07 .091 .088 .09

.117

.103 .111 .131 .110 .117 .139 .111 .117 .140 .107 .111 .137 .097 .098 .127 .080 .080 .110 .047 .033 .080

(20)

TABLE 3 b

COi1PiJTED PRESSURE DISTRIBUTION (P_ P0)/u2.

NACA

1.75

₅

= 2°

1.0

0 -.055

-.009

.037

1 .044 .085

.129

2

.07

_.115

_.151

3

.O9i

.132

.181

4

.111

.14 .196 5

.121

.14b

.206

6

.128

.148

_.212

.131

.14

.21

b

_.135

_.13u

_.22o

9

.149

.131

.24.8

= 2.0

0

_-.031

_-.005

_.021

1

_.070

_.090

_.119

2

.10

_.123

_.b4

3 127 p144

.177

4 14

.158

.19

5 15o .167

.20

6 .i65

.172

_.216 7

.172

.174

.22A 8

_.180

_.175

_.234

9

.206

.i86

.261

=

0.3 Face

Back

Fh

j

.817

1.414 _4.47

_.17

_1.414

4.47 0

-.1)0

-.155 .035

-.150

-.155

_.03

1

.0i5

_.023

_.161

_-.308

_-.263

_-.04

2

.018

.01

_.201

_-.364

_-.253

_-.054

3

.028

.0i7

.220

_-.375

-.220

_-.065 4

,028

.089

.227

-.3 5

-.17

-.06i

o 022

.08

020

.07ö

.223

.227

-.3b4

_-.345

-.13d

_-.106

-.055 .,029

.04

_.21

_-.31

_-.080

_-.O5o

.0)0

.0o7

.21h

-.28b

-.060

-.071

9

.097

.060

.233

-.263

-.042

-.119

-.O5

-.009

.O7

-.15b

-.107

-.OoO

-.19

-.1L1

-.095

-.210

-.1l

-.119

-.236

-.173

-.136

-.247

-.1

9

-.148

-.255

-.löO

-.157

-.259

-.l

-.1o4

-.266

-.10

-.172

-.283 -.1)9 -.199

-.031

-.005

021

-.13

-.105

-.083

-.17b

-.12

-.121

-.203

-.1o5 -.1

-.221

-.182

-.10

-.236

-.192

-.183

-.247

-.198

-.194

-.255

-.202

-.20

-T266

-.2O

-.21"

-.298

_-.21

_-.253

(21)

-20-TABLE c

COMPUTED PRESSURE DISTRIBUTION (P

NACA 1.75

5 - .

0.3 Face

Back

Fh

Fh j .817 1.414

4.47 .817

1.414

4.47

0 -.200

-.370

_.065

-.200

-.370

_.065

1

-.027

-.021

_.205

_-.461

_-.549

_-.072

2

.013

.085

.25

_-.55 -.531

-.108

3 4 5 6

.022

.133

.021

.151

.015

.150

.014

_.147

.2o3

.270

.20

.2o9

-.59i -.609

-.602

-.580 -.4 0

-.3i8

-.309

-.2

9

-.121

-.122

-.i18

-.n6

.0 2

.142

_.274

_-.545

-.i6

-.122

.05

.146

.295

-.507 -.149

-.152

9

.205

.170

.360

-.490

-.147

_-.246

= 1.0

C_______

0

-.085

-.014

.055 -.085

-.014

.O5

1

.035

.100

.163

-.212

-.135

-.OoO

2 .077

.138

.202

-.260

_-.179

-.104

.106

,163

.230

_-.294

-.208

-.17

4 .128

.180

.251

-.321

-.227

-.1o3 5 6

,L7

.191

.1b3

.200

.269

_.287

-.34

_-.30

-.242

_-.253

-.187

_-.209

.182

.207

.3O

_-.38ö

_-.262

_-.235

.209

.219

_.33o

-.42

-.277

-.275

9

.273

.251

.402

-.514

-.319 -.371

j=2.0

I o

i

-.046

-.00e

.0(5

.iOo

.0

.140

-.046

-.15

-.008

_-.129

-.04

.o1

2

.420

_.146

_.191

_-.226

_-.175

_-.1

3 3 4

.152

.17)

.177

.196

.223

.27

-.295

-.264

-.209

_-.234

-.1i2

-.212

5 6 .199

.213

.219

.229

.2o9

.290

_-349

-.323

-.257

-.277

-.240

-.268

7

,24

.26

.314

-.351

-.295

-.301

8

.27b

_.2b9

.34i

-.427

_.330

9 .350

_.323

_.424

-.536

_-.406

_-.464

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EXPERIMENTAL INVESTIGATION.

Experiments were carried out with a thin foil simulating the NACA 1.75 5 mean line. The foil and experimental set up are shown

in Fig. 5. The foil was made of a 1 mm thick brass sheet bent to the curvature given in Table 1. To this sheet were attached 18 brass tubes of ₃ mm outside diametre. At the central part of the foil span the tubes were pressed slightly oval with a minor dia-metre of 2 mm. This gives a total foil thickness of ₃ mm at the

central part of the span with a corresponding thickness/chord

ratio of

0.03.

At the leading and trailing edge the foil was given an elliptic thickness distribution for lO of the chord length. The pressure tubes were connected to pressure gauges as shown in Fig. 5. Due to lack of electric amplifiers, it was necessary to cut out pressure recordings from stations and

6.

The foil was tested in the towing tank and the recorded pressure distributions are given in Figs. 6 through 1. Here the recorded values are compared with those computed for the NACA _1.75 ₅ mean line of zero thickness.

The theoretical pressure

distribut. ns as given in Table ₃ and in Figs. 6 through 14 are only valid for zero foil thickness and in an ideal and non viscous fluid. Further the water surface is assumed to be flat and undisturbed. _{The experimentally recorded} pressure distributions on the other hand are influenced by the

finite thickness of the foil as well as the viscosity of the water. Concerning the first effect, we shall consider the influence of the foil thickness as being constant.

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loo

22 -Water sucked up to this level

before closing at

A

up

A T.E,

Pressure gauge

DETAIL OF FOIL

SECT/ON A-A

3mm4 brass tube pressed

oval at central part of foil

mm brass sheet

Measures in millimetres

-To pressure

gauge

To recorder

FIG. 5

L.E.

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This approximation is expected to be satisfactory since the foil thickness is very small compared with the submergence depth.

As regards the viscous effects, we note that the Reynolds' number for the tests varied between 4 1O and 6 In 7] are

described the variation of lift on 3 different profiles for a R variation from 4.2 . to 4.2 l0. The Göttingen profile Gö 625 with t/c = 0.20 showed large lift variations with R. For the flat plate at incidence less than approximately

550

the

liftvariation with R was very much less, and for the Göttingen profile Gö 417a the lift was approximately independent of R

variations inside the limitations mentioned above and for absolute incidence angles less than about 50 to 6°. The profile Gö 417a has approximately constant thickness with a tic ratio of about 0.03. We may therefore expect the lift on our foil to have been approximately independent of R variations.

In [8 _{are further described pressure variation with Rn for a} flat face - circular back profile with tic 0.10. For R variation from 0.38 . io6 to 2.6 io6 _{the pressure variation was quite evenly} distributed along the chord at an angle of attack of 2°. _Provided the same will apply for Gö 417a, we may draw the conclusion that the pressure distribution as recorded on the foil model is more or less independent of the variations in R.

The effect of the finite thickness may be estimated by con-sidering an elliptic foil in a uniform stream. The complex velocity potential is:

(27) UI

(25)

-24-where a and. b are the major and minor axis respectively.

Differentiating and putting z - ib we find the tangential velocity at the centre of the foil:

U _b

UI

(a-

--

U'

With a = 50 and b = 1.5 we obtain = 0.03 U'.

Further, applying Bernoulli's equation, we find:

LU'

U'

(U''

f _oUL

-

Z

1-_Ü-)

In these equations U' is equal to the free stream velocity U plus the horizontal velocity induced by the image system. Disregarding the induced velocity, we find

= O.O2

4

i ?'

and this will bring the recorded pressures into agreement with the theoretical ones for larger h/c values. For smaller h/c values we note that the pressures recorded on the face of the foil agree well with the theoretical results, whereas the suction recorded on the back generally is less than predicted. _{The suction top at the}

leading and trailing edge is of course due to the relatively large curvature of the foil surface at these points. _{The pressure} measure-ments at _{= 0.817,} 00, h/c _{0.3 were not successful, and due}

to the timetable for the towing basin they could not be repeated. Since no pressure recordings on a symmetrical profile cor-responding to the one tested has been carried out, we do not know

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the exact preure correction due to finite thickness in a real fluid and under a free surface. We may expect, however, that the correction given above is faIrly accurate as a mean value.

Still it is of no use to take ccount of the relatively small induced velocities in order to improve th accuracy of the thickness correction.

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-26-SUMMARY.

When regard is taken to the effects of the finite foil thick-ness, we may conclude that the shape and magnitude of the

theore-tical pressure distribution on the NACA 1.75 5 mean line has been confirmed by the experiments for larger submergence/chord ratios. At smaller submergence/chord ratios the suction on the back of the mean line is somewhat overestimated by the theory, however, the agreement is improved at higher speeds, i.e. at a depth Froude number of 4.47 or more. The depth Proude nuniber for the foils of hydrofoil boats usually rges from 5 to 10.

Moreover the pressure distribution has particular importance at very high speeds due to the cavitation danger, and this indicates that the theoretical method developed may be of special importance for investigating practical hydrofoils.

By studying the pressure distribution curves in Pigs. 6 through 14, we are able to draw two important conclusions: Firstly we observe that the pressure on the foil moves towards

the trailing edge when the submergence/chord ratio is reduced. This fact is certainly of great importance for some typos of foil control systems designed for improving the behaviour o± hydrofoil boats.

Secondly we observe that aS the speed increases, the lifting

effect of the foil is shifted from the suction side to the pressure side; the suction is reduced and the pressure is increased cor-respondingly. This agrees with the observations described in [sJ

that the image system of a vortex at high speed approaches a simple biplane system. In that case the induced velocity at the foil will

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be directed against the free stream velocity and thus lead to re-duced suction on the back and increased pressure on the face.

The two effects mentioned above are beneficial since they tend to reduce the danger of cavitation as well as air drawing,

ventilation. To avoid cavitation and ventilation is a major problem when designing subcavitating hydrofoils. However, the knowledge of

the circulation and pressure distribution on a vortex sheet in the neighbourhood of a free surface is also of major importance for

the development of supercavitating hydrofoils.

Studying the vortex distribution curves in Figs. 2, and k it is obvious that the thin, slightly cambered hydrofoil can not be substituted by a vortex at the quarter point when the sub-mergence/chord ratio is small. For the case of a flapped

hydro-foil, it is expected that the application of substitutional vortices at the quarter point of foil and flap would be misleading, and it is felt that this case can be conveniently studied by the presented vortex sheet theory.

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REFIRENCES.

[i] Bleick, W.E.:

[2] von Mises, R.: "Theory of Plight".

McGraw Hill Book Company, Inc., New York, 1945.

[3] Ausinan, J.S.:

[4] Benson, J.M, and _{"Ari Investigation of Hydrofoils} _{in the} Land, N.S.: _{NACA Tank} _{Effect of Dihedral and}

Depth of Submergence",

NACA Wartime Report, Sept. _1942.

-28-"Tables of Associated Sine and Cosine Integral Functions and of Related Complex-Valued Functions".

Technical Report No, 10, United States Naval Postgraduate School, Monterey, California, 1953.

"Experimental Investigation of the Influence of Submergence Depth upon the Wave-Making Resistance of an Hydrofoil",

University of CalIfornia, 1950.

"A Method of Calculating the Lift on

o Submerged Hydrofoils".

Norwegian Ship Model Experiment Tank Publication No. 71, 1ov. 1963.

e.

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Abbott, I.H. and "Theory of Wing Sections".

von Doenhoff, A.E.: McGraw - Hill Book Company, Inc., New York, 1949.

Riegels, F.W.: "Aerofoil Sections". Butterworth & Co. Ltd., London, 1961.

The British Shipbuilding _{"Experiments on Marine Propeller} -Research Association: Blade Sections".