Linearized theory of cavity flow past a hydrofoil of arbitrary shape

17 APR. 1973

ARCHSF

DOCUE1A bliotheek v -Onderafde - - e.ibouwkunde sehe Hogeschoo, DOCUMENTATI DATUM:

Lab.

y. Scheepsbouwkund

Technische Hogeschool

Deth

_{No. 21}

PAPERS

OF

SHIP

RESEÂÏCII

INSTITUTE

Linearized Theory of Cavity Flow Fast a Hydrofoil of Arbitrary Shape

June 1967

Ship Research Institute

Tokyo, Japan

By

a Hydrofoil of Arbitrary Shape

By Tatsuro HANAOKA Contents Introduction 2 Velocity Potential 2 Boundary Condition . 3 Integral Equation 4 Cavity Shape 7

The Solutions of the Integral Equations 8

Closure ConditionI 9

Drag 10

Hydrofoil Characteristics 11

Calculating Method of the Characteristics of the Hydrofoil of Which

Shape is Expressed by Polynomials in z 15

Closure ConditionII 20

Numerical Examples 21

Other Partially Cavitated Flow (li<li') 24

Appendix 26

Characteristic Functions 26

The Solution of a Singular Integral Equation 34

The Singular Behavior Near the Rear End of a Cavity and the

Drag Integral 35

Notation

x, z : Cartesian coordinates of the point of observation

Cartesian coordinates of singularity point V : Free-stream velocity

p : Fluid density

0 Perturbation velocity potential p : Flúid pressure

p : Vapour pressure

a Cavitation number a= 2p/(pV2)

a : Angle of attack (clockwise rotation taken as negative)

c : Half chord length

c0 Half cavity length

ô : Half cavity width

z+, z_: Upper and lower z-ordinates of the foil-cavity region Upper z-ordinate of the foil

Fig. i 2C ----d cavity .... 2C -¿2 CL, Lift coefficient

C}f Moment coefficient abOut the rear end of cavity for fully

cavitated flow and the middle point of the chord for partially cavitated flow (clockwise moment taken as

negative)

L= %/(l2-11)/(ll2) 1= -I(lÇ--l1)/(l2l)

¿= (x - x)/c0, x =(l + 1Ø/2

= (x - x0)/c, x0=(11+12)/2

The boundary-values of w and p in approaching the x-axis from

the upper or lower half-plane are denoted by w+,

p or

w_, p...

respec-tively.

Introduction

Present report is concerned with the problem of steady plane flow

of a cavitating hydrofoil, within the frame of the linearized theory.

The aim of this study is the development of a. theory which connects the gap between the theories of partially cavitated and fully cavitated

flows.

When we treat a cavity flow as perfect fluid flow, hypothetical

images are needed for various cavitation numbers. In this theory, the idea of relaxing the closure condition of a cavity is introduced. If the measured cavity-length is .introduced as a parameter, one finds better experimental agreement for lift. The th.eory is semiempirical and con-tains constants or functions to be determined by extensive experiments.

It is 'desirablé to obtain analytical prediction for cavity flow of a hydrofoil, and it may be possible, if experiments are performed

system-atically. It is hoped that the present analysis may shed some light on

future development of the theoretical prediction on a hydrofoil best fitted for cavity flow in characteristics.

1. Velocity Potential

A hydrofoil is placed in the uniform flow of a perfect fluid filling an infinite space. The stream ve'ocity is taken to be V. A' sketch of

z

the cavitating hydrofoil is shown in Fig. 1. In linearized theory, the analysis is simplified by fulfilling the surface boundary condition on the x-axis rather than on an approximate neighboring shape.

So, the velocity potential is given by

(Ï)(x, z)

5'2t1 xx'

dx'

22rV li z

+

1 51 dçt' In {(xx')2--z2)dx'

2r 110r11' dx

In these approximations, we will require that the velocity never differs too much from the free-stream velocity and that the slope of the body must be small. Therefore, we may neglect squares and higher powers of small quantities Subsequently, we obtain the linearized relation be-tween the velocity potential and the fluid pressure

from Euler's equation of motion.

2. Boundary Condition

If the median plane of the hydrofoil makes a negative angleawith

the free-stream direction, the cavity should extend along a portion of

the upper side of the x-axis. The case that the cavity extends along all portions of the suction side is teimed fully cavitated flow. So the

case that the cavity extends along a part of the suction side of the

hydrofoil may be called partially cavitated flow.

The linearized boundary condition may be stated as follows:

The velocity vector on the hydrofoil surface is parallel to the

sur-face. That is

hm

=V

dz

.+o dz dx

him d = y dz_

z.--o dz _dx

The pressure in the cavity is assumed to be. a constant,

hm pV-8--=p. for l<x<l.

dx

The condition at infinity is

(1.2)

=0.

(2.3) xoc

I

(2.1)

p.

That is (2.2)

4

(1.1) satisfies the above coñditiòn (23).

In order to hold a smooth juncture between the cavity wall and

the hydrofoil surface, the cavity should be forired so that the following

relationship is fulfilled.

do

dx x=11'

(2.4)

The condition of smooth flow at the. trailing edge is

(pp-)I=2=O-

(2.5)

The author adopts a. partly open, linearized cavity model which was introduced by A. G. Fabula [11°. That is

(2.6) The open width O is determined empirically.

To solve the mixed boundary value problem, we must ireat two integral equations simultaneously. The key of the method is to convert the two equations into one equation by elithinati.ng an unknown func-tion among two unknown fuñcfunc-tions.

3. Integral Equation

In this section, we will derive an integral equation for the solution of the bouñdary-value problem given in the above section.

Using (1.1) and (1.2), we have the following expressions

w=lirñ----

1 ,

dx'±°

(3.1) 2'rV

' xx

dx w_=Iim ÇI)

dx'----

(3.2)

2rV J z xx'

dx

p+=lim_pV!=P--

(V §12

'dx'

(3.3) x. 2 r z

or z' xx

p_=lim .pV=

pV"'

lço/4' dx'. (3.4) Ox 2 r 1 or 1'

xx

If we put

(z+z_)/2=,

(z+.z_)12=1 (3.5)

pp+=ì,

.

p-+p+=,

(3.6)

1) Numbers between square brackets refer to the iiterature listed at the end of the paper.

dx' dx

2irVJt1xx'

from (2.1) and (3.1)'(3.4). If we write

'=(zz_)/2,

='+ô,

we have from (3.8) dço

d'

+ y

do dx dx dx We also put = (z'. + z_)/2 = + ax = , + o dx dx = 2pV §12

_d/dx'

7 Z. or Z' XX dx' 2pV2 §1 _do

_i

r

i dx' CC'

where (3.7) (3.11) (3.12) (3.13)

for the convenience of following analysis.

For the case of the fully cavitated flow, (3.13) becomes

P=Z=Z*±aX (3.14)

as we may put '=0.

(i)

Fully cavitated flow (1=l, 1<1) 1f we write 2pV2 §12 d' _{dx' (3.15)}

dx' xx'

2r Z' (3.16) we get (3.17)

E=(xx)/co,

x=(l+l)/2.

(3.18)

r--6

In órder to get the solution of the boundary value problem, we

must solve two integral equations (3.7) and (3.17) of two unkflowñ

func-tions and dö/clx' simultaneously Fortunately, the analytical solution

of the integral equation Qf the same type with (3.17) is known in the field of thin-airfoil theory (see Appendix II).

When we introduce the juncture condition (2.4), it is written

If we insert the above result in (3.2) in the place of dçû/dx, two integral equations (3.7) and (4.17) are converted iñto an integral equation of .

The clear expression will be shown in the following section.,

(ii) Partially cavitated flow (l1=lÇ, l<l,)

In the case of the partially cavitated flow, it is rather convenient to the analysis to derive an integral equation of dö/dx than , because

is not continuous at the rear end of the cavity.

When we introduce Kutta's condition (2.5), the solution of the in-tegral equation (3.7) is given by

2

1,x

12 x' - l

d/dx'

dx'

V' r

xl1 J '

1,x'

x-r-x'

(see Appendix II).

Inserting (3.20) in (3.3) in the place of , we get

P4-

i

12x ç- ,/x'_l1

dI/dx' _dx'

pV2 r

x1, J ',

1,x' xx'

1 '2 dçi/dx'

dx'. (3.21)

.7rV J li xx'

As P+Pc for l,<x<l, (3.21) is written in the form

a 1 f'2' dö/dx' dx 2. r J 1 X-X'

1\/ 1,x §12' \/ x'-11

dö/dx' _dx' (3.22) x-1, i,

1,x xx

where

1,x §12 ./ x'-11

dI'/dx' _dx' r

xl1

',

1,x xx

i v/i+C

_f'

V/1_E' ' _dee.

dx r

1 J -'

i+C' c-e'

+1

§12 dz'/dx' dx'. r ,

xx

(3.22) is an integral equation of dô/dx. (3.19) (3.20) (3.23)

Li

4. Cavity Shape

Substituting (3.15) in (3.19), we get the expression of the cavity

shape. To perform the integrations, we use Poincaré-Bertrand's formula

[2]. §0 dt §0 ÇD(t, t1) dt1= îr2ço(to, t0) at-i-to a 'b dt1d ço(t,ti) dt, J a

(tt0)(t1t)

a<t0<b (4.1) and an integral formula

1

\/1 i

i

_de'

i-i

_{1-j-e' ee' x0x"+c0e'}

(

/X+1/(c0(C+X)X+i] for x"<11 or

O for ¿1<x"<l,

where X= (x0 x")/c0.

Subsequently, it is written in the form

dx dx dx

iJ1+e ç

Ji-e'

dC+2p/1

ie

j

i+e' e-e'

(i'

C12

d'

i

Ilx'

r 7T

1

1 dx' dx'

for l<x<l.

(4.3)

When l=l and l<1, the third term of the right-hand side is equal to

zero.

When we insert (4.3) in (3.2) in the place of dço/dx, the integral equation of 5 is written down as follows:

2pV

§1

xx'

dx'

2pV 1

§'

de'

p

4J1+ _VI+

11' S'2}

d'

i

\/dx'.

_(4.4)

pV 1

r

iC

'

dx xx 11x

This equation is fit for solving the problem of the fully cavitated flow as mentioned in the above section.

We can also get the lower boundary z_ of the cavity by integrat-ing (4.4) with respect to x. In such a case the second term and down-(4.2)

s

ward are to be omitted such that x<l and l<x.

Integrating (4.3) with respect to x from i,

to x, we get the

ex-pressiön of ò(x) as follows:

ò(x) ± '(x) = '(l) + C, cos' ( E) Ç'

/1-e'

dC'

2irpV2 j-i _1+e' p

° _Ç' ln dE'+PeCo 4irpV' Ji

_IE'EI

pV2

x {cos(E) 1Ç2 }+ cos-E)

_{;'5:}

x

- x' d' dx'

-

_--

1'+

1,x dx

r ¿

X tan-'

(\/l_x

tan

(e)

)

d _dx'

1,x

2 dx

When 1,<l, the. integral between the range l and 12

in the last two

terms Of (4.5) is to be omitted. We can calculate the upper boundary of the cavity by using (4.5) and the relation z+=z_+2'+2ô.

5. The Solutions of the Integral Equations

We can coñvert the integral equations in the same type as the

in-tegral equatiòn of a thin airfoil. So we readily get the solutions by applying the results of the thin airfoil theory t the integral equations.

(i)

Fully cavitated flow (l,=l, l<l)

Changing the variable from e to z, we get without difficulty, from (4.4) dz_ i x-1, c, £02 dx

2 lx

' ¿

/(x'1,)/(1x')

-i

_dx' X

(l. z) v' (l x) (x' i,)

Putting in the above equation

(1+E)L/2 = = {dz_/dxaL(1 +E)I4}(l-P(E') =,. (1 x)/(p V2) we have

A()=I

_§-,

L

/(l1,)/(l'--l)

lx

8/L2 _(5.2) Co (1±E)2+4/L2 f (5.3) (4.5) (5.1)

B(0)=

_I Ç'

4(8') dO'.

Jr J -1 8-0'

1= -/(l' - 11)1(12 - 12')

12x

8/12 C (1+0)2+4/12 (5.6)

This is also the same type as the thin airfoil theory. When we intro-duce the juncture condition (2.4), the solution is written

(5.7)

We can evaluate the cavity thickness by (5.7), if the relation between the cavity length and the cavitation number is determined.

Substitut-ing the solution (5.7) in dz/dx in (3.20), we get the lift distribution on the hydrofoil.

6. Closure ConditionI

Referring (5.4) and (5.7), we see that the hydrofoil characteristics depend only upon the relation between the cavity length and the cavita-tion number which is determined by the closure condicavita-tion, except for the foilshape.

Experience seems to indicate that satisfactory results will not be expected from only one kind of cavity models in the whole range of cavitation number. Therefore, it would be better that we pick up the cavity model or the closure condition which contains some empirical

4(0)

\/i+e

ç

/ 1-0'

B(e') do'.

Jt

1-6 J -1

1+0' 6-0'

This is the integral equation of the same type as the thin airfoil theory. If we introduce the trailing edge condition (2.5), the solution is written in the form

1 \/1 _'

_'/

_dE'

p2

7r

1+8 J-i

1-5' 8-8'

(see Appendix II).

We can evaluate the lift distribution on the hydrofoil by (5.4), if the relation between the cavity length and the cavitation number is

determined.

(ii) Partially cavitated flow (l=rl, l(l2) Putting in (3.22) I/2.(1+&)= "(x-lI)/(l2-x)

B(0)= /(l2x)(xl1) (Zo/2)/2,

4(0) = '/(l2x) (xl1) .dö/dx we have (5.4) (5.5)

lo

parameters. A partly open cavity model of which closure condition is given by (2.6) seems to be suitable for this purpose.

For the case of the fully cavitated flow, the condition is written down in the form

C0

4j1,

pde'

JrC0il

(6.1)

2pVt -1

_i+e'

₂

by using (4.5). Inserting (5.4) in (6.1) in the place of and perform-ing the integration by referrperform-ing the integral formulas shown in Appen-dix I, we have

1[+L2)

Jr C _{JrL3V'.Il+L2+l}

X

Ç' \/1±E

{(1+S)( %I1+L2_1)+2} dz* d5. (6.2)

i-i

1

(1+E)2-j-4/L2 dx

This equation (6.2) gives the relationship between the cavity length and the cavitation number for the case of the fully cavitated flow.

For the case of the partially cavitated flow, the closure condition is written

o_Çz2'dodx4Ç'

4(6) _d&.

it, dx

I

i-i

(1+6)2+4/P

Inserting (5.7) in (6.3) in the place of 4 and performing the inte-gration by referring the integral formulas shown in Appendix I, we have

ô2_Jr(7(/1+I2-1) 2'.'

c 4(1+12) 12

<'

_i-'

/1_e (1e){(1+e)11+I2_1_2J4/1±I2±flzde

i-i-e {(1+e)24/I2}2

(6.4) The equation (6.4) gives the relationship between the cavity length and the cavitation number for the case of the partially cavitated flow.

7. Drag

The drag coefficient may be defined as

(6.3)

This may be converted in

CD=

{5l2p dz_ dx+l2p dz

dx :2

p--dc

pV c ti dx ti dx t,' dx D CD= rag

-

1 ri2 _d 112

d'

_u'Z dx dx +' . (7.1) pV c _{pV c}2 p

p

, dx ', dx

-

dl dx+2 or d

_dx-2p2}

(7.2)

pVc ' dx dx

where the upper range of the integral 12 or l denotes 12

for l<1 and

l for l2<l.

Inserting (3.7) in (7.2) in the place of du/dx and (3.10) in (7.2) in the place of , we have

1 ( i _-' CD-pV2c 2pV2 dX \ 2 p dx' 11 .)Ii X_Xi 2pV2 522 or 2'

---dx

Çt2 Or 2' dz/dx' dx'-2pc2}. 7t ¿i dx .

xx'

The integrants of (7.3) being antisymmetric with respect to x and x', the first and second terms of right-hand side of (7.3) vanish, if and d/dx have not singularities in the range 11<x<12 .or l.

According to the thin airfoil theory, it is verified that the first

term vanishes, in spite of the fact that has a singularity at a leading edge in general. The second term does not vanish, because d/dx has

a singularity at the end of the cavity. The improper integral can be performed by means of the similar process to the thin airfoil theory.

When we write

dô/dx=A0(e)/ ' 1 +A(e)

(7.4)

where e=(xx)/c0, x=(l+l)/2, the final expression of the drag

coeffi-cient is written in the form

CD = irA(1)c0/ C + iö2/c (7.5)

(see Appendix III).

8. Hydrofoil Characteristics

(i)

Fully cavitated flow

When we perform the integration of (5.4), remembering (5.2), the expression for the lift distribution is written in the form

- -s/1 L2 1

-i='1i 2l+L2L1±1+L2}

+--L(1+5)+

2(4./1+L2_1)}] + 8e0 \/1- E ' \/1 ±5' 1 1 dz* d '

L'(lx)

1+5 J-i

1-5' (1+5')2+4/L2 S-5' dx'

(8.1) (7.3)

12

(see Appendix I).

Integrating (8.1) along the chord, we get the expression for the

lift coefficient CL

CL=i52

,dx C i,pV 2rc0('J1+L'-1) ( cL 16c,"

+L'-1

c(1H-L') \ 2 ) xV

_{)-' 1$}

11+E (.V'1+L2+1)(1+B')_2 dz*d. ((1+8)'+4/L')' dx

In a similar way as the calculation of CL, we get for the moment coefficient about the réar end of the cavity C

CM= 1 2c'pV' 1 2rC 8(1 + L')2c' ((2 ± L') (aL' a + cL) + 2(1 + L')"(acL)J 64c Ç' 4J1+8

i

Lc' .-i

1-8

{(1±E)'±4/L'J' (8.2) ><

rf

/1+L'-1(1( -11H-L' +1)(1+8)/2} L

/1H-L'1

L L((1+E)2+4/L'}

8.í(1±L')

x {

./1+L'( i./1+L' 1)±L'(2± /1±L')(1+S)I2}

]dz* (8.3) If we change the variable x into E by using the relation (5.2), the slope of the upper boundary of the foil-cavity region is expressed in the similar form to (5.3)

dz

_{cL(1+5) +} 1 Ç' P(5')

d'

dx 4

ir(lx) i-i E+5'+2

Inserting (5.4) in (5.3) and (8.4) in the place of P(E) and performing

the integrations, we get for the slopes of the upper surface and the

lower surface

21+L'

[1+L'-1(aL(1+E)+cJ

+ i/1+L'+1(cL(1+E)/2-2a}]+a

+

_irL'(lx)

8c,

\/_1

Ç'

\/1+E'

i

i

dz*

d'

8±1

i-'

1E'

(1+E')'-j-4/L'

EE' dx'

for 8>1 (8.5)

d

i

i

.l+L2_1[(l+)(l++3)

dx 4

E+i

_2(i+L2)

x{L+f(i+L2+i)}_2(/1_

E+3)

<

a(/i+L2+1)11

4c0

i

L

i]

rL2(lx)

'ETi

Ç'

/i+E'

dz*/dx

%"Si

_dE'

i-i

1E' (i±E')2±4/L2 EE'

E+E'+2

for 5>1.

A0(1) which is needed for the calculation of the drag, is deduced

from (8.7). Since urn ViE=2 f/(L(i+E)} by (5.2), A0(l) results

d .

2[

d

A0(i)=iim/1e--__=iim

L(i+E) dx

from(7.4). Whenx1,i+5oo, /Ei±/S+3--2/E+i, s/S+1

- E+3-+O,

_{and s15i--E+1.}

Applying these characteristics to (8.7), we get

Ai

'i+Li (

/i+L2+i

°

1+L L

2 /

Ç' + E'

i

dz*

_{d '}

7rL 3-' 1-5' (1+S')2+4/L2 dx

Inserting (8.9) in (7.5) in the place of A0(l), we get the expression of the drag coefficient.

(ii) Partially cavitated flow On using (3.20), we have dx

DZi

2f1+L2

[V/iTL2_i{aL(i+S)_)

+ %'./i+L2+i{oL(i+E)/2+2a}]+a 8c0 /5+3 Ç'

\/i+E

i

7rL2(l x) E + i i-i 1 E' (1 + E)2+ 4/L2 i dz" _dE'. (8.6)

E+E'+2 dx'

dz_/dx is known for S <1, because E <1 corresponds to x <12.

The cavity shape is obtained by integrating (8.5) and (8.6) with respect to x.

Substracting (8.5) from (8.6) and dividing it by 2, we have

(8.7)

(8.8)

14 CL=

l512d

cV ' 2 5z /x_1i

--dx-2 5'

\/1+

dl' d c '

1,x dx

-' 1 dx

where .= (x - x,)/c, x, = (i, + l,)/2.

Substituting (5.7) for dô/dx in (8.10), we get

c

I(.s/i±I2

1)

2

Ç'

/i+

dl' _dC 8

L

2(1+1') .L1

1.a dx

-x 5'

(i±e){(1+e)I" 1i+I2.+1 +2

/1+I2 _{Zd&. (8.11)}

-'

1+& {(l+e)2+4/12)2

For the moment coefficient about the middle point of the chord C, from a similar way as the calculation of CL, it follows that

CM 2V2 Ç"_4dx+ Ç %11_2 dl' d C2 J', J-' dx =215' _{v/ (1+e)2±4/12}ZdO +

5 ,""

dcc

4/s/1+I2

(Is'

s11TI'+1(2m'm')

+2 J s/1+I2_1(2m_m)}+

x{m2) (2m'

- mi") + 4- (m' m') (2m'

m2))} -(8.12)

By making use of (5.7), we can deduce the expression for the caity

shape

dò o'I'{(1+&)'+4/I'

_(2i°i"J

(1+e)'+4/I' Ç'

/i-e'

1+& Z

_de'.

(8.13)

+

2r

5'i-e'

i'-

1+9' (1+e')'+4/I' ee'

A,(1) is written in the form

Ao(1)=limf_s/1_e=sJ1212 iimf 5'ie

(8.14)

since

(8.10)

Substituing (8.13) in (8.14), we get the expression of A0(l).

9. Calculating Method of the Characteristics of the Hydrofoil of Which Shape is Expressed by Polynomials in x

We can find a simple method for the calculation of the hydrofoil characteristics, if the hydrofoil shapes are. expressed by polynomials of the form

for the partially cavitated flow.

(i)

Fully cavitated flow

Inserting (9.2) in (6.2), (8.2), (8.3), (8.9) and (8.1) in the place of

dz*/dx and performing the integrations by using the characteristic

functions shown in the Appendix I, we have

i= _2aL_

/lL2_l)_

(1+L2)/1+L2_1

x[('Vi+L2_1)ftn( 2e0

_{m2+2tn( 2c0 "ml}

n=0 \ n=0 c J CL = - 'r/L2 (s/l + L2 1) (acL/2) - 2r(1 + L2) /i + L2

- i/( /L)

X[(s1'l+L2+i)tn( 2:0 )mo-2tn( 2: )m'12]

or for the or dz* dz* ,n

(l_x

n=o

=tn

aÇ=

flow,

d,

\m m n and (9.1) (9.2) (9.3) (9.4) (9.5) (9.6) dx = b / 8c0/(cL2) 4/L2 i _(1+)2+4/L2) dx fully cavitated an C / ( where 1=(1+L2/2)/(1+L2) 'ii

= t+

i

(1-j-)2-j-4/L2 dx

d'_

dx - 1_2

r = dx (9.7) (9.8)

16

CM=ir/(8L4) . [(L2 + 2) (c(L2 1) + oL} + 2(1 + L2)312(a aL)]

_42r(1+L2)2/L4[2 _/1+L2-1 tn( ?c L2 _{i %"1+L2+1tn( 2e0} n=0 \ C / t _L( ./ 1+L2 _1)3/2 2c

) m/2

8"2 4/1±L2 n=o \ _e L8

/1+L2-1( 1+L2 +2)

16 1+L2)

2e0 'tm1

c I

i

A0(1)=

1+L2-1/ 1+L2

(a+a( s/1-j-L2+1)/(2L)} L _{'t( 2e0} 't (1)

-

_, ,, fllfl1 n=o \ C / and (9.10)

2e0 \/18

_t ( 2e0 '\"j(i)

_(')+

/1-5

pV2 lx 1+8 n=O

"k _{I %/2 /1+L2} V1

x[/1+L2_1{aL(1+5)+oi/2+ V/1+L2+1

X {aL/4(1+5)a)].

(9.11)

We can get the similar expressions of dz_/dz and dz/dx in a similar way as above.

Using the Tables I and II shown in the Appendix I, we can easily

calculate the above expressions.

If m is not a small integer, the above expressions are not suitable for the numerical calculation. The expression (9.3) is needed for this case and, for example, the lift coefficient is expressed in the form

CL r/L2( 1+L2 1)(traL/2)---ir(1+L2)

V1+L2-1 /(

íL)

>< { ( (9.12) where k 2c0 Ç" (4/L2)r(1+cosO)i

L-

4/L2 _{} dO.} (9.13)

/ o ((1+coso)2+4/L2Y ( (1+cos O)2+4/L2

It is hoped that the tables of ST will be constructed in future.

(ii) Partially cavitated flow We know the integral formulas

where or where C = -- 1-i

/1'')

dC'=b_1+b_2+ +b'2+b01

--

1

¿::

d'=h+h_1C+ . .

+hi+-in 1

135

dC

-ir Ji

2 2.4.6.. .

Inserting (9.4) and (9.5) or (9.6) in (3.23) in the piace of dl'/dx and d'/dx, and performing the integrations by using (9.14), we have

Za

_{X - ii}

Z=_a\/h2

./l2x

xl1

xl1 n=O

_i1nl2

_±oc

ir

xl1

dk= tbf_k_i+

tb_

j=k+1

jk

± s1b__1, gk= J oh1_, g.=O. J=lc+1 j=k+1

Inserting (9.15) or (9.16) in (6.4) and (8.11) in the place. of Z, we

have 4 o2

1+P

+%/±:(1+I2

_jdT°

4./1+121 r e /1+I2_1 V'1+I2 1 /7 m

ri

V2_{xl1 ,n=a} (9.15) (9.16)

2aI(/1+I2+1)

4 ö2 1+12

₊

,1114/1+12 _{d T°} 4/1+12_1 r

e ./1+P.

/l--I1

=° 7'

+

G7'T'-2

onL1)} _(9.19)

+

12s/1+12 Tj' (9.18) or

I

(9.17) (9.14)

or 18

and

c

JI(4./i_+I2_i) ira(s'1+12+i) 27rt(bn+bn+i)

2(1±12) 1+12

+ d 0)

+ 7112

f

(1)

/2 I1+i2 n=

_2s/s/i+I2

o" '

where and

c

trI(s/i+12_i)

r(s/1+I2+i)

27rt(bn+b+j)

2(1+12) 1+12 n=O +

_{s' 2 ,/J2}

+2s/i/1+I2

{0Gn')_20nL}

(9.21)

T:i)= Is' s'i+12 i(2T' T2)-2

5/ s'1+1 +1(2T' T',')

L[= is' s'i ±P 1 (2L', L2) 2

s' s'i +P+ 1 (2LV,LY,1)

= is' s'i +J2

_{+ i (2T,'' - T2).± 2}

s' s'i + P 1(2 T.

is' s/i+12+i (2

1L 2))+2 s' s/1+12 1 (2LYL')

(9.22) 4\\rÇr _(i+cos0) ii rk__\\_ji_) ) {(1+cos0)2+4/P}r I' (i+cos0)

i-io ((1+cos0)2+4/12}

where kCP denotes the binomial coefficient.

If k is not small (e.g. k>5), it is better

nO 81r2¡L . de (i+cos 0)2+4112) 8/12 (9.23) (1+cos0)2±4/12) x.ln(1+cosü)dO.

If we introduce m» and a function dèfined by

(i+cos e)' In (i +cos 0 _de

,r

_{\ j2) io}

{(i+coso)2+4/12p

T$and L$ are given by

k

T =

(- 2)kCP1flV?P p=O - k T(j)_ _(-2)kCPZ3?P .L.dr,k -pO to perform the (9.20) (9.24) (9.25)

integra-tion of (9.23) by means of numerical computaintegra-tion than to use (9.25).

If it is assumed that the hydrofoil is a flat plate and then ö=O,

it follows from the above expressions that

7=-2La

CL=-2rco/c( ../I+L2 1)

for the fully cavitated flow and

a/r=( /1+I2 1)/{21( /1+I +1)1

CL_2ra(/1±I2+1)

for the partially cavitated flow.

Geurst [3], [4], [5] solved the similar problem using the technique of conformal mapping and he assumed that the cavity is closed. His re-sults are as follows:

for the fully cavitated flow

where

c= 1+cosï

_cos2L-.

c0 2 2

and for the partially cavitated flow

o. T

a= tan

-2 2

CL

_{sin 1/2. (1+sin 1/2)} where

= c°

c 2 Since

-- =

i

tan r 1sin 1/2 o. 2 2 _1+sinh/2 CL=_7ra(1± 1 sin 1/2 F

c

l1

2 1 L2

- cos2 - and

T

sin -=

.1

.1

e0

l1

1+L2 2 2 /1+L2

for the fully cavitated flow and

(9.28) I (9.26) I (9.27) I (9.29)

20

C0

lli

12 _{- cos2 and sin} T 1

C

ll

1+12 2 2 1+I2

for the partially cavitated flow, (9.26) and (9.27) agree with (9.28) and (9.29) respectively.

10. Closure ConditionII

Since flows with cavities of finite length do not exist in the frame

of inviscid flow theory, several flow models have been proposed in an attempt to represent accurately the physical flow As they have the same boundary conditions, except for the closure condition, the various theories produce similar results, regardless of the model used. However

experimental results seem to indicate that the use of a sort of model is not physically justifiable in the whole range of cavitation number.

Espe-cially, the flow model theories seem to break down for cavity lengths

near to foil chords. It rather seems to be realistic and suitable to

aid the designer that we develop the theory of which closure conditions

are selected as a function of the cavitation number to agree with

ex-perimental data.

In this paper, the author proposes to select the cavity open width

o2 to agree with experimental data.

For the case of a flat plate, we have

a 1 %'l+L2+1 _(10.1)

2L-4/ir. (411+L2 _1)ô* 2L( 4/1+L2 +1)4 jrL2ô*

-for c0/c>1, and

%/1+P-1

21( .J1+P+1)_4/ir(1+I2)ö* for c0/c<l, from (6.2) and (6.4), where ò*=_52/(ca).

If we put

=--F1(c0/c). [2 +

and insert it in (10.1) and (10.2), we get

(10.2) (10.3) 411+L2+1

-c 2L( 1+L2 l)_L2F1{2+kF2/(1+L*)) for c0/c>1 (10.4) /1+I2_1 u 21( ,./1+I2 +i)_(1+12)F1{2+kF2I(1+L*)} for c0/c<1 (10.5)

where

F1(1)=1, F2(1)=1,

L*= /(i2l1)/jll2l

and F1, F2 and k denote empirical factors. So we have

a i

;;:- C0/C=1= 2k

k is determined by contrasting (10.6) with reliable experimental data.

11. Numerical Examples

In Fig. 2 the lengths of the cavities measured by a few researchers

c, 5-4 3 2 00

k

naD i j Theory O cz=8 is X atari'O.2 J 2.,3and 6 A 5 (flat pLate) i lberrnan (flat plate) Meli er (Symmetrical foiL) Nurnachi, Tsur,oda Chida (hydro foil O) A A A lo Fig. 2 A Experi ment 15

"(a

(10.6)

22 4.0 3.0 2.0 F,F LO 0.5

- Theory (fLat plate,

o -a Parkin(fLat plate) A _{-(a-a0J3.657Nuniachi} ITsunoda(035) \Chida / Fig. 3 Fig. 4 10 ---e 15 o-a0-a

are plotted as a function of a/(ao a), in which a0 is the angle of attack

at zero lift.

If F1 and F2 take the values indicated in Fig. 4 and k

takes 8, the equations (10.4) and (10.5) agree with the experimental

results, and then lifta curve agrees with experimental information

in outline as shown in Fig. 3. ö which is evaluated by using the values

of F1 and F2 indicated in Fig. 5 is shown in Fig. 4.

As the wake. thickness will correspond to instability of cavitation, the estimation of wake width will be important in real problem of

cavi-2

Fig. 5

_2a1(4/l+12+1)

4 ô2

1+P +2p1+12+1)

a

_c

_1+1--1

_{1+I2_1}

for c0/c<1.

We can evaluate p by the rest terms of the first and the second terms in the right-hand side of (9.7) and (9.18) or (9.19).

i

is a function of

L or I, and also it tends to a finite value, when c0/c-1, because it is

proved, through the recursions formulas, that m/(L) and T3(L) tend

to 1/L"2 as L-oc.

If we insert

ap

in (10.1) and (10.2) in the place of a, we can

carry out the evaluation for a hydrofoil with camber and thickness by the same way with the case of a flat plate.

In Fig. 6 we show a result of the evaluation with respect to a

hydrofoil expressed by the equations

Co/C

0.05 0.10 0.15

Fig. 6

'/c=0.04(0.3

l'/c=0.04(1-2).

In the calculation it is assumed that ô2=0. This hydrofoil is similar to the profile which was tested by Meijer [61. The theoretical results agree tation. The evaluated value of ô2 will be useful for the estimation of the wake width of real flow.

When the hydrofoils have camber and thickness, we may write, referring to (9.7) and (9.18) or (9.19),

a=-2aL---( 4/l+L2l)+2Lp(L),

for c0/c>1 (11.1)

24

with experimental data in outline, except for cavity lengths near to

foil chord.

In order to know the tendency of ô2, much more experiments are

wanted.

12. Other Partially Cavitated Flow (l<1)

In this section we treat the cavity flow problem that the cavitation occurs at an intermediate position of a chord. In these cases we know three kinds as shown in Fig. 7 (a), (b), (e).

we have, from (5.6)

2

(e) Fig. 7

We omit the detail descriptions regarding the case (c), because the problem can be solved by the same way with the fully cavitated flow, if we regard the length iT as the chord.

In the following analysis, it is assumed that the location of the

separation point l is already known.

(a) l2<l2 If we write

and

-

_L

§

2' 4wIX' dx'. 7rV li

xx'

If we write B(&)=±

--:

§

de' (12.2)

since dò/dx=O for 1<&(s.

Introducing a new variable 2 defined by

(&_s*)/so=2 (12.3) where (1+s)/2=s*,

(1s)/2=s,,

we have B(i)=

-I

_Ç

d2' (12.4) where B(2)B(&=soA+s*), 4(A)4(O=soA+s*).

As the equation (12.4) is the same type with the equation (5.6), the

problem. can be solved by the same way with the partially cavitated flow shown in the preceding sections.

(b)

l>l

We may write l for the upper range of the integral equation (3.7), because dI/dx*O for the range l2<x<1. The solution of the integral equation is written in the form

2\/lx tÇ1z' 4Jx'_lt du/dx'

_dx'

V2 r

xl1 J¿i

lx' xx'

We must notice that this solution is applied for the range 11<x<1. Inserting (12.5) in (3.3) in the pläce of ', we have

p+l\/l_x £l2 \/x'_ll du/dx'

_dx' pV2 r x-l1 J z

lx' xx'

(12.5) (12.6)

1 4/l_x

'2' _{Ix'-11 dz'/dx'}

dx'+1

2' d'/dx' dx' (12.7) r

xl1 J

xx'

xx'

26

J= 4/(l'

1')/(l - 1) , (1 (9)J/2 = - x)/(x - l)

B*(&)_{= 1/2(Z* - i/2) (xli), 4*((9) = (x 11)dô/dx,}

we get the integral equation of dö/dx from (12.6)

B*«9)=-If

4*((9()

(12.9)

ir

1

This is the same type with the equation (5.6). So the problem can be solved by the same way with the partially cavitated flow shown in the

preceding sections.

In order to simplify the calculation of the case that the cavitation occurs from the intermediate point of the chord; tables of new charac-teristic functions must be constructed.

The actual location of the separation point depends on several physi-. cal parameters. However, in order to predict the location, we may

adopt the rather simple condition that the pressure must be a minimum in the foil-cavity-wake region. As the minimum pressure generally oc-curs at the leading edge in the linearized theory, some contrivance must be done for the prediction of the separation point. The support of

non-linear theory will be required for the weak point of the non-linearized theory, after all.

Appendix I

Characteristic Functions

Here we shall examine the properties of' various functions defined by the equations

I / A \mCl i

(i.\i

dE

ir L2 J i-' -ViE _{{(1+E)2+4/L2}}

i( 4 ' (1+cosO) do (1.1)

YL) io

{(1+cose)2+4/Lz}Th i f A \n l i

i

Ii _L

'-'

rj

i ' \ P) .1-' 11_E2 ((1+5)2+4/L2}Th ''l 1(4yÇr _(1+cosO'Y dO' (1.2) ir \L2 J Jo ((1+cosO')2+4/L2} EcosO' Ic S(L)=I(_c__ çr (4/Ll)r(1+cosû)i

12

4/L° do (1.3) c i o _{{(1+coso)°+4/L2Y (1±cosO)2+4/L} } (12.8)

4 \\ 1

(1+E)1n(1±)

d \\ J2 J

/1E'2

{(1+E)2+4/I2 }

_1( 4

Ç (1 +cos )' ln (1+cos O) do P 1 i ((1+cosO)2±4/12}1

(i) m(L)

We introduce a function z defined by

k(a)=_ çr do (1.5)

22r Jo (a+cosO)k

where a is a complex number. If we put e°=z in the integral, we get

1

§

(2z)1c_1

dz (1.6)

2ri

{(z+a ,./a2_l)(z+a+

a2_1)}c

where the contour of the integration in complex domain is taken to

be the circle _IzI=1. We have

Ja+s/2_1I<1,

ja-1a2-1j<1

or

Ia+sIaz_1I<1,

_IaVa2lI<1

since (a+ /a2_1)(a_ /2.._1 )=1. Therefore, the integrand of (1.6) has

the only pole at the point z=a+ s/a2_1 or z=a a2_1

in the

circle in general.

Subsequently, by means of Cauchy's theorem, we get

f 2c_I ( d \k-1

(k-1)! dz)

(z+a+ /a2_1)

2=_a+ s"a-1

for Ia_a2__1I<1

(1.7)

2k1 _{( d \\k_1}

(kl)!

dz J

(z+a /a2_l)k

2=a-\

for Ia+a2_1I<l.

Hence

r1(a)= ± 1/(2 /a2_1), z2(a)= ±a/12(a2-1)3"°},

(I 8) v3(a)= ±(2a2+1)/{4(a2-1)°"2)

where the upper or lower sign is to be taken according as a /'a2_ 1 1<1

or Ia+a2_1I<1.

If a=1i2/L, we have Ia+ %h'a2_1 <1. Writing 'rk(1i2/L)=Tk(L), we have, from (1.8)

28

L /V1+L2_1

_I77T1(L)--L

ReTi(L)'%1

2(1+L2) ' 2(1-l-L2) ReT2(L) L

-- 4(1+L2)

{L2+3 4/1+L2}ImTl(L)

IT2(L)

L (L2+3+ h1+L2}ReTi(L) - 4(1+L2) L2 Re T3(L)-32(1+L2)2 (3L4+7L2+16+2(L2±4) 4/l+L2)RTj(L) L2

IT3(L)=

-32(1±L2)2 {3L4+7L2+16-2(L2+4) 4/1+1-'2}1mTl(1i) The fünction m» is derived from T as follows:

m°(L)= i2{ T1(L) 1(L)}/L=I,,4T1(L)/L

}

m'(L) = 4f T(L) + p1(L) } ¡L2 = Re8 T1(L)/L2

where denotes a conjugate complex function of T. And then

1

((

1 1 2

m°(L) =

- L2

tcosO + 1 i2/L cos O + 1 + i2/L } dO = R4T2(L)/L2+l/2m°(L)

2 ( 1 1

m(L)

jO

(cosO+1i2/L)2 (cos&+j+i2/L)2 }do =ImST2(L)/L3

jC

1 1

m°(L)=

L3 (cos O+1i2/L)3 (cos O±1+i2/L

12Cm i + L4 {(1+cos O)2+4/L} = - 1m4 T3(L)/L3 + 3/4 Çr [Ç _{1 i2/L} m3 ( )

- 7rL3 ) L (cos O + i i2/L)2

+

(cos O + 1 i2/L)3

1 i2/L _lido

I (cos O+1+i2/L)2 (cos G+1+i2/L)3 J J

+ 12 V 1+ÇQO do

rL4 30 {(1+coso)2+4/L212

= 1m2 T2(L)/L3 - Re8 T3(L)/L4.

When j)1 and n>3, it seems better to evaluate m/ by means of

recursions formulas shown in the following. A recursions formula

(1.9)

= 4/L2. (m2 m/2}

(1.12) is readily derived from (1.1). When the values of ?n$° and m are given, we can calculate m for j> 1 by the use of (1.12).

Let us derive the recursions formulas for in and If we

in-tegrate the both sides of the equation

d sine cosO-I-2k(1cosO) 2k(1cose).4/L2

de {(1+cos O)2+4/L2}k {(1+cos O)2+4/L2} ((1+cos O)2+4/L2}' with respect to O from O to 'r, we get the formula

m5°-I-(2k 1)(2m°m5») 2k{2m1m1 =0

(1.13)

because the left-hand side of the equation becomes zero. And also we can derive the formula, by the same way

8/L2 (k 1)m5°21m52+ (2k 1)(4/L2 . n5° +2m}

+4/L2 (2k-1)m-2k(4/L2 . m1+2m} =0

(1.14) from d (1+cosO)sinô de {(1+cosO)2+4/L2}'

2(1k)

₊ (4k-3) (1 +cos O) + 8(2k 1)/L2

{(1+cos O)2+4/L2}' {(1+cos O)2+41L2P

8k(2(1+cos O)+4/L2}/L2

((1+cosO)2+4/L2'

Multiplying (1.13) by 2 and adding it to (1.14), we get a recursions

formula

m'= SIL2. (k 1)m21+ (4(2k-1) (1+2/L2)+2}mj

8k(1+1/i2)m1.

(1.15)

We can derive the value of m' from m by the use of (1.15).

Eliminating m' and m' from the equations (1.13), (1.15) and the

equation in which k+1 is inserted in the place of k, we get the

re-cursions formula of If we insert k-2 in the formula in the place

of k, finally we get

8/L2 (2k-5) (k - 3)m5°23_{- {}16(k - 2)2/L2 1+4(2k-5)2 (1 +2/L2) )m°22

+ 8(k 2) ((2k-5) (1 + i/L2) + (2k-3) (1 + 2/L2) ) n°2

i6(k-2)(k-1)(1+1/L2)m°=0.

(1.16)

This is the recursions formula of m» which is used for the calculation

30 0.2 0.3 0.5 1.0 15 2.0 2.5 3.0 Table I Table II _i11'(E) -0.9 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 (i.i)

ji(S)

0.8 1.0

We can readily derive the equation

i

i

((i+')2+4/L2}(E-')

±E+4IL2 (l+S')2+4/L2 Therefore is wtitten Table III L 0.2 .975762 .952260. .929470 .907376 .885944 .865159 .845010 03 .947454 .898349 .852444 .809513 .769348 .731757 .696559 0.5 .869148 .759293 .666825 .588776 .522704 .466598 .418801 1 0 643594 435570 310730 233353 183623 150391 127279 1.5 .46S575 .254212 .159863 .113669 .0882950 .0727600 .0623440 2.0 .35Ï578 .162360 .0967444 .0686673 .0539688 0449744 .0388508 2.5 .273326 .112559 .0658620 .0471808 .0374304 .0313796 .0272084 3.0 .219205 .0830910 .0484623 .0350263 .0279498 .0235085 .0204250 L m m2 m2 m m2 m 0.2 1.457645 1.416638 1.376935 1.338491 1.301265 1.265216 1.230304 0.3 1.408341 1.322971 1.243436 1.169316 1.100220 1.035790 .975693 0.5 1.2729?7 1.084012 .926540 .795007 .684901 .592519 .514820 1.0 .892452 .549342 .351670 .235056 .164339 .120095 .0914528 1.5 .610390 .274136 .139030 .0802299 .0520496 .0370577 .0282393 2.0 .431136 .148931 .0650044 .0355540 .0230186 .0166166 .0128437 2.5 .317031 .0884555 .0352526 .0190704 .0124948 .0091:2329 .00710519 3 0 241691 0565797 0213508 0115928 00767648 00564353 00441282 - .979254 - .975594 - .99318 - .963633 - .958254 - .953073 - .948032 - .955202 - .947457 - .934210 - .922238 - .910934 - .900062 - .889502 -.889618 -.871613 -.841045 - .813613 -.787858 -.763212 -.739379 .707224 - .667909 - .602659 .545449 - .492791 - .443323 - .396355 - .568529 - .521985 - .444355 - .377713 - .317555 - .262161 - .210710 - 477215 - 427975 - 348956 - 282184 - 222825 - 169082 - 120161 - 413500 - 365243 - 288575 - 22425 - 168232 - 117927 - 0728919 - .36503 - .321039 - .247757 - .181046 - .134138 - .087S280 - -0459732 0.2 0.3 - .943099 - .938252

-

.878292 - .869051 - .933476 - .859081 - .928758- .849245 L Cj/C + L2 -1 0.5 - .716184 - .693513 - .671284 - .649439 _{02 26.0 1407263} 1.0 - .351489 - .308485 - .267196 - .227545 _{03 121111 .2098348} 1.5 2.0 - .162798 - .118276 - .0757819 - .0360340 - .0771996- .00142413 - .0398488.0268581 Q 5_1.0 5.0 _£4359433435607 2.5 - .0329831 .00145040 .0293669 .0482075 _{1:5 1.4444 .8959775} 3.0 - .01o04a8 .0198993 .0423141 .0530225 _{2.0 1.25 1.1117859} 2.5 1J6 1.3009929 L / 1+ 3.0 1. 1111 1.4704685 2cojc-(1+C)

i

l( 4 \Ç1 \/l+s'

i

n

(i+5)2+4/L2 r \ L2 / i-i

lE' {(1+5')2+4/L2}'

±

dg" (i+5')2±4/L2

EE'

4/L2.i1(E)

+ 1 ((1+ )m'+m2>}

(1+E)2+4/L2 (l+5)2+4/L2 n n

By the same way we get 4/L2.

(l+5)2±4/L2 +

(1 +E)2+41L2

((l+E)m+4/L2m'21-4/L2 m}.

(1.18) These are the recursions formulas of ji') and i.

Using the integral formulas i Ç'

i

;)- %/lEF2

dE'=(

we can easily perform the integration when n=0. Consequently, we have

, _(

i,

io ()

_{_1_1/E2_i,}

151>1 (2)

1 (2±5),

151<1

io ()=

_(2+5)_(1+E)2/E2_1,

151>1 (iii)

The recursions formulas

S(i)kI,r+1 k-1,r(j) C/\ Q) k,r(j)

S = 4/L2 (S J

are readily derived from (9.13). The other recursions formula

kJk,Q

2k(- r L2

_{)[-- k-1,2}s2 {sk'2lI_s'2l2)]

2S-3S)=2k(__9_)[2{S"2k i 1'.'ki,2J t-'k-1,1r Q(2) S2k-1,2_J]

are obtained by the same way with the case of m and

When we expand [2 4/L2 1/ ((1 + cos 0)2 + 4/L2 J ]c by binomial theorem

o, .1/ %/52_1, 151<1 IEI>i (1.17) } (1.19) (1.20) (1.21)

32

and then perform the integration of S and S term-by-term, we thus find that

S = (2c0/c)° { kGO mSÏ - kClAk 'm?1 + + (_

)IC(i))

(1.22)

where 0Cp denotes the binomial coefficient.

If k is small, this equations are applicable to the calculation of S.

(iv) X

We introduce a function 2k similar to r defined by

20(a)=--- Ç" in (2 cos 0/2) do 2r io (a-fcos 5)°

where a is a complex number. If we put e°=z in the integral, we get

A1(a)

i

_§

li+)

dz (1.24)

(z+a /a2_i )(z+a+ /a2_i)

where the contour of the integration is taken to be the circle Iz = 1.

By means of Cauchy's theorem, .we get

we have

/ ln(ia+ /a,2_i)

2 ,/ä2f

ln(ia a2_1)

22_i

Since

when I

a+ './a ii <1, we have

21(a)=r1(a) in (ia s/a2_i)

22(a) = r2(a) in (1 a - ,1a2 1)

i ± /a

i

2(a2l) Ïa s/a2_1

23(a)=r(a) in (ia Va2_i)

3a(ia Va2_i) - 'Va°1

_{8(a2i)°(a-1)}

(1.23)

for Ia_'a2_ÏI<i

(1.25)

for Ia+ Va2_ÏI<i.

If we put

a = 1 i2/I

21(1i2/I)=i11(I)= T1(I) in X

2(1 - i2/I) = A2(I) = T2(I) In X+ (I/2)° ((I. R0X 1X)

+ (ReX±IImX) }/(4(1 +12) }

23(1i2/I)=A3(I)= T3(I) in.X - (I/2) (1iI)2 [3(1 + iI/2)X

-t

( V1+12+1)/2+i (

1+I

_1)/2)]/{8(1+12)h}.

We introduce a function defined by the equation

''L'

( 4 \\ _{(1+cose)1nl2cosO/2I} de Pn P ) Jo {(1+cos0)2+4/I2} It is evident that (1.30)

This function p is obviously expressed by Ak; namely

p=4/I.IA1,

p=9/I2.R0A1,

pO)= _{4/I2ReA2+ 1/2. y(0) p'=8/I3. 1A2,}

= - 4/1 1m113 + 3/4 pO),

2/12. InÀ2 8/Ii. R0A3.

We can derive the recursioñs formulas of XP by the use of the

equations

d sinô(1+cose)ln(1+cosû)

de {(1+cos0)2+4/Iz}k

{cos0(1+coso)sin2O}ln(1+cosO)sin2ü

{(1±cos e)°±4/12P

+ 2k(1+cos 0)2 sin2 0ln (1+cos 0) ((1+cos0)2+4/I2}'

d sin .0(1+cos 0)2 in (1+cos 0)

de {(1+cos0)2+4/12}"

(1+cose)[fcoso(1+cosO)-2 sin2O} in (1-1-cos 0)sin 0]

{(1+cos0)2+4/I2}°

+

2k(1 + cos e) sin2 O In (1 + cosO)

((1+cos0)2+4/I2}°

In order to get the recursions formulas, the similar way to the case of m,9 is to be taken..

The results are written in the form

(1.31)

(1.28)

34 x'= 4/12. {X(J_2)_X(i_2)} X°+ 2(n Ï)[8 (nX (n - i)X' + 4/J2. (2n 3) (X'21 - Xi») _4/12. 2n(X» X1)

+ 4m n$

2n[8 f (n+ i)X2 nX/ }

±4/12. (2ni)(XX1)-4/I.2(n+1)(X1X2)

± 4m1 m1] - (2mi» m) =0

72)_{= (8n 6)X2 8nX1 (2n 3). 4/P. (X'1}

+2n4112 (XX1)-4m2±m"=O.

Appendix II

The Solution of a Singular Iñtega1 Equation

The integral equation

g()L "

f(E') de' (11.1)

27V J-' ¿e'

which ppears in thin airfoil theory, was studied by many researchers

(e.g. K. Shröder [7], H. Soehngen [8], J. Kondo [9]).

The general solution of integral equation (II.Ï) is given by

k 2 ,Ç'

v"iC"

J'

/ 7tS/i 7V 1C2 :i'-1

¿e'

g

where k denotes añ arbitrary constant. If f(C)1e_i=O, k is determined by

[

i_e2f(e)]i=O=k+25'

:: (E')'

Substituting this in the solution (IÏ.2), we get

f(e)_2/i+e

'

Í--e

(e')..P

r

ie J -i

i+e' e-e'

If f(C)L'=i=&; k is determined by

[ Vi_e2f(e)]O=k_

Substituting this in the solution (11.2), we get

? 4Ji_ ç'

Ji±'

'r

i+eJ_i

ir-E' E.--E'

-(i.32)

(1L2)

Appendix III

The Singular Behavior Near the Rear End of a Cavity and the Drag Integral

The drag integral (7.3) contains two repeated integrals of the form

IT

512f(x)dx §12 f(x') _dx'.

(111.1)

2'r ii

xx

where the function f(x) has a 1/ singularity with d from an end

(for example ¿2) of the range. If the interval (l2, ¿2) is excluded, the repeated integration (111.1) may be considered to be a double integral, since the function f(x) is regular over the range of integration. The

double integral for this region must be zero because of the antisym-metric character of the integral. It remains then to consider the

ex-cuded range, namely

11_L

512 f(x)dx 512 x') dx'. 2r 2 2 x x Lét x 1 + 2. Then we have

I '

Ç° f(x)dx5° f(2') d2'. 22T J'

-' 2-2'

Now, suppose near the end (2=0)

so that

2a ' _d Ç' 1

i

VL1

-

-

2.

Jo /2 Jo /2'

2-2'

Integrating with respect to 2', we get

12a Ç'ln1'±d2.

Jo

1/2/

If we put 2/e=t2, we have

4a

_{5'ln 1±t dt=ira.}

7r o

it

Consequently, we see from above result that the second term of the drag integral (7.3) contributes to drag.

36

The singular behavior of the first term of (7.3) near the leading

edge is easi]y seen to be related to the effect of the leadingedge sin-gularity on drag.

The amount of drag derived from the first term by the previous integration is to be removed by the introduction of leading-edge suction as shown in thin airfoil theory [10].

References

Fabula, A. G., "Application of Thin-Airfoil Theory to Hydrofoils with Cut-off Venti-lated Trailing Edge,' NAVWEPS Rep. 7571, China Lake, California, Sept; 1960.

Muskhelishvili N I Singular integral Equations Groningen 1953 p 56

Geùtst, J. A., ." Linearized Theory for Partially Cavitated Hydrofoils," I.S.P., Vol. 6, No. 60, 1959.

[4]. Geürst, J. A., "Linearized Theory for Fully Cavitated Hydrofoils," I.S.P;, Vol. 7, No.

65, 1960.

Geurst, J. A. and Verbrugh, P. J., "A Note on Camber Effects of a Partially

Cavitat-ed Hydrofoils," I.S.P., Vol. 6, No. 61, 1959.

Meijer, M. C., "Some Experiments on Partly Cavitating Hydrofoils," I.S;P., Vol. 6,

No. 60, 1959.

Schröder, K., "Über eine Integralgleichung erster Art der Traflügeltheorie," Sitzungs-berichte der preussischen Akademie der Wissenschaften, 1938, Phys-math. Klasse.

Soehngen, H., "Die Lösungen der Integralgleichung g(x)=---- § d' und

deren Anwendungen in der Tragflügeltheorie," Math, z. Bd. 45, 1959.

Kondo, J., "Finite Hubert Transformation and Airfoil Equation," Proc. 9th Japan

National Congress for Applied Mechanics.

Grammel, R., "Die Hydrodynamischen Grundlagen des Fluges," s. 21, Braunschweig,