APPENDIX I

THE POTENTIAL OF AN ELEMENTARY SUBMERGED

The fundamental lifting solution f o r f r e e - s u r f a c e flows i s d e r i v e d f r o m the fundamental s o u r c e - l i k e solution. If y is v e r t i c a l l y upward and the s t r e a m velocity i s c in t h e positive x-direction, the potential of a s o u r c e of s t r e n g t h m located a t (0, - f , 0 ) is, in the f o r m given by Havelock (Ref. 7):

Tr

/a oO

2 s e c a 0

S

n ( ~ - f ) c o s ( n x c o s 0)cos (nz s i n e ) dud 0

n - ⁿ

s e c 2 0

0 0

w h e r e

n o

= g/ca, r l

=

x 2 + ( y + f ) = t z a , and r a a = x2

+

( y - f ) 2

+

^z2 The f i r s t t e r m g i v e s the s o u r c e l i k e behavior a t (0, -f, 0); the second t e r m and the f i r s t i n t e g r a l (of which the principal value is t o be t a k e n ) a r e a l o c a l s y m m e t r i c d i s t u r b a n c e ; and the second i n t e g r a l is t h e s u p e r - position of f r e e s u r f a c e waves that m a k e up the wake. The e n t i r e ex- p r e s s i o n s a t i s f i e s the l i n e a r i z e d f r e e - s u r f a c e boundary condition, mxx ⁺ Xomy

=

0, on y

=

^0. Consequently d e r i v a t i v e s and i n t e g r a l s of

I h a s been pointed out t o the w r i t e r that the s u b m e r g e d lifting singu- l a r i t y h a s been p r e s e n t e d previously, f i r s t by Wu (Ref. 21, 1954) f o r t h e c a s e of

p =

0 in the p r e s e n t notation, and l a t e r by o t h e r a u t h o r s f o r g e n e r a l

P.

The appendix i s included because the a p p r o a c h i s quite different f r o m Wu's and because t h e p r e s e n t w r i t e r h a s not been able t o locate another derivation.

H a l s o s a t i s f y t h i s condition and c a n be used t o c o n s t r u c t solutions corresponding to different s i n g u l a r i t i e s a t (0, -f, 0).

The fundamental lifting solution i s a bound v o r t e x e l e m e n t of infinitesimal span c and infinite s t r e n g t h

I?

s u c h that Tc i s constant and equal t o the lift f o r c e L divided by pc, the product of density and speed, according t o the Kutta-Joukowsky lift t h e o r e m . To s a t i s f y Helmholtz' t h e o r e m s the bound v o r t e x f i l a m e n t i s continued into a

p a i r of f r e e trailing v o r t i c e s of s t r e n g t h

r

and s e p a r a t i o n a, which extend p a r a l l e l t o ' t h e x - a x i s t o infinity.

The h o r s e s h o e v o r t e x flow i s identical with t h e flow induced by a line doublet extending f r o m (0, - f a 0) t o infinity p a r a l l e l t o the x-axis, the doublet a x i s pointing opposite to the lift v e c t o r . T h i s c a n be thought of a s a p a i r of line s o u r c e s of s t r e n g t h j? and s e p a r a - tion a, o r a s a distribution of doublets of strength/unit length

re.

The potential of one doublet a t (0, -f, 0) having s t r e n g t h m e and a x i s point- ing opposite the lift d i r e c t i o n i s

a a

^'

m c ~ ( x , y, z ; ~ , - f a 0 ; ~ )

=

ma (sinp

-

^cosp

^p)

^{~ ( x ,} y, Z;O, -f, 0) (1-2)

Using the e x p r e s s i o n for H a'bove (I- 1),

By changing c o o r d i n a t e s i t i s e a s i l y e s t a b l i s h e d that

is the potential of a s i m i l a r doublet a t

( 5 ,

-4, 0).

The next s t e p i s t o i n t e g r a t e over a d i s t r i b u t i o n of s u c h doublets along the line y

=

-f, z

=

O. In o r d e r to obtain convergent i n t e g r a l s it is n e c e s s a r y t o c o n s i d e r the d e s i r e d uniform d i s t r i b u t i o n as the l i m i t - ing c a s e of a broader f a m i l y of distributions. A family of potentials i s defined by

C3

Va(x, y, z;O, -f, 0;P)

= 5

^{ema5 D(x, y,} z;5, -f, O)d5 (1-5)

0

Then the d e s i r e d v o r t e x potential i s

t - H .

-

n(!"-f)sin(xxcos 0 ) s e c

e

IT 0 pc

0

n -

^{noseca 0}

[sinpsin0sin(nzsin0)-cosp c o s ( x z s i n e ) ] dn d0

+

²

ⁿ L L,

[sinp s i n e sin(% z sin0 s e c 2 0)-cosp cos(x0z sin8sec?l)]

0 P C 0

w h e r e the s t r e n g t h h a s been identified with L / ~ C . Again, t h i s e x p r e s - sion i s t o be r e g a r d e d a s composed of the s u b m e r g e d v o r t e x r e p r e - s e n t e d i n t h e f i r s t t e r m , the l o c a l d i s t u r b a n c e of the second and t h i r d t e r m s , and the f r e e wave p a t t e r n of t h e l a s t t e r m . The f r e e wave p a t t e r n a t a d i s t a n c e d o w n s t r e a m c a n be c a l c u l a t e d f r o m the l a s t t e r m according t o

L

"/,

-x0f s e c a 0

-

- -

²ⁿ⁰

-

sin

p 5

sin0 s e c 4 0 e

pea - 72

c o s ~ ~ o s e c 2 8 ( x c o s 0

+

z s i n e ) ] dB

I T

L ^/2

- n

f s e c a 0

--

^{C O S P s e c 4} 0 e ⁰

-

^2xo

- 72

s i n

[no

s e c a 0 ( x c o s 0 + ^z s i n 8)]de (1-7)

Using t h i s r e s u l t the wave r e s i s t a n c e c a n be calculated by the m e t h o d s of Havelock (Ref. 19). It i s i n t e r e s t i n g to note t h a t for the c a s e

P =

0 (lift vector v e r t i c a l ) the wave p a t t e r n and r e s i s t a n c e a r e identical with those of a s u b m e r g e d s p h e r e whose volume i s 2 ~ / 3 n ₀

pea.

APPENDIX I1

MODIFIED METHOD O F STATIONARY

PHASE

This appendix d e a l s with the a p p r o x i m a t e evaluation of inte- g r a l s of the type

=

C

^{Y(X) e} i f ( x ) dx

(II- 1 ) with s p e c i a l attention t o c a s e s in which cp(x) h a s a z e r o a t a s t a t i o n a r y point of f(x). The notation and t r e a t m e n t a r e s i m i l a r t o L a m b ' s

(Ref. 5, Art. 241). cp(x) and f ( x ) a r e r e q u i r e d t o be analytic a t a s t a t i o n a r y point a s u c h that f ' ( a ) = 0. Then, writing

5

= x

-

^{a , we}

have

(II- 2 )

(II- 3 )

since cp(a) i s a s s u m e d z e r o .

P r o v i d e d the quotient f'lf(a)/l f " ( a )

1%

i s s m a l l , s o that the t h i r d t e r m in (11-2) c a n be neglected, the important p a r t of the i n t e g r a l , coming f r o m the neighborhood of a, i s a p p r o x i m a t e l y

1 I/

2 ^{-L 4")}

^{( a )}

^{\ ee}

zif ( a ) .

ta

. n ! d c

F o r odd n the i n t e g r a l in (11-4) v a n i s h e s by virtue of the integrand being a n odd function of

5.

F o r even n we have the i n t e g r a l , f o r a a positive integer,

which i s established by r e p e a t e d differ entiations of

with r e s p e c t t o m . The p r i n c i p a l contribution t o (11-4), then, is f r o m the lowest even d e r i v a t i v e of cp that d o e s not vanish a t a. If t h i s i s t h e d e r i v a t i v e of o r d e r 2a, we have by (11-5)

u - 1

J;;

l e i [f ( a ) f r / 4 ] ( 2 a )

cp ( a ) (11- 7 ) ( 1 2 i ) ~ a l

I

^fll(a)

1

^a+s

w h e r e the f sign i s taken according t o the sign of f"(a).

If a coincides with one of the l i m i t s of integration i n (11-l), the l i m i t s i n (11-4) m u s t be 0 t o m o r ^-00 t o 0. In that c a s e we u s e the f o r m u l a for odd n

The principal contribution t o (11-4) is now f r o m the lowest d e r i v a t i v e of cp that d o e s not 'vanish a t a. If t h i s d e r i v a t i v e is

-

even the r e s u l t i s half of (11-7). If the lowest nonvanishing d e r i v a t i v e is odd, t h e r e s u l t i s

a+1 z a a ! e if ( a ) ( 2 a t 1 )

,fll(a)~a-+-l (a).

APPENDIX I11

MAPPING O F A SOURCE DISTRIBUTION

Under a c o n f o r m a l t r a n s f o r m a t i o n a s o u r c e of finite s t r e n g t h located a t a c o n f o r m a l point of the mapping g o e s into a n equal s o u r c e a t the mapped point (Milne-Thompson, Ref. 12). To s e e how a con- tinuous s o u r c e d i s t r i b u t i o n along t h e r e a l axis, of s t r e n g t h m(z;x), i s mapped, c o n s i d e r a s m a l l r e c t a n g u l a r r e g i o n n e a r t h e z - a x i s

--

^a

two-dimensional "pillbox" of height d and length dz.

D m a p s into the closed r e g i o n A in the a-plane, s y m m e t r i c about the c - a x i s . The net efflux through 8D is d) = m ( z ; x ) 6z t 8 ( 6 z 2 ) . Since t h e s t r e a m - f u n c t i o n J( h a s the s a m e value a t corresponding points i n t h e two planes, the i n t e g r a l

6;

d) m u s t be a l s o equal t o m ( z ; x ) dz

+

0 ( 6 z a ) . Now the height d c a n be m a d e a r b i t r a r i l y small,

$A

in which c a s e the boundary a A c o n v e r g e s toward the segment 65 while the i n t e g r a l dq is constant. This shows t h e r e a r e s o u r c e s along the c - a x i s , s a y with the s t r e n g t h y(c;x) w h e r e

5

= y(z;x), s u c h that

Now taking the l i m i t a s ^{€12 -(} 0,

H e r e the s u b s c r i p t p a r t i a l d e r i v a t i v e notation i s extended t o d e r i v a - t i v e s of the mapping functions with r e s p e c t t o t h e i r c o m p l e x a r g u - m e n t s , j u s t a s in ( 6 - 12) i t w a s used f o r a d e r i v a t i v e with r e s p e c t t o x. By substitution of z

=

g ( c ; x ) and use of the identity a = y k ( a ; x ) ; x ] and i t s d e r i v a t i v e with r e s p e c t to o, 1

= ys

[g(o;x);x]g,(o;x), (111- 1 ) is put i n the f o r m

66

A P P E N D I X IV

P L A N A R WING: I N T E G R A T I O N O V E R S O U R C E S . The following i n t e g r a l s a r e required:

14

(5,

=

S

a S3(52-a2)2 ^{a3 log (}

^{f - 52)}

^df

A useful t r a n s f o r m a t i o n i s :

f = a s e c 0

,

df = a s e c B t a n 0 d0

We a l s o u s e

c

= qa 1 ~ 3 < 1 .

Then

I1 -

I become:

c0s3 8 log ( 1 - q c o s 8 ) l S q c o s 8

0

4

( q a ) = c o s 2 8 log a 2 ( s e c 2 8

-

q2) d 8

0

I5 ( q a ) = log a 2 ( s e c 2 8

-

^{q2) d8}

0

I1

i s tabulated i n Dwight (Ref. 13) No. 865.37:

-

¹

I , ( q a ) = - n s i n q

.

I2 is f i r s t w r i t t e n

then i n the second i n t e g r a l n = 8 i s substituted f o r 8 :

Now substituting q = ~2 2k ( k < 1 s i n c e q < 1) n

I2 = -log (lip)

1

^{cos 8 dB -t}

frog

( 1 - i k c o s 8 i k2)cos 8 do.

0 0

The f i r s t i n t e g r a l v a n i s h e s , and the second i s tabulated i n Dwight, No. 865.74 with m. = 1 :

I3 i s s i m i l a r l y put i n the f o r m

=

\

log (1-2kcos 9

c

k2) cos3 9 d B and, using cos3 9 = 7 1 cos3 9

+ $

c o s 9

,

Again Dwight, No. 865.74 with m. = 1, 3 :

I 5 i s b r o k e n up a s

Using Dwight No. 865.34 (with p = -qZ) and No. 865.11 :

I4

i s split up s i m i l a r l y :

Using 865.25 on t h e second,

7i- X

4

= ^{log a}

-

a ( l - 2 log 2) f 0 1 0 ~ ( 1 - ~ ~ c o s ~ 0 ) d 8

.

0

The l a s t i n t e g r a l i s c a l l e d

Now using c o s 2 8 = i c o s 28 t.

i ,

X

&a

= $ 1

1 0 g ( l - ~ ~ c o s 2 O)d8+

f

28 l o g ( l - q c o s B)d0+

$

0 0 0

log (1 4 q c o s 8) d0 The f i r s t t e r m i s evaluated b y Dwight No. 865.34:

1

+ d l -

2

^{2 log}

+-

^*

The second and t h i r d a r e put i n tabulated f o r m by substituting q = 2k 2k respectively:

q =

- lt.kz

i J

log(1-2k c o s 8

+

k2)cos 28 dB

+ i

log(1-2k cos 8 t. k2)cos28 dB

0

S

⁰

which a r e i n the f o r m of Dwight No. 865.74.

S o f i n a l l y , c o l l e c t i n g t e r m s ,

A P P E N D I X V

P L A N A R WING: SPANWISE I N T E G R A T I O N

T w o of t h e i n t e g r a l s a r e t a b u l a t e d : Dwight (Ref.13) Nos. 520..

T h e r e m a i n i n g t w o a r e

J L

=S-

^{y2 d y}

( a t y ) . - 1 t s i n 8

0

w h e r e t h e l a t t e r f o r m s h a v e b e e n o b t a i n e d b y s u b s t i t u t i n g y = a s i n 8

.

J 1

is m u l t i p l i e d i n s i d e , t o p a n d b o t t o m , b y 1

-

^{s i n} 8 :

With t h e h e l p of Dwight Nos. 452.22, 452.32

I:"

^7T

t a n 8

-

⁸

-

^{c o s 8}

-

^{s e c 8} = ( 2

-Z)

a

n' 7T

s i n c e tan(- ₂

-

^{E )}

-

^{s e c (-} ₂

-

E ) = c o t e

-

^{C S C E} = O ( E )

.

J2 is t r e a t e d s i m i l a r l y , being m u l t i p l i e d i n s i d e , t o p a n d bottom, by ( l - ~ i n e ) ~ :

Then, u s i n g Dwight Nos. 452.14, 452.24, 452.34, 480.4:

r -

.nlz

S i n c e cot3€ - - _{- ~}1 ₃

- -

₆ ^{1 -k O(E ),} c s c 3 € =

-

_{€ 3} ^{1 1} _{-I. o(E)#}

1 1

c o t E. =

7

$ O(E) a n d c s c E =

-

-k O(E), t h e quantity i n E

b r a c k e t s is O(E) a t

- -

n' ^{E ,} a n d the i n t e g r a l is finite.

2

A P P E N D I X VI

BODY O F REVOLUTION: INTEGRATION OVER SOURCES T h e following i n t e g r a l s a r e r e q u i r e d :

- -

--

1 log

3 3 ,

dg

.

I =

jia

^{( ~}- 4 a 2 ) i ²

C+ G

I1

i s i d e n t i c a l w i t h the

I1

of Appendix

,

e x c e p t f o r 2 a r e p l a c i n g a

.

Consequently,

I~ ⁼

-

^?f s i n - I

L

Za

I2

and

I3

a r e t r a n s f o r m e d by

We a l s o u s e q g/2a

.

Then the i n t e g r a l s b e c o m e

I2

i s w r i t t e n i n two p a r t s , a n d i n the s e c o n d the substitution u = s i n 0 i s made:

Using Dwight (Ref. 13) Nos. 858.541, 140.02

I3 is e x p a n d e d as follows:

w h e r e u = s i n 8 h a s b e e n u s e d i n t h e l a s t . T h e l a s t two t e r m s have i n f i n i t i e s a t t h e l o w e r l i m i t w h i c h c a n c e l e a c h o t h e r off, as t h e y m u s t ; s i n c e the i n t e g r a n d of I 3 i s c e r t a i n l y r e g u l a r e v e n a t 8 = 0. Hence, u s i n g Dwight Nos. 140.02, 152.1, a n d 432.20 :

A P P E N D I X VII

BODY O F REVOLUTION: C I R C U M F E R E N T I A L INTEGRALS A l l t h e i n t e g r a l s a r e s t r a i g h t f o r w a r d , e x c e p t i n g

F i r s t the s i n 2 8 is r e p l a c e d by 1 - c o s 2 8:

1 S c o s ~ J ₌ j ' ( 3 cos 8

-

l l c o s 3 8 i 8 c o s 5 8 ) log l - c o s e

0

- -

~ ( ~ C 8 O

-

Sl l ~ o s ~ B i 8 cos5 8) log (1 i c o s 8) dB

0

-

$3 c o s 8

-

11cos3 8 i 8 c o s 5 0) log (1-cos 8) dB

0

Now in the second i n t e g r a l 8 i s r e p l a c e d by

n -

^8, and the i n t e g r a l a s s u m e s the f o r m

J = ( 3 cos 8

-

11cos3 8 t 8 cos5 8) log (1

+

c o s 8) d8

.

0

=

f

( 3 c o s 8

-

11 cos3 8 i 8 cos5 8) log ( 2

+

² c o s 8 ) dB

0

-

log 2

f

( 3 c o s 8

-

11 cos3 0

+

8 cos5 0 ) dB

0

The second i n t e g r a l is zero. B y writing thepowers of c o s 8 i n t e r m s of the c o s i n e s of multiple angles:

J =

f

⁽

^-

^{coa 8}

^{- i}

^{cos 1 8}

^{+ i}

^{c o s 5} 8) log ( 2 i 2 c o s 8) dB

0

the i n t e g r a l i s put i n the f o r m of ~ r F b n e r and H o f r e i t e r No. 338.13a ( r e f . 20) with r =

1

:

R E F E R E N C E S

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T s a k o n a s , S.: "Effect of Appendage a n d H u l l F o r m on H y r d o - d y n a m i c Coefficients of S u r f a c e Ships.

"

Davidson L a b o r a t o r y Rep. No. 740 (1959)

L e t c h e r , J.S.: " B a l a n c e of H e l m a n d S t a t i c D i r e c t i o n a l S t a b i l i t y of Yachts Sailing C l o s e - h a u l e d . 'I J. Roy. A e r o . Soc. 69 p. 241-248;

a l s o d i s c u s s i o n p. 480-481 (1965)

Wehausen, J.V.: "Wave R e s i s t a n c e of T h i n Ships. " S y m p o s i u m on Naval H y d r o d y n a m i c s , NAS-NRC pub. 515 (1957)

L a m b , H.: H y d r o d y n a m i c s 6th Ed.

,

D o v e r (1945)

Havelock, T.H.: "Studies i n Wave R e s i s t a n c e .

"

P r o c . Roy. Soc.

London A 103 p. 571 (1923)

Havelock, T.H.: "Wave R e s i s t a n c e T h e o r y and I t s A p p l i c a t i o n s t o Ship P r o b l e m s . ^" T r a n s . S.N.A.M,E 59 p. 13-24 (1951)

Inui, T.: "Wave-Making R e s i s t a n c e of Ships. " T r a n s . S.N.A.M.E.

70 p. 283-337 (1962)

Lunde, J.K.: "On the L i n e a r i z e d T h e o r y of Wave R e s i s t a n c e f o r D i s p l a c e m e n t S h i p s . " T r a n s . S.N.A.M.E. 59 p. 24 (1951)

Havelock, T.H.: " T h e V e r t i c a l F o r c e on a C y l i n d e r S u b m e r g e d i n a U n i f o r m S t r e a m .

"

P r o c . Roy. Soc. London A 122, p. 387-95 (1928) Havelock, T.H. "The F o r c e s on a C i r c u l a r C y l i n d e r S u b m e r g e d i n a U n i f o r m s t r e a m . " P r o c . Roy. Soc. London A i57 p. 526-534 (1936) Milne- Thompson, L. M. : T h e o r e t i c a l H y d r o d y n a m i c s . 2nd edition, M a c M i l l a n (195 0)

Dwight, H.B.: T a b l e s of I n t e g r a l s a n d O t h e r M a t h e m a t i c a l Data, 4th edition, M a c M i i l a n (19 61)

Jahnke, E., and E m d e , F. : " T a b l e s of F u n c t i o n s " D o v e r (1945) A d a m s , M. C. a n d S e a r s , W.R. : "Slender -Body T h e o r y

-

Review and Extension. J. A e r o . Sci. 20 p. 85 -9 8 (1953)

16. S a c k s , A.H. :

"

A e r o d y n a m i c F o r c e s , M o m e n t s , a n d S t a b i l i t y D e r i v a t i v e s f o r S l e n d e r B o d i e s of G e n e r a l C r o s s Section. ^'I NACA T N 3283 (1954)