Contents Journal of Ship Research, Volume 10, 1966

journal of

S H I P R E S E A R C H

M e c h a n i c a l ProperHes of Metals a n d Their

C a v i t a t i o n - D a m a g e Resistance

B y A . T h i r u v e n g a d a m i a n d Sophi&i Weiring^

Detailed investigations with a magnetostriction apparatus were carried out to determine the cavitation-damage resistance of eleven metals in distilled water at 80 F. The cavita-tion-damage resistance Is defined as the reciprocal of the rate of volume loss for a given metal. Among the mechanical properties investigated (ultimate tensile strength, yield strength, ultimate elongation, Brinell hardness, modulus of elasticity and strain energy) the most significant property which characterizes the energy-absorbing capacity of the metals, under the repeated, indenting loads due to the energy of cavitation bubble collapse in the steady-state zone, was found to be the fracture strain energy of the metals. The strain energy is defined as the area of the stress-strain diagram up to fracture. The corre-lation between the strain energy and the reciprocal of the rate of volume loss leads directly to the estimation of the intensity of cavitation damage; this intensity varies as the square of the displacement amplitude of the specimen. All these conclusions are limited to the steady-state zone of damage.

Nomenclature

a = amplitude A, ₌ area of erosion

I = intensity of cavitation damage

n strain-hardening factor

r

₌

r' = correlation factor s. = strain energy

Se* = estimated strain energy s/

=

T ₌ ultimate tensile strength r/ true fracture strength

Y = yield strength

=

S I N C E t h e w o r k o f Parsons [ 1 ] ^ i n 1 9 1 9 a n d F ö t t i n g e r [ 2 ] i n 1 9 2 6 , there h a v e been m a n y a t t e m p t s t o charac-terize t h e c a v i t a t i o n - d a m a g e resistance o f m a t e r i a l s b y a single, c o m m o n m e c h a n i c a l p r o p e r t y . A l t h o u g h

^ Senior Research Scientist, Hydronautics, Incorporated, Laurel, M d .

Assistant Research Scientist, Hydronautics, Incorporated, Laurel, M d .

' Numbers in brackets designate References at end of paper. Manuscript received at S N A M B Headquarters, M a r c h 5, 1965.

H o n e g g e r [ 3 ] , i n 1 9 2 7 , d i d n o t f i n d a n y c o r r e l a t i o n be-t w e e n hardness a n d erosion resisbe-tance, G a r d n e r [ 4 ] , i n 1 9 3 2 , f o u n d t h a t t h e hardness of a m e t a l was t h e p r i n c i p a l p r o p e r t y i n d e t e r m i n i n g t h e resistance t o erosion. M a n y m o r e references m a y be c i t e d t o b r i n g o u t s i m i l a r c o n -troversies w i t h r e g a r d t o o t h e r m e c h a n i c a l p r o p e r t i e s such as y i e l d s t r e n g t h , u l t i m a t e tensile s t r e n g t h , u l t i m a t e e l o n g a t i o n a n d m o d u l u s of e l a s t i c i t y . O n e c a n get a clear p i c t u r e o f t h e m a g n i t u d e o f t h e c o n f l i c t s i n t h i s area f r o m some of t h e excellent r e v i e w articles i n t h e t e c l i -n i c a l l i t e r a t u r e [5, 6, 7 ] .

These controversies are a r e s u l t o f a n i n a d e q u a t e u n d e r s t a n d i n g o f t h e m e c h a n i s m of c a v i t a t i o n damage. R e c e n t advances i n t h i s d i r e c t i o n h a v e m a d e i t possible t o r a t i o n a l i z e some of t h e c o n f l i c t s , a n d t o propose a m e c h a n i c a l p r o p e r t y t h a t m o s t s i g n i f i c a n t l y characterizes t h e c a v i t a t i o n - d a m a g e resistance o f m e t a l s i n t h e ab-sense of c o r r o s i o n , r e l a t i v e l y speaking. I t is t h e p u r p o s e o f t h i s paper t o develop t h e logic b e h i n d such a n a r g u -m e n t , a n d t o present r e c e n t s u b s t a n t i a t i n g exiDeri-mental evidence.

One o f t h e basic paranreters i n v o l v e d i n t h e t e s t i n g o f m a t e r i a l s f o r c a v i t a t i o n - d a m a g e resistance is t h e t e s t d u r a t i o n . T h e r£|,te of loss of m a t e r i a l depends ui3on t h e test d u r a t i o n i t s e l f even t h o u g h e v e r y o t h e r test p a -r a m e t e -r is m a i n t a i n e d p-recisely c o n s t a n t . R e c e n t

Experiments on Flat-^Bottom Siomming^

An experimental investigation of rigid flat-bottom body slamming was performed at the David Taylor Model Basin by dropping a 20-in. X 26.5-in. X 0.5-in. steel plate from various elevated positions above a calm water surface. Because of the effect of the trapped air between the falling body and the water, the maximum impact pressure measured was much lower than the pressure expected, if the generally accepted acoustic pressure formula pcVo is applied.

Nomenclature

A = impact area of fluid or plate a = acceleration in general

c = speed of sound in fluid Cair = speed of sound in air

F = force acting upon falling body I = impulse in general

L = half-width of infinitely long plate

7)1 = mass of falling body 7)1/ = mass of fluid

p = impact pressure, psi

Vrrnxx = maximum impact pressure, psi

T = half period or duration of first positive pulse t = time in general

to = time at instant of impact V = velocity in general

Vo = impact velocity at time (o, fps p or pfluid = mass density of fluid

Introduction

I T has been believed g e n e r a l l y t h a t f l a t - b o t t o m slam-m i n g is a c o slam-m b i n e d acoustic a n d unsteady h y d r o d y n a slam-m i c p h e n o m e n o n . J u s t p r i o r t o t h e occurrence of i m p a c t , a l l i n t e r v e n i n g air is assumed t o be f o r c e d o u t f r o m under-n e a t h t h e p l a t e [ 1 ] . ^ T h u s , c o m p r e s s i b i l i t y of t h e f l u i d ( w a t e r i n t h i s s t u d y ) m u s t be considered. U s i n g these assumptions, a n a p p r o x i m a t e v a l u e f o r t h e m a x i m u m s l a m m i n g pressure can be c o m p u t e d as s h o w n i n t h e n e x t p a r a g r a p h s [ 2 ] .

W h e n a f l a t b o d y strUïies t h e surface of a fluid a t a n i m p a c t v e l o c i t y Fo, t h e p r o p a g a t i o n of t h e m o m e n t a r y increase o f pressure i n t h e fluid takes place a t t h e speed o f sound i n t h e fluid, designated b y c. T h e mass o f fluid accelerated i n t h e t i m e M is

' T h e opinions expressed are those of the author alone and should not be construed to reflect the official views of the N a v y Department or the N a v a l Service at large.

2 Structural Research Engineer, Structural Mechanics Labora-tory, D a v i d T a y l o r Model Basin, Washington, D . 0 .

^ Numbers in brackets designate References at end of paper. Manuscript received at S N A M E Headquarters, August 16, 1965; revised, D e c e m b e r ? , 1965.

Fig. 1 Installation of test model

711/ = pAcAt

where p is t h e mass d e n s i t y of fluid a n d A is t h e strilcing area of t h e flat b o d y u p o n t h e surface of t h e fluid.

Since t h e v e l o c i t y of t h e mass of fluid is increased f r o m zero t o Vo i n t h e t i m e At, t h e f o r c e F a c t i n g u p o n t h e f a l l i n g b o d y is t h e r e f o r e

= pAcVo

T h e pressure p, w h i c h is t h e f o r c e per u n i t area, is

p = pcVo ( 1 )

T h e d u r a t i o n of t h e compression phase is 2L/c, w h e r e L is t h e h a l f - w i d t h of a n i n f i n i t e l y l o n g p l a t e [ 3 ] .

ing Force a n d M o m e n t on bi

O b l i q u e Waves^ :

B y P u n g N i e n Hu^ a n d K i n g E n g ^

A general expression for the drifting moment about the vertical axis of an oscillating ship in regular oblique v/aves is derived from the potential theory, following a similar procedure developed by Maruo for drifting force. Explicit analytical solutions for the drifting side force and yaw moment on thin ships in long waves are obtained in terms of simple ele-mentary functions. The effect of the wave frequency, the draft of the ship, the displace-ment, and the phase angle of the ship oscillation are discussed.

W H E N a ship is o s c i l l a t i n g o n t h e f r e e surface i n re-sponse t o r e g u l a r waves e n c o u n t e r e d , t h e e x c i t i n g f o r c e a n d m o m e n t w h i c h a c t o n t h e b o d y , i n general, c a n be s e p a r a t e d i n t o t w o p a r t s ; one p a r t is o s c i l l a t o r y a n d t h e o t h e r is n o t . A l t h o u g h t h e n o n o s c i l l a t o r y f o r c e a n d m o m e n t are, w i t h respect t o t h e w a v e a m p l i t u d e , o f t h e second o r d e r as c o m p a r e d w i t h t h e o s c i l l a t o r y p a r t of t h e f o r c e a n d m o m e n t , t h e y i n d u c e a s t e a d y d r i f t t o t h e s h i p a n d are t h e r e f o r e v e r y i m p o r t a n t i n t h e s t u d y o f ship m o t i o n i n waves. T h e p r o b l e m of d r i f t was s t u d i e d b y S u y e h i r o [ 1 ] , ^ W a t a n a b e [2], H a v e l o c k [3, 4 ] , M a r u o [5, 6 ] , a n d B e s s h o [7 ] . T h e m o s t r a t i o n a l t r e a t m e n t was g i v e n b y M a r u o i n a recent paper [ 6 ] i n w h i c h he a p p l i e d t h e p o w e r f u l m o m e n t u m a n d energy t h e o r e m s t o t h e f l u i d , t o g e t h e r w i t h t h e p o t e n t i a l t h e o r y , t o o b t a i n a general expression f o r t h e d r i f t i n g f o r c e a c t i n g o n a n o s c i l l a t i n g b o d y , i n t e r m s o f t h e p o t e n t i a l due t o t h e waves a n d t o t h e i n t e r -a c t i o n b e t w e e n t h e b o d y -a n d w-aves. Once t h e v e l o c i t y p o t e n t i a l s are d e t e r m i n e d , t h e d r i f t i n g f o r c e can b e o b -t a i n e d b y s i m p l e q u a d r a -t u r e s . T h e e f f e c -t o f w a v e reflect i o n a n d reflect h a reflect o f ship osciUareflection are a u reflect o m a reflect i c a l l y i n -c l u d e d i n t h e d e t e r m i n a t i o n o f v e l o -c i t y p o t e n t i a l s . I n t h e present s t u d y , we w i l l f o U o w M a r u o ' s p r o c e d u r e b y u s i n g t h e m o m e n t - o f - m o m e n t u m t h e o r e m t o o b t a i n a general expression f o r t h e d r i f t i n g m o m e n t . A l t h o i r g h t h e expression f o r the d r i f t i n g f o r c e o b t a i n e d b y M a r u o a n d t h a t f o r t h e d r i f t i n g m o m e n t o b t a i n e d i n t h e p r e s e n t s t u d y are q u i t e general ( a p p l i c a b l e t o a n y o n c o m i n g waves w i t h t h e o n l y r e s t r i c t i o n t h a t t h e w a v e

1 Work sponsored by the Bureau of Ships Fundamental H y d r o -mechanics Research Program (S-R009-01-01), administered by D a v i d T a y l o r Model B a s i n Contract Nonr 263(21).

^ Head, Flow Phenomena Division, Davidson Laboratory, Stevens Institute of Technology, Hoboken, N . J . Presently Senior Scientist, Space Sciences, Incorporated, Waltham, Mass.

3 Research Engineer, Davidson Laboratory, Stevens Institute of Technology, Hoboken, N . J .

•* Numbers in brackets designate References at end of paper. Manuscript received at S N A M E Headquarters, October 28, 1965.

a m p l i t u d e b e s m a l l , so t h a t t h e p r o b l e m can be l i n e a r -i z e d ) , we s h a l l c a r r y o u t d e t a -i l e d c a l c u l a t -i o n s f o r t h -i n ships i n l o n g waves. I t w i l l be f o u n d t h a t t h e final ex-pressions f o r t h e d r i f t i n g f o r c e a n d m o m e n t are e x t r e m e l y s i m p l e ; s i g n i f i c a n t p h y s i c a l i n t e r p r e t a t i o n ean b e easily recognized.

General Expressions

W e consider t h e s h i p t o be floating o n t h e f r e e surface, o s c i l l a t i n g i n response t o t h e e x c i t a t i o n of o n c o m i n g r e g u l a r waves. T h e . ^ i — p l a n e o f t h e c o o r d i n a t e sys-t e m coincides w i sys-t h sys-t h e censys-terplane of sys-t h e s h i p , sys-t h e .-Cs-axis is p o s i t i v e u p w a r d a n d t h e .'!;2-axis t o p o r t , a n d t h e o r i g i n is fixed o n t h e f r e e surface a t t h e m e d i a n p l a n e o f t h e ship. T h e waves are p r o p a g a t i n g w i t h a speed c i n t h e d i r e c t i o n o b l i q u e t o t h e a;-axis a t a n angle /3. W e assume t h a t t h e w a v e h e i g h t a is s m a l l , so t h a t t h e p r o b l e m can be l i n e a r i z e d . T h e t o t a l v e l o c i t y p o t e n t i a l $, due t o t h e o s c i l l a t o r y m o t i o n of t h e fluid, m a y be expressed as * = 0 e - ' " ' ( 1 ) w h e r e co is t h e f r e q u e n c y o f o s c i l l a t i o n a n d o n l y t h e r e a l p a r t o f t h e p o t e n t i a l is t o be t a k e n . T h e p o t e n t i a l 4), a h a r m o n i c f u n c t i o n of Xj{j = 1 , 2, 3 ) , satisfies t h e b o u n d a r y c o n d i t i o n on t h e f r e e s u r f a c e Xz = 0, w h e r e k = c,'/g (3) is t h e w a v e - f r e q u e n c y p a r a m e t e r a n d is r e l a t e d t o t h e . w a v e l e n g t h X b y t h e e q u a t i o n k = 27r/X (4) T h e v e l o c i t y p o t e n t i a l o f t h e waves, s a t i s f y i n g t h e b o u n d a r y c o n d i t i o n ( 2 ) , is g i v e n b y

(jju, = ac exp {k[x3 + i{xi cos /3 -|- Xi s i n /?)]} (5)

A Linearized T w o p i m e n s i o n a l Theory for H i g h

-B y R i c h a r d P. -B e m i c k e r ^

A linearized two-dimensional theory is presented for high-speed hydrofoils near the free surface. The "direct" problem (hydrofoil shape specified) is attacked by replacing the actual foil with vortex and source sheets. The resulting integral equation for the strength of the singularity distribution is recast into an infinite set of linear algebraic equations relating the unknown constants in a Glauert-type vorticity expansion to the boundary condition on the foil. The solution is achieved using a matrix inversion technique and it is found that the matrix relating the known and unknown constants is a function of depth of submergence alone. Inversion of this matrix at each depth allows the vorticity constants to be calculated for any arbitrary foil section by matrix multiplication. The inverted ma-trices have been calculated for several depth-to-chord ratios and are presented herein. Several examples for specific camber and thickness distributions are given, and results indicate significant effects in the force characteristics at depths less than one chord. In particular, thickness effects cause a loss of lift at shallow submergences which may be an appreciable percentage of the total design lift. The second part treats the "indirect" problem of designing a hydrofoil sectional shape at a given depth to achieve a specified pressure loading. Similar to the "direct" problem treated in the first part, integral equa-tions are derived for the camber and thickness funcequa-tions by replacing the actual foil by vortex and source sheets. The solution is obtained by recasting these equations into an infinite set of linear algebraic equations relating the constants in a series expansion of the foil geometry to the known pressure boundary conditions. The matrix relating the known and unknown constants is, again, a function of the depth of submergence alone, and inversion techniques allow the sectional shape to be determined for arbitrary design pressure distributions. Several examples indicate the procedure and results are presented for the change in sectional shape for a given pressure loading as the depth of submergence of the foil is decreased.

Part 1 The Direct Problem

I N T E R E S T i n l i i g h - p e r f o r m a n c e l i y d r o f o i l c r a f t has s t i m u l a t e d t h e i n v e s t i g a t i o n of l i f t i n g surfaces o p e r a t i n g beneath a f r e e surface. A s a n a t u r a l consequence, t h e great b u l k of t h e o r y p e r t a i n i n g t o incompressible f l o w a b o u t a i r f o i l s i n a n i n f i n i t e m e d i u m has been used as a s t a r t i n g p o i n t f o r t h e h y d r o f o i l analysis. B o t h t h e classic a e r o n a u t i c a l m e t h o d s o f c b n f o r m a l t r a n s f o r m a t i o n s a n d s i n g u l a r i t y r e p r e s e n t a t i o n h a v e been a p p l i e d t o t h e h y -d r o f o i l p r o b l e m , b u t no m e t h o -d has as y e t been presente-d f o r a n " e x a c t " s o l u t i o n of a n a r b i t r a r y f o i l u n d e r a f r e e surface. V a r i o u s studies h a v e been c a r r i e d o u t w h i c h p e r t a i n t o t h e general h y d r o f o i l p r o b l e m , each i n v o l v i n g s i m p l i -f y i n g assumptions w h i c h l i m i t t h e general a p p l i c a b i l i t y . K e l d y s c h a n d L a v r e n t i e v [ 1 ] , ^ K o t c h i n [ 2 ] , a n d H a s k i n d

' Davidson Laboratory, Stevens Institute of Technology, Hobo-ken, N . J . Presently, Assistant Professor of Mechanical Engineer-ing, Stevens Institute of Technology.

" Numbers in brackets designate References at end of paper. Manuscript received at S N A M E Headquarters, October 26, 1964; revised manuscript received M a y 19, 1965.

[3] p u b l i s h e d e a r l y papers w h i c h r e p l a c e d t h e a c t u a l h y d r o f o i l b y a s y s t e m of sources a n d v o r t i c e s , l e a d i n g , h o w e v e r , t o a m a t h e m a t i c a l f o r m u l a t i o n t h a t was a l l b u t i n t r a c t a b l e . W e i n i g [ 4 ] , W a d l i n et a l [ 5 ] , a n d m o r e r e c e n t l y , S t r a n d h a g e n a n d Seikel [6 ] used t h e concept of a single s u b s t i t u t i o n v o r t e x t o represent t h e f o i l . S c h w a n eclce [ 7 ] a n d I s a y [ 8 ] c o n t i n u e d t h i s i d e a f u r t h e r a n d r e -placed t h e t h i n f o i l w i t h a v o r t e x sheet, composed o f s t a n d a r d B i r n b a u m v o r t i c i t y d i s t r i b u t i o n s , wliose s t r e n g t h was d e t e r m i n e d b y s a t i s f y i n g t h e b o u n d a r y c o n -d i t i o n s a t several c o n t r o l p o i n t s o n t h e c h o r -d l i n e . I n a l l of these papers, t h e general d e v e l o p m e n t of t h e t h e o r y f e l l s h o r t of d e s c r i b i n g t h e o v e r a l l h y d r o d y n a m i c forces a n d pressure d i s t r i b u t i o n s f o r a n a r b i t r a r y section a t a n y d e p t h . N i s h i y a m a [ 9 , 1 0 ] f o r m u l a t e d t h e p r o b l e m t o i n c l u d e t h e effects of a r b i t r a r y c a m b e r a n d t h i c k n e s s of a f o i l section a t a n y F r o u d e n u m b e r . H i s t e c h n i q u e i n v o l v e s a c o n f o r m a l t r a n s f o r m a t i o n of t h e f o i l c o n t o u r o n t o a s l i t , a n d expansion of t h e c o m p l e x v e l o c i t y p o t e n -t i a l i n a poAver series. T h e u n k n o w n c o n s -t a n -t s of -t h e v e l o c i t y p o t e n t i a l are r e l a t e d t o t h e t r a n s f o r m a t i o n c o n -s t a n t -s b y a n i n f i i r i t e -set of l i n e a r algebraic e q u a t i o n -s , a n d ^ A A R C H 1 9 6 6 9.Li

Cdvifation Phenomeri©] ©If Sterntube Bearings'

This paper contains a theoretical study of dynamically loaded journal bearings of finite length for the simultaneous journal loci and pressure distribution in the bearing. The analysis presented is applied to the case of sterntube bearings of ships with two specific aims in mind. The first of these is to firmly establish the existence of conditions permitting cavitation damage of the journal, and the second is to show how the peculiar patterns of cavitation damage observed on the journals of several ships may occur.

T H E p r o b l e m b e m g considered here is t h a t o f f i n d i n g a s o l u t i o n f o r t h e pressure d i s t r i b u t i o n a n d s n n u l t a n e o u s s h a f t l o c i of a 360deg j o u r n a l b e a r i n g s u b j e c t e d t o d y n a m i c l o a d i n g . T h e b e a r i n g is l u b r i c a t e d b y a c i i v m i f e r e n t i a l source a t one end of t h e b e a r i n g . T h e l u b r i c a f -i n g f l u -i d f l o w s o u t t h e other end of t h e b e a r -i n g . T h e b e a r i n g is considered t o be f i n i t e i n l e n g t h . T h e b e a r i n g l u b r i c a n t is w a t e r w h i c h is s u p p l i e d a t a c o n s t a n t r a t e a n d pressure. I t is assumed t h a t t h e b e a r i n g a n d j o u r n a l surfaces w i l l a l w a y s r e m a i n p a r a l l e l a n d t h a t b o t h are c o m p l e t e l y r i g i d . T h e surfaces are f u r t h e r assumed t o be p e r f e c t l y s m o o t h .

T h e p h y s i c a l m o t i v a t i o n f o r t h i s p r o b l e m comes f r o m t h e s t e r n t u b e b e a r m g of ships, w i t h i n w h i c h t h e t a i l s h a f t seemingly undergoes c a v i t a t i o n damage due t o t h e d y -n a m i c l o a d i -n g o f t h e propeller. I t has bee-n observed

[ I f i n several ships t h a t t h e t a i l s h a f t is eroded a t several

d e f i n i t e p o s i t i o n s a r o u n d i t s p e r i p h e r y a n d w i t h i n t h e confines o f t h e s t e r n b e a r i n g . T h e l u u n b e r o f damage l o c a t i o n s a n d t h e n p o s i t i o n s a r o u n d t h e p e r i p h e r y of t h e j o u r n a l v a r y d i r e c t l y as t h e n u m b e r of p r o p e l l e r blades. I f t h i s n u m b e r is e i t h e r t h r e e or five, t h e r e w i l l be e i t h e r t h r e e or five l o c a t i o n s , respectively, of t a i l s h a f t damage a n d t h e i r p o s i t i o n s w i l l be d i r e c t l y i n l i n e w i t h t h e p r o -peller blades. I f t h e n u m b e r of blades is f o u r , t h e n t h e r e w i l l be f o u r l o c a t i o n s of damage w h i c h are e x a c t l y 45 deg offset f r o m t h e l i n e of t h e p r o p e l l e r blades. T h e l o c a t i o n a l o n g t h e l e n g t h o f t h e b e a r i n g w h e r e t h i s damage o f t e n occurs is s l i g h t l y f o r w a r d of t h e q u a r t e r - w a y p o i n t of t h e b e a r i n g as measured f r o m t h e p r o p e l l e r end. W h i l e t h e p r o b l e m s t u d i e d i n t h i s e f f o r t d i f f e r s f r o m t h e a c t u a l p h y s i c a l p r o b l e m of s t e r n t u b e bearings i n t h a t i t was necessary t o m a k e c e r t a i n assumptions t o surpass some m a t h e m a t i c a l complexities, i t is f e l t t h a t i t r e p r e -sents a g o o d i n i t i a l m o d e l of t h e p h y s i c a l p r o b l e m .

1 T h i s paper represents a portion of a thesis prepared by the aiithor in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Engineering Mechanics, University of Michigan, A n n Arbor, M i c h .

^ A s s i s t a n t Professor, Department of Engineering Mechanics, University of Wisconsin, Madison, W i s .

' Numbers in brackets designate References at end of paper. Manuscript received at S N A M E Headquarters, November 23, J-yoo. T h e three m o s t i m p o r t a n t assumptions d e p a r t i n g . f r o m t h e a c t u a l p h y s i c a l s i t u a t i o n of s t e r n t u b e bearings a r e : T h e s h a f t a n d b e a r i n g are t o r e m a i n p a r a l l e l a t a l l t i m e s , t h e b e a r i n g is c o m p l e t e l y s m o o t h , a n d t h e p r o p e l l e r l o a d i n g c a n be represented b y t h e first t w o h a r m o n i c s .

T h e first of these asstimptions has t h e e f f e c t o f m a k i n g t h e film thickness a f u n c t i o n of o n l y a n g u l a r displace-m e n t a r o u n d t h e j o u r n a l . I t is believed t h a t t h e t w o m a j o r consequences of t h i s a s s u m p t i o n are a change i n t h e i n t e n s i t y of pressures o b t a i n e d a n d a s l i g h t s h i f t along t h e l e n g t h of t h e b e a r i n g of t h e r e g i o n of m i n i m u m pressures developed i n t h e l u b r i c a t i n g film. I t is n o t f e l t , h o w e v e r , t h a t t h e general pressure p r o f i l e w o u l d be s u b s t a n t i a l l y a l t e r e d b y t h i s a s s u m p t i o n . T h e second a s s u m p t i o n d e f i n i t e l y v i o l a t e s t h e a c t u a l b e a r i n g w h i c h is composed of staves spaced a r o u n d t h e b e a r i n g p e r i p h e r y . T h e m a t h e m a t i c a l c o m p l e x i t y of i n -c o r p o r a t i n g these effe-cts a t present m a k e s i t ne-cessary to assume a s m o o t h b e a r i n g surface.

C o n s i d e r i n g t h e t h i r d a s s u m p t i o n , a l t h o u g h t h e a c t u a l propeller l o a d i n g is c e r t a i n l y composed of m a n y h a r -m o n i c co-mponents, t h e first t w o of these are k n o w n t o represent t h e m a j o r p o r t i o n of t h e p r o p e l l e r l o a d i n g .

T h e analysis presented i n t h e f o l l o w i n g is essentially s u b d i v i d e d i n t o t h r e e m a j o r p o r t i o n s . T h e first p a r t consists o f t h e s o l u t i o n of t h e g o v e r n i n g f o r m of R e y n o l d s l u b r i c a t i o n e q u a t i o n f o r t h e pressure d i s t r i b u t i o n i n t h e b e a r i n g . T h e s o l u t i o n t o t h i s e q u a t i o n w i l l , h o w e v e r , c o n t a i n t w o u n l c n o w n v e l o c i t y c o m p o n e n t s d u e t o t h e t r a n s l a t i o n a l a n d r o t a t i o n a l m o t i o n of t h e j o u r n a l center i n a n o r b i t a b o u t a steady-state p o s i t i o n . T h e second p a r t is t h u s concerned w i t h t h e d e t e r m i n a -t i o n of -these -t w o u n l c n o w n v e l o c i -t y c o m p o n e n -t s . I f -t h e equations of m o t i o n f o r t h e j o u r n a l mass center are w r i t t e n , a p a i r of n o n l i n e a r , o r d i n a r y , d i f f e r e n t i a l equa-t i o n s are generaequa-ted f o r equa-these v e l o c i equa-t y c o m p o n e n equa-t s . T h e s o l u t i o n s t o t h i s p a i r of equations are o b t a i n e d b y t h e R u n g e - K u t t a f o u r t h - o r d e r m e t h o d o n a d i g i t a l c o m p u t e r . These s o l u t i o n s i n a d d i t i o n t o y i e l d i n g t h e u n l c n o w n v e l o c i t y c o m p o n e n t s g i v e t h e j o u r n a l o r b i t s . T h u s , f o r a g i v e n p o i n t i n t h e j o u r n a l o r b i t , t h e pressure d i s t r i b u -t i o n i n -t h e b e a r i n g m a y be e v a l u a -t e d f r o m -t h e s o l u -t i o n -t o R e y n o l d s e q u a t i o n . M A R C H 1 9 6 6 49

O n N!@&TiliDiear

Shop

Motions in Irregular W a v e s

It is shown that the transfer functions characterizing the nonlinear response of ships in ir-regular seas can be obtained from high order moments of the ship motions by an exten-sion of standard spectral-analysis techniques. Hence, full-scale measurements can be used to determine, for example, the coefficients of excess wave resistance and lateral drift. The method also has applications in model experiments.

T H E s t a t i s t i c a l t l i e o r y o f l i n e a r ship m o t i o n s is w e l l u n d e r s t o o d a n d has f o u n d n u m e r o u s a p p l i c a t i o n s . M o s t i n v e s t i g a t i o n s of n o n l i n e a r m o t i o n s , o n t h e o t h e r h a n d h a v e been r e s t r i c t e d t o a d e t e r m i n i s t i c e x p l a n a t i o n o f c e r t a i n effects such as t h e l a t e r a l d r i f t a n d excess resist-ance i n waves. T h e results h a v e n o t been placed w i t h i n t h e f r a m e w o r l c o f a general s t a t i s t i c a l r e p r e s e n t a t i o n . I t w i l l be s h o w n i n t h i s p a p e r t h a t t h e n o n l i n e a r t r a n s f e r f u n c t i o n s are r e l a t e d t o t h e h i g h e r order m o m e n t s o f t h e ship m o t i o n s i n t h e same w a y as t h e l i n e a r t r a n s f e r f u n c t i o n s are r e l a t e d t o t h e s p e c t r u m . Hence, a l l n o n -linear t r a n s f e r f u n c t i o n s m a y be d e r i v e d s y s t e m a t i c a l l y f r o m a h i g h e r order analysis of t h e ship m o t i o n s .

T h e usefulness of h i g h e r order m o m e n t s i n a n a l y z i n g n o n l i n e a r processes has been stressed b y Trdcey (1961). T h e m e t h o d has- been a p p l i e d p r e v i o u s l y t o d e t e r m i n e t h e n o n l i n e a r i n t e r a c t i o n s w i t h i n a n i r r e g u l a r w a v e f i e l d (Hasselmann, M u n k a n d M a c D o n a l d , 1963).

The Ship Response

L e t ^1, . . . f6 be t h e coordinates o f t h e s h i p m o t i o n : = surge v e l o c i t y MI, measured a b o u t t h e v e l o c i t y ü i n c a l m w a t e r ^2 = s w a y v e l o c i t y = heave d i s p l a c e m e n t ^4 = r o l l angle f 6 = p i t c h angle fs = y a w angle T h e v a r i a b l e s f ; are f u n c t i o n a l s o f t h e w a v e f i e l d . W e assume t h a t t h e w a v e m o t i o n is i r r o t a t i o n a l a n d ( f o r t h e present) linear, so t h a t i t can be specified b y t h e surface displacement f (x, t) = f i a n d t h e n o r m a l surface v e l o c i t y ( ö f / Ö 0 (x, t) = ^2 a t t h e m e a n f r e e surface, w h e r e x = (rui, Xi) is t h e h o r i z o n t a l C a r t e s i a n c o o r d i n a t e v e c t o r ; f 1 a n d f2 refer t o t h e u n d i s t u r b e d w a v e f i e l d i n t h e absence of t h e s h i p . I t is c o n v e n i e n t t o i n t r o d u c e t h e c o o r d i n a t e s y s t e m x = X — U'Z r e l a t i v e t o t h e m e a n s h i p m o t i o n U . E x p a n d -i n g t h e f u n c t -i o n a l s f ; -i n a T a y l o r ser-ies w -i t h respect t o f l , f j , t h e ship response m a y t h e n be w r i t t e n i n t h e gen-eral f o r m

1 Institut fur Schiffbau der Universitat Hamburg, Lammersieth, Germany, Institute of Geophysics aud Planetary Physics, Univer-sity of California, S a n Diego, Calif.

Manuscript received at S N A M E Headquarters, M a r c h 29, 1965.

m =

S -STGUt - t',

tyii'cw

a = l

+ ƒ • • -SZHi^pit - t', t

a,fl = l

t",

( x ' , « ' ) f p ( x " , t")di'dk"dt'dt" + . . . (1)

T h e kernels depend o n l y on the t i m e differences, since i n t h e c o o r d i n a t e s y s t e m x t h e ship response is i n d e p e n d e n t of t i m e t r a n s l a t i o n s . Consider n o w a w a v e f i e l d w h i c h is r a n d o m , stationar3r, a n d homogeneous. I t m a y be a p p r o x i m a t e d b y a F o u r i e r s u m i(co( - k.x) - k.x)) ₍₂₎ where w = -\- (gk) ^'^ is t h e f r e q u e n c y o f a w a v e c o m p o n e n t w i t h w a v e n u m b e r k . ( I t is m o r e c o n v e n i e n t i n n o n -l i n e a r p r o b -l e m s t o i n c -l u d e t h e c o m p -l e x c o n j u g a t e t e r m of t h e F o u r i e r s u m e x p l i c i t l y , r a t h e r t h a n use t h e r e a l -p a r t c o n v e n t i o n . ) T h e ensemble e x p e c t a t i o n values o f t h e a m p l i t u d e s a n d t h e i r cross p r o d u c t s s a t i s f y t h e r e l a t i o n s ( ^ k ) ^ 0 (Zk.Zfa) ^ 0 { Z , , M = 0 f o r k l ± k2 (3) (4) where Afc is t h e w a v e - i u i m b e r i n c r e m e n t o f t h e F o u r i e r s u m (2) a n d F(k) is t h e w a v e s p e c t r u m , d e f i n e d such t h a t E 2 | 2 k | k Fik)dk I t c a n be s h o w n t h a t i n t h e l i n e a r a p p r o x i m a t i o n a homogeneous, dispersive w a v e f i e l d r a p i d l y becomes Gaussian (Hasselmann, 1966). H e n c e t h e h i g h e r order m o m e n t s o f t h e w a v e f i e l d can also be expressed i n t e r m s of t h e s p e c t r u m . W e s h a l l r e q u i r e l a t e r t h e r e l a t i o n s (ZfaZfe^to) = (^k,^k.Zk3*) = 0 (^ki-^ka^ka^kj) = (•^ki-^ka'^ks'^ta*) — 0 (.^ki-Z'k2'^k3*-^k,*) = 0 unless h = h, ki = ki or kl = ki, ki = kz ( ^ f c Z k , Z f a * ^ k , * ) = \F{hmh){Aky (5) (6) 6 4 J O U R N A L O F S H I P R E S E A R C H

hurnal of

S H I P R E S E A R C H

\

Fracture Mechanics—A Basic Solution to Fatigue

Using Energy Principles'

By A . A. Bement^ a n d C. H. Pohler^

An energy relationship is established by using only the true stress-true strain diagram de-veloped from a simple tensile test of a material. This energy relationship is then used to develop (a) a fatigue failure equation, which results in a single nondimensional stress-cycle curve having apparent applicability to any material, (b) an equation depicting the rate of crack propagation, (c) an endurance limit equation, and (d) a method by which a stress-cycle diagram can be developed for a notched specimen. In addition, an equation is developed for determining the effect of high and low levels of residual stress on the fatigue strength of a material. Supporting theoretical development and experimental data are appended for completeness. Also included is a discussion of the application of these equations to design, for which a single stress-cycle curve has been developed, with indications of general applicability to any material. Conclusions obtained from this study are developed, indicating to the authors the desirability of a general reaffirmation and possible reorientation of the basis of fatigue tests.

" O n the strength of one link in the cable Dependeth the might of the chain.

Foreword

D E S I G N I N G f o r f a t i g u e , based o n m o d e r n concepts of " l i f e c a p a b i l i t y , " is n o t a n easy task, since t h e designer is o f t e n c o n f r o n t e d w i t h a s t r u c t u r e w h i c h , d u r i n g i t s l i f e -t i m e , m a y c o n -t a m f a -t i g u e cracks. These cracks i n i -t i a l l y grow s l o w l y u n d e r cyclic l o a d i n g , b u t , i n m a n y instances, have p r o d u c e d s u d d e n f a i l u r e of a s t r u c t u r e w i t h conse-quent loss of t i m e , m o n e y , a n d even l i f e . T o be able t o p r e d i c t f a i l u r e before final f r a c t u r e a n d t o q u a n t i t a t i v e l y u n d e r s t a n d t h e m e c h a n i s m o f t h i s c r a c k p r o p a g a t i o n i n terms of c r a c k size, stress, a n d l o a d cycles h a v e been t h e

' Philosophy and techniques contained in this paper reflect solely the personal opinions of the authors, and do not necessarily represent the official views of the Bureau of Ships or the N a v a l Service at large. Copyright M a r c h 5, 1965 ( L i b r a r y of Congress Registration No. A758001), by Arnold A . Bement and C a r l H . Pohler.

^ Submarine Structural Mechanics U n i t , H u l l Design, Scientific and Research Section, B u r e a u of Ships, Department of the N a v j ^ Washington, D . C .

' Ronald A r t h u r Hopwood, " T h e L a w s of the N a v y , " Stanza 2. Manuscript received at S N A M E Headquarters, September 7,

desires of resqarchers f o r over a c e n t u r y . D u r i n g t h i s p e r i o d l i t e r a l l y t h o u s a n d s of r e p o r t s , papers, a n d books h a v e been w r i t t e n s u m m a r i z i n g a n d t h e o r i z i n g o n t h e n a t u r e o f f a t i g u e f o r d i f f e r e n t a p p l i c a t i o n s . F r o m t h i s concerted a c t i v i t y has s t e m m e d a n ever-increasing awareness of t h e c o m p l e x i t y of f a t i g u e , b u t n o r e a l d e f i -n i t i o -n of w h a t a c t u a l l y causes f a t i g u e .

C o n s e q u e n t l y , i n some instances i t has been f o u n d necessary t o test v e r y large models [ 1 ] * a n d e v e n f u l l -scale r e p r o d u c t i o n s [ 2 ] of a c t u a l s t r u c t u r e s t o d e t e r m i n e t h e f a t i g u e s t r e n g t h of these s t r u c t u r e s . H o w e v e r , t h e c u s t o m a r y p r o c e d u r e i n d e t e r m i n i n g t h e f a t i g u e s t r e n g t h of a s t r u c t u r e is t o a p p l y engineering e x t r a p o l a t i o n t o re-sults o b t a i n e d f r o m f a t i g u e t e s t i n g of v e r y s m a l l speci-mens of t h e m a t e r i a l proposed f o r f a b r i c a t i o n . H e n c e t h e i m p o r t a n c e of these s m a l l f a t i g u e specimens is r e a d i l y o b v i o u s . T h e i n t e n t of t h i s p a p e r is t o r e e x a m i n e t h e i n d e x o r d a t u m t o w h i c h these s m a l l specimen results are r e l a t e d , b e f o r e t h e y are e x t r a p o l a t e d t o a s t r u c t u r e . D e s p i t e a general awareness t h a t no good c o r r e l a t i o n has been f o u n d b e t w e e n t h e f a t i g u e p r o p e r t i e s o f a m a t e r i a l a n d those o b t a i n e d f r o m a n o r d i n a r y (engineering) tensile test, t h e

^ Numbers in brackets designate References at end of paper.

U P S Ï ( i ( Q ] d

mÉm Sp(Qiirii

€^1(1

%MpmmQ\m\((Mn(§

H y d i r o ' l b i i i s

B y S h e i l a E v a n s W i d n a l l ^

Linearized tliree-dimensional lifting-surface theory is derived for a supercavitating hydro-foil with finite span in steady or oscillatory motion through an infinite fluid. The resulting coupled-integral equations are solved on a high-speed digital computer using a numerical method of assumed modes similar to that used for fully wetted surfaces. Numerical results for lift and moment for both steady and oscillating foils are compared with other theories and experiments. Results of these calculations indicate that this numerical solution gives an efficient and accurate prediction of loads on a supercavitating foil.

Introduction

L I N E A R I Z E D l i f t i n g - s u r f a c e m e t h o d s f o r c a l c u l a t i n g t h e l i f t d i s t r i b u t i o n o n t h i n f u l l y w e t t e d f o i l s have been re-f i n e d t o a h i g h degree o re-f accuracy a n d ere-fre-ficiency u s i n g t h e high-speed d i g i t a l c o m p u t e r . T h e i n t e g r a l e q u a t i o n r e l a t i n g t h e k n o w n b o u n d a r y values of u p w a s h v e l o c i t y

' T h i s work was performed under Contract Nonr 1841(81), Bureau of Ships Fuudameutal Hydromechanics Research Program, S-R009 01 01, administered by the D a v i d T a y l o r Model Basin. Computations were performed at the Massachusetts Institute of Technology Computation Center, Cambridge, Mass.

* Assistant Professor, Department of Aeronautics and Astro-nautics, Massachusetts Institute of Technology, Cambridge, Mass. Revised manuscript received at S N A M E Headquarters, October 11, 1965.

to t h e u n k n o w n d i s t r i b u t i o n of l i f t o n t h e f o i l surface c a n be t r e a t e d b y a n u m e r i c a l t e c h n i q u e i n w h i c h t h e l i f t d i s t r i b u t i o n is represented b y a series of modes w i t h u n -k n o w n coefficients. I n t e g r a t i o n o f these modes w e i g h t e d b y t h e k e r n e l f u n c t i o n o f the i n t e g r a l e q u a t i o n is p e r -f o r m e d n u m e r i c a l l y , a n d t h e b o u n d a r y c o n d i t i o n s are satisfied a t a n u m b e r of c o l l o c a t i o n p o i n t s . I t is n o w possible t o o b t a i n a n accurate p r e d i c t i o n o f t h e l i f t d i s -t r i b u -t i o n o n -t h i n f u l l y w e -t -t e d f o i l s i n s-teady or o s c i l l a -t o r y m o t i o n f o r a w i d e v a r i e t y o f c o n f i g u r a t i o n s i n c l u d i n g t h e i n t e r f e r e n c e effects f r o m n e a r b y boundaries such as a f r e e surface a t i n f i n i t e F r o u d e n u m b e r a n d f r o m n e a r b y l i f t i n g surfaces as i n t a i l s or F - f o i l a r r a n g e m e n t s [ 1 - 5 ] . '

' Numbers in brackets designate References at end of paper.

N o m e n c l a t u r e — — — _ _ _ —

Ob = coefficient of j i t h assumed-pressure-source mode

AR = aspect ratio

ha' = coefficient of rath assumed-pressure-doublet mode

bo root semichord

CL = lift coefficient

CM = moment coefficient

Uv) = assumed ?)-function for A P ( f , r,)

assumed f-functions for ( ö P / a r ( f , -n))

h = amplitude of heaving motion h'(x, y, t) = amplitude of unsteady

mo-tion of foil

h{x, y) = complex amplitude of foil oscillation

= assumed spanwise functions for < ö P / ö r )

= Heavyside function

h = modified Bessel function of first kind

k = reduced frequency, cobo/U

Ki,-2,3 = kernel functions defined by

equations (18), (19) and (25)

Kl = modified Bessel function of

third kind

In = chordwise assumed

f-func-tionsfor A P ( f , ?;)

[P] = matrix of pressure at x^yh due to ( ö P / ö f )

P = perturbation pressure

P = nondimensional complex

amplitude of pressure per-turbation

A p = pressure jump across foil sur-face

jump in normal derivative of pressure across foil-cavity surface S R = radius vector R = V i x - + ( y - vV + s = semispan/semichord S = surface of integration

<:f>

cavity surface (projection of) wetted surface (projection

of)

element of area on surface S free-stream velocity upwash on wetted surface of

foil

nondimensional coordinates coordinate of trailing edge of

cavity

angle of attack or amplitude of pitching motion delta function nondimensional coordinates free-stream density cavitation number perturbation-velocity poten-tial nondimensional complex amplitude of perturba-tion-velocity potential frequency of oscillation J U N E 1 9 6 6 _{l o r}

Experiments on G r a v i t y i S e c t s in

B y T . K i c e n i u k ^ mé A. J . A c o s t a ^

Experiments on the effect of a transverse gravitational field on the supercavitating flov/ past a wedge tend to confirm predictions based on linearized free-streamline theory. A small though systematic dependence upon Froude number not accounted for by the existing theory is revealed, however.

Nomenclature B = w e i g h t of l i q u i d displaced b y wedge CL = l i f t c o e f f i c i e n t = L/(pcV^/2) c = c h o r d F = F r o u d e n u m b e r = V/igcY^' g = g r a v i t a t i o n a l c o n s t a n t L = l i f t f o r c e p o s i t i v e u p w a r d s (opposite t o d i r e c t i o n oig) I = b u b b l e l e n g t h f r o m end o f wedge V = t u n n e l v e l o c i t y a = angle of attaclc Introduction

STHEET'*''^ has g i v e n a t h e o r e t i c a l a c c o u n t o f t h e ef-f e c t s oef-f a transverse g r a v i t y ef-field o n t h e ef-flow p a s t a s u p e r c a v i t a t i n g wedge. R e c e n t l y , an. o p p o r t u n i t y t o t e s t these results i n t h e H i g h Speed W a t e r T u n n e l a t t h e C a l i f o r n i a I n s t i t u t e of T e c h n o l o g y became a v a i l a b l e . F o r t h i s purpose, t w o wedges o f 6 - i n . c h o r d a n d 6 - i n . s p a n were m a d e ; one h a d a n apex angle of 7}yi deg a n d t h e o t h e r 1 5 deg. T h e wedges were m o u n t e d o n t h e f o r c e balance of t h e t w o - d i m e n s i o n a l w o r k i n g s e c t i o n i n t h e t u n n e l . " I n p r i n c i p l e a l l t h a t is necessary is t o

1 Research carried out under Bureau ot N a v a l Weapons Contract N123(60530)34767A administered by U . S. N a v a l Ordnance T e s t Station, Pasadena, Cahf.

2 Lecturer and Group Leader, Division of Engineering and Applied Science, Hydrodynamics Laboratory, von K&vm&a Labora-tory of F l u i d Mechanics and J e t Propulsion, California Institute of Technology, Pasadena, Calif.

3 Professor, Division of Engineering and Applied Science, Hydrodynamics Laboratory, von K è r m é n Laboratory of F l u i d Mechanics and Jet Propulsion, California Institute of Technology, Pasadena, Cahf.

R . L . Street, "Supercavitating F l o w About a Slender Wedge in a Transverse Gravity F i e l d , " Journal of Ship Research, vol. 7, no. 1, 1963, pp. 14-24.

' R . L . Street, "A Note on G r a v i t y Effects in Supercavitating F l o w , " Journal of Ship Research, vol. 8, no. 4, 1965, pp. 39-46.

•5 T . Kiceniuk, "A Two-Dimensional Working Section for the H i g h Speed Water T u n n e l at the California Institute of T e c h -nology," Hydrodynamics Laboratory, California Institute of Technology, Report No. E-108.10.1, December 1963.

Manuscript received at S N A M E Headquarters, December 13, 1965.

measure t h e l i f t f o r c e a t zero angle of a t t a c k i n t h e pres-ence of t h e n o r m a l g r a v i t a t i o n a l field w h e n t h e c a v i t y is established b e h i n d t h e wedge. T h e p r e d i c t e d g r a v i t y -i n d u c e d l -i f t f o r c e -is q u -i t e s m a l l ; f o r t h e wedges used -i t a m o u n t e d t o o n l y a f e w p o u n d s a t m o s t . L i f t forces due t o even s l i g h t changes i n angle o f a t t a c k a t flow velocities o f 2 0 a n d 3 0 f p s are appreciable so t h a t g r e a t care m u s t be t a k e n t o p r e v e n t m a s k i n g t h e s o u g h t - f o r g r a v i t y eft'ect b y s m a l l changes i n t h e o n c o m m g flow d i r e c t i o n p o s s i b l y i n d u c e d b y difi'erent a m b i e n t pressure, d i f t e r e n t t u n n e l v e l o c i t i e s or b y errors i n t r o d u c e d b y t h e f o r c e balance i t s e l f .

One possible source o f e r r o r was r e a d i l y a p p a r e n t w h e n base c a v i t a t i o n t o o k place b e h i n d e i t h e r wedge a t t u n n e l speeds of 2 0 t o 3 0 f p s . A s t h e fluid w i t h i n t h e w o r k i n g section is s u b j e c t t o t h e u s u a l I r y d r o s t a t i c pressure g r a d i e n t , t h e s t a t i c pressure a t t h e t o p of t h e w o r k i n g section is less b y 1.5 f t of w a t e r t h a n a t t h e w e d g e i t s e l f . A t l o w c a v i t a t i o n indices t h e fluid i n t h i s r e g i o n can also c a v i t a t e . I n v e s t i g a t i o n s revealed t h a t t h e flow a t t h e c e n t e i i i n e of t h e f o r c e balance was d i v e r t e d b y n e a r l y 0 . 1 deg w h e n t h i s h a p p e n e d . Since t h e a n t i c i p a t e d g r a v i t y efi'ect a t these t u n n e l speeds corresponds o n l y t o a n angle of a t t a c k change o f a b o u t 0 . 0 5 deg, procedures t o a v o i d t l i i s efl'ect i n p a r t i c u l a r w e r e t h e r e f o r e a b s o l u t e l y necessary. Experimental Procedures A f t e r some e x p e r i m e n t a t i o n , t h e t e c h n i q u e o f base v e n t i l a t i o n w i t h a i r was a d o p t e d . B y t h i s m e a n s t h e a m b i e n t pressure w i t h i n t h e t u n n e l c o u l d be a d j u s t e d t o a n y l e v e l desired a n d s t i l l m a i n t a i n a v e n t e d a i r b u b b l e . S u c h a v e n t i l a t e d b u b b l e s i m u l a t e s i n a l l i m p o r t a n t respects f o r t h e purposes of t h e p r e s e n t ex-p e r i m e n t , a v a ex-p o r o u s s u ex-p e r c a v i t a t i n g b u b b l e . T h e a c t u a l procedures used t o o b t a i n t h e t e s t d a t a w e r e as f o l l o w s : A wedge was m o u n t e d o n t h e f o r c e b a l a n c e a n d i t s c e n t e i i i n e was a c c u r a t e l y a l i g n e d t o t h e t u n n e l g e o m e t r i c a l c e n t e i i i n e ( w h i c h is also precisely t h e angle of zero l i f t o f a s y m m e t r i c a l h y d r o f o i l ) w i t h a p r e c i s i o n c l i n o m e t e r . T h e t u n n e l was b r o u g h t u p t o speed a t a n

J U N E 1 9 6 6

A Critical Re-Evaluation of H y d r o d y n a m i c Theories

a n d Experiments in Subcavitating Hydrofoil Flutter

-A review of relevant hydrodynamic information has been made, with emphasis on the possible cause of serious discrepancies between classical hydrodynamic theory and ex-periments for subcavitating hydrofoils. The usefulness of the best available wind tunnel and towing tank force measurements is also discussed. It is believed that these data are unreliable, and that the completion of the force calculation based on the classical lifting surface theory and subsequent flutter prediction for an actual flutter model is desirable. Some recommendations for possible future experimental research are also given.

Introduction

T H E p r o b l e m of f l u t t e r of a submerged ( s u b c a v i t a t i n g ) h y d r o f o i l has been discussed i n reference [ I ] . ' A c c o r d i n g t o t h e classical f l u t t e r t h e o r y , t h e r e is a f l u t t e r b o u n d a r y w h i c h is above a c r i t i c a l r e l a t i v e d e n s i t y p a r a m e t e r . B e l o w t h i s c r i t i c a l d e n s i t y p a r a m e t e r t h e m o t i o n is t l i e o r e t i c a U y flutter-free f o r a n u n s w e p t w i n g . T h e r e h a v e been l i m i t e d e x p e r i m e n t s r e p o r t i n g f a i l u r e s of m o d -els b e l o w t h i s c r i t i c a l d e n s i t y p a r a m e t e r , b u t t h e r e is d o u b t as t o w h e t h e r s u c h f a i l u r e s were caused b y flutter o r s t a t i c divergence. H o w e v e r , some r e l i a b l e N A C A ex-p e r i m e n t s i n F r e o n also s h o w e d discreex-pancies b e t w e e n t h e o r y a n d e x p e r i m e n t w h i c h w e r e discussed i n reference

[ 1 ] . I t was f o u n d t h a t a n exact b e a m t h e o r y was i n -s u f f i c i e n t t o reconcile t h e t h e o r y a n d experiment-s, o r t o y i e l d a c o n s e r v a t i v e r e d u c e d flutter speed near t h e flutter b o u n d a r y .

Since t h e aspect r a t i o o f t h e N A C A m o d e l considered was eight, i t was t h o u g h t t h a t t h e finite s p a n c o r r e c t i o n t o t h e t w o - d i m e n s i o n a l a e r o d y n a m i c s t r i p t h e o r y used i n these p r e d i c t i o n s m a y be r e l a t i v e l y s m a l l . F u r t h e r , t w o d i m e n s i o n a l s t r i p t h e o r y , a t least i n a e r o n a u t i c a l a p p l i -cations, u s u a l l y leads t o r a t h e r c o n s e r v a t i v e results; i.e., t h e d e r i v e d flutter speed is m u c h l o w e r t h a n t h e measured v a l u e . I t seems, t h e r e f o r e , t h a t t h e error lies i n t h e h y d r o d y n a m i c l o a d i n g p r e d i c t e d b y t h e classical t h e o r y . T h i s m i g h t be caused b y t h e c o m p l i c a t e d viscous efl'ect of a real fluid passing o v e r a n o s c i l l a t i n g f o i l .

I n p o t e n t i a l t h e o r y , t h e r e a l fluid efl'ect is u s u a l l y t a k e n

The results presented in this paper were obtained during the course of research sponsored by the B u r e a u of Ships, Department of the N a v y , under Contract No. N O b s 88599.

2 Senior Research Engineer, Department of Mechanical Sciences, Southwest Research Institute, S a n Antonio, T e x .

^ Numbers in brackets designate References at end of paper. Manuscript received at S N A M E Headquarters J a n u a r y s , 1966.

1 2 2

care o f s e m i - e m p i r i c a l l y b y r e l a x i n g t h e K u t t a c o n d i t i o n . I n reference [ 2 ] t h e p r e v i o u s e x p e r i m e n t a l d a t a l o c a t e d were i n c o n c l u s i v e i n t h e l o w reduced v e l o c i t y r a n g e (zero m e a n angle o f a t t a c k ) as c o m p a r e d w i t h p r e d i c t e d values of t h e phase l a g of t h e l i f t f o r c e .

Subsequently, h y d r o d y n a m i c spanwise l o a d i n g meas-u r e m e n t s o n a s meas-u b c a v i t a t i n g h y d r o f o i l w e r e c o n d meas-u c t e d a n d r e p o r t e d i n reference [ 3 ] . F l u t t e r f a i l u r e of a m o d e l was r e p o r t e d i n reference [4 ] . I n reference [4 ] p r e d i c t i o n of flutter based o n those measured coefficients i n reference

[ 3 ] was o v e r c o n s e r v a t i v e . P r e d i c t i o n of flutter based on i n t e r p o l a t e d w i n d t u n n e l d a t a of reference [ 5 ] was q u i t e u n c o n s e r v a t i v e f o r t h e h y d r o f o ü . T h i s w o u l d be even t h e m o r e so i f t h e Reissner-S tevens t h e o r y w e r e em-p l o y e d . O n t h e o t h e r h a n d , a n order of m a g n i t u d e esti-m a t e based o n Oseen's e q u a t i o n i n reference [ 6 ] f o u n d n e g l i g i b l e d e v i a t i o n s f r o m t h e K u t t a c o n d i t i o n i n the classical t h e o r y . I t is t h e r e f o r e desirable t o reevaluate b o t h t h e theories a n d e x p e r i m e n t s f o r flutter i n sub-c a v i t a t i n g flow i n t h e hope t h a t dissub-crepansub-cies sub-can be ex-p l a i n e d . F u r t h e r m o r e , i t is exex-pected t h a t b y such a re-e v a l u a t i o n a b re-e t t re-e r u n d re-e r s t a n d i n g of t h re-e p r o b l re-e m c a n bre-e reached a n d some reasonable r e c o m m e n d a t i o n s f o r f u t u r e research can be m a d e .

T h e i n c r e a s i n g speed of h y d r o f o i l c r a f t has l e d t o a c t i v e research i n t h e area of u n s t e a d y s u p e r c a v i t a t i n g flow, b u t s u b c a v i t a t i n g flutter m a y be s t i l l of i n t e r e s t as i t is q u i t e possible t h a t t h e c r u i s i n g speed w o u l d be m u c h l o w e r t h a n t h e m a x i m u m speed, a n d t h u s t h e flow field m a y be sub-c a v i t a t i n g . T h e r e m a y be also a p r o b l e m of passing t h r o u g h a range of s u b c a v i t a t i n g speeds i n w h i c h t h e f o i l is u n s t a b l e .

F o r t u n a t e l y , a d d i t i o n a l e x i s t i n g e x p e r i m e n t a l i n f o r m a -t i o n has been collec-ted. H o w e v e r , f o r convenience of discussion, k n o w n results i n b o u n d a r y - l a y e r t h e o r y w i l l first be s u m m a r i z e d b r i e f l y . A l t h o u g h these results are n o t d i r e c t l y a p p l i c a b l e , t h e y m a y be of qualitative value f o r

é §1

HSP K E S E A R C H i

Experimeiiifi"©]! I^esults'

The analytical method of reference [1 Y fof estimating stability derivatives, and hence stability on course, which combines Albring's empirical modifications of simplified flow theory with low aspect-ratio wing theory, is extended to take into consideration the effects on course stability of higher aspect-ratio fins as well. The method, which has been applied in the earlier report to a family of eight hulls of 0.5 block coefficient, is tested further by application to eight Series 60 forms differing in block coefficient as well as in beam, draft, and displacement-—with and without rudders; to an extreme vee modifica-tion of a Series 60 model; and to three other forms—q Mariner Class model, a destroyer, and a hopper dredge. Comparison with experimental results shows that the values of stability derivatives and indices determined by the analytical method are of the right orders of magnitude and indicate correct trends. Application to a variety of ship forms has demonstrated that the method can predict relative effects of changes in the geometry of a ship form, as well as the effects of changes in skeg and rudder a r e a .

introduction

I N a n earlier r e p o r t [1 ] , an a n a t y t i c a l m e t h o d was de-v e l o p e d f o r e s t i m a t i n g t h e f i r s t - o r d e r s t a b i l i t y d e r i de-v a t i de-v e s (static a n d r o t a r y l a t e r a l - f o r c e a n d y a w i n g - m o m e n t rates) w h i c h w o u l d i n d i c a t e t h e course s t a b i l i t y a n d t u r n i n g or steering q u a h t i e s o f ships. T h e m e t h o d was a p p h e d t o t h e case of a f a m i l y of e i g h t h u l l s of t h e same l e n g t h a n d t h e same p r i s m a t i c a n d b l o c k coefficient, b u t d i f f e r i n g i n d r a f t , beam, a n d displacement. T h e h u l l s were t h e 840 Series of t h e T a y l o r S t a n d a r d Series t y p e w i t h t h e a f t e r d e a d w o o d ( f a i r e d i n skeg) r e m o v e d . E x p e r i m e n t a l l y m e a s u r e d l a t e r a l forces a n d y a w i n g m o -ments, f r o m D a v i d s o n L a b o r a t o r y r o t a t i n g - a r m tests

1 Prepared for Bureau of Ships Fundamental Hydromechanics Research Program (S-R009-01-01). Administered by D a v i d Taylor Model Basin under Contract Nonr 263(57), D L Proiect 2803/063.

= Research Engineer, F l u i d D y n a m i c s Division, Davidson Labora-tory, Stevens Institute of Technology, Hoboken, N . J .

^ Numbers in brackets designate References at end of paper. Manuscript received at S N A M E Headquarters, December 3,

1965. ^ a t d i f f e r e n t t u r n i n g r a d i i , were a v a i l a b l e f o r these h u l l s a n d f o r t h r e e of t h e h u l l s w i t h f l a t - p l a t e skegs i n t h e place of t h e r e m o v e d deadwood. A l t h o u g h t h e a n a l y t i c a l m e t h o d is based u p o n s i m p l e concepts c o m b i n i n g s i m p l i f i e d flow t h e o r y w i t h l o w aspect-ratio w i n g t h e o r y a n d u s i n g A l b r i n g ' s [ 3 ] em-p i r i c a l m o d i f i c a t i o n s f o r v i s c i d flow, g o o d c o r r e l a t i o n was a t t a i n e d b e t w e e n t h e s t a b i f i t y d e r i v a t i v e s c a l c u l a t e d b y t h i s m e t h o d a n d those d e t e r m i n e d f r o m e x p e r i m e i r t a l d a t a . H o w e v e r , A l b r i n g ' s m o d i f i c a t i o n of t h e r o t a r y -m o -m e n t r a t e is a f u n c t i o n of p r i s -m a t i c c o e f f i c i e n t a n d , since a l l t h e h u l l s of t h e 840 Series h a v e t h e same p r i s -m a t i c (0.54), t h i s -m o d i f i c a t i o n was n o t t e s t e d f u l l y . I t Avas decided, t h e r e f o r e , t o extend a p p l i c a t i o n of t h e p r e d i c t i o n m e t h o d t o h u l l s of other p r i s m a t i c , w i t h a n d w i t h -o u t skegs -or d e a d w -o -o d a f t , f -o r w h i c h exiDcrimental d a t a were a v a i l a b l e .

Results of several straight-course tests [2] confirmed previous experience at Davidson Laboratory that entirely reliable static force and moment rates for straight-course motion can be ob-tained from rotating-arm data at sufficiently large turning radii.

S h o l l o w W a f e r a n d C h a n n e l Effoets on

W a v e Resistance

B y M a r i a K i r s c h i

Based on Sretensi<i's formula, the wave resistance in shallow water and in a channel of rectangular cross section is computed for certain mathematical ship-forms. The results are plotted as curves and compared with the wave resistance in an unbounded liquid. The use of these curves for the determination of the wave resistance from towing-tank experi-ments is illustrated through an example.

Introduction

T H E basis f o r t h i s i n v e s t i g a t i o n is p r o v i d e d b y IN'Iichell's t l i e o r y of w a v e resistance [1]^ a n d b y t h e extensions m a d e t o t h i s t h e o r y since i t s appearance ( H a v e l o c k [ 2 ] , W e i n b l u m [ 3 ] , L u n d e [ 4 ] , a n d o t h e r s ) . Of special i n -terest f o r o u r purposes are t h e papers of S r e t e n s k i [5, 6 ] , since Sretenski's f o r m u l a s are used i n t l i e present w o r k i n t h e c o m p u t a t i o n of w a v e resistance i n shallow w a t e r a n d i n a channel. A discussion of t h e wave-resistance

' Doctor of Engineering, Institut fur Schiffbau der Universitat Hamburg, Hamburg, Germany.

^ Numbers in brackets designate References at end of paper. Manuscript received at S N A M E Headquarters, December 6, 1965.

i n t e g r a l f o r a c h a n n e l was g i v e n b y W i g l e y [ 7 ] . T h e f i r s t step of t h e present w o r k was t h e s y s t e m a t i c calcula-t i o n of calcula-t h e w a v e resiscalcula-tance i n a c h a n n e l , u s i n g a n I B j \ i 650 electronic c o m p u t e r . Several results f r o m these c a l c u l a t i o n s have' already appeared [8 ] , w h e r e some of t h e c o m p u t e d wave-resistance coefhcients f „ were c o m p a r e d w i t h a n d agreed w e l l w i t h results of c o m p u t a t i o n s m a d e b y A p u k h t i n a n d V o i t k u n s k i [ 9 ] .

T h e r a n g e of calculations i n t h e present w o r k was g r e a t l y e x t e n d e d ; first of a l l , i n t h a t v a r i o u s L / i ? r e l a -t i o n s were -t a k e n i n -t o considera-tion, a n d also, i n -t h a -t -t h e wave-resistance i n t e g r a l was e v a l u a t e d f o r s h a l l o w w a t e r u s i n g Sretenski's m e t h o d s . H e r e , t h e necessary calcula-t i o n s were c a r r i e d o u calcula-t w i calcula-t h calcula-t h e h e l p of a T R 4 eleccalcula-tronic c o m p u t e r . F i n a l l y , f o r t h e same h u l l c o n t o u r s f o r w h i c h

-Nomenclature-B = beam of ship

= midship cross-sectional area

Fk = cross-sectfonal area of channel

g = acceleration of gravitj' H = water depth

I , J = integrals over wetted surface of

ship, by which symmetric and asymmetric parts ot ship's surface are taken into consideration

I*, J* = normed forms of integrals I , J In, Jn = expressions, analogous to I , J,

in formula for wave resist-ance in a channel

K = channel width

L = length of ship i?„ = wave resistance

/S,„ = shallow water T = draft of ship

V = velocity of ship or model

x,y,z = a f » = X = P <P Hz) Kx, z) FA = coordinates in horizontal, transverse, and vertical di-rections

area coefficient of waterline midship-section coefficient wave-resistance component integration variable in

wave-resistance integral density of liquid prismatic coefficient wetted ship's surface eciuation of design waterline equation of midship section g{x)-f(z) = equation of wetted

ship's surface

V

—JTy/i = Froude number

1^2 - depth Froude (g-H)

number

R* = R,„/{8/-!rpgB^TyL) = normed, dimensionless form of wave resistance according to Wein-blum [10]

To = 1/(2F")

f = x/iL/2), 7, = fix, z)/iB/2), r = z/T: normed, dimension-less coordinates

7 = 70 • A = integration variable in wave resistance integral after norming ë = 2 r / L - 7 V T o e* = coefficient of f'" in normed midship-section polynomial; function of midship-section coefficient (S 9U = functions: expressions of form

ƒ'

O n a Free-Fl@ating Ship in W a v e s

The oscillatory forces on a free-floating ship of spherical or ellipsoidal hull at zero forward speed in a head sea are evaluated for the modes of surge, heave, and pitch. The potential boundary-value problem consists of (a) the determination of the radiation potential which is due to the forced oscillation of a floating ship in a calm sea and (b) the determination of the diffraction potential which is due to the scattering of oncoming waves on a ship fixed in the equilibrium position. Upon knowing the values of the radiation potential over the immersed surface of the ship, the hydrodynamic forces arising from the impingement of oncoming waves are evaluated using the Haskind relation. Thus, the dependencies of added mass, added moment of inertia, and damping coefficients on the frequency of the forced oscillation as well as on the frequency of the oncoming wave are shown. Finally, the equation of motion governing the oscillation of the freely floating ship is obtained through the synthesis of hydrodynamic forces in the radiation and diffrac-tion problems. The frequency dependence of the two pertinent parameters occurring in the equation of motion, the amplitude ratio, and the phase lag between the ship motion and the wave motion are also established.

Introduction

O N E of t h e i n t e r e s t i n g p r o b l e m s i n ship h y d r o d y n a m i c s is d e t e r m i n i n g t h e v a r i a t i o n of t h e i n e r t i a l a n d d a m p i n g characteristics i n c o n n e c t i o n w i t h the f r e q u e n c y of a f r e e l y o s c i l l a t i n g ship i n surface waves.

W h e n t h e a m p l i t u d e fj" of t h e w a v e is s u f f i c i e n t l y s m a l l c o m p a r e d t o t h e w a v e l e n g t h X, w i t h i n t h e f r a m e w o r k of t h e l i n e a r i z e d t h e o r y , t h e fluid m o t i o n r e s u l t i n g f r o m t h e i n t e r a c t i o n b e t w e e n t h e o n c o m i n g w a v e a n d t h e ship m o t i o n m a y be described i n t e r m s of t h e v e l o c i t y p o t e n -t i a l of -t h e o n c o m i n g w a v e , -t h e v e l o c i -t y p o -t e n -t i a l of t h e scattered w a v e o n . t h e i m m e r s e d h u l l , a n d t h e v e -l o c i t y p o t e n t i a -l $ of t h e f-luid s u r r o u n d i n g a n o s c i -l -l a t i n g s h i p i n a c a l m sea. T h e r e f o r e , t h e p o t e n t i a l b o u n d a r y - v a l u e p r o b l e m con-sists of t w o p a r t s ; t h e r a d i a t i o n p r o b l e m i n w h i c h t h e p o t e n t i a l $ is sought and t h e d i f f r a c t i o n p r o b l e m i n w h i c h t h e p o t e n t i a l is also sought. I t is w e l l k n o w n t h a t b o t h p r o b l e m s can be f o r m u l a t e d u s i n g t h e concept of s u r f a c e - d i s t r i b u t e d sources. T h e n , t h e determ i n a t i o n of p o t e n t i a l s reduces t o t h e s o l u t i o n of a F r e d -h o l m i n t e g r a l e q u a t i o n of t -h e second k i n d w i t -h t -h e s t r e n g t h of sources as u i f l c n o w n . Nevertheless, t h e p r o c e d u r e necessary f o r s o l v i n g such a n i n t e g r a l equa-t i o n is r a equa-t h e r d i f f i c u l equa-t because equa-t h e k e r n e l e x h i b i equa-t s a singu-l a r character.

R e c e n t l y , t h e r a d i a t i o n p r o b l e m was f o r m a l l y t r e a t e d

1 F l i g h t Sciences Laboratorj^ Boeing Scientific Research L a b o r a -tories, Seattle, Wash.

Manuscript received at S N A M E Headquarters, December 27, 1965.

b y t h e a u t h o r [1]= f o r t h e b o d y h a v i n g t h e f o r m of a n ellipsoid or a n e l l i p t i c c y l i n d e r . I n t h i s paper, b y means of H a s k i n d ' s r e l a t i o n w h i c h is a d v o c a t e d b y N e w m a n [ 2 ] , t h e w a v e forces a c t i n g o n a ship of spherical or e l l i p -s o i d a l f o r m i n a flxed p o -s i t i o n are d e t e r m i n e d i n d i r e c t l y u s i n g t h e r a d i a t i o n p o t e n t i a l i n s t e a d of s o l v i n g f o r t h e scattered-wave p o t e n t i a l .

Once t h e d y n a m i c forces f o r t h e m o d e of heave F j , , ' a n d t h e d y n a m i c m o m e n t f o r t h e m o d e of p i t c h ft are k n o w n , r e s o l v i n g these i n t o a c o m p o n e n t i n phase w i t h t h e acceleration a n d t h e o t h e r c o m p o n e n t i n phase w i t h t h e v e l o c i t y of t h e ship (or of t h e w a v e ) , we m a y w r i t e f o r i n s t a n c e F, = - p d M ' y f - padW,? (1) ft = - pa^I^d^ — paa^H^Ö^ w i t h P = R e [ f e - ' • • " ] a n d 6 = R e [ö^e-'••"], w h e r e p is t h e d e n s i t y of t h e fluid, d denotes t h e h a l f - l e n g t h of t h e ship, a n d <T is t h e o s c i l l a t i n g f r e q u e n c y . T h e d i m e n s i o n -less q u a n t i t i e s M a n d N are d e f i n e d as t h e added mass a n d h n e a r d a m p i n g coefficient, w h i l e t h e dimensionless q u a n -t i -t i e s / a n d H are d e f i n e d as -t h e added m o m e n -t of i n e r -t i a a n d a n g u l a r d a m p i n g coefficient, respectively. T h e s e i n e r t i a l a n d d a m p i n g characteristics are s h o w n as a f u n c -t i o n of -t h e p a r a m e -t e r d-tr^/g = Ö 2 T / X = . a, g b e i n g -t h e a c c e l e r a t i o n of g r a v i t y .

^ Numbers in brackets designate References at end of paper. ' I n this work the mode of surge is also considered. However, for-the sake of a concise presentation, only for-the results will be shown.