lournal of
S H I P R E S E A R C H
Mechanical Properties of Metals and Their
Cavitation-Damage Resistance
By A . Thiruvengadam' and Sophia
Waring-Detailed investigations with a magnetostriction apparatus were carried out fo determine the cavitofion-damage resistance of eleven metals in distilled water at 80 F. The cavita-tion-damage resistance is defined as the reciprocal of fhe 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 Ihe 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 fhe displacement amplitude of the specimen. All these conclusions are limited to fhe steady-state zone of damage.
Nomenclature <•' .•iniplitiKlc iireiv of orositm / = iiilcii.sit v of c i i v i l a l i o i i iliUiiaKf H
=
s i r n i i i - l i a i ' d c i i i t i K f a c l o r r r u l e of v o l u m e los.s T' = corrt'laf i o n f a c l o ra.
.>i|iaiti cuergi,-'*
i-.-iliniali'd u t i a i n c i i c i t y S'.' I n i o s t r a i n c n e r u j ' 7'=
u l l i n i a l c Icii.sile . s l r e n n l hV
—- t r u e f r a i ' l i i r o s( r c i i n l l i )•=
y i e l i l Kl i c i i g l h < iillinial<- c l o i i u a l i o i i c l o i i K i i l i o n .-ll r i a c l i i i r IniroduclionS i . v f K (he w o r k o f I':ii-soii.'< [ 1 ] ' i l l 1910 iiiul l oltiiiK<n-|2) i l l l!)2(l, l l i c i v h a v e b c f i i m a n y iiftciiipt.s to clianic-I c r i z o l l u ! i-avi(af ioii-(lainaK<' ri'.si.slaiico of lualerial.s b y a .'iiiifflc, i-oiniuoii niochaiiical p i o i i o r l y . A l t h o u g h
' S o i i i o r I t c s c a r c l i •Scioulisl, lIv<li"oii.iiilic.S IncoiT|>oialc(I, l^aiiicl, M<1.
* As.«i.slimt I>c.<i'arcli S c i c i i l i s i , H v < l r o n a u l i c s , I i i c m p o r a l i ' i l , L a m v l , .M.I.
:> N u i i i l j i ' i s in brai-kol.s ili'.jiuiialo Itufcrcin'cs «I oiiil of )>:ipur. Maiiii-si r i p l r u c c i v e i l u l . S N ' . W l l ' ; H f a i l n i i a r l c r . s M a r c h r>, l i l f M .
Ilouegger [ 3 ] , i n 1927, d i d n o t find a n y c o r r e l a t i o n be-tween hardness a n d erosion resistance, G a r d n e r [ 4 ] , i n 1932, f o u n d that the hardness o f a m e t a l was t h e p r i i i c i | ) a l propertv i n d c t e n n i n i n g the resistance t o erosion. M a n y more references m a y be cited t o b r i n g out s i m i l a r c o i i -lroA-ersie.s w i t h regard to other mechanical p r o j i e r t i c s such as y i e l d strength, u l t i m a t e teasilc s t r e n g t h , u l t i m a t e elongation and m o d u l u s of e l a s t i c i t y . One can g e t a clear picture o f the m a g n i t u d e o f the conflicts i n t h i s area f r o m .some of the excellent review articles i n t h e t e c h -nical l i t e r a t u r e [.'>, 0 , 7 J.
These controversies arc a result o f an i i i a d o q u a t o understanding of the mechanism of c a v i t a t i o n damage. U(>ceiit advances i n t h i s d i r e c t i o n have made it possible to rationalize some of the coidlicts, and t o propo.se a. ineclianical p r o i i e r l y t h a t most s i g i i i t i c a n t l y characterizes the cavilalion-daniagc resistance of metals i n t h e ab-sen.se of corrosion, r e l a t i v e l y .siieaking. I t is (he pnrpo.se of (his paper to develop (he logic b e h i n d such a n a r g u -ment, and to present recent s u b s t a n t i a t i n g e x i i e r i m e n t a l evidence.
One of (he basic [larameters i n v o l v w l i n the t e s t i n g o f materials f o r cavitat ion-damage resistance is t h e tost duration. T h e rate of lo.-;s of m a t e r i a l depends u p o n (he lest d u r a t i o n itself even t h o u g h e v e r y o t h e r test pa-rameter is m a i n t a i n e d ])reci.-<cly cons(an(. IJecent
Expenments on Flat-Bottom Slamming'
By Sheng-Lun
Chuang-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.
N o m e n c l a t u r e .1 i i n p i u l u r e a of f l u i d or p l a l e a a c i e l e r a l ion in K c n c n i l c - s p e e d of s o u n d i n fluid
=
s p e e d of s o u n d i n a i r /'"=
f o r e e u e l i n i ; u p o n f a l l i n K b o d y I inipul.se i n K C i i e n d L l i a l f - w i d t h of i n f i n i t e l y l o n g p l a t e tn = m a s s of f a l l i n g b o d y >"!=
m a s s of fluid I'=
i m p a i t pres.sure, p s i /*,„».r
niaNiniuni i m p a c t p r e s s u r e , i>.si /*,„».r
half i>criod or d i i r a l ion o f first p o s i t i v e p u l s e=
t i m e i n g e n e r a l la l i n i e a l i n s t a n t of i m p a c t V v e l o c i t y in g e n e r a lv„
i m p a c t v e l o c i t y at t i m e /„, f p s P or pti„i,i nias.s d e n s i t y of H u i d Introduction I T has 1)00,1 l)oIiovo<l g o i . o . a l l y t h a i fl^'V'"!'"'",f'"'.V m i . i j r is a c o m b i i i o t l acott.stio a m i t m s l e a d y h y d i o d y i . a m . c l.honomonon. .lust p . i o r t o t h e occun-ence of ""Pac a l l i i i t o r v e n i t i g a i r is assutued t o be f o r c e d o u t f ' o m J " ' ^ ^ ' " v.eatl. the p l a t e [ I ] . ^ ' T h u s , eonMne.-sH.b.hty o f t h e f l u i d ( w a t e r i n his s t u d y ) m u s t be considered U s i n g the.se 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 . m u n s l a m m i n g p i U u r e can be e o m i m t e d as .shown m t h e n e x t ' ' ' W h e l l ' l r t l a t h o t l y s t r i k e s t h e s u r f a c e o f a f l u i d a t an i m p a c t v e l o c i t y \ \ , t h e p r o p a g a t i o n of t h e " ^ « ' " ^ " t a r y increase o f pressure i n t h e fluid t a k e s ,)lace a t the speed o f s o u . i . l i n t h e fluid, designated b y c. T h e maiis o f fluid accelerated i n t h e t i m e At is • T h e o p i n i o n s e x p r e s s e d u r e t h o s e of t h e a u t h o r a l o n e ^ a u d s h o u l d not b e . o n s t r a e d K . r e f l e c t t h e o l h e i u l v i e w s of t h e Nilv.V n c - p a n n i e n l o r t h e N a v a l St!r>-Ke at l a r g e . . - S t r i i i t n r a l l i e s e a r c h K i i g i i i e e r , S t r u c t u r a l M e c h u i i i c s l^nbora-t o r y , l J u v i d T a y l o r M o d e l B a . s i n , W a s h i n g l^nbora-t o n , 1). C . ^' N n m h e r s in' b r u c k e l s d e s i g n a t e R e f e r e n c e s a l e n d of p a p e r . .Nlaiiu.s.iijit r e c e i v e d at S N A M K H e u d i | u a r l e r s , A n g u s f 10, ]'.)I1.5; r e v i s e d , D e c e m b e r / , l!lG."). Installation of test m o d e l VI f = p A c A fwhere p is (he mass d e n s i t y o f fluid a n d A is t h e s t r i k i n g area o f the flat b o d y u j i o n the s u r f a c e o f the fluid.
Since (he v e l o c i ( y o f t h e mass o f fluid is increased f r o m zero t o Fo i n t h e t i m e At, t h o 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 therefore
T h e i)ro.sstue j), w h i c h is t h e f o r c e per u n i t area, is
•p = pcV'o (1)
T h e d u r a t i o n o f t h o comino.s.sion iihase is 'IL/c, whore L is the h a l f - w i d t h o f an i n h n i t e l y l o n g p l a t e
Drifting Force and Moment on Ships in
Oblique Waves'
By Pung Nien Hu' and King Eng"
A general expression for the drifting moment about the vertical axis of an oscillating ship in regular oblique waves 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 s h i p is o s c i l l a t i n g o n t h e f r e e surface i n r e -sponse t o r e g u l a r Avavcs e n c o u n t e r e d , t h e e x c i t i n g force 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
separated i n t o t w o j j a r t s ; one p a r t is oscillator}' 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 force a n d
m o m e n t are, w i l h rcsiiect t o t h e w a v e a m i ) l i t u d e , of t h e second o r d e r as c o m p a r e d w i l h t h e osciHatoiy p a r t o f t h e foree a n d m o m e n t , t h e y i n d u c e a steady d r i f t l o t h e .ship a n d a r e t h e r e f o r e v e r y i m i i 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 o p r o b l e m o f d r i f t w a s s t u d i e d b,v S u y e h i r o [1],* W a l a n a b e [ 2 ] , H a v e l o c k [3, 4 ) , .Maruo [0, ( i j . ' a n d IJe.ssho
[7 ] . T h o most r a t i o n a l t r e a t m e n t w a s given b y M a r u o i n a recent i)apor [ti) i n w h i c h he ai)i)lied t h e p o w e r f u l m o m e n t u m a n d energy theorems l o t h e f l u i d , together w i t h t h o p o t e n t i a l t h e o r y , l o o b t a i n a general exjircssion f o r t h e d r i f t i n g f o r c o a c t i n g o n an o s c i l l a t i n g b o d y , i n t e r m s o f (h<' j i o t o n t i a l d u e 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 between t h e b o d y -a n d w-avos. Once l l i e v e l o c i t y I i o l e i i 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 force can ho o b -t a i n e d b y s i m p l e ejuadra-turos. T h e effec-t of wave reflect i o n a n d reflect h a reflect o f s l i i i i o s c i l l a reflect i o n are a u l o m a l i e a l l y i n -c l u d e d i n t h o d e t e r m i n a t i o n o f v e l o -c i t y ))otentials.
I n t h o present s t u d y , w e w i l l f o l l o w Alaruo's procedure b y u s i n g t h e m o i n o i i t - o f - m o m e i i l u m theorem l o o b t a i n a general oxiJio.ssion 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 u g h t h o oxprc-jsion f o r t h e d r i f t i n g force o b t a i n e d b y -Maruo a n d t h a t f o r t h e d r i f t i n g n i o i n e i i t obtained i n t h e p i c s o i i t s t u d y a r e C | u i t e general ( a j i i i l i c a b l o 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 that t h e wave
' W o r k .sponsored b y I h c H u i c a u of Ship.s KniidamcMilul H y d r o -iiKii lmnii-s l i c s c i M c l i I ' r o K n n n ( ! S - l ! O I I ! M ) l - O j ) , u d m i i i i s l e i c d b v 1 );ivi<l T a y l o r .Model H a s i n C . ' o n l r a f l N o n r 2(>:5(21).
- H c a < l , F l o w I'hcnonnMia D i v i s i o n , D a v i d . s o n b ; i h o r a l o r y S i f ' v t i i i s I n s l i l u t i - of T ( ' c l u i o l o g y , l l o b o k c n , N . . J . I ' r c s e n l l y fioiiior Si'i(Milis(, .Spall'.Scii'Mii'S, I n c o r p o i a l c d , V V a l l h a m , .\ta.ss,
I f c s c a n l i I ' j i K i i K ' c r , D a v i d . s o n I , ! i l i o r a l o i y , S l c v c n s I n s t i l n i o of T c i - h i i o l o t i y , l i o l i o k c n , N . . J .
' N v n n l i c r s i n b r a c k e t s d e s i g n a t e Itefererices a l e n d it! p a p e r . . M a n u s c r i p l r e i e i v e d al . S X A . M I - ; lleadcpiarler.s, O c t o b e r 2.S I!)().").
a m i i l i t u d e be .small, so t h a t t h e p r o b l e m can he liiieav-i ü c d ) , wc sJliiieav-iall c a r r y o u t d e t a liiieav-i l e d c a l c u l a t liiieav-i o n s f o r t h liiieav-i n ships i l l l o n g waves. I t w i l l be f o u n d t h a t t h o f i n a l ex-l)rcs.sioiis f o r t h o d r i f t i n g forco a n d m o m o n t a r c e x t r e m e l y s i m p l e ; s i g i i i l i c a n t p h y s i c a l i n t o r p r o t a t i o i i can h c ea.-^ily recognized. General Expressions W e consider t h e s h i p t o be f l o a t i n g o n t h o f r e e surface, o.scillating i n response t o t h e o x c i l a t i o n o f o n c o m i n g rogular wavos. T h e .Vi — r.-i i i l a i i o o f t h o c o o r d i i i a l o sy.s-toiu coincides w i t h t h e c c i i t o r p l a i i o o f t h o .-ihip, t h e .rj-axis is positive u p w a r d a n d t h e . i v a x i s 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 s u r f a c e a t t h e i i i c d i a i i p l a n e o f t h e .ship. T h o wavos 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 l i l i i i u o t o t h e .r-axis a t a n a n g l e jS. W e assume t h a t t h e w a v e height a is s m a l l , .so t h a t t h e p r o b l e m c a n b e l i i i c a r i z c i l . T h e t o t a l v e l o c i t y i i o t o n t i a l * , d u o t o t h o o s c i l l a l o r y m o t i o n of t h e fluid, m a y bo expressed as * = 0 c - ' ' " ' ( 1 ) where us is t h e f r e c i u e i i c j ' o f o s c i l k i t i o i i a n d o n l y t h e real p a r t o f tho i i o t o n t i a l is t o be t a k e n . T h o i i o t e n t i a l 0 , a h a r m o n i c l u i i c t i o i i o( .Vj(j = 1, 2 , 3 ) , satisfies t h e b o u n t l a r y c o n d i t i o n _ j,^ = 0 ( 2 ) 0.1:3 on t h e free s u r f a c e .T:, = 0 , w h o i c A- = io^d (3) is t h o w a v c - f r o c i u o i i c y p a r a m o t o r a n d is r e l a t e d l o I h o w a v e l e n g t h X b y t h e e ( i u a t i o i i A- = 27r/A (4) T l i o v e l o c i t y p o t e n t i a l o f t h e w a v e s , s a t i s f y i n g t h o 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
= «c c x p \k \.r., + i(.t; cos li + .i-.j s i n f))]\ (ö)
A Lmearized Two-Dimensional Theory for
High-Speed Hydrofoils Near the Free Surface
By Richard P. Bernicker'
A linearized two-dimensional ttieory is presented for higti-speed hydrofoils near the free surface. The "direct" problem (hydrofoil shape specified) is attacked by replacing the actual foil wilh vortex and source sheets. The resulting integral equation for fhe strength of the singularity distribution is recast into an infinite set of linear algebraic equations relating fhe unknown constants in a Glauert-fype vorflcify expansion to the boundary condition on fhe foil. The solution is achieved using a matrix inversion technique and it is found that the matrix relating Ihe 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 deplh-to-chord ratios and are presented herein. Several examples for specific camber and thickness distributions ore given, and results indicate significant effects in Ihe force characteristics at depths less than one chord. In particular, thickness effects cause a loss of lift at shallow submergences which moy be an appreciable percentage of the total design lift. The second port 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 info an inflnite set of linear algebraic equafions relafing fhe constants in a series expansion of the foil geometry fo the known pressure boundary conditions. The matrix relating fhe known and unknown constants is, again, a function of the depth of submergence alone, and inversion techniques allow fhe sectional shape to be determined for orbifrary design pressure distributions. Several examples indicate the procedure and resulfs ore presented for fhe change in sectional shape for a given pressure loading as the depth of submergence of Ihe foil is decreased.
Part 1 The Direct Problem
I N T E K E - S T i n h i g h - p c i f o n n a n c e liychofoil craft, has s t i m u l a t e d the i n v e s t i g a t i o n of l i f t i t i g .surfaces operating beneath a free .surface. A s a n a t u r a l consequence, (lie g r c i i t h u l k of t h e o r y p e r t a i n i n g to iticoinpressible 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 t i i t e m e d i u m h:is been u.'iod 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 the cla.s.sic a e r o n a u t i c a l m e t h o d s of c o n f o r m a l transformations atid s i n g u l i i r i t y representation have been itpplicd l o the h y -d r o f o i l p r o b l e m , h u t no m e t h o -d has t i s y e l been pie.sciile-d f o r tin " e . \ a e l " s o l u t i o n o f an a r b i t r a r y f o i l under a free surface.
V a r i o u s studies h:ive been ctirricd out which pertain to I 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 a.s.sitmptions w h i c h l i m i t the general t i p p l i c a b i l i l y . Keldy.sch a n d Ivavrentiev [ l ] , = K o t c h i n [ 2 ] , and H a s k i n d
' D a v i d s o n U i h o r a t o r v , S l e v o u s I n s t i t u t e tif T e r l n i o l o K V , l l o b o -k e n , N . J . P r e s e n t l y , A s s i s t a n t P r o f e s s o r of .Met l i a u i e a l lOngiiieer-i n g , S t e v e n s h lOngiiieer-i s t lOngiiieer-i l u t ' c of TCCIUIOIOKV.
= N t i i a h e r s i n b r a c k e t s d e s i R i i a l e H e f e r e i i c e s at e i u l of p a p e r . . M a n u s c r i p t r e c e i v e d a t S N . \ . M K l l e a d q i i a r t o r s , O c t o b e r •J('>, l ! l ( ) 4 ; r<!vise<l i i i a i i u s c r i p t receive<l M a y 1!>, IDli.'i.
[3] published early papers w h i c h replaced t h e a c t u a l h y d r o f o i l b y a sy.sfem of .'sources and vortices, leading, however, t o a mathematical f o n n u l a t i o n ( h a t was a l l h u t intractable. W c h i i g [ 4 ] , W a d h n et al [ö], a n d m o r e recently, Strandhagen and Scikel [0] used the concept o f a.single s u b s t i t u t i o n yort(>x to represent the f o i l . S c h w a i i -cckc [7] and I.say [8] continued this idea f u r t h e r a n d re-placed the t h i l l " f o i l w i l h a vortex .sheet, compo.>scd o f standard 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 , who.sc s(length Wiis determined b y s a l i s f y i i i g (hc b o u n d a r y c o n -ditions at .scvend control points on the c h o r d line. I n all of these papers, the general d e v e l o p m e n t o f t h e Iheory fell short of describing Ihe overall h y d r o d y n a m i c forces and pres.surc di'^liibutioiis f o r an a r l i i t r a r y section at any d e p t h . Nishiyania ('.), 10) l o r i i u i l a t c d the p r o b l e m to include (he cfTecIs of a r b i t r a r y cainber tind thickness of a foil .section at any Froude number. H i s tochiii([ue involves a coiiforinal t r a n s f o r m a t i o n of the f o i l c o n t o u r onto a .slit, and expansion of the complex v e l o c i t y po(en-tial i n a power .scries. T h e u n k n o w n constants of I h o v e l o c i t y potential are related to the t r a n s f o r m a t i o n c o n -stants "by an i n f i n i t e set of linear algebraic oquatioiis, a n d
Cavitation Phenomena of Sterntube Bearings'
By P. G .
Kessel-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 cose of sterntube bearings of ships with two specific aims in mind. The first of these is fo firmly establish fhe existence of conditions permitting cavitation damage of fhe journal, and fhe second is to show how the peculiar patterns of cavitation damage observed on fhe journals of several ships may occur.
T H E i n o h l o i n b e i n g coiisideied here is t h a t of finding a s o l u t i o n f o r t h e pres.surc d i s t r i b u t i o n and .«iiinultiuicous s h a f t l o c i o f a ;{GOdeg j o u r n a l bearing .subjected to d y n a m i c l o a d i n g . T h e l i e a r i n g is lubricated b y a c i r c u m -f e r e n t i a l source at one end o -f the bearing. T h e lubricat-i n g flulubricat-id Hows o u t t h e other oud o f the bearlubricat-ing. T h e b e a r i n g is considered t o be l i n i t e i n length. T h e bearing l u b i i c a n f is w a t e r w h i c h is supplied a l a constant rate a n d pressure. I I is assumcil t h a t the hearing and j o u r n a l surfaces w i l l a l w a y s r e m a i n parallel anil t h a t b o t h arc c o m i ) l c t c I y r i g i d . T h e surfaces arc f u r t h e r as.sumed l o be l i c r f c c t l y s m o o t h .
T h e phy.siical m o t i v a t i o n f o r this problem comes f r o m t h e s t e r n t u b e b e a r i n g o f .ships, w i t h i n w h i c h the (ailshaft s e e m i n g l y undergoes c a v i t a t i o n damage due l o the d y -n a m i c l o a d i -n g o f the ])roi)elle.r. I t has bee-n observed
[1 i n .s(>vcrai ships (hat I h c 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 its periphery and w i t h i n the c o n f i n e s o f t h e s t e r n bearing. T h e number of damage l o c a t i o n s a n d ( h e i r posdioiis around I h c peri|)hery o f (he j o u r n a l v a i y d i r e c t l y as the number o f propeller blades. I f t h i s n u n d j c r is c i t h e r three o r five, there w i l l be either t h r e e o r f i v e l o c a t i o n s , respectively, of tailshaft damage a n d ( 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 in line w i t h the p r o -peller blades. I f t he n u m b e r o f blades is four, then (here w i l l be f o u r l o c a t i o n s o f damage which are exactly 4.5 deg o f f s e t f r o m ( l i e l i n e o f (he propeller blades. The localion a l o n g t h e l e n g t h o f t h e bearing where this damage o f t e n occurs is s l i g h t l y f o r w a r t i o f the quarter-way j i o i n t of the b e a r i n g as measured f r o m the iiropoller end.
W h i l e the p r o b l e m studied i n this e l ï o r t differs f r o m t h e a c t u a l p h y s i c a l i n o h l e m o f sterntube bearings i n that i t w a s iieccs.sarj' t o m a k e c e r t a h i assumptions t o surpass some n i a t h c m a l i c a l complexities, it is f e l t t h a t it reprc-•sciifs a g o o d i n i t i a l m o d e l o f the jihysical iiroblem.
' T h i . s ])iiiicr r e p i c s p i i L s a p o r l i o n of a I h f s i s ]tii'j)aicd l>y l l i c a u t h o r i n p a r t i a l fiiirdlniont of the r e q u i r e m e n t s for the degi-ee of D o e l e n - o f I ' h i l o s o p h v , E n g i n e e r i n g Mcehaiiie.s, U n i v e r s i t y of M i e h i g a i i , A i m A r h o r . ' . M i c l i . ' A s s i s l a i i t l>rofe.s.sor, D e p a r t m c i i l of K i i g i n e e r i i i g Mei-liaiiics, U n i v e r s i t y of W i s c o n s i n , M a d i s o n , W i s . ' N u m b e r s i n b r a i ' k e t s de.sigiiale l i e f e r e n c c s a l e n d of p a p e r . M a n u s c r i p t r e c e i v e d a l S N A . M I O Headtpiarler.s, X o v e m b e r
um.
The three most i m p o r t a n t as.sumjitions d c | i a r t i n g f r o m the actual jilij-sical s i t u a t i o n o f sterntube Ijcarings a r c : Tho s h a f t and bearing are t o remain i i a r a l l c l a t a l l t i m e s , the bearing is c o m p i c l c l y smooth, a n d the p r o p e l l e r loading can be reiiresentcd b y t h e first t w o harmonics.
T h e first of these assumptions has the effect o f m a k i n g the film thickness a f u n c t i o n o f o n l y angular displace-m e n t around the j o u r n a l . I t is believed t h a t the t w o m a j o r coiLscqueiicesof this assmnption arc a change i n t h e iiitoiisity of pressures o b t a i n e d and a .slight s h i f t a l o n g t h e lengtli of the h e a l i n g o f the region of m i n i m u m pressures developed i n (hc l u b r i c a t i n g f i l m . I t is n o t fell , h o w e v e r , t h a t (he general pre.s.surc profile w o u l d bo s u b s t a n t i a l l y altered by this a.ssumpt ion.
T h e second assumption d c f i i i i l c l y \-iola(cs t h e a c ( u a l bearing w h i c h is compo.'!od o f staves spaced a r o u n d t h e bearing periphery. T h e i n a l h o n i a t i c a l c o m p l e x i t y o f i n -corporating these effects at prc.s(>iit makes i t iicce.s.sary to a.s.suiiic a s m o o t h bearing surface.
Considering the t h i r d assumiition, a l t l i o n g h the 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 o f m a n y h a r -monic conipononts, (ho firs( t w o o f these are k n o w n t o represent the m a j o r p o r t i o n o f the propeller l o a d i n g .
T h e analysis presented i n the f o l l o w i n g is e.s.sentially subdivided "hito throe m a j o r i i o r l i o n s . T h e first j u i r t consists of the s o l u t i o n o f the g o v e r n i n g f o r m o f R e y n o l d s h i b r i c a t i o n equation f o r the p r t w u r c d i s t r i b u t i o n i n t h e bearing. T h e solution (o this e q u a t i o n w i l l , however, contahi t w o u n k n o w n v e l o c i t y components due t o t h e translalioiial and r o t a t i o n a l m o ( i o n o f (ho j o u r n a l center in iui orbi( abou( a steady-state j i o s i t i o n .
T h e .second j i a r t is thus concerned w i t h the d e t e r m i n a -t i o n o f -these -t w o u n k n o w n v e l o c i -t y c o m i i o i i e n -t s . I f -t h o equations o f m o t i o n f o r t h e j o u r n a l mass center a r c w r i t t e n , a p a i r o f nonlinear, o r d i n a r y , d i f f e r e n t i a l e q u a -tions arc generated f o r these velocity components. T h e solutions to this pair o f e q u a l i o i i s arc o b t a i n e d b y t h e I h m g o - K u t t a f o u r t h - o r d e r m e t h o d on a d i g i t a l c o m i m t o r . These solutions i n a d d i t i o n t o y i e l d i n g the u n k n o w n velocity components give the j o u r n a l o r b i t s . T i n t s , f o r a given p o i n t i n the j o u r n a l o r b i t , the pressure d i s t r i b u -tion i n the bearing m a y be evaluated f r o m t h e .solu-tion t o Roynoltis equation.
On Nonlinear Ship Motions in Irregular Waves
By K. Hasselmann'
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 fhe ship motions by on exten-sion of standard spectral-analysis techniques. Hence, full-scale measurements con 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 .slati.sHcal t l i c o i y o f l i n e a r .sljip niotion.s is well understood a n d has f o u n d iitiinerous a p p l i c a t i o n s , ^^ost investigations o f 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 hatifl have 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 certain effects such as t h e l a t e r a l d r i f t a n d exces.s rcsi.st-ance i n waves. T h e results have n o t been placed w i t h i n t h e f r a m e w o r k o f a general s t a t i s t i c a l r c i n c s e n t a t i o n . I t w i l l be shown i n t h i s p a p e r t h a t t h e nonlinear 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 o r d e r m o n i c n t s o f the shi|) m o t i o n s i n t h e .«amc w a y as t h e linear 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 i j c c t r u m . Hence, a l l n o n -linear tran.sfer f u n c t i o n s m ; i y be d e r i v e d s \ ' s t c m a t i c a l l y f r o m a higher o r d e r anah'sis o f t h e ship m o t i o n s .
T h e asefulness o f higher o r d e r 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 T u k e j ' (1901). T h e m e t h o d has been a p j d i e d p i eviou.sly t o d e t e r m i n e t h e n o t i l i n c 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 i i h u ' w a v e f i e l d (Ha.s.selimiim, . M u n k t i n d ^NlacDonald, 1903). T h e S h i p R e s p o n s e L e t f l , . . . fe be t h e c o o r t l i n a t c s o f t h e .<hip m o t i o n : f l = surge v e l o c i t y M I , n i c a s i i r e d i i b o i i t t h e v e l o c i t y U i l l c a l m w a t e r $2 = s w a y v e l o c i t y ti^ f.t = heave disi)lacemeiil $4 = r o l l angle ff, = p i t c h angle f« = y a w angle
T h e v a r i a b l e s f , arc f u n c t ioiials o f t h e w a v e Held. W c 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 pr<-.sent) linear, .so t h a t i t can be specified b y t h e .surface displacc>ineiit f (x, I) • - f i a m i t h e n o r m a l s u r f i u c v e l o c i t y ( ö f / d / ) ( x , I) = f., at t h e mean f r e e surface, where x -Of,, .rs) is t h e h o r i z o n t a l C a r t e s i a n <-oordinatc v e c t o r ; •(•, a n d f., r<-fcr 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 o f t h e s h i p . . _ I l is c o n v e n i e n t l o i n t r o d u c e t h e c o o r d i n a t e sy.sf c m x X \J t r e l a t i v e t o t h e mean s h i p m o t i o n U . h x p a i u l -i n g I h e f -i m c l -i o -i -i a l s f , -i n a T a y l o r ser-ies w -i t h respect l o r i . h, lh<- .ship rcspmi.se m a y t h e n b e w i i t l c i i in t h e gen-erai f o r m V . « . t i u u v > v. \ n-. \ n u \ v v.t r . t - , , v U> > u ^ ; U u\ V V. \ \ i« : U y V b V r - l r s , V U l V C l -' H v iif l'uViforn'i-.». S; i u t V i e g o . ( . ' a l i i . M u n u> . vipi r . M c i v . - d at S N A M K l l c u d q u u r n - i s , M a t c h J'.l, l ! l l . . , . UD = S f t a u i - ('• i ' ) U i ' , i')'ii'<ii' n - 1 + ƒ • • • f l l f l t a . i t - - I", k', X " ) f „ U.lJr I ( X ' , r ) f / x " , I")f/X'<li"(ll'<lt" + . . . ( 1 ) T h e kernels depend o n l y o n t h e t i m e differences, .siin e i n t h e coordinate .system X t h e .ship response is i i i d e i i c n d e i i t of t i m e translations. Consider now a w a v e f i e l d w h i c h is r a n d o m , s t a t i o n a r y , a n d homogeneous. I t m a y be a p | ) i o x i m a t e d h y a F o u r i e r s u m (-') where o. = -f- ( ( / A ) " - i s the 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 wave 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 -linear problems t o i n c l u d e the c o m p l e x c o n j u g a t e t e r m of the l o u r i e r s u m e x i i 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 i i l i t v i d c s and t h e i r cro.ss i i r o d n c t s s a t i s f y the r e l a t i o n s = 0 ( Z M ) = 0 f o r k , ± k , < Z M ) = ^/^(k)(AA-)-^ (4) where M is t h e w a v e - n u m b e r i n c r e m e n t o f t h o 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 = | : 2 | ^ , M = ƒƒ ""J / ' ' ( k ) ( / k I t can hc s h o w n t h a i i n t h e linear a p p r o x i m a t i o n •, homogeneous, <lisix-rsivc w a v e field ' - j P i d l y l i e c o i u , ; Caussiaii (Ha.s.sclmanii, l » b b ) . Ht-iicc the h i g h e r ,nxhn-m o ,nxhn-m e n t s o f t h e w a v e lick can also be expressed i „ t e , , n s of t h e s p e c t r u m . W e s h a l l rccpmo l a t e r t h e r e l a t i o n s
{Z^,ZM = <^fic.Zk=^k.*) = 0
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lournal 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 . B e m e n t - a n d C. H . P o h l e r
An energy relationship is established by using only fhe true sfress-frue strain diagram de-veloped from a simple tensile test of a material. This energy relationship is then used to develop (o) a fatigue failure equation, v/hich results in a single nondimensional stress-cycle curve having apparent applicability to any material, (b) an equation depicting the rate of crock propagation, (c) an endurance limit equation, and (d) a method by which o stress-cycle diagram con be developed for a notched specimen. In addition, on 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 ore developed, indicating fo the authors the desirability of a general reaffirmation and possible reorientation of the basis of fatigue tests.
" O n fhe strength of o n e link in the cable Dependeth
the might of the chain."" F o r e w o r d
D E S I G . V I . N - G f o r f n t l K U c b:i<ctl on modern 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 the dcsigi.ei-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 H u r c w l i i i ' l ' . d u r i n g its life-t i m e , m a y c o n life-t a i n falife-ti-^ue craelcs. These cracks i n i life-t i a l l y g r o w s l o w l y n m l e r r-yclTc loading, b u t , i n many instances, haye j n o d u c e d .sudden f a i l u r e of a s t n i c t u i v w i t h coiise-«lueiit loss o f t i m e , m o n e y , and oven life, l o be able o predict f a i l u r e b e f o r e fin-i'l f r a c t u r e and to ([uaiititatiyely iindcr.stand the m e c h a n i s m o f this crack iiropagatioii m l e r m s o f crack size, stress, and load c.Veles I K U C been the
> l'liil..s>.|)l.y a i u l I , . , . < , „ e s . o n l a i i i c d i " l l ' ' ^ I'-'l'^'r ' c l l e j l M , l . . l y I h c , K . i s . . „ : , l o , , i „ i ; ; . a u l h . . . a n d d n n - " i ' ^ ' ^ i ^ ' ' ^ ' ^ , .,r|)ie.-=cnl i h c o l l i . i ; , ! v i o u s „ f I h o Huroiiu of S u|'-^ " r HM; N a v a l .S.M vi.-o a l l a r g e . C o j . v r glit \ 1 i v l i Ö, lOli'^ (l->l'n..y <'f < •""f'';r l i c g i s l r a l i n n N o . A T . l S i . o t r^ A r n o l d A . Uon.oiil a n d ( a i l 11. I ' o h l e r . ' • ' V s u ' b i u a r i i i c . S i r , , , . ! , , , , , ) \ | o , . h n n . s I ' n i l , ' ' u l l D.'sig", S.-iealili.. a u . l K e s c a i v l i S t v l i o n , l ^ u i a a Ship-S 1 ) c | « " " ' » - " ' " f -^''vy, ' ' ' ï S d ' A H i i m : 110P.V00.,. " T h e b a w s o f J ^ ' ^ : H -i V l a -i . U M -i -i p I -i e , e -i v e . l a l S N A . M I C l l e . -i d . -i u a -i l e ' - . • M>lunl.e-i , , l i l l M . desires of researchei-s f o r o \ c r a c e n t u r y . D u r i n . g t h i s period l i t e r a l l y thousands of reports, papers, antl booivs have been w r i t t e n .summari^iing and t h e o r i z i n g o n t h e nature o f f a t i g u e f o r d i f f e r e n t applications. 1;™'» ^ concerted a c t i v i t y has stemmed » »
awareness o f the ('omplexitv o f f a t i g u e , b u t n o i t a l t l c h -n i t i o -n of w h a t a c t u a l l y cairscs f a t i g u e
Coii.setiucntly. m some instances n na.
necessary to tes, very large motlels H i * " ' l ^ ; " " .^eale reproductions | 2 ] of actual - ^ t ^ ' - ' ^ ' - ^ - j ^ ; ; .'^^^^^
<l'e f a t i g u e .strength of tlie.-^c structures.
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•-nils obtained f r o m f a t i g u e ti>stiiig o l v e r y ••-'''*'' • ' ' nu-iis of (he m a t e r i a l pr..posctl f o r fabricat 1011. H< > t « iniporlaiice of the.se small fati.iinc specimens 1., u a t t i l v
obvious. , 1
, T h e intent of this paper is •
« \ i ' ; ; ; ' ; ^ : u e d
; l ' ' f ' n u t o which these .small spcciiiien i v s i d t s - " ^ n u ^ iH'lore they are extrapolated t o a s l r t i c l n r e . . ; general a warei.c.ss t hat no gotul .-o- rclat 10.1 h a - • ' [ ••elwcen the f a t i g u e properties of a ' , obtained f r o m
anOidi.iary
(cngin.>cring) tensile tc.st. t l u' N u m h c i x i n b i a e k c l s d e s i g i i a l e l l e f e r e n . es at e n d o f p a p ' i
-JUNE 1 9 6 6
Unsteady Loads on Supercavitating Hydrofoils
of Finite Span
By Sheila Evans Widnall'
Linearized ttiree-dimensional lifting-surface tiieory is derived for a supercavitating hydro-foil wilh 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 sfmitar to that used for fully wetted surfaces. Numerical results for lift and moment for both steady and oscillating foils ore compared wilh 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.
Iniroduclion
hi.\K.\nii'-i:i> liftiiig-.siiifuco iiii'lhotl.s for ciilctiiatiiijr tltc l i f t <ii.stiihiition on t l i i n f u l l y w c t t w l foil.s have been re-f i n e d t o it h i g h degree o re-f accuracy and ere-fHcieticy using the iiigh-.spced t h g i t a l computer. T h e integral cr|ualion r e l a t i n g the k n o w n h o u n d a r y values of upwash velocity ' Tlii.< « i i r k w a s pcrforniLHl u n d e r C i n i t r a e t X n i i r I . S 4 l t . S i ) , B u r e a u o f .Ships K i i i i d a i i i e i i l a l I l y d r o i n e t h a n i e s l i c s e a r e h P r o g r a m , S-Ull()'.» 01 I U , a d m i i i i s l e r e . 1 b y the D a v i d T a y l o r M o d e l B a s i n . C o m p u t a t i o n s w e r e p e r f o r m e d a l the M a s . s a r l i i i s e t I s b i s l i i m e of T e e l i n o l o R y ( N i n i p i i l a l i o n C e n t e r , C a m b r i d g e , .Ma.ss.
= A s s i s t a n t Brofe.s.sor, D c p a r l m e i i l of ..\eroiiaulies a u d .-Vslro-n a u l i c . s , Ma.s.sai h u s e l l . « I .-Vslro-n s l i t u t e of T e c h .-Vslro-n o l o g y , C a m b i i i l g e , M a s s .
H c v i s c d i n a i i i i s c r i p t r e c e i v e d a l . S N . \ . M l ! ; H c a d q u a r l e r . s , O c t o b e r 11, l!l().5.
lo the unknown d i s t r i b u t i o n of l i f t on the f o i l surface can be treated by a mimerical technique i n w h i c h the l i f t di.sI r i b u l i o i i is represented b y a .series of modes w i t h u n -known coefficient.^, h i t e g r a t i o n of tlie.so modes w e i g h t e d by the kernel f u n c t i o n of the integral e q u a t i o n is per-formed numerically, and the boundary c o n d i t i o n s are satisfied at a number of collocation points. I t i.s now possible to obtain an accurate prediction o f the l i f t di.s-I r i b u t i o i i on t l i i u f u l l y w e t t e d foils i n steady or o s c i l l a l o i v m o l i o i i f o r a wide v a r i e t y of configurations i n c l u d i n g the interference effects f r o m nearby boundaries such as a free surface at i n f i n i t e Froude 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 tails or F - f o i l arrangements [ F ö].»
' N i u n l i e i s in b r a c k e t s d e s i g n a t e l i e f e r e n c c s at e n d of p a p e r . • N o m e n c i a t u r e -« „ = c o e l l i c i e n i . of ; i l l i a.ssunicd-p r e . s s i i r e - a o i i r c c m o d e . l / i ' = a s j i e c t r a t i o ll,,' — c o e l l i c i e n i of j i t h a s . s u m c d -p r e . s s i i r e - d o u b l e l m o d e lia = root .seniichoi'd
I'l. = lift i-oeHii-ient ('M = i n o i n e n l e o e l l i c i e n l
= a.ssiinied ))-fuiii tion for .)) fy(f) = a s s u m e d f - f i i i i c t i o n s for <ö/Vc-)f(f, >;)) a = ainpliliKle of h e a v i n g m o t i o n U'(j',li,l) — a m p l i t u d e of u n s t e a d y mo-t i o n o f f o i l ''(.fl .V) = c m i i p l e x a m | i ] i t i i d e of foil os<'illntion
/'I >;.')i>')^) = a s s u m e d s | ) a n H i.se f i i n c l i o n s f<ir (dJ'/dt) ) = I l e a v y s i d e f i i i i c l i o i i / i = i i i o d i l i e d Be.ssel f u i i c l i o i i of lii-st k i n d /.' = r e d u c e d f i e i p i e i i i y , a ' f i j / r A ' l . j . i = k e r n e l f u i i c l i o n s defined b y e q u a t i o n s ( I S ) , (1!)) a n d
A'l = nuxlified Be.s.scl f i i n c t i o u of i h i r d k i n d
= c h o r d w i s e a.ssiimed f f i i i i i ' -tions for A / ' ( f , i/) j / ' l = matri.'c of pie.ssiire at J i ' / j d u e to ( ö / ' / ö f ) / ' = p e r l n i b a t i o n p r e s s u r e / ' = noiidimciisioiial coinple.\ a n i p l i l i i d e of l u e s s i i i e p c i -l u r -l i a -l i o i i i i / ' = p r e s s u r e j u m p a c r o s s foil s u r -face j u m p in n o r m a l d e r i v a l i v e o f p r e s s u r e acni.ss f o i l - c a v i i y Sl I i f ace .S R = r a d i u s v e c t o r R = .•! = .semispaii/seiiiichoi'd .S' = s u r f a c e of i n t e g r a l ion Sc = c a v i t y .surface ( p r o j e c t i o n o f ) ,<!„. = w e t t e d .surface ( i > r o j e c t i o n o f ) <IS = e l e m e n t o f a r e a o n s u r f a c e .S (• = f i e e - s t r e a m v e l o c i l j ,y) = upwa.sli o n w e l l e d s u r f a c e of foil j-^ ij, z = i i o i i d i i i i c i i s i o i i a l c o o r d i n a t e s . V , ; = c o o r d i i u i l e o f t i a i l i i i g e d g e o f I ' a v i l y (Ï = a n g l e of a t t a c k o r a i n p l i l u d e of | ) i l c h i i i g m o t i o n aix) = i l e l t a f u i n ' t i o n t , ij. f = n o n d i m e n s i o n a l c o o n i i i i a i c s p = f r o e - s t r e a i u i l e n s i t y a = c a v i t a t i l u m b e r ^ = p e r t u r b a t i o i i v e l i i c i t y p o i e n -t i a l y- = i i i i i i d i n i e i i s i o n a l i - o i n p l c x a n i i i l i t i l d e of p e r l u r l i a -l i o n - v e -l o c i t y i m -l e n -l i a -l CO = f i e q i i c i i c v of o s c i l l a t i o n J U N E 1 9 6 6 107
Experiments on Gravity Effects in
Supercavitating Flow
By T. Kiceniuk- and A . J . Acosta-'
Experiments on ttie efFect of a transverse gravitational field on the supercavitating flow past a wedge tend fo confirm predictions based on linearized free-streamline Iheory. A small though systemotic dependence upon Froude number not accounted for by fhe existing theory is revealed, however.
Nomenclalure
Ii = wi-iglit o f l i q u i d di.splacod by wodgo
('l. = li f t coi'flicitMit = L/iP'-y-/-)
V = c b o r t l
F = F i o u d o i i u i u b c r = V/igi)'^'
(/
=
g t a v i t a t i o t i a l fou.stantL = l i f t f o r c o p o s i t i v e ujnvards (opposite to dirocliou o f g )
I = b u b b l e Iciigith f r o i u etid of wedge
r = t u u i i e l v e l o c i t j '
a = atigle of a t t a c k
Iniroduclion
ÖTISKKT*'" has g i v e n a theoretical account of the ef-f e c t s o ef-f a t i i u i s v e r s c g r a v i t y ef-field oti the ef-flow past a s u p e r c a v i t a t i n g winlgo. Kocently, an o p p o r t t m i t y to t e s t those re.sults in tho H i g h Sliced Water T u n n e l at the C.-alifornia I i i . s t i t u t c of Technologv became available. J'^or t h i s purpose, t w o wedges of ti-in. chord and Q-'in. s p a n w e r e m a d e ; one h a d an apex angle of 7} > deg and t i i e o t h e r 1.") deg. T h e wedges wore mounted on tho f o r c e balance o f the two-diineiisioiial w o r k i n g .<cctioii i n t h e t u n n e l . " I n p r i n c i p l e all that is noces.sary is to
' I t c s c a i c h cjii i i e d m i l i i i i i l r i - U n r o a i i of N a v a l WeaiKiiis C o i i i r a c t Nl'2;K0ll."):iO):j47ti7A a d m i i i i . s l c r c d b y V. ,S. N a v a l ( I r d n a i i c c TeM S t a t i o n , I ' a s a d c n a , C a l i f . ' bc<;Hii-ur a n d ( i r o i i p h e a d e r . D i v i s i o n of KiininceriiiK a n d A p p l i e d . S c i e n c e , I t v d r o d v i i a n i i e s b a l i o r a l o r y , v o n K A r i n i l n b a l i o r a -l o r y of F -l u i d .Mec-liauic.s a m i .-let P r o p u -l s i o n , C a -l i f o r n i a b i s t i t n i e of T e c h n o l o g y , l':is:i(l,.n:i, C a l i f .
' Proft'.s.s<n-, D i v i s i o n of lOiiKiiieeriiiK a n d -Vpplied Science, n y d r o d . \ " i i a i n i c s I . ; i l i o r : i l o r v , v o n K ï t r i i u t n b n b o r a t o r v of F l u i d . \ f e c l i a i i i c s IMKI . | , . | P r o p u l s i o n , C a l i f o r n i a I n s t i t u t e o f ' r e d i n o l o g y , P a s a d e n a , C a l i f .
* 11. b . Slre<'i, • S i i p e r c a v i t . i l i i i n F l o w .Mmiit a .^lender Wedge in a T i a i i s v i ' i s e ( i i a v i l v F i e l d , " .hiiininl iif Ship Jltsioith, v o l . 7, n o . I , liHi:!, p p . 1 4 - 2 1 .
' 1!. b . S l i c ' e l , ".-V N o t e o n C r a v i l y IvlTecIs i n S i i p e i c a v i l a l i n g M o w , " ./(iiiiiinl «/'.S7i//j /{IXIiinli, v o l . S . no. 4, l<ll>.">. pp. ; i ! l - 4 ü .
« T . K i c e i i i i i l i , ' " A ' r w o - D i i i i e i i s i o i i a l W o r k i n g Section for the H i g h .Speed W a t e r T i i i i i i e l a l the C a l i f o r n i a l i i s t i l i i t e of T e d i -n o i o g y , " l l v i h ' o i l v i i a i -n i c s b a l i o r a l o r y , C a l i f o r -n i a I -n s l i l i i l e of ' r e i - l i i i o l . i g y , l i e p o i l . \ o . F . - I I I S . l l l . l , D e c e m b e r l!Hi;!.
. M a i u i ^ c i i p i r e c e i v e d a l . S N . V . M F l l i n i d i p i a r l e r s , Deceinlun- l o , lilti.'i.
measure the l i f t force at zero angle o f a t t a c k i n the pres-ence of tho n o r m a l g r a v i t a t i o n a l field when the ca\ i t y is established behind the wedge. T h e p r w l i c t e d g r a v i t y -iiidiiccd l i f t force is quite s m a l l ; f o r the wedges used i t amounted t o o n l y a few pounds at most. L i f t forces duo to even sligiit changes i n angle of a t t a c k at flow velocities of 20 a n d 30 f p s aiT appreciable .so t h a i groat care m u s t be taken to p r e v e n t masking t h e s o u g h t - f o r g r a v i t y effect b y small changes in tho o n c o m i n g flow direction iiossibly induced b y d i f f e r e n t ambient pressure, different t u n n e l "velocities o r b y errors i n t r o d u c e d h y t h e forco balance itself.
One possible source of error was readily apparent w h e n base c a v i t a t i o n t o o k place behind cither wedge at t u n n e l spcetls of 20 t o 30 f p s . A s the fluid w i t h i n t h e w o r k i n g section is subject t o the usual h y d r o s t a t i c pressure gradient, the static prcssun- at the l o p o f t h e w o r k i n g section is less h y 1.-5 f t of w a t e r t h a n at t h o wedge itself. A t low c a v i t a t i o n indices t h e fluid i n this region can also cavitate. Investigations revealed t h a t the flow at the ccntorline of tho force balance was d i v e r t e d b y n o a r l y 0.1 deg when this happened. Since tho a n t i c i i i a t c d g r a v i t y effect at these t u n n e l .speeds corresponds o n l y to a u angle of a t t a c k change of about 0.0.-> deg. procedures t o avoid this effect i n p a r t i c u l a r were therefore a b s o h i t e l j -iicccs-sary.
Experimental Procedures
.Viler some o x p e i i m c i i t a l i o n . the t o c h i i i q i i c of base ventilation w i t h a i r was adopted. Hy this moans the anibient prcs.surc w i t h i n Ibe t u n n e l could be a i l j u s t o d to any level d c s i n t l and s t i l l i n a i i i l a i i i a v e n t e d a i r b u b b l e Such a v e n t i l a t e d bubble .simulates in a l l important r c s p w l s f o r the purposes o f the preseiil ex-perinioiit, a vaporous s i i p c r c a v i t . i t i n g bubble. Tlu> actual proccdiin-s used to €>btaiii the lost thita wore as follows: A winlge was i t i o u i i l c d on l l i e force balance and its coiilcrliiie was a c c i i i a t o l y aligned to the t n i i i i c l geoinetrical cciitcrliiic ( w h i c h is also precisely the angle of m o l i f t o f a s y m m e t r i c a l h y i l r o f o i U w i t h a precision cliiioincter. T l i c " t i i i m c l was brought u p to spocil at an
A Critical Re-Evaluation of Hydrodynamic Theories
and Experiments in Subcavitating Hydrofoil Flutter
By Wen-Hwa Chu=
A review of relevant hydrodynamic informotion 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 fhe force calculofion based on fhe classical lifting surface theory and subsequent flutter prediction for on actual flutter model is desirable. Some recommendations for possible future experimental research ore also given.
Iniroduclion
T H E p r o b l e m o f f l u t t e r o f a submerged (sidx a v i t a t i i i g )
h y d r o f o i l has been discussed i n reference [ 1 A c c o r d i n g to the classical flutter t h e o r y , there is a flutter b o m i d a r y w h i c h is a b o v e a c r i t i c a l r e l a t i v e d e n s i t y parameter. 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 h e o r e t i c a l l y fluttci-frec f o r a n u n s w c p t w m g . I b c i c have been l i m i t e d c x p e r i m c i i t s r e p o r t i n g failures o f m o d -els below t h i s c r i t i c a l d e n s i t y i i a r a m c t c r , b u t l l i e i c is d o u b t as t o w h e t h e r such f a i l u r e s were caused b y Hut t c i o i s t a t i c divergence. H o w e v e r , some «'l'^i'.'''^.
p c r i m e n l s i n F r e o n tdso s h o w e d discrepancies I'et^^^ " t h e o r y and e x p e r i m e n t w h i c h were discus.scd in r c f c i c i c
[11. " 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 wa.s m -sufficient t o reconcile t h e t h e o r y a n d ' ^ • ^ P " " " 7 , ' * ' ' ' ' ° y i e l d a c o n s e r v a t i v e reduced f l u t t e r speed near the H u t t e i
'Tnl^^e
aspect r a t i o o f t h e X A C A m o d e l c m ^ d e i e dwas e i g h t , i t was t h o u g h t t h a t t h e finite s,,im e o n e t u n o the 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 l f ^ ' y ff^^.^'! 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 . I "/.
d i m c n s i o . i a l s t r i p t h e o r y , at least i n " e r o i u u i t i c a p p l . cations, u s u a l l y leads t o r a t h e r eonHcrvativc res ts^^i.c^^ t h e t l e r i y t l flutter s p e d is m u c h l o w e r t h t u . " f ' v a l u e . I t seems, t h e r e f o r e , t h a t t h e
^^^o^"J,''^
h y d r o d y n a m i c l o a . l i n g p r e d i c t e d b y ' ^ ^ ^ ^ ^ ^ ^T h i s m i g h t be cause.! h y t h e complu-aU-d viscous c f l . < t a real flui.l pas.sing o v e r an o s c i l l a t i n g f o i l .
l l , „ „ t e i i l i u l I h e o i y . t h e n ai f b n . l . f f c c t is usuan> t A c n . • n „ . p i - . . . , . l e d i l , . b i . . . . p e r
I ' l l l ' ^ ^ ^ ' l I'^^lJ^l'^,.;'^^
of t h e N a v y , u n . l . r C ' o i i . r a e t N o . N O b s SS...1.».« S e n i o r i i e . s e a r e b ICnilio.'e.-, I l e i K . r l . o . n t of M.^ehaiiical S.i.'l..-e>, Souilnv.'.-t K . ' s e a r e h l i i s l i t n . . ' . S a i l A n l m i i o , l e x .
' N > i i i i l » ' i > i l l b i a e k e l s d i ' s i K i i a l e Hef.-reii.-es at e n d of p a p e r . . M a n u s c r i p l l e c e i v . - d at . S N A . M K I l . ' a d ( | U a r t e i s . l a i n t a i y .'i, I'.lliti
care of .scmi-cmpirically b y relaxing the K u t t a c o n d i t i o n . I n reference [21 the previous experimental d a t a l o c a t e d were incoiiclii.sive i n t h e low reduced v e l o c i t y range (zero mean angle of a t t a c k ) as compared w i t h p r e d i c t e d v a l u e s of the phase l a g of the l i f t force.
Sub.seciucntly, l i y . l r o d y n a m i c spanwise l o a d i n g meas-ureiiieiits o n a s u b c a v i t a t i n g h y d r o f o i l were c o n d u c t e d and r e i i o r t c d i n reference [:?]. I ' l u t t e r f a i l u r e o f a m o d e l was reiiortiHl i n reference (4 j . I n reference (1 ] p r e d i c t i o n of flutter based on lho.se measured cocHi.-ieiits i n reference
[81 was o v c r c o n s c i v a l i y e . I'rediction o f flutter btiscd o n 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 [."il was q u i t e uncoiiscrvatiye for the h y d r o f o i l . T h i s w o u l . l be even t h e m o r e so i f the Rei.ssiierSte\eiis t h e o r y w e r e c m -j i l o y c d . O n the other hand, an order of m a g n i t u d e esti-mate based o n Osecn's equation i n reference [Gl f o u n d negligible d e v i a t i o n s f r o m tho K u t t a c o n d i t i o n i n the dassi.-al t h e o r y . I t is therefore desirable to reevalu-Ue b o t h the theories a n d oxpcrimeiits f o r flutter i n s u h -c a v i t a t i i i g How i n t h e hope t h a t dis-crepan-cies -can bo ex p l a i n e d . F u r t h e r m o r e , i t is expected t h a t b y such a r o -e y a l i i a t i o n a b-ett-er u n d -e r s t a n d i n g o f t h -e i n o i j l o m c-m 1K> reached a n d some reasonable recommendations f o r f u t u r e research ciin be made.
T h e increasing speed of h y d r o f o i l c r a f t has l e d to activ.> research i n the area of unsteady s u p e r c a y i t a t i i i g flow h u t H u b c a v i l a t i n g flutter m a y he stiU of interest as \ i is ó n , , , ; possibU- t h a t t h e c n i i s u i g speed w . n i l . l lie mu.-l, Unvin- In n t h e m a x m u m , s,,c.-.|, an.l (lu,s Ihe (low field n u i y lie suh-e a v i t a t . t i g . I hsuh-ersuh-e ntay hsuh-e also a p r o b l suh-e m of passing t h r o u g h a range ol s t , i . . , t v . b t t i „ g speo.ls i n w h i . - h the foU
IS i i n s l a l i l i ' .
I ' o r H i i i a l e i y , a d i l i i i o n a l <>xistitig e x p c r i i n e n t a l i n f o r m a -tion Jiiis been colloct.-d. H o w e v e r , f o r cotivenience of di.scii.ssioii, k i i m v i i r.-siilts i n b o u i i d a r y - h i y o r t h e o r y w i l l first be .stimmariy.ctl l i r i o l l y . A l l i i o u g l i llio.so results arc not d i r e c t l y applicable, t l i c y m a y \n'(t( iiiKililtiliic mine f o r
lournal of
S H I P R E S E A R C H
Estimation of Stability Derivatives and Indices
of Various Ship Forms, and Comparison With
Experimental Results'
By W. R. Jacobs^
The analytical method of reference [ 1 ] ^ for estimating stability derivatives, and hence stability on course, which combines Albring's empirical modifications of simplified flow Iheory with low aspect-ratio wing theory, is extended to lake into consideration the effects on course stability of higher ospect-roho 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 opplicofion 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—a Mariner Class model, a destroyer, and a hopper dredge. Comparison with experimental results shows that fhe 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 fhe method can predict relative effects of changes in fhe geometry of a ship form, as well as fhe effects of changes in skeg and rudder area.
Iniroduclion
I.s an oailiei- i f i « n l (1), att aiialylic-al mothod was dc-v c l o p o d f o r cslimatiiip; t l i f f i i s l - o r d c r .slahility dpridc-validc-ve.s (.sliitic a n d r o t t i r y lateral-force ami yawiiig-momenf rales) w h i c h w o n i d indicate the course s t a b i l i t y and l u r n i n g o r steering (lualilies of ships. The method was a p p l i e d t o the case of a f a m i l y of eight hulls of the .same h ' l i g f h a m i the .same pristnatic and block coellicieni, b u t d i l f e i h i g i n d r a f l , beam, and displacenient. T h e hulls were I h e 840 Series of the 'J^iylor Standard Series t y j i c w i t h I h e a f t e r dcailwootl (faircd-in .skeg) removed. Ex-p e r i m e n t a l l y measured lateral I'orces and y.-iwing nio-inent.s, f r o m D a v i d s o n L a b o r a l o i y rolating-arni tests
' I ' l c p a i o l fur B u r e a u of .Ships I'uMilaiiieiilal l l y d r o m e e h a i i i i s He.sc-iiri-h r r o g r a m (.S-I!ll()'.l-0I-lll). . \ d i n i i i i s i e r é i l h y D a v i d • P a v h i r . M o d e l Ba.sin u n d e r ( ' o n l i a i i N o n r :^(i:!(ö7), D b P r o j e c t 2S(i:{/l)li;!. I t e s e a i e l i ICngineer, K l i i i d D v n a i n i i s D i v i s i o n , I l a v i d s o n l . a l i o r a -n a y , S t e v e -n s I -n s t il u l e of T e c h i u i l o j i y , H o l i o k e -n , N . J . N i i n i b e i - s in b i a i k e l s d e s i g n a t e l i e f e i o m es al end of p a p e r . . M a i i i i s e r i p l r e c e i v e d al . s N . \ . M I O I l e a d i p i a r l e i s , D e c e i i i h e r .!. I '.Hi.').
at dilfereuf t u r n i n g radii, were available for the.se hulls and f o r three of the hulls w i t h flat-plate skcgs i n the jilacc of the removed dcttdwood.*
.\Ithough the analytical method is ba.scd uiioii s i m i j l e concepts combining simplified flow theoiy w i t h l o w aspectratio w i n g theory aiul using jMbring's [.S] e m -pirical modificalioiis f o r vi.scid flow, good correlation was attained between the s t a b i l i t y deri\-ativcs calculated b y this method anil Iho.sc determined f r o m cx|icrimental data. Ihnvever, .Mbring's inodifictition of the r o l a r y -nioment rale is a f u n c t i o n of prismatic coeflicient and, since all the hulls of the 840 Series have the same plas-matic (0..54), this moditication was not tested h i l l y . I t was decided, therefore, to extend application of the prediction method to hulls of other prismatic, w i t h and w i t h -out skcgs or deadwood a f t . f o r w h i c h experimenlal data were available.
' U e s u l l s of s e v e r a l s t i a i g h t - c o u i s e tests (Jl conlirnied p r e v i o u s cxpiTieiice at D a v i d s o n l..:il)oraiory tliiil e i i l i i e l y reliable s i a l i c force a n d iiioiiieiil r a l e s for s i r a i g l i l c o i i i s e m o i i o n c a n be o b -l a i n e d froiii i n -l a i i n g - . n n i d a -l a a -l s i i H i c i e n -l -l y -large -l i i r n i i i g r a d i i .
shallow Water and Channel Effects on
Wave Resistance
By Maria Kirsch'
Based on Sretenski's formula, the wove 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 wove resistance from towing-tonk experi-ments is illustrated through on example.
I n t r o d u c t i o n
T H E basis f o r t l i i s i n v e s t i g a t i o n is p r o v i d e d by M i c h e l l ' s t h e o r y of w a v e re.sistaiiee ( I I'- a n d by t h e extensions m a d e t o t l i i s t h e o r y sinee-its api)earance (Haveloi-k (21, W e i n b l u i n L n n d e ! 4 ] , a n d others). ^ O f special i n -terest f o i our purposes are the papers of Sreteiiski (."), l i ] , .since 8reteiiski's f o r m u l a s air- used i n Ihe iire.seiil w o r k i n t h e c o m i i u l a l i o n o f wave resistance i n .shallow water and i n a <diamiel. . \ discussion of the wave-re.si.staiicc^
' l)<.<|.,r of K n g u i c i r i i i g . l u s l i l u t f u r S c h i l l b a u <h r t i i i v c r s i l a t HMinliiirt;, I b n u b i u K . <'.crniauy.
= N ' n m b - r s in b r a . k c i s d c s i g n a l c l i c f c i c n i c s a l c u d of p a p e r . -M.iioisciipt r c < c i v c . l a l . S N ' . \ . \ I | - ; IIca<h|U.iiici-.~, D e c e m b e r (i,
l i l l M .
integral f o r a cliaiinel was g i v e n by AVigley 17]. T h e first step of the pre.seiil work was the s y s t e m a t i c calcula-t i o n of f h e wave resiscalcula-tance in a channel, u s i n g an I B M li;")0 electronic computer. Several results f r o m tlic.sc calculations have already appeared |8 |. where .s<iine of the ï-oiiiputed wave-resistance CTiefficicnts were c o m p a r e d w i t h and agri'cd well w i t h results of c o i i i p i i l a l ions made b y . V p i i k l i l i n and \ ' o i l k i i n s k i | ! ) | .
T h e range of calculations i n the present w o r k was g r e a t l y exlen<led; lirst of a l l . i n that v a r i o u s / , ' 7 i r e l a
-tioiis were laken i n t o c o i i s i d e r a l i o n . anil also, i n t h a i Ihe wave-resist a iii'c integral was e v a l i i a l c d f o r shallow water using Srelciiski's iiictliods. Here, the ncc'cssary calcula-tions were <-arricd out w i t h the lii-lp of a T I N electronic conipiiler. |-'inally, for the same h u l l c o n t o u r s f o r w h i c h
N o m e n c l a t u r e It = b e a m of s h i p /•' - = m i i l s h i p cros.~.-sectioiial a r e a /•"; = c r o s s - s e c t i o t i a l a r e a of c h a n n e l If = a c c e l e r a t i o n o f i j r a v i l y ff = w a t e r d e p t h f . J = int e g r a l s . i v c r wet I ed s u r f a c e of s h i p , b y w l i i i h s y n u n e t r i c a m i a s y i i i n i e l r i i ' p a r t s of s h i p ' s s u r f a i e a r c l a k e n into c o n s i d e r a t i o n / ' . . / * = i i o i i n e i l f o r m s of i n l e g r a l s f, ./ / . . . / , = I'xpri-.ssioii:-. a n a l o g o u s l o / . . / , ill h i r n i u l a for w a v e r e s i l -aii'-e in a c h a n n e l l\ — c h a n n e l w i d i li / . = h ' l i g l h of s h i p f t , = w a v e resist a I ic<-.S'.. = s h a l l o w w a t e r 7' =^ i l r a f i of s h i p t ~ velocit y of . h i p or nioflel J-. !i, z = c o o r d i n a l c s ii, h o r i z o n l a l , lraiisver.se, a n d v e r t i c a l d i -rections a = a r e a coellicieni of w a l e i l i n e li - inid.ship-.-ei lion coellii ienl f » = w a v e - r e s i s i a n c e coiiiponeni
X = i n t e K i a t i o n v a r i a b l e in w a v e -re<islaiic<> i o l e p r a l
p = i l e n s i t y of | i ( | u i i l >f = lii isniatic coeHicient il = w e t t e d .ship's .surface f/Cj-) = e i | i i a t i o i i of design w a t e r l i n e lilzl = e q u a t i o n of m i d s h i p s e c t i o n ,fU,z) = l / ' j ' b / ^ ; ) ^ c i i u a l i o i i of w e t t e d ship's s u r f a c e t'J-l.l' ^ = I*ICIIHU- iiiiiiilicr (<j-ff> iiiiinlier I : = i l c | i l h r i o i i d e /.'* = f,\/(>i/irpijH-r-/f:) = i i o n u e d , i l i m e n s i o n l c s s f o r m of w a v e r e s i s l a n c e a c c o r d i n g l o W e l i i -b h i m 1101 > = l / ( 2 F - ^ i f = xHl.n). 1) - A J - , s ) / ( H / 2 l ,
