Contents Journal of Ship Research 1965 (Mag-8)

JOURNAL

OF

SHIP

UA y. Sdieepsböüwkundc

Tecfinische Hogeschool

Defft'

RESEARCH

volume 9, number 7

JUNE 1965

Performance of a Propeller in a W a k e and the Interaction of

Propeller and Hull by Quentin Wald 1

A Theorem on Buckling and Structural StifFness

by William P. Vafakos 9

Elastic Vibration Characteristics of Cantilever Plates in W a t e r

by Ulric S. Lindholm, Daniel D. Kano, Wen-Hwa Chu, and

H. Norman Abramson 11

The Virtual Inertia of Propellers Under L o a d . .by D. H. Norrie 2 3

Asymptotic Solution of the Laminar Boundary L a y e r Equations

for a Non-Newtonian Fluid . . . .by L van Wiingaarden 3 7

Two-Dimensional Supercavitating Plate Oscillating Under a

Free Surface by C. S. Song 4 0

Theoretical investigation of a Peristaltic MagnetoFluid D y

-namic Induction Compressor—2

by Joseph L. Neuringer, Eugene h^gotsky, James H. Turner,

and Robert M. Haag 5 6

Axisymmetric Buckling of Ring-Stiffened Cylindrical Shells

Under Axial Compression by Thein Wah 6 6

Publications of the Technical and Research Committees . . . 7 4

An Announcemênt

Spring Meeting H Y D R O F O I L S Y M P O S I U M , complete with

D I S C U S S I O N , to be a v a i l a b l e in book form 7 3

Published quarterly by

THE SOCIETY O F NAVAL ARCHITECTS

A N D MARINE E N G I N E E R S

lournai of

SHIP R E S E A R C H

Hydrostatic Tests of Two Prolate Spheroidal Shells

By John J . H e a l e y '

Two machined models were collapsed under external hydrostatic pressure to determine fhe elasfic buckling sfrenglh o f complefe prolate spheroidal shells. The test results demon-strated that collapse pressures 40 percent greater than predicted by available theory con be achieved for a prolote spheroidal shell with a major to minor axis ratio of 3.0 and a thickness to diameter ratio o f 0.015.

Nomenclature

Ii = m o d u l u s o f <'la.sl i c i l y /( = (hick noss

n = l e n g t h o f .semi-major axis

// = lenK(h o f .scmi-miiioi axis-(' = l*ois.s(>n's I'iUio

l> = d i a m e l o r o f model = 2/»

/( = i i u m h p i ' o f c i r c u m f e r e i K i a i lohcs or waves

Introduction

IviTTLE w o r k has heeii done on (lie piobleui of the elaslic s l a i i i l i l y o f p r o l a t e spheroidal shells, and (hero are n o e x p e r i m e u l a l d a l a (o support (he few (heories--* w h i c h aro a v a i l a b l e . T h i s la<'k of experinietital evidence is significaiM inasmuch as (he prola(e .spheroid has heeu disregarded f o r p a r d c u l a r pie.ssuie ve.'isel applicadoits because o f (he p o o r s ( i u c ( u r a l efiiciency predicted h y a v a i l a b l e (heory.

T h e Mu.'ih(ari f o n n u l a f o r (he liydros(atic b u c k l i n g pi(>.ssuie o f a p r o l a t e .spheroidal shell is given hy

1.21/i/(=

/>> = _•2a- f o r V = 0..{ ( I )

Since (he sphere and (lie semi-inliiii(e circular <yliiider are (he <'x(rente ca.ses o f (he prohHe spheroid. i( is reason-able (o expec( (hat e<iua(ioii (1) reduces l o reasonreason-able solutions f o r these shapes. When (} = /». .Mushtari's solu(ion y i e l d s ( l u ' <-la.«sical sniall-(ieflec(ion-buckling ' S l l - i l i ' l i u i i l I t e s e a r c h ICnuiiu'cr. I t i i v i d Tii.vliir M m l o l U:isin. W i i . s l i i i i K l ' i i i , l > . , , „ ,. , ^ , ' K l i .M .Mu.-ililari i i i " ! I v . I . ( l a l i i n n v . N n n l m c i r T l i c o r v nf T i i i i i l-:iii'slic S l i c l l s , ' " K a z a n , T a l k i i i n i ' i z d a l . lll.'iT, p. ;«1S. 3 It ( i . S i n k i n , " O n l l " ' T l u D r y e f S l a l i i l i l y of a i ' r o l a l e S p l i o -r o i d a l S l u ' l l m i d o -r ("(li-rnnn I C x l e -r n a i -r -r o s s n -r e . " D a v i . l T a y l i i -r Modet Ha.-in T r a i i . - i l a l i i n i : n 7 , .Faiiiiarv l ! | l i ( . . M a i m s i i i p l r e r e i v e d a l S N . V . M K l l c a d . p i a r U M N N o v e n i l i e r 111. I ' l t i l .

(•(liialioii f o r (he perfect spliere.-" H o w e v e r , wlien 0 » / ) , (he p i c t u r e predieletl by e<iuatioii (1) approac hes zero and iio( (lie b u c k l i n g pre.ssure o f a s e i u i - i i i f i i i i ( e cylinder. T h e r e is reason to believe, l l i e r e f o r e , l h a l (iie collapse pre.ssure calculated f r o m ecpialiou (1) m a y be conservalive f o r a large range o f values f o r (he r . i l i o

a/b.

T h e e . x p l o n i i o i y s l u d y repori ed i n this paper \va-< u i i d e r l a k e n (o p r o v i d e .•^onie e x p e r i n i e i K a l evidence o n (he b u c k l i n g s t r e n g t h o f p r o l a t e spheroids and (o i-lu-ck the v a l i d i l y o f .Mushtari's s o l u l i o n . T h e lesls showed (hat eollap.se pressures 40 percenl g i e a ( e r ( h a i i (hose predicted b v e<iua(ion (1) could be o b t a i n e d f o r t h e p a r l i c u l a r geometry tested. I t is felt ( h a t even h i g h e r ia(ios o f e x p e n m e n t a l to (heorelical pre.ssure w o u l d be o b l a i i i e d f o r t h i c k e r and or less spherical p r o l a t e spheroidal shells.

Description of AAodels

T w o complete p r o l a t e .spheroidal shells w i ( h a b = .1.0 and li/D = 0.01.") were machined f r o m 7()7.")-T(i a l u m i n u m bar s l o c k w i d i a n o i n i i i a l p r o p o r l ional l i m i ( of ()0,00() psi. ^'o^lng's m o d u l u s , as d e t e i i n i n e d b y opdcal strain-gage measurements, was 10.8 X 10'' psi. I n desijrning (he models, stress levels were l i m i t e d l o 2."),000 psi a( the pressure predie(ed b y . M i i s h i a r i i u order ( o ensuie (ha( (he stre.s.ses a l f a i l u r e w o u l d b(> well below the p r o p o r t i o n a l l i m i l o f (he nwiferial. .\ Poisson's ra( io o f 0..S was assumed in .all c a l c u l a l ions.

T h e lirsl m o d e l . I ' H I . contained a mechani.'al c i r c u m -ferenlial j o i i K a l d i e eiiuator. T h e resulls wero <-n-c o u r a g i i i " despite ( h e <-n-c r i ( i <-n-c a l lo<-n-cation o l (he j o m l and a second model was f a l . r i a l w l . T h i s motl(>l. I1{L>. . . o n

-• S Ï ' T i m e s l u - i i k . . a n . l .1. M - <!civ. Tlirmu » ƒ /;/«.-./<. Slahilil!,. M r C r a v v - l I i l l » i « . k ( • . . n i | i a n y . I n . .. N.'W N . . r k . N . ^ ., s. . . I

e<liI i l l l l , (!l(i[, p . i t l "

-S E P T E M B E R 1 9 6 5

Unsfeady Lifting Surface Theory for a Marine

Propeller of Low Pitch Angle With Chordwise

Loading Distribution'

By S . T s a k o n a s - and W. R. J a c o b s '

This study IS third in a series of investigations applying the unsteady lifting-surface theory to the marine propeller case. In the present investigation, the surface integral equation IS solved for a mathematical model where the chordwise loading is taken as the first term of Birnboum's lift distribution (flat-plate chordwise distribution), in conjunction with Glouerfs lift operator, which, in essence, satisfies the chordwise boundary conditions by a weighted average. It is shown that this model is an improvement over the modified Weissinger model used previously in this series, because it contains as a nucleus the exact two-di-mensional solution, and thus it provides a sounder basis for determining the three-dimen-sional effects. The blade-loading is determined for a propeller operating in flow disturb-ances induced by the presence of a hull and by the blade-camber and incidence-angle effects. The stationary loading obtained by the present model is less than that obtained by the modified Weissinger model, whereas the nonstationary loading is slightly larger. The results of numencal calculations are applied to the problem of propeller vibratory thrust and torque, and comparison is made with previous theoretical and experimental values. Conclusions o f the eadier studies as fo the dependence of loading on the im-portant parameters—blode-areo ratio, aspect ratio and pitch—ore confirmed by the present results.

Introduction

T H E basic iHohlom lo w l i i c l i the unsleady lifliuK-.siu-faee t h e o i y is applied i n the marine propeller ease is pred i e l i o n o f f l i c inpredueepred v i b r a t o i y forces anpred the pred i s t r i b u -•ioiis o f Ihcse forces on t h e iiiopeller it.self as well as on n e a r b y boundaries. I t is .scari'cly necessary (o empha-size l h e i m p o r t a n c e of t h i s problem, w h i c h i.s relevant, to the problems of h u l l v i b r a t i o n , underwater acoustics, and fi.vdroclastie i i i s t a l i i l i t y of l i f t i n g surfai^es. Bccau.se of picwenl lendeiicies t o w a r d hicreased speed, it is of para-i para-i para-i o u u t para-i m p o r l a n c e to o b t a para-i n accurate predpara-ictpara-ions of l o a d i n g d i s t r i b u t i o n s o u propeUer-blade l i f t i n g surfaces r o l a l i u K i n i m s l e . i d y f l o w behind a water-borne vehicle, 1\V t a k i n g cogniz.iiice not onl.v of realistic flow conditions b u t also of (he spatial g e o m e t i y of (he l i f l i i i K .surface.

T h e use of two-dimonsioiial unstead.v-flow t h e o i y , a p p l i e d i n a .stripwise manner, Ii.as proved not o n l y i n -a d e q u -a t e b u t i n c o r r e d i n the m -a r i n e propeller c-ase, de-Rjiitc t h e inclusion o f semi-empirical correction factors to accounl f o r (he ( l i i c e - d i m e i i s i o n a l i ( y of (he flow lield.

' T l i i s ."ittidy "Jis c.n ried nut nl D a v i d s d i i L a l m r a l o r y , .Sicvoiis I n s l i i u t e (if Tecliiiiikit-'y m i ' l c r U i i r c a u nf .ships r u i i d a n u ' i i l a l H v d r o m e c h a i i i c a Ito.-oarcli I V o i a a n j ( S - l i O U i l - O I - l l l , C M i t r , , , . ! N . i i i r 2(!.tf;W)), admiiii.'^fcicd b y l)avi<l 1 a y l n r M m l e l Ha.sln.

' H e a d , F l u i d D v i i a n i i . s D I V M K I I I , n u v i d . s . m I . a l u . r a l o r v , . S l o v e n s I n s l i l u l e of Teeliii"li>Ky. H e l m k o n , N . .1.

» S e n i ( i r lie.iearoh Knuinci'i. I ' a v i d . s d i i b a t i i . r a l o i y , S l o v e n s I n -fililiile .if Teoliuol<.K,v, U n h o k c i i . N . J .

M a n i i s . i i p t i v c e i v o d a l S N . V . M I ' , H e a d . i u a r l o i s , .luiio 2 . l O l H .

I n .search of more realistic reiire.sentations of propeller geometry and flow conditions, Da\-idsou L a b o r a t o i y o f Sle\-ens Inst if u i e of T e c l m o l o g y has conducted a .series of investigations a p p l y i n g the i m s t e a d y l i f t h i g - . s u i f a c e t h e o i y derived b y the acceleration-potential m e t h o d t o the marine propeller case.

T h e three-dimensional integral e q i i a f i o u f o r t h e screw pi'opeller i u stead.v flow was derived b y S p a i e n b e r g [1 ],•> who related (he imlvuown propcfler l o a d i n g t o the k n o w n v e l o c i t y d i s t r i b u t i o n o u the blades. I n t h e fir.st o f t h e present series of studies at D a v i d s o n L a b o r a t o i y , Shioiri and Tsakonas [ 2 ] extended his w o r k to t h e case of unsteady flow, paralleVmg H a n a o k a ' s .study [:5].

Since there is l i t t l e hope of a d i r e c l .solution of t h e resulting surface integral e i p i a t i o n w i t h i t s i n t e g r a l k e r -nel h a v i n g " H a d a m a r d " t y p e singularities, Reference

[2] uses t h e Weissinger a p p r o x i m a t i o n w h i e h t r a n s f o r m s the .surface i n t e g r a l equation i n t o a l i n e i n t e g r . i l ecpiation. I l l this m a ( h e m a ( i c a l model (he .surface-lif( d i s l r i b u l i o n is replaceil b.v a line d i s t r i b u t i o n along t h e 1/4-cliord line, w i l h the control | i o i i i l s (aken along t h e .•V4-chord line. .\s a m o d i f i c a l i o n of lhe W e i s . s i i i g e r m o d e l , t h e chordwise b o u n d a i y coiidit ions are .sal islied b y a weight ed average over l h e c h o r d . T h i s m e t h o d is k n o w n l o give the exacl l i f t i n l l i o (wo-<linieii.sioiial s l e a d y - s ( a l e c.nse and also i n the Ihree-dimeusioual steady-slate i-ase

' N m u l i o r - ' in l u a . koi.-- d e s i t t n a l e liofereii.'os a( e m l nf p a p e r .

Some New Measurements on the Drag of

Cavitating Disks

By G . J . Klose' ond A . J . A c o s t a '

As part of an experiment on unsteady flow past a cavitating circular disk, it was necessary to moke calibrating measurements of ttie drag on disks in steady flow. The measurements were made for greater cavitation numbers than have been previously recorded, and show that the d r a g coefficient is essentially linearly dependent upon cavitation number up to values of this parameter as high as 1.3.

Nomenclature A — area nf tl'iak

Cj - tlraj; eoclli' iP'il = ip^'f^

C,% •= (',, f(,r <r = (I

tl = d i s k ( l i a n i e l e r

rf^ = t u n n e l d i a m e l o r /> = d r a g f i n v o p , = p r e s s n r e in l a v i l y

= pres.sure in i m d i s l i n l i e d apprnac liiiit: H " "

V = v c l o c i l y of i m d i s l i i r l i o d approai-liinfr flow p = fluid den.-iily

, ƒ'» " V'

a = oavilation nuailier = ^- p j "

T H E in-cseut expciimciit.s were carried out i u the high¬ speed water l u i u i e l o f the I l y d r o d y i i a n i i e s L a b o r a i o r y at the C a l i f o r n i a I n s l i l u l e of T e c l m o l o g y (1 ] ' . H i e model disks were s u m i o r l e d f r o m l h e d o w n s t r e a m side on a .slem w h i c h , i u t u r n , was a t t a c h e d t o the s l r u l sup-port of the three-component f o r c e balance [2] of the tunnel. T h e strut was surrounded b y another s t r u t , which .shielded the a d i v e strut f r o m the force of Ihe f l o w i n g water. T i n ; pie.s.sure w i t h i n the c a v i t y was measured b y a piezometric t u b e c o m n u m i c a l i n K w i t h the c a v i t y behind the di.sk. V e n t i l a t i n g a i r could also be i i i l r o d u c c d I h r o u g h the ba-^e of the shielding s l r u l to v a r y the c a v i l v pie.s.sure. T h e three disks used had

diauKlers of 2 I 4 i n . , :i>4 a n d 4».t i n . , t h e t u n n e l w o r k -injr section being 14 i u . d i a .

T h e lesl i.rocedure eonsisled of s e l l i n g the tunnel velocity a l the desired value a n d then decreasing l h c t u n n e l pre.<suiv u n t i l a long, clear c a v i t y was established. W i t h t i n ' How .slidiilii^ed, a l l measuiemetils were laken al llu-. moiiienl l l u ; force l)alaiic<- ha<l reached a stead\' reading. I n c i d e n t a l l y , it was d i l l i e i i l l l o establish steady conditions i u l l i e t u n n e l f o r these large disks excepi

relatively u< av i l i e -'eliok^'d'" e o i u l i i i o n .

I ' i g . 1 shows a t o p vi(!W o f the c a v i l y behind 2 ! , , -in. d ï s k . T i l l - t i p i n o j e c l i n g f r o m the f r o n t of the disk

' H y d r o d y i i a i o i r s I.aliornloiy, K;Miiian balnaal<ay of K l u i d

.\tei-liaiii<s a n d Jel P r o p u l s i o n , C a l i f o r n i a I n s l i l u l e of 'roclniolnny, P a s i i d e u a , C a l i f .

•' N u m b e r s iu l a a r k c l s d(.siKiialo U e f e r o m c s a l en<l of p a p e r . M u u u s i r i p l r o K . i v o d n l S.N'A.MK l l e a d ( | U a r l e r s , N'ovenda.r !», 1 i l l i t .

FI'K- 1 Top view of Cavity behind the 2 l.| in. circular disk.

was used o n l y f o r i j h o t o g r a p h i c p u r p o s e s a n d n o l f o r t h e d a l a runs reported h e n a n .

T u n n e l velocities rangetl f r o m 27 t o :{2 f p s f o r t h e 2 l . j - i n . d i s k , f r o m 18 t o 24 f p s f o r t h e : i ' . i - i n . d i s k , a n d f r o m 14 to I'J f p s f o r t h e 4-*4-in. d i s k . Vapcn- c a v i t i e s were used w i t h t h e 2 . l . | - i i i . d i s k , a i r - s u p i ) o r t e d c a v i t i e s w i t h Ihe I L ' . j - i n . d i s k , a n d b o l h a i r - s u i i i i o r l e d a n d v a p o r cavil ics w i t h the 4-*4-in. di.sk. .Vll h o u g h t h e c a v i t y pressure v a r i e d f r o m 0.08.") t o 0.10.") f l (abs) f o r v a p o r cavities a n d f r o m 0.472 t o 0.828 f l H g ( a b s ) f o r n i r -s u p p o r f e d cavitie-s, f h e met h o d of e -s t a b l i -s h i n g t h e c a v i t i e -s , namely, i i u m p i i i g d o w n t h e l u i i n e l a m b i e u l prc.s.sure u n t i l a n essentially <-hoking cavit.v was e s t a b l i s h e d , caused the c a v i t a t i o n l u i m b e r s l o f a l l i n a n a r r o w r a n g e f o r e a e h of the disks. T h e How c o n d i t i o n s t h e n w e r e s i m i l a r f o r all the d a l a p o i n i s f o r eaeh d i s k regardless o f t h e w a y

Ihe c a v i l y was f o r m e d .

B<'fore discussing lli(^ r e s u l t s o f t h e present i n v e s t i g a -l i o u , reference s h o u -l d be m a d e to ) ) r e v i o i i s m e a s u r e m e n -l s . l-'ig. 2 .shows p o i n t s o b t a i n e d b y U e i c l u i r d l w i l h v a i i o r cavities i n a f i e e - j e l t u n n e l 1:^1, l i y K e r m e e u w i t h v a p o r cavities i u the high-speed w a l e r U m n e l u l (."I T a n d b y f V N e i l l w i l h a i r c a v i t i e s i n t h e f r e e - s u r f a c e w a t e r t u n n e l at ( • I T H I , a n d b y iMsenberg a n d P o n d w i l b " p a r t i a l " v a p o r c a v i l ies at D a v i d T a y l o r .Model B a s i n 1.") |. ( I ' n d e r p a r l i a l c a v i l a l i o n coiidition.s, t h e c a v i l y a p p e a r s s m o o t h and o p a i p i e l o l h e e y e ; L u l h i g h - s p e e d p l i o l o g i a p h y reveals l h a t the c a v i t y is i u f u e l l i l l e d w i t h a r a p i d l . v p u l s a t i n g v a p o i - w u t e r m i x t u r e a n d has a < p i i t e i r r e g u l a r envelope, o n l y t h e a v e r a g e o f w h i c h is seen l>y t h e e y e . ) T h e d a l a p o i n t s f o r K e i c h a r d l , K e r m e e u a n d ( ) ' X < - i l l w e r e r e p l o l l e d f r o m O ' X i i l l |4 |, w h i l e ICisenberg a n d P o i u l ' s were l a k e n f r o m l h e i r r e i i o r l |."i|. . M l h o u g h I h e r e is

Uniform Oval Reinforcing

Rings—Power-Series Expansions

By William P. V a f a k o s - and Louis Rostand '

Power-series expansions are employed in the analysis of oval reinforcing rings which ore arbitrarily located on the inside or outside of oval cylindrical shells of prescribed cross section. The expansions ore in terms of two small geometric parameters which ore fixed by specifying the cross section of the ring and the major and minor-axis lengths of fhe oval. Analyfical solutions ore presented for fhe case of a uniform radial load, A graphical comparison is made with fhe results of a relatively lengthy energy solution. In addifion, fhe effect of inside or outside rings is displayed in graphs, where fhe results ore plotted versus the major-to-minor axis ratio.

T H E analy.sis of r i n g - i v i i i f o r c i ' t l oval cylindrical sliclls consisis, i n {loncral, of l w o .separate .solutions, one for f he shell a n d a n o l l i e r l o r the r i n j ; . These .sohilions are m a t c h e d b y ('(pialing the displaeemenls and i n t e r a c l i o i i loads al lhe line of conlaci of r i n g and shell. I'or the case of an applied h y d i o s l a l i c pressure, sohilions which are applicable t o the s h e l l , ' and corresponding sohilions f o r the o v a l r e i n f o r c i n g rings,-" have been obtained b y ap-p l y i n g the l l i e o r e i n of the n i i n i n i u m o f t h e l o l a l ap-p o l e n i i a l . T h e nmiH-iieal <-aI<-iilalions which iniisl be performed i n f i e a l i n g a spei-ilii- example are. however, exlremely l e n g t h y , a n d i n m a n y instances are sensilive enough to l i u i u - a l i o n a n d round-off eriors l o reciuiie extensive use of a high-specfl di.gilal computer.

l l is l h e obje(-l of the present paper to present a new and nui(-li simpler apiu oacli for the analysis of the d o u b l y .synmictric o v a l rciiil'orciiig rings (-ousideied earlier.-' 'I'his new analysis, iu which numerical results ean be ob-l a i i i e d a&ob-lt;-&ob-lt;-ui-ateob-ly b y the use of a sob-lide ruob-le aob-lone, foob-lob-lows closely the m e t h o d of p a i a m e l r i c expansions presented i n au earlier report."^ l i o w e v e r , the method i.s here

e.\' e.\'Ie.\'his r e s e a r c h w a s j i i i n l l v spniisMreil l>y lhe O l l i r e nf N a v a l I t e -.se.in h a n i l l h e H i i i e a n uf .Sliips iiiuler C n n l i a i l N o . N o n r s:i'.ll 14), I ' n i j e c l N o . .NU IHi4lt>7. 1.. U o s l a i u l w a s also ( l a r l i a l l y s u p p o r l e i l liv Ihi N a t i o n a l .'SciiMice l''oim(lalion i i m l e r ( h a u l N S K C --_'|.S7(i.

- A s s o i i a l e P r o f e s s o r of .Xpplied .Mei liaiiics, I ' o l y l e i i i n i c I n s l i l u l e of B r o o k l y n , H r o o k l y n , N . V .

' Kese.iVeli ' r e i h n i r i a i i , I ' o l v l e i h i i i r l i i s l i t u l e o r l i i d o k h ii, l i i d o k -l y i i , N . V .

' W . I ' . N'afakos, K . f t o i i i a n o . a m i .1. K e m p i i e r , --('lamped S l i o r l O v a l C v l i n d r i c a l S h e l l s C m l e r l l y d i o s l a i i i - P r e s s u r e , " .loiiniiil of . l / m - v w r r ,S'i/<7/e..s, v o l . ' J U , l!Hi'-'. pp. | : t l 7 lo.'w: t'ormerly 1*1 It.Vb l i c p o r l .No. .V.I4, I V i l y U ' c l i u i r l i i s i il u i e of H r n o k l y n , .liuie H l l i l .

•'• W . I ' . \ a f a k o s , " . \ i i a l y s i s of f n i f o r m D e e p O v a l Iteinfureiiij; H i i i C s , " . l o i u \ \ i . i i r S i n e U K S K M I C I I , v o l . 7. . \ p r i l I!lli4, pp. '_M 'AS; f o r i i i c r l y P I H . M . I l e p o r l N o . I i 7 s . I ' l i l v l e c l m i c l u s l i i u l e of H m o k -I v n , -I ' V l i r u . i r v M l l i l .

" W . 1'. V a f a k o s a n d N . Nis.sel, " P a r a n i c l n c K x p a u s i o u s fur O v a l I t e i l l f o r i iliu Hinns,"" .Inuninl of llu- EmjiiHiriiHi Mirhriiiii.-'

Diit.^iou, . l . S T / i , v o l . !lll. N o . i:.M.-i, r i o i e e d i i i K s P.iper 4111)'.', l i l l i l ,

p p . 1 ' i l l : f o r m e r l v P I H . M - H e p m i No. (i.s4, I ' o l y l e c h i i i r l i i s l i n u e of I S r o o k l y i i , . M a r r h I ' . ) ' . ! .

. M a i i i i s i r i p l r e c e i v i i l a l S N . V . M l ' . I l e a d i p i a r l e r s , D i l o l i e r 2 ! l , MUi4.

leiuled to enable lhe anal.ysis ol a r b i l r a r i l y located, i.e., inside or outside, oval r e i n f o r c i n g rings.

-Vnalylical sohitions. w h i c h are |>re.senied f o r l h e case of an applied u n i f o r m radial-line load, are shown l o be i n good agreemenl w i l h mnnerical resulls p i e v i o u s l y puhli.shcd.-'' I n a d d i t i o n , a p a i a m e l r i c .study is made i n a series of graphs w h i c h show l h e v a r i a l i o u of the d o m i n a n t forces, moments, strains a n d displaeemenls i u t h e o v a l ring w i l h increasing n o u c i r c u l a r i l y , and l h e e f f e d on i h i s variation o f lhe location of the reinfon-ing r i n g ; i.e., t h e effect of an inside or outside r i n g .

Basic Assumptions and Equations

T h e assimi])! ions and governing eipial ions applii-al)le t o f h i s anal.vsis are identical t o those employed in an earlier l i n g analysis' and are summarized i n lhc l o l l o w i n g .

T h e local c u r v a t u r e l / i of the u n d e f o r m e d , d o u b l y .symnielrie, line of coiiia<-t lietween r i n g and .shell of p e i i m e l e r /.„ = 'ITTI;, (herein called the reference line) is

speellied so as t o s i i n i i l i f y the more d i f l i e u l l shell a n a l y -sis;* i . e . , i l isassumed that

l / r = ( l / r „ ) l l + 4 cos ( l V ' - . . ) l (1) w h e r e * is a I'ircumfei-ciiliai coordinaie. F i g . 1. a n d ? is a

paranieler whii-h li.xes the m a j o r - l o - m i n o r axis r a t i o (/)/«)• A discussion' of the dependence of li/a upon t shows l h a l 1.00 l>/n ^ 2.0(i for 0 < ^- < 1. Values of ^ ' > I are nol considered Iwcause l h e y resull iu i-efereuce lines w h i c h are not convi-x o u t w a r d a l every p o i n t . A n y point of lhe r i n g is located h y specifying the radial c o o r d i -naie £, Fig. 1, in a d d i l i o i i to lhe p e r i m e l i l e a l coorthnaleA-.

l l is assumed f u r i h e r t h a i lhe rings have a u n i f o r m cross seclion which is s y i i i i n e i r i c a l l y disposed w i l h re-specl lo lhe cenlroidal plane, l h a l all loads are a p p l i e d in ihis plane. Fig. - . anrI l h a l b u c k l i n g or i w i s l i n g of l h e ring o u l of i h i s plane does nol «icciir.

; I'". H o m a m i a m i .1. K e n i p n e r . "•.Sue^ses in S h o r t .Nom i n - i i l a r C v l i n d r i c a l Shell.- I iider U u e r a l P r e s s u r e . " .lonrniil of Aii/iliii/

.\ini,<iiiir.i, v o l . 'Jii. Traits. .{SMi:. v o l . S 4 . S e r i e - F.. ÜIIW. p p . litHI (174; f o r n u ' l l v I ' l H . V I . l ! e | i o r l N o . 41.-i. r o l y t c c l m i c I n s l i i u t e

or'Hiook'lvii,,Iuly"l!l.">S.

An Experimental Determination of Pressure Forces on

a Ship Model From an Analysis of the W a v e Profile

By Grant Lewison^

A method for the analysis of the wove profile alongside a ship model is presented in a new form. The pressure distribution over an axisymmetric model is computed, and wove resistance and vertical forces are derived from it. The application of the method to other model forms and also to full-scale ships is discussed.

W H E N a .ship model advances steadily i n t o s l i l l w a t e r , a p a l l e r n o f waves w h i c h moves w i t h the model is created. 'J"he p a t l e r n may be measured, and several uu'lhods f o r its anal.ysis have been jiropo.scd [1--S)- i n o r d e r to determine (he wave resisliuice of (he model, de-line<l as (he par( of (he (otal resis(auce (ha( .••upplics euerji.v (o (he wave pa((eru. There is uo general agree-nien( (ha( one par(i<'ular n u ' l h o d of meas\uemeut atid anal.y.sis is m a n i f e s l l v superior to (he o d i e i s , and all of (hem rccpiire complex measuring apparatus and data re-d u c t i o n .

T h i s paper describes how (he wave resislauce ma.v be ealcula(ed couveniendy f r o m au anal.vsis of (he wave p r o f i l e along (he side of (lie model. T h i s prolile was re-c o n l e d phologiapliire-call.v for an axi.<vnune(rire-c model at dilTeren( speeds. A graphical inediod of anal.vsis, i n -veu(ed b.v ( i u i l l o l o u |4 j , was then u.-^cd (o compute (he pre.s-sme d i s ( r i b u i i o n over (he w e d c d surfaee of (he mod(4, and b.v i n ( e g r a l i o n , (he re.sis(ance was obtained. A d i g i t a l couipu(er was emplo.ved f o r (he anal.vsis and (he m e d i o d was (heieb.v made a practical po.<sibili(.v f o r (he lirst (ina-. .\s a check upon (he accurac.v of (he w o r k , the v e r d c a l fences and p i f c h i n g mcMuenlsduc to (he pressnre lield were al.-o computed and compared w i d i d i r e c l meas;ureinents of these (|uaii(i(ic.s.

T h e resuhs show (piile good agrcemeid w i d i (hose ob-t a i n e d f r o m o d i e r ineduids and, because (he melhod is s i n i p h ' and also could be applied lo full-scale ships, it apiieais worlh.y o f adoption as a slandard (echniipie i n model les(s.

' D e p i i r l n i e n I of ICnuineeriiiK, l i i i i v e r s i l y iif C a n i b r i d n e : I're.s-e n l l v . l ) I're.s-e | i i u l n i I're.s-e u l of N i i v i i l .\rI're.s-eliilI're.s-e< t u r I're.s-e , UnivI're.s-er.sil v of C i i l i f o r n i a .

U c r k e l u v , C-alif.

= K i i n i t i e r s in l i r a c k e l s ili'.sijruale Hi-feronees a l e n d of paper. M a i H i s e r i p t r e c e i v e d af S N . W I I C t t e a d t p i a r l e r s , N o v e m b e r 21, l'.ll>4: r e v i s e d n i a n u s e r i p t received . \ p r i l 2il, l'M\h.

Experimental Work

The model was axisynimelric [.")) and was l o w e d axially. I t was attached r i g i d l y to the t o w i n g carria,ge and sinkage and t r i m were thus p r e v e i i l e d . I ' h o l o g i a p l i s were taken f r o m d i e .-^ide of the t o w i n g t a n k , above the w a l e r .surface, at ei.gliteeu speeds i n the range o f f r o u d e number 0.1.") < .f < 0.41. F r o m a comparison between p n i i l s al o u e - f i f l h life .size, of the s l a l i o n a r y model and of die model al .speed f ' , the wave profile was recorded at. intervals o f I ' A i n . as a .series o f 117 ordinates. w i t h au accuracy estimated al belter t h a n "2 percenl. T h i s was po.ssible because each o r d i n a l e was o b t a i n e d b.v d i r e c l iiieasuremeul. and was independent of i l s neighbors. T h u s the experimeulal i n f o r m a t i o n was at least as ac-cural e as lhat derived f r o m w a v e - p a t t e r n measuremenls.

Theoretical Analysis

T h e .ship model is a.s,siiiiicd to be ' I h i n ' enough f o r the l i n e u i z a d o u s of M i c h e l l ' s theory [O] t o be v a l i d . W e .seek l o f i n d the pressure d i s l r i b u l i o n over the cenler plane of the model and as.sume t h a i this holds also at d i e model surface. T h e pressure d i s l r i b u l i o n is l o be ile-fined by a series of conlour lines o f eonslani pre.ssure. o f which i h e f i r s l , corresponding to atmospheric (or zero) pressure, is the wave prolile.

W e define an o n g i n i u the u n d i s t u r b e d w a t e r .surface amidships, and .r is positive f o r w a r d . .(/ positive to star-board and z posit ive downwards. T h e model is o f lengl h

•2L: each section is circular, of radius i ( . r ) . a m i Ihere is

fore-aud-afl s y m m e t r y , so r(.r) = li-.v). F o l l o w i n g O u i l l o l o u (41, we consider a f u n c d o n i r ( . r . - ) l h a l obeys

the relation

dz 0 c).r=

i u the i n t e r i o r of the f l u i d near l o l h e m o v i n g model.

.Nomenclature.

_{1', "}

e

g

_{r =}

A Note on Short-Time Prediction of Ship Motions

By John F. Dalzell'

The prediction of aircraft-carrier, landing-ramp motions a short time into Ihe future, is o problem o f importance in that new or improved methods lead directly f o increased safety and extended operofionol capability. This note presents a possibly different point o f view of this problem and the results of some preliminary work which indicate that further study of fhe feasibility of this approach may be warranted.

PIJECI.SE prediction of the v e r t i c a l position of the l a i i d -itiK ramp of an a i r c r a f t carrier i n a seaway is of enormous advantage w i t h re.sjiect t o hetterineiit of landing safety through automated l a n d i n g guidance sy.stems. Even an improvement i n c a p a b i l i t y such t h a t rislc to p i l o t and etiuipment can be m a i n t a i n e d a t present levels in seaways of greater severity can be translated d i r e c t l y i n t o i m -proved operational etTectiveiiess.

I t should he noted t h a t the usage of the w o r d predic-t i o n herein is n o predic-t i n predic-the .sense o f average a m j i l i predic-t u d c s , periods, a n d so o n , it is i n t h e d e t e r m i n i s t i i t sense: t h a t is, knowing the present and past h i s t o r y of motions, w h a t w i l l he the ship's a t t i t u d e A ' .seconds hencx-? T h e jire-diction of the a t t i t u d e of a ship a t some i n s t a n t i n the f u t u r e lias been the subject of considerable w o r k a l t h o u g h few |)uhlished accounts are available. One of these few is contained in a work b y .1. T . Fleck. " T h i s is a s t r a i g h t -f o r w a r d ( i n i)riiiciple) a i i p l i c a t i o u o-f Wiener prediction methods developed d u r i n g W o r l d W a r I I . These metliods use only the past t i m e h i s t o r y of the m o t i o n to be iiredicted and i n v o l v e c o m p u t a t i o n of power spectra and construction o f an o p t i m u m prediction filter w i t h projierties dictated b y the .solution to an integral etpia-t i o n . T h e ina.\innini ahead predicetpia-tion i n etpia-t e r v a l whicli was thought acceptable i n the cited reference was (i sec. I t is clear f r o m t h i s work t h a t considerable e l f o r t must bc expended in order to realize a f e w seconds of prediction f i m e . b u t i t is e<|iially clear f r o m the l a y descriptions

' S d U l l i w e s t U e s e u r e l i I n s l i l u l e , D e p a r t m e n l nf . M e c l i a n i c a l .Sciences, S a n .\iitciiiiii, T e . \ .

-.1. T . F l e r k . " S l m r l - T i m e I'reilii l i m i iif l h e M i i l i m i nf S h i p m W a v e s , " ,S'/i/;;.v awl W'mis, C i i u n e i l n n W a v i ' lie.searcli a n d T h e .Siieiely nf N a v a l . \ i i h i t e c l s a m i M a r i n e I'.iinineers, lil'i.").

.MaiiUKcript rei c i v e d a l S N / V M F H e a d i p i a r l e r s , N i i v e m l i e r U i , I m i l .

of l a u d i n g o i i e r a l i o i i s o n a i r c r a f t e a i i i c i s l I u i l l a n d i n g a j e t on au a i n a a f l c a r r i e r is a i l i l l i c u l t . a n d ( l a i i g i ' r o i i s business, l e t i u i r i n g close t i m i n g aii<l an o p e r a t i o n w h e r e small prediction times m a y be o f v a l u e .

A d i f f e r e n t approach'' r e s u l t i n g f r o m a basie s t n d y o f ship motions, i m p l i e d t h e j i o . s s i b i l i t y of o b t a i n i n g a b o u t sec predictions of scaled-up t o w i n g t a n k e x p e r i u i e i i t s h u l d i d not elaborate o n l h e general f e a s i b i l i t y . T b i s a | ) -proacli i n v o l v e d i n e a s t u i n g t h e i n c i d e n l w a v e s o n t h e ship s u f f i c i e n t l y " u p w a v e " .so t h a t w h e n t h e l u o p e r t i e s o f the .ship are c o n i b i u c d w i t h t h e d i s p e r s i v e p r o p e r t i e s o f waves a s h o r t - t i m e pre<lictioii " l i l t e r " can be v i s u a l i z e d easily. T h i s m e t h o d does not t l i r e c t l y i n v o l v e t h e w a v e s p e c t r u m , b u t is p r e d i c a l c f l i n the p r a c t i c a l sense o n a n a b i l i t y t o measure waves at .some t l i s t a i i c e a w a y f r o m t h e

LCCi of t h e .ship.

T h e p r i m a r y d i f f e r e n c e between t h e t w o p r e d i c t i o n methods c i t e d is t h a t t h e f i r s t u.«es t h e m o t i o n t o be p r e -d i c t e -d as i i i | ) u t t o the | ) i e -d i c t o r s y s t e i i i , t h e s e c o n -d uses the elevation o f incident waves as i n p u t . I t is 1 he p u r p o s e of t h i s note to present some p r e l i m i n a r y r e s u l t s o f a paper d e m o n s t r a t i o n o f t h e second c o n c e p t .

A n a l y s i s

.-Vs m e n t i o n e d i n the i i i t r o d u e t i o i i , t h e c o n c e p t o f n s i n g wave elevation a.saii i n p u t Lo a p r e d i i t o r scheme i n \ - o l v e s a kiiowletige of w a v e e l e v a t i o n a l .some d i s t a n c e f r o m t h e ship. T h i s alone m i g h t r u h ; o u t p r a c t i c a l e x p l o i t a t i o n f o r an a r b i t r a i y ship b u t s i i u e a n a i r c r a f t c a r r i e r is rarely, i f ever, alone, t h e presence o f o t h e r s h i p s c a p a b l e of keeping s t a t i o n o n t h e c a r r i e r cau be a s s u n i e i l . I n a d d i t i o n , the pos.sil)ilily of a l i e l i c o p l e r k e e p i n g s t a t i o n at

" .\1. F a i i c e v , •'(."heck n n l h e l . i i i e a r i l y nf a S h i p . M o t i o n S y s l e m , " D a v i d s o n U d i o i a l o r y , T e c h n i i a l . M e n i o r a n d i i n i T . M - I ' J l i , ' S l e v e n s I n s l i l i i l e o f T e c h n o l o g v , H o b o k e n , N . .1.. N o v e m b e r l i l l i l .

II = di.Hlaiice f r o m l . ( ' ( ! l o ulerii (as.siimed e q u a l l o Hill fl f o r a i n i l l l - f l s h i p ) I) •= ili.slaiice of H a v e i n e a s i i r e m e n l f r o m \,C,i\ 1/ = g r a v i l a l i o n a l a e c e l e r a l i o n l\^T) = a kernel f u n c l i o n lt(l) = vi^rlical Klern m o l i o n / « i m e V = .sliiji v e l o c i t y l l i/tü) = l i a i i K f e r f u n c t i o n lieliveeii w a v e e l e v a l i o n n i e a s i i r e d I)

.Nomenclature.

a

Forces on a Submarine Hull Induced by the Propeller

By George Chertock'

The induced-flow fleld generofed by o propeller operating in the wake of a submarine can exert forces on the submorine in some arbitrary mode of motion in two ways: Either by pressures applied to the propeller blades and resulting in the conventional propeller thrust or by pressures applied to the hull surface. The ratio of these forces is shown, by theoretical analysis, to depend most strongly on the flow velocity which fhe particular hull mode of motion would generate at fhe propeller, to be slightly dependent on the shape of fhe propeller, and fo be negligibly dependent on fhe magnitude or distribution of fhe submarine wake. The derived relations are valid for both fhe inertial and quasi-steady components of the induced-pressure field. The predominant factor in the force ratio is evaluafed by a numerical technique which is applicable fo any surface of revolution moving in almost any pattern of motion. As one applicalion, it is shown thaf the steady longitudinal force on fhe hull is 8 to 12 percent of the steady component of fhe propeller thrust, in good agreement with measurements of fhe thrust-deduction factor. As another opplicafion, it is shown that a parficular vibratory component of fhe hull force, fending to vibrate Ihe hull in fhe accordion-vibrafion mode, is ó to 8 percenl of the same component of the propeller thrust.

A Pitoi'ELi.EK o p i ' i a t i i i g in the wake of a .subinaiine generales a pre.ssure lield whieh ean exert forees on the subinarine in two ways. One force is the direct axial llirusl on the propeller which is Irausferrcd I h r o u s l i Ihe propeller shaft lo lhe l l i r u s l hlock. T h e other force comes from distribiiled pressures actiiiK on the outer surface of I he- hull. This paper e\-aliialcs I he rat io of t he 1 wo forces hy a melhod which compares the forces w i l h o u l calcii-l a calcii-l i i i g calcii-lheir .separate vacalcii-lues.

The inelhod of ilie analysis is to express each o f the I wo forces in lerius o f the induced pressures a l I he surface of the pif)pcller blades, and l o derive Ihe ratio o f llic forces in a form which depends separately on the shape of lhe propeller and on the charaeterislics of the sub-marine h u l l , and is negligibly dependent on the charae-I c r i s i i c s o f the induced flow.

The Kcnc-ral analysis is applied to l w o p a r l i c u l a r cases of praciicul importance. I n the hrst case, the .steady, or lime-avei-aKc conipoiieiil of lhe l o n g i l u d i i i a l hull force is coinpiilcd as a iVaclioti o f lhe steady componenl of the direcl propeller I h r t i s l . T h i s is relafed l o the f a m i l i a r problem of lhe l l i r u s l d c d u c l i o n .

I n lhe second ca.se, l h e h u l l force is laken lo he lhe generalized force l e n d i n g l o v i b r a l e l h c hull in lhe s i m -plcsl l o t i K i l u d i i i a l . or •'accordion" v i h r a t i o n mode. T h i s v i l i i a l i o i i tiimle is i m p m t a n t because U \s easily cxciicd

b y periodic. loiigitniVmal forees at the t h r u s t hlock or hy periodic pressures on l h e a f l p o r l i o i i of the h u l l , and also liecau.se i l s re.soiiaiil frc<piency is w i l h i n the normal f r e ipieney range of a subinarine propeller. Hence lhe u n -slciidy, periodic compotienl o f this v i b r a t o r y hull IVnee

' D . i v i d ' I ' a y k i r M o i f c l Ha.siii, W a s l i i i i K l ' m , I ) . ('. .Miiiiii.scriiii recciveil .'it .S.VA.MIC He;elr|ii.'irler!', .N'ovemljer 10, P l l i t

is cvalualed as a f r a c l i o i i of l h e unsleady, periodic c o i n -poiient of lhe direct propeller t h i u s l .

I n fael Ihe present anal.vsis, as firsl (h'veloped a n d cir-culated as an inlernal m e m o r a n d u m o f the M o d e l Hasin in .'Vugusl l!l(i:{, applied onl.v l o the unslead.v <-ompoueuls of Ihe propeller l l i r u s l . Kub.scfpicnlI.v i l was .sugKcstctl by Hicsliii | l ) - ' l h a l l h e ratio o f Ihe u n s l e a d y forces should be "analogous" l o l h e r a t i o o f the s l e a d y foices in the l l i r u s l dcduclion p r o h l c n i , a n d lhe analysis t h e n was extended lo a p p l y to Ihe sleady c o m p o t t e n t s o f the propeller I h r u s l .

Induced Pressure Field and Forces

T h e two forces w i l l be analyzed in l e t n i s of t h e p e r l u i -baiion v e l o c i t y a n d the p e r l u r b a l i o i i pressure o f l l i e f l o w induced by the propeller a n d Ihe s u b i n a r i n e h u l l .

We I l e a l the w a l e r as incoiupre.ssible a n d i n v i s c i d , a n d we assuine l h a l l h e propeller is on the l o n g i t u d i n a l axis of l h e submarine, movittfi; f o r w a r d w i l h s l e a d y v c l o c i l y v.iand r o l a l i n g w i l h a n g u l a r velocit.v lo. T h e n a p o i u l on lhc propeller al veelor dislance r f r o m t h e h u b moves w i t h v e l o c i t y v,, -|- (o X r .

Denote b y qu(r, /) t h e f l o w V C U M I I V , Ve\al\ve t o the earth, w h i c h is i i u l n c e i i l i y t\\e m o l i o n of lUe submaviue

if the propeller were r e m o v e d . T h a i is, i f n d e i i o l e s a local n o r m a l l o the surface, then

n q , ) = n-Vo ( I ) al l h e .submarine h u l l , w h i l e qi, is e o i i l i n u o u s a t t h e p o s i

-tion o f the propeller blade. Also, d e n o t e b y q , ( r , 0 t h e

nil'lilioiial f l o w v e l o c i t y , r e l a t i v e t o t h e e a r t h , d u e l o l h e

presence a n d m o t i o n of the propeller. H e n c e

- Xtiiiiliers in braekels tlcsinnale Uefereiiees al eml of pa|)er.

1 2 2

Unified Lifting-Line Theory of Fully Wetted Hydrofoils

By Tetsuo N i s h i y a m a '

Unified lifting-line theory is developed for the hydrofoil of fully-submerged ond surface-piercing type. The indirect and direct problems are discussed in some detail in relation fo fhe effect of fhe Froude number. The disturbing velocity potential is derived from fhe linearized boundary condition on fhe free wafer surface. Then from the boundary condition on the hydrofoil, a basic integral equation is obtained f o r fhe distribution o f circulation over the span, from which the lift and resistance can be computed readily. Some numerical examples of practical interest ore shown f o r fhe characteristics and optimum condition in a specified condition of operation for fhe fully-submerged and surface-piercing hydrofoils.

Introduction

Rr.vuHAi, f.lu'oretical n'.xfiirclics liavo bt-cn i n u d f f o r the f n l l y - w c t f o d h y i h o f o i l o f fully-submerged and also sur-faee-picreing t y p e f r o m the standpoint of l i f t i n g - l i n e eoneepls ( W u 1!).>1.= K a p l a n l!l.').")aud HWO, Breslin 19.')7, X i s l i i y a n m lilö!) and IflliO). However, at the presenl stage o f d e v e l o p m e n t , i l appears l h a l Ihose theoretical t r c a l m e n l s are not o f gcMieralily enough to be al>le (o exaniine s y s l e n n i l i c a l l y every ease o f practical interest; a t least t h e t w o m a i n types i n the foregoing. F u r t h e r -m o r e , siuwi d a l a are very few f o r the surtace-pierciug l.vpe c o m p a r e d w i l h iho.se f o r Ihe fully-submerged type, the d e s i r a b i l i t y o f a d d i l i o m d d e l a i l c d I r e a l m e n l o f the f o r m e r has been rising rapidl.v.

T h e presenl paper is aimed l o develop l h e unilicd l i f t i n g - l i n e t h e o r v b.v tre.itiug f h e t w o n\ain t.vpes o f h.vdrofoils.

I n Ihe first p a r i o f l h e paper, a method o f calculating t h e h y d r o d . v n a m i c clnnaclcristics is prcsenled and Ihen i l s appi'f)prialene.ss is c o n f i r m e d f r o m comparisons w i l h exp(>rimeut.al d a t a . I n a d d i t i o n , several i n l c r e s l i n g f e a l u r e s peculiar l o the h y d r o f o i l , i n p a r t i c u l a r l h c effects o f the free w.alersurface on the eharaclcristics, are elearlv e x p l a i n e d tln-ough the resulls o f numerical calculation.

I n f h e second p a r t , a design |)roblem is treated i n some d e t a i l ; the o p t i n n n n d i s t r i b u t i o n o f c i r c u l a t i o n and m i n i n m m resistatu:e o f extreme i m p o r l a n c e i n h y d r o f o i l design are e l a i i f i e d f o r the h.vdrofoils of b o t h types in t i n -g i v e n c o n d i t i o n o f o j i e r a l i o n .

Disturbing Velocity Potential

A s usual i n f h e linearized theory, we stand on the a s s u m p t i o n t h a i l h e d i s t u r b i n g velocil.v is small com-paretl w i l h the s p e e d o f the h y d r o f o i l . T h e n we can s u b s l i l u l e the h.vdrofoil b.v a b o m i d v o r l e x , or a l i f l i n g

l i n e .

. \ o w we l a k e l h e o r i g i n o f coordinaie on the s l i l l waU-i-lev<-l j u s l over the midsj>an w i l h e.r-axis in the direction

I l-'iirnlt.v of I'.nniiicerinti, Toliokn I'niversilv. Seuiliii, Japan. .Manosiripl received al SNA.MF lle.•llh^llarlers, Aii(;u.sl ;i, l!l()4.

= lieference.s are listed .alplialielically in Ihe Hil)li<iKraphy al Ihc end of Ihe paper.

o f advance, « ï - a x i s v e r t i c a l l y u])wards and ij-axis i n the direi-liou of span, as shown i n F i g . 1. T h e m i d s p a n is

- b / 2

l^if-. I C o o r d i n a t e

located al submergence d e p t h f and the angle o f i u

-t-liualion is / J t o the h o r i z o n t a l plane.

Assuming t h a t the h.vdrofoil o f span b advances w i t h constani speed 1' at the d i s t a n c e . ƒ f r o m the o r i g i n i n an i n f i i i i l e fluid, then the v e l o c i t y p o t e n t i a l ean be expressed b y X dK j " sec ec-x^'^f-'"' + "^"rfö ƒ''•" r(„)rfn ƒ ° c - + COS {K(y - v»>)\dK

+ ' Ï -

f "

'•('jW')

r

(IK f l a n 0

/

r(>))f/i;

+

wlu-ri-/ = sin ti. I» = 1^

Cl = X cos 0 + (!/ — VII') t*'" Ö

(2)

S E P T E M B E R 1 9 6 5

lournai of

SHIP R E S E A R C H

A Sea Spectrum for Model Tests and Long-Term

Ship Prediction

By J . R. Scott'

This paper outlines one-dimensional spectrum terminology, and suggests a spectrum suit-able OS o standard for model and ship-prediction work. This spectrum is represented by

for — 0.26 < to — a-o < T .65 = 0 elsewhere, in which a>fl = 3 . 1 5 r - ' - f 8.987-=

to is angular frequency in radians per sec

S(to) is energy density appropriate to on angular frequency base H is significant wave height in feet

too is spectrum peak frequency

r is average period of fhe waves.

In applicafions where wind speed is specified in place o f T the following supporting rela-tions may be used, provided fhe specified wind speed is more than 15 knots:

to,,-' = 0.03H + 1.35 H = 0.82 W - 5.8

in which W is wind speed in knots

.STATf.sTics o f w.xvo l i o i g l i t aiid p o i i o d Iiavc beou c o i -Inctod o i l tli(> variou.s w o r l d sea roufo.s, and aro available iu aMa.s f o n n . 'I'lie m a i n objeet o f tlii.s papor i.s to pre-sent a one-diinensional .sea .siieetrum whieh i i e n n i l s these s t a t i s t i c s t o be e.x-pre.s.sed realistically i n energy density t e r m s ; such expression is t l i e p r i n c i j i a l step towards f h e i r u.se f o r a.s.se.ssiiig (he l o i i g - t e n n d i s t r i b u t i o n of ship b e h a v i o r .

A secondary o b j e c t is the r e l a t i n g of (he s p e c l r n m w i t h w i n d speed wliere possible, w i t h (he view of an.swering f|Ut\stions a b o u t ship belun-ior i n specified hipih winds. Sueli a r e l a t i o n defines c o i u p l e f e l y a s))cirtrum in tenns o f w i n d speetl, and .such spectra m a y bc produced i n model expei-iment t a n k s f o r (he s ( u d y o f a v a r i e t y o f (yiies o f m o d e l resjion.se. I ' o r jnirposes o f eomi>ari.soii be(weeii inotlel results o b f a i n e d in dilTcrcnf (anks i( is desirable ( h a t each esbiblishineul shonid use s l a i u l a r d specda, w h i c h .should closely represent average sea conditions.

Spectrum Terminology

T h e expression

' . S e n i o r I'liysii-i.st, A ieker.s L i n i i l e i l , S I . .-\ll)ilii.s, H e r l . s . ICnulinul. . \ l ; i n i i . s i r i j ) l reci'ivecl n l S X . A M I C Itejnliiiiiirler.s, S e j i l e n i b c r .S, f!Ui4.

V = Ë .'/. = S «. t'OS (to,/ + C,) ( 1 )

represeiKs (he displacement o f a p o i n t p e r f o r m i n g .suiiul-(aneouslv n simple h a r m o n i c oseilladm.s^ o f a . n p l K u d e

angular f r e q u e n c y a-,- a n d phase c, (/ - b • • . .n). T h e 7M.aiis (a' to,) arc .saitl to define the s p e c t r u m o f Iht^ displacemeiK. ICvidendy these n pa.r.s m a y be expres.sed iu anv f u n c t i o n a l f o r m w h i c h is c o n v e i u e n l , a n d a n y o t h e r f o r m ' m a v bc d e r i v e d f r o m .sncli exiiressioii Because ,s the .sum o f n .separate oscillations, a n y o f ( le q u a n i t i e s

( f a single-A-alucl f u m - t i o n o f ti.) is a d d i i v e (o f o r m a conipan.hle q u a n t i f y ap|.lieahle <o the displacement tis a whole. I t is convenient t o seU-ct a t u n c t i o n of a, w i i c h leads to u-seful pl.vsical results f u n c t i o n s a, a n d ar are the o n l y ones w h i c h p r o v i d e q u a n t i t i e s cai>able ot .simplei.hy.sical i i i t e r p r e f a t i o n . É « , rc|.re.sents fh,_ m a x i

-m u -m po.ssible di.splace-ment o f (he p o i n t , f

p r o p t i r l i o n a l t o the energy o f the m t i t i o n . I n (he •• i .pl -calion envi.sagcd o n l y the .second ol fhese t , u a n t i l i e s leads to useful re.sul(.s, a n d i t is n o w genera pra.;l.ce (o use (he ( l u a i . t i t v ar-/-2 as t h e a m | i l i t i i d e I m i c t i m i m a s p e c t i u m .

D E C E I W B E R 1 9 6 5

Supercavitating Propellers—Momentum Theory'

By M a r s h a l l P. Tulin^

Based entirely on momentum and other simple considerations a reasonably complete picture of the one-dimensional pressure and velocity fields associated with heavily super-cavitating propellers and drag disks has been constructed. It is shown that the flow is often retarded while approaching a heavily supercavitating propeller, so that the so-called ideal efficiency of such a propeller takes on values in excess of unity. The ideal efficiency actually increases wilh decreasing blade covitationol efficiency; however, the net efficiency at Ihe same lime decreases. This retardation causes o reduction of Ihrusl deduction, or even a change in its sign. Charts are provided from which the inflow speed may be estimofed readily. The effect of water-tunnel boundanes is discussed, particularly the phenomenon of choking.

T H E s u p e r e a v i l a t i n g propeller was in(roduec<l i n t o marine teehnolog.v l).v Soviet Aeadamieian V . L . Pos-d u n i n c L a t e r , researeli at the D.aviPos-d T a y l o r M o d e l B a s i n , p a r t i c u l a r l y on the subject of elticicut su|iercavifafin.g-l)ladc seclions, led to the adoption i n western countries o f supercavitatinp; propellers f o r high-speed p l a n i n g c r a f t a n d h y d r o f o i l boats [ 4 - 7 ] . T h e use of such pro]iellers is expanding a t the jiresent time and w i t h the increase i u size and speeds of contemplated h y d r o f o i l c r a f t t h e y w o u l d seem to i)lay an increasingly more i m -))ortant role i n m a r i n e technolog}'. I n .in earlier jiaper, [8], t l i e a u t h o r discussed the history, operating cliarac-teristics, and mechani.sm of ojieration of .supercavitating j)ro))ellers. T h e present paper m a y be regardetl as a companion t o t h i s earlier w o r k .

O u r understanding of the hydrod.ynamics of .super-c a v i t a t i n g propellers is q u i t e imperfe.super-ct. W e have not

' .Suiiportcil l u i J e r C o n l r a e t Nniir-:M3.")(()I)), O l l i c e of X a v a l I T c s o a r e h , N a v \ - D e p a r t m e n l .

' H y d r o n a u t i e s , I m o r p o r a t c d , b a i i i e l , M d .

M a n u . s e r i p t r e c e i v e d a l S N . \ M B Headquarter.';, AugU-st .5,1!10.").

quantitativcl.v understood well enough man.y aspec ts of their design and operation. T w o hydrod.vnamic eft'ects have been (larticularly ignored or shrouded m i n y s t e r y . These aie (i) the interference botween su]ierca\-itafing blades and their cavities [ 8 ] , and ( i i ) the effect o f the cavities on the inflow to the inojieller. T h i s paper is specificall.v devoted t o the l a t t e r effect.

I t is i m p o r t a n t to have accurate knowledge of t h e i n -flow speeds a t the propeller disk i u order t h a t the e f i e c t i v e angles of attack at which the blade eloirients operate ma.v be calculated. For a s u b c a v i t a t i n g propeller t h i s i n f l o w is generated almost entirel.v b.v the v o r t e x wake shed behind the propeller as a consequence of its t h r u s t i n g .action. T h e i n f l o w speed to a t h r u s t i n g subcavil.ating piopeller is always greater t h a n the relative free-stream speed and increa.^es w i t h increasing t h r u s t . A r o u g h idea of this i n f l o w speed ma.v be g o t t e n f r o m m o m e n t u m t h e o i y (Froude-Rankine): |)ieci.se calculations i n c l u d i n g

' Numbers in braiket.s desiniiale lieference^ .it end of pai)er.

Nomenclature

=

=

-tl area of circle eircMiii.scribinn pro- e.ivifalion \ = advance ratio (axial inflow- spued/ peller {<lisk urea) M .slagnafion-prcssure change aero-ss blade relative rotational speed) 1 raiisverse area of downstream flow actuator di.sk p = fluid densily

outside of choked jaopeller /.-, tlirtisf cocflicieiil, T/pn-D' Siiliscripls I), 1, '2, 3 ami -1 when ii.snd slips! renin /. bladc-elcinciit lift ill coiiiicctioii wilh f', and refer lo -Iv,

=

Iraii.s-flow in jels wilhin choked pro- T flinist vi'ise lo How: peller sli|isli-eam V flow- speed 0—fur tipslream

Al

=

₌

=

=

on disk area and tipslream speed .Siibscripl tiiax refers lo iii:i\iiniim allow-thrust eoenicient, T/lpUo'A, V = propeller or l)l:ide cavilat ional ef- able value.

A Riemann-Hilbert Problem for Nonlinear,

Fully Cavitating Flow'

By B. E. Larock' a n d R. L. Street'

A nonlinear problem for a flot-plafe hydrofoil is solved by using conformal mapping and the Riemann-Hilbert solution to a mixed-boundary-value problem in an auxiliary half-plane. Expressions ore developed for plate length, lift and drag coefficients, and cavity configuration. All are given in terms of nonsingular quadratures for prescribed cavitation number and attack angle. The results are in quantitative agreement with experimental data and other nonlinear theories.

I r is well kiiowiv t h a t f u l l y c a v i t a t i n g flow.s arise, i u general, as a consequence of either va|)0r c a v i t a t i o n or v e n t i l a t i o n [ I ] . * V a p o r c a v i t a t i o n occurs i n flows when the local pressure approaches the fluid va|)or pressure, w h i l e v e n t i l a t i o n occurs w h e n a low-pre.ssurc region i n a flow is e.xi)o.sed to a source of air or other gas. These phenomena occur i n t h e operation of highspeed h y d r o -foils and underwater ve.ssels, a n d o n subsurface missiles and )>ropeller systems t h a t are i n e s c u t l y being designed and constructed i n hicreasingly large munbers.

' T h e w o r k was e a r n e d out u n d e r I h e B u r e a u of &liip.=i I' u n d a -m e n f a l f f v d r o -m e e h a n i e . s I t e s e a r e l i P r o g r a -m S-l{(109-fl_l-Ul, OHico of N a v a l fte.<eareh C o n t r a e f N o n r 2 2 . ï ( ö ( j ) . l i e p r o d i i e t i o i i m wtiole or h i part i s p e r m i t t e d f o r a n y purpo.se of t h e l i u t e d . S t a l e s C o v e r n -""="l{'eseareli .Assistant, l i e p a r t m e i i t of C i v i l iMighieeriiig, S t a n f o r d U i i i v e r . s i t v , S t a n f o r d , C a l i f . r i

' . W s l a i i t Profe.ssor, D e p a r t m e n t of C i v i l t - i i g i n e c r i n g , S t a n f o r d U n i v e r s i t y , S t a n f o r d , C a l i f .

' Nuiiih"ers i n b r a c k e t s d e s i g i i a l e l i c f e r c n c e s a t e n d of pape;;-^ pape;;-^ a n l l s c r i p I r e c e i v e d a t S N A M I C H e a d q u a r t e r s , J u l y l o , l O W .

Under the s t i m u l u s of t h i s j i c t i v i l y , research w o r k e r s are c o n t i n u i n g their e f f o r t s t o o b t a i n reliable e x p e r i m e n t a l d a t a and develop suitable theories t o a i d t b e designer. E x p e r i m e n t a l investigations have been u u d c r l i t k c n l o v e r i f y oxi.sting theories or t o d e m o n s t r a t e t h e i r inade-quacy, to j n o v i d e d a t a as a ba.sis f o r e n g i n e e r i n g design, and to s t u d y the influence of factors s u c h as s u r f a c e ten-sion and f m i t e asjiect r a t i o [2, 3, 4 ) . O t h e r s h a v e w o r k e d t o review, summarize, a n d collate t h e r e s u l t s o f the various theoretical areas [5, ( i , 7 , ] i n w h i c h new I h e o r e t i -Cid developments are id.so being p r o d u c e d . I n p a r t i c u l a r , new flow models have been proiiosed [ti ], w h i l e efl'orls are being made t o reduce t h e c o m i i u t a t i o n a l c o m p l e x i t i e s of nonlinear theories [ 8 ] . A t t e m p t s to replace some o f t h e heuiistic correction factors of linearized t h e o r y w i t h more rigorous t h e o r y are .succeeding, [e.g., [9, 10, I I , 121, and the effects of g r a v i t y a n d surface tension, as w e l l as s t a b i l i t y questions, c o n t i n u e t o be s u b j e c t s o f t h e o r e t i c a l interest and s t u d y [13, 14, 1.5]. As a c o n t r i b u t i o n t o w a r d these goals, t h i s p a j i e r

.Nomenclature.

j = noniiegal ive integer

p l a l e l e n g t h l i f l o r d e r of m a g n i l u d e of a i p i a i i -l i t y p r e s s u r e p r e s s u r e in c a v i l y pre.s.suro u l i n f i n i t y m a g n i l t t d o of v e l o c i t y i n u g i i i t u d c of v e l o c i t y on c a v i l y s u r f a c e s o l u l i o n lo Hilbert prolik^iii lief ) = real i)art of c o m p l e x n u m b e r

u p p e r h a l f - p l a n e v a r i a b l e nose of p l a t e , point li, in l b e

f-plaiie How v e k i c i l y in j - - d i i e c t i o i i at i n f i n i t y lp -{- = c o m p l e x p o t e n t i a l c o o r d i n a i e a x e s in phy.sical p l a n e

Zih

ll,