Granville P.a Modified Froude Me.dec.1974.JSR

7/22/2019 Granville P.a Modified Froude Me.dec.1974.JSR

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Vol. 18, N o. 4, Dec. 1974, pp . 215-223

Journal o fShip R esearch

l S . G r a n v i l l e 1T h e F r o u d e m e t h o d for ext rapolat ing the resistance o f t o w e d s h i p m o d e l s t o f u l l - s c a l e c o n d i t i o n shas b e e n m o d i f i e d w i t h r e s p e c t t o the prediction of v iscous r e s i s t a n c e f ro m a n e q u i v a l e n t f la t -p l a te res i s t an ce . A fo r m f a c t o r for extrapolat ion is t o b e d e t e r m i n e d f r o m a b o u n d a r y - l a y e r ca l c u -l a t io n o f a u n i q u e l y d e f i n e d e q u i v a l e n t b o d y o f r e v o l u ti o n . E x a m p l e s o f f o r m f a c t o r s c a l c u l a t e dfo r th e Lue), a .shfon and a h u l l w i th a b lo ck c oe f f i c ie n t o f 0 .8 a re s how n to be qui te sat isfactory .

I n tro d u c t i o nTIlE VALIDITY of the Froude method for dete rmin ing the re-

he present time due in part to more detailed measurements ofcomponents. Wavemak ing resistance and vis-en shown [115 to in ter act with each o thermbe r but also of Reynolds number and, conversely,

er but also of Froude number. This contradicts the sim-

The validi ty of the Frou de method [3] has, however, hade measure of uncer tai nty from its very beginni ng in theh century even though it led to the present era oftanks. Despite its defects, the Froude method pro-

and turbines. Model measurements could be extrapolatedconditi ons in a reasonable fashion. This was dueliam Froude, who realized around 1868 that ships travelingnly affected b y two differentresistance: that arising from wavemaking as the hull

between the moving hull surface and the water. Froudee by towing the model at aed which was reduced to give a wave patter n similar to th at atle; in modern parlance, the full-scale Froude num ber.

~Naval Ship Research and Development Center, Bethesda,Numbers in brackets designate References at end of paper.Manuscript received at SNAME Headquarters June 21, 1973.

eosity by scaling in terms of an equi valent plank resistance of thesame area, length, and speed. Froude determined such plankresistance from towing tests. Since the basic tenets of theFroude analysis are physically correct, they have stood the testof time. However, changes continue to occur in procedures forimproving the calculation of the change in viscous resistancefrom model to full-scale conditions.The adve nt of dimensional analysis in engineering analysis atthe beginni ng of the twent ieth cen tury led to the present state-ment of the Frou de method in terms of dimensionless ratios suchas resistance coefficients, Froude numbers, and Reynolds num-bers. The idea of the equival ent plank resistance of Froudewas transformed to th at of the equi valent flat-plate frictiona lresistance coefficient as a function of a length Reynolds number.Power-law relations for flat-plate resistance developed frompower-law similarity laws for turbulent boundary layers provedinadequate at first to cover the required range of Reynoldsnumbe r from model to full scale. Then improved similari tylaws by von KrmAn and by Prandtl for turbulent boundarylayers led to the logarithmic Sehoenherr formula for flat plates,which did prove adequate to cover the necessary range ofReynolds numbers. Viscous resistance could then be interpretedas the sum of the equivalent flat-plate resistance and a constantform resistance. In practice, const ant form resistance wasusually added to wavemaking resistance, and the sum wastermed residual resistance.The construction of smoother ship hulls, part icular ly by weld-ing instead of riveting, led to a new difficulty for the Froudemethod. For some ships, correlation of resistance from full-scaletrials and model tests led to negative roughness allowances,which, however, are physically impossible. As a way out,Hughes [4] proposed that the form resistance be considered notas a constant but ra ther as proportional to the flatplate resistance.This leads to a lower predicted full-scale resistance and usuallyeliminates the negative roughness allowance. The ratio of formresistance to flat-plate resistance is termed the form factor and isdetermined experimentally from the t otal resistance of the modelat low speeds, where the wavemaki ng resistance is negligible.

1974 215


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U n f o r t u n a t e l y , i t i s d if f ic u l t t o m e a s u r e a c c u r a t e l y t h e l o w - s p e e dr e s i s ta n c e o f m o d e l s s i n ce t h e d y n a m o m e t e r r e a d i n g s a r e a t t h e i rl o w e s t p r e c i s io n a n d t h e t u r b u l e n t s t i m u l a t i o n u s e d t o e l im i n a t el a m i n a r f l ow m a y n o t b e a d e q u a t e a t l o w s p e e d s , e s p e c i a l l y f o rs m a l l m o d e l s . T o h e l p r e m e d y t h i s, P r o h a s k a [ 5] h a s p r o p o s e da n e x t r a p o l a ti o n p r o c ed u r e b a s e d o n a n a s s u m e d v a r i a t i o n o fw a v e m a k i n g r e s is t an c e w i t h F r o u d e n u m b e r .H u g h e s [ 4] a l s o i n t r o d u c e d a n u n f o r t u n a t e c o m p l i c a t i o n t oh i s m e t h o d b y p r o p o s i n g l o w e r v a l u e s fo r f i a t - p l a t e r e s i s t a n c e a ti n f i n i t e a s p e c t r a t i o d e d u c e d f r o m t e s t s w i t h t o w e d p l a t e s o ff i n i te a s p e c t r a t i o . T h e v a l u e s a r e b a s e d o n a n a s s u m e d e x t r a p o -l a t i o n p r o c e d u r e w h o s e v a l i d i t y h a s r e c e i v e d u n f a v o r a b l e c r i t i -c i sm [2] .A m o r e r e c e n t d i f f i c u l t y [ 6 ] w i t h t h e F r o u d e m e t h o d h a so c c u r r e d w i t h t h e d e v e l o p m e n t of v e r y f u l l s h ip s . H e r e , t h eH u g h e s f o r m - f a c t o r p r o c e d u r e g i v e s a n a p p a r e n t n e g a t i v ew a v e m a k i n g r e s i s t a n c e .F u r t h e r m o r e , r e c e n t d i r e c t m e a s u r e m e n t s o f v i sc o u s r e s i s t a n c eb y w a k e s u r v e y s a n d o f w a v e m a k i n g r e s i s t a n c e f r o m w a v ep a t t e r n s s h o w t h e c o m p l e x i n t e r a c t i o n b e t w e e n t h e t w o r e s is -t a n c e s [ 1 ]. B y c o m p a r i s o n , p r e s e n t - d a y p r o c e d u r e s b a s e d o ns i m p l y m e a s u r i n g th e t o t a l re s i s ta n c e w i t h a d y n a m o m e t e r a n de x t r a p o l a t i n g b y t h e p r e s e n t F r o u d e m e t h o d s e e m a l m o s t a r ch a i c .H o w e v e r , e v e n a d o p t i n g t h e m u c h m o r e d i f f ic u l t a n d e x p e n s i v ed i r e c t m e a s u r e m e n t s o f t h e r e s i s t a n c e c o m p o n e n t s o f t h e m o d e l

b y w a k e s u r v e y s a n d w a v e c u t s w o u l d s t i l l l e a v e t h e p r o b l e m o fe x t r a p o l a t i o n t o f u l l sc a l e.I n a n e f f o rt t o s a t i s f y t h e l a t e s t f i n d in g s o n t h e n a t u r e o f s h i pr e s i s ta n c e , a n d s t i ll k e e p a n d e v e n i m p r o v e t h e s i m p l e F r o u d em e t h o d o f m e r e l y m e a s u r i n g t h e t o t a l r e s i s t a n c e o f t h e m o d e la t t he f u l l - s c a l e F r o ud e nu m b e r , t h e f o l l ow i ng p roc e ~ t ur e i s nowp r o p o s e d :1 . A n e w s e p a r a t i o n o f r e s i s ta n c e c o m p o n e n t s is m a d e t oa c c o m m o d a t e c u r r e n t f i n d i n g s o n t h e i n t e r a c t i o n s b e t w e e nw a v e m a k i n g a n d v i s co u s r e s is t a n c es . F r o u d e - n u m b e r - d e p e n d e n tc o m p o n e n t s a r e s e p a r a t e d f r o m v i s c o u s r e s i s t a n c e , l e a v i n g ar e s i d u e d e p e n d e n t o n l y o n R e y n o l d s n u m b e r . T h e e f fe c t o fR e y n o l d s n u m b e r f r o m m o d e l to f u ll s c a l e i s t h e n a c c o u n t e d f o rb y t h e c h a n g e i n v i sc o u s r e s i s ta n c e a t z e r o F r o u d e n u m b e r .2 . T h e v i s c o u s r e s i s ta n c e o f t h e s h i p a t z e r o F r o u d e n u m b e ri s t o b e a p p r o x i m a t e d b y t h e v i s c o u s r e s i s ta n c e o f a n e q u i v a l e n tb o d y o f r e v o l u t i o n w h i c h is to b e o b t a i n e d b y a b o u n d a r y - l a y e rc a l c u l a ti o n . C u r r e n t m e t h o d s of t h r e e - d i m e n s i o n a l b o u n d a r y -l a y e r c a l c u l a t i o n s f o r a s h i p h u l l a r e n o t a s w e l l d e v e l o p e d a s t h es i m p l e r a x i s y m m e t r i e m e t h o d s f o r a b o d y o f r e v o l u ti o n .3 . T h e r a d i i o f t h e e q u i v a l e n t b o d y o f r e v o l u t i o n a r e d e -f i n e d b y a d u a l c o n s i d e r a t io n t o g e t a s c l o se a s p o s s i b le to b o t ht h e p r e s s u r e d i s t r ib u t i o n a n d t h e s u r fa c e a r e a o f th e a c t u a l s h i p .4 . T h e v i s c o u s r e s i s ta n c e o f t h e s h i p a t z e r o F r o u d e n u m b e ri s a l s o t o be de f i ne d a s a f o r m f a c t o r r e l a t e d t o t he f i a t - p l a t er e s i s t a n c e b u t w i t h t h e a d d i t i o n o f a c o n s t a n t t o a l l o w f o r t h ea d d e d p r e s s u r e r e s i s t a n c e f r o m f l o w s e p a r a t i o n a t t h e s t e r n a n df r o m t h e b i lg e s . T h i s is a n i m p r o v e m e n t t o t h e b a s i c H u g h e s

m e t h o d a n d w a s fi rs t su g g e s te d m a n y y e a r s a g o b y L a n d w e b e r[ 7 ] i n a no t he r c on t e x t .5 . T h e n e w f o r m fa c t o r is t o b e d e t e r m i n e d f r o m t h e c a l -c u l a t e d v i s c o u s r e s i s ta n c e o f t h e e q u i v a l e n t b o d y o f r e v o l u t i o nd e f i n e d b y d u a l c o n s i d e r a t i o n s . A s i m p l e p o w e r - la w a n a l y s i si s u s e d i n t h e b o u n d a r y - l a y e r c a l c u l a t i o n s i n c e i t p r o v i d e s af o r m f a c t o r i n d e p e n d e n t o f R e y n o l d s n u m b e r .6 . T h e n e w f o r m f a c t o r t h e n d e f i n e s a l in e o f e x t r a p o l a t i o nw h i c h i s u n i q u e f o r t h e p a r t i c u l a r h u l l b ei n g c o n s i d er e d . R e -s i d u a l r e s i st a n c e a s a f u n c t i o n o f F r o u d e n u m b e r i s d e t e r m i n e df r o m t h i s l i n e o f e x t r a p o l a t i o n t o p r e d i c t f u l l -s c a l e t o t a l r e s is t a n c e .A s e x a m p l e s , f o r m f a c t o r s w e r e c a l c u l a t e d f o r t h e L u c y A s h t o na nd a hu l l o f b l oc k c oe ff i c ie n t 0 .8 . Th e r e s u l t s a r e m o s t r e a s on -a b l e .

Proposed separation of resistance componentsT h e t o t a l r e s i s t a n c e o f s u r f a c e v e s s el s t o s t e a d y f o r w a r dm o t i o n i n a c a l m s e a d e p e n d s p r i m a r i l y o n t h e v i s c o u s c h a r a c -t e r is t ic s o f t h e w a t e r m e d i u m a n d t h e w a v e m a k i n g c h a r a c t e r -i s t i c s o f th e a i r - w a t e r i n t e r f a c e .F o r g e o m e t r i c a l l y s i m i l a r h u l l s, t h e t o t a l r e s i st a n c e o r d r a gD T i s ~ f u n c t i o n o f s i z e ~ s i n d i c a t e d b y l e n g t h L , t h e f o r w a r dv e l o c i t y U ~ , t h e g r a v i t a t i o n a l f i el d c o n t r o l li n g w a v e m a k i n g a si n d i c a t e d b y t h e a c c e l e r a t i o n d u e t o g r a v i t y g , a n d t h e p h y s i c a lp r o p e r t ie s of th e w a t e r m e d i u m - - d e n s i t y p a n d k i n e m a t i c

v i s c o s it y v - - o rDr = f [L, U~, g , p , v] (1 )

B y d i m e n s i o n a l a n a ly s i s , t h e s e v a r i a b l e s m a y b e g r o u p e d i n t od i m e n s i on l e s s r a t i o s : a c oe f f ic i e n t o f t o t a l re s i s t a nc e CT w h i c hi s a f u n c t i o n of F r o u d e n u m b e r F a n d R e y n o l d s n u m b e r R , w h e r eD TCT ~-- ~pUJS

U ~F - -a n d

U ~ LR ~ - YF o r h y d r o d y n a m i c c o n s i d e r a ti o n s , t h e c o e ff ic i en t of r e s i s ta n c e i sde f i ne d i n t e r m s o f w e t t e d s u r f a c e a r e a S i n s t e a d o f L 2. Th i si s pe r m i s s i b l e s i nc e S / L 2 i s a c o n s t a n t f o r g e o m e t r i c a l l y s i m i l a rs h a p e s . T h e n

C T = f [ F , R ] (2 )A c oe f f ic i e n t o f v i s c ous r e s i s t a nc e C o is de f i ne d a s

D vCv ~ 1 2~ p U ~ Sw he r e D v i s t he r e s i s t a nc e a t t r i b u t e d t o v i s c ous e f f e ct s . A l s o , ac oe f f ic i e n t o f w a ve l n a k i ng r e s i s t a nc e C w i s de f i ne d a s

DwC w ~ - { p U 2 Sw h e r e D ~ i s t h e r e s i s ta n c e a t t r i b u t e d t o w a v e m a k i n g , a n d

D r = D ~ + D w (3 )CT[F, R] = Cv[F, R] + Cw[F, R] (4 )

I n t e r a c t i o n b e t w e e n v i s c o u s a n d w a v e m a k i n g r e s i s t a n c e i sa c c o m m o d a t e d b y m a k i n g b o t h c o m p o n e n t s f u n c t i o n s o f Fa n d R .B a b a [8 ] r e c e n t l y d i s c o v e re d a n a d d i t i o n a l c o m p o n e n t o fv i s c ous r e s i s t a nc e , C ~ ,2, w h i c h r e s u l t s f r om t he b r e a k i ng o fw a v e s a t t h e b o w , e s p e c i a l ly f o r f u l l s h i p s. T h i s c o m p o n e n tg i v e s r i s e t o s y m m e t r i c v e l o c i t y d e f e c t s w h i c h a p p e a r p o r t a n ds t a r b o a r d , w e l l o u t b o a r d o f t h e c e n t e rl i n e . T h e w a v e b r e a k i n gr e s i s t a nc e , a l t hough m e a s u r e d a s a v i s c ous w a ke , i s a r e s u l t o f aw a v e b u i l d u p o n t h e b o w a n d h e n c e , a s s h o w n b y B a b a , i sp r i m a r i l y a f u t c t i o n o f F r ~ u d e n u m b e r . T h e w a v e b r e a k i n gr e s i s t a nc e C ~ .2 m a y be s p l i t o ff f r om t he v i s c ous r e s i s t a nc e t o g i ve

C ~ , , [ r , R ] = C ~ [ F , R ] - C , , ~ [ F ] ( 5 )w h e r e C~.I[F, R] i s t h e v i s c o u s r e s i s t a n c e d e t e r m i n e d f r o m am o m e n t u m s u r v e y o f t h e u s u a l w a k e n e a r t h e c e n t e r l i n e o f t h es h i p .M e a s u r e m e n t s o f C ~ ,~ s h o w u n d u l a t i o n s a b o u t s o m e m e a nl i ne [9 , 10 , 11 ]. I t i s now p r opos e d t h a t a n un du l a t i ng c o r n -

2 16 J O U R N A L O F S H I P R E S E A R C H


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4/9

U ~

- - o . -

I - -

~ " - - - - - - ~ A KE~ ~

Fig . 1 B o u n d a r y l a y e r o n a b o d y o f r e v o l u t io n

CT [F - -~ O, Rm]]CH -- CF [Rm ] 1 (14)Hence the difficulties mentioned earlier for determiningCT[F "~ O, Rm] at low speeds also apply to kn. Prohaska [5] hasa method to improve this.

3 . Com bi na t i on o f c ons t an t f o r m r e s is t anc e and f i a t -p l a t edependence. The third possible method is ascribed to Land-weber [7]. HereQ0 [R ] = (1 + k,)C~[R] + k~ (15)

where kl and k~ are constants independent of R or F. However,both kl and k2 are func tions of geometric shape. Also, ob-viously k1 ~ kg .The factor of constant form resistance k2 takes care of pressuredrag due to flow separation such as occurs on blunt sterns.More recently it has been discovered that there is an additionaldrag caused by vortices generated by flow over the bilges [13, 14].Probably most ships have this added separation drag, whichmay now be included in the constant form resistance factor k2.The principal difficulty with the Landweber proposition is itslack of procedure for im plementa tion either experimental ly ortheoretically. One further simplification results for the Froudeextrapolation method which requires only the change in C,.0.The k2 factor then drops out sinceC,,0[R~I - C~,0[R~]= (1 + k~)(CF[Rm] - CF[R~]) (16)There still remains the problem of evaluating/:~. The calcu-lation of the viscous resistance of an equivalent body of revolu~tio n is now proposed as a method for evaluatin g k~ for indiv idualhulls.

Eq u i v a l e n t b o d y o f r e v o l u t io nThe viscous resistance at zero Froude number Cv,0 is to beevaluated from the viscous resistance of an equivalent body ofrevolution. In principle, Cv.o should be obtained by a three-dimensional boundary-l ayer calculation. Since such methodsare still being developed and are of great intr icacy and question-able quality, it is believed that the much simpler axisymmetrie

boundary- layer calculation on a body of revoluti on is sufficientlyaccurate for extrapolation purposes.In general, a bounda ry layer develops under the influence of thepressure field acting on its surface. The equ ival ent body ofrevolution is now defined to accommodate both the pressurefield and the actual surface area of the hull. The pressure fieldis to be obtained from an equivalent body of revolution whoseradius dis tribut ion ra [/] gives a cross-sectional area d istribu tionequal to twice that of the ship A [l], orrA[l] = ~2 A[ /] (17)

where 1 s the long itud inal position. This is obviously the radiusof a half circle giving the same cross-sectional area as the hull

at the given longitud inalposition 1. The radius of the equival entbody of revolution rp required for the boundary- layer calcula-tions to accommodate the wetted surface area of the ship isobtained from the perimeter P of the hull such thatrp [ l] = PI l l (18)

71"

This is obviously the radius of a half circle having the sameperimeter as the hull at the given longitudin al position l.The cross-sectional area A and perime ter P of a ship are easilyobtained from the hull geometry as given in a body plan or atable of offsets.F o r m f a c t o r f r o m b o u n d a r y - l a y e r c a l cu l a t io n o f an o n s e p a r a t i n g e q u i v a l e n t b o d y o f r e v o l u ti o n

The new form factor kl is to be evaluated from a calculationof the viscous resistance of an equivalent body of revolution;see Fig. 1. Since separation drag has been included in the ot herform factor k2, the viscous resistance of the equivalent body ofrevolution is to be calculated without separation. Granville[15] has designed a simple method for streamlined bodies ofrevolution, that is, without separation, wherein a constantboundary-l ayer shape parameter is used to integrate a differentialequation. Since shape parameters increase greatly at separation,the use of a constant shape parameter means that kl includes nocontribut ions from separation drag. The power-law analysisused by Granville also gives a form factor independent ofReynolds number, which is what is desired here.Now the met hod described by Granvil le is to be further simpli-fied by making an appropriate allowance for the skirt frictionof the thick bounda ry layer on the tail. If the simple thi nbound ary- laye r precedm'e for skin frict ion is used for the wholebody including the tail, the resulting error can only be smallsince the t ail cont ributes a small surface area and consequentlya small skin friction to the tot al drag. The simplification ismade as follows:For bodies of revolution, a momentum area ft is defined for athin boundary layer:

= rO (19)where r is the cross-sectional radius of the body and 0 is the usualtwo-dimensional moment um thickness. Now a flat-plate skinfriction formula is used for the t urbu lent skin friction

rw

whererw = wall shearing stressU = velocity outside the boundar y layer~', m = const ants

: )18 J OURN AL OF S HIP RE S E ARCH


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lds numbe r up to full scale. Constant values of ~" and m. In terms of momen tum area for the thin boundary layer:

rw ~"p U 2 (_~)~ (21)

tum equation [15]boundary -layer development from nose to tail:

d ~ ~ d U r ~d-~ + (h + 2) U ds r (22)p U 2

the meridian coordinate starting at the nose. Usingskin friction relation, equation (21), and keeping h constant,h is the case for nonseparating bounda ry layers, provides anThe nondimension al solution for the whole body with aturb ulen t boufldary layer from nose to tail is

~A \L~ / , = R ~

is the angle of the meridian contour with reference todr / d l , and subscript e refers to con-The coefficient of viscous resistance for the nonseparatingof revolution (C~) is obtained [15] from(97r ~ ~ (24)( C ~ ) - S / L 2

( ~ / L ~ ) ~ is the momentum area ratio downstream atAs shown by Granville [15]:

is a constan t. A choice of q > 1 is more in accordancethe well-known formula of Young, who= 1.Now the form factor 1 + k~ is to be the ratio of the viscousv) to t hat of the equivalen t flat plate C F such that

1 + k~ = (Co~) (26)CF

U / U ~ = 1 .

+ k ~ = I f 01 ( ? ) l + m ( U ~ (l+ m ) (h + 2 )-m\ ~ / seco~ dL 1 / r p \ l + m 1-~m/ U \ 1 - hX . - -. ~ (27)\vo/~

Here re has been substituted for r to provided the equivalentradius to give the same surface area as the ship. The term( U / U ~ ) [ I / L ] is obtained from a calculation of the potentialflow about a body of revolution defined by ra which gives thesame displacement as the ship. Equa tion (27) represents twosimple quadratures which may even be evaluated b y the Simpsonrule.N t r m e r i c a l r e s u l t s

N u m e r i c a l v a l u e s f o r c o n s t a n t sNumerical values are required for the constants in the calcu-lation of the form factor in equation (27). The Sehoenherrformula is selected as the reference flat-plate resistance formulabecause it has proved reliable. However, since a general pro-cedure is being proposed here, the prac tition er is free to chooseany fl at-p late formula. Granville [16 ] derived a local skinfriction formula ~,,/pU~[Ro] from the Schoenherr formula inorder to perform boundary -layer calculations. This relationhas a logarithmic term and has to be fitted by a power law,equa tion (20), to suit the present analysis. A power-law fit[15] gives m = 0.1686 and ~" = 0.006361. Granvi lle [15] alsorecommends values of h = 1.4 and q = 7.

C a l c u l a t i o n s fo r L ucy AshtonThe former ferry L u c y A s h t o n has been the subject of intensiveinvestiga tion with respect to resistance. Different-sized modelshave been towed to investigate scale effect. The full-size vesselhas even been ex ternally propelled by aircraft engines to elimi-nat e interference from marine propellers. Like any ferry, L u c yA s h t o n is rather broa d; the ratio of half-breadth to draft is10.03:4.65. The block coefficient is 0.685, and the length-to-breadth ratio is 9.From the body plan [17], the offsets for the eq uivalen t bodyof revolution were determined from the cross-sectional areadistribu tion and from the wetted perimeter as shown in Fig. 2.The cross-sectional area equiva lent radii were used to calculatethe pressure distribution for potential flow by the well-knownDouglas program [18]. Figure 3 shows the more useful velocity

ratios U / U ~ instead of pressure distribution plotted againstrelative axial distance l / L . Since physically U / U ~ cannot goto zero at the tail as required by potential flow, an adju stme nt ismade as shown in Fig. 3. This adju stme nt was made arbitrarily ,but it can be done by repeating the calculation for the originalbody plus the displacement thickness. Using these U / U +values and the radii from the wetted perimeter r p / L in equation(27) gives kl = 0.043. Co nn et al. [17] give a value of k,~ = 0.08based on the Schoenherr line. This agrees with the propositiontha t kl ~< kw. Conn et al. [17] correlated the geosyms of L u c yA s h t o n with either the usual constant difference method or theHughes method with k n = 0.08 within the experimental scatter.Hence a value of kl = 0.043 is very satisfactory. The full-scaleship had a rough hull, and the extrapolations led to reasonableroughness allowances. Comparison of full-scale resul ts isshown in Fig. 4 using an enlarged scale.M o o r - M o d e l 6 8 4

An example of a full merchant ship is Model 684 for whichMoor [19 ] gives geometrical and resistance characteristics.Here the block coefficient is 0.08. The 26-ft draf t was investi-gated. The radii of the equivale nt body of revolution are showniu Fig. 5 and the corresponding velocity distributi on is given inFig. 6. The calculat ed kl = 0.185 while, relative to the Schoen-

In general the total resistance of a full-scale ship is given byequations (8) and (16), or1 9 74 2 19


6/9

B /2- - = 0 . 06 53L H A L F - B E A M / L E N G T H R A T I O

%x

5

, =

cc / /

/ 1

H~ - = 0.0244

rp- - P A R A M E T E RL

rA- - C R O S S - S E C T I O N AL A R E AL

D R A F T / L E N G T H R A T I O k \ \ \ \ \ ~L U C Y A S H T O N

B L O C K C O E F F I C I E N T = 0 .6 8 5

0.1 0. 2

F i g . 2

I I I I I0.3 0.4 0.5 0.6 0.7R E L A T I V E A X I A L D I S T A N C E F R O M N O S E

LE q u i v a l e n t b o d y o f r e v o l u t i o n f o r L u c y A s h f o n

I 10 , 8 0 . 9 1. 0

1.03

1.02

1.01

1.00

0.99

0.96

0.97

( L E F T - H A N D S C A L E )

0,96 I I I I I I0.1 0.2 0.3 0.4 0.5 0,6 0.7R E L A T I V E A X I A L D I S T A N C E FR O M N O S E ~(~L

F i g . 3

A D J U S T E D F O ~ - 1 ' 2V I S C O U S, E F F E C T S1

- 1 . 0S T E R N

( R I G H T - H A N D "S C A L E )

-o8 ,= ,

o- -0.6- 0 . 4

- 0 . 2

I I o0.8 0.9 1.0

L u c y A s h t o n - -o u t s i d e v e l o c i t y d i s t r i b u t i o n f o r e q u i v a l e n t b o d y o f r e v o l u t io n d e f i n e d b y c r o s s -s e c t i o n a la r e a

C T [ F . , R f l = C T[ F ~, R ~ ] - - ( 1 + k ) ( C ~ [ R m ] - C ~ [ R f l ) ( 2 S)where k is an y form factor. This form factor may be also con-sider~d an extrapola tion factor. For the three methods givenhere

k = 0, cons tant form factorZ20

k = k~, Hughes methodk = kl, proposed method of this reportThe extrapolation of the model results of Moor-Model 684for the three methods is shown in Fig. 7. The linearized resis-tance diagram [2] is used here where A is substit uted for Froudenumb er F as the independent variable. As shown by Granville

J O U R N A L O F S H I P R E S E A R C H


7/9

2.9

%x

2.8

2.7

2 . 6

Z 5

2.4 ,o~

2.3

2"22.17.9

F i g . 4

L U C Y A S H T O NFULL- SCALE 190 .5 - FOO T SH IP

A = 1.213 x 109 FULL- SCALE TESTS

A L U M I N U M P A I N T W I T H F A I R E D S E AM S( E Q U I V A L E N T S A N D R O U G H N E S S ~ 0 .0 0 23 I N C H )

/ / /

I I I I8.1 8.2 8.3 8.4log R

C o m p a r i s o n o f f u l l - s c a l e p r e d i c t i o n s f o r L uc y Ashfon

I I8.0 8.5 8.6

i. ~ i4 Ia: I

Iw 3 - ! II

ltlI I I I

B P E R I M E T E Rrp ~L r ~A CR OS S- SECTION AL AREA

B / 2 / L = 0 .0 6 72 H A L F -B E A M / LE N G T H R A T I O " ~ " t

- - = ,H 0 0 6 3 6 D R A F T / L E N G T H R A T I O

I

BLOCK COEFFIC IEN T = 0 . S

1 0 0.1 I I I I I I

M O O R - M O D E L 6 8 4

o , 8.2 0.3 0.4 0.5 0.6 0.7R E L A T I V E A X I A L D I S T AN C E F R O M N O S E LFig. 5 Equivalentbody of revolution or Moor-Model684

\t

IIIIII11

0.9

i \t1.o

1 9 7 4 2 2 1


8/9

1.12

1 1 o - / i1.08 - - !

= 1 = 8 I1 . 0 6 1

1.04

1.02

1.00

AD JUSTED FORV I S C O U S E F FE C T S - -

.+ . . . . . .

S T E R N(R IGHT- HAN D SCALE}

% ,{ , -

I I I I I0.1 0.2 0.3 0.4 0.5 0.6 ,1~ 0.7 0.8 0.9R E L A T I V E A X I A L D I S T A NC E F R O M N O S E - -L

- 1 . 2

- 1 . 0

- 0 . 8 m

_o.e:

- 0 . 4

--0.2

01.0

F ig . 6 M o o r - M o d e l 6 8 4 - - o u t s id e v e l o c i t y d is t r i b u t i o n f o r e q u i v a l e n t b o d y o f r e v o l u t i o n d e f i n e d b y c ro s s -s e c t i o n a l a r e a

6 16 - FOO T MO D EL M O O R - M O D E L 6 84

4 ~ ~ 400-FO O T S H IP+ 7 + ~ _ + - ' 2 7 - - - . . 2 u G , ~ s - A : 3 0 9 x lO '. . . . o ~ - ~ ; x : ~ ~ o ~ F,~~ _ " ~ U To t ~ R A~"~ ~ C T o R ,+ C O N S T A N T F O R M - ~ . I

7 , , i- - c 7 - - - _ _ . _ . _ _ _ "

1 - -

3.5C F x 103

3,4 3.3 3.2 3.1 3,0 2.9 2.8 2.7 2.6 2,5 2.4 2,3 2,2 2.1 2,0 1.9 1.8 1.7 1,6 1.5J l h h ~I I In n l [ I n l l a I n i t n l t l d n I I t h n t l t l n I I4 5 6 7 8 9 10 2 3 4 5 6 7 8 910 8 2 3 4 5 6 8 109R E Y N O L D S N U M B E R R

F ig . 7 L i n e a r iz e d r e s i s t a n c e d i a g r a m f o r M o o r - M o d e l 5 84

[ 2] , a n o n d i m e n s i o n a l l e n g t h A i s re q u i r e d b y t h e n o n d i m e n s i o n a ls y s t e m , t h a t i sCT = ) [ A , R ] ( 2 9 )

w h e r eA ~--- L3/2g1/2 R

v FF o r t h e l i n e a ri z e d r e si s t an c e d i a g r a m

C T = f [A, CF] (30)222

T h e p r e d i c t e d f u l l -s c a l e r e s i st a n c e s o f t h e M o o r - M o d e l 6 8 4 a r es h o w n m o r e c l e a r l y i n F ig . 8 . I t i s s e e n t h a t t h e p r o p o s e dm e t h o d g i v e s re s u l t s t h a t f a l l i n b e t w e e n t h o s e o f t h e . m e t h o d s i nu s e . T h i s s e e m s d e s i ra b l e t o i n v e s t i g a t o r s w h o c o n s i d e r t h er e s u l t s o f t h e c o n s t a n t f a c t o r m e t h o d t o o h i g h a n d t h o s e o f t h eH u g h e s m e t h o d t o o lo w .C o n c l u s i o n

T h e p r o p o s e d m o d i f ic a t i o n o f t h e F r o u d e m e t h o d f o r e x t r a p -d a t i n g t h e r e s i s t a n c e o f m o d e l s t o f u l l- s c a le s h i p s i s w e l l -J O U R N A L O F S H I P R E S E A R C H


9/9

3 .6

. 8 F u l l- s c a le r e s i s ta n c e f o r M o o r - M o d e l 6 84c t e d b y t h r e e m e t h o d s

3.4

3.2

3. 0%x

2.8

2.6

2.4

2. 2

/. " I L l

A : 3 s x . , / , ' , 1- . , / , ' 1

i j IO S ED E Q U I V A L E N T BoDY OF ~- , . . . . . / JP R O P . . . . . O,185"1 ~ '~ JREVO LUTIO~ ~..~ .. . . . .~ - - - - . . .. .. . s - -

2 . o I I I [ ] I I I I l ] ]8 .6 2 8 . 6 4 8 . 6 6 8 . 6 8 8 .7 0 8 .7 2 8 .7 4 8 .7 6 8 .7 8 8 .8 0 8 .8 2 8 .8 4 8 .8 6 8 .8 8 9 . 0 0LO G R

in present-day fluid mechanics. The boundary- layeres the physical justification for Froude' s originalng components. Viscous effects tha t give tangenti al stressesthe viscous drag are due mostly to the flow in the boun daryto the hull. Pressure effects from waves formed out-

e the boun dary layer at the free surface produce the wave-drag.The boundary-layer concept is utilized further in this reportline of extrapolation for residual resistance. Each

vidual geometry.The numerical results from the two examples treated here areLike all new methods, the full benefit a n dty can come only from fur ther usage.A c k n o w l e d g m e n t

The work described in th is report was funded by theromechani es Research Program, SR-023-0101, Work Uni t

R e f e r e n c e s1 Paffett, J. A. H., "The Components of Resistance," Report ofance Committee, 13th Interna tional Towing Tank Conference,2 Granville P. S. "The Viscous Resistance of Surface Vesselshe Skin Frictaon of Fla t Plates, T r a n s . S N A M E , Vol. 64, 1956.3 Todd, F. H., "Resistance and Propulsion," Chapter VII ofles of Nava l Architecture, J. P. Comstock ed., S N A M E 1967.4 Hughes, G., "Friction and Form Resistance in Turbulen trmula tion for Use in Model and ShipTr ans . I N A, Vol. 96, 1954.5 Proh~ska, C. W., " A Simple Method for the Evaluat ion of the

Form Factor and the Low Speed Wave Resistance," Proceedings of11th Interna tional Towing Tank Conference, Tokyo, Japan, 1966.6 Couch, R. B. and Moss, J. L., "Application of Large Pro-truding Bulbs to Ships of High Block Coefficient," T ra n s. S N A M E ,Vol. 74, 1966.7 Landweber, L., Discussion of "Skin Friction Resistance andthe Effects of Surface Roughness" by F. H. Todd, T ra n s. S N A M E ,Vol. 59, 1951, p. 360.8 Baba, E., " A New Component of Viscous Resistance ofShips," Jou rna l of Society of Naval A rchitects. of Jap an, Vol. 125,June 1969.9 Townsin, R. L., "Viscous Drag from Wake Survey Measure-ments in the Wak6 of a Luc y As h t on Model," Tr ans . RI N A, Vol. 110,1968.10 Corm,J. F. C. and Ferguson, A. M., "Results Obtained with aSeries of Geometrically Similar Models," Tr ans . RI N A, Vol. 110,1968.11 Amfilokhiev, W. B. and Conn, J. F. C., "Note on the Inter -action between the Viscous and Wavemaking Component Resis-tances," T r a n s . R I N A , Vol. 113, 1971.12 Emerson, A., "The Calculations of Ship Resistance: AnApplication of Guillo tin's Method," T r a ns . R I N A , Vol. 109, 1967.13 Tatinelaux, J., "Experimental Investigation of the DragInduced by Bilge Vortices," Schiffstechnik, Vol. 17, No. 87, May1970, p. 37.

]4 Sasajima, H. et al., "O n Stern Flow Field of Full Ship Formsand Induced Drag Due to Bilge Vortices," Journal o f Socie ty o fNaval Architects of Japan , Vol. 128, Dec. 1970, p. 43.15 Granville, P. S., "The Calculation of the Viscous Drag ofBodies of Revolution," David Taylor Model Basin Report 849,July 1953.16 Granville, P. S., " A Method for the Calculation of the Tur-bulent Boundary Layer in a Pressure Gradient," David TaylorModel Basin Report 752, May 1951.17 Conn, J. F. C. et al., "B.S.R.A. Resistance Experiments onthe Lucy Ashton: Part II--The Ship Model Correlation for theNaked Hull Condition," Trans . INA, Vol. 95, 1953.18 Smith, A, M. O. and Pierce, J., "Exact Solution of the Neu-mann Problem, Calculation of Non-Circulatory Plane and AxiallySymmetric Flows about or within Arbitrary Boundaries," DouglasAircraft Company Report ES-26988, April 1958.19 Moor, D. I . , " ' T h e . ~ o f Some 0.80 CB Forms," T r a n s. R I N A ,Vol. 102, 1960.E M B E R 1 9 74 2 23

Granville P.a Modified Froude Me.dec.1974.JSR

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