Determination of the viscous drag of a ship model

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VISCOUS DRAG OF A SHIP MODEL

b y

K. T . S. T z o u a n d L . L a n d w e b e r

I

p C ^ i ^ + + w^2) =

+1

PU2 ( 3 )

The i n c r e a s e i n f l u x o f f l u i d across t h e downstream s e c t i o n CD due t o t h e replacement o f u by u^^ may be a t t r i b u t e d t o a volume d i s t r i -b u t i o n o f source d e n s i t y q which, -by Gauss's flux-theorem, s a t i s f i e s t h e r e l a t i o n

lur ƒ q d ^ • /u " (h)

the volume i n t e g r a l extending over t h e volume o f t h e wake from t h e body t o the c o n t r o l s e c t i o n «. A p p l y i n g t h e L a g a l l y theorem, we o b t a i n f o r t h e t o t a l f o r c e on t h e sources

D^ = - W ƒ q u^ d ¥• • (5)

The use o f the exact form o f Gauss's f l u x theorem i n (k) and u^ i n s t e a d o f U i n ( 5 ) , which makes t h e a p p l i c a t i o n o f t h e L a g a l l y theorem a l s o exact, was proposed by Landweber [9] i n response t o c r i t i c i s m s o f previous

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analyses by Sharm^a [ l O ] , Hence, a p p l y i n g t h e mean-value theorem and ( U ) , we f i n d

= - P

!^

- (6)

where u^ i s a mean value o f u^ over t h e r e g i o n bounded by the c o n t r o l surfaces,.

For t h e t o t a l f o r c e on t h e body together w i t h these sources on the upstream s i d e o f s e c t i o n CD, we o b t a i n by a p p l y i n g t h e momentum theorem,

°w ^ °s = /s ^^o * P"^^^ ^o • k^H P^^^'^.S. (T) o

E l i m i n a t i o n o f D and D from ( l ) , ( 2 ) , (6) and (T) y i e l d s

^v - P * P l ^ ^ ^ " - - dS (8) S u b s t i t u t i n g f o r p^ by equation ( 3 ) , we o b t a i n ^ = "fij^o ^. * ^ ' ^ ^ * u 20^) -v^2 _ as (9) o r , n e g l e c t i n g v^^ + ^v = '^fJ^o - ^ - u ) ( u ^ + u - 2 ^ ) ] d S (10) Here, H and H are the undisturbed t o t a l head and t h e t o t a l head i n t h e

o

wake, r e s p e c t i v e l y s an.d y = Pg i s t h e s p e c i f i c weight o f water.

Expression (10) i s q u i t e s i m i l a r t o t h a t given i n [k] and used by Wu [ 5 ] ,

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These are seen t o d i f f e r o n l y i n t h a t u^ i s , r e p l a c e d by U. Since ^ cannot be r e a d i l y e s t i m a t e d , i t i s convenient t o r e p l a c e i t by u^ i n ( 1 0 ) , which then becomes

°v = y / J H ^ - H - ^ u ^ . - u)2]dS (12)

This i s t h e form used t o c a l c u l a t e the v i s c o u s dreig from the present s e t of measurements.

EQUinment

Towing tank and s h i p model. The Iowa I n s t i t u t e o f ^ ^ d r a u l i c Re-search towing tank i s 10 f e e t wide, 10 f e e t deep and 300 f e e t l o n g . I t i s equipped w i t h a c a b l e - d r i v e n c a r r i a g e and can be operated a t constant

speeds up t o 2k f e e t per second. The d r i v e c o n s i s t s o f a 6.00-rpm, 3-phas.e, 25-HP i n d u c t i o n motor, coupled t o a s e r v o - c o n t r o l l e d e l e c t r i c d r i v e - A d i g i t a l tachometer i n d i c a t e s t h e c a r r i a g e speed t o an accuracy o f '* 0.01 f o o t pér second.

The s h i p model was attached beneath the c a r r i a g e , w h i l e e i t h e r the p i t o t rake o r the probe was set on a t r a i l e r 6 f e e t behind t h e s t e m o f the s h i p model. The c a r r i a g e and the t r a i l e r r i d e ón round s t a i n l e s s -s t e e l r a i l -s 2.25 Inche-s i n diameter, -set on both -s i d e -s o f t h e channel.

A 10-foot S e r i e s 60 s h i p model o f 0.60-block c o e f f i c i e n t was used f p r t h i s experiment. A row o f p i n s o f 1/8-inch diaineter and 1/10-ineh height f i t t e d at 1/2-inch spacing 1 f o o t behind the bow p r o f i l e was oa-ployed f o r t u r b u l e n t s t i m u l a t i o n . The s h i p model has the f o l l o w i n g chartic-t e r l s chartic-t i c s

Waterline l e n g t h 10.17 f t Wetted s u r f a c e area 17.6k f t ^ Displacement 2T5 l b

P i t o t rake and manometer. The p i t o t rake c o n s i s t s o f an a r r a y of p i t o t tubes extending from a wooden board o f s t r e a m l i n e d form, 1/2-foot wide, 1/iO-foot t h i c k and k-llk f e e t l o n g . I t c o n t a i n s 17 t o t a l - h e a d tubes and l 8 piezometric-head tubes mounted a l t e r n a t e l y at a spacing o f

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1/10 f o o t . The t i p s o f the t o t a l - h e a d tubes and the s i d e holes o f the piezometric-head tubes are a l i g n e d t o be i n the same transvérse s e c t i o n . The rake can be set at v a r i o u s e l e v a t i o n s below t h e water s u r f a c e , t o t a k e measurements over the wake area i n t r a n s v e r s e planes behind the model. A l s o towed by the t r a i l e r are f o u r a d d i t i o n a l t o t a l - h e a d t u b e s , two near each w a l l , mounted a t a l e v e l o f 2-1/2 f e e t below the undisturbed water s u r f a c e , t o measure t h e t o t a l head outside the wake.

P l a s t i c t u b i n g connects the 35 tubes oh the rake and the h addi-t i o n a l addi-t o addi-t a l - h e a d addi-tubes addi-t o 39 g l a s s addi-tubes on addi-t h e a i r - w a addi-t e r d i f f e r e n addi-t i a l manometer board. The manometers are mounted on a 2-foot by 3-foot board w i t h a 1-1/2-inch diaméter brass pipe at the i ^ p e r end, and 39 manifolds t o which g l a s s tubes are a t t a c h e d at t h e i r t o p s . Two h o l l o w c y l i n d e r s , 6-inches i n diameter and 11-inches l o n g , are connected t o the brass pipe as an a i r chamber. The a i r chamber i s i n s u l a t e d t o a v o i d temperature changes during the t e s t . A r e v e r s i b l e vacuum d e v i c e was used t o suck water up i n t o the g l a s s tubes t o a d e s i r e d h e i g h t above t h e tank.

A l*0-watt f l u o r e s c e n t l i g h t and a 35-nmi camera were set at a d i s t a n c e o f k f e e t from t h e manometer f o r i l l u m i n a t i n g and photographing the m e n i s c i i n the manometer tubes a g a i n s t a scaJLe on the board..

S u r f a c e - p r o f i l e probe and t r a v e r s i n g mechanism. F o r measuring the height o f the f r e e s u r f a c e above the wake, a commercieil "ceroc" w i r e w i t h a ceramic, h e a v y - t e f l o n c o a t i n g was s e l e c t e d . I t has s u f f i c i e n t

s e n s i t i v i t y , low s u r f a c e - t e n s i o n e f f e c t s , a l i n e a r output s i g n a l and prac-t i c a l l y insprac-tanprac-taneous response [11]. Since prac-the i n s u l a prac-t i o n o f prac-the prac-t e f l o n w i r e W i l l s l o w l y absorb water, t h i s causes a g r a d u a l d r i f t o f the output

s i g n a l . To e l i m i n a t e the d r i f t , a 2lt-hour p r e l i m i n a r y immersion t i m e , suggested by Kobus [ 1 2 ] , was used.

The t r a v e r s i n g mechanism which t r a n s p o r t s the probe across the tank during the t e s t , d e s c r i b e d by Kobus [ 1 2 ] , i s mounted on t h e t r a i l e r at a d i s t a n c e o f 6 f e e t behind t h e s t e m o f the s h i p , A 1/50-HP motor s u p p l i e s the power f o r a s m a l l c a r r i a g e which t r a n s p o r t s the probe from one side, o f the tank t o the o t h e r , at a constant speed o f 0.5 f p s .

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The signals from the probe àré • t r a n s m i t t e d t o a Sanborn r e c o r d -e r . A mark-er, d-esign-ed f o r r -e c o r d i n g -ev-ery foot o f t r a v -e l , 'giv-eis à con-tinuous r e c o r d o f probe p o s i t i o n versus surface e l e v a t i o n . A v e r t i c a l v e r n i e r s c a l e i s a v a i l a b l e f o r s t a t i c c a l i b r a t i o n o f t h e probe.

Procedure . ^

Totalhead arid piezometrichead measurement. The p i e z o m e t r i c -head tubes were c a l i b r a t e d before t h e t e s t by towing t h e p i t o t rake under the water sxirface without t h e s h i p inodel. A c c o r d i n g t o Wu's i n v e s t i g a -t i o n [ 5 ] 9 -t h e c a l i b r a -t i o n o f -the -t o -t a l - h e a d -tubes and a c o r r e l a -t i o n f o r turbulence i n the wake were not necessary. ,

Measurements were taken a t t h e t r a n s v e r s e plane 6 f e e t behind the s t e r n o f the model at speeds o f 3 . 0 , 3 . 5 i ^ . 0 , U.5, 5 ^ 0 , 5 - 5 and 6 . 0 f e e t p e r second. F o r each speed, t h e p i t o t rake was s e t a t t h e depths o f

0 . 0 2 5 , 0 . 0 5 0 , Ö . O 7 5 , 0 . 1 0 , 0 . 1 5 , . . . » 0 . 5 0 , 0 . 6 0 , . . . , 0 . 9 and 1 , 0 f o o t

below the undisturbed s u r f a c e .

Photographs were taken inîmediately before each r u n and i ^ e n t h e reading on t h e d i g i t a l tachometer i n d i c a t e d a n e a r l y constant c a r r i a g e v e l o c i t y which d e v i a t e d by l e s s than * O.Oi foot per second from t h e

des-i r e d speed. E des-i g h t mdes-inutes was s e l e c t e d as an adequate des-i n t e r v a l between runs.

Transverse s u r f a c e p r o f i l e measurement. The probe was c a l i

-b r a t e d s t a t i c a l l y -by moving t h e capacitance w i r e along t h e v e r t i c a l v e r n i e r s c a l e . A dynamic c a l i b r a t i o n was a l s o p e r f o m e d by running t h e probe a t v a r i o u s speeds without t h e s h i p model.

A l l measurements were taken a f t e r t h e c a r r i a g e had a t t a i n e d t h e d e s i r e d speed. At each speed, two runs were made. A f t e r c o r r e c t i n g f o r t h e n u l l r e a d i n g from t h e undisturbed water and f o r t h e d e f l e c t i o n a s s o c i

-ated w i t h the speed o f t r a v e l , the readings o f the Sanborn r e c o r d e r were converted t o s u r f a c e e l e v a t i o n s by means o f t h e c a l i b r a t i o n curve. The r e s u l t s a r e shown i n Table 1 and F i g . 2 .

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E f f e c t o f temperature change on the manometer r e a d i n g . I ^ t us d e f i n e V, P, and T as t h e volume, absolute pressure,and a b s o l u t e temper-ature i n the a i r - s u c t i o n chamber o f the manometer systan. Let the sub-s c r i p t o r e f e r t o the i n i t i a l c o n d i t i o n , and R t o the gasub-s consub-staht. Then, we have the p e r f e c t - ^ a s law

f

= ^ = H (13)

o But

P = P^ - Yh_a {lh)

V = - ah (15 )

where P denotes t h e absolute pressure o f the atmosphere, a the t o t a l c r o s s - s e c t i o n a l area o f the g l a s s t u b e s , h the height o f the water column i n a g l a s s tube.

S u b s t i t u t i n g equation (lk) and (15) i n t o ( 1 3 ) , we have

(P„ - Yh)(V^ - ah) = RT ( l 6 ) a o

l E f f e r e n t i a t i n g equation ( l 6 ) gives

dh _ R

dT " [2Yah - (yV^ + aP.)] ^-^'^

Since the q u a n t i t y ayah i s s m a l l i n cpmparison w i t h the q u a n t i t y f + aP ), the expression f o r ^

o a diT

(yV + aP ), the expression f o r ^ may be w r i t t e n , approximately,

dh _ R a r ~ • yV aP

o a By s u b s t i t u t i n g the apisropriate d a t a , we o b t a i n

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The above a n a l y s i s shows t h a t the temperature has â pronounced e f f e c t on the a i r - s u c t i o n manometer system. Therefore, i n t h i s s t u d y , the a i r chamber was i n s u l a t e d and the g l a s s tubes and p l a s t i c t u b i n g were a l l p r o t e c t e d by a board t o keep the wind from blowing across them i n t h e course o f a t e s t . Furthermore a low-heat f l u o r e s c e n t l i g h t , p l a c e d k f e e t from the manometer board, was used i n s t e a d of the incandescent f l o o d

l i g h t s p r e v i o u s l y srrployed f o r photographic purposes. .

Method o f A n a l y s i s

Viscous-drag formula i n terms o f measurable q u a n t i t i e s and c a l i b r a t i o n c o e f f i c i e n t s . The r e f i n e d v i s c o u s - d r a g formula has been given i n ( 1 2 ) . Let us d e f i n e t h e c a l i b r a t i o n c o e f f i c i e n t s of the p i e z o m e t r i c -head tubes by

C = (19) E - ^

Y

where -Ç- i s the measured value o f the p i e z o m e t r i c - p r e s s u r e head. These c a l i b r a t i o n c o e f f i c i e n t s can be obtained by running the c a r r i a g e at v a r i o u s speeds without the s h i p model. By means o f t h i s c a l i b r a t i o n c o e f f i c i e n t , the v e l o c i t y i n the wake can be c a l c u l a t e d from t h e data obtained i n a t e s t run by the formula

u = ^ g C ( H - ^ ) ( 2 0 )

For t h e value o f u^ i n ( 1 2 ) , the assumption i s made, f o l l o w i n g r e f e r e n c e [It],, t h a t t h i s .is equal t o Ug, t h e value o f the v e l o c i t y u at t h e edge of the wake. Then equation ( 1 2 ) becomes

K = y / J ( S o - - - '^^'JdS ( 2 1 )

o r , expressing Ug and u i n terms o f the measured p i e z o m e t r i c pressures p l and p' at t h e corresponding p o i n t s , we o b t a i n by ( 2 0 )

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where C_ i s the c a l i b r a t i o n c o e f f i c i e n t o f the p i t o t tube at the edge o f t h e wake. This l a s t expression gives D^ i n terms o f measured quantities..

E f f e c t o f e r r o r i n i n the viscous-drag formula. From

equa-t i o n ( 2 1 ) , w i equa-t h iL, = i^g(H r- — ) and u - »^g(H - • ^ ) , we o b equa-t a i n J!i O Y V

D _{= Y}, [2 A h - ^ ) ( H - -E) - (H - ^ ) - (H - £)]dS

w / O Y Y Y Y

h.

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D i f f e r e n t i a t i n g equation (23) w i t h respect t o gives

3D ] = Y

1 /

31 P f H - - ^ i o Y dS = y f

r

^ ) d s

(2l|)

Since the order o f magnitude o f ~- i s 1 wé o b t a i n UE

o

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where oi i s the r e g i o n o f i n t e g r a t i o n .

The above r e l a t i o n s h i p shows t h a t the v i s c o u s drag i s v e r y s e n s i -t i v e -t o -the undis-turbed -t o -t a l head. The e r r o r i n -t h e v i s c o u s drag i s prop o r t i o n a l t o t h e e r r o r i n t h e t o t a l head as w e l l as t o the area o f i n t e g r a -t i o n . I f -the area o f i n -t e g r a -t i o n i s l a r g e , even a s m a l l e r r o r i n -the un-d i s t u r b e un-d t o t a l heaun-d w i l l cause a l a r g e e r r o r i n the v i s c o u s un-drag- Theref o r e , s p e c i a l care must be taken i n t h e s e l e c t i o n o Theref t h e r e g i o n o Theref i n t e g r a -t i o n and -the de-termina-tion o f -the undis-turbed -t o -t a l head.

I n t h i s study, the mean v a l u e o f t h e f o u r undisturbed t o t a l - h e a d readings was used. As t o the s e l e c t i o n o f t h e r e g i o n o f i n t e g r a t i o n , i t must i n c l u d e t h e wake, but extend as l i t t l e as p o s s i b l e beyond It,

As was shown by Wu [ 5 ] , the plane o f a wake t r a v e r s e should not be so c l o s e t o t h e s t e m o f a s h i p model t h a t the t r a n s v e r s e components o f

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the v e l o c i t y i n the wake are not n e g l i g i b l e , but not so f a r t h a t t h e v a r i a t i o n s i n the' t o t a l head and pressure axe too s m a l l f o r accurate measurement. Nor should the measuring plane be i n t h e r e g i o n o f the waves r e f l e c t e d from t h e tank w a l l s . Accordingly.,, a t r a n s v e r s e plane

6 f e e t behind the s t e m o f the model was s e l e c t e d f o r measuring t h e r e -q u i r e d i n f o r m a t i o n , and an area o f two f e e t i n w i d t h , bounded above by the f r e e surface and below b y a l i n e 1 foot below t h e undisturbed f r e e s u r f a c e , was s e l e c t e d f o r t h e r e g i o n o f i n t e g r a t i o n .

I n t e r p o l a t i o n o f t h e data. Since t h e t o t a l - h e a d tubes and t h e piezometrichead tübés were mounted a l t e r n a t e l y , a Upoint Lagrange i n t e r -p o l a t i o n method w€is used f o r determining t h e i n t e r -p o l a t e d values o f t h e t o t a l head a t t h e l o c a t i o n s o f the p i e z o m e t r i c head t u b e s , and t h e values o f t h e p i e z o m e t r i c head a t t h e l o c a t i o n s o f t h e t o t a l - h e a d tubes.

The formulae used are

and

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where t h e t o t a l head i s measured a t t h e p o i n t s yj.g» ^ J - 1 ' ^J+1' and t h e p i e z o m e t r i c pressure a t Y j . g ' yy ^J+2' ^J+U'

Treatment o f untraversed wake r e g i o n . I n making t h e wake s u r v e y , i t was not p o s s i b l e t o t r a v e r s e t h e wake t o t h e a c t u a l l o c a t i p n o f t h e f r e e s u r f a c e . F o r e s t i m a t i n g t h e c o n t r i b u t i o n t o t h e v i s c o u s - d r a g i n t e g r a l from the untraversed r e g i o n , a p a r a b o l i c e x t i r a p o l a t i o n formula was used. From equation ( 2 2 ) , l e t us d e f i n e a v i s c o u s - d r a g i n t e n s i t y by

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where the s u b s c r i p t s i r e f e r t o the v e r t i c a l p o s i t i o n and J t o the h o r i z o n t a l p o s i t i o n o f a p o i n t . Then, from the known values o f G^j , and at the l e v e l s z^ =-0.025, =--0..050 and z^ "-0.075 foot below the undisturbed water s u r f a c e , we can c a l c u l a t e the viscousdrag i n -t e n s i -t y a-t -the a c -t u a l l e v e l of -the f r e e s u r f a c e from -the p a r a b o l i c formula

G = Cq - C^z + CgZ^ (29)

where z = 0 i s at the undisturbed l e v e l o f the f r e e surface and the curve G(z) passes through the t h r e e p o i n t s (z.,, G^ . ) , ( z ^ , G^ ) and

( z - , G- ). The viscous-drag i n t e n s i t y at t h e f r e e s u r f a c e can then c a l c u l a t e d frbm (29) u s i n g f o r C^, C^, Cg the values Cq = 3 G ^ . 3 G g * G 3 ^1 = à ( 5 G ^ - 8 G , . 3 G 3 ) ^2 ^ - 2^2 * (30) where h = 0.025 f t .

Data p r o c e s s i n g and r e s u l t s . A m i c r o f i l m reader was used f o r reading the t o t a l - and piezometric-head data from 35 mm f i l m . These data were t h a i s u b s t i t u t e d i n t o formula (22) t o c a l c u l a t e the v i s c o u s drag.

For e v a l u a t i n g the viscous-drag i n t e g r a l , the t r a p e z o i d a l r u l e was employed f o r the area from the f r e e s u r f a c e t o 0.1 f o o t below i t s

un-d i s t u r b e un-d l e v e l - , anun-d Simpson's o n e ^ t h i r un-d r u l e was u t i l i z e un-d f o r the un-deeper areas. The i n t e g r a t i o n area w was d e f i n e d as the area i n the transverse plaiie enclosed by the water surface and the r e g i o n where the reading

(H^ - H) i s p r a c t i c a l l y zero.

A l l the data p r o c e s s i n g was accomplished on the 701*4 IBM com-p u t e r . The r e s u l t s are shown i n Tables 2 through 8 and i n F i g . 3.

D i s c u s s i o n o f the R e s u l t a n t Viscous-Drag Curve

As shown i n F i g , 3, the r e s u l t a n t curve o f the c o e f f i c i e n t o f viscous drag versus Froude number f o r a S e r i e s 60 model i s above t h a t o f

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Wu [ 5 ] , but shows a s i m i l a r sinuous t r e n d i n the neighborhood o f F = 0,25. This t r e n d seems t o be c o n s i s t e n t w i t h the f i n d i n g s o f Webster [7] and Chow [8] which have a l r e a d y been mentioned i n the I n t r o d u c t i o n .

On a model as f i n e as the S e r i e s 60, 0.60 b l o c k form, i t i s un-l i k e un-l y t h a t separation wouun-ld occur, as has been v e r i f i e d e x p e r i m e n t a un-l un-l y at the Davidson Laboratory. N e v e r t h e l e s s , even when the flow does not separate, the e f f e c t o f an adverse pressure gradient near the s t e r n would be t o t h i c k e n the boundary l a y e r , which would r e s u l t i n an i n c r e a s e i u t h e pressure defect at the s t e r n . Since the p i e z o m e t r i c - p r e s s u r e gradient i s adverse when a s t r e a m l i n e a t the f r e e s u r f a c e i s c l i m b i n g t o the c r e s t o f a wave, one would expect t o f i n d a c o r r e l a t i o n between the p o s i t i o n of the wave c r e s t neax the s t e m and the v i s c o u s drag. T h i s e f f e c t may be a m p l i

-f i e d by the secondary -flow i n the boundary l a y e r at a wave c r e s t , the pres-ence o f which was conjectured by Chow [8] and v e r i f i e d by Tzou [ 1 3 ] .

At Froude numbers l e s s than about 0.25, the present r e s u l t s f o r C^ are i n much c l o s e r agreement than those o f Wu C5] w i t h the ITHC c o r r e

-l a t i o n -l i n e , as i s eiIso shown i n F i g . 3. Since -l o n g experience w i t h s h i p models a t low Froude numbers and w i t h double models i n wind t u n n e l s has

i n d i c a t e d t h e reasonableness o f t h i s c o r r e l a t i o n l i n e when wavemaking i s n e g l i g i b l e , t h i s improved agreement i s an i n d i c a t i o n t h a t the m o d i f i c a -t i o n s and refinemen-ts used -t o o b -t a i n -the presen-t r e s u l -t s have no-t been i n v a i n ,

Conclusions

1. A refinement o f the B e t z - T u l i n formula f o r v i s c o u s drag was ob-t a i n e d by r e p l a c i n g Beob-tz's approximaob-te f o i i n u l a f o r ob-the f o r c e on ob-the wake

sources by an exact one.

2. Temperature has a pronounced e f f e c t on manometer systems o f the a i r - w a t e r d i f f e r e n t i a l t y p e . M i n i m i z i n g temperature v a r i a t i o n i n the a i r chamber during a run i s e s s e n t i a l .

3. The viscous-drag i n t e g r a l i s s e n s i t i v e t o an e r r o r i n the un-d i s t u r b e un-d t o t a l heaun-d H^. I t i s important t o un-determine t o a h i g h degree of accuracy and t o extend the r e g i o n o f i n t e g r a t i o n o f t h e

(16)

viscous-drag i n t e g r a l o u t s i d e o f the wake as l i t t l e as p o s s i b l e ,

k. The r e s u l t a n t curve o f the c o e f f i c i e n t o f v i s c o u s drag versus

Froude number i s higher than the curve which was obtained by Wu [ 5 ] , but shows a s i m i l a r sinuous t r e n d i n the neighborhood o f a Froude number of 0.25, T h i s , together w i t h other evidence t h a t has been adduced, confirms t h a t there i s a marked v a r i a t i o n of the v i s c o u s drag w i t h Froude number.

(17)

References

[ l ] M. F. T u l i n , "The Separation o f Viscous Drag and Wave Drag by Means o f the Wake Survey," DTMB Report 772, J u l y 1951.

[2] A. B e t z , "A Method f o r the D i r e c t Determination o f Wing-Section Drag," HACA TM 337, November

1925-[3] J.- Wu, "The Separation o f Viscous from_ Wave-Making Drag o f Ship Forms," Jour, o f Ship Research, v o l - 6, no. 1, June 1962.

[k] L. Landweber and J . Wu, "The Determination o f Viscous Drag o f Sub-merged and F l o a t i n g Bodies by Wake Surveys," Jour., o f Ship Research, v o l . 7, no. 1, June 1963.

[5] Wu, "Technique o f Determining the Viscous Drag o f a Model by Means o f a Wake Survey," Ph.D. D i s s e r t a t i o n , The Ifaiv. o f Iowa, August 1963-[6] K, Key, " V e r i f i c a t i o n o f Method o f Determining t h e Viscous Drag o f

a Ship Model," M.S. T h e s i s , The Univ. o f Iowa, January 1965« [7] W, C. Webster, "Study o f t h e Boundary Layer on Ship Forms," ^ d r o

-n a u t i c s . I -n c . Co-ntract P r o p o s a l t o S.N,A.M.E.., February

1965-[8] S. K. Chow, "Free-Surface E f f e c t s on Turbulent Bomdary-Layer Sepa-r a t i o n , " IIHR RepoSepa-rt, The Univ. o f Iowa, MaSepa-rch

1965-[9] L.. Landweber, "The B e t z - T u l i n Method f o r O b t a i n i n g Viscous Drag," Report o f Resistance Committee, Proceedings o f 11th I n t e m a t i o n a l Towing Tank Conference, Tokyo, 1966.

[10] S, D. Sharma, "Untersuchungen über den Zähigkeits- und Wellenwider-stand von S c h i f f e n mit besonderer Berücksichtigung i h r e r Wechsel-wirkung," I n s t i t u t für S c h i f f b a u der Universität Hamburg, S c h r i f t Nr. 20kk, September 1961*.

[ u ] M. D. Pearlman, "Dynamic C a l i b r a t i o n öf Wave Probes," M.I.T. Report, 1963.

[12] H. Kobus, " A n a l y t i c a l and Experimental Study o f Eggers' R e l a t i o n -s h i p between Tran-sver-se Wave P r o f i l e -s and Wave R e -s i -s t a n c e , " Ph.D-D i s s e r t a t i o n , The Ifeiv. o f Iowa, August

1965-[13] K. Tzou, "Secondary Flow Near a Simulated Free S u r f a c e , " M.S. T h e s i s , The Univ. o f Iowa, June 1966.

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rTable 1

(KPS

UarACE O.IS.TUK'JANCË / f P T l AT .dFT BEHlND-STEftN

Y(l-T 3^000 B.SOO 4,eO0 4.500 5.000 5.500 6.00:0

- 2 . 4 0 -^».20 -^.00 -UHO, -1.60 -1 . 40 -1*20 -1.00 -0.t-:o -Q.oO -0,40 -0.20 -0.00 ü.?0 Ü.40 0.6:0 0.80 1.00 1.20 U 4 0 1.60 1.60 2.0Ü 2-20 2,40 0.002 0.002 ü.002 0.0 02 O.Oöl O-OOO 0.000 - 0 . 0 0 1 -O.OOO -Ü.OGu - 0 . 0 0 1 -Û.0G1 -O.OCl -Ü.001 - 0 . 0 0 ? - 0 . 0 0 2 - o . n o i -O.OOI - 0 . 0 0 0 - 0 . 0 0 0 0.000 0.001 o.böi 0.0C2 0.002 0.003 0.001 0.000 o-óoo -0.001 -0.001 -0.001 -0.002 -ü.002 •ü.001 -0.001 -0.000 -0-000 0.000 0.000 0.001 0.00] 0.001 0.001 •O.000 •0.001 -0.003 •0.0 04 •0.004 •0._004 0.001 0.Ü03 0.Ô07 0.009 0.009 0.009 0.Û10 OwOiO 0.009 Ô.009 O. 008 0.008 0.007 0.00 7 0.007 0.008 0.008 0.003 ü.009 0.008 0.010 0.011 0.010 0.010 O..010. - 0 . 0 0 2 - 0 . 0 0 4 - 0 . 0 0 9 -Ö.OIÜ - 0 . 0 1 2 - 0 . 0 1 3 - 0 . 0 1 3 - 0 . 0 1 2 - 0 . 0 1 1 - 0 . 0 1 1 -O^OIO -O.OIO - 0 . 0 0 9 -O.OIO - 0 . 0 1 1 -0.O12 -O.012 - 0 . 0 1 4 - 0 . 0 1 5 - 0 . 0 1 7 - 0 . 0 2 0 - 0 . 0 1 9 - 0 . 0 1 7 - 0 . 0 1 5 -O.OIO ,0. 03 9 O. 041 O. 042 0.04Ü O . 039 0.036 0.034 0.032 0-029 0.026 0.024 0.022 0.022 0.021 0-021 0-022 0.025 0.027 0.027 0.027 Ö-032 0.036 0.040 0.043 O-045 0.035 0.039 0-Ô43 Ô.043 0.042 0-036 Q.034 0-032 0.031 0.031 0-030 0.029 0.029 0^029 0.029 0.029 0-029 0.027 0.025 0.030 0.032 0-034 0.032 0.031 0.023 •0.044 -0.O40 •0.028 -0.013 0.005 0.018 0.024 0.050 0.O34 0.037 Ö.038 0.039 0.041 0.042 0.041 0.040 .0.040 0.042 0é043 0-041 0-037 0,026 0.009 •0.006 •0.016

(19)

he 10 O ^ N O 0''0'N m ^ ^ O co a O fM w D-o-o'Coe.ooooc-eooeooo'O'oo

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l ^ ( n^ e — — 9 <£i<^ ^ O ' ^ ' ^ ^ v d O o - c D K P N i e i

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(20)

O _ j O I ^ !^ ^ f^"C^ frt ^ Irt ^ t^'t^ (rt frt (rt fP" ff^ f e o ^ m ^ l M r t J M g D t f ^ r > l« r t < C 4 r( N( ^ ^ ' ^ • a D C D eo B• l r ^ • • • * ' * ' ^ ^ u M ^ ^ • > 4 • * * • • • • t f ^ ^ r • * • * >*• bFn"ifW(rt(^(rt»rt(rt«(rtrt(rt<rt(rtfrt trt e»o^'*^i'^*'^'rt<ooih«^-ir-« (M M e cc cr> e œ•^ft^ftlftlr•(ft«ft•r<*(*•^rt(rt•l^l^ft^(^lf^*.^tf^ • 4 * *.':'.« • . • .• • • I, . • • • . O ^ (rt ^ rt rt. rtrtrtrtrtrtrtrtrtrtrtrtfrtirtrt i . ' c T i r T i f v. f T ^ i r i f* i r t f * ^ r t r t r t r t r t r t ' r t rt rt irt rt rt JO> CC.O" <M Ift « «• rt C • f - O m O I T ' t ' J ' f N J e O » ' ^ * • O l * ifi m IT i r . ' i f r t w M 4-, « ; i r Ift ift * iri ift o o M o . » 'f- r- h- ^ h- » ' . ^ tsj —< © . p 9 o

* VI i r iri i P r * ^ ^ •^.•t * - * . i f t (ft Ift i f t - * - * i r

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