Velocity field around ship hull in roll motion

1.MEI íaO

ARCHAE

Report

of

Department of Naval Architecture

University of Osaka Prefecture

No. 00406

October

,

1979

Velocity Field around Ship Hull in Roll Motion

by

Yoshiho IKEDA**

,

Toshihiro FUJIWARA***

Yoji HIMENO****

_{and Nono TANAKA*****}

Lab.

_{y. Scheepsbouwk.j,}

Tecinsche

_Hogsc0o

'Dell

f

*

'

published in the Journal of The Kansai Society _{of Naval Archtects}

Japan , No.17]. (1978)

Research Assistant _{, University of Osaka Prefécture} Graduate student

Associate Professor

Report of Department of Naval Architecture, University of

Osaka Prefecture , No.00406 , October 1979

Department of Naval Architecture University of Osaka Prefecture

4-804 Mozu-Umemachj , Sakai-shi Osaka 591 Japan

Velocity Field around Ship Hull in Roll Motion

by Yoshiho IKEDA

,

Toshihiro FUJIWABA

Yoji HIMENO

and Nono TANAKA

Summary

The unsteady flow velocity near the hull of two-dimensional cylinders in roll motion is measured using a hot wire anemometer. The measured values are found to be in good agreement with the theoretical ones which are expre-ssed as a composite form of the solutions of the innér viscous flow and the outer potential flow. The comparison of the skin friction between theory and experiment which is determined by the measured velocity gradient at the wall also shows an agreement in case of the cylinder of a midship section. Applying the theory to an ordinary three-dimensional ship form, the theoreti-cal value of the friction damping is found to be almost the saine to that of Kato's formula.

In addition, from the measurements of the velocity profile for cylinders with small bilge radius, it is found that the size of the separation bubble

is unexpectedly small, and that the velocity just outside of the bubble is considerably large. The eddy making component of roll damping is also

discu-ssed with the aid of the measured velocity.

-Introduction

The authors have recently proposed a prediction method ]) ,2) _{of roll}

damping in which the damping is divided into five components, that is, the friction, wave-making, eddy-making,lift and bilge-keel components. The authors have obtained practical prediction formulas for these components. These works have made a considerable achievement on establishing a practical prediction method of roll damping of a ship. However there still remain

several points to be studied,for this prediction method including several empirical formulas is not always valid for all kinds of hull forms. Therefore

it seems to be necessary more to study the detail _{on each component damping.}

From this point of view, the velocity field around a ship hull in roll motion is investigated in detail on the purpose of obtaining basic information of viscous components, the friction and eddy-making components of a naked hull. From the velocity measurement using hot-wire anemometer, it is found that the velocity field around a ship hull in roll motion can be expressed by Stokes' theory on oscillating boundary layer flow. The friction component is calculated theoretically. Furthermore, some discussions on the relation between flow

field and the eddy-making damping are made.

Calculation of Velocity Field around Rolling Hull

The velocity profile around a ship hull in roll motion is calculated by composing the outer potential flow and the inner Stokes' viscous-flow solution for an oscillating flat plate.

2.1. Calculation of Outer Potential Flow

In order simplify the calculation, suppose a Lewis form cylinder rotating

with a constant angular velocity in an unbounded fluid (Milne-Thomson4). The

mapping function for Lewis form is described as follows

.!,La2 a2

M

where z=x1+iy ,

=iee

and M is a magnification fact9r. The velocity

components u1 and y1 in x1 and Yl directions at the point (x11y1) on z-plane

corresponding to arbitarary points (a,) on-p1ane, can be obtained by the

following equation.

-a1+iv1=

2ic,M1+02a2

(2)

In order to check the difference between the calculated results by Eq. (2) and the exact ones with free surface effect, the húlÏ surface velocity was calculated by Eq. (2) and the Urse.1l-Tasai metho&). The comparisons between them are shown in Fig.l through Fig.3.

oìi

applying the UrseIl-Tasai method, the

velocity on a hull was obtained numerically by differentiating the calculated

velocity potential on the hull surfaöewith respect to the girth length.

The calculated tangential velocity at the moment 0=0 at the three points (Lewis

angle. fl=900,500100) on a Lewis-form cylinder: (H0 (=b/d)el.25 , c=l..0) is shown

in Fig.l. The obtained values by the. UrséÏl-Tasai method depend on roll freqúenôy. near free surface ( fl=90°) and gradually approach to the value of Eq. (2) in high frequency range. On the hull bottom, however, the difference

between thém is small. The results of Takaki's singularity method5show the

similar tendency with those by the Ursell-Tasal method Examples for the case

of midship section are shown .in Figs.2 and .3 , in which the calculation were

carried at thé 0-marked points. In Fig.2 , the velocity vector calculated at.

each point is also shown , and the tangential velocity components u is plotted

on the coordinate .noal to the hull-urface. The difference of te ,resúlts

between the methods is large near free surface , but decreases at the. bilge and

the-bottom. The tangential fluid velocity at the bilge-is 1.7 to 1.8 times of

the velocity of the surface. As shown in Fig.3 , the calculated velocity

distributions on the bow-shape section by these methods are in. fairly good agreemént in the range of high frequency.

Froni these comparisons , we can safely say that the calculated results by Eq. (2) are accurate enough for practical usage except for the neighbourhood of free surface.

2.2. Calculation of Velocity Prof-ile in Boundary Layer

Viscous velocity field near a hull can be obtained using Stokes'

approxima-tion Then this solution and the outer potential flow mentioned above are

6

cOmpoSed in terms of- the compositeexpansion of the singular purturbation method. Disregarding the effect of the curvature , the basic equation for the viscous

velocity field around the two-dimensional section can be expressed as follows,

a0 a0 8u

.N+u

av a0 a0 -

+U+0

lap Ju 8u

_V\1+î

lap Jo 8'v

pap+'\

axz+_ap

where x and y represent respectivèly thé coordinates along the girth and the

normal direction to the surface , u and y the velocity in x and y directions, p the préssure- and V the coefficient of the k-ineÌnatic viscosity. Though it is difficult to solve Eq. (3). exactly , several approximate solutions have been obtained. One.. of these solutions was obtained by Stokes in the case of an oscillating flat plate , where basic equation is given in the form.

$u a'0 .

()

In this' case , the, flow is. pa.ralle.l with the flat plate. Thé Eq.(4) can be.

called as a kind of Stokes' approximation because the nonlinear inertia term is neglected., and can also b considered as boundary layer approximation because

and a2u/x2 are neglected. The solution .of Eq. (4) satisfying the

boundary cOnditions. , that is , u=U0cosWt .at y=0 and uo at

y=

, is well known.

u=Uaecos(cst-ky)

.

(5)

where u0 denOtes the velocity amplitude of the. flat plate notion and.k=(w/2V)1. As it is said that Eq. (5) is also useful in the presence of pressure gradient8, we may. apply Eq. (5) to the calculation of the velocity profile around ship hull in roll motion.

Substituting the calculated velocity amplitude of potential flow on the hull surface into- uo in Eq. (5) the inner solution u of Eq. (5) and the outer

J

solution can be composed. Then we can obtain the relativê tangential velocity

at any point near the hull as follows,

uñ()=u(y)+urp&(0)$1-ecos('t-

ky)_un,t(0)

(6)

=u(y)-u(0)e'cos(wt-ky)

(7)

The first term of Eq. (6) is the outer solution , the second term the inner solution and third term the comxrcn part. Suffixes r,p andt of u represent the relative

velocity ,, the theoretical. value of potential flow and the tangential component respectively. The term Út(y) thus represents the tangential components of the

relative velocity obtained bythe potential

_flow

theory.

The comparison between Eq.(7) and experiement will be shown in the next

section.

3. Measurement and Discussioñ of Relative Velocityear hull

The velocity field-near the hull is measured by a hot-wire anemometer.

The models are three two-dimensional cylinders , each of which has different bilge

radius R , and named Model A (R/B=O.147 , B: beam of a cylinder) , Ñodel B (R/B= 0.0357) and Model C (R/B=O.0) . The principal particulars are shown in Table 1. Model A is a midship-section shape of Series 60

, CBO.6

ship form , Model B

a midship-section shape of a full ship form and Model C a rectangular section

shape. Sinòe it is difficult to measure the velocity in water by hot-wire

anemometer because of the presence of dust and temparature change , the

measur-ements were carried out in air in which the cylinders were hanged in air and

flat plates were installed at the free surfàce plane. The locations of velocity measurèments are shown in Fig.4. The roll mot-ion of the cylinder was given by a forced roll equipment The hot wire was traversed in the y-direction through

the small holes of the model , and L-shape probe shown in Fig.5 was used in order to get rid ôf the effect of these small holes ôn measured results

The calibration was done at a point far from the hull , and examples are shown in Fig.6. Since the curves of the calibration change with temparature

and individual probe , calibrations were carried out as often as possible.

At first we beg-in with the comparisons between experimental

results of Model A and theoretical ones. Fig.7 is an example of out-put signal

of the hot-wire at A2 point of Model A. In this figure , onlythe data in the

region of 0=-00 to 0o (for example a to b in Fig.7) are meaningful. Otherwise the effect of small holes of a hull and a support of a hot-wire might arise. As seen from Fig.7 , the velocity at a certain distance from the hull surface has

the same phase as. hull xxtion. The phase difference near the hull surface (for example y=lmm) becomes large because of the viscous effect.

The experimental results at each point of Model A are shown in Figs.8 to 12.

In these figures ., the measured. velocity at 0=0 are shown by O-mark , the outer

solution of Eq. (2) by solid line , the theoretical values of Eq. (7) by

broken line and the velocity of motion of the hot-wire by chain line with

one dot. The value of Urt by Eq. (7) increases with the distance from the hull surface and has a maximam at y=5 to 8nun , and then gradually approaches the theoretical value of the outer potential flow. Except the. measured results at A3 point just downstream of. the bilge , the agreement between the experiment ànd

the calculation is fairly good in all the range of y , particularly at the

neighbourhood of the hull surface. Therefore the ábove-mentioned viscous solution of the Stokes' approxmation is useful for calculating the viscous velocity field around rolling hull with round bilge.

On the contrary , the velocity profile at A3 point (Fig.lO) has a small

3u/ay

value near the hull surface , and looks like the velocity profile of

sepa-rated flow. In addition, in high frequency region , the velocity at the boundary

layer edge is . larger than the theoretical one , the reason of which is believed to be due to the increase of the outer potential flow by a separation bubblè.

Then, the results of Model B with small bilge radius are shown in Figs.13

to 17. The veloäity at B1 and B2 points behind the bilge is zero in the range of y=O to 4mm due to the separation.. But the separation bubble is unexpectedly

small , and its length seems to be àbout 20 to 40mm in this case.

to the case of Fi-g.1Ô , there is a fast flow at y=lO to 20mm. Iñ Fig.18 , the

output signal of the hot-wiré at B1 point is showi in order to see the

flow field in detail. From Fig.18 , the velocity at y=lOrnin is quite large and the velocity at y=25mm shows strong turbulence.. We can also see that in the

profile of the output signal near thp hull surface (y=3mm) a peak appears regularly

in the early stage of one swing motion and then the velocity becomes zero. These regular peak seems to be caused by the potential flów which is dominant at the early stage of the motion of a body from rest , or the viscous flow created

by this potential flow. This flow vanishes when the eddies are generated. From

the velocity profile shown in Fig.15 , the flow separation does not seem to occur

at B3 point ., but at y=2Omm in Fig.19 the two' different kinds of flow appear intermittently The cause of the appearance of this phenomenon perhaps seems to be the change of the outer flow velocity by the uncertainty of the size of the separation bubble. The velocity profile at B4 point shown in Fig.16 does not show any high-speed region and is similar to that of Model A. There is also a velocity defect in the region far from the hull surface (B5 point in

Fig.17) , resulting that a positive pressure acts on the hull surface in front of bilge keel.

The output signal of the hot-wire at C3 point ôf Model C with rectangular cross section is shown in Fig.20. As analogous to'the experimental results at B1 and B2 point of Model B , there is a high speed flow region with strong

turbulence at y=2Omm as seen in Fig.20. After the velocity near the hull surface

reacilesa peak atan eaÈly stage of themotion , the output signal increases again. This tendency is different from that of Model B The velocity increase after the first peak seems to represent the reverse flow in the separation bubble. But we can not get the accurate values of this reverse flow velocity because the present hot-wire anemometer gives only one directional velocity accurately as was stated previously. The measured velocity profïles at the instant 0=o for Model C are shown

in Figs.2l to 26,which show a similar tendency to the case of Model B. The experimental results mentioned above can be summarized as follòws.

In,the case of the hull shape with a comparatively large bilge radius like Model A, the theoretical flow velocity calculated by Eq. (7) is in fairly good agreement with the experimental results except just behind the bilge. On the contrary,

in the hull shape with a very small bilge radius like Model B , the separation bubble takes place , where the flow velocity is almost zero inside the bubble and becomes quite large at the bubble edge. In upstream of the bilge , the flow

velocity remains small even at the point far from the hull. In the case of a rectangular hull shape as Model C , the tendency of the flöw velocity profile is similar to the case of Model B except the reverse flow in the separation bubble.

Consider the cases in water and the scale effect of the velocity field. Since the viscous flow around a rolling hull can be expressed by the theory of Stokes' approximation , the boundary layer thickness ô is in pröportion to

17

Therefore ô in water is 1/3 times of that in air . Without considerating the

transition of flow , ô of the actual ship is times of that of the model

(1/A is a scale factor of the model) , so that the boundary layer of àn actual ship in roll motion is relatively thinner than that of the model. For example in the case of X=l00 , 6/L value of the actual ship is about 1/30 times of the

model.

4. Calculation of Frictional Roll Damping

13

There have so far been studied by Kato12 arid Tanaka on the friction roll damping at zero advance speed. Kato pointed out that the friction of an oscill-at-ing circular cylinder has the same Reynolds-number dependency as the Brasius' formula for a plate in steady-laminar flow , and empirically defined an effective

Reynolds number for an oscillating cylinder. Furthermore he proposed empirical formulas of the friction roll damping of a ship both in turbulent and laminar cases using the effective Reynolds number, an effective radius and an effective wetted surface area. Tanaka's study was the first theoretical work on the

friction roll damping applying unsteady viscous flow theory. The subject of his study is to calculate the friòtion damping of a rolling circular cylinder using Eq. (4) as the basic equation , the roll damping with decay and the transition of the flow were discussed in detail.

From Eq. (5) the frictiònal stress on the hull surface can be expressed as,

(8)

As Ur t(°)=°o (where r represents a radius of a circular cylinder and 00 the

amp1iude of roll motion) in the case of a circular cylinder , the equivalent

linear damping coefficient BF can be obtaiñed as ,

B, = pSkr3 v= O ;7O7pSr'5v (9)

where S denotes the surface area of a ciróular ëylindér. The equation (8)

with the results by Tanaka-3 , while BF by Kato's formüla Is exprêssed

B, 0.78 7pS !

(lo)

From the comparison between Eq. (9) arid Eq. (10) , both equatidns have the saine form , but the coefficiefit of Eq. (10) is about 10% larger than that çf_{Eq. (9).}

Then, applying Eq (8) to the cross-section forms of an actual ship , the friction roll damping coefficient BF an be obtained as,

,ße4f5'r.rurp!(0)cosã,dx...(il)

where x is the coordinate along girth

fréé surface , and rf is the distance betweén the roll axis and the hüll _surfáce.

which i zerO at the kéel and xS1 at

af 15 an angle between the tangential line and the direction of motion at each hull tirfacepoints. _{In calculating the potential flow urpt(0) on the hull} surface , the Urseli-Tasai method was used here,

, for the alculation by Eq. (2)

has an error near free sürfàce.

At first the comparison between the calculated results of the locàl shear stress on the rolling hull surface by Eq.(8) and the experimental results using ineasuredvelocity profile is shown in Fig.27. The cefficientCf was obtained

from dUr/df at y=0 which was calculated by fitting the data in the region from y=0 to y=5inm to a quadratic equation by the least _{squares method. Thé agr} ee-ments between the calculated and the éxperimental results are

ood ecept A3

point where the separatión séems to occur. ..

From these results _{, we can say that Eq (11) can be used for the prediction} of the friction roll damping of an actual ship , because the

1w

1 region like

at A3 point will not be so wide even for a hull with a very small bilge radius as expected.from the small separation bubble measured fôr Model B and Nòdel C.

Now we proceed on the comparison between the formula of Kato and Eq (11) The comparison for a two-dimensional cylinder , Model A is shown in Fig 28 in

which the ordinate is the non-dimensional damping _coefficient VB/2g/pVB2)

and the abscissa is the non-dimensional

_{circular frequency ¿3('7g)}

_The value of Kato's formula is proportional to the square root of frequency , while Eq. (11) has a little hump-and-hollow due to the effect df free surfacé. The

agreement between them is fairly good , and may even be curious considering the

fact that the value by Kato's formula is abolit 10% larger than that of Eq. (11) as mentioned above.

The reason comès from thé difinition of the equivalent radius r _{which is determined from the tangential flow velocity on the hull.}

The longitudinal distribution of the friction roll damping of

th

ship form Series 60 _{, CB=O.6 calculated by Eq.(ll) is shown in Fig.29 , where}

BF dénotes BF for a cylinder with unit length _{At the aft and fore part of a ship hull}

F -is small because of the moment lever and the wetted surface area are siciall enough _{Integrating BF along the ship lengh}

, we can obtain

F for Series 60,

CBO.6 which is compared with Kato's formula in _{Fig.30. The agreement is ood',}

5. Discussion on Eddy-Making Damping of Naked Hull

Eddy-making damping of a naked hull which is due to the pressure. created by fluid viscosity has an important role in roll damping The authors have proposed the prediction method of eddy-making damping on the basis of the experimental results of many two-dimensional cylinders ànd some theoretical considerations.

In this chapter , however , we can discuss this problem from the view point of velocity profile mentioned above in order to obtain fundamental information First of all we begin with the experimental results of the pressure on a rolling hull. Model A was forced to roll , and. the pressure on the hull was measured by a pressure sensor attached on the hull surfacé. The locations of the sensor were so selected that the effect of the viscosity becomes large and the effect of free surface is small , as shown in Fig 31 The amplitude of the velocity-phase

pressure d was obtained from the pressure record at 00 and that of the

acceleration-phase pressure a was obtained from the record at O=±Oo (where the static pressure was taken off from the pressure record). The non-dimensional

pressure coefficients P(=P/pgbO) and P(=P/pgbeo) are shown in Fig 31

The values of P at A1 point are different from the theoretical results of the

Ursell-Tasai method , the reason of which seems to be due to the viscous effect while the values of dr at A2 and A3 points are in good agreement with the

theoretiòal ones. These facts show that the region where the viscous effect appears on a rolling naked hull is unexpectedly small On the contrary , the

measured values of ar are in fairly good agreement with the theoretical ones which is the same conclusion as the the usual opinions.

Consider the relation between the pressure. on the hull surface due to viscosity and the velocity field around a rolling hull. In the following discussion , we assume that the pressure is constant along y axis through the separation bubble on the basis of the experimental results in steady flow'4', so that the pressure on the hull coincides with the one at the outer edge of the separatipn bubble. With this assumption w can get the pressure p on the hull surface from Eq. (3) as follows.

=

_pf!dx4pUf

(12)

where U denotes the velocity at the outer edge of the separation bubble , and the function of time and locations. Write for the velocity potenUal at the

outer edge of the separation bubble , the Eq. (12) becomes

a

In considering roll ment , it is convenient to use the pressure difference at

the symmetrical points of the both sides of the hull The pressure difference can be expressed as,

4p=p,,_,,, -p,,__,, (14)

where _a denotes a Léwis angle at arbitarary point. In the absence of boundary layer separation , _{coincides with the velocity potential on the hull surface} and are symmetrical in both sides , so that the following relations exist.

f

k k at/,,._

(a\

k a1_- \

With these relations, Eq. (14). becomes,

4p=-2p()

_at

As seen from Eq. (16) , the second term of Eq. (13) disappears. In the presence of

separation , however , _{there is a separation bubble only at one side in}

symmetri-cal points of a hull as shown in Fig.32 so that the absolute values of the

(15)

velocity potential at the symmetrical points of the both sides are different and then the second term of Eq.(13) has a certain value as follows.

ria

f3\

4p=

p

pJ(8\2

(açb\2

21

The second term of Eq. (17) is due tO the changé of the potential flow by the

-separation bubble..

Defining p2 as the second term and using the \relocity U at the oüter edge of the separation bubble , can be expressed as,

4p = - O.5p (-uL , - UL (18)

Then , if there is' a separation bubble at the point 11a as"shown in Fig.-32 , the

value _{is higher than that of the. potential flow along the hull surface and}

Ur_n

is 1oser to that of the potential f 10w. Thè value U7-_ obtained from

i 'a

.

the experimental results at B1 and B2 points of Model B and at C1 point of Model C shown in Fig. 33 are proportional to the roll frequency. Furthermore- , the value of _{is approximately proportional to the frequency near the bilge as shown}

in Fig.l Finally _{èxpEessed as Eq. (18) arè proportional to the square of} the roll frequency. _{Hence the eddy-making roll damping of a naked hull is} proportional to the square of "the roll frequency. This is the same results of

the authors' measurements of this componenti-).

The calculated P2 of Model B and Model C by Eq. (19) taking U= as measured and assuming to be the velocity of motion (=rO) are shown in Figs 34 and 35 The comparison between measured 4P2 in water arid the'calculated one by Eq.. (18)

which is also shown in Fig.35 is reasonable. Though the measured is higher than 'the calculated one on the surface far away from the bilge , this difference

will become smaller ,

considering that U_

upstream of the bilge is smaller

than the velocity of motion as seen from Fig26. _{The eddy-making roll damping}

coefficient CR calculated from the experimental results of ₂ of Model B is 2.3, while the measured CR is about 3.,2P If considering the positive pressure acted on the surface upstream of the bilge , the agreement will be improved. Through these discussions it can be said that the eddy-making roll damping of 'a nàked

hull is mainly'caused by the pressure decrease ,i.e.,the velocity increase at the edge of the separation bubble.

('7)

6. Conclusions

As a fundamental study on calculations of the flow field on the eddy-making damping are obtained.

the friction roll damping ; the measurements and

around a rolling hull are carrd out. A disbussion

also made. The following conclusion's can be The velocity field around a rolling hull with a relatively large bilge can be expressed as a composite form of the solutions of the inner viscous by Stokes' solution and the outer potential f-low.

The thickness 'of the boundary layer is proportional actual ship, has much thinner thickness than the model.

The Stokes' solution for the local skin friction is

the experiment except for the region just downstream The, Stokes' solution for an actual 'hull forms is

Kato's formula.

The existance of the separation bubble is confirmed form with small bilge radius , and it is found that the bubble is relatively short unexpectedly.

The eddy-making roll damping estimated from the measurement of the. outer-edge velocity of the separation bubble is f the saine order with the measured one.

-7-in good agreement. with of the bilge.

in good agreements with experimentally for a hull

length of the. separation to vÇ575 sO that

radius flow

Reference

Y.Ikeda, Y.Himeno and N.Tanaka : On Eddy Making Component of Roll Daitping Force on Naked Hull , Jour. Soc. of Naval Arch. Japan, Vol.142 (1977) ,p.54 Y.Ikeda, Y.Himeno and N.Tanaka : Components of Roll Damping of Ship at Forward Speed , Jour. Soc. of Naval Arch. Japan, Vol.143 (1978) ,p.113 F.Tasai and M.Takagi : Theory and Calcutation Method of Ship Responce in Regular Wave , Proceeding of the Simposiun on Seakeeping , Soc. Naval Arch. Japan (1969) 'p.1

L.M.Milne-Thomson : Theoretical Hydrodynamics , 5th edition , Macmillan &

Co. Ltd. (l968),p.258

M.Takaki : Calculation of Velocity Field around a TwcI-dimensiora1 Cylinder by Singularity Method , Report of SR161 committee No.13-2 (1977)

M.Van Dyke : Perturbation Methods in Fluid Mechanics , Acd. Press (1964) for example, L.Resenhead _{: Laminar Boundary Layer , Oxford University Press}

(1966), p.381

G.K.Batchelor : An Introduction to Fluid Dynamics , Cambridge University Press (1970), p.353

I.Tanaka,y.Himeno and N.Matsuxnoto : Calculation of Viscous Flow Field around Ship Hull with Special Reference to Stern Wake Distribution , Jour, of The Kansai Soc. of Naval Arch.,Japan, No.150 (i973) ,p.l9

S.Taneda _{: Visualization of Viscous Flow around Bodies, Proceeding of the} Simposium on Viscous Resistance, Soc. Naval Arch.,Japan (l973),p.4O

I.Imai : Hydrodynamics, Syokabo Co. (1973) ,p.292

H.Kato : On the frictional resistance to the rolling ships

, Jour. of Soc.

Naval Arch. Japan , Vol.102 (1958). p.115

I.Tanaka _{Viscous Damping of Rotational Oscillations of a Cylinder} Technical Note No. 719 , Stevens Institute of Technology (June, 1964)

T.Ota and M.Itasaki _{A separated and Reattached Flow on a Blunt Flat Plate,}

0.5

1.0

Fig. _{1. Tangential velocity on hull surface}

of Lewis form cylinder (H0=l.25,

Iodimoriea1

_{0.-1.25 a -0.9831}

-.36i1oga

bi

--r 0-1.20J -rmoi .mttmd

-, 11a

r cotathrg cylirr

- r trgra1 e1ty of 51m hull -, r 1mr1a a1uth locity r

a rotathig cylir

Fig. 2. Distribution of velocity on hull

surface.

Tab e 1. Particulars of _models.

Fig. 3. Distribution of velocity on hull

surface.

-9-Out put 0.0 11.825er I

4-

1 model A I I menee 60.c0-0.6 I I midship mection I ..-..-I-. lie-1.232 a -0.977 L .2 A5 1A41A 6.12er 14.0er 14.0cm 1.5mm 3.0mm 4. 5.0mm .0mm model D H,1.25 0 -0.9986 model c H,1.25 a -1.0

Fig. 4. Locations of velocity measurement..

f

FIg. 5. Sketch of hot wire.

O : mea

ed (t30c)

4 r measured (t.ioc) roll mxii

0.0 _{5.0 (ml/eec) 10.0}

0.0 -e' 0.1 Ur 0.1 0.0 0.0 0.0 0.1 0.0 u' 0.2 0.1 a A1

angle4

-

4

Lan

V

eut put eut put y-l. 0=

Fig. 7. Example of out put signal of hot

wire (A2 point of Model A).

O: eOaOured C O,lOdeg j

(calculated by eq. (a).

0.o 0.1

o

-T2.0se R(-eth2/p)3.fl5gQ 11rt (calculated byeq. (7)) O T1.2eec -R,5.193xj05

o-T0.9eec R,6.923*1Ó! »4.593*10' T0.9aeø IlG.l24z1O' Oi eiauurod ('0s10deg O

Fig. 8. Velocity profile near hull (A1 point

,of Model A).

pu-l.

756x10'-'A2

0.0

O0 - 10 20 30 ₄₀

50 y (mc)

Fig. 9. Velocity profile near hull (A2 point

of -Mädel A). '1 Ur (rn/eec .0.1

00

0.1 0.0 Ur 0.-1 0. 0.0 Ur 0.'l 0.0 Ur 0.1 0.1

00

0.0 0. Ur 0.1 T2.Osec Re2.927X1OS T0. 9sec Be6. 503x1O Ur O o measured 0010deg (e/sec) O Urpt Urt O 0.0. 0.0 10 20

Fig. 10. Velocity profile near hull (A3 pÌnt of Model A). 'O measured C e2xoa9 T2.0sec R0e3.116x10'

rpt

A3 30 40 y(mm)

-ö-Urt T=1.2sec R5. 103x103 T0.9sec

R6923x10'

-Q-.

0.0 10 20 30 40 y(mm)

FIg. 11. Velocity profile near hull (A4

point of Model A).

T0.9sec

R'6-.503x10° Ur O: measured' C OolOdeg

3C-R2. 927*10'

0-.

rpt

rt

T=1. 2séc Re4 . 878x10'

-.-o-0.000 _{10 20} 30 40 yCuos)

Fig. 12. VelocIty profile near hull (A5

point of Model A).

o

T1.2sec

Rj4. 878x10' 0.1

0.2 Or ('..c3 0.0 0.2 tar 0.1 0.0 Ut 0.3 0.2 o 0.1 0.0 0.2 tar _O Wa.c) 0.1 tar 0.3 0.2 0.1 0.0 0.0 0.0

i

o o o 00 o 0

Fig. 13. Velocity profile near hull (B2

pòint of Modèl B). 0

8 8 6

.1

50 50 ₁₀₀

y()

BA

-11-0.2 Or 0.1 0.4 0.3 0.2 0.1 0.0 0.0 0.0 00 100 y(3 0.0 o o tata _'

[J

50 _{100 y()}

FIg. 15. VelocIty profile near hull (B8 point of Model B)

34

Fig. 14. VelocIty profile near hull (B2 _{Fig. 16. Velocity profile near hull (B4} point of Model B). -. point of Model B).

000

00

a¼.

L_._0-T-2.Oooa O a aoured (11 p. 0,-10oJ 0.2 tar Ware) 0.1 o o Ut O

90

-u 2.0aoa O a aaaured (11. . 0,.1OoJ

00

o

00..

_O o 0 0.2 _o o T-1. 6oec O o 0.1 0.0 o T.1.Eec o

o o0

o Or 0.0 ₀₀ °:urnd cmii . e 0.2 ₀ T-2. Ooec O a aBured(U enp,0,1g) 0.1 o T-i.6G00

_;:

0.0 o 50 y(mm) 100

00

Ur

a

Fig. 18. Example Vof out put signal of hot wire (B2 point of Model B).

Oo 0.0 0 0 measured I e=1ode0) o i

000

Ti.6ec

A £A AA A a

VVV ViV

Fig. 19. Example of out put signal of hot

wire (B3 point of Model B)

-12-Fig. 20. Example of out put signal of hot

wire (C2 point of Model C).

o_o 0.2 0

----0---4--g

0.2 0.1 Q -

-00 0.4 0.3 b _t o -0 reree fia., o o o 0.00.0 50 - T-2.Oeec O measoxed ( 0.-lOdeg) T-1.6aes y( 3.00

FIg. 21. Velocity profile near huU (C1 point of Modél C). ¡ OUt put

14411k

1.Oooc I

J,AJV\jv

"

L

-Y -Y

Out put y7Orm

AALÄA.

I Outpu

Vi6secV

V V

ltput

V

L

joutput

-1azin-:t

:11 V ¡ '-6nm A

0AI .kiA t

0.2 0.1 o

_vo

O medel 0 T1. Osec La5 0.0 0.0 50 y(nsü) -- too.

Fig. 17. Velocity profile near hull (B5

o point of Model B). 0.1 0.0 mr 0.2 O T-i. 08es 0.1

[ o 0.2r o .110 I --I o o

__.0-G

o.i L

0-T.1. 6cec 0.OJ

000

0---o

_L.o_?_--0

6 0.0 0.0

000

O ?2.0ge

rever.. f icv° meaSured C Oe.iOdag)

0

-T1.0ec

00

0.0 _y(

100

Fig. 22. Velòcity profile near hull (C3

point of Model C). 0.2 or u.

o°8

.!_L--0.1 T-2.Oaec Os reasured C e.-loaeq) 0.0 u O o 0.0

00

-8

o..._.-0.2 ?-1.Ogec 0.2 0.1 0 V.1.0.50

Fig. 4. Velocity profil, fleer hull

(C polsi o2redel C)

FIg. 24. Velocity profile near hull (C4 point of Model C). C3 0.0 1 [ 71=) lOO 6.2 IIea.54(0.10 la/nc _7.2.0.50 0 1 _o so - yi ioo Fig. 25. Velocity profile near hull (C5

point of Model C). Ó.Ools 1.1.0.50 o ( o-10g) 1.2.05.5 0.1 0.0 eri t

----t-

_o I

000 0 0

Io 0.0,_1 ut 0.2 0.1

---o----.-

_{00 0}

o o o t'-1.3.40 model A Dp -: Kato a formula -i alculated by eq. (10) 0,lOdeg 0. 1.139x10"m1/eec -or-o Fo 0.0 .0 00 71=) - 100 --Fig. 26. Velocity profile nearhull (Ce

point of Model C).

Fig. 27. Shear stress distribution on ship

hull surface in rolling motion.

0.0

0.0 _{0.5 1.0}

Fig. 28. Frictional damping coefficient b

for two-dimensional cylinder.

LI L L 0-1 0. O 0.2 0.1 o0 _T-1.050m 0 0 o 0.0 ut -Q 0.0 0.0 50

y()

100

Fig. 23. Velocity profile near hull (C3

point of Model C). 0.2 0 0.1

..

0.3 0.2 0.1 u,

°

30 00 0.2 Vr (a/eec 0.1 0.0

0.001 Seriea60.C50.6 0.69 Ol0deg V 1.l39xl0 mt/sec A.P.

Fig. 29. Longitudinal distribution of frictional roll damping coefficient.

0.0

F.p.

Fig. 30. Frictional damping coefficient bF ' for Series 60, CB=O. 6 model at.zero

advance speed. -.

O t eeured t 0..504.J

t UretlI-?e.ej nethod

Fig. 31. Hull surface pressure on

two-dimensional cylinder.

Fig. 32 Schematic view of separating flow.

-14-0.5 0.25 o 11.2cm 0.0 measured t C1 point of model C t S point of model B 2 point ofmodel. B 2.0 4.0 1.0 2.0 3.0 dp2/0.5p(rue0)° e 6.0 (red/tte)

Fig. 33. Velocity at the edge of separation

bubble.

modelS I.

O calculated by eg.(18)I using measured

-L

velecity O.

Fig. 35. Comparison of calculated and mea-sìi±èd préssure difference at bottom

hull surface. o

Op

0.0 -0.3 P.00 0.0 .0-.0.j P2 -0.1 -0.2 0.0 p -dr 0.0 '-0.1 -- oQ '0.1. 2o___8s5e_q_o__

;:.

ON\

0.0 0.5 1.0 0

Fig. 34. Premure distribution near bilge

circle calculated from measured

velocity at outer edge of

sepa-ration bubble.