5 JUNi 1960
ARCH I EF
Report
of
Department of Naval Architecture
University of Osaka Prefecture
No. 00401
December
,1978
*
On Roll Damping Force of Ship
-Effect of Friction of Hull and Normal force
of Bilge
Keels-by
**Yoshiho IKEDA
,Yoji HIMENO
****
and Noria TANAKA
* published in the Journal of The Kansai Japan, No.161 (1976)
**
Research Assistant , Univ. Osaka Pref.***
Associate Professor****
Pröf essorLb. y. Scheepsbouwkumle
Technische Hogeschool
DellE
Report of Department of Naval Architecture, University of
Osaka Prefecture , No.00401 , Deceiriber 1978
Department of Naval Architecture University of Osaka Prefecture
Mozu-Umemachi , Sakai-shi Osaka 591 Japan
On Roll Damping Force of Ship
Effect of Friction of Hull and Normal Force of Bilge Keels
-by Yoshiho IKEDA , Yoji HIMENO
and Nono TAÑAKA
SUMMARY
This paper presents an. atempt of calculating frictional roll damping force on the basis of three-dimensional unsteady turbulent boundary layer theory. The results of the calculation show fairly good agreement with experiments and also with the Kato-Tamiya's empirical formula.
Furthermore, a formula for ròll damping due to the normal force of the bilqe keels is proposed on the basis of the recent studies on the drag coefficient of oscillating bluff bodies. It is found that the drag coefficient of the normal
force of the bilge keels depends only on the roll amplitude,or so called period parameter, and that the normal force affects not only a nonlinear roll damping but also a linear ones. The effect of advance speed on the normal-force damping of bilge keels is also found to be quite small.
.1. Introduction
Recently, the theory of ship motion has been developed rapidly, so that the ship motions can be calculated by the strip theory even in the ,nitiaI stage of ship design. On the roll motion,howeven, the difference between the values by the theory and the ones measured is still large,which is mainly caused by the diffic-ulty of the treatments for the viscous effect and the forward speed effect on the roll damping. Although several empirical formulas for actual. ships have been proposed 1) these effects are not always taken Sufficiently into account.
Therefore, it seems to be unavoidable at present stage to take an experimental value into the roll damping term for calculating the ship roll motion in sufficient
accuracy.
In orden to establish a more reasonable prediction formula for the roll
damping, it is necessary to study all the aspects of the damping in detail. Some studies on these components were carried out separately
2)7)
The authors have also been studying recently on the components mainly due to the fluid viscosity, namely, the frictional, the bilge keel, the eddy making and the lift components. In this report the authors deal with the frictional damping at forward speed and the damping due to the normal force of bilge keels.Although in an actual ship, the ratio of the frictional component to the total roll damping is quite small, it can not always be negligible in case of small ship model, so that an error might possibly occur when one would estimate the roll motion of an actual ship with the roll-damping value obtained from the experiment on a small model. In the absence of ship speêd, Káto derived a formula for the frictional damping from a skin friction law of a flat platè in steady flow intro-ducing a concept of effective Reynolds number. The effect of advance speed on this component was also studied by Tamiya and et al , who established a simple formula based on the theoretical modeling and the experimental results for rolling ellipsoid models. It should be noted that they. used a form of the unsteady boundary layer theory in their analysis for the fast time In order to inquire the validity of their formulas, the authors carry out the calculation of the frictional roll damping of ellipsoid bodies at frward speed using the more detailed form of the three-dim-ensional turbulent boundary layr equations.
The bilge keel effect on the roll damping is also important.Tanaka, one of the authors, studies this effect dividing it into two components after Bryan, the normal force component and the surface-pressure one , and proposed empirical formulas for predicting these components. Also, Kato proposed a fórmula on the basis of the Tanaka's experiment and a theoretical consideration. In these ages the problem of the normal force of the bilge keels was usually considered on the analogy of the draq of a flat plate in steàdy flow.
In recent days, however, many works on the unsteady drag of the oscillating cylinders have been carried out. In this paper, the authors propose a new formula for estimating the roll damping component due to the normal force of the bilge keels oh the basis ôf the results f these recent works. -
-1-2. Frictional Roll Dampingat Forward Speed
2.1. Theoretical Calculation of Frictional Roll Damping. at Forward Speed
Consider here än ellipsoid rolling stèadily about the longItudinal axis and moving forward at constañt advance speed, and let the coordinate system be as shown in Fig i In this figure, u,v and w are the velocity components in the boundary layer in x,y and z directions respectively , and U the velocity of flow at the Outer edge of the boundary layer.
As the basic equations wé:can take the boundary layer equations in x and z directions and the continuity equation We assume that the term of au/at can be neglected in a small oscillation considered here, the basic equations can be written as the forms,
&u au dU 1 a u
+ V = U +
-8x 3y dxpay
-+
at ax ayPay
i a av--(ru)+=O
rax aywhere TX and Tz denote the shear stresses in x and z directions respectively in the boi.indary layer. The boundary condition can be expressed as follows,
at y=O
u=O, w=-W(=-rg0CO31)
at j-ô u=(J, w=Owhere 6 represents the bount1ji layer thickness, W the velocity of the hull in roll direôtion, w the circular frequency and O the roll amplitude.
Eq (1) can be solved independently since it involves no term of w The integral method which is usually used oñ solving the steady boundary layer problem, can also
be adopted for solving, this unsteady problem.
Integrating Eq (1) and Eq (2) along y from the surface (y=O O) to the edge of the boundary layer (y=6), we can obtain the momentum. integral equations as follows,
-+ (H-+ 2)--- !
dx Udx pU'
(5)
where 62.is the momentum thickness, H the shape factor (=61/62 ó the displace-ment thickness) and T and Twz the frictional stresses in x and y directiòns respectively.
In the calculation-of Eq.(4), we can use the power law as velocity profile, the entrainment equation and the Ledwieg-Tiliman' s skin friction formula For solving Eq. (5), the velocity profile of w should be needed.Since we have no experimental data of this profile at present -tithé, we assume this profile- can be represented by the Mager model which is often adopted in the calculations of three-dimensional turbulent boundary layer in steady flow. TO beg-in with, ve áonsider the case of ari oscillating external flow, and assume as follows,
ß ßocos(ót±)
(6)
As shown in Fig.2 ,Ct thé angle between x axis(positive) and the directIon of the
outer potential flow, and ß the angle between the directions of the outer potential flow and the shear stress at wall. We note that B oscillates with the amplitude
o
and the phase difference c from that of the motion W. FrOm Eq. (6) the velocity profile w for an rolling ellipsoid takes the following forth.
-
-(7)
- ß=ß0cos(,j+)
The frictional stress in z direction is given'asj
t,,,=tw:
(ß+a)=o
Cos (wt+T) (8)from that of the velocity of motion. By integrating the value T0 over the surface of. an ellipsoid, the frictional roll damping can be obtained. Rearranging Eq. (5) by substituting Eqs.(7) and (8), we can get the following differential equations
for o and
E.
dß de
cose fl0sin
e = Cß0 + C 9
dx dx dß0 de
sin e +ß0cos e_d_ CiPo + C,9, dx
Eq. (9) can be solved numerically with, the specified initial values of I3 and E
by means of Runge-Kutta-Gill method.
It may be reasonable to take zero as the initjal values of and E because at the leading edge of the body the boundary layer is quite thin. But if these assumptions would be taken, Eq. (9) would become uncertain. Therefore we assume here only that =0.O, and from Eq. (9) we. obtain the forms,
dx (10)
e= tan'(C22/C2)
so that we can assume dE/dx=0.0 at the initïal point.
Figs.3 and 4 show the local skin frictïon coefficient c directed to the main flow, the crosswise skin friction coefficient c (=2T0/p(rwe0)2) of the amplitude
T0 of the frictional stress Twz and the phase. difference y between the outer edge velocity and the frictional stress. The value c.f decreases toward the downstream, and the separation occurs near the end of the ellipsoid. The coefficient Cf0 and the phase lag y have their maxima, but locations of these maxima do not coincide with the stations of the maximum cross section. These positions slide downward both with the advance speed and with the roll period. The reason is because the
rate of growth of becomes slow with advance speed and roll period.
In Fig.5 the calculated results of the frictional damping coefficient BF are
shown. It is found from these results that the effect of the roll period on the frictional damping-grows as advance speed increases. These results are also in agre-ement with the experimental values by. Tamiya et al 8)
. In the same figure, the calculated values by the Kato's formula are also plotted. The values of the present solution is smaller than the one of the Kato's formula at low forward speed. The
reason for this seems because the velocity profiles-used 'in this paper may not be-appropriate in the low speed range.
As seen from Eq.(9), is proportional to the roll amplitude 0, and E is independent of 0, so that from Eq.(8) T is proportional to 00 and finally the
frictional roll damping coefficient BF is independent of 8. In other wards, the roll damping turns out to be linear. This result coincides with the Tamiya's
analysis entirely.
The effect of the centrifugal force can be taken in the basic equations as
follows,
&u 8
du w°dr
18v,-U+Vj u+T±
(11)8w 8w 8w
uw dr
i 8r
In this case Eq. (ll)can n6t be solved independently, because Eq.(ll) coritainsthe: unsteady term. Integrating Eq. (11) with time t from O to T(=2Tr/w) , we get a
time-independent equation which can be solved separately from Eq. (12). Through the calculated resúlts shown in Fig.4 we can conclude that the effect of the
centri-fugal force is negligible.
2.2. Experiment and Discussion.
The free roll test was carried out for an ellipsoid (L/B=5.0) model, the dimensions of which are shown in Table 1. The roll axis were kept to coincide with the longitudinal axis, and the roll period were varied in four values between T=0.99 and l.93sec by changing the positions of the center of gravity-.
On analyzing the test data, the equatièn of roll motion is considered as the following linear form.
(l+A)+BF+CO=O
(i:-3-The frictional roll damping coefficient BF can be obtained from the energy loss AE
in one swing of free roll.
BF T 4E
TfwGf49\
e
i-)
And the non-dimensional form is defined as the forni.
'
Tif
(15)
The term O in Eq. (14) represents the decrease .of the roll amplitude in one swing,
and V the. displacemént vblume.
The comparison between the calculated results and the experimental ones is shown in Fig.6. The broken line in the figure stands for the results of the Kato-Tamiya's method which predicts at first thé frictional roll damping at zero advance speed by the .Kato's formula and then multiplies the Tamiya's function expressing the effect of advance speed. The experimental resùlts at zero forward speed are ingood.agreement with the Kato's formula. The present calculation are also in fairly good agreement with the experiment èxcept for the low-advance-speed range. The Tamiya's formula is given as the multiplier,
BF/BFQ=1±a2
(2=U/wL, a=4.1)
where BFO represents the BF valúe at zero advance speed. The estimated esults by the Kato-Tamiya's formula are also ïn approximate agreement with the experimental
ones. These results say that the effect of advance speed on the frictional roll
damping depends only on the parameter X and the error whidh may be caused by their assumptions seems to be included in the empirical constant
a.
Although the presentcalculation can give a better prédiction in the high speed range than the Kato-Tamiya,'s method, it is rather tedious and complicated. Furthermore1 considering the ratio of the frictional component to the tOtal roll. damping decreases with advance speed, we can safely conclude that the Kato-Tamiya's method is sufficiently accurate in practical use. The empirical constant a is about 5.8 for the present experiments, but it may not be necessary to change the original value 4.1.
3. Normal Force Component of Bilge Keel
In this chapter, a formula for the roll damping due to the normal force of the bilge keels is deduced on the basis of the recent studies on the drag coefficient of oscillating flat plates, and the predicted values by the formula are compared with the experimental ones. The effect of the advance speed on this component is also studied experimentally.
3.1. Formula of Normal Force of Bilge Keels
The drag F acting on a bluff body ïn steady flow can be expressed as,
F=pCnSU2
. (1)where S represents the maximum cross-plane area and U the velocity of the incoming flow. At high Reynolds number, the drag coefficient is almost independent of
Reynolds number and it depends only on the shape of the bluff body.
In unsteady motion, however, the value of the drag coefficient CD varies with the kind of motion. In a constant-acceleration motion, the flow pattern around the body depends on the.parameter Ut/D ( t; time, D; the representative length )9) In an oscillating motion, it has been clearified by Keulegan and Carpenter 10) that the drag coefficient CD depends also on the similar parameter, the so called period para-meter UmaxT/D (=2TrA/D), where Umax represents the maximum speed in the oscillatory
motion, T the period and A the amplitude. In other wards, the drag coefficient CD in oscillating motion is indepéndent of the frequency of the motion, but depends
both on the shape of the body and on the amplitude of the motion. This characteristics has been confirmed by the experiments by Sarpkaya et al , Paape et al]-2) and Shih
For applying the results of these experiments to the normal fOrce of the.bilge keels, we can replace D in the period parameter with twice of the breadth of the bilge keel bBK , so that the period parameter for the bilge keels is expressed as
TrrOm/bBK.
The measurement of the drag coefficient of the bilge keels which is attached on an ellipsoid (Fig.9) were carried out on free roll tesl. The drag coefficient CD can be related to the energy loss EE in one swing roll as follows,
_3
4Eb,1r3ûìO,
where 'BK denotes the length of the bilge keel, r the distance from the roll axis to the bilge keel, 0m the mean roll amplitude andw the circular frequency. The results of the measurement are shown in Fig.7, in which we can see that the drag coefficient of the bilge keels is in good agreement with those of oscillating flat plates if the period parameter is suitábly taken as above mentioned. Therefore we can safely say that the effect of the hull boundary layer on the bilge keel is not so large. In
the same figure, the result of the drag of the bilge keel attached on a two-dimen-sional model (midship sectïon, B=O.25m, d=O.lm, bilge radius=O.03m) is also shown, and it is in good agreement with that of the oscillating flat plate. And the effect of the bilge radius on the bilge keel drag seems to be small except for the case of small bilge radius as concluded by Tanaka
From these experimental results, the drag coeff-icient CD of the bilge keel can be fitted to the following empirical function,
( b,,
C=22.59)-1 2.40
(19)4< rr em<20
and then the energy loss E in one swing due to a pair of the bilge kèels takes
the form.
4E cu (20)
The extinction coefficients a and b (O=aOm+bOm2) óan also be expressed throuth the relation E=WGMOmO as the forms.
3OPIbL,r'c,î r WGM
b= 3.2,rPlb, r2w2
180WGM
From Eq. (21) it is found that the damping due to the normal force of the bìlge keels consists of not Qnly the nonlinear damping (coefficient b) but also the linéar damp-ing (coefficient a) which is proportional to the square of the breadth of the bilge kee.l because the drag coefficient CD contains a part in inverse -proportion to the period parameter TrrOJbBK. In Fig.8 , the extinction coefficients in Eq. (21) are compared with the measured, and the agreement seems to be good
3.2. Comparison with Results of Forced Roll Test
The forced roll tests of the ellipsoid with a pair of the bilge keels (bBK=lOmm 1BK=600mm, Fig.9) in which the roll axis were adjusted to the longitudinal axis of the ellipsoid, were carried out. In this case, the measured roll damping moment consists of two components, that ïs, the frictional component and the còmponent due to the normal force of the bilge keels. Assuming that the frictional component is not influenced by the bilge keels, the damping moment due to the normal force of
- the bilge keels is derived by subtracting the damping measured for naked hull from
the one with the bilge keels. The experimental results are shown in terms of the non-dimensional equivalent roll damping coefficient N From (19) the
predic-tion- fórmula of BN can be expressed as the form.
BN=-Prlw(22.5 br+24ObrO)
(18)
The agreement between the measured and the predicted values shown in Fig.lO is fairly
good. For the case of the bilge keels attached on a circular cylinder measured by äsai and Tàkaki 15) , the agreement is also good as shown in Fig.11.
Fig.l2 shows the results of the measurement on the damping coefficient BN due to thé normal forcé of the bilge keels at forward speed. This figure indicaté us that the effect of the advance speed on the damping moment due to the normal force of the bilge keels is quite small, so that we can safely use the present formula
(Eq. (22)) even in the presence of forward speed.
4. Conclusions
The frictional roll damping and the damping due bilge keels are treated throuth experiments and some The following is the conclusions obtained.
The calculated results of the frictional damping dimensional unsteady turbulent-boundary-layer theory with the experimental ones.
The Kato-Tamiya's formula for predicting the frictional roll damping is confirmed to be accurate for practical usage.
A new formula for the roll damping due to the normal force of the bilge keels
is deduced.
It turns out from the present formula that the damping moment due to the normal force of the bilge keels consists of both nonlinear damping moment and linear one.
The effect of advance speed on the damping due to the normal force of the bilge keels can safely be neglected.
Acknowledgment
The present study was carried out as a part of the research workes of the l6lth Research Committee of the Shipbuilding Research Association of Japan. The authors should like to express their gratitudes to Prof. Shoichi Nakamura,Osalça University, of the chairman of this committee and to the committee members by whom many fruitful guidance and discussions were given.
A part of this study was supported by the Scientific Research Fund of the Mini-stry of Education. The computers at University of Osaka Prefecture was used for the calculations.
Soc.. of Naval Arch.
H.Sasajima Effect Arch. Japan , No.86
Part 1 - Part 4
to the normal force of the theoretical considerations.
at forward speed using three-are in a fairly good agreement
Re feren ces
Y.Watanabe,S.Inoue and T.Murahashj The modification of Rolling Resistance Full Ships , Jour. of Soc. of Naval Arch. of West Japan , No.27 (1964) ,p.69 T.Hishida Studies on the wave-making resistance for rolling of ships
-- JapanNo.85,86,87,89 (1952,1954,1955,1956)
of Bilge Keel in Rolling of Ship Jour. Soc.
(1954) , p.285
N.Tanaka A study on the bilge keel,
(1957,1958,1959,1961)
A.Yumuro and I.Mizutani : A.study on Anti-Rolling
Engneering Review, Vol.10, No.2 (1970') 'p.107
H.Kato : On the frictional resistance to the rolling Arch. Japan, Vol.102 (1958), p.115
H.Kato Effects on bilge keels on the rolling of- ships ,Jour. Soc. of Naval Arch. Japan,Vol.1l7 (i965) ,p.93
S.Tamiya and T.Komura Topics on Ship rolling Characteristics with Advance Speed, Jour. Soc., of Naval Ar-ch. Japan, Vol.132 (1972), p.159
T.Taneda Obsearvatjon of Viscous Flow around Bodies , Proceeding of the Syimposiurn on Viscous Resistance, Soc.. of Naval Arch. Japan (1973), p.35
10) G.H.Keuiegan and L.H.Carpenter Forces on cyilinders and plates in an oscillat-ing fluid, J.Res. Nat.Bur.Standards ,Lx No.5 (1958), p.,423
il) T. Sarpkaya and O.Tuter Periodic FlOw About Bluff Bodies Part 1 Forces on Cylinders and Spheres in a Sinusoidally Oscillating Fluid, Naval Postgradu-ate School, Carif. ,Rep.,No. NPS-59 SL 74091 (1974)
12) A.Paape and H.N.C.. Breusere : The Influence of Pile Dimension on Forces Exerted by Waves , Proc.
l'i-h
Conf. Coastal Eng. ASCE, Vol.2 (i969) ,p.840Fins (2), Ishikawajima-Harima ships, of Naval No.l0i,l03,105,109 fo r Jour.
Jour. Soc. of Naval
3i
C.C.Shih and H.J.Buchanan : The drag on oscillating flat plates in liquids at low Reynolds nunibers, Journal of Fluid Mechanics, Vol. 48 ,Part 2 (1971), p.229 N.Tanaka Bilge Keel , Proceeding of the Syimposiuin on Seakeeping , Soc. Naval
Arch. Japan (1969) 'p.143
Japan Ship Research Association, SR161 Committee : Ori the prediction of Ship Performance in Wave , Rep.No.257 (1975)
-7-Table 1. Particulars and test coñditiòns.
Fig. Ï. Coordinate system.
. Fricej003]. stre0
Fig. 2. Schematic view of velocity profile.
0000 0004 coo .0002 o FP AP
Fig. 3. Boundary layer solutions at constant frequency (EdO. 604). 0.2 Y (rad) O -0. 2 TP
without ccntrjfucal effect
C-)
N
woth ccntrjfuuj effoct
Fig. 4. Boundary layer solutions at constant
advance speed (R=1. 32x106). Length L 1. 50m Breàdth 13 0.30rn Draft d 0. 15m Displacement 35. 34kg ¿a (=w2 d/g) =0.617 GM 0.0373m =0.387 =0. 0238m
-
-=0. 308 -- =0. 0175m =0.162 =0.0122m u 0 .00 06 .00 04 C10 .0002 o 0.006 0.004 C0 0.002
0.001 L Kto-Taflhiyas method 0.0005 o o o o measured à Kato's formula present method o
Fig. 5. Calculated results of frictional roll
damping coefficient. r,10.617
(.:d)
L 106 Bl.OS X 10-R5.26X i°
present method Kato's formula o 9-0.5tô(a )
ad o oFig. 6. Frictional roll damping coefficient of ellipsoid model.
measured mean value by Keulegan et al,Shih et al and Paape et al.
present formul
0°
o o o
mark b5 X number test method model
o O A 0 1.Ocm.x i 1.Ocmx2 1.5cm
0.7cinxl
free roll
ellipsoid0 0.9cm x i
press. dif,.
2-dim. c1iñder5 10 15 20
UsaxT 2bBK bBx S
Fig. 7. Drag coefficient of bilge keels.
o 0.1 0.2 0.3 F.1 0 0.1 0.2 0.3 F
0.001
0.0005
0.002 0.004 0.001 0.-002 a
r
5mm o. b5present cal.
10mm 15mm b8Fig. 8. Extinction coefficient a and b.
- present
0.006v formula
3 0.5 1.0 - Cd 1.5
Fig. 11. Damping- coefficient due to thé
normal force of bilge keels.
0.005. -. A ---o-u-
-o -ø13.5°
O A. : CdO 309 D :E=O.l67 Ao eid'
measured o . by lasai et a].. U Ø.] 0.2 - 0.3 0.4Fig 12. Effect of advance spedon the
normal force of bilge keels.
0.005
Fig. 9. Ellipsoid model with bilge keels.
BN
o measured at forced roll test
$o13.5
-0 0.1 0.2 0.3 0.4 0.5
te Fig. 10. Damping coefficient due to the
