Hydrodynamic forces acting on a vertical circular cylinder oscillating with a very low frequency in waves

Ocean Engng, Vol. 15, No 3, pp. 271-287, 1988.

0029-8(t18/88 $3.00 + .00 Pergamon Press plc

Printed in Great Britain,

H Y D R O D Y N A M I C F O R C E S A C T I N G ON A V E R T I C A L CIRCULAR CYLINDER OSCILLATING WITH A VERY L O W F R E Q U E N C Y IN W A V E S WATARU KOTERAYAMAa n d MASAHIKO NAKAMURA Research Institute for Applied Mechanics, Kyushu University 87, Kasuga 816, Japan Abstract--Hydrodynamic forccs acting on vertical cylinders forced to surge in very low frequencies in regular waves are investigated with the view of studying the viscous damping force acting on a moored floating structure undergoing the slow-drift oscillation in irregular waves. Experiments show that the drag coefficients of the cylinder oscillating with low frequencies in waves are quite differcnt from those of the cylinder oscillating in a still water. NOMENCLATURE

A A,,, B,,

K Kci, Ks, Kw

Projected area of cylinder nth-terms of Fourier expansions of Morison equation See the equations (13) and (5) nth-terms of Fourier expansion of measured forces Added mass- and drag coefficient for slow oscillation obtained from two harmonic forced surging test or slow oscillation tests in regular waves Added mass- and drag coefficient for fast oscillation obtained from two harmonic forced surging test Added mass- and drag coefficient of cylinder obtained from simple harmonic forced surging tests in low frequency in a still water Added mass- and drag coefficient of cylinder obtained from simple harmonic forced surging tests in high frequency in a still water Inertia coefficient of cylinder oscillating in low frequency in waves Inertia coefficient of cylinder fixed in waves Drag coefficient obtained from two harmonic forced surging tests analysed by using independent flow concept Diameter of cylinder Submerged depth of cylinder Inline force acting on cylinder Inertia force due to slow- and fast-oscillation Lift and drag force acting on cylinder Inertia force due to fluid motion acting on cylinder Acceleration of gravity Wave height Wave number (K=to2/g) Keulegan-Carpenter number for slow-, fast- and wave motion (Kc = UjTc/D),

M N RL,Rs,Rw

Mass of cylinder Integer (N=tos/wL or tow~toe) Reynolds number for slow-, fast- and wave motion (Re= ULD/v), (Rs-UsD/v),

T L , Ts-,Tw UL, Us, Uw

Period of slow-, fast- and wave motion Maximum velocity of slow-, fast- and wave motion Reduced velocity (ULTs/D or ULTw/D)

a,, b,

C.., Cl.. CAN, CI).% CAI.O, CI)I.O CA),O, CDS() CMs, CMso D d F F~,L, /~),.s

F~., F~ F..w g

Hu

(Ks = UsTs/D), (Kw = UwTw/D)

(Rw = UwD/v)

Ur

271

272

If'fr Y-Z Y- K~ El, E2

gA F

P 031. , {.0,,', 03 W

V

WA'IARU KOfERAYAMA a n d MASMtl/~O NAKAMLIRA

Relative velocity of cylinder to fluid particle Cartesian coordinates Amplitude of slow and fast motion See Equations (15) and (17) Phase difference of slow motion to fast motion and wave m¢)tion Semi-amplitude of wa~c Kinetic viscosity Water density Circular frequency of slow-, fast- and wave motion Volume of cylinder. INTRODUCTION

IN THE estimation of the line tension of moored systems such as semi-submersible platforms and other floating ocean structures, slow drift oscillation is one of the most important problems. Investigations on the slow-drift oscillation have been concentrated on nonlinear wave drift forces. The amplitude of the slow drift oscillation of a moored structure depends on not only the magnitude of the wave drift force but also that of the damping force acting on the structure. Recently several investigations (Wichers and Sluijs, 1979; Nakamura et al., 1986) on the wave damping force were carried out and they concluded that the wave damping force acting on a moored structure undergoing the slow drift oscillation in waves is larger than that acting on the structure oscillating in a still water. There is a need to upgrade the tools for the estimation of the viscous damping force acting on the floating structure performing the slow drift oscillation, lnoue (1983), A n d o and Kato (1984) found that the viscous damping force acting on the structure oscillating in waves is larger than that in a still water. Experimental and theoretical studies are needed for having an understanding of why the viscous damping force increases in waves and the applicability of the Morison equation (Morison et al., 1950) to present problem should be discussed in the studies. Many experimental investigations on wave forces acting on a fixed cylinder (Keulegan and Carpenter, 1958) or the hydrodynamic forces acting on the cylinder oscillating in a still water were carried out, and several investigations (Chakrabarti and Cotter~ 1984; Koterayama and Nakamura, 1986) on the hydrodynamic forces acting on the cylinder surging at the same frequency as the incident waves were carried out. They found that the Morison equation extended to the relative motion concept is applicable in these cases. Moe and Verley (1980), Koterayama (1984) showed that the one-drag coefficient Morison equation is not valid in describing the hydrodynamic forces acting on a vertical cylinder in waves and current. In this report, an experimental study on the hydrodynamic forces acting on the cylinder oscillating with combined low frequency and wave frequency is carried out in order to discuss the applicability of the Morison equation to the two frequencies flow field problem with the view of studying the viscous damping force acting on the floating structure undergoing the slow drift-oscillation. MODEL Model experiments were carried out at the Experimental Tank for Sea Disaster ( L x B × d = 8 O m x 8 m x 3 m ) I of Research Institute for Applied Mechanics of Kyushu University. The mechanical part of a forced surging apparatus and models are shown

Slow-drift oscillation

273

in Fig. 1. T o e l i m i n a t e the e n d effect of t h e c i r c u l a r c y l i n d e r o n the h y d r o d y n a m i c forces, d u m m y c y l i n d e r s are set at b o t h e n d s of t h e m o d e l c y l i n d e r a n d t h e slits b e t w e e n the m o d e l c y l i n d e r a n d a d u m m y c y l i n d e r are c o v e r e d with t h i n r u b b e r . T w o c i r c u l a r c y l i n d e r m o d e l s of w h i c h d i a m e t e r are 0 . 1 m a n d 0 . 1 2 m are u s e d , t h e y are s h o w n in Fig. 1. EXPERIMENTS E x p e r i m e n t a l conditions for M O D E L

I and M O D E L

II are shown in Tables 1 a n d 2.

d-oscillation

,paratus

Dyn,

~-

Model

2m

Model

I

Model

IT

FIG. I. Models. TABLE 1, EXPERIMENTAL CONDITIONS OF FORCED OSCILLATION TESTS IN STILL WATER

Model exp.

T~.

Yr.

I Slow osc.

9.6 14.4

0.1-0.5

K,.

Rxl0 4

Ts

Ys

Ks

R~xl0 ~

U,

0.46-2.3 6.28-31.4 0.31-1.5

I Fast osc. I Slow-fast II Slow osc. II Fast osc. II Slow-Fast

9.6 14.4

0.1-0,5

6.28-31.4

1.7- 8.4 3.0-15.0 0.1-0.5

5.1-26.1

6.0-15.0

0.2-0.5

10.2-26.1

1.2-1.6 0./12-0.20 1.1-12.6 0.51)- 5.9 0.02 1.25 11.46-2.3 1.2, 1.6 0.03 1.88 1t.511-1.3 0.8-5.2 /).31-1.5 1.2, 1.6 0.04 2.51 11.5-3.5 5.3 2.9

2.9

2.0 0.75 1.5

0.02-0.3 1.1-15.7 1).9 -13.1 0.02 l. 1 0.04 2.4 2.3

1.3 2.6

274

~VATARU KOIERAYAMAand MASAtlIKO NAKI,~MURA

"FABL[~

Model exp. I Wave test l Slow osc. in waves

7)

9.6 14.4

2.

EXPERIMENfA[

('ONI)IIIONS

OI,

SI.O~,

OS(

II.LAIION

ItS'IS

IN

\VA\IS

Y~

0.28-31.4

0.141.5

U.4~w2._; 0.31-1.5

.2. I.<+ (UJt-¢) (1172

1.0,-2.0

+2, l.i~ l.~

If.lib9-11.172

1.(~-2.(~ ~I+(l-7 [ ?

(

0 t4!3

I.()-3 li

I>t+` ] "

1.5

0,12

211

2 ~

11.(~7-I ~

H.~ 5 2

II

Wave test II Slow osc. in waves 6.0-15.0

10.2-26.1

0.2-0.5

2.9

7+

We carried out two series of experiments, namely series of Slow-fast oscillation test and series of Slow oscillation test in waves. The series of Slow-fast oscillation test are simple harmonic forced surging tests in low frequencies (Slow osc.), in high frequencies (Fast osc.), two harmonic forced surging tests (Slow-fast) as shown in Table 1. The Slow oscillation test in waves is wave force measurements on fixed cylinder (Wave test) and simple harmonic surging tests in waves (Slow osc. in waves) as shown in "Fable 2. Table 1 contains experimental conditions for two harmonic forced surging Icsts: the period TL, amplitudes Yt+ of the forced surging motion in low frequencies, Keulegan-Carpenter number KA (=UrTxjD, Uc --:2=YA,'T~ ) and Reynolds number RL (= ULD/v), where D is the diameter of the cylinder and v is the kinematic viscosity. It contains also the values of periods Ts, amplitudes Y~ of the forced surging test in high frequencies, Keulegan-Carpenter number K~ (UsT~/D, U~==2~rYs/T~), Reynolds number Rs and the reduced velocity U~ (--U~T~/D)+ An cxperimemal study (Koterayama, 1984) showed that the reduced velocity U, ( = U Ts/D) is the dominant parameter on wave forces acting on a vertical circular cylinder with a constant forward velocity U. In the present study, the reduced velocity defined by UA and T~ are expected as a dominant parameter. S l o w - f a s t oscillation YL= 0.5 rn TL = 14.4 sec to/ Ys=0.O3m T s = t 6 sec

Force v

Q ,7 #~t

v

v

v

5

~

-

-

~

v

v - ~ v

v

~'

-

10

15

20

25 t(sec)

10

15

20

25 t(sec)

Lift force

u_ ,~

5

1

FIG. 2. E x a m p l e of m o t i o n and m e a s u r e d force time histories in two h a r m o n i c forced surging test, +

Slow-drift oscillation

275

In Fig. 2, we show an example of records of the motion and measured forces in two harmonic forced surging tests in order to facilitate the understanding of the experimental method. Table 2 contains experimental conditions for the series of forced surging tests in waves, that is, Y,, Kt,, and Rz, it also contains periods Tw, double-amplitude Hu. of the incident waves, Keulegan-Carpenter number Kw (Uw,Tw/D), Reynolds number Rw (=UwD/V) and the reduced velocity U~ (=ULTw/D) where Uw = rrH~e kd/Tw, K~.-(2rr/Tw)e/g, d is the immersed depth of the model cylinder. In Fig. 3, an example of records of the motion, incident waves and measured forces in forced surging tests in waves are shown. ANALYSIS OF RESULTS OF THE T W O - H A R M O N I C FORCED S U R G I N G TESTS Data of the measured forces, the motion of the cylinder, and the surface displacement of waves are digitized with a sampling frequency 100 Hz and they are expanded in a Fourier series. The method of analysis is as follows; take 0Y horizontal and on the undisturbed water surface and 0Z positive upward. The surging displacement Y, velocity Y and acceleration Y are written as

Y

-

YL sin oJ/7 + Y~ sin(ms, t + ~j)

f = YLOL cOS tOLl + Ys,~oscos(mst + e,)

(1)

]P = - YMo~_sin m,t - Ysm~ sin (to s,t + e,) where e~ is the phase difference between the Slow oscillation and the Fast oscillation. ~o,_ is 2rdTt. and ms is 2rr/Ts. Wave

E!f Hw=0.139mTw:l.6sec V V •V V Slow oscillation YL = 0.5m TL =14.4 sec

E~I

~Lo/Force ~rA.A

~ ~.AAA~AAA

~ ~A A

91

a~_ Lift force o

I-

-.~ -

,~F 5

----^=

-

10

-

- v~nv

15

A-^

- .-

20

-

-~^v~

-

25 ~s~)

FIG. 3. Example of wave, motion and measured force time histories in forced surging test in waves.

276

WAFARU KO'IERAYAMAand MASAHIK{) NAKAMURA

Horizontal inline force F is written as the sum of the added mass- and mass-torce FAL due to Slow oscillation, FAS due to the Fast oscillation and the drag l\~rce 12, b~ the modified Morison equation in terms of relative velocity.

=

(O~CAI

- +

M)Yzoo~= sin (.+t

+ (pVCA.s. + M) Ys.to~-sin (tost +- e~) 1

- 2 pA Co [ Yt.to/. cos eoz.t + Ys.o~ cos (cost + • ~)] × IYzeo~. cos o~tt + Y.~~s cos (o~st + •~ )i

(2)

where P is the water density, V the volume of the cylinder, M the mass, A the projected frontal area, Cat. the added mass coefficient for the slow oscillation, C~s the added mass coefficient for the fast oscillation and Cz~ the drag coefficient. For the simplicity of the data analysis to, is selected as tos = Ne0L (N is an integer) in our experiments. The drag force Fo is expanded as a Fourier series. l

Fz~ : - 2 pACt){A i cos tort q ,4 ~ cos 3tot t + . + Ax 2 cos[(N-2)tozt+

N

N

"~

, -~

•']

+ A,.,, cos (Ncozt + el) +AN~2c°s[(N+2)

t°tt+

(3) N+2 N •'1+ ....

}"

The numerical analysis shows that even terms and odd sine terms are much smaller than the odd cosine terms, therefore they are omitted from Equation (3). The measured horizontal inline force F~p is expanded as Fcxp = at) + a l COS oat t +

+ be sin 2tOrt + . .

b~ sin totJ +

a 2 cos

2tot.t

• + ax cos (Ntott + e~)

+b,vsin(Ntott+ej)+a.+lcos[(N+l)tott+

N+l N el]+ ....

(4)

Upon comparing Equations (2), (3), (4), we have Co, CAL and CAs. We have plural values of Co by comparing Equation (3) with Equation (4). The dominant terms in Equation (3) is A~-term and AN-term, Coc is the drag coefficient obtained from A~term in Equation (4) and Cos is the drag coefficient obtained from AN term,

Slow-drift oscillation

277

a, V

al

pA tOl-

2 pAtOL Y~Aa

AN

(5) bi M CA L = -, -pVox/. YL pg

bx CAS =

--,

pgtO~.Ys,

M pV

where ocYZLA,=A~, o ~ . ~ . A u : A x . In case the values of CoL coincide with those of CDs, we may conclude that the Morison equation is suitable when extended to the case of relative motion. We discuss on this point later. ANALYSIS OF RESULTS OF THE FORCED S U R G I N G TESTS IN WAVES The forced surging motion Y, velocity ~" and acceleration ~" are given by Y = Y~. sin tozt ri~toL cos tod

~" :

(6)

~" = - YLto~. sin colt . Displacement g of the water surface at the central axis of the cylinder is written as = gA sin ( K Y + towt + e2)

(7)

where e, is the phase difference between the surging motion and incident waves,

tow= 2rr/ Tw. The orbital velocity u and acceleration it of a fluid particle at the depth z on the axis of the cylinder are written as u = -tow¢A e -K: sin ( K Y + towt + e2)

]

--tOW¢A e -K= cos ( K Y + tOwt + ~2) . I

h

(8)

The values of u and iJ vary with z as shown in Equation (8), however, the length of the model cylinder is short as compared with the incident wave length and the error caused by using the constant value of u and ~ at the midpoint of the model cylinder instead of the variable value along z axis is less than five percent. The force F acting on the cylinder is written as the sum of the added mass- and mass-force FAt., inertia force FM due to the orbital acceleration of a fluid particle, drag force FI~. F=

FAL + FM + FD

: (o~TCAL +

M) to~ Y L

s i n t o l I -- DVCMsto~V~A e - K ' l

1

× cos(KY + towt + ¢2) - }pACo[tOtYL cos tOid

+ tOWgAe-KJ sin ( K Y + tOwt + ~2)]ItOLYLCOStOd + tOwgAe -KJ sin(KY + tOwt + %)1

(9)

where Ca4s is the inertia coefficient. In Equation (9), the convective acceleration force is neglected because it does not contain tOL- or tO~-frequency term which we are concerned with.

278

WAIARU KOII-RAYAMAand MASAttlKO NAK~',MUR,'~

For the simplicity of the data analysis, to; is selected as to s=Nto;~ (N is an integer} in our experiments. Since Y is dependent on time t as shown in Equation (o), f~~ as well as f.,, is non~ linear. Tc is much larger than T~, therefore /:~; does not contain the Io~ frequency terms. The drag force /;), and inertia force /:'w is expanded as 1 F,~ = - 2PACt~{A~ costoct + A~cos3to;t * , . ,

* B~ ~ sin [(N-- 2) to;t + + Bx ,sin [ ( N - l)toct +

N-2 N ~21 N- 1

+ B,~,sin (Ntoct + ~e) +B,,~.+,sin[(N+ 1)to, t +

N +-1 N tel...',

.

(10)

FM = pgCM.sto~. ~Ae ,~d{... _~ A x e COS [ ( N - 2 ) t o t t + + A,,v I c o s [ ( N - 1)co;t +

N-I N ~-~]

+A,~cos(N~oct+e2)+Ax,~cos[(N+l)toct+ + A , ~ . ~ e c o s [ ( N + 2 ) toz.t+

N-2 N %]

N+I N ~2]+- • .}.

N+I N e2] (11)

The measured horizontal inline force F,:~p is expanded as; Fcxp = ao + al cos toll + b~ sin o2;j + a2 cos 200it + aN cos (Noatt + %)

+ b2 sin 2coLt + . . .

+ bN sin (Ntott + e2) + . . . . .

(12)

We get the plural values of CD and CMS by comparing Equations (5), (6) and (7). We denote the drag coefficient obtained from the first term by CDL and that obtained from the Nth term by Cos. The experiments and the theory show that the values of CMS obtained from N - l t h term, Nth term and N + l t h term coincide each other within 10% error, therefore we may determine the value of CMS by using only Nth term. We obtain CDc, Cos, CAl. and CMS a s follows; aj

bl

M

2PA toT. Yec ,41

bN CDs= i

72PAo~w~Ae

Ka

BN

aN Cgs = pV~O2~Ae~KdA~

(13)

Slow-drift oscillation

279

where toZ ~ Y~A)=A ). If the value of Cos coincides with the value of CDc, the Morison equation shown in Equation (4) is valid in this case. We will discuss this point later. RESULTS

AND

DISCUSSIONS

The results of the forced surging tests are shown in Figs 4 and 5. In Fig. 4, the drag coefficients Cr~t,o obtained from the forced surging tests in a still water are shown. On the model I, the periods TL of the forced surging tests are constant. On the Model ll, the Reynolds numbers RL are constant. The results of experiments show that the value of C;)LO depends on not only KL but also Rt. The Reynolds number for Model 11 are much greater than Model I due to the water temperature. In Fig. 5, the added mass coefficients CAt~O are shown as a function of K~.. The values of CAZ.O are not remarkably dependent on the Reynolds number as Ct>t_o. The drag coefficients Cos() obtained from the fast forced surging tests and the wave force measurements are shown in Fig. 6. The results of the wave force are scattered. In these experiments the Keulegan-Carpenter number is small and the drag force is much smaller than the added mass force or the inertia force, therefore the experimental errors have a great effect on the value of the drag coefficeint. We can not discuss quantitatively on the drag coefficient in the range of small K - C number in this study.

20 CDLO

1

1.0

V

. . . . . . .

Model V TL 9.6

/

I 14.4 1.7~ L4 3.0-15.0

I

10

20

30 • KL

Fro. 4. Drag coefficient C , ~ , obtained from slow oscillation tcst.

I ] 9.6 14.4 1.7~8.413.0~15C [RLXl0"~ 0.46-2.3 0.31~1.5 5,3 2.9 Model TL

1.0

'~

[,

.-o-I---~

---v--[-~-

CALO

0.5

0

I

10

I

[

20

30 )

FIG. 5. A d d e d mass coefficient

KL

CA,.O obtained from slow oscillation test.

WAI'ARUKOTF.RAYAMA and MASAtllKONAKAMURA

280

!~ill

10 @

C~o

~,

......

~

:

t

..

/•

LW.ve force rr~osurement] • 7 " )

0L-_ .................................

5

FIG. 6. Drag coefficient

C,~o

l

CMso-I.0 J

10 -~

_ .........

15

Ks.Kw

(';,s,, obtained from last oscillation test and wave test.

%~%

.

-E;- [3--

b.

0

L

t____

10

5

....

FIG. 7. Added mass coefficient

15 * Ks, K w

CAS() obtained from fast oscillation test and inertia coefficient (M~,, obtained from wave test.

"Is o n d

fost

Ksof

1.2 seC

4O

[]

G,

i.25 1.88

2.51

]

i.6 sec

o a

6

o

t~

A

2.0 (5

Lx I

.....

10 FIG. 8. Drag coefficient

o I___ 20 KL

i

--

30

C'm. obtained from slow-fast oscillation test analysed by using Independent Flow Concept.

In Fig. 7, the added mass coefficient CASO and the inertia coefficient CMso are shown. In these figures, CASO is compared with CMso --1, In case of Model II, the value of CASO agrees with that of CMso --1. In case of Model I, the value of Cnso is smaller than that of CMso - 1 when K,., K,, < 5.

Slow-drift oscillation

281

These values of CDLo, CALO, Ct)so, CMso and Caso obtained from simple harmonic surging tests and wave force measurements are compared with CDc, CAL, Cz)s, CMs and CAS obtained from two harmonic surging tests and forced surging tests in waves. In this paper, we adopt the Morison equation extended to the relative velocity concept in which the drag force is assumed to be proportional to the square of the relative velocity as shown in Equations (2) and (9), on the other hand Moe and Verley (1980) proposed the Independent Flow Field Model which is based on the superposition of two independent flow fields and two term drag forces corresponding to each flow field. The drag force is written in the Independent Flow Field Model as follows; for the two harmonic oscillation tests 1

t

1

,

Fo = }pACocy2L ~o7.cos ~oLtlCos~oLti + ~pAfosY~so~2cos (COst+ ej)[cos(tost + ~l)l for the forced surging tests in waves 1

,

FD = 2PACDL Y~ o3~ cos (octI COS~oztl 1

- }pAC'Ds tO2w~2e-2Kdcos(KY + tOwt+ e2) (14)

x Icos(gY + o~.,t + ~2)1

where C~c is the drag coefficient for the slow oscillation, C~,s is that for the fast oscillation or the wave motion. In Fig. 8, examples of the drag coefficient C ~ obtained from two harmonic surging tests by using the Independent Flow Field Model are shown. The solid line in the figure indicates CDLO which is the drag coefficient obtained from the single harmonic surging tests. The results of C[u. are different from those of Ct, LO in the region of KL < 20 and they vary widely with the amplitude of the fast oscillation. It is difficult to estimate the drag forces in two frequencies flow field problem from the figure obtained by using the Independent Flow Field Model. 1.0

• ,:X----

CDL

T

0.5

RL=2.9xlO ~, Rs=2.4xlO 4 l's =O.?Ssec, Ks =1.05 0 rs = 1.5 sec, Ks= 2.1 ,~ rw= 1.5 sec, Kw= 2.1 • <3-

0

¢0

0

o

-

- O

3'0

210 KL

FIG. 9. Drag coefficient

Cm. obtained

from slow-fast oscillation test vs KI. number.

\VAIARU K()TERAYAMAand MASAHIKO NAKAMURA

282

We return to the relative velocity model again. In Fig. 9. the experimental results of

(]/)t_ of Model II obtained by using the relative velocity model are shown as a function of KL. In the experiments, the Reynolds number R~ is constant. This figure shows that values of Ct)r depend clearly on Ks or 7~. In Fig. 10, C~)z. are shown as a function of the reduced velocity U,. ( -/J'~ I,,D) Each values of CTt~. are divided by Ct)t,, which are obtained from the simple harmonic surging tests at the same K~. number. In the figure, the solid line is the mean line of experimental results, and the broken line is the results of the forced surging tests in a steady flow ( K o t e r a y a m a , 1984). White circles, triangles and squares indicate the results on Model 1 and the black circle and triangle indicate the mean values on Model 11. In the case of Model 1, experiments are carried out in the constant Reynolds number. therefore, /J~ is constant in the experiments and the reduced velocity U, depends on only the period Ts of the Fast oscillation and wc obtain two experimental resuhs as a function of the reduced velocity from eight experimental results shown in Fig. ~). The solid line coincides with the broken line at /_!,< 3. At U, =-4.0, the wdues of ( ;,/'('/,t.~, in a steady flow reach to 2.0 but these obtained from two harmonic forced surging tests is less than 1.2. In a steady flow, the w)rtex shedding from the cylinder due to the steady flow is resonant with the oscillation of the cylinder. The resonance effects in two harmonic surging tests, however, are much smaller than those in a slcad,, flow. The vortex shedding caused by the slow oscillation resonate with the fast oscillation at the instant when the reduced velocity defined by the velocity of the slo,a ~,scillation and the period of the Fast oscillation (:-U~T's/D) coincides with the reciprocal of the Strouhal number. In the range of U, 4.(i (the reciprocal of the Strouhal rmmber in the experimental range of the Reynolds number is between 4.0 and 5.0, lhc .resonance p h e n o m e n a occur four times during one cycle of the slow oscillation and the values ol Ctn/C~,,.o are greater than unity. The denominator for Cz,L is greater than that for C D L O , therefore the drag force on the cylinder undergoing the two-harmonic oscillation is much greater than that on the cylinder surging in a still water. The Strouhal number of a circular cylinder depends on the Reynolds number and the critical value ( U, =4.0) r-

fi~a~--ITL

vs

"

144see

,'

, 2 s , c 1 1 . 6 s , c 1.2s,;; 1 ~

1.5~ Ks=1.25 o 4 ~ ~---~-[-~---

d

0.5

___-

"

=

® - ~--1 v----+---]

Model I - -~s~:1.s~.Kg~2J

COL

"i 9 ( sec c i

,j,

' ,

-

"--_

7"Exp. in '--. steady flow a. g..2- . . . . . . . ~

v

[3

]

0

®1.0 0

FIG. Ilk Drag coefficient

Ct,i

i

20

i

~0

i

40 U T~ 50 ) Dr= D

for slow oscillation obtained from slow-fast oscillation tesl.

Slow-drift oscillation

: 283

of the reduced velocity varies with the Reynolds number. In the range of Ur less than 3.0, the value of CDc/CoLo is smaller than unity and the drag force on the cylinder undergoing two-harmonic oscillation is smaller than that on the cylinder surging in a still water. In this region, the period T~ of the vortex shedding induced by the slow oscillation is longer than the period of the Fast oscillation and the vortex formation is disturbed in the presence of the Fast oscillation. In Fig. 11, CDc obtained from simple harmonic surging tests in waves are shown. The values of CoL shown in Fig. 11 coincide with those shown in Fig. 10. The orbital velocity of a fluid particle varies with the depth, and phenomenon in waves is supposed to be more complicated than those in the two-harmonic surging tests in which the velocity relative to the cylinder is uniform in the direction of the depth. Experimental TL 14.4 s e c 9.6 s e c T__v; 1.2sec 1.6sec 1.2sec 1.6sec Kw .6 Kw = 2.2

o t, ~ Model

1.5 Kw = 2.6

CO._~L

6 L 71

® v o

~ Tw =l.5sec, Kw=2.1 •

CDLO

T

/

,~

,"'-Exp. in / flow

<~

steody

,(5'

1.0

3--

0.5

I

r

I

I

1.0

2.0

30

40

5.0

- Ur= ULTw

D

Flo. 11. Drag coefficient C/)~. for slow oscillation obtained from slow oscillation test in regular short waves. Model I TL 14.4 s e c Ts 1.2sec 1.6sec Ks=1.25 o 6 Ks=1.88 4 x Ks=2.51 ~ ~ Modet H

3.0

13

CAt.

~

~AO

9.6 s e c 1.2sec 1.6sec ® v ~' 0 ,5

~ 0 I's=O.75sec,Ks=l ~ .05e~ Ts=l.Ssec,Ks=2.1 •

CAL0

T 2.0 6

1.C

I

1.0

I

2.0

1

3.0

~

[

I

4.0 5.0 • Ur: ULTs D

FIG. 12. A d d e d mass coefficient CAL for slow oscillation obtained from slow-fast oscillation test.

284

WA'IARU KOTERAYAMAand MASAItlKO NAKAMURA

studies (Chakrabarti and Cotter, t984; Koterayama and Nakamura, 1986) show that the Morison equation extended to the relative velocity concept can be applied on the hydrodynamic forces acting on the cylinder oscillating in the same frequency to the incident waves. The experimental results of the present study and the above mentioned study suggest that the results obtained in this investigation are applicable to the problem of an ocean structure undergoing the slow drift oscillation and the surging motion.in waves because the motion is decomposed into the two harmonic oscillations in still water and the simple harmonic oscillation in waves. In Figs 12 and 13, the added mass coefficients ( ~ t for the slow oscillation arc shown. These figures shows that C~,L. obtained from two-harmonic forced surging ,~csts and those obtained from surging tests in waves are much larger than those obtained from

I

3.0!-

I

Model

TL

14.4 sec

]w

9.6 sec 1.2$ec 1.6see

1.2see 1.6see

KwL~--1.6_

o

~

IKw=2.6

n

el

[__ Model II ~L Tw=l.Sse% Kw=2.1

, i

I

~

1.0i

v

®

"

i

v

0 [

J

---1.0

2~0

~

30

6

o .....

6]

I

,2;- ......... - T

,

~, v

•

=

v ± ...............

4.0 510 - - - + U r : U~)Tw

FIG. 13. A d d e d m a s s c o e f f i c i e n t ( . ~ , f o r s l o w o s c i l l a t i o n o b l a i r l e d l r o m s l o w o s c i l l a t i o n test in r e g u l a r s h o r t waves.

I

Ts

I

Ks=L25

CAs

!

CAS0

[ ~

A,

K s=2.51

~

'

Model I 14.4 s e ¢ 1.2 sec 1.6 sec

o o

Model

9.6 sec 1.2 sec 1.6 se(

(b

o

6

~

0

L '~

~

--

~

",~7-Exp. in s t e o d y " flow

0 t

21o

1.o

J

3'.o ----',

FIG. 14. A d d e d m a s s c o e f f i c i e n t

CAs

",

I

4.0 u,:.;s.o, U r = ""b

. . . . .

"" ......

for fast o s c i l l a t i o n o b t a i n e d f r o m s l o w - f a s t o s c i l l a t i o n test.

Slow-drift oscillation

285

surging tests in a still water. In estimating the period of the slow drifting oscillation of a moored structure, we should note this fact. In Fig. 14, the ratios of the added mass coefficients CAS for the fast oscillation obtained from two-harmonic forced surging tests to the added mass coefficient CASO resulted from simple harmonic forced oscillation tests are shown. The solid line indicates the mean value of experimental results. The broken line represents the ratios of the added mass coefficient obtained from the forced surging tests in a steady flow to CA,s,o. In the range of U, <3.0, the results of two harmonic forced surging tests coincide with those of simple harmonic forced surging tests in a steady flow. In the range of U,.>4.0, the results in a steady flow shows that CAS becomes minus, but CA.S-resulted from two harmonic forced surging tests does not take such an extreme value. In Fig. 15, inertia coefficients CMs obtained from simple harmonic forced surging tests in waves are shown. The broken line shows the results of wave force measurements in a steady flow. The tendency of the results is quite similar to the case of CAs. The experiments in a steady flow show that the values of CAs and CMs become minimum at U~=6.0 and they increase with U~ at U~>6.0. In this investigation, the experimental range of U~ is smaller than 6.0, and in order to know the tendency of C~.s, or C.~4~s,at Ur>6.0, another experimental study is needed. In Fig. 16, the drag coefficient for the fast oscillation obtained from two-harmonic forced surging tests are shown. The experimental errors have a great effect on the value of Cz)s, therefore we can not discuss the value of C~)s in detail. The values of C~)s scatter between 0.2 and 0.8 in the experimental range of U~, on the other hand, the values of Cruz. resulted from Fig. 4 and Fig. 10 are between 0.3 and 1.7. Apparently the value of CoL is larger than that of Ct)s. This fact shows that the one-drag coefficient Morison equation is not suitable for the present problem. The drag coefficient CDz~ is crucial for the designs of mooring systems for an ocean platform in respect of the slow drift oscillation. The value of CDL obtained by using the independent flow field model depends on the amplitude and period of the small fast oscillation or waves in addition to the Keulegan-Carpenter number KL of the slow oscillation. It is inconvenient to the designer of the mooring system that the value of Cz)r depend on two parameters. Here

TL

Tw 1.2 sec 1.6 sec Kw=l.6 o 6

CM~-I.0

CMSO-I.0 l

Model I 14.4 sec

1.0

~1

~ _ ~ _

"l

! 9.6 sec t2sec i~

Kw:Z2

t,

,k

v

lKw=2.6

a

6

o

Model

K

1.6sec t~ 6

0.5

0

[ 1.0

I 2.0

~ 3.0

@~-~Exp. in s t e o d y flow I J 4.0 " ' - .5.0 , Ur =ULTw . . . . . . . D

FIG. 15. Inertia coefficient CMs obtained from slow oscillation test in regular short waves.

286

WA'f'ARU KOTERAYAMA a n d MASAH1KO NAKAMURA

T~ 14.4 sec 9.6 Ts 1.2see 1.6see 1.2 sec 'l.6secl Ks=l,25 o 6 • ~ Ks=1.88 zx A v . ~, i Ks=2.51 o r~ O ' ~ Model I[ ....... i Ts=O.75sec,Ks=1.05. Ts=1.5sec,K~.~2.1~ ~ f ~ : . ~

I •

1-0i 6

,

-~-

ol

~.2

E

,~ 2

A '~1.0

. ~

2.0

x

p

.

in

® o

3.0

4.0

510

......~" Ur= U~.Ts

D

FIG. 16. D r a g c o e f f i c i e n t (',~.s for lasl o s c i l l a t i o n o b t a i n e d f r o m s l o w - f a s t o s c i l l a t i o n test

we propose a model for representing the force on the cylinder undergoing the slow drift oscillation as it oscillates in the wave frequency in waves as follows:

F = (pVCAL + M ) ~ . Y~sin oJ~t +(pVC,~, + M)Ysoa., sin(co.,t + ~ ) _

9VCM.sO~,~Ae-Ka cos(KY ~- c.,,,t ~- e~) 1

v

-

- 2PA Cos ~oZy2LA sCOSo~,t 1

- 2PACz~s~O~,Y~ ,4,s-cos(oJ..t + e)

(15)

where definitions of letters are the same as those in Equations (1) and 14), e is the phase difference between the Fast motion of the cylinder and the incident wave, The physical ground of Equation (15) is on the time dependent drag coefficient Co(t). Time dependent drag coefficient C~(t) is expanded as a Fourier series, N

CD(t) = Ct,r, + E K

CDt,- cos(Ktoi_t * et,.) .

(16)

I

By substituting Equation (16) instead of the constant C'z~ into the Morison equation extended to the relative motion concept and carrying out some reduction to select terms which contributes to the energy dissipation, we get the Equation (15). The values of AL and Y~As are obtained as follows; Put V~ for the relative velocity of the cylinder to the fluid particle and the drag force Fn is written as 1

Fo = 2PA C,~V,]V~[ where

Vr = Y1~Ol.cos oaLt + Yso~.,COS(oJ.,t + el) + oJw~Ae~Kasin ( K Y + oJ.,t + %) = Ytsot. cos ooLt + Y~. U)wcos(o.,t + ~) .

~17)

Slow-drift oscillation

287

W e can use CDC, CDS o b t a i n e d f r o m the p r e s e n t investigations w h e n we d e t e r m i n e as follows:

OJ2L~L A L = 1

V~lVrlcOs toz t dt

,-ff

~

¢j)2 Y~As -- 1 ~

(18) +

J. T h e h i g h e r o r d e r t e r m s of d r a g forces are o m i t t e d in E q u a t i o n (15) b e c a u s e they are not so i m p o r t a n t for e s t i m a t i n g the m o t i o n of an o c e a n structure. R e l a t i o n s b e t w e e n CDL, CAL, CAS and CDLO, CAI.O, CASO are f o u n d in the p r e s e n t w o r k and we can e s t i m a t e values of CDL , CAL , CAS from those of CDt~o, CAt, O, C A s o . CONCLUSIONS T h e h y d r o d y n a m i c force acting on the c y l i n d e r f o r c e d to surge in very low f r e q u e n c i e s in w a v e s a n d the h y d r o d y n a m i c forces acting on the c y l i n d e r f o r c e d to surge in two h a r m o n i c m o t i o n s in low and high f r e q u e n c i e s are i n v e s t i g a t e d e x p e r i m e n t a l l y . T h e d r a g coefficient CDL and the a d d e d mass coefficient C A t for the slow oscillation, the d r a g coefficient CDs and the a d d e d mass coefficient CAS for the fast oscillation, the i n e r t i a coefficient CMS for the wave m o t i o n were o b t a i n e d . T h e y were c o m p a r e d with the d r a g coefficient Ct.,LO a d d e d mass coefficient CAL,::,, CASO o b t a i n e d f r o m simple h a r m o n i c f o r c e d surging tests in a still w a t e r and the inertia coefficient C g s o o b t a i n e d from w a v e force m e a s u r e m e n t s . T h e m a i n conclusions o b t a i n e d are as follows: (1) T h e value of COL d e p e n d s mainly on the r e d u c e d velocity of U,. (2) T h e values o f CAS a n d C g s d e c r e a s e with /.Jr at U r < 6 . 0 . T h e t e n d e n c i e s e x p r e s s e d in (1) a n d (2) are very similar to those f o u n d in e x p e r i m e n t s in a s t e a d y flow. (3) T h e M o r i s o n e q u a t i o n using the one c o n s t a n t drag coefficient is not s u i t a b l e for two f r e q u e n c i e s flow field p r o b l e m . It is practical to use E q u a t i o n (15) for d e s c r i b i n g the d r a g force acting on the c y l i n d e r oscillating in two f r e q u e n c i e s . REFERENCES ANoo, S. and KATO,S. 1984. On the hydrodynamic forces at the low frequency motion of moored floating structures (in Japanese). Proc 7th Ocean Engng Symp. Society of Naval Architects of Japan, Tokyo. CHAKRABARTI,S.K. and ComER, D.C. 1984. Hydrodynamic coefficients of a mooring tower. J. Energy Resource Technol. 106, 449-458. INOUE, Y. 1983. A study on the deep sea mooring (in Japanese). Report of the Shipbuilding Research Association of Japan, No. 362. KEULEGAN, G.H. and CARPENTER,L.H. 1958. Force on cylinders and plates in an oscillating fluid. J. Res. natn. Bur. Stand. 60, (5) 423-440. KOrERAYAMA,W. 1984. Wave forces acting on a vertical circular cylinder with a constant forward velocity. Ocean Engng 11 (4), 363-379. KOTERAYAMA,W. and NAKAMURA,M. 1986. Wave forces acting on a moving cylinder. 5th Int. OMAE Syrup., Tokyo. MOE, G. and VERLEY, R.L.P. 1980. Hydrodynamic damping of offshore structures in waves and currents. OTC Paper 3798. MORISON,J.R. et al. 1950. The force exerted by surface waves on piles. Trans. Am. Inst. Min. rnetall. Engrs Petrol Trans. 189, 149-157. NAKAMURA,S., et al. 1986. On the increased damping of a moored body during low-frequency motions in waves. 5th Int. OMAE Symp., Tokyo. WICHERS, J.E.W. 1982. On the low-frequency surge motions of vessels moored in high seas. O.T.C. Paper No. 4437.