Loading and response of a small diameter flexibly mounted cylinder in waves

VScheepsbojaJuj

and Structures (1988) 2, 479-501

Technische Hogeschool

Deift

LOADING AND RESPONSE OF A SMALL

DIAMETER FLEXIBLY MOUNTED

CYLINDER IN WAVES

A. G. L. BORTHWICK

Department of Civil Engineering, University of Salford, Salford M5 4WT, U.K. AND

D. M.

HERBERTI-Hydraulics Research Limited, Wallingford, OX1O 8BA, U.K.

(Received 11 December 1987 and in revised form 4 April 1988)

Wave loading on flexibly mounted vertical cylinders is of particular interest to offshore engineers since the dynamic responses of slender tubulars in fixed jacket platforms, marine risers and compliant structures require accurate estimation. This paper describes a series of experiments and an analytical study concerning the behaviour of a spring mounted small diameter cylinder in waves. In-line and transverse forces and cylinder response displace-ments are examined with particular emphasis on the differences between resonant and non-resonant cases. The influence of vortex shedding on the displacement locus of the cylinder is considered. Force coefficients from the flexibly mounted cylinder are found to be larger than those for the same cylinder but with a rigid mounting. The force coefficients

exhibit dependence on the orbital shape parameter Q for local KeuleganCarpenter numbers, KC, larger than 14. At lower KC values, the force coefficients become

influenced by the ratio of wave frequency to natural frequency. From spectral analyses of the force and displacement data the dominant frequencies are identified in accordance with the theoretical predictions.

1. INTRODUCTION

ADVANCES IN RESEARCH allied to the increasing experience of offshore engineers have led to many improvements in the understanding of wave loading on structures. Even

so, catastrophic failures such as those of Texas TowerNo. 4 in 1961 and the Alexander Kielland in 1980have highlighted the need for further research, particularly regarding

vortex-induced fluctuating transverse forces.

This paper concentrates on the wave-induced loading and response of a flexibly

mounted vertical cylinder which is representative of a slender or compliant element in

an offshore structure. Fixed steel jacket platform substructures and marine risers consist of small diameter circular cylinders whose diameter is less than one-fifth the

wave length. The individual slender cylinders may be prone to vortex-induced

transverse vibrations at frequencies which are integer multiples of the wave frequency.

In-line vibrations which are at odd multiples of the wave frequency may also be important as regards to the fatigue life of the structure. Flexible structures such as articulated columns and guyed towers have been proposed for deep water applications where

depths exceed 150m. These compliant structures are designed to move with the wave

motions, thus reducing the magnitude of the forces experienced. Accurate prediction

t Formerly at Department of Civil Engineering, University of Salford.

0889-9746/88/050479 + 23 $03.00 © 1988 Academic Press Limited

V

.4

480 A. G. L. BORTHWICK AND D. M. HERBERT

of the stresses caused by the response motions is essential from strength and fatigue

points of view. The response motions themselves must satisfy health and safety criteria laid down by the Certifying Authorities.

The starting point for all offshore structural analysis is the estimation of the forces generated by the fluid loading. It is usual to consider the total force divided into in-line

and transverse components relative to the direction of the incident velocity vector.

Morison et al. [1] proposed that the in-line force per unit length F1 on a vertical cylinder

may be given by the linear addition of drag and inertia terms. The drag term is

proportional to the square of the water particle velocity and is related to the

momentum deficit caused by separation in the boundary layers. The inertia term is related to the acceleration vector which is colinear with the velocity vector for a

vertical cylinder in waves. Morison's equation, modified to allow for relative motion of the cylinder, may be expressed as

jrD2 jrD2

FJ=pDCD(ui)u iI+p----.(CM 1)(à I)+p

a, (1)

in which F1 is the in-line force per unit length, p the fluid density, D the cylinder diameter, CD and CM are drag and inertia coefficients, u and a are horizontal water

particle velocity and acceleration respectively, and I and i are the horizontal velocity

and acceleration of the structure. The predominant frequency of the in-line force

predicted by equation (1) is the same as that of the oscillating flow. It should be noted

that the maximum force at any elevation occurs before the wave crest passes the

vertical axis of the cylinder. Moreover, this maximum force does not act

simul-taneously at each elevation but occurs at the bottom of the cylinder before occurring at the top.

As a result of wake asymmetry linked with vortex shedding, an oscillating lift force is experienced which may be expressed as

FL = pDCL(t)(u )2 (2)

where FL is the lift force per unit length and CL(t) is a time-dependent lift coefficient. The lift force oscillates at the vortex shedding frequency f which may, if close to the natural frequency of the cylinder fN, "lock-on" to fN. This synchronized fluidstructure interaction leads to increased correlation and strength of vortices in the near wake.

Equations (1) and (2) are empirical and the force coefficients CD, CM and CL(t)

depend on a large number of variables. These include the Reynolds number Re,

KeuleganCarpenter number KC, surface roughness, ratio of wave frequency to

natural frequency fIfN, an orbital shape parameter Q expressed as the ratio of the

vertical to the horizontal first harmonic velocity components and time. It

_{is not}

practicable to consider the Morison force coefficients CD and CM as varying with time, and so time-averaged values are generally used. This is not entirely satisfactory, since the method of averaging itself influences the value of the coefficients obtained. The lift coefficient is not represented as a time-averaged value, since the fluctuating nature of the lift force would render the average near zero. Root-mean-square methods, Fourier techniques, etc., have been used in order to determine the amplitude and occasionally the phase of the fluctuating lift coefficient. Substantial research has been undertaken to evaluate the force coefficients CD, CM and CL(t) over ranges of Reynolds numbers,

KeuleganCarpenter numbers and surface roughnesses for steady unidirectional

incident flow and harmonic flow past a rigid cylinder. Sarpkaya and Isaacson [2J have produced an extensive review of this research.

Dynamic analysis of the equation of motion shows that the response is a function of the force coefficients, frequency ratio fJfN, mass ratio Mr = Mel pD2 where Me is the

effective mass, KeuleganCarpenter number, damping ratio N' time and the wave

depth parameter kd where k is the wave number and d is the water depth. The sheer

number of parameters involved serves to indicate the complex nature of wave loading and structural response. Although there are several detailed summaries of the dynamic behaviour of flexible structures in oscillating flow such as given by Blevins [3], Hallam et al. [4] and Sarpkaya and Isaacson [2], the subject has not received as much attention

as that of fixed structures in waves. It is the purpose of this paper to examine the

dynamic fluidstructure interaction of a spring-mounted cylinder in waves. This paper

continues the work of Bullock [5] who obtained force coefficients from the same

cylinder when rigidly mounted in waves.

2. RELATED STUDIES

This review indicates recent developments in research concerning the behaviour of

flexibly mounted cylinders in oscillatory flows. It is sensible to divide the experimental research into two categories; namely, two-dimensional analogies and laboratory wave

flume tests.

McConnell and Park [6, 7] measured the lift forces and responses of a vertical

cylinder oscillated in still water. They remarked that, since the response of an

elastically mounted cylinder has a feedback effect on the fluid loading, forces on

elastically mounted cylinders in oscillating flows cannot be predicted using data from

fixed cylinders or flexibly mounted cylinders in steady flow. At KeuleganCarpenter numbers, KC, below 32 a detached vortex is unable to move away fast enough from

the responding cylinder before it returns and the vortex is again involved in the

fluidstructure

interaction. At "lock-on" the

lift

force and structural

response

increase owing to synchronized vortex shedding as the fluidstructure motions form

closed path Lissajous figures. For KC >32 the vortices have more time to move away and dissipate leading to disintegration of the Lissajous patterns. McConnell and Jiao

[8] determined the Morison drag and inertia coefficients from the in-line forces

measured on an elastically mounted cylinder. At small KeuleganCarpenter numbers

they found that the drag and inertia coefficients were primarily dependent on the ratio of natural frequency to driving frequency IN/f..,. McConnell and Jiao suggested that the

in-line force coefficients are highly coupled with the transverse response.

Very few investigations have examined the behaviour of a flexibly mounted cylinder in waves. Sawaragi et al. [9] classified the flow patterns around a vertical cantilevered

Rubble beach Perspex window

Splash hood in channel wall

Piston type wave maker

Water I m deep

4.5 m

Figure 1. Wave channel.

Hairlock spending beach

482 A. G. L. BORTHWICK AND D. M. HERBERT

cylinder in waves in terms of the surface KeuleganCarpenter number, SKC, for

1 <SKC < 30. Later Sawaragi et al. [10] considered the response displacement loci as a

function of frequency ratio fNIf. For fN/f < 1-1 the displacements formed a nearly

straight line in the in-line direction. Figures of eight, double ellipses and triple or long

ellipses occurred for 1-1<fN/f<1-67, l-67<fN/f<2-5 and 2-5<fN/f<3-3

respec-tively. Sawaragi et al. found that, as long as SKC >4, resonance occurred in both

directions for fN/f,. = 1, 2 and 3.

Zedan et al. [111, Isaacson and Maull [12], Zedan and Rajabi [13], Angrilli and Cossalter [14] and Hayashi [15] examined the wave-induced dynamic response of vertical cylinders in terms of the surface KeuleganCarpenter number SKC, wave

depth parameter kd, frequency ratio and mass ratio. In particular, Angrilli and

Cossalter noted that transverse resonance occurred at a surface reduced velocity

SI",. = 17.54 for fNIfW = 1 and SKC = 175, at SVr = 5-75 for fNIf. = 2 and SKC = 1F5,

at SI", = 5-93 for fN/fW 3

and SKC = 17-8 and

SI",. = 8-98

for fN/f = 4 and

SKC=35-9.

3. LINEARIZED MODEL

A simplified form of the modified Morison loading equation is used to predict the

in-line fluid loading and response of the cylinder. The equation of motion is rewritten

in a form suitable for linearization and appropriate analytical treatment. It is also

required that the structural characteristics be altered by the added-mass of fluid

entrained by the cylinder and fluid damping related to simplification of the non-linear drag term. Although this approach is undesirable with regard to the prediction of the cylinder responses it does give an indication of the relative importance of dimension-less groups such as mass ratio Mr, frequency ratio fIfN, etc. In this case the drag and inertia coefficients CD and CM are assumed to be constant with depth.

Assuming that the cylinder and mounting act together as a two degree-of-freedom system, the effective mass, stiffness and damping may be determined via modal analysis as described by Blevins [3]. Here the spring-mounted cylinder is idealized as a rigid

rod pivoted at a point a distance h above the floor of the tank. Figures 2 and 3

illustrate the in-line and transverse wave-induced displacements of the idealized system. The line of action of horizontal forces, effective mass, etc., is taken to act a

S.W L

Figure 3. Definition sketch of transverse response.

distance L above the pivot at the level of the free surface when the cylinder is

immersed in still water of depth d. The axis z1 is measured upwards from the still water level. The elevation of the wave surface above the still water level and directly above

the pivot point is given by ij, whereas h denotes the elevation at which the cylinder

pierces the wave surface.

For a pivoted rod the modal shape parameter lp(z) is defined as

p(z)=z/L,

(3)

where z is measured vertically upwards from the pivot point. The in-line

displace-ments, velocities and accelerations at still water level are given by X(t), X(t) and

1(t).

Similarly in the transverse direction the responses at still water level are Y(t), Y(t) and V(t), respectively. The corresponding displacement responses x(z, t) and y(z, t) at any elevation z are determined from

x(z, t) = p(z)X(t), y(z, t) = ip(z)Y(t). (4)

The velocities and accelerations are obtained in a similar manner.

Using modal analysis the instantaneous values of effective structural mass and

effective stiffness in air are calculated from

Jm(z,t)ip2(z)dz

and

2 m(z, t)giji(z)/Ldz Kea - L

fL ip(z)dz

where m(z, t) represents the distribution of structural mass per unit length with

distance z above the pivot as a function of time, K95 is the angular stiffness of the helical spring pivot, L the distance from the pivot to the still water level, LT the

distance from the pivot to the top of the cylinder and g the acceleration due to gravity.

(6)

A CYLINDER IN WAVES

=

where a is the damping ratio of the cylinder in air and is composed of structural

and

material damping only. It is implicitly assumed _{here that ais approximately equal}

to

the damping ratio of the cylinder in vacuum. The effective force is calculated assuming that the waves are of small amplitude in deep water and that _{may be neglected. The} equation of motion in the in-line direction is

MeaX + eoX + KeaX= Fex, ₍₈₎

where the effective force F, for small displacements is given by F(z, t)1jJ(z) dz JL pgrD2_{X(t)ip(z) dz}

cx ₍₉₎

J,2(Z dz

_J 4'2(z)

0 ₀

in which F(z, t) is given by equation (1). In order to make the analysis feasible _the structural inertia component and the buoyancy

_{term in F are transferred to the}

left-hand side of equation (8), which is rearranged to give

MX + CeaX + KX

_{= F,}

₍₁₀₎

where and Kew are the values of effective_{mass and stiffness in water and}

is the

modified effective force. In this context, the effective mass in water is defined as

Mew=Mea+Caf,

(11)

where Ca is the added-mass coefficient. The natural frequency of the system in _{water is}

given by

1 1K

fNw=-x)m

(12)

in which the effective stiffness in water Ks,. is the sum of the effective _{stiffness in air} and an additional term due to the restoring force from buoyancy. Thus

1L pgrD2

Jo 4L

p(z)dz

= K0 + _L

.10

In equation (10) the modified effective force is expressed as

JL

{ PDCDIu - (u

- ) +

JrD CMa}tP(z)dz

F'1

1p2(z) dz

j,2(z)th

0

The non-linear drag term in equation (14) may be rewritten neglecting terms of order2

according to Penzien [16], as

!u±J (ui)=u juj -2(IuP)i,

₍₁₅₎

where (ful) represents the time average of ju'.

484 _{A. G. L. BORTHWICK AND D. M. HERBERT}

The effective structural damping Cea is thus obtained from

The second term in equation (15) may be viewed as resulting from fluid damping. Letting

j

2(IuI)iip(z)dz

4fxMer.'fN

= pDC

_L (16)

J

the fluid damping ratio fx may be derived once the time average (IuI has been

determined from the Fourier series expansion of the horizontal particle velocity given by linear wave theory as

kag cosh[k(z +

u = _w cos(wt), (17)

cosh(kd)

where k is the wave number, a is the wave amplitude, and w is the _{wave frequency.}

After some rearrangement of equation (16) the fluid damping ratio fx is expressed by

CDSKC

/f\

_f1

fx

= 2Jt2Mr cosh(kd)'

in which CD is the Morison drag coefficient, SKC is the surface KeuleganCarpenter number, M is the mass ratio including added mass, fW/fN the ratio of wave frequency to cylinder natural frequency in water, kdthe wave depth parameter and f1 a function which describes the velocity distribution with depth. The parameter ft is given for the pivoted rod by

f

=

_{{Cosh(kh)[_cOsh(kL)+}

(+)sinh(kL)]

+

sinh(kh)[(1-

+ )cosh(kL)

- -

sinh(kL)]. Rewriting the effective damping coefficient as follows

Cea = 4Y(lM,fNW,

where

= 1 ₍₁

PD2Ca\]1'

Lkea

\

4Mea )i

the total damping ratio becomes

Tx1+fx.

(22)

The equation of motion may therefore be written as follows

MewX +4JtTxMewfNwX + 4Jr2Mf,X = F, (23) where prD2

F='°

4 CmÜ}4(Z)dZ

jip(z)dz

0 (18) (24)

(30) 486 A. G. 1. BORTHWICK AND D. M. HERBERT

Substituting for the water particle kinematics using equation (17) and its time

derivative, it is possible to express the effective in-line force F in non-dimensional form as a Fourier series; thus,

.2Cmf3

F

CDf2

'Y C1 cos(2jrf..,t)

SKC cosh(kd)sin(2JrfWt) (25)

pDu.

cosh2(kd)

in which U,,,, is the maximum particle velocity at mean water level,

(i

2

C1 = --(

-

_{f2_4)}

(26)

f2=tcosh2(khp)[_+

31

L2 L sinh(2kL)_4k cosh(2kL)_8k2

_+8kj

1 1

+ sinh(2khp)EL cosh(2kL) sinh(2kL)1 2k 4k2 ] + sinh2(khp)EL sinh(2kL) cosh(2kL) L2 1 1 (27) 4k 8k2 4 and

f3={cosh(kh)[Lsinh(

cosh(kL) 11 k k2 +k21 IL cosh(kL) sinh(kL) 1 +sinh(khp)L _{k k2} (28)

The functions f2 and f3 express the effect of the wave particle kinematic distributions with depth on the pivoted rod. On examination of equation (25) it is obvious that the

in-line force has dominant components occurring at odd multiples of the wave

frequency f.

If the still water level displacement X is expanded as a Fourier series, equation (23) may be solved analytically to give

X SKC / f \21sKCCof2

sin(2JrjfWt) +

(i

- i--- )cos(2JrJf..t)]MFi

D8V2MrfNw) lCOsh2(kd)1=1,3,5 _{fNw f NW'}

7r2CMf2

[(i

cosh(kd)

- 2

f Nw)sin(2JTIWt)

- 2Tx

fN cos(2Jrfwt)]MF1}, (29) where the dynamic magnification factor MF1 is given by

1 MF1

= [(i

j2f22

(2iTXf-)2]

Thus the in-line displacements are a function of the surface KeuleganCarpenter

number SKC, mass ratio M,, frequency ratio f.JfN, wave depth parameter kd, the

effective total damping ratio the drag coefficient CD, inertia coefficient CM and by

inference the surface reduced velocity SV,. Equation (29) also indicates that the

dominant frequencies of the in-line response are at odd multiples of the wave

frequency f.

In order to isolate the important parameters which control the transverse loading and cylinder responses, the transverse equation of motion is linearised in a similar

6

manner to that of the in-line model. The effective mass, stiffness and damping

coefficient are again taken to be equal to their zero displacement values and the

cylinder is assumed to pierce the water surface approximately at the still water level.

These assumptions are reasonable for small displacements of the cylinder in small

amplitude waves. The equation of motion may thus be written as

Me,Y+CeaY+KeaYFy,

(31)

pDCL(t)(u

- pDCa9}(Z) dz

JL pgD2

Y(t)p(z) dz

Fey _{L L} . (32)

f

dz

J

0

2)

The added-mass, inertia and buoyancy terms are moved to the left hand side of

equation (31) which is rearranged, assuming that the structural velocities are small, to

give

MewY + 42TdIMewfNwY + 4Jr2MewfwY = F, (33)

pDC,(t)u2ip(z) dz where

(34)

L

dz

The lift coefficient CL(t) is time-dependent and fluctuates at a frequency related to

vortex shedding. Chakrabarti et al. [17] have defined

CL(t) = CL(j)sin[2Jrjf,t + 4(j)], (35)

i I

where CLe(f) is the depth invariant lift coefficient of the jth harmonic and 4(j) is the

phase angle of the jth harmonic force. An alternative definition based on a

quasi-steady model of regular vortex shedding was proposed by Bearman et al. [18] for a fixed cylinder in oscillatory flow. This may be modified for a cylinder free to respond

giving

CL(t) =CL cos{f

2r u - ij (S/D) dt

+ (36)

in which CL is the amplitude of the transverse force coefficient, ₀ is the phase angle of

the transverse force and S is the Strouhal number. In the quasi-steady model CL, ₀

and S are assumed constant over each half cycle.

Equation (35) is simpler to apply than equation (36) and so is used in the analytical

model described here. However, equation (36) describes the bursting nature of the

transverse force and response oscillations as the vortex shedding process is impulsively

started, builds up to a maximum and decreases each half cycle and so is more

appropriate physically than equation (35).

Substituting equations (35) and (17) into equation (34), the effective transverse force may be given in non-dimensional form as

F f2 CL(J){sinE(j - 2)2:rfwt_{+ 0(1)]} 2pDu,,, cos (

)=

+ 2 sin[2Jrjft + çb(j)] + sin[(j + 2)24wt + (j)]}. (37) 487 where _L

F=

r

488 A. G. L. BORTHWICK AND D. M. HERBERT

For KeuleganCarpenter numbers above 4. the lift force is due mainly to vortex

shedding and so, if a constant pattern of n vortices are shed per wave, it is reasonable

to suppose that CLe(J) has a maximum when j = n. In this case the dominant

frequencies of the lift force will be at (n - 2)fW, nf,. and (n +2)fW. Hence, at least three

major peaks would be expected in the multipeaked spectrum of lift force with

additional significant peaks occurring in cases where the vortex shedding process varies from cycle to cycle.

Substituting equation (37) into (33) the following expression for

the

non-dimensionalized displacement Y/D is obtained

Y(SKC)2(f2

f2

Cj.e(f)

D 8t2M, \fNWI cosh2(kd)1=1

[1

_{- (j}

2)2f]sin[(j

- 2)2JrfWt + 4(I)] - 2(1 -

2)f-

cos[(J - 2)2Jrft +

[

[(i

-

-

2)2)

+

(2(i -

2)-)]

2[(1 _iP)sin(2rJfWt

+ 4(j)) - 2j1-cos(2vjft +

r/

2ç22 _/

_f

\2

L(1_4) +(2i)

_fNW

_f

[i_

(j +

2)2]sin[(j

+ 2)2jrft +

(j)] - 2(j + 2)- cos[(j + 2)2JrfWt

± J

[(1_(j±2)2)2± (21(j±2)L)2]

f Nw fNw

(38)

Equation (38) indicates that the transverse response is a function of the surface

KeuleganCarpenter number SKC, the mass ratio, M,, frequency ratio fW/fNW, a wave

depth parameter kd, a parameter related to the wave particle distribution with depth

f2, the damping ratio and the effective lift coefficient of the wave harmonics. The expression also shows that the transverse response spectrum should be similar to that of the transverse force spectrum.

4. EXPERIMENTAL ARRANGEMENT

The tests were carried out in a wave flume 396 m long x F26 m wide x F36m deep,

the layout of which is shown in Figure 1. At one end of the flume is a piston-type wave paddle which is activated by a servo-controlled hydraulic ram. The paddle was able to

generate both random and regular waves, but in these tests only regular waves were

used. Wave generation was controlled by a Texas Instruments 960A computer which sent a sinusoidal electrical signal to the hydraulic ram. The required frequency for the desired wave was directly obtained using the computer's internal clock. The specified

wave height was produced as follows. The computer estimated the amplitude of the

electrical signal needed for a given wave and monitored the actual wave produced. If the wave amplitude was not within prescribed limits then the amplitude of the electrical signal was changed. This continued iteratively until the wave was the correct shape.

At the downstream end of the flume there is a spending beach fabricated from

x

+

impermeable sheets covered with a 100 mm thick layer of Hairlock. The beach profile had previously been adjusted so that the reflection coefficient was minimized to a value between 2 and 5% for the range of wave frequencies used in the tests.

The test cylinder was placed in the centre of the flume 40 m from the toe of the spending beach. It was constructed from aluminium tubes 38 mm in diameter and connected to the floor of the wave flume by a spring mounting designed to limit the

maximum deflection of the top of the cylinder to ±200 mm. Forces were measured on a

50mm long element of the cylinder using a double channel KistlerMorse DSK force

transducer. Depending on the test, the centre of the element was positioned so that it was either 03m, 0-5m or 07 m below still water level. The cylinder interior was kept

as dry as possible by sealing joints with PTFE tape. In contrast, the instrumented section, which was independent of the cylinder (apart from its connection with the

force transducer), was always full of water during tests. The force transducer was able

to measure forces in two perpendicular directions normal to the cylinder axis. Each

transducer channel was linked to a Fylde359amplifier and a 20 Hz third order low-pass

Butterworth filter. Dampness of the connections affected the performance of the force transducers and so each transducer was encased in a plastic tube which was then filled with vaseline and sealed with a rubber bung.

A 600 mm long Meclec capacitance wave gauge was placed next to the cylinder at the test section so that passing waves could be monitored. A vernier scale was attached to the wave gauge for calibration purposes. Output from the wave gauge was converted to an analogue signal by an interface device.

Deflections of the top of the cylinder were measured in both in-line and transverse directions using a system developed by Bakeret al. [19] at the University of Salford. A

plastic cross piece with orthogonal arms was screwed to the top of the cylinder. Two of

the neighbouring arms contained probes from a signal processor circuit which were

immersed in water baths. Each bath consisted of a perspex tank with two aluminium

plates mounted parallel to each other and connected to the driver circuit of the

deflection measuring system. The output signal from the driver circuit is directly proportional to the probe displacement and so the position of the cylinder can be

determined at all times.

Before each test series commenced the force transducers were calibrated by holding the cylinder horizontal, balancing the Fylde amplifiers and hanging weights from the centre of the element. Once an acceptable slope and intercept had been obtained from

linear regression analysis, the cylinder was rotated through 180° and the calibration

repeated. The two slopes were then averaged for greater accuracy. The deflection

measuring system was calibrated using a vernier scale by moving the cylinder through known distances. Likewise the wave gauge was calibrated using its vernier scale. At the start of each test the computer recorded a zero intercept reading for each channel.

The wave forces involved in the tests are non-repetitive even though the wave

motion is highly regular. It was therefore decided that the signals from each test should be recorded over at least 200 waves. Over the range of tests8192 readings were taken

per channel with sampling rates from 10 to 25 Hz. Each test began with five minutes

wave generation before the output from the wave gauge was recorded and the standard deviation of the water surface elevation calculated. If the standard deviation was not

within the required limits the signal to the paddle was adjusted until the required

accuracy was reached. Another 100 waves were generated before the outputs from the wave gauge, force transducer and deflection measuring system were recorded. Once all the data were collected the paddle was stopped for five minutes in order to let the water settle down before another test commenced. The data acquired by the computer were

fN ('b)

Measured values

Predicted values

Effective mass in air,

M (Kg/rn) 3.97

Effective mass in water, 514 5-18

M (Kg/rn)

Mass ratio, 3-56 358

Mr

Effective stiffness in air, 189 157

K (Kg m2/s2/rad)

Effective stiffness in water, 178 175 (Kg m2/s2/rad)

Damping ratio in air,

Fluid Damping,

0-021

0010 0007, 0-014 Damping ratio in water,

Effective damping in air,

0-031 0028, 0-035

1-05

C (Kg/mis)

Effective damping in water,

-

1-69

Ce.,. (Kg/mis)

Natural frequency in air, 0-988 1-000

fN ('b)

Natural frequency in water, 0-937 0-925

490 A. G. L. BORTHWICK AND D. M. HERBERT

written to magnetic tape using a Pertec magnetic tape machine and later transferred to a Prime main-frame computer for processing.

Wave particle kinematics were measured via a Laser Doppler anemometer in an

earlier series of tests by Bullock [5] in the absence of the cylinder for exactly the same

wave conditions.

5. RESULTS

Before conducting the main test programme the structural characteristics of the

spring-mounted cylinder in

air and still water were determined as described by

Borthwick et al. [20]. Table 1 summarizes the measured and predicted values of the

structural characteristics. The effective mass was calculated via modal analysis after

each component part

of the cylinder was weighed and

its length measured. Experimental values of the effective stiffness in air and still water were determined

from the corresponding angular stiffness obtained by applying a horizontal force at the

top of the cylinder and measuring the resultant displacements. Damping ratios and

natural frequencies in air and water were determined experimentally via free vibration

tests.

Table 2 gives a summary of the wave test programme. The tests were conducted at

sub-critical Reynolds numbers in the range 2000 Re 10400. A total of 128 tests were undertaken, all of which were processed to obtain in-line and transverse force

coefficients. A representative sample consisting of 24 tests was analysed in detail giving temporal force and displacement plots and the corresponding spectra.

TABLE 1

TABLE 2 _{Test cases} Test _series number Depth of instrumented section (m below S.W.L.) Wave frequency ft,. _(Hz) Range of wave height, H (mm) Wave depth parameter, kd Frequency parameter, p (= Re/KC)

Reynolds number range,

Re Frequency ratio I 2 3 4 5 6 7 8 03 03 03 05 05 05 07 07 0713 0510 0358 0637 0468 0291 0658 0446 1103-2144 651-2206 481-2008 1216-2234 707-2234 396-1697 1584-2127 707-2212 211 124 079 174 110 062 183 103 9031 6460 4355 8069 5928 3686 8335 5649 4800-10,300 3100-10,400 2300-9500 4300-8300 3100-10,400 2000-8500 4000-5800 3200-9300 0761 0544 0382 0680 0499 0311 0702 0476

492 A. G. L. BORTHWICK AND D. M. HERBERT

In the majority of cases the in-line force and cylinder displacement data exhibited repeatability for successive waves. Figure 4(a) illustrates the in-line

force and

displacement data for a non-resonant case when the frequency ratio f/fN = 0-761, orbital shape parameter Q = 0-9, wave depth parameter kd = 2-108 and the surface

KeuleganCarpenter number SKC = 16- 1. The period of each cycle is approximately

constant though there is evidence of amplitude modulation due to slow drift of the

cylinder. Vortex shedding at wave crests may also be discerned corresponding to small

secondary peaks or discontinuities in the in-line force trace. It should be noted that,

although there may be repeatability for many waves, the mode of vortex shedding may

alter, a period of transition occur and then a slightly different but repeatable in-line

force cycle become established as reported by Bullock [5].

The transverse force and displacement data are generally irregular, with large

differences in amplitude for non-resonant cases where the frequency ratio fJfN was

not an integer sub-multiple. An example is shown in Figure 4(b). The transverse forces fluctuate with apparently random amplitudes due to the complex interaction between vortex shedding and the cylinder response in the oscillating flow. The process is further complicated by the three-dimensionality of the wave flow which implies that vortices

may be shed in cells due to variation in local KeuleganCarpenter number KC along the length of the cylinder as discussed by Zedan and Rajabi [13]. Even so, there are cases where the transverse force and response behaved in a repeatable manner and

exhibited regular bursts of activity. Figure 5 presents force and response data when

f/fN = 0-382,

2 =

05, kd

= 0-786 and SKC = 11-8.

It indicates that up to three

vortices are shed per wave cycle. The trough of every second wave corresponds to the largest in-line and transverse forces and displacements. The vortex shedding pattern

has an effect on the in-line forces activating a third harmonic response component.

Although this is a case where the frequency ratio fJfN,.. has a non-resonant value, the

E N

z-03m

jiiiIsitiI

ItI

,n,!",,tl

!!tl!lj

I'?'?

I I I I I I i i i i I I I Ill

_lIIIjIlIIII

- I I I I i i I I I I I I 0 4

6

-iL 0

lii

I I I I I i i I i i

lit 1 1 I

Ii iii Ii

ii!

z-03 m

0 0 20 30 40 50 0 0 20 30 40 50

(a)

f(s)

(b)

Figure 4. (a) In-line and (b) transverse force and displacement traces with time; f,jfN,. = 0-761, Q = 09,

kd=2108, SKC= 161, SV,= 122. 4 2 0

2

4

-6

05 IcL 0.0 E z N Ic 00 05

05

I t I I I I I I J I I i ii I -0 0 20 30 40 50

transverse force and response amplitudes are larger than the in-line ones. The

transverse forces show a periodic bursting pattern very similar to that observed by

Bearman et al. [18].

In certain cases where resonance occurred as the frequency ratio fJfN approached

an integer submultiple value, the forces and responses exhibit large amplitude

repeatability in both the transverse and in-line directions. Figure 6 illustrates the forces and responses when fJfN = 0499, Q = 05, kd = 1.101 and SKC = 11.7. The amplitude of the transverse force is of similar magnitude to that of the in-line force.

Here, as the cylinder undergoes resonant vibrations, the transverse displacements are substantially larger than the in-line displacements of the cylinder.

The combined in-line and transverse loci of force and response give an overall picture of the dynamic effects experienced by the cylinder. In accordance with the

observations made by Sawaragi et al. [10], the displacements form patterns which may

be classified by the frequency ratio fIfN. This is also the case for the vectorial

distribution of forces incident on the cylinder. For 087 >fjfN. > 06 the forces and

displacements are predominantly in-line with the direction of the wave train. The force

distribution shows that the maximum transverse force occurs near the extremes of

in-line force corresponding approximately to vortex shedding at the wave crests and troughs. Representative force and displacement distributions are shown in Figure 7 for

fIf. = 0761, Q = 09, kd = 2108 and SKC = 161. The locus of displacement tends

to be smoother than that of the forces and forms elongated ellipses or figures of eight, with the long axis in-line with the wave direction.

For non-resonant cases in the range O6 >fIfN.,. > 04 the displacements generally

form double ellipses, as shown in Figure 8. The force distributions tend to be similar

to that shown in Figure 7. Close to resonance, as the frequency ratio fIf

approaches 05, the force distributions form a clearly defined U-shaped pattern. For a

493

(a) f(s) (b)

Figure 5. (a) In-line and (b) transverse force and displacement traces with time; fJfN,, =O382, Q =O5,

kd= O786, SKC = 118, SV, = 452.

zO3 m

/

U .4 I iii 111111111 1111111111 2 C

2

3

z-03 m

0 II

2

-I I I I I I I i i i I i i I i i i I

2 E z z o

-2

-3

_ii

I

iii

I

Ii

I I IL I

-

j j

ii ii

I Ij i I j

ii liii

I (a)

-6--Figure 7. Force and displacement patterns;f.jfN,.,=0761, Q=09, kd=21O8, SKC=161, SV=122.

E z I I: N -2 -3 0 Id

f(s)

(b) z=-O5 111111111111 I I I

fli'

_ii

iiilflflifflffll lilt Ifllflui _'flU!

II I ii 111111

illi

ii

0 [0 20 30 40 50 0 [0 20 30 40 50

Figure 6. (a) In-line and (b) transverse force and displacement traces with time; fIfN =_{0499, Q= 05,}

kd= 1.101, SKC- 117, SV= 583.

surface KeuleganCarpenter number SKC of 115 and a surface reduced velocity of 58

approximately, a condition of perfect resonance occurs. Two vortices are shed per

wave; one at the wave crest, the other at the trough. The cylinder displacements form a

double ellipse distorted into a horseshoe shape, as shown in Figure 9. The cylinder

motions form a closed loop Lissajous figure through a process similar to that described by McConnell and Park [7]. It would seem that a residual vortex, which has detached

previously from the cylinder, is in the correct position to interact with the cylinder

when it returns to the proximity of the vortex, thus increasing its strength. As a

consequence, a strong and consistent response pattern is formed, with the cylinder

rotating about the residual vortex or vortices.

This explains the fluid-dynamic

amplification of in-line and transverse forces and responses at resonance.

2 =_0.3 m

Pr

Figure 8. Force and displacement patterns; f,./fN = 0544, Q = 07,kd= 1239, SKC = 114, SV, = 622.

-,.

,

-

.

I, "

-3

-3-Figure 9. Force and displacement patterns; f/f = 0499, Q = 05, kd = 1-101, SKC = 11-7, SV, = 5-83.

The Wave-induced force and cylinder response characteristics become more

compli-cated in the frequency ratio range 04 > fIfN. > 0-3. Up to three vortices are shedper

wave cycle and so the displacement patterns form triple ellipses or highly complex

combinations of double and triple ellipses. A typical example is given in Figure 10. Morison drag and inertia coefficients were determined from a least-squares fit of the

Figure 10. Force and displacement patterns; f,./fN = 0382, Q = 0-5.kd = 0-786, SKC = 11-8. SV, = 452.

A CYLINDER IN WAVES 495

3....

-05

rn

-496 A. G. L. BORTHWICK AND D. M. HERBERT

modified Morison equation to the measured in-line force and displacement data. The

inertial effect of the effective mass of the transducer element was included in the

modified Morison equation, as follows:

F.r,,, + (Met - prD2)I = PDCD luIl (uI) + CM prD2(à 1),

(39)

in which Fim is the measured in-line force per unit length, is the effective mass of

the transducer element, x and i are the measured in-line structural velocities and

accelerations, and u and a are the measured particle velocities and accelerations. From the least squares analysis the values of CD and CM averaged over each wave cycle were determined from

AA

-CD = A2A6A5A3 and and A6 prD2 (a

and the summations apply over each wave cycle.

The maximum value of the lift coefficient CLm. was calculated each wave period from

2 pD(Iu - XIm)

where FLmOX is the maximum value of the transverse force per unit length and u _1lmax

is the maximum value of the relative in-line velocity in the wave cycle. Mean values of CD, CM and CLm, were then obtained by averaging over all the waves in the test.

Figure 11 illustrates the variation of the force coefficients CO3 CM and CLm with local KeuleganCarpenter number KC in terms of the orbital shape parameter Q. For local KeuleganCarpenter numbers greater than 14 the drag and lift coefficients are

dependent on the orbital shape parameter, with larger values of C0 and CLmax

occurring for reduced ellipticity. This is reasonable because, for the largest value of Q, the wave particle orbits are nearly circular and vortices shed from the cylinder in cells

along its length. The flow is highly three-dimensional in nature, with the cellular

vortices having spanwise phase differences relative to each other. The lack of

A1A5 - A2A4 CM

AA A2A6'

where A1 =

[Fxm lu - x (u - x) + (Met - prD2)I lu - II (u 1),

A2 = pD

- 112 (u -1)2

A3= pJrD2>. (a I)Iu 1I(u I),

A4 =

[Fr,,,(U 1) + (M1 - prD2)I(a 1)],

A5 = pD

lu - II (u - i)(a 1)

FLmax

4 2 Kc 3 2 Kc Kc

Figure 11 Force coefficientsversitslocal KeuleganCarpenter number as a function of the orbital shape

parameter Q (0. 0-9; 0, 0-7; , 0-5;+, 0.3).

correlation in the vortices leads to a marked reduction in their effective strength and

hence the lift force experienced by the cylinder. For low values of orbital shape

parameter the flow becomes almost planar harmonic, with the vortices exhibiting a greater degree of spanwise correlation. The lift and drag forces therefore increase.

For local KeuleganCarpenter numbers below 14 the drag coefficients lose their

dependence on the orbital shape parameter Q. The lift coefficients are more scattered but are still somewhat dependent on Q at low values of KC. Throughout the range of

local KeuleganCarpenter numbers considered here, no discernible trend may be

detected in the inertia coefficients with orbital shape parameter.

Comparison with the fixed cylinder data presented by Bullock [5]indicates that the Morison in-line force coefficients CD and CM are generally larger for the flexibly

mounted cylinder in the same flow. This has important repercussions for offshore engineers who therefore should not use force coefficients from fixed cylinder data

when designing compliant structures in a wave environment.

In order to investigate further the behaviour of the force coefficients with frequency ratio individual graphs were plotted of CD, CM, and CLm. against a surface

KeuleganCarpenter SKC for each frequency ratio. Figure 12 illustrates typical

results. For the non-resonant cases wherefw/fN =0761 and O382 the drag and inertia

coefficients did not vary with SKC. However, for the nearly resonant case where

fJfN =

O499, the drag coefficient data exhibited at

least two peaks; one at

SKC= 117 and SV =5-83 corresponding to perfect resonance, the other at SKC=

16-8 and SVr=8-4. The first peak corresponds closely to that obtained by Angrilli and

Cossalter [14] who noted that transverse resonance occurred at SKC

=11.5 and

20 30

10

0

0 4

A. G. L. BORTHWICK AND D. M. HERBERT

-I 1 I 09 4 _1'0',v076t kd = 2I08 0 4 05

.

'YN kd r 1 101 2

0 ot!l5

S.

c4

£2 0 5 0382 kd= 0786 2

-kd 0.619 -

Lo.

_S

'I!1 !!!S

2

-

!I

0 oO 0000000000o

0₀ -0 4 8 12 16 20 2 SKC

Figure 12. Force coefficients versus surface KeuleganCarpenter number. S CD;0, CM; A, CLm.x.

SVr

= 575 for

fIfNw= . Although there were no tests corresponding exactly to

fJfN

= some evidence of resonance may be seen in the drag coefficient data for

fwIfN = 031 in Figure 12. Peaks may be discerned at SKC = 118 andSVr= 37 and at SKC = 186 and SVr

= 6. The second peak again compares favourably with the

corresponding values of SKC = 178 and SV = 593 obtained by Angrilli and Cossalter

[14].

The inertia coefficients presented in Figure 12 are similar to the drag coefficients in

that they are remarkably constant with SKC for non-resonant cases, but exhibit

variations as_f,.,IfN approaches or . Although it is more scattered,the lift coefficient

CLmaX

behaves in a similar fashion to the drag coefficient, confirming that the

transverse response and drag coefficient are coupled at resonance as proposed by

McConnell and Jiao [8].

In all cases the dominant in-line force and response spectral components coincided with the wave frequency. The maximum spectral energy of the transverse forces and

displacements tended to be concentrated at a value equal to the average number of

vortices shed per wave, which was approximately equal to 2 for 06 > fW/fNW> 04 and

3 for 0.4_>fIfN,..

>03.

I0 10°

10-. _io

Figure 13. (a) In-line and (b) transverse force and displacement spectra; = 0-310, Q = 0-3, kd = 0-619, SKC = 17-7, SV, = 5-49.

A CYLINDER IN WAVES

defined narrow-band spike at the wave frequency. The corresponding transverse force

and response spectra were fairly broad band with major peaks at f., 2f,, and 3f. The

dominant peak in the transverse force spectrum occurred at 2ff., indicating that two

vortices were usually shed per wave although the process was by no means consistent. The major transverse response component occurred at the wave frequency, close to the natural frequency of the cylinder.

Similar in-line force and response spectra were obtained for all cases within the frequency range 06 >fWIfN, > 04. The in-line spectra contained greatest energy at the wave frequency

and significant components at 2f, 3f, 4f,, and 5f,,. The high

energy content at f, 3f,. and 5f,

is predicted by equations (25) and (29). Secondary

waves generated by the paddle were partly responsible for the spikes at 2f and 4f. These peaks may also have been influenced by coupling between the vortex-induced transverse and in-line fluidstructure interactions. In the transverse spectra the largest peak occurred at 2f, corresponding to two vortices being shed per wave. A peak at f

was influenced by differences instrengths of the vortices, depending on when they were

shed during the wave cycle. Other significant components were

observed at 4f and

6f,. This is in broad agreement with equations (37) and (38), which predict significant

transverse forces and responses at (n - 2)f, nf and (n + 2) f where n is the number

of vortices shed per wave.

Figure 13 shows typical force and response spectra obtained for a frequency ratio in

the range 04 >fIfN. > O3. In this case fIfN... = 0.31, kd =

0.619 and SKC = 17.7. The in-line spectra have major peaks at f and 3f,.. with smaller peaks at 5f and 7f,, as

predicted by equations (25) and (29). Maximum transverse spectral energies are

concentrated at 3f. with subsidiary peaks at f. and 5f indicating that three vortices are

0 2 4 6

500 A. G. L. BORTHWICK AND D. M. HERBERT

usually shed per wave. In both the in-line and transverse spectra, further peaks also

occur at 2f.,. 4f,.. and 6f, which may be due either to paddle vibration effects or else to changes in the mode of vortex shedding where four vortices are shed per cycle.

6. CONCLUSIONS

This paper has described a series of experiments in which the forces and responses of a spring mounted small diameter cylinder were measured in regular waves generated in a laboratory flume. Theoretical models of the in-line and transverse loadings show that

the dynamic behaviour of a flexibly mounted cylinder depends on the ratio of wave frequency to natural frequency fJfN, the surface KeuleganCarpenter number SKC,

mass ratio M,, the wave depth parameter kd, an effective total damping ratio -, the

force coefficients CD, CM and CL(J) and functions which describe the effective

distribution of velocities along the length of the cylinder. From the theoretical analyses it was predicted that the in-line forces and responses have frequencies at odd multiples

of the wave frequency f. whereas the transverse forces and responses have at least

three dominant components at frequencies equal to (n - 2)f,,, nf, and (n + 2)f,.,, where n is the number of vortices shed per wave in cases where the vortex shedding process is well defined and repeatable.

Measured in-line forces and displacements were generally repeatable, with a

dominant component at the wave frequency. Secondary peaks corresponding to vortex shedding were noticeable on the force and displacement traces. The transverse forces

and displacements were often irregular, with large amplitude variations for

non-resonant cases. Nevertheless there were cases where the transverse loads and responses were characterized by regular bursting patterns similar to those discussed by Bearman et al. [18] for a fixed cylinder in a planar harmonic flow. At resonance, the frequency

ratio f..,/fN approached an integer submultiple value and the in-line and transverse

forces and displacements became amplified and highly repeatable as the cylinder

interacted with residual vortices that had detached previously from the cylinder. The force coefficients CD, CM and CLmax exhibited dependence on the orbital shape parameter for local KeuleganCarpenter numbers KC larger than 14. At lower KC

values the force coefficients became influenced by the frequency ratio fJfN,.

A comparison with fixed cylinder data [5] indicated that fixed cylinder values of in-line force coefficients are generally lower than those for a flexibly-mounted cylinder due to force amplification caused by the fluidstructure interactions. The maximum value of

drag coefficient occurred at perfect resonance when f/fN

SKC 11.7 and SV = 583, in close agreement with observations by Angriuli and Cossalter [14] and

Zedan et al. [11]. Coupling between the drag coefficient and the transverse response was evident in waves similar to that obtained for an oscillating cylinder by McConnell and Jiao [8].

Spectral analyses indicated that the dominant in-line force and response coincided with the wave frequency f. Other significant components occurred at odd harmonics of the wave frequency as predicted by the theoretical linearized analysis of Morison's

equation. Spectra of transverse force and response usually contained several peaks

related to vortex shedding, again in accordance with the theoretical model.

ACKNOWLEDGEMENTS

This work forms part of the research programme of the Marine Technology

501

SERC, Department of Energy and the Offshore Industry. It was funded through the auspices of Marinetech North West-the North Western Universities Consortium for

Marine Technology. The authors would also like to thank Dr G. N. Bullock of

Plymouth Polytechnic for his advice in conducting the experiments described herein.

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