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E L S E V I E R

Hydrodynamic damping of a cylinder in still water and in

a transverse current

J.R. Chaplin*, K. Subbiah

Department of Civil Engineering, City University, London EC IV OHB, UK Received for publication 28 A p r i l 1998

Abstract

Laboratory measurements are presented o f the hydrodynamic damping o f smooth and roughened circular cylinders i n s t i l l water and i n the presence o f a transverse cun-ent at a Stokes parameter /3 o f 1 6 6 9 0 0 and at Keulegan Carpenter numbers i n the range 0 . 0 0 1 - 0 . 2 . The experiments were carried out b y a p p l y i n g a k n o w n external harmonic f o r c e at constant amplitude to the elastically-mounted submerged cylinder, and observing its response. For the smooth cylinder, levels o f hydrodynamic d a m p i n g i n still water were f o u n d to be approximately t w i c e those predicted by W a n g . B e l o w , a certain velocity, a transverse current had very l i t t l e effect on the damping, but over a range o f stronger currents i t j u m p e d to a new constant level about 7 0 % greater than the still water value. A surface roughness o f k/d = 0.00079 generated an increase i n damping o f about 3 5 % i n still water, but o n l y 5% i n a current. © 1998 Elsevier Science L t d . A l l rights reserved.

1. Introduction

Predictions o f the response o f compliant offshore systems to high frequency excitation rely on knowledge o f hydrodynamic damping as w e l l as o f the excitation. Natural f r e -quencies o f tension leg platforms (TLPs) i n heave, p i t c h and r o l l are typically three or more times the peak frequency o f the wave spectrum, and w h i l e the linear part o f the excita-tion can be predicted w i t h reasonable confidence, equally important nonlinear contributions are subject to much greater uncertainty. A c c o r d i n g l y , considerable e f f o r t is directed at the problem o f nonlinear surface interaction solutions, w h i c h may have to be extended to high orders, or become f u l l y nonlinear, to capture essential elements o f resonant excitation referred to as springing and ringing. M u c h less attention has been paid to the other part o f the problem, that o f predicting the damping.

Uncertainty over hydrodynamic damping i n this context derives chiefly f r o m the fact that very f e w measurements have been made i n the appropriate flow regime. T y p i c a l cross-sectional dimensions o f the pontoons and columns o f TLPs are i n the range 1 0 - 3 0 m , and typical periods o f oscillation f o r high frequency motions are around 3 s. These figures indicate values o f the Stokes parameter (3 —fd^lv o f order 1 0 ^ where ƒ is the frequency o f oscillation, d is a characteristic dimension, and v is the kinematic viscosity.

* Corresponding author

T y p i c a l amplitudes o f springing and ringing response are measured i n centimetres, indicating Keulegan Carpenter numbers {K = m o t i o n amplitude X litld) less than 0.01. Other applications include the damping o f m o o r i n g lines and tethers, where values o f (3 may be two or three orders o f magnihide lower, and Keulegan Carpenter numbers two or three orders o f magnitude higher. There is an acute shortage o f data on hydrodynamic damping i n these regimes.

W h e n a T L P undergoes springing or r i n g i n g oscillations, deeply submerged parts o f i t may be oscillating i n still water, but i t is more l i k e l y that they do so i n the presence o f a current, or wave motion. This raises the question o f whether hydrodynamic damping is necessarily increased i n these conditions, and i f so, by how much. W h e n the oscillation takes place i n the presence o f a current, a third dimensionless parameter can be identified, namely the reduced velocity U, = current spted/fd. This is a measure o f the ratio o f the displacement o f a particle i n the undisturbed flow over the period o f one oscillation to the c y l i n -der's diameter. The ratio o f the cylin-der's m a x i m u m speed to the speed o f the current is K/Uj. I n a current o f 1 m/s, the columns and pontoons o f a T L P w i l l have reduced velocities o f the order o f 0.3, w h i c h is considerably l o w e r than the range o f values that have been studied previously i n con-nection w i t h the vortex induced oscillations o f a compliant cylinder i n a cross-flow.

Well-established sources o f data on the M o r i s o n loading on a cylinder i n oscillatory or wave flow (or on a 0141-1187/98/$ - see front matter © 1998 Elsevier Science L t d . A l l rights reserved

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252 J.R. Chaplin, K. Subbiah /Applied Ocean Research 20 (1998) 251-259 combination o f an oscillatory flow and a current) extend

d o w n to Keulegan Carpenter numbers o f around 5 (e.g. Sarpkaya and Isaacson [ 1 ] ) . I n the interval between 5 and 1, only a h a n d f u l o f measurements can be f o u n d i n the literaUire, notably M o e and Verley [ 2 ] , Sarpkaya [ 3 ] , D e m i r b i l e k et al. [4], Takahashi et al. [5], Bearman and M a c k w o o d [ 6 ] , and D o w n i e et al. [ 7 ] . B u t i n afl these cases the lowest Keulegan Carpenter number was greater than the range o f predominant interest here, and most o f them refer to the regime where the drag coefficient o f a cylinder oscillating i n still water does not vary much w i t h and where the loading is strongly influenced by separa-tion and vortex shedding. This is quite d i f f e r e n t f r o m the behaviour o f the flow at still smaller ampliUides, where f r o m theory [8] the drag coefficient can be expected to be inver-sely proportional to K.

W a n g ' s solution [8] is f o r the laminar flow over a cylinder undergoing harmonic oscillation at small amplitude i n a fluid otherwise at rest, and extends the theory developed by Stokes [ 9 ] . W h e n expressed i n the f o r m o f drag and added mass coefficients, (and using the approximation cosÖ|cosö| ~ (8/37r)cosö), Wang's result gives

_3 r 37r 2K ( 7 r / 3 ) - 2 + ( , r / 3 ) - M m • 4 ( 7 r / 3 ) - 2 + ( 7 r / 3 ) - 2 ( 1 ) (2) A t large values o f /? the first term i n each series becomes dominant and the product of the drag coefficient and the Keulegan Carpenter number is approximated closely b y

Cd/f = 26.24/^/8 (3)

The o n l y experimental results that o f f e r comparisons w i t h Eq. (3) seem to be those of Sarpkaya [3] (1035 < /3 < 11240, K > 0.3), Otter [10] {(i = 61400, K > 0.03) and Bearman and Russell [11] {(3 < 60000, i r > 0.003). Denot-ing the theoretical result Eq. (3) as W, results o f these experimental investigations conespond approximately to CiK = 5W, W and 2 W respectively, pointing to a consider-able margin o f uncertainty i n any predictions o f hydro-dynamic damping o f a cylinder oscillating i n s t i l l water, quite apart f r o m possible scale effects.

T h i s article describes new measurements at h i g h /3 and l o w K, not o n l y i n s t i l l water but also i n steady currents i n the d i r e c t i o n n o r m a l to the c y l i n d e r ' s axis and the d i r e c t i o n o f o s c i l l a t i o n . The experiments described here were carried out w i t h (3 at 166 900, and at values o f K d o w n to 0.001. Section 2 describes the arrangement o f the experiments, i n w h i c h measurements were made o f the response o f a elastically mounted submerged c y l i n d e r to a harmonic f o r c e excitation. The p r o b l e m o f extracting the h y d r o d y n a m i c d a m p i n g f r o m the measurements is analysed i n Section 3, and results f o r smooth and r o u g h -ened cylinders i n s t i l l water and i n a current are presented i n Sections 4 - 6 .

2. E x p e r i m e n t a l arrangements

I n the range o f small Keulegan Carpenter numbers o f interest here, i t is impracticable to measure the hydro-dynamic damping on a body undergoing f o r c e d oscillations, even though the kinematics o f the m o t i o n may be identical to that i n the intended application. A s K is reduced, the perceived loading is increasingly dominated by inertia forces, and that small part w h i c h is i n phase w i t h the v e l o -city cannot be separated w i t h any confidence. A c c o r d i n g l y , i n this w o r k we f o l l o w e d a different approach i n w h i c h the cylinder was elastically mounted on springs o f l o w stmc-tural damping, and subjected to k n o w n oscillating external forces. The hydrodynamic damping was deduced f i ' o m measurements o f response, based on a knowledge o f the tare damping, i.e. those forces associated w i t h the structural support system and the cylinder's endplates.

The experiments were canied out i n a tank 55 m l o n g , 1.75 m wide, w i t h a water depth o f 1.75 m , equipped w i t h a can'iage on w h i c h was b u i l t the r i g shown i n F i g . 1. The r i g consisted first o f a f r a m e w o r k constructed f r o m 50 m m steel tubular bars w i t h vertical members reaching d o w n f r o m the carriage at the sides o f the tank, and horizontal members at the base a similar distance above the tank floor. W i t h i n this structure ( f o r m i n g a cube o f sides 1.63 m ) , the test cylinder was supported across the flume at about half depth b y a system o f f o u r tension springs at each side.

The t w o test cylinders were made f r o m r i g i d P V C tubing w i t h a smooth suiface o f outside diameter 0.323 m . Their lengths, between 0.91 m diameter endplates, were 1.493 and 0.707 m , and their masses (including the endplates) were 54.5 and 41.1kg. The long cylinder was filled w i t h water-resistant f o a m and had a submerged natural frequency close to 1.60 Hz, corresponding to (3 = 166 900. The internal dis-placement o f the short cylinder was designed so that its submerged natural frequency w o u l d be the same.

The apparatus f o r applying harmonic external forces to the test cylinder made use of t w o servocontrolled D C motors i n the control systems sketched i n F i g . 1. The motors drove carbon fibre pushrods connected to the cylinder through load cells and articulated joints. They were controlled i n such a way as continuously to m i n i m i s e the d i f -ference between a force command signal and the force actually applied through the pushrods to the cylinder. I n this way an excitation was achieved that was not signifi-cantly affected by the cylinder's position or velocity. Con-tinuous recordings were made o f the forces i n the pushrods and their displacements, the latter provided by position encoders on the output shafts o f the motors. The axial s t i f f -ness o f the pushrods was extremely h i g h , so that the flex-i b flex-i l flex-i t y o f the lflex-inkage was essentflex-ially that o f the load cells, less than 0.01 m m / N . A x i a l forces i n the pushrods were w e l l below I O N . The structural damping o f the system, measured i n free oscillations i n air w i t h the pushrods dis-connected, was close to 0.04%.

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_ leSOinm Carriage " m 1 M-Mean water - U L flum e wal l level 323mm cylinder flum e wal l flum e wal l 1 / flum e wal l flum e wal l 1498mm between endplates flum e wal l flume floor CROSS-SECTION 1450mm Carriage Reaction frame from 50mm steel tubes

ELEVATION SHOWING THE SUPPORT SPRINGS

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254 J.R. Chaplin, K. Subbiah/Applied Ocean Research 20 (1998) 251-259 vertical and horizontal excitation to be applied to the

cylinder, and measurements were made o f both vertical and horizontal oscillations. However, o w i n g to a lack o f horizontal rigidity i n the carnage, the latter were generally less successful and i n the experiments presented here the excitation and the dominant part o f the response were both i n the vertical direction. I n some cases the m o t i o n o f the cylinder included a small horizontal component (probably because o f a slight n o n - u n i f o r m i t y i n the springs), making its path a very shallow ellipse. T o study the impoilance o f this m o t i o n i n the results, t w o additional tests were caiTied out. I n the first we added a horizontal oscillatory component to the exciting force, i n quadrature w i t h that i n the vertical direction, so as to drive the cylinder i n an elliptical m o t i o n . AtK = 0.127, up to an orbit eUipticity o f 25% (more than five times greater than that w h i c h ever occurred w i t h no horizontal excitation), there was no measurable change i n the hydrodynamic damping i n the vertical direction. I n a second, separate set o f tests, horizontal wires were added to the support system to prevent any m o t i o n i n that direction and measurements o f hydrodynamic damping o f vertical oscillations were repeated. A g a i n , there was no significant change, suggesting that small horizontal excursions were unimportant.

Oscillations o f the cylinder inevitably generated waves at the surface, and i t is w o r t h estimating the magnitude o f its wavemaking resistance. This can be done using the first-order solution f o r a horizontal cylinder beneath waves by O g i l v i e [12], f r o m w h i c h (using O g i l v i e ' s notation) it can be shown that the damping force per unit velocity per u n i t length is

Trpo)d Si-B

4(1 +Si) (4)

regai'dless of the direction o f oscillation. For the present conditions a solution o f this theory f o r a frequency of 1.6 H z gives = 5.5 X lO-\ s j = 2.72, 5 , = 4.9 X 10~'. The resulting value o f c„, corresponds to a level o f damping o f 4.5 X 10"^% o f critical. This is several orders o f magnitude smaller than the viscous damping, and was ignored throughout.

3. Theoretical considerations 3.1. The theoty of the experiments

The theoretical basis o f the experiment is simply that of the single-degree-of-freedom system w i t h an external har-monic force, i n w h i c h the response is characterised by a dynamic magnification, a, and a phase lag, 0 , between the force and the response. Measurements o f a and (/> are i n principle sufficient to provide an estimate o f the damping, but i f the damping is light i t has a significant effect on the response only at frequencies very close to resonance. I t was therefore necessai-y f o r the system to be excited i n this

range, though the natural frequency was not k n o w n i n advance and had to be considered a f u n c t i o n o f amplitude.

The cylinder's equation o f m o d o n is

{m + C^m')x + [2^,^J{m + C^m')l<: + c^^]x + kx = Fcoswt (5) where m is its mass, m' its displaced mass, Ca, the added mass coefficient, f ^ , the structural damping, k, the stractural stiffness, c^, the total hydrodynamic damping force per unit velocity, and F the amplitude o f the externally-applied force. I n the experiments the excitation frequency, co, was the natural frequency observed i n small amphtude oscilla-tions i n still water, and this was used to define a reference added mass coefficient C„n,

CO = (6)

I n E q . (5) and c^, are unknowns, to be f o u n d f r o m mea-surements o f the dynamic magnification and phase lag. I n doing this, i t was reasonable to assume that inside the damp-ing term the added mass coefficient was not significantly different f r o m C^q ( w h i c h was always very close to 1). Separating the in-phase and quadrature components o f Eq. (5) then leads to expressions f o r the added mass coefficient and the hydrodynamic damping:

W — CaO cose/)

,^^:=,^,„'^^l±^(sin0-2ar,)

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where M = m/m'. I f the observed hydrodynamic damping is to be attributed solely to drag on the cylinder, the con'e-sponding drag coefficient is given by

87;7co

C,K-- 3 7 r ^ M + Q o , .

8 aM (9)

As CiK is an appropriate damping factor, the experimental results are mostly expressed i n these terms.

I n the experiments, representative values f o r these para-meters were: C^o = 1, M = 0.4, a = 80, 4> = 110°, co =

1.6rads/s, = 0.0004, i m p l y i n g = 1.006, CiK = 0.4 (uncorrected f o r damping contributions due to the endplates, discussed below). I n such conditions the magnitude o f the second term i n the parentheses o f E q . (9) is about 7% o f the first, leaving the equation quite w e l l conditioned f o r the purposes o f evaluating C,K.

3.2. Endplate corrections

I n the absence o f endplates, the flow around the ends o f the cylinder w o u l d influence the damping, though this effect might become insignificant i f the cylinder were very long. Some measurements o f damping were made i n still water without endplates, and, as described below, i n some cases

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255 the end effects d i d seem not to be very important. B u t

otherwise the cylinders were fitted w i t h circular endplates w i t h chamfered edges. Unfortunately, the force on the end-plates because o f their m o t i o n i n their o w n plane was then a significant proportion o f the total, and this section describes the steps that were taken to isolate the damping force on the cylinder alone.

A measure o f the damping force on each endplate during oscillations i n still water can be obtained f r o m laminar flow theory. I n these conditions the amplitude of the damping force per unit area on a plate oscillating harmonically i n its o w n plane w i t h peak velocity V and frequency w is pV^JuivIl [13]. Over the outer surfaces of the endplates (of diameter D) i t seems reasonable to apply this result directly, w i t h the peak velocity equal to that o f the cylinder. Over their inner surfaces (annuli o f inner and outer dia-meters d and D respectively), the oscillatory fluid motion is not u n i f o r m , but its distribution can be approximated as the potential flow around the cylinder. It is easy to show on this basis that when the shear stress acting i n the direction o f the cylinder's m o t i o n is integrated over the annulus, the result is identical to the force acting on the same portion o f t h e outer faces o f the endplates where the flow is assumed to be u n i f o r m . It f o f l o w s that the total damping force per unit velocity caused by both endplates is theoretically

C „ e = C0(7r/3) - 2 T T j 3 {É^ - i ) (10) where E = Did > 1. For a cylinder o f length L, this

force w o u l d appear as an increase i n the damping factor CiK o f

A Q / r = ^ ( . / 3 ) - H ( £ (11)

The approach that was used to eliminate endplate and other effects i n most cases was to repeat the measurements w i t h cylinders o f two different lengths, but otherwise similar aiTangements. F o l l o w i n g Bearman and Russell [11] we refer to this as the ' l o n g - s h o r t ' method. I f the perceived drag coefficients (computed f r o m the measurements w i t h o u t regard to the loading on the endplates and elsewhere) are C d j and C(j2 f o r cylinders o f lengths Li and L2, then the true drag coefficient f o r the cylinder alone is

Q 2 L 2 — Qi^^i

L2-U (12)

and the contribution that the tare damping makes to the perceived drag coefficient on a cylinder o f length L is

A Ch = L1L2

( C d 2 - Q i ) (13)

Calculating the damping on the cylinder f r o m the difference between t w o sets o f measurements naturally has a detrimen-tal effect on the accuracy o f the result. A measure o f the overaU error was obtained by taking representative values f o r the parameters i n Eq. (9) f o r each case, assigning to each

one an uncorrelated variability w i t h a normal distribution and a standard deviation o f 5% (thought to be a consei-vative estimate o f the uncertainties involved), and computing the standard deviation o f the final outcome. This was about 16%.

3.3. Morison's equation

I n analysing measurements o f hydrodynamic damping made i n the presence o f a current, i t is h e l p f u l to refer to the 'relative velocity' f o r m o f M o r i s o n ' s equation model. For the case o f a cylinder undergoing small amplitude oscil-lations w i t h velocity u i n a transverse current o f speed U, this gives

F= -CiypduVu^ + ii^ . (14) f o r the damping force per unit length, where C^v is a drag

coefficient that is not much different f r o m that w h i c h w o u l d con-espond to the steady drag on a stationary cylinder i n a current o f speed U. The relationship between C^K and C j v is then

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taking only the amplitude o f the fundamental frequency component o f the expression i n braces. The l i m i t s o f E q . (15) are

CéK^ Y^dvUr as Y " ^ " " '

and CiK C^yK as § ^ 0 (16) A

4. Oscillations i n still water

Some typical results f r o m a test i n still water are shown i n F i g . 2, w h i c h contains: (a) the vertical force applied to the spring-mounted cylinder though the pushrods; (b) the force spectrum; (c) the corresponding response; and (d) the response spectrum. Though the command signal that was played out to each motor's control system was a single harmonic o f constant ampliUrde, the actual force i n the pushrods had lai-ge components at higher frequencies, as shown i n F i g . 2(a) and (b). The loop gain was set as h i g h as possible w i t h o u t introducing instability, and the h i g h frequency force components were probably generated by the rapid transient corrections that can be seen i n F i g . 2(a) to occur twice i n each cycle. Though the cause o f this behaviour is not clear, i t seems reasonable to assume that since they were w e l l away f r o m the natural frequency, the resulting forces d i d not significantiy affect the cylinder's response to the resonant excitation. This is supported b y the response spectrum i n F i g . 2(d), i n w h i c h the peak at the natural frequency is five orders o f magnitude greater than any other.

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256 J.R. Chaplin, K. Subbiah /Applied Ocean Research 20 (1998) 251-259

Frequency (Hz)

Frequency (Hz)

Fig. 2. Measurements at A" = 0.127: (a) the excitation; (b) the excitation spectrum; (c) the response; (d) the response spectrum.

I n F i g . 3, drag coefficients f o r the still water tests, uncor-rected f o r endplate forces, are plotted against the Keulegan Carpenter number. The importance o f endplate forces is evident i n the difference between perceived drag c o e f f i -cients f r o m the l o n g and short cylinders. The product CiK changes only slowly w i t h K, and as described above can be treated directly as a hydrodynamic damping factor. For the long cylinder, the mean value over the interval 0.01 < 0.04 is 0.203. That f o r the short cylinder is 0.268, i m p l y i n g by Eq. (13) that the total tare damping o n the l o n g cylinder was equivalent to a contribution to CiK o f 0.058. The part o f this that is due to shear stress on the endplates can be predicted theoretically (assuiiung laminar boundary layers) by evalu-ating E q . (11) f o r each cylinder i n turn and subtracting the

Aspect ratio 4.64 o 2.19 X

0.01 0.1

Fig. 3. UncoiTected drag coefficients for the long and short cylinders.

result f o r the long cylinder f r o m that f o r the short one. The outcome is 0.109 - 0.052 = 0.057. T o this must be added a contribution o f 0.003, w h i c h corresponds to the structural damping o f 0.04%, m a k i n g a total o f 0.060. This is i n good agreement w i t h the measured value o f 0.058.

The values given above f o r CiK i n the interval 0.01 < 0.04 i m p l y a corrected result o f CiK = 0.145, about 2.2 times greater than the theoretical prediction Eq. (3). I n d i v i -dual measurements f r o m the l o n g cylinder were similarly corrected by E q . (12), interpolating data f r o m the short cylinder measurements f o r the same K i n each case. The results are plotted as solid symbols i n F i g . 4. A l s o shown are measurements at values o f K d o w n to 0.001 that have been corrected f o r endplate damping using the theoretical result E q . (11), and f o r structural damping by further sub-tracting 0.003 f r o m CiK i n each case. I n this range o f very small amplitudes the results o f the l o n g - s h o i t method were more scattered and have been omitted f r o m F i g . 4. Over the interval 0.001 < 0.01 the mean value o f CiK was f o u n d to be 0.122, or 1.9 times the theoretical result E q . (3). Bearman

0.2 0.1 0.0 Endplate corrections: Long-short • Theoretical o No endplates +

• ••

2 . 2 f K + 0 . 0 8 i r 26.24|3-'«=r 0.001 0.01 0.1 K

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and Russell [11] similarly f o u n d a factor o f about 2 between theory and measurement at values o f |3 up to 60 000.

The theory mentioned here refers to two-dimensional laminar flow, w h i c h was shown b y H a l l [14] to be subject to a three-dimensional instability previously observed by H o n j i [ 1 5 ] , and refen-ed to as the H o n j i instability [ 3 ] . A t (3 — 166 900, H a l l ' s prediction o f the critical amplitude f o r the onset o f the instability is K — 0.29, significanfly larger than the value o f K i n the present experiments. I n the force measurements o f Sarpkaya [3] at j8~1000, the instability is linked w i t h a j u m p i n the drag coefficient, though the latter was observed at rather lower Keulegan Carpenter numbers. I n the absence o f visualisations o f the flow at higher p, i t may be conjectured that the departure o f the present results f r o m the theory is caused by similar structural changes i n the boundary layer.

A l s o shown i n Fig. 4 is a set o f results obtained f r o m measurements using the long cylinder without endplates. These were corrected as described before f o r the effects o f structural damping, and, using laminar boundary layer theory as before, f o r the effect o f the oscillatory shear stress over the cylinder's plane ends. The latter amounts to a con-tribution o f 0.003 i n CiK. A t small K these measurements are i n reasonable agreement w i t h those obtained w i t h the endplates i n place, but suggest that f o r K>Q.l the damping is sigiuficantly increased by the effects o f flow around the cyhnder's ends.

A t larger values o f K, Ci can no longer be expected to be inversely proportional to K o w i n g to the onset o f vortex shedding. Bearman and Russell [11] argue that f o r this reason a contribution to o f O.OSisT can be expected. As shown i n Fig. 4, where dK = 2.2W + O.OSAT^ is plotted as a

1.0 o.H 1 i 1 1 \ 1 — i — 1 0.6 0 O 0.04 • ' 0.1 • 0.2 O M — -0.3 • 0.4 •!• 0.3 0.5 A 0.6 X 0 2 _ ^ ^ 0.1 U = 0.04 .-' £/.= 0 0.01 0 1 K

Fig. 5. Damping factors for the smooth cylinder oscillating in a transverse current. Broken lines are calculated from the relative velocity Morison equation with a constant drag coefficient of I . l .

broken line, this effect becomes significant only beyond the range o f the present data.

5. Transverse oscillations in a steady current

I n these experiments the cylinder was excited i n the vertical direction w h i l e the carriage was driven at constant velocity along the channel w i t h reduced velocities between 0.04 and 0.6. As before, measurements were made w i t h cylinders o f t w o lengths, and the effects o f smafl horizontal osciUafions on damping i n the vertical direction were shown to be insignificant i n separate tests w i t h horizontal restraining wires. B u t since the level o f background mechanical noise was higher, reliable data could not be collected f o r K below about 0.02.

The results are plotted i n F i g . 5. A t a reduced velocity

U; = 0.04 and f o r K < 0.07, the damping factor is

indis-tinguishable f r o m the still water results, suggesting that the cun'ent has not significantly disrapted the oscillatory bound-ary layer o n the cylinder. B u t at the same amplitudes, a cuiTcnt o f — 0.1 clearly does have an effect, and i n these conditions the damping is about 15% greater than i n still water. A t Ur = 0.04 and 0.1 the damping increases w i t h

K, and i n this respect (though i n no others) the results are

qualitatively compatible w i t h the implications o f the rela-tive velocity M o r i s o n equation. For current speeds i n the range 0.3 < [7^ £ 0.6 the damping is almost constant and independent o f K, at CiK 0.24, about 7 0 % greater than that i n still water.

The broken lines i n F i g . 5 represent the predictions o f the relative velocity M o r i s o n equation calculated f r o m E q . (15) w i t h a drag coefficient Cdv = E l appropriate to the Reynolds numbers o f the steady flow. For reduced velocities greater than 0.2, M o r i s o n ' s equation, when applied i n this way, leads to severe overestimates o f the damping; at

0.3 0.2 0.1 0.0 O O O 0.02 < i s : < O.I ( A ) • O.I < i s : < 0 . 2 (B) 0.0 0.2 0.4 0.6 u.

Fig. 6. Damping factors for the smooth cylinder oscillating in a transverse current plotted as a function of reduced velocity. Broken lines represent postulated step changes for the specified ranges of Keulegan Carpenter number.

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258 J.R. Chaplin, K. Subbiah/Applied Ocean Research 20 (1998) 251-259 • • > • 0 o 0.1 • 0.3

*

2.2w+o.08;£:'/ 0.6 X K

Fig. 7. Damping factors for the roughened cylinder; k/d = 0.00079.

U, = 0.6, the measured damping was only one t h i r d o f the

prediction. Clearly the assumption i m p l i c i t i n M o r i s o n ' s equation, that the oscillatory wake can be treated as quasi-steady, is seriously wrong. I t seems rather that once the current exceeds a certain threshold (here = 0.3 at the most), the flow has undergone a step change, and that further increases i n current (up to at least = 0.6) have l i t t l e effect on the organisation o f the boundary layer. For the purposes o f predicting the loading i n these conditions i t is more appropriate to consider the oscillatory part o f the motion to be independent o f the steady current.

The critical reduced velocity depends also on the ampli-tude o f the m o t i o n ; measurements shown i n F i g . 5 suggest that a.tK= 0.2 the change has already occurred at 11^ = 0.1. This is illustrated i n F i g . 6 where mean damping factors over the ranges 0.02 < 0.1 and 0.1 < 0.2 are plotted against the reduced velocity. Postulated step changes are sketched i n at A and B .

6. T h e effect of surface roughness

I n a separate set o f experiments the cylinder was covered i n abrasive paper o f grade P60, having an average grain size o f 0.255 m m . T h i s corresponds to a relative roughness o f

k/d = 0.00079, or to a roughness height o f about 10% o f

the n o m i n a l thicloiess, 5<i(7r/3)-"^ o f the Stokes layer i n laminar oscillatory flow. I n steady flow the thickness o f the laminar boundary layer on a cylinder at separation is o f the order o f 3dRe~"\ or about 3 m m at a Reynolds number Re o f 10^ (corresponding to the m a x i m u m velocity 0.31 m/s: = 0.6). I t is also o f interest to relate the roughness height w i t h the fliickness o f the laminar sublayer beneath a turbulent boundary layer i n the same conditions. This cannot be calcu-lated w i t h any precision i n the unsteady and non-uniform flow around the cyhnder, but f o r a flat plate i n water at 0.31 m/s the thickness o f the laminar sublayer on a smooth surface can be estimated at around 0.2 m m [ 1 3 ] .

N o short cylinder tests were carried out w i t h the roughened cylinder. The raw measurements were corrected by subtract-i n g the tare dampsubtract-ing obtasubtract-ined f r o m the dsubtract-ifference between results f o r the long and short smooth cylinders i n the same flow conditions. The results are plotted i n Fig. 7, where i t can be seen that the mean damping factor f o r the rough cylinder i n still water at K < 0.1 is about 3W, or about 35% greater

than that f o r the smooth cylinder. As before, at the smallest Keulegan Carpenter numbers a current o f = 0.1 seems to produce little change i n the damping, though again its effect increases at larger ampHtudes. For stronger currents the roughness appears to have little effect; at f / , = 0.3 and 0.6 the mean damping factor is 0.26, only about 5% greater than the con-esponding result f o r the smooth cylinder.

7. Conclusions

Measurements were made o f the hydrodynamic damping o f smooth and roughened circular cylinders undergoing con-stant oscillations o f smafl amplitude at /? = 166 900, i n stifl water and i n a steady transverse cunrent. The results are presented i n terms o f the product C^K f o r w h i c h the theore-tical result by W a n g f o r still water oscillations (26.24/^/3 at high 13) is 0.0642 (here referred to as W). A n assessment o f uncertainties i n the experiments points to an overall margin o f error i n the region o f ± 1 6 % .

For the smooth cylinder, hydrodynamic damping i n still water at amplitudes corresponding to Keulegan Carpenter numbers between 0.001 and 0.2 was f o u n d to be almost constant at between approximately CiK = 0.12 and 0.14 (1.9H^and 2.2W). B e l o w a certain current speed and oscilla-tion amplitude, the damping o f transverse oscillaoscilla-tions i n a steady current was the same as i n s t i f l water. B u t at a certain current speed ( w h i c h depended also on K), C j / i : j u m p e d to about 0.24 (3.8W) and was thereafter almost constant. For results averaged over 0.02 < 0.1 this change took place at a reduced velocity U, « 0.075, and f o r 0.1 < 0.2, at « 0.3. For Ur> 0.15 M o r i s o n ' s equation, when used i n the relative velocity f o r m , seriously over-predicted hydrodynamic damping f o r a l l values o f K tested (up to 0.2).

The measured hydrodynamic damping f o r a cylinder w i t h surface roughness Idd = 0.00079 i n still water was about

CiK = 0.19 (3W), or about 35% greater than that f o r a

smooth cyUnder i n the same conditions. B u t i n a steady current the damping j u m p e d to a level very close to that o f the smooth cylinder.

Though the measurements described here were made at probably the highest value o f /3 yet achieved i n the labora-tory, still there is a very large gap between these and f u l l scale conditions. For this reason, and because we do not have a good understanding o f scale effects i n this flow, i t w o u l d be unwise to assume that the present results can be applied quantitatively i n practice. Rather, they indicate qualitative features that may appear at f u l l scale. There is scope f o r f u r t h e r useful research i n some o f these areas, and, using techniques adopted i n this w o r k , possibiflties o f rais-i n g 13 strais-ill hrais-igher under laboratory condrais-itrais-ions.

Acknowledgements

This w o r k f o r m s part o f the research programme 'Uncertainties i n Loads on O f f s h o r e Structures', sponsored

(9)

259 by EPSRC (Grant GR/J23990) through M T D L t d and j o i n t l y

f u n d e d w i t h : A k e r Engineering A m o c o ( U K ) Exploration Company; B P Exploration Operating Co. L t d . ; B r o w n and Root L t d . ; Exxon Production Research Company; Health and Safety Executive; Shell U K E x p l o r a t i o n and Produc-t i o n ; SProduc-taProduc-toil; Texaco B r i Produc-t a i n L Produc-t d . The auProduc-thors are indebProduc-ted Produc-to Richard Y e m m and N i g e l MacCarthy f o r their w o r k on the design and construction o f the experiment.

References

[ I ] Sarpkaya T, Isaacson M . Mechanics of wave forces on offshore struc-tures. New York: Van Nostrand, 1981.

[2] Moe G, Verley RLP. Hydrodynamic damping of offshore structures in waves and currents, paper OTC3798. I n : Proceedings of the OTC, 1980:37-44.

[3] Sarpkaya T. Force on a circular cylinder in viscous oscillatory flow at low Keulegan-Carpenter numbers. Journal of Fluid Mechanics

1986;165:61-71.

[4] Demirbilek Z, Moe G, YttervoII PO. Morison's formula: relative velocity versus independent flow fields formulation for a case repre-senting fluid damping. In: Proceedings of the 6th O M A E Conference, Vol. 2, 1987:25-31.

[5] Takahashi K, Tsuruga H, Endoh K. In-line forces on oscillating bodies

in a fluid flow. Joumal of Chemical Engineering of Japan I988;21(4):405-410.

[6] Bearman PW, Mackwood PR. Measurements of the hydrodynamic damping of oscillating cylinders. In: Proceedings of the 6th Interna-donal BOSS Conference, London, 1992:405-414.

[7] Downie MJ, Graham JMR, Zhao Y - D , Zhou C-Y. Viscous damping of a submerged pontoon undergoing forces combined motions. Inter-national Joumal of Offshore and Polar Engineering 1995;5(4):241-250. [8] Wang C-Y. On high frequency oscillating viscous flows. Journal of

Fluid Mechanics 1968;32:55-68.

[9] Stokes GG. On the effect of the intemal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society 1851;9:8-106.

[10] Otter A. Damping forces on a cylinder oscillating in a viscous fluid. Applied Ocean Research 1990;12:153-155.

[11] Bearman PW, Russell M P . Measurements of hydrodynamic damping of bluff bodies with application to the prediction of viscous damping of TLP hulls. In: Proceedings of the 2Ist Symposium of Naval Hydro-dynamics, Trondheim, Norway, 1996.

[12] Ogilvie TF. First- and second-order forces on a cylinder submerged under a free surface. Joumal of Fluid Mechanics 1963;16:451-472. [13] Schlichting H . Boundary layer theory, 7th edn. New York:

McGraw-H i l l , 1979.

[14] Hall P. On the stability of unsteady boundary layer on a cyhnder oscillating transversely in a viscous fluid. Joumal of Fluid Mechanics 1984;146:347-367.

[15] Honji H. Streaked flow around an oscillating circular cylinder. Joumal of Fluid Mechanics 1981;107:509-520.

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