Applied Ocean
Research
E L S E V I E R Applied Ocean Research 21(1999) 311-328 . www.elsevier.com/locate/apor
On the estimation of Morison force coefficients and their predictive
accuracy for very rough circular cyhnders
J. Wolfram^'*, M . Naghipour^
Depai tment of Civil and Offshore Engineering, Heriot-Watt University, Edinburgh EH 14 4AS, UK Received 28 August 1998; received in revised form 7 July 1999
Abstract
This paper makes an assessment of the various method that may be used to analyse experiment data on the force experienced by a circular cylinder in waves and combined wave and current flows to estimate drag and inertia coefficients for use in Morison's equation. Most of the widely used techniques are considered together with a weighted least squares approach for time domain analysis. A set of data obtained from expeiiments on heavily roughened circular cylinders of diameters 0.513 and 0.216 m in the Delta wave flume at De Voorst in Holland i n waves and simulated current has been analysed in turn by all these techniques. The experiment data was split into two halves. The first was used for the analyses and the second was used to assess the predictive accuracy of Morison's equation. Using the force coefficients obtained from the different analysis techniques coiTesponding predicted force time series were constructed using the particle kinematics measured in the second parts of the data sets. These predicted time series were then compared with the corresponding measured force time histories. The root mean square en'or and the bias in the estimation of maximum force in each wave cycle are used as measures of predictive accuracy and as a basis for comparing the efficiency of the different analysis techniques. It was found that the weighted least square method generally gave the best predictive accuracy, but only by a small margin. © 1999 Elsevier Science Ltd. A l l rights reserved.
Keywords: Morison's equation; Surface roughness; Keulegan-Carpenter numbers; Hydrodynamic force coefficients
Nomenclature CD drag coefficient fnnJ(^-^pL>"Tms)' t o t ^ l force coeflficient C M inertia coefficient D cylinder diameter £ƒ eiTor term E mean square error E[x] expectation o f x ƒ Morison force
/e estimated or predicted M o r i s o n force /m measured M o r i s o n force
/rms root mean square force g acceleration due to gravity i V = T
I m denotes imaginary part o f f u n c t i o n
k index of w e i g h t i n g term and roughness height K C (ii^T/D), Keulegan-Caipenter number A^D drag term
* Corresponding author.
' Total Oil Marine Professor of Offshore Research and Development. ^ Fonnerly PhD student at Heriot-Watt University.
linearised drag term inertia term
A'^ number o f observations
N^y number o f waves o f above average height R Dean's reliability ratio
Re denotes real part o f f u n c t i o n S-^(cS) one-sided energy spectra S^y(cS) cross-spectral density o f x and y t time
T • time period o f wave cycle u horizontal wave particle velocity
i/m m a x i m u m horizontal water particle v e l o c i t y i n wave cycle
«TOis root mean square horizontal wave particle velocity il horizontal wave particle acceleration
amplitude o f horizontal wave particle v e l o c i t y at frequency <u
U u n i f o r m steady current velocity IJi mean value
p density o f water cr standard deviation U) wave frequency (x) time average o f x Ixl modulus o f X 0141-1187/99/$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved.
3 1 2 /. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328 1. Introduction
The domain o f M o r i s o n ' s equation [28] covers cylinders w i t h diameters up to around 20% o f wavelength. Thus the hydrodynamic loading on jacket structures and many other c y l i n d r i c a l components such as pipelines, risers and u m b i -licals f a l l w i t h i n the M o r i s o n regime. W h e n assessing the m a x i m u m hydrodynamic loading on a jacket structure the designer w i l l usually have at his disposal the estimated m a x i m u m wave height and coiTesponding period f o r the structures proposed location and estimated values o f C D and C M to use i n M o r i s o n ' s equation. He may w e l l w i s h to have a measure o f the eiTor or uncertainty associated w i t h this prediction particularly i f doing a structural relia-b i l i t y assessment or a risk analysis. The sources o f uncer-tainty, or error, i n the predicted m a x i m u m force can be split into three parts. The first is associated w i t h the prediction o f the extreme wave and its characteristics, i n c l u d i n g height, period and surface profile. The second is associated w i t h the p r e d i c t i o n ' o f the particle kinematics beneath the wave suiface i n the field o f the structure; and the third is asso-ciated w i t h the predictive accuracy o f M o r i s o n ' s equation given the force coefficients chosen.
This paper is p r i m a r i l y concerned w i t h the last o f these sources o f en^or. I t is not practically possible to dissociate the m o d e l l i n g eiTor intrinsic to Morison's equation f r o m that associated f r o m the force coefficients used w i t h i n i t ; and here they are considered together. There has been a consid-erable volume o f experimental research undertaken to esti-mate the force coefficients i n Morison's equation. The early results obtained f r o m small scale experiments and i n steady flows have now been largely discarded i n the hght o f results f r o m larger scale experiments, at more appropriate Reynolds numbers, i n oscillatory and wave flow regimes (e.g. Refs. [1,33,15]). Cylinder surface conditions more representative o f the marine growth coverage f o u n d offshore have also been used i n some experiments (e.g. Ref. [37]). Notable experiments undertaken offshore to measure forces i n the M o r i s o n regime include those at the Christchurch Bay T o w e r o f f the south coast o f England [5] and at the Ocean Test Structure (OTS) i n the G u l f o f M e x i c o [22]. W h i l s t these are, i n principle, the most realistic experi-ments, the resulting force coefficients show considerable scatter associated w i t h the difficulties o f accurate simulta-neous measurements o f force and particle Idnematics at the same location i n uncontrollable conditions. As a result o f all these activities, there are available i n the literature, values o f C D and C M f o r a variety o f surface roughness conditions and Keulegan-Carpenter numbers. However there are f e w measures of their predictive accuracy when used i n M o r i -son's equation.
This paper addresses the question of predictive accuracy o f M o r i s o n ' s equation and its relationship w i t h the analysis method used to obtain force coefficients f r o m experimental data. This has been done i n the context o f a particular set o f data obtained w i t h two heavily roughened circular cylinders
o f diameters 0.513 and 0.216 m , that have been tested i n long-crested random waves i n the Delta wave flume at De Voorst i n H o l l a n d . The experiments were at a large scale w i t h Reynolds numbers up to 5 X 10^ and both cylinders had a suiface roughness coefficient (k/D) o f 0.038 that is typical o f the hard f o u l i n g conditions f o u n d i n the N o r t h Sea. These experiments are described i n more detail i n Section 2.
The data f r o m each random wave experiment has been divided into two parts. The first has been used f o r the deter-mination o f force coefficients i n the context o f Morison's equation by a variety o f different niethods that are listed i n Section 3. The different method produced somewhat differ-ent force coefficidiffer-ents f r o m the same set o f data. I n each case these force coefficients have then been used i n Morison's equation to predict the force time history f o r the second half o f the set o f data based on the particle kinematics measured i n this part o f the data. These force predictions are then compared w i t h the measured force time histories and the discrepancies used as measures o f predictive accuracy.
The two halves o f each data set had the same n o m i n a l J O N S W A P spectmm (Joint N o r t h Sea Wave Project) (as described i n the Section 2) but i n each case the actual sample wave spectra and the i n d i v i d u a l time series f o r the t w o halves were quite different i.e. there is no repetition i n the time series. Thus the tests o f predictive accuracy were independent o f the analysis process to a much greater degree than when the analysed time series is refitted w i t h a recon-structed time series using M o r i s o n ' s equation. Ideally the data used f o r testing predictive accuracy should be obtained f r o m a number o f quite different wave spectra and the f i t to a variety o f different force time series examined. U n f o r t u -nately such experimental data were not available.
I n order to estimate the predictive accuracy a measure is needed o f how w e l l the predicted force maps onto the recorded force. One measure w o u l d be the root mean square e n w normalised by some f u n c t i o n o f the magnitude o f the recorded signal. This w o u l d measure the quality o f the mapping at all points o f the time series. H o w e v e r i t is the m a x i m u m magnitude o f the force i n v o l v i n g a single extreme wave that is o f interest i n the ultimate limit-state design assessment. I n the fatigue limit-state i t is the range o f the force produced b y each wave that is of concern. Thus the measure o f predictive accuracy used here is based o n the difference between the measured m a x i m u m ƒ„, and the predicted m a x i m u m f o r c e / e i n each wave cycle.
I n a statistical sense a good estimator should be unbiased and o f m i n i m u m variance. Thus t w o parameters may be used to assess how w e l l a predicted force time series compares w i t h the coiTesponding measured force time series. The chosen non-dimensional parameters are the normalised mean error ( M N E ) or bias and root mean square eiTor ( R M S E ) :
J. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328 313 wave maker n carriage fixed pile
a _
beach
Fig, 1, Schematic diagram of fixed and mobile cylinders in the large wave flume.
and
R M S E = 100.
i
A ^ w è L (fm)/ J ' 1 f r (fm)/ - (fe),- f(2)
where N^, is the number o f waves o f above average height. A s these parameters can be unduly influenced when ƒ„, is small and the absolute error is large i t was thought desirable not to consider small waves and their con-esponding forces when predicting the measured time series. Eor jacket type offshore structures the ultimate loading involves very large waves and most o f the fatigue damage also occurs i n larger waves; and so the ability to predict the forces due to small waves is o f little practical interest. Therefore i t was decided to see h o w w e l l the measured force due to waves o f above average height could be predicted.
The Keulegan-Caipenter number ( K C — d e f i n e d i n the nomenclature) f o r the tubular members o f an offshore struc-ture varies not only w i t h the height and period o f each wave but also w i t h the diameter o f the various members and their location relative to the free surface. I n the design o f offshore structures i t is usual to ignore the variation o f hydrodynamic coefficients w i t h K C and simply to use the same single values f o r the drag and inertia coefficients f o r the whole structure. That approach is reflected i n the analysis under-taken here where the emphasis has been on determining a single mean value f o r each coefficient f r o m each random wave experiment. I t is these single values that are then used i n the prediction o f the force time histories. Furthermore, variations o f force coefficients w i t h K C are not readily discernible f r o m frequency domain analysis o f random wave data. As one o f the objectives o f the study was an assessment o f the relative merits o f the vai-ious time domain and frequency domain analysis techniques a common basis f o r comparison was required and hence the mean values o f C D and C M have been used. The other objective was to give some measure o f the uncertainty and bias involved i n using M o r i s o n ' s equation f o r the prediction o f in-line forces that w o u l d be helpful i n structural reliability calculations and structural assessments and therefore f o l l o w i n g typical design practice was desirable.
Section 2 o f the paper describes the experiments that were undertaken i n the D e l f t Hydraulics Laboratories ( D H L ) l o n g wave flume at De Voorst i n H o l l a n d as part o f an E C / EPSRC funded project. Section 3 overviews the various methods f o r the prediction o f f o r c e coefiicients f r o m experi-mental data. Section 4 looks i n detail at the time domain analysis approaches and Section 5 at the frequency domain
techniques. Section 6 presents a discussion o f the results f r o m the analysis o f the experimental data using the various approaches and their associated limitations. E i n a l l y some conclusions are drawn.
2. Description of the experiments
D u r i n g September and October 1993 a series o f experi-ments were undertaken to examine the wave loading on t w o large scale circular cylinders i n the D e l f t H y d r a u l i c Labor-atory's Delta wave flume i n the Netheriands ( D H L ) . This flume is 230 m long, 5 m wide, 7 m deep and during the tests, was filled w i t h water to a depth o f about 5 m . The waves were generated by a programmable, hydraulically driven, piston type wave-maker and their energy was dissi-pated at the other end o f the flume thi-ough the use o f a 1:6 sloping concrete beach. This f a c i l i t y is capable o f generat-ing regular and random waves w i t h a range o f periods o f about 3 - 1 0 s and wave heights up to about 2 m over most o f the range o f periods.
For the random wave experiinents the J O N S W A P [12,21] spectrum was used and the results presented i n this paper are f o r experiments i n long crested random waves w i t h a signif-icant wave height o f 1.5 m and a peak period o f 5.9 s. D u r i n g the random wave experiments occasionally i n d i v i -dual wave crests over-topped the flume i m p l y i n g a wave crest elevation o f more than 2 m . The larger waves were visibly non-linear w i t h more sharply peaked, and sometimes breaking, crests.
For simulating the effects o f cun-ent and combined wave/ cunent flows the flume is equipped w i t h an 8 m by 6 m towing carnage that can attain a steady velocity o f 1 m/s and runs on a set o f rails on the top o f the flume walls. The m a x i m u m t o w i n g distance is dependent upon the test set up. For these experiments the caniage speeds were ± 1 m/s (for the larger cylinder) and ± 0.5 m/s ( f o r the smaller cylinder) and the t o w i n g distance was approximately 110 m . F i g . 1 shows a schematic longitudinal section o f the flume w i t h a cylinder mounted on the m o v i n g caniage, a fixed cylinder, the beach and the wave-maker.
The t w o vertical cylinders used f o r the experiments had base diameters o f 0.21 m (small) and 0.5 m (large) and were mounted i n turn on the t o w i n g can-iage and at fixed loca-tions i n the flume. B o t h cylinders were manufactured f r o m stainless steel and were covered w i t h the roughness pattern that was originally developed by W o l f r a m [38] w h i c h has been shown to simulate w e l l the effect o f hard marine
314 J. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328 MWL large pile 1.5m 2.5m
\!/
Fig. 2. Schematic diagram of cylinders in the flume showing the force sleeve positions.
growth. This roughness consists o f a pseudo-random aiTangement o f three different sizes o f right square pyramids w h i c h i n each case have heights o f the same dimensions as the base. Tlie roughness elements were cast i n fibreglass on semi-circular cylindrical shells w h i c h were strapped around the cylinder g i v i n g an effective roughness ratio (k/D) o f 0.038 and corresponding effective diameters f o r the large and small rough cylinders o f 0.513 and 0.216 m , respec-tively [26]. The schematic general elevation o f the cylinders and location o f the force sleeves i n the cross sections o f f l u m e is shown i n F i g . 2.
Measurements o f the wave force time series f o r b o t h cylinders were obtained using strain-gauged force measur-ing sections, each half a diameter i n length and capable o f measuring both the in-line and transverse forces. The small cylinder had five forces sleeves; one centred 0.5 m above the mean water level ( M W L ) and f o u r w i t h centres at distances o f 0.5, 1.5, 2.5 and 3.5 m below the M W L as shown i n F i g . 2. The large cylinder had two forces measur-ing sleeves centred at distances o f 1.5 and 2.5 m b e l o w the M W L . The measurements at the force sleeves at position 2 (see F i g . 2) on both the small and large cylinders were used f o r the analyses described i n this paper. These sleeves were sufficiently close to the water surface to give reasonably large measured forces but were always submerged during the experiments ensuring continuity o f the force measure-ments.
W h e n mounted on the carriage the cylinder was r i g i d l y fixed to the caniage at the top and had a horizontal plate
r i g i d l y attached at the bottom. Braces were fitted between the cylinder and the b o t t o m support so that fiie estimated natural frequency o f the system as a whole was about 4 H z . This frequency is some 16 times higher than the highest wave frequency tested but sometimes some vibration energy was input to the cylinder b y the m o t i o n o f the caniage and was evident i n the force signal traces [ 2 6 ] .
The ambient flow velocity was always measured at the longitudinal location o f the cylinder and at the elevation o f f o r c e sleeves using either electromagnetic flow meters ( E M F ) or perforated ball velocity meters. ( P V M ) . The E M F meters are very sensitive to magnetic fields and the output signals had to be filtered to reduce noise. The E M F meters were generally used except f o r the large m o b i l e cylinder and f o r the upper force sleeves on the small fixed cylinder where P V M s were used. The original f o r m o f the P V M was first used by Bishop [7] i n wave force investiga-tions at the second Christchurch Bay Tower. These instru-ments have also been used to measure particle velocities i n three-dimensions i n waves at laboratory scale b y Chaplin and Subbiah [16]. The signals f r o m the P V M s represent the Morison-type forces on the ball and these have been cali-brated i n oscillatory flow. The wave surface elevation was also measured continuously at the same sections o f the wave flume as the axes o f the cylinders using surface-following capacitance-type wave probes. The analogue voltages f r o m the flow meters, wave probes and load cells were digitised using an A / D converter at a sampling frequency o f 40 H z .
The enors associated w i t h the measurements o f force,
Table 1
Details of all experiments analysed (R is Dean's reliability ratio and KC is Keulegan-Carpenter number)
Run no. Current (m/s) Pile diameter (mm) No. of waves R KC
Min Mean Max Min Mean Max
1 0 513 291 0.03 0.74 1.46 0.2 5.5 17.5 2 1 513 130 0.72 3.79 15.5 1.92 13.5 25.3 3 - 1 513 102 3.22 5.66 24.8 1.43 17.4 32.3 4 0 216 286 0.08 1.72 3.21 0.4 12.8 37.9 5 0.5 216 152 0.99 3.63 6.0 0.75 21.8 45.9 6 - 0.5 216 141 0.63 5.18 10.5 1.92 22.9 57
J. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328 315
3^ 5?
S(0}) (second part of data)
S(m) (first part of data)
Runl
0.3 0.4 0.5 Frequency (Hz)
0.6
Fig. 3. Comparison between surface elevation spectra in the two halves of the random wave experiment for the tixed large pile.
water particle velocity and wave surface elevation have not been quantified e x p l i c i t l y i n a f o r m a l manner. B u t the proce-dures used are w e l l established and the techniques have been employed f o r over ten years satisfactorily, h is consid-ered that the measurement errors are small compared to the predictive enors that are the subject o f this paper.
The details o f the six experiment runs considered here are given i n Table 1 f r p m w h i c h i t can be seen that there were experiments w i t h a current i n the wave direction, w i t h a current opposing the wave direction and w i t h no cunent f o r both the small and large cylinders. The cuirents were achieved by translating the cylinder on the m o v i n g carriage either away f r o m the wave-maker (i.e. cuiTent i n the wave direction) or towards the wave-maker (i.e. cunent opposing the wave direction). Several such translations needed to be patched together to produce a complete m n o f 10 m i n duration f o r each experiment condition. The experiments w i t h the stationaiy cylinders were nominally o f 30 m i n duration.
W h i l e the two cylinders and their roughness patterns were geometrically similar, the cuiTent speeds were not i n exactly the same scale relationship and, as the same spectrum was used throughout, a l l the experiments are distinctly different rather than being scaled up (or down) versions o f one another. The small cylinder was originally introduced into the programme to extend the range o f K C investigated. The time series conesponding to the wave spectmm was simu-lated by a sum o f 200 sinusoidal components so the first and second halves o f each experiment m n were independent and had significantly different sample spectra; as can be seen f r o m F i g . 3 w h i c h is a typical example.
3. Overview of methods for estimating force coefficients A w i d e variety o f approaches have been used to analyse experiment data to determine drag and inertia coefficients.
The authors discuss below all the techniques that have been used, at least to their knowledge. H o w e v e r they have not considered approaches w h i c h seek to m o d i f y the f o r m o f M o r i s o n ' s equation, using systems identification and other techniques, by the addition o f further terms (see f o r example Refs. [34,39]). Such modified f o r m s o f M o r i s o n ' s equation are s t i l l the stibject o f research and are not c t m e n t l y used i n practice.
The various techniques considered here can be cate-gorised according to type and those that have been used i n the present study are identified as M e t h o d 1, M e t h o d 2, etc. i n the f o l l o w i n g list.
1. T i m e domain techniques: W a v e b y wave analyses
F i t t i n g at m a x i m u m velocity and acceleration (Evans) Fourier averaging
Bearman, Chaplin et al. [2] (Method 1) K l o p m a n and Kostense (Method 2) [24] Ordinary least squares ( M e t h o d 3) W e i g h t e d least squares
W h o l e record analyses
Mean square method (Bishop and Shipway) Ordinary least squares ( M e t h o d 4)
Weighted least squares (Method 5)
M e t h o d o f moments (Pierson and Holmes) ( M e t h o d 6) 2. Frequency domain techniques
Linear models (i.e. w i t h a linearised drag term) Cross spectral density (force and particle velocity) ( M e t h o d 7)
Cross spectral density (force and surface elevation) ( M e t h o d 8)
3 1 6 J. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328
Non-linear models
Cubic drag model o f Bendat and Piersol [3] ( M e t h o d 10)
I n wave by wave analysis CQ and C M are assumed constant over each wave cycle and the output o f the analysis can be presented as a plot against K C . The alternative is to assume that C D and C M are constant over the whole record and to analysis this as a single entity. I n this latter case no e x p l i c i t measure o f the variance C D and C M is obtained. I n random waves the resultant average values o f C D and C M obtained f r o m these two approaches w i l l be different even i f essen-t i a l l y essen-the same analysis essen-technique is used. This is because i n the f o r m e r case each wave has an equal w e i g h t i n g w h e n f i n d i n g the mean whereas i n the latter case the influence o f each wave on the outcome w i l l be proportional to its wave period (or more specifically the number o f data points w i t h i n the wave cycle).
The frequency domain analysis methods aU use the entire record and use spectra and cross spectra i n various ways to estimate the force coefiicients. I n some cases the force coeffi-cients are obtained as functions o f frequency and i n others as single values. M o s t o f the analysis techniques examined here involve linearisation of the drag term. However one non-linear approach i n v o l v i n g a cubic expansion is examined.
M o s t o f the methods listed above use the velocity t i m e series measured beside the f o r c e sleeve on the test c y l i n d e r to estimate particle kinematics. Occasionally, both offshore and i n the laboratory this is not available and the particle Idnematics must be estimated directly f r o m the suiface elevation by a wave theory. M e t h o d 8 i n the above list incorporates this directly i n the analysis but this additional aspect could be included i n any o f the methods using random linear wave theory as discussed later. M o s t o f the methods have been extended to include the effects o f a steady cuirent o f u n i f o r m depth-wise p r o f i l e and the con-e-sponding expressions f o r the coefficients are also presented in the f o l l o w i n g sections.
The enor associated w i t h any estimate o f t h e force coefiicient w i l l depend i n pai-t on the assumptions i m p l i c i t i n the analysis technique used. I t w i l l also depend upon the data and whether these are weU or poorly conditioned f o r resolving C M and CD. Dean [ 19], i n the context o f least squai-es time domain analysis, suggested that much o f the scatter i n the reported coefficients can be due to poorly conditioned data. He presented a criteriafor evaluating the suitabihty o f data f o r determining C M and C D i n either t i m e or frequency domain analysis. This involves calculating the f o l l o w i n g 'reliability ratio'
R = 2 (3)
where () indicates time averaging o f the enclosed quantity over the wave cycle. He showed w i t h a particular example that i f the mean square error is o f the order o f 10% o f the measured force, then the e n o r i n estimated values o f C M and
C D can be on the order o f 85 and 50%, respectively. Dean suggested that data w i l l be well-conditioned f o r evaluating both C M and C D together when 0.25 < 7? < 4 and f o r C M only when Q<R< 0.25 and f o r C D only when R> A. The mean and range o f the r e l i a b i l i t y ratio f o r a l l the experi-ments analysed here are presented i n Table 1 and as w i l l be seen later some o f the data are poorly conditioned f o r the estimation o f C D and some f o r C M .
4. T i m e domain analysis techniques
The general f o r m o f M o r i s o n ' s equation f o r a vertical cylinder adapted to allow f o r the presence o f a cunent U is given below together w i t h its shorthand f o r m .
ƒ = 0 . 5 C D P D ( « - f U)\{u + U)\ + 0.25CMPTTD^U
= Kuiu + U)\{u + U)\ + KMII. (4)
Here ƒ is the M o r i s o n force, p is water density. C M and Co are the drag and inertia coefficients, respectively, and u and w are the horizontal components o f water particle velocity and acceleration, respectively. I n the absence o f cunent the term U is removed.
4.1. Wave by wave analyses
The simplest approach is to find the points o f zero accel-eration where the corresponding force is purely drag and calculate C D directly. S i m i l a r l y at the points where the velo-c i t y is zero the forvelo-ce w i l l be purely inertial and C M velo-can be calculated (e.g. Ref. [20]). However this approach uses very little o f the data and may be subject to significant enors i f there is noise on the velocity and force time series. I t is not w i d e l y used and is not considered here.
Keulegan and Carpenter [23] introduced a method to find the inertia and drag force coefficients i n each cycle using Fourier or time averaging [ 1 3 ] . This basic approach is widely adopted and methods based on i t have been devel-oped and used b y Bearman et al. [1,2], Bishop [6], IClopman and Kostense [24] and Davies [18].
I n the method used by Bearman et al. [2] f o r the case without current, M o r i s o n ' s equation is m u l t i p l i e d by u and the time average taken over a wave cycle. Assuming that the time-averaged cross product (iiii) is zero over a wave cycle this yields the f o l l o w i n g expression:
_ (ƒ»)
S i m i l a r l y m u l t i p l y i n g M o r i s o n ' s equation by ii, taldng the time average over each wave cycle and assuining is zero yields:
_ m
0.25pTTD2(«2) • ^6)
The approach i m p l i c i t l y assumes that the velocity and accel-eration time series are orthogonal. O n essentially the same basis Klopman and Kostense [24] modified this approach
J. Wolfram, M. Naghipoiir / AppUed Ocean Research 21 (1999) 311-328 317 and m u l t i p l i e d M o r i s o n ' s equation first by ii\ii\ and then b y ic
before time averaging to give the f o l l o w i n g pair o f equations. and C M 0.5pD{i/) ( f u ) 0.25p7TD2(;i2) (7) (8)
Here term just the terni (H«|M|) is assumed to be zero. B o t h o f these Fourier averaging approaches can easily be extended to include the effect o f a u n i f o r m steady current U. However the time averages considered above to be zero w i l l no longer be so and the pair o f equations obtained after time averaging must be solved simultaneously. The equivalent expressions f o r Eqs. (5) and (6) are then f o u n d to be:
(fill + U)){i?) - (ƒ»)((» + U)ii) 0.5pD((|(n + U f ) { i i ^ ) + {{li + U)\{ii + U)\ii))
(9)
C M —
+ ;7)|^) - {f{u + U)){ii{ii + U)\{u + ^)|)
0.25pi^D\{iP-){\iii + u f ) - {ii{ii + U)){ii(,u + U)\{ii + U)\))
(10) For Eqs. (7) and (8) they become:
^ ^ (ƒ(» + U)\u + U\){ih - (fii){{ii + ;7)|(» + U)\u) " 0 . 5 p D ( ( ( ( H + UfXii^} - « | M + U\iu + U ) i i ) f )
(11)
C . (ƒ«)((» + U f ) - ( f { i i + U)\u + U\){ii{,ii + U)\u + U\) 0.25pTTD^iiii^Xiii + U f ) - ({|M + U\{u + U ) i i ) f )
(12) I n the derivation o f Eqs. ( 9 ) - ( 1 2 ) no assumptions or restric-tions have been placed on the velocity or acceleration time series or their time averages. Hence the shape o f the wave is unrestricted and i t w i l l be noticed that when [/ = 0 (i.e. there is no cun'ent) these equations are different f r o m the correspond-ing Eqs. ( 5 ) - ( 8 ) i n that there is a second term i n both the numerator and the denominator o f Eqs. ( 9 ) - ( 1 2 ) . These terms involve the time averages which B e a m a n et al. [2] and K l o p m a n and Kostense [24] assumed were zero i n their approaches. M o r e interestingly i t is f o u n d that Eqs. (11) and (12) are identical to those obtained using a least-squares approach and as w i l l be seen later the least-squai'es approach
yields better results. So although these terms are, generally, comparatively small they do have some effect and as they are readily calculated there is a httle point i n not including them. The least-squares approach can be applied i n a wave by wave analysis or to a whole record or, indeed, i n the frequency domain. I n the time domain CQ and C M are chosen to m i n i m i z e the sum o f the squares o f the difference between the measured and the estimated f o r c e at each measurement point. T h i s yields t w o simultaneous equations that can be solved to give the f o h o w i n g expressions:
Cn =
o . 5 p D { X " ' * Z ; ' ^ - ( I " w | » | ) ^ } '
(13)
(14)
The corresponding equations f o r the case w h e n there is a steady u n i f o r m cunent are obtained simply by substituting U + ll f o r ll on each occurrence o f ii i n the Eqs. (13) and (14). The least-squares method can be applied to any length of time series. There need not be a whole number o f waves or even a whole wave cycle.
I f a single wave cycle is considered then f o r , say, the particle velocity {l/N)Y.ii = (u), and s i m i l a r l y f o r all other time averages over a wave cycle. Thus Eqs. (13) and (14) are seen to be identical to Eqs. (11) and (12) when U = 0, f o r wave b y wave analysis. Furthermore, i f the second terms i n both the numerators and denominators o f Eqs. (13) and (14) are considered negligible then these equations become the same as those used b y K l o p m a n and Kostense; Eqs. (7) and (8) above.
I n the least-squares approach all data points i n i h e measured force time series have an equal influence on the determination o f CQ and C M . This may be thought a l i m i t a -tion when there is httle interest i n predicting forces close to zero and m u c h more interest i n predicting m a x i m u m forces. T o ensure more emphasis is given to finding f o r c e c o e f f i -cients w h i c h w i l l predict m a x i m u m forces accurately a weighted least-squares approach where the weights are related to the magnitude o f the force can be adopted. Thus at each p o i n t i n the t i m e series the difference between the measured and predicted f o r c e is m u l t i p l i e d hyf'^, where k is a positive index. T h i s means that those parts o f the t i m e series where the measured force is small have very l i t t l e infiuence on the estimated values o f C D and C M but w h e n the magnitude o f the force is large the influence is consider-able. The expressions f o r the force coefficients w h e n there is a steady current U then become:
X / V ( » + u)\u + u \ Y . f v - I . f f ü Z f i i ' + u)\u + u\t' 0.5pD{ Zf'^iu + Uf Zf'^ii^ - QLf'iKi' + U)\u + (7|)2} '
l / V » l / ' ( » + U)' - I / V ( » + U)\u + U l T f ' i i j u + U)\i, + u\
0.25PTTD^ Z/2'--i',2 ^^/S'-TM + Uf - (Zf'^iiiu + U)\u + { / | ) 2 }
(15)
318 /. Wolfram, M. Naghipoiir / AppUed Ocean Research 21 (1999) 311-328 Table 2
Values of Co and CM from the analysis methods in the random waves for a fixed large pile—Run 1
IVlethod of analysis CD (mean) a^i CM (mean) o-c,„ MNE RMSE Time doinain: wave by wave
1. Bearman et al. [2] 1.55 1.01 2.04 0.17 5.28 11.57
2. Klopman and Kostense [24] 1.38 1.11 2.04 0.17 6.78 12.31
3. Ordinary least-squares 1.88 1.54 2.08 0.15 1.32 11.41
Time doinain: whole record
4. Ordinary least-squares (k = 0) 1.57
-
2.04 - 5.08 11.51 5. Weighted least-squares k = 1 1.54-
2.14-
1.23 10.82 K = 2 1.52-
2.22 - - 1.64 11.28 A: = 3 1.50 - 2.28 - 4 . 1 8 12.24 6. Method of moments 1.73-
2.06 _ 2.78 11.08 Frequency domain7. Cross spectra—force and paiticle vel. 1.41 0.40 1.93 0.44 11.03 14.70
8. Cross spectra—force and surface elevations 1.57 0.69 1.95 1.08 8.70 13.24
9. Least-squares fit to spectra 2.26
-
1.87-
1.94 14.5410. Cubic model of Bendat and Piersol 1.42 ' 0.58 1.93 0.44 10.93 14.63
For a wave by wave analysis, i n the l i m i t when k is very large, C D and C M w i l l be f o u n d such that the reconstructed time series exactly matches the original but only at the points o f m a x i m u m and m i n i m u m ( m a x i m u m negative) force. However when predicting the m a x i m a and m i n i m a for a different time series large values o f k are f o u n d to produce poor results. W h e n ^ = 0 then this method clearly becomes the same as the simple least squares method.
As the wave b y wave analysis usually involves simple averaging o f the f o r c e coefficients f o u n d i n each wave cycle some o f the weighting effects w i l l be reduced and very large waves w i l l have no more influence on the final outcome than very small ones. So i n the study here the authors have applied this approach predominantly to whole experiment records.
4.2. Whole records time series analysis
Some techniques require the variability o f iiTegular waves i n a whole time series to work at a l l . One such is the mean square method developed by Bishop [ 6 ] and
Bishop and Shipway [7] to analyse the data collected i n the offshore experiments at the Christchurch B a y Tower. This approach involves estimating moments f r o m the data and then using a regression analysis. I t is u s e f u l where cycle by cycle analysis is not possible due to an u n k n o w n phase lag between the force and velocity measurements; as occurred at the Christchurch Bay T o w e r experiments. However w i t h scatter i n the data i t can lead to some negative estimates o f C D , as discussed by Davies [ 1 8 ] , and i t is not used here. The techniques used i n this study f o r whole record time domain analysis are the weighted least-squares approach (outlined above) and the method o f moments.
I n the weighted least-squares approach the authors have used various values o f k i n Eqs. (15) and (16) and have f o u n d that this can be optimised i n an iterative manner to give a m i n i m u m predictive eiTor i n the peak f o r c e regions. Results f o r various values o f k f r o m 0 (corresponding to ordinary least squares) to three are presented i n Tables 2 ¬ 7 and f o r these sets o f data a value o f two gives, o n average, the best results.
The method o f moments was introduced by Pierson and
Table 3
Values of and CM from the analysis methods in the random waves for a mobile large pile—Run 2
Method of analysis CD (mean) cr.i CM (mean) a,^ MNE RMSE Time domain: wave by wave
1. Bearman et al. [2] 2. Klopman and Kostense [24] 3. Ordinary least-squares Time domain: whole record 4. Ordinary least-squares (k = 0) 5. Weighted least-squares *: = 1 /t = 2 /t = 3 6. Method of moments Frequency domain
8. Cross spectra—force and surface elevations 9. Least-squares fit to spectra
1.32 0.37 1.49 1.32 0.36 1.50 1.51 0.37 1.72 1.42 - 1.59 1.44 - 1.82 1.44 - 1.98 1.45 . - 2 . 1 3 1.46 - 0.53 1.10 0.49 2.34 1.14 - 2.32 0.56 15.18 16.69 0.57 15.54 17.01 0.72 2.82 8.44 8.74 11.50 7.29 10.49 6.26 9.85 5.36 9.35 7.80 11.01 1.31 24.81 25.55 22.6 23.4
J. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328 3 1 9
Table 4
Values of Co and CM from the analysis methods in the random waves for a mobile large pile—Run 3
Method of analysis CD (mean) O-cd CM (mean) MNE RMSE
Time domain: wave by wave
1. Bearman et al. [2] 1.45 2. Klopman and Kostense [24] 1.44 3. Ordinary least squares 1.54 Time domain: whole record
4. Ordinary least squares (k = 0) 1.48 5. Weighted least squares k = I 1.47
k = 2 1.46 k = 3 1.45 6. Method of moments 1.55
Frequency domain
8. Cross spectra—force and surface elevadons 1.43
9. Least squares fit to spectra 1.55 0.33 0.32 0.30 0.05 1.36 1.37 1.69 1.55 1.62 1.63 1.64 1.68 1.43 1.68 0.62 0.60 1.81 0.27 1.28 2.06 • 4.84 0.89 0.14 0.34 0.69 5.54 2.40 5.54 7.50 7.60 9.14 7.55 7.40 7.36 7.36 9.58 7.68 9.58
Holmes [32] and used by M u g a and W i l s o n [29]. I t starts b y assuming that the velocity and acceleration, ii and ii, are independent normal random variables w i t h zero mean. Then expressions are obtained f o r the expected values, E\f^] and £ [ ƒ * ] f r o m M o r i s o n ' s equation i n terms o f Ku and KM and the moments o f (( and it. These expected values are then estimated w i t h the means o f the square and the f o u r t h power o f the measured force, denoted here by 1x2 and p,4, and the resultant pair o f simultaneous equations is solved to y i e l d the f o l l o w i n g expressions:
and
3KUIJI
(17)
(/.4 - 3 i 4 f '
780-1 (18)
For the coiTesponding case o f a co-existing steady current
T u n g and Huang [35] found: Ml K and 2 a ^ [ ( l + r ^ ) Z ( y ) + yzm KM — f4 - Klati3 + 6y + y") (19) (20)
i n w h i c h /x 1 is the mean value o f the measured force, y =
IXJo-,, and Z ( y ) = J^z(x) d r and z{y) =
(2TT)"°-^ e x p ( - 0 . 5 - ) ^ ) .
As w e l l as the assumption o f a n o r m a l distribution f o r wave particle velocity and acceleration, the method o f moments also requires long records and significant irregu-larity to allow the accurate estimation o f the higher moments. However the method is robust i n that any u n k n o w n phase lags between f o r c e and velocity measure-ments w i l l not influence the results p r o v i d e d the time series is long enough.
A f e w points can be made to summarise the time domain analysis techniques. The least squares method requires no
Table 5
Values of Co and CM from the analysis methods in the random waves for a fixed small pile- -Run 4
Method of analysis CD (mean) (mean) MNE RMSE
Time domain: wave by wave
1. Bearman et al. [2] 2.05 1.44 1.91 0.40 - 18.3 27.76
2. Klopman and Kostense [24] 1.95 1.34 1.91 0.40 - 13.5 23.75
3. Ordinary least squares 1.95 1.34 1.91 0.40 - 13.5 23.75
Time domain: whole record
4. Ordinary least squares (k = 0) 1.61 - 1.93 - 2.30 15.41
5. Weighted least squares k = I 1.55 - 2.22 - 0.83 14.48
k = 2 1.51 - 2.35 - - 1.56 14.79
k = 3 1.49 - 2.57 - - 4.43 16.14
6. Method of moments 1.46 - 2.45 - 0.92 14.47
Frequency domain
1. Cross spectra—force and particle vel 1.74 0.15 1.70 0.52 5.67 16.95 8. Cross spectra—force and surface elevations 1.66 0.51 2.31 2.4 - 5.14 16.25
9. Least squares fit to spectra 1.85 - 1.84 - - 7.81 20.03
320 J. Wolfram, M. Naghipow / Applied Ocean Research 21 (1999) 311-328 Table 6
Values of CD aud CM from the analysis methods in the random waves for a mobile large pile—Run 5
Method of analysis Co (mean) a,. CM (mean) cr,,,, MNE RMSE Time domain: wave by wave
1. Bearman et al. [2] 1.53 0.35 1.56 0.42 - 6 . 1 1 13.75
2. Klopman and Kostense [24] 1.47 0.34 1.56 0.44 - 2.48 12.14
3. Ordinary least squares 1.60 0.28 1.61 0.48 - 10.9 16.85
Time domam: whole record
4. Ordinary least squares (k = 0) 1.48 - 1.74 - 3.88 12.62
5. Weighted least squares k = { 1.44 - 2.07 _ - 2.34 11.98
k = 2 1.42
-
2.22 - - 1.83 11.80k = 3 1.41 - 2.31 _ - 1.83 11.79
6. Method of moments 1.71
-
2.56 _ - 22.3 26.4Frequency domain
8. Cross spectra—force and surface eievations 1.23 0.67 2.19 3.03 10.53 14.67
9. Least squares fit to spectra 1.47 - 1.59 - - 2.26 12.06
specific assumptions concerning the data and can be applied to any waves regular or irregular, linear or non-linear, and to a record o f any (sensible) length. The Fourier averaging methods require whole wave cycles and assume that velo-city and acceleration time series are orthogonal. The method o f moments further requires that the velocity and accelera-tion t i m e series are zero mean random processes w i t h Gaus-sian distributions. Finally the weighted least squares approach allows emphasis to be placed on force maxima at the expense o f smaller values o f force.
The data analysed here were obtained i n waves that were significantly non-linear and i n some cases sufficiently steep to have breaking crests (like real storm waves!). The results o f the analysis f o r the six experiments are shown i n Tables 2 - 7 and, not surprisingly, show that the analysis methods making fewer assumptions give better predictive accuracy. As accuracy is measured i n term o f ability to predict maxima i t is not suiprising that the weighted least squares gives on average (see Table 8) the best results.
5. F r e q u e n c y domain methods
Frequency domain methods may be divided into t w o
categories; those that i n v o l v e the linearisation o f M o r i s o n ' s equation, and those that are non-linear i n some way. Three o f the f o r m e r type and one o f the latter are considered here. N o t considered here are models that depart f r o m the o i i g i n a l f o r m o f M o r i s o n ' s equation by adding additional terms. There have been many such attempts to improve M o r i s o n ' s equation most recently using systems identification techniques (see f o r example Refs. [34,39]).
A l l the frequency domain techniques assume that the water particle velocity is a random Gaussian process w h i c h , i n the absence o f cuiTent, is also a zero mean process, and further, that u\u\ i n the drag term can be approximated by a p o l y n o m i a l yielding:
(21) TT
as the linear approximation, and
u\ll\ = licr^,' (22)
as the cubic approximation. See Refs. [3,9,10,11] f o r more details. I n the presence o f a u n i f o r m current a similar p o l y -nomial fitting procedure can be used and then the mean o f Table 7
Values of Co and CM from the analysis methods in the random waves for a mobile large pile—Run 6
Method of analysis CD (mean) CM (mean) MNE RMSE
Time domain: wave by wave
1. Bearman et al. [2] 1.04 0.34 1.56 0.46 14.99 17.31
2. Klopman and Kostense [24] 1.05 0.32 1.56 0.46 14.17 16.64
3. Ordinary least squares 1.16 0.25 1.76 0.81 4.38 10.71
Time domain: Whole record 10.71
4. Ordinary least squares (k = 0) 1.21 — 1.74 _ 1.06 10.06
5. Weighted least squares k = 1 1.21 - 1.86 _ 0.35 10.25
k = 2 1.21 - 1.94 - 0.14 10.34
k = 3 1.21 - 1.94 — 0.06 10.34
6. Method of moments 1.15 - 2.99 _ - 0.10 12.58
Frequency domain
8. Cross spectra—force and surface elevations 1.80 0.81 1.63 0.50 - 44.0 46.2
J. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328 321
Table 8
Values of CQ and C.^ from the analysis methods in the random waves (averaged in the six runs)
Method of analysis 'oMNE 'oRMSE CD (mean) CM (mean)
Time domain: wave by wave
1. Bearman etal. [2] 10.19 2. Klopman and Kostense [24] 9.09 3. Ordinary least squares 5.56 Time domain: whole record
4. Ordinary least-squares (k = 0) 3.66 5. Weighted least-squares k = 2 2.03 6. Method of moments 6.57 Frequency domain
8. Cross spectra—force and surface elevations 15.93
9. Least-squares fit to spectra 24.14
Averages 10.14 15.76 14.91 12.83 11.44 10.90 14.19 20.6 30.80 16.67 1.49 1.44 1.42 1.46 1.51 1.51 1.47 1.82 1.54 1.65 1.66 1.80 1.77 2.04 2.05 1.98 1.79 1.86
the water particle velocity is no longer zero but equal to the current speed U. This yields [10,14]
«I = (TIIH - i \ 2 Z { i ) - 1) + l y z m
+ 2a-„[7(2Z(y) - 1) + 22(y)]w (23) f o r the linear approximation where the various terms have been defined above after E q . (20). I f the current is zero then 7= 0 , and 2(7) = l/V27Tand the expression above reduces to the linear approximation i n E q . (21).
5.1. Linear models
There are a variety o f related analysis methods based on the use o f spectra and cross spectra that have been devel-oped by Borgman [9,11] and Bendat and Piersol [3] among others. The first considered here involves the cross-spectra o f measured f o r c e and wave particle velocity and is the linear counterpart o f the non-linear model considered i n the Section 5.2. I n the linear model M o r i s o n ' s equation becomes;
F{t) = KMiKt) + Ko^j - (Tuiiit) = KMÜ(t) + Kuii(t). (24)
B y assuming random linear wave theory an expression is obtained l i n k i n g M o r i s o n force and wave particle velocity. T a k i n g Fourier transforms the corresponding frequency domain expression is f o u n d w i t h force and velocity l i n k e d by the transfer f u n c t i o n H„f{co) f r o m w h i c h the force c o e f f i -cients can be estimated as f o l l o w s :
K^(co) = pDCoico)^ - a-,, = Re[//„y(a))] (25)
KM(OJ) = C M ( W ) = (26)
The transfer f u n c t i o n i(a))can be estimated f r o m (see f o r example Refs. [4,31]):
S„f{co) 5 „ ( ^ )
(27)
where ^ „ ( w ) is the one-sided spectral density f u n c t i o n o f t h e input i.e. measured particle velocity and S„f{a)) is the cross-spectral density o f the input and output i.e. measured parti-cle velocity and measured force.
Borgman [11] proposed another type o f linear cross-spec-tral density model f o r use when particle kinematics have to be infeiTed f r o m the wave surface elevation. Linear random wave theory and the linear dispersion relationship are used to establish time series f o r the wave particle kinematics f r o m the wave suiface elevation time series. The cross spec-tra i n the expression below (obtained by Borgman) are then computed.
(28)
Each o f the cross spectra i n this equation has a real and imaginary part y i e l d i n g t w o equations that can be solved simultaneously to provide the f o l l o w i n g expressions f o r the force coefficients as functions o f frequency:
C D ( W )
C M ( W )
IT 2 / C ^ i i ö , / - Q-quC-nf 2 (
(29)
(30)
Here C and Q indicate the real and imaginary parts o f cross-spectra i n E q . (28).
I n the case where there is current as w e l l as waves then Eq. (28) may be written as:
S^ico) = 4(z(7) + \y\Ziy))Kua„S^,X<^) + KuS^d^o), (31) and the drag coefficient is then given by:
C D ( « ) =
1 C-qiiQnf QniiC-rf
2pDc7„(z(7) + | y Z ( 7 ) ) I C^.;Q^„ - Q^,C^,. l (32) T h e presence o f the cuiTent does effect the equation f o r the inertia coefficient.
322 / Wolfram, M. Naghipow / AppUed Ocean Research 21 (1999) 311-328 A l t h o u g h they may not appear so at first sight Eqs. (29)
and (30) are very similar to Eqs. (25) and (26) and this method (denoted as M e t h o d 8) is quite similar to the one above ( M e t h o d 7). I n both methods the drag and inertia coefhcients are obtained as functions o f frequency, rather than Keulegan-Carpenter. Here the results have been aver-aged over the frequency range f r o m 0.1 to 0.4 H z (where f r o m Eig. 3 most o f the energy is seen to lie) to provide single values o f Co and C M to be used i n the prediction study. The predictive accuracy of these two methods may be j u d g e d f r o m the results presented i n Tables 2 - 7 and summarised i n Table 8. Overall both are seen to be s i g n i f i -cantly poorer than the time domain approaches.
B o r g m a n [11] also proposed a least-squares frequency domain method f o r determining force coefficients using a linear model based on the f o l l o w i n g equation f o r power spectral densities.
Sfico) = ^ ^ S „ i c o ) + Kl,S,(oj). (33) The spectra f o r the particle kinematics can be obtained
f r o m the suiface elevation or, alternatively, they may be computed more directly, and usually more accurately, f r o m the velocity time series. However f o r the analysis here they have been computed f r o m the wave suiface eleva-t i o n eleva-time series.
The coefficients and are estimated such that the predicted force spectrum Sficj) f r o m Eq. (23) is the best least squares fit to the measured force spectrum. This yields the f o l l o w i n g equations:
where is the number o f discrete frequencies at w h i c h the spectram is computed f r o m the time series. Note that in this single value of each of the force coefficients is obtained.
W h e n there is a current i n addition to waves then Eq. (33) may be shown [25] to take the f o r m :
S(oj) = 16(z(y) + \y\Ziy)fKlof,S„(a^) + KI,S,-XCO) (36) where the terms have the same meaning as i n Eqs. (19) and (20). The coiTesponding pair o f equations f o r the force coef-ficients, similar i n f o r m to Eqs. (34) and (35), is readily obtained. I n this method it is w o r t h noting that no account is taken o f the phase between the force and particle
kinematic time series. I n this respect the method is similar to_ those o f Bishop [6] and Pierson and Holmes [32] and hence may be useful when there is an u n k n o w n phase lag between force and velocity measurements. However the overah the predictive accuracy o f this method (Method 9) is seen to be poorest of a l l i n Table 8.
5.2. Non-linear models
Non-linear frequency domain models have been consid-ered theoretically by Bendat and Piersol [ 3 ] . These were applied to small scale experiments by Vugts and Bouquet [36], and to large scale experiment data, obtained i n the De Voorst wave flume, by B l i e k and K l o p m a n [ 8 ] . M o r e recently non-linear extensions to M o r i s o n ' s equation have been considered by W o r d e n et al. [ 3 9 ] , among others, using systems identification techniques applied to a variety o f experiment data.
The model considered here is shown i n E i g . 4 and was originally developed b y Bendat and Piersol. I t uses a cubic approximation i n w h i c h the terms Zi(w) and T^{co) represent the transfer f u n c t i o n of the linear and non-linear parts o f M o r i s o n ' s equation. To restrict this model to the usual M o r i -son f o r m TI{M) is considered to be purely imaginary (i.e. w i t h 90° phase), coiTesponding to the inertia term and T^ico) purely real corresponding to the drag term. The case f o r doing this is discussed by Vugts and Bouquet. The corre-sponding f o r m o f M o r i s o n ' s equation is then:
f i t ) = KuiKt) + Kuium + Ku2n0)\ (37)
(34)
(35)
where
'^M — > J^Di — ^ . .^^02 — and
The corresponding frequency doinain expression is f o u n d to be
F(ü)) = icoK^iiiü)) + Kua{3o^,u{co) + u\o})) (38) 2 _ 1=1
M
X S,icüi)S,M) X S,M)Sfico,) - X Slico,-) X S,;io}dSf(co!)
,=I 1=1
Kh = (7r/8)^
X S„(üJi)S,;(ü>d) - X Sl(coi) f Slico,) \ i = i / ;=i / = i N N N N
Y^S,XoJi)S,(oj^)Y,S,;{a>i)Sfico;) - Y^slicoi)Y^S,XcOi)Sf((Oi) 1=1 ,'=1
1=1
X 5„(«,)S,(a>,.) - X Slico^) X Sl(coi) 1=1
J. Wolfram, M. Naghipoiir / Applied Ocean Research 21 (1999) 311-328
r, L . n(l)
u(t)
ac7 3
(X
Fig. 4. Non-linear model for Morison's equation after Bendat and Piersol.
323
fit)
where ii' (co) is a triple convolution o f ii{(ü) w i t h itself (see Refs. [3,39] f o r f u l l and interesting discussions o f this and related models). C M can be estimated directiy f r o m the imaginary part o f the transfer f u n c t i o n i n exactly the same w a y as described f o r M e t h o d 7 using Eqs. (25) and (26) above.
T o estimate C ^ the non-linear term must first be estimated and to do this accurately requires a long time series; as noted b y Vugts and Bouquet and B l i e k and K l o p m a n . There are various possible procedures f o r obtaining the estimate (see Ref. [ 8 ] ) and the one used here is that described i n Bendat and Pierson [ 3 ] . B r i e f l y the velocity cubed time series is Eourier transfoimed and the corresponding spectral density 5„3(w) f o u n d . Then the f o l l o w i n g expression is used to esti-mate Tn(cti) : Sfiw) = a^\Uco)\''[na%i(o) + 5„3(cü)}; whence Ku(cj) = R e [ r „ ( a ; ) ] . (39) (40) The variation o f these force coefficients w i t h frequency is seen to be significant i n Eigs. 5 and 6 f o r the lai'ge and small cylinder, respectively, i n the case where there is no cuiTent. I t is interesting to note the variation o f C D w i t h frequency is quite different f o r the t w o cases reflecting, perhaps, the
different ranges o f K C at each frequency. However as can be seen f r o m Tables 2 and 5 there is no improvement i n the predictive accuracy. A s a result the authors d i d not consider it w o r t h applying this non-linear model to cases where cuirent was present. The extension to the case w i t h current is explained i n Bendat and Pierson [ 3 ] .
The frequency range over w h i c h the averages f o r CD{Ü}) and C M ( W ) are computed w i h affect the predictive accuracy, as is obvious f r o m Figs. 5 and 6. I t is interesting to note that f o r a given cylinder diameter K C is directiy proportional to wave amplitude and thus the square root o f the wave energy spectral density; i f the random wave is considered as the sum o f sinusoidal waves w i t h one wave f o r each elemental frequency range. So the variation o f K C w i t h frequency is directiy related to the spectral shape and the same K C values w i l l occur twice, once at the l o w frequency end and once at the h i g h frequency end. I n principle some weighted average could be used to reflect tiiis f a c t and also the distribution o f energy i n the wave spectrum. However the authors have not done this as they consider that generally frequency domain methods are not really worth pursuing i n the context o f estimating the M o r i s o n force coefficients. T o summarise, the frequency domain methods make more assumptions than the time domain methods. M o s t notable is the linear-isation (or cubic approximation) o f the drag term. I n addi-tion all frequency domain methods assume random linear
.? 3.5 u •a a 2.5 0.2 Frequency
324 J. Wolfram, M. Naghipow /Applied Ocean Research 21 (1999) 311-328 •a c 0.15 0.2 0.25 Frequency (Hz) 0.3 0.35 0.4
Fig. 6. Variation in CQ and CM with frequency using the non-linear model (Method 10) for Run 4. wave theory, and i t is seen i n the time domain methods
when this assumption is relaxed predictive accuracy improves. Finally the variation o f Co and C M w i t h frequency is o f no apparent practical value; indeed i t is a nuisance. I t is therefore not surprising that time domain methods appear to have greater predictive accuracy.
6. Discussion of the results
The principal objective o f this paper is to examine the efficiency of the various method o f analysis rather than to present an extensive set o f experiment data and so only a subset o f a larger programme o f experiments are considered. The results of the other experiments i n the same programme, on smooth cylinders and those w i t h slight roughness are presented i n Ref. [ 2 7 ] .
Before examining the efficiency o f the various methods f o r estimating CD and C M f r o m random wave data it is interesting to l o o k at some plots o f the total force coefficient Cf (defined i n the nomenclature) against K C The results obtained w i t h no current are shown i n Fig. 7 f o r the small cylinder. The inference that can be drawn is that at l o w K C
without current a large number o f coiTespondingly small waves are needed to estimate a mean value o f Cf w i t h statis-tical accuracy. Importantly, given the scatter i n Cf at l o w K C then there w i l l be at least similar scatter i n C D and C M irrespective o f how the total force is split into the corre-sponding component parts. Interestingly the effect of intro-ducing a positive cun-ent is to reduce not only the scatt:er i n Cf at l o w K C values but also its average magnitude as can be seen f r o m F i g . 8. A negative cun-ent reduces the average magnitude even further and across a wider range o f K C as can be seen i n F i g . 9. Similar effects are observed f o r the large cylinder.
Dean's reliabihty ratio R is shown f o r the lai-ge and small cylinders, respectively, i n Figs. 10 and 12. The conespond-i n g plots o f CD and C M (estconespond-imated usconespond-ing least squares wave by wave analysis) against K C are shown i n Figs. 11 and 13. These clearly show the effect o f the reliability ratio upon the scatter i n the estimated force coefficients. For example the generally higher R values f o r the smaller cylinder are reflected i n greater scatter i n the corresponding C M values when compared to those f o r the larger cylinder. So not only is there scatter f r o m wave to wave i n the total non-dimensional force, but the data are often not w e l l conditioned
45 40 35 30 25 20 15 10 5 —s — • Run4 --1 1 1 =>• «'Sö 63 <=><=> 10 15 20 25 30 35 H : :
J. Wolfram, M. Naghipow / AppUed Ocean Research 21 (1999) 311-328 325 7 6 5 U 4 3 2 1 0
Fig, 8. Variation of Cj- with KC in random waves for the mobile small pile, U = +0,5 m/s.
to allocate this total force into component parts. Thus iiTe-spective o f the analysis method employed to predict CD and C M the scatter endemic i n the data ensures that there w i l l inevitably be uncertainty i n the estimates found.
Tables 2 - 7 show the mean values o f C D and C M obtained using the various analysis methods f o r the six experiment runs. The standard deviations quoted f o r the wave by wave analyses (Methods 1 - 3 ) express the wave to wave variability; whereas those f o r the frequency domain analyses ( M e t h -ods 7 , 8 and 1 0 ) express the variation w i t h frequency across the range f r o m 0 . 1 to 0 . 4 H z . A l s o shown i n these Tables are the con-esponding mean bias and standard enor when these coefficients are used f o r predicting the second, unanalysed, part o f the measured force time series. There are several points worth noting about these results.
1 . Inespective o f the method o f analysis both CD and C M are reduced by the presence o f a current. Interestingly the lai-ger reduction occurs f o r the lai-ger pile when the cunent is i n the same direction as the waves but the reverse is_ true f o r the small pile. I t is not immediately clear w h y this should be the case but i t is w o r t h remembering that the range o f K C is quite different f o r the t w o piles and hence so w i l l be the effects o f wake re-encounter when the wave flow reverses i n each wave cycle. For the wave by wave analysis methods i t is noticeable that the stan-dard deviation o f the C D reduces significantly i n a l l cases w i t h the addition o f cunent but f o r the C M values the reverse is true. This may be explained by the increased
magnitude o f the drag term relative to the inertia term when a cunent is present and thus the proportional reduc-t i o n i n any relareduc-tive error when esreduc-timareduc-ting i reduc-t f r o m reduc-the reduc-toreduc-tal force. T u r n i n g to predictive accuracy f e w clear trends occur. The R M S E tends to reduce somewhat i n most cases when cunent is added but the bias ( M N E ) shows no particular trend.
2 . I n the time domain, the wave by wave analyses give similar results as w o u l d be expected i n hght o f the s i m i -larity among these methods highlighted i n Section 4 . The least squares method (Method 3 ) comes out slightiy ahead overall (but not consistentiy so) i n terms o f predic-tive accuracy as might be expected as i t makes fewer assumptions. Given the improvement seen i n the whole record analysis by weighting the least squares it was decided to do a wave by wave analysis using this approach, w i t h k = 2, and the results are summarised i n Table 8; f r o m w h i c h i t is seen to be generally superior to a l l the other methods. The time domain, whole record, least squares analysis yields similar results to the least squares wave by wave analysis. H o w e v e r the method of moments (Method 6 ) is seen to be significantly poorer and f o r the small fixed pile gave a bias o f — 2 2 % and a standard e n o r o f 2 6 . 4 % ; a plausible explanation f o r w h i c h is beyond the authors.
3. The frequency domain methods generally give poorer predictive accuracy than the time domain methods. This can be explained by the additional assumptions i m p l i -cit i n approximating the drag term and the necessarily
326 J. Wolfram, M. Naghipow / Applied Ocean Research 21 (1999) 311-328
Fig. 10. Variation of Dean's reliability ratio R with KC for the fixed lai-ge pile.
arbitrary clroice o f spectral w i d t h over w h i c h tp average the frequency dependent force coefficients. I t is surpris-ing that the cross spectral method i n v o l v i n g the surface elevation (Method 8) is marginally better than the one i n v o l v i n g the water particle velocity (Method 7) f o r both the fixed cylinders (see Tables 2 and 5), as particle kine-matics predictions f r o m surface elevation using linear theory are o f t e n poor. W h e n a current is present the latter approach is seen to give very variable results (see parti-cularly Table 7) i m p l y i n g a lack o f numerical robustness. The frequency domain least-squares approach suffers i n a similar, but more extreme manner (see Table 7 again). I n this case the high frequency cut-off was f o u n d to have a significant effect upon the denominators i n Eqs. (34) and (35).
4. The introduction o f a non-linear term into the frequency domain analysis makes little difference f o r the large pile case as can be seen i n Table 2; whereas f o r the small pile the result is a very distinct decrease i n predictive accu-racy both i n terms o f R M S E and bias. This may, i n part.
be due to the relative short length o f the records o f around 15 m i n or 140 waves. However the results obtained by B l i e k and K l o p m a n and Vugts and Bouquet w i t h addition o f non-linear terms were not much more encouraging than those obtained here; despite i n the former case o f having a record length o f 189 nun.
The results presented here are f o r very rough cylinders. The question anises w o u l d similar results f o r predictive accuracy have been obtained w i t h smooth cylinders. I n the authors' v i e w the ranking o f the analysis methods w o u l d not have been altered because this is largely dictated by the assumptions and approximations i m p l i c i t i n each o f the methods as discussed above. The predictive accuracy w o u l d probably be similar, but not the same. Even at the largest scale so f a r employed f o r model experiments on smooth circular cylinders i n random waves there is doubt concerning the flow regime. Particularly i n the smaller waves there is the possibility that the flow is not f u l l y turbu-lent throughout the wave cycle. This is l i k e l y to lead to
JV JK- JVM-1 < K C < JVM-1 8 , mean(iaj)=5.5, mean(CD)= JVM-1.88, R u n l ^ JV 10 12 14 KC
1<KC<18, mean(KC)=5.5, mean(C^,)=2.08, Runl
10 12 14
KC
1 6
16
J. Wolfram, M. Naghipom-/ Applied Ocean Research 21 (1999) 311-328 327
Fig. 12. Variation of Dean's reliability ratio R with KC for fixed small pile.
greater uncertainty i n the prediction o f peak forces, particu-larly when all the data are used i n the analysis.
One o f the p r o b l e m f a c i n g someone undertaking a r i s k assessment or a structural r e l i a b i h t y calculation f o r an offshore j a c k e t structure is to estimate the uncertainty i n the wave/cuiTent loading predicted by M o r i s o n ' s equation. The values o f M N E and R M S E quoted i n Table 8 probably represent the l o w e r bounds o f the uncertainty that i t is possible to achieve currently. One p r o b l e m w i t h laboratory experiments is that i n order to achieve acceptably high Reynolds numbers the range o f K C is lower than that w h i c h occurs i n practice so there is inevitably some extrapolation. However this error is thought to be small (particular f o r the small cylinder data considered here) as at large K C drag coefficients generally approach their steady flow values asymptotically.
Space does not permit presentation and discussion o f a l l the results obtained f r o m the analyses undertaken and the reader is referred to Ref. [30] f o r further details.
7. Conclusions
I t is clear that the method used to analyse experimental data i n terms o f M o r i s o n ' s equation has a significant effect on both the force coefiicients obtained and their predictive accuracy. I t is f o u n d that no single method is consistently better under a l l circumstances but on average the wave by wave weighted least squares method gives both the lowest bias (2%) and root mean square error (11%) as can be seen i n Table 8. The other time domain analysis methods whilst a little poorer worked reasonably satisfactorily. The frequency domain techniques were found to be less satisfac-tory and, on occasions, were not robust and so are not recommended unless there are particular reasons f o r their adoptions.
The force coefficients obtained by the various methods varied significantly but there was a clear trend w h i c h showed that the addition o f current significantly decreased the drag coefiicient and to a lesser extent the inertia coeffi-cient. Eor K C values above around 10 use o f mean values
1<KC<38, mean(CD)=1.85, mean(KC)= 12.8 -1 Run4 10 15 20 KC 25 30 35
