An Experimental Investigation of Higher Harmonic Forces on a Vertical Cylinder_99

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An experimental investigation of higher harmonicforces on a vertical cylinderbyMorten Huseby and John Grue

Mechanics Division, Department of MathematicsUniversity of Oslo, Norway

First and higher harmonic wave loads on a vertical circular cylinder are investigatedexperimentally in a wave tank of small scale. The incoming waves are (periodic) Stokeswaves with wave slope up to 0.24. A large set of waves which are long comparedto the cylinder radius are calibrated. The first seven harmonic components of th emeasured horizontal force on the cylinder are analyzed. The higher harmonic forcesare significantly smaller than th e first harmonic force for all wave parameters. Themeasurements are continued until the wave amplitude is comparable to the cylinderradius, where the second, third and fourth harmonic forces become of th e comparablesize. Comparison with existing perturbation and fully non-linear theories show, with afew exceptions, an overall good agreement for small and moderate wave amplitude. Afully non-linear theory agrees with the experiments even up to th e seventh harmonicforce for a part of the amplitude range. For th e large amplitudes the theoretical modelsmostly give conservative predictions. It is important that the distance from the wavemaker to the cylinder is large in order to avoid parasitic effects in the incoming wavefield. The limited width of the wave tank is no t important to the results except whenclose to resonance.

1 IntroductionTension-leg and gravity-based platforms constructed by vertical cylinders may havea resonance period of up to a few seconds. These platforms may in high sea-statesexperience that responses of considerable amplitude very suddenly are generated at theresonance period of the platform. This is a concern with respect to extreme loading.The generation of such responses, so-called 'ringing', is characterized by a resonantbuild-up during a time-interval of the order one wave periodo The wave period whenthis occurs may be several seconds, typically about fifteen seconds. This is severaltimes longer than th e resonance period of the platform. The generation mechanism ofth e higher harmonic wave loads leading to ringing of offshore structures is no t yet wellunderstood.

In recent time several attempts have been made to analyze this problem. The investigations are both theoretical and experimental. On the theoretical side perturbationmethods have been developed under the assumption of incoming Stokes waves. Theobjective has been to capture the wave loads up to the third harmonic component(Faltinsen, Newman & Vinje, 1995; Malenica & Molin, 1995; Newman, 1996a). Fullynon-linear methods have also been developed to analyze this problem (Cai & Mehlum,1996; Ferrant, 1998). Several model tests and small scale experiments have been undertaken, primarily in focused waves or irregular waves (Grue, Bj0rshol & Strand, 1993;Stansberg et aL, 1995; Chaplin, Rainey & Yemm, 1997).

To check the soundness and th e domain of validity of the most adequate numerical1


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calculation methods for ringing analysis, it may be desirable to have available a se tof experimental force measurements in periodic waves. A direct comparison of th efirst few harmonic force components can then be performed. This is the motivationof the present investigation. Small scale experiments are undertaken with the purposeto measure th e horizontal higher harmonic wave loads on a slender vertical circularcylinder in a wave tank. A relatively large range of wave amplitudes is investigated forwaves which are long relative to the cylinder radius.

We have chosen to work with incoming Stokes waves in deep water. This corresponds to th e assumptions in the perturbation methods. Moreover, th e velocity fieldof the incoming waves has only one frequency, up to a relatively large wave slope. Thehigher harmonic wave forces are then caused by the presence of the cylinder in th ewaves. We have spent much effort to document the incoming wave field at th e positions in th e wave tank where th e force measurements are carried out. The incomingwaves become periodic, after a transient leading part, an d are in th e beginning no tdisturbed by parasitic waves. We carry out the force measurements in this period oftime. At a later time, second harmonic parasitic waves appear in the wave field. Ourmeasurements of these waves agree with those of previous investigations, e.g. Schiiffer(1996).

When the calibration of th e incoming waves is finished, one of the two cylinderswhich are used in the experiments is mounted into th e tank. The force measurementsare carried ou t at a position of ten to twenty wave lengths from the wave maker. Sinceth e incoming waves are long relative to the cylinder radius, the first harmonic diffractedwave field is small. The higher harmonic diffracted waves are quite visible, on the otherhand.

As mentioned, the force measurements are performed when the wave motion atth e cylinder has become periodic. We shall find that the first harmonic force alwaysdominate the higher harmonic forces. The latter are, however, of significant size, andwe are able to analyze the complex force components up to th e seventh harmonic.We compare th e force measurements with available theoretical results. These includeprimarily the first, second an d third harmonic forces. A fully nonlinear theory byFerrant (1998) may, however, produce results for any of the harmonic forces. Wecompare our measurements with his computations an d find rather good agreement up toth e seventh harmonic force, for a part of the wave amplitude range of the experiments.

The ratio between the width of th e wave tank to th e cylinder diameter is 6 - 8 inthis investigation. We find that the limited width of the wave tank is no t important toth e results, however, except when close to resonance. This means that a comparisonbetween the experiments and the theoretical models is relevant, where in the lattercase th e effect of tank walls are no t included. During the course of th e work we havefound that th e distance from the wave maker to th e cylinder should be great, in orderto avoid parasitic effects in the incoming wave field.

The paper is organized as follows: Following the Introduction, the experimentalset-up is described in 2. The incoming wave field is described and documented in 3.The analysis of th e force measurements and comparisons with theoretical models andother relevant experiments are presented in 4. In 5 certain oscillations of the forcewith respect to the wave slope are discussed. The oscillations are due to (unwanted)parasitic effects of th e incoming wave field, which take place either when the recordingsare performed in a late time window, or when the cylinder is close too the wave maker.The effects of viscosity and separation is also commented on in this paragraph. Finally,6 is a conclusion.

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wave maker. The purpose is to study how the waves develop as they propagate downth e wave tank.

We primarily present force measurements for th e small cylinder at the position 12.41m from the wave maker, and for the large cylinder at the position 15.41m from th ewave maker. At these positions, the incoming waves ar e measured at 10 sub positions,for 16 different amplitudes pe r wave frequency, giving a total of 1280 measurements ofth e incoming waves. The wave field is examined at different times after the start-up ofth e wave maker.

3.1 How to avoid the parasitic wavesA common problem in this type of experiments, is th e so called parasitic waves, i.e. th efree second harmonic waves which originate at the wave maker. Rere we circumventthis problem by performing the measurements before th e parasitic waves have reachedth e cylinder. Below we show experimentally that th e parasitic waves are absent for arelatively large portion of the time of an individual measurement. We perform all ofth e force measurements in this time window.

We measure the components of the second harmonic wave elevation as explained inGrue (1992). We assume that th e second harmonic component of the waves r(2)(x, t)is on th e form

(2)where w an d k satisfy th e dispersion relation w2 = gk(l +O(A2P))tanhkh, 2w andk2 satisfy (2W)2 = gk2tanh k2h, an d where 9 is th e acceleration due to gravity. Furthermore af2) is the amplitude of the locked second harmonic Stokes wave, a)2) is th eamplitude of the free second-harmonic wave, to determines the phase of the free waverelative to th e locked wave, an d x is th e distance to the wave maker. For all waveparameters in th e present investigation, we have w2 ~ gk(l + A2P) and 4w 2 ~ gk2,which means in practice that the waves are deep water waves.

The second harmonic Fourier component of the wave elevation is obtained by1 rOTi(2)(x) = 10T Jo r(2)(x,t)exp(-i2wt)dt (3)

By measuring i(2)(x) at two positions, Xl an d Xl + we obtain the amplitudes ofth e individual components of the second harmonic wave elevation for a time intervalof 10 wave periods. We then have (Grue (1992) eqs. 12 an d 13):

af2) = I i n ( ~ ~ x ) I I i ( 2 ) ( x d - expi4k6.x i(2)(XI +a)2) = . 1 1i(2)(xd _ expi2k6.x i(2)(XI +I m ( k ~ x ) 1

(4)

(5)Since the parasitic waves travel at half the speed of the main waves, they arrive at th ecylinder at twice the time compared to when the leading part of th e wave train arrives.We will then have a time window where the parasitic waves have no t yet reached th ecylinder, there are no refiections from the beach, and the waves have become reasonablyperiodic. We have found that this time window begins at about 10-15 wave periodsafter the leading part of the wave train has reached the cylinder. An illustration of

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without any large oscillations in the wave height (upper plot in figure 6). When wechoose a time window after the parasitic waves have reached th e measuring position,we get oscillations of the wave height as we move further down in th e wave tank (lowerplot in figure 6). We note that the oscillations at the position closest to th e wave maker(upper plot in figure 6) are due to the fact that when the measurements are done thisclose to the wave maker, it is impossible to find a long enough time window (withperiodic waves) where the parasitic waves have no t reached the measuring position.

We observe that th e amplitude of the waves de crease as a result of viscosity, byabout 0.6% per wave length. The frequency of th e waves is measured to be exactly th esame at all of the 40 measuring positions along the wave tank.

From these extensive measurements we can conclude that force measurements maybe carried ou t in a reasonably long time window, where the incoming waves are quiteperiodic. The surface elevation may be regarded as close to pure Stokes waves up toAk ~ 0.2, i.e.

r(t) = Acos(kx - wt) + af2) cos(2(kx - wt)) + af3) cos(3(kx - wt)) +... (6)where af2) ~ (1/2)A2 k an d af3) ~ (3/8)A3P, according to Stokes theory. This alsoindicates that the incoming waves are represented by the velocity potential

(7)which determines the corresponding velocity an d pressure fields. Rere w2 = gk(l +A2P) an d y denotes the vertical coordinate with y = Oat the initially calm free surface.

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4 The ForceThe first seven harmonic components of the force are presented. The amplitude of th en-th harmonic force is made non-dimensional dividing by pgAnR(3-n). The phase of th en-th harmonic force is compared to the phase of th e incoming waves. Due to the factthat the force measurements are much more accurate than th e wave measurements,we have chosen to use th e phase of th e first harmonic force as a reference for th ephase of the higher harmonic forces. We note that the phase of the incoming waves,at a given position in the tank, changes as the wave amplitude increases, due to (nonlinear) amplitude dispersion. The phase of the resulting force on the cylinder changesaccordingly.

The force measurements are compared with theoretical models. The Morison equation may be applied to compute the first harmonic force. Molin (1979) and laterNewman (1996b) have employed a second order theory for calculation of the secondharmonic wave force. Malenica & Molin (1995), hereafter referred to as M&M, havedeveloped a third order theory using a perturbation expansion in th e wave slope. Athird order theory using a perturbation expansion in both the wave slope and th e wavenumber has been developed by Faltinsen, Newman & Vinje (1995), hereafter referredto as FNV. Ferrant (1998) has derived a fully non linear theoretical and numericalmethod to calculate all higher harmonic components of th e force. The incoming wavesin our experiments which are pure Stokes waves, correspond to th e assumptions in th etheoretical methods.

For a very limited span of parameters, we also compare our measurements withexperiments performed by Stansberg (1997).

While the experiments are carried out in a wave tank with a limited breath b, allth e theories assume an unbounded horizontal fluid domain. However, with a ratio bjRin the interval from 6.3 to 8.3, we find that the limited breadth of th e tank is no timportant to the investigation, except close to resonance, when standing cross wavesare generated at the cylinder. We present no results for wave parameters close toresonance.

4.1 First harmonic forceTo obtain the first harmonic force we apply the Morison equation, and find that FI =21fpgAR2 lr / 2 A drag force due to th e laminar boundary layer may be estimated to beapproximately O.11fpgAR2 for th e two cylinders. Figure 8 shows a comparison of ourmeasurements, linear theory and Ferrant's (1998) non-linear theory, for the amplitudeof the first harmonic force. The amplitude seems to be in good agreement with theory,even though it seems that the measured force is slightly larger than th e theory. Thephase of the first harmonic force (figure 9) is also predicted well by theory. We notethat the amplitude and phase of th e force are both constant for the whole range ofwave slopes, even when Ak has become quite large, and the expected range of validityof perturbation theory is exceeded.

4.2 Second harmonic forceFor th e second harmonic force we compare our measurements with the second ordertheory by Molin (1979), Newman (1996b) and with the fully non-linear theory byFerrant (1998). We observe that the amplitude of th e second harmonic force (figure

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10) decreases as th e wave number increases. The measured forces are smaller thanth e theoretical forces for all wave slopes. For large wave slopes, the measured force issignificantly smaller than th e theoretical results. The theory by Ferrant predicts th ede crease of th e force, observed in the experiments, even though the measured force issmaller than this theory predicts. The phase of th e second harmonic force (figure 11)is, however, quite constant as the wave slope increases, an d is well predicted by theory.

4.3 Third harmonic forceFor all but the smallest wavenumber, the measurements show that the amplitude ofthe third harmonic force is nearly constant as th e wave slope increases. There is goodagreement between th e measurements and the theoretical predictions of IF31 by FNV,M&M and Ferrant (figures 12 to 16). We note that the FNV theory provides a verysimple result for the third harmonic force, i.e. F3/ pgA3 = 27r(kR)2lr / 2. Fo r the phaseof F3 , the measurements agree with M&M's and Ferrant's results. The FNV theorydoes, however, predict a value of the phase which is significantly different from ourmeasurements. The phase predicted by FNV does no t vary as a function of the wavenumber.

We note that the third and higher harmonic forces are extremely small when th ewave slope is small. For example, F3 is 100 to 400 times smaller than FI when Ak ~0.05. This introduces difficulties in measuring and extracting the higher harmoniccomponents from a much larger signal. As we see in sorne of the figures, this causessorne experimental scatter in th e results, for small Ak. Because of this we are no t ableto measure the third harmonic force for wave slopes smaller than Ak = 0.05. Furtherdetails of how the higher harmonic forces are extracted from the time series are givenin Appendix B.

4.4 Fourth harmonic forceThe measurements of th e fourth harmonic force are displayed in figures 17 and 18. Forone of the wave numbers we compare with th e theory by Ferrant. Because of th e smallsize of the fourth harmonic force, we can not give experimental data for wave slopes lessthan about 0.1. We see that our measurements are in surprisingly good agreement withFerrant's theory, both for the amplitude and the phase of the force. We further notethat th e value of 1F41, to a rough approximation, behaves as 1F41 / pgA4R- I ) rv kR, forkR ~ 0.315 and moderate wave slope.

4.5 Higher harmonic forcesWe also display measurements of th e amplitude and phase of the fifth, sixth and seventhharmonic force components for one wave number (kR = 0.245) in figures 19 to 21. Forthese increasingly higher harmon ics we require the waves to be increasingly larger beforewe are able to measure th e very small force components. We compare our measurementswith th e non-linear theory by Ferrant. His results could no t be obtained for waveslopes larger than Ak = 0.145, and our measurements for these higher harmonics couldno t be performed for wave slopes much smaller than this. The common domain ofth e measurements and the calculations is then quite small. The trend of the resultshowever, show good agreement, taken into account the small size of th e measuredforces.

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4.6 Other experimentsStansberg (1997) has performed measurements of the amplitude of the second andthe third harmonic force on a cylinder in periodic waves. He has performed measurements for several cylinders in a towing tank which is 80 m long, 10 m wide and 10 mdeep. The draught d of the cylinders was 1.44 m and 0.94 m, and the radius 10 cm,16.3 cm and 31.3 cm. We compare our measurements with those of Stansberg whereth e non-dimensional wave number kR and draught diR have a size relevant to ourmeasurements, i.e. we compare our measurements with parameters kR = 0.166 anddiR = 15 or 20 (R = 4 cm or 3 cm) to Stansberg's measurements with parameterskR = 0.149 and diR = 8.8 (R = 16.3 cm). As we see in figure 22, our measurementsare in relatively good agreement with the measurements performed by Stansberg.

I t is interesting to see that both our and his measurements give a decreasing amplitude of the third harmonic force (bottom plot in figure 22) for small to medium waveslopes. For Ak larger than 0.2 th e measurements indicate that the amplitude of th ethird harmonic force is constant.

The other results of Stansberg for diR smaller than 8.8, show smaller values of 1F21and 1F31 than obtained in our experiments. We believe that this is due to the limiteddraught of th e cylinders, in Stansbergs experiments.

5 Oscillating force and effects of viscosity5.1 Oscillating forceAs mentioned, when we perform measurement at the position 12.41m from the wavemaker, we first wait for the waves to become periodic (10-15 time periods from the firstlarge wave has reached th e cylinder). We may then perform measurements over a timewindow of 10 wave periods before th e wave field is disturbed by unwanted effects. Theamplitude of e.g. th e second harmonic force will then look like the one displayed in th eupper plot in figure 23. When the force measurements are carried out in a later timewindow, we obtain an oscillation in the second harmonic and higher harmonic forces(bottom plot in figure 23). The force oscillates around a mean value which correspondsto the force in an earlier time window (upper plot in figure 23). After the changeof character, the measured force willlook the same as time increases, except for sornesmall fiuctuations which appear when the time elapsed is so long that we get refiectionsfrom the beach and th e wave maker.

The same behavior of th e measured forces also occur when the cylinder is situatedat other positions in the wave tank. We have particularly investigated th e higherharmonic forces when th e cylinder is placed close to the wave maker, at a distanceof 6.33 m. An example for th e second harmonic force is shown in figure 24. Thesame oscillations, but with a smaller frequency, occur at a late time window. At thisposition, minor oscillations are also present at an early time window. This is due toth e fact that it is impossible to find a sufficiently long time window this close to thewave maker, where the incoming waves can be regarded as pure Stokes waves. Similaroscillations occur also for th e other higher harmonic forces. We note that the time ofth e change of character in th e force measurements coincides with the time when th esecond harmonic parasitic waves arrive at the cylinder. This may possibly indicate thatth e parasitic waves are the cause of the oscillations of the force. To document this,one would have to perform th e same experiments with waves that are measured to be

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absolutely free from second harmonic parasitic waves at all times. We note, however,that other non-linear effects may lead to oscillations in the second an d higher harmonicforces. Our investigation gives, on th e other hand, no evidence on this point.

As we have seen, there exists a time window in which we can perform force measurements without getting oscillations in the force as th e wave slope increases. Thelength of this time window will be proportional to the distance from the wave maker toth e cylinder. I f we perform the measurements too close to th e wave maker, we are no table to find a time window of sufficient length where we can assume that th e incomingwave field is pure Stokes waves. Thus we will expect to always get oscillations in th eforce measurements if th e cylinder is too close to the wave maker. In our experiments,we required a distance of at least 10 wave lengths from th e wave maker to the cylinder.

All of th e measurements of the force that are presented in th e previous section areconducted at a distance of 12.41 m or 15.45 m from the wave maker. We were thenable to use a suitable time window, so that th e problems discussed in this section donot relate to the force measurements previously presented.

5.2 Viscous effectsWhen th e wave height is large compared to th e cylinder diameter, th e value of th eKeulegan-Carpenter number KC and the Reynolds number Re become important parameters in the problem. In th e present experiments 1 < KC < 3.6, an d Re rv 20000,for th e large waves. Sarpkaya (1986, figure 3) obtains that the viscous drag force on acircular cylinder is very small for the values of KC an d Re in the present experiments.We also note that we have no t observed fiow separation in th e experiments.

6 Conel isionWe have performed an extensive se t of experiments with a vertical circular cylinderin periodic waves. The purpose has been to measure the first and higher harmonichorizontal wave loads on th e cylinder. We have also performed an extensive se t ofmeasurements of the incoming waves, without the cylinder present, to document thatth e wave field is pure Stokes waves in a relatively long time window. This is true forwaves with wave slope up to about 0.2. For larger wave slope, the incoming wave fieldcontains disturbances in addition to the Stokes waves. We note that th e (periodic)Stokes waves are present in a relatively early, but relatively long time window, at theselected measurement positions. At a later time window, second harmonic parasiticwaves appear, originating from th e wave maker. Our recordings of these waves agreewith previous results by e.g. Schiiffer (1996). Parasitic waves become very early presentwhen the recording position is close to the wave maker.

Experiments with two cylinders of different radii have been undertaken. The largercylinder is placed further from the wave maker than the smaller one, such that th edistance relative to th e cylinder radius is the same. The waves are also correspondinglylonger for th e larger cylinder than for the smaller, so that the waves travel the samenumber of wave lengths down th e wave tank before they reach th e cylinders. The onlydifference between the measurements performed with th e two different cylinders is th eratio of th e tank width to th e radius of the cylinder, which is 6.3 for th e larger cylinderan d 8.3 for the smaller one. The results indicate that a limited width of th e wave tankis no t important to th e investigation, except close to resonance, where results are no t

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given, however. This is also confirmed by the comparisons with th e theoretical models,where th e latter assume a fluid which is unbounded horizontally. We have, on the otherhand, found that it is important to perform th e measurements with th e cylinder at asufficiently large distance from the wave maker, to avoid misleading results. In ourexperiments the distance to the wave maker is more than te n wave lengths.

The n-th harmonic force is made non-dimensional with respect to pgAnR3 -n, andit s phase is measured relative to the elevation of th e incoming waves. The measuredcomplex non-dimensional first harmonic force is found to be constant for all wave slopesinvestigated, i.e. for Ak up to 0.24. We find that the first harmonic force is ratherwell predicted by the Morison equation. The non-dimensional second harmonic forceis always less than 10% of the non-dimensional first harmonic force. Second ordertheory (Molin, 1979; Newman, 1996b) is found to be in reasonable agreement with th emeasurements for most of the wavenumbers, for small and moderate wave slope. Thesecond harmonic force decreases with increasing wave slope and becomes significantlysmaller than what is predicted by second order theory when Ak is around 0.2. Thephase of the second harmonic force is reasonably constant as function of th e waveamplitude and is in good agreement with second order theory for all wavenumbers.

The measured third harmonic force has a magnitude which agrees rather well withthe perturbation theories of M&M and FNV. This is true for all investigated wavenumbers when the wave slope is small or moderate. The measured 1F31 is somewhat smallerthan what is predicted by the aboye mentioned theories when the wave slope is large,however. The measured phase of F3 is in good agreement with th e theory by M&M,for all wavenumbers, but differs from the predictions by the FNV theory. Our resultsfor the amplitude of the second and third harmonic forces are found to compare wellwith a set of the measurements performed by Stansberg (1997), for cylinders of largerscale in a relatively wider wave tank than ours.

The measured fourth harmonic force has an amplitude which to a rough approximation behaves as 1F41/(pgA4R- 1 ) rv kR when kR < 0.315. This force component hasa different behavior when the wavenumber is larger than 0.315, however. We find thatth e phase of the fourth harmonic force varies with respect to th e wavenumber, but isrelatively constant as function of the wave slope, like for the lower harmonic forces. Wealso present measurements of the fifth, sixth and seventh harmonic forces on the cylin-ders when the wavenumber is kR = 0.245. The magnitude of these (non-dimensional)forces exhibit a pronounced decay with increasing amplitude and become very smallwhen Ak is larger than 0.2. The phase of these forces are relatively constant in th ewhole wave amplitude range. From the few comparisons we have carried out, it seemsthat the non-linear theory by Ferrant (1998) predicts the measured higher harmonicforces well, in a limited wave amplitude range. It would be interesting to know if thisor other non-linear methods can predict the other measured results that are includedin this investigation.

For three of the wavenumbers, th e measurements are carried out with incomingwave amplitude increasing up to the radius of the cylinder. At this value of the waveamplitude, the second, third and fourth harmonic forces all become of equal size. Thefifth, sixth and seventh harmonic forces are, however, always smaller than the lowerharmonics.

We finally note sorne other observations from the experiments than what are de-scribed in this investigation. In th e early part of the force histories, where the incomingwaves are no t yet periodic, rather intense higher harmonic forces may take place whenth e wave elevation is sufficiently large. The latter higher harmonic forces seem to be

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much more pronounced than those due to incoming Stokes waves. The problem ofhigher harmonic wave loads in transient wave trains, leading to ringing, is currentlyunder investigation in the Hydrodynamic Laboratory at the University of Oslo.

Acknowledgements. The authors are grateful to DI. Pierre Ferrant for exchange ofscientific results and to MI. Arve Kvalheim and MI. Svein Vesterby for their skillfultechnical assistance. This work was supported by the Joint Industry Project DeepWater Analysis Tools - "DEEPER". We are grateful for the support of the individual sponsors, including The Research Council of Norway, Norsk Hydro, Statoil, SagaPetroleum, Mobil, Petrobras, Aker Engineering, Kvcerner Oil & Gas, Offshore Design,Brown & Root, J. Ray McDermott, Umoe Technology, ETPM SA, AMECRC and DetNorske Veritas.

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A Instabilities from the set-upThe input to the wave maker is a pure sinusoidal movement. Depending on th e precisionof the equipment that is used to move the wave maker, we will introduce a narrow sideband in the amplitude spectrum of the movement of th e wave maker, and therebyintroduce a narrow side band in the amplitude spectrum of the wave elevation. Inour experiments this possible side band in the wave elevation is so narrow that wecan not measure it using a discrete Fourier transform over 10 wave periods. This sideband would however expose it self at sorne point, as a modulation, if th e wave tankwas long enough for th e waves to keep on propagating. The extent to which this sideband gives rise to instabilities, will depend on the length from the wave maker to themeasuring position, the breadth of the tank, the frequency of the waves, the accuracyof the equipment that controls th e wave maker and the accuracy of the wave tank (i.e.if the walls of the wave tank are completely parallel). We have no t observed side bandinstabilities in the present experiments.

B Sources of inaccuracy in the analysisThe analysis of the measurements is complicated by th e fact that the higher harmonicforces are much smaller that th e first harmonic forces. Typically the third harmonic canbe from 100 to 400 times smaller than the first harmonic. It is then difficult to calculateth e higher harmonics with high accuracy. The result that we want to obtain from th eanalysis of the measurements, is the amplitude spectrum. I f we had measurements overan infinite time period, we would calculate this using the discrete Fourier transform(DFT) to obtain F(w) = DFT(f) = : : ~ ( X ) f[n]e- inw . We do however, only havemeasurements over a finite time domain. The DFT will then be a smeared versionof th e DFT over infinite time. The DFT of the time limited signal can be writtenF = ::r! f[n] = : : ~ ( X ) f[n]w[n], where w[n] is a rectangular time window that has th evalue 1 for n E [O, N] (where we have measurements), and is equal to O everywhereelse. This expression can be written as the convolution

(8)Rere W (w) is the DFT of w [n] from - 00 to 00 (figure 7). The Fourier transform of th ewindow has the form of a main lobe and several smaller side lobes. The width of theseside lobes is a direct function of th e length of the time window, thus on the assumedfrequency of the waves. I f the frequency of th e waves is such that the time window is ofth e exact same length as an integer multiplied with th e wave period, then the breadthof the side-lobes is such that the side-lobes will not contribute to the second or higherharmonic amplitudes that we find using the DFT (W(w)=O for = 1,2,3, . . . ). I f onth e other hand the frequency of the waves is no t as prescribed, so that the length of th etime window is not an integer times the wave period, then the side-lobes of the DFTof the windowing func tion may introduce significant errors. These errors are normallyvery small, since the side lobes are small compared to the main lobeo What makesanalysis like the one in this work so sensitive, is that the higher harmonic componentsmay be of comparable size with th e side lobes th e window convolved with the firstharmonic. This may lead to that F(w) may be 100% different from F(w), which isvery unfortunate. While it is F(w) that results from our analysis of th e measurements,

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[4] FALTINsEN, O. M., NEWMAN, J. N. AND VINJE, T. (1995) Nonlinear wave loads on a slender vertical cylinder.J. Fluid Mech. 289, 179-199.

[5] GRUE, J. (1992) The nonlinear water waves at a submerged obstacle or bottom topography. J. Fluid Mech. 244,455-476.

[6] GRUE, J. , BJ0RSHOL, G. AND STRAND, 0. (1993) Higher harmonic wave excit ing forces on a vertical cylinder.Institute of Mathematics, University of Oslo Preprint, No . 2. ISBN 82-553-0862-8, 28pp.

[7] MALENlcA, S. AND MOLIN, B. (1995) Third-harmonic wave diffraction by a vertical cylinder. J. Fluid Mech.302, 203-229.

[8] MOLIN, B. (1979) Second-order diffraction loads upon three dimensional bodies. Appl. Ocean Res. 1, 197-202.[9] NEWMAN, J. N. (1996A) Nonlinear scattering of long waves by a vertical cylinder. In: Waves and nonlinear

processes in hydrodynamics. Grue. J. , et al (Editors). Kluwer 1996. pp 91-102.[10] NEWMAN, J. N. (1996B) The second-order wave force on a vertical cylinder. J. Fluid Mech. 320, pp . 417-443.[11] SARPKAYA, T. (1986) Force on a circular cylinder in viscous oscillatory ftow a t low Keulegan-Carpenter numbers.

J. Fluid Mech. 165, pp . 61-71.[12] SCHAFFER, H. A. (1996) Second-order wave maker theory fo r irregular waves. Ocean Engng. Vol. 23 No. 1 pp .

47-88[13] STANSBERG, C. T . (1997) Comparing ringing loads from experiments with cylinders of different diameters - An

empirical study. Proceedings, Vol. 2, the 8th Conference on th e Behaviour of Offshore Structures (BOSS '97)(held in Delft, The Netherlands), Pergamon/Elsevier Science, London, UK , pp . 95-109.

[14] STANSBERG, C. T., HUSE, E., KROKSTAD, J. R. AND LEHN, E. (1995) Experimental study of nonlinear loadson vertical cylinders in steep random waves. Proceedings, Vol. I, th e 5t h ISOPE Conference (held in The Hague,The Netherlands), ISOPE , Golden, Colorado, USA, pp . 75-82.

19


20/36


21/36

0.5

1

1.5

2

2.5

0 0.05 0.1 0.15 0.2

kR=0.245

Ferrant(1998)R=3cm

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.378

R=4cm

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.166

R=3cmR=4cm

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.204

R=3cmR=4cm

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.315

R=3cmR=4cm


22/36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2

kR=0.245

Ferrant(1998)R=3cm

00.10.20.30.40.50.60.70.80.9

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.378

R=4cm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.166

R=3cmR=4cm

0

0.1

0.2

0.3

0.4

0.50.6

0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.204

R=3cm

R=4cm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.315

R=3cmR=4cm


23/36

0

0.5

1

1.5

2

2.5

3

0 0.05 0.1 0.15 0.2

kR=0.245

Ferrant(1998)R=3cm

-0.5

0

0.5

1

1.5

2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.378

R=4cm

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.166

R=3cmR=4cm

0

0.5

1

1.5

2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.204

R=3cm

R=4cm

-0.5

0

0.5

1

1.5

2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.315

R=3cmR=4cm


24/36

0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.166

FNV(1995)M&M(1995)

R=3cmR=4cm

-3

-2

-1

0

1

2

3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.166

FNV(1995)M&M(1995)

R=3cmR=4cm


25/36

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.204

FNV(1995)M&M(1995)

R=3cmR=4cm

-3

-2

-1

0

1

2

3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.204

FNV(1995)M&M(1995)

R=3cmR=4cm


26/36

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.05 0.1 0.15 0.2

kR=0.245

FNV(1995)M&M(1995)

Ferrant(1998)R=3cm

-3

-2

-1

0

1

2

3

0 0.05 0.1 0.15 0.2

kR=0.245

FNV(1995)M&M(1995)

Ferrant(1998)R=3cm


27/36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.315

FNV(1995)M&M(1995)

R=3cmR=4cm

-3

-2

-1

0

1

2

3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.315

FNV(1995)M&M(1995)

R=3cmR=4cm


28/36

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.378

FNV(1995)M&M(1995)

R=4cm

-2

-1

0

1

2

3

4

5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

kR=0.378

FNV(1995)M&M(1995)

R=4cm


29/36

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2

kR=0.245

Ferrant(1998)R=3cm

0

0.4

0.8

1.2

1.6

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.378

R=4cm

0

0.05

0.1

0.15

0.2

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.166

R=3cmR=4cm

0

0.05

0.1

0.15

0.2

0.25

0.3

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.204

R=3cmR=4cm

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.315

R=3cmR=4cm


30/36

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2

kR=0.245

Ferrant(1998)R=3cm

0

0.5

1

1.5

2

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.378

R=4cm

2.4

2.5

2.6

2.7

2.8

2.9

3

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.166

R=3cmR=4cm

2

2.5

3

3.5

4

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.204

R=3cmR=4cm

1

1.5

2

2.5

3

3.5

4

0.08 0.1 0.12 0.14 0.16 0.18

kR=0.315

R=3cmR=4cm


31/36

0

0.05

0.1

0.15

0.2

0.25

0.3

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

kR=0.245

Ferrant(1998)R=3cm

-3

-2

-1

0

1

2

3

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

kR=0.245

Ferrant(1998)R=3cm


32/36

0

0.05

0.1

0.15

0.2

0.25

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

kR=0.245

Ferrant(1998)R=3cm

-4

-3

-2

-1

0

1

2

3

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

kR=0.245

Ferrant(1998)R=3cm


33/36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

kR=0.245

Ferrant(1998)R=3cm

-3

-2

-1

0

1

2

3

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

kR=0.245

Ferrant(1998)R=3cm


34/36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2 0.25

Stansberg(1997):kR=0.149,R=16.3cmNewman(1996):kR=0.149

H&G:kR=0.166,R=3cmH&G:kR=0.166,R=4cm

Newman(1996):kR=0.166

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25

Stansberg(1997):kR=0.149,R=16.3cmFNV(1995):kR=0.149

M&M(1995):kR=0.149H&G:kR=0.166,R=3cmH&G:kR=0.166,R=4cm

FNV(1995):kR=0.166M&M(1995):kR=0.166


35/36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2

kR=0.166

R=3cm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2

kR=0.166

R=3cm


36/36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2

kR=0.166

R=3cm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2

kR=0.166

R=3cm

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An Experimental Investigation of Higher Harmonic Forces on a Vertical Cylinder_99