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Wave impact forces on a horizontal cylinder Prasa, Sundar 1994

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WAVE IMPACT FORCES ON A HORIZONTAL CYLINDERbySundar PrasadB.Eng., University of Delhi, 1986M.Tech., Indian Institute of Technology, Madras, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1994© Sundar Prasad, 1994In presenting this thesis in partial fulfilment of the requirements for an advanced degree at theUniversity of British Columbia, I agree that the Library shall make it freely available forreference and study. I further agree that permission for extensive copying of this thesis forscholarly purposes may be granted by the Head of the Department or by his or herrepresentatives. It is understood that copying or publication of this thesis for financial gain shallnot be allowed without my written permission.Department of Civil EngineeringThe University of British Columbia2324 Main MallVancouver, B.C. V6T 1Z4CanadajL)L/S/C//IIAbstractImpact forces due to wave slamming on structural elementsof offshore platforms have beenknown to reach very high magnitudes andcontribute to accelerated fatigue of members and jointsdue to the resulting dynamic response. Results ofpreviously reported theoretical analyses varyby as much as 100% with regards to the peakvalue of the slamming coefficient, andexperimental verification of these results hasbeen difficult due to the significant amount ofscatter in the data reported by several investigators.The present thesis investigates the slammingforce due to non-breaking and breaking waveimpact on a fixed horizontal circular cylinderlocated near the still water level.A numerical model which is based on a combinationof slamming, buoyancy, drag, and inertiaforce components has been developedin order to predict the time history of the vertical force ona fixed horizontal cylinder in waves. The model hasalso been modified to include the effects ofdynamic response and cylinder inclination.In addition, an approach based on an impulsecoefficient is proposed for estimatingthe maximum dynamic response of an elastically supportedcylinder.Experiments have been carried outin the wave flume of the Hydraulics Laboratory of theDepartment of Civil Engineering at theUniversity of British Columbia in order tomeasure thevertical force on a horizontal test cylinderfor a range of regular (non-breaking) waveconditionsand cylinder elevations. The datahas been analyzed to obtain the corresponding slammingandimpulse coefficients, as well asthe impulse rise-time and duration. Correctionsto the measuredcoefficients to account for buoyancy,dynamic response and free surface slope are indicated.Thecoefficients exhibit a considerabledegree of scatter, even when the variouscorrections are takeninto account. However, the degreeof scatter of the impulse coefficient isnotably less than that111of the slamming coefficient. The results for the maximum slamming coefficient C0agree withthose of recent studies which observe that C0 may be closer to 2it thanthe generally acceptedvalue of it. A limited number of tests have also been performed for thecase of an inclinedcylinder, and the effect of tilt on the maximum slamming force and rise-timeis examined.The numerical model for the rigid horizontal cylinder has been used to determine the variationof the maximum non-dimensional vertical force in regular waves asa function of the governingnon-dimensional parameters. Statistics of the maximum force obtainedfrom simulations inrandom waves are compared to corresponding results derivedfrom available analyticalexpressions, and indicate reasonable agreement inthe case of a narrow-band spectrum. Thetemporal variation of the vertical force predicted by the numerical modelis also compared to thatof the measured force in regular non-breaking waves. In general, theagreement is quite good forboth a horizontal and inclined cylinder. The application of the numericalmodel to an estimationof a member’s response in a prototype situation is illustrated. It isseen that the approach basedon the impulse coefficient is relatively simple, and appears tobe effective in estimatingmaximum responses for conditions under which the methodis applicable.Experiments have also been carried out in order to measure the impactforces due to plungingwave action on a horizontal circular cylinderlocated near the still water level. The vertical andhorizontal components of the impact force on the cylinderdue to a single plunging breaker havebeen measured for three elevations of the cylinder, and six locationsof wave breaking relative tothe horizontal location of the cylinder. A video recordof the impact process has been used toestimate the kinematics of the wave and plungingjet prior to impact. The force measurementshave been corrected for the dynamic response of thecylinder, and analyzed to obtain slammingcoefficients and rise times. It is observed that thecylinder elevation and the wave breakinglocation relative to the cylinder havea significant effect on the peak impact force. Themagnitude of the impact force due to a breaking wave is4 to 20 times greater than that due to aregular non-breaking wave of similarheight and period. In addition to the fluid velocity, thecurvature of the water surface has anoticeable effect on the peak impact force.ivTable of ContentsAbstract iiTable of Contents ivList of Tables viiiList of Figures ixList of Principal Symbols xvAcknowledgements xixINTRODUCTION 11.1 General 11.2 Literature Review 41.2.1 Wave Force on a Horizontal Cylinder in the Splash Zone 41.2.2 Forces on Horizontal Cylinders due to Breaking Waves 101.3 Scope of Present Investigation 111.3.1 Numerical Modelling 121.3.2 Experiments on Slamming in Regular Waves 131.3.3 Experiments on Slamming in Breaking Waves 13THEORETICAL FORMULATION 152.1 Dimensional Analysis 15V2.2 Hydrodynamic Force on a Rigid Horizontal Cylinder.182.2.1 Buoyancy Force182.2.2 Slamming Force192.2.3 Inertia Force252.2.4 Drag Force262.2.5 Combination of Force Components272.3 Hydrodynamic Force on an Elastically Supported Horizontal Cylinder302.3.1 Response of an SDOF System toImpact Loading 302.3.2 Cylinder Response to Wave Impact Loading352.3.3 Modelling Slamming as an Impulse392.4 Slamming Force on an Inclined Cylinder422.5 Water Particle Kinematics in Waves452.5.1 Regular Waves452.5.2 Effects of Free Surface Slope462.5.3 Random Waves482.6 Computational Considerations50EXPERIMENTAL STUDY533.1 Test Facilities533.1.1 Wave Flume543.1.2 Wave Generation543.1.3 Cylinder model563.1.4 Data Acquisition and Control58vi3.2 Dynamic Characteristics of the Test Cylinder 613.3 Horizontal Cylinder in Non-Breaking Waves 633.4 Data Analysis 653.4.1 Noise Filtering Techniques 653.4.2 Determination of the Instant of Slamming 673.4.3 Analysis of Wave and Force Records 693.5 Inclined Cylinder in Non-Breaking Waves 713.6 Horizontal Cylinder in Breaking Waves 723.6.1 Generation of the Breaking Wave 723.6.2 Measurement of Force and Breaking Wave Profiles733.6.3 Analysis of Breaking Wave Impact Force 74RESULTS AND DISCUSSION784.1 Slamming Forces in Non-Breaking Waves794.1.1 Raw Data from Horizontal Cylinder Experiments794.1.2 Slamming Coefficients from Horizontal Cylinder Experiments824.1.3 Impulse Coefficients from Horizontal Cylinder Experiments864.1.4 Tests on Inclined Cylinder884.2 Numerical Simulation894.2.1 Regular Waves904.2.2 Random waves924.2.3 Comparison with Experimental Observations954.2.4 Practical Application98vii4.3 Breaking WaveImpact.1004.3.1 Cylinder Elevation I1014.3.2 Cylinder Elevation II1024.3.3 Cylinder Elevation III1044.3.4 Slamming Coefficients due to Breaking WaveImpact 105CONCLUSIONS1075.1 Wave Slamming on a Horizontal Cylinder1075.1.1 Experimental Study1075.1.2 Numerical Modelling1105.2 Plunging Wave Impact on a Horizontal Cylinder1115.3 Recommendations112References115Tables120Figures127viiiList of Tables1.1 Peak slamming coefficient C reported in earlier experimental studies.4.1 Properties of regular waves used in slamming experiments.4.2 Peak slamming coefficients and related parameters estimated from multiple slammingevents in a regular wave test. T = 1.5 see, H = 18.4 cm, h = 0.5 cm.4.3 Summary of test conditions and principal results from slamming tests in regular waves.4.4 Summary of impulse coefficients and related parameters estimated from slamming testsin regular waves.4.5 Summary of observations from impact tests in breaking waves.ixList of Figures1.1 Photograph of non-breaking wave impact on horizontal test cylinder.1.2 Photograph of plunging wave impact on horizontal test cylinder.1.3 Comparison of analytical and experimental results for the slamming coefficient C as afunction of relative submergence s/a (Greenhow and Li, 1987). 1, experiments ofCampbell and Weynberg (1980); 2, ellipse theory of Fabula (1957); 3, von Karman(1929); 4, semi-Wagner; 5, Wagner’s flat plate approach (1932); 6, Taylor (1930);7, semi-von Karman; 8, semi-Wagner; 9, Wagner’s exact body approach.2.1 Definition sketch for a fixed cylinder.2.2 Regimes of cylinder submergence.2.3 Variation of dimensionless buoyancy force with relative submergence s/a.2.4 Variation ofCwith submergence s/a - selected results. , von Karman;— — ——, Wagner; , Taylor; — — — —, Campbell and Weynberg;-,Miao; — — — ,Armand and Cointe.2.5 Variation of inertia coefficientCmwith relative submergence s/a. ,Taylor’ssolution(Cmo= 2.0); — — — , approximations forCmo= 2.0 and 1.7.2.6 Sketch of free surface elevation and corresponding vertical wave force over one wavecycle. , model I; — — — , model II. (a) complete submergence,(b) partial submergence.2.7 Proposed variation of combinedC+Cdcoefficient with relative submergence s/a forCd= 0.8.,C+Cd;, Taylor’s solution for C.2.8 Definition sketch of a single degree of freedom (SDOF) system.2.9 Representation of an idealized impact force withTd/Tr= 1.0.x2.10 ResponseFt/F0 of SDOF system toan applied impulsive force with different values ofT/f. , applied force and response for impact withTd/Tr = 1.0;applied force and response for impact withTj1’r = 2.0. (a) Tr/Tn= 0.2, (b)Tr/Tn = 2.0.2.11 Dynamic amplification factorF0/Fand relative rise timeTtITr as functions of Tr/Tnforan applied impulsive force with different values ofTd/Tr. , = 0;,C=0.02;—-—-,C=0.05. (a)and(b),Td/Tr=1.0;(c)and(d),TdITr = 2.0.2.12 Definition sketch for dynamically responding cylinder.2.13 Definition sketch for the computation of the impulse coefficient. , combinedimpulsive and residual force; , residual force.2.14 Definition sketch for wave impact on an inclined circular cylinder.2.15 Definition sketch for impact due to a sloping water surface.2.16 Variation of the free surface slope correction factors C/C( )and CIC1-— -)for an experimentally measured wave( )of period T = 1.1 sec,and height H = 17 cm.3.1 Photograph of wave flume in the Hydraulics Laboratory.3.2 Photograph of computer controlled wave generator.3.3 (a) Photograph of test cylinder assembly, (b) Sketch of the experimental setup.3.4 Block diagram of wave generation and data acquisition equipment.3.5 Record of the vertical force due to free vibration of the cylinder inducedby an appliedstep force of -19.6 N (2kg).3.6 Spectral density of free vibration record in Fig. 3.5.3.7 Early stages of the measured vertical force on the test cylinder due toa typical waveslamming event. , indicates individual force samples.3.8 Time histories of free surface elevation and vertical force duringa slamming event.T = 1.4 sec, H = 22.8 cm, h = 0.5 cm.xi3.9 Time series of vertical force( )and corresponding local variance( )used to detect the onset of slamming.3.10 Flow chart showing steps in analysis of experimental data.3.11 Frame of video record defining the wave breaking locationxb.3.12 Record of the horizontal force due to free vibration of the cylinder induced by an appliedstep force of 9.8 N (1 kg).3.13 Spectral density of free vibration record in Fig. 3.12.3.14 Comparison of corrected horizontal force( ),with recorded horizontal force( ),and applied step force( ).3.15 Impact force on the horizontal test cylinder due to plunging wave (h = 8.7 cm,Xb = 36 cm). (a) , recorded vertical force component; , recordedhorizontal force component. (b) , recorded vertical force component;, corrected horizontal force component.4.1 Time histories of the free surface elevation and vertical force over a 10 sec duration for awave of low steepness. T = 1.8 sec, H = 13.5 cm, h = 0.5 cm.4.2 Time histories of the free surface elevation and vertical force over a 10 sec duration for awave of medium steepness. T = 1.5 see, H = 18.4 cm, h = 0.5 cm.4.3 Time histories of the free surface elevation and vertical force over a 10 sec duration for awave of large steepness. T = 1.1 see, H = 17 cm, h = 0.5 cm.4.4 Time histories of free surface elevation and vertical force during a slamming event.T = 1.1 see, H = 17.5 cm, h = 0.5 cm.4.5 Time histories of free surface elevation and vertical force during a slamming event.T = 1.4 see, H = 16.5 cm, h = 0.5 cm.4.6 Time histories of free surface elevation and vertical force during a slamming event.T = 1.6 see, H = 16.8 cm, h = 0.5 cm.4.7 Time histories of free surface elevation and vertical force during a slamming event.T = 1.5 see, H = 13.9 cm, h = 0.5 cm.XII4.8 Time histories of free surface elevation and vertical force duringa slamming event.T = 1.5 sec, H = 18.4 cm, h = 0.5 cm.4.9 Time histories of free surface elevation and vertical force duringa slamming event.T = 1.5 sec, H = 22.9 cm, h = 0.5 cm.4.10 Time histories of free surface elevation and vertical forceduring a slamming event.T = 1.5 see, H = 18.4 cm, h = 4.5 cm.4.11 Time histories of free surface elevation and vertical forceduring a slamming event.T = 1.5 see, H = 18.4 cm, h = -4.5 cm.4.12 Correction factors for peak slamming forceand rise time as a function of the observedrise time ratioTiT. ,FIF0; ,Tt/Tr.4.13 Probability density histogram of C0( )based on data collected from entireset of experiments. , log-normal probability density.4.14 Probability density histogram of C( )based on data collected from entireset of experiments. , log-normal probability density.4.15 Probability density histogram of C basedon data collected from entire set ofexperiments.4.16 Comparison of slamming force time historiesfor different cylinder inclinations, T = 1.2see, H = 19.3 cm. , 0 = 0°; ,0 = 4.8°; - — -, 0 = 9.6°.4.17 Comparison of slamming force time historiesfor different cylinder inclinations, T = 1.8see, H = 17.8 cm. , 0 = 0°;, 0 = 4.8°; - -, 0 = 9.6°.4.18 Time histories of free surface elevation andsimulated vertical force for &alg = 0.05,c02H/g = 0.6, and different cylinder elevations., Model I; — - — -,Model II.4.19 Distribution of the non-dimensional maximum vertical forceas a function of cylinderelevation and co2HJg. , Model I;, Model II. (a) o2a/g = 0.005,(b) &a/g = 0.01, (c) o2aJg = 0.05, (d) w2alg = 0.1.4.20 Variation of the non-dimensional maximumvertical force as a function of w2a/g fordifferent values ofo2HJg. , Model I;, Model II.xli’4.21 Spectral density( )and corresponding amplitude spectrum( )withH = lOm, and T = 14.3 sec, used in the numerical simulation ofrandom waves.(a) Narrow-band spectrum, (b) Pierson-Moskowitz spectrum.4.22 Segment of numerically simulated time seriesof (a) free surface elevation and, (b) non-dimensional vertical force for a narrow-band spectrum.4.23 Segment of numerically simulated time seriesof (a) free surface elevation and, (b) non-dimensional vertical force for a two-parameter Pierson-Moskowitzspectrum.4.24 Comparison of probability density of forcemaxima on a horizontal cylinder located ath = 0. - — -, analytical prediction (Isaacsonand Subbiah, 1990);numerical simulation, method A;, numerical simulation, method B.(a) narrow-band spectrum, (b) two-parameter Pierson-Moskowitzspectrum.4.25 Comparison of vertical force predictedby rigid cylinder model( )with themeasured force( ).T = 1.2 sec, H = 15.2 cm, h = 0.5 cm.4.26 Early stages of slamming forcepredicted by dynamic cylinder model( )compared with rigid cylinder model estimate(— - -)and the measured force( ). T=1.2sec,H=15.2cm,h=0.5cm,Tn=290Hz,Tr=20msec.4.27 Comparison of vertical force predictedby rigid cylinder model( )with themeasured force( ).T = 1.5 sec, H = 18.4 cm, h = 0.5 cm.4.28 Photograph showing mass of water suspendedfrom test cylinder after recession ofincident wave.4.29 Comparison of vertical force predictedby rigid cylinder model( )with themeasured force( ).T = 1.5 sec, H = 18.4 cm, h = -4.5 cm.4.30 Early stages of slamming forcepredicted by dynamic cylinder model( )compared with the measured force( ).T = 1.5 sec, H = 18.4 cm, h = -4.5 cm,T = 290 Hz,Tr= 18 msec.4.31 Comparison of force on inclined cylinder predictedby numerical model( )with the measured force( ).T = 1.8 sec, H = 17.8 cm, 0 = 4.8°.4.32 Comparison of force on inclinedcylinder predicted by numerical model( )with the measured force( ).T = 1.2 sec, H = 15.2 cm, 0 = 9.6°.xiv4.33 Predicted time history of the vertical force on the cylinder for the example application.total force on rigid cylinder; , buoyancy force component;--, mid-span cylinder response for fixed end condition; , mid-spancylinder response for pinned-end condition.4.34 Sequence of video frames showing plunging wave impact on horizontal test cylinder.h = 4.7 cm,Xb = 25.5cm.4.35 Digitized profiles of plunging wave in the vicinity of the horizontal test cylinder.h = 4.7 cm,Xb = 36 cm.4.36 Time histories of recorded vertical force( )and corrected horizontal force( )on horizontal test cylinder due to breaking wave impact. h = 4.7 cm,x, = 25.5 cm.4.37 Sequence of video frames showing plunging wave impact on horizontaltest cylinder.h = 8.7 cm,Xb = -3.5cm.4.38 Sequence of video frames showing plunging wave impact on horizontal test cylinder.h= 8.7 cm,Xb25.5cm.4.39 Digitized profiles of plunging wave in the vicinity of the horizontaltest cylinder.h = 8.7 cm, x, = 25.5 cm.4.40 Time histories of recorded vertical force( )and corrected horizontal force( )on horizontal test cylinder due to breaking wave impact. h = 8.7 cm,Xb = 25.5 cm.4.41 Sequence of video frames showing plunging wave impacton horizontal test cylinder.h = 8.7 cm,xb =49 cm.4.42 Sequence of video frames showing plunging wave impacton horizontal test cylinder.h = 12.7 cm,Xb= 25.5 cm.4.43 Digitized profiles of plunging wave in the vicinityof the horizontal test cylinder.h=l2.7cm,xb= 17cm.4.44 Time histories of recorded vertical force( )and corrected horizontal force( )on horizontal test cylinder due to breaking wave impact. h = 12.7 cm,Xb = 25.5 cm.xvList of Principal Symbolsa cylinder radius.Across-section area of the cylinder.A1 immersed area of cylinder cross-section.cx angle subtended at the cylinder axis by the water surface.c wave celerity.C damping coefficient of single-degree-of-freedom system.Cddrag coefficient.C1 impulse coefficient.C impulse coefficient corrected for effects of free surface slope.Cminertia coefficient.Cmoinertia coefficient for large value of cylinder submergence.Csslamming coefficient.C0maximum value of the slamming coefficient.C maximum slanmiing coefficient corrected for effects of free surface slope.C0 maximum slamming coefficient corrected for effects of free surface slope and buoyancy.C0 maximum slamming coefficient corrected for effects of free surface slope, buoyancy, anddynamic amplification.C5÷j combined slamming and drag coefficient.d water depth.xviD diameter of cylinder.angle of inclination of the free surface during impact.f0 peak frequency of wave spectrum.maximum value of the vertical force due to non-breaking wave impact.Fb buoyancy force.Fd drag force.F inertia force.F0 maximum value of the applied force.Fr Froude number.F slamming force.F measured force.F0 maximum value of the measured force.maximum value of the resultant force due to breaking wave impact.maximum value of the horizontal force component due to breaking wave impact.maximum value of the vertical force component due to breaking wave impact.direction of the maximum resultant force due to breaking wave impact.g gravitational acceleration.h elevation of cylinder measured from the still water level.H wave height.Hsignificant wave height of wave spectrum.free surface elevation.vertical velocity of free surface.xviivertical acceleration of free surface.k wave number.K stiffness of single-degree-of-freedom system.L cylinder length.m added mass.M mass of single-degree-of-freedom system.v kinematic viscosity of water.8 inclination of the cylinder axis.r displacement response of the cylinder.velocity of the cylinder.acceleration of the cylinder.Re Reynolds number.r radius of curvature of the breaking wave-front prior to impact.p density of water.s cylinder submergence.S(Ospectral density of free surface elevation.T wave period.Td decay-time of the applied force.T duration of the impulse.T natural period of vibration.Tr rise-time of the applied force.T rise-time of the measured force.v water particle velocity normal to the free surface.angular frequency of non-breaking wave.Mdamped natural angular frequency of vibration.o natural angular frequency of vibration.xb location of wave breaking.non-dimensional damping ratio.xviiixixAcknowledgementsThe author would like to acknowledge several individuals who assisted in various phases ofthe project. The author would like to thank his research advisor, Dr. Michael Isaacson, for hishelp and encouragement throughout the course of this study, and his invaluable insight andcritical appraisal during the preparation of the thesis. The expertise of Mr. Kurt Nielsen,Mr. Ron Dolling and Mr. John Wong of the Department of Civil Engineering, in assembling thetest cylinder, and setting up the instrumentation and signal cables is gratefully acknowledged.The advice of Mr. Dan Pelletier of the Hydraulics Laboratory at the National Research Councilof Canada in the customization of the real-time control and data acquisition systems andsoftware for the purposes of this study is deeply appreciated. The author would also like tothank his colleagues and friends for their help and support.Financial support in the form of a University of British Columbia Graduate Fellowship and aresearch assistantship from the Department of Civil Engineering is gratefully acknowledged.1Chapter 1INTRODUCTION1.1 GeneralThe phenomenon of hydrodynamic impact or ‘slamming’ manifests itself in many forms.These usually involve either a rapidly moving body entering a water surface, or a moving watersurface striking a body. One of the earliest reported studies of this phenomenon was promptedby the occurrence of impact on the floats of seaplanes during landing (von Karman, 1929). Asimilar type of impact can cause snatch loading of crane hoisting cables during the installation ofsubsea modules. Ships are subjected to impact when the bottom of the vessel hits the water witha high velocity, or when oncoming waves slam the bow above the waterline. Wave slammingcan occur on the underside of the deck between the two hulls of a catamaran or semisubmersibleplatform, and on horizontal piers and docks. Members of jacket platforms which wouldnormally not be situated in the splash zone during the operating life of the structure may still besubjected to slamming forces during towing and launch operations. In addition to impact due toregular waves on horizontal or near-horizontal structures near the water surface, verticalstructures such as the columns of offshore platforms and sea-walls are also subjected to impactloads due to breaking waves. Sloshing of fluid inside a storage tank can lead to very highslamming pressures on the walls of the tank.Wave impact forces are highly dynamic and are characterized by their large magnitudes andshort durations. In cases where the entire length of a horizontal bracing is simultaneously struckby a rising wave, the impact forces can be much higher than the hydrodynamic force experienced2by the member when it is fully submerged and subjected to a flow of similar velocity. Suchforces can cause local damage to structural elements. Even in cases where impact forces maynot be larger than the static loads on the structural element, dynamic stresses due to impact andthe possibility of increased fatigue stressing of joints due to such loads may contribute tostructural failure and consequently determine design criteria for the member.A common case of hydrodynamic impact corresponds to a horizontal cylinder located near thewater surface which undergoes intermittent submergence due to incident waves, and attention isfocussed on this particular case. An example of wave slamming on a horizontal cylinder isillustrated in Figure 1.1. Some of the readily observable features of this phenomenon are thesignificant distortion of the free surface adjacent to the cylinder and air entrainment due to theeffects of the accompanying splash. The wave slamming force on a horizontal member isgenerally taken as proportional to the square of the water impact velocity and involves the use ofa slamming coefficientCs which varies with cylinder submergence. The usual approach toestimating wave slamming force on horizontal members is given by:Fs= CspDw2(1.1)where p is the fluid density,F is the wave slamming force per unit length of the member, C is aslamming coefficient, w is the water particle velocity normal to the surface of the member and Dis the diameter of the member. In factCmay vary with time after the onset of slamming, and itsmaximum value, which occurs after a very short rise-time subsequent to water-cylinder contact,is of primary interest in design. This value is designated here as C. A number of theoreticaland experimental studies have been carried out in order to establish appropriatevalues of theslamming coefficient for the common case of a horizontal circular cylinder. Experimentalobservations have yielded values ofC0which exhibit a considerable degree of scatter, rangingfrom about 1.0 to 6.4, although a value of it is generally recommendedin design codes (Miller,1977, Sarpkaya, 1978, Campbell and Weynberg, 1980,Miao, 1988). There are a number ofdifficulties in carrying out experimental studies on slamming. The veryshort durations involved3make it a highly dynamic process, and the response of the member due to the load can changethe magnitude of the impact. Furthermore, the rise-time associated with the slam force dependson various factors such as air entrainment, compressibility, cylinder roughness, cylinderinclination and motion of the cylinder. The variability of the rise-time can cause a significantvariation in the dynamic amplification of the impact force resulting in appreciable scatter in theobserved force.A theoretical basis for the slamming force formulation has also been investigated by severalauthors. During the early stages of impact, the slamming force is associated with the rate ofchange of momentum of the fluid, and can thereby be expressed in terms of theadded mass ofthe partially submerged member. However, there are a number of complications in extendingsuch a development beyond the initial stages of impact, due in part to water level variationsaround the partially submerged member.In addition to the slamming force, other force components also contribute significantly to thevertical force after the onset of impact. These are the buoyancy force, an inertia force associatedwith the fluid’s acceleration, and a drag force component associated with flow separation effectsand dependent on the fluid’s velocity. The coefficients used to determine the magnitude of theinertia and drag forces may vary with the extent of submergence, the cylinder size andthe waterparticle kinematics.Figure 1.2 illustrates a typical plunging wave impact ona horizontal test cylinder. Themechanism of plunging wave impact is quite different from a regular wave impact, with largerhorizontal water particle velocities, increased air entrainmentand turbulence. It is well knownthat breaking wave impact on a structure in the splash zone givesrise to higher local pressuresand forces in comparison to the impact due to non-breaking waves of comparable height andperiod. A theoretical or numerical treatment of the dynamics of breaking waves and theirinteraction with structures presents significant difficulties due to the strongly nonlinear nature ofthe problem which is essentially a transient two-phase turbulent process. Carefully conducted4experiments are a source of useful data that can aid in understanding the breaking wave impactprocess, and help to estimate forces that could occur under similar conditions in the field.The objectives of the present study are to examine existing models of wave slamming on aslender horizontal cylinder, conduct experimental investigations to observe the impact force onan instrumented cylinder subjected to non-breaking regular waves and compare the observeddata with previously reported values. It is intended that the influence of variables such ascylinder elevation and orientation with respect to the water surface will also be studied in theexperiments. This thesis also presents the results from an experimental study of the forces due tothe impact of a plunging or deep-water breaking wave on a slender horizontal cylinder. Theintent of these objectives is to help verify or suggest modifications to analytical models in orderto enable designers to predict more accurately operating and design conditions to which anoffshore structure and its structural components may be exposed.1.2 Literature Review1.2.1 Wave Force on a Horizontal Cylinder in the Splash ZoneTheoretical StudiesThe impact force on a horizontal cylinder due to a rising water surface is givenby the rate ofchange of momentum of this flow, which in turn dependson the properties of the cylinder’svertical added mass. The added mass varies with the levelof submergence up to the stage whenthe cylinder is submerged 3 to 4 diameters below the free surface. This quantityalso depends onthe frequency associated with the flow, and in thecase of impulsive flows, it is the infinitefrequency limit that is of interest in determiningthe magnitude of the impact force. Anexpression for the added mass maybe developed on the basis of potential flow theory. Taylor(1930) solved the above problem using a conformalmapping technique and thereby derived aclosed-form expression for the vertical added mass of apartially submerged cylinder. Numericaltechniques such as a source distribution method(Garrison, 1978) and the Frank close-fit method5(Faltinsen et al., 1977) may also be used to provide estimates of the added mass. These methodsyield a C0 value of about 3.1.Several theoretical techniques have been used to modify the above approaches in order toaccount for the local deformation of the free-surface near the cylinder during the early stages ofpenetration. Wagner (1931) and Fabula (1957) proposed modifications, which include a‘wetting correction’, in order to account for the piled up water when calculating the wetted widthof the body, and a ‘drag correction’ which include quadratic terms in the expression for pressure.Both approaches yield a C0 value of 2n, although Fabula’s (1957) technique predicts a morerapid decay of the slamming force with submergence. Detailed numerical and experimentalresults concerning the water impact of a circular cylinder were given in an EPRI report (1978)based on research connected with boiling water reactors. The four numerical models describedin this report correspond to; an explicit Lagrangian method (Gross, EPRI 1978), a boundaryintegral method (Geers, EPRI 1978), a finite element method (Marcal, EPRI 1978),and anincompressible Eulerian fluid method (Nichols and Flirt, EPRI 1978). The results of three of thenumerical simulations were in good agreement with the experimental results and indicateda Cvalue of 2n.Cointe and Armand (1987) used the method of matched asymptotic expansions to solve theboundary value problem for water-cylinder impact. They too conclude that C is 2it rather thant as given by some of the earlier theories. Greenhow and Li (1987) reviewed a number ofdifferent formulations for evaluating the added mass of a horizontal circular cylinder movingnear the free surface and conclude that the effects of free surface deformation on the slammingcoefficient is significant and must be included in any theoretical treatment. They recommendtwo different methods to calculate the added mass for small and large cylinder submergencerespectively which both indicate a C0 value of 4n13. Figure 1.3 shows a comparison ofCspredicted by a number of theoretical and experimental results presented in their paper whereinthe abcissa corresponds to the relative submergence of the cylinder and is denoted by s/a where s6and a are the submergence and cylinder radius respectively. It is seen that there are considerabledifferences between the various models, especially with respect to the value of C0.In additon to the impact force on the cylinder, other components contributing to the verticalforce include the buoyancy, drag and inertia forces. The buoyancy force at any instant isassociated with the submerged volume of the cylinder and can be readily determined. Incontrast, the drag force due to separation effects cannot be easily quantified, especially duringthe initial stages of submergence. In addition, both the drag and impact force components arefunctions of the square of the relative velocity between the water and the cylinder, and hencecannot be differentiated in any experimental data. The inertia force component is a function ofthe relative water acceleration and is a function of the added mass of the cylinder.The vertical hydrodynamic force on a horizontal cylinder subjected to wave slamming may besimulated numerically by the use of a suitable model which incorporates the four forcecomponents identified above without introducing abrupt changes in either the magnitude or rateof change of the total hydrodynamic force. Although the rise-time of the slamming force is notimportant if the cylinder is assumed rigid, a realistic situation will involve an elasticallysupported structure which will respond to the applied force. The magnitude of response dependson the dynamic characteristics of the structure and determines the force transmitted by thestructure to its supports. Since the rise-time of the slamming force is not accounted for in thecurrent added mass models it is usually introduced by increasing the slamming force linearlyfrom zero at the instant of impact, to a peak value after a specified delay.The Morison equation (Morison et aL, 1950) is a numerical model for estimating fluid forceson a fixed body in an unsteady flow. It is based on the assumption that the force can be given bythe linear superposition of a drag force which is dependent on the square of the velocity and actson the projected frontal area, and an inertia force which is dependent on the acceleration and thevolumetric displacement. The formula estimates the magnitude of these force components byusing two parameters known respectively as the drag coefficientCdand the inertia coefficient7Cm,whose values must be chosen mainly on the basis of empirical data. In the case of surfacepiercing vertical cylinders, the flow regime over which this formulation is applicable generallycorresponds to the case where the diameter of the cylinder is less than 20% of the wavelength.A modified form of the Morison equation which includes the buoyancyforce component on apartially submerged horizontal cylinder was proposed by Dixon et al.(1979a) and expressions todescribe the force spectra in narrow-band and wide-band Gaussianseas were also formulated(Dixon et al., 1979b, Easson et al., 1981). These studies neglect the effect of slamming forces onthe cylinder. Kaplan and Silbert (1976) described a mathematical model for slamming on a rigidcylinder and presented impact force statistics from a simulation in random waves.Miller (1977)developed a computer model for the vertical wave force on an instrumentedcylinder whichbehaves as a two degree-of-freedom dynamic system. The slammingforce was assumed toincrease linearly to a peak value over a specified rise-time, andto decrease linearly to zero over aspecified decay-time. The drag and inertia components were assumedto act only when thecylinder was fully submerged and were computed from constantdrag and inertia coefficients.The resulting combined force exhibits discontinuities in magnitude and the buoyancy componenthad to be reduced when the water surface was receding in orderto avoid an unrealistic dynamicresponse during that stage.Arhan et al. (1978) modelled the response of a horizontal elastic cylinderwith clamped endssubjected to wave slamming. They assumed that the slamming force risesinstantaneously uponimpact to a peak value and then decreases linearly to zero whenthe cylinder was partiallysubmerged by half its radius. The maximum deflection and stress on the cylinder werecomputed by numerically integrating the equation of motion. Miao (1990) computed thebending stress on a flexible cylinder subjected to water impact. The hydrodynamic force wascomputed as the sum of the momentum, drag and buoyancy forces and the equation of motionwas solved for various end fixity conditions using the mode superposition approach. Computedbending stresses compared well with his experimental observations of slamming on a cylinderdriven into still water. Isaacson and Subbiah (1990) considered the application of a suitable8force formulation to the case of a cylinder subjected to wave impact in random waves. Byrestricting the analysis to force maxima in waves with narrow-band spectra, they were able toprovide analytical results for the probabilistic properties of the force maxima.Experimental StudiesExperimental investigations on slamming have been conducted ina variety of ways. Differentmethods have been used to induce slamming -- a model cylinder drivenat a constant velocitythrough a still water surface; a fixed horizontal cylinder subjectedto slamming forces in a U-Tube; or a fixed horizontal cylinder subjected to slamming forcesdue to waves. In addition,field observations of impact forces have included measurements from a horizontal brace of theOcean Test Structure (Kaplan, 1979), and full-scale slamming on British Petroleum’s West Soleplatform (Miller, 1980).Dalton and Nash (1976) conducted tests to observe impact forces on a horizontal cylinder in awave tank and reported values of C0 ranging from 1.0to 4.5. However, they report that thesevalues are derived from force peaks that occur after the cylinder hasbeen submerged by 8 to 12diameters and observed that significant wave forces associated with regular wave trains are notof an impulsive nature, which indicates that their measurements maynot have included the earlystages of the slamming force.Miller (1977) identified loading regimes associated with wave slamming and describedslan-iming tests on a cylinder in waves. Tests conducted for 3 cylinder elevations indicated anaverage value of C0 of 3.6, although there was appreciable scatter in the results. Millerconcluded that this was consistent with the theoretical value of it and attributed the observedhigher value to dynamic amplification effects. The effects of the slamming forcerise-time onthe dynamic response were illustrated by simulating numerically the vertical wave force using adynamic analogue and comparing the computed force traces with the experimental records.9Faltinsen et al. (1977) conducted experiments with elastic horizontal circular cylinders thatwere driven with constant velocity through an initially calm free surface and reported C valuesranging from 4.1 to 6.4. The experimental data was compared with results from a dynamicmodel which computed the slam load from potential theory, and it was observed that theoreticalpredictions were lower than the experimental values. Sarpkaya (1978) measured thehydrodynamic force on a cylinder subjected to slamming in a U-Tube. He observed that thedynamic characteristics of the measuring system play a significant role in the observed impactforce. He reported experimental observations for C as 3.17 ± 0.05, and so concluded that C0was essentially equal to the theoretical value of it.Campbell and Weynberg (1980) measured spanwise and circumferential pressures in additionto the vertical force on horizontal and inclined cylinders driven through a still water surface.They observed that the slamming force was predominant for tests involving a Froude number(Fr = w/Jji5 where w is the fluid velocity,g is the gravitational acceleration, and D is thecylinder diameter) higher than 0.6. They also indicated that the slamming force was masked bythe dynamic response of the force transducer and that scatter in the observed data was the resultof variable rise-times that were sensitive to small variations in the slope of the cylinder. It wasalso noted that drips from the cylinder had a significant effect on the response. Theysummarized their results by proposing an empirical equation thatrelates the slammingcoefficient and cylinder submergence, and which usesC0= 5.15. This equation is independentof the Froude number and was not corrected for buoyancy effects.Kaplan (1979) presented results from an analysis of impact force data collected from ahorizontal brace of the Ocean Test Structure in the Gulf of Mexico. The force measurementswere recorded at a time interval of 0.1 sec. after being low-pass filtered using a 3 Hz cutofffrequency. The low sampling rate and high degree of filtering make it difficult to perform aquantitative analysis of the slamming force characteristics. The data was analyzed by comparingit to a synthetically generated slamming force record that had also been low-pass filtered at 3 Hz.C values between 1.88 and 5.11 with a mean value of 2.98 were reported.10Miller (1980) reviewed the results of slamming tests done by various investigators who used avariety of experimental techniques. The reported values of C0 varied from 0.4 to 6.9. Miao(1990) reported results from experiments on a 1.52 m long flexible horizontal cylinder whichwas driven at a constant velocity through a stationary water surface. He proposed an expressionfor the variation ofCswith submergence which indicates a C50 value of 6.1, and also concludedthat for typical truss members in heavy seas, the dynamic amplification of the response and theinduced stresses ranged from 0.3 to 0.6 due to the short impulse times observed in theexperiments. Table 1.1 provides a summary of results from earlier experimental studies onslamming forces on a horizontal cylinder.1.2.2 Forces on Horizontal Cylinders due to Breaking WavesStudies of breaking wave impact on structures are especially pertinent tothe design of offshoreplatforms since these may give rise to the largest environmental loads during the operating life ofthe structure. The complex nature of the problem and the difficulty of obtaining precise controlover wave breaking in laboratory experiments has been the main reason for the scarcity ofliterature available on this topic. Impact forces due to a breaking wave have been observed to betwo to four times larger than those due to non-breaking wave of comparable amplitude, and theforce rise-time is significantly smaller.Studies of wave impact forces due to breaking waves have largely related to vertical piles andwalls, and vertical plates (e.g. Kjelclsen and Myrhaug, 1979, Kjeldsen, 1981, Sawaragi andNochino, 1984, Kjeldsen et al., 1986, Basco and Niedzwecki, 1989, Chan and Melville, 1989,and Zhou et al., 1991). These have shown that the most severe impulsive forces are due toplunging waves and that these act at elevations above the mean water level. Measurements ofimpact pressures on vertical plates due to plunging waves have shown that impulsive pressuresmay occur over a range of horizontal locations relative to the plunging wave location (Chan andMelville, 1988, Chan et al. 199la, Chan et al. 1991b, and Zhou et al., 1991). These pressures11may be spread over a vertical distance of about half the wave height, and peak impulsivepressures range from 3 pc2 to 30pc2,where c is the wave celerity.Vinje and Brevig (1981) used a numerical time-stepping procedure to simulate breaking waveimpact on a horizontal cylinder and a vertical wall. Since their study did not include the effectsof air entrainment, it is not possible to apply their model to most situations where the dynamicsof trapped air plays an important role in determining the impact pressures. Based onexperiments with vertical and inclined plates, Kjeldsen (1981) observed that the magnitude ofshock pressures is a function of the steepness of the wave-front, and that a plate tilted forward at45° to the horizontal which is struck from below experienced larger impact pressures than avertical plate. He also noted that scaling laboratory results to prototype conditions may bedifficult due to the differing magnitudes of air entrainment in the wave crest for the two cases.Easson and Greated (1984) performed experiments with plunging wave impact on a horizontalcylinder located at different elevations above the still water level. They observed that the peakimpact force and rise-time changes with the vertical location of the cylinder. The influence ofthe measuring system’s dynamic response on the impact force was not examined in their study.In a recent work related to plunging wave impact on a large horizontal cylinder, Chan (1993)measured pressures on the upwave face of the cylinder for different cylinder elevations and wavebreaking locations relative to the cylinder axis. Peak pressures were found to vary from 4pc2to33pc2 and were affected by factors such as the local wave profile and the amount of entrainedair.1.3 Scope of Present InvestigationThis study addresses three areas relating to wave impact on a slender horizontal cylinderlocated near the still water level which undergoes intermittent submergence in the presence ofwaves. These are, (i) Development of a numerical model to estimate the vertical force on ahorizontal cylinder using available expressions for the different force components, and proposingmodifications to include the effect of dynamic response and cylinder inclination; (ii) Conducting12slamming experiments in regular non-breaking waves in order to determineC as well as relatedparameters, and to compare the experimental observations with the estimatesof the numericalmodel; and (iii) Conducting experiments on slamming due to a plunging wave inorder todetermine the maximum impact force and corresponding slamming coefficients.1.3.1 Numerical ModellingThe vertical component of the wave force on a horizontal cylinderlocated in the zone ofintermittent submergence is made up of four components:the impact force, the inertia force, thedrag force and the buoyancy force. Although previous studies havedealt with the nature of thesecomponents, a numerical model which involvesa combination of these components in aconsistent manner and in varying ways is presented inthis investigation. The simulation of thetotal wave force variation without slope discontinuities during thedifferent stages of cylindersubmergence is one of the main features of this model.In addition to estimating the applied force on a rigid cylinderin regular and random waves, themodel is suitably modified so as to predict the appliedand transmitted force in the case of acylinder that responds dynamically to the applied force. This dynamicforce model includes theeffect of the finite rise-time associated with theslamming force. The effect of cylinderinclination on the hydrodynamic force is also examined.Results from regular wave simulationsare used to obtain typical traces of force variation in awave cycle for various cylinder locations, and to also providethe non-dimensional peak force ina wave cycle as a function of the other governing non-dimensionalvariables of the problem. Inthe case of random waves, statistics of peak forcedue to waves synthesised from narrow-bandand Pierson-Moskowitz spectra are computed andcompared with corresponding results derivedfrom closed-form expressions.131.3.2 Experiments on Slamming in Regular WavesAs already indicated, alternative theoretical predictions of the slamming coefficient showsignificant differences, especially with respect to C0. In the offshore industry, therecommended value of C0 is 3.5 although several theories indicate that C0 is as high as6.3 (2it). Since both safety and economy are primary concerns in design, it is important thateither value of C0 be backed up by reliable experimental observations.The value of the peak slamming coefficient C and related parameters have been determinedby conducting experiments on a slender horizontal cylinder subjected to slamming in regularnon-breaking waves. Tests are conducted for a range of wave heights and periods, and differentcylinder elevations. The vertical force on the cylinder and the water surface elevation at thecylinder location have been measured and subsequently analyzed to provide the value of thepeak slamming coefficient immediately after impact, the impact force rise-time, as well as otherrelevant parameters. A new approach to the prediction of wave slamming effects is developed.This is based on an impulse coefficient C, and values of C1 are obtained from the experimentalrecords. Experimental tests which are intended to examine the effect of cylinder inclination onthe slanmiing force are also presented.The experimental force records are compared with the predictions of the alternative numericalmodels for the cases of a rigid horizontal cylinder, a dynamically responding horizontal cylinder,and a rigid inclined cylinder. Suitable modifications to theory are suggested so that results fromthe numerical model more closely match the experimental data.1.3.3 Experiments on Slamming in Breaking WavesThe test cylinder has also been used to study the impact force due to a breaking wave, which isalso referred to as a ‘plunging’ wave in this study. Impact pressures due to a plunging wave areknown to be the highest among the various types of breaking waves. A single breaking wave isgenerated using a frequency and amplitude modulated wave packet consisting of 30 prescribed14sinusoidal components. Tests are carried out for 3 cylinder elevations, and 6 different locationsof wave breaking. Both the horizontal and vertical components of the impact force aremeasured, and are analyzed to determine the resultant peak force and the corresponding rise-time. A video record of the wave impact is used to estimate the kinematics of the wave-frontprior to impact. The video records are also used to examine the influence of the geometry of theplunging wave on the characteristics of the impact force. The impact forces are compared withthose obtained due to a regular non-breaking wave impact. The issues governing theapplicability of these results to large scale situations are also discussed.The importance of wave slamming in offshore design is aptly illustrated by Attfield (1975)who refers to the case of British Petroleum’s WB West Sole platform in the southern North Sea:“Wave slam on horizontal members in the splash zone was underestimated on theWest Sole platforms to be more than three times the normal wave loads allowedThese factors caused overstress of members in the splash zone and in fact onehorizontal brace just below water level fell off in September 1972.”He also notes that inspection of tubular members in West Sole’s other structures revealed fatiguecracks, and that design faults were more apparent in the splash zone than underwater.In this context, the present thesis is intended to provide relevant numerical and experimentalresults of wave slamming on horizontal cylinders and consequently contribute to a betterunderstanding of the problems associated with wave slamming.15Chapter 2THEORETICAL FORMULATIONVarious considerations influence the force on a horizontal cylinder subjected to impact inwaves. This chapter identifies these and describes a suitable formulation in order to simulatenumerically the variation of hydrodynamic force for a given regular or random wave train.2.1 Dimensional AnalysisA definition sketch of the problem which is initially under investigation is shown in Fig. 2.1.Uni-directional waves propagate past a fixed slender horizontal circular cylinder of radius a,whose lower surface is at a distance h above the still water level (SWL). The water surfaceelevation at any instant is given by i which subtends an angle ot at the cylinder axis, and thecorresponding cylinder submergence and immersed cross-section area are given by s and Arespectively.It is appropriate initially to examine the vertical force on the basis of dimensionalconsiderations. The maximum force per unit length acting on the cylinder in regular wavesdepends on the cylinder radius a, the cylinder elevation h, the wave angular frequency o, thewave height H, the gravitational constantg, the still water depth d, the fluid density p. thekinematic viscosity v, and the bulk modulus B. A dimensional analysis indicates that the nondimensional vertical force can then be expressed as a function of dimensionless parameters in thefollowing manner:__hco2aHCF = = f(, ,, kd, Re, Ca) (2.1)pga2g g16h/H is a dimensionless cylinder elevation, w2a/g is a frequency parameter,co2HJg is similar to thewave steepness, kd is a depth parameter, where k isthe wave number related to w by thedispersion relationship, Re is a Reynolds number, and Ca is a Cauchy number basedon theimpact velocity and E. Alternatives to Eq. 2.1 are of course possible.In particular, it may alsobe useful to consider the wave height to cylinder radius ratio H/a,which is similar to theKeulegan-Carpenter number, and if required this maybe adopted in place of the steepnessparameter. Equation 2.1 may be simplified by assumingdeep water waves and ignoring theinfluence of viscosity and compressibility so that:F hoaoHpga2= f(,-, -j—)(2.2)The vertical force per unit length acting on the cylinderis generally modelled as thecombination of components correspondingto an impulsive slamming force; a time-varyingbuoyancy force; and drag and inertia forces based on the vertical fluidvelocity and accelerationrespectively. The hydrodynamic force per unit length on the cylinderis thus expressed as thesum of four contributions:F=Fb+Fs+Fd+Fj (2.3)whereFb is the buoyancy force, F is the slamming force,Fdis the drag force, and F is theinertia force. It is instructive to consider theratios of the magnitudes of these four forcecomponents in the context of the dimensionless parametersgiven in Eq. 2.2, by examining theusual representation of each component in termsof empirical coefficients and the differentvariables describing the wave train andthe cylinder. The ratios of the magnitudes of theslamming, drag and inertia forces to the buoyancy forceare thereby given respectively as:= 0 [&H/g]= o[02]= 0[(w2a/g)()2] (2.4)= o[(2]= 0[ro2a/g)(D2J17The buoyancy force on a fully submerged member is proportional to the volume of fluiddisplaced by the member and is independent of the wave conditions. The inertia to buoyancyforce ratio depends primarily on the wave steepness; while the drag and slamming forcecomponents both depend in a similar way on the wave steepness and frequency parameters.It is also useful to consider the values of the dimensionless parameters which may occur intypical situations. For waves with periods varying from 5 to 20 sec, wave heights extending to20 m and the cylinder diameter varying from 0.1 to 2 m, the parameter ranges are approximately:0.001 — 0.20—0.9Furthermore, h/H within the range between ±0.5 is of interest in the present context, so thatslamming on the cylinder does occur. More specifically, on the basis of a sinusoidal waveprofile, the conditions under which no submergence, only partial submergence, partial andcomplete submergence, and only complete submergence occur, can be expressed in terms of h/Hand HJa as follows:No submergence: > 0.5Only partially submerged: 0.5, and + 2 > 0.5Partial and complete submergence: - 0.5 < + 2 0.5h aCompletely submerged: + 2 - 0.5For convenience, this is indicated in Fig. 2.2, which shows the ranges of h/H and H/a for whichno submergence, partial submergence and complete submergence occur. The figure indicateshow partial submergence is less likely to occur for larger values of H/a.182.2 Hydrodynamic Force on a Rigid Horizontal CylinderThe various components comprising the vertical force on a horizontal cylinder have beenidentified in the preceding section, and each of these is now examined in detail.2.2.1 Buoyancy ForceThe buoyancy force per unit length of the cylinder is given by:Fb(t) =pgA(s) for 0 s/a <2(2.5)1..pgA for s/a2where At(s) is the sectional area of the immersed portion of the cylinder, s = - h, as indicated inFig. 2.1, and A = 1ca2 is the sectional area of the cylinder, s/a 2 corresponds to completesubmergence. At(s) may be expressed as:Ai(s) = 0.5 a2 (a - sina) (2.6)where a is as shown in Fig. 2.1. The buoyancy force for partial submergence can thus beexpressed as:Fb(t) = p ga2 (a - sin a) (2.7)The angle a and s are related by:a = 2 cos1 (1- )for 0 s/a 2 (2.8)The variation of the dimensionless buoyancy force F/pga2with the relative submergence s/ais shown in Fig. 2.3. It is noted that the above expressions are valid under the assumption thatthe free surface is horizontal and there is no run-up on the sides of the cylinder.192.2.2 Slamming ForceThe earliest treatment of the water impact problem was by von Karman (1929) who examinedthe impact force on the floats of seaplanes during landing. The basic simplifying assumptionsused for a theoretical treatment of this problem are that the fluid is incompressible and inviscid,the flow is irrotational, the body is rigid, and that surface tension is negligible. The force on astationary object subjected to impact from a rising water surface is given by the rate of change ofmomentum of the fluid in the vicinity of the object. The impact force is thereby expressed as:Fm(t) =(m w) (2.9)atwhereFm(t)is the force due to the change of momentum, m is the added mass of the body and wis the vertical fluid particle velocity. The added mass m is a hydrodynamic property of the bodyand the flow geometry such that an acceleration of the body gives rise to a hydrodynamic forcem5 resisting this acceleration.In the case of a horizontal cylinder, von Karman (1929) suggested that the effects of spray andwater rise up the sides of the cylinder during the initial stages of impact were negligible, and thatthe flow-field about the cylinder is similar to that for a flat plate of the same width as theinstantaneous waterplane section (shown as a dashed line in Fig. 2.1). For a two-dimensionalflow around a flat plate, the flow field is symmetric both fore and aft of the plate. For a plate ofwidth 2b and length L, the added mass is given by pitb2L. However, since there is no fluidabove the equivalent flat plate in the case of the cylinder, the added mass per unit length of thecylinder is half that of the plate and is given by:m=pitb2(2.10)where b is the immersed half-width of the cylinder. Eq. 2.10 can be expressed in terms of thesubmergence s as:m = pirs(2a-s) (2.11)20Instead of treating the above problem as equivalent to flow around a flat plate, Taylor (1930)used the method of conformal mapping to solve for the unbounded flow around a circular lens.This solution can be used to derive an expression for the vertical added mass m per unit length ofa partially submerged cylinder (0 s/a 2), and is given as:1 r2it3(1—coscx) it . 1= pa23(2it cx)2+ -(1—cosa) + smcx— cx (2.12)After full submergence (s/a > 2), the added mass of the cylinder can still be evaluated usingpotential flow theory but using different methods. The velocity potential for irrotational flow inan inviscid unbounded fluid due to a body (a horizontal cylinder near the free surface in thepresent case) undergoing small amplitude harmonic oscifiations may be expressed as:6(x,y,z,t) = E,j 4(x,y,z)e)t(2.13)j=1where is a complex amplitude with respect to the j-th mode of motion( j= 1, ..., 6,corresponding to surge, sway, heave, roll, pitch and yaw, respectively),Øjis the potentialfunction associated with each mode of motion, and 0 is the oscillation frequency. The linearizedboundary condition at the free surface is given by:fr =atz=0 (2.14)ga2For the case of slamming considered here, the hydrodynamic loading is a high frequencyphenomenon so that the free surface boundary condition can be expressed as the high frequencylimit of Eq. 2.14:atz=0,o—oo(2.15)The corresponding potential flow problem can be solved by a source distribution method, or aFrank close-fit method as described by Faltinsen et al. (1977). The solution can in turn providethe added mass of the cylinder. Greenhow and Li (1987) have given results for the vertical21added mass of a cylinder when it enters or exits a free surface. Their expression, which is alsovalid when the cylinder is partially submerged is given by:f(1 - g2) 1(1 + mj)K2(m1) E(m1)K(m1)—1. q2[123it2 It2—— 2 (1 + rn)K2(m)+ 2E(m)K(m)]— 1}(2.16)where E(m1)and K(m1)are the complete elliptic integrals of the first and second kindsrespectively (see Abramowitz and Stegun, 1970), m1 is the elliptic integral parameter, and mj,m2 and m are related to m1 through:mj = 1-rn1m = 1-m(2.17)K(m2)1K(m)K(m) = 2 K(m)Also:K(m’)q = exp[ItK(m1)]= exp[ - a0] (2.18)a0 is a parameter describing the cylinder position and is given as:sinh2a0= s(s-2a) = (1- q2)(2.19)Reverting to the definition of the impact force as the rate of change of fluid momentumasgiven in Eq. 2.9, it is possible to expand the expression as:aw amFm(t)= rn— + w— (2.20)at atEquation 2.20 can be rewritten as:amFm(t)= m w + —w2 (2.21)as22since s = wt. Eq. 2.21 has two terms which are a function of the fluid acceleration and velocityrespectively. The acceleration term is a component of the inertia force which is discussed in thenext section. For a situation where the flow is non-accelerating, the only force component is dueto the velocity-squared term. This term is designated as the slamming force and is alsoexpressed in terms of a slamming coefficientCsas:F(t) = C p Dw (2.22)where D(=2a) is the diameter of the cylinder. C5 may hence be expressed as a function of therate of change of added mass:=-— (2.23)pD asExpressions for the added mass m, given by Eqs. 2.11 or 2.12 for partial submergence and byEq. 2.16 for both partial and complete submergence, may be used to develop correspondingexpressions for the slamming coefficient. In the case of von Karman’ s added mass model,Cisgiven by:C = ——pit(a-s) = t(1-) (2.24)pDWagner (1931) used a technique similar to the above, but he proposed that the equivalent flatplate be extended between the spray roots at the sides of the cylinder. He determined thelocation of the spray roots by using the vertical velocity of the upwash predicted by potentialflow theory for a flat plate. Wellicome (Campbell et al., 1977) used Wagner’s model to integratethe pressure distribution over a flat plate, and derived an expression for the variation ofC:C2t225S— (1+1.5 s/a). )The expression for C on the basis of Taylor’s model is given by:1 F2n3 / sinc 2(1 - cosx) it= + ) +- sma + cosct - 1 (2.26)-‘ (2it-c (2it-a)3sin23which is applicable for a between 0 and 2it (s/a =0 to 2). Figure 2.4 showsa comparison of thebehaviour ofCspredicted by the above expressions. According to von Karman’s model, theslamming coefficient at impact has a value of it which decreases linearly with increasingsubmergence till it drops to 0 when the cylinder is half submerged and remains zero as s/aincreases beyond 1 since the flow is past a constant projected width(=D). Wagner’s wettingcorrection doubles the value of C to 2it, but predicts a very gradual rate of decay for C. Thefeatures of Taylor’s solution are that C5 = it at the instant of impact, s/a = 0; C5 reaches aminimum value of 0.298 at s/a = 1.555; andC increases to a value of 1.351(=iu3/9 - 2it/3) at theinstant of complete submergence, s/a = 2. Previous experimentaldata (e.g. Sarpkaya, 1978,Campbell and Weynberg, 1980) indicate that a maximum inCnear the instant of completesubmergence as predicted by Eq. 2.26 is unrealistic.In comparison, expressions for the variation of C with cylindersubmergence estimated fromtests involving a horizontal cylinder impacting calm water at a constant velocity have beengiven by Campbell and Weynberg (1980), and Miao (1988)respectively:=i+5sia+0275(51*’(2.27)C = 6.1 exp (-6.2 s/a) + 0.4 (2.28)The variation of the above expressions is also shown in Fig. 2.4. The expressions in Eqs. 2.27and 2.28 have not been corrected for buoyancy effects which were estimatedto be less than 5%of the total force during the initial high load phase of the slam. It is seen that the aboveexperimental curves for C predict a value ofC0which is larger than the generally acceptedvalue of it and which also exhibits a faster rate of decay with increasing submergencethan do theanalytical expressions.In an extensive review of several analytical methods to determine theadded mass of a cylindermoving near the free surface, Greenhow and Li (1987) present results forthe variation of C withcylinder submergence which are shown in Fig. 1.3. They emphasize that the assumptions24entailed in the derivation of Eq. 2.16 become unreliable after the initial stages of impact (s/a 1)due to the possibility of local particle accelerations no longer being large compared withg andsignificant displacement of the free surface from z = 0 for high entry speeds. The methodswhich use corrections to account for the deformation of the free surface during impact, and theexact shape of the wetted cylinder, yield different estimates of the peak value C0 ranging from itto 2it.In the most recent analytical treatment of the impact problem, Cointe and Armand (1987) usethe method of matched asymptotic expansions to derive a second-order solution for theslamming force on a cylinder. They justify the wetting correction proposed by Wagner (1931)which modifies the first-order estimate ofCsfrom it to 2it.As mentioned in Chapter 1, experiments have indicated a finite rise-time during which theslamming force increases from zero to a peak value, though the theoretical models forCimplythat the peak force occurs instantaneously. The compressibility of the air between the cylinderand the water surface, aeration, and compressibility of waterare some of the factors which caninfluence the rise time of the slamming force. The presence of air bubblesalso decreases theeffective density of the water, and the corresponding velocity of sound in water, whichin turnmay lead to reduced hydrodynamic pressure in vicinity of the cylinder anda smaller slammingforce. In the absence of a suitable model to account for these effects, the peak impactforce onan infinitely rigid cylinder entering an incompressible fluid is assumed to havea zero rise-timesince this does not affect the magnitude of the transmitted force.It is not possible to use Eq. 2.22 in the case of cylinder exit, i.e. when the water surface isreceding from the cylinder. Greenhow (1988) analyzed the cylinder entry and exit problems, andpresented results from a numerical model based on potential flow theory with exact nonlinearfree surface boundary conditions. He observed that while forces predicted by conventionalslamming theory showed fair agreement with his results in some cases, there was a ‘wide andfascinating’ diversity of free-surface flows for various values of Froude number and starting25positions of the cylinder. These results thus preclude the general use of added mass theory. Inthe case of cylinder exit, the simulations predicted the lifting of water above the cylinder and thesubsequent formation of thin layers; the draw-down and rush-up of the free surface beneath thecylinder, and the rush-up terminating in localized breaking (also seen in his experiments); andthe formation of large regions of strong negative pressure on the cylinder surface. A largeincrease in upward force on the cylinder due to the rush-up was predicted by one of thecalculations, and was also observed in pressure measurements in a corresponding experiment,though the author states that further experimental work is needed to confirm this effect.In the absence of an accepted theoretical model or sufficient experimentaldata, the force due toadded mass effects when the wave recedes from the cylinder has not been considered in thisstudy.2.2.3 Inertia ForceThe inertia force F1 acting on the cylinder is associated with the vertical fluid acceleration ‘and is expressed as:F(t)=p A1ii+ m ii (2.29)where m is the added mass of the partially submerged cylinder (Eq. 2.12 or 2.16). The first termin Eq. 2.29 is referred to as the Froude-Krylov force which corresponds to the inertia of the fluidvolume displaced by the cylinder, and the second term is the momentum component as derivedin Eq. 2.21. The inertia force can be written in terms of an inertia coefficientCmas:F1(t)= Cm p Ac (2.30)where C =A1/A+m/pAis the corresponding inertia coefficient based on the areaAratherthanA.On the basis of Eq. 2.16, the corresponding variation ofCm with immersion s/a based on mgiven by Taylor (i.e. Eq. 2.12) is shown in Fig. 2.5. At complete immersion,s/a = 2, the inertia26coefficient reaches a value of 1.645(2/6),and for a deeply submerged cylinder, s/a-+ oo,theinertia coefficient approaches a value of 2.0 corresponding to the case of an unbounded fluid.Cm is within 2%of this value for s/a 4.1. The value ofCm at s/a= 00is designated as C.2.2.4 Drag ForceThe fourth component in Eq. 2.3 is the drag force. For a completely submerged cylinder, thedrag force component of the Morison equation is given by:Fd(t) = CdpDw wI (2.31)whereCdis the drag coefficient and w is the vertical fluid velocity.The drag force acting during partial submergence is less well known, and corresponding valuesof the drag coefficient are not available. During the early stages of impact, flow separationshould not occur and the drag force rises from a value of zero at the instant of impact to themagnitude given by Eq. 2.31 after full submergence. It may be possible to extend this to the caseof partial submergence by utilising the instantaneous immersed width of the cylinder in place ofD, but the variation of the corresponding drag coefficient would still be unknown. It is notedthat Eq. 2.31 has the same form as the slamming force in Eq. 2.22, both being proportional to thesquare of the water particle velocity, and it is expected that these two components will undergo asmooth transition from one to the other during partial submergence, and from the partiallysubmerged stage to the fully submerged stage.The semi-von Karman method of determining the slamming coefficient (Greenhow and Li,1987) indicates that beyond s/a = 0.6,Cis approximately constant at a value of 0.7, which is therecommended value for s/a> 0.15. They justify this conclusion based on the agreement of thesemi-von Karman method with the experimental data of Campbell and Weynberg (1980).However, experimental data obtained by measuring the vertical force on a horizontal cylindersubjected to slamming cannot directly distinguish between the contributions of the slamming anddrag components during the stage of partial submergence (0 < s/a < 2). For this stage, it may27thus be useful to consider the drag and slamming coefficients in combination,Cd+Cs,ratherthan separately.2.2.5 Combination of Force ComponentsThere are a number of difficulties in attempting to superpose thefour force componentsdescribed above in order to describe the variation of the total force with time. Firstly,a suitablemodel for the slamming coefficient has to be selected. If present industry practice ofadoptingC50 = it is considered, Taylor’s model (Eq. 2.26) may be a possiblechoice. However, thevariation of C5 indicated in Fig. 2.4 is clearly unreasonable beyond the initial stagesof impact.Likewise, the theoretical variation of the inertia coefficientwith submergence indicates a valueof C = 2.0 for a fully submerged cylinder far from the free surface, which maynot beconsistent with values ofCmobserved in experimental studies. Furthermore, the variation of thedrag force due to partial submergence is also unclear, although it is expectedthat in combinationwith the slamming force, the variation is continuous.Certain options are possible for describing the force variation withtime and two such modelsare adopted here. Model I is based on the approach used by Isaacson and Subbiah (1990),whowere concerned only with the force maxima during any particular wave.They assume that thevariation of the vertical force from the instant of impact to the instantof complete submergenceis linear with time. This leads to considerable simplicity in theforce formulation since theslamming force contribution is evaluated only at the instant of impactusing Eq. 2.22 with aspecified value of C; and the Morison equation with constantdrag and inertia coefficients,together with the buoyancy force, is used during the fully submergedstage. The time variationof the vertical force over one wave is indicated as a solid line in Fig. 2.6(a) forthe cases ofpartial and complete submergence. As an aid to interpreting thebehaviour of the force, thefigures include horizontal and vertical lines which indicate thecylinder location andcorresponding instants of impact and complete submergence. This showshow the total verticalforce varies linearly with time from the maximum slanmilng forceat the instant of wave-cylinder28contact to the value at the instant of complete submergence. In a similar fashion, as the freesurface recedes from the cylinder, the total vertical force drops with a linear variation from thevalue at the instant of complete submergence to zero. Isaacson and Subbiah avoided the casewhen the cylinder is only partially submerged during the passage of the wave(h < H/2 < h + 2a),and it is appropriate here to extend their model to include this range also. In order to do so inareasonably simple manner, a corresponding approximation is made for this range, which leadstothe force variation shown in Fig. 2.6(b). The vertical forceat the instant of maximumsubmergence (‘ = H12), corresponds to inertia and buoyancy only. However, although themagnitude of the inertia force at maximum submergence is unclear, itis assumed that the inertiaforce at this instant can be determined from the specified value ofCmused in conjunction withthe submerged cross-sectional area of the cylinder. The vertical force isthen assumed to varylinearly from the instant of impact, where the force is due to slammingand is given by Eq. 2.22,to the instant of maximum submergence, where the force is made up ofbuoyancy and inertiacomponents only, and then to vary linearly again from this instantto zero at the instant when thecylinder is no longer in contact with the wave.An alternative approach, denoted here as Model II, attemptsto model the variation of thevertical force during the stage of partial submergence using modifiedexpressions for the inertia,drag and slamming force as indicated below. As mentionedin Section 2.2.4, it may be useful toconsider the combined slamming and drag forces in orderto account for the velocity-squaredbased force during partial submergence. This is achieved by assumingthat the sum,C+Cd,varies as Eq. 2.23 from the instant of impact to the pointat which this combined coefficientreaches a specified value ofCd.After this point, the combined coefficient remains constant atthe value ofCd until thewave recedes from the cylinder. A possible variation of C +Cd,henceforth designated asCs+d,is shown by the solid line in Fig. 2.7 for the case when C isdetermined using Taylor’s model (dashed line), andCd= 0.8. The division of this combinedforce into slamming and drag components need not be of concern,but for convenience one mayconsider this to account for C decreasing from its initial peak valueofC0 to Cd,whereupon itcontinues decreasing to zero and the drag coefficient thenbegins increasing from zero to the29specified value in such a way that the sum of the two coefficients remains constant. For typicalvalues ofCd (0.6 to1.0), this transition occurs near s/a = 1, when the cylinder is approximatelyhalf submerged, which might be expected, as the use ofCs basedon potential theory is thenprone to inaccuracy, while the drag force builds up as flow separation commences.The velocity-squared force is then evaluated using Eq. 2.22 withC+d used in place of C5. When the freesurface recedes from the cylinder, it is assumed that slanmiing force is absent and as the cylinderbecomes partially submerged, the drag force is computed by using the maximum immersedwidth of the cylinder in place of D in Eq. 2.31 together with the specified valueofCd.The inertia force is evaluated using Eq. 2.30. Instead of usinga constant value ofCmaftercomplete submergence as in Model I, the inertia coefficient is now assumedto vary withsubmergence as a function of the added mass, as indicated earlier. Eq.2.12 is used to obtainvalues of added mass during partial submergence. However, in orderto facilitate thecomputation of the dimensionless added mass rn/pa2 after complete submergence,and also inorder to incorporate a variation which approaches an empirical inertia coefficientother than thetheoretical value of 2.0, a three parameter exponential curve has been adoptedto describe thevariation of inertia coefficient with s/a. The dimensionless added massvariation after fullsubmergence is expressed as:m s s= - 2exp(-2)2cc(2.32)pawhere 2q,2and 23 are parameters which are chosen such that the inertia coefficientCm (1 +m/pA) reaches a specified value Cmo at s/a= cc,and the variation of added mass and itsgradient with respect to s/a are continuous at s/a = 2, on the basis of Eq. 2.12.The parametersmay thereby be derived as:= it(C - 1)= it(C- 2/6)exp(-23) (2.33)(n2/9 - 2/3)=(Cmo30In the particular case of potential theory whenCmo= 2, the parameters ?i, 2and areapproximately it, 12.57 and 1.21 respectively. Fig. 2.5 shows the predicted variation ofCmwiths/a as dashed lines for cases corresponding toCmo = 1.7 and 2.0.The above expressions for the variation ofCm after full submergence are valid forCmo 1.645. If a simulation requires that the value of Cmo be less than 1.645, the model uses anapproach similar to that in the case of the variation ofCs÷d with increasing submergence. Oncethe value ofCm predicted on the basis of Eq. 2.12 reaches the value of Cmo during partialsubmergence, it is held constant at that value with increasing submergence. A similar method isused when the free surface recedes from the cylinder.2.3 Hydrodynamic Force on an Elastically SupportedHorizontal CylinderThe hydrodynamic vertical force on a rigid cylinder subjected to wave slamming has beenformulated in Section 2.2. The expressions for the various force components have been derivedon the assumption that the cylinder and its support structure do not respond to the applied waveforce. In the case of an elastically supported cylinder subjected to wave slamming, the dynamicresponse alters the magnitude of the applied force since the fluid particle kinematics relative tothe cylinder are modified due to the cylinder vibration. The force transmitted by the cylinder toits supports is also affected due to its dynamic response and this force maybe higher or lowerthan the applied force, depending on the length of the slamming impulse and the naturalfrequency of the cylinder. The governing variables and the modified numerical model for waveslamming on an elastically supported cylinder are examined in the following sections.2.3.1 Response of an SDOF System to Impact LoadingThe cylinder and its support structure can be modelled as a single-degree-of-freedom (SDOF)system with a mass M, a stiffness K and a damping coefficient C. The schematic representation31of such a system is shown in Fig. 2.8. The equation of motion of this system when subjected to atime varying force F(t) is given by:M + C i + K r = F(t) (2.34)where r, i and I are the time varying cylinder displacement, velocityand accelerationrespectively. The natural frequency of this system, denoted byo and the damping coefficient Ccan be expressed in terms of a non-dimensional damping ratio denoted as:rk=C= C(2.35)2McoThe applied force F(t) in Eq. 2.34 has the same general form as given in Eq.2.3. In the presentstudy, it is assumed that the force transducer in the support structure determines the dynamiccharacteristics of the cylinder. If the applied force were to be measuredby such a configuration,the measured force which is also termed as the transmitted force is given by:Ft(t) = K r(t) (2.36)In cases where the damping force component is also assumed to bea part of the transmittedforce, the right-hand side of Eq. 2.36 is given by Kr(t) + Ci(t). Themeasured force is hence afunction of the natural frequency and damping of the system. For the case of an applied loadF(t), the response of an SDOF system starting from rest is givenby:r(t) =fF(t) exp[-C(t - t)]sin[wd(t - t)Jdt (2.37)Od0whereodis the damped natural frequency of the system and is given by oj\J 1-C2and t is adummy variable of integration.32Let us consider the idealized case of an impact load F(t) which increases linearly from 0 to apeak value F0 in a rise-time Tr and then drops linearly to 0 in a decay-timeTd as shown inFig. 2.9. During the stage when the force is increasing, F(t) can be expressed as:F(t)=0 t Tr [Stage A] (2.38)The response of the system to this load can be determined using Eq. 2.37. If the system isinitially at rest, the response during the stage when the force is increasing (0 tTr), isgivenby:r(t) =j-[t — + exp(-Ccot)cos( Ojt)+ (2C2— 1)sin(odt)cOn O)COd(t) = — exp(-CcOt) (cos(okIt)+csin(codt)J](2.39)When the force is decreasing (Stage B), the system response can be evaluated in terms oft’ = t- Tr,since that simplifies the expression for the applied force to:F(t) = F0 [1f]TrtTr + Td [Stage B] (2.40)The response in Stage B can now be given as the sum of the the free vibration due tor and (att = Tr) from Eqs. 2.39, designated as r0 and respectively and the forced response due to theapplied force in Eq. 2.40. The displacement r(C) in Stage B is hence:( (F0+ C0)nro) .r(t’) = exp(- Ccot’)r0 cos( COdt’) + srn((Odt’)I+COd)KTd[+ TdJ (1 — exp (-Co)nt’) COs(COdt’)) — t +(1—2C2—CcOnTd)exp (-wC)sin(cOdt’)] (2.41)33The motion of the system after the force has dropped to zero, shown as Stage C in Fig. 2.9, canbe evaluated in a manner similar to that for Stage B, wherein the system displacement andvelocity at the end of Stage B (at t’= Td) are used to determine the free vibration response inStage C.It can be seen from Eqs. 2.39 and 2.41, that the transmitted forceF(t) given by Eq. 2.36 is afunction of the dynamic characteristics of the system and that its magnitude at any instant maynot be the same as that of the applied force. Since the initial stages of the wave slamming forceare similar to the impact force analyzed above, it is useful to compare the transmitted force(which would be measured by a typical force transducer) with the actual applied force. The twomain parameters of interest are the measured peak forceFt0immediately after impact and thecorresponding rise-timeT.The ideal measuring system should report the same force as theapplied force at every instant and an examination of Eqs. 2.39 and 2.41 indicates that such ameasuring system would have a natural frequency o such that oT is very large. The effect ofthe damping ratio on the measured force using systems with high natural frequency is notsignificant, especially for typical damping values between 5% and 10%.As mentioned earlier, measurements indicate that there is a finite rise-time during which theslamming force increases from zero to a maximum value. Sarpkaya (1978) showed thatthe rise-time and the natural frequency of an elastic cylinder can cause the apparent maximumvalue ofCs to vary between 0.5it and 1.7n. In order to examine the influence of the dynamiccharacteristics of the system and the rise-time of the slamming force on thetransmitted (ormeasured) forceF, two idealized types of impact force are considered. In the case of the firstimpact, the rise-timeTr is the same as the decay time Td, and in the second case, Td = 2 Tr.Fig. 2.10(a) illustrates both load cases and the corresponding force measured by a system withTr/Tn = 0.2 (where the natural period T=2itko) and = 0.05. Fig. 2.10(b) shows the resultsfor the case whenTrITn = 2.0. It is observed that there is a significant difference between theapplied and recorded force for the case where T1.iT = 0.2. The measured force maximum ishigher than the applied force peak due to dynamic amplification and the apparent rise-time to34reach the force maximum is larger than Tr. Obviously this is not a desirable configuration for anexperimental study of impact force. For the case whereTrITn = 2.0, Fig. 2.10(b) shows that themeasured force is a better estimate of the applied force, though there are differences due to theoscillatory nature of the response.The effect of the variation ofTriTnfor various values of (0, 0.02, 0.05) on the dynamicmagnification factorF0fF and the relative rise-timeTtiTrare summarized in Fig. 2.11 for bothaforementioned loading cases. The desirable properties of a measurement system for recordingimpact forces can be interpreted from these figures. It is observed in Fig. 2.11(a) that forTrIl’smaller than 0.1, FfF0is less than 0.6 and decreases toO asTr/Tnapproaches 0. This is becausea system that has a large natural period in comparison to the rise and decay time of the impactforce does not have adequate time to ‘see’ the force. Consequently the response is smaller thanthat for a static load and the measured force is lower than the applied force. It is seen that F/F0approaches a value of 1 in an oscillatory manner asTr/Tnincreases beyond 1. Hence if a systemis chosen such that T/f is at least 2 and preferably greater than 3, the measured force maximumwill be a fairly good estimate of the applied force maximum. A similar requirement is evident inFig. 2.11(b) which shows that the relative rise-timeTtiTr also approaches a value of 1 as Tr/Tnbecomes large. It is also observed that the effect of damping on the measured peak force andrise-time diminishes with increasing values ofTrITn.Figures 2.11(c) and (d) illustrate the variation ofF0fFandTtITr for the case where Td = 2Tr.The trends are seen to be similar to those in Figs. 2.11(a) and (b) respectively. Although theduration of the impact is longer in this case, the amplification factorF0/F is seen to increasemarginally by about 7% atTr/Tn 0.5. Consequently, a method which corrects the measuredforce for the effects of dynamic amplification, based on the assumption that the slamming forcecomponent can be represented as an impact whereTd = Trwill be reasonably accurate even if theactual decay timeTdis 100% larger than its assumed value.352.3.2 Cylinder Response to Wave Impact LoadingThe slamming, inertia and drag forces on the cylinder are functions of the water particlevelocity and acceleration. In the case of a cylinder which responds dynamically to the appliedload, the nature of the loading changes since the motion of the cylinder modifiesthe relativevelocity and acceleration between the water and the cylinder. The problem hasto be solvediteratively, since the response at any instant is a function of the applied force whichin turn is anonlinear function of the relative water particle velocity (and thus a nonlinear function of theresponse). A numerical model which accounts for the dynamic behaviour of the cylinder andexpresses the applied hydrodynamic load as a function of the relative water particle kinematics isdescribed.A definition sketch for the case of an elastically supported horizontal cylindersubjected towave slamming is shown in Fig. 2.12. The equilibrium heightof the cylinder axis from the stillwater level (SWL) is given by h. The cylinder displacement from its equilibriumposition due toan applied force is denoted by r. Both h and r are measured positive upwards. If the cylinder issubjected to hydrodynamic forces due to an incident wave train, the equation of motion of thecylinder is given by:M + C i + K r = L(Fb + F+ F1+ Fd) (2.42)where the terms on the left side of Eq. 2.42 are as in Section 2.3.1,the 4 force terms on the rightside have been described for the case of a rigid cylinder in Sections 2.2.1 - 2.2.4,and L is thelength of the cylinder. For the case of a dynamically responding cylinder, thebuoyancy force Fbis given by Eq. 2.5 where the cylinder submergence:s=r - (h+r)(2.43)Since the cylinder is located near the still water level, the following relationsfor the appliedforce are based on the assumption that the vertical components of water particlevelocity andacceleration are given by ij and i1 respectively (see Section 2.5.1). Equation2.9 can be rewritten36to account for the relative water particle velocity and hence the force per unit length of thecylinder due to the rate of change of fluid momentum is given by:F(t) = [m (i - F)] (2.44)atEquation 2.44 can be expanded in a form similar to Eq. 2.21:Fm(t)= m (i- )+ (i - F) (2.45)as atwhich can be further simplified to:Fm(t) = m(ii- )+(iF)2(2.46)asThe first term in Eq. 2.46 is the momentum component of the inertiaforce and the second term isthe slamming force which can be written in terms of the slamming coefficientCsas:F(t) = CpD (i - F)2 (2.47)Equation 2.29 for the inertia force per unit length is rewrittento account for the relativeacceleration:F(t)=p A1fi+ m (ij-i) (2.48)The drag force is similarly given by:Fd(t) = Cd p D (i - F) I - F I (2.49)Equation 2.42 can now be expressed as:M + CF + Kr = L [Fb+cSpDF)2+ pAi + m(i-)+ CdpDO-i)l1-FI] (2.50)37As described in Section 2.2.5, Model II combines the hydrodynamic force due to the slammingand drag components into a single force that is proportional to the squareof the relative velocityand a combined coefficientCs÷d.Equation 2.50 can be rewritten for Model II as:r tD2..(M+ml)r + Cr + Kr = L[Fb+ CmP11+ CS+dpD(1-F)I1-H] (2.51)The variations of the coefficientsCm and Cjhave been discussed in Section 2.2.5.Equation 2.51 is nonlinear and can be solved in the time domain using any suitable numericaltechnique. The Newmark iteration method with a linear accelerationformulation has been usedin this study. The solution begins at a time step when the water surfaceis below the initiallystationary cylinder, since the applied force and the responseare known to be zero. Theacceleration 1 at the next time step is assumed to be thesame as that at the previous time step(i.e.t+t = t)and this value is used in the first step of the iteration to obtain the velocity i anddisplacement r at t+At:=+tAt +‘yAt(t÷,.t - t)rt+At = rt+ tAt + (At)2[ +(t+& - )](2.52)where At is the time step size andyand f3 are coefficients which have the values 0.5 andrespectively for the linear acceleration method. Sincethe values of, ij andT1are known attime N-At, the right hand side of Eq. 2.51 can now be evaluated andis designated as Arevised estimate fori+ iscalculated using Eq. 2.51:1rt+t= (M + m 1)[F+t- Crt÷t- Krt+t1(2.53)This value of1t+Atis used in Eq. 2.52 to obtain more accurate values of andrt÷t and theiteration continues until the value of calculated by Eq. 2.53 converges.38The choice of zt is very important in order to run a simulation and interpret results correctly.The time interval chosen should be small enough such that all quantitieswhich are functions oftime are accurately modelled (i.e. frequency components in the inputwave force and in theoutput response oscillation are represented without aliasing) andthe Newmark iteration processconverges, and large enough to avoid lengthy computationtimes and redundant data. Thecriteria used to select a suitable value of zt are discussed inSection 2.5.3.In the case of a cylinder which does not respond to an applied force, the rise-timeof the impactforce has not been considered in the analysis since itdoes not influence the magnitude of thetransmitted force. It has been shown in Section 2.3.1that in the case of a compliant cylinder, thetransmitted force is a function of the cylinder’s naturalfrequency and the rise-time of the appliedforce. Consequently, the assumption that the slammingcoefficient reaches a peak value of C(=it or higher value depending on the choice of model forC) atthe instant of impact may leadto unrealistic predictions of the transmitted force. A numerical modelwhich can simulatevarying impact force rise-times allows fora more realistic simulation of the wave slammingforce and consequent dynamic response of compliantcylinders. There is no explicit formulationfor the force rise-time in Eq. 2.50 and unlike thecase of a rigid cylinder wherein it is possible tosimulate a linear variation of force using interpolationbetween known force magnitudes at anytwo points in time, the hydrodynamic force ona vibrating cylinder is not known beyond onetime step due to the interdependent natureof the force and cylinder response.However, it is known that the dominant contributionto the total force on the cylinder duringthe initial stages of impact is due to slamming.Consequently a pseudo-linear (since thebuoyancy and inertia force components add a little nonlinearityin the early stages of immersionand the relative velocity changes slightly during therise-time) increase of the applied impactforce can be introduced by modifying the coefficientCs-i-’i,such that it increases linearly from 0at the instant of impact to the peak value of C overa specified rise-timeTr:Cs+d = C(t-t)tj t tj+ Tr (2.54)39where t1 is the instant of impact. After t = ti +Tr, Cs+dvaries in the same manner as it would fora rigid cylinder, wherein the submergence s/a is assumed to increase from0 after t = Tr. Thevalue of Trchosen for a simulation should be such thatCs+dreaches its peak value of C0 beforesignificant immersion of the cylinder (s/a 0.4) and is typically between5 and 30 msec.2.3.3 Modelling Slamming as an ImpulseSections 2.3.1 and 2.3.2 have shown possible techniques of numericallymodelling waveslamming on a horizontal cylinder which respondsto the rapidly varying load, and ofdetermining effects such as the dynamic amplification of theapplied force. This treatmentrequires detailed information on the characteristics of the slammingforce variation with timeduring the early stages of impact. However, past experimentalevidence has proven that it isquite difficult to obtain reliable estimates of the slamming coefficientdue to several factors suchas air entrainment, cylinder inclination, and dynamic response. Thesecause appreciable scatterin the peak value of the slanmiing coefficient reportedby several investigators.The time scale of the slamming force on horizontalmembers of offshore platforms is in therange of 0.01 - 0.2 sec. If the natural periodof such a member is in the range 0.5 - 1 sec, themaximum response of the member is usually governedby the magnitude of the slammingimpulse, rather than the precise form of the applied loaditself. Hence, as an alternative to theconventional approach involving the slamming coefficient,it may be more convenient to definethe slamming event using an impulse coefficient which combines theimpact force and rise-timeinto a single dimensionless quantity.A typical time history of the measured slamming forceis sketched in Fig. 2.13. As indicatedin the figure, the force F may be considered to bemade up of two components: an impulsiveforce F, which rises rapidly to a maximum and then fallsto zero over a duration Ti; and aresidual forceFr which may be taken to increase steadily over this duration, starting from zero,and which is associated approximately with drag,inertia and buoyancy components. Forconvenience, T1 may be taken to correspondto the instant at which the overall force F reaches a40minimum as indicated in the figure. The measured force record using a test cylinder, usuallyincludes oscillations at the natural frequency of the cylinder. In such a case, the minimum forcewhich is used to define the end of the impulse should be identified after the record is first filteredso as to remove these oscillations, since this would correspond more closely to the applied force.On the basis of the above representation, the impulse I on the cylinder, which is associated withthe impulsive force F1, may be written as:T1= fFdt0T11= JFdt — TF(T) (2.55)0The second term on the right-hand side of Eq. 2.55 corresponds to the area of the shaded triangleshown in Fig. 2.13. An impulse coefficient may now be defined as:C1=(2.56)pa2wLwhere w is the vertical water particle velocity at the instant of impact, and L is the length of thecylinder.Since it is assumed that the impulsive force F is due to the slamming force discussed inSection 2.2.2, it is possible to estimate a theoretical impulse coefficient from the closed-formexpression forCs. It is seenin Figs. 1.3 and 2.4 that the more plausible models derived for thevariation ofCreach a minimum value by the stage when the cylinder has been submerged byhalf, i.e. s/a = 1. As discussed in Section 2.2.5, this is consistent with the notion thatconventional added mass theory can provide a fair estimate of the slammingcoefficient onlyduring the early stages of impact. As the submergence increases, flow separation effects begin toinfluence the hycirodynamic force, and it is difficult to separate the drag force component fromthe impulsive force. Hence it is assumed that the theoretical impulse coefficient can be41estimated between the limits of 0 s/a 1. The theoretical impulse is expressed in a mannersimilar to Eq. 2.55:T11I= $O.5CpDw2Ldt = pa2wLJC(sIa) d(sla) (2.57)The theoretical impulse coefficient corresponds to the area under the C curvebetween s/a = 0and 1. The expressions for C given in Section 2.2.2 have been integrated to yieldthe followingvalues of C:C1 = 1.571 von Karman (based on Eq. 2.24)C1 = 3.833 Wagner (based on Eq. 2.25)C1 = 1.57 1 Taylor (based on Eq. 2.26)These can be compared to C1 obtained from experimentally derived expressionsforC:C = 1.412 Campbell and Weynberg (based on Eq. 2.27)C1 = 1.382 Miao (based on Eq. 2.28)It is important to note that the above definition of C is based on the slammingforce which is afunction of the square of the velocity. This must be kept inmind when interpreting the abovevalues of C derived from experimental observations which include buoyancyeffects. Thecontribution of buoyancy to the impulse, denotedlb.is a function of the square of the Froudenumber, and is given by:=aiL(2.58)This indicates a decrease of 0.1 to 0.2 in the C values derived from theexpressions given byCampbell and Wenberg (1980) and Miao (1988), leading to an averageestimate of aboutC1 = 1.2.42The response r of an SDOF system with mass M and natural period Tto an impulse I ofduration T (<<Ta) is given as (e.g. Clough and Penzien, 1975):r=sin (oat’)(2.59)Mcowhere o = 2it/T and C is time measured after the end of the impulse.Thus the maximumresponse due to the impulse I is given as JIM co and this occurs at timeC = TI4. In order toobtain the overall maximum response, the response at this instant due to theresidual forceFrisalso required. However, such a calculation may itself require a time-steppingprocedure in orderto account for the dynamic response of the cylinder. Once the two componentresponses havebeen obtained, they may then be summed to obtain the overall maximumresponse. This can inturn be used to determine quantities such as the maximumtransmitted force or stress.It is expected that the impulse coefficient derived from experimental observationswill exhibitless scatter since it is not dependent on a single observationof the peak force. However, theapplicability of the impulse coefficient in the context of engineeringdesign is confined tosituations where the impulse duration T is reasonably small in relation tothe natural period Tof the cylinder (e.g. a brace), such that the dynamic responseand stresses may be determined bythe impulse magnitude rather than the precise timehistory of the loading.2.4 Slamming Force on an InclinedCylinderThe previous sections have described the specific caseof a horizontal cylinder subjected towave impact. In general, the member’s axis or thewater surface may not be exactly horizontalwhich leads to an oblique impact. The flow in thissituation is no longer two-dimensional and ismore complex. Miller (1977) modelled this case usinga strip approach wherein the slammingforce is determined by adding the contributions of severalincremental slanuning loads on thinelements of the cylinder. The definition sketch for inclinedimpact is shown in Fig. 2.14. Acylinder of length L is inclined at an angle e to the still waterlevel and its axis is oriented normal43to the direction of wave propagation. The height of the lower edge of the cylinder from the stillwater level is denoted by h.When the cylinder is subjected to an incident wave train, the initial contact is made at the loweredge and as the water level rises, its submerged length increases from 0 to L. This leads to amore gradual increase in the hydrodynamic force in comparison to the case of a horizontalcylinder. The submerged length of the cylinder is given by:1for01(h+Lsine)= sine(o°<e<9o°) (2.60)L for1> (h + L sine)The submergence normal to the cylinder axis varies along the submerged length and is expressedas:s(y) = (L-y)tane= -h-ysine(0yL)(2.61)cos ewhere y is measured from the lower edge of the cylinder.The force on the cylinder is determined by analysing the force on a strip of width dysubmerged by an amount s(y) (see Fig. 2.14). The buoyancy force component normal to thecylinder axis on the strip is:dFb=p gAi(s) cosO dy (2.62)where At(s) is given by Eq. 2.6. The water particle velocity and acceleration normal to thecylinder axis are given by wcose and *cos e respectively. The slamniing force on the strip canbe expressed as:dF = C(s) pD (w cosO)2dy (2.63)whereCs(s) is the slamming coefficient. Similarly, the expression for the inertia force on thestrip is given by:44D2dF= Cm(S) pn-4--(’’ cose) dy (2.64)whereCm(s)is the inertia coefficient. The elemental drag force may be expressed as:dFd Cd(s) pD (w cos9)2dy (2.65)The four force components on the strip can be integrated along the submerged length of thecylinder to yield the total force at any instant:F=dF+ dF + dF + dFd (2.66)Eq. 2.60 can be rewritten in a summation form to yield:F = (PAi(s)coso +[C+d(s)]pD(wcos8)2+ Cm(5)P(.uiCOSO)) (2.67)where N is the number of strips into which the lengthL isdivided, the submergence s isdetermined from Eq. 2.61, and y=(jL)JN. Thevariations of the coefficientsCs+dandCmwithsubmergence have been discussed in Section 2.2.5.Wave impact on an inclined cylinder results in a smaller peakforce and longer rise-time incomparison to the case where the cylinder is horizontal. The effect ofthe longer rise-time isinconsequential for rigid cylinders, but may actually lead to a largerdynamic response in thecase of a compliant cylinder if its natural period is comparable to the rise-time asseen in Section2.3.1. Hence it may be necessary for a designer to determine whetherthe decrease in peakimpact load due to the inclination of a cylinder has been offsetby increased dynamicamplification caused by a corresponding increase in the rise-time.45It is noted that in reality, wave impact on an inclined cylinder results in a three-dimensionalflow problem which is not being solved here. The flow in the vicinity of the cylinder, and theassociated hydrodynamic force change significantly with increase in cylinder inclination, and thestrip method described above is not expected to result in good estimates of the force for values of0 greater than200.2.5 Water Particle Kinematics in WavesThe hydrodynamic force due to wave impact has been expressed in terms of the water particlevelocity and acceleration at the free surface during the early stages of impact, and at the cylinderaxis after complete submergence of the cylinder. Linear wave theory is used to determine thefree surface elevation r and the corresponding water particle kinematics.2.5.1 Regular WavesFor a regular wave train the water surface elevation1at the cylinder location is given by:H=cos ot (2.68)where H is the wave height, o is the wave angular frequency(=2it/T where T is the waveperiod) and t is time. The linear dispersion relation, which relates w and the wave number k, isgiven by:o2= gk tanh(kd) (2.69)where d is the water depth.The hydrodynamic force in the vertical direction is associated with the vertical water particlevelocity and acceleration. On the basis of linear wave theory, these may be expressed as:w=-wG(z)sinCot1(2.70)Hw = - - G(z) cos Cot46whereG1— sinh[k(d+z)j(2 71)— sinh[kd]and z is measured upwards from the still water level. In the present case, the value of zcorresponds to the vertical location of the cylinder axis. Since Eqs. 2.70 are valid only for zbetween -d and 0, several methods have been described to evaluate the water particle kinematicsin the vicinity of the free surface based on alternative extensions to the expressions given bylinear wave theory. Chakrabarti (1988) has summarized various modifications to Eqs. 2.70 inorder to extrapolate the evaluation of water particle kinematics above the still water level. Forthe present investigation, the kinematics above the still water level are taken as the values at thestill water level itself. This assumption is valid for small amplitude waves since elevationsrelative to the still water level of the order of wave height are considered and thus it is reasonableto assume that G(z) 1 within the precision of linear wave theory. Hence w and i are replacedby i andflrespectively.2.5.2 Effects of Free Surface SlopeThe development of the slamming force which has been described in the previous sections isstrictly valid only for a horizontal free surface. If instead the free surface is assumed to impactthe cylinder at an angle 6 to the horizontal as shown in Fig. 2.15, the waterparticle velocitynormal to the free surface is v = cos 6, and the vertical component of the slaniming force perunit length is then given by:12F=CpD v cosö=CpDi2cos3ö (2.72)On the basis of a wave train which is assumedto propagate steadily, it can be shown that theangle 6 is a function of the vertical velocity of thefree surface:47= -tan6=--i (2.73)where c is the wave celerity and x is measured along the direction of wave propagation.Suitable corrections to the slamming and impulse coefficients obtained from measurements ofthe vertical force can now be developed by applying the result for tanö in Eq. 2.73 to Eq. 2.72.On this basis, it may be shown that the slope correction factor for the slamming coefficient isgiven by:E (213/2i;=L’JJ(2.74)where C is a slamming coefficient derived on the basis of v instead ofi. A similar correctionfactor can be obtained for the impulse coefficient in Eq. 2.56 to provide:= 1+(2.75)Figure 2.16 illustrates the behaviour of these correction factorsfor the case of the steepest non-breaking wave used in the experiments which has a period T = 1.1sec, and height H = 17 cm.Only the portion of the wave record varying from the lowestpoint in the wave trough to thehighest point in the wave crest is shown since the correction factors are applicable for a risingwater surface which causes slamming forces. It is seen that the factors deviate significantly fromthe value of 1.0 in the zero-crossing zone of the wave where i reachesa maximum. Themaximum value of C/Cs is about 1.17 and that of CIC1 is 1.11.It has been pointed out earlier that the slamming process is governedby a variety of factors,many of which are difficult to control, and which accountfor considerable scatter inexperimental data. Because of this, it may well turn out that the abovecorrections for the freesurface slope effects in general may be an unnecessary refinement.482.5.3 Random WavesThe extension of the numerical simulation to the case of random waves described by aspecified spectrum may also be made. The two cases considered here correspond to a narrow-band spectrum and the two parameter Pierson-Moskowitz spectrum.The form of the narrow-band spectrum adopted here is given by:=cos[ (f - fo)] for f0 - f f0 +(2.76)0 otherwisewhereH is the significantwave height, f0 is the peak frequency, andfrrepresents the frequencyrange over which the spectrum is non-zero.The two-parameter Pierson-Moskowitz spectrum is instead given as:S1(f)=exp[...f4](2.77)For either spectrum, a time record of the free surface elevation may be represented as asuperposition of regular wave trains with different amplitudes, frequencies and phases:NfT1(t) =A1 cos (2nft- )(2.78)i=1where A, f and e are respectively the amplitude, frequency and phaseof the i-th regular wavecomponent, and Nf is the number of frequencies used in generating the random wave record.The randomness in‘rI(t)can be incorporated in various ways. The method adopted here,known as the deterministic spectral amplitude model (Tuah and Hudspeth,1982), or the randomphase spectrum method (Funke and Mansard, 1984), assumes that the the phases E in Eq. 2.78are random and uniformly distributed between 0 and 2n, while the amplitudesA1 are determinedfrom the discretized spectral density as:A= .J 2S1(f) zS.f (i = 1 to Nf) (2.79)49The spectrum of the resulting time series of ii will then exactly match the target spectrum.Analogous to Eqs. 2.70, expressions for the vertical velocity w and acceleration i for the caseof a random wave train are given respectively by:Nfw = = - 2icf A1 sin (2itft-ej)i=1(2.80)Nf* =1 = -(2itf A cos (2itft-1=1There are a number of interrelated parameters which must be selected in order to carry out anumerical simulation of,f, andfl.These include the frequency interval zS.f, the number offrequency values Nf, the time step size the duration of the wave record T, and the numberof data points in the wave record N. Relations between these parameters are:Af = (2.81)AT= 1(2.82)2N fAfN = 1 (2.83)Eq. 2.82 represents the Nyquist criterion which relates the maximum frequency NfAf that can beresolved from a signal which is discretized with a time intervalAT.Tmust be sufficiently large to ensure that the number of waves in the record is sufficient forcarrying out a probability distribution analysis; and Af is determined directly from Eq. 2.81. Inthe case of a narrow-band spectrum, the frequency range of the spectrumfralso needs to beselected and may be related to the spectral width parameter of the wave record. The number ofdiscretized frequencies with non-zero coefficients is approximatelyfr/At502.6 Computational ConsiderationsThe numerical simulation of slamming force deals with an event which occurs over very smalltime scales in comparison to the wave period. The time step size zt between successive valuesof the computed vertical force should be chosen so as to represent accurately the variation of theforce with time and simultaneously minimize the computational effort required. Although thefrequency components in a typical wave train giving rise to the vertical force may be relativelylow, there is a very rapid variation of the vertical force in every wave cycle associated with theonset of slamming. This corresponds to the presence of significant high frequency componentsin the force record. The choice of a particular time step size depends in part on the specificproblem under consideration i.e. whether the cylinder is rigid or compliant, the cylinder ishorizontal or inclined and whether the waves are regular or random.In the case of regular waves interacting with a rigid horizontal cylinder, the choiceofzS1 tobeused in the simulation is influenced mainly by the need to model the rapid variation in verticalforce immediately after the instant of impact. The maximum slamming force isassumed here tooccur instantaneously, so that it is important that the instant of slammingbe present in thesample values since the vertical force reaches a critical value at this point. The value ofzt isdetermined so as to ensure that the partially submerged stage of thecylinder is adequatelyrepresented in the simulation. Specifically, the difference in waveelevations between twosuccessive time steps in the simulation when the free surface is travellingthe fastest, is taken as0. 1D. Thus for deep water waves:= ftJ(2.84)Typically this choice leads to between 50 and 2500 time steps in each waveperiod. The instantof impact can be determined by linear interpolation between successive time stepsfor which thecylinder is above the water surface at the former instant and partially submergedat the latter.The hydrodynamic force on the cylinder is then computed at this time step also.51A random wave record synthesized using the Nyquist criterion of Eq. 2.82 may not havesufficient resolution to determine the instant of slamming using linear interpolation described forregular waves, since in cases where the cylinder diameter is appreciably smaller than thesignificant wave height, it is possible that successive values ofrimight correspond to nosubmergence and complete submergence respectively. In order to treat such occurrences, andsince a smaller zt interval is required only for periods of partial submergence, a cubic splineinterpolation technique is used to introduce additional sample points within the correspondingtime interval. The cubic spline method preserves rapid changes in slope especially in thevelocity and acceleration records. Once values at smaller intervals have been obtained, linearinterpolation is used to obtain the instants at which slamming occurs.When a computation of the dynamic response of the cylinder to the wave force is required, thefactors that govern the choice ofAtare the frequency components in the incident wave train, thenatural frequency of the system, the Newmark iteration convergence criterion and the frequencycomponents in the hydrodynamic force. The chosen value ofztis the smallest obtained from aconsideration of these four factors and is usually governed by the need to correctly represent therapid variation of the hydrodynamic force immediately after impact.Unlike the case of the rigidcylinder, the technique of interpolation to determine the exact instantof slamming cannot beapplied since the position of the cylinder is not fixed. The value of At should be sufficientlysmall so that successive samples when the cylinder is dry and thenpartly submerged are suchthat s/a 0 for the latter sample and the slamming force peak that occursimmediately afterwater-cylinder contact is not underestimated by a large amount. In addition,if the slammingforce is assumed to reach its peak value instantaneously, the value ofAt used in the simulationwill imply a rise-time of Hence a zero rise-time conditioncannot be simulated in the casewhen the dynamic response of the cylinder is considered.It has however been shown in Section 2.3.1 that a zero (or near zero)rise-time condition for atypical slamming event will not produce significantdynamic response implying that themagnitude of the dynamic stresses will be low andnot be of interest in design situations. For52typical rise-times between 5 and 25 msec, a value ofLS.tsthat ensures at least 2 samples within therise-time duration is adequate to accurately compute the dynamic response. The simulation isconducted at equally spaced points in time in contrast to the rigid cylinder case wherein the ztvalue varies depending on the stage of the cylinder submergence.The above techniques are applied in a similar way for the case of the inclined cylinder, whereonly the rigid cylinder case has been examined.53Chapter 3EXPERIMENTAL STUDYA significant component of this study on wave slamming relates to an experimentalinvestigation in which an instrumented cylinder has been subjected to slamming forces in non-breaking and breaking waves. The horizontal test cylinder was fixed at different elevations nearthe still water level and the vertical force was measured for incident regular waves of variousheights and periods. The same test set-up was also used to measure the horizontal and verticalimpact forces due to a plunging wave which could be generated to break at different locationsalong the wave flume. The tests have been conducted in the wave flume of the HydraulicsLaboratory of the Department of Civil Engineering at the University of British Columbia. Thisflume is equipped with a wave generator which is capable of producing regular and randomwaves.The following sections describe the equipment and methods used in the experimental set-up forwave generation and data acquisition. The techniques of signal conditioning and data analysisused to derive pertinent information from the measurements are also discussed.3.1 Test FacilitiesThe apparatus used for the experimental study of the slamming force can be classified underthree separate functional headings. These are: the wave generation equipment, the instrumentedcylinder which is subjected to wave slamming, and the equipment used for data acquisition.543.1.1 Wave FlumeA photograph of the wave flume in which the experiments have been conducted is shown inFig. 3.1. The flume is 40 m long, with a 15 m long test section, 0.62 m wide,and operates at anominal water depth of 0.55 m. The flume has a plywood beach of 7.7 m lengthset at a slope of1:14, and which is covered by a mat of synthetic hair. A large holding tankat the end of thebeach also helps to absorb and dissipate incident wave energy. Thereflection of wave energyfrom this beach is quite low, although it has not been measured in this study.3.1.2 Wave GenerationA photograph of the wave generator is shown in Fig. 3.2. The generatorconsists of anelectrically powered servo-motor which drives a linearactuator connected to a hinged wavepaddle. The rotation of the servo-motor shaft is converted to a translatory motionof a thrust rodby the linear actuator which has a usable operating stroke of ± 25 cm. The wavepaddle can beconfigured to operate in three modes, namely a hinged flapper,a combination flapper-piston anda piston articulation. These enable the reproduction of velocity profiles rangingfrom shallow todeep water waves. Waves with periods between 0.8 and 2.5 sec and wave heightsof up to about25 cm can be generated in this flume.The wave generator is controlled by a Digital Equipment Corporation (DEC) VAXstation3200computer running the VMS operating system. The GEDAP (GeneralizedExperiment control,Data acquisition and Analysis Package) library of softwareand the associated RTC (Real TimeControl) programs developed at the Hydraulics Laboratoryof the National Research Council ofCanada (Miles, 1989, Pelletier, 1989) allow the user tocontrol the wave generation and dataacquisition processes. The software allows for three regular wavegeneration options: asinusoidal wave train with specified height and period;a sinusoidal wave train of maximumsteepness for a specified wave height and a sinusoidal wave train of maximumsteepness for aspecified wave period. Random waves can also begenerated either from a target wave spectrumbased on a theoretical parametric spectral model (e.g. Pierson-Moskowitz,JONSWAP etc.) or55from a prototype spectrum obtained from either a wave hindcasting procedure or from full-scalemeasurements at sea.Synthesis of Wave Generator Drive SignalsThe computer programs for the synthesis of the wave generator control signals are based onlinear theory (Havelock, 1929) which relates the rotation or displacement of the waveboard tothe water surface elevation at the front of the waveboard. Factors such as the propagation of thewave train from the waveboard to the test section, the articulation mode of the wave paddle, andthe dynamics of the servo-system are all taken into account in the creation of the driving signal.The process involves the generation of a desired free surface elevation time series at aspecified location in the wave flume. This wave record is generated at a sampling interval of0.05 sec (20 samples/sec). The transfer function between the paddle motion and the resultingwater surface elevation, along with the wave generator calibration relating the paddle motion tothe applied voltage are used to compute another time series which is subsequently used as thedrive signal. A DEC AAV1 1 digital-to-analog (DIA) converter combined with an analog activelow-pass filter is used to convert the drive signal to a continuous voltage which is sent to thecontroller of the servo-motor. The highest frequency of waves (with a minimum wave height of5 cm) that can be generated in this flume is limited to about 3.5 Hz due to the maximum velocityof 0.61 mlsec which can be achieved by the actuator thrust rod.If the water surface elevation at the test area in the wave flume is measuredwith a wave probewhen waves are generated with the above signal, the variation of the free surface elevation withtime will usually not be the same as that of the desired wave train. Some of thefactors that causethe differences include limitations of the transfer function used to approximate the servodynamics of the wave machine controller, limitations of linear wave theory in calculating thewaveboard motion and nonlinear wave propagation effects whichmay occur as the wave travelsfrom the wave machine to the wave probe. In the case of regular waves,the generated signal canbe modified with a suitable amplification factor givenby HdIHm, whereHd isthe desired wave56height, andHm is the measured wave height. This amplification factor is applied to generateanother time series of the driving signal, and the procedure is repeated until the desired waveheight is obtained at the test location.3.1.3 Cylinder modelA photograph of the test cylinder assembly is shown in Fig. 3.3(a), and a correspondingschematic diagram is shown in Fig. 3.3(b). The PVC cylinder has an external diameter of 4.2 cmand is 30 cm long. It is expected that the low length/diameter ratio of 7.14 willreduce thelikelihood of errors due to wave skew and also limit the flexural response of thecylinder. Thesealed ends of the cylinder are connected by aluminium tubes to the sensing plate of the forcetransducer which is located 34 cm above the cylinder axis. Since the aluminiumtubes transmitthe force on the cylinder to the dynamometer, they have to be shielded fromthe effects of theincident waves in order for the dynamometer to read the force on the cylinderonly. This isachieved by means of plastic tubes which are connected to the back faceof the dynamometer,and which act as sleeves in keeping the force transmitting members dry.The dynamometer is attached to a large frame spanning the width of the flumeand fixed to thelaboratory floor. The connection of the dynamometer to the frame is viaa solid steel rod suchthat the cylinder-transducer assembly can be easily moved vertically andalso pivoted about thevertical and horizontal planes. In this position, the cylinder axis is located8.7 m from the wavepaddle.Measurement of Hydrodynamic Impact ForcesThe variables which affect the vertical force on a horizontalcircular cylinder in waves havebeen discussed in Section 2.1. Some of these can be readily quantifiedand controlled in alaboratory. However, other factors such as air entrainment in the water surface,compressibilityof the fluid on impact with the cylinder, water run-up on the sides of the cylinder,the possibilityof not having a perfectly smooth cylinder surface or a perfectly circularcylinder cross-section,57minor deviations in cylinder alignment in the horizontal and vertical planes, and residualvibration from the laboratory floor due to the wave generator are difficult to measure and theirinfluence on the hydrodynamic force is unknown. It is expected though, that the relativedeviations in the measured force due to these factors will be small, and that the results from thisexperimental study will be useful from an engineering standpoint.Section 2.3 examines the requirements of the force measuring system for this particularapplication and points to the necessity of making an assumption regarding the nature of the rise-time of the slamming force. This is difficult since the rise-time is one of the unknowns in thisstudy. Various methods can be used to measure hydrodynamic impact on a cylinder. Onepossibility is to measure the dynamic response of the test cylinder with an accelerometer;compute the velocity and displacement of the cylinder from the acceleration record and then useEq. 2.34 to determine the applied force on the cylinder. This method may suffer from difficultiesrelating to noise, and the unknown effects of the added mass and hydrodynamic damping withincreasing cylinder submergence. The use of several pressure transducers located on the surfaceof the test cylinder has been reported in previous studies. The measurements of pressure on thecylinder surface are integrated to determine the total force on the cylinder. The advantages ofthis method include the relative absence of dynamic effects, and extremely fast response times.However among the limitations are the requirements for elaborate instrumentation andsignificant data storage capacity. A dynamometer can be used to obtain good measurements ofslamming forces in waves if the experimental rig is designed to meet the necessary dynamiccriteria. The total force on the cylinder is available as a single signal, and it is possible to carryout longer tests in waves and generate smaller amounts of data in comparison to an experimentinvolving pressure transducers.The most important factor to be considered when designing a model to measure impulsiveforces is that the system should have a very high natural frequency of vibration in comparison tothe dominant frequencies which are present in the dynamic force. There are two different modesof vibration that can influence the measured slamming force in the above cylinder-dynamometer58configuration. The cylinder may vibrate in a flexural mode between its end supports and theoverall cylinder and force transmitting member assembly may oscillate as a lumped mass withthe dynamometer providing the stiffness component. The flexural mode of vibration can bereduced to a very large extent by using a rigid cylinder with a low length-to-diameter ratio. Thereduction of the overall dynamic response necessitates that the cylinder and force transmittingmembers be light and the dynamometer have a high stiffness in order to accurately measure therapidly varying applied force. The damping of the model should also be low in order tominimize phase errors between the applied and measured force.There are contradictory requirements in the need for a model that is quite stiff, so that thedynamic amplification of the impulsive force is minimal, and the rise-time is measured correctly.In order for the system to be stiff, the dynamometer must have a high stiffness, which for lowload levels implies that the strain gauges in the dynamometer will not produce a significantchange in voltage, leading to a weak signal which is not much higher than the background noise.Conversely, if a dynamometer with a lower stiffness is employed so that a high level signal isobtained for the applied force, the natural frequency of the system is low and will lead to themeasured slamming force being significantly influenced by dynamic effects.In this study, the careful use of high-gain signal amplifiers, low-noise cables and judiciousnumerical filtering techniques have made it possible to use a reasonably stiff dynamometer andobtain good results from the force measurements.3.1.4 Data Acquisition and ControlIn addition to the physical factors which affect the magnitude of the slamming force, theaccuracy with which this highly dynamic force is measured and stored is dependent on theinstrumentation, which includes the dynamometer, amplifiers, power supplies, signal carryingcables, analog to digital conversion hardware and the controlling computer. After the force isconverted to a voltage by the dynamometer, the signal is amplified, digitized and stored asnumerical data for further analysis. The accuracy of the sampled data corresponding to the force59depends on the linearity of the transducer and amplifiers and their ability to remain drift freebetween calibration and the actual tests. Furthermore, noise due to stray electromagnetic fieldsand the AC power line frequency of 60 Hz can get mixed with the signal from the dynamometerstage, and during signal amplification and transmission. Standard low-pass noise filtering of thisdata is not possible because the slamming force may have significant frequency components inthe same range as the interfering noise.Figure 3.4 shows a block diagram of the apparatus used for acquiring water surface elevationand force data during the experiments. The main components are the wave probe, forcetransducer, amplifiers, signal cables, analog-to-digital (A/D) converter and the computer used tocontrol the experiment and simultaneously acquire data.The wave probe is based on a design of the Hydraulics Laboratory of the National ResearchCouncil of Canada. It is a capacitance-type ‘bow-string’ sensor consisting of a loop of wirestretched on one side of a metal frame. The wire loop sensor is connected to an amplifierdesigned to convert the change of capacitance to a measurable change in voltage. This devicehas a linearity better than 98.5% and a resolution better than 1 mm, the latter being limitedmainly by the meniscus, and under wave action, by the run-up.The dynarnometer used for measuring the force on the cylinder is a single-ring multicomponent transducer (Advanced Mechanical Technology Inc. SRMC3A-l00) with six outputchannels. These channels correspond to the forces in three orthogonal directions(F’, F and F7)and moments about those three axes(M’,M andMt).The instrument uses strain gauges toperform the force and moment measurements. The gauges are configured in four-arm bridges toprovide high thermal stability. TheFchannel of the dynamometer is used for the measurementof the vertical force in this study. At a rated stiffness of 2.63x106N/rn, the peak load capacity ofthis channel is 225 N and the sensitivity is 54 .tV/N for an excitation of 10 V.The output voltage from theF channel of the dynarnometer is very low level for typical waveslamming forces and has to be amplified before it can be recorded. The amplifier used for this60application is a Pacific Instruments Model 8255 Transducer Conditioning Amplifier whichprovides excitation, balance, calibration, amplification and filtering functions. The switchselectable low-pass filter provides steps of 1 Hz, 10 Hz, 100 Hz, 1 kHz, 10 kHz and wideband(no filtering). The amplifier can be set for a maximum gain of 2500. However, it was found thata gain of 2500 was not adequate to obtain a good signal to noise ratio for the experiments. Theforce signal is hence amplified in two stages. The first stage consists of an amplifier built in-house which is powered by lead-acid batteries and provides a gain of 1000. By virtue of itsrunning from a DC power source, this amplifier adds very little noise to the signal from thetransducer. This preamplified signal is then further amplified by a factor of 5 using the PacificInstruments amplifier. The amplified force signal is transmitted by shielded cable to the A/Dconverter.The analog-to-digital converter is used to discretize a continuously varying voltage to a seriesof digital values spaced at a pre-determined time interval. The two properties of the converterwhich determine how accurately the digitized force signal represents the original continuouswaveform are the resolution and the sampling rate. The resolution of the converter is defined asthe smallest difference in input voltage which can be measured. The DEC ADQ32 12 bit A/Dconverter used in this study has a maximum input range of ± 10 V and programmable gainsG = 1, 2, 4 or 8. The usable input voltage range is hence determined from the gain setting forany particular channel asVrange= ± 10/G V. The objective is to maximize the resolution andhence minimize the quantization noise in the sampled signal. In the experiments with non-breaking waves, the force signal did not exceed ± 2 V, and the gain for that channel was set to 4yielding aVrange of ± 2.5 V. The resolution in this case is given by zV = 5/212= 0.00 122 V.This corresponds to a resolution of 1.26 x io N, based on the calibration constant of thedynamometer.The sampling rate which is defined as the number of analog-to-digital conversions made in onesecond is determined on the basis of the frequency components in the input force signal. TheAID converter used in this study is capable of a peak sampling rate of 200,000 samples/sec when61using only one channel. However, the sampling rate should be chosen so that the input force iscorrectly recorded, and the size of the data files obtained from the experiments are notunnecessarily large. The sampling interval has to satisfy the Nyquist criterion which states thatLtmax= 1/(2fmax),whereZtmaxis the maximum allowable time difference between successivesamples to prevent aliasing andmaxis the highest frequency component in the signal. Thisimplies that there must be at least 2 samples per cycle of the highest frequency componentpresent in the acquired signal. It has been noted earlier that the slamming force signal hassignificant high frequency components and that low-pass filtering of the signal should be done ata cut-off frequency beyond which no force information is present. In order to determine thesampling rate for the measurement of the force signal, it is necessary to know the highestfrequency which must be retained in order to accurately describe the variation of the slammingforce.A preliminary measurement of the slamming force was made in which the cylinder was placedjust touching the still water level, the amplifier low-pass filter cut-off was set to 10 kHz and thesampling rate was 20 kHz. It was observed that the rise-time for a slamming event due to a waveof 1.2 sec period and 0.2 m height was typically 8 msec. The rise-time and magnitude of theslamming force peak were not modified when the low-pass filter on the amplifier was reducedfrom the 10 kHz setting to a 1 kHz cut-off. There however was a noticeable difference in theforce trace when the filter setting was further decreased to 100 Hz. It was thus decided tooperate the amplifier filter using a 1 kHz cut-off. The sampling rate was then fixed at 2.5 kHz tosatisfy the Nyquist criterion and prevent aliasing errors.3.2 Dynamic Characteristics of the Test CylinderThe dynamic characteristics of the cylinder were examined using a free vibration test whichinvolved applying a step load on the test cylinder using a weight suspended by a steel wire thatwas then cut using an acetylene torch. The resulting force transmitted to the dynamometer wasmeasured and is shown in Fig. 3.5. The cylinder-dynamometer combination oscillates at the62predominant frequency of 290 Hz as can be seen from the spectral densityof the record shown inFig. 3.6. The other minor vibration components presentat frequencies of 75 Hz, 130 Hz and190 Hz which do not seem to be related to the dominant frequency, maybe caused by vibrationof the support structure and cross-talk from the other channelsof the dynamometer. Quantitativemeasurements of these effects including the levelof background noise were not obtained. Thefirst force peak is reached in 2.4 msec which providesan indication of the transient responsecapabilities of the test cylinder assembly. The dampingfactor is evaluated from the freevibration trace using the following relationships:Rf=1n[-]m+n__________(3.1)_____\IR+47t2n2wheremandm+nare amplitudes of the free vibration force traceafter m cycles and m+n cyclesrespectively. The average damping factor in air wasdetermined to be about 2%.Figure 3.7 shows the early stages ofa typical wave slamming record which has only beenfiltered with a 1 kHz low-pass filter at the amplifierstage. It can be seen that when the freesurface is below the cylinder, the measured forceis not zero but consists of backgroundelectromagnetic noise and vibration.At the instant of slamming, the force on the cylinderbeginsto increase rapidly and reaches a maximum after15 msec. This rise-time is not the same for allslamming events and varies to quite an extent. Theindividual digitized samples are also shownas dots on the force trace. It is evident that the samplingrate is high enough to record accuratelythe increase of the slamming force. It isalso seen that there are some oscillations superposedonthe force trace as it increasesto the first maximum, which is comparableto the simulatedbehaviour shown in Fig. 2.10(b). This indicates thatthe cylinder-dynamometer assemblyis ableto respond quickly to the applied force(see Section 2.3.1), and that the measured peak forceandrise-time are good estimates of theactual values.633.3 Horizontal Cylinder in Non-Breaking WavesThe computer used for data acquisition runs software which controls the A/D converter interms of setting up the channels to be used, the gain for each channel, the sampling rate and thelength of time for which data is collected. The wave probe and dynamometer were calibrated bya program which relates known values of water surface elevation or force to correspondingvoltages read by the A/D converter. Since both the devices showed linear characteristics overthe range of interest, the calibration constant and zero offset were the only parametersdetermined from the calibration process.The computer runs two intercommunicating processes concurrently. One of the processessends the control signal to the wave machine and the other acquires data from the A/D converter.This allows for wave generation and data acquisition to be synchronized so that sampling isstarted at a specified time interval after the wave machine is activated. The delay betweenactivating the generator and starting measurements is usually set between 15 to 20 sec whichallows for the generator to ramp up to full amplitude and the wave train at the test site tostabilize. It is hence possible to conduct an experiment and store the measured wave record atthe test site, and repeat the experiment to store the corresponding slamming force on thecylinder, with both records having the same starting time with reference to the instant that thewave generator was activated.The experiments were carried out in two stages. The first stage consisted of generating regularwaves of different heights (10 to 23 cm) and periods (ito 1.8 sec) in the flume and storing themeasured wave train at the test site at a sampling rate of 2500 samples/sec. Although thissampling rate is very high for acquiring data from a wave probe, it is necessary to sample thewater surface elevation at the same rate as the wave force due to a data acquisition software andhardware limitation which introduces a varying phase difference between observed values of thesame signal which has been sampled at different rates. This error is not important in cases wheredata with low frequency content is being digitized, but it is not acceptable when events are64occurring which have time scales in milliseconds. Once the wave record was acquired, it wasconverted to physical units (metres) using the calibration information, numerically filtered toremove noise, and resampled at a lower rate of 50 samples/sec to conserve storage space on thecomputer.The cylinder model was not present at this stage and the wave probe was placed coincidentwith the axis of the cylinder. It is noted here that the wave record measured at the cylinderlocation in the absence of the cylinder will be different from the wave record observed when thecylinder is present in the flume. The wave field in the vicinity of the experimental cylinder isslightly modified due to runup and reflection effects which influences the water surface elevationand hence the computed water particle kinematics. However, the experimental results obtainedfor the force coefficients should be applicable in design situations where the incident,undisturbed wave conditions are used to estimate the water particle kinematics. Consequentlythe incident, undisturbed water surface elevation at the cylinder locationhas been measured inthis experimental study. To ensure that a particular wave train was repeatable over separateexperiments, multiple measurements of the same wave train were obtained and analyzed forvariability. The repeatability was excellent. Data for eachtest run for a particular wave heightand period was recorded for 20 sec.In the second stage, the wave probe was removed and the cylinderwas placed in the flume.The orientation and elevation of the cylinder werechecked to ensure that the axis wasperpendicular to the walls of the flume, and that the lower surface of thecylinder was horizontal.The waves corresponding to the stored regular wave trains were generatedagain and the verticalforce on the cylinder was measured at a sampling rateof 2500 Hz and stored. Tests wereperformed for 3 different elevations of the cylinder, h= 0.5 cm, -4.5 cm, and 4.5 cm.653.4 Data AnalysisThe following section describes the techniques used to analyze the wave record andcorresponding slamming force records obtained from the tests in regular waves in which thecylinder axis was horizontal and oriented parallel to the wave crests. Section 3.4.1 examines theeffects of noise in the force record and the techniques used to filter unwanted components out ofthe signal. Sections 3.4.2 and 3.4.3 describes how the filtered wave and force records areanalyzed to determine quantities such as the slamming coefficient and the rise-time.3.4.1 Noise Filtering TechniquesMost analyses of experimental data usually involve filtering the data to remove unwanted noisewhich can affect the results. This is a fairly simple procedure which can either be implementedin the form of analog filtering by dedicated circuits during the data collection stage, or by usingdigital filtering algorithms after the data has been digitized and stored. The treatment of correctnoise removal is especially important in this study since the slamming force typically hasfrequency information which is in the same range as some electrical noise. Any filteringtechnique used for analyzing such data must ensure that valid force information is not removedalong with the interfering noise.A variety of sources can produce noise. The power supply lines that run in the laboratorygenerate electromagnetic noise which shows up as a 60 Hz signal of unknown magnitude mixedwith the main signal. Electromagnetic noise is also generated from other devices (e.g. motors)within the laboratory. Thermal noise of a random nature and spread over a range of frequenciesis generated within the transducers and circuitry of the amplifiers. The dynamometer which isquite sensitive picks up mechanical noise from its floor supports. In addition, quantization noiseis introduced during the analog-to-digital conversion process, but can be mitigated to a largeextent by using the full range of the A/D converter for a given signal.66A significant amount of effort has been spent in this study in attempting to minimize the pickup and transmission of noise during the experiments. A high signal-to-noise ratio cannot beachieved in this study since the stiffness requirements for the dynamometer result in low outputvoltages as discussed earlier in Section 3.1.3 and 3.1.4. Although the dynamometer output isamplified by a factor of 5000, any noise present before amplification is also increased and theadvantages of amplification are evident only in the case of noise picked up during transmissionof the signal from the amplifiers to the A/D converter. The objective in the present set-up hasbeen to minimize noise before and after amplification by using stable power supplies, highquality transducers, amplifiers and shielded signal cables.It has been stated in Section 3.1.4 that the slamming force is not modified if the low-pass filteron the amplifier is set at a cut-off of 1 kHz. This enables the removal of all noise components inthe force signal which have frequencies greater than about 1100 Hz, since the filter has a finiteroll-off slope.Figure 3.8(a) shows the variation of the water surface elevation due to a regular wave of heightH = 22.6 cm and period T = 1.4 see, and also indicates the location of the test cylinder. Theslight difference between the actual elevation of the cylinder, h = 0.5 cm, and the elevationindicated in Fig. 3.8(a) is due to the wave record being measured in the absence of the cylinder,whereas the presence of the cylinder causes local variations of 3 to 5 mm in the water surfacewhich can cause impact to occur slightly earlier or later than the instant in the wave record when= 0.5 cm. The corresponding vertical force on the cylinder is shown in Fig. 3.8(b). It can beseen that the signal-to-noise ratio is much higher for the wave record than for the force record.The analysis of these records begins with numerical (digital) filtering in order to remove thosecomponents of noise that have not been removed by the analog filters in the amplifiers of thedynamometer and the wave probe.Digital filtering of time series data can be done in two ways. A non-recursive filteringtechnique using an FFT-based algorithm provides low-pass, high-pass, band-pass or band-stop67(notch) filtering capabilities. The other method, known as polynomial filtering is recursive andis based on simple interpolating polynomials which act as low-pass filters. Polynomial filteringis carried out by fitting a curve of a chosen order to a window of data within the main data series.This method weights the individual datum by its neighbouring data to substantially reduce noise.The polynomial filter used to process the water surface elevation and force data is a GEDAPprogram LPK_FILT which provides low-pass filtering of an input time series by convolving thedata with a 41 point Kaiser filter that allows for variable ripple attenuation levels. Theattenuation is inversely related to the roll-off slope of the filter. Convolution in the time domainrather than multiplication in the frequency domain saves on computer memory requirementssince the frequency domain technique needs a filter of length equal to the size of the input timeseries.3.4.2 Determination of the Instant of SlammingThe slamming event shown in Fig. 3.8(a) shows that the vertical force begins to increaserapidly at about t = 18.25 sec. This estimate is made visually and is based on the rapid rise of theforce beyond the ambient noise component which itself ranges from -0.05 N to 0.05 N. Thecorrect determination of the instant of slamming is important in this study since it affects thecorrect estimation of parameters such as the cylinder submergence and the associated buoyancyforce component, as well as the rise-time of the slamming force. Due to the volume of data to beprocessed, it is not efficient to fix visually the instants of slamming in a force record, and hencean accurate and stable slamming detection algorithm is developed which can be used to processthe measured force.Since the elevation of the cylinder h is known for every experiment, and the wave recordcorresponding to the measured slamming force is also known, the instant of slamming can befixed as that when the water surface elevation becomes greater than the value of h. Thistechnique was tried unsuccessfully, as it was observed that the force signal was sometimes still68zero or was already rising rapidly when a slamming event was indicated by the wave record asseen earlier in Fig. 3.8.It is also not possible to associate the onset of slamming when the measured force increasesbeyond a small threshold value, due to the significant ambient noise level when the cylinder isabove the free surface. Consequently it is necessary to use some property of the force recordwhich is not affected by the noise, but which changes when slamming occurs. One suchparameter is the local variance which is a measure of the variance of the sampled data over achosen interval of time in the force record. The local variance of the force as a function of timet, with a sampling window of width t,, is given by:a2(t,t) = F2 (t,t)- (F)2(t,t) (3.2)where F2(t,t) is the movingaverage of the square of the force record, (F)2(t,t)is the squareof the moving average of the record anda2(t,t) is the local variance. The behaviour of the localvariance of a small section of the slamming force record in Fig. 3.8(b) is shown in Fig. 3.9. Itcan be seen that while the force record is affected by background noise, the local variance isquite stable and begins to increase only with the onset of slamming. The window size and thethreshold beyond which an increase in the local variance indicates the onset of slamming werechosen after analysing several slamming records with various values of t,. A window sizet, = 0.004 sec (10 samples) and a variance threshold of 0.01 N2 has been used to detectslamming events in this study.The method has been optimized in the case of a force record for regular waves. A visualinspection is first used to determine the points in time within which the first slamming eventoccurs. The algorithm scans this region of the record, determines the instant when the variancethreshold is exceeded and marks that point as the beginning of a slamming event. The search isthen moved ahead by one wave period and the procedure is repeated for all the slamming eventsuntil the end of the force record.693.4.3 Analysis of Wave and Force RecordsFigure 3.10 shows a flow diagram of the various stages involved in the analysis of the forceand wave records. The records are initially subject to digital low-pass filtering to remove highfrequency noise components which were not removed by the amplifier. The cut-off frequencyfor the force record has been set at 400 Hz since a spectral analysis of the slamming forceindicates that there is no significant energy component in the record beyond 400 Hz. Thefiltering does not affect the peak magnitude or the rise-time of the slamming force. The waverecords which have already been stored at a lower sampling rate of 50 Hz are then low-passfiltered at a cut-off frequency of 6 Hz to remove any stray data points in the record which mayaffect the water particle velocity values. The filtered force record is then processed using thelocal variance algorithm (see Section 3.4.2) in order to identify the slamming events in therecord. Since the wave record has only 50 samples per second as compared to 2500 samples/secfor the force, a cubic spline based representation of the record is used to determine the freesurface elevation and velocity at any instant corresponding to a given point in the force record.The resulting time histories of the water surface elevation and the corresponding vertical forceon the cylinder are analyzed to obtain the slanmiing coefficients and other relevant parameters.The main parameter of interest is the peak slamming coefficient which corresponds to themaximum force observed immediately after impact and is estimated as:= F0(33)0.5pLDi2whereFt0is the peak value of the total force immediately after impact, and rj is thecorresponding vertical water particle velocity at the free surface. There are possible correctionswhich may be applied to the above estimate of the maximum slamming coefficient.The value of i is corrected for the effects of the free surface slope as discussed in Section 2.5.2(Eq. 2.74) and used to obtain a modified value of the peak slamming coefficient designated as70C. This is generally expected to result in an increase in the value of C0 by 5% to 10%,depending on the steepness of the wave train used in the test.Secondly, because of the finite rise-time taken to reach the the maximum slamming force, thebuoyancy force component is not zero at that instant. This may be estimated by assuming thatthe cylinder submergence at any instant is s=- h, and may thereby be subtracted from themeasured maximum force prior to estimating the slamming coefficient, which is now denoted asC0. It is noted that the assumption of a level free surface after impact is implicit in thisoperation. In reality, the actual contribution of the buoyancy force component will be less thanthe computed estimate since photographs of wave impact have shown that the freesurface in thevicinity of the cylinder is disturbed to a certain extent dependingon the velocity of the impact.The dynamic response of the cylinder may be accounted for approximately usingthe resultspresented in Section 2.3.1 for a triangular impulse havingequal rise and decay times. Acorrected estimate of the rise-timeTr of the applied force is determined using the observed valueof the rise-time for a particular slamming eventT, and the known natural period T and dampingof the measuring system. The ratio ofTr/Tn is then used to determine the dynamicamplification factor to obtain a corrected value of the slammingcoefficient. This coefficientwhich has been corrected for the effects of free surface slope,buoyancy and dynamic effects isdenoted by C.The impulse coefficient C1 defined in Eq. 2.56 (see Section 2.3.3) is alsoevaluated for thevarious tests. Since the correction for the free surface slopecan also be applied in this case, amodified value of the impulse coefficient denoted as C is determined using Eq.2.75.Additional parameters of interest which are reported include the Froudenumber at impactbased on the vertical velocity of the free surface the Froude numberbased on the slopecorrected free surface velocity v/Jj the observed rise-timeT,the rise-time corrected for71dynamic effectsTr,the relative cylinder submergence at the instant of peak impact force(VnTr/a),and the relative cylinder submergence at the end of the impulse (vTIa).Since the vertical force on the cylinder is measured over a duration of 20 sec, there may be 10to 20 slamming events in each force record, depending on the wave period. A statistical analysisof the above parameters is therefore carried out, after removing the highest and lowest values ofthe peak slamming force from that record. The mean, standard deviation, minimum andmaximum quantities for the parameters are determined for every combination of wave height andperiod.The wave records are analyzed separately by a zero-crossing method to determine the exactwave height and period that is realized during the tests.3.5 Inclined Cylinder in Non-Breaking WavesIn addition to the slamming tests on the horizontal cylinder in regular waves, a number ofexperiments were performed in which the cylinder axis was tilted by rotating the cylinderdynamometer assembly in the vertical plane. The cylinder axis was aligned normal to thedirection of wave propagation as in the case of the horizontal cylinder tests.The objective of these tests is to observe the behaviour of the slaniming force on the inclinedcylinder and to compare this to that for the horizontal cylinder for the same wave condition. Inaddition, the predictions of the theoretical model for slamming on inclined cylinders (seeSection 2.4) are also compared with the experimental results.Tests were conducted for two inclinations of the cylinder axis, e = 4.8° and 9.6°. Thecorresponding height of the lower edge of the cylinder measured from the still water level wash = 0.5 cm and -2.0 cm respectively. The force normal to the cylinder axis (in the vertical plane)was measured in regular waves with the same heights and periods used in the horizontal cylindertests.723.6 Horizontal Cylinder in Breaking WavesThe last phase of the experimental study on wave slamming consists of a set of tests on theforces due to breaking wave impact on a fixed, horizontal circular cylinder located near the freesurface. The two-dimensional case of uni-directional waves propagating in a directionorthogonal to the cylinder axis is considered. The vertical and horizontal components of theimpact force on the cylinder due to a single breaking or ‘plunging’ wave were measured for threeelevations of the cylinder, and six locations of wave breaking relative to the horizontal locationof the cylinder. The measurements have been corrected for the dynamic response of thecylinder, and analyzed to obtain impact coefficients and rise-times. A video record of the impactprocess was also obtained in order to estimate the kinematics of the wave and plunging jet priorto impact, and provide information on the effects of the shape of the wavefront on the observedimpact force.3.6.1 Generation of the Breaking WaveThe experimental facilities and the test cylinder have been described in Section3.1. A singleplunging wave was generated during each run of the experiments, using a procedure similar tothat described by Chan and Melville (1988). This involves the generationof a frequency andamplitude modulated wave packet consisting of a number of prescribed sinusoidalcomponents.The phases of the individual wave components were chosen on the basis of linearwave theory togive rise to a summation of crest elevations at a desired wave breaking location.In the presentstudy, the wave components have been generated with the following characteristics:Period of centre frequency component 1.1 secLength of centre frequency component 1.81 mFrequency bandwidth 1.11 HzNumber of frequency components 30Due to nonlinear interactions between the various components, the wavecrest breaksapproximately 1 m before the estimated location of breakingbased on linear theory. The wave73generation program was hence used to compute a wave which would break 1 m downwave of thecylinder location, so that the actual wave breaking in the experiments occurred in the vicinity ofthe cylinder. The actual location of breaking is determined by video records made in the absenceof the cylinder, which would otherwise interfere with the plunging crest of the wave. Thehorizontal location of wave breaking is characterized by xb which is defined as the distancebetween the cylinder axis and the point in the wave flume where the plunging jet from the crestof the breaking wave touches down on the free surface. The value of xj, is negative if the jettouches down before reaching the cylinder. The determination of the wave breaking locationdepends on a visual estimate from video records of the experiments, and consequently reportedvalues ofxmay have an error of ± 2 cm. A typical example is shown in Fig. 3.11 forx = -3.5 cm.The experiments were carried out for 6 values ofxranging from -3.5 cm to 49.0 cm, whichcover a variety of impact conditions, ranging from a wave that has already broken beforereaching the cylinder, to one that is in the early stages of steepening when it impacts the cylinder.3.6.2 Measurement of Force and Breaking Wave ProfilesSince it is well known that breaking waves cause significantly higher impact forces thanregular waves, and that the horizontal force is a major component of the total force due tobreaking wave impact, the test cylinder assembly was modified to ensure that the larger forcesdid not interfere with the proper operation of the dynamometer. The plastic sleeve tubes whichshield the connecting members between the cylinder and dynamometer were disconnected fromthe support frame, and rigidly clamped to the walls of the wave-flume. This ensured that thelarge horizontal forces would not cause the sleeve tubes to move and disturb the forcetransmitting members.In addition to the vertical force channel of the dynamometer which was already configured forthe non-breaking wave slamming tests, another channel of the dynamometer was connected to anindependent amplifier in order to measure the horizontal force. The channel which senses the74moment about the dynamometer’s Y-axis (parallel to the cylinder axis) was calibrated tomeasure the horizontal force on the cylinder. A free vibration test was conducted to determinethe dynamic characteristics of the cylinder in the horizontal direction. Figure 3.12 shows thetrace of the horizontal force measurement made by applying a step load in the horizontaldirection using the method described in Section 3.2. A comparison ofFigs. 3.5 and 3.12indicates that the cylinder-dynamometer assembly has different dynamic characteristicsin thevertical and horizontal directions. Figure 3.13 shows the spectrum of the horizontal freevibration record which indicates the presence of only one dominant componentat 29 Hz. Theaverage damping determined from the free vibration trace wasabout 2%.Experiments were conducted for 3 cylinder axis elevations above the still waterlevel, h = 4.7,8.7, and 12.7 cm, and for each of these elevations, tests were carried outfor 6 horizontallocations of wave breaking. The slamming force due to a regular non-breaking waveof periodT = 1.2 sec and height H = 20.5 cm was also measured for each cylinder elevation.For each test,the force records were sampled at the rate of 10,000 samples/sec over a durationof 2 sec. Thesignals were obtained after low-pass filtering at a high cut-off frequency (1000 Hz)to ensure thattransient force components were not lost.The tests were recorded on videotape at 30 frames/sec anda shutter speed of 1/1000 sec inorder to estimate parameters such as the breaking wave location, the wavekinematicsimmediately prior to impact, and also to provide a visualization of the impact process.3.6.3 Analysis of Breaking Wave Impact ForceThe effect of the dynamic response of the cylinder-dynamometersystem on the measured forceis expected to be more severe in the case of the horizontalforce component, due to the lownatural frequency of the measurement system incomparison to the higher frequency componentsthat are present in the rapidly varying impact force.Typical traces of the variation of themeasured vertical and horizontal components of the impact force(h = 8.7 cm, x = 25.5 cm) areshown in Fig. 3.15(a) which indicates that while thevertical force rises rapidly immediately after75impact, the corresponding horizontal force increases more gradually and shows a strongoscillatory component during and after the initial stages of impact. This causes the measuredmaximum horizontal force to differ from that of the applied force and also introduces a phasedifference between the applied and measured force as discussed in Section 2.3.1.In order to correct the measured horizontal force for dynamic effects, the equation of motion ofthe cylinder and its support structure in the horizontal direction can be expressed as:M’x+C’x+K’x=F’ (3.4)where M’, C’, and K’ are the effective mass, damping, and stiffness respectively in the horizontaldirection; x, * and x denote respectively the horizontal displacement, velocity and acceleration,and F’ is the horizontal component of the applied force. The natural frequency for the horizontalmode of vibration, and damping ratio are defined as = ‘JK’/M’ and ‘ = C’/2M’o respectively.For a known time history of the applied force, F’(t), the response x(t) of the SDOF system whichis initially at rest is given by Eq. 2.37 and is repeated here for convenience:x(t) =fF’(t)exp[-C’o(t- t)] sin[cid(t - t)] dt (3.5)MCOdowhere Oj = 1- C’2, and t is a variable of integration. The force measured by thedynamometer is given by F(t) = K’ x(t) and can therefore be expressed as:F(t)=2 fF’(’r) exp[-’(t - t)Jsin[od(t-t)] dt (3.6)It is now possible to determine the unknown applied force F’(t) from the measured force F (t)and the dynamic characteristics of the measuring system, using a numerical method whichexpresses the integral in Eq. 3.6 as a summation and solves for F ‘(t) at every time step.If the measured force is digitally recorded at a pre-determined sampling rate such that the timeinterval between successive samples is t, the applied force may be derived from Eq. 3.6 as:76F’(O)= F(At) (1- ‘2)(3.7)Q)dAt exp(-CoAt) sin(O)dAt)F’(iAt)= F((i+1)At) (1- C’2)(‘)dAt exp(-Co4t) sin( (OdAt)i.- 1F’(jAt)exp(-C’o(i-j)At) sin(Od(i-j)At)—fori=ltoN-1 (3.8)exp(- oAt)sln(oidAt)where N is the total number of points in the measured force record.The method has been verified by its application to the time series ofFt(t) generated by the freevibration test described earlier, and the results are shown in Fig. 3.14. Values of= 182.2 rad/sec (f = 29 Hz) andC= 2% have been used. The applied step loading is shownby the dashed line, and the corresponding recorded horizontal force is shown by the thin dottedline. It is seen that although the corrected force trace shown by the solidline in Fig. 3.14 isaffected by the presence of residual dynamic components, it is still a reasonable estimate of theapplied step load. The correction method is also applied to the horizontal force measured due tobreaking wave impact as shown in Fig. 3.15(a). Figure 3.15(b) shows the corrected horizontalforce component derived from the recorded force time series, and also repeatsthe vertical forcetrace shown in Fig. 3.15(a) for the purpose of comparison. The correctedforce trace inFig. 3.15(b) shows an oscillation before the beginning of the actual impactwhich is associatedwith the numerical low pass filtering (125 Hz cut-off) performed on therecorded horizontalforce record. This was applied in order to remove high frequency noisebefore the application ofthe dynamic correction method, since there was a tendency for noise componentsin the recordedforce to amplify and add to the predicted maximum force.The cut-off frequency of 125 Hz wasselected after examining the predicted value of the maximum appliedhorizontal force usingprogressively lower cut-off frequencies (e.g. 400, 250,150 Hz). The predicted maximum wassignificantly affected when a cut-off of 100 Hz was used.77Although the free vibration record of the vertical force indicates a dominant frequency of290 Hz, the presence of other frequency components at 75, 130 and 190 Hz rules out theapplication of the above correction method to the recorded vertical force, since these will beamplified by varying degrees and result in an incorrect estimate of the maximum slammingforce. The peak value of the recorded vertical force has been corrected using the same techniquedescribed in Section 3.4.3 for the regular wave tests. The peak values of the corrected verticaland horizontal force are used to obtain the magnitude and direction of the maximum resultantforce on the cylinder due to the breaking wave, which is reported along with the correspondingrise-time.There are certain assumptions implicit in the application of the above technique to the presentproblem. The response x(t) and its derivatives modify the applied hydrodynamic force F(t),whereas it is assumed here that the influence of cylinder motion on the applied force is minimal,and that the results derived from the above method will be usable from an engineeringstandpoint. The treatment of the cylinder-dynamometer assembly as an SDOF system alsorepresents a simplification, but this is not expected to have a significant effect on the estimate ofthe applied force.The water particle velocity in the plunging wave immediately prior to impact is estimated byanalyzing a series of still images digitized from a video record of the impact. A series of 4successive images with a time interval of 1/30 sec are captured over a duration spanningapproximately 1/15 sec before impact to 1/30 sec after impact. The profile of the wave-front ineach image is digitized for further analysis. The celerity of the wave which is constant for all thetests is a representative quantity which is determined from the wave profiles. The velocitynormal to the moving water surface adjacent to the cylinder is also determined for cases wherethe surface is relatively undisturbed. Since the impact force on the test cylinder is dependent onthe contact area between the water surface and the cylinder during the early stages of impact, theradius of curvature of the wave surface adjacent to the cylinder is also determined from the videorecords.78Chapter 4RESULTS AND DISCUSSIONThe results of the experimental and numerical investigations which have been described arepresented and discussed in this chapter. The first part of this chapter focuses on the experimentalresults from slamming tests in non-breaking waves of various heights and periods, and differentelevations of the cylinder axis near the still water level. The parameters of interest estimatedfrom the measured force records include the maximum value of the slamming coefficientimmediately after impact, the corresponding rise-time, the impulse coefficient, and thecorresponding impulse duration. The various corrections applied to these coefficients due to theeffects of buoyancy, free surface slope, and dynamic amplification are also presented. Statisticalproperties of these estimates from different experiments are indicated. The effects of cylinderinclination on the maximum force and rise-time are examined by comparison with the forcesobserved on a horizontal cylinder subjected to the same incident waves.The application of numerical models to estimate the vertical wave force acting on a section of afixed, rigid horizontal cylinder located near the free surface, and the effect of variations of thegoverning non-dimensional parameters on the magnitudes of themaximum vertical force arediscussed. Numerical simulations for a horizontal cylinder subjected to slamming in randomwaves have been used to estimate the probability density of the maximumforce and the result iscompared with analytical predictions. Time series of the numerically predictedforce arecompared with experimentally observed data for the cases of both horizontal and inclinedcylinders. An example application of the slamming model to practical situations is alsopresented.79The final section of this chapter discusses the experimental results for impact forces due to abreaking wave. The maximum horizontal and vertical force components, along with theresultant force and the rise-time are presented for several cases which involve differentelevations of the cylinder and various wave breaking locations. The geometry of the waveprofile in the vicinity of the cylinder immediately prior to impact is shown and its effect on theimpact force is examined. Slamming coefficients based on the local particle velocity and thewave celerity are indicated. The issues of applying these results at prototype scales are alsodiscussed.4.1 Slamming Forces in Non-BreakingWavesTests have been conducted in non-breaking regular waves for a horizontal cylinder at threedifferent elevations and an inclined cylinder at two different tilt angles. The component of thehydrodynamic force normal to the cylinder axis in the vertical plane was measured, along withthe water surface elevation at the cylinder location in the absence of the cylinder. In the case ofthe tests on the horizontal cylinder, the force data has been analyzedto obtain various parameterssuch as the peak slamming coefficient. The force data obtained from the inclined cylinder testsare used to illustrate the effects of axis tilt on the magnitude and temporal variation of theslamming force.4.1.1 Raw Data from Horizontal Cylinder ExperimentsTests have been conducted for 16 regular wave conditions with a range of wave periods(T = 1.0 to 1.8 sec) and wave heights (H = 9.8 to 22.9 cm). The wave conditions along with thecorresponding wave steepness and celerity for all the tests are listed inTable 4.1. For the casewhere the cylinder elevation h = 0.5 cm, the water particle velocity at the instant of impact isnearly a maximum and the corresponding acceleration is closeto zero. Six additional tests havebeen conducted for cylinder elevations h = -4.5 cm and4.5 cm.80Since the vertical force on the cylinder was measured over a duration of 20 sec, there aremultiple slamming events in each force record, and Fig. 4.1 shows an example of the free surfaceelevation and corresponding vertical force traces over a duration of 10 sec for an incident wavehaving a low steepness H/L = 0.036, and for h = 0.5 cm. Similar traces are shown in Figs. 4.2and 4.3 for incident waves of moderate (H/L = 0.063) and large steepness (H/L = 0.102)respectively. When the cylinder is out of the water, the measured force is seento consist ofbackground electromagnetic noise which is present even at frequencies below the 1 kHz cut-offsetting of the amplifier. These figures indicate that the slamming force maxima vary somewhatfrom one event to the next. This variation is caused by random disturbances onthe water surfacedue to reflection, splashing and drips after each cylinder submergence event, and tends toincrease with wave steepness. It is observed that this influences only the slammingforce andcorresponding rise-time, and that the vertical force after the initial stagesof slamming isessentially the same for every submergence event. The variability of theslamming forcemaxima observed in Figs. 4.1 - 4.3 emphasizes the difficulty in obtaining consistent estimates ofthe slamming coefficient.Figures 4.4 - 4.11 show time histories of the vertical force and correspondingwater surfaceelevation over a single slamming event for 8 differenttests. As an aid to interpreting theseresults, the figures include horizontal and vertical lines which indicate the cylinderlocation andcorresponding instants of impact and complete submergence. Figures4.4, 4.5, and 4.6correspond to three tests characterized by a change in waveperiod (with a slight difference inwave height); Figs. 4.7, 4.8 and 4.9 correspond to a change in wave heightfor the same waveperiod; and Figs. 4.8, 4.10 and 4.11 correspond to changes in cylinder elevationfor the samewave height and period.A comparison between Figs. 4.4, 4.5, and 4.6 for thecase of waves with H 17 cm, andT = 1.1, 1.4, and 1.6 sec respectively, indicates thatthe slamming force maximum which occursimmediately after wave impact, decreases with increasing wave period, whereasthe subsequentmaximum in the force which occurs after full submergenceis less variable. It is also observed81that the magnitude of the peak slamming force drops rapidly from a value that is more thandouble the maximum force which occurs after full submergencedue to a combination ofbuoyancy, inertia and drag force contributions, to one which is about half of the overallmaximum force. The change in the maximum slamming force is expected on account of areduction in the water particle velocity at impact. Figures 4.7, 4.8 and 4.9, for impact due towaves with T = 1.5 see, and H = 13.9, 18.4 and 22.9 cm respectively, show that boththe peakslamming force and the maximum force after complete submergence increase with anincrease inwave height which also corresponds to a larger impact velocity atthe same cylinder location.The nature of the force variation after impact changes since the magnitudeof the drag forcerelative to the inertia force is affected by the change in wave height. Finally, Figs. 4.8, 4.10, and4.11 are used to illustrate the distinctly different force recordsobtained with a wave ofT = 1.5 see, H = 18.4 cm, and cylinder elevations h = 0.5, 4.5 and-4.5 cm respectively. Whenthe cylinder is lowered from h = 0.5 to -4.5 cm, Fig. 4.11 shows thatthere is a marked reductionin the peak impact force, since the impact velocity is reduced, and the significant hydrodynamicforces occur after full submergence. In contrast, there is a smaller reduction inthe peakslaniming force when h = 4.5 cm due to a relatively smaller decreasein impact velocity near theupper portion of the wave which displays vertical asymmetry in theform of shallow troughs andsteep crests.All the time histories of the vertical force show the presence of residual vibrationcomponentsafter the initial wave impact. The natureof the oscillation is very similar to that shown inFig. 3.5. The slamming force contains components at frequencies whichare higher than thosepresent in the case of the force on a continuously submerged member.In the case of structuralelements of offshore platforms, slamming forces may induce dynamicresponse of the memberresulting in dynamic stresses. These stresses may be significantespecially if the frequencycontent of the force is close to the natural frequency of the member,and consequently indicatesthe need for dynamic analysis of the member during itsdesign. In addition to the magnitude ofthe dynamic stress, the occurrence of frequent stress reversals will leadto fatigue damage of themember and its support connections, and must also beconsidered as a design criterion.824.1.2 Slamming Coefficients from Horizontal Cylinder ExperimentsThe time histories of the water surface elevation and the corresponding vertical force on thecylinder have been analyzed to obtain the slamming coefficients and other relevant parameters.Since the vertical force on the cylinder was measured over a duration of 20 sec, the number ofslamming events for any one test ranges from 10 to 20, depending onthe wave period. Everyforce record was analyzed after removing the highest and lowest values of the peak slammingforce from the data. As discussed in Section 3.4.3, a maximum slamming coefficientC0 can beestimated from the observed value of the peak slamming force F andthe corresponding valueofi at that instant. The various corrections to this value of C, namely thefree surface slope,buoyancy, and dynamic amplification corrections have also been described inSection 3.4.3.Table 4.2 illustrates the effects of the different corrections applied to thedata obtained from atypical experiment using a wave with T = 1.5 sec, H = 18.4 cm, and cylinder elevation ofh = 0.5 cm. The value of i realized for the successive slamming events is seento be reasonablyconstant at about 0.39 rn/sec. However, the corresponding peak slamming coefficientC0 showsa larger degree of scatter. The coefficient of variation (C.O.V.), which is definedas the ratio ofthe standard deviation to the mean, is used to quantify the degree of scatterin the data. Theindividual values of C8 vary from 4.36 to 6.63, and it has a mean value andC.O.V. of 5.16 and0.16 respectively. The effect of the free surface slope correction is reflectedin the values of thewater particle velocity normal to the free surface v and the corresponding slammingcoefficientC0 which is determined on the basis of v. It is seen that in comparison to 1 andC, there is aminor reduction of about 2.5% in the value of v and the mean valueof C0 increases by 6% to5.47 respectively. The degree of scatter remains unaffected. Thecontribution of the buoyancyforce to the above estimates of the peak slamming coefficient can bedetermined by the relativesubmergence of the cylinder (s/a)0at the instant of occurrenceof the maximum force. The valueof (s/a)0varies between 0.29 and 0.44, and has a meanand C.O.V. of 0.35 and 0.14 respectively.This indicates that the cylinder is submerged to approximately one-sixth ofits diameter when thepeak slamming force occurs. It is noted that photographicstudies of cylinder entry into water83have shown that the free surface is significantly displaced during the early stages of impact, andthat the assumption of a level free surface will generally lead to an overestimation of thebuoyancy force contribution. The modified slamming coefficient C which is derived afterremoving the buoyancy force component is also shown in Table 4.2. It is seen that the averagevalue of C decreases by about 10% to 4.91 in comparison to C, while its C.O.V. of 0.18 isabout the same, and individual values vary from 3.96 to 6.51.Finally, the dynamic response of the cylinder may be accounted for approximately using theresults presented in Section 2.3.1 for a triangular impulse and which is shown in Fig. 4.12 in amodified form. The correction is carried out as follows:by assuming a natural periodT = 3.45 msec (f = 290 Hz) and a damping ratio = 2%, the observed rise-timeTis used withthe results in Fig. 4.12 to estimate the corresponding actual rise-timeTr of the applied force.This is then used with the measured force maximum F (with the buoyancy componentremoved) to obtain the corresponding applied force maximum F0. Since the natural frequency ofthe test cylinder is quite high, it is able to respond quickly to the rapidly increasing slammingforce. Consequently the corrections to the observed rise-time and vertical force are small for thecase shown in Table 4.2, and vary between 0 and 5%. The value ofTtvaries between 17.2 and25.6 msec, and has an average of 20.1 msec. The corrected valueTr has an average value of19.4 msec. The C.O.V. for these parameters is 0.13. Theslamming coefficient C whichincludes the dynamic correction is seen to decrease slightlyin comparison to C, and variesfrom 3.87 to 6.43 with a mean value of 4.81. The cumulativeeffect of the 3 corrections is thusseen to reduce the average value of the peak slamming coefficient byabout 7% in this particularexample. However, the magnitude of the correction varies with the wave conditionsused in thetest. It is also noted that the various corrections do not contributeto any significant change inthe degree of scatter exhibited by the different slamming coefficients.A summary of the results from all 22 experiments with non-breakingwaves incident on thehorizontal cylinder located at 3 different elevations is presented inTable 4.3. Since the waterparticle velocity at impact is the primary quantity whichdetermines the magnitude of the84slamming force, the non-dimensionalvalues of i/.Jjthat wereobserved in the various tests areindicated. The mean valueof this parameter varies from0.37 for the wave with theleaststeepness, T = 1.8 sec, H =13.5 cm, to 0.78 for the caseof the relatively steep waveofT = 1.2 sec, and H = 19.3 cm. TheC.O.V. ofi/’Jjis the lowestamong all the other quantitiesmeasured in this study, and variesfrom 0.02 to 0.14. This indicatesthat the waves generated inthe flume showed good repeatability,with the steeper wavesdisplaying slightly larger variationin the impact velocity for successiveslamiriing events.The mean value of C for eachof the 22 cases varies from3.04 to 7.79, and has been plottedin turn as a functionof different combinationsof the parameters indicated inEq. 2.2. Thesereveal no apparent trendof C with these parameters.However, it is possiblethat the absenceof any such trend maybe due to the limited rangeof values of the parameters,the relatively lownumber of observations made,and the possibility ofa complex dependence on severalof theparameters.The C.O.V. of C varies from0.09 to 0.36 and increaseswith the steepness of the incidentwave. This trend isexplained by the likelihoodof increased disturbances on thewater surfacedue to the higher water particlekinematics in steeper waves.Individual values of C0vary from1.86 to 11.7 and the overallmean of C0 is 4.87.It is noted that Campbell andWeynberg (1980)reported a value of C= 5.15 based on their experiments.Figure 4.13 shows the probabilitydensity histogram ofC0 based on a total of 311 slamming events recordedin the 22 tests. Thedata is seen to followa log-normal distribution whichis shown as a solidline in the figure. Theexpression for the log-normalprobability density for C0 isbased on its median value,C, andthe standard deviationofln(C0),denoted a:— 1 (1n(Cso/o)”2p(C) — ex -0.5(4.1)Csoa’[ LLawhere = j.t exp(a/2),and a2 = 1n(y2 + 1). Herep. is the mean ofC0(4.87 in the presentstudy), and ‘ is the coefficientof variation of C0(0.38 in the present study). Fübrböter(1986)85observed that the maximum pressure due to plunging wave impact on a sloping beach alsofollows a log-normal distribution.The impact velocity corrected for the slope of the free surface is shown as vbIjb and itsmagnitude is always smaller than the corresponding value of i/-Jj As expected, themagnitude of correction is generally small ( 9%) and increases with wave steepness. The meanvalue of this quantity varies from 0.36 to 0.75. The corresponding mean value of C0 is 3 - 12%larger in comparison to C0,and varies from 3.14 to 8.26, with its C.O.V. expected to be thesame as that for C.The measured rise-timeTis an important parameter since it provides an estimate of thedynamic amplification of the impact force. The average value ofTtfor the tests varies from 11.6to 37.1 msec and its C.O.V. ranges from 0.13 to 0.36. For the case h = 0.5 cm,T tends toincrease withv11b1j which may be attributed to the slower rise of impact pressures due toincreased presence of entrained air and local disturbances as a consequence of larger impactvelocities. Based on the mean value ofT observed in each test, the ratio TIT varies from 3.36to 10.8, and the corresponding correction factorsTtiTr vary between 1.07 and 1. The correctedrise-timeTr varies from 10.8 to 37.1 msec for the 22 tests.Miller (1980) indicated that, under ideal conditions of water impact on a horizontal cylinder,the rise-time of the slamming force may be associated with the propagation of a pressure wavewithin the water, and is of the order of the time taken for the wave to travel a distance 0.2D, i.e.Tr = O[0.2D/c], where D is the cylinder diameter and c is the velocity of sound in water. For thecase of water with no entrained air, c 1500 mIs. This corresponds to a rise-time of order0.005 msec, based on the above estimate. Lundgren (1969) has shown that the speed of sound inwater is strongly affected by the presence of air bubbles and decreases to about 40 mis for 10%entrained air content, which would lead to a longer rise-time of 0.2 msec. These rise-timeestimates are significantly shorter than those observed in the present study as reported above,which indicate that the effects of compressibility may be relativelyminor in comparison to otherfactors which influence the rise-time.86The cylinder submergencevnTr/a when the slamming force reaches a maximum isseen to benearly the same as the quantity(s/a)0determined from the water surfaceelevation record for thetests, where s= - h. The value of vnTr/a ranges from 0.13 to 0.80, and itsC.O.V. varies from0.11 to 0.41 which is very similarto the scatter shown byT. This quantity also exhibits anincreasing trend with respectto v/Jjfor reasons similarto those attributed to the trend shownbyT (and hence Ti). Although not shown in Table 4.3, thequantityvTt/a which was evaluatedfor every slamming event (311samples) indicates an overallmean cylinder submergence (s/a)0of 0.36. This correspondsto an average cylinder submergenceof 7 mm when the peak slammingforce occurred.The mean value of the maximumslamming coefficient correctedfor buoyancy effects Cranges from 3.07 to 7.72,with its C.O.V. varyingfrom 0.07 to 0.44. Figure 4.14shows theprobability density histogramof C0 values based on 311 slammingevents observed in 22 tests.The overall mean of C0is 4.68 which is 4% smaller thanthe corresponding value ofC, andits C.O.V is 0.44. Figure 4.14also shows the log-normal probabilitydensity variation for C0based on its mean and C.O.Vas defined in Eq. 4.1. It is seenthat the comparison betweenthehistogram and the theoretical curvefor C0 is fair, althoughnot as good as that shownfor C0(Fig. 4.13).Finally, the effectof the measuring system’s dynamicresponse on the slammingcoefficient isestimated using the mean valueofTobserved in the 22 tests,which yields correspondingvaluesof the peak force correctionfactorF0/F that vary from 1 to 1.04. The peak slammingcoefficient corrected for dynamiceffects C0 varies between2.92 and 7.60.4.1.3 Impulse Coefficientsfrom Horizontal CylinderExperimentsThe use of an impulse coefficientto describe slamming hasbeen discussed in Section2.3.3.The data obtained fromthe non-breaking regular wavetests on the horizontal cylinderhas beenanalyzed to obtain estimatesof the impulse coefficientC1 and related parameters whicharesummarized in Table 4.4.The mean value of C1 estimatedfrom each of the 22tests using i as87the impact velocity, varies from 0.26to 1.36, with an overall mean of0.73; and the C.O.V. of Cvaries from 0.04 to 0.41. Themean value of the impulse coefficientbased on the impact velocitycorrected for free surface slopeC varies from 0.26 to 1.47, and is higherthan the correspondingvalue of C1 by between 1%and 9%. In general, C exhibits trendswhich are similar to those ofC1, and has similar C.O.V.s. Figure 4.15shows a probability densityhistogram based onestimates of C from every slammingevent, and these observationsdo not appear to fit any ofthe common probability distributionmodels. The overall meanof C is 0.77. The degree ofscatter for C in relation tothat for C is of particular interest,and the present results indicatethat the C.O.V. of C is less thanthat for C by about 25% on average.In some cases (e.g. forT = 1.2 see, H = 12.1 cm; andfor T = 1.4 see, H = 16.5 cm)the C.O.V is reduced by as much as76%. There were only 4 tests in whichthe C.O.V. of C showsan increase. The estimates ofthe impulse coefficient didnot show any observable trendwith respect to incident waveconditions or cylinder elevation.The dimensionless impulse durationvT/a indicates the relative submergenceof the cylinderwhen the impact is assumedto have ended. Its mean valuevaries from 0.29 to 1.15, and exhibitsless scatter than the dimensionlessrise-timevnTr/a. The C.O.V. of vT/a ranges from 0.05 to0.21, which on average is lessthan the C.O.V of vT/a by about54%. The overall mean valueof vT1/a is 0.69 which correspondsto a submergence of about one-thirdof the cylinder’sdiameter at the end ofthe slamming impulse.The reduction in scatter ofthe impulse coefficient in comparisonto that of the slammingcoefficient suggests that theformer quantity maybe relatively useful in estimating thedynamicresponse of cylindrical elementssubjected to wave slamming.However, it is emphasizedthatthis approach is only useful forcases where the impulse durationT1 is reasonably small inrelation to the natural periodT of the system, such thatthe system’s response maybedetermined by the impulse magnituderather than the precise timehistory of the loading.884.1.4 Tests on InclinedCylinderA number of tests have been conductedby tilting the test cylinder’s axisat different angleswith respect to the still waterlevel and measuring thehydrodynamic force normalto the cylinderaxis in the vertical plane. Theobjective of these tests is solelyto observe the changes in thecharacteristics of theimpact force due to the slopeof the cylinder’s axis, and anexhaustiveanalysis of the measured forcehas not been conducted in thisstudy.Tests have been conductedfor two inclinations of the cylinderaxis 0 = 4.8° and 9.6° and threewave conditions. In thecase where 0 = 4.8°, the heightof the lowest edge of the test cylinderfrom the still water level wash = 0.5 cm (see Fig. 2.14),and for 0 = 9.6° this was insteadh = -2.0 cm. It is expectedthat the initial impact velocityfor these 2 cases is similarto thoseobserved during the horizontalcylinder tests with h =0.5 cm. In general, the force recordsmeasured during thesetests confirm the experimentalobservations of Campbell andWeynberg(1980) who conductedimpact measurements on acircular cylinder which was driveninto a stillwater surfaceat various angles of inclination.They found that the slammingforce increasedmore gradually andthat the corresponding peak loaddecreased with inclination,although thepresence of disturbanceson the water surface sometimestended to induce significant dynamicresponse.Figure 4.16 comparesthe slamming force on thetest cylinder for 0 = 0°, 4.8°and 9.6°, and anon-breaking regular wave withT = 1.2 sec, and H= 19.3 cm which is one of therelatively steepwaves used in this study. Thedynamic response of the cylinderis still evident inthe forcetraces, although it isless severe for the case of0 = 9.6°. In comparison tothe case of thehorizontal cylinder where theimpact force reaches a maximumof 11.3 N in 12.6 msec, thepeakforce drops to7.9 N and the corresponding rise-timeincreases to 36 msecwhen 0 = 4.8°. Thereis no evidence of anyslamming force maximumin the case where 0= 9.6°, and the rate ofincrease of the normalforce is quite gradual.In the case of a wavewith medium steepness,T = 1.5 sec, H = 18.4cm, the peak impact forcedecreases from 6.2 Nto 2.5 N, and the rise-time89increases from 18 msec to 55 msec when 0 changes from 0° to 4.8°. There is no evidence ofslamming in the case of 0 = 9.6°. The final test was conducted with a wave of T = 1.8 sec, andH = 17.8 cm which has relatively low steepness, and the force traces for the 3 inclinations arecompared in Fig. 4.17. The dynamic effects are less severe due to the reduced wave steepness.It is seen that the slamming force is not the dominant component of the hydrodynamic force, andthat it is barely evident even in the case of 0 = 4.8°. An interesting feature of thesemeasurements is that although the rise-time of the slamming force increases to 40 msec when0 = 4.8°, compared to 10 msec for the horizontal cylinder case, the corresponding peak force hasnearly the same value of 2 N in the two cases. This may be caused by the buoyancy forcebecoming the dominant component after the early stages of impact, and offsetting anydifferences in the relatively small slamming force seen in this case. In the case of 0 = 9.6°, theforce rises very gradually in a manner similar to the other wave conditions.The observations discussed above are for individual cases taken from multiple slammingevents observed in an experiment. These also tend to show scatter in a manner similar tohorizontal cylinder tests due to the presence of disturbances on the water surface, although thescatter decreases with increasing cylinder inclination which results in reduced slamming forces.However, the trends for the reduction in the slamming force and increase of rise-time are quiterepeatable.4.2 Numerical SimulationThe numerical method for estimating the various components which makeup the vertical waveforce acting on a section of a rigid horizontal circular cylinder locatednear the free surface,taking account of wave slamming and intermittent submergence has beendiscussed in Chapter 2.Two numerical models, namely Models I and II which essentially differin their treatment of thevertical force during the partially submerged phase, have beenused to obtain time histories ofthe vertical force on a cylinder in simulated regular and random waves. The effect of variations90of the governing non-dimensional parameters on the magnitudesof the maximum force areexamined, and results from alternative models for evaluating thetime histories of the force inregular waves are compared. Corresponding resultsfor random waves are also shown.The force predicted by the numerical model is compared withexperimental observations forboth cases of a rigid and dynamically responding cylinder.A practical application of the variousmethods described is also illustrated.4.2.1 Regular WavesAs indicated in Chapter 2, the governing non-dimensionalparameters in this study are oa/g,co2H/g and hJH, since deep-water conditionsare assumed and the influence of Reynolds numberis only considered through the choiceof the empirical force coefficients. The influence of theseparameters on the vertical forceis investigated. Force coefficient values ofC0 = it, Cd= 0.8,andCm = 1.7 (Cmo = 1.7 for Model II) have been adopted for this investigation. Although recentstudies and the results of the present experimental investigationindicate a higher value of C, avalue of it is being used to examine theproperties of the dimensionless maximum vertical force.The variation of the inertia coefficient, andthe combined slamming and drag coefficient(C÷j)with submergence in Model II isassumed to be as shown in Figs. 2.5 and 2.7 respectively.Figure 4.18 shows the variation of the dimensionlessvertical force F/p ga2 over one wave cycleforo2aJg = 0.05, o2HJg = 0.6 andtwo cylinder elevations, h/H = -0.3 and 0.3, and comparesthepredictions of Models I and II. Itcan be observed that significant differences in theforcepredicted by the two models occur inthe partially submerged stage, with Model I predictingalarger force than Model II. This is dueto the different assumptions made in the two models withrespect to the variation of the hydrodynamic force duringpartial submergence. In Model II,thecombined slam and drag coefficient usedto determine the velocity-squared force decreasesrapidly after the instant of slamming,while the inertia coefficient begins increasing fromzero.The hydrodynamic force determined using thesevarying coefficients is less than the linearvariation of the force given by ModelI. Model II also predicts a negative forceas the wave91recedes from the cylinder which isnot shown by Model I. This differenceis associated with theinertia and drag forces computed in ModelII when the cylinder is partially submergedas the freesurface recedes. The forcedue to this contribution is significant whenthe cylinder is above themean water level since boththe inertia and drag loads are downwardand oppose the buoyancyforce.From the aspect of engineeringdesign, the maximum force on thecylinder over a wave cycleis of particular interest. Thishas been computed for a range of waveconditions and is shown indimensionless form in Figs.4. 19(a)-(d) as a function of the dimensionlesscylinder elevationh/H, for four values of steepnessw2H/g (0.2, 0.4, 0.6 and0.8) and for four values of thefrequency parameter co2a/g(0.005, 0.01, 0.05 and 0.1).The values of H/a are also indicatedinthe figure. The solid anddashed lines correspondto the predictions of ModelsI and IIrespectively. The closeagreement overall betweenthe two predictions provides somejustification for the use of thesimpler Model I by Isaacsonand Subbiah (1990). The maximumforce with respect to cylinder elevationis seen to occur at h/H= 0 for most values of thesteepness and frequencyparameters. Consequently, adesigner should avoid placing horizontalstructural elements inclose proximity to the still water level.A number of features of Fig. 4.19can be explained by considering thestage of the wave cyclewhich gives rise to the peakforce. For the largest valuesof H/a, which are containedin Figs.4.19(a) and 4.19(b), the peak forceover most of the elevationrange is due to slamming. Thisgives rise to the parabolic-typevariation of peak force withh/H, with maxima near h/H =0.This is because i at the instantof impact is greatestat h/H = 0. At lower valuesof h/H (< -0.4),the slamming force is absent,or is small since i at the instantof impact is then also small.Thepeak force then occurs after fullsubmergence, at a phase whichdoes not vary with h. Sincetheflow kinematics are assumed tobe constant with elevationnear the mean water level,thismaximum force is seen not tovary with cylinder location.For the lowest values ofH/a, which are contained in Figs.4.19(c) and 4.19(d),a slightdisparity in the peak force predictionsof the two models isapparent. This discrepancyarises92because the force maxima then occur mostly during partial submergence, wherethe predictionsof the two models differ the most. Furthermore, both models thenpredict the maximum force toincrease fairly uniformly with decreasing h/H. This is because for this range ofconditions (largew2a/g and small (02H/g or H/a), the buoyancy force becomes significant incomparison with theother components (see Eq. 2.4), and for incomplete submergencethis increases with decreasingh/H. The curves in Fig. 4.19(d) do not reach a maximumvalue since partial submergence stilloccurs until h/H < -0.5 - 2a/H, which is below the range of h/H shownin the figure.The largest maximum force with respect to variationsin cylinder elevation was seen in Fig.4.19 to occur in most cases near h = 0, and is ofparticular interest. Figure 4.20 shows thislargest force in non-dimensional form as a function ofthe frequency parameter o2a/g for twovalues of wave steepness o)2HJg (0.2 and0.6). The predictions of both Models I and II areincluded, but are virtually indistinguishableso that the agreement between the two modelspredictions of this largest peak force is verygood.The figure indicates that the non-dimensional maximumforce becomes constant for smallervalues of H/a, corresponding to increasing values of co2aJgfor any particular value ofo)2HJg. Bycomparing these results with those of Fig. 4.19,it may be seen that for larger values of H/a thelargest force occurs at h/H = 0 and is associatedwith slamming, whereas for lower values of H/athe largest force occurs at lower elevationsof the cylinder and is associated with buoyancy,inertia and drag during partialor full submergence. In fact, for the selected valuesof theempirical coefficients C,CdandCm, the largest force for H/a < 6.7 is due to buoyancy andinertia only, and these give rise toa non-dimensional force which is independent of&a/g. Thiscorresponds to the observed ranges in Fig.4.20 over which the nondimensional force is constantwith respect to &a/g.4.2.2 Random wavesAs discussed in Section 2.5.3, resultsfor random waves have been obtained both fora narrowband spectrum as well as for a two-parameterPierson-Moskowitz spectrum. For bothcases,93conditions corresponding to a significant wave heightH = 10 m, a peak period Tp = 14.3 sec,and a cylinder diameter D =40 cm have been used. On the basis of the results shown for regularwaves, it is expected that both Models I and II will give similar predictionsof the force maxima,and consequently only Model II has been used throughout.The spectral density and corresponding amplitude spectrum for both the narrow-band andPierson-Moskowitz spectra are shown in Fig. 4.21. For the narrow-band spectrum,the simulatedwave and force time series were obtained with a record lengthTr= 2860 sec, a time intervalAt = 0.559 sec, a frequency range corresponding tor’o= 0.1 (see Eq. 2.76) and a frequencyinterval Af = 0.00035 Hz. These values correspond to approximately200 waves, and a totalnumber of data points N = 5120. However, this isincreased to approximately 8000 pointsbecause of additional points used for durations of partial submergence.In the case of thePierson-Moskowitz spectrum, the time series were obtained withTr = 1500 sec, At = 0.25 sec,and Nf= 1500.Portions of the simulated time series of the free surface elevation andthe correspondingvertical force for the cylinder located at the still water level,h = 0, are shown in Fig. 4.22 for thecase of the narrow-band spectrum, and in Fig. 4.23 for the case ofthe Pierson-Moskowitzspectrum. In the latter case, the vertical force is seento contain a relatively large number ofpeaks for each period of cylinder immersion. Furthermore,for this case the largest force maximaare not necessarily associated with waves which have the largest individualheights taken in thesame order. Indeed, the largest dimensionless peak force is34.5, and corresponds to a waveheight of 12.8 m, based on successive zero-upcrossings,or a height of 17.2 m based onsuccessive zero down-crossings. The dimensionless maximum force forregular waves withthese two heights would be 26.1 and 51.5 respectively.Thus the extension of regular waveresults to the random wave case, which may well be appropriate fora narrow-band spectrum,may not be applicable for a broad-band spectrum.Various statistics of the maximum force can be obtained from time seriessuch as thosedescribed above. In this study, the simulated data correspondingto Figs. 4.22 and 4.23 have94simply been used to obtain the corresponding probability densities of the peak force, and thesedensities are compared to the analytical predictions of Isaacson and Subbiah (1990) in Fig. 4.24.Isaacson and Subbiah’ s results are based on the assumption of a narrow-band spectrum, but it isof interest to examine the adequacy of their predictions for spectra of more general shape so thatthe comparison for the broader band spectrum is also given. A possible limitation of theirapproach may be the assumption of a single force peak within each wave. In orderto attemptaccounting for this, the probability densities from the simulations have been obtained on thebasis of all the force maxima in each record, denoted method A and shownas a solid thin line inFig. 4.24; and a single (largest) force maximum for every wave (definedby successive zeroupcrossings), denoted method B and shown as a solid thick line in Fig. 4.24. These two methodsyield quite different probabilities, since the latter omits the smaller secondary force peaks withineach wave which may be present.Figure 4.24(a) indicates that for the case of the narrow-band spectrumthe agreement betweenthe analytical prediction and the simulation results based on a single force maximumfor eachwave is quite reasonable; whereas the simulation results based on all the peakforces yieldspoorer agreement. This is because the simulated force record contains secondarypeaks for somewaves, as indicated in Fig. 4.22(b), and the analytical predictionsdo not apply to these.Discrepancies between the predictions and either of the simulation results alsoarise because ofsample variability, in that the simulation results pertain to only one force record,whereas theanalytical predictions correspond to the average of many force records.Figure 4.24(b) indicates that for the case of the broader band spectrum theagreement betweenthe analytical prediction and the simulation results based on either method isgenerally poor, inthat the theory predicts a relatively large number of force peakswithin a relatively narrow peakforce range. As before, some of this discrepancy arises partlybecause of the secondary peakswhich occur within each wave, andpartly because of sample variability. Even so, the pooragreement indicates that the analytical model cannot be extended directlyto the case of a broadband spectrum. This is consistent with the numerical illustration given earlierwhich relates tothe single largest peak force.954.2.3 Comparison with Experimental ObservationsIt is of interest to compare the vertical force predicted by the numerical model for the case ofthe test cylinder in regular waves with corresponding experimental observations. This makes itpossible to test the validity of the various assumptions made in the formulation of Model IIwhich relate to the variation of the combined slamming and drag coefficient, and the inertiacoefficient during partial submergence. The predictions of the models for both the rigid anddynamically responding horizontal cylinder are presented.Preliminary results from the rigid cylinder model simulations indicate that the variation of theslamming coefficient during the early phases of partial submergence corresponds more closely toCampbell and Weynberg’ s (1980) expression derived from their experimental observations(Eq. 2.27), which is based on aC0 value of 5.15 and has a faster rate of decay in comparison toTaylor’s model as seen in Fig. 2.4. It is also simpler to use in comparison with Taylor’s modeland allows straightforward changes to the value ofC0used in the simulation. Theimplementation of this expression is similar to that described in Section 2.2.5 wherein thecombinedCs+dcoefficient is initially determined using Eq. 2.27 during partial submergence ofthe cylinder, until it reaches a value ofCdwhereupon it remains constant at the value ofCdforthe remainder of the wave cycle. The variation of C predicted by Eq. 2.27 reaches a minimumvalue of 0.74 at s/a = 1.3 and then begins rising gradually with increasing submergence. Thisprevents the use of the combinedCs+dmodel for values ofCdlower than 0.74, and Eq. 2.27 hasbeen modified to accommodate lower values ofCd.This is achieved by assuming that the valueofCreaches the value ofCdat the instant of complete submergence (s/a = 2). The modifiedversion of Eq. 2.27 for this criterion is hence given by:r 1 (0.5Cd 1= C[1 + 9.5s/a15.15- O.O25Js/aJ (4.2)whereC0is the desired value of the peak slamming coefficient, andCdis the drag coefficientwhich is less than 0.74.96The inertia coefficient is taken to increase from zero to the desired value of Cm and is then heldconstant. The values ofCdandCmused in the numerical model are based on estimates derivedfrom a least squares fit method applied to the fully submerged portion of the experimentalvertical force record. The corresponding experimental wave record is used to obtain the waterparticle velocity and acceleration values which are used in the numerical model.Horizontal CylinderFigure 4.25 compares the measured vertical force on the test cylinder located at h = 0.5 cm, forthe case of incident waves with period T = 1.2 sec and height H = 15.2 cm, with the prediction ofthe numerical model for a rigid cylinder based on C0 = 5.15, Cd = 0.7, and Cm = 0.6. It is seenthat the significant differences between the predicted and observed force occur during thepartially submerged stage. The predicted slamming force reaches a maximum instantaneously,while the experimental record shows a rise-time of 20 msec. However, the nature of thepredicted vertical force variation compares very well over the entire length of the experimentalrecord. The dynamic model described in Section 2.3.2 incorporates the effects of cylinderresponse to the applied force and also allows the inclusion of a specified rise-time. The resultfrom a simulation which uses the test cylinder’s natural frequency of 290 Hz, a damping value of2%, and a rise-time of 20 msec, for the same cylinder location, wave conditions, and forcecoefficients as above, is illustrated in Fig. 4.26. Only the initial 0.25 sec of impact are shown forbetter detail, along with the segment of the corresponding measured force record. It is seen thatthe predicted force matches the measured force quite accurately. While the former quantityshows an oscillation at the natural frequency superposed on the general trend of the force, thelatter shows seemingly random oscillations which are due to the presence of other modes ofvibration (see Fig. 3.6). The oscillations in the predicted force trace are damped outapproximately 0.3 sec after impact, and the subsequent behaviour of the force is the same as thatpredicted by the rigid cylinder model.Figure 4.27 compares the predicted and measured vertical force for incident waves withT = 1.5 sec, and H = 18.4 cm, and the cylinder located at h = 0.5 cm. The values of the force97coefficients chosen are C = 5.15, andCd = Cm = 0.8. It is seenthat while the trends of thepredicted and measured force agree quite well during the initial phase of slanmiingand partialsubmergence, they show markedly different behaviour after full submergenceof the cylinder.This is possibly due to the choice of the inertia coefficient which is seen tohave values whichare close to zero or negative in some cases when estimated using the least squaresfit method. Inthe present case, the least squares method estimates ofCdand C are 0.7 and -0.1 respectively.The discrepancy in the predicted force consequently arisessince aCmvalue of 0.8 is used.It is also seen in Figs. 4.25 and 4.27 that the vertical force is generallyoverpredicted by thenumerical model during the early stages of slamming. Thismay be due to the larger buoyancyforce estimated by the numerical model based onthe assumption of an undisturbed free surfacein the vicinity of the cylinder while the actual submergenceof the cylinder is usually lesser dueto the formation of spray roots. There are indicationsof a slightly larger negative (downward)force which persists for a longer duration in theobserved force record as the free surface recedesfrom the cylinder. Video records of the slamming event show that evenafter the free surfacedrops below the level of the cylinder, there is a mass of waterwhich remains suspended from thecylinder for a longer length of time as seen in Fig. 4.28. Thismay be the cause of the observedbehaviour of the negative force which is not predictedby the numerical model.Figures 4.29 and 4.30 present comparisons of themeasured force record for the case of thecylinder located at h = -4.5 cm, in incident waves of T= 1.5 sec, and H = 18.4 cm, with thepredictions of the rigid and dynamic cylinder models respectively.The values ofC0,Cd,andCmused in the simulation are 5.15, 1.25 and 0.5 respectively.A rise-time of 18 msec is alsoused in the dynamic model. It is seen that the correspondencebetween the observed andpredicted force is very good.Inclined CylinderThe numerical model for the force on an inclined cylinderlocated near the still water level hasbeen described in Section 2.4. The behaviour of the variousforce components is modelled in a98manner similar to the above case of a horizontal cylinder and the components of the waterparticle kinematics normal to the cylinder axis in the vertical plane are used to estimate theslamming, drag and inertia force.Figure 4.31 compares the early stages of the measured and predicted forcedue to a slammingevent in waves of period T = 1.8 sec, and height H = 17.8 cm, on the test cylinder which wasinclined at 8 = 4.8°, and had a clearance h = 0.5 cm. The force coefficientsused in thesimulation are C = 6.18,Cd = 0.8, and C= 0.5. It is seen that there is an indication of a smallforce peak approximately 0.08 sec after the instant of slamming, and the force rises graduallyuntil the cylinder is fully submerged. The predicted values agree fairly well with the observedforce, although the numerical model predicts a slightly larger maximumat the instant when thecylinder becomes fully submerged.Figure 4.32 presents another comparsion of the measured and predicted force overone wavecycle for the case of the test cylinder inclined at 8 = 9.6°, located at h = -2.0 cm andsubjected towaves with T = 1.2 sec, and H = 15.2 cm. Force coefficients C0= 5.15,Cd= 0.7 andCm0.6are used in the simulation. The two force records are seen to comparevery well, although theobserved force is seen to rise slightly faster during the earlystages of cylinder immersion.The examples shown above indicate that the numerical modelsfor predicting the vertical forceon a horizontal or inclined cylinder located near the free surfacegenerally yield results whichcompare well with experimental data. The method for combining the slamming,buoyancy, drag,and inertia force components works reasonably well when used with appropriatevalues of theforce coefficients.4.2.4 Practical ApplicationIn order to illustrate the various approaches indicated here in the contextof a typical oceanengineering application, consideration is given to an example correspondingto a cylinder ofdiameter 40 cm, length 16.0 m, and wall thicknessof 12.5 mm, located with its lower surface at99the still water level, h = 0 cm, and subjected to a wave train of period T = 12sec and heightH = 8 m. Two configurations of the cylinder have been analyzed based on differentend fixityconditions. It is also assumed that the dynamic response of the cylinderdue to the appliedslamming force is predominantly in its fundamental mode.The first case in which the cylinder is assumed to haveboth ends rigidly fixed yields a modalstiffness value of 7.16 MN/rn, and a modal massof 1906 kg. This corresponds to a naturalperiod of 0.1 sec. Force coefficients have been assumedas follows:Cd= 0.8,Cm= 1.2,C0 = 4.5. Figure 4.33 shows the predicted time series of the appliedvertical force on thecylinder assuming it to be rigid (shown as a solidline), where it is seen that the maximumvertical force of 63.1 kN occurs at the instant of impact.The buoyancy component of the force(dashed line), and the mid-span response of the cylinder(chain-dashed line) to the applied forceare also shown in Fig. 4.33. The maximum mid-spandisplacement of the cylinder due thisapplied force is estimated to be 12.1 mm. If the maximumforce of 63.1 kN is treated as a staticload, the corresponding mid-span deflectionfor the cylinder with both ends fixed is seen to be11.5 mm. This indicates that in this example, the severityof the impact force is amplified due tothe dynamic response of the cylinder. It is alsonoted that the mid-span displacement of thecylinder due to its self-weight would be 3.4 mm.The second case involves the treatment of the abovecylinder with pinned supports. Thecorresponding modal stiffness decreases to 0.67 MN/rn, andthe modal mass is also reduced to953 kg, which results in a larger natural period of 0.23sec. The static mid-span displacementdue to the cylinder’s self weight is 17.0mm. The dynamic response of this system to the appliedforce described above is also shownin Fig. 4.33 as a dotted line. The maximum mid-spanresponse in this case is estimated at 38.7mm. If the maximum force of 63.1 kN is treated as astatic load, the corresponding mid-span deflection for the cylinderwith pinned end conditions isseen to be 57.4 mm. This indicates that the effect of theimpact force is attenuated by about 33%since some of the applied force is being resistedby the inertia of the cylinder.100Since the natural period of the cylinder in the second case is sufficiently large in comparison tothe duration of the impulse, it is possible to estimate the maximum mid-span response of thecylinder as the sum of the maximum dynamic response due to the slamming impulse, and thestatic displacement due to the more gradually varying vertical force after the end of the impulse.Although the maximum response of a single degree of freedom system to an applied impulseoccurs at T/4 sec after the end of the impulse, where T is the natural period of the system, themaximum response of a cylinder subjected to wave impact will usually occur later due to thepresence of the increasing buoyancy, drag, and inertia force components, and in general may beassumed to occur between TI4 and T /2 sec after the end of the impulse. This is evident in theabove example where the maximum response of 38.7 mm occurs 0.12 sec after impact which isslightly larger than Tfl12 sec. On the basis of an impulse coefficient C1 = 0.7, which correspondsclosely to the predicted applied force time series, and the vertical force on the cylinder at TI3 or0.08 sec after the end of the impulse, which is seen to be 24 kN at t = 3.12 sec in Fig. 4.33, themaximum displacement is instead predicted to be 43.7 mm. This shows that the impulseresponse approach may be used to obtain a quick estimate of the member response under certainconditions. It is emphasized that the present example relates to fundamental mode motions only,and that higher mode motions, with relatively high natural frequencies, would also be excited bythe wave slamming.4.3 Breaking Wave ImpactResults from the experimental study of breaking wave impact on a horizontal cylinder arepresented in this section. Tests have been conducted for three differentelevations of thecylinder, and six different horizontal locations of wave breaking. Both horizontal and verticalcomponents of the force on the cylinder were recorded, and then analyzedto obtain the peakforce and rise-time after applying corrections for dynamic response of the cylinder. Althoughthe force records illustrated in this section show the corrected horizontal force trace, thecorresponding vertical force records are uncorrected since the dynamic correction is carried outonly for the peak value of the vertical force.101The corrected rise-time for the peak horizontal force component is within 1 msec of the rise -time for the vertical force component in all the tests. This is evident in the force traces shown inFigs. 4.36, 4.40, and 4.44 where the vertical and horizontal force peaks are seen to occur nearlysimultaneously. This allows the estimation of the peak resultant force from the vertical andhorizontal force components by assuming that they occur at the same time. The peak values ofthe corrected horizontal and vertical components of the impact force, the peak resultant force andits direction, and the corresponding rise-time are presented in Table 4.5.4.3.1 Cylinder Elevation IThe first set of tests were carried out with the lowest cylinder position (h = 4.7 cm). Figure4.34 presents a series of images that show breaking wave impact for the case Xb = 25.5 cm. Thefirst two frames show the plunging jet of the wave in the formative stage. The 3rd frame showsthe initial impact to be due to the lower face of the wave and not the spilling front of the wave.The last frame shows the cylinder just after complete submergence. Although the jet has startedcollapsing and there is evidence of splash, the water in the vicinity of the cylinder is relativelyundisturbed. The wave profiles have been digitized and a typical example corresponding to thecase where the wave breaks atxi, = 36cm is shown in Fig. 4.35 which also indicates the locationof the cylinder with respect to the breaking wave. The horizontal and vertical axes of Fig. 4.35correspond to the horizontal and vertical dimensions in the vicinity of the test cylinder, and thefour curves which represent the wave-front are spaced at time intervals of 1/30 sec from left toright. These curves are used to estimate quantities such as the impact velocity and curvature ofthe water surface.The observed vertical force and the corrected horizontal force record obtained for the testwhereXb = 25.5 cm are shown in Fig. 4.36, where it is seen that the peak horizontal forcecomponent of 17 N, and the peak vertical component of 21 N (16.9 N after correction) occurnearly simultaneously at 3.8 msec after impact. The normal velocity of the water surface prior toimpact is estimated from Fig. 4.35 to be 0.63 rn/sec and the radius of curvature of the surface isquite large (>20 cm). The vertical force component shows a temporal variation similar to that102obtained from regular wave impact, although the horizontal force component in the present caseis also quite significant on account of the slope of the water surface at the instant of impact.Table 4.5 summarizes the peak forces observed for 6 different values ofxb.For all the tests(except for x = -3.5 cm), the peak horizontal and vertical force components show an increasingtrend as the wave breaking location progresses from upwave to downwave of the cylinder. Thepeak resultant force increases from 12.6 N to 30.5 N as the breaking location advances from7.5 cm to 49 cm. The direction of the resultant peak force also changes from 53° to 36°, as thehorizontal force component becomes more significant. This trend is possibly due to theincreasing steepness of the lower face of the wave at the location of the cylinder with anincreasing value of xb. The rise-time to the peak force is quite similar for all these cases andvaries from 2.6 msec to 4.4 msec. The test for the case ofx= -3.5 cm is different from theother cases in this set as the wave has already broken before the occurrence of impact. There is alarge amount of splash and disturbance on the water surface and consequently it is difficult toobtain a good estimate the water surface velocity in this case. The peak horizontal forcecomponent has a magnitude of 15 N, but the peak vertical component is directed downward andhas a magnitude of 4.5 N. The resultant force of 15.7 N is higher than the peak force in the caseofxb= 7.5 cm. This force is directed 17° below the horizontal indicating that the downwardvelocity component of the plunging jet contributes to the impact force. The peak impact forcedue to a regular wave (T = 1.2 sec, H = 20.5 cm) is mainly due to the vertical force component,while the horizontal component has a more gradual rate of increase. The peak resultant impactforce of 7.2 N is significantly lower than the breaking wave impact forces seen in these tests, andalso shows a larger rise-time of 15.1 msec.4.3.2 Cylinder Elevation IIThe impact forces for this location of the cylinder (h = 8.7 cm), were the highest among all thetests. Figures 4.37, 4.38 and 4.41 show a series of images which illustrate the 3 different typesof breaking wave impact that occur in this set. Figure 4.37 shows the impact for the case of= -3.5 cm. It can be seen in the 1st frame that the plunging jet of the wave has already broken103onto the lower face of the wave, and that there is significant amount of splash. The 2nd and 3rdframes show the onset of impact which appears to be initially due to the jet and theaccompanying splash. The last frame shows that even after complete submergence, there is a lotof residual air entrainment and disturbance in the vicinity of the cylinder. It is not possible toestimate the kinematics of the wave from the video records in this case.The next series of images shown in Fig. 4.38 are for the most severe impact case correspondingto wave breaking at xb = 25.5 cm. The first three frames show the development of the jet of thebreaking wave, and indicate evidence of slight air entrainment in the upper concave face of thewave. The 4th frame indicates that the impact is quite strongand that it arises from the mainbody of the wave after the forward tip of the wave has passed over the cylinder. The digitizedprofiles corresponding to this case are shown in Fig. 4.39, and it is seen that theradius ofcurvature of the wave under the jet is approximately 1.7 cm which is comparableto the cylinderradius of 2 cm. This leads to a larger contact area at the instant of impact,and consequently alarger force. The time series of the force components for thistest are shown in Fig. 4.40.The impact due to a wave breaking atxb= 49 cm is shown in Fig. 4.41. In contrast with thetwo previous examples, the jet in the wave crest has not developed prior to impact,the impactitself does not disturb the flow significantly, and the presenceof splash and air entrainment isevident only after the cylinder is fully submerged. The radius of curvature(9.7 cm) of the watersurface at the instant of impact is also noticeably larger than in the previous case,x = 25.5 cm.There is a decrease in the peak horizontal force observed in this test.A summary of the results obtained from the 6 breaking wave tests for this cylinder elevation isincluded in Table 4.5. The peak resultant force increases from 32.1 Nto 63.7 N as the wavebreaking location advances from -3.5 cm to 25.5 cm. It is observed that thisincrease in the peakforce is associated with a decrease in the radius of curvature of the wave-front,and a lessdisturbed water surface prior to impact. The peak force decreases asx increases beyond25.5 cm, even though there seems to be a slight increase in velocity at impact. This couldpossibly be attributed to the increasing radius of curvature of the water surface at impact. The104horizontal force component is dominant for all the tests in this set, and consequently thedirection of the peak force is between 14° and 18° above horizontal for all thetests. The casewherexb = 17 cm yields force data that does not conform to the general trend seen for the other5 cases. A possible explanation for this may be due to the transition froma relatively disturbedimpact where the wave has nearly broken(xb = -3.5 and 7.5 cm), to a less disturbed and moredefined impact (xb = 25.5, 36 and 49 cm). The rise-times for the peak forceare very similar,ranging from 1.6 to 1.8 msec, except for the case ofxb = 17 cm where the rise-time is 2.9 msec.The results from the regular wave impact test yielded a peak slammingforce that isapproximately 20 times less than the largest impactforce due to a breaking wave. The peakhorizontal and vertical force components were equal at 2.3 N, and yielded a peakresultant forceof 3.3 N that had a rise-time of 6.1 msec.4.3.3 Cylinder Elevation ifiThe last set of tests has been conducted with the elevationof the cylinder axis at 12.7 cm abovethe still water level. At this elevation, the cylinder issubjected to impact only by the crest of thebreaking wave. Fig. 4.42 shows a series of images for oneof the tests in this set for the casexb= 25.5 cm. The first two frames in the figure show the developingjet at the wave crest andthe onset of some spilling on the front of the wave. The 3rdand 4th frames show the impact ofthe convex tip of the jet with the cylinder. The digitized breaking wave profilesfor the test inwhich the wave breaks at x = 17 cm are shown inFig. 4.43. Figure 4.44 shows the forcerecords corresponding to wave breaking atx = 25.5 cm. The corrected peak forces and rise-times for the tests in this set are presented in Table 4.5. The velocityof the water surface atimpact cannot be estimated whenx = -3.5 and 7.5 cm, as the wave crest has already broken.The observed peak forces are then 27.8 N and 24.8 N, respectivelyand the force vector isdirected about 24° above the horizontal. For the remaining teststhe velocity at impact increasesmarginally from 1.7 mlsec to 1.8 mlsec and is about0.3 rn/sec faster than the celerity of thewave. The jet which develops at the wave crest hasa convex profile. For x = 17, 25.5, and36 cm, the peak forces show an increasing trend(23.3, 23.4 to 31.1 N respectively). In the latter105two tests the peak force vector is directed approximately 30° below horizontal. The rise-timesfor all the tests are relatively similar and vary between 2 to 3.2 msec.The tests in regular wavesdo not indicate the presence of a slamming force for this cylinder elevationand may beexplained by the low water particle velocity at impact. In this case, the peak horizontalandvertical force components of 2.5 N and 2.8 N respectively occurafter full submergence, andyield a maximum resultant force of 3.8 N.4.3.4 Slamming Coefficients due to Breaking WaveImpactThe plunging wave impact forces observed at the three different cylinder elevations showsignificant differences. The dynamics of a plunging wave are considerablymore complex thanthose of a regular non-breaking wave. The geometryof the wave at impact in the vicinity of thecylinder depends on the vertical and horizontal location ofthe cylinder. The presence ofentrained air and splash affect the local water particlekinematics and pressures, andconsequently influence the magnitude of the impact force. Ananalytical treatment whichaccounts for all these factors is still not feasible using currentlyavailable techniques. It may bedifficult to formulate a single closed-form expressionsimilar to Eq. 2.22 which can predict theimpact force due to the various types of wave impactseen in this study.Although the force measurements in thisstudy are reasonably accurate, the method used toestimate the water surface velocity at impact has limitations.The video equipment used in thisstudy captures images at the rate of 30 frames/see,whereas a rate of 80 to 100 frames/sec isdesirable in order to record a rapidly moving breakingwave and determine velocities andaccelerations from the images with reasonable accuracy.However, the test results reported inthis study may provide a useful indication of the generalproblem.Table 4.5 shows that the water particle velocityin the vicinity of the cylinder varies with wavebreaking locationxb,and the cylinder elevation h. If these velocitiesare used to determine apeak slamming coefficient based on Eq. 2.22, the values obtainedshow some scatter and do notreveal any obvious trends. At the lowest cylinderelevation of h = 4.7 cm, the peak slammingcoefficient C based on the local particle velocity variesbetween 8.0 and 10.7. For the next106case (h = 8.7 cm), the slamming coefficient varies from 1.9 to 10.1. At the highest elevation(h = 12.7 cm), the slamming coefficient vanes from 1.3 to 1.5. A constant velocity equal to thecelerity of the breaking wave (1.5 mlsec) has also been used to obtain the peak slammingcoefficient based on the test that yields the largest resultant force at each of the three cylinderelevations. The largest values of C0 thus obtained are 2.2, 4.5 and 2.2, for the low, middle andhigh cylinder elevations respectively. It is seen that the most severe impacts occur when thecylinder is located above the still water level at approximately two-thirdsof the wave crestelevation. The combination of large impact velocities and smaller radiusof curvature of thewave-front at this elevation can cause impact forces which are significantly larger thanthoseoccurring for non-breaking wave conditions.The issue of using results from model studies of breaking wave impactat prototype scalesremains unresolved. Three different types of impact pressures can becaused by breaking waves.These are defined as ventilated shocks, compressionshocks, and hammer shocks, and it isexpected that all the breaking wave impacts seen in the present studyare of the ventilated typewherein any air present between the wave-front and thecylinder is able to escape during impact.Kjeldsen (1981) states that the absolute dimensions and coalescing characteristicsof entrainedair bubbles differ in sea water and fresh water, and thatthe damping effect on shock peakscaused by bubble populations with different absolutebubble sizes is unknown, which generallyprecludes a viable model law for ventilated shocks. Althoughpotential flow theory has beenused to obtain a good approximation of the development of a plungingbreaker (Vinje andBrevig, 1981), the inability to predict air entrainmentand the formation of foam makes itdifficult to estimate impact pressures and forces during theearly stages of a ventilated impact.Lundgren (1969) suggests that peak shock pressuresand impulses for ventilated shocks mightbe scaled according to Froude’ s law, but states thatthis will yield conservative results since it isexpected that relative air entrainment for this kind ofimpact is expected to be higher in the sea incomparison to laboratory experiments. This conclusionis supported by Alfred FUhrböter (1986)who compared results for wave impact and run-upfrom experiments at 1:10 model andprototype scales.107Chapter 5CONCLUSIONSThe primary objective of this thesis has been to examine the impact loads due to wavesinteracting with a horizontal circular cylinder located near the still water level. Numericalmodels have been developed to predict the vertical force on such a cylinder, and these have beenextended to the case of a dynamically responding cylinder, and an inclined cylinder.Experiments have been conducted to measure the vertical force on a horizontal cylindersubjected to slamming in regular non-breaking waves. Some experiments have also been carriedout with an inclined cylinder. The observed data has been analyzed to obtain peak slammingcoefficients and related parameters. The numerical models have been used to examine thebehaviour of the maximum vertical force in regular and random waves, and the predicted forcehas also been compared with experimental observations. Experiments to measure the force dueto plunging wave impact on a horizontal cylinder have been conducted for different cylinderelevations and wave breaking locations. The conclusions of the present investigation arepresented in two separate sections which represent different aspects of this study.5.1 Wave Slamming on a Horizontal Cylinder5.1.1 Experimental StudyExperiments have been conducted to measure the vertical force due to waves incident on aslender horizontal cylinder located near the still water level. The water surface elevation and thecorresponding force have been measured over a duration of 20 sec whichincludes several108slamming events. This data has been analyzed to determine the maximum slamming coefficientC, the rise-time for the peak slamming force and related parameters. Corrections to themeasured coefficients to account for buoyancy, free surface slope and dynamic response and freesurface slope are indicated.Results from tests in regular waves of various periods and heights and with different cylinderelevations indicate that these coefficients exhibit a considerable degree of scatter, whichhas alsobeen reported by earlier experimental investigators. The maximum slamming coefficient C(with no corrections applied) for each of the 22 tests has mean values which vary from 3.04 to7.79, and exhibits no noticeable trend with either the incident wave characteristics or theelevation of the cylinder. It is possible that the absence of any such trend in the valuesof Cmay be due to the limited range of values of the parameters, the relatively low number ofobservations made, and the possibility of a complex dependence on severalof the parameters.The C.O.V. of C, varies from 0.09 to 0.36 and increases with thesteepness of the incidentwave. This trend is explained by the likelihood of increased disturbanceson the water surfacedue to the higher water particle kinematics in steeper waves. Individual valuesof C0 vary from1.86 to 11.7 and the overall mean of C, is 4.87. This compares well with theexperimentalresults of Campbell and Weynberg (1980) who reporteda mean value of C = 5.15. Theprobability density histogram of C based on a total of 311 slamming events recordedin the 22tests is seen to follow a log-normal distribution.The average value of the measured rise-timeT for the tests varies from 11.6 to 37.1 msec andits C.O.V. ranges from 0.13 to 0.36. There is a tendencyforTtto increase with the impactvelocity v/Jj which may be attributed to the slowerrise of impact pressures due to increasedpresence of entrained air and local disturbances. The rise-timeTrcorrected for dynamic effectsvaries from 10.8 to 37.1 msec for the 22 tests. An averagecylinder submergence (s/a)0 of 0.33(7 mm) was noted when the peak slamming force occurred.109The maximum slamming coefficient corrected for buoyancy, dynamic response and freesurface slope, denoted C0,varies between 2.92 and 7.60 with an overall mean value of 4.29, andexhibits no noticeable reduction in scatter in comparison to C.A method of defining slamming as an impulse has been demonstrated, and a correspondingimpulse coefficient C1 has been estimated from the experimental data. The mean value of C1estimated from the 22 tests varies from 0.26 to 1.36, with an overall mean of 0.73; and theC.O.V. of C1 varies from 0.04 to 0.41. The mean value of the impulse coefficient based on theimpact velocity corrected for free surface slope C varies from 0.26 to 1.47 with an overall meanof 0.77, and is higher than the corresponding value of C1 by between 1% and 9%. In general, Cexhibits trends which are similar to those of C1, and has similar C.O.V.s. The degree of scatterfor C is less than that for C by about 25% on average. The estimates of the impulsecoefficient did not show any observable trend with respect to incident wave conditions orcylinder elevation.The dimensionless impulse duration vT/a which indicates the relative submergence of thecylinder at the end of the impact has a mean value which varies from 0.29 to 1.15, and exhibitsless scatter than the quantityvnTr/a by about 54%. The overall mean value of vT/a is 0.69which corresponds to a submergence of about 1/3rd of the cylinder’s diameter at the end of theslamming impulse.Some experiments in regular waves have also been conducted for the test cylinder inclined at4.8° and 9.6° with respect to the horizontal. In general, the force records measured during thesetests indicate that the slamming force increases more gradually and that the corresponding peakload decreases with inclination, which is similar to the experimental observations of Campbelland Weynberg (1980). There is no obvious evidence of a slamming force maximum for thelarger tilt angle of 9.6°, and the rate of increase of the normal force is quite gradual which alsoresults in lower dynamic response of the cylinder. While these force measurements also tend toshow scatter in a manner similar to horizontal cylinder tests, the repeatability improves withincreasing cylinder inclination which results in smaller slamming forces.1105.1.2 Numerical ModellingA numerical model based on the use of a slamming coefficient for simulating the vertical waveforce acting on fixed, rigid, horizontal circular cylinder located near the still water level isdescribed. Only the two-dimensional case of unidirectional waves propagating in a directionorthogonal to the cylinder axis is considered. The model attempts to account for the limitationsof current theory regarding the variation of the slamming coefficient with cylindersubmergence,and treats the vertical force during the partial submergence phase in alternate ways.The modelhas also been extended to include the case of a horizontal cylinder which responds dynamicallyto the applied wave force, and an inclined cylinder.Model I (Isaacson and Subbiah, 1990) which uses a C0 value of it and simply assumes alinearvariation of the vertical force during partial submergenceof the cylinder, and Model II whichalso uses a C value of it but a modifiedCs+dcoefficient, are used to study the variation of themaximum force as a function of the governing non-dimensional parametersin the problem. It isseen that although there are notable differences during periods of partial submergenceof thecylinder, both models predict maximum forces which are very closeto each other, whichindicates that a detailed modelling of the force variation during the stageof partial submergencemay be avoided in certain casesFor the case of random waves, the time variation of the force is illustratedboth for a narrow-band spectrum and a two-parameter Pierson-Moskowitz spectrum. The probabilitydensities ofthe force peaks are compared with available analytical predictionswhich are based on theassumption of a narrow-band spectrum. These probability densities havebeen obtained using allthe force peaks in a record, as well as using a single force peak foreach wave, so that thepresence of secondary peaks during each wave is accounted for.The latter method providesbetter agreement with the predictions. In the case of a narrow-band spectrum,the agreement isquite reasonable, with discrepancies due in part to sampling variability.For the case of a broaderband spectrum, the agreement is quite poor and indicates that the analyticalpredictions cannot bedirectly extended to the case of a broad-band spectrum.111The predictions of the numerical model have also been compared with the experimental data,and results from the simulations indicate that the variation of the slamming coefficient during theearly phase of partial submergence is more accurately predicted by Campbell and Weynberg’ s(1980) expression which is based on a C value of 5.15 and has a faster rate of decay incomparison to Taylor’s model. It is seen that the proposed method for combining the slamming,buoyancy, drag, and inertia force components works reasonably well when used with appropriatevalues of the force coefficients, and that the model is able to predict a variation of the verticalforce which compares well with the experimentally observed force over the entire wave cycle.The dynamic model which also incorporates a rise-time parameter in the slamming force is seento perform equally well in predicting the wave force and the resulting response of the cylinder.It is noted that a finite rise-time used in the simulation will give rise to a larger dynamic responsefor the same value of C0 used with a zero rise-time. Reasonable agreement between simulationsand experiments is seen in the case of the inclined cylinder which validates the strip method forestimating the slamming force.The various methods including an approach based on an impulse coefficient for estimating themaximum response of a cylinder subjected to wave impact, are illustrated in the context of atypical ocean engineering application. In addition to the smaller scatter shown by theexperimental estimate of C in comparison to C, the use of the impulse coefficient method isrelatively simple, and appears to be effective in estimating maximum structural elementresponses for conditions under which this method is applicable.5.2 Plunging Wave Impact on a HorizontalCylinderAn experimental study of impact forces due to plunging wave action on a slender horizontalcylinder has been conducted for different cylinder elevations and wave breaking locations. Inaddition to the horizontal and vertical force components on the cylinder,a video record of theimpact has been obtained for the different tests. The maximum valuesof the force componentshave been corrected for the dynamic response of the cylinder, andthe resultant peak force along112with the impact velocity and radius of curvature of the wave-front whichare estimated from thedigitized video images, are reported.As expected, the peak impact force is found to depend onthe vertical and horizontal locationof the cylinder relative to the breaking wave. The impact is more severefor cases where theradius of curvature of the water surface at impact is similar to theradius of the cylinder. Inaddition to a significant horizontal force component, the magnitude ofthe impact force due to abreaking wave is considerably larger than that due toa regular non-breaking wave of the sameheight and period. The peak slamming coefficient C0,determinedfrom the recorded force andthe corresponding local water surface velocity at impact,exhibits considerable scatter, varyingfrom 1.3 to 10.7. If instead, the wave celerity is used asa basis for calculating C0,the largestvalues of C0 obtained are 2.2, 4.5 and 2.2 for the low, middleand high cylinder elevationsrespectively.No currently accepted scaling law is available for extendingresults from model tests onbreaking wave forces to prototype scales due to the differing characteristicsof the entrained airin the two cases. However, it is expected that theuse of Froude’ s law will lead to results on theconservative side when using model results at large scales.5.3 RecommendationsEstimation of Design LoadsThe results of the present investigation lendsupport to earlier studies which conclude that themagnitude of C0 is larger than the generallyaccepted value of t which has been used in design.The nature of the wave slamming phenomenon issuch that any departure from ‘ideal’conditions, including the presence of disturbanceson the water surface, entrained air,compressibilty, cylinder roughness, cylinder inclination,and non-normal approach of incidentwaves tend to result in a reduction of the severity ofthe slamming force. Other factors such ascylinder motion, and presence of fouling can result in elevatedvalues of the slamming113coefficient. It is currently not possible to use theoretical techniques which can account for theeffects of some of these factors.In the absence of detailed full-scale experimental data, it is recommended that designersuse aC value between 5 and 6 when calculating slamming forces ona horizontal cylinder in non-breaking regular waves. In cases where the dynamic response ofthe element is an issue, a timedomain simulation to determine the forces and design stresses may be warranted. Themodifiedexpression for the combined slamming and drag coefficient based onCampbell and Weynberg’ s(1980) results used along with the Morison equation and suitableforce coefficients is seen toyield reasonable estimates of the vertical force variation withtime. For structural elements withnatural periods greater than 0.5 sec, the use of an impulse coefficientC1 between 0.8 and 1.2 canalso provide a quick estimate of the dynamic response ofthe element.The data from the present study shows that the most severe impactforces occur when theunderside of the cylinder is close to the still water levelfor the case of non-breaking waveimpact, and the underside of the cylinder is approximatelytwo-thirds of the wave crest elevationabove the still water level for the case of plunging waveimpact.Further StudyThere are several avenues which can be explored further in thestudy of wave slamming oncylinders. In addition to a closer examination of the effect of cylinder tilton the slamming force,oblique wave impact in the case where the wave crest approachesthe cylinder at an angle is aproblem which merits further study. Laboratory experimentson wave slamming, especially testsinvolving Froude numbers higher than 1.5, which accurately measurethe impact velocity usinglaser-doppler velocimetry can also help to provide better estimatesof the slamming coefficient.Data on the statistical properties of the impactforce in experiments using random waves willalso be of use to designers.114Since the slamming force is sensitive to a number of parameters which are difficult toreproduce in a laboratory, a valuable contribution to the present understanding of the subject willinvolve reliable full-scale slamming force data on a platform element in typical sea states.Earlier reported full scale studies have suffered from problems including excessive filtering offorce data and the use of inadequate sampling rates. Currently available instrumentation andcomputer based data collection methods should allow an easier implementation of fieldmeasurements.Forces due to breaking wave impact on a horizontal cylinder are seen to beup to 20 timeslarger than those due to regular waves of the same period and height. Data fromthe presentstudy suggests that the maximum force is strongly dependent on the cylinder elevationand wavebreaking location. Accurate measurements of the curvatureand velocity of the breaking wavefront will allow better correlation between these parameters andthe maximum impact force. Theunavailability of an accepted scaling law for this problem points to the necessityof conductingsimilar experiments at large scales which will serveas an important extension to the datagathered in the present study.115ReferencesAbramowitz, M., and Stegun, I. A. (1970). Handbook of Mathematical Functions, 9th edn.,Dover Publications, New York.Arhan, M., Deleuil, G., and Doris, C. G. 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(1981). “Numerical Calculation of Forces from Breaking Waves,” Proc.mt. Symp. on Hydrodynamics in Ocean Engineering, Trondheim, Norway, Vol. 1,pp. 547 -565.von Karman, T., and Wattendorf, P. L. (1929). “The Impact on Seaplane Floats DuringLanding,” TN 321, U.S. National Advisory Committee for Aeronautics, Washington,October.Wagner, H. (1931). “Landing of Seaplanes,” TN 622, U.S. National Advisory Committee forAeronautics, Washington, May.Zhou, D., Chan, B. S., and Melville, W. K. (1991). “Wave Impact Pressures on VerticalCylinders,” Applied Ocean Research, Vol. 13, No. 5,pp.220 - 234.120InvestigatorIType of TestILIDIFroude No.IC0Arhan et al. (1978) Drop test 3.3 - 10 0.6 - 2.6 2.4 - 6.9Campbell and Weynberg Drop test 6- 16 1.9 - 5.7 3.5 - 6.5(1980)Schnitzer and Hathaway Drop test 3.6 1.7(referred to by Miller, 1980)Sollied (referred to by Drop test 1.1 - 1.5 4.1 - 6.4Miller, 1980)Watanabe (referred to by Drop test 0.5 1.4 - 2.1 6.2Miller, 1980)Souter et al. (referred to by Drop test 3.6, 4.6 1.6- 5.1 5.2 - 6.8Miller, 1980)Faltinsen et al (1977) Drop test 23- 37 0.4 - 1.2 4.1 - 6.4Miao (1988) Drop test 0.4, 2, 300.1 - 3.8 Avg. 6.1Bergren (referred to by Regular waves 40 - 100 1 - 3.1Miller, 1980)Canham (referred to by Regular waves 150.35 - 0.6Miller, 1980)Dalton and Nash (1976) Regular waves 201 - 3.5 1 - 4.5Holmes et al. (referred to by Regular and 24- 1080.3 - 3.4 0.4 - 2.9Miller, 1980) Irregular WavesMiller (1977) Irregular wave 150.3 - 1.5 1 - 8 (Avg.packet 3.5)Webb (referred to by Miller, Ocean waves30 3 -41980)Sarpkaya (1978) Rising water4.5 - 12 0.5 - 1.3 3.2surface (U-tube)Table 1.1 Peak slamming coefficient C0 reportedin earlier experimental studies.121Period HeightSteepnessCelerityT (see) H (cm) c (mis)1.0 9.8 0.064 1.531.0 14.6 0.096 1.531.1 13.8 0.079 1.611.1 17.0 0.102 1.601.2 12.1 0.058 1.741.2 15.2 0.073 1.741.2 19.3 0.092 1.741.4 16.5 0.062 1.891.4 22.6 0.085 1.891.5 13.9 0.048 1.941.5 18.4 0.063 1.941.5 22.9 0.078 1.941.6 16.8 0.053 1.991.6 22.4 0.070 1.991.8 13.5 0.036 2.061.8 17.8 0.048 2.06Table 4.1 Properties of regular waves used in slamming experiments.‘ñ(mis)C0v11(mis)C0(s/a)0C0T(msec)Tr(msec)C00.375.060.365.320.334.7619.618.84.630.374.710.364.960.444.1325.624.94.100.396.220.386.590.336.0918.417.76.070.414.880.405.190.364.6619.618.84.530.407.200.397.630.307.1817.216.97.050.396.630.387.020.336.5118.818.16.430.404.050.394.300.513.4227.627.13.370.404.870.395.170.314.7217.216.94.640.394.690.384.960.294.4419.218.54.340.406.280.396.660.366.1120.019.45.930.394.360.394.620.403.9622.822.13.870.414.520.404.810.304.3717.216.94.290.394.570.384.840.404.1922.421.74.13Mean0.395.160.385.470.354.9120.119.44.81C.O.V.0.030.160.030.160.140.180.130.130.18Mm.0.374.360.354.620.293.9617.216.93.87Max.0.416.630.387.020.446.5125.624.96.43Table4.2Peakslammingcoefficientsandrelatedparametersestimatedfrommultipleslammingeventsinaregularwavetest.T=1.5see,H=18.4cm,h=0.5cm.LJh=0.5cm1.09.80.491.014.60.731.113.80.731.117.00.711.212.10.451.215.20.701.219.30.781.416.50.531.422.60.711.513.90.411.518.40.611.522.90.721.616.80.501.622.40.661.813.50.371.817.80.420.02 0.08 0.11 0.06 0.04 0.07 0.09 0.03 0.04 0.04 0.03 0.04 0.04 0.04 0.02 0.065.27 6.51 4.14 7.32 7.79 5.09 5.12 6.65 4.15 4.71 5.16 3.75 3.04 3.55 3.38 3.520.10 0.29 0.25 0.21 0.17 0.18 0.33 0.19 0.36 0.16 0.16 0.24 0.11 0.35 0.09 0.130.48 0.70 0.70 0.68 0.45 0.68 0.75 0.52 0.69 0.41 0.60 0.70 0.49 0.65 0.36 0.425.59 7.31 4.66 8.26 8.11 5.56 5.72 6.99 4.53 4.84 5.47 4.06 3.14 3.79 3.44 3.6014.722.2 26.8 20.714.020.1 20.114.831.316.820.1 37.118.921.912.6 11.60.18 0.30 0.19 0.31 0.19 0.26 0.36 0.24 0.33 0.13 0.13 0.16 0.20 0.32 0.20 0.2514.121.6 26.3 20.313.6 19.5 19.5 14.231.016.3 19.537.1 18.2 21.311.9 10.80.210.200.460.340.560.220.420.330.190.180.400.290.440.370.230.250.660.330.200.130.360.110.800.150.270.190.420.300.130.190.140.215.18 6.69 3.85 7.67 7.72 5.02 5.17 6.59 3.55 4.31 4.91 2.91 2.57 3.16 3.07 3.290.12 0.32 0.30 0.24 0.17 0.22 0.35 0.20 0.44 0.17 0.18 0.38 0.15 0.42 0.09 0.145.17 6.62 3.77 7.55 7.60 4.87 5.02 6.57 3.55 4.16 4.76 2.91 2.53 3.14 2.92 3.28Table4.3Summaryoftestconditionsandprincipalresultsfromslammingtestsinregularwaves.1.117.00.750.143.930.310.724.4520.00.3219.40.430.343.890.323.781.416.50.540.054.200.150.534.4218.10.3117.60.280.353.870.203.861.518.40.750.073.240.130.723.5316.50.2815.90.350.303.110.182.991.616.80.550.024.800.070.555.0312.40.1911.70.200.204.720.074.52h=-4.5cm1.422.60.550.083.720.160.543.9221.50.3320.90.340.413.400.193.381.518.40.410.044.540.140.414.6621.10.2320.70.260.253.950.183.92Table4.3(contd.)Summaryoftestconditionsandprincipalresultsfromslammingtestsinregularwaves.h=4.5cm125T H C1C vT/a(see) (cm) Mean MeanIC.O.V. MeanIC.O.V.h=0.5cm1.0 9.8 0.650.68 0.05 0.480.081.0 14.6 1.081.15 0.19 0.880.131.1 13.8 0.73 0.800.15 0.90 0.131.1 17.0 1.36 1.470.19 0.85 0.131.2 12.1 0.74 0.760.04 0.41 0.051.2 15.2 0.84 0.890.14 0.74 0.151.2 19.3 1.211.30 0.27 1.00 0.151.4 16.5 0.900.93 0.05 0.54 0.121.4 22.6 0.80 0.840.30 1.11 0.161.5 13.90.53 0.54 0.15 0.430.071.5 18.4 0.720.75 0.12 0.680.061.5 22.9 0.550.58 0.41 1.150.141.6 16.8 0.350.38 0.12 0.480.081.6 22.4 0.640.67 0.30 0.760.151.8 13.5 0.260.26 0.04 0.290.061.8 17.8 0.330.33 0.10 0.340.07h=4.Sem1.1 17.0 0.931.01 0.26 0.900.121.4 16.50.61 0.63 0.060.57 0.171.5 18.4 0.600.64 0.09 0.740.141.6 16.8 0.710.73 0.05 0.500.07h=-4.Scm1.4 22.60.37 0.47 0.190.51 0.211.5 18.4 0.340.35 0.27 0.400.15Table 4.4 Summaryof impulse coefficients and relatedparameters estimated from slammingtests in regular waves.Xb(cm)x(N)z(N)Tr(msec)IF(N)•(deg)v(mlsec)(cm)h=4.7cm-3.515.0-4.52.615.7-17——7.57.510.14.412.6530.45>2017.012.513.43.618.3470.52>2025.517.016.93.824.0450.63>2036.021.618.52.628.4410.75>2049.024.817.73.930.5360.75>20Regularwave4.55.615.17.251h=8.7cm-3.531.08.51.632.115——7.528.07.11.728.9141.44.817.019.0-14.12.923.7-371.42.825.561.018.41.863.7171.01.736.056.017.01.858.5171.16.749.050.016.31.852.6181.259.7Regularwave2.32.36.13.345h=12.7cm-3.525.610.82.027.823——7.522.79.92.024.824——17.023.03.73.023.391.7convex25.518.3-14.63.223.4-391.7convex36.027.3-14.83.031.1-281.8convexRegularwave2.52.825.23.848Notation AAF,Fpeakhorizontalandverticalimpactforcecomponentsrespectively.=[()2+()2jO.5=-1(/) normalvelocityofthewatersurfacepriortoitscontactwiththetestcylinder.rwradiusofcurvatureofthewatersurfacepriortoitscontactwiththetestcylinder.reasonableestimatesofvorrnotpossiblebecausewatersurfacewastoodisturbed.Table4.5Summaryofobservationsfromimpacttestsinbreakingwaves.127Fig. 1.1 Photograph of nonbreaking wave impacton horizontal test cylinder.Fig. 1.2 Photograph of plunging wave impact on horizontaltest cylinder.1282w6iJ95.4 5VCs3II31I6IIt0 Q2 Q4Q6 0.8 1.214 1.6 18 2 22 2.4ass/aFig. 1.3 Comparison of analytical and experimental results for theslamming coefficientCsas afunction of relative submergence s/a (Greenhow and Li, 1987). 1, experiments ofCampbell and Weynberg (1980); 2, ellipse theory of Fabula (1957); 3, von Karman(1929); 4, semi-Wagner; 5, Wagner’s flat plate approach(1932); 6, Taylor (1930);7, semi-von Karman; 8, semi-Wagner; 9, Wagner’s exact body approach.1290.80.60.40.20.0—0.2—0.4—0.6—0.8D = 2aFig. 2.1 Definition sketch for a fixed cylinder.HJaFig. 2.2 Regimes of cylinder submergence.100Ti1 1000.0 0.2 0.4 0.6 0.8Fig. 2.4 Variation ofCswith submergence s/a - selected results. , von Karman;— — ——, Wagner; , Taylor; — — — — , Campbell and Weynberg;, Miao; — — — , Armand and Cointe.1300.0 0.5 1.0 1.5 2.043Fb2pga2102.5Fig. 2.3 Variation of dimensionless buoyancyforce with relative submergence s/a.7654Cs321s/as/a1.0 1.2 1.4 1.6 1.8 2.0Cm1.0Fig. 2.5 Variation of inertia coefficientTaylor’s solution(Cmo = 2.0); —Cmo= 2.0Cmo1.7Cmwith relative submergence s/a.— —, approximations for C0 = 2.0 and 1.7.2.01.51310.50.00 1 2 3 4 5s/aIIIIIINII::,:IIIIIIFig. 2.6 Sketch of free surface elevation and corresponding vertical waveforce over one wavecycle. , model I; — — — , model II. (a) complete submergence,(b) partial submergence.132(a)11F11Ft(b)t133Cs+Cd3.53.02.52.01.51.00.50.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.61.8 2.0s/aFig. 2.7 Proposed variation of combinedCs+Cd coefficient with relative submergence s/a forCd=O.8. ,C5 +Cd; ,Taylor’ssolutionforC5.KIMIF(t)r(t)Fig. 2.8 Definition sketchof a single degree of freedom(SDOF) system.F134TrTd tFig. 2.9 Representation of an idealizedimpact force withTd/Tr = 1.1351.41.21.00.80.60.40.201.21.00.80.60.40.20Fig. 2.10 ResponseFIF0 of SDOF system to an applied impulsive force with different valuesofTr/Tn. , applied force and response for impact withTd/Tr= 1.0;, applied force and response for impact withTd/Tr = 2.0. (a) Tr/Tn = 0.2,(b)Tr/Tn = 2.0.0 0.1 0.2 0.3 0.4 0.5 0.60.7u-rn0 1 2 3 45 6 7tJT1361.61.41.21.0F0JF 0.80.60.40.200.043Tt/Tr 2103.0Fig. 2.11 (a) Dynamic amplificationfactor FfF0and, (b) relative risetimeT/T as functionsofTr/Tn for an applied impulsive force withTd/Tr = 1.0. , = 0;,C=O.02; —-—-,C=O.05.0.5 1.0 1.52.0 2.5 3.0Tr/Tn0 0.5 1.01.5 2.0 2.5Tr/Tn137F0IF1.81.61.41.21.00.80.60.40.20.00.043Tt/Tr 2100.03.0Fig. 2.11 (c) Dynamic amplificationfactor F/F0and, (d) relativeriseofTr/Tn for an applied impulsive force withTd/Tr = 2.0.,=0.02;—-—-,C=0.05.timeTtII’r as functions= 0;0.5 1.0 1.52.0Tr/Tn3.00.5 1.0 1.52.0 2.5TrIfn138FFig. 2.12 Definition sketch fordynamically responding cylinder.Fig. 2.13 Definition sketchfor the computation ofthe impulse coefficient.combined impulsive andresidual force;, residual force.D = 2atInitial locationSWL110.t139Fig. 2.14 Definition sketchfor wave impact on an inclinedcircular cylinder.SWLFig. 2.15 Definition sketch forimpact due to a sloping watersurface.1400.121.20.08 -- 1.1/‘..— /0.04 -.......-— //0-V -0.9-0.04 --0.08I I II I0.89.7 9.8 9.910.0 10.1 10.210.3t (see)Fig. 2.16 Variation of thefree surface slope correctionfactorsCIC( )and CIC(— - — -)for an experimentallymeasured wave( )of periodT= 1.1 see, andheightll=17 cm.141Fig. 3.1 Photograph ofwave flume in the Hydraulics Laboratory.Fig. 3.2 Photograph of computer-controlledwave generator.142(a)(b)Wave flumeFig. 3.3 (a) Photograph of test cylinder assembly, (b) Sketch of the experimental setup.1434Low-pass filterWave probeCylinder andforce transducer assemblyAmplifiersLow-pass filtersjAnalog to digital converterDigital to analog converterFVAXstation 3200 computerrunning RTC & GEDAP softwareWave generatorIntelligent servo positionerFig. 3.4 Block diagram of wave generationand data acquisition equipment.144g202.1 2.2 2.3 2.4t (sec)24.016.08.00.0—8.0—16.0—24.02.0 2.5Fig. 3.5 Record ofthe vertical force due to free vibration ofthe cylinder inducedby an applied stepforce of -19.6 N (2kg).0.60.5I100.0 200.0 300.0 400.0f (Hz)500.0Fig. 3.6 Spectral density of free vibration record in Fig.3.5.1456.06.0,—. 4.002.00.0—2.07.07.1Fig. 3.7 Early stages of the measured vertical forceon the test cylinder due to a typical waveslamming event. •, indicates individual force samples.7.02 7.04 7.06 7.08t (sec)g01461(m)0.20.10.0—0.110.06.06.04.02.00.0—2.0I I I(a)faz____NI__ I I-__I I(b)J—I I18.0 18.2 18.4 18.6 18.819.0t (sec)Fig. 3.8 Time histories of free surface elevation and verticalforce during a slamming event.T = 1.4 sec, H = 22.8 cm, h = 0.5 cm.147z0.00.50.250.10.050.0—0.05—0.118.15Fig. 3.9 Time series of vertical force( )and corresponding local variance( )used to detect the onset of slamming.—0.2518.17 1 8.19 18.21 18.23t (sec)—0.518.25148Measure wave record Measure vertical forceat cylinder location: on test cylinder:2500 samples/sec 2500 samples/sec____‘IResample wave recordLow-pass filterat a lower rate: forcerecord:50 samples/sec 400HzLow-pass filter Determineonset ofwave record: slammingevents using6 Hz variance algorithm‘IFit cubic-spline toCalculate cylinderwave recordsubmergence andbuoyancy component‘ICompute particle velocityDetermine peak force,slamming coefficients,at free surface:1and vrise times etc.Estimate impulsecoefficents andimpulse durationReport statistics forresults from multipleslamming eventsFig. 3.10 Flow chart showing steps inanalysis of experimental data.149IT_1.11&:tz::r‘wgrV:flai_I—I’>--4_taL IFig. 3.11 Frame of video record defining the wave breaking locationxb.150201.4 1.8 2.2 2.6(see)12.08.04.00.0—4.0—8.01.0 3.0Fig. 3.12 Record of the horizontal force due to free vibration of the cylinder induced by an appliedstep force of 9.8 N (1 kg).2.0i ::0.00.0 20.0 40.0 80.0 80.0f (Hz)100.0Fig. 3.13 Spectral density of free vibration record in Fig. 3.12.20I.15140.030.020.010.00.0—10.0—20.0—30.0—40.02.48 2.52 2.56 2.62.64 2.68t (see)Fig. 3.14 Comparison of corrected horizontal force( ),with recorded horizontalforce( ),and applied step force(g1520.02 0.04 0.06 0.08 0.1t (sec)40.030.020.010.0C0.0—10.0—20.00.060.050.040.030.020.010.00.0—10.00.0 0.1Fig. 3.15 Impact force on the horizontal test cylinder due to plunging wave (h = 8.7 cm,xb = 36 cm). (a) , recorded vertical force component;recorded horizontal force component. (b) , recorded vertical forcecomponent; , corrected horizontal force component.0.02 0.04 0.06 0.08t (sec)0.061) 2 00.0400—0.04 —0.085. 4. 3. 2. 1. 0.—1.04.86.88.610.812.8t(see)Fig.4.1Timehistoriesofthefreesurfaceelevationandverticalforceovera10secdurationforawaveoflowsteepness.T=1.8see,H=13.5cm,h=0.5cm.14.80.120.0—0.04 —0.087.5 6.0 4.5a)3.001.5 0.0—1.55.015.0Fig.4.2Timehistoriesofthefreesurfaceelevationandverticalforceovera10secdurationforawaveofmediumsteepness.T=1.5sec,H=18.4cm,h=0.5cm.0.06 0.047.09.011.013.0t(sec)0.120.0—0.04 —0.0816.0 14.0 12.0 10.0‘—‘8.0ti) 26.0C4.0 2.0 0.0—2.02.012.0Fig.4.3Timehistoriesofthefreesurfaceelevationandverticalforceovera10secdurationforawaveoflargesteepness.T=1.lsec,H=l7cm,h=0.5cm.0.08 0.044.06.08.010.0t(sec)1560.120.080.040.0—0.04—0.08—0.1210.08.06.04.02.00.0—2.012.012.8 13.0Fig. 4.4 Time histories of free surface elevation and verticalforce during a slamming event.T = 1.1 sec, H = 17.5 cm, h = 0.5 cm.12.2 12.4 12.6t (see)1570.120.080.04I‘— 0.0—0.04—0.08—0.126.05.04.0g3.0U2r.22.01.00.0—1.015.3 16.1 16.3t (see)Fig. 4.5 Time histories of free surface elevation and vertical forceduring a slamming event.T = 1.4 see, H = 16.5 cm, h = 0.5 cm.15.5 15.7 15.901580.120.080.040.0—0.04—0.086.05.04.03.0.01.00.0—1.012.6 13.6t (sec)Fig. 4.6 Time histories of free surface elevation and vertical force during a slamming event.T = 1.6 see, H = 16.8 cm, h = 0.5 cm.12.8 13.0 13.2 13.41590.10.0750.050.0250.0—0.025—0.055.04.03.0g2.01.00.0—1.012.7 13.5 13.7t (see)Fig. 4.7 Time histories of free surface elevation and vertical force duringa slamming event.T = 1.5 see, H = 13.9 cm, h = 0.5 cm.12.9 13.1 13.31607.06.05.04.03.02.01.00.0t (see)15.2Fig. 4.8 Time histories of free surface elevation and vertical force during a slamming event.T = 1.5 see, H = 18.4 cm, h = 0.5 cm.0.120.080.040.0—0.04—0.08g0)2C—1.014.2 14.4 14.6 14.8 15.0:\It (sec)Fig. 4.9 Time histories of free surface elevationand vertical force during a slamming event.T = 1.5 sec, H = 22.9 cm, h = 0.5 cm.161N0.160.120.060.040.0—0.04—0.088.06.04.02.00.0—2.0 -18.6 18.8 19.0 19.219.4 19.65.04 03.02 0t (see)12.2162Fig. 4.10 Time histories of free surface elevation and vertical force duringa slamming event.T = 1.5 see, H = 18.4 cm, h = 4.5 cm.0.120.080.040.0—0.04—0.0 B1.00.0—1.011.2 11.4 11.6 11.8 12.0zC0.121630.080.040.0—0.04—0.087.06.05.0__4.03.02.01.00.0—1.09.6 10.4t (sec)Fig. 4.11 Time histories of free surface elevation and vertical force duringa slamming event.T = 1.5 sec, H = 18.4 cm, h = -4.5 cm.9.8 10.0 10.2 10.61641.5 5-41-j‘I-3Ft0JF Tt/Tr-20.5- / ‘——— 1I I I I0.0 0.5 1.0 1.5 2.0 2.5 3.0T/TFig. 4.12 Correction factors for peak slamming force and rise time asa function of theobserved rise time ratioT/T. , FfF; , TIl’.0.3 I I I0.25ci0.151•0.10.05 .0.0- —:0.0 4.0 8.0 12.0 16.020.0C0Fig. 4.13 Probability density histogram of C0( )based on data collected from entireset of experiments. , log-normal probability density.Fig. 4.15 Probabilityexperiments.density histogram ofCrbased on data collected from entire set of1650.250.20.150.10.050.00.0 4.0 8.0 12.016.0C,IsoI20.0Fig. 4.14 Probability density histogramof C0( )based on data collected from entireset of experiments. , log-normal probabilitydensity.2.01.5I1.00.50.00.0 0.4 0.8 1.2 1.62.0Ct2.42C1)2CFig. 4.17 Comparison of slamming force time histories fordifferent cylinderT = 1.8 see, H = 17.8 cm. , 0= 00;, 0 = 4.8°;0=9.6°.inclinations,16612.010.08.06.04.02.00.0—2.012.85 12.95 13.05t (see)13.15Fig. 4.16 Comparison of slamming force time histories fordifferent cylinder inclinations,T = 1.2 see, H = 19.3 cm. , 0 = 0°; , 0 = 4.8°;— - -,9=9.6°.6.05.04.03.02.01.00.0—1.017.4 17.5 17.617.7t (see)17.8543210—15432101670.60.40.2a0.0H—0.2—0.4—0.6I I I I I I- h/H=0.3 -II.I”IIFpga2Fpga20.0—1—0.50.5t/rFig. 4.18 Time variation of free surface elevation and simulated verticalforce for co2a/g = 0.05,o2H/g = 0.6, and different cylinder elevations. , Model I; — - — - —,Model II.1680.6(a)0.40.2h00H0.2 0.4 0.6o)2H/g = 0.8—0.2(40) (80) (120) (H/a= 160)—0.4—0.60 20 40 60 80 100 120/pga20.6(b)0.40.2hH0.0—0.20.2 0.4 0.6cx2H/g = 0.8(20) (40) (60) (H/a =80)—0.4—0.60 10 20 30 40 50 60Ipga2Fig. 4.19 DistributiOn of the nondimeflSiOflal maximum vertical forceas a function of cylinderelevation ando2H/g.——, Model I;— ——, ModelII. (a) o2alg = 0.005,(b)o2aJg=O.Oi.1690.6I I I I I(c)0.40.2 -.1100-H—0.2- 0.2 0.4 0.6co2H/g = 0.8(4) (8) (12)(H/a = 16)—0.4 -—0.6I I I II0 2 4 6 810 12F/pga20.6I I I I I(d)0.4N0.2-\\0-0.2-\o.2 \0.4 0.6 o2H/g = 0.8\(2) \(4)(6)(H/a=8)-0.4-N—0.6II \0 1 2 34 5 6fIpga2Fig. 4.19 Distribution of the non-dimensional maximumvertical force as a function of cylinderelevation and o2HJg. , Model I; — —— —, Model II. (c) o2aJg = 0.05,(d) co2alg=0.l.170p210_0.00 0.02 0.04 0.06 0.08 0.10(O2aIgFig. 4.20 Variation of the non-dimensional maximum vertical forceas a function of co2a/g fordifferent values ofco2HJg. , model I; — — —, modelII.1711.5‘ II (a).A AA A40001>I-,ES.— A AF200-0I I I I0.04 .06 .08.1f (Hz)1I I III I III I I I III(b).8100 -ES.-,50-II.20 ‘I I00 .2 .4 .6 .81f (Hz)Fig. 4.21 Spectral density( )and corresponding amplitude spectrum(z\)withH = 10 m,and Tp = 14.3 sec, used in the numerical simulation of random waves.(a) Narrow-bandspectrum, (b) Pierson-Moskowitz spectrum.1728642i (m) 0—2—4—6—81900210016128F/pga240—419002100t (sec)Fig. 4.22 Segment of numerically simulated time series of (a) freesurface elevation and, (b) nondimensional vertical force for a narrow-band spectrum.1950 2000 2050t (sec)1950 2000 2050Fig. 4.23 Segment of numerically simulatedtime series of (a) free surface elevationand, (b) non-dimensional vertical force fora two-parameter Pierson-Moskowitz spectrum.173300105Ti(m) 0—5—10403020F/pga2 100—10—20350 400 450t (see)500300 350 400450t (see)5001741.0II • I I(a)0.8-40.6-p(F/pga2)0.40.2JI4_0.0 — I--L0 2 4 68 10 12/pga21.0 II I I III I I I I I I I(b)0.8--!0.6--p(F/pga2)!\0.4--.IF/pga2Fig. 4.24 Comparison of probability densityof force maxima on a horizontal cylinderlocated ath =0. — - — - —, analytical prediction(Isaacson and Subbiah, 1990);numerical simulation, method A;, numerical simulation, method B.(a) narrow-band spectrum, (b) two-parameterPierson-Moskowitz spectrum.20.3 0.4 0.50.6 0.7 0.8175765‘ 410-10.20.9t (see)Fig. 4.25 Comparisonof vertical force predictedby rigid cylinder model( )with themeasured force( ).T = 1.2 see, H = 15.2 cm,h = 0.5 cm.76543210—10.25 0.300.35 0.400.45t (see)0.50Fig. 4.26 Earlystages of slamming forcepredicted by dynamiccylinder model( )compared with rigidcylinder model estimate(— - -)and the measured force( ).T = 1.2 see, H = 15.2cm, h = 0.5 em, T =290 Hz,Tr= 20 msee.-100.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9(see)176Fig. 4.27 Comparison of vertical force predicted by rigid cylinder model( )with themeasured force( ).T = 1.5 see, H = 18.4 cm, h 0.5 cm.Fig. 4.28 Photograph showing mass of water suspended from test cylinder after recession ofincident wave.765432z0.4 0.6 0.8 1.01.2177765C210—10.21.4t (sec)Fig. 4.29 Comparison of vertical forcepredicted by rigid cylinder model( )with themeasured force( ).T = 1.5 sec, H = 18.4 cm, h = -4.5 cm.765__43210—10.50Fig. 4.30 Early stages of slamming forcepredicted by dynamic cylinder model( )compared with the measured force( ).T = 1.5 sec, H = 18.4 cm,h=-4.5cm,Tn=29OHz,Tr= l8msec.0.25 0.300.35 0.40 0.45t (see)U0Fig. 4.32 Comparison of force oninclined cylinder predicted by numerical model( )with the measured force( ).T = 1.2 see, H = 15.2 cm, 9 = 9.6°.0.2 0.3 0.4178543U210—10.10.5t (see)Fig. 4.31 Comparison of force oninclined cylinder predicted by numerical model( )with the measured force( ).T = 1.8 see, H = 17.8 cm, 9 = 4.8°.432100.1 0.2 0.3 0.4 0.5t (see)0.6 0.7 0.8 0.9179804070 - :‘ ‘.. ..•. - 3560- I - 3050- // .- 2540- .‘\2030- 1520-1010- /\ ,‘ /\/\/\/\/\,‘ 50___\,/\/\/0—10I I I II LI2.9 3.0 3.1 3.23.3 3.4 3.5 3.6 3.7 3.8t (sec)Fig. 4.33 Predicted time history of the verticalforce on the cylinder for the exampleapplication. , total force on rigidcylinder; , buoyancy forcecomponent; - -, mid-spancylinder response for fixed-end condition;, mid-span cylinder response for pinned-endcondition.Fig. 4.34 Sequenceof video frames showingplunging wave impacton horizontal test cylinder.h=4.7cm,xb=25.5cm.Fig. 4.35 Digitizedprofiles of plungingwave in the vicinityof theh = 4.7 cm,Xb = 36 cm.horizontal testcylinder.1801510(cm)50-55 1015 2025(cm)30181Fig. 4.36 Time historiesof recorded vertical force( )and corrected horizontal force( )on horizontal test cylinderdue to breaking waveimpact. h = 4.7 cm,Xb = 25.5 cm.Fig. 4.37 Sequenceof video frames showingplunging wave impacton horizontal test cylinder.h = 8.7 cm,Xb= -3.5 cm.2024.020.016.012.08.04.00.0—4.00.0 0.020.04 0.060.08t (sec)0.1I Ii I____I IiII1 11182Fig. 4.38 Sequenceof video frames showingplunging wave impacton horizontal test cylinder.h = 8.7 cm,Xb= 25.5 cm.(c)c_5 1015 2025 30(cm)Fig. 4.39 Digitized profilesof plunging wave inthe vicinity of the horizontaltest cylinder.h = 8.7 cm,Xb = 25.5 cm.i1 1I I II III I 1 1I1I I II 1111 11510(cm)50-5183g075.060.045.030.015.00.0—15.00.0 0.02 0.040.06 0.08 0.1t (sec)Xb= 25.5 cm.Fig. 4.40 Time histories of recorded vertical force( )and corrected horizontal force( )on horizontal test cylinder due to breaking waveimpact. h = 8.7 cm,Fig. 4.41 Sequence of video frames showing plungingwave impact on horizontal test cylinder.h = 8.7 cm,Xb =49 cm.184Fig. 4.42 Sequenceof video frames showing plungingwave impact on horizontaltest cylinder.h = 12.7 cm, Xj = 25.5cm.I5 10 1520 2530(cm)Fig. 4.43 Digitized profilesof plunging wave inthe vicinity of the horizontaltest cylinder.h = 12.7 cm,Xb = 17 cm.I I I I I________I tI I2015(cm)105018520.010.0C—10.0—20.00.00.1Fig. 4.44 Time histories of recordedvertical force( )and corrected horizontal force( )on horizontal test cylinder due to breakingwave impact. h = 12.7 cm,0.02 0.04 0.080.08t (sec)Xb= 25.5 cm.

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