NUMERICAL SIMULATION O F W A V E GROUPING E F F E C T S ON MOORED STRUCTURES by VENKATA RAMI REDDY PEMMIREDDY B.E.(Hons.), B.LT.S. Pilani, India M.Tech, K.R.E.C, Surathkal, India A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1992 © Venkata Rami Reddy Pemmireddy, 1992 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of the department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering, The University of British Columbia, 2324 Main Mall, Vancouver, B.C., Canada, V6T 1Z4. Date: AtA^toAt ZC ^1 Abstract The present thesis investigates wave grouping effects on the low-frequency drift force and the low-frequency surge response of a moored structure. The smoothed instantaneous wave energy history, SIWEH, which may be characterized by a number of parameters, is considered as an acceptable approach to describe wave groupiness. A numerical simulation of wave records is carried out for a specified wave spectral density and SIWEH incorporating various wave grouping parameters. The low-frequency drift force and the surge response of a large moored circular cylinder, modeled as a single degree of freedom system, is estimated on the basis of Pinkster's approximate method, which involves the use of computed drift force coefficients. Results show that the wave grouping parameters have a significant effect on the low-frequency drift force and surge response. It is also observed that the low-frequency drift force can be related to the SIWEH in terms of a transfer function. A statistical analysis of simulated low-frequency surge motions is performed, and the calculated extreme response is compared with the predictions of three theoretical models, namely the Gaussian, Exponential, and three-parameter methods. If the surge response has a nonGaussian distribution, Stansberg's three-parameter model predicts extreme amplitudes reasonably well. The low-frequency drift force on a slender fixed circular cylinder is calculated using both Pinkster's approximate method and in the time domain using the Morison equation. Pinkster's approximate method underestimates the force, but it is computationally more efficient. The time domain approach is also used to estimate the low-frequency drift force on a moored slender circular cylinder and the results are compared to those for a fixed cylinder. It is observed that cylinder motion tends to reduce the drift force significantly. Table of Contents Page Abstract ii Table of Contents iii List of Tables v List of Figures vi List of Symbols ix Acknowledgments 1. Introduction 1 1.1 General 1 1.2 Literature Review 2 1.2.1 Effect of Wave Grouping on Floating Structures 2 1.2.2 Wave Drift Forces on a Large Structure 3 1.2.3 Low-Frequency Drift Motion of a Large S tructure 5 1.2.4 Statistics of Low-Frequency Motions 7 1.2.5 Wave Drift Forces on a Slender Structure 8 1.3 2. xii Scope of The Present Investigation 8 Numerical Simulation of Wave Group Activity 10 2.1 Wave Group Definition and Measures 10 2.1.1 SIWEH 10 2.1.2 Groupiness Factor 12 2.1.3 SIWEH Peak Frequency 12 2.1.4 Broadness Factor 12 2.2 Simulation of Wave Group Activity 13 2.2.1 Synthesis of a SIWEH from its Spectral Density 13 2.2.3 Generation of Wave Records with Different Wave Grouping 16 3. 4. Wave Drift Forces on a Moored Large Structure 17 3.1 Mean Drift Force in Regular Waves 17 3.2 Drift Force in Irregular Waves 18 3.3 Low-Frequency S urge Motion 20 3.4 Statistics of Low-Frequency Surge Response 22 3.4.1 Extreme Values of Non-Gaussian Slow-Drift Response 23 3.4.2 Slow Drift Response Prediction Model 24 Wave Drift Forces on a Moored Slender Structure 26 4.1 Time Domain Approach 26 4.1.1 29 4.2 5. 6. Simulation of Wave Kinematics Pinkster's Method 30 Results and Discussion 32 5.1 Synthesis of a Grouped Wave Train 33 5.2 Low-Frequency Drift Force and Response of a Large Cylinder 35 5.4 Low-Frequency Surge Response Statistics 39 5.5 Low-Frequency Drift Force on a Slender Cylinder 40 Conclusions 43 6.1 45 Recommendations for Further Study References 46 Tables 51 Figures 56 List of Tables Table 5.1 Characteristics of simulated sea states. Table 5.2 Statistics of the low-frequency drift force and surge response for a moored large circular cylinder. Table 5.3 Comparison of calculated maximum response amphtude of a moored large circular cylinder with theoretical predictions (damping ratio = 1.89 %). Table 5.4 Comparison of calculated maximum response amplitude of a moored large circular cylinder with theoretical predictions (damping ratio = 12.36 %). Table 5.5 Statistics of low-frequency drift force for the fixed slender circular cylinder. List of Figures Figure 1.1 Sketch of time histories of surface elevation and second-order drift force in a regular wave group. Figure 1.2 Definition sketch of the motions of a floating body. Figure 2.1 SIWEH example; (a) a wave record, (b) the corresponding SIWEH and (c) the spectral density of the SIWEH. Figure 5.1 Surface elevation time series for wave records with different values of G F for the case fp = 0.2, Hg = 2.0 m, y = 1-0, fgw = 0.02 Hz, C = 0.2 (a) Gp = 0.6, (b) Gp = 1.0 and (c) corresponding SIWEH spectra. Figure 5.2 SIWEH and SIWEH spectra for wave records with different values of fgw for the case fp = 0.2, Hg = 2.0 m, y = 1.0, Gp = 0.6, C = 0.2 (a) SIWEH time series and (b) SIWEH spectra. Figure 5.3 SIWEH spectra and surface elevation time series for wave records with different values of C for the case fp = 0.2, Hg = 2.0 m, y = 1.0, Gp = 0.6, fsw = 0.02 Hz. Figure 5.4 Definition sketch for the moored large circular cylinder. Figure 5.5 Variation of the mean drift force coefficient with frequency for a moored large circular cylinder. Figure 5.6 Calculated added mass coefficient as a function of frequency. Figure 5.7 Calculated radiation damping coefficient as a function of frequency. Figure 5.8 Time histories of (a) surface elevation, (b) low-frequency drift force and (c) low-frequency surge response, for waves with different values of Gp (fp = 0.2, Hs = 2.0 m, y = 1.0, fsw = 0.02 Hz, C = 0.2, fn = 0.0158). Figure 5.9 Time histories of (a) surface elevation, (b) low-frequency drift force and (c) low-frequency surge response, for waves with different values of fsw (fp = 0.2, Hs = 2.0 m, y = 1.0, Gp = 0.6, C = 0.2, fn = 0.0158). Figure 5.10 Time histories of (a) surface elevation, (b) low-frequency drift force and (c) low-frequency surge response, for waves with different values of Ç (fp = 0.2, Hs = 2.0 m, Y = 1.0,GF = 0.6, fsw = 0.02 Hz, fn = 0.0158). Figure 5.11 Effect of different values of Gp on (a) spectra of the low-frequency drift force and (b) spectra of the low-frequency surge response (fp = 0.2, Hg = 2.0 m, Y = 1-0, fsw = 0.02 Hz, C = 0.2, fn = 0.0158). Figure 5.12 Effect of different values of fsw on (a) spectra of the low-frequency drift force and (b) spectra of the low-frequency surge response(fp = 0.2, Hs = 2.0 m, Y = 1.0, G F = 0.6, C = 0.2, fn = 0.0158). Figure 5.13 Effect of different values of C, on (a) spectra of the low-frequency drift force and (b) spectra of the low-frequency surge response (fp = 0.2, Hg = 2.0 m, Y = 1.0, fsw = 0.02 Hz, fsw = 0.02 Hz, fn = 0.0158). Figure 5.14 The influence of groupiness factor Gp on (a) the low-frequency drift force and (b) the low-frequency surge response of a moored large circular cyUnder. Figure 5.15 The influence of SIWEH peak frequency fsw on (a) the low-frequency drift force and (b) the low-frequency surge response of a moored large circular cylinder. Figure 5.16 The influence of broadness factor C, on (a) the low-frequency drift force and (b) the low-frequency surge response of a moored large circular cylinder. Figure 5.17 The effect of mooring stiffness on the low frequency surge response of a moored large circular cylinder. Figure 5.18 Probability distributions of the SIWEH compared to Gaussian distributions for (a) test 5 and (b) test 14. Figure 5.19 Probability distributions of the low-frequency surge response compared to Gaussian distribution for test 5 with (a) damping ratio 1.89% and (b) damping ratio 12.36%. Figure 5.20 Definition sketch for the moored slender circular cylinder. Figure 5.21 Variation of the mean drift force coefficient with frequency for a fixed slender circular cylinder. Figure 5.22 Comparison of the low-frequency drift force spectra on fixed cylinder calculated by Pinkster's method with Time domain approach using Morison equation for (a) test 1, (b) test 2 and (c) test 3. Figure 5.23 The influence of cylinder motion on the low-frequency drift force (a) test 1 and (b) test 3. List of Symbols The following symbols are used in this thesis: Ar = amphtude of the reflected wave B = spectral band width parameter CD(f) - mean drift force coefficient Cd, Cm. Ca = drag, inertia, and added mass coefficients d = water depth D = diameter of the cyhnder E(t) = smoothed instantaneous wave energy history (SIWEH) f = frequency in Hz fp = peak frequency of the wave spectrum fr = normalized frequency fsw = SIWEH peak frequency fn = natural frequency F = mean drift force F(t) = low-frequency drift force F[ ] = Fourier transform F'l[] = inverse Fourier transform Gp = groupiness factor Hs - significant wave height k = mooring stiffness m = structure mass mo = zeroth moment of the wave spectral density meo = zeroth moment of the SIWEH spectral density p = hydrodynamic pressure rms = root-mean-square value Q(t) = smoothing or window function SE(f) = SIWEH spectral density SpCf) = spectral density of low-frequency drift force Sx(f) = spectral density of low-frequency surge response S-q(f) = wave spectral density t = time T - wave period Tp = peak period of the wave spectrum. Tr = length of the wave record VR.Ve = radial and tangential components of the fluid velocity Wg = width of the wave group spectrum Wx - width of the response spectrum | i = added mass Ox = rms value of surge response T = time shift Ç = broadness factor of the SIWEH spectrum Y = spectral peak enhancement factor p = water density CO = wave angular frequency X, = damping coefficient •n(t) = water surface elevation Acknowledgments The author would like to express his profound sense of gratitude to his advisor Dr. Michael Isaacson, for his expert guidance and encouragement throughout the preparation of this thesis and his invaluable help offered during the course of study. The author would also like to express his appreciation for the cooperation and support shown by his colleagues. In particular, the author wishes to thank Shin-ichi Aoki, Sundar Prasad, and Enda O'Sullivan for their help. Financial support in the form of a Research Assistantship from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Chapter 1 Introduction 1.1 General In recent years, problems concerning the mooring of floating structures have gained considerable attention in the context of the loading and discharge of large tankers in open seas and moored structures used in the exploration and exploitation of the seabed. Reliable cost effective systems can be built by accounting for factors at the design stage that have a significant bearing on system performance. In this context, the mooring system which keeps a floating vessel on station is one of the major components that influences the design of a floating system. In the design of such a system, the properties of the wave field interacting with the structure and the response of the structure to the specified wave field must be correcdy understood. In general, mooring system design is based on two major load cases: extreme loads associated with extreme environmental conditions, and operational loads associated with more common environmental conditions. For most offshore locations, waves are a primary source of loading on the mooring system. In particular, large low-fi-equency horizontal motions and associated mooring loads form a dominant part of the system behavior and loads under extreme wave conditions. The natural frequencies of a mooring system are usually of the order of 0.01 Hz, which is well below the peak frequencies of the waves themselves. Oscillations in the horizontal plane therefore take place within two distinct frequency ranges, one at wave frequencies and the other over a lower frequency range close to these natural frequencies. Excitation at these low frequencies is induced by nonlinear effects of irregular waves. The nonUnearities in the wave loading process giveriseto a wave force which has both mean and low-frequency components, as indicated in Fig. 1.1, which are collectively known as 'wave drift forces'. Wave drift forces are generally proportional to the square of the wave height, while their lowfrequency components are generally due to those of wave groups corresponding to alternating sets of high and low waves. Large low-frequency motions of a moored structure may arise when the dominant frequency of wave groups is close to the naturalfrequencyof the mooring system. This has led to increasing interest in the effect of wave groups on the low-frequency response of moored structures. The above comments relate to physical aspects of wave loads and the system's response. Such knowledge is fundamental to a proper understanding of the processes involved. Furthermore, for the design of a mooring system which is dominated by dynamic effects that are random, such as in the case of wave drift forces, knowledge of the statistical properties of the motion and mooring loads is also essential. The focus of the present work is on a description of wave grouping characteristics of a wave train and on the effects of these on the resulting response of a moored structure. 1.2 Literature Review 1.2.1 Effect of Wave Groupin g on Hoatin g Stmctures The tendency of ocean waves to appear in groups has received increasing interest from coastal and ocean engineers. A notable early study of wave groups was carried out by Goda (1970). It has been suggested that groups of high waves affect coastal structures such as breakwaters and pipelines (Burcharth, 1979) and influence the response of floating vessels (Pinkster, 1974). Hsu and Blenkam (1970) pointed out that slow drift oscillations of vessels and mooring forces are related to sequences of high waves in random seas, and Ewing (1973) noted that ships can capsize or be damaged by severe motions caused by high wave groups. The influence of wave grouping on the low-frequency motion of a moored structure and characteristics of the corresponding nonlinear hydrodynamic forces are generally investigated by model tests and numerical simulation. Such results show that the sequence of the incident waves has a significant influence on the low-frequency part of the sway motion and force (Spangenberg and Jacobsen, 1980, Sawaragi et al., 1988). 1.2.2 Wave Drift Forces on a Large Struchire As stated earlier, wave loads on large floating structures in random seas consist of a first-order component at wave frequencies, as well as a second-order low-frequency component. The firstorder wave forces and motions in regular unidirectional waves can usually be determined from well established procedures based on three-dimensional linear diffraction theory (e.g. Sarpkaya and Isaacson, 1981). Such first-order results can be extended to random, short-crested waves by using linear superposition principles (Isaacson and Sinha, 1986). Investigations into the nature of the wave drift forces have generally stemmed from the need to include these effects in the total load on structures moored by anchor lines or cables (e.g. Pinkster, 1979). Such investigations were initially prompted by the fact that observed horizontal motions of vessels moored in irregular waves and associated mooring line forces exhibit low-frequency components which cannot be explained by linear wave theory alone, van Oortmerssen (1976) demonstrated that the low-frequency horizontal motions of a vessel moored to a jetty in irregular waves were in many cases directiy attributable to the second-order wave drift force, thus underlining the necessity for reUable means of determining this part of the total wave force. Hsu and Blenkam (1970) presented an approximate method for calculating the drift force in irregular waves from a knowledge of the mean drift force in regular waves. The method assumes that an irregular wave train can be considered as a succession of half-periods of regular waves. The slowly varying drift forces are then obtained from the mean drift force which the structure would experience in a regular wave with corresponding amplitudes and periods of each halfperiod. Analytical methods of estimating drift forces in regular waves can be categorized into two main areas, the far-field and near-field approaches. Maruo (1960) developed expressions for the in-line and transverse components of the drift force in regular waves by considering changes in the momentum of the fluid. The approach is sometimes termed as the 'far-field' approach, since the behaviour of the fluid potentials in the far-field is used to evaluate the forces. Newman (1967) extended Maruo's method to include the computation of the mean yaw moment. The results were evaluated using slender body theory and assuming an infinite water depth. Faltinsen and Michelsen (1974) generalized the expressions forfinitewater depth and used the three-dimensional source distribution method to evaluate the required potentials. As an alternative to the far-field approach, Pinkster and van Oortmerssen (1977) presented expressions for the mean drift forces and moments by a 'near-field' method based on a direct integration of the pressures over the wetted surface of the structure. This approach shows exphcitiy the different components of the drift force, and can also be used to evaluate vertical drift forces. This approach is computationally more expensive than the corresponding far-field method, but it can be used to calculate mean and low-frequency vertical components of the wave force. The low-frequency force involves a quadratic transfer function describing the force caused by interacting pairs of regular wave trains with different frequencies. Newman (1974) and Pinkster (1974) had earlier proposed an approximate procedure based on Hsu and Blenkam's approach for estimating low-frequency forces, in which the quadratic function is replaced by the mean force acting in regular waves at the mean wave frequency. The calculated slowly varying motions based on these methods have been found to agree fairly well with Remery and Herman's (1971) experimental data for a rectangular barge. This approximate procedure is widely used in design. It is computationally efficient and the required mean force data are more readily available from experiments or theory. Pinkster's method is found to underestimate the moment in the low-frequency range, emphasizing the importance of including a term due to the moving body condition in the evaluation of the second-order wave effect. An exact second-order theory for predicting the slowly varying second-order hydrodynamic force on a floating structure in irregular waves was formulated by Matsui (1976). Green's second identity was exploited to derive the complete expression for the second-order hydrodynamic forces due to the second-order velocity potential involving first-order quantities and the second-order undisturbed potential alone. It was shown that the contribution of the second-order velocity potential to the slow drift moment can be significant in irregular seas with longer mean wave period. 1.2.3 Low-Frequencv Drift Motion of a Large Structure In addition to excitation at wave frequencies, a moored vessel with a low fundamental frequency may also undergo low-frequency resonant oscillations in waves. The wave spectrum itself does not usually possess sufficient energy to excite such a response. Different nonlinear mechanisms may be responsible, including nonlinear stiffness terms in the equation of motion, or second-order wave drift forces. A nonlinear stiffness may often be present through the mooring system. The second-order wave forces in moderate sea conditions can induce large amplitude resonant motions in spite of their small magnitudes. They can be generally divided into a lowfrequency part, known as "slowly varying wave drift force" and a high-frequency or sumfrequency part. The difference-frequency forces can excite resonant oscillations in surge, sway and yaw of moored structures and in heave, roll and pitch of floating structures with small waterplane area. A well known approach for the analysis of slow-drift motions was originally proposed by Pinkster (1974). In this approach, the second order wave excitation forces in the frequencydomain without the effect of slow drift motions were calculated and then the motion equations in thetime-domainwere solved by means of a suitable time-stepping algorithm. A structure moored in irregular head waves undergoes wave frequency motions and large lowfrequency surge motions induced by drift forces. Figure 1.2 shows a definition sketch of a floating body indicating its six degrees of freedom. The six equations of motion corresponding to the six degrees-of-freedom can be separated to provide a corresponding set of equations for the low-frequency components. In the case of moored vessel primarily undergoing surge motion, the system can be analyzed using a single degree of freedom model and is described by the following equation of motion: (m + li) x(t) + X k(t) + k x(t) = F(t) (1.1) where m is the mass of the vessel, [i is the added mass, X is a damping coefficient made up of viscous damping, wave drift damping, and wave radiation damping, k is the stiffness, F(t) is the low-frequency wave drift force and x(t) is the low-frequency surge motion. For a linear mooring system k is a constant, whereas for a nonlinear system k varies with x. The added mass and viscous damping are generally frequency dependent quantities. However, the resultant motion is generally sufficiently narrow banded to allow the assumption of constant values of the added mass and damping coefficient. For the purpose of predicting the surge motions, the added mass and damping values can be obtained on the basis of calculations and model tests. Wichers and van Sluijs (1979) carried out free oscillation tests for the surge motion of a model tanker in still water and subjected to regular waves. The results revealed an additional lowfrequency damping term due to the presence of the waves, often referred to as wave drift damping coefficient. This is defined as the wave drift force per unit small forward velocity of a body moving in waves (Nossen et al., 1991). In many sea states, this wave drift damping is of the same order of magnitude as viscous forces, and may even be the dominant damping effect. Nossen et al (1991) developed a numerical method to compute the velocity potential and the first-order and mean second order wave forces onfloatingbodies with small forward speed in three dimensions. The method is based on applying Green's theorem and linearizing the Green's function and velocity potential with respect to forward speed. The wave drift damping coefficient is readily obtained from the mean second-order drift force. Zhao et al. (1988) and Zhao and Faltinsen (1989) have presented a numerical method to calculate the wave drift damping coefficient. They applied a boundary element method with Rankine sources close to the body for the flow in the vicinity of the body, and have matched this to the flow in an outer regime where a multiple expansion is used. 1.2,4 Statistics pf Low-Frçqygnçy Motions The statistics of the second-order forces and motions are of ultimate interest in the design of mooring systems. Vinje (1983) derived various closed-form expressions for the probability distributions of the motions based on alternative assumptions. Hineno (1988) developed an approximate method for the determination of the distribution of the response maxima for a system with a weakly nonlinear response in irregular waves. The method included the effect of the finite bandwidth of the response spectrum. For slow drift responses, it has been shown by several authors (Naess, 1986, and Stansberg, 1991) that the drift forces and corresponding drift motions are strongly non-Gaussian. Under the assumption that the equations of motion are linear, with time-invariant parameters, and decoupled and that the excitation is limited to second-order slowly varying drift forces, Naess (1986) derived a closed-form expression for the probability density function of the slow drift motions. The basic assumption of the analysis is that the force and response processes can be modeled as a secondorder dynamic system. The theory developed assumes that the slowly varying phenomenon can be modeled as a quadratic dynamic system, i.e. a system characterized by a quadratic transfer function. A simple extreme value estimation method for non-Gaussian slow drift responses based on the bandwidth of the slow drift response spectrum is given by Stansberg (1991). He presented a three-parameter statistical model corresponding to a simplified version of Naess's analysis, where, in addition to the standard deviation and the number of oscillations in a time series, the bandwidth of the response spectrum is a required input parameter. A comparison between simulations based on this model and experiments showed a reasonably good agreement. 1.2.5 Wave Drift Forces on a Slender Structure The above theories are all based on potentials in the absence of a viscous drag force. However, for slender structures the drag force is an important component of the mean and low-frequency drift forces and should be taken into account. A semi-submersible rig is a particular example of a structure with slender members, and a number of theories have been proposed for calculating the wave drift forces on such a structure. Karppinen (1979) presented a computational method based on potential theory and the assumption that the elements of the rig such as the columns and floaters are slender and do not interact hydrodynamically. Pijfers and Brink (1977) and Ferretti and Berta (1980) also make use of the assumption of slendemess and the absence of hydrodynamic interaction of the elements of a semi-submersible. The hydrodynamic force in each element is determined by the use of the Morison equation and the relative velocity between the fluid and the elements. The total force is found by a summation over the elements. The drift force is defined as the mean value of the total force averaged over a wave period. Results of calculations indicate that viscous effects arising from the drag term in Morison's equation are significant and that the drift forces are proportional to about the third power of wave height. In general, most research on drift forces and low frequency motion has related to large structures and very few studies have been carried out with respect to slender structures. 1.3 Scope of The Present Investigation The objective of the present study is to investigate numerically the influence of wave grouping characteristics on the low-frequency drift forces and motions of both moored large and slender structures subjected to irregular waves. In order to achieve this, the study incorporates the following components: (i) A numerical simulation of wave records with specified spectral densities and wave grouping characteristics. (ii) An examination of the influence of wave grouping parameters, including the groupiness factor, peak frequency of the SIWEH, and broadness factor, on the response of a moored large circular cylinder. (iii) A statistical analysis of slow drift forces and motions of a moored large circular cylinder in order to verify the validity of a slow drift response prediction model. (iv) An examination of the influence of wave grouping parameters on the drift force on a fixed slender circular cylinder, and the influence of cylinder motion on the drift force on a moored cylinder. Chapter 2 Numerical Simulation of Wave Group Activity 2.1 Wave Group Definition and Measures Wave groups correspond to a run of higher than normal waves appearing consecutively in a sea state and are frequently observed at sea. Wave groups have an important impact on a wide range of coastal and offshore activities and consequently suitable measures of wave groupiness should be included in wave data collection programs and should be taken into account appropriately in coastal and offshore design. A number of approaches have been proposed to describe wave groupiness. One measure which has become widely accepted is the SIWEH introduced by Funke and Mansard (1980). SIWEH is an acronym for the "Smoothed Instantaneous Wave Energy History" and is a continuous function of time representing the intensity of wave grouping. This in turn, can be characterized by a number of parameters including the groupiness factor, the SIWEH peak frequency, and the broadness factor of the SIWEH. The various measures indicated above, together with methods of their estimation, are now indicated. 2.1.1 SIWEH The time varying level of wave group activity can be described by the variation of wave energy density with time, such that wave energy here is defined as a moving average of the square of the water surface elevation over a specified duration. The smoothed instantaneous wave energy history E(t) is defined as: E(t) = ^ J T]\t+x) Qi(T)dT for Tp < t < Tr-Tp (2.1) where 2Tp is the duration over which the averaging is performed, Tr is the duration of the wave record, ri(t) is the water surface elevation, and Qi(t) is a general smoothing or window function, with Q = 1 corresponding to no smoothing. In fact, Tp is generally taken as the peak period of the wave spectrum, so that the averaging is typically carried out over a duration of two characteristic waves. For the initial and final portions of the record, Eq. 2.1 should be modified to take account of the corresponding limits of integration. Thus we have ^(^^ = {Tp + t) 1 Ql^'^) for 0 < t < Tp (2.2) Tr-t 9 E(t) = (Tp + (Tr - t)) Tl^(t+x) Qi(x) dx for Tr-Tp < t < Tr (2.3) x=-Tp A rectangular smoothing function corresponding to Q = 1 is the simplest and most direct possibility, but the Bartlett window is generally considered to be superior in providing a smoother time series of E(t). This is given as: Q,« J ' - % . 0 for -Tp s . < Tp ^2 otherwise Since the SIWEH E(t) corresponds to a time-varying signal, it can be used to develop a corresponding spectral density function SE(f), which is considered to be an important descriptor of wave grouping. Fig. 2.1 illustrates the foregoing and indicates (a) a wave record Ti(t), (b) the corresponding SIWEH E(t), and (c) the spectral density S^(f) of the SIWEH. 2.1.2 Groupiness Factor As previously mentioned, the SIWEH is a continuous function of time and it is appropriate to develop suitable characteristic parameters of the SIWEH. Perhaps the single most useful such parameter is the groupiness factor Gp. This is defined as the standard deviation of E(t) and normalized with respect to the mean E: [E(t)-E] dt T1/2 (2.5) Since the variation of E(t) about its mean must equal the area under the SIWEH spectral density, the groupiness factor, Gp may also be expressed as: Gp (2.6) where meo and mo are the zeroth moments of the SIWEH spectral density SE(f) and the wave spectral density Sj^f), respectively. 2.1.3 SIWEH Peak Frequencv The SIWEH peak frequency fsw is defined as a characteristic frequency of wave group occurrence and is taken to correspond to the peak frequency of the SIWEH spectrum. In Fig. 2.1, the peaks in the SIWEH E(t) correspond to the occurrence of wave groups, so that 1/fsw corresponds to a suitable repetition period of the wave groups. 2.1.4 Broadness Factor The broadness factor ^ describes the width of the spectral peak within the SIWEH spectral density, and thus is analogous to a damping ratio. In the present context, it may be defined in terms of a normalized parametric form of the SIWEH spectral density SE(f), whereby SE(f) is given in terms of Ç as: Ç SE(fr) = ir (2.7) V(i-frY.4cVVI7 where SE(fr) is the SIWEH spectral density as a function of normalized frequency, fr defined as fr = f/fsw> and C is a scaling factor. The spectral peak becomes narrower as C, becomes smaller, with C, typically varying from 0.1 to 1.0. Furthermore, the SIWEH becomes more regular for smaller ^ because the spectral width is smaller, so that the regularity of the envelope of grouped waves increases with decreasing 2.2 Simulation of Wave Group Activity A suitable procedure for synthesizing a wave record corresponding both to a specified spectral density as well as a specified SIWEH has been described by Funke and Mansard (1980). In most cases, the SIWEH spectral density rather than the SIWEH itself may be specified. In such a case, one would initially synthesize a SIWEH record and then use this together with the specified wave spectral density to synthesize the wave record itself. These procedures are briefly summarized in turn. 2.2.1 Synthesis of a SIWEH from its Spectral Densitv A specified SIWEH spectral density SE(0 can be used to generate a family of different SIWEH records, all of which have the same SIWEH spectral density but with different phase spectra corresponding to more or less randomly selected phases. In this case, the SIWEH is expressed as a Fourier series: N E(t) = E n=l En cos(nCût - Yn) (2.8) where E is the mean and En and Yn are the Fourier amplitudes and phases respectively, and co is the wave angular frequency. This corresponds to an inverse discrete Fourier transform which may be denoted: E(t) = F-l(En,Yn) (2.9) It is required to obtain En and Yn in terms of the specified quantities, so that these can be applied to Eq. 2.8 to obtain E(t). The Fourier amplitudes En can be expressed in terms of the specified SIWEH spectrum (Funke and Mansard, 1980) as: (2.10) where SE(nAf) is the amplitude of the SIWEH spectral density atfrequencynAf, Af = l/Tf, and Tj is the repetition period of the SIWEH. In order to apply the discrete inverse Fourier transform, a suitable phase spectrum defining Yn must be developed. It is common practice to create this by selecting the phases from a random number generator which has a uniform distribution over the interval -7C to 7t. However, the selection of the phases Yn is not quite so simple. Such a phase spectrum applied to the inverse Fourier transformation can be shown to give rise to a Gaussian amplitude probability density function, whereas the SIWEH function is itself highly non-Gaussian (Funke and Mansard, 1980). This imposes a severe limitation on the ability to synthesize such a function through inverse Fourier transformation. Funke and Mansard have suggested an iterative procedure to overcome this difficulty. An initial estimate of the SIWEH, denoted E(l)(t), is obtained by the inverse Fourier transform corresponding to Eq. 2.9, with Yn initially obtained so as to have a uniform distribution, and the Fourier amplitudes En given by Eq. 2.10. The resulting time function E(l)(t) is then clipped below a value of -amo, where a is some arbitrary value, so that the resultant time signal, denoted E(2)(t), is now distorted with its troughs having values greater than or equal to -amQ. A subsequent forward discrete Fourier transform of this clipped function E(2)(t) will produce a new set of ampUtudes and phases: [Bn(2), Yn(2) ] = F [E(2)(t)] (2.11) Although the new amplitude spectrum Bn^^) is incorrect, it is expected that the new phase spectrum y^P") is a better approximation of the unknown phase spectrum than was the initial phase spectrum Yn^^^- Consequently the discrete inverse Fourier transform is now applied to [Bn^l), Yn^2)] to obtain a modified time function, which is again clipped as before to provide E(3)(t). The resultant time function will now be non-Gaussian. However, this will still contain values of -amo affected by the clipping procedure. It is therefore necessary to repeat this operation of clipping and transforming until the clipping procedure no longer has an effect on the time function. 2.2.2 Svnthesis of a Wave Record from a SIWEH The wave record ri (t) may be obtained by an inverse discrete Fourier transform of regular waves of amplitudes An and phase angles En deriving from: N Tl(t) = X n=l (2.12) An COS(nCûot - En) The amplitudes An are known for the specified wave spectral density Syff): (2.13) The phases En are commonly taken as uniformly distributed from 0 to 27i, but in the present case must be precisely established so as to produce the desired SIWEH E(t). In order to create a phase spectrum, which will indeed correspond to the specified SIWEH E(t), an iterative procedure described by Funke and Mansard (1980) is used. In order to develop an initial approximation, the wave record is first taken as: Tlo(t) = VË(t)sin[cûpt-^e(o)(t)] (2.14) where cOp is the peak frequency of the wave spectrum, and e(o)(t) is estimated very approximately in terms of E(t). This is used to obtain a corresponding SIWEH Eo(t). This is compared to the original SIWEH to provide a correcting function C(t) defined as : The use of Co(t) provides a correction to Tio(t) which represents a first approximation to the wave record: Tll(t) = Co(t)Tio(t) (2.16) This is Fourier transformed to provide corresponding amplitudes and phases A^^^ and e^^'^. A^^^ is replaced by the precise values An obtained from Eq. 2.13, and an inverse discrete Fourier transform on An and e^^^^ is then applied to obtain r\^^\t). This procedure is repeated so as to obtain successive sets of En which leads to further improvements. 2.2.3 Generation of Wave Records with Different Wave Groupin g On the basis of the foregoing procedure, described in detail by Funke and Mansard (1980), a time series Ti(t) of a grouped wave train can be synthesized from a specified wave spectral density ST^(f) and a specified SIWEH spectral density S^(f). The GEDAP (Generalized Experiment control. Data acquisition and Analysis package) analysis program WAVE_GEN 2D is used for this purpose. GEDAP is a general purpose software system developed at the Hydraulics Laboratory of the National Research Council of Canada for the analysis and management of laboratory data, including real-time experiment control and data acquisition functions. WAVE_GEN 2D is a random wave generation program which includes wave train synthesis for specified wave spectra and wave grouping characteristics. A Digital Equipment Corporation (DEC) VAXstation 3200 computer running VMS (Virtual Memory System) version 5.3 as the operating system is used to run the GEDAP programs. Chapter 3 Wave Drift Forces on a Moored Large Structure The force on a large structure subjected to irregular waves can be decomposed into (i) firstorder oscillatory forces at wave frequencies, (ii) high frequency second-order forces associated with sums of the component wave frequencies, and (iii) low-frequency second-order forces associated with differences of the component wave frequencies. Although the second-order lowfrequency forces, better known as the slowly varying wave drift forces, are small in magnitude, they may act at frequencies close to the resonant frequencies of a structure and hence induce significant resonant motions. The mean and low-frequency second-order wave drift forces acting on a structure moored in waves are usually of interest with respect to mooring loads and horizontal motions. 3.1 Mean Drift Force in Regular Waves The hydrodynamic force on a floating body in regular waves may be resolved into an oscillatory part and a constant part, the latter being known as the mean drift force. Maruo (1960) shows for the two-dimensional case of an infinitely long horizontal cylinder fixed in regular waves with its axis perpendicular to the wave direction that the mean drift force per unit length is given by: F^^pgAr^ (3.1) where p is the mass density of water, and Ar is the amplitude of the reflected waves scattered by the body in a direction opposite to that of the incident wave propagation, and which is directly proportional to the incident wave amplitude. More generally, the mean drift force per unit length, in regular waves can be written in the form: F = 2PgCDA- (3.2) where A is the incident wave ampUtude and CD is a mean drift force coefficient. The mean drift force coefficient CD may be determined experimentally, or in some cases by calculation. For a given structure it is generally a function of wave frequency. Newman (1974) has shown that the mean horizontal drift force on the structure may be expressed in the form: F = J J [pcos 0 -I-PVR(VRCOS0 - Vesine)]Rd0 dz (3.3) where p is the first-order hydrodynamic pressure, V R and Ve are the radial and tangential components of fluid velocity, the bar denotes a time average over a wave period, and the integration is over the surface Soo of an infinitely large vertical circular cylinder surrounding the structure. Newman's formulation is valid for infinite water depth. Faltinsen and Michelsen (1974) have extended the method to finite water depth. It is noted that in evaluating Eq. 3.3 it is only necessary to know the first-order velocity potential. 3.2 Drift Force in Irregular Waves An irregular, long-crested sea state may be considered as the superposition of a large number of regular uni-directional wave trains: N r|(t) = X A i «^os (coi t -I- eO (3.4) i=l where. Ai and £[ are respectively the amplitude and phase of the component wave trains, ei has the opposite sense to that in Eq. 2.12 in order to maintain consistency with the literature. In order to develop a suitable expression for the drift force (Pinkster, 1975), Eq. 3.4 may be expressed as an amplitude modulated signal with central frequency cOr, varying amplitude A(t), and phase e(t). Equation 3.4 may be written in the form: Ti(t) = A(t)cos(cort + e(t)) (3.5) It may be shown that on the basis of Eqs. 3.4 and 3.5, A(t) and e(t) are given as: A(t) = .N N LI I A i Aj cos {(0)i - 0)j) t + (Ei - Ej)} 1=1 j=l 1/2 J (3.6) and N X A i sin [(coi - COr)t + Ei] tan[E(t)] = ^ (3.7) X A i cos [(COi- CÛr)t + Ei] i=l If the wave record has a narrow-band spectrum, corresponding to small differences between the highest and lowest frequencies and the central frequency then the amphtude A(t) and phase E(t) are of a slowly varying form. From Eq. 3.5, a slowly varyingfrequencyco(t) may be defined as: 9e(t) = 0)r + ^ p m (3.8) From Eqs. 3.7 and 3.8, cû(t) may be expressed as: N N S S A i A j Û)jCOS[(C0i - ©j)t+ (Ei - Ej)] œ(t) = ^ S r S ^ S (3.9) E A i AjCOS[(CÛi - COj)t -1- (Ei - Ej)] i=l j=l which, after regrouping terms, becomes: N 1 m = N COi -I- CDi 1 AiAj(-^^—l)cOS[(Cûi-C0j)t-H(Ei-Ej)] '^'^ ^ I IAiAjCOS[(C0i-(ûj)t+(Ei-Ej)] i=l j=l (3.10) If the wave amplitude A(t) and the frequency co(t) are slowly varying quantities, then the wave drift force per unit length may be approximated from Eq. 3.2 as: F(t) = ^pgCD[a)(t)]A^(t) (3.11) According to Pinkster (1975), by substituting A(t) as given in Eq. 3.6, Eq. 3.11 reduces to: F(t) = i pg X S Ai Aj CB(^^~^) ^ i=l j=l ^ cos[(cûi - coj) t + (ei - ej)] (3.12) Equation 3.12 can be rewritten by using the Fourier coefficient of the water surface elevation, ri*(cû) as: F(t) = Re[^ pg I ^ 1=1 I Ti*(a)i) TVVJ") C o C ^ i f ^ ) exp Rcoj - coj) t ]] j=l ^ (3.13) where the overbar indicates the complex conjugate. Once the complex amplitudes T)* are known, an inverse Fast Fourier Transform (IFFT) can be used to compute F(t) more efficiently than through direct summation. It is seen from Eq. 3.12 that the drift force is also of a slowly varying form since it contains frequencies that are differences of the frequencies cOi and cOj. The total wave drift force in irregular waves also contains a steady part corresponding to the portion of Eq. 3.12 when i = j . Thus the mean drift force is given as: F = ^ p g I Ai^CD(cùi) ^ i=l 3.3 (3.14) Low-Frequency Surge Motion The response of a moored structure subjected to irregular waves consists of small amplitude high frequency surge, heave, and pitch motions and large amplitude low-frequency surge motions. The high frequency motions are related to the individual wave frequency components in the wave train. The low-frequency surge motion is concentrated around the naturalfrequencyof the moored vessel. The surge motions of a structure moored by a system with linear restoring characteristics, which is treated as a single degree of freedom system, can be described by the following equation of motion: (m + \i((ù)) x(t) + À(œ) k(t) + k x(t) = F(t) (3.15) where m is the mass of the structure, | i is a frequency dependent added mass, X is a low-frequency damping coefficient including the effects of viscous damping, wave drift damping, and wave radiation damping, k is the mooring stiffness, F(t) is the low-frequency wave drift force, and x(t) is the low-frequency surge motion. The wave drift and radiation damping are frequency dependent, whereas the viscous damping is generally considered to be constant. The frequency dependent added mass and wave radiation damping coefficients may be calculated by treating the corresponding linear wave diffraction problem. As indicated above, the damping coefficient can be separated into three components, so that , ,(v) X= X -(w) +X (r) +X , where X (V). . . (w). is a viscous damping coefficient. A, . is a wave dnft damping coefficient and X^ ^ is the wave radiation damping coefficient. The wave drift damping coefficient .(w) A is associated with the potential flow field corresponding to the incident wave field interacting with the slowly moving structure. To a first approximation, the corresponding damping force may be taken as proportional to the low-frequency velocity of the structure and to the square of the wave amplitude. Thus the damping coefficient X X^^^ = a((ù) (i) k(t) may be expressed as: (3.16) where a(Cû) is a proportionality constant which may be obtained by solving the wave diffraction problem for a structure with a small horizontal velocity. By considering a change in reference frame, this is related to the force due to a combined wave and slow current field past a fixed structure (e.g. Isaacson and Cheung, 1992). By using the numerical method given by Zhao et al. (1988), it is possible to calculate the wave drift damping for any wave length. Since the calculation of the wave drift damping is not the main objective of the present work, the wave drift damping coefficients adopted here for a vertical surface-piercing cylinder moving in head waves are approximated from available results (Zhao and Faltinsen, 1989). Once the added mass | i , damping coefficient X, and low-frequency exciting force F are known, the surge response x(t) can be calculated by solving Eq. 3.15. Since the equation is linear, it is computationally more efficient to solve the equation of motion in frequency domain. If the motion is periodic with frequency oa, the excitation force F(t) and response x(t) can be written as F(t) = FQ exp(-itot) and x(t) = XQ exp(-icot), where FQ and XQ are the complex amplitudes, and Eq. 3.15 then becomes: [-(ù'im+^id))) - icoXica) + k]xo = FQ (3.17) The response in the frequency domain is given by: xo(cû) = H(cû)Fo((û) (3.18) 2 -1 where H(o)) = [-co (m-l-|j,(a))) - ico À,(co) + k] . Once H(co) is known, the response in the time domain can be calculated using an Inverse Fast Fourier Transform (IFFT). 3.4 Statistics of Low-Frequency Surge Response In the final analysis, the information required is the statistics relating to the most probable maximum values of the motions and loads for a given set of sea conditions. The surge motion of a moored structure in irregular waves is often dominated by a significant low-frequency contribution excited by nonlinear wave drift forces. As a result, the evaluation of design values for extreme motion responses must include slow-drift responses as a major contribution. 3.4.1 Extreme Values of Non-Gaussian Slow-Drift Response In design, the maximum value of the slow drift displacement is required, since this will correspond to the largest forces in the mooring lines. For displacements which have a Gaussian distribution, the theory of extreme values may be applied, such that the expected value of the maximum of the displacement Xmax, is given approximately by: xmax = CJxV2 In Nc (3.19) where Ox is the standard deviation of the surge response x, Nc is the number of cycles in the specified duration and is given by Nc = Tr/T^, where Tr is the duration of the record and Tz is the zero-crossing period of the record. For a lightly damped narrow-band motion, Tz corresponds approximately to the natural period of the system. In fact, the above approach will generally be unsuitable since it has been indicated by several authors (Newman, 1974, and Naess, 1986) that the wave drift force is strongly non-Gaussian, so that the resulting motions will also generally be non-Gaussian (Naess, 1986, and Stansberg, 1991). Consequently, extreme values of the motion may be significantly larger than those estimated using Eq. 3.19, which is based on the assumption of a Gaussian process. An empirical alternative has been suggested by Stansberg (1991). In this rough approximation, the extreme slow-drift response Xmax should instead be estimated on the basis of an exponential distribution of the amplitudes: Xmax = C^xlnNc (3.20) For many cases, however, Eq. 3.20 is too conservative and a more accurate three-parameter model has been recommended (Stansberg, 1991), such that in addition to Ox and Nc, the band-width of the response spectrum also influences the maximum response. This is considered further below. 3.4.2 Slow Drift Response Prediction Model Stansberg (1991) presented a three-parameter statistical model based on Naess' (1986) analysis to estimate the extreme values of the slow drift response. The three-parameter model is based on the following assumptions: (1) The drift forces vary little over the frequency range of the input wave spectrum, so that the forces are assumed proportional to the square wave envelope. (2) The dynamics of the moored system are described by a linearized system model. Assuming that the largest maxima are statistically independent of each other, the maxima may be assumed to follow an exponential distribution asymptotically. Apartfromthe standard deviation and number of oscillations, the model involves an additional parameter B which is defined as the ratio between the width of the wave group spectrum W G and that of the slow drift response spectrum Wx, B = WQAVX. The spectral width Wx of the drift response is given by: Wx = (3.22) J[Sx(f)]2df o oo where mo = J [Sx(f)] df, and a corresponding definition applies to W Q . According to this o model, the corresponding expected extreme amplitude among Nc maxima is given by: ^max = o^x[(—)ln Nc + ( ^ ) (1 + In do) - V B ] <7x CTx (3.21) The parameters in Eq. 3.21 can be completely determined with the known standard deviation CTX, of the surge response and the spectral width ratio B : where = Xo = 2ax V B / (B+1) (3.22) do- (3.23) UlUa-XM] m=l [(B-1)/(B+1)]"'. However, it is observed that unless B is very large, the resulting extreme value behaviour is roughly determined by the exponential model. The low-frequency drift force and response of a large moored cylinder have been calculated using the theoretical methods described above and corresponding results are presented and discussed in Chapter 5. For the present. Chapter 4 describes corresponding approaches for a slender moored cylinder. Chapter 4 Wave Drift Forces on a Moored Slender Structure Theoretical analyses of the mean and low-frequency drift forces on large floating structures have been provided by many authors on the basis of potential flow theory, and generally good agreement with the experimental results has been found for structures such as ships. Potential flow theory is based on the assumption of an inviscid fluid, and although this may be reasonable for ships or barges, it is no longer valid for structures such as a semi-submersible rig, which contain many slender structural elements. For such elements, flow separation and associated drag forces arise, and cannot be disregarded. In order to examine this wave loading regime, the case of a moored slender circular cylinder in irregular waves is considered here. The low-frequency drift force for such a structure can be calculated using two different approaches: (i) atimedomain approach based on Morison's equation for irregular waves. (ii) Pinkster's approximate method. These are considered in turn. 4.1 Time Domain Approach The Morison equation continues to form the basis of wave loading and response predictions for slender structural members. In this formula, the hydrodynamic force on a fixed cylinder is expressed as the sum of a drag force and an inertia force. The drag force is proportional to the square of the fluid velocity, while the inertia force is proportional to the fluid acceleration. For a moving cylinder, the Morison equation is extended to account for the velocity and acceleration of the cylinder itself. Thus the force on a circular section is given as: 2 F = ^pDCd(u-x)lu-il+ p ^ ^ C m U 2 -p^^CaX (4.1) where p is the density of the fluid, D is the cylinder diameter, and Cj, Cm and Ca are the drag, inertia and added mass coefficients respectively. In the expression for the fluid force u and u are the ambient flow velocity and acceleration respectively. For a vertical circular cylinder at the water surface, which moves initially as a single degree of freedom system, this can be applied by integrating the force over the immersed portion of the cylinder, to yield: mx + X x + lcx = Fi(t) + Fd(t) - ^ix (4.2) Tl(t) u(z)dz Fi(t) = p ^ — C m ^ z=-h - (4.3) Tl(t) ({i(z)-x) l{i(z)-ildz Fd(t) = 5 p D C d ^ z=-h (4.4) 2 where | l = p (TI D /4) Ca . m, X, k are the equivalent mass, damping constant, and stiffness of the single degree of freedom system, respectively, x is the cylinder velocity, and dots denote time derivatives. Ti(t) is the instantaneous water surface elevation and h is the draft of the cylinder. For piuposes of assessing the alternative formulations the cylinder is restrained in the vertical direction (i.e. zero heave), even though this may be somewhat unreahstic. Wave drift forces on such a structure may arise because of two kinds of nonUnearities. One is associated with the drag term (a - x)lu - xl in the Morison equation, and the other is due to the varying water surface elevation. In a deterministic analysis of a fixed structure it is relatively straightforward to include the drag nonlinearity, since x = 0 and this will only involve only u, which is known. However, in the case of a floating structure the nonlinearity now involves both the incident flow velocity u as well as the unknown response x. The cylinder can be discretized in the vertical direction, and with known velocities and accelerations at different elevations the Morison loading can be calculated by integration of forces from the bottom of the structure to the instantaneous water surface elevation. Nevertheless Eq. 4.2 can be solved in the time domain using a suitable numerical technique. In the present study, the Newmark iteration method (Newmark, 1959) with a linear acceleration formulation has been used. The method proceeds as follows. With the acceleration xat an advanced time step t + At initially assessed to be the same as that at the previous time-step, xt+At = xt. the corresponding velocity x and displacement x at are given by: Xt+At = xt -I- xtAt + YAt(xt+At - xt) Xt+At = Xt + ktAt + (At)2 ^ + p (xt + At - Xt) (4.5) where At is the time step size, y = 0.5 and P = 1/6. These initial values of x and x are used in Eq. 4.2 to obtain a corresponding value of x at the same advanced time t + At. In doing this, Eq. 4.2 is re-written as: .. Fi(t) + Fi(t) - Xx - kx X = (4.6) This revised value is applied to Eq. 4.5 to obtain revised values of x and x and the iteration continues until the value of x calculated by Eq. 4.6 converges. Since the interest here is on the wave drift force, the force time series obtained on the basis of the above procedure is used in turn to provide the wave drift component. In order to do so, a Fast Fourier Transform is performed on the force time series, and the low-frequency force component can then be separated from the remaining force component at the higher frequency. 4.1.1 Simulation of Wave Kinematics The water particle velocities and accelerations, which are required in Eqs. 4.3 and 4.4, at the instantaneous displaced position of the structure, are computed on the basis of linear wave theory. Once a surface elevation time series has been obtained, corresponding time series of the particle velocity and acceleration can readily be obtained by treating the specified record as periodic, and using a Fourier analysis approach. Thus thefreesurface elevation may be expressed as: N Tl(t) = X (^n COS Cunt - bn sin Cunt) (4.7) n=i where an and bn are the Fourier coefficients obtained from a direct Fast Fourier Transform (FFT) of a specified time series, cOn = ncoo, cOo - 2K/Tt, Tr = M At, is the record length. At is the time- step size, M is the number oftimesteps in the record and N = M/2 -i-1. The corresponding velocity and acceleration at an elevation z (measured upwards from the still water level) at any instant are then given on the basis of linear wave theory as: N 1^ cosh ^ yr j \\ u(x,z,t) = 2, COn—^j^^j^^^j^y^ [anCOS (knX-COnt) - bnSin (knX-COnt)] (4.8) N ù(x,z,t) = X con^'^-^^^^^^^^ [an sin(knx-cùnt) - bn cos(knx-cûnt)] (4.9) where kn is related to cOn, through the linear dispersion relationship: con^ = gkntanh(knd) (4.10) Thus, a specified free surface elevation record is first used to obtain the corresponding Fourier amplitudes, an and bn, and these are in turn used to construct the corresponding horizontal velocity and acceleration at any specified elevation, z. The kinematics above the still water level are extrapolated from the values at still water level using the partial derivative of the kinematic variable at the still water level. For any kinematic variable v, this can be expressed as: v(z, t) = v(0, t) + z ^ (0, t) 4.2 for z > 0 (4.11) Pinkster's Method Pinkster's approximate method, explained in Section 3.2, can be used to calculate the lowfrequency drift force in irregular waves from the calculated drift force coefficients for regular waves. The mean drift force can be defined as the time average of the hydrodynamic forces acting on the structure. The mean drift force coefficient can be obtained by determining the mean hydrodynamic forces over one period of encounter. In general the hydrodynamic force on a cylinder can be calculated using the Morison equation given by Eq. 4.1. The mean drift force F in regular waves is computed as: t-i-T FM(t)dt (4.12) where T is the wave period and F M (t) is the time varying force obtained on the basis of the Morison equation. It turns out that the inertia force does not contribute to the mean drift force. This is because there is no nonlinearity inherent in the inertia force which giveriseto the drift force; and secondly, the free surface effect can be shown to give no contribution. This is because T] is proportional to cos(Cût), il is proportional to sin(a)t), so that the second order term proportional to r\ u(o) has a zero mean. The drift force coefficient CD(f) is defined as: where f is the wave frequency and A is the amplitude of the of the incident waves. The low-frequency force is calculated in terms of C D by using Pinkster's approximate expression given by Eq. 3.12. Pinkster's approximate expression is widely used to calculate the mean and low-frequency drift forces in irregular waves based on the mean force data for regular waves determined either from experiments or theory. The calculated low-frequency motions based on this approach have been found to agree fairly well with the experimental data for large structures reported by various researchers (e.g. Remery and Herman, 1971, and Sawaragi et al., 1988). However, none of the reported cases compare numerical results based on this approach with results from the complete nonlinear solution for the case of slender structures by using the Morison equation in time domain. This completes a description of the theoretical approaches for a slender cylinder. Numerical results for both large and slender cylinders are now presented. Chapter 5 Results and Discussion The results of a numerical study of wave groups and the effect of wave grouping on the forces on moored large and slender structures and the response of a large structure are presented and discussed in this chapter. Initially, results relating to the numerical simulation of wave records corresponding to different wave grouping parameters are presented. These wave records are then used to calculate the lowfrequency drift force for a moored large circular cylinder, using Pinkster's approximate method described in Section 3.2. The low-frequency response of a large cylinder is calculated in the frequency domain on the assumption that the motion corresponds to a single degree of freedom system. In order to examine the effect of the various wave grouping parameters, results are presented for conditions in which each wave group parameter is varied in mm. Finally, a statistical analysis of the low-frequency response of a moored large circular cylinder is presented. Extreme values of the motions are predicted using the slow drift response prediction model described in Section 3.4.2. Corresponding results are then presented for a moored slender circular cylinder, for which the low-frequency force is calculated using the two approaches described in Chapter 4. In one approach, the time history of the force is calculated by using the Morison equation for irregular waves. The total force is determined by integrating the force distribution on the cylinder up to the instantaneous water surface elevation. The drift force arises mainly because of changes in the wetted surface of the cyhnder and the nonlinear drag term in the Morison equation. The lowfrequency drift force is then extracted from this force record by a Fourier Transform technique. In the second approach, the mean drift force coefficient in regular waves is first calculated using the Morison equation. These regular wave results are then applied to Pinkster's approximate method in order to obtain the corresponding low-frequency force in irregular waves. Corresponding results of the two methods are then compared. 5.1 Synthesis of a Grouped Wave Train Time histories of the free surface elevation have been simulated for a number of representative wave conditions. The conditions were selected with respect to a base case corresponding to the following parameters: Wave spectrum: Significant wave height, Hs = 2.0 m Peak frequency, fp = 0.2 Hz Peak enhancement factor, Y = 1.0 The peak enhancement factor Y is the ratio of the maximum spectral density to that of the corresponding Pierson-Moskowitz (P-M) spectrum. Thus the spectrum with Y = 1.0 corresponds to P-M spectrum. SIWEH spectrum: Peak frequency, fsw = 0.02 Groupiness factor, Gp =0.6 Spectral broadness factor, C, = 0.2 The various conditions simulated are indicated in Table 5.1. These conditions are intended to represent changes to this base case such that the influence of the various parameters in turn can be examined, while keeping the remaining wave spectrum and SIWEH spectrum characteristics constant. The conditions examined are as follows: (a) The effect of the groupiness factor Gpis examined by changing its value to include G F = 0.6, 0.7, 0.8, 1.0 (tests 1-4). (b) The effect of SIWEH peakfrequencyfsw is investigated by changing its value to include fsw = 0.02, 0.04, 0.014, 0.01 (tests 1, 5-7). (c) The effect of spectral broadness factor Ç is examined by changing its value to include C = 0.2, 0.4, 0.6, 0.8 (tests 1, 8-10). (d) The effect of peak frequency fp is examined by changing its value to include fp = 0.05, 0.1, 0.2 Hz (tests 1, 11,12). (e) The effect of significant wave height Hs is examined by changing its value to include Hs = 2.0, 4.0, 5.0 m (tests 1, 13 ,14). (f) The effect of the peak enhancement factor y is examined by changing its value to include y = 1.0, 3.3 (tests 1, 15). The value of y = 3.3 corresponds to JONSWAP spectrum. The numerical parameters used in the simulations are as follows: Length of the wave record, Tr = 1200 sec Maximumfrequency,fmax = 10 fn Frequency interval, Af = 0.0009765 Hz Together with the various peak frequencies fp = 0.05, 0.1, 0.2 Hz, which have been used above, these values correspond to a wave record discretized at M = 512, 1024, 2048 points and with a time step size At = 2.34375 sec, 1.17188 sec and 0.58594 sec respectively. Program PARSPEC of the GEDAP package generates the wave spectrum at equally spaced frequency increments from 0 to fmax- Program SYSI first generates the SIWEH spectrum for a specified SIWEH peak frequency and broadness factor, and adjusts the scale of the SIWEH spectral density so as to satisfy the desired values of the groupiness factor and the mean value of the desired SIWEH. The mean value of the SIWEH corresponds to the variance of the wave spectrum. Once the SIWEH spectral density has been computed, the program then generates the corresponding SIWEH record, again using the iterative method described in Sect. 2.2.1. Finally, the program SYW synthesizes the wave record itself from the computed SIWEH and wave spectrum. Figure 5.1 shows two wave records (Fig. 5.1a and 5.1b) and the corresponding SIWEH spectra (Fig. 5.1c) having the same wave spectral density but with different groupiness factors Gp (tests 1 and 4). The magnitude of waves in wave groups is seen to be relatively high for the wave record with the higher groupiness factor (Fig. 5.1b). Although both records correspond to the same wave spectrum, their SIWEH spectral densities differ in magnitude (although similar in shape). Figure 5.2 shows two SIWEH records E(t) (Fig. 5.2a) and the corresponding SIWEH spectra (Fig. 5.2b), for wave records having the same wave spectral density but different SIWEH peak frequencies (tests 5 and 7). The SIWEH record corresponding to the higher SIWEH peak frequency exhibits a higher number of peaks. This demonstrates that the reciprocal of the frequency l/fgw corresponds to the repetition period of the wave groups, since the SIWEH maxima correspond to the occurrence of wave groups. The SIWEH spectra shown in Fig. 5.2b exhibit more noticeable differences, such that the SIWEH with the higher peak frequency has a smaller peak spectral magnitude and contains higherfrequencycomponents. Figure 5.3 shows SIWEH spectra for wave records with the same groupiness factor and SIWEH peakfrequency,but with different broadness factors Ç, (tests 1, 8-10). The shape of the SIWEH spectra is different for different broadness factors, with the figure indicating how the SIWEH becomes narrower for smaller values of ^, and accordingly, the regularity of the envelope of the grouped waves then increases. Figure 5.3b and 5.3c shows two wave records with C = 0.2 and 0.8. 5.2 Low-Frequency Drift Force and Response of a Large Cylinder Low-frequency drift force and response records have been obtained for the moored large circular cylinder shown in Fig. 5.4. The cylinder has a diameter D = 20 m, freeboard s = 2 m. draft h = 10 m and the water depth is 30 m. The centre of gravity is 6 m above the base of the cylinder (i.e. 4 m below the still water level). The mass of the cylinder is 3220 xlO^ kg. The mooring stiffness in surge is varied from 30 kN/m to 90 kN/m in order to examine its effect on the low-frequency surge drift response. There are no other mooring stiffness components. The low-frequency drift force on the cylinder has been calculated on the basis of Eq. 3.12 for the different wave records obtained in Section 5.1. In applying this, the mean drift force coefficient Cj){f) is obtained using the computer program WELSAS2 (Isaacson, 1985), which provides a solution for wave loads on a large offshore structure on the basis of wave diffraction theory. The mean drift force coefficient was calculated at a number offrequenciesextending up to 0.6 Hz, and the results are shown in Fig. 5.5, which shows the corresponding variation of force with frequency. Values at higherfrequenciesthan 0.6 Hz have been estimated by an extrapolation from Fig. 5.5. The calculated low-frequency drift force time series is applied to a single degree of freedom system simulating the moored cylinder, defined by the equation of motion given in Eq. 3.15. In order to derive the response from Eq. 3.15, values of the added mass and damping coefficient are required in addition to the low-frequency wave drift force. The added mass and radiation damping coefficients are calculated using the computer program WELSAS2. Figs. 5.6 and 5.7 show respectively the computed added mass coefficient and radiation damping coefficient as functions of frequency. The wave drift damping coefficient is also required, but since its calculation is complicated and involves a substantial computational effort, approximate values have been used in the present study. These have been estimated from the results of Zhao and Faltinsen (1989) for a vertical surface-piercing cylinder moving in head waves. Their results apply to a restrained cylinder with draft to radius ratio h/a = 3. Finally, the viscous damping coefficient required in Eq. 3.15 has been neglected for the sake of simplicity. Since the main purpose of this study is to investigate the effect of wave grouping on the response, rather than to calculate the exact response of the moored cylinder, such approximations are considered to be justified. In addition, the mooring stiffness in surge, k is taken as 50 kN/m, which results in a natural frequency fn of 0.016 Hz. The low-frequency surge response is calculated by solving the single degree of freedom equation in the frequency domain. Figures 5.8 to 5.10 show the first 140 sec of the 20 min time histories of surface elevation, as well as the corresponding low-frequency drift force and surge response, for different values of Gp, fgw and C, as indicated in the figures. The variation of the low-frequency drift force involves large values when wave grouping is significant. Higher values of the low-frequency force and surge response are observed for larger values of Gp (Fig. 5.8b and 5.8c). Figure 5.9 shows that the representative zero-crossing period of the low-frequency force time series corresponds to the peak period of the SIWEH spectrum. The broadness factor Ç has no significant effect on the lowfrequency force, although the response is higher for smaller Ç as shown in Fig. 5.10. The spectral densities of the low-frequency drift force and response are calculated using the GEDAP program VSD. VSD is a general purpose program which uses a Fourier analysis technique to calculate the spectral density of a time series. Before taking a Fourier Transform of the data signal, the program first multiplies the signal by a trapezoidal window in order to reduce leakage. The Fourier Transform is then obtained and the periodogram resulting from this operation is smoothed using a simple moving average filter to provide the spectral density function. Figure 5.11 shows spectra of the low-frequency drift force and surge response for different values of the groupiness factor Gp. Although the shapes of the force spectra are similar, the magnitude of the spectral peak increases as Gp increases (Fig. 5.1 la). A similar trend is also seen in the case of a SIWEH spectrum, since it is derived from the square of the wave elevation time series, and the low-frequency force is directly proportional to the square of the wave amplitude. The surge response spectrum also follows a similar trend. Figure 5.12 shows spectra of the low-frequency drift force and the low-frequency surge response for different values of the SIWEH peak frequency fsw The peakfrequenciesof the lowfrequency force spectra correspond to the respective SIWEH peak frequencies, and the magnitude of the peaks of the force spectrum increase as fgw becomes smaller. The surge response spectrum shows a relatively large peak when fsw = 0.014 Hz, since this is close to the natural frequency of the system (0.01583 Hz). The effect of changes in Ç is indicated in Fig. 5.13. The peak of the force spectrum increases and becomes narrower as C, decreases (Fig. 5.13a). The peak frequency of the force spectrum does not exactly correspond to the SIWEH peak frequency and those differences lead to the significant differences in the response spectrum, depending the naturalfi-equencyof the system. In Figures 5.14 to 5.17, the magnitudes of the force and motion are represented in the form of variance ratios mop/moTi and mox/moT], where moF, niox and mo^ are the variances of the lowfrequency drift force, low-frequency surge response and surface elevation, respectively. The figures show these plots as functions of Gp, fsw /fn» and C, in tum. The low-frequency force and surge response increase with increase in Gp (Fig. 5.14). Fig. 5.15 shows that the surge variance peaks in the vicinity of fsw/fn = 1-0, although there is no significant variation in the force variance itself. Figure 5.16 shows that the response variance is influenced by the broadness factor ^, although the force variance is not. Finally, Fig. 5.17 shows that the response values decreased with increase in the mooring stiffness. In all the cases the maximum response variances correspond to fsw/fn = 1.0 as expected. The influence of the significant wave height, peak frequency of the spectrum and peak enhancement factor on the low-frequency drift force and response have also been examined. Computed results of the root-mean-square (rms) and the maximum amplitude (max) of the lowfrequency force and surge response of the large circular cyhnder are shown in Table. 5.2. The rms force values of the force for tests with significant wave heights 2.0, 4.0 and 6.0 m are 15.9, 63.6 and 143.0 kN, respectively (tests 1, 13 and 14). This indicates that the low-frequency force is proportional to the square of the significant wave height. Similarly, the rms response, which is equal to 1.15, 4.60 and 10.34 m, respectively for Hs = 2.0, 4.0 and 6.0 m, is also proportional to the square of the wave height. The peak frequency of the wave spectrum fp also significantly influences the low-frequency force. The rms values of the low-frequency force are 15.9, 8.0 and 0.7 kN respectively for fp = 0.2, 0.1, 0.05 (tests 1,12 and 11). The low-frequency forces are small for smaller peak frequencies of the wave spectrum as the mean drift force associated with smaller frequencies is very small. The low-frequency response also follows a similar trend. The peak enhancement factor y does not show any significant effect on the low-frequency force. The rms values of the low-frequency force for y =1.0 and 3.3 are 15.9 and 16.1 kN respectively. 5.4 Low-Frequency Surge Response Statistics A statistical analysis has been performed on the low-frequency surge motions in order to calculate the root-mean-square values, maximum value Xmax and the number of slow drift oscillations. The GEDAP program STATl is used to calculate the mean, standard deviation, rms, minimum and maximum of a time series record. The number of slow drift oscillations for a given response time series is calculated using the GEDAP program ZCA. This performs a time domain zero-crossing analysis on a time series, and changes the inter-sample spacing of the input signal to ensure that the sampling rate is high enough for accurate zero crossing analysis. It also uses local parabolic curve fitting to define the peaks and troughs in the signal. The results of the statistical analysis on the low-frequency surge response for different wave conditions are presented in Table 5.3. The program STAT2 has been used to calculate the probability density of the low-frequency response. The probability densities of the square wave envelope time series (i.e. the SIWEH) for tests 5 and 14 are compared with the equivalent (same mean and standard deviation) Gaussian envelope in Fig. 5.18. Figure 5.19 shows a similar comparison for the low-frequency surge response . The probability distribution of the SIWEH is distinctly non-Gaussian since the SIWEH is a nonlinear function of the water surface elevation which is itself well represented by a Gaussian distribution. It has been postulated (Stansberg, 1991) that the square wave envelope of a Gaussian distributed rime series is exponentially distributed. The low-frequency surge response is reasonably well represented by the Gaussian distribution (Fig. 5.19a). This is due to the low system damping which is about 1.89%. Stansberg (1991) concludes that the distribution of the low-frequency response of a linear system subjected to non-Gaussian excitation should approach a Gaussian distribution as the damping approaches zero. The slow drift extreme response statistics have been calculated using the three-parameter prediction model of Stansberg (1991) (Eq. 3.22) as well as the Gaussian and exponential models. Tables 5.3 and 5.4 show a comparison of the extreme amplitudes calculatedfromthe response time series and the predictions of the three models, for damping ratios of 1.89% and 12.36% respectively. The Gaussian model predicts the extreme ampUtudes reasonably well for the smaller damping ratio of 1.89%. This is in agreement with the earlier observation that the response approaches Gaussian distribution as the damping of the system approaches zero. The exponential model predicts extreme response amplitudes reasonably well for smaller values of the spectral width ratio B, which corresponds to a response spectrum with a relatively large spectral width. The three-parameter model performs well for B values ranging between 4.73 and 4.91. The value of B becomes smaller for the same input spectrum, i.e. the response becomes non-Gaussian (Fig. 5.19b), when the damping ratio is 12.36% and the three-parameter model predicts the extreme amplitudes reasonably well (Table 5.4). This shows that as damping increases, the response becomes non-Gaussian and that the three-parameter model is suitable for the case of a non-Gaussian response. 5.5 Low-Frequency Drift Force on a Slender Cylinder Low-frequency drift force results have also been obtained for the moored slender circular cylinder shown in Fig. 5.20. The cyhnder has a diameter D = 2 m, freeboard s = 2 m, draft h = 10 m and the water depth d = 30 m. The centre of gravity is 6 m above the bottom of the cylinder (i.e. 4 m below the still water level). The mooring stiffness in surge is taken as k = 3.2 kN/m. This corresponds to a natural period of 20 sec in surge. The low-frequency drift force has been calculated using both the methods described in Sect. 4.1 and 4.2: the Morison equation applied directly to the wave record; as well as Pinkster's approximate method used in conjunction with the Morison equation applied to regular waves. The Morison equation is used to calculate the force in regular waves by Eq. 4.1 with the cylinder assumed to be fixed. As described in Sect. 4.1.1, the computations are carried out by the use of linear wave theory to describe fluid kinematics, and involve integrating the wave force up to the instantaneous water surface elevation. Drag and inertia coefficients Cd = 1 and Cm = 2, respectively, are assumed at all segments along the length of the cylinder. In applying the latter method, Eq. 3.14 is used to calculate the low-frequency drift force. The mean drift force coefficient CoCf) is calculated by using Eq. 4.13 for frequencies up to 1 Hz. Figure 5.21 shows the calculated mean drift force coefficient as a function of wave frequency. Characteristic values of the low-frequency force calculated for various tests using Pinkster's method as indicated above are given in Table 5.5. The results for tests 1-4 indicate that the rms value of the low-frequency force increase with an increase in groupiness factor Gp. Tests 1,5-7 show that a decrease in the SIWEH peakfrequencyincreases the rms force. Although this trend is the same as for the case of the large cylinder, the differences are too small to reach any definite conclusions. Results for the remaining tests indicate that the low-frequency force shows similar trends with ^, Hg and fp as in the case of the large circular cylinder. Finally, tests 1 and 15 indicate that the rms force is smaller for the more peaked spectrum y = 3.3 in comparison to the case when y =1.0. The low-frequency wave loads have also been computed using the Morison equation applied directly with the irregular wave record for the cases of a fixed cylinder and moored cylinder. As described in section 4.1.1, the wave kinematics are calculated based on linear wave theory; the cylinder velocity and acceleration are calculated by solving the equation of motion given by Eq. 4.2 using the Newmark iteration method; and the low-frequency force is separated from the calculated force time series by performing a Fast Fourier Transform. In Fig. 5.22 the spectra of the low-frequency drift force are shown based on the results of Pinkster's method and the time domain approach using the Morison equation. It can be clearly seen that the low-frequency spectra calculated by both the methods are similar in shape but the magnitudes are different. Pinkster's method involves only the drift force coefficients in regular waves and consequendy underestimates the low-frequency force. This is in agreement with the experimental results of Pinkster and Huijsmans (1982) for a semi-submersible. Pinkster's method is based on the assumption that the wave record has a very narrow-band spectrum. The inaccuracies in the method are due to the fact that the wave spectrum is not narrow to support such an assumption. However, considering the substantial simplification achieved by adopting Pinkster's method, it will stiU be quite useful for peak drift force prediction purposes. Figure 5.23 shows the influence of the cylinder motion on the low-frequency force spectrum. It can be observed that the cylinder motion leads to a significant reduction in the calculated low-frequency drift force. Chapter 6 Conclusions This thesis has considered the effects of wave groups on the low-frequency drift force and the low-frequency surge response of a moored structure. Initially, characteristic parameters describing wave group activity have been identified, and wave records have been simulated incorporating these parameters. The SIWEH, introduced by Funke and Mansard (1980), is considered to be an acceptable approach to describe wave groupiness, and may be characterized by the groupiness factor Gp, the SIWEH peakfrequencyfsw and the SIWEH broadness factor C,. The magnitude of waves in a wave group increases with increase in groupiness factor; the SIWEH peak frequency corresponds to a representative frequency of wave group occurrence; and the SIWEH broadness factor controls the regularity of the envelope of waves in a group, with a more regular envelope as the broadness factor becomes smaller. For a large floating structure represented by a single degree of freedom system, the estimation of the low-frequency drift force, low-frequency surge response and statistics of this response is described on the basis of Pinkster's approximate method for calculating the low-frequency drift force from regular wave results. Corresponding calculations have been carried out for various simulated wave records. The low-frequency drift force reaches large values when wave grouping is significant. An increase in the groupiness factor Gp increases the magnitude of the force spectrum, whereas the force spectrum shape remains unaltered and is similar to that of SIWEH spectrum. The predominant frequency of the low-frequency force time series corresponds to the SIWEH peak frequency. As the broadness factor Ç decreases, the peak of the force spectrum increases and the spectral bandwidth becomes smaller. Since both the SIWEH and the lowfrequency drift force are proportional to the square of the wave amplitude, there is a predictable relationship between these quantities. The low-frequency surge response increases, as does the corresponding force, with an increase in the groupiness factor. The surge response magnitude peaks when fsw is close to the natural frequency of the system. The rms values of the low-frequency force and response are proportional to the square of the significant wave height. The peakfrequencyof the wave spectrum fp is found to have a significant influence on the low-frequency force. The low-frequency forces are small for smaller peak frequencies of the wave spectrum, as the mean drift force associated with smaller frequencies is very small. The low-frequency response also follows a similar trend. The peak enhancement factor Y does not show any significant effect on the low-frequency force. A statistical analysis of the low-frequency surge motions has been carried out. The probability distribution of the SIWEH and the corresponding low-frequency force is non-Gaussian since they are nonlinear functions of the water surface elevation which itself follows a Gaussian distribution. The low-frequency surge response is reasonably well represented by a Gaussian distribution for a damping ratio of 1.89%. This shows that the low-frequency response of a Unear system with low damping subjected to non-Gaussian excitation, approaches a Gaussian distribution. For the same input spectrum, the low-frequency surge response becomes non-Gaussian as the damping increases from 1.89% to 12.36%. The three-parameter model given by Stansberg (1991) predicts the extreme amplitudes quite well for this non-Gaussian response. The low-frequency force on a fixed slender cicular cylinder calculated using Pinkster's method applied to regular wave results has been compared with the results of the time domain approach using the Morison equation applied directiy to an irregular wave record. The low-frequency spectra calculated by both the methods are similar in shape but the magnitudes are underestimated by the former. The discrepancy in the results obtained using the two methods is due to Pinkster's method being based on the assumption of a narrow-band wave spectrum. However, considering the substantial simplification achieved by adopting Pinkster's method, it should still be quite useful for peak drift force prediction purposes. The low-frequency force on a slender cylinder shows similar trends with Gp, C. Hg and fp as in the case of the large cylinder. The low-frequency force increases with a decrease in the SIWEH peak frequency. The rms value of low-frequency force for the JONSWAP wave spectrum (y =3.3) is smaller than that for the P-M spectrum (y =1.0). The low-frequency drift forces are also calculated for a moored slender circular cylinder and it is seen that the force is significandy reduced due to the oscillations. 6.1 Recommendations for Further Study There are several areas in which the work described in this thesis can be extended. Force and response simulations have been carried out here only for surge motions and extensions to all six degrees of freedom should be performed. Further work is needed to study the effects of multidirectional waves on wave grouping and on the resulting low-frequency drift forces and response. In the present study, the wave drift damping results for the large cylinder have been approximated. However it is important to include a more reliable estimation of this damping and this requires an analysis of forward speed effects in treating the diffraction / radiation problem. Although some research has been carried out on this problem, it is important that this be continued. The work can also be extended to include a system with nonlinear moorings. This leads to considerable difficulties in the simulation, and can giveriseto chaotic motions. References Burcharth, H.F., 1979. "The Effect of Wave Grouping on On-Shore Structures". Coastal Engineering, Vol. 2, pp. 5-99. Chakrabarti, S.K. 1982. "Experiments on Wave Drift Force on a Moored Floating Vessel". Proceedings of the Offshore Technology Conference, Houston, Paper No. OTC 4436, pp. 693-709. Chakrabarti, S.K. and Cotter, D.C. 1984. "Nonlinear Wave Interaction With a Moored Floating Cylinder". 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Funke, E.R. and Mansard, E.P.D. 1979. "On the Synthesis of Realistic Sea States in a Laboratory Flume". NRC Division of Mechanical Engineering,Technical Report LTR-HY66, National Research Council of Canada. Goda, Y., 1976. "On Wave Groups". Proceedings of the International Conference on Behavior of Off-Shore Structures, BOSS' 76, Trondheim, Vol. 1, pp. 115-128. Hsu, F.H. and Blenkarn, K.A. 1970. "Analysis of Peak Mooring Forces Caused by Slow Vessel Drift Oscillations in Random Seas". Proceedings of the Offshore Technology Conference, Houston, Paper No. OTC 1159, pp. 135- 146. Hineno, M. 1988. "A Calculation of the Statistical Distribution of the Maxima on Non-Unear Responses in Irregular Waves". Proceedings of the Society of Naval Architects of Japan , May. Isaacson, M. 1984. "Recent Advances in the Computation of Nonlinear Wave Effects on Offshore Structures". Specialty Conference on Computer Methods in Offshore Engineering, CSCE Annual Conference, Halifax, N.S, pp. 439-453. Isaacson, M. 1985. "Wave Effects on Large Offshore Structures of Arbitrary Shape". Coastal I Ocean Engineering Report, Department of Civil Engineering, University of British Columbia, Vancouver, Canada. Isaacson, M . and Sinha, S. 1986. "Directional Wave Effects on Large Offshore Structures". Journal of the Waterway, Port, Coastal, and Ocean Engineering, ASCE, Vol. 112, No. 4, pp. 482-497. Isaacson, M. and Cheung, K.F. 1992. "Second Order Wave Diffraction Around Two- Dimensional Bodies by Time-Domain Method". Applied Ocean Research, Vol. 13, No. 4, pp. 175-186. Isaacson, M. and Cheung, K.F. 1992. "Simulation of Wave-Current Interactions With a Large Offshore Structure". Proceedings of the Conference Civil Engineering in the Oceans V, ASCE, College Station, Texas, in press. Johnson, R.R., Ploeg, J. and Mansard, E.P.D. 1978. "Effects of Wave Grouping on Breakwater Stability", Proceedings of the 16th International Conference on Coastal Engineering, Hamburg, vol. 3, pp. 2228-2243. Karppinen, T., 1979. "An Approach to Computing the Second Order Steady Forces on SemiSubmersible Structures". Laboratory, Report No. 16. Helsinki University of Technology, Ship Hydrodynamics Loken, A . E . and Olsen, O.A. 1979. "The Influence of Slowly Varying Wave Forces on Mooring Systems". Proceedings of the Offshore Technology Conference, Houston, Paper No. OTC 3626, pp. 2325-2331. Matsui, T. 1986. "Analysis of Slowly Varying Wave Drift Forces on Compliant Structures". Proceedings of the Fifth OMAE Conference, Tokyo, Japan, Vol. 1, pp. 289-296. Marthinsen, T. 1983. "Calculation of Slowly Varying Drift forces" Applied Ocean Research, Vol. 5, No. 3, pp. 141-144. Maruo, H. 1960. "The Drift of a Body Floating on Waves". Journal of Ship Research, Vol. 4, No. 3, pp. 1-10. Naess, A. 1986. "On the Statistical Analysis of Slow-Drift Forces and Motions of Floating Offshore Structures". Proceedings of the Fifth OMAE Conference, Tokyo, Japan, Vol. 1, pp. 317-329. Newman, J.N. 1967. "The Drift Force and Moment on Ships in Waves". Journal of Ship Research, Vol. 11, No. 1, pp. 51-60. Newman, J.N. 1974. "Second-Order, Slowly Varying Forces on Vessels in Irregular Waves". Proceedings of the International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London, England, pp. 182-186. Newmark, N.M. 1959. "A Method of Computation for Structural Dynamics". ASCE Journal of Engineering Mechanics Division, No. 85, pp. 67-94. Nwogu, O. 1989. "Analysis of Fixed and Floating Structures in Random Multidirectional Waves". Ph. D. Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada. Nwogu, O. and Isaacson, M . 1991. "Drift Motions of a Floating Barge in Random Multidirectional Waves". Journal of Offshore Mechanics and Arctic Engineering, Vol. 113, pp. 37-42. Pijfers, J.G.L. and Brink, A.W. 1977. "Calculated Drift Forces of Two Semisubmersible Platform Types in Regular and Irregular Waves". Proceedings of the Offshore Technology Conference, Houston, Paper No. OTC 2977, pp. 155-164. Pinkster, J.A. 1975. "Low Frequency Phenomena Associated With Vessels Moored at Sea", Society of Petroleum Engineers Journal, AIME, Paper No. SPE 4837, pp. 487-494. Pinkster, J.A. 1976. "Low Frequency Second Order Forces on Vessels Moored at Sea". Proceedings of the 11th symposium on Naval Hydrodynamics, University College, London, pp. 603-615. Pinkster, J . A . 1979. "Mean and Low Frequency Wave Drifting Forces on Floating Structures". Ocean Engineering, Vol. 6, pp. 593-615. Pinkster, J.A. and Huijsmans, R.H.M. 1982. "The Low Frequency Motions of a Semi- Submersible in Waves". Proceedings of the Third International Conference on Behavior of Off-Shore Structures, MIT, Cambridge, Massachusetts, pp. 447-466. Pinkster, J.A. and Wichers, J.E.W. 1987. "The Statistical Properties of Low Frequency Motions of Nonhnearly Moored Tankers". Proceedings of the Offshore Technology Conference, Houston, Paper No. OTC 5457, pp. 317-331. Pinkster, J.A. and Van Oortmerssen, G. 1977. "Computation of the First and Second Order Wave Forces on Bodies Oscillating in Regular Waves". Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, Berkeley, pp. 136-156. Remery, G.F.M. and Hermans, A . J . 1971. "The Slow Drift Oscillations of a Moored Object in Random Seas". Proceedings of the Offshore Technology Conference, Houston, Paper No. OTC 1500, pp. 829-836. Sawaragi, T., Aoki, S. and Masayuki, T. 1988. "Effects of Wave Grouping on the Low Frequency Motion of a Moored Rectangular Vessel and the Characteristics of Nonlinear Hydrodynamic Forces". Coastal Engineering in Japan, Vol. 31, No. 1, pp. 167-182. Sarpkaya, T. and Isaacson, M. 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold, N.Y. Spangenberg, S. and Jacobsen, B.K. 1980. 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"The Influence of Waves on the Low Frequency Hydrodynamic Coefficients of Moored Vessels". Proceedings of the Ojfshore Technology Conference, Houston, Paper No. OTC 3625, pp. 2313-2324. Zhao, R., Faltinsen, O.M., Krokstad, J.R. and Aanesland, V. 1988. "Wave- Current Interaction Effects on Large-Volume Structures". Proceedings of the International Conference on Behavior of Off-Shore Structures, BOSS'88, Trondheim, Vol. 2, pp. 623638. Zhao, R., Faltinsen, O.M. 1989. "Interaction Between Current, Waves and Marine Structures". Proceedings of the Fifth International Conference on Numerical Ship Hydrodynamics, Hiroshima, Japan, pp. 513-527. Il Test Peak Significant Groupiness SIWEH peak Broadness Peak wave height, fi-equency, enhancement factor, Gp frequency, factor, C Hs(m) f.w(Hz) fn(Hz) factor, Y Ibase case 2.0 0.2 1.0 0.6 0.02 0.2 2 2.0 0.2 1.0 0.7 0.02 0.2 3 2.0 0.2 1.0 0.8 0.02 0.2 4 2.0 0.2 1.0 1.0 0.02 0.2 5 2.0 0.2 1.0 0.6 0.04 0.2 6 2.0 0.2 1.0 0.6 0.014 0.2 7 2.0 0.2 1.0 0.6 0.01 0.2 8 2.0 0.2 1.0 0.6 0.02 0.4 9 2.0 0.2 1.0 0.6 0.02 0.6 10 2.0 0.2 1.0 0.6 0.02 0.8 n 2.0 0.05 1.0 0.6 0.02 0.2 12 2.0 0.1 1.0 0.6 0.02 0.2 13 4.0 0.2 1.0 0.6 0.02 0.2 14 6.0 0.2 1.0 0.6 0.02 0.2 15 2.0 0.2 3.3 0.6 0.02 0.2 Table 5.1 Characteristics of simulated sea states. Test Hs (m) fp (Hz) Y 1 2.0 0.2 2 2.0 3 low-frequency force, F(t) (kN) low-frequency surge response, x(t) (m) nns max rms max 1.0 15.9 61.6 1.15 2.92 0.2 1.0 17.9 59.3 1.29 3.44 2.0 0.2 1.0 20.4 113.3 1.77 3.97 4 2.0 0.2 1.0 25.3 117.2 1.91 4.66 5 2.0 0.2 1.0 15.8 46.8 0.52 1.10 6 2.0 0.2 1.0 15.9 49.9 1.44 3.46 7 2.0 0.2 1.0 15.5 58.6 0.98 2.31 8 2.0 0.2 1.0 15.9 52.2 0.96 2.49 9 2.0 0.2 1.0 16.0 58.1 1.15 2.67 10 2.0 0.2 1.0 15.8 50.2 0.78 1.85 11 2.0 0.05 1.0 0.7 4.2 0.03 0.09 12 2.0 0.1 1.0 8.0 46.2 0.33 0.95 13 4.0 0.2 1.0 63.6 246.3 4.60 11.59 14 6.0 0.2 1.0 143.0 553.5 10.34 26.65 15 2.0 0.2 3.3 16.1 56.9 1.09 2.78 Table 5.2. Statistics of the low-frequency drift force and surge response for a moored large circular cyUnder. Test No. No. of oscillations Spectral widtli ratio, M Calculated Xmax (m) Predicted Xmax ('^) 3-para. Gauss. Exp. 1 base case 17 4.91 2.92 3.00 2.74 3.25 2 17 4.83 3.43 3.37 3.07 3.67 3 17 5.14 3.96 4.62 4.21 5.01 4 17 4.91 4.70 4.98 4.54 5.41 5 17 9.52 1.10 1.43 1.24 1.48 6 17 3.64 3.46 3.8 3.43 4.09 7 16 2.41 2.31 2.61 2.32 2.73 8 17 6.24 2.49 2.52 2.29 2.72 9 17 7.73 2.67 3.05 2.73 3.25 10 17 7.86 1.85 2.09 1.87 2.22 11 16 1.83 0.09 0.09 0.08 0.09 12 17 2.07 0.95 0.90 0.78 0.94 13 17 4.91 11.60 12.01 10.98 13.01 14 17 4.91 26.70 26.95 24.61 29.31 .5 17 4.73 2.81 2.84 2.59 3.09 Table 5.3 Comparison of calculated maximum response amplitude of a moored large circular cylinder with theoretical predictions (damping ratio = 1.89 %). Test No. of oscillations Spectral width ratio, M Calculate d Xmax (m) Predicted Xmax (m) 3-para. Gauss. Exp. 1 (base case) 21 2.71 1.11 1.28 1.10 1.36 2 21 2.97 1.28 1.53 1.33 1.64 3 21 2.38 1.67 1.65 1.40 1.72 4 21 2.94 2.32 2.43 2.11 2.59 5 24 3.55 0.60 0.65 0.56 0.71 6 19 1.90 1.30 1.43 1.21 1.47 7 16 1.18 1.36 1.23 1.04 1.24 8 21 3.20 1.13 1.15 1.00 1.23 9 21 3.49 1.08 1.07 0.95 1.17 10 21 3.46 0.91 0.97 0.86 1.06 Table 5.4 Comparison of calculated maximum response amplitude of a moored large circular cylinder with theoretical predictions (damping ratio = 12.36 %). Test low-free uency force, F(t) (kN) rms +ve max - ve max Remarks 1 428 1700.5 872.7 Base case 2 491 2224.5 1037.3 Influence of Gp 3 518 2485.8 997.3 4 593 2808.0 1023.2 5 415 1308.2 909.1 6 558 4098.6 845.3 7 569 3097.3 811.6 8 440 1750.5 883.8 Influence of C 9 472 2156.1 881.9 >? 10 458 2603.2 898.2 55 11 75 332.9 105.5 Influence of fp 12 149 745.5 255.7 55 13 1714 6802.0 3490.9 Influence of Hg 14 3858 15376.0 7853.0 95 15 332 1370.1 736.7 Table 5.5. Influence of fsw Influence of y Statistics of low-frequency drift force for the fixed slender circular cylinder. Figure 1.1 Sketch of time histories of surface elevation and second-order drift force in a regular wave group. Figure 1.2 Definition sketch of the motions of a floating body. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 f (Hz) Figure 2.1 SIWEH example; (a) a wave record, (b) the corresponding SIWEH and (c) the spectral density of the SIWEH. (a) 300 (b) 300 (a) 0.00 Figure 5.1 0.20 Surface elevation time series for wave records with different values of Gp for the case fp = 0.2, Hs = 2.0 m, y = 1.0, fsw = 0.02 Hz, C = 0.2 (a) Gp = 0.6, (b) Gp = 1.0 and (c) corresponding SIWEH spectra. Figure 5.2 SIWEH and SIWEH spectra for wave records with different values of fsw for the case fp = 0.2, Hs = 2.0 m, y = 1.0, Gp = 0.6, C = 0.2 (a) SIWEH time series and (b) SIWEH spectra. Figure 5.3 SIWEH spectra and surface elevation time series for wave records with different values of C, for the case fp = 0.2, Hs = 2.0 m, y = 1-0, G F = 0.6, fsw = 0.02 Hz. > 5 12 m VVWW- vwvw- 10 m 20 m Figure 5.4 30 m Definition sketch for the moored large circular cyhnder. f (Hz) Figure 5.5 Variation of the mean drift force coefficient with frequency for a moored large circular cylinder. f (Hz) Figure 5.6 Calculated added mass coefficient as a function of frequency. f (Hz) Figure 5.7 Calculated radiation damping coefficient as a function of frequency. Gp = 0.6 Gp = 1.0 1 0 1 20 1 40 1 1 60 1 1 1 80 100 120 140 80 100 120 140 t (sec) 120 0 20 40 60 t (sec) Figure 5.8 Time histories of (a) surface elevation, (b) low-frequency drift force and (c) low-frequency surge response, for waves with different values of Gp (fp = 0.2, Hs = 2.0 m, Y = 1.0, fsw = 0.02 Hz, C = 0.2, fn = 0.158). Figure 5.9 Time histories of (a) surface elevation, (b) low-frequency drift force and (c) low-frequency surge response, for waves with different values of fsw (fp = 0.2, Hs = 2.0 m, Y= 1.0, Gp = 0.6, C = 0.2, fn = 0.158). Figure 5.10 Time histories of (a) surface elevation, (b) low-frequency drift force and (c) low-frequency surge response, for waves with different values of ^ (fp = 0.2, Hs = 2.0 m, y = 1.0,GF = 0.6, fsw = 0.02 Hz, fn = 0.158). 0.00 1 r 0.06 0.08 0.14 f(Hz) Figure 5.11 Effect of different values of Gpon (a) spectra of the low-frequency drift force and (b) spectra of the low-frequency surge response (fp = 0.2, Hs = 2.0 m, Y = 1.0, fsw = 0.02 Hz, Ç, = 0.2, fn = 0.158). — fsw = 0-04 ---fs^ = 0.02 fsw - 0-014 ---4^=0.01 H / ^ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 f(Hz) Figure 5.12 Effect of different values of fsw on (a) spectra of the low-frequency drift force and (b) spectra of the low-frequency surge response (fp = 0.2, Hs = 2.0 m, Y= 1.0, Gp = 0.6, ^ = 0.2, fn = 0.158) 200 N 5 150- 100X on 50- 0.00 T 0.06 0.08 0.14 f(Hz) Figure 5.13 Effect of different values of C on (a) spectra of the low-frequency drift force and (b) spectra of the low-frequency surge response (fp = 0.2, Hs = 2.0 m, Y= 1.0, fsw = 0.02 Hz, fsw = 0.02 Hz, fn = 0.158) Figure 5.14 The influence of groupiness factor Gp on (a) the low-frequency drift force and (b) the low-frequency surge response of a moored large circular cyhnder. Figure 5.15 The influence of SIWEH peak frequency fsw on (a) the low-frequency drift force and (b) the low-frequency surge response of a moored large circular cylinder. Figure 5.16 The influence of broadness factor Ç on (a) the low-frequency drift force and (b) the low-frequency surge response of a moored large circular cylinder. 20. - - - k = 30kN k = 50 kN - - - k = 60 kN k = 90 kN 15- 10- 5- 0.0 1 0.5 1 1.0 1 1.5 1 2.0 1 2.5 1 3.0 3.5 r 4.0 fsw/^n Figure 5.17 The effect of mooring stiffness on the low frequency surge response of a moored large circular cylinder. E(mO Figure 5.18 Probability distributions of the SIWEH compared to Gaussian distributions for (a) test 5 and (b) test 14. (a) (b) Figure 5.19 Probability distributions of tiie low-frequency surge response compared to Gaussian distribution for test 5 with (a) damping ratio 1.89% and (b) damping ratio 12.36%. T 12 m 10 m 30 m 2m Figure 5.20 Definition sketch for the moored slender circular cylinder. f (Hz) Figure 5.21 Variation of the mean drift force coefficient with frequency for a moored slender circular cylinder. Pinkster's approach Time domain approach 20- Pinkster's approach Time domain approach 5 - / N 0.00 0.02 0.04 0.06 0.08 0.10 f(Hz) 20- Hnkster's approach Time domain approach / f ta ^ / y 0.00 \ 0.02 '— 0.04 r 0.06 0.08 0.10 f(Hz) Figure 5.22 Comparison of the low-frequency drift force spectra on fixed cylinder calculated by Pinkster's method with Time domain approach using Morison equation for (a) test 1, (b) test 2 and (c) test 3. Figure 5.23 The influence of cylinder motion on the low-frequency drift force (a) test 1 (b) test 3.
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Numerical simulation of wave grouping effects on moored structures Pemmireddy, Venkata Rami Reddy 1992
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Title | Numerical simulation of wave grouping effects on moored structures |
Creator |
Pemmireddy, Venkata Rami Reddy |
Date Issued | 1992 |
Description | The present thesis investigates wave grouping effects on the low-frequency drift force and the low-frequency surge response of a moored structure. The smoothed instantaneous wave energy history, SIWEH, which may be characterized by a number of parameters, is considered as an acceptable approach to describe wave groupiness. A numerical simulation of wave records is carried out for a specified wave spectral density and SIWEH incorporating various wave grouping parameters. The low-frequency drift force and the surge response of a large moored circular cylinder, modeled as a single degree of freedom system, is estimated on the basis of Pinkster's approximate method, which involves the use of computed drift force coefficients. Results show that the wave grouping parameters have a significant effect on the low-frequency drift force and surge response. It is also observed that the low-frequency drift force can be related to the SIWEH in terms of a transfer function. A statistical analysis of simulated low-frequency surge motions is performed, and the calculated extreme response is compared with the predictions of three theoretical models, namely the Gaussian, Exponential, and three-parameter methods. If the surge response has a non- Gaussian distribution, Stansberg's three-parameter model predicts extreme amplitudes reasonably well. The low-frequency drift force on a slender fixed circular cylinder is calculated using both Pinkster's approximate method and in the time domain using the Morison equation. Pinkster's approximate method underestimates the force, but it is computationally more efficient. The time domain approach is also used to estimate the low-frequency drift force on a moored slender circular cylinder and the results are compared to those for a fixed cylinder. It is observed that cylinder motion tends to reduce the drift force significantly. |
Extent | 3220617 bytes |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-12-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050487 |
URI | http://hdl.handle.net/2429/3044 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1992-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
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