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In-line forces on a slender structure subjected to combined waves and currents Hughes, Brian R. 1988

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I N - L I N E F O R C E S O N A S L E N D E R S T R U C T U R E S U B J E C T E D T O C O M B I N E D W A V E S A N D C U R R E N T S Brian R. Hughes B.A.Sc. University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF A P P L I E D 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 April 1988 © Brian R. Hughes, 1988 In presenting this thesis in partial fulfilment 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 my 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 1956 Main Mall Vancouver, Canada Date: Abstract The present investigation considers the hydrodynamic forces acting on a slender struc-ture subjected to a combined wave and current flow regime. The experimental aspect of the study measured the in-line peak-to-peak forces on a vertical cylinder mounted in a wave-current flume. Although there were some inconsistencies in the data, the general trend indicated a substantial increase in the force with a positive underlying current and a less pronounced increase for a negative current. A numerical analysis of the problem evaluated Morison's equation using the current-invariant force transfer coefficients and flow kinematics obtained through Stokes Fifth Order Wave Theory. The results of this analysis revealed a trend qualitatively similar to that found experimentally. The important distinction between the results obtained through the experimental investigation and those obtained numerically was the consis-tent over-prediction observed in the numerical analysis. ii Table of Contents Abstract ii Acknowledgement viii 1 I N T R O D U C T I O N 1 1.1 General 1 1.2 Literature Review 3 2 T H E O R E T I C A L B A C K G R O U N D 6 2.1 Unidirectional Flows Past a Rigid Cylinder 6 2.1.1 Steady Flow of a Real Fluid Past a Rigid Cylinder 6 2.1.2 Accelerating Flow of an Ideal Fluid Past a Rigid Cylinder . . . . 7 2.1.3 Harmonically Oscillating Flow of a Real Fluid Past a Rigid Cylin-der 8 2.2 Wave Theory Development 10 2.3 Wave and Current Interactions 13 3 P R E S E N T I N V E S T I G A T I O N 16 3.1 Objectives 16 3.2 Description of Wave Conditions 17 3.3 Current Flow Conditions 19 3.4 In-Line Force Distribution 20 4 E X P E R I M E N T A L M E T H O D O L O G Y 23 iii 4.1 Experimental Apparatus 23 4.2 Flow Kinematics 24 4.3 Measured Force Distributions 25 4.4 Data Acquisition 29 5 N U M E R I C A L M E T H O D O L O G Y 31 5.1 Modified Morison Equation 31 5.2 Force Transfer Coefficients 32 5.3 Flow Kinematics 37 6 R E S U L T S A N D DISCUSSION 41 6.1 Experimental Results 41 6.2 Comparison of Experimental and Theoretical Force Distributions . . . . 44 6.3 Effect of Current Magnitude on In-line Forces 45 7 C O N C L U S I O N S A N D R E C O M M E N D E D F U R T H E R S T U D Y 47 7.1 Conclusions 47 7.2 Recommendations for Further Study 48 Bibliography 50 iv List of Tables 5.1 Flow Kinematics Measured in the Laboratory 39 6.2 Dimensionless Wave Parameters Calculated from Experimental Data . . 42 v List of Figures 1(a). Wave Flow and Test Cylinder Definition Sketch 54 1(b). Combined Wave and Current Definition Sketch 55 2. Elevation View of Wave-Current Flume 56 3. Test Cylinder Mounted in Flume 57 4. Test Cylinder Natural Frequency as a Function of Submerged Length . . 58 5. Detail of Test Cylinder Construction 59 6. Detail of Instrumented Rod 60 7. Schematic Diagram of Data Acquisition Equipment 61 8. Typical Measured and Idealized Velocity Profile 62 9. Ranges of Suitability of Various Wave Theories 63 10. In-Line Force Distribution for Vr ~ 0.0 (/? ~ 1000) 64 11. In-Line Force Distribution for Vr ~ +0.4 (0 ~ 1000) 65 12. In-Line Force Distribution for Vr ~ +0.6 (0 ~ 1000) 66 13. In-Line Force Distribution for Vr ~ +1.0 (0 ~ 1000) 67 14. In-Line Force Distribution for Vr ~ -0.4 (0 ~ 1000) 68 15. In-Line Force Distribution for Vr ~ -0.6 (0 ~ 1000) 69 16. In-Line Force Distribution for Vr ~ -1.0 (0 ~ 1000) 70 17. In-Line Force Distribution for Vr ~ 0.0 (f3 ~ 410) 71 18. In-Line Force Distribution for Vr ~ +0.3 (/? ~ 410) 72 19. In-Line Force Distribution for Vr ~ +0.6 (/? ~ 410) 73 20. In-Line Force Distribution for Vr ~ +1.2 (/? ~ 410) 74 21. In-Line Force Distribution for Vr ~ -0.3 (0 ~ 410) 75 vi 22. In-Line Force Distribution for Vr ~ -0.5 (/? ~ 410) 76 23. In-Line Force Distribution for Vr ~ -0.7 (/? ~ 410) 77 24. Variation of Peak Pile Force with Current Magnitude (/? ~ 1000) . . . . 78 25. Variation of Peak Pile Force with Current Magnitude (/? ~ 410) . . . . 79 vii Acknowledgement The work described in this thesis was performed at the University of British Columbia employing the facilities available in the hydraulics laboratory and graphics laboratory within the Department of Civil Engineering. Financial aid was provided through a research assistantship from the Natural Science and Engineering Research Council of Canada. The author wishes to thank Dr. M. de St. Q. Isaacson for his support and advice offered in the supervision of the present research. Also, the author wishes to thank his family and friends, and in particular Josephine McGee, for their unlimited encouragement and patience. viii Chapter 1 I N T R O D U C T I O N 1.1 General As the demand for additional energy resources continues its rapid growth, offshore structures are being placed in deeper waters and are subjected to more severe envi-ronments. Many fixed offshore platforms are presently located in waters well over 150 metres deep and future platforms have been designed for placement in waters up to 350 metres deep. Primary loads for which an offshore structure is designed include wave, current, ice and earthquake. As one might expect, the hydrodynamic loads as-sociated with these deep water platforms have reached levels substantially higher than previously imagined. In an attempt to accurately predict the hydrodynamic requirements of such struc-tures, the design procedures employed in their development have correspondingly be-come more sophisticated and complex. A good example of this is the recent interest in the alteration of the hydrodynamic forces resulting from a co-existing wave and current flow regime and the ability to accurately predict these forces. The following thesis will focus on the details of various combined wave and current flow regimes and the hydrodynamic forces these flows may induce on offshore struc-tures. In particular, the accuracy of predicted force distributions acting on fixed slender stuctures subjected to a co-existing flow regime shall be investigated. Historically, the presence of currents in the ocean environment has been realized, 1 Chapter 1. INTRODUCTION 2 but the exact hydrodynamic effects resulting from the wave-current interaction remains relatively poorly understood. This lack of understanding can be attributed to the complexity of the problem and the closely related lack of experimental information. The interaction between wave-only environments and offshore structures is fairly well researched and understood. This interaction process is generally classified into two distinct flow regimes. If the structure is considered large in relation to the oribital fluid motion, analysis involves wave diffraction calculations and is often referred to as the large structure flow regime. This subject is not dealt with in the present investigation. Rather, the analysis associated with the interaction of waves and structures which are considered small in relation to the fluid motions, the slender structure flow regime, shall be more closely examined. Briefly, the presence of a fixed slender structure in a harmonically oscillating fluid will force the fluid particles to deviate from their regular orbital paths and give rise to flow separation. This in turn will generate in-line and transverse forces acting on the structure. The in-line forces are of interest in the present study and are commonly estimated using a two-term, semi-empirical formula known as the Morison Equation. With the superposition of a current, which may vary temporally and spatially, the water particle motions, vortex shedding, and resulting forces tend to become increas-ingly complex. At present, the most common method of predicting the hydrodynamic forces generated by this combination of waves and currents is through the use of a modified Morison equation. In recent years there has been a considerable amount of controversy over the most representative form of modification applied to the already semi-empirical equation, or if, in fact, any form of the Morison equation is truly appli-cable to a combined wave-current analysis. This thesis concentrates on the hydrodynamic force variation along the length of a fixed, vertical, surface-piercing, slender cylinder subjected to various uniform co-linear Chapter 1. INTRODUCTION 3 currents superposed on different monochromatic wave trains. The present investigation examines the wave-current interaction process through a series of laboratory experi-ments and the corresponding conventional numerical analysis. The results from the two procedures will then be examined with emphasis on the modification of wave-induced forces due to the presence of currents and the ability of a modified Morison equation to accurately predict these values. 1.2 Literature Review Excellent reviews on the many aspects of combined wave-current interactions with structures are offered by Thomas (1979), Sarpkaya and Isaacson (1981), Chakrabarti (1985), and Isaacson and Baldwin (1987). Although the present thesis concentrates on uniform currents imposed on regular waves and the resulting interaction with fixed slender structures, information on related topics such as non-uniform currents, obliquely approaching currents, irregular or random waves, inclined members, flexible structures, and large structures can be found in the papers listed above. Regardless of the specialized subject area of these and related papers, the majority of authors invariably emphasize the need for continuing experimental work in order to test existing and proposed wave-current theories. The fact that this need exists was a primary motive in the decision to include both experimental and numerical analysis in the present study. Another important topic raised throughout the literature was the tendancy for the Morison equation to overpredict the combined wave-current forces when using con-ventional force transfer coefficients. In a paper by Sarpkaya and Storm (1985), the authors point out that the use of conventional current-invariant force transfer coef-ficients implies that these coefficents are independent of the biased vortex shedding Chapter 1. INTRODUCTION 4 inherent in co-existing flows. Through a series of laboratory experiments Sarpkaya and Storm along with several other researchers have shown that in fact the force transfer coefficients for a combined flow regime can vary substantially from their no-current values. With a better understanding of the combined flow regime it should be possible to estimate the hydrodynamic forces with improved accuracy. Sarpkaya and Isaacson (1981) attribute the complex nature of wave-current inter-actions with slender structures to several pertinent factors. Firstly, there presently exists no exact analytical solution for even the most idealized wave-current interactive regime. Secondly, there remains considerable uncertainty with respect to the correct form and actual validity of the Morison equation. Thirdly, with waves and currents tending to be highly irregular and omnidirectional in the ocean, realistic water particle motion and resultant vortex shedding can be largely assymmetrical. Finally, it has been practically impossible to carry out a systematic field investigation for a practical range of Reynolds number, Keulegan-Carpenter number, relative current velocity, and current gradient. In order to try and better understand the non-linear effects of combined waves and currents acting on slender structures, numerous laboratory experiments have been performed. Sarpkaya and Isaacson (1981) offer an extensive review of these. The most important thrust of these investigations has been the evaluation of modified drag and inertia coefficients to be employed in the Morison equation for wave-current flow fields. Several of the more relevant studies will be briefly described here. Moe and Verly (1980) measured the damping rate of the oscillation amplitudes for a pendulum-mounted cylinder subjected to uniform and constant fluid velocities. Evaluation of the drag and inertia coefficients from this data revealed an overall decrease for the drag coefficient and increase for the inertia coefficient relative to the no-current case. Chapter 1. INTRODUCTION 5 Iwagaki, Asano and Nagai (1983) performed a set of experiments with two small vertical cylinders in a wave flume with recirculating flow. Force transfer coefficients were evaluated by two approximate methods, and then expressed in terms of various modified Keulegan-Carpenter numbers. Through the introduction of a new Keulegan-Carpenter number based on relative fluid displacement, they were able to correlate the results quite well. Similar experiments were performed the following year, Iwagaki and Ansano (1984), in which they used flow visualization techniques and their newly defined Keulegan-Carpenter number to investigate further in-line and transverse forces. Most notably, experiments performed by Sarpkaya and Storm (1985) reveal sig-nificant differences in the force transfer coefficients due to the presence of currents. In general they found an overall decrease in the drag coefficient, particularly in the drag-inertia regime; while the inertia coefficient tended to decrease in the inertia domi-nated regime, increase considerably in the drag-inertia regime, and decrease slightly in the drag dominated regime. These variations agree, at least qualitatively, with those found by Moe and Verly (1980) and Iwagaki et. al. (1983). Sarpkaya and Storm (1985) conclude that with the use of these modified force coefficients the Morison equation can predict combined wave and current forces as adequately as the original Morison equation predicts the wave-only forces. It is important to note that due to wake biasing, the changes in the force transfer coefficients occur even for relatively small current magnitudes and thus should not be ignored. With this brief introduction to the non-linear effects currents impose on waves and the associated hydrodynamic forces, the following chapter will discuss the pertinent theoretical aspects of this phenomenon. C h a p t e r 2 T H E O R E T I C A L B A C K G R O U N D This section is not intended to act as a rigorous development of the theoretical back-ground of wave-current interactions with structures; rather, it will briefly explain the basic theories and introduce the relevant equations. Firstly, a short description of the fundamental flows past a rigid cylinder will introduce the concepts of drag and inertia forces and their contributions to the overall force. Secondly, linear wave theory will be summarized along with a short extension to non-linear wave theory. Finally, combined wave and current flow regimes and the conventional methods of analysis will be dis-cussed. If more detail is desired, it should be noted that the theory summarized here has been extensively researched by numerous individuals and excellent references may be found throughout the literature. 2.1 U n i d i r e c t i o n a l F l o w s P a s t a R i g i d C y l i n d e r 2.1.1 S t e a d y F l o w o f a R e a l F l u i d P a s t a R i g i d C y l i n d e r A uniform flow of a viscous fluid past a rigid slender structure will tend to separate in the form of eddies which are shed from alternating sides of the body and carried downstream in what is referred to as the vortex street. The momentum of the flow is consumed by both wall shear and by the alternating pressure gradient, giving rise to lift and drag forces. The lift force acts in a transverse direction and is not of interest in the present discussion. The drag force is comprised of a fluctuating component and 6 Chapter 2. THEORETICAL BA CKGRO UND 7 a steady component both acting parallel to the flow direction. The fluctuating drag force has been found to be relatively small, Fung (1960). The steady component of the drag force is commonly expressed as: FD = ^pDCdu2 (2.1) where p is the fluid density, D is the cylinder diameter, Cd is the drag coefficient, and u is the flow velocity. The actual value of Cd depends upon the body shape and the Reynolds number. 2.1.2 Accelerating Flow of an Ideal Fluid Past a Rigid Cylinder The flow of an ideal fluid assumes no viscous effects and thus experiences no flow separation and hence the cylinder would not be subjected to any lift or drag forces. An accelerating flow, however, will induce an in-line hydrodynamic force referred to as the inertia force. The inertia force is commonly expressed as: Fj = CmPV^ (2.2) where Cm is the inertia coefficient, V is the immersed volume of the body and thus pV is the mass of the displaced fluid. The inertia force is often regarded as being the sum of two distinct components. The first component, known as the Froude-Krylov force, Fk, is due directly to the force field which causes the fluid particles to accelerate with time. du Fk - „V- (2.3) The second component, Fd, is associated with the added mass of the body and develops as a result of the deviation of the fluid path around the structure. Combining Chapter 2. THEORETICAL BACKGROUND 8 equations (2.2) and (2.3), Fd is given by: Fd = (Cm - l)pV du ~di (2.4) The term (Cm — l)pV is often referred to in the literature as the "added mass" of the body. 2.1.3 Harmonically Oscillating Flow of a Real Fluid Past a Rigid Cylinder When considering a viscous fluid accelerating past a rigid cylinder it is necessary to account for both flow separation and the ever changing flow field around the cylinder. In effect, harmonic oscillations of a real fluid past a rigid cylinder can be viewed as a combination of the above two reference flows. One of the most important parameters associated with this type of flow is one which describes the amplitude of the fluid particle motions relative to the structure dimension. For example, with relatively small fluid particle motions little or no flow separation will have time to develop before the flow reverses direction, thus drag effects will be minimal and the flow may be considered as inertia dominated. On the other hand, if the fluid particle motions are large relative to the cylinder diameter, flow separation will be of prime importance giving rise to a drag dominated flow regime. Obviously there exists a mid-range of particle motion amplitudes such that drag and inertia effects are both substantial. As one might expect such a range is referred to as a drag-inertia flow regime. Keulegan and Carpenter (1958) examined the drag and inertia dominated flow regimes described above and introduced a dimensionless parameter, commonly referred to as the Keulegan-Carpenter number, which can be expressed as: K = D (2.5) Chapter 2. THEORETICAL BACKGROUND 9 where Um is the maximum fluid particle velocity in the harmonic flow, T is the period of oscillation and D is the cylinder diameter. The Keulegan-Carpenter number can now be used to define the above three flow regimes. That is, there exists an inertia dominated flow for K < 8, a drag dominated flow for K > 25, and, of course, drag-inertia flow for 8 < K < 25. A widely recognized approach to the problem of evaluating the forces acting on a slender structure subjected to a harmonically oscillating flow was first proposed by Morison, O'Brien, Johnson, and Shaaf (1950). The semi-empirical formula developed by Morison et al. simply states that the total in-line force acting on the body is the sum of the drag and inertia force components introduced in sections 2.1.1 and 2.1.2 respectively. That is: FTOTAL = FDRAG + FINERTIA This formula has come to be known as the Morison equation, and for a rigid vertical cylinder in two dimensional oscillatory flow is most commonly expressed as: ^/ 1 i i nD2 „ du F = -pDCdu\u\ + P—Cm— (2.6) where F' is the total in-line force per unit length. It is important to note that this formula is empirically based and thus the drag and inertia coefficients, Cd and Cm, do not in general take on the same values as for the two reference flows. In fact, these values can vary significantly for even the simplest oscillating flow conditions depending upon the Keulegan-Carpenter number (^^-) and the Reynolds number ( ^ ^ ) . Extensive laboratory and field investigations have been performed for a wide range of Keulegan-Carpenter and Reynolds numbers to provide a substantial data base for the selection of the appropriate force coefficients. However, there remain various flow conditions, such as the co-existing flows discussed here, where the selection of these coefficients is far from certain. Chapter 2. THEORETICAL BACKGROUND 10 The Morison equation has been verified experimentally by numerous researchers, and with the appropriate force transfer coefficients, and Cm, agrees well with mea-sured force data. For more complicated flow conditions, such as those involving inclined members, interference effects, flexible members, and wave-current interactions, the form of the Morison equation and selection of the appropriate coefficients correspondingly becomes more complex. 2 . 2 Wave Theory Development A brief summary of the results of linear and non-linear wave theories will be con-sidered at this time. The classical development of the various wave theories is well documented, and the reader is referred to Stokes (1847), Lamb (1945), and Skjelbreia and Hendrickson (1960) for more detail. The simplest approach in the analytical description of a two dimensional wave train is to seek a linear solution to the problem by assuming the wave height, H , is much smaller than both the wavelength, L , and the water depth, d (H « L and H << d). This approach is found to represent a first approximation to the exact solution and is commonly referred to in the literature as any of the following: Linear Wave Theory, Small Amplitude Wave Theory, Sinusoidal Wave Theory, or Airy Wave Theory. There are many literary sources which offer a detailed account of the steps involved in the Linear Wave Theory development and supply the salient results in their entirity (see for example Sorenson, 1978 or Sarpkaya and Isaacson, 1981). Expressions for the linear dispersion relation, free surface elevation, horizontal component of water particle velocity and horizontal component of water particle acceleration are provided here. Dispersion Relation: = j tanh(/cd) (2.7) Chapter 2. THEORETICAL BA CKGRO UND 11 Free Surface Elevation: rj = — cos(kx — uit) (2-8) 2 Horizontal Particle Velocity: TTH cosh(k(d + z)) T sinh(fcd) Horizontal Particle Acceleration: du _ 2n2H cosh(fc(of + 2)) It ~ ~Y2 sinh(A;d) cos(kx - ui) (2.9) sin(fcx - ui) (2.10) where u = y 1 is the angular frequency, A; = ^ is the wave number, g is the gravitational constant, and all other variables are as illustrated in Figure 1. The water particles follow closed elliptical paths with an eccentricity dependent on the value of kd which is a measure of the water depth to wave length ratio. In many applications the first approximation given by the linear wave theory is adequate, however, for situations where more accurate results are desired, successive approximations can be obtained by relaxing the small amplitude wave restriction and using a perturbation procedure in the analysis. The application of this perturbation procedure, in which the variables describing the flow are evaluated as power series in terms of a small perturbation parameter, led to the development of what is now referred to as Stokes Wave Theory. The accuracy of this analysis is directly dependent upon the number of approxima-tions evaluated (i.e. the number of terms retained in the power series). Stokes Fifth Order Wave Theory, in which a five-term power series is analysed, was developed by Skjelbreia and Hendrickson (1960) and is widely used in the ocean engineering pro-fession. Again, numerous literary sources exist which provide extensive detail in the development of this theory see Skjelbreia and Hendrickson, 1960 and Sarpkaya and Isaacson, 1981), however, for convenience only expressions for the dispersion relation, Chapter 2. THEORETICAL BACKGROUND 12 free water surface, horizontal particle velocity and horizontal particle acceleration are listed here. Dispersion Relation: Free Surface Elevation: 5 kr) = ^2r)'ncos(n0) (2.12) n=l Horizontal Particle Velocity: u 5 — = ^2 n<^'n cosh(nfcs) cos(n#) (2-13) C n=l Horizontal Particle Velocity: 5 = n V n cosh(nA;s) sin(rc0) (2.14) 3u at w c „=i where s is the distance above the seabed (s = d + z) and 9 is equivalent to [kx — tut) found in the Linear Theory expressions. For a given design wave, the parameters A and kd are to be determined before the above equations can be applied. These quantities may be obtained through an iterative evaluation of the following pair of equations. ^ ( A + B33X3 + (B35 + B55)X5) = |j (2.15) Jfcdtanh(ifcd)(l + C^2 + C 2 A 4 ) = 4 T T 2 - ^ (2.16) The coefficients B,j and C,j are known functions of kd, given by Skjelbreia and Hen-drickson (1960). Expressions for the remaining variables, r)'n and <f>'n, can also be found in Skjelbreia and Hendrickson (i960). This fifth order wave theory is used in the numerical aspect of the present investi-gation. It is interesting to note that as a result of development of the theory by several Chapter 2. THEORETICAL BACKGROUND 13 individual researchers, there now exists numerous versions of the Stokes Fifth Order Wave Theory. Fenton (1985) investigated the separate versions and discovered errors in some of the results. The Fifth Order Theory employed in the present study is believed to be free of such errors, but if further detail on the subject is desired the reader is referred to Fenton (1985). 2.3 Wave and Current Interactions A wide variety of combined wave and current flow regimes are found in practice. The simplest and most fundamental regime involves a regular wave train propagating in the same direction as a steady uniform current. More complex interactions involve oblique waves, non-uniform currents, random waves, and wave-induced currents. As a logical first step in the investigation of wave-current interactions, this thesis concentrates on the colinear superposition of regular waves and steady uniform currents. In the original development of the wave theories discussed in Section 2.2, it was necessary to assume the absence of any underlying current before a solution relative to the fixed reference frame (x, z) could be found. This assumption implies that the time averaged horizontal particle velocity at any one location is zero. The horizontal particle velocity, u', in a reference frame moving with the waves, (x',z), can be expressed in terms of the corresponding particle velocity, u, and wave speed, c, relative to the fixed reference frame. Now if a steady uniform current is superposed in the same direction as the wave train propagation, u' does not change and is now expressed as: u = u — c (2.17) u = [u + V)- cc (2.18) Chapter 2. THEORETICAL BACKGROUND 14 where V is the current velocity and c c is the wave celerity in the presence of the current. Note that u is still just the horizontal particle velocity due to the oscillatory component of the flow. In this moving reference frame the wave theory solution is identical to the solution with no superposed current indicating there is no change in the wavelength or wave number due to the change in reference frames. Transferring the solution back into the fixed reference frame then leads to expressions for wave speed and angular wave frequency in the presence of currents given by: cc = c + V (2.19) OJC = u + kV (2.20) The linear dispersion relation, equation (2.7), may be applied to obtain expressions for c and u and hence the above equations can be rewritten as: cc = sj^tanh(kd) + V (2.21) OJC = \Jgk tanh(fcd) + kV (2.22) The most obvious consequence of the addition of an underlying current is the alter-ation of the fluid particle motions and velocities. Clearly, the fluid particles no longer follow closed elliptical orbits, rather they will move in a cycloidic manner with a par-ticle travelling a greater distance in the downstream direction than in the upstream direction. If the current was not present, the expected fluid particle velocities are given by linear wave theory as: u = Umcos(ut) (2.23) where Um is the maximum particle velocity, given by linear theory as: _ 7rHcosh(k(d + z))  U m ~ T sinh(fcd) [ Z - 2 4 } Chapter 2. THEORETICAL BACKGROUND 15 With the addition of an in-line current of velocity V, the fluid particle velocities are now described by: uc = V + Um cos(wi) (2.25) It is evident that the fluid particle velocities will follow a trend similar to that of the fluid particle motions. That is, as a wave passes any point the velocities will be greater in the downstream direction than in the upstream direction. This unbalanced nature of the particle motions and velocities will clearly lead to a larger number of vortices being shed from the downstream edge of a body than its upstream edge and hence a somewhat biased wake. The severity of biasing depends upon the relative magnitudes of the current and wave velocities. The ratio of current velocity to wave velocity will be referred to as the reduced velocity, defined as Vr = jf-. In addition to this alteration of fluid particle velocities, Hogben and Standing (1975) identify two additional modifications to the flow characteristics. Firstly, the authors discuss the change in the wave celerity, c c, as indicated in equation (2.19) of this thesis. Secondly, and perhaps less obvious, is the tendency for the structure to generate standing waves in much the same manner that a ship moving through still water will create waves. For the slender structure investigated here, the generated waves and any wave-making resistance felt by the body, are considered negligible. As one might expect, the wake biasing inherent in this flow regime will alter the characteristics of the fluid-structure interaction and the resultant hydrodynamic forces. It is this variation in hydrodynamic loading associated with wave-current flow fields that is investigated here. The following chapter discusses the objectives of the present investigation and provides additional detail on the wave and current flow combinations particular to the study. Chapter 3 P R E S E N T I N V E S T I G A T I O N It is beneficial at this point to discuss various aspects of the present investigation in fur-ther detail. This chapter will examine the objectives of the study, offering a description of the specific wave-current combinations and expected in-line force distributions. The two methods of analysis, experimental and numerical, will then be detailed in Chapters 4 and 5 with emphasis on experimental apparatus, data acquisition, and selection of the appropriate values to be used in the numerical analysis. 3.1 Objectives The present investigation was undertaken to determine the force distributions acting on a slender cylinder subjected to various wave and current combinations. This type of cylinder is a common structural element found in many offshore structures in the form of support legs and braces, submarine risers, cables and pipelines. Although in practical applications these elements may be inclined, flexible, or subjected to irregular wave climates, the theory involved in predicting the hydrodynamic loads are very similar and are, in general, simply extensions to the basic theory studied here. The wave-current flow combinations considered here consist of two regular wave trains of different periods superposed with currents of varying magnitude and direction. The slender cylinder tested is a rigid, vertical, surface piercing structure of diameter D. Comparisons of the measured force distributions for the various flow fields pro-duced in the laboratory are analysed in terms of the effects of current magnitude. In 16 Chapter 3. PRESENT INVESTIGATION 17 addition, a numerical procedure is carried out employing "conventional" engineering design methods for identical flow conditions, and these results are compared with the corresponding experimental data. As a method of calibrating the experimental and numerical models, tests were performed for flow conditions of current-only, wave-only, and finally superposed waves and currents. This calibration process also proved extremely helpful in evaluating the quality of the experimental data. 3.2 Description of Wave Conditions It is convenient to classify wave conditions as being either "deep water waves", "in-termediate depth waves", or "shallow water waves" depending upon the relationship between the water depth and the wave length or wave period. This classification of waves takes advantage of certain approximations, valid over specific ranges of kd, for the hyperbolic functions found in several linear wave theory equations. These approx-imations are as follows: For kd > 7r: sinh(fcd) ~ cosh(&<£) ~ ^ exp kd tanh(kd) ~ 1 For kd < ^: sinh(A;d) ^ tanh(kd) ~ kd cosh(kd) ~ 1 The first range, kd > n, corresponds to deep water waves in which the magnitude of the water particle orbits diminishes exponentially with depth, until, at a depth of approximately j , the particle motion may be considered negligible. For this reason Chapter 3. PRESENT INVESTIGATION 18 deep water waves are sometimes described as being unable to "feel" the bottom. The second range, kd < fg, on the other hand, represents shallow water waves in which the horizontal water particle motions do not vary significantly from the water surface to the seabed. In this case the waves are said to "feel" the bottom. Obviously, there exists a range of kd ( ^ < kd < IT ) corresponding to intermediate depth waves in which the fluid particle motions at the seabed are notably less than those at the water surface yet cannot be considered negligible. The three distinct wave types can alternately be defined in the following manner: Shallow Water Waves: d < 0.0025 gT* Intermediate Depth Waves: Deep Water Waves: d 0.0025 < —— < 0.08 d > 0.08 gT* In the present investigation the water depth was held approximately constant at 0.50 metres and the periods of the two waves examined were 0.75 seconds and 1.5 seconds respectively. Substituting these values into the expression where g is the gravitational constant taken to be 9.81 j^;, one produces — 0.0906 and = 0.0227. Comparing these values with the limits outlined above, one can plainly see that the two wave periods used in this study, Ti = 0.75 seconds and Ti — 1.5 seconds, correspond to a deep water wave and an intermediate depth wave respectively. The values of the wave heights selected for the above two wave climates were chosen on the basis of preliminary laboratory observations indicating relatively smooth and regular wave profiles. In both cases a wave height of approximately 0.06 metres (6 Chapter 3. PRESENT INVESTIGATION 19 centimetres) was deemed satisfactory. Later, with the superposition of relatively large currents, noticable changes to the wave profile were observed. 3.3 Current Flow Conditions For each of the two wave conditions described in section 3.2, experimental measure-ments and numerical results are obtained for six distinct superposed colinear currents in addition to the results obtained for the wave-only flow conditions. Three of these currents travelled in the same direction as the waves, while the remaining three opposed the wave propagation. Selection of the current magnitudes was based on the maximum fluid particle veloc-ities, Um, at the water surface due to the wave-only oscillations as given by linear wave theory. Using equation (2.24) with values of H — 0.06 m, 2\ = 0.75 sec, d = —z — 0.50 m, and (kd)i — 3.14 for the first wave studied; and H = 0.06 m, T 2 = 1.5 sec, d = —z — 0.50 m, and (kd)2 — 7.77 for the second wave studied produces values for the maximum particle velocity at the surface as follows: First Wave (Tx = 0.75 s) : Umi = 0 .294- (3.26) Second Wave (T2 = 1.5 s) : (3.27) Attempting to cover the entire range, from 0.0 to ±1.0, of relative current velocities (Vr), the following current velocities, V, were selected for the present study: First Wave {Tx = 0.75 s): , m V{- ±0.10— with Vr ~ 0.3 V? si ±0.15 m with Vr ~ 0.5 s Chapter 3. PRESENT INVESTIGATION 20 Vr ~ 0.8 Vr ~ 0.3 Vr ~ 0.5 Vr ~ 0.8 where a positive sign indicates a current travelling in the same direction as the waves, and a negative sign indicates a current opposing the waves. It should be noted that these values are only intended to be taken as approxima-tions, and in fact, in the experimental work will vary slightly due to the difficulties encountered in setting exact current flow velocities. To maintain continuity between the experimental and numerical procedures the actual values used in the model tests are measured and are then incorporated into the numerical analysis. It is also important to note that a constant value for the current velocity is assumed to extend from the water surface to the seafloor, even though in the ocean, and, in fact, in the laboratory this is generally not the case. For the experimental work, velocity measurements were made along the water column for the various current-only conditions, and it was the averaged values of these measurements that were then used in the remaining calculations. 3.4 In-Line Force Distribution A slender circular cylinder in a purely harmonic flow will clearly be subjected to some sort of periodic loading which will oscillate about the zero or no-load axis. The ex-pression for the in-line forces associated with this type of flow has been introduced in Chapter 2 of this paper as the Morison equation: F' = ^PDCdu\u\ + P^-cJ^ (3.28) TTt V? ~ ±0.23— with s Second Wave (T2 = 1.5 s): V, 1 ~ ±0.05— with 2 s V, 2 ~ ±0.10— with 2 s TTt V, 3 ~ ±0.15— with Chapter 3. PRESENT INVESTIGATION 21 If an underlying current of velocity V is superposed on this harmonic flow, intuitively there will be some alteration in the resultant hydrodynamic forces. The in-line forces would continue to follow some periodic pattern, but would no longer oscillate about the no-load axis. The current component of the flow introduces a supplemental steady force, which when added to the purely harmonic response, results in a loading response curve that is shifted vertically. The magnitude, and direction of this shift are functions of the current velocity and flow direction respectively. This current component of the force may in fact contain some high frequency fluctuations associated with the additional vortex shedding arising for large current velocities, but in most instances this effect could be considered negligible. Methods for taking this additional force into account vary widely, but generally involve some speculative extension of the Morison equation (3.28) and require experi-mental verification. A widely accepted modification to the Morison equation involves replacing the wave velocity, u, with the vectoral sum of the wave and current velocities. F' = \PDCdc{u + V)\u + V\ + P ^ C m c ^ (3.29) where V is the current velocity and Cdc and Cmc are the force transfer coefficients in the presence of currents. In most past applications equation (3.29) has been employed with Cdc = Cd and Cmc = Cm for simplicity. Recent work, including the present study, have indicated that when dealing with a co-existing flow regime, in general Cdc 7^  Cd and Cmc ^ Cm. It has been argued that equation (3.29) may not be applicable to the combined wave and current flow regime since there are no fluid-mechanical reasons for its justification, yet in recent years there has been experimental verification of this equation provided appropriate force transfer coefficients are used. (See Sarpkaya and Storm, 1985). Chapter 3. PRESENT INVESTIGATION 22 A second, perhaps less accepted, form of the Morision equation involves the in-troduction of a third term to account for the additional steady force component. For example, C = QmSfDUi = ~ C*\C°SW I « * ( * ) + Cj^ Bin(fl) (3.30) where Vr is the reduced velocity parameter, and Cd and Cm are given by their Fourier averages. Sarpkaya and Storm (1985) found that the drag coefficient, Cd exhibits unrealistically large values and that neither of the coefficients are at all similar to their no-current values. For these reasons, equation (3.30) was deemed as an inappropriate representation of the in-line forces generated in a co-existing climate. In the present investigation equation (3.29) will be employed to provide a numerical prediction for the in-line forces associated with the various wave-current combinations considered. The force transfer coeffiecients will be assumed to be equal to their no-current values (Cdc = Cd and Cme — Cm). These results will then be compared with the laboratory measurements to illustrate the accuracy or inaccuracy of the predictions. Chapter 4 E X P E R I M E N T A L M E T H O D O L O G Y 4.1 Experimental Apparatus The experimental aspects of the present study were performed at the University of British Columbia employing the facilities available within the hydraulics laboratory in the Department of Civi l Engineering. In particular, the laboratory wave flume with its recirculating flow capabilities, as illustrated in Figure 2, was used to generate the combined wave and current flow regimes as required. The flume is approximately 20 metres long, 0.60 metres wide and can accommodate water depths up to 0.70 metres. Waves are created using a bottom-hinged wave paddle at the upstream end of the flume which is driven by a variable speed motor through a crank arm of adjustable stroke. This system enables the user to set the desired wave period and wave height. Directly downstream from the paddle the floor of the flume slopes upward and a sequence of wire mesh filters are located to absorb any high frequency irregularities in the generated waves. The waves propagate through the test section of the flume and are then absorbed by an artificial hair matting beach. It was important to employ a porous material in the design of the beach structure to allow for a continous flow of fluid in either the upstream or downstream direction. That is, to allow for either a forward or reverse current flow. The test section of the flume is approximately 3 metres long, starting approximately 7.5 metres downstream from the wave paddle. Within the test section measurements 23 Chapter 4. EXPERIMENTAL METHODOLOGY 24 were made of the steady current velocities, the dynamic water surface elevation, and the dynamic force distribution acting on the test cylinder. 4.2 Flow Kinematics Generating the current in the forward direction was achieved using the two 5 horsepower pumps shown in Figure 2 which carry water from the tailbox at the downstream end to outlets just in front of the wave paddle at the upstream end. The outlets are designed to distribute the flow as uniformly as possible, but again the current flow passes through the same filtering system as described for the waves to remove any irregularities. In order to generate a reverse current it was necessay to pump water from the laboratory's underground sump into the tailbox while simultaneously draining the flume at the upstream end. Without any automatic valves to balance the flows, establishing and maintaining the desired flowrate proved to be a somewhat tedious endeavour. A stilling well was installed on the flume to aid in setting the reverse flows and to ensure that the system remained balanced during the experiment. This arrangement along with intermittent calibration runs made it possible to maintain a relatively steady flowrate. Current velocity measurements were made using a Ott velocity probe and meter. Measurements were taken under current-only conditions at the start of each test run to ensure the desired current was initially set, and repeated at the end of the test to verify a steady flowrate was maintained during the run. Velocity measurements were performed at approximately ten equally spaced locations along the water column, allowing for the estimation of the fluid particle velocty profile. A simple average of the measured values was calculated and used in the subsequent analysis. The water surface elevation was measured using a Robertshaw wave probe with its output directed to the data acquisition system. Calibration of the wave probe output Chapter 4. EXPERIMENTAL METHODOLOGY 25 signal was achieved by simply correlating various known still water depths with the digital output displayed on the computer. This wave probe system appears to have behaved very well, and it is believed the readings obtained are accurate to within ±0.5 millimetres. In the reverse current flow regimes, several short "calibration" runs were performed during each test run to ensure the pumping/draining arrangement remained balanced. This involved turning off the wave paddle, waiting for a relatively steady state, taking wave probe measurements, and making any necessary adjustments. Unfortunately, it was noted that the signals radiating from the wave probe and its cabling tended to interfere with the signals collected from the force measuring device. Attempts made to minimize the interference were met with little success, and therefore, to ensure accurate data measurement it was decided that the two instruments should not be used simultaneously. It was later realized that this prevented any direct time correlation between the wave profile and the measured force data leading to difficulties in determining the relative phase between successive signals. 4.3 Measured Force Distributions The hydrodynamic loading on the "active" length of the test cylinder was measured using an instrumented rod onto which the test cylinder attached with the whole system clamped to a fixed overhead frame. The relevance of referring to the "active" portion of the test cylinder is clarified through a discussion of its construction characteristics. Figure 3 shows the cylinder mounted in the wave flume and indicates that the "active portion" of the cylinder is separated from the "inactive portion" by a gap of approx-imately 1 millimetre to avoid any interaction between them. It is important to note that the forces measured through the instrumented rod represent only the forces which act on the active portion of the cylinder with the inactive portion present merely to Chapter 4. EXPERIMENTAL METHODOLOGY 26 simulate a continous pile. The instrumentation on the rod consisted of two strain gauges mounted approxi-mately 20 centimetres apart and aligned in such a manner as to measure the in-line strains of the rod. The signals from these gauges are then passed through a Wheat-stone bridge completion circuit and manipulated to produce the total force on the active portion of the cylinder. The dimensions and material that were used in the design of the test cylinder were chosen on the basis of availability and required minimal resonant vibrations. Reso-nance occurs when the frequency of excitation (i.e. the waves) approaches the natural frequency of the cylinder. The waves studied in this investigation, with periods of 0.75 seconds and 1.5 seconds, have frequencies of 1.33 hertz and 0.67 hertz respectively and thus a test cylinder with a natural frequency considerably higher than this was desirable. The natural frequencies of various cylinders were determined in and out of water simply by impacting the cylinder and recording the resultant vibrations on an oscil-loscope. Finally, a 1.0 inch (2.54 cm) diameter solid aluminum cylinder with a total length of 60 centimetres was deemed adequate. Figure 4 shows a curve of submerged length versus natural frequency for the chosen test cylinder. Unfortunately, it was later noted that the superposition of large current velocities produced an eddy shed-ding frequency which approached the natural frequency for cylinders longer than about 30 centimetres. This induced significant dynamic oscillations and is discussed further in Chapter 6. The cylinder itself, as illustrated in Figure 5, is in fact made up of 21 individual segments, each 2.54 centimetres in diameter and 3 centimetres in length. Each segment was drilled and tapped to enable connections between successive segments to be made simply by screwing one segment into the next. (See segment detail in Figure 5). This Chapter 4. EXPERIMENTAL METHODOLOGY 27 arrangement made it possible to incrementally change the length of the active portion of the test cylinder by simply removing a segment from the active portion and adding it to the inactive portion. Clearly, by starting with the entire cylinder being "active" and incrementally de-creasing the active length (by 3 cm increments), until the entire cylinder becomes "inactive", while taking force measurements for each increment, it is possible to mea-sure the complete force distribution along the pile. Theoretically, it should be possible to subtract successive real time force signals and evaluate a dynamic force per unit length response signal. However, due to difficulties experienced in simultaneous mea-surement of the wave characteristics and the force response, the time phase between successive signals could not be accurately determined and hence the subtraction could not be performed. On the other hand, it was possible to estimate the peak-to-peak force acting on successive active lengths, and by subtracting these values a distribution of maximum in-line forces could be evaluated. It was this procedure that was followed in the data analysis. Figure 6 provides closer detail of the rod and shows the strain gauges mounted at locations A and B, a distance / apart (/ = 20 cm in the present study). The distance from gauge location B to the point of load application, P, is in general some unkown distance, b. From strucural theory, the relationship between the strain, e, and the moment, M, at any point along the rod is given by: _ My _ Mw 6 = ~EI = 2EI where w is the diameter of the rod, E is Young's Modulus of the rod and I is the moment of inertia of the rod. The moments induced at locations A and B due to the load applied at location P are given by MA = F(l + 6) and MB = F{b) respectively. Substituting these expressions into the strain equation above leads to equations for the Chapter 4. EXPERIMENTAL METHODOLOGY 28 strains at locations A and B as follows: and, which can be rewritten as: w Ae = eA - eB = ——Fl A 2EI „ 2EI F = — - A c wl This final expression indicates that by recording the difference in the two strains, the total force acting on the test cylinder can be deduced regardless of its point of application along the cylinder. This result also made calibration of the output signal a relatively simple task. Briefly, with the rod supported in a horizontal manner, a series of known weights were suspended along its length and the correlation between the applied force and the digitally displayed output was evaluated. At the time of this calibration process there was considerable uncertainty in the accuracy of the strain gauges and their ability to reproduce results. That is, it appeared that the zero, or no-load, output signal could vary significantly between tests if any of the test apparatus had been disturbed or jarred. The process of removing a segment from the active portion of the cylinder and adding it to the inactive portion could in fact disturb the apparatus sufficiently to alter the zero in the output signal. As a result, it was concluded that the absolute force values between successive tests coud not be evaluated. On the other hand, it was found that the zero remained relatively constant during each individual test and therefore the measured peak-to-peak force values were considered to be realiable. It was these values that were then used in the subsequent data analysis. Chapter 4. EXPERIMENTAL METHODOLOGY 29 The experimental apparatus described in this section is commonly found in most hy-draulic laboratories and has been used successfully in numerous experiments in the past. The Ott velocity probe and meter are entirely self-contained, and through the use of a calibration equation, velocity measurements are directly obtainable. The Robertshaw wave probe and the two strain gauges produce analogue output signals requiring con-siderable more attention. The digitization and collection of these signals will be looked at in more detail in the following section. 4.4 Data Acquisition The Robertshaw wave probe and the two strain gauges are supplied with a constant DC voltage input and as the water level fluctuates or a strain is applied to the gauges, there will be a variation in the voltage across the particular device. It is this variation in the signal that represents a change in the measured quantities and is returned from the devices as output. These analogue signals can be recorded directly using an oscilloscope and chart recorder, as was done in Buckingham (1982), or alternately, can be converted to a digital output and recorded in a numerical format as was done in the present study. A schematic representation of the equipment employed in the data acquisition pro-cess is illustrated in Figure 7. A signal conditioning unit reads in the wave probe signal directly and accepts the strain gauge signals after they have passed through a Wheatstone bridge completion circuit. From the signal conditioning unit the signals are transmitted into an IBM Personal Computer where an analogue to digital conversion card performs the actual signal digitization. Commercially available data acquisiton software interacts with the analogue to digital hardware enabling the user to spec-ify the desired sampling rates, duration of sampling, channels to be sampled and any applicable scale factors. Chapter 4. EXPERIMENTAL METHODOLOGY 30 A simple FORTRAN calling program was developed to invoke the desired data acquisition routines. Sample durations for each increment of pile length were set at 10 seconds with a sampling rate of approximately 90 samples/second. This produced records which contained responses for approximately 12 wave periods with the 0.75 second wave, and approximately 7 wave periods with the 1.5 second wave. It was later realized that longer sample durations may have been beneficial in the analysis of the experimental data. Scale factors obtained from the calibration procedures for the wave probe and strain gauges were applied as the signals were recorded, converting the digitized voltage readings directly to water surface elevation and in-line force values respectively. The results were then written to computer disks and later transferred to the mainframe computer at UBC for storage and subsequent analysis. Electronic filtering of the signals was not employed at the time of data acquisition since digital filtering could be performed during the analysis process with no risk of filtering out potentially valuable raw data. Upon examination of the actual raw signals and their frequency compositions, using a Fast Fourier Transform program, digital filtering was also deemed unnecessary for the subsequent data analysis. This system of data acquisition proved to be very efficient and enabled the bulk of the data to be collected within a two week period. Also, having the data in a numerical format stored in computer files made a remarkable difference in the ensuing data analysis. Chapter 5 N U M E R I C A L M E T H O D O L O G Y As alluded to earlier in this thesis, a primary objective of the investigation is a com-parison of the results obtained in a series of laboratory experiments with the results which are obtainable from the corresponding numerical procedures. At this point a description will be provided of the conventional engineering practice employed in the prediction of the hydrodynamic forces acting on a rigid slender cylinder subjected to combined wave and current flow regimes. The numerical analysis employed in the present study has found wide acceptance in the engineering profession, though more recently it has been suggested that overly conservative results may be produced (Sarpkaya and Storm, 1985). 5.1 Modified Morison Equation Various possible modifications to the original Morison equation have been briefly de-scribed in section 3.4 of this paper. Equation (3.29), reproduced below, is used in the present study to predict the in-line force per unit length, F ' , acting on a cylinder of diameter D subjected to waves and currents. F' = \PDCdc(V + u)\V + u\ + p ^ C ^ (5.31) In the above expression V is the current velocity as measured in the laboratory and u and ^ are the wave particle velocities and accelerations respectively which will be predicted through numerical procedures. 31 Chapter 5. NUMERICAL METHODOLOGY 32 It is ordinarily assumed, and in fact recommended by the American Petroleum Institute (see Sarpkaya and Storm, 1985), that Cdc and Cmc have constant, current-invariant values equal to those employed in the wave-only flow regimes. That is, Cdc = Cd and Cmc = Cm. It is this point that has in recent years has been an area of controversy. Several prominent researchers, Moe and Verly (1980), Iwagaki et al. (1983) and Sarpkaya and Storm (1985), have argued that the values of the drag and inertia coefficients can vary substantially as a result of the biased vortex shedding associated with the superposition of waves and currents. Each of these researchers have performed laboratory tests in conjunction with numerical analysis to support their hypothesis. (See section 1.2). In the present investigation, the numerical procedures follow the conventional prac-tice of evaluating equation (5.31) using the current-invariant values of Cd and Cm. Comparisons are then made between the force distributions measured in the labora-tory and those obtained using equation (5.31). Clearly, to maintain the correct relationship between the experimental and numer-ical methods, the variables in equation (5.31) must match the actual values measured in the laboratory. The cylinder diameter, D, and the density of water, p, are obvious constants and are taken as D = 0.0254 m and p = 998.2 ^ respectively. Selection of the appropriate force transfer coefficients and estimation of the flow kinematics, on the other hand, each require closer examination. 5.2 Force Transfer Coefficients A simple dimensional analysis of the combined flow regime considered here reveals that the time-invariant force transfer coefficients are functions of a Keulegan-Carpenter Chapter 5. NUMERICAL METHODOLOGY 33 number, a Reynolds number, and a current velocity parameter. 77 T U D V Cdc = / i ( ^ p , — , y H = h{K,Re,Vr) (5.32) L) V U M U T U D V Cmc = / 2 ( ^ - , — , — ) = h{K,Re,Vr) (5.33) It is important to note that each of the three terms, K, Re, and Vr, in the above relations are a function of the maximum water particle velocity, Um, which in turn, is dependent upon the depth, z, at which it is evaluated, as disclosed in equation (2.24) of this thesis. In the present investigation the three parameters, K, Re, and Vr, were evaluated at the still water level (2 = 0), at mid-depth (z = — | ) , and at the seafloor (z = —d). The difference in the calculated values was so slight it was felt that their dependence on depth could be ignored, and the parameters as evaluated at the still water level could be employed in the ensuing analysis. For periodic flows it is felt that the Reynolds number may not be the most suitable parameter due to the fact that both the Keulegan-Carpenter and Reynolds numbers contain the term Um. For this reason a new parameter, the "frequency parameter", defined as the Reynolds number divided by the Keulegan-Carpenter number (/? = f^- = ^f), is commonly introduced. With respect to dimensional analysis, the Reynolds number and the frequency parameter are equally valid as a dimensionless parameter. The frequency parameter, /?, however, will take on a constant value for a series of experiments performed with a cylinder of diameter D in water of uniform and constant temperature (constant u) if the wave period, T, is kept constant. Thus, replacing Reynolds number with the frequency parameter in the above expressions yields: Cdc = fx(K,(5,Vr) (5.34) Cmc = f2(K,(3,Vr) (5.35) Chapter 5. NUMERICAL METHODOLOGY 34 In an attempt to reduce the number of governing parameters from three to two, numerous researchers have suggested various definitions of the Keulegan-Carpenter and Reynolds numbers. The idea behind these alternate definitions is to possibly eliminate the reduced velocity parameter, V r , by including it in the K and Re (or equivalently, (3) definitions. A partial list of suggested Keulegan-Carpenter and Reynold number definitions is shown below. (a) Conventional Definition: XT U M T r> UMD K = ~D~ ; R E = — (5-36) (b) Iwagaki and Asano (1984): K+ = K{1 + \Vr\) ; Re+ = Re{l + |7 r |) (5.37) (c) Iwagaki and Asano (1984): K, = K T \Vr - cos{9)\dB, forVr < 1 (5.38) * 00 D2 Re3 = KA——) (d) Sarpkaya and Storm (1985): Km = K(l + \Vr\)2 ; Rem - Re{l + \Vr\)2 (5.39) There has been some experimental work performed with results which support the various K and Re definitions above (equations 5.37 - 5.39) allowing for the elimination of the third VT parameter. However, the ranges over which the definitions are valid have been found to be restrictive, Sarpkaya and Storm (1985). For this reason, the conventional definitions (equation 5.36) for K and Re (or /?) along with the additional reduced velocity parameter, Vr, are retained as the governing parameter definitions. Chapter 5. NUMERICAL METHODOLOGY 35 In the present investigation the simplistic assumption that Cdc — Cd and Cmc = Cm implies that the coefficients are independent of the current velocity and hence the Vr parameter falls out of relations (5.34) and (5.35). Cdc = Cd = f1(K,(3) (5.40) Cmc = Cd = fx(K,(3) (5.41) Many laboratory investigations have been performed to determine the relationship between the force transfer coefficients and the two parameters, K and (3 for the wave-only flow regimes. Particularly useful are the results obtained by Sarpkaya (1976) in which the drag and inertia coefficients were evaluated for ranges of K, Re, and (3. It is results such as these which are commonly used to obtain appropriate drag and inertia coefficients, for particular values of K, Re, and (3, to be employed in the Morison equation. In the present investigation, the values generated for K, Re, and /? are as follows: First Wave = 0.0906): K ~ 5.9 Re ~ 5500 (3 ~ 1000 Second Wave ( ^ = 0.0227): K ~ 10.3 Re ~ 4200 (3 ~ 410 These values were calculated using the values for wave height, wave period, and water depth as measured in the laboratory. The Keulegan-Carpenter and Reynold numbers listed above were evaluated for the maximum water particle velocity, Um, at the still water level as discussed earlier in this section. Evidently, the values obtained for K, Re, and /? are relatively small in relation to values commonly found in practice, making the prediction of the force transfer coeffi-cients a rather difficult task. Combining the results listed in various papers (Sarpkaya Chapter 5. NUMERICAL METHODOLOGY 36 (1976), Sarpkaya and Isaacson (1981), and Sarpkaya(l985)) and extrapolating this data to match the characteristics in the present study, preliminary drag and inertia coeffi-cients were chosen as follows: First Wave ( ^ = 0.0906): Cd = 1.60 and Cm = 2.05 Second Wave ( ^ = 0.0227): Cd = 1.85 and Cm = 1.60 Trial numerical calculations were performed to test the suitability and sensitivity of the chosen coefficients. As one might expect with the relatively low Keulegan-Carpenter numbers, indicating inertia dominated flows, the numerical calculations tended to be considerably more sensitive to changes in the inertia coefficient than in the drag coef-ficient. Preliminary comparisons were made between the experimental and numerical results employing various combinations of force transfer coefficients for the two pure wave flow regimes to ensure the selection of the most appropriate coefficients. Conse-quently, the coefficients as listed above were deemed adequate on the basis of the best match between the experimental and numerical force distributions. As discussed in section 5.1 the numerical generation of these maximum in-line force distributions employ the modified Morison equation (5.31) with the appropriate vari-ables. Having selected the drag and inertia coefficients as above, the remaining variables to be defined are the current velocities and the wave velocities and accelerations. The numerical prediction of these flow kinematics is the topic of the following section. Chapter 5. NUMERICAL METHODOLOGY 37 5 . 3 F l o w K i n e m a t i c s A typical current velocity profile measured in the laboratory using the Ott velocity probe is illustrated in Figure 8. The depth-invariant velocity profile used in the present study as an approximation to the actual profile is indicated in the figure by a dashed line. The value of this uniform velocity has been evaluated simply as the mean value of the individual velocity measurements along the water column. Even though actual velocity profiles in the ocean environment generally show more pronounced variation with depth than those found in the laboratory, it is not uncommon to evaluate an equivalent uniform current velocity profile based on individual measurements as was done here. The spatial and temporal randomness inherent in the ocean makes other "more realistic" profiles either inadequate or overly complex. Prediction of the water particle velocities and accelerations associated with har-monic wave motion was considerably more involved than the process outlined above for current velocity estimation. Several different wave theories are readily available including Linear Wave Theory and Stokes Wave Theory, as discussed in section (2.2), as well as Stream Function Wave Theories, Cnoidal Wave Theories, and others. (See Sarpkaya and Isaacson, 1981). Comparisons of the various the various theories indicate that particular theories are more suitable for different flow conditions. Figure 9 indi-cates the ranges of ^ and for which the various wave theories have been found to be suitable. For the two waves studied here, evaluation of the and variables reveals that the Stokes Wave Theories are suitable. Consequently, a FORTRAN program was developed to call the "STOKSV" rou-tines (Stokes Fifth Order Wave Theory) available on the mainframe computer at the University of British Columbia. The interactive program allows the user to enter the wave parameters required by the STOKSV routines which then calculates the various Chapter 5. NUMERICAL METHODOLOGY 38 flow kinematics associated with the harmonic flow regime. The flow kinematics re-turned include angular wave velocity, water surface elevation, and the horizontal and vertical particle velocities and accelerations at some location (x, z) and some time t. By incrementally increasing the time variable the periodic profile of the different flow kinematic variables can be generated over one complete wave cycle. Further detail on the theory behind the S T O K S V routines can be found in Skjelbreia and Hendrickson (1960). The wave parameters required by S T O K S V include the water depth, d, the wave height, H, the wave period, T, and the spatial and temporal coordinates to be con-sidered. Table 5.1 lists the various wave parameters as measured in the laboratory for the fourteen different flow combinations investigated. Note that Table 5.1 contains two columns for the wave period: Tc, which is the wave period actually measured in the laboratory; and T, which is an equivalent no-current wave period. For the pure wave flow regime, Tc = T; for the combined wave and current flow regimes, however, Tc ^ T. It is this no-current value, T, which is required by S T O K S V to generate the oscillatory flow kinematics, u(z,t) and associated with the oscillatory portion of the flow. Equation (2.20), as reproduced below, indicates how the desired wave period, T, depends upon the measured wave period, Tc. UC = OJ + kV (5.42) Sustituting uc = ~ and u — jr one gets: 27r 2TT , T , . . Y c = Y + k v ( 5 - 4 3 ) where Tc and V are measured in the laboratory, and T and k are both unknown. To obtain a first approximation to the desired value, T, the linear dispersion relation, equation (2.7), can be used as the second equation providing a relation between T and Chapter 5. NUMERICAL METHODOLOGY 39 Flow Water Wave Wave Current Wave Condition Depth Height Period Velocity Period d(m) H(m) Tc (s) V (m/s) T(s ) A l 0.504 0.048 0.771 0.0 0.771 A2 0.503 0.054 0.758 +0.101 0.818 A3 0.502 0.065 0.767 +0.159 0.859 A4 0.501 0.059 0.763 +0.232 0.892 A5 0.502 0.056 0.763 -0.099 0.693 A6 0.504 0.048 0.763 -0.159 0.642 A7 0.502 0.037 0.770 -0.233 0.568 B l 0.505 0.068 1.463 0.0 1.463 B2 0.501 0.064 1.483 +0.060 1.530 B3 0.501 0.059 1.458 +0.104 1.538 B4 0.500 0.049 1.462 +0.161 1.584 B5 0.505 0.067 1.467 -0.062 1.418 B6 0.504 0.068 1.475 -0.099 1.396 B7 0.502 0.072 1.480 -0.150 1.357 Table 5.1: Flow Kinematics Measured in the Laboratory A;. Equation (2.7) is reproduced here for convience. u2 = gk tanh(kd) (5.44) or, again substituting u = Y o n e g e t s : ( — )2 = gjfctanh(JW) (5.45) Clearly, equations (5.42) and (5.44) (or equivalently, equations (5.43) and (5.45) can be solved simultaneously using an iterative procedure. This process was carried out for each of the wave-current combinations investigated and produced the no-current wave periods, T, as listed in Table 5.1 . It should be noted that a more accurate prediction of T may have been possible using the dispersion relation from Stokes Fifth Order Wave Theory rather than that from Linear Wave Theory but was judged unnecessary for the present investigation. Chapter 5. NUMERICAL METHODOLOGY 40 With the no-current wave periods as calculated above, and the remaining wave parameters as measured in the laboratory, a complete flow kinematics "map" was generated through a rapid succession of calls to the STOKSV routines. Water particle velocities and accelerations were evaluated at the midpoint of each three centimetre pile segment for each of the 75 time steps into which the wave period was divided. The FORTRAN program then proceeded to construct a similar force per unit length "map" by evaluating the modified Morison equation (5.31) with the constants as defined in sections 5.1 and 5.2, and the velocities and accelerations as returned from the calls to STOKSV. An integration process was then invoked to generate the total in-line force acting on any specified submerged pile length for each time step within the wave period. The maximum and minimum force values were then retrieved numerically and used to determine the peak-to-peak in-line force over the wave period. These peak values were evaluated for each incremental change in the submerged pile length and for all fourteen flow regimes tested in the laboratory. The results of the numerical analysis were then written to a computer file in a format suitable for comparison with the experimental data. A graphical representation of the results obtained by the two methods was chosen as the most suitable on the basis of clarity and convenience. These results and the discussion of them is provided in the following chapter. Chapter 6 R E S U L T S A N D DISCUSSION 6.1 Experimental Results The laboratory data collected for each of the combined flow regimes investigated con-sisted of current velocities, dynamic water surface elevations, and most importantly, in-line forces. Measurements of the current velocities and water surface profiles were performed to provide a quantitative definition of the actual flow condition tested in the laboratory, allowing for identical flow conditions to be simulated in the numerical analysis. The current velocities and relevant wave parameters deduced from the wave profile measurements are listed in Table 6.1 in dimensionless format. As a result of the unforeseen alterations in the wave period and wave height due to the underlying cur-rent, there is a considerable variation of the non-dimensional parameters within each series of tests. The hydrodynamic force data was recorded directly as the in-line force variation with time for a ten second sample duration. For the pure wave flow regimes, the periodic fluctuations of the force response was well defined allowing the average peak-to-peak force to be estimated with good accuracy. As the superposed current magnitude was increased (in either the positive or negative directions), an additional higher frequency fluctuation in the response became apparent. This additional fluctuation tended to become more pronounced for larger current magnitudes and longer pile lengths. In fact, substantial oscillatory motions of the test cylinder were observed for several of the 41 Chapter 6. RESULTS AND DISCUSSION 42 Flow Condition d H d V um P Figure A l 0.086 0.0952 0.0 830 10 A2 0.077 0.1074 +0.446 790 11 A3 0.069 0.1295 +0.606 750 12 . A4 0.064 0.1178 + 1.015 702 13 A5 0.107 0.1116 -0.350 930 14 A6 0.125 0.0952 -0.607 1000 15 A7 0.1586 0.0737 -1.001 1130 16 B l 0.024 0.1347 0.0 440 17 B2 0.022 0.1277 +0.334 420 18 B3 0.022 0.1178 +0.631 420 19 B4 0.020 0.0980 + 1.193 410 20 B5 0.027 0.1317 -0.313 450 21 B6 0.026 0.1349 -0.487 460 22 B7 0.028 0.1424 -0.690 470 23 Table 6.2: Dimensionless Wave Parameters Calculated from Experimental Data more extreme flow combinations. It is believed that the superposition of the current induced this high frequency response by increasing the strength and number of vortices shed from the test cylinder. On the average these attribute fluctuations had a frequency of approximately 6.5 Hz for both series of tests = 0.0906 and -4fi — 0.0227). The curve of test cylinder natural frequency versus submerged length, as illustrated in Figure 4, discloses a possible resonant response could be expected for * ~ —0.8 where the cylinder's natural frequency approaches 6.5 Hz. The increased vortex shedding generated by the majority of the combined flow com-binations induced only "sub-harmonic" resonant responses. Truly resonant responses did occur, however, for Vr greater than approximately 1.0 and | less than approximately -0.6 . This phenomenon will be further discussed later in this section. Whether sub-harmonic or truly harmonic, the presence of these high frequency fluctuations made the selection of the actual peak-to-peak forces more difficult. In an attempt to maintain consistency in the analysis, the fifth largest observed maxima and Chapter 6. RESULTS AND DISCUSSION 43 minima were employed in calculating the peak-to-peak value for all test data. These force values were then non-dimensionalized by dividing through by ^pDU^ and then plotted as a function of the dimensionless depth parameter * (where z is the distance from the still water level and d is the water depth). Figures 10 through 23 illustrate the variation of the maximum in-line force as a function of depth for the four-teen co-existing flow regimes tested. The actual values derived from the experimental work are indicated with the data markers and the solid line gives the best fit to the data points using a cubic spline smoothing procedure. The dashed line in each figure represents the theoretically predicted force distributions employing the conventional numerical methods outlined in Chapter 5. Values for ^j, jf~, and (3 are included within the title of each figure. It is interesting to note that the resonant responses referred to earlier in this section are evident in Figures 13, 16 and 20 which represent the experimental results for test conditions with a reduced velocity, Vr, greater than one. In fact, in Figure 20 it appears that the entire resonant peak is present. That is, resonance appears to start at a depth of | ~ —0.60, reaches a maximum at ^ — —0.80, then returns to the non-resonant pattern at | ~ —0.95 . These results agree well with the resonant response predicted earlier for | ~ —0.80 when the test cylinder's natural frequency equals frequency of the supplemental fluctuations observed for large current magnitudes. The plotting routine employed to generate these plots was unable to produce a suitable smoothed curve for the resonant values, so just the data points are shown. It is also felt that the quality of the resonant data is highly questionable due to the fact that substantial alterations in the hydrodynamics arise when a structure begins to oscillate. In effect the structure can no longer be considered as rigid, and the water particle kinematics close to the body will now become highly dependent on the motion of the structure itself. In depth analysis of this phenomonen is beyond the scope of Chapter 6. RESULTS AND DISCUSSION 44 this thesis. Also of interest is the difference in the shape of the forces profiles between the first series of tests (Figures 10 - 16) and the second series (Figures 17 - 23). The difference in these profile shapes can be explained by first recalling that the first wave tested ( ^ 2 = 0.0906) is a deep water wave, while the second wave (-^j — 0.0227) is an intermediate depth wave. The profiles illustrated in the first seven figures indicate a substantial variation in the incremental increases of the in-line force values. For small values of | there are very large increases in the force, while for larger values of | , the incremental force increases are much less substantial. This type of force distribution is characteristic of deep water waves. The remaining seven figures, on the other hand, show fairly constant incremental increase of the in-line forces with | . A force distribution of this type is characteristic of intermediate and shallow water waves. Overall, the experimentally measured and numerically predicted curves follow simi-lar profile shapes, but rarely do the two curves share identical values. The discrepency between the measured and predicted curves shall be examined in the following section. 6.2 Comparison of Experimental and Theoretical Force Distributions Figures 10 through 23 illustrate the experimental and numerical force distributions along the submerged pile length for the fourteen flow conditions investigated. The results for the pure wave regimes, shown in Figures 10 and 17, indicate fairly good agreement between the two curves, particularly for the second wave (Figure 17) where the curves could be considered equal. Figure 10 discloses an apparent under prediction of the forces by the numerical methods. The discrepency between the two curves in Figure 10 is less than ten percent and, in fact, experimental data falls on either side of the theoretical curve indicating it to be within acceptable limits. The analytical Chapter 6. RESULTS AND DISCUSSION 45 reasons for this underprediction are not clear. Figures 1 1 - 1 6 and 18 - 23 show the experimental and numerical results for flow regimes consisting of combined current and harmonic flows. Firstly, one should note that for the majority of cases, the numerically predicted curve lies wholly to the right of the experimentally derived curve. This manifests a consistent overprediction of the in-line forces by numerical methods. In three instances (Figures 21, 22 and 23) where a negative current is superposed on the second set of waves (^ r = 0.0227), the two curves have a tendency to cross at small Other than these three instances, the numerical overprediction referred to by past researchers (See section 1.2) is clearly evident. There is no obvious distinction in the severity of over-prediction between the first and second data sets. That is, given the data generated in the present investigation it is difficult to say if the numerical methods are more accurate for deep water or intermediate depth waves. There does appear to be a relationship between the degree of over-prediction by numerical methods and the magnitude (and direction) of the superposed current. Com-paring the force distributions for increasing positive and negative current magnitudes, reveals a clear variation in the distance separating the experimental and numerical curves with respect to current velocity. The severity of the over-prediction tends to be greatest for the larger positive superposed currents and least for the smaller negative currents. The variation of in-line forces with the superposed current magnitude and direction is discussed further in the following section. 6.3 Effect of Current Magnitude on In-line Forces Figures 24 and 25 were generated using the in-line force values at ^ = —0.5 taken from Figures 10 - 23 for the various flow combinations. These two plots illustrate Chapter 6. RESULTS AND DISCUSSION 46 the variation in both the measured and predicted forces with current magnitude. The experimental data in Figure 24 does not include force values for > 1.0 since resonant v m vibrations have made them unsuitable. Although there are some anomolies present in the experimental data, in general the experimental and theoretical curves follow similar patterns. Over most of the range of jj-, the experimental curve lies below the theoretical curve again illustrating the tendency towards numerical over-prediction. The pattern of over-prediction discussed in the previous section, indicating most severe error for large positive currents and least severe error for small negative currents, is clearly shown in Figure 25. In both figures the experimental curves tend to exhibit a local maximum close to JJ- = 0.0, with the maximum being more pronounced in the first figure (^j = 0.0906) than in the second figure (^y = 0.0227). The presence of this local maximum is difficult to explain and is not apparent in the theoretical curves. As the current increases in both the positive and negative directions, the curves follow a more expected pattern showing increased forces for larger current magnitudes. The overall maximum force tends to occur for largest positive current. C h a p t e r 7 C O N C L U S I O N S A N D R E C O M M E N D E D F U R T H E R S T U D Y 7.1 C o n c l u s i o n s The present investigation examined the effect of superposed currents and waves on the in-line force distributions along a vertical slender cylinder. Laboratory and numerical analysis were performed for currents of varying magnitude and direction superposed on deep water and intermediate depth waves. Results obtained by both procedures manifest significant changes in the peak-to-peak in-line forces due to the presence of the underlying current. The theoretical consequences indicate increased forces generated by both positive and negative currents with the more substantial increases induced by the positive currents. Experimental results were less coherent but were at least qualitatively similar to those obtained numerically. The most notable outcome of the study was the consistent over prediction of the measured forces by the conventional numerical methods. This overprediction has a tendency to be most pronounced for the larger postive currents and less salient for the negative current magnitudes approaching zero. It is believed the discordance in the results attained from these two methods is primarily the consequence of employing the current-invarient values for the drag and inertia coefficients in the modified Morison equation (5.31). Unfortuately, the current dependant coefficients could not be evaluated in the present study as a result of the inability to accurately determine the phase of 47 Chapter 7. CONCLUSIONS AND RECOMMENDED FURTHER STUDY 48 the response. The presence of an underlying current induces substantial alterations in the vortex shedding patterns around a submerged body, As a result, the reasonably well under-stood wave loading problem becomes considerably more complex. The conventional methods for determining hydrodynamic loads resulting from a combined wave and current flow regime are clearly inappropriate. Alternate methods employing current-dependent drag and inertia coefficients have seen limited success, but further experi-mental investigated are required. A better understanding of the various effects inherent in a combined flow regime is essential since rarely will one discover an ocean environment where current and other random disturbances are not also present. 7.2 Recommendations for Further Study In regards to the present investigation, it would be interesting to examine the actual flow kinematics of co-existing regimes in more detail. Measured quantities for the flow velocities and accelerations, made perhaps using laser dopier equipment, could be entered directly into the Morison equation for an additional method of predicting the in-line force distributions. Clearly, a method for accurately determining the phase of the structure response would be beneficial in any forthcoming investigations. Also more sophisticated instrumentation, less susceptible to interference and drift effects, would be advantageous. Overall, there remains numerous aspects of combined wave and current flow regimes where further investigation is essential. For even the fundamental conditions, such as those studied here, there is an obvious need for additional experimental, numerical and field data to aid in the understanding of the interaction process. Beyond that, one Chapter 7. CONCLUSIONS AND RECOMMENDED FURTHER STUDY 49 may wish to investigate the more complex permutations of the problem including non-uniform currents, obliquely approaching currents, irregular or random waves, inclined members and flexible structures. Bibliography Bearman, P.W., Downie, M . J . , Graham, J .M.R . and Obasaju, E .D . 1985. "Forces on Cylinders in Viscous Oscillatory Flow at Low Keulegan-Carpenter Numbers", Journal of Fluid Mechanics, Vol. 154, pp. 337-356. Buckingham, W.R. 1982. "Forces on a Cylinder Due to Waves and a Colinear Cur-rent", M A S c Thesis, Dept. of Civi l Engineering, University of British Columbia, Vancouver, Canada. Chakrabarti, S.K. 1985. "Recent Advances in High Frequency Wave Forces", Proc. Fourth Int. Offshore Mechanics and Artie Engineering Symposium, A S M E , Vol. 1, pp 125-141. Fenton, J .D. 1985. "A Fifth-Order Stokes Theory for Steady Waves",ASME Jour-nal of Waterway, Port, Coastal and Ocean Engineering, Vol . I l l , No. 2, pp 216-234. [5] Hamming, R . W . 1977. "Digital Filters",Bell Laboratories and Naval Post-Graduate School, Prentice-Hall, New Jersey, 07632. [6] Hedges, T.S. 1979. "Measurement and Analysis of Waves and Currents", in Me-chanics of Wave Induced Forces on Cylinders, Shaw, T . L . (ed.), Pitman, London, pp 249-259. Hogben, N . and Standing, R . G . 1975. "Experience in Computing Wave Loads on Large Bodies", Offshore Technology Conference, Houston, Texas, O T C 2189, Vol. 11, pp. 413-431. Isaacson, M . de St. Q. 1985. "Recent Advances in the Computation of Nonlinear Wave Effects on Offshore Structures" Canadian Journal of Civil Engineering, Vol. 12, pp. 439-453. Isaacson, M . de St. Q. and Baldwin, J .F . 1987. "Combined Wave-Current Effects on Structures", Department of Civi l Engineering, University of British Columbia, Vancouver, Canada [10] Iwagaki, Y . , Asano, T. and Nagai, F . 1983. "Hydrodynamic Forces on a Circular Cylinder Placed in Wave-Current Co-exixting Fields", Memoirs of the Faculty of Engineering, Kyoto University, X L V , pp. 11-23. [11] Iwagaki, Y . and Ansano, T. 1984. "Hydrodynamic Forces on a Circular Cylinder due to Combined Wave and Current Loading", Proceedings Coastal Engineering, Houston, Texas, Vol. 3, pp. 2857-2874, Chpt. 191. [12] Kaplan, P. and Dummer, J . 1986. "Analysis of the Effect of Currents on Wave Forces Measured on an Offshore Structure at Sea",Proc. Offshore Technology Con-ference, Houston, Texas, Vol. 1, pp. 527-536, No. 5143. 50 Bibliography 51 [13] Keulegan, G . H . and Carpenter, L . H . 1958. "Forces on Cylinders and Plates in an Oscillating Fluid" , J. Res. Nat. Bureau of Standards, Vol . 60, No. 5, pp. 423-430. [14] Kruijt , J .A . and Van Oorschot, J .H. 1979. "Interaction Between Waveand Current Forces on the Concrete Piers of the Easter Scheldt Storm Surge Surge Barrier", in Mechanics of Wave Induced Forces on Cylinders, Shaw, T . L . (ed.), Pitman, London, pp. 684-703. [15] Le Mehaute, B . 1976. "An Introduction to Hydrodynamics and Water Waves", Springer-Verlag, Dusselsorf. [16] Massie, W . W . 1979. "Hydrodynamic Forces in Waves and Currents", in Mechanics of Wave Induced Forces on Cylinders, Shaw, T . L . (ed.), Pitman, London, pp.324-333. [17] Matten, R . B . , Hogben, N . and Ashley, R . M . 1979. "A Circular Cylinder Oscillating in Still Water, In Waves and in Currents", in Mechanics of Wave Induced Forces on Cylinders, Shaw, T .L . (ed.), Pitman, London, pp. 475-489. [18] Moe, G . and Verley, R .L .P . 1980, "Hydrodynamic Damping of Offshore Structures in Waves and Currents", Offshore Technology Conference, Houston, Texas, Vol. 3, P - O T C 3798. [19] Morison, J.R., O'Brien, M.P . , Johnson, J .W. and Schaaf, S.A. 1950. "The Force Exerted by Surface Waves on Piles", Petrol. Trans., A I M E , Vol. 189, pp.149-154. [20] Peregrine, D .H . 1976. "Interaction of Water Waves and Currents", Advances in Applied Mechanics, Vol. 16, pp. 9-117. [21] Sarpkaya, T. 1976. "In-Line and Transverse Forces on Smooth and Sand-Roughened Cylinders in Oscillatory Flow at High Reynold's Numbers" ,Naval Post-graduate School Technical Report No. NPS-69sl76062, Monterey, C A . [22] Sarpkaya, T. 1985. "Force on a Circular Cylinder in Viscous Oscillatory Flow at Low Keulegan-Carpenter Numbers", Journal of Fluid Mechanics, Vol 165, pp. 61-71. [23] Sarpkaya, T. and Isaacson, M . de St. Q. 1981. "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold, New York. [24] Sarpkaya, T. and Storm, M . 1985. "In-line Force on a Cylinder Translating in Oscillatory Flow", Applied Ocean Research, Vol. 17, No. 4, pp. 188-196. [25] Skjelbreia, L . and Hendrickson, J .A. 1960. "Fifth Order Gravity Wave Theory", Proc. 7th Coastal Engineering Conference, The Hague, pp. 184-196. [26] Sorensen, R . M . 1978. "Basic Coastal Engineering", Wiley, New York. [27] Stokes, G . G . 1847. "On the Theory of Oscillating Waves", Trans. Camb. Phil. Soc, Vol. 8, pp. 441-455. Bibliography 52 [28] Thomas, G.P. 1979 "Water Wave-Current Interactions: A Review", Mechanics of Wave Induced Forces on Structures, Shaw, T .L . (ed.), Pitman, London, pp. 179-203. [29] Thomas, G.P. 1979 "Water Wave-Current Interactions: An Experimental and Numerical Study", Mechanics of Wave Induced Forces on Structures, Shaw, T .L . (ed.), Pitman, London, pp, 179-203. F I G U R E S 53 54 Incident Waves "Wave Speed = c Wave Period, T = L/c Free Surface Elevation, T) Water Depth, d Wave Length, L [«._ Cylinder Diameter, D Wave Height, H Figure 1(a): Wave Flow and Test Cylinder Definition Sketch 55 Underlying Current, V d Figure 1(b): Combined Wave and Current Flow Definition Sketch Wave Paddle (f Closed System i i Flow Filter Screens Test Section Closed System Beach Underground Sump Underground Sump Figure 2: Elevation View of Wave-Current Flume cn CD 57 Inactive Portion Figure 3: Test Cylinder Mounted in Flume 1 1 I 1 1 1 I 1 1 1 I 0.0 4.0 8.0 12.0 16.0 20 Natural Frequency (Hz) Figure 4: Test Cylinder Natural Frequency as a Funct ion of Submerged Length. 59 Active Portion Inactive Portion Figure 5: Detail of Test Cylinder Construction. 60 Strain Gauge (^) w-Strain Gauge ( B j Clamping Arrangement Applied Force Active Portion Inactive Portion -TV-Figure 6: Detail of Instrumented Rod Velocity Meter Wheatstone Bridge Circuit Signal Conditioning Personal Computer •with A/D Board Ott Velocity Probe Power Supply Instrumented Rod and Test Cylinder Robertshaw Wave Probe Figure 7: Schematic of Data Acquisition Equipment. — Measured Profile - - Idealized Profile -- + L + r + - /+ i ' 1 — 1 i 1 1 1 i 1 1 1 1 1 1 ' i ' 1 1 r 0.10 0.12 0.14 0 16 0 18 0.20 . Current Velocity, V (m/s) Figure 8: Typical Measured and Idealized Current Velocity Profiles. 63 0.05 0.02 \-0.01 0.005 h H 0.002 h 0.001 0.0005 0.0002 0.0001 0.00005 0.001 0.002 0.005 0.01 0.02 d 0.05 0.1 0.2 Figure 9: Ranges of Suitability for Various Wave Theories as Suggested by Le Mehaute (1976). 64 F 0.5/oDU*' F i g u r e 10: I n - l i n e Force D i s t r i b u t i o n for V r ~ 0.0 ( B ~ 1000 - - d/gT 2 = 0 .086 — H / d = 0 .0952 ) 65 F 0.5pDU* Figure 11: In-(§ ~1000 -line Force Distribution for Vr~ +0.4 - d/gT2 = 0.077 -- H/d = 0.1074 ) F 0.5pDU* Figure 12: In-line Force Distribution for Vr—h0.6 ( B ~ 1000 -- d/gT2 = 0.069 -- H/d = 0.1295 ) \ N\ \ + Measured Force Values \ \ \ s Cubic Spline Smoothing  \ \ \ Theoretical Prediction \ ^  \ \ \ \ \ \+ v \ \ \ \ VA > \ \ \ \ \ + \ v \ \ \ \ 1 V \ \ \ \ \ \ + ! 1 + \ \ \ \ + \ \ \ \ \ \ \ + \ \ \ \ \ + \ \ \ + \ \ I I \ + , 1 ' 1 ; • 1 ; 1 1 0.0 0.8 1 6 2.4 3.2 F 0.5pDU2 Figure 13: In-line Force Distribution for Vr~ +1.0 ( 0 ~ 1000 -- d/gT2 = 0 064 -- H/d = 0.1178 ) 68 F 0.5pDU£ Figure 14: In-line Force Distribution for Vr~ -0.4 ( B ~ 1000 d/gT2 = 0.107 -- H/d = 0.1116 ) Figure 15: In-line Force Distribution for Vr~ -0.6 ( 0 ~ 1000 -- d/gT2 = 0.125 — H/d = 0.0952 ) d o \ \ \ \ + Measured Force Values V\ \ \ \ \ \ \ V" v Cubic Spline Smoothing Theoretical Prediction ci -1 \ \ I \ 1 \ + I 1 1 I \ \ 1 \ + 1 \ \ \ M • + 6 -i 1 \ I I ( \ 1 1 \ + + to \ 1  -1 \1 I \ V 1 ( 1 t 1 \ + + + CO b -1  1 \ t 1 I 1 + + t + o + —: ' I ' I : 1 1 1 i 0.0 0.8 1.6 2.4 3 2 F 05pDU2 Figure 16: In-line Force Distribution for Vr~ -1.0 ( 8 ~ 1000 d/gT2 = 0.159 -- H/d = 0.0737 ) Figure 17: In-line Force Distribution for Vr~ 0.0 ( 0 ~ 410 -- d/gT2 = 0.024 -- H/d = 0.1347 ) o Figure 18: I n - l i n e Force Dis t r ibu t ion for V r ~ +0.3 ( p ~ 410 - - d/gT2 = 0.022 - - H/d = 0.1277 ) 73 o I 1 1 1 1 ! 1 ! ! 1 1 1 ' 1 1 • i 0.0 0.4 0.8 1.2 1 6 2.0 F 0.5pDU* Figure 19. I n - l i n e Force Dis t r ibut ion for V r ~ +0.6 ( /? ~ 410 - - d/gT? = 0.022 - - H/d = 0.1178 ) F 05pDU2 Figure 20: In-line Force Distribution for Vr~ + 1.2 ( B ~ 410 -- d/gT2 = 0 020 -- H/d = 0.0980 ) i ' ' : 1 ' 1 1 ! ' 1 r 0.0 0.4 0.8 1.2 16 F 0.5pDU2 Figure 21: In-line Force Distribution for Vr~ -0.3 ( 0 ~ 410 -- d/gT2 = 0.026 -- H/d = 0.1317 ) 76 o d o \ ~r\ \ \ \ \ \ \ M + \ \ \ \ + Measured Force Values Cubic Spline Smoothing Theoret ical Pred ic t ion 'V w i \\ - f \ \ \ \ \ \ \ \ \ \ v x \ \ \ \ \ V \ + ^ \ \ \ \ -t o-i \ \ \ + v \ \ \ \ \ \ \ \ \ \ \ \ \ \ to o -1 \ \ \ \ \ \ - H \ \ \ \ \ \ \ +\ \+ \ \ \ \ \ \ \ \ CO o -1 + \ + \ + \ \ \ \ \ \ \ \ \ \ \ i \ \ \ \ \ o 1 ! 1 1 r — 1 ! r-— 1 1 ' — I ' !-^ 1 1— 0.0 0.4 0.8 1.2 1.6 2.0 F 0.5pDU 2 Figure 22: I n - l i n e Force Dis t r ibu t ion for V r ~ -0 .5 ( 8 ~ 410 - - d/gT2 = 0.026 - - H/d = 0.1349 ) 1 1 1 1 1 1 1 1 1 1 i ' 1 1 1 ' 1— 0.0 0.4 0.8 1.2 1.6 2.0 •F 0.5pDU* Figure 23: I n - l i n e Force Dis t r ibu t ion for V r ~ -0.7 ( /S ~ 410 - - d/gT2 = 0.028 - - H/d = 0.1424 ) — Experimental Measurement — Theoretical Prediction — — i — i — i — i — i — i — i i i i i i i i i i i -1.2 -0.8 -0.4 0.0 0.4 0 8 1.2 V um Figure 24: Variation of Peak Pile Force with Current Strength for /S ~ 1000 ( d/gT2 = 0.080 — H/d = 0.10 — z/d = 0.50 ) F 05pDU2 f — Experimental Measurement / r / - - Theoretical Prediction / y / / / / / / / / / / / / / s / / + / s / s "1 1 1 ' ' 1 ' 1 ' 1 1 ' ' 1 1 I 1 1 1 1 I 1 1 ' 1 I 1 1 1 ' I -1.2 -0.8 -0.4 0.0 0 4 0 8 1 2 V Figure 25: Variation of Peak Pile Force with Current Strength for 6 ~ 410 ( d/gT2 = 0.025 - - H/d = 0.13 - - z/d = 0.50 ) CD 

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