Advanced Series on Ocean Engineering — Volume 21
WAVES AND WAVE FORCES ON COASTAL AND OCEAN STRUCTURES
Robert T. Hudspeth
World Scientific
WAVES AND WAVE FORCES ON COASTAL AND OCEAN STRUCTURES
ADVANCED SERIES ON OCEAN ENGINEERING Series Editor-in-Chief Philip L- F Liu (Cornell University)
Vol. 9 Offshore Structure Modeling by Subrata K. Chakrabarti (Chicago Bridge & Iron Technical Services Co., USA) Vol. 10 Water Waves Generated by Underwater Explosion by Bernard Le Mehaute and Shen Wang (Univ. Miami) Vol. 11 Ocean Surface Waves; Their Physics and Prediction by Stanislaw R Massel (Australian Inst, of Marine Sci) Vol. 12 Hydrodynamics Around Cylindrical Structures by B Mutlu Sumer and Jorgen Fredsee (Tech. Univ. of Denmark) Vol. 13 Water Wave Propagation Over Uneven Bottoms Part I — Linear Wave Propagation by Maarten W Dingemans (Delft Hydraulics) Part II — Non-linear Wave Propagation by Maarten W Dingemans (Delft Hydraulics) Vol. 14 Coastal Stabilization by Richard Silvester and John R C Hsu (The Univ. of Western Australia) Vol. 15 Random Seas and Design of Maritime Structures (2nd Edition) by Yoshimi Goda (Yokohama National University) Vol. 16 Introduction to Coastal Engineering and Management by J William Kamphuis (Queen's Univ.) Vol. 17 The Mechanics of Scour in the Marine Environment by B Mutlu Sumer and Jsrgen Fredsoe (Tech. Univ. of Denmark) Vol. 18 Beach Nourishment: Theory and Practice by Robert G. Dean (Univ. Florida) Vol. 19 Saving America's Beaches: The Causes of and Solutions to Beach Erosion by Scott L Douglass (Univ. South Alabama) Vol. 20 The Theory and Practice of Hydrodynamics and Vibration by Subrata K. Chakrabarti (Offshore Structure Analysis, Inc., Illinois, USA) Vol. 21 Waves and Wave Forces on Coastal and Ocean Structures by Robert T. Hudspeth (Oregon State Univ., USA) Vol. 22 The Dynamics of Marine Craft: Maneuvering and Seakeeping by Edward M. Lewandowski (Computer Sciences Corporation, USA) Vol. 23 Theory and Applications of Ocean Surface Waves Part 1: Linear Aspects Part 2: Nonlinear Aspects by Chiang C. Mei (Massachusetts Inst, of Technology, USA), Michael Stiassnie (Technion-lsrael Inst, of Technology, Israel) and Dick K. P. Yue (Massachusetts Inst, of Technology, USA) Vol. 24 Introduction to Nearshore Hydrodynamics by lb A. Svendsen (Univ. of Delaware, USA)
Advanced Series on Ocean Engineering — Volume 21
WAVES AND WAVE FORCES ON COASTAL AND OCEAN STRUCTURES
Robert T. Hudspeth Oregon State University, USA
\jjp World Scientific N E W JERSEY
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Contents
Preface
xv
1
Introduction
1
2
Mathematical Preliminaries
5
2.1 2.2
Introduction Symbols, Functions and Linear Operators 2.2.1 Landau Order Symbols 0(e) and o(e) (Nayfeh, 1973, Chapter 1.3 and Olver, 1990, Chapter 12.1.1) 2.2.2 Heaviside Step Function U(x — £) 2.2.3 Kronecker Delta 8mn Function and Dirac Delta 5(JC — £) Distribution 2.2.4 Levi-Civita Symbol eijk (Arfken, 1985) 2.2.5 Gamma Functions r » (Andrews, 1985) 2.2.6 Error Functions Erf (•) and Erfc(«) (Barcilon, 1990, p. 351) 2.2.7 Gradient Vector Operator V(») 2.2.8 Curl Vector Operator m = V x (•) 2.2.9 Laplacian Operator V2(«) = A (•) 2.2.10 Stokes Material Derivative Operator D(»)/Dt 2.2.11 Leibnitz's Rule for Differentiation of Integrals with Parameters (Hildebrand, 1976, Chapter 7.9) 2.2.12 Signum (sign ±) Function
5 5
vu
5 7 7 7 8 9 10 10 11 11 13 13
Contents
Vlll
2.3
2.4
2.5
2.6
Properties of Series 2.3.1 Power Series (Hildebrand, 1976, Chapter 4.1) 2.3.2 Function Series 2.3.3 Maclauren and Taylor Series (Hildebrand, 1976, Chapters 4.1 and 7.5) 2.3.4 Binomial Expansion (Wylie and Barrett, 1982, p.938) Elementary and Special Functions (Hildebrand, 1976, Chapter 10.2) 2.4.1 Trigonometric and Hyperbolic Identities 2.4.2 Euler's Constant yE (Barcilon, 1990, p. 346) 2.4.3 Bessel Functions (Hildebrand, 1976, Chapters 4.8 to 4.10) 2.4.4 Orthogonal Polynomials Linear Ordinary Differential Equations (Hildebrand, 1976, Chapters 1.1 to 1.11) and Operational Calculus (Friedman, 1956) 2.5.1 Initial and Boundary Data (Stakgold, 1979) 2.5.2 First Order Linear Ordinary Differential Equations (Hildebrand, 1976, Chapter 1.4) 2.5.3 Variation of Parameters and the Duhamel Convolution Integral (Hildebrand, 1976, Chapter 1.9) 2.5.4 Properties of Linear Differential Operators £(•) (Hildebrand, 1976, Chapter 1.7) 2.5.5 Method of Frobenius (Hildebrand, 1976, Chapter 4.4) 2.5.6 Method of Undetermined Coefficients (Hildebrand, 1976, Chapter 1.5) Sturm-Liouville Systems (Morse and Feshbach, 1953, Chapter 6.3; Hildebrand, 1976, Chapter 5.6; Oates, 1990, Chapter 3.6.5 and Benton, 1990, Chapter 6.6)
13 13 14 15 17 18 18 20 21 26
31 32 34 35 39 40 45
48
3
Fundamentals of Fluid Mechanics
53
3.1 3.2
Introduction Conservation of Mass (Continuity Field Equation)
53 55
IX
3.3
3.4 3.5 3.6 3.7
Momentum Principle 3.3.1 Inertial Forces 3.3.2 Surface Stresses 3.3.3 Body Forces 3.3.4 Navier-Stokes Equations (Lamb, 1932) 3.3.5 Euler's Equations (Lamb, 1932) Mechanical Energy Principle Scaling of Equations Dimensional Analyses Problems
57 58 63 69 69 70 70 71 81 84
4
Long-Crested, Linear Wave Theory (LWT)
85
4.1 4.2 4.3
85 87
4.7
Introduction Dimensional Boundary Value Problem (BVP) for LWT Solutions to Dimensional Boundary Value Problem (BVP) for Long-Crested, Linear Wave Theory (LWT) 4.3.1 Wave Celerity C and Computing the Eigenvalue k from the Frequency Dispersion Equation Eulerian Kinematic Fields and Lagrangian Particle Displacements Eulerian Dynamic Fields, Energy and Energy Flux Conservation Principles for Long-Crested Linear Waves Wave Transformations for Long-Crested, Progressive Linear Waves: Shoaling and Refraction Problems
5
Wavemaker Theories
149
5.1 5.2
Introduction Planar Wavemakers in a 2D Channel 5.2.1 Computation of the Eigenvalues Kn by the Newton-Raphson Method 5.2.2 Rate of Decay of Evanescent Eigenmodes: n > 2 5.2.3 Orthogonality of Orthonormal Eigenfunctions V„(K„,z/h)
149 154
4.4 4.5 4.6
97 105 113 121 140 144
170 176 111
Contents
X
5.8
5.2.4 Evaluation of Coefficients Cn by Wavemaker Vertical Displacement xiz/h) 5.2.5 Determination of Wave Amplitude from Wavemaker Motion 5.2.6 Hydrodynamic Pressure Force and Moment (Added Mass and Radiation Damping) Circular Wavemakers 5.3.1 Determination of Wave Amplitude from Wavemaker Motion 5.3.2 Hydrodynamic Pressure Force and Moment (Added Mass and Radiation Damping) Double-Actuated Wavemaker Directional Wavemaker Sloshing Waves in a 2D Wave Channel Conformal and Domain Mapping of WMBVP 5.7.1 Conformal Mapping 5.7.2 Domain Mapping Problems
6
Nonlinear Wave Theories
309
6.1 6.2
Introduction Classical Stokes: The Method of Successive Approximations Traditional Stokes: Lindstedt-Poincare 4th Order Perturbation Solution 6.3.1 Traditional Stokes: Stokes Drift Method of Multiple Scales (MMS) Stream Function Solutions Breaking Progressive Waves Second-Order Nonlinear Planar Wavemaker Theory Chaotic Cross Waves: Generalized Melnikov Method (GMM) and Liapunov Exponents Problems
309
5.3
5.4 5.5 5.6 5.7
6.3
6.4 6.5 6.6 6.7 6.8 6.9
180 185 189 201 219 231 242 251 268 279 280 301 306
310 316 361 371 394 400 405 423 492
XI
7
Deterministic Dynamics of Small Solid Bodies
7.1 7.2 7.3
495
Introduction Small Body Hypothesis (Morison Equation) Drag dFd and Inertia dFm Forces 7.3.1 Inertia Forces 7.4 Comparison Between a Fixed Cylinder in Accelerating Flow and an Accelerating Cylinder in Still Fluid 7.4.1 Accelerating Cylinder in Still Fluid 7.4.2 Fixed Cylinder in an Accelerating Flow 7.5 Maximum Static-Equivalent Force/Moment (Fixed-Free Beam) 7.6 Parametric Dependency of Force Coefficients Cm and Cd 7.6.1 Relative Importance of dFm (z,, t) and dFd (n, t) 7.6.2 Computing the Force Coefficients Cm and Cd 7.6.3 Methods of Analyses 7.6.4 Linearized Drag Force 7.6.5 Laboratory U-Tube Data 7.6.6 Ocean Wave Data 7.7 The Dean Eccentricity Parameter and Data Condition 7.7.1 Dean Error Ellipse and Eccentricity Parameter E (Geometric) 7.7.2 Amplitude/Phase Method (Geometric) 7.7.3 Matrix Condition Numbers (Numerical) 7.8 Modified Wave Force Equation (WFE, Relative Motion Morison Equation) 7.8.1 Articulated Circular Cylindrical Tower: SDOF System 7.8.2 Two Semi-Immersed Horizontal Circular Cylinders: MDOF System 7.9 Transverse Forces on Bluff Solid Bodies 7.10 Stability of Marine Pipelines 7.11 Problems
495 498 505 508
8
Deterministic Dynamics of Large Solid Bodies
619
8.1
Dynamic Response of Large Bodies: An Overview
619
509 510 510 515 520 520 522 5 24 530 532 535 538 541 548 555 559 563 573 599 605 615
Contents
Xll
8.2
636 638 643 662
8.7 8.8
Linearized MDOF Large Solid Body Dynamics 8.2.1 Kinematic Body Boundary Conditions (KBBC) 8.2.2 Dynamic Body Boundary Conditions (DBBC) Froude-Kriloff Approximations for Potential Theory 8.3.1 Froude-Kriloff Load in 2-D Cartesian Coordinates 8.3.2 Froude-Kriloff Load in Circular Cylindrical Coordinates Diffraction by a Full-Draft Vertical Circular Cylinder Reciprocity Relationships Green's Functions and Fredholm Integral Equations 8.6.1 Orthonormal Eigenfunction Expansion of Green's Function for 2D Wavemaker Wave Loads Computed by the FEM Problems
9
Real Ocean Waves
719
9.1 9.2 9.3
Introduction Fourier Analyses Ocean Wave Spectra 9.3.1 Generic Four-Parameter Wave Density Spectrum 9.3.2 Wave and Spectral Parameters Computed from Spectral Moments mn 9.3.3 Multi-Parameter Theoretical Spectra 9.3.4 Spectral Directional Spreading Functions 9.3.5 Confidence Intervals for FFT Estimates Probability Functions for Random Waves 9.4.1 Gaussian (Normal) Probability Distribution 9.4.2 Rayleigh Probability Distribution 9.4.3 Distribution of the Maxima Wave Groups 9.5.1 Resolving Incident and Reflected Random Wave Time Series
719 720 730 740
8.3
8.4 8.5 8.6
9.4
9.5
663 667 672 682 688 699 706 716
1M 752 754 759 760 765 772 780 790 803
Contents
9.6
Xlll
Random Wave Simulations 9.6.1 Conditional Wave Simulations 9.7 Data Analyses: An Example from Hurricane CARLA 9.8 Random Wave Forces on Small Circular Members 9.8.1 Probability Density Function p(Y) (pdf) and Covariance Function CfTfT(t) for Nondeterministic Wave Force per Unit Length for a Small Vertical Circular Pile 9.8.2 Stochastic Response of Space-Frame Offshore Structure 9.9 Frequency Domain Input-Output Transfer Functions 9.10 Problems
808 813 823 831
Bibliography
867
Author Index
909
Subject Index
917
832 843 854 864
Preface
The modest number of topics on waves and wave forces that is reviewed comes primarily from three sources: (1) from four one-quarter in length (10 weeks) graduate courses taught at Oregon State University (OSU) in the USA; viz. (i) linear waves, (ii) nonlinear waves, (iii) random real ocean waves and (iv) wave forces on structures; (2) from Masters and PhD theses written for the Departments of Civil, Construction and Environmental Engineering and of Mathematics at OSU; and (3) from manuscripts co-authored by the author. Because of these limited resources for the topics reviewed, the topics are intended primarily as a reference for graduate classes on waves and wave loads; although it is hoped that some practicing coastal/ocean engineers may also find these topics of some value. The emphases are (1) on the fundamental physics of the dynamics of fluids and of semi-immersed Lagrangian solid bodies that are responding to wave induced loads; (2) on the scaling of dimensional equations and boundary value problems in order to determine a small dimensionless parameter, e say, that may be evaluated to linearize the equations and the boundary value problems in order to obtain a linear system; (3) on the replacing of differential and integral calculus equations with algebraic equations that require only algebraic substitutions instead of differentiations and integrations; and (4) on the importance of comparing analytical and numerical computations with data from laboratories and/or nature. More extensive treatments of the topics reviewed here are given from the point of view of naval architecture by Wehausen and Laitone, Surface Waves in Handbook of Physics, Fluid Dynamics III (1960) and from the point of view of coastal and ocean engineering by Mei in The Applied Dynamics of Ocean Surface Waves (1989).
XV
XVI
Preface
Chapter 1 introduces the topic of wave forces on coastal and ocean structures and attempts to motivate the reasons for studying the theories that are reviewed. Chapter 2 reviews the mathematical methods employed in the remaining chapters; and is motivated by the format used by Morse and Ingard, Theoretical Acoustics (1968) Chapter 1.2. The pedagogy used by the author in the four graduate courses cited above is to formulate problems on waves and wave loads on coastal and ocean structures in generic mathematical nomenclature; and to then search the mathematical literature for the solutions to the generic mathematical problem formulated rather than associating a generic mathematical solution method to a particular wave or wave load problem on a specific topic or in a specific chapter. By this pedagogy, it is hoped that readers will look for generic mathematical solutions to wave and wave load problems in the mathematical literature where the solution methods are compactly archived by specific mathematical nomenclature. This will expedite solutions and make all problems on waves and wave loads generic to their mathematical solutions. Chapter 3 reviews the fundamental laws for the conservation of mass, momentum and energy for an Eulerian fluid field by the differential element method in Cartesian coordinates. The derivations are limited to the differential element method in Cartesian coordinates; but the corresponding results from arbitrary integral control volumes and from tensor analyses are also included for completeness but without rigorous derivations. The similarities of non-uniqueness between the scaling of equations and dimensional analysis are included in order to stress the importance of analyses by both of these methods for numerical and analytical computations and by the subsequent essential comparison between numerical and analytical results with data from laboratories and/or nature. Chapter 4 begins the surface gravity wave analyses by reviewing longcrested linear wave theory (LWT) applying real-valued elementary transcendental functions. Differential control volumes in Cartesian coordinates are employed for deriving the equations for the boundary value problem for longcrested linear waves and the subsequent solutions for the conservation of mass, momentum and energy flux for both progressive and standing linear waves. Algorithms for computing the propagating eigenvalue (wave number) for the well-posed Sturm-Liouville boundary value problem that is formulated in Chapter 2 are reviewed.
Preface
xvii
Chapter 5 reviews a variety of 2D and 3D wavemaker theories applying complex-valued elementary transcendental functions. The emphasis of the wavemaker theories reviewed is the application of the wavemaker theory to the radiation potential for computing forces and moments on large semi-immersed Lagrangian solid bodies that are reviewed in Chapter 8. There is a deliberate and repeated effort to connect this review to the radiation potential for large body forces and moments in Chapter 8. In addition, integral equations for the coefficients of the eigenfunctions, for the forces and moments on wavemakers and for computing wavemaker power are replaced by algebraic equations that only require algebraic substitutions and eliminate all integral computations. Two-dimensional planar and double-articulated wavemakers are analyzed; and algorithms for computing the evanescent (or "local" wave components, John, 1950, p. 48) eigenvalues are reviewed. Both amplitude-modulated (AM) and phase-modulated (PM) circular wavemakers are reviewed. Sloshing waves in 2D channels and directional wavemakers in 3D basins are reviewed. Finally, the mapping of 2D wavemakers by both conformal mappings and by domain mapping are reviewed. Chapter 6 reviews several nonlinear wave theories. Stokes 2D nonlinear waves are reviewed first by the classical (but extremely tedious) Stokes method of successive approximations and, subsequently, by the traditional Stokes-Lindstedt-Poincare perturbation method. In the traditional StokesLindstedt-Poincare perturbation method there are two methods that may be applied to suppress resonant forcing. One method is to expand the wave celerity C in a perturbation expansion according to C = ^2,m=0 em (m + \)C. Another option is to expand the radian wave frequency co in a perturbation expansion according to co = Yln=o e"ftJn- Both of these methods are reviewed. Specifically, the first method of expanding the wave celerity C is applied in the traditional Stokes perturbation expansion in Chapter 6.3; and the second method of expanding the radian wave frequency co is applied in the nonlinear planar wavemaker theory in Chapter 6.7. In particular, the review of the Stokes-Lindstedt-Poincare traditional perturbation method introduces the method of replacing differential calculus with algebraic substitutions. The numerous repeated term by term differentiations required in the boundary conditions (as many as 75 differentiations at fourth order!) are replaced by six relatively simple algebraic equations that many times are identically
XV111
Preface
equal to zero! The more modern methods of multiple scales (MMS) and the numerical stream function theory are reviewed briefly. Wave breaking is also reviewed briefly. An extension of the 2D wavemaker theory to weakly nonlinear waves and to Stokes drift in 2D wave channels is included. An analysis of weakly damped cross-waves with surface tension is analyzed by the generalized Melnikov method for nonlinear nonautonomous Hamiltonian systems with contact (canonical) transformations computed by an extension of the Herglotz algorithm to nonautonomous transformations. Chapter 7 begins the formal analyses of deterministic wave loads on structures by reviewing the dynamics of small Lagrangian solid bodies responding to loads computed from the Morison equation and from the Modified Wave Force equation (relative motion Morison equation). The parametric dependencies of the inertial Cm and drag Cd force coefficients in the Morison equation are reviewed and are connected to the scaling and dimensional analyses methods reviewed in Chapter 3. The linearized wave force equation (relative motion Morison equation) is applied to analyze both a single-degree-of-freedom (SDOF) articulated tower and a semi-immersed, three-degree-of-freedom (MDOF) double pontoon system. The linearization of the relative motion wave loads in these two examples makes it possible to separate the wave loads into terms proportional only to the body motions and to terms proportional only to loads on fixed bodies. Because of this linear decomposition of the wave loads, deterministic wave loads on fixed small bodies need not be treated as a separate topic in this review because the wave loads on fixed bodies are identical to the exciting wave loads on dynamically responding small bodies to the first linear approximation! Finally, the chapter concludes by reviewing the transverse lift forces on small bodies and the stability of bottom laid marine pipelines through the surf zone. Chapter 8 reviews the deterministic dynamics of large Lagrangian solid bodies responding to linear wave loads computed by potential wave theory and the linear progressive wave potentials derived in Chapter 4. Similar to the analyses in Chapter 7, the linear decomposition of the wave loads on large bodies reduces to uncoupled boundary value problems for (1) a scattered wave potential that computes the exciting wave loads on a fixed large Lagrangian body and to (2) a radiated wave potential that computes the restoring or wavemaker loads on an oscillating (wavemaker) body in otherwise still water. This linear decomposition in two separate and uncoupled boundary value problems is the direct consequence of the equalitarian treatment of
Preface
xix
the kinematic and dynamic boundary conditions between the Eulerian fluid field and the large Lagrangian solid body. Because the independent variables of the Eulerian fluid field (viz. space and time) and of the Lagrangian solid body (viz. particle and time) are not the same, the two boundary conditions on the boundary between the Eulerian fluid field and on the Lagrangian solid body must be treated with care. Specifically, the kinematic boundary condition converts the Lagrangian particle velocities of the semi-immersed solid body to Eulerian field velocities by the vector dot product between the time-dependent Lagrangian body velocities and the spatially-dependent unit normal to the Lagrangian solid body. Then the spatial dependencies of the dynamic Eulerian pressure field are integrated around the submerged portions of the semi-immersed Lagrangian solid body so that the wave loads are transformed to only time-dependent Lagrangian variables in the Lagrangian dynamic boundary condition. The boundary value problem for the radiated wave potential is connected repeatedly to the wavemaker theories in Chapter 5. Froude-Kriloff approximations for wave loads in both Cartesian and circular cylindrical coordinates are reviewed as well as the MacCamy-Fuchs diffraction theory for a full draft vertical cylinder. Reviews of the reciprocity relations and the Green's functions as resolvent kernels in Fredholm integral equations are reviewed. Wave loads computed by the Finite Element Method (FEM) complete the chapter on wave loads on large Lagrangian solid bodies. Chapter 9 reviews the non-deterministic wave theories and wave loads. Fourier analyses of stochastic processes are reviewed and an application of the finite Fourier transform (FFT) to measured wave data from Hurricane CARLA is given. Generic 4 parameter and multi-parameter wave spectra are reviewed. Gaussian and Rayleigh probability theories are applied to the time series of wave profiles and wave heights, respectively. Wave groups are analyzed by the Hilbert transform and then applied to a stability analysis of coastal rubble-mound breakwater structures with comparisons to laboratory data. This analysis requires an algorithm that is capable of resolving incident and reflected times series (with wave phases as well as wave amplitudes) in order to apply the Hilbert transform and to compute the groupiness of the incident time series. Algorithms for random wave simulations by digital computers for both deterministic spectral amplitude (DSA) and nonderministic spectral amplitude (NSA) random wave simulations are reviewed along with conditional random wave simulations. Random wave forces on a prototype space-frame structure in the Gulf of Mexico are computed by the linearized
XX
Preface
relative motion Morison equation from Chapter 7 applying nondeterministic random waves and are compared with data from Hurricane CARLA on the same space-frame structure. The structural model also includes a dynamic soilspring response algorithm. Finally, the frequency domain transfer functions for Eulerian field variables and for Lagrangian body motions are reviewed. A Bibliography that contains a relatively large number of references that are not cited explicitly in the main chapters concludes this review. I have benefitted enormously over the past 30 years from my harmonious collaborations with my colleagues of the College of Engineering, the Department of Mathematics and the College of Oceanic and Atmospheric Sciences at OSU in the USA; from colleagues in laboratories and institutions in Indonesia, Japan, Korea, Poland, Spain and Taiwan; as well as graduate students in both the Department of Civil, Construction and Environmental Engineering and the Department of Mathematics at OSU. In addition, I must also acknowledge my mentors from many academic and governmental institutions that have educated me. Although I feel obligated to cite each of them individually for their contributions to this review; it would require another review of equal length; but I do wish to acknowledge and to express my sincere appreciation individually to each of them for their contributions. However, it is virtually impossible to eliminate all of the theoretical and typographical errors in a review with the number of equations and theories that are reviewed here; and I, alone, bear all of the responsibility for these errors. I have, however, made repeated efforts to proof both the theories and equations; and I apologize here in advance for those errors that have survived all of my scrutiny. In addition, some of the materials that are reviewed such as replacing differential and integral calculus with algebraic substitutions in Chapters 5 and 6 and the applications of the generalized Melnikov method as well as the generalized Herglotz algorithm to analyze chaotic cross-waves in Chapter 6 are relatively new ideas (at least to the author) that are among many others in this review that almost surely require further elaborations and/or clarifications. I solicit these elaborations, clarifications and corrections and I welcome all comments, criticisms, suggestions and corrections. The coastal and ocean waters that cover our planet are wonderful areas to enjoy both for recreation and to benefit from their productivity. May some of the readers of my modest efforts here benefit from these efforts and help all of us to enjoy the beauty and the productivity of our coastal and ocean waters.
Chapter 1 Introduction
Ocean and coastal engineering is the application of science in an ocean and coastal environment in order to meet the needs of the people. Although one of the important needs of people is the economic feasibility and the environmental impact of ocean and coastal projects, this particular aspect of ocean and coastal engineering will not be addressed in this review. This does not reflect an attitude that is intended to diminish the importance of these two very important aspects of ocean and coastal engineering, but rather a lack of expertise in reviewing these two important topics. This limited review will, instead, focus on the physical and mathematical aspects of ocean and coastal engineering. One scenario for the engineering design of an ocean or coastal project is given in Fig. 1.1. Each element of the design begins with a statement of the principles of physics that govern that particular element. These principles of physics are then translated into mathematical problems that must then be solved Site analysis and synthesis
Facility analysis and synthesis
Site and facility evaluation
Loads analysis
Operational requirements Mission requirements
Structural concepts and dynamics Site requirements
Listing of specifications
Candidate preliminary
Foundation and mooring concepts
Evaluation of design to stated design objectives
Evaluation of design to stated constraints
Installation procedures
Fig. 1.1. Scenario for an ocean or coastal engineering project. 1
Final Design
2
Waves and Wave Forces on Coastal and Ocean Structures
either analytically or numerically. If a solution is obtained, it then remains to interpret these results physically. For this reason, the pedagogy in this review is to begin each problem analyzed by a statement of the principles of physics that govern the problem being analyzed. Only after the principles of physics have been identified will the mathematical formulation and solution technique be addressed. The intent is to identify the mathematical formulation and solution technique in a generic manner. This approach is intended to encourage a pedagogy of classifying the mathematical models and solution techniques required to solve engineering problems in a generic manner. This sequence emphasizes the principles of physics first and the mathematical details second. In addition, by attempting to classify the mathematical models generically rather than in a problem-specific manner, a problem solution technique (PST) may be illustrated that may make it more comfortable for engineers to use applied mathematics that have, perhaps, been developed or refined in fields other than that of their own training and education. This pedagogy will hopefully make it possible for engineers to be more comfortable with applied mathematics from a wide variety of different technical fields. In an effort to illustrate the value of this approach, a rather brief selection of the limited mathematics applied in this review is assembled in Chapter 2 in a generic manner. This illustrates the approach by referring to this collection repeatedly throughout the review when the mathematical models are required, rather than to a specific problem where a particular mathematical model or solution technique is first encountered. It will also have the advantage of not imbedding in isolated sections important mathematical operations or functions that are applied repeatedly; and then forcing the reader to search out at some later time the desired mathematics by referring to another engineering topic instead of to the appropriate generic mathematical model. This approach will hopefully avoid giving the false impression that the mathematics are problem-specific; an impression that is not at all related to the way that mathematical models are employed in real engineering applications. This particular pedagogy may facilitate the ability of engineers to apply mathematical techniques that may have been developed or refined in technical fields other than their own. An obvious impediment to engineers extracting mathematical techniques from other technical fields is that engineers have been trained to approach problems from a basic understanding of the governing principles of physics. Grazing through journals or references in other technical
Introduction
3
fields looking for mathematical techniques that may be applicable to an engineering problem at hand may, at first, be uncomfortable because of a lack of understanding of the principles of physics applied in the foreign technical field. However, because the mathematical techniques are not related uniquely to any unique principle of physics, but are rather more generic; it then makes sense to look for generic mathematical techniques in these foreign technical fields even though the principles of physics in that particular foreign technical field may not be well-understood. The first step required to compute the magnitudes of the wave-induced loads on a structure is to estimate the effect of the structure on the wave field. The two extreme examples of this effect of wave-structure interaction are described by two separate theories: 1) small body or Froude-Kriloff theory; and 2) large body or diffraction theory. The distinction between wave loads on large and small structures depends on both the wave steepness parameter k A and the Stokes parameter S = (kA)/(kh)3, (Stokes, 1849). The distinction between wave loads on large and small structures may be observed physically; viz. 1) what may be observed, and 2) what may not be observed. In the small body or Froude-Kriloff theory, a relatively small structural member is assumed to have no sensible effect on the wave field and waves propagating past the structure remain essentially unmodified by the presence of the structure. Physically, flow separation around the body and the formation of a wake behind the member may be observed due to the effects of the fluid viscosity; while no waves will be observed radiating away from the structure. The Morison equation is applied to compute these loads from a deterministic wave theory in Chapter 7. In the large body theory, a relatively large structural member will exhibit a significant and noticeable effect on the incident wave field; and a scattered (from a fixed body) and a radiated (from a freely floating body) wave may be observed. Potential theory is applied to compute the loads from a deterministic wave theory on large bodies in Chapter 8. Finally, loads on both small and large structures computed from random wave theory are reviewed in Chapter 9.
Chapter 2
Mathematical Preliminaries
2.1. Introduction A number of mathematical operations and functions, as well as their associated properties, are applied repeatedly throughout this review. For this reason, these operations are collected in this chapter and are briefly outlined for the convenience and ease of reference later. This chapter is not intended to serve as a treatise for defining these mathematical operations and functions. For a more thorough treatment of these topics, the references identified with each operation, function or property should be consulted.
2.2. Symbols, Functions and Linear Operators A number of symbols and functions frequently encountered in coastal and ocean engineering are summarized below.
2.2.1. Landau Order Symbols 0{e) and o(() (Nayfeh, 1973, Chapter 1.3 and Olver, 1990, Chapter 12.1.1) It is often necessary to estimate the relative magnitude of terms. For example, approximating a function f(x) by an asymptotic series and then estimating the error associated with truncating the expansion at the nth term in the series. The residual of the nth term in an asymptotic series for a function f(x) as the independent variable x -> c is usually given in terms of the Landau symbols O (e) and o (e) that are commonly referred to as Big O and small o, respectively. 5
6
Waves and Wave Forces on Coastal and Ocean Structures
Order Big 0(») If an asymptotic series for the function / ( x ) is approximated by N
fix) = ^2 Snix), n=0
and if there exists a positive-definite number 0 < A < + 0 0 , say, that is independent of x, then \fix)\
< A\gN(x)\,
and the ratio
fix)
lim
\x\eX• c; or, equivalently, x—>-c
f(x) = 0[gN(x)],
x -• c e X
and f(x) is said to be the "Big Oh" of gNix) as x ->• c. For example, the Taylor series approximation in Sec. 3 for sin(x) for small x is x3 x5 sin(x) = x - —- + — + sin(x) =
0(x7),
0(x).
These are some of the most common applications of the Big 0(») notation.
Order small o(») If the function fix) is asymptotic to gNix), say; and if the ratio
fix) = 0 gNix) x—>c implies that f(x) = o[gNix)] as x -^ c; or, equivalently, lim
/ ( x ) = o[gjv(x)],
x ^ c 6 X,
/ ( x ) is said to be the "small Oh" of gNix) as x -» c. For example, the Taylor series approximation in Sec. 3 for sin(x) and exp(—x) for small x are sin(x) = o(l), exp(—x) = o(l),
x -> 0 x —>• 0.
These are some of the most common applications of the small o(«) notation.
7
Mathematical Preliminaries
2.2.2. Heaviside Step Function U(x — £) The Heaviside step function U(x — £) is a generalized function from operational calculus (Lighthill, 1964) and is only meaningful when applied in integral equations. The function may be applied to formally change limits of integrations by turning functions on and off like a light switch (vide., wavemaker shape function and eigenfunction expansions in Chapter 5 and probability distributions in Chapter 9). The Heaviside step function U(x — i-) is defined as X >
U(x-$)={]>
1
0,
(2.1)
* 0.
(2.6a)
Integration by parts gives T(a + 1) = aV(a),
(2.6b)
and for an integer argument a = n, r(n + 1) = n\,
n is an integer.
Special values of the Gamma function are: T(l) = 1;
r ( l / 2 ) = VJr,
r(0) = oo.
(2.6c)
Mathematical Preliminaries
9
Asymptotic Formula: r(a + l ) ~ V 2 l m ( — — ) , Vexpa/ Stirling s Formula for integer n\:
a -+ +00.
(2.6d)
n! = T{n + 1) ~ V27r« ( — — ) , n -» +00. (2.6e) \expn/ Generalized Incomplete Gamma function T{a,ZQ,Z\): The generalized incomplete Gamma function T(a, zo, zi) is defined as fZl -1 I xa ' e x p ( - x ) d x = r(a,zo) ~ r(a,zi), Jzo where F(a, z) is the incomplete Gamma function. Incomplete Gamma function F(a,z) The incomplete Gamma function T(a, z) is defined as r(a,zo,zi)
-
1 y(a,z)= ta~"lexp(-t)dt, (a,z) = / " V Jo
(2.7)
ta~lexp(-t)dt,
(2.8a)
a > 0.
(2.8b)
2.2.6. Error Functions Erf (•) and Erfc(») (Barcilon, 1990, p. 351) The Error Function Erf (•) is defined by the integral 2 Erf(z) = —
fz / exp(-? 2 )dr.
(2.9a)
V 7T JO
The Complementary Error Function Erfc(«) is defined by 2 f00 Erfc(z) = 1 - Erf (z) = — / exp(-f 2 ) dt. V ^ Jz The £>ror Function Erf (•) may be expanded in the series
Erf (z) = -== E *Jn
(
,r9)ln
„ n!(2« + 1)
(2.9b)
(2.9c)
10
Waves and Wave Forces on Coastal and Ocean
Structures
with an asymptotic series expansion given by Erf (z) =
2exp(-z) 2 ^
E
2nz2n+l (2n + l)!!'
(2.9d)
«=o where the odd-factorial (2n + 1)!! = (1 x 3 x 5 x • • • x 2n + 1). 2.2.7. Gradient Vector Operator V (•) The gradient or nabla (a Greek word for harp) vector operator V(») is a pseudo vector because it does not commute with (•) and it represents geometrically the rate of change of (•) in the direction of the coordinates of the gradient operator. In Cartesian coordinates {x, y, z) with basis vectors {ex,ey,ez}, the gradient vector operator is given by
„, ,
L
3
^3
^ d
(2.10a)
(•),
and in circular cylindrical coordinates {r,9,z} with basis vectors {er, e$, ez) by
™={i'h+t>7h+i'hV-)-
(210b)
2.2.8. Curl Vector Operator w = V x (•) The curl vector operator V x (•) is a pseudo vector because it does not commute with the vector (•) and it represents geometrically a rotation vector that is perpendicular to the plane of rotation. In Cartesian coordinates {x, y, z] with basis vectors {ex,ey,ez}, the curl vector operator is given by
m(mx,
my, TUZ) = V x (•) = curl(») =
&x
£y
&z
d
d
d_
dx
dy
dz
(•)x
(»)y
(«) Z
(2.11a)
where (•), in the determinant represents the /th scalar component of the vector (•). For example, the ey rotation vector component in Cartesian coordinates is perpendicular to the [x, z] plane and has a magnitude of Wy
=
3(.)x
3(.) z
dz
dx
11
Mathematical Preliminaries
In circular cylindrical coordinates {r,9,z} with basis vectors {er,ee,ez}, the curl vector operator is given by ree 3 (2.11b) zn(mr,mQ,mz) = V x (•) = curl(i) = 3r 36> Yz (•)r r{»)e («)z where (•), in the determinant represents the /th scalar component of the vector (•). 1
2.2.9. Laplacian Operator V 2 ( . ) = A(») The scalar operator symbols V2(») = A(») are referred to as Laplacian operators or the Laplacian. In Cartesian coordinates (x,y,z), the scalar operators are written as 3 2 (.) 3 2 (.) 3 2 (.) V (.) = A(.) = _dxV2 + _dyV2 + 3z2 ' and in circular cylindrical coordinates {r, 0, z] as ,
2
1 3 / 3(.) V'(.) = A(.) = - — i r r d r \ dr
+
1 3 2 (.) . 32(«) - 2 30 2 + dz 2 '
2.2.10. Stokes Material Derivative Operator
(2.12a)
(2.12b)
D(»)/Dt
Stokes material surfaces appear in both Lagrangian particle and Eulerian field problems. Stokes material surfaces are written as scalar functions of space and time S{x,y,z,t) and the Stokes material derivative of a Stokes material surface is the total derivative of the function for the surface. In Cartesian coordinates (x,y,z), the Stokes Material Derivative Operator D(»)/Dt for a Stokes material surface S(x, y, z, t) is written as DS(x,y,z,t) —
=
\3 -1 I g7 + q(x'y'z' ^ [ S(x'y'z' ^ 3 3 — + u(x,y,z,t) — at ox 3 3 + v(x, y, z,t)— + w(x, y, z, t) — S(x,y,z,t), dy dz (2.13a)
12
Waves and Wave Forces on Coastal and Ocean Structures
and in circular cylindrical coordinates S(r, 0, z, t) as DS(r,e,z,t) Dt
-+q{r,9,z,t).V\S{r,0,z,t) 3 3 — +ur(r,e,z,t)— +ue(r,9,z,t)dt dr +
uz(r,0,z,t)
dz
19 — r 88 (2.13b)
S(r,6,z,t),
where the velocity vector q(x, y, z, t) = {u{x, y, z, t), v(x, y, z, t), w(x, y, z, t)} &n&q{r,G,z,t) = {ur(r,6,z,t),ue(r,6,z,t),uz{r,6,z,t)}. For an Eulerian field vector V(x,y,z,t) in Cartesian coordinates, the Stokes Material Derivative Operator is
DV(x,y,z,t) Dt
=1 \d
dt
+ q(x, y, z, 0«V [ V(x, y, z, t)
3 3 — + u(x,y,z,t)—- + v(x,y,z,t) at ox + w(x,y,z,t)-r-
\
3 — By
V(x,y,z,t),
(2.13c)
and in circular cylindrical coordinates V(r, 0, z, t) as
DV{r,6,z,t) Dt
— at
+q(r,d,z,t).V }v(r,6,z,
t)
3 3 13 — + ur{r,e,z,t)— + ue{r,d,z,t)— dt dr r d6 +
3 1 uz(r,e,z,t)—\V(r,6,z,t).
(2.13d)
Mathematical Preliminaries
13
2.2.11. Leibnitz's Rule for Differentiation of Integrals Parameters (Hildebrand, 1976, Chapter 7.9)
with
When a function F(x) is defined by an integral where the integrand has a parameter f(x,%) such as *PW rPto
F(x)= 00= /I
f(x,t=)di;,
(2.14a)
Ja(x)
then the derivative of the function F{x) requires the differentiation of the integral on the RHS in Eq. (2.14a) according to dF(x) _ d_ A rPM r^ /(*,!)• oo, then the series converges when p < 1 and diverges when p > 1. If the limit L exists, then the series given by Eq. (2.16) converges when \x — Xo| < L A
p = lim
(n+l)
A-n
n—>-oo
\(x - x 0 ) |
= L\(x
-XQ)\,
A
L = lim n—>-oo
(n+\) An
and the series diverges when \x — ^ol > L~ When the limit L exists and is finite, then there is an interval of convergence for the series that is symmetric about xo given by (xo - L~l, XQ + L _ 1 ). The series converges inside this symmetric interval and diverges outside of it. The distance R = 1/L is called the radius of convergence. However, when on the radius of convergence, the test fails and the series must be tested separately there. 2.3.2. Function
Series
Sometimes the power series defined by Eq. (2.16) may be identified as a sequence that converges to an elementary transcendental or special function. These series are then called function series and may be defined by
fix) = Y^an4>ni*) n=0
(2.17)
Mathematical Preliminaries
15
Each function (j)n{x) may itself be represented by a power series of the form given by Eq. (2.16). The function series given by Eq. (2.17) may be applied to expand the function f(x) in an interval [a,b]. The Weierstrass M test is applied to test for uniform convergence in the interval [a, b]. 2.3.2.1. Weierstrass M Test oo
The Weierstrass M test states that the function series 2_] 4>n OO defined in the n=0
interval [a,b] converges uniformly in x if there exist a sequence of positive 00
numbers Mk such that | ^ ( J C ) | < Mk for all k and V ] Mfc converges. it=o This is an important test for the convergence of the characteristic or eigenfunction expansions (vide., Sec. 2.6). 2.3.3. Maclauren and Taylor Series (Hildebrand, 1976, Chapters 4.1 and 7.5) Taylor series expansions may be applied repeatedly in two important ways: • to derive governing differential equations in one of the 13 separable coordinate systems applying infinitesimal differential elements (Morse and Feshbach, 1953) • to find the roots of nonlinear algebraic equations applying Newton's iterative method (Hildebrand, 1976, Sec. 7.10) InEq. (2.16), define _ dnf(x
= x0)/dxn
_
f(-n)(x=x0)
A function of one or more variables may have the following power series expansion about an arbitrary point x = XQ: oo
/(*>=£
fM(x=x0)i
,~
(*-*o)"
(2-18)
«=0
if the derivatives f^n\x=xo) exist at X=XQ. When xo = 0, the series expansion in Eq. (2.18) is called a Maclauren series.
Waves and Wave Forces on Coastal and Ocean Structures
16
In the derivation of governing differential equations by infinitesimal elements, it is often convenient to expand the dependent variable f{x) about the center of the infinitesimal element centered at XQ. This convention implies that the differential element shrinks to zero uniformly in all dimensions (i.e. |JC — JCOI = Ax —> 0).
If the values for f(x) are known at some point xo for all points |JC| < |xo|, an estimate for the value of/(x) at a differential distance |x — xo| = Ax away from xo may be obtained from Eq. (2.18) from the leading term (n = 0) as fix) = /(xo + Ax) ~ /(x 0 )
(2.19)
that states simply that for afirstapproximation of / (x) the function is a straight line with zero slope (i.e. df/dx = 0 at xo for n > 1 in Eq. (2.18)). However, because all of the previous values of the function / ( x ) are known for all points up to and including xo, the first approximation whereby the function f{x) is a straight line with zero slope may be improved by using the rate of change (or slope) of the function with respect to x evaluated at xo (not at x). This previous knowledge of the history of the behavior of the function fix) may now be exploited in order to improve the previous estimate of the value of fix) at the point x by applying a straight line with a non-zero slope. The magnitude of the slope df/dx at xo, however, must be estimated from data at point xo and not at point x. An improved estimate for the function fix) based on prior knowledge of its behavior for |x| < |xo| may be obtained from fix) = /(xo + Ax) = /(xo) ± dfixQ)
= /(xo) ± (d~P\
dx.
(2.20a)
(2.20b)
\dxJx=xQ Retaining only the linear terms in = 0 and 1) in the expansion, the Taylor series becomes fix) = /(xo) ± t3/-) dx
\ /x=x0
(x - xo) + Oix - x0)2.
(2.21)
17
Mathematical Preliminaries
f(x)
XQ X 0 +AX
Fig. 2.2. Linear correction in a Taylor series.
The sign (±) of the differential correction to the first approximation that the function f(x) is a constant depends on the signum (x — xo) in Eq. (2.15). Specifically, i f x ( ^ ) , then (x - * O ) ( < Q ) a n dthe sign (±) of the differential correction (x — xo) to the constant linear approximation f(x) is (^ 0 ). This linear approximation for the function f(x) may be applied repeatedly throughout the derivations for differential elements. The extension of Eq. (2.21) to the functions of several independent variables is simply fix + xo, y + yo, •••) = f(*o, yo, • • •) ±
3A dxjx
(x - xo)
xo y = yo
'3/ dy Jx = xo y = yo
(2.22)
2.3.4. Binomial Expansion (Wylie and Barrett, 1982, p. 938) The binomial expansion for p real and when |e | < 1 on the unit disk may be written as
2!
=' + £ n=l
3!
,(,-l)(,,-2)...0,-
+
l) t . |
«! (2.23a)
18
Waves and Wave Forces on Coastal and Ocean Structures
or, equivalently, P!
(l+e)p= Y
n= 0 ^
e"
'
where the binomial coefficient is defined by Pl n\(p — ri)\
p\ KnJ
that is valid for all real p in the unit disk |e| < 1. If |e| > 1, then the series is valid only if p is a nonnegative integer. This is a very useful expansion for approximating roots, such as (1 _ e ) l / 2 ^ 1 - | + 0(e 2 ),
|C|
( 2 4 4 )
dx
and Eq. (2.43) is to be interpreted as \ciJp(x) + c2J-Py(x), Zp = \ [ciJn(x) + c2Yn(x),
p£ integer > 0
(2.45a, b)
n = integer > 0,
where Jp(x) is the Bessel function of the first kind of order p defined by J±P(x) = J2 Jn(x) = J2
/uli
\,
' I ' \
'
P # integer > 0,
(2.46a)
,
p = integer > 0,
(2.46b)
where Yp(x) is the Bessel function of the second kind of order p defined by (COS/?7T)./p(x) -
Yp(x) =
r sin/?7r
J-p{x)
— , P ^ integer > 0,
(2.47)
22
Waves and Wave Forces on Coastal and Ocean Structures
(log- +yEJ Jn(x) (n-ifc-l)!(jc/2)< 2 *- ,, > - k=0 E k\ 00 2fc+n r l 1 i r ( _ nl W r 1 _Lx^k+n 1 (JC/2) 2 2^\ ) Zwn+1 "+" 2-,m=\ m m + k=0 L J jfe!(ifc + /i)! j""
Y„(x) = TC
1
n = integer > 0,
(2.48)
where Euler's constant YE is defined in Eq. (2.4le). Bessel functions of the first and second kind may be linearly combined to form complex-valued functions (cf. Hildebrand, 1976, Chapter 4) known as Bessel functions of the third kind or Hankel functions of the first and second kinds, respectively, defined by H{pl\x) = Jpix) + iYpix),
(2.49a)
Hp2\x)
(2.49b)
=
Jp{x)-iYp{x)
If *Jd/s is imaginary and b = r = 0, a Eq. (2.42), then od y dy -,
s = 1, c — —p2, d = — 1 in ,
* T 4 + * T - - ( * +p )y = o, dxl dx and Eq. (2.43) is to be interpreted as I c\Ip(x) + c2I-p(x), ZP = I c\In(x) + C2Kn(x),
p ^ 0 or integer > 0 n — integer.
(2.50)
(2.51a)
If the argument of the Bessel function Jp(t) is pure imaginary (t = ix, cf. Hildebrand, 1976, Sec. 4), then the modified Bessel function of the first kind of order p is to be defined by 00
k=0
(t/2)2k+p
(2.52)
k\(k + p)\
If p is not equal to zero or a positive integer, then the general solution to the modified Bessel equation (2.50) is given by y = Zp(ix) = c\Ip(x) +
c2I-p(x).
(2.53)
23
Mathematical Preliminaries
When p is equal to zero or a positive integer, a second solution known as the modified Bessel function of the second kind of order n (or Kelvin function) may be defined by Kn(x) = ( | ) in+l[Jn(ix)
+ iYn(ix)] = ( | ) in+lH^(ix)
(2.54)
that may be applied to obtain a general solution to Eq. (2.50) for n > 0, given by y = Zn(ix) = c\In{x) + c2Kn(x).
(2.55a)
If p is not equal to zero or to a positive integer, then
=
„ j /.,w-/,W \ 2/
sin /?7T
2.4.3.1. Generating Functions and Identities (Wylie and Barrett, 1982, p. 594) Define: f = exp i (0 + — J = exp i (—) exp iO = i exp i0,
(2.56)
then the Jacobi-Anger expansion for the generating function for the Bessel function of the first kind of order m Jm (z) is given by oo ,{(z/2)(?-(l/J))}
= J2 $mjrn(z)\ ^ # 0 ra=—oo
=
^
im expim9Jm(z)
m = — oo
= ^
(2 - 8mo)im cosmOJm(z).
m=0
(2.57a, b)
24
Waves and Wave Forces on Coastal and Ocean Structures
The generating functions in Eqs. (2.57a, b) are useful for expanding waves in circular cylindrical coordinates; viz., expikx = exp ikr cos 0 — exp ikr
im exp(im8)Jm(kr)
^
,'ew + e~i0x~i
= ^ ( 2 - Sm0)im
-oo
cosm6Jm(kr).
m=0
(2.57c, d) 2.4.3.2. Asymptotic Approximations as Function of the Arguments (Hildebrand, 1976, Sec. 4.9 and Abramowitz and Stegun, 1965, Sec. 9) For small values of the argument x (x —> 0): 1
J„(x)
c
2Pp\"
> '
2P(p-iy.
YP(x)
J—p\x) ~~p
1 xp,„
2Pp\
. . I-P(x)
Kp{x)~2P-\p-\)\x-p
(/>#«),
(2.58a, b)
Y0(x)~-logx
(2.59a, b)
(-/>)!
x~p (p^O),
7T
IP(x)
2P
It
2P
(-P)! (p^O),
(p^n),
K0(x)~-logx.
(2.60a, b)
(2.61a, b)
For large values of the argument x (x -> oo): / 2
/
IX
pit \
/Pw
, _
/ 2
. /
/OTX
(2.62a, b) (2.63a)
expx
V27TJC
JT i 4/>2-l ~ ' — exp — x \ 1 H *«(*) 2x "~r { 8x
7T
(4p 2 - l)(4p 2 - 9) h 2!(8x)2
(4^ 2 - l)(4/72 - 9)(4/>2 - 25) 3!(8x)3
+ ••
(2.63b)
25
Mathematical Preliminaries
H
?M~JJ;'*-,[X-I-T\-
(2.64a, b)
2.4.3.3. Derivative Formulas (Hildebrand, 1976, Sec. 4.9) The derivative formulas for Bessel functions are given by the following:
-^-[xpyp(ax)] dx
= (±)axPyp-1(ax)
d_ [x py (ax)] p dx
j +' -,
y = J
' Y>U y = K,
*
(2.65a,b)
•> y = J,Y,K,H\y
p
= (±)ax
H
yp+\{ax)
+,
y = I, (2.66a, b)
d dx yp{ax) = (±)ayp-i(ax)
d —yp(ax)
p yP(ax)
p = (±)ay
p+i(ax)
+
y = J,Y ,I,H(*) ) y = K, (2.67a, b)
+.
3
-yp(ax)
+,
y-- = J,Y, K,H$) y-- = 1, (2.68a, b)
Zp+i(ax)
= —Z p (ax) - Zp-i(ax)
| z = J,Y,H\2K
(2-69)
Relationships for obtaining Bessel functions of higher orders applying recursive relationships are given by Ip+i(ax) Kp+i(ax)
=
2p
Ip(ax) + /p_i(ax), ax 2p = —K p (ax) + Kp-\{ax). ax
(2.70) (2.71)
26
Waves and Wave Forces on Coastal and Ocean Structures
2.4.3.4. Wronskians (Hildebrand, 1976, pp. 177-178) The Wronskian W[y\ (x), yi(x)] of any two linearly independent solutions to Bessel's equation (2.42) is of one of the following forms: d yi(x)—y2(x)
d C - yi{x)-—yx{x) = —,
J (x) — Yp(x) - Yp{x) dx d Jp{x) — J-P(x) - J-p{x) dx Ip(x)-^Kp(x)
— Jp{x) = — , dx Ttx d 2 — Jp(x) = smpx, dx TTX
- Kp(x)j-Ip(x)
= -K
d d 2 Ip(x) — I-P(x) - I-P(x) — Ip(x) = smpx, dx dx rtx Hpl)(x)-£-H™(JC) y dx F 2.4.4. Orthogonal
_ Hf (x)^-Hpl\x) = -i — . F dx F Ttx
(2.72) (2.73) (2.74) (2.75) (2.76) (2.77)
Polynomials
A number of orthogonal polynomial expansions are encountered frequently in coastal and ocean engineering applications. The nth order orthogonal polynomial may be defined as a solution to the following second-order ordinary differential operator: £(.) = [D{S(x)D} + R(x)D + G(x)](.) where the ordinary differential operator D = d/dx. 2.4.4.1. Legendre Polynomials Pn(x) (Barcilon, 1990, Sec. 7.6.1) Modify Eq. (2.78) to [D{S(x)D] + R(x)D + G0OK-) and define the coefficients by S(x) = l-x2,
R(x) = -2x,
Q(x) = n(n + 1).
(2.78)
27
Mathematical Preliminaries
Polynomial expansion: •2k
"v
(2.79)
£-, 2nk\(n - k)\(n - 2k)\
/fc = 0
where the upper limit [n/2] means the greatest integer part of the argument; i.e. [2.1] = 2, [3] = 3; etc. The first seven terms in the expansion are P0(x)
= 1,
PI(JC) = x,
= \Ox2
P2(x)
P 3 (x) = ±(5x3 - 3JC),
-
1);
P 4 (*) = |(35x 4 - 30x2 + 3);
P 5 (x) = |(63x 5 - 70x3 + 15x),
P 6 (x) = T U 2 3 1 * 6 ~
315
*4 +
Rodriques' recurrence formulas:
(n + l)P„+i(x) = (2n + l)xP„(x) -nP„_i(jc), (1 - x 2 ) D P „ ( x ) = -nxPn(x)
+
nPn-i(x).
Generating function: oo
1
Vl - 2tx + t2
= J2t"Pn(x),
\t\ (")• x*-2fc \(n-2k)\ 2*ik!(w-2Jk)!
(2 82)
31
Mathematical Preliminaries
where the upper limit [n/2] means the greatest integer part of the argument; i.e. [2.1] = 2, [3] = 3, etc. and where the first six terms in the expansion (2.82) are H0(x) = l,
Hi(x) = 2x,
H4(x) = 16x4 - 48x2 + 12,
H2(x) = 4x2-2,
tf3O0
= 8x3 - 12x,
H5(x) = 32x5 - 160x3 + 120x.
Rodriques' formula: ffB(x) = ( - i ) V 2 D " ( r ' 2 ) . Recurrence relations: Hn+\{x) = 2xHn(x) D(Hn(x)) =
2nHn-i(x),
2nHn-i(x).
Generating function:
n=0
Orthogonality—completeness: /
e
x
Hm(x)Hn(x)
dx = 2nn\-sfrt8mn.
J—oo
Explicit expressions for many orthogonal polynomials are tabulated by Hochstrasser (Abramowitz and Stegun, 1965, pp. 774-775).
2.5. Linear Ordinary Differential Equations (Hildebrand, 1976, Chapters 1.1 to 1.11) and Operational Calculus (Friedman, 1956) One of the fundamental methods applied to obtain analytical solutions to linear partial differential equations (PDEs) is to transform them into linear ordinary differential equations (ODEs). For this reason, several elementary analytical methods for solving linear ordinary differential equations are reviewed.
32
Waves and Wave Forces on Coastal and Ocean Structures
A linear differential operator l cos(cot - 0),
(2.99a)
with the initial data y(t0) = Y0,
Dy(t0) = Y0.
(2.99b, c)
The coefficients in Eq. (2.99a) are related to the physical parameters in structural dynamics by Ko(a)d) is the undamped (damped) natural frequency, Ast is the static displacement and Fo is the amplitude of the exciting force. (Clough and Penzien, 1975). The homogeneous solutions to Eq. (2.99a) are u(t) = {exp — (t;a>ot)} cos (a>dt),
v(t) = {exp — (f coot)} sin (coat), (2.100a, b) with the Wronskian W[u(r), V(T)] given by W[u(r), v(t)] = cod exp - ( 2 f co0r),
(2.100c)
and the convolution kernel (or unit impulse response function, Clough and Penzien, 1975) given by exp — (c~a>o(t — r)) h(t,r) = —-—^-^ sm [cod(t-t)]. COd
(2.100d)
38
Waves and Wave Forces on Coastal and Ocean Structures
The integral equation solution to Eq. (2.99a) from Eq. (2.98) is (l - (co/coo)2) cos (cot - P) + (2;co/COQ) sin (cot - P) y(t) = &st-
(1 - (co/coo)2) +(2$co/coo)2
- AJf
exp - ({coo (t - t0)) 2
2 V l - ? j ( l - (co/coo)2)2 + (2$co/coo)2}
yi-S2+co/coo\
K2 + ( 7 l - K2 - co/coof]
x cos (cod (t - to) - cot0 + P) + k / l -t}
- co/coo\
t,2 + (y/l - f 2 + co/coo) cos (cod (t - to) + coto P) ? 2 + ( 7 l - K2 - o/coo)
sin (cod (t - to) - coto + P)
$2 + (yi-$2 + oo/co0ysin (cod (t -\
Yo
to) + coto - P)
exp - (t;coo (t - to)) {cod cos (cod (t - to)) + ?&>o sin (cod (t - t0))}
COd
-\
(2.100e)
exp - ($coo (t - to)) sin (cod (t - t0)).
The steady-state (to • oo) component of Eq. (2.100e) may be written in a more familiar form by the method of undetermined coefficients given below in Sec. 2.5.6: y(t)
= D [(aV^o)2] A s , cos(a>f - a),
(2.101a)
where co/coo = frequency ratio (Clough and Penzien, 1975, p. 53) and where -1/2
D [(w/wo) 2 ] = { [l - (co/coo)2]2 + (21; (co/coo))7
(2.101b)
39
Mathematical Preliminaries (b) 180 160 140 15> 120 •8 100
0
1
0.5
1.5
12. co0
1° T
2
Fig. 2.3. The dynamic magnification factor D[(o) ] and phase angle e. a — arctan
tan fi + tan e 1 - tan ft tan e J '
f 2^(CO/COQ) \
e = arctan " I 1 - (w/wo)22 , (2.101c,d) I 1 - (co/coo) I cos e = D [(w/w 0 ) 2 ] ( l - (co/coof) , sin e = D [(w/« 0 ) 2 ] (2? (W^o)) • (2.101e,f) The total solution in Eq. (2.100e) reduces to the initial conditions in Eqs. (2.99b, c) when to —• 0. The dynamic magnification factor D [(W/WQ) 2 ] and phase angle e are shown in Fig. 2.3. 2.5.4. Properties of Linear Differential Operators (Hildebrand, 1976, Chapter 1.7)
£(•)
When the coefficients a„ in Eq. (2.83a) are constants and £()>H) = 0, the homogeneous operator may be factored uniquely into a polynomial given by N
2
^>D
+
§
+
e « > ' y = o,
(2.103)
¥
where § = x — XQ and where • oo
(2.104a) n=0
>(£)] G(f)
•
00 =
£G»* B
•
(2.104b)
n=0 oo
R($)\
£*«*"
(2.104c)
. n=0
where RQ = 1 is assumed to be given by the following power series expansion: A
y(£) = £
m§ m+s
(2.105)
m=0
where s must be determined from the indicial equation (2.108) derived below. Substituting Eq. (2.105) into Eq. (2.103) gives 00
£(y) = £
OO
£
[Rn(s + m){s + m-l)
+ Pn(s + m) + Qn]
m=0n=0
x Am^m+n+s-2)
= 0.
(2.106)
The following change of variables m + n = k transforms £(y) into k
•c(y) = £
/(*)A* + (1 - 8k0) £
fe=0
g„(s + A0A*_„ ^(k+s-2)
n=l
=
0>
(2.107a)
where /(*) = ^ + Co - 1)* + of Eq. (2.107a) results in the following general recurrence formula: k
f(s + k)Ak = - (1 - Sk0)J2sn(s
+ k)Ak-n
(2.109)
n=\
that determines all values of the coefficients Ak in terms of AQ. Example: (Hildebrand, 1976, p. 88): Consider again the following linear homogeneous equation of motion for a single-degree-of-freedom (SDOF) damped harmonic oscillator that is now written as: D2y + 2pDy + (a2 + p2)y = 0,
(2.110a)
where the coefficients in Eq. (2.110a) are related to the physical parameters in structural dynamics in Eqs. (2.99d-h) by P = -± = Scoo, K = ^L, (a2 + p2) = co2 = ^ ,
Ccr = 2VK^i,
a2 = a% = (4(1 - !2),
(2.110b, c,d) (2.110e,f)
where Cv(Ccr) is the structural (critical) damping, £ is the damping ratio, K is the structural stiffness, m is the structural mass, and &>o (&> 0 :
(s + 2 + m)(s + 1 + m)Am+2 = -26(s + 1 + m)Am+i - (a2 + yS2)Am,
(2.112c)
that are the recurrence formulas. The roots to the indicial equation (2.112a) are s\ = 0 and S2 = 1. Because the two roots s\ and $2 differ by a positive integer constant, the solution becomes an exceptional case (cf. Hildebrand, 1976, Sec. 4.4). There are two possibilities for the Frobenius series given by Eq. (2.111): (i) Eq. (2.108) is valid only for the larger exponent (i.e. S2 = 1), or (ii) there are two linearly independent solutions for the smaller (i.e. si = 0). The second possibility implies that there are two arbitrary coefficients in the series (Ao and B\, say). Because of the possibility (ii), the exponent s\ = 0 will be investigated first in order to see if there are two linearly independent solutions to Eq. (2.110a). The solution given by Eq. (2.111) must now be written by the following linear combination: oo
y(t) = J2Aktk =Ho(0 + "l(0,
(2.H3)
where the uo(t) solution will be given in terms of the AQ coefficient and the u\(t) solution will be given in terms of the B\ coefficient. s = 0 : Note that because s = 0 is a root, the recurrence formula in Eq. (2.112b)forthecoefficientof^_1 may not be divided by s. This means that the coefficient A i is also arbitrary; and a logical choice would be s A \ = — ^AQ. Note also that the coefficient A i must be multiplied by dimensional parameters such as a or fi in order for the power series expansion to remain dimensionally homogeneous. However, this particular choice for A\ would not be very good for small values of the coefficient B that is related to the damping ratio f. A better choice would be a linear combination of the two arbitrary coefficients
43
Mathematical Preliminaries
AQ and B\(A\ =aB\— /3AQ, say). Each of the remaining coefficients A„ may now be expressed in terms of Ao and B\ for n > 1. The first ten coefficients computed from Eqs. (2.112) are listed in Table 2.5. The denominators for both ano and bn \ in Table 2.5 were combined in a factorial notation that corresponds to the magnitude of the powers of a and /3 in the numerator in order to see more readily the final solution as a series of well-known elementary transcendental functions.
Table 2.5. Frobenius coefficients An = ano^O + a
An
1
1
-P
2 3
5
(V
a2 2!
p2 2!
+
2!
8
p a2 2! 2!
a 4! 4! a 6!
a5
3! 2!
5!
2
4
4
/J a 2 4! 2!
3 4
a 6!
/S a 3! 4!
8
2
5
2
a
+
6
P a
P1 a2 7! 2!
a10 10! +
3
+
£7
a1
p2 a5
a1
p3 a5
6
4
a
8!
P5 a— 5!
+
T\ ~ ~T\ 1\ 9
2
a 9!
7
p a 2! 7!
P6 a3
p9 9!
/J2 a 8 2! 8!
jS6 a4 6! 4!
p
P3a3 3! 3!
p4 a 3
ap5 ~6T
P5 a3
fi1
7!
4
5
£4
6!
a p a p a P a2 + ~8!~ ~ ~2\ ~6\ ^4T ~4\ ~ !~62! 8
4
p2 a 3
a5 —P 5!
P6 +
,8 a 5! 2!
6
^3
^3!-a3T
f
£ a 2! 4!
+
a3
p4 4!
+
y+p3»2
9
10
a3 £2 ha— 3! 2!
3!
6
7
-Pa
2
6
6
&„1
0
4
4
/
n0
0
bn\B\.
~5\ 1) ~~a — 4
p a5 4! 5!
+
P%
ha— 6! 3! 8!
/J4 (-I0 m y> (-1)"(«Q2" m=0
(2.114a)
= exp(-/J?)cos(af)-
n=0
The power series solution for u\(t) is _ («Q 3
«l(Q _ I
( and ifr by first defining
49
Mathematical Preliminaries
(Guenther and Lee, 1996): F = ^V(j),
(2.122c)
and then substituting Eq. (2.122c) into the Gauss divergence theorem (2.122a) to obtain Green's first identity given by [ff[xfrV24> + Vir.V(l)]dV=
fjir^-dS,
D
(2.123a)
dB
where - _ 30 V0.n = — . dn A symmetric form of Eq. (2.123a) may be obtained by redefining Eq. (2.122c) as F = cj>V^,
(2.123b)
then substituting Eq. (2.123b) into the Gauss divergence theorem (2.122a) and then subtracting from Eq. (2.123a) to obtain
111 WV2(f) - (/)V2f]dV = [J D
30 x// dn
dB
difr (p— dS, dn
(2.123c)
that is the Green's second identity. Finally, Green's third identity may be obtained by integrating by parts Laplace's equation from Sec. 2.2.9 (or simply let xfr = 1 in Eq. (2.123a)) to obtain
l[[v2(PdV=[[—dS. D
(2.123d)
dB
Boundary value problems (BVP) that are defined by partial differential operators in one of the 13 separable coordinate systems (Morse and Feshbach, 1953, Sec. 5) may be solved analytically by the method of separation of variables (Hildebrand, 1976, Sec. 9.3). This method may result in a well-posed Sturm-Liouville system; and solutions to a well-posed Sturm-Liouville system are frequently computed numerically by Fredholm integral equations and the boundary integral element method (BIEM).
50
Waves and Wave Forces on Coastal and Ocean Structures
Regular Sturm-Liouville Systems A regular well-posed Sturm-Liouville system consists of the following essential elements (Benton, 1990, Sec. 6.6): (i) A homogeneous linear ordinary differential operator defined by £{4>) + [A.W(JC)0] = {D(p(x)D) + q(x)4>) + [kw(x)] = 0, (2.124) where the ordinary differential operator £)(•) = d{»)/dx. (ii) A self-adjoint homogeneous linear ordinary differential operator £*(4>) where q(x) = Dp(x) in Eq. (2.124) so that £*() = £($) (iii) (iv) (v) (vi)
(2.125)
A weighting function w(x). A parameter k whose value is not specified a priori. A finite interval [a, b] with b — a = I. Homogeneous boundary data where n = 1 and y = 0 in Table 2.3.
These six elements are essential and must be specified in order for the system to be classified as a well-posed regular Sturm-Liouville system (Benton, 1990, Sec. 6.6). Because both the forcing and boundary data are homogeneous, the only non-trivial solution to the system defined by elements (i)-(vi) (i.e. a solution other than the trivial one of = 0) depends on the characteristic or eigenvalue k that must be determined from the boundary data. Solutions that depend on the parameter k are called characteristic or eigenfunctions. If a solution exists, it may be expressed by a set of orthogonal characteristic functions (/}n(x, kn) that depend implicitly on the characteristic or eigenvalue kn in the following series form:
y(x) - ^2an(j>n(x,kn).
(2.126)
n
The eigenfunctions (pn (x,kn) are orthogonal with respect to the weighting function w(x) if the boundary data satisfy certain conditions. To determine what these conditions are, consider two eigenfunctions (f>\ {x,k\) and $2 (x, A.2) where the eigenvalues l i / I 2 (Benton, 1990, Sec. 6.6). Substitute each eigenfunction into Eq. (2.124), cross multiply by the other eigenfunction and
51
Mathematical Preliminaries
subtract each Eq. (2.124) to obtain the Lagrange identity (Guenther and Lee, 1996, Sec. 7.2) (/>2(x,k2){£((t>i(x,ki)) +
\iw(x)i(x,ki)^2(x,X.2)dx = p(b)W(b)-p(a)W(a). (2.128) Ja The Wronskian W(») is evaluated at the boundaries x = a and b and is W(x) = 4>i(Di(a,ki) = 0,
A ^ ( M , - ) + BD• 0 shrinks to zero, Eq. (3.27) reduces to lim ( T « = zzx),
(3.28)
AV-s-0
that implies that the off-diagonal stresses in Eqs. (3.25) are symmetric. The stress matrix in Eqs. (3.25) is a generic stress matrix for the differential volume element in Fig. 3.7; and the stresses may now be applied specifically to a fluid volume by expanding the stresses in the constitutive equations that are appropriate for a fluid. Daily and Harleman (1966, Chapters 5.2 and 5.3) derive constitutive equations for a strain rate dependent fluid stress tensor by an analogy to a strain dependent stress in an elastic solid where the dynamic fluid viscosity is analogous to the shear modulus of elasticity for a solid. This derivation that is analogous to stresses in an elastic solid avoids the requirement for two Lame constants (Stokes hypothesis, 1845); and the corresponding controversial requirement that the second viscosity that amplifies only dilational rate straining is a negative-definite fluid property (Lamb, 1932, Chapters 325 and 326; Schlichting, 1979, Chapter 3e; Karim, 1953 and Lieberman, 1949). Many derivations for the constitutive equations for the total fluid stress attempt to maintain an analogy between the definition for the hydrostatic fluid pressure and the definition for the hydrodynamic fluid pressure. If the static fluid pressure is defined as the negative compressive normal stresses in a fluid
66
Waves and Wave Forces on Coastal and Ocean Structures
in the absence of fluid motion, then an analogous definition for the hydrodynamic fluid pressure would be the negative compressive normal stresses in a fluid in the absence of velocity gradients (i.e. zero strain rate). This definition would result in a total normal fluid stress component given by ixx = ~P + (*xx, tyy = -p + oyy,
xzz = -p + o\z,
(3.29a,b,c)
where oij is the deviatoric normal fluid stress tensor that is due solely to viscosity and to velocity gradients. The total fluid stresses T,; in Eqs. (3.29) are a linear sum of an isotropic pressure component —p and a non-isotropic velocity gradient component on that is called the deviatoric stress tensor (Batchelor, 1983, p. 142). The definition for the hydrodynamic normal fluid stress in Eqs. (3.29) implies that it is very difficult to measure the fluid pressure p because it requires either the velocity gradients in a moving fluid or the fluid velocity to be equal to zero at the pressure measuring device. The Stokes law of viscous friction for a Newtonian fluid relates the deviatoric shear stress components to the rates of angular deformations. By analogy to the stress in an elastic solid, the components of the normal deviatoric stress tensor aa in Eqs. (3.29) may be written by the Stokes law of viscous friction according to axx=2ixl
{du 1 - A — --V.q\,
(3.29d)
oyy = 2n( — --V.q), ~dy ~ 3
(3.29e)
azz=2n(^-l-W.qy
(3.29f)
and the total normal stress T,-,- in Eqs. (3.29a-c) in a fluid is given by /du 1-; ^ rx;c = - p + 2/zl — --V.q),
(3.29g)
Tyy = -p + 2n(j--jV.q),
(3.29h)
rzz = -p + 2fi(^-jV.q),
(3.29i)
Fundamentals of Fluid Mechanics
67
or in the more compact tensor notation rijSa - -pSa
+2/x
du,
i,j = 1,2,3.
S
" ~ 3
dxj
(3-29J)
The three remaining symmetric tangential surface stresses are given by du dv (3.29k)
(3.291) (3.29m)
or, again, in the more compact tensor notation as xu = ix
du/
du,
i
. .
dx + -7Tdxi ) ,
i,j = 1,2,3.
(3.29n)
In general, the complete stress matrix f may be decomposed into an isotropic component of the normal pressure and a deviatoric velocity gradient component according to: T =
-p 0 0
0 -p 0
0 " 0 -p_ dv
! ~i(du v -^ \9x~3 7
+ HX
dv du dx 9v dw du -J- — dx 9z
2
dx (dv dv dz
—
4-
-j-
du dy 1- A dw dy
dw
du _|_ — dx dz dv dw dz dy 1-. ^ /(dw 2 V '9 \dz 3 (3.30a)
or in tensor notation as Xn fxA lj = — —pSij ~FulJ + ~r l^^i],
i,j 1,2,3, i, J = — >-,^,->,
(3.30b)
where the deviatoric strain rate component A,y is due entirely to the velocity gradients. Summing the normal and the tangential surface stress forces on the differential fluid element in Fig. 3.7 in all three coordinate directions gives in
68
Waves and Wave Forces on Coastal and Ocean Structures
each coordinate direction, respectively (3.31a)
Fsx = Ftfx + FTX, d2u Fsx =
dx
+ /*
3^2
u 3 /du + —— 3 dx \dx Fsy = F^y + Fjy, FSy =
Fsz
+
FNZ
dz
9^2
3?
dw 1
dy
AxAyAz,
dz
d2v
d2v
dx2
dy2
d. 2v\ dz2J
dv
dw
dx
dy
dz
AxAyAz,
(3.32b) (3.33a)
Fjz,
+M
(3.31b) (3.32a)
d2W
d2w
d2w
dx2
dy2
dz2 J
IX 3 (du
+3
d2u
du
dp Fsz
d2u
dv 1
9/> - T - + A* /z 3
+
dz \dx
dv
dw\
dy
dz / .
AxAyAz,
(3.33b)
or, again, in the more compact tensor notation dp FSi =
3
dxi
(dm (dui
dxj \dxj
du
2
dxi
3 dxjy
ij
duk\
i,j,k
= 1,2,3. (3.34a)
For incompressible fluids V »q = dui/dxi = 0, i = 1,2,3 so that the normal component of the stress tensor xij is equal to the isotropic pressure p plus the rate of dilational deformation straining according to tii = -pSu
dut + 2/z—, dxj
i = 1,2,3,
(3.34b)
i,j,k
(3.34c)
and Eq. (3.34a) reduces to dp Fsi--
dxi
dui dxj
dxj'
= 1,2,3.
69
Fundamentals of Fluid Mechanics
3.3.3. Body Forces The only body force per unit volume (weight density) that will be considered will be that given by gravity according to FB = ~Pgez = -pg = -pgVz
(3.35)
= - y Vz.
where y — pg. 3.3.4. Navier-Stokes Equations (Lamb, 1932) Combining Eqs. (3.21, 3.31-3.33 and 3.35), the momentum balance is
p ^ + p (q . V) q = -y V U + z ) + /x ( v . v ) q,
(3.36)
that is the Navier-Stokes equation for an incompressible, Newtonian fluid. Alternate dimensional forms of the Navier-Stokes equations are the following: Vector (Milne-Thomson, 1968, p. 643): Dq
=
Dt dq
dq_ + dt
( } . V ) q = -V
P , -. IP
P
_ 1
+ - V | i / | 2 - t f x ( v x t f ) = - V - + g»x ~dt .p
(3.37a)
-Vx(vx?), (3.37b)
where the following vector identities are applied (Hildebrand, 1976, p. 284): (i)
(«.V)i = S v ( j . } ) - j x ( V x j ) ,
(3.38a)
(ii)
V x ( V x J ) = V(V,?)-(V.V)« = V(V.?)-V2i
(3.38b)
Tensor: dui dt
dut dx J
1 dp P dXi
\x i3 +
d2U{
p dxj
dx
-,
ij
= 1,2,3.
(3.39)
For some applications it is often more appropriate to express the momentum principle by the Reynolds transport theorem. Both the linear and angular
70
Waves and Wave Forces on Coastal and Ocean Structures
momentum are given by (Hudspeth, 1991): d [ p(j?\dV+(f dt Jcv(t) V x qj
i
cs Jc
f
PL \r
Jcv(t)
.idS /
x n
PU
p(J
Jcs
\(q.n)dS
\r x q)
cs
rL ^ ) dS \r x n, (3.40a) (3.40b)
-)(\g\z)dV.
\rxVt
where the top term in each parenthesis (•) is linear momentum and the bottom term is angular momentum. 3.3.5. Euler's Equations (Lamb, 1932) Many dynamical problems for free-surface gravity waves may be approximated by the irrotational flow of an inviscid fluid (i.e. /J, = 0) given by: Dq
- + g,x (3.41) P Integration of Euler's equation along a streamline yields the Bernoulli equation that is applied repeatedly in water wave theories. Dt
3.4. Mechanical Energy Principle Forming the vector dot product between the fluid velocity q{x,y,z,t) = q(x,t) and the fluid momentum balance in Eq. (3.36) for an incompressible fluid (i.e. V • q(x, y, z, t) = 0) gives (Landau and Lifshitz, 1987) 1 d
n 2"87 ^
- + gz + — P
2
,q + ^q.V2q. P
(3.42)
Applications of Eq. (3.42) are reviewed by Phillips (1977). When the fluid is assumed to be inviscid (i.e. ^ = 0) and the flow irrotational (i.e. V x q{x, y,z, t) = 0), the momentum principle in Eq. (3.36) together with Eq. (3.38a) reduces to Euler's equation in Eq. (3.41) given by dq(x,y,z,t) ; at
1- _ A ~p(x,y,z,t) 2 + -V|?(x,v,z,OI = - V I
+ gz
(3.43)
71
Fundamentals of Fluid Mechanics
Because of the inviscid fluid and irrotational flow assumptions, the fluid velocity vector field may be obtained from the negative gradient (Chapter 2.2.7) of a scalar velocity potential (x, y, z, t) according to q(x,y,z,t) = -V$(x,y,z,t),
(3.44)
and substituted into Eq. (3.43) to obtain -,/
d$(x,y,z,t)
*{
Tt
+
\V$>(x,y,z,t)\2 p(x,y,z,t)
2
+
+8Z
-^— )=°
\
(3 45)
'
that may be integrated along a streamline to obtain the unsteady Bernoulli's equation: d(x,y,z,t) \V{x,y,z,t)\2 = rz gz + Q(t), (3.46) p dt 2 where the Bernoulli constant Q(t) must be a function only of time because of the spatial integration of Eq. (3.45) along a streamline (Eagleson and Dean, 1966). p(x,y,z,t)
3.5. Scaling of Equations Every constitutive relation for the forces due to viscous stresses requires an empirical coefficient (e.g. the dynamic viscosity //, in the Stokes law of viscous friction defined in Eqs. (3.29d-f) in Sec. 3, the friction coefficient / in the Darcy-Weisbach equation, the drag force coefficient Cd in the Morison wave force equation for small bodies in Chapter 7, etc.). The parametric dependency of these empirical coefficients must be determined in order to compactly correlate their parametric dependencies on a single graph (e.g. the Moody diagram for the friction coefficient / in the Darcy-Weisbach equation). The two methods for establishing this parametric dependency are: (1) scaling of the equations and (2) dimensional analyses. In order to illustrate both the non-uniqueness of scaling equations and the different dimensionless scaling parameters that may result from this non-uniqueness, the scaling of the momentum equations for both Eulerian viscous fluid fields and for Lagrangian solid bodies are reviewed. Because there are usually more dimensional variables identified in the dimensional analysis method than there are dimensional
72
Waves and Wave Forces on Coastal and Ocean Structures
constitutive relations applied in the dimensional equations that approximate the physics of a process, the number of dimensionless parameters generated by dimensional analysis will almost always be greater than the dimensionless parameters obtained from scaling equations. Some of the dimensionless parameters obtained from dimensional analysis may be evaluated in order to justify omitting some of the constitutive relations in the equations that approximate the physics of a process. Both the dimensional analyses and the scaling of equations are required when analyzing fluid structure interactions. Momentum for Eulerian viscous fluid fields: Navier-Stokes equations: The equations for fluid momentum given by the Navier-Stokes equations (3.36) in Sec. 3.3.4 and the equations for the dynamic response of a solid Lagrangian body by Eq. (2.99) in Chapter 2.5 are scaled. The order of magnitude of the contribution to the physics from each term in the Navier-Stokes equations may be determined by scaling the equations. The dimensional variables denoted by tildes (•) in the Navier-Stokes equations (3.36) in Sec. 3.3.4 may be scaled by the following non-unique variables: q a)A
P=
p - p =-r> p{o)A)2
o ^ g ^ ^ x 9 = — = ez, x =-, \g\ b
(^\~ t=\-r)t, \ b J
where b is a characteristic dimension of a structural member, A is the wave amplitude, a> = 2n/T is the radian frequency of the wave, and po is the reference pressure. Substituting these dimensionless variables into the NavierStokes equations (3.36) in Sec. 3.3.4 yields ^ + ($.V)$ = - V p - - ^ + -i(V.V)J, (3.47a) at t r n where the following dimensionless ratios (parameters) are defined as: _ ptiAb A*
2
_ (a)A)2 \g\b
and where R is the Reynolds parameter and F2 is the Froude parameter. The dimensionless Navier-Stokes equation (3.47a) demonstrates that at high Reynolds parameter (R 3> 1) viscous stresses may be neglected in comparison with the inertial forces on the LHS. However, turbulent Reynolds shear stresses originate from the convective inertial terms on the LHS and they must be
73
Fundamentals of Fluid Mechanics
considered at high Reynolds parameters. A similar conclusion may be reached by an order of magnitude comparison of the characteristic scales in Table 3.1 in Sec. 3. It is instructive at this point to emphasize some subtle but important points in the choice of scaling variables. First, the radian frequency u> and wave amplitude A were selected to scale the velocity, rather than the maximum fluid velocity Um . Second, two length scales were defined: (1) a wave amplitude length scale A, and (2) a structural characteristic length scale b. Note also that both the local and convective acceleration terms are of the same order 0(1) in Eq. (3.47a) from this non-unique choice of scaling. The non-uniqueness of selecting scaling variables may be illustrated by selecting the wave number k = In/X to scale lengths and gravity y gk to scale time, yielding the following dimensionless scaling: q=
—,
Ajgk
p =
—,
g = — =ez,
^A
x = kx,
t = tJ gk.
\g\
The dimensionless Navier-Stokes equations (3.36) now become ^ + (kA)(q . V)q = -Vp ot
- ^ + i(V • Vtf, kA R
where the Reynolds parameter is now given by R = pJgk/jlk2
(3.47b) —
Jg/k/vk
that is independent of the wave amplitude A. The nonlinear perturbation parameter kA now amplifies the convective acceleration term in the fluid inertia on the LHS of Eq. (3.47b) and the hydrostatic body force on the RHS is amplified by (kA)~l. Finally, the non-uniqueness of selecting scaling variables may again be illustrated by selecting the wave number k = 2TT/X to scale lengths, gravity J gk to scale time and the maximum amplitude of the fluid velocity Um to scale the fluid motions so that the dimensionless scaling is given by the following:
74
Waves and Wave Forces on Coastal and Ocean Structures
The dimensionless Navier-Stokes equations (3.36) now become
dq
( ft ~ \ ^ ~ ^
( ft ~ \ -
-£+ \J-Um \(q»V)q = -\J-Um dt 8
\v* )
v
ft
g
vk2 -. -> _
V p - j l 4 - + - 7 =(V.V)fl.
)
v*tf» y^ (3.47c)
The scaling examples in Eqs. (3.47) illustrate the difficulty involved in selecting the scaling variables. The goal of scaling is to make each of the differential terms in Eqs. (3.47) (such as the convective acceleration term (q • V)$) order unity so that the contributions to the physics that is approximated by each term in the differential equation may be determined by the magnitude of the dimensionless coefficient that multiplies each differential term in the total equation. For example, the contribution of the convective acceleration term in Eq. (3.47b) to the physics of the fluid inertia is 0(kA) that is a small quantity compared to unity. Consequently, the convective acceleration term may be neglected to the first order of approximation compared to the local acceleration term with this non-unique choice of scaling variables. However, after comparing numerical values computed from Eqs. (3.47) with data it may be necessary to re-evaluate the non-unique choice of scaling variables and to re-scale the differential equation with a different set of non-unique scaling variables. The selection of scaling variables is non-unique and is analogous to the selection of repeating variables in dimensional analyses that is reviewed in Sec. 6 below. The final selection of scaling variables will only be possible after a satisfactory comparison of computed numerical values with data. Scaling equations is a difficult but critically important first step in any analysis in fluid mechanics. Momenta for small Lagrangian solid bodies: damped harmonic oscillator The dimensional differential equation governing the dynamic response of the semi-immersed Lagrangian solid body with two-degrees of freedom illustrated in Fig. 3.9 is given by
\mXi(t) J (/5(0]
jb s Xi (t) 1 , ( ^ * 1 (0) \0s&5(t)\ i*©5(F) J
=
( ^l (t) 1 \M5(t)\'
(3.48a) (3.48b)
where m is the dimensional body mass per unit length, / is the dimensional polar mass moment of inertia per unit length, k>s, fis a r e the dimensional structural damping coefficients per unit length, K, k are the dimensional
Fundamentals of Fluid Mechanics
75
Fig. 3.9. Definition sketch for the translational and rotational motions of a semi-immersed Lagrangian solid body.
combinations of structural, hydrostatic and mooring stiffnesses per unit length (vide., Chapters 7.8 and 8.2), X\{t),X\(i),X\{i) are the dimensional translational displacement, velocity and acceleration, respectively and ©5(f), ©5(F), ©5(f) are the rotational displacement, velocity and acceleration, respectively, where the overdots (•) and ('•') denote ordinary temporal derivatives of Lagrangian solid body motions according to (i) =
dt '
('•') =
dt2 '
(3.48c,d)
and where F\ (J) , M5 (?) are the dimensional wave-induced hydrodynamic pressure force and the moment per unit length, respectively, on a semi-immersed Lagrangian solid body of characteristic length dimension D = 2b that are given by the following component forms of the dimensional relative motion Morison equation (Modified Wave Force Equation, Chapter 7.8): ut(zo,t) sout(zo,i)
-Cap
\u(zo,t) -Xi(t)\[u(zo,t) - Xi(f)] | (zo,t) - So®5(t)\so[u(zo,t) - so&5(t)]\ (3.49a,b)
76
Waves and Wave Forces on Coastal and Ocean Structures
where the subscript (•), denotes the partial temporal differentiation of an Eulerian field variable according to 8(.)
(•)r = ~jf,
(3.49c)
where the inertia coefficient Cm — 1 + Ca (vide., Chapter 7.4), Ca is the added mass coefficient and so is the dimensional moment arm measured from the point of rotation to the unit length of the member at vertical elevation zo (vide., Fig. 7.4 in Chapter 7.2), u (zo, t), iit (zo, t) are the dimensional horizontal water particle velocity and acceleration, respectively, at a vertical elevation zo measured down from the free surface (vide., Chapter 7.2). Scaling of the dimensional hydrodynamic pressure force, and the moment per unit length F\(t), M$(J) in Eqs. (3.49a,b) is reviewed for small bodies in Chapter 7 and for large bodies in Chapter 8. Scaling Eqs. (3.49a, b) is by the following time and length scales: ~7
fi
^°
^
CA
u
\
T = cot, so = —, zo = —, (3.50a,b,c) h h where u> — 2n/f, and f is the dimensional wave period. The dimensional Lagrangian solid body motions are scaled by the following scales: Xi(t) XI(T) = - ^ , 05(r) =
05 (0
T7T'
• Xi(t) Xi(r) = - ^ - , •
05(r) =
.. Xi(t) Xi(t) = ^ - ,
@5(f)
..
7I7^'
®
5(T)
®5(t)
= TJirh'
(3.50d,e,f)
(3-5°8'h'i)
A/b (A/b)a> (A/b)co2 where A (= H/2) is the dimensional wave amplitude, H is the dimensional wave height, b is a characteristic body dimension, and scaling the Eulerian water particle motions (vide., Eqs. (7.126 d, j) in Chapter 7.8.2) by the following scales where ao is an arbitrary phase angle and where Um(zo) is the dimensionless maximum amplitude of the horizontal water particle velocity measured at zo: M(ZO, r)
= —^r-^— = UM(zo) COS(T - a0) Aw
utizo, r) =
'„ ' Aco1
= -UM(zo) sin(r - a0).
(3.50j) (3.50k)
77
Fundamentals of Fluid Mechanics
These scales modify Eqs. (3.48a,b) to the following form of the dimensionless damped harmonic oscillator in Eq. (2.99a) in Chapter 2.5.3 but with non-constant, time-dependent and nonlinear coefficients: (^I(T)J
I ©500 J
k i l (Zi[K(zo,r),Xi(r)]J \coih\ fXi(r) K5 J lZ5[M(Z0,T),©5(T)]j [C05h\ \@5(T)
\h Xl(T)
(3.51a) (3.51b)
CO'
+ *4©5(r)
where the dimensionless damping coefficient Z, (i = 1,5) is a function of the dimensionless horizontal water particle velocity U(ZQ,T) and the dimensionless velocities of the Lagrangian solid body X\[x) and &${x) (vide., Eqs. (3.49a,b)). Consequently, the damping term is nonlinear and timedependent so that the equations of motion of Eqs. (3.51) are nonlinear ordinary differential equations (ODE) with non-constant, time-dependent coefficients. The wave-induced forcing terms on the RHS of the modified wave force equations (3.49a,b) may be linearized to obtain linear ODE's with constant coefficients (vide., Chapter 7.8). The dimensionless coefficients without tildes (•) in Eqs. (3.51) are defined by
bscr
-CdDAco
(3.52a,b)
-Cab~sl»U
(3.52c,d)
2JKm{\ + inCa)
(3.52e)
2j£I(l+ix5Ca)
(3.52f)
Dh P
Pscr
[Pscr\
k
pTtD2
Am
CO
m{\ + ix\Ca)
2^2 J5TZDZSQ
CO
a
(3.52g,h) (3.52i,j)
/ ( l + H5Ca) J
4/ \co\h\ \(»5h\
=
CO
5h CO
(3.52k) (3.521)
78
Waves and Wave Forces on Coastal and Ocean Structures
[ZI[W(ZO,T),XI(T)]
|H(ZO,T)-Xl(T)|
lZ 5 [w(zo,r),0 5 (r)]
W(ZQ,T) - JO®5(T)|
(3.52m) (3.52n) The dimensionless exciting force/moment, on a fixed Lagrangian solid body on the RHS of Eqs. (3.51) are defined by F f (T) 1 _ [Mf(r)J
Ai(r)a)ffc cos(r - a0 + £ I ( T ) ) | A 5 ( T ) ^ c o s ( r - a 0 + ^5(T))|'
(3.53a) (3.53b)
where the dimensionless static displacements A, (r)for i = 1,5 are defined by #! £ (r) (3.52c)
AI(T) A5(T)
(3.52d) where the dimensional exciting force/moment amplitudes £j ( T ) / M ^ ( T ) and phase angles JS,-(T) for / = 1,5 are defined by ^(r) = ^
J r D 2
^
M ( g o )
V ^ + C^2Zf[M(zo,r),Xl(r)L
CM
0100 =
Mf(r) =
(3.53e) (3.53f)
C^ZI[II(ZO,T),XI(T)]
/ p7TD2sobUM(zo)\ \
4
/
c 2 +cj/ir 2 z 5 2 [ M (zo,r),e5(T)], (3.53g)
Ps(r) =
c»» Q J firZ 5 [ M (zo,r) ) 0 5 (r)] 2A _ lib
H
(3.53h) (3.53i)
TtD
where K is the Keulegan-Carpenter parameter that is evaluated in Chapter 7.7. Scaling of the momentum equations illustrates the need both to scale the differential equations and to perform dimensional analyses. Scaling the momenta in an Eulerian fluid in Eqs. (3.47) and the momenta in a Lagrangian solid body in Eqs. (3.51) identifies only three dimensionless force parameters, viz., the Reynolds parameter R, the Froude number Fr and the Keulegan-Carpenter
Fundamentals of Fluid Mechanics
79
parameter AT. The reason that the Reynolds parameter if and the Froude parameter Fr are not given explicitly in Eqs. (3.51) is because the relative motion form ofthe Morison equation (Modified Wave Force equation, Chapter 7.8) only considers the inertia of the Eulerian fluid particles on the LHS of Eqs. (3.47) that would have been at the location of the Lagrangian solid body if the small body were not at that location. The dimensional analysis that is reviewed in Sec. 6 below identifies many more dimensionless parameters that may be obtained from dimensional analysis. Model and prototype scaling of Lagrangian momentum equations: The dynamic equations of motion for Lagrangian solid bodies must be analyzed for both model and prototype bodies (Sarpkaya and Isaacson, 1981). Denoting the dimensional (without tildes (•)) dynamic response of a model Lagrangian solid body by the subscript "m" in Eqs. (3.48a,b) yields mmXm(t)
bSmXm(t)
F£ cos(cot - oro)
KmXm(t) +
MS. cos{cot — ao) (3.54a,b) that may be transformed to the dimensionless equations of motion for a damped harmonic oscillator in Eq. (2.99a) in Chapter 2.5.3 according to l/«©m(Oj
lj85 m ©«(0}
i*m©m(0j
'Xi m (Ol . f 2 0 ^ o m X i m ( 0 ] . [<w§Xi m (0] ®sjt)\ + |2?5mno(B©51B(0| + ^0 _©5 m (0 Aimcolmcos(o)t - a 0 ) A5mQl
(3.54c) (3.54d)
cos(cot-a0)
where the dimensionless damping ratios, undamped natural frequencies and static displacements are defined by \bSm/bcrm
2*fKn m„
\Psm/Pcrm
I
Km/mm Km 11m
F
m/Km 1
M%/icm
(3.54e,f) (3.54g,h) (3.54i,j) (3.54k,l)
Denoting the dimensional dynamic response of a prototype Lagrangian solid body by the subscript"/?" and substituting the model to prototype scale ratios for the mass, for the damping coefficient, for the structural stiffness, for the
80
Waves and Wave Forces on Coastal and Ocean Structures
structural motion, for the force and the moment loads and for time that are defined by \bsA
m/fTlp
\bSm/bSp
=
Kr
1 h J " 1 /,
ml *p
Kr
Ff IF*
[XljIXlm/Xl, l©5 r J
Km/Kp I K mlKp J (3.55a, b, c)
M
\t
\®5m/®5p
\MUM5
E
{tr} = {tm/tp}.
X
(3.55d,e,f)
The scale ratios from Eqs. (3.55a-d) transform Eqs. (3.54a, b) to the following dimensional prototype equations of motion: bSrXlr'
^-)mpXh(t)
+
—p-)Ip®5„(t)
\bspXip(t)
tr
\Psp®5p(t)
tr
\KrXXrKpXXp(t)
+
I Kr®5rKp®5p(t) ,
[ Ff Ff cos(cot — «o)
(3.56a)
yM^M^
(3.56b)
cos(atf — ao)
and Eqs. (3.54c,d) to the following dimensional prototype equations of motion:
Ir®5r tz
+
©5D(0
tr
XlrKrco20pXlp(t)}
+
J F^A^co2^ cos(a;f - a 0 ) 1 (3.56c)
&5ricrQ20p&5p(t)\
~ [MlA5pn20pCOs(cot
- a o ) j ' (3.56d)
Dividing Eqs. (3.54a,c and 3.56a,c) by KrX\r and Eqs. (3.54b,d and 3.56b,d) by Kr@sr, and then comparing equivalent terms in Eqs. (3.54a,b and 3.54c,d) to (3.56a,b and 3.56c,d), respectively, requires that
\
bs
\( ')\
\Krt}) .
=
•
[ U ) J {-) \K t J r r
F
l 1
(3.56e,f,g)
KrX\r •
=
•
(3.56h,i,j) *>©5r
81
Fundamentals of Fluid Mechanics
Similarly, the damping ratios f,y and undamped natural frequencies a>or and Qor may be computed from \bsjbcrr
(OZ
\Psr/Pcrr
co: Or.
(3.56k,l,m) (3.56n,o,p)
A thorough review of the similarity analyses of physical systems is given by Hansen (1964).
3.6. Dimensional Analyses There are three essential physical similarities that must be preserved in physical modeling. These three similarities between model and prototype are: (1) geometric similitude (geometric shapes must be similar), (2) kinematic similitude (velocity vectors must be similar), and (3) dynamic similitude (force vectors must be similar). Table 3.2 lists the most common dimensionless parameters applied in fluid flow modeling. The force ratios in the Formula column are defined below. Dimensional analysis illuminates the parametric dependencies of the hydrodynamic pressure forces applied to compute loads on both large and small Lagrangian bodies. Assume that the wave-induced, hydrodynamic pressure force per unit length F depends upon the following variables: F = f\{[h, H,X, T], [m,i-,g,b,e, E], [p, fi,a]},
(3.57)
Table 3.2. Dimensionless fluid force parameters and their corresponding force ratios. Dimensionless Parameter Cauchy: C Euler: P Froude: F2 Reynolds: R Stokes: Svg Weber: W Keulegan-Carpenter: K Strouhal: St Frequency: /J
Formula 2
pU /E pU2/2Ap U2/gb Ub/v vU/gb2 pbU2/o UT/b b/UTs b2/vT
Force Ratio inertia/elastic inertia/pressure inertia/gravity inertia/viscous viscous/gravity inertia/surface tension wave period parameter Vortex shedding frequency: fst R/K
i/TSt
82
Waves and Wave Forces on Coastal and Ocean Structures
where the wave field variables [h, H, X, T] are: h is the water depth, H is the wave height, A. is the wavelength, and T is the wave period; the structural variables [m, %,g,b, e, E] are: m is the structural mass density, £ is the amplitude of the Lagrangian body motion, g is the gravitational constant, b is a characteristic dimension of the body, e is the surface roughness, E is the Young's modulus of elasticity; and the fluid variables [p, \i, a] are: p is the fluid mass density, ju,(= vp) is the dynamic viscosity of the fluid, v is the kinematic viscosity of the fluid and a is the surface tension of the fluid. The Stokes dispersion parameter derived in Chapter 6.2 is defined by S = (H/2X)(X/h)i that is a measure of the free-surface nonlinearities H/2X and of the shallow water nonlinearities (A./TO3 for surface gravity waves. The Stokes nonlinear ratio parameter S demonstrates that the parametric dependence of the hydrodynamic pressure force per unit upon the wave period T need not be considered explicitly but is included in the wavelength X dependency by the frequency dispersion equation in Eqs. (4.15) in Chapter 4.3 for linear waves (Stokes, 1847 or Ursell, 1953a). Selecting p,g, and b as the non-unique repeating variables yields the following dimensionless wave force parametric dependency: pg&3 I b' b ' b ' V b' pZ?3' b' b' pgb' pu ' pgb2 J If the classical dimensionless hydrodynamic parameters in Table 3.2 are given by the Froude parameter F2r and Reynolds parameter R are defined by Fr = UIsfgb and R = pbU//x where U is the maximum amplitude of the water particle velocity, then it is easy to show that the Stokes parameter Svg in Table 3.2 (that is different from the Stokes dispersion nonlinear ratio parameter in Chapter 6.2) is equivalent to Svg = F2/R = vU/gb2. This parameter is applied by Garrison and Chow (1972); and it may be applied to demonstrate that model test data may not, in general, be scaled to prototype conditions by both Reynolds and Froude similitude. The only restriction applied up to now is that the waves must be strictly periodic. These results, however, are not restricted to simple linear sinusoidal flows. Keulegan and Carpenter (1958) identified the following variables for a wave force per unit length in their laboratory experiments for standing wave forces on a horizontal cylinder: F = f3{t,T,Um,D,p,n},
(3.59)
where t is time; Um is the amplitude of an oscillatory wave velocity measured at the node of a standing wave; D is the cylinder diameter; and v is the
Fundamentals of Fluid Mechanics
83
kinematic fluid viscosity. Selecting p, D and T as the non-unique repeating variables, they obtained a dimensionless force per unit length expressed by
P pD
3T-2
|i : M;££l /4 ^[T' D ' fiT
F
f t UmT
,3.60a)
UmD)
ft
)
where the LHS of Eq. (3.60a) is divided by K2 and the third ratio in the function ft, in Eq. (3.60a) is multiplied by K in order to obtain the parametric equation (3.60b). They found no dependence on the Reynolds parameter R = Um D/v in their experiments; but they did find a dependence on a period parameter K = UmT/D (now called the Keulegan-Carpenter parameter K). Sarpkaya (1975) reached a similar conclusion after analyzing data from U-tube experiments that harmonic forces correlate only with the KeuleganCarpenter period parameter K and not with the Reynolds parameter R. A subsequent re-evaluation of his data by Miller (1975) and by Garrison (1982), as well as by Sarpkaya (1977), eventually illuminated the true Reynolds parameter dependency of these U-tube data (vide., Chapter 7). The difficulty of conducting experiments that must be modeled simultaneously by both Reynolds and Froude similitude may be illustrated by considering the ratios of the Reynolds parameter R to the Froude parameter Fr for both the model (subscript m) and prototype (subscript p) data; viz. Rm
Frm
— =
Rp
(Ub/V)m = 1,
F„
{U/Vgb)m = -,
{Ub/v)p
==r—,
(j.ola,b)
(£//ViE),
/E(i)(*-) 1/, _ 1 , JEMl
= i,
(3.6l0,d)
where the subscript r = model(m)/prototype(p) ratio for the variable (•),-. Consequently, in order to scale the model data to prototype designs by both Reynolds R and Froude Fr scaling, Eqs. (3.61) imply that for any scale model with a length ratio less than unity (i.e. br = bm/bp < 1) that the experiment must be conducted either in a centrifuge with a fluid when vr = 1 (i.e. gr = gm/gp > 1) or with a super-fluid when the gr — 1 (i.e. vr = vm/vp < 1). If water at standard atmospheric conditions is used in a wave flume with a free surface or in an oscillating U-tube on earth, the data may not be scaled simultaneously by both the Reynolds R and the Froude Fr parameters.
84
Waves and Wave Forces on Coastal and Ocean Structures
3.7. Problems 3.1. 3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
Draw the equivalent figure to Fig. 3.5 for the angular deformation without rotation for the x—y plane and derive the formula (3.18c) for {h}. Draw the equivalent figure to Fig. 3.5 for the angular deformation without rotation for the y—z plane and derive the formula (3.18b) for {/}. Draw the equivalent figure to Fig. 3.6 for the angular rotation without angular deformation for the x—y plane and derive the formula (3.19c) fortf}. Draw the equivalent figure to Fig. 3.6 for the angular rotation without angular deformation for the y—z plane and derive the formula (3.19b) for{§}. Draw the equivalent free-body-diagram to Fig. 3.8 for the side ABCD of the differential volume AV = AxAyAz in Fig. 3.7 and prove that the tangential surface stresses xyz and rzy are symmetric. Draw the equivalent free-body-diagram to Fig. 3.8 for the side AEFB of the differential volume A V = AxAyAz in Fig. 3.7 and prove that the tangential surface stresses rxy and xyx are symmetric. Keulegan and Carpenter (1958) applied the Buckingham Pi theorem to obtain the following parametric dependency for the force per unit length of Eqs. (3.60) in Sec. 3.6: >DUl
h
/ UmT T' D
UmD
h
y,K,R\.
(3.60b)
Verify their result by the Buckingham Pi theorem. 3.8. What are the requirements to scale simultaneously model data to prototype data by both Reynolds and Froude similitude? 3.9. Draw the equivalent figure to Fig. 3.4 for the elongation of a differential fluid element due to linear, dilational deformation in the y-direction; and prove that the time rate of strain in the y-direction may be computed from A^/(Ay/A?) = dv/dy = {b}. 3.10. Draw the equivalent figure to Fig. 3.4 for the elongation of a differential fluid element due to linear, dilational deformation in the z-direction; and prove that the time rate of strain in the z-direction may be computed from Af z /(AzA0 = dw/dz = {c}.
Chapter 4
Long-Crested, Linear Wave Theory (LWT)
4.1. Introduction Linear wave theory (LWT) is the most commonly applied description for wind-generated surface gravity waves. LWT assumes that the height of the wave H is small compared to the wavelength X (small amplitude assumption H/2X) and that the depth h is not small compared to the wavelength X (finitedepth assumption h/X). Small amplitude LWT was originally developed by Airy (1845); and, consequently, LWT is also referred to as the Airy wave theory. In spite of the restriction of the small amplitude ratio H /2X(x,z,F)
1
V$(x,2,r>
92
Waves and Wave Forces on Coastal and Ocean Structures
Scaling the BVP in Eqs. (4.5) The dimensional LWT BVP is solved by the method of separation of variables. The variables that are separated are the Eulerian independent variables of space x, z and time t. Because the dimensional CKDFSBC in Eq. (4.5h) is evaluated at the instantaneous location of the unknown dimensional free-surface z = rj{x,T), this boundary condition is a function of x. In order to eliminate this spatial dependency on x in the CKDFSBC and to avoid having to evaluate Eqs. (4.5h and 4.6a) at the unknown location z = fj {x, t), the CKDFSBC is expanded in a Maclauren series (vide., Chapter 2.3.3) about the free-surface equilibrium location at z = 0. The dimensional BVP in Eq. (4.5) may now be assumed to be periodic in both the dimensional horizontal x coordinate over a dimensional interval of periodicity given by the dimensional wavelength A. shown in Fig. 4.1 and in time t over a dimensional interval of periodicity given by the dimensional wave period f, i.e., In = —, CO =
2TC
—,
$(x + i,z,t)
=
(x,z,t),
(4.6f,g)
where k = 2TT/X is the dimensional wave number and (x,z,t) ^ '- = 0, dz r
m
CKDFSBC
m
rj (x,t) d ^ ml dz"
m=0
32
t > 0,
. 3
W 3i
Q(x,z,t) =
_--V*(JC,z,f).V 1 = 0.
(4.7b)
I = -h.
+8
(4.7a)
V<J>(i,z,F) (4.7c)
With the spatial periodicity assumption in the dimensional horizontal x coordinate and the temporal periodicity assumption in the dimensional time t, the dimensional BVP in Eqs. (4.7) for the dimensional scalar velocity potential
93
Long-Crested, Linear Wave Theory (LWT)
? ) = V z O(x,z,0 = 0, |*| < oo, - 1 < z < 0,
t > 0, (4.10b)
3(x,z,t) = 0, ?y(x,0 =
|JC|
< oo,
|x| < oo,
9$(x,z,0 -, dt
z = -l, z = 0, \x\ < oo,
f > 0,
(4.10c)
f > 0, z = 0,
(4.10d) £ > 0, (4.10e)
p(x,z,t) =
d$(x,z,t) dt '
|x| < oo,
- 1 < z < 0,
? > 0.
(4.1 Of) The dimensionless LWT BVP in Eqs. (4.10) is scaled and linearized by setting the dimensionless parameter e = kH/2 « 0; and, consequently, solutions to the dimensionless LWT BVP in Eqs. (4.10) are restricted to small oscillations of the dimensionless free-surface displacement r](x,t) about the equilibrium free-surface z = 0. Because the dimensionless LWT BVP in Eqs. (4.10) is now posed in the infinite rectangular plane —1 < z < 0 and |JC| < oo with kh = 0(1) and with homogeneous boundary conditions in Eqs. (4.10c,d) imposed on values of the dimensionless vertical coordinate z that are constants (viz., z — — 1 and z = 0), the dimensionless LWT BVP in Eqs. (4.10) may now be solved by the method of separation of variables (Carrier and Pearson, 1968, Chapter 6; Hildebrand, 1976, Chapter 9.3; Ince, 1956, Chapter 9.41). The method of separation of the independent variables will result in an homogenous ODE in the vertical coordinate z with homogeneous boundary conditions on the horizontal boundaries z = — 1 and z — 0 that is a well-posed Sturm-Liouville system (vide., Chapter 2.6) that will have eigenfunction solutions with eigenvalues (Carrier and Pearson, 1968, Chapter 6; Hildebrand, 1976, Chapter 5.6; Ince, 1956, Chapter 9.41).
97
Long-Crested, Linear Wave Theory (LWT)
4.3. Solutions to Dimensional Boundary Value Problem (BVP) for Long-Crested, Linear Wave Theory (LWT) For deterministic wave designs of offshore facilities, the design engineer specifies a wave height H, a wave period f and the water depth h. The wavelength A. is an unknown eigenvalue of the well-posed Sturm-Liouville system defined by Eqs. (4.10); and its value depends on the theory applied to solve the system and so it cannot be specified. It is important to remember this point and to not present the wave theory computations in either tabular or graphical formats that are functions of the unknown wavelength; e.g. pressure forces as a function of kh = 2nh/k. In order to evaluate the parametric dependencies of the kinematic and dynamic LWT field variables on the design specifications of wave height H, wave period f and water depth h, the dimensionless LWT BVP in Eqs. (4.10) must be solved in dimensional form (but without the tilde (•) notation for dimensional variables as illustrated in Fig. 4.1). In dimensional form without the tilde (i) notation, the LWT BVP in Eqs. (4.10) is given by q(x,z,t)
= u(x,z,t)ex
+
w(x,z,t)ez
a
0.4
i
i
0.8
i "-•'
1.2
1
'
1.6
1
I
2.0
1
2.4
I
1
2.8 3.0
kh --koh=0.1 — koh=2.0
koh=0.6 — TANH(kh)
Fig. 4.3b. Solutions to linear frequency dispersion equation (4.26a)
guess" and then added to the initial "first guess" estimate. The solution is computed by iteration. Let koh=Xl=X-> represent the initial "first guess" (j = 1) estimate. A Taylor Series expansion about this initial "first guess" estimate is given by F(XJ + 8XJ) = F(XJ) +
dF(X->)
—-8XJ dXJ
•T
+ 0(SXJ)2
= 0.
Retaining only the linear correction gives the correction 8XJ =
F(XJ) 3F(XJ)/dXJ
(4.28a)
108
Waves and Wave Forces on Coastal and Ocean Structures
The (j + 1) estimate for the jth root XJ may be determined iteratively from XJ+1=XJ+SXj,
(4.28b)
where [(k0h)/(XJ)2
+ sech2(XJ)]
The iterations are terminated when the 8 {khy iterative correction is acceptably small (1.0 E — 4, say, Marquardt, 1963). The geometric properties of this iterative method involve constructing tangents to the solution (vide., Sokolnikoff and Redheffer, 1966, pp. 657-659); and, consequently, the initial estimate should be taken near the interval inside of which the root is known to be. Therefore, for the propagating mode, the initial estimate should be the deep water wave number ko = 2/g; i.e. X1 =k0h.
(4.28d)
The final jth iterative solution XJ = kh to the transcendental equation F(XJ =kh) has been identified in LWT as the wave number defined by k=2ir/k where k is the wavelength or interval of horizontal periodicity. Because the numerical value of kh must be computed from a well-posed Sturm-Liouville system in Eqs. (4.11 b-d) or, equivalently, an eigenvalue problem in the vertical z coordinate direction, equivalence of the eigenvalue k to the wave number 2n/k requires that a pseudo-horizontal boundary condition of periodicity in Eq. (4.6e) must be introduced into the boundary value problem (viz., 0 < y = uzh/g = k0h < 1, (4.30a)
where pi = -0.42806826
41 = -0.59553493
P2 = 0.09392834
^ = 0.16262861
p 3 = -0.00269417
93 = -0.01497505
that has a maximum error of 0.0002%, or for the same range of y, the more simpler approximation X
= 1 ^ / ^ = feA> ° < y = « 2 V S < 1, (4.30b) 1 - (y/6) that has a maximum error of 0.04%. Young and Sobey (1980) suggest a "Table-Look-Up" method that may be even more computationally efficient when several thousand wave numbers must be computed (e.g. for shoaling and refraction problems covering large coastal areas). The table is initialized with uniformly spaced values of kh = X to obtain initial values for k^h = co2h/g — y. Uniformly spaced values of y may then be determined in reverse by linearly interpolating between the
110
Waves and Wave Forces on Coastal and Ocean Structures
10 20 30 40 SO 60 70 80 90 100
COMPUTATIONS (10"3)
Fig. 4.4. Comparison of CPU times (after Young and Sobey, 1980).
Table 4.1. Comparison of CPU times in seconds between the Table-Look-Up Method and the Pade Approximation Method for determining 60 000 values of the propagating wave number (Young and Sobey, 1980). Method Table-Look-Up Pade Approximates
Low (0.10%) (sees) 18.6 20.5
High (0.01%) (sees) 20.0 27.5
evenly spaced values of X used to initialize the table. Figure 4.4 compares the CPU times for the Newton-Raphson method with those required from either Pade approximates or the Table-Look-Up method as a function of the number of computations. Table 4.1 compares the CPU times for 60 000 computations for both high (0.01%) and low (0.10%) accuracies. The CPU times for the Newton-Raphson method are totally unacceptable for these numbers of computations. Classification of relative water depths h/X: deep-, finite- and shallow-: The relative water depths h/X for deep-, finite- and shallow-water depths are determined from the asymptotic approximations of the hyperbolic functions as a function of the water depth to wavelength ratio h/X. Table 4.2 summarizes the limiting relative water depth ratios h/X for each of the three relative water
111
Long-Crested, Linear Wave Theory (LWT)
Table 4.2. Summary of relative water depths h/X based on asymptotic values of hyperbolic functions. Hyperbolic Function
Deep-water {h/X > 1/2) [Approx]
sinh(27r/z/A)
(2nh\
1 26XP(
— )
1
/2nh\
*(%) -0.2%
Shallow-water (h/X < 1/25) [Approx] 2nh
€(%)
1.1%
cos\\(2nh/X)
2^{—)
-0.2%
1.0
3.1%
tanh(27r/i//l)
1.0
-0.4%
2nh ~X~
2.1%
depth regions. An error e based on the asymptotic approximations is defined by {[True] - [Approx]} [True] where [True] is the true hyperbolic value and [Approx] is the asymptotic approximate value of the hyperbolic function. Based on the summary in Table 4.2, the following three relative water depths h/X may be determined for LWT: Shallow-water h/X < 1/25 h/X0 < 1/100
Intermediate-water 1/25 < h/X < 1/2 1/100 < h/X0 < 1/2
Deep-water h/X > 1/2 . h/X0 > 1/2
The relative depth h/X definitions in Table 4.2 may now be applied to illustrate that linear surface gravity waves in shallow-water (h/X 1/2) and finite- (l/25 9999; WAVE NUMBER NOT CONVERGING' WRITE(*,*)' koh = ',koh,' kh = ',X STOP END IF END IF XKH = X END IF RETURN END
BothMATHEMATICA™ and FORTRAN computer algorithms that compute the evanescent (i.e. negative separation constants k2 < 0) eigenvalues may be found in Chapter 5.2.
4.4. Eulerian Kinematic Fields and Lagrangian Particle Displacements The Eulerian kinematic field variables for LWT that are identified in Fig. 4.1 are the Eulerian kinematic (motion) fields of the horizontal water particle velocity u(x, z, t) and the vertical water particle velocity w(x, z, t). The independent variables for these Eulerian fields are space x, z and time t. Eulerian fields are continuous functions of both space x, z and time t. The Lagrangian water particle displacements for LWT that are identified in Fig. 4.1 are the horizontal water particle displacement £(xo, zo, 0 and the vertical water particle displacement t, (xo, zo, t) about a fixed equilibrium spatial position xo, zo that may be computed from the Eulerian velocity fields. The independent variables
114
Waves and Wave Forces on Coastal and Ocean
Structures
for these Lagrangian displacements are the water particle and time t; and they represent small displacements about a fixed equilibrium spatial position xo, zoThese Lagrangian water particle displacements are continuous functions of a water particle with respect to time t about a fixed equilibrium spatial position xo, zo; and, consequently, they will not be continuous functions of space x, z as are the Eulerian velocity fields. The kinematic Eulerian velocity fields may be computed from the standing surface gravity wave potentials Eqs. (4.20a-d) for finite-water depth and from Eqs. (4.24a-d) for deep-water. The kinematic Eulerian velocity fields may be computed from the progressive surface gravity wave potentials in Eqs. (4.22a,b and 4.23a,b) for finite-water depth and from Eq. (4.24e,f) for deep-water. The spatial gradients (vide., Chapter 2.2.7) of these same equations will provide the Eulerian velocity fields that may be integrated with respect to time t about a fixed equilibrium position xo, zo in order to obtain the horizontal £(xo,zo,0 and vertical £(xo,zo,0 Lagrangian water particle displacements, respectively. The horizontal ^(xo, zo, 0 and vertical f (xo, zo, 0 Lagrangian water particle displacements may be estimated for both deep- and finite-water depths and for both standing and progressive surface gravity waves from the following Taylor series expansions (vide., Chapter 2.3.3) about an equilibrium position XQ, ZO'§(x0 + Ax,zo + Az,0 = £(x 0 ,zo,0 + dl-(x,z,t) + dz /
M(X 0 ,ZO,
d%(x,z,t) 3x
Az + XQ,ZQ
x)dx + O
f (xo + Ax,zo + Az,0 = t, (x0,zo,t) +
+
di;{x,z,t) dz
/
Ax Az
di;(x,z,t) 3x
(4.31a) Ax
*o,zo
Az + • • • . *o.zo
w{xQ,zo,x)dx + O •
Ax *0>Z0
Ax Az
(4.31b)
115
Long-Crested, Linear Wave Theory (LWT)
Standing surface gravity waves: Finite-water depth: The horizontal u (XQ, ZO, t) and vertical w (XQ, ZO, t) water particle velocities required for the integrals in Eqs. (4.31) for standing surface gravity waves in finite-water depth may be obtained by substituting Eqs. (4.20a-d) into Eqs. (4.2) according to U\A,
-d$>(x,z,t) )— dx
0
(4.32a) (4.32b) (4.32c) (4.32d)
cos(fcx) cos(cot) — sin(fcx) sm(cot) sin(&x) cos(cot) — cos(kx) sin(cot)
(4.32e) (4.32f) (4.32g) (4.32h)
-dT)
f J
- sin(fcxo) sin(a>?) cos(£xo) cos(ft)?) cos(fcxo) sin(cot) - sin(A;xo) cos(a>?)
sin(fur) COS(ftJT)
dx
sin(a)T) (4.33a) (4.33b) (4.33c) (4.33d)
116
Waves and Wave Forces on Coastal and Ocean Structures
S(x0,zo,t)
f
w(xo,zo, x)dx
H gk siahk(zo + h) 2 co cosh kh
cos(£xo) — sin(£xo) sin(A:xo) /' — cos(fcxo)
H gk sinh&(zo + h) cosh kh 2 w7
cos(fcxo) sm(oot) sin(fcxo) cos(oot) sin(£xo) sin(?) = -r expfcoz cos(fco^) cos(cot) 2 co sin(&ox) sin(cot) w(x,z,t) =
(4.35a) (4.35b) (4.35c) (4.35d)
-3 dco = gtanh kh + dco dk CG
kh cosh2 kh
dk,
kh g tanh kh 1 + 2co tanh kh cosh2 kh C 2kh 2kh i + = co 1 + sinh 2kh sinh 2kh 2k Cn
(4.60d)
where the wave celerity C = co/k = k/T. The wave group velocity may be interpreted as the speed of advance of the wave energy in a group of waves as illustrated by wave beats in Fig. 6.8 in Chapter 6.4. The relation between the wave group velocity CG and the wave
Waves and Wave Forces on Coastal and Ocean Structures
Table 4.5. Summary of wave group velocity to wave celerity ratios CQ/C based on asymptotic values of hyperbolic functions and relative water depths. Relative water depth
CG/C = n
h 1 Deep-water: — > A ^
A
2kh
h
• t. ! Finite-depth: — < — < !v 25 A 2 h 1 Shallow-water: — < — A 25 J
1 + sinh 2kh 1
celerity C in deep-, finite- and shallow-water are summarized in Table 4.5 after applying the asymptotic approximations for the hyperbolic functions from Table 4.2. The group velocity is equal to the wave celerity in Table 4.5 in shallowwater h/X < 1/25 because shallow-water waves are non-dispersive where C = sfgh and because the wave celerity does not depend on the wave number k. Conservation of energy flux or wave power between two vertical planes that are perpendicular to the direction of wave propagation requires that
BICGM
(4.61a)
=E2Cc2b2.
Substituting Eqs. (4.57c and 4.60d) into Eq. (4.61a) gives the following ratio between the wave height Hi at location 2 nearshore and wave height H\ at location 1 offshore:
i +
N =
{KS}KR,
i +
2k\h\ sinh 2k\ h\ Ikjhi
sinh Ikih 2 (4.61b)
139
Long-Crested, Linear Wave Theory (LWT)
where the shoaling coefficient K$ = {•} term in curly brackets in Eq. (4.61b) is due to the change in bathymetry between the two locations; viz.,
2k\h\
1 + sva\a.2k\h\
KS =
(4.61c)
2kihi
"\J
1 + sinh 2&2/12
and the refraction coefficient KR is due to the change in the wave direction; viz.,
b\ bi
(4.61d)
In many ocean engineering design applications, Eq. (4.61b) is applied to a deep-water site at an arbitrary location in deep-water offshore and to a design site nearshore. If site " 1 " is a deep-water site in Eqs. (4.61) (i.e. subscript " 1 " = "0") and site " 2 " is a design site nearshore, then Eqs. (4.61) may be modified as follows: E0CGofco = EC G &, "
H Ho
CGo fb~o~ V CGi b
2k0h0 sinh 2koho _
Mo
i
(4.62a)
VT
"1
2 k h
,
'
^0"
• J b
sinh 2kh _
\
= {KS}KR, "
N
KR
F
2koho sinh2&o/io 2kh ' 1H sinh 2kh _
vV*
\
Mo
K» -
(4.62b)
V b•
[
V
2 i
2JkA
/*oA "
1
V V2^
(4.62c)
sinh 2kh _ (4.62d)
The transformations due to the shoaling Ks and to the refraction KR coefficients are discussed separately in Sec. 6 below.
140
Waves and Wave Forces on Coastal and Ocean Structures
General conservation principle for surface gravity waves: Whitham (1974) outlines very eloquently the general conservation principles for surface gravity (as well as many other) waves. For a wave density property P and flux density Q(P), the general conservation principle is analogous to the Reynolds transport theorem and is given by 9P
3Q(P)
In many applications of classical mechanics (e.g. Goldstein, 1980), aproperty that satisfies Eq. (4.63) is the energy per radian frequency E/co that is called wave action. Substituting wave action for propagating surface gravity waves into Eq.(4.63) results in 9 /E\ dt \coJ
3 /E dx \co
\ _ J
where the average energy density E per horizontal surface area for a propagating surface gravity wave in both finite- and deep-water depths is given by Eq. (4.57c) and the group velocity Co by Eq. (4.60d). Because the average energy density E per horizontal surface area is averaged over one wave period T, the energy density is independent of time and the first temporal derivative in Eq. (4.64) is identically equal to zero. This result implies that no waves may accumulate in a volume of fluid in an interval of time as discussed in the conservation of wave period with respect to depth in Sec. 3. Consequently, the spatial derivative in Eq. (4.64) is also equal to zero so that the flux or transport of wave action by the group velocity must be a constant E/co = constant and Eq. (4.61a) follows. Related to this conservation of wave action is the conservation of wave phase that may be computed by Eq. (4.63) in the form of — + Vco = 0 dt that is reviewed by Dean and Dalrymple (1991) and by Mei (1989).
(4.65)
4.6. Wave Transformations for Long-Crested, Progressive Linear Waves: Shoaling and Refraction The conservation of energy flux derived in Sec. 5 yields two separate effects in Eq. (4.61b): viz., (1) shoaling effects Ks due to changes in
Long-Crested, Linear Wave Theory (LWT)
141
the bathymetry and (2) refraction effects KR due to changes in the direction of wave energy propagation as a result of changes in bathymetry. Each of these effects may be analyzed separately. Because most applications of Eq. (4.61b) involve the changes in wave properties from deep- to shallow-water, the wave properties are normalized or referenced to deep-water conditions and are illustrated in Fig. 4.11. The combined effects of shoaling and refraction KSKR for straight and parallel bottom contours are illustrated in Fig. 4.13. Shoaling Ks'. The shoaling coefficient Ks is given by Eq. (4.61c) where the ratio n in Eq. (4.60b) is given by
that is illustrated in Fig. 4.11 over two decades of the dimensionless relative water depth h/Xa, where XQ = gT2/2n. Again, the abscissa in Fig. 4.11 may be computed directly from the design specifications of wave height H, wave period T and water depth h whereas the dimensionless water depth h/X requires a wave theory in order to compute the wavelength X. For this reason, wave properties should always be graphed or tabulated as functions of the design specifications h/X$ and never h/X. The ratio of the wave celerities C/CQ = X/XQ = tanh kh is also illustrated in Fig. 4.11 along with the dimensionless water depth h/X. Recall that in shallow-water that h/X < 1 /25 and that the tanh kh «a kh from Table 4.2. This may be observed in Fig. 4.11 where the curves C/Co = X/XQ = tanh kh and h/X are seen to be parallel for /2/A.0 < 0.1. The ratio (C/Co)/(h/X) for a fixed value ofh/Xo is approximately equal to lit for values ofh/Xo < 0.06 that corresponds to values of h/X < 1/10. The relative wave steepness (H/X)/(HO/XQ) is equal to K^(Xo/X) and is illustrated in Fig. 4.11. When KR — 1, the shoaling coefficient Ks in Eq. (4.62c) is equal to H/Ho and is illustrated in Fig. 4.11 by the curve identified as Ks- It is interesting to observe from this curve in Fig. 4.11 that as a long-crested wave propagates from deep-water towards finite-water depth that when it first encounters the effects of finite-depth at h/Xo = 1/2 that the deep-water wave height HQ initially begins to decrease as the shoaling coefficient Ks is less than unity in the dimensionless water depth interval 0.06 < h/Xo < 0.5. It is not until
Waves and Wave Forces on Coastal and Ocean Structures
142
rfctfcU -1--t-1-H
0.01 0.01
Trrn ~rrrn I I I 11
0.1 h/X0
Fig. 4.11. Dimensionless linear wave transformation properties for waves normally incident from deep-water (subscript "0") with Kg = 1 and lit/Xo = a> /g = £Q-
the dimensionless water depth h/Xo < 0.06 that the wave height begins to increase in value above the deep-water value and the wave eventually breaks nearshore (vide., Chapter 6.6). The shoaling coefficient Ks in Eq. (4.61c) from linear wave theory may also be written as a function of deep-water conditions with Ai = A,o and A.2 = A. as ,2
xi =
^_o
A-o
;JI\ll ++ _2kh_] 2kh 1 _
2kn'
l ^ sinh2*^ J
(4.67)
where n is given by Eq. (4.60b). Refraction KR. Wave ray refraction may be computed from Snell's law (vide., Morse and Feshbach, 1953 or Morse and Ingard, 1968) that may be written for long-crested surface gravity waves as sinoto sina(x) C(x) Co (4.68) sina(i) =
sinao, A.Q
where the subscript "0" refers to deep-water conditions that are illustrated schematically in Fig. 4.12. Wave energy is transported in the direction of the wave rays perpendicular to the wave crests or wave fronts and there may be no flux or transport of wave energy along the wave crests or wave fronts in
143
Long-Crested, Linear Wave Theory (LWT)
a
o \ j Deep ocean
Rayt Depth contours'
AT _ _«M
,3
h
4!
n
Fig. 4.12. Snell's law for idealized, straight and parallel depth contours (adapted from Dean and Dalrymple, 1991).
long-crested waves. Because the wave period is conserved (vide., Sec. 3), there may be no accumulation of waves between the depth contours in a time interval r as was demonstrated in Sec. 5 by the conservation of energy action principle and, consequently, the width between wave rays l(x) measured parallel to the depth contours in Fig. 4.12 must be a constant; viz., I (x) = constant. The refraction coefficient KR may now be expressed in terms of the angle a (x) between the wave fronts and the depth contours in Fig. 4.12 (or, equivalently, between the wave rays and the shore normal); i.e.
KR =
Ib0 b(x)
I ^ocosao £(x) cosa(x)
cosao cosa(x)'
(4.69)
where the distance x is measured perpendicular to the depth contours and positive offshore in Fig. 4.12. For irregular offshore bathymetry, digital computer algorithms are available for constructing wave refraction diagrams. For depth contours that are straight and parallel to a coastline, the product of KSKR may be computed from the product of Eqs. (4.61c and 4.69) and is illustrated in Fig. 4.13.
144
Waves and Wave Forces on Coastal and Ocean Structures
O.0O0I
0.00O2
0.0004 0.0006
0.001
0.002
0.004 0.006
0.01
0.02
0.04
0.06
0.1
h/gT2 Fig. 4.13. Combined shoaling and refraction coefficient KSKR for straight and parallel bottom contours (Shore Protection Manual, 1984).
4.7. Problems 4.1.
A measured progressive wave has a wave height H = 4 ft. (1.22 m) and period T = 12 sec. Find the maximum values for Umax, Umax, £max, and £max at z = 0, -h/2, and -h in a total depth h = 100 ft. (30.48 m). What are the values of u, w, §, and £ at z = 0, -h/2, and -h for 6 = kx - cot = 45°? How long will it take for the wave crest to travel a distance x = 146 ft (45 m); = 584 ft, (178 m); and one wavelength = A.? 4.2. An electromagnetic current meter records a maximum horizontal water particle speed of 5.5 ft/sec. (1.7 m/sec) for a progressive wave with a period T = 9.5 sec. at mid-depth in a total water depth h = 75 ft. (22.9 m). What is the wave height H of the linear progressive wave being recorded? 4.3. Two pressure sensors are located as shown in the diagram. The pressure amplitudes recorded for a progressive wave with a period T = 8 sec. by Sensors #1 and #2 are 272 psf (13 kPa) and 320 psf
145
Long-Crested, Linear Wave Theory (LWT)
(15.32 kPa), respectively. What are the depth h, wave height H, and wavelength k, of the progressive wave?
TSWL
^Sn . h
Sensor #2
Sensor #1
25 ft
£5^
Problem 4.3. (after Dean and Dalrymple, 1991)
4.4.
Substituting the frequency dispersion relationship (2n/T)2 = g(2n/k)t&nh(2Tth/k) from Eq. (4.15a) in Sec. 4.3, verify that the horizontal and vertical water particle velocities, respectively, for a progressive linear wave given by Eqs. (4.37b and 4.37d) in Sec. 4 may be transformed to the following: Hit cosh[2jr(z + h)/k] (x cos 2n\ T sinh(2nh/k) \k
u= w=
Hit sinh[27r(z + /i)A] . (x — sin in I T sinh(27r/i/A.) \X
t T t T
and that the horizontal and vertical water particle accelerations, respectively, for a progressive linear wave given by Eqs. (4.50b and 4.50d) in Sec. 5 may be transformed to the following: du(x,z,t) dt dw(x,z,t)
=2H
„ „ / ^ \ sinh[27t(z + h)/k] = —2H\
4.5.
n\ 2 .cosh[27z(z + h)/k] . (x , —I sin 2TT ( : \TJ sinh(2nh/X) \X —
:
„ (x C0S27T
t T t\ ).
dt \TJ smh(2jth/k) \k Tj A linear progressive wave with the following deep-water characteristics propagates towards the coast: HQ = 5 ft. (1.52 m), T = 15 sec. At a particular nearshore site with a depth h = 15 ft. (4.57 m) a refraction diagram indicates that the spacing between the wave orthogonals is one-half the spacing in deep-water b(x) = bo/2.
Waves and Wave Forces on Coastal and Ocean Structures
(a) Compute the wave height H and wavelength k at the nearshore site. (b) Compute the wave power being propagated through one foot (m) of width. (c) Assuming that one-half the energy flux in deep-water is lost to friction, compute the wave height H. A linear progressive wave with the following deep-water characteristics is propagating towards the shore in an area where the bottom contours are all straight and parallel to the coastline: HQ = 20 ft.; T = 15 sec. The bottom is composed of a sand of 1/8 in. diameter. If a water particle velocity of 1 ft/sec is required to initiate sediment motion, what is the greatest depth in which sediment motion can occur? (after Dean and Dalrymple, 1991) A deep-water progressive linear wave with the following deep-water characteristics is propagating towards a straight coastline with parallel bottom contours and bottom slope V : H = 0.05, H0 = 30 ft (9.14 m), 7 = 12 sec. At a nearshore station, a wave height H = 32 ft (9.75 m) is measured. What is the water depth at this site? For the two design wave generation areas shown, which of the two wave generation areas (#1 or #2) would produce the highest wave at the design site where the water depth is h = 20 ft? What would be the wave heights H, wavelengths k and wave directions a at the design site from each of the two wave generation areas? Assume straight and parallel bottom contours and that waves from the two wave generation areas do not occur simultaneously. V v
AREA #1 H 0 =12ft T=8sec <xo=20°
Problem 4.8.
147
Long-Crested, Linear Wave Theory (LWT)
4.9.
The harbor entrance shown is designed for the following deep-water wave conditions: H$ — 20 ft. (9.14 m), T = 18 sec. The side walls are vertical. What must be the slope shown between A and B be if the design wave height at Station B is H = 15 ft (4.57 m) if bK = 300 ft (91.44 m), hA = 40 ft (12.19 m), hB = 26 ft (7.92 m)? Assume as a first approximation that the rate at which wave energy is propagated into the harbor is simply (P}? = EAQJA&A-
^\^^^ B A
:\\\>
(x, z, t) are given first. Next, the fully nonlinear WMBVP is scaled in order to determine a small dimensionless parameter e, say, that will make it possible to linearize the WMBVP. The classical dimensional 2D planar wavemaker theory is reviewed in Sec. 2 for two types of double articulated planar wavemakers. The sway X2O) displacement of a full-draft piston wavemaker and the roll @5(0 rotation of a hinged wavemaker are connected directly to the sway displacement and the roll rotation of the semi-immersed, large Lagrangian solid body that is shown in Fig. 5.2 and that are reviewed in Chapter 8. This connection to the radiation BVP for large Lagrangian solid bodies in Chapter 8 illustrates the direct relationship that exists between the WMBVP that is reviewed here and the construction of radiated wave potentials by the radiation BVP that is reviewed in Chapter 8. In the review in Sec. 2, integral calculus formulae for computing the integrals that are required to compute the coefficients of the eigenseries for the fluid motion as well as the integrals required to compute forces and moments on the wavemaker are replaced by algebraic formulae. For example, an integral equation that is required to compute the nth eigenseries coefficient Cn for the nth orthonormal eigenfunction tyn(Kn, z/h),
153
Wavemaker Theories
say, from a wavemaker shape function x (z/h) may be computed symbolically and expressed by a dimensionless algebraic formula In(ce, P, b, d, Kn) that is given by Cn=h
I
x(z/h)V„(z/h)d(z/h)
= In(a,p,b,d,Kn).
(5.2)
The coefficient in Eq. (5.2) may then be computed very efficiently by substitution into the algebraic formula given by In(oe, fi, b, d, K„) in all subsequent applications such as computing forces and moments on the wavemakers or the power required to generate waves by a wavemaker. Replacing integral calculations with algebraic substitutions into algebraic formulae is a very efficient method for reducing repetitive integrations of the same function. In Sec. 3, the dimensional theory for both amplitude-modulated (AM) andphase-modulated (PM) circular wavemakers are reviewed. In Sec. 4, a dimensionless theory for double-actuated planar wavemakers is reviewed. Because the derivation in Sec. 4 is dimensionless, it provides an opportunity to compare a derivation of a dimensionless wavemaker theory with a derivation of a dimensional wavemaker theory for 2D planar wavemakers that is reviewed in Sec. 2. In Sec. 5, a dimensional directional wavemaker theory for relatively large wave basins that is based on a WKBJ approximation (Morse and Feshbach, 1953, Chapter 9.3) is reviewed. In Sec. 6, a theory for sloshing waves due to the transverse motions of a segmented wavemaker in a relatively narrow wave channel is reviewed. In Sec. 7, 2D planar wavemakers are mapped to a unit circle and to a fixed rectangular domain and both the linear and nonlinear wavemaker solutions are computed numerically. Conformal mapping is applied to map the WMBVP to the unit disk; while domain mapping is employed to map the fully nonlinear, time-dependent WMBVP to a fixed rectangular domain. NonlinearWavemakerTheory. With the exception of the numerical domain mapping theory reviewed in Sec. 7, the wavemaker theories that are reviewed are linear wavemaker theories. Chaotic instabilities generated by 2D planar wavemakers that are called cross-waves are a strongly nonlinear phenomena that may be evaluated by the generalized Melnikov method (GMM) and by the largest Liapunov characteristic exponent. The GMM is adopted from analyses by the Floquet theory for the parametric excitation of non-linear dynamical systems. Because cross-waves are strongly nonlinear waves, their review is
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Waves and Wave Forces on Coastal and Ocean Structures
given in Chapter 6.8. Because the GMM computes Smale horseshoes from the recurring transverse intersections of the stable and unstable manifolds about a separatrix, the WMBVP for chaotic cross wave instabilities is reviewed by constructing a Lagrangian density for the WMBVP; and then obtaining a Hamiltonian by the Legendre transform from which the fixed points may be computed that are connected by a separatrix that is required for applications of the GMM. The WMBVP that is reviewed in Chapter 6.8 includes both surface tension and dissipation.
5.2. Planar Wavemakers in a 2D Channel The Havelock (1929) linear wavemaker theory for planar wavemakers is reviewed because wave laboratories are frequently employed in ocean and coastal engineering applications to obtain empirical data. This powerful theory is not only the basis for a variety of practical ocean and coastal engineering applications that take place in laboratory wave flumes; but it is also the boundary value problem for the radiated wave potential that is applied in computing the forces on large Lagrangian solid bodies by potential theory in Chapter 8. Havelock (1929) applied Fourier integrals to develop a theory for forced surface gravity waves in water of both infinite and finite depth. Both planar and circular wavemakers were treated. Kennard (1949) also applied Fourier integrals but included the initial value problem. Biesel and Suquet (1953) obtained explicit linear solutions for the wave motions generated by both a piston and a hinged wavemaker. Hyun (1976) extended the work of Biesel and Suquet (1953) to include hinged wavemakers of variable draft. The solution presented by Hyun (1976) for a hinged-type wavemaker of variable-draft was extended by Hudspeth and Chen (1981a) within the limits of linearized potential wave theory to wave flumes having two constant depth regions that are connected by a gradually sloping transition region. Design curves for the dimensionless wavemaker gain function, the dimensionless hydrodynamic pressure moment, moment arm, and wavemaker power for this type of wave flume geometry were presented. Experimental verifications of those design curves for monochromatic waves were obtained at the Oregon State University O.H. Hinsdale-Wave Research Laboratory (OHH-WRL) and were reported by Hudspeth et al. (1981b). Experimental verifications of random forces on hinged wavemakers have also been reported by Hudspeth and Leonard (1980).
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Wavemaker Theories
/7777777777777777777777777777/
\VsVs^VAV>/.
Fig. 5.3. (a) Definition sketch for a Type I planar wavemaker. (b) Definition sketch for a Type II planar wavemaker.
Wave heights predicted by these wavemaker theories have been verified experimentally for a piston wavemaker by Ursell et al. (1960), Galvin (1964), and Keating and Webber (1977); and for a hinged wavemaker of variable draft by Galvin (1964), Patel and Ionnaou (1980), and Hudspeth et al. (1981b). Two generic planar wavemaker configurations that may be described in Cartesian coordinates are shown in Figs. 5.3a and b. The displacements of the planar wavemakers in these two figures may be expressed by £(z, t) and they generate wavey, two-dimensional, irrotational motion of an inviscid, incompressible fluid in a semi-infinite channel of constant still water depth h. The fluid motion may be obtained from the negative gradient vector of a scalar velocity potential (x,z,t),
(5.3a)
where the two-dimensional gradient vector operator (Eq. (2.10a) in Chapter 2.2.7) is 3(.). V(.) = -V-e* 9 ( . )e. z. dx dz The total pressure field P(x,z,t) may be computed from the unsteady Bernoulli equation (4.3) by P{x,z,t) = p{x,z,t)
~
P
+ ps(z)
ld$>(x,z,t)
1
dt
1
2
Vd>(x,z,0
+ Q(t)
pgz,
(5.3b)
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Waves and Wave Forces on Coastal and Ocean Structures
where Q(t) is the Bernoulli constant, and the free surface elevation r){x, t) for zero pressure by . 1 \d<S>{x,ri,t) ( n(x,t) — — { Sl
dt
1 2
V$>(x,r],t)
+ Q(t)
)•
x > %(ri,t), z = rj(x,t).
(5.3c)
The scalar velocity potential $(x, z, t) must be a solution to the continuity equation given by the Laplace equation V 2 $ = 0,
x>%(z,t),
-h
30
1 (kA)
**>
3<J> +
«
\si
x>(ksmfj,t) (5.5e)
+ Q(t),
and the total pressure P(x,z,t) from the unsteady Bernoulli equation Eq. (5.3b) according to P(x,z,t)
=
3$ 1 (kA) ~dl~2 (kh)2
i>(*S)£(z,0,
m
*F
fcA -l