backend.orbit.dtu.dki Preface This thesis is submitted as partial fulfillment for the degree of Doctor of Philosophy at the Technical University of Denmark …

A Deterministic Combination of Numerical and Physical Models

for Coastal Waves

Haiwen Zhang

Technical University of Denmark Department of Mechanical Engineering

Maritime Engineering December 2005

Published in Denmark by

Technical University of Denmark

Copyright Haiwen Zhang 2005

All rights reserved

Maritime Engineering

Department of Mechanical Engineering

Technical University of Denmark

Studentertorvet, Building 101E, DK-2800 Kgs. Lyngby, Denmark Phone +45 4525 1360, Telefax +45 4588 4325

E-mail: [email protected]

WWW: http://www.mek.dtu.dk/

Publication Reference Data

Zhang, Haiwen

A Deterministic Combination of Numerical and Physical Models for Coastal

Waves

PhD Thesis Technical University of Denmark, Maritime Engineering

December 2005

ISBN 87-89502-66-3 Keywords: deterministic combination, numerical model, physical model,

coastal waves

i

Preface This thesis is submitted as partial fulfillment for the degree of Doctor of Philosophy at the Technical University of Denmark (DTU), Lyngby, Denmark. This work has been carried out over the time period July 1, 2002 till December 31, 2005 in the Department of Mechanical Engineering (MEK), DTU, also in DHI Water & Environment (DHI), Hørsholm, Denmark. The period from February 1, 2005 to July 31, 2005 was spent as an employee at DHI on leave from this study. The study has been supervised by Dr. Harry B. Bingham (MEK), and Dr. Hemming A. Schäffer (DHI). I do appreciate their great help during the period of study. This study was supported financially by the Danish Technical Research Council (STVF grant no. 26-01-0043). Their support is greatly acknowledged. Finally, many thanks to colleagues at MEK and DHI, friends and family for encouragement and support. Haiwen Zhang Lyngby, Denmark December, 2005

ii Preface

iii

Executive Summary A deterministic combination of numerical and physical models for coastal waves The main objective of this study is to make a deterministic combination of numerical and physical models for coastal waves. For this purpose, an ad hoc unified wave generation theory for wave flumes and basins is developed which combines linear fully dispersive wavemaker theory and nonlinear shallow water wavemaker theory. Numerical models and physical models are two main approaches to study water wave problems in coastal engineering. Numerical models are often used for large areas, but they are typically unable to capture highly nonlinear physics including wave breaking. Physical models are suitable to simulate complex nonlinear processes near the shore or near fixed or floating structures, but they are restricted by the scale and the size of the facility. The limited extent of the physical model often prohibits that the offshore boundary is located in sufficiently deep water for the incident waves to be well described by standard, parameterized wave spectra. As this is typically the only available incident wave description, the limited size of the model facility often precludes important local phenomena like refraction and diffraction. The integrated use of the two approaches offers an attractive alternative to using either alone. A suitable combination is a physical model focusing on the complex part of the problem near the shore or near structures and a numerical model for the surrounding wave transformation. Typically, the combination has been to use a numerical model for the determination of the wave conditions at the boundary of the physical model. In the traditional combined modelling approach, the data transfer between the two models is only made on a stochastic level through bulk parameters such as significant wave height and peak period. This study focuses on a deterministic combination of a numerical and a physical model. The data transfer between two models is thus on a deterministic level with detailed wave information like a time series of surface elevation transferred along the wavemaker. No attempt is made to obtain a two-way coupling, where reflected waves can enter the numerical model from the physical model. Based on linear fully dispersive wavemaker theory and nonlinear shallow water wavemaker theory, an ad hoc unified wave generation method is developed for wave flumes and basins. It accounts for shallow water nonlinearity and compensates for local wave phenomena (evanescent modes) near the wavemaker. For small amplitude linear waves, the fully dispersive wavemaker theory is recovered. For shallow water waves, it is consistent with nonlinear long wave generation. This approach offers a deterministic link between the numerical and physical models. In the combined model, various numerical models can be used for the numerical wave computations in the far field. In this study, a Boussinesq model MIKE 21 BW is

iv Executive Summary

chosen. Segmented, piston-type wavemakers and the associated control system with active absorption provide the interface between the numerical and physical models. The wavemakers are controlled for simultaneous wave generation and active absorption by an Active Wave Absorption Control System for wave flumes (DHI AWACS) and wave basins (DHI 3D AWACS). Three different types of wavemaker control are offered by the DHI (3D) AWACS. Position mode is compatible with the general approach to nonlinear wave generation. However active absorption is not included. Single mode is a traditional approach for active absorption. The weakness of it is inconsistent nonlinear wave generation. A third method termed dual mode has been developed recently. This allows for consistent nonlinear wave generation in combination with active absorption. The active absorption appears as a linear perturbation on the nonlinear wave generation. The control signals in dual mode are provided by the ad hoc unified wave generation. In combined wave flumes, some cases on regular and irregular, linear non-shallow and nonlinear shallow waves are tested. All the measurements in physical flumes match the numerical calculations well. The results are not very sensitive to the positioning of the wavemaker in the combined model, i.e. it is not critical where the physical model takes over from the numerical model. The dominant error in the combined model is due to the limitation of the wave theory (i.e. Cnoidal wave theory, or Boussinesq equations) rather than the ad hoc unified wave generation procedure. For the wave basin tests, we first eliminate the numerical model to test oblique nonlinear waves generated by the 3D unified wave generation method. This is done by substituting the numerical model by Cnoidal wave theory and Stream Function theory, respectively, in order to test the unified wave generation method. Both can reproduce oblique nonlinear waves successfully in general. The difficulty in obtaining oblique nonlinear waves of constant form is mainly due to the limitation of the facility on the width of the paddles. To test the deterministic combined model of multidirectional waves in basins, three irregular cases are tested, which include oblique irregular waves propagating on constant water depth, directional irregular waves propagating on constant water depth, and irregular waves behind a breakwater after propagating up a slope. The time series of surface elevation measured in the physical basin match the numerical calculations well. Thus, the deterministic combination of numerical and physical wave models is considered to be successful. Finally, highly nonlinear wave generation in wave flumes is discussed as spin-off in this study. An approximate Stream Function wavemaker theory which is based on Stream Function wave theory and the ad hoc unified wave generation method, is compared with linear wavemaker theory, second-order Stokes wavemaker theory and Cnoidal wavemaker theory. The advantage of the approximate Stream Function wavemaker theory is obvious in shallow and intermediate water depth, especially for highly nonlinear shallow water waves. Furthermore, an improved Stream Function wavemaker theory is devised which is based on Stream Function theory for progressive waves, and the linear theory for evanescent modes. This one-step approach matches the previous approximate Stream Function wavemaker theory for linear waves, as well as for nonlinear shallow water waves.

v

Synopsis Deterministisk kombination af numeriske og fysiske modeller for bølger ved kyster

Hovedformålet med dette studium er at lave en deterministisk kombination af numeriske og fysiske modeller for bølger nær en kyst. Til dette formål er der udviklet en ad hoc forenet bølgegenereringsteori for bølgerender og bassiner, som kombinerer lineær fuldt dispersiv bølgegenereringsteori og ikke-lineær bølgegenereringsteori for fladvandsbølger. Numeriske modeller og fysiske modeller udgør to hovedtilgange til studiet af vandbølgeproblemer indenfor kystingeniørområdet. Numeriske modeller bliver ofte anvendt i forbindelse med store områder, men de er typisk ikke i stand til at fange den stærkt ikke-lineære fysik såsom bølgebrydning. Fysiske modeller er egnede til at simulere komplekse ikke-lineære processor nær ved en kyst, men de begrænses af modelskala og størrelse af forsøgsfaciliteten. Den begrænsede størrelse af den fysiske model forhindrer ofte, at dybdvandsranden kan lægges på tilstrækkelig dybt vand til at de indkomne bølger kan beskrives fyldestgørende ved standard, parametriserede bølgespektra. Da dette typisk er den eneste tilgængelige beskrivelse af de indkomne bølger, udelukker den begrænsede størrelse af modelfaciliteten ofte vigtige lokale fænomener som refraktion og diffraktion. Den integrerede brug af den fysiske og numeriske fremgangsmåde udgør et attraktivt alternativ til at bruge metoderne hver for sig. En velegnet kombination opnås ved en fysisk model, der fokuserer på den komplekse del af problemet nær ved en kyst eller nær ved konstruktioner og en numerisk model for den omgivende bølgetransformation. Typisk har kombinationen bestået is at anvende den numeriske model til at bestemme bølgeforholdene ved randen af den fysiske model. I den traditionelle tilgang til den kombinerede modellering, er dataoverførslen mellem de to modeller kun foregået på stokastisk niveau gennem helhedsparametre såsom signifikant bølgehøjde og peak periode. Dette studium fokuserer på en deterministisk kombination af en numerisk og en fysisk model. Dataoverførslen mellem de to modeller foregår således på et determinisk niveau med detaljeret bølgeinformation såsom tidsserier af overfladeelevation overført langs med bølgemaskinen. Der gøres ikke forsøg på at opnå en to-vejs kobling, hvor reflekterede bølger kan løbe fra den fysiske model til den numeriske model. Baseret på lineær fuldt dispersiv bølgegenereringsteori og ikke-lineær bølgegenereringsteori for fladvandsbølger, udvikles der en ad hoc forenet bølgegenereringsmetode for bølgerender og bassiner. Denne tager højde for fladvandsikke-linearitet og kompenserer for lokale fænomener (evanescent modes) nær bølgemaskinen. For lineære bølger, reduceres metoden til fuldt dispersiv bølgegenereringsteori. For fladvandsbølger er metoden konsistent med ikke-lineær

vi Synopsis

generering af lange bølger. Denne metode giver mulighed for en deterministisk forbindelse mellem den numeriske og den fysiske model. I den kombinerede model kan der anvendes diverse modeller for de numeriske beregninger af fjernfeltet. I nærværende studium er der valgt en Boussinesq model, Mike 21 BW. Segmenterede bølgemaskiner af stempeltypen og det tilhørende kontrolsystem med aktiv absorption udgør interfacet mellem den numeriske og den fysiske model. Bølgemaskinerne styres med samtidig bølgegenerering og aktiv absorption vha. Active Wave Absorption Control System til bølgerender (DHI AWACS) og bølgebassiner (DHI 3D AWACS). DHI (3D) AWACS kan give tre forskellige typer bølgemaskinestyring. Positionsstyring er i overensstemmelse med den almindelige metode til ikke-lineær bølgegenerering. Imidlertid tillader denne ikke aktiv absorption. "Single mode" er den traditionelle metode for aktiv absorption. Svagheden ved denne teknik er, at den er inkonsistent for ikke-lineær bølgegenerering. En tredje metode, der kaldes "dual mode" er blevet udviklet for nylig. Denne metode tillader konsistent ikke-lineær generering kombineret med aktiv absorption. Den aktive absorption optræder som en lineær perturbation på den ikke-lineære generering. Styresignalerne i dual mode leveres af den ad hoc forenede bølgegenereringmetode. I kombinerede bølgerender testes forskellige tilfælde af regelmæssige og uregelmæssige, lineære og ikke-lineære fladvandsbølger. Målingerne i den fysiske rende stemmer overens med de numeriske beregninger. Resultaterne er ikke særlig følsomme overfor placeringen af bølgemaskinen i den kombinerede model, dvs. det er ikke kritisk, hvor den fysiske model tager over fra den numeriske model. Den dominerende fejl i den kombinerede model skyldes begrænsningen af bølgeteorien (dvs. Cnoidal bølgeteori eller Boussinesq ligningerne) snarere end den ad hoc forenede bølgegeneringsmetode. I forbindelse med tests i bølgebassin, elimineres den numeriske model i første omgang for at teste skrå ikke-lineære bølger genereret vha. den forenede 3D genereringsmetode. Dette gøres ved at erstatte den numeriske model med de respektive teorier for Cnoidale bølger og "Strømfunktionsbølger" med det formål at teste den forenede bølgegenereringsmetode. Generelt fører begge teorier til vellykket reproduktion af skrå ikke-lineære bølger. Vanskeligheden med at opnå skrå ikke-lineære bølger af konstant form skyldes hovedsagelig facilitetens begrænsning i kraft af bredden af bølgemaskineflapperne. For at teste den kombinerede deterministiske model for retningsspredte bølger i bassiner, testes der tre irregulære tilfælde, heriblandt skrå, uregelmæssige bølger på konstant vanddybde og uregelmæssige bølger bagved en bølgebryder efter at bølgerne er løbet op ad en skråning. Tidsserier af overfladeelevation målt i det fysiske bassin passer godt med de numeriske beregninger. Således kan den deterministiske kombination af en numerisk og en fysisk model anses for at være vellykket. Endelig diskuteres stærkt ikke-lineær bølgegenerering som en sidegevinst af dette studium. En tilnærmet "Strømfunktions" bølgemaskineteori baseret på "Strømfunktions bølgeteori" og den ad hoc forenede bølgegenereringsmetode er sammenlignet med lineær bølgegenereringsteori, anden ordens Stokes bølgegenereringsteori og Cnoidal bølgegenereringsteori. Fordelen ved den tilnærmede

Synopsis vii

Strømfunktions-bølgegenereringsteori er tydelig på fladt vand og mellemstor vanddybde, især for stærkt ikke-lineære fladvandsbølger. Ydermere, er en forbedret Strømfunktions-bølgegenereringsteori anvist, som er baseret på Strømfunktionsteori for fremadskridende bølger og den lineære teori for evanescent modes. Denne et-trins metode svarer til den tidligere tilnærmede Strømfunktionsbølgegenereringsteori i tilfælde af lineære bølger eller ikke-lineære fladvandsbølger.

viii Synopsis

ix

Contents

Preface i

Executive Summary iii

Synopsis (in Danish) V

Contents iX

Symbols Xiii

1 Introduction 1 1.1 Overview and Background….…………………………….ΩΩΩΩ..ΩΩ....1

1.2 Approaches……………….....…………………………….Ω…ΩΩ..ΩΩ....4

1.3 Thesis Outline…………...….…………………………….ΩΩΩΩ..ΩΩ….5

2 Linear Wavemaker Theory 7

2.1 Governing Equations…………………………………….ΩΩΩΩΩΩΩ....7

2.2 SolutionsΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩ….ΩΩΩΩΩΩΩ.….9

2.2.1 Progressive wavesΩΩΩΩΩΩΩΩΩ….ΩΩΩΩΩΩΩ….…10

2.2.2 Evanescent modesΩΩΩΩΩΩΩΩΩ….ΩΩΩΩΩΩΩ…….10

2.2.3 Complete solutionsΩΩΩΩΩΩΩΩΩ….ΩΩΩΩΩΩΩ…...11

2.2.4 Solutions in complex notation ΩΩΩΩΩΩΩΩΩ….ΩΩ….....13

3 Cnoidal Wavemaker Theory 17

3.1 Normal Cnoidal WavesΩΩΩΩΩΩΩΩΩ….ΩΩΩΩΩΩΩ……....17

3.1.1 Solution of normal Cnoidal waves..Ω….ΩΩΩΩΩΩΩ…….....17

3.1.2 The generation of normal Cnoidal waves....Ω….ΩΩΩΩΩΩ....18

3.2 Oblique Cnoidal WavesΩ….ΩΩΩΩΩΩΩ……...Ω….ΩΩΩΩΩ…19

3.3 Generation of Oblique Cnoidal WavesΩ….ΩΩΩΩΩΩΩ……............20

x Contents

3.3.1 Governing equationsΩ.….ΩΩΩΩΩΩΩ……...Ω….ΩΩΩΩ.20

3.3.2 Numerical solutionsΩ.….ΩΩΩΩΩΩΩ……...Ω….ΩΩΩΩ..21

4 Active Wave Absorption Theory 27

4.1 Governing Equations in Fourier SpaceΩΩΩΩΩ……...Ω….ΩΩΩΩ..27

4.2 Single Mode Absorption…..Ω….ΩΩΩΩΩΩΩ……...Ω….ΩΩΩ…29

4.3 Dual Mode Absorption......Ω….ΩΩΩΩΩΩΩ……...Ω….ΩΩΩΩ...29

4.4 Nonlinear Wave Generation in Dual ModeΩΩ………...Ω….ΩΩΩΩ..30

4.5 Real-time Realization of Active AbsorptionΩΩΩ……...Ω….ΩΩΩ....31

5 A Deterministic Combined Model for Wave Flumes 33

5.1 Unified Wavemaker Theory for Wave FlumesΩΩΩ……...Ω….ΩΩΩ..33

5.1.1 Linear wave generationΩΩΩ……...Ω….ΩΩΩΩ…….……….34

5.1.2 Nonlinear shallow water wave generationΩ...ΩΩ……...Ω….Ω..38

5.1.3 Ad hoc unified wave generation.......ΩΩ……...Ω….ΩΩΩΩ….38

5.2 Tests of the Deterministic Combined Model ………………….…………...39

5.2.1 Linear wave caseΩ..ΩΩ……...Ω….ΩΩΩΩ…………………..40

5.2.2 Nonlinear shallow water wave cases…...…………………………..43

5.2.3 Irregular waves propagating on constant water depth……………...57

5.2.4 Irregular waves propagating on variable water depth....…………...63

5.3 Summary and Conclusions………………………………………………...67

6 A Deterministic Combined Model for 3D Wave Basins 69

6.1 Unified Wavemaker Theory for 3D Wave Basins……….………………...69

6.1.1 Linear wave generation…..………………………………………...69

6.1.2 Nonlinear shallow water wave generation………………………....73

6.1.3 Ad hoc unified wave generation…………………………………...73

6.2 Tests of the Unified Wave Generation…………………………………….74

6.2.1 Oblique linear wave case...………………………………………...74

6.2.2 Oblique Cnoidal waves.…………………………………………....76

6.2.3 Oblique Stream Function waves……..…………………………….83

6.3 Tests of the Deterministic Combined Model………………………………87

6.3.1 Oblique irregular waves propagating on constant water depth…....87

6.3.2 Directional irregular waves propagating on constant water depth...92

Contents xi

6.3.3 Irregular waves behind a breakwater after propagating up a slope…97

6.4 Summary and Conclusions……...………………………………………...105

7 Stream Function Wavemaker Theory for Highly Nonlinear Waves in Wave

Flumes 107

7.1 Approximate Stream Function Wavemaker Theory ……………………....107

7.2 Experimental Validation…...………………….…………………………...109

7.2.1 Pure wave generation………………..…………………………….110

7.2.2 Wave generation with active absorption …...…………………......118

7.3 Improved Stream Function Wavemaker Theory ………………………….123

7.4 Summary and Conclusions……………………………………………......128

8 Summary, Conclusions and Recommendations 131

8.1 Summary and Conclusions….……………………………………………..131

8.2 Recommendations for Future Work ………………………………………133

References 135

A Boussinesq-type Equations in Mike 21 BW 141

B The Elliptic Functions Related to Cnoidal Waves 143

C Stream Function Wave Theory 145

xii Contents

xiii

Symbols The symbols used in this thesis are generally explained when they are first introduced. The following list contains the main symbols used. Some symbols defined only locally have not been included.

Roman Symbols abs superscript for absorbing re-reflected waves A complex amplitude of waves AI complex amplitude of progressive waves AI,0 complex amplitude of waves at the paddle front B complex amplitude of U c phase velocity c0 Biésel transfer function cj transfer function for normally emitted waves of evanescent modes cs real transfer function for evanescent modes c.c complex conjugate cn Jacobian elliptic function dt time step

dx grid spacing in x direction dy grid spacing in y direction e0 transfer function for oblique waves ej transfer function for oblique waves of evanescent modes eva superscript for evanescent modes E the complete elliptic integral of the second kind ErrH error of wave height ErrRMS root-mean-squared error E[xk(t)] mean value of time series xk(t) f the function of wave board motion type fd decaying function fr ramp function F Fourier-domain recipe for active absorption in single mode g acceleration of gravity gen superscript for generating target progressive waves

xiv Symbols

h still water depth H wave height Hm0 significant wave height i imaginary unit I subscript for the target, progressive, incident wave, j integer exponent, subscript, denoting j’th mode k wave number, k=2p/L k0 wave number in the propagation direction.

kj length of the wave number vector jkuur

ks real wave numbers for evanescent modes kx wave number in x direction

kxj x-component of jkuur

ky wave number in y direction

kr

wave number vector (kx ,ky)

jkuur

the complex wave number vector

K the complete elliptic integral of the first kind li subscript for use of linear wavemaker theory L wave length L0 wave length in deep water Lx wave length in x direction Ly wave length in y direction m index of zero and positive integer m elliptic parameter N The order of wave harmonics ODE ordinary differential equation p superscript for progressive wave P depth-integrated velocity in x direction

P time average of P over wave periods PDE partial differential equation Q depth-integrated velocity in y direction r parameter in improved Stream Function wavemaker theory R subscript for progressive reflected waves Re Real part of RR subscript for progressive re-reflected waves Rxy correlation function s superscript for use of evanescent modes sw superscript for use of shallow water wave theory t time t0 effective width of the impulse response function T wave period

Symbols xv

Tmin the minimum period Tp peak period Tr parameter in fr u horizontal particle velocity in the x direction U depth averaged particle velocity in x-direction Ur Ursell number v horizontal particle velocity in the y direction V depth averaged particle velocity in y-direction w vertical particle velocity x horizontal Cartesian coordinate

xr

spatial vector (x,y) x0 mean paddle position in the unified model X wave board position X0a amplitude of wave board position in physical space Xa complex amplitude of wave board position in Fourier space Xsw wave board position in shallow water y horizontal Cartesian coordinate z vertical Cartesian coordinate

Greek Symbols a wave direction a attenuation factor aj complex wave direction according to evanescent modes g parameter on JONSWAP frequency spectrum g m impulse response function corresponding to Gm

G transfer function for evanescent-mode correction Gm modified transfer function of G

h free surface elevation hI surface elevation of progressive wave hI,0 surface elevation at the paddle front

,0PIη surface elevation of progressive waves at the paddle front

0evaη evanescent modes at the paddle front

hmin trough elevation lm impulse response function corresponding to L m L transfer function for dispersion correction Lm modified transfer function of L rxy correlation coefficient function sa, sb parameters on JONSWAP frequency spectrum sx ,sy standard deviation

xvi Symbols

f velocity potential w angular frequency, w=2p/T wc characteristic angular frequency of the high-pass filter

1

Chapter 1 Introduction 1.1 Overview and Background Physical model tests and numerical models are two major tools for solving various coastal engineering problems, and studying wave transformation and interaction between waves, currents and marine structures. Both approaches have their strengths and weaknesses. The main strengths of numerical models include that they are easily applicable to large-scale flow fields; they do not require expensive buildings, areas and equipment; there is much flexibility in application, adaptation, maintenance and extension; and they do not suffer from scale effects like physical models. But the numerical models are not able to adequately treat highly complex and nonlinear processes such as wave breaking near the shore or near structures; the governing equations are formulated by approximating the prototype conditions. On the contrary, the main strengths of physical models include that, they are capable of simulating complex nonlinear processes and interactions which are hardly accessible to mathematical formulation; and they do not suffer from approximation error like numerical models. But the weaknesses of physical models include the limitation of the scale and size of the facility, high costs and long preparation time, and the limitation of flexibility in variation of geometric and material properties (Oumeraci, 1999). Furthermore, the limited extent of the physical model often prohibits that the offshore boundary is located in sufficiently deep water for the incident waves to be well described by standard, parameterised wave spectra. As this is typically the only available incident wave description, the limited size of the model facility often precludes important local phenomena like refraction and diffraction. Therefore, the integrated use of physical and numerical models which utilizes their advantages and eliminates the disadvantages at the same time could offer an attractive alternative to using either alone. A suitable combination is a physical model focusing on the complex part of the problem near the shore or near structures and a numerical model for the surrounding wave transformation. Boussinesq methods are widely used for predicting the propagation of nonlinear waves in coastal engineering. The fundamental issue of Boussinesq-type equations is the elimination of the vertical coordinate by introducing a polynomial approximation of the vertical distribution of the flow field, while accounting for non-hydrostatic

2 Chapter 1. Introduction

effects due to the vertical acceleration of water. This principle was initially introduced by Boussinesq (1872). During the past 20 years, the international use of Boussinesq-type equations has steadily increased, and commercial numerical models based on this idea have also been developed to solve practical engineering problems. Boussinesq-type equations are basically an expansion from the shallow water approximation to the fully dispersive and nonlinear water wave problem. The introduction of Padé approximant greatly improves the dispersion and shoaling characteristics of the equations, making them an attractive tool for general coastal applications (reviewed by Madsen and Schäffer 1998). Madsen et al.(1991), Madsen and Sørensen (1992) and Nwogu (1993) enhanced the accuracy of the linear dispersion relation by incorporating a Padé [2,2] expansion in dimensionless wave number kh of the Stokes linear dispersion relation in the models, which makes the range of applicability of such models up to khºp with phase celerity errors restricted to about 5%. Later on, Padé [4,4] dispersion has been incorporated (Schäffer and Madsen 1995) and this makes the range of applicability up to khº6. Along with the improvement in linear dispersion and shoaling properties, the accuracy of the nonlinear terms has been improved to cover larger and larger relative depth. Recently fully nonlinear models being accurate to khº40 are available (Madsen et al. 2002). But these accurate methods are still computationally expensive as an engineering tool for large areas. For physical wave models, wave generation is a main issue. Reference is made to the book by Dean and Dalrymple (1984, chapter 6), the review by Svendsen (1985), and the book by Hughes (1993, chapter 7). The foundation of wavemaker theory for mechanical wave generation was presented by Havelock (1929). After that, Biesel and Suquet (1951, 1954) described theoretical investigations corresponding to linearized Stokes theory and practical analysis of different wave generator types. Which found the basis for today’s wave generation in hydraulics laboratories. Numerous authors have contributed to the derivation of nonlinear wavemaker theory for unidirectional waves. Fontanet (1961) was the first to present a complete second-order theory in Lagrangian coordinates for the waves produced by a sinusoidally-moving plane wavemaker. Madsen (1971) developed an approximate solution for suppression of spurious superharmonics in regular waves generated in fairly shallow water. Flick and Guza (1980) analysed the second-order regular wave field generated by a first order control. Hudspeth & Sulisz (1991), and Sulisz & Hudspeth (1993) derived the fully dispersive second-order wavemaker theory for regular waves. Schäffer (1993, 1996) and Moubayed & Williams (1993) presented the full second-order wavemaker theory including both subharmonic and superharmonic interactions. For nonlinear long waves, Hammack and Segur (1974) presented the generation of solitary waves theoretically and experimentally. Goring (1979, Goring and Raichlen 1980) developed a Cnoidal wavemaker theory for studying the propagation of a tsunami wave onto a shelf, and presented a general method of long wave generation by matching the velocities of the wave board and the water particles of the desired wave as the board moves. This is the basis of the shallow water wavemaker theory. Second-order wavemaker theory and Cnoidal wavemaker theory have been two major wavemaker theories for nonlinear waves in non-shallow water depth and shallow water depth, respectively. However, they are inadequate for highly nonlinear waves. Stream Function wave theory is a numerical high order wave theory for fully nonlinear waves. It was first used by Chappelear (1961). Later, this approach was

1.1 Overview and Background 3

developed by Dean (1965), Chaplin (1980), Rienecker and Fenton (1981), and Fenton (1988). This wave theory has become a popular theory for nonlinear steady waves in constant water depth. With Stream Function wave generation, the regular highly nonlinear waves may be reproduced with higher accuracy. In this study, an approximate Stream Function wavemaker theory for shallow and intermediate water depth will be developed as a spin-off. As to directional wave generation, first-order theory for the generation of oblique waves by two wavemaker segments between lateral walls was studied by Madsen (1974) who solved the Laplace equation in three dimensions with non-flow condition on the sidewalls. Gilbert (1976) examined oblique waves generated by a segmented wave generator and showed that spurious waves were generated in the case of a finite segment width of the wavemaker. Later on, Takayama (1984, 1987) and Dalrymple and Greenberg (1985) used different methods for analysing monochromatic waves generated by a single finite-width wavemaker. Suh and Dalrymple (1987) took the first steps towards second-order directional wavemaker theory. Recently, Steenberg & Schäffer (2000), Schäffer & Steenberg (2003) developed the complete second-order wavemaker theory for the generation of multidirectional waves in a semi-infinite basin based on the theory for wave flumes (Schäffer, 1996). Wave absorption for the reduction of the reflection effects is another issue for physical wave models. Active absorption which refers to the use of a wavemaker as a moving boundary controlled to absorb waves impinging on it is an attractive alternative to traditional absorption by passive absorbers. Reference is made to the review by Schäffer and Klopman (2000) and the book by Hughes (1993, chapter 7). Milgram (1970) made the first approach to active wave absorption in a wave flume. For many years, active absorption has been applied in a number of hydraulic laboratories including DHI water and Environment (DHI). The approach used at DHI since the middle of 1980s has been to measure the surface elevation by one or more gauges mounted on the front of the wave board in order to provide a hydrodynamic feedback. The original idea was to use an electric circuit for the absorption loop. Later on, the system called Active Wave Absorption Control System (DHI AWACS) was developed using digital recursive filters (Schäffer 1994, 1999) for wave flumes. Later on, the DHI 3D AWACS was developed for wave basins (Schäffer 1998, 2001). Recently, Schäffer and Jakobsen (2003) presented a new method for nonlinear wave generation and active absorption with the paddle position and the corresponding surface elevation at the paddle as control signals. Combinations of physical and numerical models have been referred to as composite or hybrid modelling and discussed by Kamphuis (1995, 1996, 2000), Watts (1999), Schäffer (1999) and others. In some studies, a combination of numerical and physical modelling has been applied. Figure 1.1 shows a sketch of a combined model (Gierlevesen, et al. 2003). Typically the approach has been to use numerical modelling for the determination of wave conditions at the boundary of the physical model. In the traditional combined modelling approach, the data transfer between the numerical and the physical model is only on a stochastic level through the wave spectrum or even as bulk parameters describing the wave condition to be generated by the wavemaker in the physical model ( Kofoed-Hansen, et al. 2000, and Gierlevsen, et al. 2003). The present study aims at a deterministic combined model in two horizontal dimensions.


Figure 1.1: Example of a combined numerical and physical model (Gierlevsen, et al. 2003)

1.2 Approaches The main objective of this study is to make a deterministic combination of numerical and physical models for coastal waves. The data transfer between two models will thus be on a deterministic level with detailed wave information like time series of surface elevation transferred along the wavemaker. Considering the accuracy and the computational expense of numerical methods, we choose a Boussinesq model (Mike 21 BW, Madsen and Sørensen, 1992) for the numerical wave computations in the far field (see Appendix A). A wavemaker and the associated control system with active absorption provide the interface between the numerical and physical models. A one-way model coupling is used and thus waves absorbed by the wavemaker are not fed back in the numerical model. Piston-type 2D and segmented 3D wavemakers are controlled for simultaneous wave generation and active absorption by DHI AWACS and DHI 3D AWACS for wave flumes and wave basins, respectively. A new heuristic unified wave generation method is devised particularly for this study, which accounts for shallow water nonlinearity and compensates for local wave phenomena (evanescent modes) near the wavemaker. This ad hoc unified wave generation offers a deterministic link between numerical and physical models. Three different types of wavemaker control are offered by the DHI (3D) AWACS. In position mode, the control signal is a time series of wavemaker paddle position. This mode is compatible with the general approach to nonlinear wave generation. However active absorption is not included in position mode. Single mode is a traditional approach for active absorption, in which the control signal is the incident progressive wave elevation. The weakness of single mode is an inconsistent nonlinear wave generation. A third method termed dual mode has been developed recently (Schäffer and Jakobsen, 2003). This allows for active absorption in combination with consistent nonlinear wave generation. The active absorption appears as a linear perturbation on the nonlinear wave generation. The control signals in dual mode are time series of wavemaker paddle position and the corresponding surface elevation at the moving paddle. These two control signals are provided by the ad hoc unified wave generation method developed here.

1.3 Thesis Outline 5

1.3 Thesis Outline In the following, an outline for the remaining chapters of this thesis will be given. Chapter 2 describes linear wavemaker theory, Chapter 3 gives Cnoidal wavemaker theory. Chapter 4 introduces the methods of active absorption used in DHI AWACS. Chapter 5 presents the ad hoc 2D unified wave generation method, and then the combined numerical and physical models in flumes using this approach. The ad hoc 3D unified wave generation method, and the application of it including the combined model in two horizontal dimensions are shown in Chapter 6. Chapter 7 presents an approximate Stream Function wavemaker theory for highly nonlinear waves in flumes with experimental validation based on Stream Function wave theory and the ad hoc unified wave generation method as a spin-off. Furthermore, an improved Stream Function wavemaker theory is discussed in Chapter 7. Finally, summary and conclusions are given in Chapter 8.


7

Chapter 2

Linear Wavemaker Theory In this chapter some general definitions on waves and wave generation are introduced. As the basic wavemaker theory, 3D linear fully dispersive wavemaker theory is described.

2.1 Governing Equations A three dimensional irrotational flow in a homogeneous, incompressible, and inviscid fluid is considered. Define the particle velocity components in the x, y and z direction respectively by (u,v,w)=(φx,φy,φz) with the velocity potential φ(x,y,z,t), where t denotes the time, in a Cartesian coordinate system (x,y,z). The definition sketch is shown in Figure 2.1. The governing equation for the velocity potential in the fluid domain is the Laplace equation by mass conservation, + + =0 in the fluidxx yy zzφ φ φ (2.1)

Figure 2.1: Definition sketch.

The fluid domain is bounded by a free surface, a horizontal bottom, a wavemaker and a lateral boundary. Let z=η(x,y,t) be the surface elevation. Then the function that describes free surface is

( , , , ) ( , , )F x y z t z x y tη= − (2.2)

8 Chapter 2. Linear Wavemaker Theory

The kinematic free surface boundary condition is

( , , , )

0 at ( , , , ) 0 DF x y z t

F x y z tDt

= = (2.3)

thus, at ( , , )z t x x y y z x y tφ η φ η φ η η= + + = (2.4)

The pressure on the free surface should be uniform along the wave form. Therefore the Bernoulli equation gives the dynamic free surface boundary condition. It is

2 2 21( ) 0 at ( , , )

2t x y zg z x y tη φ φ φ φ η+ + + + = = (2.5)

where, g is the acceleration of gravity. The linearized forms of free surface boundary conditions are,

at 0z t zφ η= = (2.6)

0 at 0tg zη φ+ = = (2.7) Eliminating the surface elevation by combining equations (2.6) and (2.7) gives 0 at 0.z ttg zφ φ+ = = (2.8) The bottom boundary condition is the usual no-flow condition, 0 at z z hφ = = − (2.9) where, h is the still water depth. For the lateral boundary condition, the velocity potential should have a limitation as x or y tends to infinity. At the wavemaker x=0, a kinematic condition must be satisfied. Let the wave board position X=X(y,z,t), then the boundary condition at the wavemaker is

at ( , , )x y y z z tX X X x X y z tφ φ φ− − = = (2.10)

The linearized form of the above is

at 0x tX xφ = = (2.11) The position of the wave board is given as 0( , , ) ( ) ( , )X y z t f z X y t= (2.12) where f(z) describes the type of wave board motion, such as,

2.2 Solutions 9

1 for - ( - ) 0

( ) 0 for - -( - )

zh d z

f z h lh z h d

+ ≤ ≤= + ≤ <

(2.13)

Here z=-(h+l) gives the centre of rotation (-h<l≤ ∞ ) (see Figure 2.1), d is the elevation of the hinge over the bottom. Figure 2.2 shows the different types of wavemakers considered. If the centre of rotation is at or above the bottom (l≤0, c and d cases in Figure 2.2), d ≥ 0, i.e. d=-l. If the centre of rotation is at or below the bottom (l ≥ 0, a, b and c cases in Figure 2.2), then d=0 and the last case in (2.13) becomes irrelevant. Thus the restriction that either d=-l or d=0 (for l ≥ 0) has been made, which ensures that only continuous shape functions f(z) are considered.

Figure 2.2: Types of wave board motions: (a) translatory (piston-type) and

(b)~(d) rotational with the centre of rotation (b) below, (c) at, (d) above the bottom.

2.2 Solutions Since it is possible to superpose wave components to make linear irregular waves, it is only necessary to analyse monochromatic waves. We consider an infinitely long continuous wavemaker and let the paddle position at still water level for each of the wave components be given

0( , , ) ( )sin( )a yX y z t X f z t k yω= − (2.14)

where X0a is the constant paddle amplitude at still water level, w is the wavemaker frequency, ky is the wave number in y direction. Then, the boundary condition (2.11) yields, 0(0, , , ) ( ) cos( )x a yy z t X f z t k yφ ω ω= − (2.15)

For solving the above governing equations, separation of variables is used. The velocity potential φ(x,y,z,t) can be expressed as,

( , , , ) Re ( ) ( ) ( ) i tx y z t G z H x M y e ωφ = (2.16)

Substituting (2.16) into (2.1) gives

''( ) ''( ) ''( )

0G z H x M y

G H M+ + = (2.17)


2.2.1 Progressive waves Let

2 2''( ) ''( ) ''( )0,

G z H x M yk k

G H M= > + = − (2.18)

Under this condition, due to the boundary condition of (2.9), and imposing a restriction that yk k≥ , the solution is

2 20 0( , , , ) cosh ( )sin( )p y yx y z t C k h z k k x k y tφ ω= + − + − (2.19)

Here adding the subscript “0” on k in (2.18) to denote a progressive wave, k0 is the wave number in the propagation direction. Substituting the solution of (2.19) into (2.8) yields,

20 0tanhgk k hω = (2.20)

which is the dispersion relation for a progressive wave (see Figure 2.3).

2 4 6 8k h

0. 2

0. 4

0. 6

0. 8

1T an h H k h L1 ê H kh L

Figure 2.3: Graphical representation of the dispersion relation for a progressive wave with w2h/g=1.0.

2.2.2 Evanescent modes Let

2 2''( ) ''( ) ''( )0,

G z H x M yk k

G H M= − < + = (2.21)

Under this condition, and due to the boundary condition (2.9), the solution is,

2 2

( , , , ) cos ( ) cos( )s yk k x

s s yx y z t C e k h z k y tφ ω− += + − (2.22)

Adding the subscript “s” on k in (2.22) to denote evanescent modes which are due to the mismatch between the shape of the progressive wave velocity profile and the shape function f(z) in (2.13). Here ks is real and ks >0. Substituting the solution of (2.22) into (2.8) yields,

2.2 Solutions 11

2 tans sgk k hω = − (2.23)

which is the dispersion relation for the evanescent modes. Figure 2.4 shows that there are an infinite number of solutions of ks to the dispersion relation. Each solution will be denoted as ksj, where j is a non-zero integer.

2 4 6 8 10kh

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1 −Tan Hkh L1

" " " "" ""kh

Figure2.4: Graphical representation of the dispersion relation for the evanescent modes with w2h/g=1.0.

2.2.3 Complete solutions When k is equal to zero, there is also a possible solution of the velocity potential to Laplace Equation, but it is not suitable to the problem of waves due to the boundary conditions. Thus, the complete solution should be the combination of the progressive wave solution and the evanescent-mode solution.

2 2

2 20 0

1

( , , , ) cosh ( )sin( )

cos[ ( )]cos( )sj y

p y y

k k x

j sj yj

x y z t C k h z k k x k y t

C e k h z k y t

φ ω

ω∞

− +

=

= + − + −

+ + −∑ (2.24)

For solving the coefficients Cp and Cj in (2.24), the orthogonality property of trigonometric functions is utilized. By using the dispersion relations of (2.20) and (2.23), we can get,

0

0cosh ( )cos[ ( )] 0sjhk h z k h z dz

−+ + =∫ (2.25)

0

01 02

0 001 02 0

0

01 02

cosh ( ) cosh[ ( )]

2 sinh(2 ), for

4 =

0 , for

hk h z k h z dz

k h k hk k k

k

k k

−+ +

+ = = ≠

∫ (2.26)


0cos[ ( )]cos[ ( )]

2 sin(2 ), for

4 =

0 , for

sj smh

sj sjsj sm

sj

sj sm

k h z k h z dz

k h k hk k

k

k k

−+ +

+=

≠

∫ (2.27)

By means of the boundary condition (2.15),

00

02 20 0 0

0 2 220 00

( )cosh[ ( )]

sinhcosh [ ( )]

a

h y ap

yh

X f zk h z dz

k k X cC

k k k hk h z dz

ωω−

−

+−

= =−+

∫

∫ (2.28)

00

2 2

0

0 2 22

( )cos[ ( )]

sincos [ ( )]

asj

h sj y a sjj

sj y sjsjh

X f zk h z dz

k k X cC

k k k hk h z dz

ωω−

−

− ++ −

= =++

∫

∫ (2.29)

where, c0 and csj are the real-quantity transfer functions as

0

0

0 0 0 20

0 0 0 0 0 0 0

0 0 0

( )cosh[ ( )]

sinhcosh [ ( )]

4sinh [cosh cosh ( )sinh ( )sinh ]

( )(2 sinh2 )

h

h

f z k h z dz

c k hk h z dz

k h k d k h k d l k d k h l k h

k h l k h k h

−

−

+=

+

− − + + +=+ +

∫

∫ (2.30)

0

0 2

( )cos[ ( )]

sincos [ ( )]

4sin [cos cos ( )sin ( )sin ] =

( )(2 sin2 )

sj

hsj sj

sjh

sj sj sj sj sj sj sj

sj sj sj

f z k h z dz

c k hk h z dz

k h k d k h k l d k d k l h k h

k h l k h k h

−

−

+=

+

− − + + − ++ +

∫

∫ (2.31)

For piston-type of the wave board motion (Figure 2.2.a) with the expression f(z)=1, the transfer functions are,

20

00 0

2

4sinh (a)

2 sinh 2

4sin (b)

2 sin 2sj

sjsj sj

k hc

k h k h

k hc

k h k h

=+

=+

(2.32)

Using the free surface boundary condition (2.7), the wave height η(x,y,t) is obtained as,

2.2 Solutions 13

2 2

2 20 0 0

1

1( , , ) | cosh cos( )

cos sin( )sj y

t z p y y

k k x

j sj yn

x y t C k h t k k x k yg g

C e k h t k y

ωη φ ω

ω

=

∞− +

=

= − = − − −

+ −∑ (2.33)

Substituting (2.28) and (2.29) into the above and by means of dispersion relations (2.20) and (2.23), it yields,

2 2

2 200 0 02 2

0

0 2 21

( , , ) cos( )

sin( )sj y

a y y

y

k k xsja sj y

nsj y

kx y t X c t k k x k y

k k

kX c e t k y

k k

η ω

ω∞

− +

=

= − − −−

+ −+

∑ (2.34)

2.2.4 Solutions in complex notation The solution can be rewritten in the compact notation provided by a complex representation. Then we have,

( )0

1( , , ) . .

2yi t k y

aX y z t iX e c cω −= − + (2.35)

( )0

0

cosh ( )1( , , , ) . .

2 coshjj i t k xa

jj j

k h zigXx y z t e e c c

k hωφ

ω

∞− ⋅

=

+ = +

∑r r

(2.36)

( )0

0

1( , , ) . .

2ji t k x

a jj

x y t X e e c cωη∞

− ⋅

=

= +

∑

r r

(2.37)

2 tanhj jgk k hω = (2.38)

provided that 2 2 2

xj j yk k k= − (2.39)

Where, c.c. denotes the complex conjugate of the preceding term. jkr

=(kxj,ky) is the

complex wave number vector, jjk k≡r

denotes the length of the wave number vector,

and xr

=(x,y) is the horizontal position vector. Eqs. (2.36)~(2.38) include both the progressive wave and the evanescent modes. k0 is the real solution for the progressive wave, and an infinity of purely imaginary solutions kj are for evanescent modes, where ikj=ksj >0, (j=1,2….). The coefficients ej are called transfer functions and defined as


1

cosj

j j jxj j

ke c c

k α= = (2.40)

where cj are the transfer functions for normally emitted waves (ky=0). For piston-type wavemaker, it is given by,

24sinh

2 sinh 2j

jj j

k hc

k h k h=

+ (2.41)

which is the same expression as (2.32a) except for the subscript. For j=0, it gives the real quantity c0 for the progressive wave which is the Biesel transfer function, and e0 is either real or imaginary depending on the choice of ky. For j=1,2…, cj is imaginary and cj=-icsj in (2.32), and ej is always imaginary.

For 0yk k≤ (serpent wave length longer than or equal to length of progressive wave),

kx is real. The wave field generated has a progressive part

( )1( , , ) . .

2i t k x

I Ix y t A e c cωη − ⋅= +r r

(2.42)

where AI is a complex amplitude given by

0 0I aA e X= (2.43)

and e0 is real. When a complex amplitude of paddle position Xa is given, i.e.

0a aX iX= − in (2.35), we have

0I aA ie X= (2.44) In this case ( , ) (cos ,sin )x yk k k α α= (2.45)

and

0 0

1

cose c

α= (2.46)

Here subscript “0” is omitted on k and a for j=0. a is the wave propagation direction (see Figure 2.5).

Figure2.5: Definition of wave number vectors. For j=1,2… in (2.36)~(2.38), kxj is imaginary corresponding to evanescent modes. Generalizing to complex aj as defined by

2.2 Solutions 15

sin siny j jk k kα α= = (2.47)

we have

2 2

2 2

22

cos 1 for j=0sin

sincos 11 1 for j 1

xjj

j j

j

k kk

k kk

αα αα

≤= = − = + ≥ ≥

(2.48)

For a piston-type wavemaker, the Biesel transfer function c0 and the sum of the

evanescent-mode transfer functions 1

n

jj

i c=∑ with different n are shown in Figure 2.6.

While c0 reaches its asymptotic value of 2 at non-dimensional frequency /h gω =2,

the evanescent-mode transfer function 1

n

jj

i c=∑ increases continuously with increasing

frequency. For the evanescent modes, the sum of transfer function increases as n

increases. It appears that 1

n

jj

i c=∑ at n=20 matches well with that at n=200

when /h gω <4. Furthermore, 1

n

jj

i c=∑ is greater than c0 when /h gω reaches to a

certain high value. For example, 20

1j

j

i c=∑ > c0 for /h gω >3.4, which means the

amplitude of the local disturbance at the wave board exceeds the amplitude of the progressive wave. This demonstrates the problem of generating high frequency waves with a piston-type wavemaker.

1 2 3 4 5ωè!!!!!!!!!!!h ê g

0.5

1

1.5

2

2.5

c0

ij=1

n

cj

1

51020

200

Figure 2.6: The transfer function c0 and the total evanescent-mode transfer function 1

n

jj

i c=∑

with different n for a piston-type wavemaker.

The transfer function e0 and the total evanescent-mode transfer function 1

n

jj

i e=∑ for

several wave propagation directions for a piston-type wavemaker are shown in Figure 2.7. Here and in the following we choose n=20 for the evanescent modes. When a increases, the transfer function e0 increases while the evanescent-mode transfer

function ji e∑ decreases. For yk k> (serpent wave length shorter than length of


progressive wave), even the mode for j=0 is evanescent and no progressive component is generated.

1 2 3 4 5ωè!!!!!!!!!!!h ê g

1

2

3

4

0°30 °

45 °

60 °

e0

0°

30 °45 °60 °iÚej

Figure 2.7: The transfer function e0 and the total evanescent-mode transfer function ji e∑

for several directions for a piston-type wavemaker.

For the progressive waves, the depth-averaged horizontal particle velocity U(x,y,t) is defined as

01

( , , ) ( , , , )h

U x y t u x y z t dzh −

= ∫ (2.49)

where u(x,y,z,t) can be obtained by the progressive part of velocity potential expressed in (2.36), it yields,

( )0

1 cosh ( )( , , , ) . .

2 coshi t k xa

x x

igX k z hu x y z t k e e c c

khωφ

ω− ⋅+ = = +

r r

(2.50)

By means of the dispersion relation, we get

( )0

1 cosh ( )( , , , ) . .

2 sinhi t k x

a

k z hu x y z t iX c e c c

khωω − ⋅+ = +

r r

(2.51)

Substituting it into (2.49) yields,

( )01( , , ) . .

2i t k x

a

cU x y t i X e c c

khωω − ⋅ = +

r r

(2.52)

Let BI denotes the complex amplitude of U(x,y,t), then we get,

0I a

cB i X

khω= (2.53)

17

Chapter 3 Cnoidal Wavemaker Theory Linear wave theory is applicable to small-amplitude waves. When the wave height is finite, the nonlinearity of the wave should be considered. Cnoidal waves are an analytic nonlinear long wave solution. Goring (1979) showed how waves can be generated according to Cnoidal wave theory. In this chapter, wavemaker theory for normal and oblique Cnoidal waves is described. 3.1 Normal Cnoidal Waves 3.1.1 Solution of normal Cnoidal waves Considering Cnoidal waves of height H and period T propagating in the x direction with water depth h. The surface elevation ηI(x,t) is given by 2

min ( , )I Hcn mη η θ= + (3.1) where cn(q,m) is a Jacobian elliptic function of argument q and the elliptic parameter m, see Appendix B for definitions (Milne-Thomson, 1965). The argument q is the combination of x and t as the following expression,

2 ( )( )t x

K mT L

θ = − (3.2)

here L is the wave length in the propagating x direction. Furthermore, minη (<0) is the trough elevation given by,

min

1 ( )[ (1 ) 1]

( )

E mH

m K mη = − − (3.3)

where K(m) is the complete elliptic integral of the first kind, E(m) is the complete elliptic integral of the second kind.

18 Chapter 3. Cnoidal Wavemaker Theory

The elliptic parameter m (0≤ m<1) is the solution to

2

23

16( )

3

HLUr mK m

h= = (3.4)

Where, Ur is the Ursell number, which is an important parameter to show the ratio of

the nonlinearity (H

hε = ) and dispersion (

h

Lµ = ) of waves.

The phase velocity of the Cnoidal wave is obtained by,

[1 ( )]H

c gh A mh

= + (3.5)

in which, c is the phase velocity, and c=L/T. The function A(m) is,

2 3 ( )

( ) 1( )

E mA m

m mK m= − − (3.6)

Thus, the wave length L is given by,

[1 ( )]H

L T gh A mh

= + (3.7)

3.1.2 The generation of normal Cnoidal waves In order to determine the wave paddle position of the wavemaker for a given Cnoidal wave, match the velocity of the wave plate at all positions with the corresponding velocity of the particles under the wave. For long waves, the velocity can be considered as the velocity averaged over the depth, ( , )U x t , since the horizontal particle velocity is almost uniform over the depth. Which yields,

( )

( ( ), )dX t

U X t tdt

= (3.8)

where, X(t) is the wave paddle position for a piston-type wavemaker. For waves of permanent form, the velocity averaged over the depth is obtained as the following by means of continuity,

( , )

( , )( , )

c x tU x t

h x t

ηη

=+

(3.9)

Thus, in terms of the plate velocity,

3.2 Oblique Cnoidal Waves 19

( ) ( ( ), )

( ( ), )

dX t c X t t

dt h X t t

ηη

=+

(3.10)

This is a general equation for evaluating the trajectory of the wave paddle for long waves. For cnoidal waves, the parameter m is obtained by using (3.4) first. Next we can get the other parameters such as E(m), K(m), A(m) and also minη , L. Then X(t) can be obtained using (4.1) and (4.10). For getting a solution which compares well to reality, the condition should be µ=h/L<0.10. Figure 3.1 shows the time series of paddle position X(t) for H=0.2m, T=2.3s and h=0.4m. Figure 3.2 shows the corresponding surface elevation of the water wave η(X(t),t).

1 2 3 4tHsL

-0.15

-0.1

-0.05

0.05

0.1

0.15

X

Figure 3.1: Wave paddle position of the given Cnoidal waves.

1 2 3 4t H s L

-0.05

0.05

0.1

0.15

η

Figure 3.2: Surface elevation η(X(t),t).

3.2 Oblique Cnoidal Waves To consider oblique Cnoidal waves in x-y Cartesian coordinates, we define a as the angle between wave direction and the x-coordinate (see Figure 3.3). Then, we have

cos

, sin

x

y

k kk k

k k

αα

== =

r (3.11)

in which, k is the wave number in the propagating direction, kx and ky are the wave numbers in the x and y direction, respectively. The surface elevation of the oblique Cnoidal wave η(x,y,t) can be described by


2min( , , ) ( , )

cos sin( , , ) 2 ( )( ) 2 ( )( )

x y

x y t Hcn m

t x y t x yx y t K m K m

T L L T L L

η η θα αθ

= + = − − = − −

(3.12)

where, L is the wave length in the propagating direction, and Lx and Ly are the wave lengths in the x and y direction respectively.

2 2

, cos sinx y

x y

L LL L

k k

π πα α

= = = = (3.13)

3.3 Generation of Oblique Cnoidal Waves 3.3.1 Governing equations Define X(y,z,t) as the wave paddle position in the x direction. The kinematic wave maker boundary condition is at ( , , )x y y z z tX X X x X y z tφ φ φ− − = = (3.14)

where, f(x,y,z,t) is the velocity potential. For a piston-type wavemaker, the wave paddle position could be considered as X(y,t), since the horizontal particle velocity is almost uniform over the depth for long waves. In this case, the boundary condition (3.14) can be rewritten as,

( , ) ( , )

( ( , ), , ) ( ( , ), , )X y t X y t

v X y t y t u X y t y tt y

∂ ∂+ =∂ ∂

(3.15)


The corresponding velocities of the particles under the wave at the wave paddle can be expressed by means of continuity,

kr

kx

ky

a

X(y,t)

Dy

y

x

3.3 Generation of Oblique Cnoidal Waves 21

( , , )( , , ) cos

( , , ), at ( , )

( , , )( , , ) sin

( , , )

c x y tu x y t

h x y tx X y t

c x y tv x y t

h x y t

η αη

η αη

= + = = +

(3.16)

Combining (3.15) and (3.16), we can obtain a partial differential equation (PDE) for the wave paddle position X(y,t),

( ( , ), , ) ( , ) ( , ) ( ( , ), , )

cos sin ( ( , ), , ) ( ( , ), , )

c X y t y t X y t X y t c X y t y t

h X y t y t t y h X y t y t

η ηα αη η

∂ ∂= ++ ∂ ∂ +

(3.17)

with the initial condition: ( , ) 0 at 0X y t t= = (3.18) and the periodic boundary condition: (0, ) ( , ) yX t X L t= (3.19)

3.3.2 Numerical solutions A test was performed with the following conditions: wave height H=0.20m, wave period T=2.3s, water depth h=0.4m, wave directions a=p/4. In the numerical calculation, we choose time step dt=T/300, grid spacing dy=Ly/100. 3.3.2.1 Transient solution Using the Mathematica software package to solve (3.17) directly with initial condition (3.18), boundary condition (3.19) and the expression of the surface elevation (3.12), we get the solution of wave paddle position, which is shown in Figure 3.4 and Figure 3.5. From the graphs we see that there is an offset in the solution. This is due to the motionless initial condition. It is not a suitable solution for the practical generation of waves in a physical experiment due to the asymmetry. We therefore look for a periodic solution without offset in order to satisfy the limitations of practical wave paddle motions and generate waves conveniently.


0

0.5

1

1.5

2

t

0

2

4

6

y

-0.2-0.1

0

0.1

0.2

X

0

0.5

1

1.5

2

t

Figure 3.4: Wave paddle position by solving (3.17).

1 2 3 4 5 6y

- 0.2 - 0.1

0.1

0.2

X t=0

1 2 3 4 5 6 y

-0.2

-0.1

0.1

0.2

X

t=5T/10

1 2 3 4 5 6y

- 0.2 - 0.1

0.1

0.2

X

t=T/10

1 2 3 4 5 6 y

-0.2

-0.1

0.1

0.2

X

t=7T/10

1 2 3 4 5 6y

- 0.2 - 0.1

0.1

0.2

X t=3T/10

1 2 3 4 5 6 y

-0.2

-0.1

0.1

0.2

Xt=9T/10

(a) profiles at different t


0.5 1 1.5 2t

- 0.2 - 0.1

0.1 0.2

X y=0

0.5 1 1.5 2 t

-0.2

-0.1

0.1

0.2

X

y=5Ly/10

0.5 1 1.5 2t

- 0.2 - 0.1

0.1

0.2

X y=Ly/10

0.5 1 1.5 2 t

-0.2

-0.1

0.1

0.2

X

y=7Ly/10

0.5 1 1.5 2t

- 0.2 - 0.1

0.1 0.2

X y=3Ly/10

0.5 1 1.5 2 t

-0.2

-0.1

0.1

0.2

X

y=9Ly/10

(b) time series at different y

Figure 3.5: Variation of wave paddle position. 3.3.2.2 Periodic solution In order to avoid the offset problem, we simplify the PDE to an ordinary differential equation (ODE) by means of an assumption and then solve the ODE.

Assuming

( , ) ( ) ( )y

TX y t X t y X t

L= − = $ (3.20)

where, y

Tt t y

L= −$ . We have

, ( )y

T

t y Lt t

∂ ∂ ∂ ∂= = −∂ ∂∂ ∂$ $

(3.21)

Then the expression of surface elevation (3.12) can be rewritten as,


2min( , ) ( , )

( , ) 2 ( )( )x

x t Hcn m

t xx t K m

T L

η η θ

θ

= +

= −

$

$$

(3.22)

and the PDE (3.17) becomes an ODE,

( ( ), ) ( ) ( ( ), )

cos (1 sin ) ( ( ), ) ( ( ), )y

c X t t X t T c X t t

Lh X t t t h X t t

η ηα αη η

∂= −+ ∂ +

$ $ $ $ $

$ $ $ $ $ (3.23)

Solving (3.23) with (3.22) and the initial condition (0) 0 X = by finite-differences, we

can get the solution ( )X t$ , which is shown in Figure 3.6.

1 2 3 4

-0.1

-0.05

0.05

0.1

Figure 3.6: Time series of wave paddle position with respect to .t$

Transferring the wave paddle position ( )X t$ into X(y,t) using the assumption (3.20). Figure 3.7 shows the variation of wave paddle position X(y,t) with respect to y and t respectively. This solution X(y,t) has no offset either respect to y or to t, and has a permanent shape. When t (or y) varies, the shape of the variation of wave paddle position with respect to y (or t) is the same except for the different phase.

1 2 3 4 5 6y

- 0.2

- 0.1

0.1

0.2 X t=0

1 2 3 4 5 6 y

-0.2

-0.1

0.1

0.2X

t=5T/10

1 2 3 4 5 6y

- 0.2

- 0.1

0.1

0.2 X t=T/10

1 2 3 4 5 6 y

-0.2

-0.1

0.1

0.2X

t=7T/10

t$

( )X t$


1 2 3 4 5 6y

- 0.2

- 0.1

0.1

0.2 X

t=3T/10

1 2 3 4 5 6 y

-0.2

-0.1

0.1

0.2X

t=9T/10

(a) profiles at different t

0.5 1 1.5 2 t

- 0.2

- 0.1

0.1

0.2 X

y=0

0.5 1 1.5 2 t

-0.2

-0.1

0.1

0.2 X y=5Ly/10

0.5 1 1.5 2 t

- 0.2

- 0.1

0.1

0.2 X

y=Ly/10

0.5 1 1.5 2 t

-0.2

-0.1

0.1

0.2 X y=7Ly/10

0.5 1 1.5 2 t

- 0.2

- 0.1

0.1

0.2 X

y=3Ly/10

0.5 1 1.5 2 t

-0.2

-0.1

0.1

0.2 X

y=9Ly/10

(b) time series at different y Figure 4.7: Variation of wave paddle position.

3.3.2.3 Validation of the periodic solution As we know the solution of the PDE has an offset due to the motionless initial condition. The solution of the ODE has no offset by means of an assumption, and this solution is suitable for practical wave generation. Therefore we try to use the ODE solution as the initial condition of the PDE to solve the PDE. Here we use an explicit

finite difference method with a central difference scheme for ( , )X y t

y

∂∂

in the field, a


forward scheme on the left side boundary and backward scheme on the right side boundary. The discrete version of (3.17) in the field is the following,

, 1, 1, ,, 1 ,

, ,

[ , , ] [ , , ]( cos sin )

[ , , ] 2 [ , , ]

i j i j i j i ji j i j

i j i j

c X idy jdt X X c X idy jdtX X dt

h X idy jdt dy h X idy jdt

η ηα αη η

+ −+ −= + −

+ + (3.24)

Figure 3.8 shows the solution of the PDE using the ODE solution as the initial condition. From the graph we see that there is no offset in the snakelike solution. Comparing this final PDE solution with the initial condition (i.e. ODE solution), the maximum relative error is 0.64%. That means the ODE solution satisfies the PDE (3.17), and also confirms that the assumptions are reasonable. Therefore, the ODE solution is suitable for the generation of oblique Cnoidal waves.

100

200

300

time steps20

40

60

80

100

grids in y

-0.1-0.05

00.050.1

X

100

200time steps

Figure 3.8: PDE solution using ODE solution as initial condition (dt=T/300,dx=Ly/100).

27

Chapter 4 Active Wave Absorption Theory Active wave absorption refers to the use of a wavemaker as a moving boundary controlled to absorb waves impinging on it. Wavemakers equipped with a control system for simultaneous wave generation and active absorption are often called absorbing wavemakers or active absorption systems. Active absorption systems are either used to generate waves and at the same time prevent re-reflections of waves reflected somewhere in the facility, or as pure wave absorbers instead of traditional passive absorbers. Their absorption ability is typically much better at low frequencies than that of passive absorbers. They have also some other attractive features. First of all, spurious re-reflection of outgoing waves is largely avoided. Another important feature is resonant oscillations in the facility are suppressed (see Schäffer, 2000). Nowadays active absorption systems are widely used in hydraulic laboratories. Active absorption in a physical wave flume or basin requires a hydrodynamic feedback mechanism, which provides information about the waves to be absorbed. For this purpose we only consider the commonly used method, where the hydrodynamic feedback is provided by surface elevation gauges integrated in the paddle front. Two methods of active absorption described in this chapter are particularly used in the DHI (3D) AWACS. 4.1 Governing Equations in Fourier Space Based on linear wavemaker theory introduced in Chapter 2, we can write up the governing equations for the active absorption problem in Fourier space. Time series of surface elevation and paddle position along the wave paddle in a 3D basin are denoted h(y,t) and X(y,t) respectively, while A(ky,w) and Xa(ky,w) denote the equivalent complex Fourier amplitudes:

2D FourierTransform

( , ) ( , )yy t A kη ω⇔ (4.1)

28 Chapter 4. Active Wave Absorption Theory

2D FourierTransform

( , ) ( , )a yX y t X k ω⇔ (4.2)

For the wave flume case, ky vanishes and only a one-dimensional Fourier transform is needed. The quantities A and h carry the following subscripts: “I” for the target, progressive, incident waves, “0” for waves measured right at the paddle front, “R” for progressive reflected waves and “RR” for progressive re-reflected waves, see Figure 4.1 for definitions. All surface elevation and equivalent amplitudes are taken at x=0, the mean paddle position. Using linear wavemaker theory and further assuming full re-reflection on the wave paddle when the wavemaker is at rest, the following equations apply 0I a RRA ie X A= + (4.3)

0 01

( )a j R RRj

A iX e e A A∞

=

= + + +∑ (4.4)

R RRA A= (4.5) here e0 and ej are oblique-wave transfer functions with the same definitions as in Chapter 2.2.4. We have

1

for 0,1,2...cosj j

j

e c jα

= = (4.6)

2 2

2 2

22

cos 1 for 0sin

sincos 11 1 for 1

xjj

j j

j

jk k

kjk k

k

αα αα

≤ == = − = + ≥ ≥

(4.7)

2 2 2

xj j yk k k= − (4.8)

kj satisfies the linear dispersion relation generalized to complex wave number. 2 tanh for 0,1,2...j jgk k h jω = = (4.9)

For the wave flume case, ej reduce to cj. c0 is the Biesel transfer function with a real value and cj ( 1j ≥ ) is the purely imaginary transfer function for the j’th evanescent mode. For a piston-type wavemaker, we have

24sinh

for 0,1, 2...2 sinh 2

jj

j j

k hc j

k h k h= =

+ (4.10)

If no reflections occur, then 0R RRA A= = , (4.3) and (4.4) are consistent with standard linear wavemaker theory. (4.3) gives the amplitude of the incident waves in terms of the paddle amplitude with the imaginary unit giving a 90 degree phase shift. (4.4) gives the amplitude including evanescent modes at the wave paddle. Conversely, if

4.2 Single Mode Absorption 29

0aX = then (4.3) expresses that incident waves are due to re-reflections alone, while (4.4) indicates that both reflected and re-reflected waves contribute to the amplitude measured at the wave paddle.


4.2 Single Mode Absorption Solving (4.3)~(4.5) with respect to aX yields 0(2 )a IX A A F= − (4.11) where

01

jj

iF

e e∞

=

= −−∑

(4.12)

This is the Fourier-domain recipe for active absorption. In addition to the hydrodynamic feedback giving wave elevation measured at paddle front 0A , this method needs only one quantity, the target progressive wave elevation

IA , to be specified. Thus it is called single mode. In the present formulation, the inevitable delay in the position servo loop of the wavemaker control is neglected for simplicity. 4.3 Dual Mode Absorption An attractive alternative to single mode absorption is the method of dual mode for improved wave generation. It is achieved by splitting the paddle position into a generation part and an absorption part. We write, gen abs

a a aX X X= + (4.13)


Here, genaX is the paddle position in case of no reflection, which is for generating the

target progressive waves. absaX is the absorption part of the paddle position for

absorbing re-reflected waves. Assuming absaX is expressed as

,0( )abs

a I IX A A F= − (4.14)

in which ,0IA is the expected wave elevation at the paddle front including target

progressive waves and evanescent modes without re-reflection. Eq. (4.14) gives a relation between the absorption part of the paddle position abs

aX and the deviation

between the measured and the expected elevation at the paddle front ,0( )I IA A− . The

deviation can be thought of as the re-reflected part of the wave elevation. This relation relies on the Fourier-domain recipe of the single mode F, which means it is based on linear wavemaker theory. In addition to hydrodynamic feedback in the form of wave elevation measured at paddle front 0A , this method needs two quantities: the generation part of paddle

position genaX and the expected elevation amplitude ,0IA . Thus, we denote this dual

mode. If gen

aX and ,0IA are evaluated according to linear wavemaker theory, we have

0

gen Ia

AX

ie= (4.15)

and

,0 010

( )II j

j

AA e e

e

∞

== +∑ (4.16)

In combination with (4.13) and (4.14), they satisfy equations (4.3)~(4.5) and are theoretically equivalent to the method of single mode. 4.4 Nonlinear Wave Generation in Dual Mode In the dual mode method, the generation part of paddle position gen

aX can be computed offline according to any wavemaker theory. The same applies to the expected elevation amplitude ,0IA , which would be taken at x=0 for linear wavemaker

theory, while it would be evaluated at x= genaX for any nonlinear theory.

In the case of no reflection, the measured elevation would be equal to the expected one, and the correction to the paddle position (4.13) would vanish. With reflections, only the deviation between the measured and the expected elevation would be processed by the active absorption procedure as a linear perturbation abs

aX on the

nonlinear wave generation genaX . For small reflections, this is consistent. For large

4.5 Real-time Realization of Active Absorption 31

reflections, an improvement over the traditional single mode method would be expected though the whole procedure would not be fully consistent with regard to nonlinearity. 4.5 Real-time Realization of Active Absorption It is necessary to transform the Fourier-domain recipe to the time domain. Since this is not the focus in this project, it is not presented here. See the reference (Schäffer 2001 and 2003) for the details.


33

Chapter 5 A Deterministic Combined Model for Wave Flumes In this chapter, a deterministic combination of a numerical model and a physical model for waves in flumes is presented. In order to fulfil the combination, based on the linear wavemaker theory and Cnoidal wavemaker theory descried in Chapter 2 and Chapter 3, a new unified wave generation method is devised first, which combines the linear fully dispersive wavemaker theory and the method of nonlinear long wave generation while compensating for local wave phenomena (evanescent modes) near the wavemaker.

5.1 Unified Wavemaker Theory for Wave Flumes Time series of surface elevation, depth-averaged horizontal particle velocity, and paddle position at the wave paddle are denoted h(t),U(t), and X(t) respectively in the time domain, while A(w), B(w) and Xa(w) denote the equivalent complex Fourier amplitudes in frequency domain:

FourierTransform

( ) ( )t Aη ω⇔ (5.1)

FourierTransform

( ) ( )U t B ω⇔ (5.2)

FourierTransform

( ) ( )aX t X ω⇔ (5.3) Here t is time and w is angular frequency. The quantities A and h carry the following subscripts: “I” for the target, progressive, incident waves, “0” for waves measured right at the paddle front, and “I,0” for the expected wave elevation at the paddle front.

34 Chapter 5. A Deterministic Combined Model for Wave Flumes

5.1.1 Linear wave generation Considering the horizontal wave flume case, according to linear fully dispersive wavemaker theory in Chapter 2, the paddle position amplitude relates to the progressive wave amplitude as 0 ( ) ( )a Iic X Aω ω= (5.4) where i is the imaginary unit showing a 90 degree phase shift, and c0 is known as the Biesel transfer function. For a piston-type wavemaker, we have

2

0

4sinh

2 sinh 2

khc

kh kh=

+ (5.5)

With B denoting the complex amplitude of the depth-averaged velocity, mass conservation yields,

( ) ( )IB Akh

ωω ω= (5.6)

Eliminating ( )IA ω from (5.4) and (5.6), we get ( ) ( )ai X Bω ω ω= Λ (5.7) where

2

0

(2 sinh 2 )

4sinh

kh kh kh kh

c kh

+Λ ≡ = (5.8)

Eq. (5.7) is the same as (2.53). Using the full dispersion relation for progressive waves as 2 tanhgk khω = (5.9) we can obtain the transfer function L on angular frequency. Figure 5.1 shows L

versus non-dimensional angular frequency /h gω . Since the limit on the frequency of the paddle motion has been traditionally chosen as

max1 2.5f f Hz≤ = for physical wave flume experiments. Also since the Boussinesq

model providing the incident waves is only accurate up to 3kh ≈ ( / 3)h gω ≈ , we

damp the high frequency response by replacing L with a modified Lm using a decaying function ( )df ω . ( )m df ωΛ = Λ (5.10)

5.1 Unified Wavemaker Theory for Wave Flumes 35

1 2

1

21 tanh[2 ]( )

2df

ω ωωπ

ωω

+−−

= (5.11)

where, w1 and w2 are parameters relevant to max1min(2 , 3( / ) )f g hπ . Assuming the maximum available frequency for physical experiment is due to the accuracy of Boussinesq model. The limit of the non-dimensional frequency is thus

chosen as 3 . Figure 5.2 shows the decaying function with respect to the non-

dimensional frequency ( / )df h gω when choosing 1 (1 3) /g hω = + and

2 (2 3) /g hω = + . The modified transfer function Lm with respect to the non-

dimensional frequency is shown in Figure 5.1. Both L and Lm tend to unity when the frequency tends to 0 ( 0)kh → . Thus the dispersion correction vanishes in the long-wave limit.

1 2 3 4 5 6ωè!!!!!!!!!!!hê g2.5

5

7.5

10

12.5

15

17.5

Λ

Λm

Figure 5.1: The transfer function for dispersion correction L and the modified transfer

function Lm versus non-dimensional frequency.

1 2 3 4 5 6ωè !!!!!! !!!!!h êg0.2

0.4

0.6

0.8

1

fd

Figure 5.2: The decaying function with respect to non-dimensional frequency. Eq. (5.7) may be rewritten as two equations

( ) ( ) (a)

( ) ( ) (b)

swa

swa m a

i X B

X X

ω ω ωω ω

=

= Λ (5.12)


where superscript “sw” indicates the use of shallow water wave theory for obtaining the paddle position from the depth-averaged particle velocity at the mean paddle position. Eq. (5.12.b) gives a dispersion correction leaving the shallow water limit. The wave paddle position in the time domain can be expressed as

( )( )

swdX tU t

dt= (5.13)

0

0

( ) ( ´) ( ) ´t

swm

t

X t X t t t dtλ−

= −∫ (5.14)

where lm(t) is the impulse response function corresponding to Lm(w) for the dispersion correction, see Figure 5.3. Here t0 reflects the effective width of the impulse response function. The effective half width of the impulse response function is about

0 6g

th

≈ (5.15)

In real practice, the dispersion correction is made in a discrete convolution by the Discrete Inverse Fourier transform (Press et al., 1989) using discrete a ( )swX ω and Lm(w) .

−8 −6 −4 −2 2 4 6 8tè!!!!!!!!!!!gê h

−1

−0.5

0.5

1

1.5

2

2.5λm

Figure 5.3: Impulse response function lm(t) for dispersion correction.

For active absorption with dual control the surface elevation at the moving paddle is also required. Due to the mismatch between the shape of the wave paddle and the vertical profile of the horizontal velocity of progressive waves, an evanescent wave field exists at the paddle front. Therefore an evanescent-mode correction to the progressive wave field is needed. From linear wavemaker theory, we obtain, ,0 ( ) ( ) ( )I I aA A Xω ω ω= + Γ (5.16)

where

5.1 Unified Wavemaker Theory for Wave Flumes 37

1

jj

i c∞

=Γ ≡ ∑ (5.17)

and

24sinh

2 sinh 2j

jj j

k hc

k h k h=

+ (5.18)

Here kj is purely imaginary corresponding to evanescent modes, and satisfies the linear dispersion relation generalized to complex wave numbers, 2 tanhj jgk k hω = (5.19)

The transfer function G is shown in Figure 5.4 when 20 evanescent modes are considered. G is modified to Gm by damping at high frequencies as done for L , also see Figure 5.4. ( )m df ωΓ = Γ (5.20) Both G and Gm tend to zero when the frequency tends to zero. Thus the evanescent-mode correction vanishes in the long wave limit.

1 2 3 4 5 6ωè!!!!!!!!!!!hê g0.5

1

1.5

2

2.5

Γ

Γm

Figure 5.4: The transfer function for the additional evanescent modes G and the modified

transfer function Gm versus non-dimensional frequency.

The surface elevation at the moving paddle in the time domain is expressed as

0

0

,0 ( ) ( ) ( ´) ( ´) ´t

I I m

t

t t X t t t dtη η γ−

= + −∫ (5.21)

here gm(t) is the impulse response function corresponding to Gm(w), see Figure 5.5. Here t0 is in accordance with (5.15).


−8 −6 −4 −2 2 4 6 8tè!!!!!!!!!!!gê h

−1

−0.75

−0.5

−0.25

0.25

0.5

0.75

1γm

Figure 5.5: Impulse response function gm(t) for evanescent-mode correction.

5.1.2 Nonlinear shallow water wave generation For nonlinear shallow water waves with small dispersion such as Cnoidal waves, the horizontal particle velocity is almost uniform over depth. The time-domain relation between the depth-averaged velocity and the paddle position is given directly as

( )

( ( ), )sw

swdX tU X t t

dt= (5.22)

with the initial condition (0) 0swX = . This approach was used successfully by Goring (1979) first for Cnoidal waves. This step captures the nonlinearity of the numerical model, but corresponds to the shallow water limit for the wave generation. Eq. (5.22) matches the result from linear shallow water theory (5.13) except that here U is taken at the actual moving paddle position rather than at its mean position. For active absorption with dual control, the associated surface elevation at the moving paddle is also needed, ,0 ( ) ( ( ), )sw

I t X t tη η= (5.23)

5.1.3 Ad hoc unified wave generation

Given the lack of a universally valid theory, we propose an ad hoc combination of the linear fully dispersive wavemaker theory and the method of nonlinear long wave generation. We compute the shallow water paddle position from (5.22), and then compensate for the disregarded dispersion using (5.14). For vanishing dispersion this procedure reduces to nonlinear long wave generation, while fully dispersive linear wavemaker theory is recovered in case of vanishing nonlinearity.

In order to avoid a slow drift of the paddle, a small term proportional to the paddle signal is added to the differential equation. This has the effect of a first order high-pass filter (Humpherys, 1970). Let wc denote the characteristic angular frequency of this filter, then the unified wave generation is altogether governed by

5.2 Tests of the Deterministic Combined Model 39

( )

( ) ( ( ), )sw

sw swc

dX tX t U X t t

dtω+ = (5.24)

followed by the dispersion correction

0

0

( ) ( ´) ( ) ´t

swm

t

X t X t t t dtλ−

= −∫ (5.25)

For dual mode active absorption, the expected surface elevation at the moving paddle

,0 ( )I tη is needed. Without the evanescent-mode correction, we have

,0 ( ) ( ( ), )P

I t X t tη η= (5.26)

where the superscript “p” on the quantity ,0 ( )P

I tη indicates that only progressive waves

are accounted for. With the evanescent-mode correction, the surface elevation at the moving paddle is

,0 ,0 0( ) ( ) p evaI It tη η η= + (5.27)

0

0

0 ( ´) ( ) ´ t

evam

t

X t t t dtη γ−

= −∫ (5.28)

where 0evaη is the correction part for the evanescent modes.

In this unified wave generation, ( , )U x t , ( , )x tη can be from a numerical wave propagation model or some wave theory such as Cnoidal wave theory. 5.2 Tests of the Deterministic Combined Model In the combined model, the Mike 21 BW 2D module is chosen for the numerical model. As an extension of the numerical flume, the physical tests are made in a 0.75m wide, 1.20m deep, and 23m long flume. The ad hoc unified wave generation is applied as the link between the numerical model and the physical model. The flume is equipped with a piston-type wavemaker with an electric drive system including a brushless AC motor and an integrated linear drive/bearing system. The wavemaker is controlled by the DHI AWACS with three different control modes. These control modes are single mode with active absorption, dual mode with active absorption and consistent nonlinear wave generation, and position mode with nonlinear wave generation but without active absorption. Two control modes of active absorption have been introduced in Chapter 4. Some tests on regular and irregular, linear non-shallow and nonlinear shallow waves in flumes are presented.


5.2.1 Linear wave case First of all, we use linear fully dispersive waves to test the method of unified wave generation taking account into the dispersion and evanescent-mode corrections. The length of the simulated wave flume is 160m. The numerical simulation is based on the constant water depth h=0.7m, wave height H=0.05m, wave period T=1s, which gives the wave length L=1.55m and the relative water depth kh=2.83. The time step is taken as dt=0.01s, the grid spacing is dx=0.1m. The internal wave generation is set at x=10.2m. Figure 5.6 shows profiles of surface elevation h and P flux (Uh) at a certain time between x=100m and x=110m extracted from the numerical results of Mike 21 BW. Surface elevation [m]

P flux [m^3/s/m]

101.0 102.0 103.0 104.0 105.0 106.0 107.0 108.0 109.0 110.0x (m)

-0.04

-0.02

0.00

0.02

0.04

[m]

Figure 5.6: Profiles of h and P flux at a certain time between x=100m and 110m. Defining two relative errors for a general time series ( )ky t compared with the expected time series xk(t), ErrRMS gives the root-mean-squared error, and ErrH is the error of wave height,

2

1

( ( ) ( ))

(a)max( )

max( ) min( ) [max( ) min( )] (b)

[max( ) min( )]

N

k ki

RMSk

k k k kH

k k

y idt x idt

NErrx

y y x xErr

x x

=

−

=

− − −=−

∑

(5.29)

Compared with the linear wave theory, ErrRMS =1.0% and ErrH =-0.04% on h, and ErrRMS =1.0% and ErrH =3.79% on P flux at x=93.2m where is chosen as the mean paddle position for calculating X(t). The numerical solution includes higher order waves due to Boussinesq model itself. We utilize the unified wave generation method to calculate wave paddle position for the control signals of the physical model. Since it is a pure linear wave case, the average velocity in (5.24) could be at the mean paddle position, and in order to


remove the effect of mean P flux created by errors in the numerical simulation of Mike 21 BW, (5.24) is modified to

0

( )( ) ( , ) (a)

( , ) ( , )( , ) (b)

swsw

c

dX tX t U x t

dt

P x t P x tU x t

h

ω

+ =

− =

%

%

(5.30)

Here P(x,t) and h(x,t) are numerical calculations, x0 is the mean paddle position, x0=93.2m for this case. cω should be much lower than the frequency of the waves to

serve as a high-pass filter, 2 / 30 c Hzω π= is chosen. We use the explicit forward Euler (1st order) scheme and central (2nd order) scheme to discretize (5.30a) for the numerical simulation, respectively. The initial time for

( )swX t is chosen as a certain time when 0( , )U x t% arrives at a crest value, it corresponds

to ( ) 0swX t = . The same time step is chosen as the simulation by Mike 21 BW,

dt=0.01s. The time series of ( )swX t is obtained as shown in Figure 5.7. Since the Biesel transfer function c0 has a limit for shallow water,

0 for 0c kh kh→ → , according to linear wavemaker theory, we have

0

( ) sin (a)2

( ) sin (b)2

li

swli

HX t t

c

HX t t

kh

ω

ω

=

= (5.31)

where, ( )liX t denotes the paddle position by linear wavemaker theory, and ( )sw

liX t is the paddle position with the shallow water limit by linear wavemaker theory. We get ErrRMS =2.65% and ErrH =3.74% on ( )swX t compared with ( )sw

liX t using 1st

order scheme. When changing 2nd order scheme, ErrRMS =3.10% and ErrH =3.79%. The 2nd order scheme has no advantage for this case. The expected wave paddle position X(t) is obtained by means of the dispersion correction using (5.25). Practically, the dispersion correction is made in a discrete convolution by the Discrete Inverse Fourier transform of discrete list a ( ) ( )sw

mX ω ωΛ in the frequency domain. For this regular linear wave case T=1s, h=0.7m, we get

/ 1.68h gω = and 1.48.mΛ = Λ = The comparison of X(t) with dispersion correction

and ( )swX t is shown in Figure 5.7. The difference between them is considerable.

Comparing X(t) with the theoretical solution ( )liX t , ErrRMS =2.60% and ErrH =3.71% using 1st order scheme, and ErrRMS =3.05% and ErrH =3.58% using 2nd order scheme. Since we use numerical calculations of Mike 21 BW to calculate time series of paddle positions ( )swX t and X(t), the above errors of paddle positions include the errors of the


numerical model (Mike 21 BW), besides the procedure error of solving (5.30a). In order to exclude the error of numerical model and see the procedure error, we use the velocity field of linear wave theory to calculate paddle positions. Replacing 0( , )U x t% in (5.30a) with the theoretical U(0,t) to calculate time series of

paddle positions ( )swX t and X(t), and then comparing them with theoretical solutions

in (5.31). ErrRMS =2.22% and ErrH =43.3 10−− × on ( )swX t , ErrRMS =2.22% and ErrH

=-0.11% on ( )X t using 1st order scheme. ErrRMS =44.7 10−× and ErrH =

46.58 10−× on

( )swX t , ErrRMS = 58.9 10−× and ErrH= 41.26 10−− × on ( )X t using 2nd order scheme, respectively. Thus the dominant error is due to the numerical model during the whole procedure of calculation.

Figure 5.7: The expected wave paddle positions X(t) with dispersion correction,

and the paddle position in shallow water ( )swX t .

The surface elevation at the moving paddle is required for the dual mode control system in the physical model. Eq. (5.26) ~ (5.28) are utilized for that. For this regular linear wave case, 0.8.mΓ = Γ = Surface elevations at the moving paddle are shown in

Figure 5.8. ,0 ( )I tη is the final control signal for the physical model. The correction

part for the evanescent modes 0 ( )eva tη has a different phase from the progressive part

,0 ( )pI tη (90 degree shift). According to the linear theory, we have

1,0

0

( ) (cos sin )2

jj

I

i cH

t t tc

η ω ω

∞

== +∑

(5.32)

Comparing the calculated solution with this theoretical solution, ErrRMS =1.89% and ErrH =3.66% using 1st order scheme, and ErrRMS =1.97% and ErrH =3.58% using 2nd order scheme. Here the errors of the numerical model are again involved. Through this linear wave case, the procedure for the combined model is verified.


Figure 5.8: Surface elevations at the paddle: ,0 ( )I tη is the expected,

0 ( )eva tη is the evanescent modes, and ,0 ( )pI tη is the progressive part.

5.2.2 Nonlinear shallow water wave cases

Next, we test nonlinear shallow water waves for the application of the unified wave generation method. The combined model is run with Cnoidal wave input to the numerical model.

The length of the simulated wave flume is 160m. The numerical simulation is based on the constant water depth h=0.4m, wave period T=2.3s. In order to test waves with various nonlinearity, we choose a group of wave height: H=0.12m, 0.16m and 0.20m for flume tests. Table 5.1 shows the corresponding wave lengths and relative water depths kh according to Cnoidal wave theory.

Table 5.1: wave conditions at h=0.4m, T=2.3s according to Cnoidal wave theory _______________________________________________________________________________

Wave Height H (m) 0.12 0.16 0.20 _______________________________________________________________________________

Wave length L (m) 4.48 4.61 4.76 kh 0.561 0.545 0.528 Ur (HL2/h3) 37.63 53.13 70.80 _______________________________________________________________________________

The grid spacing is taken as dx=0.1m for each wave height condition. The time step is taken as dt=0.01s. Figure 5.9 shows profiles of surface elevation h and P flux at a certain time between x=130m and x=160m extracted from the numerical calculations of Mike 21 BW using H=0.20m. Since the numerical calculations are not exactly periodic due to errors of the numerical simulation, they are modified to the periodic zero-mean values, as

1

0

1ˆ( , ) ( , ), for 0 (a)

ˆ ˆ( , ) ( , ) ( , ) j=0,1,2... (b)

N

j

x t x t jT t TN

x t jT x t x t

η η

η η η

−

=

= + ≤ ≤

+ = −

∑%

(5.33)


where, N is the number of periods simulated in the numerical model. P flux is also modified to the periodic zero-mean value ( , )P x t% in the same way as done for h. Figure 5.10 shows the comparison of the modified surface elevation from the numerical model with the theoretical Cnoidal wave for H=0.20m. As we know Stream Function theory (See Appendix C) is much more accurate for fully nonlinear waves, and Stream Function wave is also shown in Figure 5.10 to compare. Since Cnoidal waves are the input of Mike 21 BW, the deviation of the numerical calculation from the Cnoidal wave shows the error made by numerical model. The difference of shapes between Cnoidal wave and Stream Function wave is obvious. Among the three solutions, the Stream Function wave has the highest crest, and the numerical calculation has the lowest trough.

130 135 140 145 150 155 160x (m)

-0.10

0.00

0.10

0.20

[m] Surface elevationP flux [m^3/s/m]

Figure 5.9: Profiles of h and P flux between x=130m and 160m for H=0.20m.

Figure 5.10: The comparison of h among the modified numerical calculation by Mike 21 BW

(BW), Cnoidal wave theory (CN) and Stream Function theory (SF) for H=0.20m. Figure 5.11 shows the relative errors of the numerical calculation ( , )x tη% as done for linear wave case in (5.29) compared with the Stream Function wave and the Cnoidal wave, respectively. The numerical calculation is closer to the Cnoidal wave. The errors of Cnoidal wave compared with Stream Function wave is also shown in Figure 5.11. The deviation of Cnoidal wave from Stream Function wave is quite large although they have almost the same wave height. The higher nonlinearity, the larger errors they have. Cnoidal wave theory is thus inadequate for highly nonlinear waves.


Figure 5.11: The relative errors of surface elevations.

For nonlinear waves, the wave paddle position can be calculated by (5.24). We modify U(x,t) using the modified periodic zero-mean numerical calculations ( , )P x t% and ( , )x tη% from Mike 21 BW, this yields,

( )( ) ( ( ), ) (a)

( , )( , ) (b)

( , )

swsw sw

c

dX tX t U X t t

dt

P x tU x t

h x t

ω

η

+ =

= +

%

%%

%

(5.34)

For simulating (5.34), the mean paddle position is set at x0=133.2m in the whole unified model, and 2 / 30 c Hzω π= is chosen. The time step is taken as dt=0.01s. Since the grid spacing is taken as dx=0.1m in the numerical calculation, not many grid points are involved in the range of the moving paddle for the nonlinear term

( ( ), )swU X t t% . We thus use a ‘spline’ method instead of linear ‘interpolation’ to smooth the nonlinear distribution of velocity around the moving paddle. The expected wave paddle position X(t) is obtained by the dispersion correction (5.25) away from

the limit of shallow water. For T=2.3s, h=0.4m, we have / 0.55h gω = and

1.0024.mΛ = Λ = The comparison of X(t) and ( )swX t for H=0.20m is shown in Figure 5.12. The deviation is small enough to make the dispersion correction unnecessary. This is in accordance with the wavemaker theory for shallow water waves.

Figure 5.12: The comparison of X(t) and ( )swX t for H=0.20m.

Figure 5.13 shows the comparison of ( )X t calculated by the modified solution of Mike 21 BW with those using linear wavemaker theory, Cnoidal wavemaker theory,

0.0%

2.0%

4.0%

6.0%

8.0%

0.12m

0.16m

0.20m

H

Err

RM

S BW compared with CN

BW compared with SF

CN compared with SF

-1.0%

0.0%

1.0%

2.0%

3.0%

0.12m

0.16m

0.20m

H

Err

H

BW compared with CN

BW compared with SF

CN compared with SF


and solving (3.10) with solution of Stream Function theory (see Chapter 7) for H=0.20m. The linear wavemaker theory gives the largest range of paddle position. On the contrary, Stream Function theory provides the smallest range, but the highest nonlinearity. ( )X t by Mike 21 BW and the Cnoidal wavemaker theory are matched in general between the other two theories. The deviation of Cnoidal wavemaker theory from Stream Function wavemaker theory is quite large. Cnoidal wavemaker theory is thus inadequate for highly nonlinear waves.

Figure 5.13: The comparison of ( )X t calculated by modified solution of

Mike 21 BW (BW), linear wavemaker theory (Linear), Cnoidal wavemaker theory (CN), and Stream Function theory (SF) for H=0.20m .

The surface elevation at the moving paddle ,0 ( )I tη is obtained by (5.26) ~ (5.28).

Figure 5.14 shows the comparison of ,0 ( )I tη , the progressive part ,0 ( )pI tη , the

evanescent-mode part 0 ( )eva tη , and the elevation at the mean paddle position 0( , )I x tη

for H=0.20m. The deviation of ,0 ( )pI tη from 0( , )I x tη is considerable. ,0 ( )I tη has much

more nonlinearity than 0( , )I x tη . 0 ( )eva tη is much smaller than ,0 ( )pI tη . For T=2.3s and

h=0.4m we have 0.008mΓ = Γ = . The ratios of wave height of 0 ( )eva tη to the wave

height of ,0 ( )pI tη are 3.91% for H=0.12m, 5.09% for H=0.16m, and 6.45% for

H=0.20m, respectively.

Figure 5.14: The comparison of ,0 ( )I tη (total), 0( , )I x tη (eta(x0,t)), ,0 ( )p

I tη (eta(X(t),t)), and

0 ( )eva tη (evanescent modes) for H=0.20m.


For the experimental validation, we make tests using the three different control modes discussed in introduction and Chapter 4. The control signals have been obtained as above. In the flume, time series of surface elevation are measured by three wave gauges at 1.0m (gauge 1), 4.4m (gauge 2), and 8.7m (gauge 3) from the mean paddle position. All the analyses of the measurements are based on 5 periods of surface elevation measured at each gauge before reflected waves arrive. Furthermore, the phase was shifted for each signal to get a wave crest at t=0 for better comparison of wave shapes. First of all, we make tests using linear wavemaker theory in position mode for two cases as references. Figure 5.15 shows time series of surface elevation measured at three different gauges in the flumes compared with the theoretical solution by Stream Function theory. It is clear to see the shape of waves is very unstable in the flume. Next, we generate waves using the modified calculations of Mike 21 BW. The dispersion correction and evanescent-mode correction are considered in the control signals as described before. Figure 5.16 shows the measurements of waves in the flumes for H=0.12m and 0.20m. The modified calculation of Mike 21 BW is also shown in each graph to compare. The measurements match the numerical results well in general for H=0.12m at 3 gauges. When wave height increases, at gauge 4.4m both dual mode and position mode give higher elevation peaks than expected numerical results. Thus the reproduction of quite nonlinear waves is unstable in the flume for highly nonlinear waves.

(a) H=0.12m

(b) H=0.20m


Figure 5.15: Measured surface elevation of waves generated by linear wavemaker theory.

(a) H=0.12m


(b) H=0.20m

Figure 5.16: Measured surface elevation of waves generated by modified calculations of Mike 21 BW compared with the numerical calculation (BW).

Figure 5.17 gives the relative errors of the measurements at three gauges for each wave height, compared with the numerical solutions as done for calculations of surface elevation in Figure 5.11. The minus on ErrH means the wave height measured is smaller than the desired value. The big variation of ErrH from gauge to gauge (as e.g. for H=0.20m) demonstrates the non-constant form of the waves in the flume. For


these regular waves, the attenuation factor a is also calculated and shown in Figure 5.18. Referred to Newland (1984), it is defined as

00 2

( )( ) y xy

xyx x

Rσ τα ρ τ

σ σ= = (5.35)

Here, 0( )xyρ τ is the correlation coefficient function of time series yk(t) and the target

one xk(t) at lag-time t=t0 when two time series are synchronized. 0( )xyR τ is the

correlation function, and sx and sy are standard deviations. They are expressed as

00

( )( ) xy

xyx y

R τρ τ

σ σ= (5.36)

( ) [ ( ) ( )]xy k kR E x t y tτ τ= + (5.37)

2[( ( ) [ ( )]) ]x k kE x t E x tσ = − (5.38)

where [ ( )]kE x t is the mean value of xk(t). For the finite number of observed values,

1

1[ ( )] ( )

N

k ki

E x t x iN =

= ∑ (5.39)

0.0%

5.0%

10.0%

15.0%

20.0%

0.12m 0.16m 0.20m

H

Err

RM

S single mode

dual mode

position mode

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

0.12m 0.16m 0.20mH

Err

H

single mode

dual mode

position mode

Figure 5.17: The relative errors of measurements using results of Mike 21 BW .


0.7

0.8

0.9

1

0.12m 0.16m 0.20m

H

Att

enu

atio

n f

acto

r

single mode

dual mode

position mode

Figure 5.18: Attenuation factors using results of Mike 21 BW.

The errors become larger as the wave height increases with the higher nonlinearity. Among three groups of attenuation factors a in three different modes, factors of single mode are always smaller than the corresponding factors of other two modes as expected due to the inconsistent nonlinear wave generation. The factors a in position mode are the largest among three modes (except for the case H=0.20m at gauge 3). Thus, position mode is in general the best mode for the nonlinear wave generation in physical model. However it has the severe disadvantage of fully re-reflecting waves in case a structure or a beach is typically located. We also notice that dual mode gives slightly better results than position modes when H=0.16m and H=0.20m. The reason is probably that in dual mode the system gets a second chance for making slight corrections to the paddle signal in case the measured surface elevation does not exactly match the expected one. The real advantage of dual mode is the active absorption with consistent nonlinear wave generation. To investigate the reason for the mismatch between the physical results and the numerical calculation of Mike 21 BW, we eliminate the numerical model by testing waves with the same conditions (T=2.3s, h=0.4m, H=0.12m, 0.16m and 0.20m) but generated by Cnoidal wavemaker theory in physical flume. The measurements of surface elevation are compared with the theoretical Cnoidal waves for H=0.12m and 0.20m in Figure 5.19. The same phenomenon happens as before, the peaks of the surface elevation measured in dual mode and position mode are still much higher than the theoretical elevation at the gauge x=4.4m for H=0.20m. Also Figure 5.16 and Figure 5.19 are quite similar for each case. For these cases, it is clear that the results in single mode are much worse than the results in other two modes, especially for highly nonlinear waves. This is expected due to the inconsistent nonlinear wave generation of single mode. The relative errors and the attenuation factors of the measurements compared with the theoretical Cnoidal waves are calculated as shown in Figure 5.20 and Figure 5.21, respectively.


(a) H=0.12m


(b) H=0.20m Figure 5.19: Measured surface elevation of waves generated by Cnoidal wavemaker theory

compared with the theoretical solution (CN).


0.0%

5.0%

10.0%

15.0%

20.0%

0.12m 0.16m 0.20m

H

Err

RM

S single mode

dual mode

position mode

-15.0%

-5.0%

5.0%

15.0%

25.0%

35.0%

0.12m 0.16m 0.20mH

Err

H

single mode

dual mode

position mode

Figure 5.20: The relative errors of measurements using Cnoidal wavemaker theory.

0.7

0.8

0.9

1

1.1

0.12m 0.16m 0.20m

H

Att

enu

atio

n f

acto

r

single mode

dual mode

position mode

Figure 5.21: Attenuation factors using Cnoidal wavemaker theory.

In general the results using solutions of Mike 21 BW and those using Cnoidal wavemaker theory are quite similar, although the results by Cnoidal wavemaker theory are a little better in dual mode and position mode, with a little larger attenuation factors and bit smaller errors compared with those using numerical calculations. This indicates that possible errors in the numerical wave flume (Mike 21 BW) have a little effect in this case. The mismatch between the surface elevation measured at gauge x=4.4m and the Cnoidal wave theory for H=0.20m shows that Cnoidal (wavemaker) theory is inadequate for highly nonlinear waves. To further investigate the reason for the difficulty in obtaining these quite nonlinear waves of constant form in the physical wave flume, we make tests generating waves using Stream Function theory. This method is investigated in detail in Chapter 7. The measurements of surface elevation are compared with the theoretical Stream Function wave in Figure 5.22 for H=0.12m and 0.20m. The relative errors and the attenuation factors of the measurements compared with the theoretical solution are shown in Figure 5.23 and Figure 5.24, respectively.


(a) H=0.12m


(b) H=0.20m Figure 5.22: Measured surface elevation of waves generated by Stream Function wavemaker

theory compared with the theoretical solution (SF).

The results of single mode are still bad as expected. The measurements of surface elevations in dual mode and position mode match the solutions of Stream Function theory very well at each gauge for each case. Especially at the gauge 4.4m for H=0.20m, the results are improved a lot comparing with the results either by Cnoidal


wavemaker theory or by calculations of Mike 21 BW. The attenuation factors in these two modes are much close to unity, and the errors for dual and position modes are much smaller than the corresponding errors using Cnoidal wavemaker theory or calculations of Mike 21 BW. Compared with dual mode, position mode gives better results, especially for the highly nonlinear waves. This is because there is no any disturbance in the control system for pure wave generation. Therefore, for these shallow cases, the surface elevation of highly nonlinear waves is reproduced very well by the Stream Function wavemaker theory. The dominant error of the wave reproduction is due to the limited accuracy of Mike 21 BW or Cnoidal wave theory, rather than the ad hoc unified wave generation.

0.0%

5.0%

10.0%

15.0%

0.12m 0.16m 0.20m

H

Err

RM

S single mode

dual mode

position mode

-20.0%

-15.0%

-10.0%

-5.0%

0.0%

5.0%

0.12m 0.16m 0.20m

H

Err

H

single mode

dual mode

position mode

Figure 5.23: The relative errors of measurements using Stream Function wavemaker theory.

0.7

0.8

0.9

1

1.1

0.12m 0.16m 0.20m

H

Att

enu

atio

n f

acto

r

single mode

dual mode

position mode

Figure 5.24: The attenuation factors of measurements using solutions of

Stream Function theory . 5.2.3 Irregular waves propagating on constant water depth With the purpose of testing the dispersion correction and the evanescent-mode correction, we now turn to a rather deep water case for irregular waves. The simulated wave flume is 160m long with a constant water depth h=0.7m. The irregular incoming wave conditions are a JONSWAP frequency spectrum, with a significant wave height


Hm0=0.05m, a peak period Tp=1.2s, and the relevant shape parameters, g=3.3, sa=0.07, sb= 0.09. The spectrum is truncated omitting periods smaller than 0.95s considering the limit of Mike 21 BW model. This gives a range of relative water depth of the irregular waves kh between 2.02 (for Tp=1.2s) and 3.13 (for Tmin=0.95s). The truncated spectrum is rescaled in the numerical model. The time step is taken as dt=0.01s, the grid spacing is dx=0.1m. The internal wave generation is set at x=10.2m in the numerical model. For irregular waves, the time series of surface elevation and the depth-averaged velocity are dependent on the location in the simulated flume, so that the data transfer between numerical and physical models and the time series of wave paddle position may depend on the location of the mean paddle position in the flume. Therefore, we set two fixed wave gauges x1=100m and x2=103.4m in the whole simulated flume, and choose two different locations, x0=96m and x0=99m for the data transfer between the numerical and physical models. Thus x0 is the mean paddle position for the physical flume. See Figure 5.25 for the sketch of the combined model. Since the procedure is the same for two different x0, in the following description of the procedure before physical tests, all the results are presented for x0=96m.

Figure 5.25: Sketch of the combined model.

We set the mean paddle position of the physical model at x0=96m first, and make t=250s in the numerical model at the start point to transfer data between numerical and physical model. Time series of surface elevation and the depth-averaged velocity at x0 extracted from calculations of numerical model are shown in Figure 5.26.

Figure 5.26: Calculations at x0=96m extracted from output of numerical model.

Paddle position

X

Wave gauge

Simulated flume by numerical model: 160m

X

Wave

gauge

h=0.7m x0=99m

x0=96m


To calculate wave paddle position, we make t0=250.66s to be the initial time for X=0 as U(x0,t0) arrives at the first crest. The flux of the numerical calculation is also modified to zero-mean in order to remove some errors of numerical model. Then (5.24) yields,

( )( ) ( ( ), ) (a)

( , ) ( , )( , ) (b)

( , )

swsw sw

c

dX tX t U X t t

dt

P x t P x tU x t

h x t

ω

η

+ =

− = +

%

% (5.40)

We choose 2 /10 c Hzω π= for the high-pass filter in this case, and the total simulation period for the paddle position is about 10min. The expected wave paddle position X(t) is obtained using (5.25) with dispersion correction. The comparison of ( )swX t and X(t) in an extracted short time period is shown in Figure 5.27 corresponding to the numerical calculation in Figure 5.26. Note that in the following graphs, the time scale is in accordance with time in numerical model. The deviation between ( )swX t and X(t) is considerable.

Figure 5.27: The comparison of ( )swX t and X(t).

The surface elevation at the moving paddle ,0 ( )I tη is obtained for the requirement of

dual mode control system. The comparison of progressive wave part ,0 ( )pI tη and the

progressive wave elevation at the mean paddle 0( , )I x tη is shown in Figure 5.28 in the

short period. The deviation of ,0 ( )pI tη from 0( , )I x tη is almost zero, which means

waves are nearly linear. The correction for the evanescent modes 0 ( )eva tη is calculated

by (5.28). Figure 5.29 shows the comparison of ,0 ( )pI tη , 0 ( )eva tη and ,0 ( )I tη during the

extracted short time period. The evanescent modes are substantial.


Figure 5.28: Time series of ,0 ( )p

I tη and 0( , )I x tη .

Figure 5.29: The comparison of surface elevations at the moving paddle:

progressive wave ,0 ( )pI tη , evanescent modes 0 ( )eva tη , and the expected elevation ,0 ( )I tη .

As we know the dual mode has advantages over the other two modes of wave generation, so in this case we will present wave flume tests in dual mode using control signals calculated above. Two wave gauges are set at 4.0m (gauge 1) and 7.4m (gauge 2) far from the mean paddle position in the flume. Figure 5.30 shows the comparison of the surface elevation measured at gauge 1 with the numerical solution in an extracted short time period. Figure 5.31 shows the measurement at gauge 2 and the comparison of it with the numerical calculation. The measurements in the flume and the numerical calculation are matched very well. For the irregular waves, we calculate the correlation coefficient function rxy(t) expressed in (5.36) of the surface elevation measured in the whole 10-min period with the target numerical solution. Figure 5.32 shows the correlation coefficient function of the surface elevation measured at gauge 1, and the auto-correlation coefficients of the numerical calculation for comparison, as an example. The zero-lag correlation coefficient is the maximum value 0.987, and the shape is matched well with auto-correlation coefficients of the target numerical calculation. This means the measurement is synchronized with the corresponding target numerical calculation. At gauge 2, the correlation coefficient 0.968 also arrives at zero lag time. Both correlation coefficients at gauge 1 and 2 are very close to unity. Thus the numerical


model is passed to the physical flume very well for this irregular wave case. We notice that the agreement at gauge 1 is a little better than that at gauge 2. It is reasonable because the gauge 1 is closer to the wavemaker.

Figure 5.30: The comparison of surface elevation measured at gauge 1 (dual mode)

with the numerical calculation (BW) choosing x0=96m.



Figure 5.32: Correlation coefficient function at gauge 1 choosing x0=96m.


We now repeat the test but with the mean paddle position shifted 3m to x0=99m in the flume. Two wave gauges are moved accordingly to 1.0m (gauge 1) and 4.4m (gauge 2) far from the mean paddle position in the flume, in order to measure elevation time series at the same distance from the up-wave boundary of the numerical flume as before. The comparison of measurements and the numerical calculation in the extracted short period at gauge 1 is shown in Figure 5.33. Figure 5.34 shows the measurement at gauge 2 compared with the numerical calculation. The correlation coefficients are 0.994 at gauge 1 and 0.985 at gauge 2 for the 10-min duration of physical test. Note that the gauge positions were chosen to leave the distance to the up-wave boundary of the numerical model unchanged. Thus the target (BW) at gauge 1 in Figure 5.30 and Figure 5.33 are identical, the target (BW) at gauge 2 in Figure 5.31 and Figure 5.34 are also identical.

Figure 5.33: The Comparison of surface elevation measured at gauge 1(dual mode)



with the numerical calculation (BW) choosing x0=99m. Thus, all the measurements in physical flume match the numerical calculation very well. The measurement is closer to the numerical result when the wave gauge is closer to the mean paddle position. This is due to the error of the numerical model. However, the combined model is almost independent of the mean paddle’s location. Therefore,


the combined model is not sensitive to where the physical model takes over from the numerical model. 5.2.4 Irregular waves propagating on variable water depth The last example considers irregular waves propagating up a slope from a deep plateau to a shallow plateau. See Figure 5.35 for the sketch of the combined model. At the end of the slope the nonlinearity is very high. The length of the simulated wave flume is 160m. The physical model is set at flat shallow water depth, h=0.4m. The internal wave generation of numerical model is set at x=10.2m with the deep water depth 2.6m. The slope is 1/50 and the shallow water depth is 0.4m. The irregular incident wave conditions are a JONSWAP frequency spectrum, with a significant wave height Hm0=0.12m, a peak period Tp=3s, and the relevant shape parameters, g=3.3, sa=0.07, sb= 0.09. The spectrum is truncated omitting periods smaller than 2.6s. The truncated spectrum is rescaled in the numerical model. The time step is taken as dt=0.01s, the grid spacing is dx=0.1m. Figure 5.36 shows profiles of flux and surface elevation at t=180s extracted from the output at the part of the flume with flat shallow water depth h=0.4m, where the physical model is set. After shoaling at flat shallow water depth waves turn to irregular nonlinear long waves. We set two fixed wave gauges at x=128.2m and x=131.6m in the whole simulated flume. For the data transfer between numerical and physical models, we choose two locations x0=125.2m and x0=127.2m as the mean paddle positions for physical flume tests. The mean paddle position of the physical model is set at x0=125.2m first in the whole simulated flume. We make t=60s in the numerical model at the start point to transfer data from numerical model to physical model. Then for calculations of wave generation, the initial time is chosen t0=60.28s when U(x0,t) arrives at the first crest. Time series of surface elevation and the depth-averaged velocity at the mean paddle position extracted from the numerical calculations are shown in Figure 5.37.

Figure 5.35: Sketch of the combined model.

Paddle position

X

Wave gauge

Simulated flume by numerical model: 160m

X

Wave gauge

h1=2.6m

h2=0.4m

1:50

1:50

x0=127.2m

x0=125.2m


Surface elevation [m] P flux [m^3/s/m]

125 130 135 140 145

-0.10 0.00 0.10 0.20 0.30

Figure 5.36: Profiles of P flux and surface elevation at flat water depth h=0.4m at t=180s.

Figure 5.37: Numerical calculations at x0=125.2m in a short period.

We choose 2 / 20 c Hzω π= for the high-pass filter effect, and the total simulated

period is about 10 min for calculating wave paddle position. Time series of ( )swX t in shallow water is solved using (5.40). The expected wave paddle position ( )X t is obtained by the dispersion correction. The maximum value of deviation between ( )swX t and ( )X t is 3.6ä10-3m which occurs at X(t)=0.253m. The comparison

of ( )swX t and ( )X t in a short period is shown in Figure 5.38. The deviation is small enough to make the dispersion correction omitted. This is in accordance with the real phenomenon of nonlinear long waves.

Figure 5.38: Comparison of ( )swX t and X(t) choosing x0=125.2m.


For another control signal in dual mode, Figure 5.39 shows the comparison of ,0 ( )pI tη

with 0( , )I x tη in a short period. There is a little deviation when the nonlinearity is

high. 0 ( )eva tη is obtained by evanescent-mode correction. Its maximum value is

9.3ä10-3m. The comparison of ,0 ( )pI tη , 0 ( )eva tη and ,0 ( )I tη in an extracted short time

period is shown in Figure 5.40. The values of 0 ( )eva tη are small enough to be omitted

comparing with ,0 ( )pI tη . It means the evanescent-mode correction is not necessary in

this case. This is in accordance with shallow water waves.

Figure 5.39: Comparison of ,0 ( )p

I tη and 0( , )I x tη .

Figure 5.40: Comparison of surface elevations: progressive wave ,0 ( )p

I tη , evanescent modes

0 ( )eva tη , and the expected elevation ,0 ( )I tη .

We again take wave flume tests in dual mode. Two wave gauges are set at 3.0m (gauge 1) and 6.4m (gauge 2) far from the mean paddle position x0=125.2m in the flume. Figure 5.41 shows the surface elevation measured at gauge 1 compared with the numerical calculation in an extracted short time period. Figure 5.42 shows the comparison of measurements with the numerical calculation at gauge 2. Although deviations occur especially for the higher waves, the match between the numerical results and physical results is quite good considering the very high nonlinearity of the waves. The Ursell number is about 71 for a wave at Hm0 and Tp, and thus about twice that for the highest waves in the wave train. In fact, some wave breaking occurred in


the physical fume on the down-wave side of the wave gauges Wave breaking was not accounted for in the numerical model. For the smaller nonlinear waves, the deviation between physical and numerical results is smaller. The correlation coefficients are 0.947 at gauge 1 and 0.941 at gauge 2 for the entire 10-min duration.

Figure 5.41: Comparison of surface elevation measured at gauge 1 with the numerical

calculation (BW) choosing x0=125.2m.

Figure 5.42: comparison of surface elevation measured at gauge 2 with the numerical

calculation (BW) choosing x0=125.2m. Now we repeat the test with the mean paddle position at x0=127.2m. The two wave gauges are moved accordingly to 1.0m (gauge 1) and 4.4m (gauge 2) far from the mean paddle position in the flume. Figure 5.43 shows the measurements at gauge 1 and gauge 2, respectively, compared with the numerical calculations. The correlation coefficients are 0.942 at gauge 1 and 0.948 at gauge 2 in the entire duration.

5.3 Summary and Conclusions 67

(a) at gauge 1

(b) at gauge 2

Figure 5.43: Measurements compared with the numerical calculation (BW) when x0=127.2m. 5.3 Summary and Conclusions In this chapter, an ad hoc unified wave generation theory for wave flumes has been devised, that accounts for shallow water nonlinearity and compensates for local wave phenomena (evanescent modes) near the wavemaker. For small amplitude linear waves, the fully dispersive wavemaker theory is recovered. For shallow water waves, it is consistent with nonlinear long wave generation. Any model can be used to provide the numerical side in a combined model, but results have been presented here using Mike 21 BW model. A wave flume with a piston-type wavemaker and DHI AWACS control system were utilized for the physical model. The link between numerical and physical models was provided by the ad hoc unified wave generation theory. Based on the results of the tests carried out using the deterministic combination of numerical and physical model, the following conclusions can be drawn:


Cnoidal wavemaker theory is inadequate for highly nonlinear waves. But highly nonlinear waves in shallow water wave flumes are reproduced very well using the ad hoc unified wave generation and Stream Function theory. Thus the dominant error is due to the limitation of Cnoidal wave theory or low-order Boussinesq equations, rather than the ad hoc unified wave generation method. The measurements in physical flumes matched the numerical calculations well. As an extension of the numerical model, the physical model is insensitive to the location of the mean paddle position set in the numerical part for the data transfer between two models. Therefore, the deterministic combination of numerical and physical models in wave flumes is considered successful.

69

Chapter 6 A Deterministic Combined Model for 3D Wave Basins In this chapter, a deterministic combination of a numerical model and a physical model for 3D waves basins is developed which is based on the unified wavemaker theory for wave flumes developed in Chapter 5, the 3D linear wavemaker theory and Cnoidal wavemaker theory described in Chapters 2 and 3, respectively. In the combined model, Mike 21 BW is again chosen for the numerical model. The physical tests are made in a nearshore wave basin with a segmented 3D wavemaker at DHI. The link between numerical and physical models is provided by an ad hoc 3D unified wavemaker theory which is devised here. 6.1 Unified Wavemaker Theory for 3D Wave Basins Time series of surface elevation, depth-averaged horizontal particle velocity in the x direction, and paddle position at the wave paddle are denoted h(y,t),U(y,t), and X(y,t) respectively in physical space, while A(ky,w), B(ky,w) and Xa(ky,w) denote the equivalent complex Fourier amplitudes:

2D FourierTransform

( , ) ( , )yy t A kη ω⇔ (6.1)

2D FourierTransform

( , ) ( , )yU y t B k ω⇔ (6.2)

2D FourierTransform

( , ) ( , )a yX y t X k ω⇔ (6.3)

6.1.1 Linear wave generation According to linear fully dispersive wavemaker theory, the paddle position amplitude is related to the progressive wave amplitude by

0 a Iie X A= (6.4)

70 Chapter 6. A Deterministic Combined Model for 3D Wave Basins

where i is the imaginary unit showing a 90 degree phase shift. e0 is a transfer function defined as

0 0

1

cose c

α= (6.5)

where a is the wave propagation direction, c0 is the Biesel transfer function. With B denoting the x-component of the complex amplitude of the depth-averaged velocity of the progressive wave, we have

cosIB Akh

ω α= (6.6)

Eliminating AI yields ai X Bω = Λ (6.7) where

2

0

(2 sinh 2 )

4sinh

kh kh kh kh

c kh

+Λ = = (6.8)

This transfer function L is the same as the function (5.8) for wave flumes (see Figure 5.1). This indicates that L is not related to the wave propagation direction. The wave paddle position in physical space thus can be done as described in Chapter 5.1.1, and expressed as,

( , )

( , )swX y t

U y tt

∂ =∂

(6.9)

0

0

( , ) ( , ´) ( ) ´t

swm

t

X y t X y t t t dtλ−

= −∫ (6.10)

where lm(t) is the impulse response function corresponding to the modified Lm(w) (see Figures 5.2 and 5.3). In practice, the dispersion correction is again made in a discrete convolution by the Discrete Inverse Fourier transform using discrete

a ( , )swyX k ω and Lm(w) in Fourier space.

For active absorption with dual control the surface elevation at the moving paddle is furthermore required. An evanescent-mode correction to the progressive wave field is needed due to the existence of evanescent wave field at the paddle front. From linear wavemaker theory, we have, ,0I I aA A X= + Γ (6.11)

where

1

jj

i e∞

=Γ ≡ ∑ (6.12)

The transfer function ej has been defined in (2.40), see Chapter 2.2.4.

6.1 Unified Wavemaker Theory for 3D Wave Basins 71

Figure 6.1 shows the 2D transfer function G with respect to nondimensional wave

number and frequency (kyh, /h gω ). Figure 6.1 includes case of yk k≤ (serpent

wave length longer than or equal to length of the progressive wave), and also

yk k> (serpent wave length shorter than progressive wave). The curve on the surface

in the graph shows the critical value yk k= . For progressive waves yk k≤

corresponding to the lower right side of the critical curve.

0

2

4

6

wè!!!!!!!!!h êg

0

2

4

6

8

kyh

0

1

2G

0

2

4wè!!!!!!!!!h êg

Figure 6.1: The transfer function for the evanescent modes G .

Because of the frequency limititation of the facility and the deep water limit (kh<3) of Mike 21 BW, we modify G(ky,w) using decaying functions ( )df ω of (5.11) and

( )d yf k in two dimensions (ky, w), respectively.

( , ) ( , ) ( ) ( )m y y d d yk k f f kω ω ωΓ = Γ (6.13)

Here we choose the same expression of ( )df ω for ( )d yf k ,

21

1

21 tanh[2 ]

( )2

y yy

yd y

k kk

kf k

π

+−

−= (6.14)

where, ky1 and ky2 are parameters relevant to the maximum value of kyh (=3) due to the deep water limit of Mike 21 BW. 1 4yk h = and 2 5yk h = are chosen here.

Figure 6.2 shows the modified 2D transfer function for the evanescent modes

Gm(kyh, /h gω ). For normally emitted waves, Gm is exactly the same as the 1D modified transfer function Gm shown in Figure 5.4.


The surface elevation at the moving paddle in physical space is expressed as

0 0

0 0

,0 ( , ) ( , ) ( ', ) ( ', ) ' ´t y

I I m

t y

y t y t X y y t t y t dy dtη η γ− −

= + − −∫ ∫ (6.15)

where gm(y,t) is the 2D impulse response function corresponding to Gm(ky,w), it is shown in Figure 6.3. In practice, we also make the evanescent-mode correction in a discrete convolution by means of the 2D Discrete Inverse Fourier transform using discrete ( , )yX k ω and Gm(ky,w) in Fourier space.

0

2

4

6wè!!!!!!!!!h ê g

0

2

4

6

8

kyh0

0.51

1.5

2

Gm

0

2

4wè!!!!!!!!!h ê g

Figure 6.2: The modified transfer function for the evanescent modes Gm(kyh, /h gω ).

Figure 6.3: 2D impulse response function gm with respect to nondimensional t and y.

6.1 Unified Wavemaker Theory for 3D Wave Basins 73

6.1.2 Nonlinear shallow water wave generation For nonlinear shallow water waves with small dispersion such as Cnoidal waves, the horizontal particle velocity is almost uniform over the depth. The time-domain relation between the depth-averaged velocity from the numerical model and the paddle position is given directly by:

( , ) ( , )

( ( , ), , ) ( ( , ), , )sw sw

sw swX y t X y tV X y t y t U X y t y t

t y

∂ ∂+ =∂ ∂

(6.16)

with known initial and boundary conditions. The approach has been described in Chapter 3 for Cnoidal waves. This step captures the nonlinearity of the numerical model, but corresponds to the shallow water limit for the wave generation. We see that (6.16) may be simplified to linear shallow water wave generation (6.9) by taking U at mean paddle position instead of the actual moving paddle position, and omitting the nonlinear second term on the left-hand side. For active absorption with dual control, the associated surface elevation at the moving paddle is also needed: ,0 ( , ) ( ( , ), , )sw

I y t X y t y tη η= (6.17)

6.1.3 Ad hoc unified wave generation The ad hoc unified 2D wave generation method of Chapter 5.1.3 can be readily generalized to 3D by combining linear fully dispersive wavemaker theory and the method of nonlinear long wave generation for 3D wave basins described above.

In order to avoid a slow drift of the paddle, we also add a small term proportional to the paddle signal to the differential equation to simulate the effect of a first order high-pass filter, as was done for wave flumes in (5.24). Let wc denote the characteristic angular frequency of this filter, then the unified wave generation is altogether governed by

( , ) ( , )

( ( , ), , ) ( , ) ( ( , ), , )sw sw

sw sw swc

X y t X y tV X y t y t X y t U X y t y t

t yω∂ ∂+ + =

∂ ∂ (6.18)

followed by the dispersion correction

0

0

( , ) ( , ´) ( ) ´t

swm

t

X y t X y t t t dtλ−

= −∫ (6.19)

For dual control active absorption, the expected surface elevation at the moving paddle is also needed. We have

,0 ( , ) ( ( , ), , )pI y t X y t y tη η= (6.20)


here on the right side, h(x,y,t) is the field from the numerical wave propagation model or wave theory. With the evanescent-mode correction, the elevation at the moving paddle is

0 0

0 0

,0 ,0 0

0

( , ) ( , ) ( , ) (a)

( , ) ( ', ´) ( ', ) ' ´ (b)

p evaI I

t yeva

m

t y

y t y t y t

y t X y y t t y t dy dt

η η η

η γ− −

= +

= − −∫ ∫ (6.21)

where the superscript ‘p’ on the quantity ,0 ( , )pI y tη indicates that only progressive

waves are accounted for in (6.20). 0 ( , )eva y tη is the evanescent modes corrected.

6.2 Tests of the Unified Wave Generation Before testing the deterministic combined model, we eliminate the numerical model to test the unified wave generation using linear wave theory, Cnoidal wave theory and Stream Function theory, respectively. 6.2.1 Oblique linear wave case In order to verify the unified wave generation theory for 3D waves, we test with an oblique linear fully dispersive wave first. The water depth is h=0.7m, the wave period is T=1s, the wave height is H=0.05m, and the wave direction is a=p/4. The relative water depth is given as kh=2.83, k is the wave number in the propagation direction. The wave length in y direction is given Ly=2.195m. Based on the linear wave theory, we have

( , , ) cos( cos sin ) (a)

2( , , ) ( , , )

( , , ) cos cos (b)

Hx y t t kx ky

c x y t x y tU x y t

h kh

η ω α α

η ωηα α

= − − = =

(6.22)

For calculating the wave paddle position using shallow water limit, (6.18) is linearized first by taking U at mean paddle position instead of the actual moving paddle position, and omitting the second term on the left-hand side. Furthermore, in order to avoid the offset problem discussed in Chapter 3.3.2, we modify U(x,y,t) with a ramp function fr(t) here.

sin( )

2( )

1

rrr

r

t t TTf t

t T

π <= ≥

(6.23)

Thus, (6.18) is modified to

6.2 Tests of the Unified Wave Generation 75

0

( , )( , ) ( , , ) ( )

swsw

c r

X y tX y t U x y t f t

tω∂ + =

∂ (6.24)

where x0 is the mean paddle position, 2 / 30cω π= Hz and Tr=15s are chosen. This is followed by the dispersion correction (6.19) to extend from the shallow water limit. We use explicit central scheme to discretize (6.24) for the numerical simulation. The grid spacing is taken as dy=0.2m and the time step is dt=0.01s. Figure 6.4 shows the expected paddle position X(y,t) after the dispersion correction. Figure 6.5 shows comparison of time series of Xsw(Ly,t) with X(Ly,t), and profiles of Xsw(y,T) with X(y,T). The dispersion correction is substantial.

Figure 6.4: The expected paddle position, dy=0.2m, dt=0.01s.

Figure 6.5: The comparison between Xsw and X.

For dual mode active absorption, the surface elevation at the moving paddle is required. The progressive part is given by (6.20) for X=0 due to the linearity. The correction part for the evanescent modes is calculated by (6.21b), as shown in Figure 6.6.


Figure 6.6: The surface elevation corrected for evanescent modes, dy=0.2m, dt=0.01s.

Based on the 3D linear wave (wavemaker) theory, we have the theoretical solutions of wave paddle position and the surface elevation of evanescent modes as,

0

1

0

( , ) sin( sin ) (a)2

( , , ) sin( sin ) (b)2

li

jjeva

li

HX y t t ky

e

i e H

x y t t kye

ω α

η ω α

∞

=

= − = −

∑ (6.25)

For this given wave, 1

0.62jj

i e∞

=≈∑ . Comparing the computational X(y,t) with the

theoretical Xli(y,t), the maximum ErrRMS(y) and ErrH(y) are 0.10% and 0.19%, respectively. Here, ErrRMS(y) and ErrH(y) are defined as (5.29). Comparing the correction of evanescent modes with the theoretical ( , , )eva

li x y tη , the maximum ErrRMS(y) and ErrH(y) are 0.02% and 0.07%. These verify the procedure for oblique linear waves. 6.2.2 Oblique Cnoidal waves For the oblique Cnoidal wave cases, the same wave conditions are chosen as the tests in flumes (see Chapter 5.2.2) except for the oblique propagation. The water depth is h=0.4m, the wave period is T=2.3s, the wave heights are chosen as H=0.12m, 0.16m and 0.20m with different nonlinearity. The wave direction is a=45±. As done for (6.24), (6.18) is modified with the ramp function fr(t) in (6.23) for U(x,y,t) and V(x,y,t) for the calculation of Xsw(y,t),

( , ) ( , )

( ( , ), , ) ( ) ( , ) ( ( , ), , ) ( ) sw sw

sw sw swr c r

X y t X y tV X y t y t f t X y t U X y t y t f t

t yω∂ ∂+ + =

∂ ∂(6.26)

in which U(x,y,t) and V(x,y,t) are expressed as,


( , , )( , , ) cos (a)

( , , )

( , , )( , , ) sin (b)

( , , )

c x y tU x y t

h x y t

c x y tV x y t

h x y t

η αη

η αη

= + = +

(6.27)

where h(x,y,t) is expressed as the following based on Cnoidal wave theory described in Chapter 3,

2min( , , ) 2 ( )( ), (a)

2 2; (b)

cos sin

x y

x yx y

t x yx y t Hcn K m m

T L L

L LL L

k k

η η

π πα α

= + − −

= = = =

(6.28)

Eq. (6.26) is followed by the dispersion correction (6.19) to leave the shallow water limit. For the numerical integration, we use explicit central schemes for both

/ and /x y∂ ∂ ∂ ∂ to discretize (6.26). The grid spacing is taken as dy=Ly/40 (see Table

5.1 for L), the time step is dt=T/100=0.023s, 2 / 30cω π= Hz and Tr=5T are chosen. The 3D wavemaker utilized at DHI includes 36 segments, and the width of each is 0.5m. In order to meet this distance, dy is changed to 0.5m using ‘Spline’ interpolation method in the output of the control signals. Figure 6.7 shows the expected wave paddle positions X(y,t) with dispersion correction for H=0.12m. The comparisons of time series of Xsw(Ly,t) with X(Ly,t), and profiles of Xsw(y,T) with X(y,T) are shown in Figure 6.8 for H=0.12m. The deviation is small enough to be omitted. This is in accordance with Cnoidal wavemaker theory.

Figure 6.7: X(y,t) with dispersion correction for H=0.12m, dy=0.5m, dt=0.023s.


(a) time series at y=Ly.

(b) profiles at t=T, dy=0.5m.

Figure 6.8: Comparison between Xsw and X for H=0.12m. For dual mode active absorption, ,0 ( , )I y tη is obtained using evanescent-mode

correction. Figure 6.9 shows ,0 ( , )I y tη for H=0.12m. The comparison of ,0 ( , )pI y tη ,

0( , , )x y tη , and the evanescent modes corrected ,0 ( , )evaI y tη is shown in Figure 6.10

through time series of ,0 ( , )pI yL tη , 0( , , )yx L tη , and 0 ( , )eva

yL tη as well as profiles of

,0 ( , )pI y Tη , 0( , , )x y Tη , and 0 ( , )eva y Tη for H=0.12m. The evanescent-mode part is

much smaller than the progressive part. These two parts have 90± phase shift.

Figure 6.9. ,0 ( , )I y tη for H=0.12m, dy=0.5m, dt=0.023s.


(a) time series at y=Ly.

(b) profiles at t=T, dy=0.5m.

Figure 6.10: Comparison of ( ( , ), , )X y t y tη , 0( , , )x y tη , and ,0 ( , )evaI y tη for H=0.12m.

We make all the tests in a wave basin with a segmented 3D wavemaker at DHI. The 36-segment 3D wavemaker is of piston-type with vertical hinges between the segments providing a linear segmentation of the paddle front. The paddle width is dy=0.5m with 1.2m height and the maximum stroke is 0.6m. The maximum difference of paddle position dX between two neighbouring paddles is 0.1m. Precision control of each actuator is achieved using a brushless AC servomotor with a ball screw transmission and encoder feedback. The 3D wavemaker is controlled by the DHI 3D AWACS with three different control modes, single mode, dual mode and position mode. The section available for the present tests is about 8.5m long in the x direction, and 19.5m wide in the y direction. At the down-wave boundaries of the basin, some passive wave absorbers are installed. The up-wave boundary along the x direction is made up by a guide-wall. The experimental set-up is sketched in Figure 6.11, which also shows the location of the wave gauges for the first set of tests. First, we make some tests using 3D linear wavemaker theory in position mode for two cases (H=0.12m and 0.20m) as references. Figure 6.12 shows time series of surface elevation measured at four different gauges in the basins compared with the theoretical surface elevation by Stream Function theory. For these oblique waves with wave direction a=45±, the measurements at gauge 1 and 3 are in phase, so are measurements at gauge 2 and 4. Note that in these and following figures for oblique regular waves, the mean value of surface elevation measured are removed in order to


leave some unexpected effects of physical condition. The results are poor as expected. Even wave breaking occurs in the basin for H=0.20m.

Figure 6.11 Sketch of the test set-up.

Note: the basin coordinate here is specially for the facility and physical tests.

(a) H=0.12m

(b) H=0.20m

Figure 6.12: Measured surface elevation of waves generated by 3D linear wavemaker theory.

Next, we make tests for generating oblique waves using control signals of 3D Cnoidal wavemaker theory in three different modes, respectively. For H=0.20m, only test in position mode is made. When active absorption is involved for this case, either in

Y1Y36 3D segmented wavemaker

20 15 10 5

5

0.6

0

Passive wave absorber


Gauge 2 Gauge 1

Gauge 3 Gauge 4


single or dual mode, the maximum dX between two neighbouring paddles is out of the limitation of the facility. The measurements of surface elevation for H=0.12m in three modes, and H=0.20m in position mode, are compared with the theoretical Stream Function waves, respectively, in Figure 6.13. The results using Cnoidal wavemaker theory are much better than using linear wavemaker theory. Comparing the measurements in different modes for H=0.12m, dual mode and position mode reproduce oblique waves better than single mode. The measurements match the theoretical solutions fairly well in these two modes in general for this nonlinear wave case. For the case H=0.20m, the wave shape is unstable at different gauges. The crest part of waves reproduced match the theory better than the trough part.

(a) H=0.12m


(b) H=0.20m

Figure 6.13: Measured surface elevation of waves generated by Cnoidal wavemaker theory.

Figure 6.14 shows the relative errors ErrRMS and ErrH of the measurements (involving 3 periods) at four gauges for each wave height quantitatively, using Cnoidal wavemaker theory, compared with the theoretical solution of Stream Function wave theory. It is clear to show the advantage of the dual and position modes over single mode. However, the results of oblique waves generated are worse than the results of wave flumes in Chapter 5.2.2 using Cnoidal wavemaker theory.

0%

5%

10%

15%

20%

0.12m 0.16m 0.20m

H

Err

RM

S single mode

dual mode

position mode

-5%

0%

5%

10%

15%

20%

25%

30%

35%

0.12m 0.16m 0.20mH

Err

H

single mode

dual mode

position mode

Figure 6.14: The relative errors of measurements using Cnoidal wavemaker theory.


6.2.3 Oblique Stream Function waves

Now we turn to oblique waves with Stream Function theory doing the same procedure as Chapter 6.2.2. To calculate wave paddle position, in (6.26), h(x,y,t) can be expressed as the following based on Stream Function theory,

( )1

( , , ) cos ( ) (a)

2 2cos ; sin (b)

N

j x yj

x yx y

x y t A j t k x k y

k k k kL L

η ω

π πα α

=

= − − = = = =

∑ (6.29)

where 1...NA are constants for a given wave. For the numerical simulation of control signals, the same conditions are chosen as before. Figure 6.15 shows the comparison of X(y,t) using Cnoidal and Stream Function wavemaker theories through the time series of X(Ly,t) and the profiles of X(y,T) for H=0.12m. The amplitude of X(y,t) using Stream Function theory is smaller than using Cnoidal theory, which is the same as in Figure 5.13 for wave flumes. Also the wave length along the y direction is a little smaller given by Cnoidal theory than by Stream Function theory. This shows the different phase velocities. Phase velocity c=1.948m/s given by Cnoidal theory, and c=1.937m/s by Stream Function theory.

(a) time series of X(Ly, t)

(b) profiles of X(y, T), dy=0.5m

Figure 6.15: Comparison of X(y,t) between Cnoidal and Stream Function wavemaker theories for H=0.12m.


For dual mode control, ,0 ( , )I y tη is obtained with evanescent-mode correction as was

done for oblique Cnoidal waves. The comparison of ,0 ( , )I y tη between Cnoidal and

Stream Function wavemaker theories is shown in Figure 6.16 through the time series of ,0 ( , )I yL tη and the profiles of ,0 ( , )I y Tη for H=0.12m. The values at crest and

trough given by Stream Function theory are a little larger than those given by Cnoidal theory. Also the wave length along the y direction is little smaller using Stream Function theory than using Cnoidal theory due to the different phase velocities.

(a) time series of ,0 ( , )I yL tη

(b) profiles of ,0 ( , )I y Tη , dy=0.5m.

Figure 6.16: Comparison of ,0Iη between Cnodial and Stream Function theories for H=0.12m.

We repeat the tests of nonlinear wave generation as done for oblique Cnoidal waves, but using control signals of 3D Stream Function wavemaker theory. The measurements of surface elevation for H=0.12m in three modes, and H=0.20m in position mode, are compared with the theoretical Stream Function waves in Figure 6.17. They are much better than the measurements using linear wavemaker theory shown in Figure 6.12, but similar to the results using Cnoidal wavemaker theory in Figure 6.13. The general appearance of nonlinear waves is captured by each mode. Dual and position modes give better results than single mode. However the wave shapes are still unstable in the wave basins. The relative errors of all the measurements are shown in Figure 6.18.


We have seen that, in Chapter 5, the advantage of Stream Function wavemaker theory is distinguished for wave flumes, compared with Cnoidal wavemaker theory. However, here for wave basins, there is no advantage using 3D Stream Function wavemaker theory. There are many possible reasons for that. For example in the control system, the effect of tiny time-lag between updated control signals of neighbouring paddles. In measurement system, the accuracy of calibration factors for wave gauges. In the physical condition, the disturbance of still wave level, the tiny change of water depth and so on. In the wave generation theory, the dispersion correction and the evanescent-mode correction are based on linear theory. All these may create some errors. We think the main reason is due to the width of 3D segmented wavemaker which is dy=0.5m. This grid spacing is too coarse for the generation of highly nonlinear waves. For instance, the wave length in the y direction is Ly=6.29m for H=0.12m, h=0.4m, T=2.3s, a=45± given by Stream Function theory. The fixed grid spacing gives only 13 grid points per wave length for the primary harmonics. For the higher order harmonics, grid points per wave length are fewer. The linear paddle front segmentation makes the nonlinearity incoherent. Thus in general, both Cnoidal and Stream Function wavemaker theories are considered to reproduce oblique nonlinear waves successfully. The difficulty in obtaining these nonlinear waves of constant form in wave basins is mainly due to the limitation of the facility on the width of paddles.


(a) H=0.12m

(b) H=0.20m

Figure 6.17: Measured surface elevation of waves generated using Stream Function wavemaker theory.

Figure 6.18: The relative errors of measurements using Stream Function wavemaker theory.

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

35%

0.12m 0.16m 0.20m

H

Err

H

single mode

dual mode

position mode

0%

5%

10%

15%

0.12m 0.16m 0.20m

H

Err

RM

S single mode

dual mode

position mode


6.3 Tests of the Deterministic Combined Model The unified 3D wave generation is utilized as a deterministic link between the numerical and physical models. Mike 21 BW is again chosen for numerical calculations. The physical models are made with the same facilities and set-up as done for oblique regular waves. See Figure 6.11 for the sketch. Three tests on oblique and multidirectional irregular waves in wave basins are presented. 6.3.1 Oblique irregular waves propagating on constant water depth Based on the rather deep water case in Chapter 5.2.3, we use the same wave conditions but for oblique waves with 45± wave propagating direction in order to test the dispersion correction and the evanescent-mode correction. The simulated wave basin is 100m long (x direction) and 30m wide (y direction) with a constant water depth of h=0.7m. The irregular incoming wave conditions are synthesized from a standard JONSWAP frequency spectrum, with a significant wave height of Hm0=0.05m, a peak period of Tp=1.2s, and the relevant shape parameter, g=3.3, sa=0.07, sb=0.09. The spectrum is truncated omitting periods smaller than 0.95s. The truncated spectrum is rescaled in the numerical model to retain the specified Hm0=0.05m.. As we know that the combined model is not very sensitive to where the physical model takes over from the numerical model in Chapter 5, here the 18m-long wavemaker in the physical wave basin is arbitrarily placed from y=6m to 24m at x=65.6m, where x=0 coincides with the up-wave boundary of the numerical model there. Thus, x0=65.6m is the mean paddle position for the data transfer from numerical to physical model, see Figure 6.19 for a sketch of the combined model.

Figure 6.19: Sketch of plan view and vertical cross-section of the combined model.

In the numerical model, the time step is taken as dt=0.02s, and the grid spacing is dx=dy=0.2m. Figure 6.20 shows contour plots of surface elevation in respect to x and y at t=110s, and in respect to t and y at the mean paddle x0=65.6m, extracted from the output of numerical calculations.

100m X wave gauge

h=0.7m

numerical

wave

maker

30m 65.6m

18m

6m

•

Physical model

x

y


(a) Contour plot of h(x,y) at t=110s

(b) Contour plot of h(y,t) at x0=65.6m

Figure 6.20: Contour plots of surface elevation extracted from output of the numerical model.

To calculate the wave paddle position in shallow water Xsw(y,t), (6.26) is solved with the calculations of Mike 21 BW, U(x,y,t) and V(x,y,t). We make t=40s in the numerical model to be the initial to transfer data between the numerical and physical models. For this case, we choose 2 /10 c Hzω π= for the high-pass filter effect, and Tr=3s for the ramp function in (6.23). The time step and the grid spacing are taken the same as in the numerical model, i.e. dt=0.02s, and dy=0.2m. The total simulation period for the paddle position is 5min. Then dy is changed to 0.5m using ‘spline’ for the control signal as before to fit the width of paddles. The expected paddle position X(y,t) is obtained by means of dispersion correction and shown in Figure 6.21 through a contour plot extracted from the calculation. The comparison between Xsw(y,t) and X(y,t) is shown in Figure 6.22 through time series at y=10m and profiles at t=110s, which indicates that the dispersion correction is substantial for this case.


Figure 6.21: Contour plot of X(y,t).

(a) time series of Xsw(t) and X(t) at y=10m

(b) profile of Xsw(y) and X(y) at t=110s

Figure 6.22 Comparison of Xsw(y,t) and X(y,t). For dual mode active absorption, ,0 ( , )I y tη is required. The comparison between

,0 ( , )pI y tη , h(x0,y,t), and 0 ( , )eva y tη is shown in Figure 6.23 through time series at y=10m

and profiles at t=110s. ,0 ( , )pI y tη and h(x0,y,t) match well which shows the linearity of

waves in this case. Compared with ,0 ( , )pI y tη , the evanescent modes corrected

0 ( , )eva y tη is substantial.


(a) time series of of h(X(t),t), h(x0,t) and 0 ( )eva tη at y=10m

(b) profile of of h(X(y),y), h(x0,y) and 0 ( )eva yη at t=110s

Figure 6.23: Comparison of ,0 ( , )pI y tη , h(x0,y,t), and 0 ( , )eva y tη .

We make physical tests in dual mode and position mode, respectively. The test set-up including the locations of four wave gauges is the same as for regular oblique waves shown in Figure 6.11. The paddle ‘Y36’ near the guide wall corresponds to y=6m, as well as ‘Y1’ corresponds to y=24m in this case. The time series of surface elevation measured at gauge 1 in two modes are compared with the numerical calculations in Figure 6.24. The match is quite good although deviations occur for high waves. Dual mode gives higher waves than position mode for this case. The measurements at other three gauges in dual mode are compared with numerical calculations in Figure 6.25. In dual mode, the correlation coefficients are 0.982, 0.975 , 0.956 and 0.965 for gauge 1 to 4, respectively, in the whole 5-min period of the physical test. In position mode, the respective coefficients are 0.964, 0.950, 0.952 and 0.957. All the measurements in physical basin match the numerical calculation quite well. Dual mode gives a little larger correlation coefficients at each gauge. The measurements are closer to the numerical results when the wave gauges are closer to the mean paddle position which is the same as in the flume.


Figure 6.24: Surface elevation measured at gauge 1 compared with

the numerical calculation (BW).


Figure 6.25: Surface elevation measured at gauge 2~4 compared with the numerical

calculation (BW). 6.3.2 Directional irregular waves propagating on constant water depth This example considers a rather deep water case for directional irregular waves based on the last oblique irregular waves case in Chapter 6.3.1. The simulated wave basin is identical to the one used before (see Figure 6.19 for the sketch) except for the wave direction. In this case, the main wave direction is 30±. The maximum deviation from the main wave direction is chosen as 30±. The directional distribution is expressed as

4cos ( 30 )α − ° . The same conditions as in Chapter 6.3.1 to be chosen for the numerical calculations and the computation of control signals. Figure 6.26 shows contour plots of surface elevation in respect to x and y at t=190s, and in respect to t and y at x0=65.6m, extracted from the output of numerical calculation. The total simulation period for the wave paddle position is 10min. Figure 6.27 shows a contour plot of paddle position with dispersion correction. The comparison between Xsw(y,t) and X(y,t) is shown in Figure 6.28 through time series at y=16m and profiles at t=190s, which indicates that the dispersion correction is again substantial in this case.


(a) contour plot of h(x,y) at t=190s

(b) contour plot of h(y,t) at x=65.6m

Figure 6.26: Contour plots of surface elevation extracted from output of the numerical model.





Figure 6.28: Comparison of Xsw(y,t) and X(y,t).

,0 ( , )I y tη is calculated for dual mode active absorption. Figure 6.29 shows the

comparison of ,0 ( , )pI y tη , h(x0,y,t), and the evanescent modes corrected 0 ( , )eva y tη ,

through time series at y=16m and profiles at t=190s. ,0 ( , )pI y tη and h(x0,y,t) match

well, which shows waves are linear as expected. Also the evanescent modes corrected can not be neglected in this case.

(a) time series of of h(X(t),t), h(x0,t), and evanescent modes 0 ( )eva tη at y=16m


(b) profile of of h(X(y),y), h(x0,y), and evanescent modes 0 ( )eva yη at t=190s

Figure 6.29: Comparison of ,0 ( , )pI y tη , h(x0,y,t) , and 0 ( , )eva y tη .

Physical tests are made in dual mode and position mode, respectively. The test set-up is identical to the one in Figure 6.11. Figure 6.30 shows a snapshot of the wave field in the physical basin compared with a snapshot of the surface elevation extracted from numerical calculation in the physical model’s area at t=44s. In the picture of numerical model, the left boundary coincides with the physical wavemaker. The calm area in the upper right of the lab picture reduces the equivalence with the numerical model a bit. This is due to the guide wall and the associated lee zone and diffraction.

(a) snapshot of physical wave filed

(b) snapshot of numerical surface elevation in physical model’s area at t=44s

Figure 6.30: Comparison of physical and numerical models’ snapshots.


The time series of surface elevation measured at gauge 1 are compared with the numerical calculations in Figure 6.31 using position mode and dual mode, respectively. The match is quite good although deviations occur, as the last oblique irregular case. Position mode gives smaller waves than dual mode. Using dual mode the measurements at other three gauges are compared with numerical calculations in Figure 6.32. In dual mode, the correlation coefficients are 0.887, 0.887, 0.893 and 0.885 for gauges 1 to 4, respectively, in the whole 10-min period of the physical test. In position mode, the respective coefficients are 0.911, 0.900, 0.914 and 0.906.





calculation (BW). 6.3.3 Irregular waves behind a breakwater after propagating up a slope To test directional nonlinear shallow water waves, the last example considers irregular waves propagating up a slope from a deep plateau to a shallow plateau with a breakwater, see a sketch of the combined model in Figure 6.33. The simulated


numerical wave basin is 175m long (x direction), 50m wide (y direction). The internal wave generation of the numerical model is set at x=13m with the deep water depth 2.6m. The slope is 1/50 and the shallow water depth is 0.4m. The breakwater is set on the shallow plateau, from x=128.25m to x=128.75m, y=0 to y=25m. The irregular incoming wave conditions are synthesized from a standard JONSWAP frequency spectrum, with a significant wave height of Hm0=0.09m, a peak period of Tp=3s, and the relevant shape parameter, g=3.3, sa=0.07, sb=0.09. The incoming waves propagate along x direction. The spectrum is truncated omitting periods smaller than 2.6s. The truncated spectrum is rescaled in the numerical model to retain the specified significant wave height. Based on the flume case in Chapter 5.2.4, we know that at the end of the slope waves are irregular nonlinear long waves. Then waves turn to irregular nonlinear long waves with slight directional spreading behind the breakwater due to diffraction. The 18m-long wavemaker in the physical wave basin is set from y=15m to 33m and x=139.5m in the whole unified model. Thus x0=139.5m is the location of the mean paddle position for the data transfer from numerical to physical model. In the numerical model, the time step is taken as dt=0.02s, and the grid spacing is dx=dy=0.25m. Figure 6.34 shows numerical calculations through contour plots of surface elevation in respect to x and y at t=169s, and in respect to t and y at the mean paddle x0=139.5m.

Figure 6.33: Sketch of plan view and vertical cross-section of the combined model.

(a) contour plot of h(x,y) at t=169s

175m

X Wave gauge

numerical model

wave maker50m

128.25m

18m

15m

•

h1=2.6m h2=0.4

1:50

25m

139.5m

Physical model

x

y


(b) contour plot of h(y,t) at x=139.5s

Figure 6.34: Contour plots of h extracted from the output of numerical calculation.

To calculate wave paddle position, we choose 2 /10 c Hzω π= to simulate the high-pass filter effect, Tr=3s for the ramp function in (6.26), dt=0.02s, and dy=0.25m. The total simulation period is 4-min. Then, dy is changed to 0.5m easily for the control signal to meet requirement of the facility. A contour plot of X(y,t) with dispersion correction extracted from the calculation is shown in Figure 6.35. For y<22m, X(y,t) are very tiny because of wave diffraction. The comparison between Xsw(y,t) and X(y,t) is shown in Figure 6.36 through time series at y=30m and profiles at t=169s. Two time series match well at y=30m. Compared with Xsw(y) at t=169s, X(y) give a little deviation for y>30m. This is due to the quite high nonlinearity of waves. Thus the dispersion correction can be ignored for this shallow water wave case.





Figure 6.36: Comparison of Xsw(y,t) and X(y,t). For dual mode control, the comparison of ,0 ( , )p

I y tη , h(x0,y,t), and 0 ( , )eva y tη is shown

in Figure 6.37 by the time series at y=30m and profiles at t=169s. The deviation of

,0 ( , )pI y tη from h(x0,y,t) shows the nonlinearity of waves. It becomes much larger as y

increases for y>25m in Figure 6.37b. This indicates the wave nonlinearity is quite high in this area at that moment. As we know the breakwater is set from y=0 to 25m. Thus for y>25m, the effect of breakwater weakens gradually as y increases. Compared with ,0 ( , )p

I y tη , the evanescent modes are very tiny. The evanescent-mode correction

can be neglected for this case, which is in accordance with the reality of shallow water waves.


(a) time series of of h(X(t),t), h(x0,t), and 0 ( )eva tη at y=30m

(b) profile of of h(X(y),y), h(x0,y) and 0 ( )eva yη at t=169s

Figure 6.37: Comparison of ,0 ( , )pI y tη , h(x0,y,t) and 0 ( , )eva y tη .

In the physical model, the test set-up is identical to the one used before except for the location of wave gauges, see Figure 6.38. The paddle ‘Y36’ corresponds to y=33m in the combined model, and ‘Y21’ corresponding to y=25m which is parallel to the end of the breakwater. Figure 6.39 shows a snapshot of the wave field in the physical basin, compared with a snapshot of the numerical calculation in the physical model’s area at t=205s. Two pictures match well. The shape of nonlinear long wave is shown clearly on the upper side. On the lower side, the surface is almost still due to the breakwater.

Figure 6.38: The sketch of the test set-up.

Y21Y36

20 15 10 5

0.60

Passive

wave absorber


Gauge 2 Gauge 1

Gauge 4

Gauge 3

Gauge 5 Gauge 6

Y1

5

3D segmented wavemaker


(a) snapshot of wave field in physical basin

(b) snapshot of numerical surface elevation in the physical model’s area at t=205s Figure 6.39: The comparison of snapshots of numerical and physical models.

The time series of surface elevation measured at gauge 1 in position mode and dual mode are compared with the numerical calculation in Figure 6.40. The measurements match the numerical calculation well in both modes, but with slightly lower peaks in the physical model. Dual mode gives a little better results. The measurements at other gauges in dual mode are compared with numerical calculations in Figure 6.41. As expected, the wave amplitudes decrease gradually from gauge 1 to 3 and from gauge 4 to 6 as y decreases. This is due to the wave diffraction. Which also introduces small phase changes bwteen the signals at gauges 4 to 6 and 1 to 3, respectively. In dual mode, the correlation coefficients are 0.975, 0.965, 0.938, 0.979, 0.970 and 0.910 for gauges 1 to 6, respectively, in the whole 4-min period of the physical test. In position mode, the respective coefficients are 0.968, 0.940, 0.923, 0.968, 0.959 and 0.864. All the measurements in the physical basin match the numerical calculation quite well. Dual mode gives a little larger correlation coefficients at each gauge for this case compared with position mode. The measurements are closer to the numerical results when the wave gauges are closer to the mean paddle position.







calculation (BW) .

6.4 Summary and Conclusions In this chapter, an ad hoc unified wave generation theory for 3D wave basins has been generalized from the 2D method in Chapter 5. This 3D wave generation method again accounts for shallow water nonlinearity and compensates for local wave phenomena (evanescent modes) near the 3D wavemaker. A deterministic combination of numerical and physical models in 3D wave basins has been presented. In the combined model, Mike 21 BW was chosen for the numerical calculation. A wave basin with segmented 3D piston-type wavemaker and DHI 3D AWACS control system were utilized for the physical model. A deterministic link between the two type of models was provided by the ad hoc unified 3D wave generation theory. Based on the results of the tests carried out, the following conclusions can be drawn: Cnoidal and Stream Function wavemaker theories can reproduce oblique nonlinear waves in wave basins. The difficulty in obtaining oblique nonlinear waves of constant form is mainly due to the limitation of the facility on the width of paddles. For oblique and multidirectional irregular wave cases, all the measurements in physical basins matched the numerical calculations well. Therefore, the deterministic combination of numerical and physical models for wave basins is feasible either with or without active absorption.


107

Chapter 7 Stream Function Wavemaker Theory for Highly Nonlinear Waves in Wave Flumes In this chapter, an approximate Stream Function wavemaker theory for highly nonlinear regular waves in flumes is presented first. This theory is based on Stream Function wave theory and the ad hoc unified wave generation method presented in the Chapter 5. Physical wave flume experiments are made with various wave periods and wave heights for different relative water depth and different nonlinearity with and without active absorption, respectively. We consider the pure wave generation first. For the shallow water case T=2.3s, H=0.20m, h=0.4m, waves generated by linear wavemaker theory, second-order Stokes wavemaker theory, Cnoidal wavemaker theory, and approximate Stream Function wavemaker theory, respectively, are compared. For the other rather shallow water cases, tests are made by second-order wavemaker theory, modified Cnoidal wavemaker theory with dispersion correction, and approximate Stream Function wavemaker theory. For non-shallow cases, this new wave generation approach is compared with second-order wavemaker theory. The advantage of the new wavemaker theory is supported by the results of the experiments. Considering active absorption, single mode and dual mode using Cnoidal and Stream Function wavemaker theories, respectively, are compared. Furthermore, an improved Stream Function wavemaker theory is devised for fully nonlinear regular waves in flumes. This one-step approach is based on the distribution of horizontal velocity given by Stream Function theory for progressive waves. The evanescent modes are evaluated by linear theory. 7.1 Approximate Stream Function wavemaker theory In Chapter 5.1, we have made an ad hoc unified wave generation method given by,

( )

( ) ( ( ), )sw

sw swc

dX tX t U X t t

dtω+ = (7.1)

followed by the dispersion correction:

0

0

( ) ( ) ( ) ´t

sw

t

X t X t t t dtλ−

= −∫ (7.2)

The depth-averaged horizontal particle velocity U(x,t) can be expressed by means of continuity,

108 Chapter 7. Stream Function Wavemaker Theory in Wave Flumes

( , )

( , )( , )

c x tU x t

h x t

ηη

=+

(7.3)

Here, h(x,t) may be the solution of Stream Function theory. For dual mode active absorption, the expected surface elevation at the moving paddle is obtained in (5.26)~(5.28). In Chapter 5, the transfer function L for the dispersion correction and G for evanescent-mode correction are modified to damp the high frequencies above

/h gω º 3 in order to match the deep-water limit of the Boussinesq model. Here, we modify it with the following in order to make higher frequency involved in the reproduction.

/ 7 / 7

; = for / 7

; = for / 7

m m

m mh g h g

h g

h gω ω

ω

ω= =

Λ = Λ Γ Γ ≤

Λ = Λ Γ Γ >

(7.4)

The paddle position X(t) by Stream Function wavemaker theory is compared with Cnoidal, second-order Stokes and linear wavemaker theories for T=2.3s, H=0.20m, h=0.4m, see Figure 7.1. In this case, kh=0.54 and Ur( 2 3/HL h≡ )=66.9 using Stream Function theory. Comparing with linear wavemaker theory, the nonlinearity of three other theories is clear due to the asymmetry. Stream Function wavemaker theory has the highest order of nonlinearity but the smallest range of the paddle. The second-order Stokes wavemaker theory has the largest range. The deviation of Cnodial wavemaker theory from the Stream Function wavemaker theory is substantial. Note that here Cnoidal wavemaker theory is the traditional theory without dispersion correction, but the dispersion correction was made in Stream Function wavemaker theory. For this rather shallow water case, the effect of dispersion correction is inconsiderable. The maximum relative error of Xsw(t) compared with X(t) by Stream Function wavemaker theory is only about 2.3%.

Figure 7.1: The comparison of wave paddle position by Stream Function wavemaker theory (SF), traditional Cnoidal wavemaker theory (CN), second-order Stokes wavemaker theory

(2nd), and linear wavemaker thoery for T=2.3s, H=0.20m, h=0.4m.

7.2 Experimental Validations 109

7.2 Experimental Validation All the cases (see Table 7.1) are tested with h=0.4m. Various wave periods and wave heights are chosen for different relative water depth h/L0 and different nonlinearity of waves. Here L0=(g/2p)T2 is the wave length in deep water according to linear theory. The beat length between free and bound second harmonics according to second-order Stokes theory is also shown in the Table. Figure 7.2 shows all the cases in terms of the ratio of wave height to water depth versus the ratio of wavelength to water depth. Numerical investigation of the limiting wave height for non-breaking regular waves were made by Williams (1981), see also Fenton (1990) where a convenient rational fit to the data is given. Figure 7.2 also includes the Williams/Fenton results, the practical maximum wave height in physical tests using linear wavemaker theory synthesized by Massel (1996), and the proposed demarcation line between Stokes and Cnoidal theories of Fenton (1990). Since we are not looking for the limit of the wave height for the present study, the highest wave cases for different T in Table 7.1 do not necessarily show the obtainable limit. In the flume, the set-up is identical to the one used in Chapter 5.2.2. In the analyses here, minor offsets from zero mean elevation were removed from the signals.

Table 7.1 Cases tested in the flume with h=0.4m T(s) /h gω h/L0 Beat length(m) H(m)

2.3 1.8 1.6 1.4 1.1 0.9

0.75 0.6

0.55 0.70 0.79 0.91 1.15 1.41 1.69 2.11

0.048 0.079 0.10 0.13 0.21 0.31 0.45 0.71

11.51 4.88 3.25 2.11 1.06 0.65 0.44 0.28

0.08, 0.12, 0.16, 0.20, 0.22, 0.24 0.08, 0.12, 0.16, 0.20, 0.22 0.08, 0.12, 0.16, 0.20, 0.22 0.08, 0.12, 0.16, 0.20 0.08, 0.12, 0.16 0.04, 0.08, 0.10, 0.12 0.04, 0.06, 0.08 0.04, 0.06

Figure 7.2: The region of cases in Table 7.1, compared with Williams line which is the limit of the highest steady waves in practice, the proposed Demarcation line between Stokes and

Cnoidal wave theory, and Hmax/h by Massel.


7.2.1 Pure wave generation In this section, we make all the tests in position mode for the pure wave generation in order to testify the approximate Stream Function wavemaker theory. 7.2.1.1 Rather shallow water waves First of all, for the shallowest water wave case (T=2.3s, H=0.20m, h=0.4m) discussed above, we compare test results for four different methods: Stream Function wavemaker theory, traditional Cnoidal wavemaker theory, second-order Stokes wavemaker theory, and linear wavemaker theory. The paddle positions of these four methods have been shown in Figure 7.1. In the case of linear wavemaker theory, Figure 7.3 shows phase aligned time series of surface elevation measured at the three locations in the flume along with the theoretical surface elevation by Stream Function theory. The attempt to reproduce a regular wave in the flume by linear wavemaker theory is very poor as expected for this quite highly nonlinear wave case. Using a 2nd order control signal, the nonlinear wave shape is captured to some extent as shown in Figure 7.4, but the deviation from the theoretical solution is still large. The traditional Cnoidal wavemaker theory provides a further improvement of the wave shape, see Figure 7.5, but especially gauge 2 (4.4m from the mean paddle), it gives higher peaks than the theoretical Stream Function solution. Finally, Figure 7.6 shows how the approximate Stream Function wavemaker theory provides a regular wave of almost constant form that matches the theoretical Stream Function solution very well.

Figure 7.3: Time series of surface elevations using linear wavemaker theory for T=2.3s,

H=0.20m, h=0.4m, compared with the theoretical solution (SF theory).


Figure 7.4: As for Figure 7.3, but using second-order Stokes wavemaker theory.

Figure 7.5: As for Figure 7.3, but using traditional Cnoidal wavemaker theory.

Figure 7.6: As for Figure 7.3, but using Stream Function wavemaker theory.

Keeping the period and water depth at T=2.3s and h=0.4m, we now turn up the wave height to H=0.24m, the highest wave in Table 1. In this case, H/h=0.6 and Ur=83.8 based on the wave length from Stream Function theory. For this case we concentrate on Stream Function wavemaker theory and traditional Cnoidal wavemaker theory,


respectively, as linear and second order wave generation result in wave breaking already at H=0.22m. Figure 7.7 shows the measurements for Cnoidal wavemaker theory. Some wave breaking occurs as indicated by the reduced peak at the last wave gauge. The results of Stream Function wavemaker theory are shown in Figure 7.8. The wave shape is steady, no breaking occurs and the measurements match the theoretical solution very well at each gauge in the flume. The average wave height of the measurements at three gauges is 0.233m using Stream Function wavemaker theory. This corresponds to H/h=0.58, which is larger than the limit of Hmax/h=0.53 given by Massel (1996) for T=2.3s and h=0.4m. However, since Massel’s analysis was based on a synthesis of experimental results using linear wavemaker theory, there is no contradiction here.

Figure 7.7: As for Figure 7.5, using traditional Cnoidal wavemaker theory,

but for T=2.3s, H=0.24m, h=0.4m.

Figure 7.8: As for Figure 7.6, using Stream Function wavemaker theory but for T=2.3s,

H=0.24m, h=0.4m.

Cnoidal wavemaker theory may be improved by applying the dispersion correction as done for Stream Function wavemaker theory, and this will be done from now on. This correction has virtually no influence for the case of T=2.3, but gives small improvements for the shorter waves.


For the range of rather shallow water cases T=2.3, 1.8 and 1.6s in Table 7.1, we make tests using three different methods: Stream Function wavemaker theory, modified Cnoidal wavemeker theory, and second-order Stokes theory, respectively. Comparing time series of surface elevation measured in the flume with the theoretical solution of Stream Function theory, ErrRMS and ErrH at the three gauges are calculated for each wave height. Figure 7.9 and Figure 7.10 show ErrRMS and ErrH for T=2.3s, 1.8s and 1.6s, respectively. The big variation of ErrH from gauge to gauge (as e.g. for T=2.3s, H=0.24m by Cnoidal wavemaker theory) demonstrates the non-constant form of the waves in the flume as shown in Figure 7.7.

Figure 7.9: The relative errors ErrRMS, for rather shallow wave cases.

T=1.6s

0%

5%

10%

15%

20%

25%

30%

0.08m 0.12m 0.16m 0.20m 0.22m

H

Err

RM

S 2nd

CN

SF

T=1.8s

0%

5%

10%

15%

20%

25%

30%

0.08m 0.12m 0.16m 0.20m 0.22m

H

Err

RM

S 2nd

CN

SF

T=2.3s

0%

5%

10%

15%

20%

0.08m 0.12m 0.16m 0.20m 0.22m 0.24m

H

Err

RM

S

2nd

CN

SF


Figure 7.10: The relative errors ErrH, for rather shallow wave cases.

It is obvious that Stream Function wavemaker theory gives the best results among the three wavemaker theories with almost all errors being less than or around 5%. Especially for the highly nonlinear waves, the errors are still quite small when the other two wavemaker theories are not applicable due to wave breaking. Comparing the other two theories, the Cnoidal wavemaker theory gives better results than second-order wavemaker theory clearly for each wave height when T=2.3s. The errors go up as the wave height increases. Wave breaking takes place for H=0.22m using second-order wavemaker theory, but for H=0.24m using Cnoidal wavemaker theory. When T=1.8s the advantage of Cnoidal wavemaker theory becomes weak, and for T=1.6s it is superseded by second-order wavemaker theory. For high waves, wave breaking takes place at H=0.20m using Cnoidal wavemaker theory for these two periods, but at H=0.22m using second-order wavemaker theory. As the period decreases (waves become shorter), the results of Cnoidal wavemaker theory deteriorates. This is expected as the wave condition gets closer to the critical water depth of the application of Cnoidal wave theory. Figure 7.11 shows time series of surface elevation for T=1.6s, H=0.22m, and h=0.4m using second-order wavemaker theory. A clear improvement is seen in Figure 7.12

T=1.6s

-30%

-25%

-20%

-15%-10%

-5%

0%

5%

10%

0.08m 0.12m 0.16m 0.20m 0.22m

H

Err

H

2nd

CN

SF

T=1.8s

-30%-25%-20%-15%-10%-5%0%5%

10%15%20%

0.08m 0.12m 0.16m 0.20m 0.22m

H

Err

H

2nd

CN

SF

T=2.3s

-30%-25%-20%-15%-10%-5%0%5%

10%15%20%25%30%

0.08m 0.12m 0.16m 0.20m 0.22m 0.24m

H

Err

H

2nd

CN

SF


which shows the results of Stream Function wavemaker theory. For the Stream Function case, the average wave height of three gauges measured is 0.208m corresponding to a ratio of wave height to water depth H/h=0.52.

Figure 7.11: Time series of surface elevation measured using second-order wavemaker

theory, for T=1.6s, H=0.22m, h=0.4m.


7.2.1.2 Non-shallow water waves For the non-shallow water cases: T=1.4, 1.1, 0.9, 0.75 and 0.6s, the tests are made with Stream Function wavemaker theory and second-order Stokes wavemaker theory, respectively. The Cnoidal theory is not available for these cases since the relative water depths exceeds the Cnoidal limit. Figure 7.13 and Figure 7.14 show time series of surface elevation for T=0.9s, H=0.10m, and h=0.4m using second-order wavemaker theory, and Stream Function wavemaker theory, respectively. The difference between the quality of the two time series of surface elevation is not very clear for this non-shallow water case. However, as shown in Figure 7.15, looking at the relative errors ErrRMS and ErrH, the Stream Function approach is clearly superior to second-order wavemaker theory.


Figure 7.13. Time series of surface elevation measured using second-order wavemaker

theory, for T=0.9s, H=0.10m, h=0.4m.

Figure 7.14. As for Figure 7.13, but using Stream Function wavemaker theory.

Figure 7.15. Relative errors ErrRMS and ErrH for T=0.9s, H=0.10m, h=0.4m.

Figure 7.16 shows ErrRMS and ErrH for all the wave height tested for all non-shallow water cases. The errors increase as the wave height becomes larger. When T=1.4s, 1.1s, and 0.9s, using Stream Function wavemaker theory, almost all the errors ErrRMS and ErrH are smaller than the corresponding ones by second-order wavemaker theory. They are less than or around 5% for most of the cases at all three gauges, except for ErrH on T=0.9s, H=0.12m which is 7.8% on the average. When T=0.75s with H=0.04m and 0.06m, the results by Stream Function theory are still better than second-order wavemaker theory with the value of errors around 5%. However, for the

0%

2%

4%

6%

8%

10%

12%

1 2 3

gauge

Err

RM

S

-12%

-10%

-8%

-6%

-4%

-2%

0%

1 2 3

gauge

Err

H

2nd

SF


case with large wave height T=0.75s, H=0.08m, the errors are relatively large when using Stream Function wavemaker theory. The average ErrRMS at the three gauges is 9.2%, and the average ErrH is -10.9%. Also for the deep water cases T=0.6s, both of wavemaker theories give quite large errors, especially for H=0.06m. The reason for this deterioration of the approximate Stream Function wavemaker theory is that the dispersion correction to the paddle position is based on linear wavemaker theory. In shallow water, this method is consistent with nonlinear long wave generation since the transfer function L tends to unity when the relative water depth tends to zero. This has been verified by the tests of the rather shallow water cases in the above section. However for nonlinear waves in non-shallow water, this dispersion correction based on linear theory is inadequate and the error increases with nonlinearity.

T=0.75s

0%

5%

10%

15%

0.04m 0.06m 0.08m

H

Err

RM

S

T=0.9s

0%

5%

10%

15%

0.04m 0.08m 0.10m 0.12m

H

Err

RM

S

T=0.75s

-20%

-15%

-10%

-5%

0%

0.04m 0.06m 0.08m

H

Err

H 2nd

SF

T=0.9s

-15%

-10%

-5%

0%

0.04m 0.08m 0.10m 0.12m

H

Err

H 2nd

SF

T=1.1s

0%

5%

10%

15%

0.08m 0.12m 0.16m

H

Err

RM

S

T=1.1s

-10%

-5%

0%

5%

0.08m 0.12m 0.16m

H

Err

H

2nd

SF

T=1.4s

0%

5%

10%

15%

0.08m 0.12m 0.16m 0.20m

H

Err

RM

S

T=1.4s

-10%

-5%

0%

5%

0.08m 0.12m 0.16m 0.20m

H

Err

H

2nd

SF


Figure 7.16. Relative errors ErrRMS and ErrH.

7.2.2 Wave generation with active absorption In this section, we consider wave generation with active absorption. Some rather shallow water wave cases at T=2.3s and non-shallow water cases at T=1.4s in Table 7.1 are picked for the physical tests. For the cases at T=2.3s (except for H=0.08m), we make tests using Cnoidal wavemaker theory and Stream Function wavemaker theory, in single mode and dual mode, respectively, compared with position mode which has been done in the above. Note that the control signals of Cnoidal wavemaker theory in dual mode are modified with dispersion and evanescent-mode corrections as done for Stream Function wavemaker theory. The corrections have no effect on single mode control signals of either wavemaker theory, since only (theoretical) progressive wave surface elevation is required for it. For the cases at T=1.4s, tests are made using only Stream Function wavemaker theory in single mode and dual mode to compare. Figure 7.17 and Figure 7.18 show surface elevation measured at three gauges in the flumes using control signals in single mode and dual mode by two different theories, respectively, for the case T=2.3s, H=0.24m, h=0.4m. See Figure 7.7 and Figure 7.8 for the results in position mode as references. Comparing the results in two different modes for active absorption, the advantage of dual mode is obvious. Either Cnoidal or Stream Function wavemaker theory in single mode reproduces waves very poorly in the flumes. Also wave breaking takes place in the flumes respectively. Looking at measurements in dual mode. Both of Cnoidal and Stream Function wavemaker theory reproduce highly nonlinear waves with rather steady forms. The results of Stream Function wavemaker theory are getting better. Comparing dual mode with position mode shown in Figure 7.7 and Figure 7.8, we see that using Cnoidal wavemaker theory, dual mode gives fairly steady waves, meanwhile the reproduction of waves in the flume is inconstant as well as wave breaking takes place in position mode. The reason of poor quality in position mode has been mentioned before, which is due to the inaccuracy of the theory for highly nonlinear waves. The reason of that dual mode gives better results than position mode is probably that, when the wavemaker theory is inadequate for highly nonlinear waves, in dual mode the system gets a second chance for making slight corrections to the paddle signal. Using Stream Function wavemaker theory, the quality of reproduction in dual mode is as good as in position mode for this case. An interesting thing is that, in single mode, Stream Function wavemaker theory reproduces nonlinear waves worse than Cnoidal wavemaker theory. This is probably due to that Stream Function wave theory includes higher nonlinearity than Cnoidal theory. In single mode, the specified surface elevation is transformed to a paddle position via the active absorption system based on linear wavemaker theory.

T=0.6s

0%

5%

10%

15%

0.04m 0.06m

H

Err

RM

S

T=0.6s

-25%

-20%

-15%

-10%

-5%

0%

0.04m 0.06m

H

Err

H

2nd

SF


This linearity used in the system creates larger deviation from the expected nonlinearity of Stream Function theory than that from Cnoidal theory.

Figure 7.17: As for Figure 7.7, using Cnoidal wavemaker theory, but with active

absorption for T=2.3s,H=0.24m, h=0.4m.



Figure 7.19 and 7.20 show the comparison of relative errors ErrRMS and ErrH at three gauges for each wave height at T=2.3s in three control modes using Cnoidal wavemaker theory and Stream Function wavemaker theory, respectively. These errors demonstrate the advantage of Stream Function wavemaker theory in dual mode and position mode obviously. They also testify the advantage of dual mode in each wavemaker theory comparing with the corresponding single mode. Looking at Cnoidal wavemaker theory, ErrRMS increases as the wave height increase, as well as ErrH varies strongly from gauge to gauge when wave height is high in position mode. In dual mode, the results are improved for high waves. ErrRMS values and the variance of ErrH are smaller than in position mode. Now turn to Stream Function wavemaker theory, the results of position mode are always better than in dual mode. Thus we can say when the control signal of paddle position match well the expected one, the system in dual mode gives some disturbance instead of correction. But the most advantage of dual mode is the nonlinear wave generation with active absorption, which is applicable for nonlinear wave field with highly reflective structures.

Figure 7.19: Relative errors ErrRMS for T=2.3s.

T=2.3s

0%

5%

10%

15%

20%

0.12m 0.16m 0.20m 0.22m 0.24m

H

Err

RM

S SF position

SF dual

SF single

T=2.3s

0%

5%

10%

15%

20%

0.12m 0.16m 0.20m 0.22m 0.24m

H

Err

RM

S CN position

CN dual

CN single


Figure 7.20: Relative errors ErrH for T=2.3s.

Figure 7.21 shows surface elevation measured using control signals in three different modes by Stream Function wavemaker theory for the non-shallow water wave case T=1.4s, H=0.20m, h=0.4m. The result in single mode is as poor as expected. The measurements in dual mode and position mode match the theoretical results well. Compared with single mode, the advantage of dual mode and position mode are obvious. The result in position mode is better than that in dual mode. The main reason of that is the evanescent-mode correction to the surface elevation at the moving paddle are based on linear wavemaker theory. Figure 7.22 shows the comparison of relative errors ErrRMS and ErrH at each gauge for each wave height at T=1.4s in three different modes using Stream Function wavemaker theory. As mentioned before, the results of single mode are the worst among three modes for all the cases. Although the relative errors ErrRMS and ErrH in dual mode are a little larger than those in position mode, they are still relative small with values § 5%, except for the highest case H=0.20m with large ErrH about 10% average.

T=2.3s

-30%

-20%

-10%

0%

10%

0.12m 0.16m 0.20m 0.22m 0.24m

H

Err

H

SF position

SF dual

SF single

T=2.3s

-30%-25%-20%-15%-10%

-5%0%5%

10%15%20%25%30%

0.12m 0.16m 0.20m 0.22m 0.24m

HE

rr H

CN position

CN dual

CN single


Figure 7.21: Time series of surface elevation measured using Stream Function wavemaker

theory, with (SF single mode, and SF dual mode ) and without active absorption (SF position

mode) for T=1.4s, H=0.20m, h=0.4m.

Figure 7.22: Relative errors ErrRMS and ErrH for T=1.4s.

T=1.4s

-25%

-20%

-15%

-10%

-5%

0%

5%

0.08m 0.12m 0.16m 0.20m

H

Err

H

SF position

SF dual

SF single

T=1.4s

0%

5%

10%

15%

20%

25%

30%

0.08m 0.12m 0.16m 0.20m

H

Err

RM

S SF position

SF dual

SF single

7.3 Improved Stream Function Wavemaker Theory 123

7.3 Improved Stream Function Wavemaker Theory In the previous approximate Stream Function wavemaker theory, the paddle position is obtained by two steps using the ad hoc unified wave generation method. The paddle position with shallow water limit is obtained first. Then dispersion correction is to compensate the deviation from shallow water limit. In which the dispersion correction is based on linear theory. Also for pure generation of nonlinear waves, evanescent modes are not involved. The advantage of the approach over second-order wavemaker theory thus vanishes for deep water wave cases. We hope to devise a one-step wave generation method directly from Stream Function wave theory. In which, higher nonlinearity will be included, also evanescent modes are considered. For the progressive highly nonlinear waves, Stream Function theory gives a solution. The surface elevation h, the velocity potential φ, and velocity component in x direction u are expressed in Fourier functions as

1

( , ) cos[ ( )]N

jj

x t A jk x ctη=

= −∑ (7.5)

1

cosh[ ( )]( , , ) cos[ ( )]

cosh( )

Njp

j

B jk z hx z t jk x ct

jk jkhφ

=

+= −∑ (7.6)

1

cosh[ ( )]( , , ) cos[ ( )]

cosh( )

Np

jj

jk z hu x z t B jk x ct

jkh=

+= −∑ (7.7)

where c is the phase velocity of the wave, and 1... 1..., N NA B are constants for a given wave of

height H, water depth h, period T.

Based on the wavemaker theory, the wave paddle position X=X(z,t), the lateral boundary condition is

at ( , )x z z tX X x X z tφ φ− = = (7.8)

For a piston-type wavemaker, 0zX = , thus

( , , ) ( , , ) at ( ) p evat xX u x z t u x z t x X tφ= = + = (7.9)

Here, evau is the velocity component of evanescent-modes in the x direction. The depth-averaged velocity U of progressive wave is expressed as

1

1 1 sinh[ ( )]( , ) ( , , ) cos[ ( )]

cosh( )

Np

jjh

jk hU x t u x z t dz B jk x ct

h h jk jkh

η ηη η =−

+= = −+ + ∑∫ (7.10)


Making (7.9) multiplied by cosh[ ( )]

(m 0,... )cosh ( )r

mk z hN

mkh

+ = , respectively. Here r is a

parameter related to the dispersion and nonlinearity. Then integrating them from –h to h along z direction:

( ) ( ) ( ) at ( )p eva evat t

h h h

X dz X h u u dz U h u dz x X tη η η

η η− − −

= + = + = + + =∫ ∫ ∫ (7.11)

cosh[ ( )] cosh[ ( )]

( ) at ( )cosh ( ) cosh ( )

p evat r r

h h

k h z k h zX dz u u dz x X t

kh kh

η η

− −

+ += + =∫ ∫ (7.12)

......

cosh[ ( )] cosh[ ( )]

( ) at ( )cosh ( ) cosh ( )

p evat r r

h h

mk h z mk h zX dz u u dz x X t

mkh mkh

η η

− −

+ += + =∫ ∫ (7.13)

......

cosh[ ( )] cosh[ ( )]

( ) at ( )cosh ( ) cosh ( )

p evat r r

h h

Nk h z Nk h zX dz u u dz x X t

Nkh Nkh

η η

− −

+ += + =∫ ∫ (7.14)

Summarizing these equations (7.11) to (7.14) on both sides which yields,

1

1 0

cosh[ ( )][( ) ]

cosh ( )

cosh[ ( )] cosh[ ( )] ( ) ( , , ) ( , , )

cosh ( ) cosh ( )

N

t rm h

N Np eva

r rm mh h

mk h zX h dz

mkh

mk h z mk h zU h u X z t dz u X z t dz

mkh mkh

η

η η

η

η

= −

= =− −

++ + =

+ ++ + +

∑∫

∑ ∑∫ ∫(7.15)

Rewriting (7.15) as

* *

1

sinh[ ( )][( ) ] ( ) ( , ) ( , )

cosh ( )

N

t p erm

mk hX h U h U X t U X t

mk mkh

ηη η=

++ + = + + +∑ (7.16)

where

*

1

1 1

cosh[ ( )]( , ) ( , , )

cosh ( )

cosh[ ( )]cosh[ ( )]cos[ ( )]

cosh( ) cosh ( )

Np

p rm h

N Nj h

rj m

mk h zU X t u X z t dz

mkh

jk h z mk h z dzB jk X ct

jkh mkh

η

η

= −

−

= =

+= =

+ +−

∑∫

∫∑ ∑ (7.17)

in which


cosh[ ( )]cosh[ ( )]

1 sinh[2 ( )][2( ) ] for

4

1 sinh[ ( )( )] sinh[ ( )( )][ ] for

2 ( ) ( )

h

jk h z mk h z dz

jk hh m j

jk

k j m h k j m hm j

k j m k j m

η

ηη

η η

−

+ + =

+ + + = − + + + + ≠

− +

∫

(7.18)

*

0

cosh[ ( )]( , ) ( , , )

cosh ( )

Neva

e rm h

mk h zU X t u X z t dz

mkh

η

= −

+=∑ ∫ (7.19)

We may evaluate ( , , )evau x z t using linear theory

( )

1 0

( )( , , ) cos( )cos[ ( )( )]

2 sin[ ( ) ]sk n xeva s

sn s

c nHu x z t e t k n h z

c k n h

ω ω∞

−

=

= +∑ (7.20)

Here c0 is known as the Biesel transfer function, cs(n) (n=1,…¶) are real transfer functions for evanescent modes, w is 1st order angular frequency. For a piston-type wavemaker, we have

22

0

4sin [ ( ) ]4sinh , ( )

2 sinh 2 2 ( ) sin[2 ( ) ]s

ss s

k n hkhc c n

kh kh k n h k n h= =

+ + (7.21)

Where ks(n) (n=1,…¶) are the real wave numbers corresponding to evanescent modes, and satisfy the linear dispersion relation, 2 tans sgk k hω = − (7.22)

Thus *eU can be expressed as

( )*

10

0

sin[ ( ) ]2( , ) cos( )

2 ( ) sin[2 ( ) ]

cos[ ( )( )]cosh[ ( )]

cosh ( )

sk n Xse

n s s

sNh

rm

k n hHU X t t e

c k n h k n h

k n h z mk h z dz

mkh

η

ω ω∞

−

=

−

=

= ⋅+

+ +

∑

∫∑

(7.23)

in which

2 2

cos[ ( )]cosh[ ( )]

cosh[ ( )]sin[ ( )] sinh[ ( )]cos[ ( )]

( )

s

h

s s s

s

k h z mk h z dz

k mk h k h mk mk h k h

k mk

η

η η η η−

+ + =

+ + + + ++

∫ (7.24)


The paddle position X(t) can thus be obtained by solving (7.16) directly. The parameter r is to make U(h+h) be the dominant term comparing with corrected parts

*pU and *

eU , as well as to make Xt(h+h) lead on the left-hand side.

For nonlinear shallow water waves, O( *

eU )<<O( *pU ), we have

*

1

( )

sinh[ ( )][( ) ]

cosh ( )

pt N

rm

U h UX

mk hh

mk mkh

ηηη

=

+ +=

++ +∑ (7.25)

Eq. (7.25) matches (7.1) (when wc is ignored), but with a correction for higher nonlinearity. Figure 7.23 shows X(t) when r=2, compared with Xsw(t) in shallow water limit by solving (7.1), previous approximate Stream Function wavemaker theory with dispersion correction (7.2), and linear wavemaker theory for T=2.3s, H=0.12m, h=0.4m. The improved theory match the previous approximate Stream Function wavemaker theory. Figure 7.24 shows the comparison of X(t) with r=2 for T=2.3s, H=0.24m, h=0.4m. The new approach has higher nonlinearity than the previous clearly.

Figure 7.23: The comparison of wave paddle position by solving (7.1) with (7.2)

(approximate SF), solving (7.1) in shallow water limit (Xsw), solving (7.16) when r=2 (improved SF), and linear wavemaker theory for T=2.3s, H=0.12m, h=0.4m.

Figure 7.24: As for Figure 7.23, but for T=2.3s, H=0.24m, h=0.4m.


For an intermediate water wave case, T=1.4s, H=0.16m, h=0.4m, the paddle position using the new approach with r=2 is shown in Figure 7.25 compared with the others as for Figure 7.23. The new approach again matches the previous approximate theory.


For linear fully dispersive waves, O( *

eU )ºO( *pU ). Figure 7.26 shows X(t) with r=0

for a linear case T=0.6s, H=0.002m, h=0.4m. The new theory matches the previous approximate approach, as well as linear wavemaker theory.


For deep or rather deep nonlinear waves, the parameter r should vary with dispersion and nonlinearity. The regularity has not been worked out yet. In order to show the possibility of improvement using this new approach, Figure 7.27 shows measured surface elevation in the physical flume using the new approach, previous approximate Stream Function, respectively, compared with the solution of Stream Function wave theory for T=0.6s, H=0.06m, h=0.4m. The control signal of improved Stream Function wavemaker theory is used for H=0.04m with r=0, actually. It may possible to give an accurate control signal by adjusting the parameter r.


Figure 7.27: Measured surface elevation using previous approximate Stream Function (upper)

and improved Stream Function (lower) wavemaker theories, respectively, compared with Stream Function wave for T=0.6s, H=0.06m, h=0.4m.

7.4 Summary and Conclusions In this Chapter, an approximate Stream Function wavemaker theory for highly nonlinear waves in the flumes has been developed first. This theory is based on Stream Function wave theory, and the ad hoc unified wave generation method. For rather shallow water waves, Stream Function wavemaker theory reproduces highly nonlinear waves with much higher accuracy than second-order Stokes and Cnoidal wavemaker theories. Particularly, this approach can generate higher nonlinear waves with constant form than the other wavemaker theories. For non-shallow water waves, the advantage of it is still clear when compared with the results of second-order Stokes wavemaker theory provided that the nonlinearity is moderate. For highly nonlinear waves in deep water, the approximate Stream Function wavemaker theory deteriorates and the advantage over second-order wavemaker theory vanished. The main reason is that the dispersion correction to the paddle position is based on linear wavemaker theory. As to the wave generation with active absorption, using traditional single mode, both of Stream Function and Cnoidal wavemaker theories reproduce nonlinear waves


poorly in the physical flume. The results of Stream Function wavemaker theory are worse. The reason is that single mode active absorption system is based on linear wavemaker theory. The linearity used in the system creates larger deviation from the expected nonlinearity of Stream Function theory. In dual mode, Stream Function wavemaker theory reproduces highly nonlinear waves well for rather shallow water waves, and non-shallow water waves with non-high wave height. When the measured surface elevation does not match the expected one well, such as using Cnoidal wavemaker theory, dual mode is helpful to avoid harmonic generation in the flumes. But when they match well like using Stream Function wavemaker theory, dual mode give a little disturbance instead of correction to the paddle signal. For highly nonlinear waves in non-shallow water depth, the inaccuracy becomes large comparing with pure wave generation. The main reason is that the evanescent-mode correction to the surface elevation at the moving paddle are based on linear wavemaker theory. Altogether, the approximate Stream Function wavemaker theory either with or without dual mode active adsorption is successful to generate highly nonlinear in shallow and intermediate water flumes. This was supported by the experiment results. Furthermore, an improved Stream Function wavemaker theory has been discussed, which is based on the distribution of horizontal velocity given by Stream Function theory for progressive waves. In which, the evanescent modes are evaluated by linear theory. This one-step approach matches the previous approximate Stream Function wavemaker theory for linear waves, as well as nonlinear shallow water waves. However, more work on the relation of the parameter r with dispersion and nonlinearity need to be done for (rather) deep water nonlinear waves.


131

Chapter 8 Summary, Conclusions and Recommendations In this study, several topics related to wave generation have been touched, including the main topic of deterministic combination of numerical and physical models. This chapter presents a summary of some important findings from this thesis, and also gives recommendations to future work. 8.1 Summary and Conclusions Based on the linear fully dispersive wavemaker theory and the nonlinear shallow water wavemaker theory, an ad hoc unified wave generation theory for wave flumes has been devised in Chapter 5. It accounts for shallow water nonlinearity and compensates for local wave phenomena (evanescent modes) near the wavemaker. For small amplitude linear waves, the fully dispersive wavemaker theory is recovered. For shallow water waves, it is consistent with nonlinear long wave generation. Using this unified wave generation method as a link between numerical and physical models for wave flumes, a deterministic combination of two models has been presented. In the combined model, Boussinesq model Mike 21 BW was chosen for the numerical calculation. A wave flume with piston-type wavemaker and DHI AWACS control system were utilized for the physical model. The measurements in the physical flume matched the numerical calculations well for some cases on regular and irregular, linear non-shallow and nonlinear shallow water waves. The combined model is not very sensitive to the positioning of the point at which the physical model takes over from the numerical model. For the nonlinear shallow water wave generation, Cnoidal wavemaker theory is inadequate for highly nonlinear waves. But highly nonlinear waves in shallow water wave flumes are reproduced very well using the ad hoc unified wave generation and the Stream Function theory. Thus the dominant error is due to the limitation of Cnoidal wave theory or Boussinesq equations, rather than the ad hoc unified wave generation. Therefore, the deterministic combination of numerical and physical models in wave flumes is successful. Based on the ad hoc 2D unified wave generation, the 3D unified wave generation theory for wave basins has been developed in Chapter 6. This 3D wave generation still accounts for shallow water nonlinearity and compensates for local wave

132 Chapter 8. Summary, Conclusions and Recommendations

phenomena (evanescent modes) near the 3D wavemaker. Then a deterministic combination of numerical and physical models in wave basins has been presented. In the combined model, Mike 21 BW was chosen for the numerical calculation. A wave basin with segmented 3D piston-type wavemaker and DHI 3D AWACS control system were utilized for the physical model. The link between numerical and physical models was offered by the ad hoc unified 3D wave generation theory. All the measurements in physical basins matched the numerical calculations quite well for some oblique, directional, linear non-shallow and nonlinear shallow, irregular wave cases. Therefore, the deterministic combination of numerical and physical models in wave basins is also successful. For nonlinear shallow water wave generation, Cnoidal wavemaker theory and Stream Function wavemaker theory can reproduce oblique nonlinear waves in wave basins successfully in general. The difficulty in obtaining oblique nonlinear waves of constant form is mainly due to the limit of the facility on the width of paddles. As a spin-off in this study, fully nonlinear wave generation in wave flumes has been discussed in Chapter 7. The approximate Stream Function wavemaker theory, which is based on Stream Function wave theory, and the ad hoc unified wave generation method, are compared with second-order Stokes wavemaker theory and Cnoidal wavemaker theory. For rather shallow water waves, the approximate Stream Function wavemaker theory reproduces fully nonlinear waves with much higher accuracy than second-order Stokes and Cnoidal wavemaker theories. Especially, this theory can generate higher nonlinear waves with constant form than the other wavemaker theories. For non-shallow water waves, the advantage of this theory is obvious compared with the results of second-order Stokes wavemaker theory when the nonlinearity of waves is not high. For highly nonlinear waves in deep water, its advantage is not distinguished, the main reason is that the dispersion correction to the paddle position is based on linear wavemaker theory. Considering wave generation with active absorption, using the traditional active absorption single mode, both of the approximate Stream Function wavemaker theory and Cnoidal wavemaker theory reproduce nonlinear waves poorly in the flumes. The results of Stream Function wavemaker theory are worse. The reason is that single mode active absorption system is based on linear wavemaker theory. In dual mode which is for nonlinear wave generation and active absorption, Stream Function wavemaker theory reproduces fully nonlinear waves well for rather shallow water waves, and non-shallow water waves with non-high wave height. When the measured surface elevation does not match the expected one well, such as using Cnoidal wavemaker theory, dual mode is helpful to avoid harmonic generation in the flumes. But when they match well like using Stream Function wavemaker theory, dual mode gives a little disturbance instead of a correction to the paddle singnal. For highly nonlinear waves in non-shallow water depth, the inaccuracy becomes large comparing with pure generation. The main reason is that the evanescent-mode correction to the surface elevation at the moving paddle are based on linear wavemaker theory. Altogether, the approximate Stream Function wavemaker theory either with or without dual mode active adsorption is successful to generate fully nonlinear in shallow and intermediate water flumes. This was supported by the results in experiments.

8.2 Recommendations for Future Work 133

Furthermore, an improved Stream Function wavemaker theory is devised, which is based on Stream Function theory for progressive waves, and the linear theory for evanescent modes. This one-step approach matches the previous approximate Stream Function wavemaker theory for linear waves, as well as nonlinear shallow water waves. 8.2 Recommendations for Future Work Some recommendations for further studies in the areas touched upon in this thesis are provided here. The following are some ideas for future work. Firstly, the study on the parameter r in improved Stream Function wavemaker theory could be made. The parameter r related to dispersion and nonlinearity has much effect on the amplitude of paddle position. The new wavemaker theory may be developed if the relation of r with the dispersion and nonlinearity is worked out. After that, some physical tests on fully nonlinear wave generation from shallow to deep wave flumes need to be made for the validation of this theory. This approach is derived directly from the horizontal velocity of Stream Function wave theory for the progressive wave. It has much nonlinearity compared with the previous approximate Stream Function wavemaker theory in which the dispersion correction is based on linear theory. Also the evanescent modes are considered in the new approach. Thus the reproduction of steady nonlinear waves in flumes should be improved using the new approach. As the improved Stream Function wavemaker theory is for reproducing regular nonlinear waves, it is not applicable to wave generation of irregular waves. The ad hoc unified wave generation method is still the only way for irregular nonlinear wave generation. In this method, the paddle position is calculated by two steps, and the second-step dispersion correction is based on linear theory. The author think it may be improved if the paddle position is calculated directly from the velocity distribution of irregular nonlinear waves which is obtained from numerical calculations. The experience of improved Stream Function wavemaker theory may be used for reference. In either improved Stream Function wavemaker theory or ad hoc unified wave generation method, the evanescent-mode is evaluated by linear wave theory. For nonlinear deep water waves, the effect of this becomes large. The study on the higher order of evanescent modes might also be interesting.

134 Chapter 8. Summary, Conclusions and Recommendations

135

References Biesel, F., Suquet, F., 1951. Les Appareils Generateurs de houle en laboratoire. La houille blanche, 6, Nos. 2, 4, and 5; and 7, No.6. Biesel, F. Suquet, F., 1954. Laboratory wave-generating apparatus. St. Anthony falls hydraulic laboratory Proj. Rep. No. 29. Boussinesq, J.V., 1872. Theorie des ondes et des remous qui se propagent le long d’un canal rectangularie horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55-108. Chaplin, J.R., 1980. Developments of stream function theory. Coastal Engineering, 3(3), pp 179-206. Chappelear, J.E., 1961. Direct numerical calculation of wave properties. Journal of Geophysical Research, 66, pp 501-508. Dean,R.G., 1965. Stream function representation of nonlinear ocean waves. Journal of Geophysical Research, 70(18), 4561-4572. Dean, R. and Dalrymple R.A., 1984. Wavemaker theory. In:Water wave mechanics for engineers and scientists. World Scientific, Singapore. Dalrymple, R.A., Greenberg, M., 1985. Directional wave makers. In: Dalrymple, R.A. (Ed.), Physical Modelling in Coastal Engineering. A.A. Balkema, Rotterdam, pp 67-81. Flick, R.E. and Guza, R.T., 1980. Paddle generated waves in laboratory channels. J. WatWays, Port, Coastal Ocean Div., ASCE 106 (WW1), pp 79-92. Fontanet P., 1961. Theorie de la generation de la houle cylindrique par un batteur plan. La Huille Blanche 16, pp3-31 (part 1) and pp174-196 (part 2). Fenton, J.D., 1988. The numerical solution of steady water wave problems. Computers and Geosciences, 14(3), pp 357-368. Fenton J.D., 1990. Nonlinear wave theories. In D.M. Le Mehaute, B. & Hanes, editor, The Sea, pp 3-25.

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140 References

141

Appendix A Boussinesq-type Equations in Mike 21 BW MIKE 21 BW software is the Boussinesq model we choose for the numerical wave computations in the far field. It is suitable for simulation of the propagation of directional wave trains travelling from deep to shallow water. Mike 21 BW is based on the numerical solution of time domain formulations of Boussinesq-type equations which include nonlinearity as well as frequency dispersion. The frequency dispersion is introduced in the momentum equations by taking into account the effect of vertical accelerations on the pressure distribution. The dimensional enhanced equations (originally derived by Madsen et al, 1991, and Madsen and Sørensen, 1992) read 0t x yP Qη + + = (A.1)

2

1( ) ( ) 0t x y x

P PQP gd

d dη ψ+ + + + = (A.2)

2

2( ) ( ) 0t y x y

Q PQQ gd

d dη ψ+ + + + = (A.3)

where subscripts x,y, and t, denote differentiation with respect to space and time, d is the total water depth, h is the still water depth, h is the surface elevation, P and Q are the depth-integrated velocity components, and y1, y2 are the Boussinesq terms defined by

2 31

1( ) ( ) ( )

31 1 1

( 2 ) ( )3 6 6

xxt xyt xxx xyy

x xt yt xx yy y xt xy

B h P Q Bgh

hh P Q Bgh Bgh hh Q Bgh

ψ η η

η η η

= − + + − +

− + + + − + (A.4)

2 32

1( ) ( ) ( )

31 1 1

( 2 ) ( )3 6 6

yyt xyt yyy xxy

y yt xt yy xx x yt xy

B h Q P Bgh

hh Q P Bgh Bgh hh P Bgh

ψ η η

η η η

= − + + − +

− + + + − + (A.5)

142 Appendix A. Boussinesq-type Equations in Mike 21 BW

where the value of the coefficient B=1/15, which gives the linear dispersion relation,

22

22

11

151 1

1 ( )3 15

k h

κϖ

κ

+=

+ + (A.6)

where, k=kh. This is interpreted as a Padé [2,2] expansion in k of the linear dispersion relation of Stokes,

2

2

tanh( ).

Stokes

k h

ϖ κκ

=

(A.7)

Figure A.1 shows the relative phase velocity compared with Stokes’ target solution obtained by (A.6) and (A.7). The error is restricted to 2.5% for 3κ < .

0.5 1 1.5 2 2.5 3κ

1.01

1.02

1.03

1.04

1.05c ê c Stokes

Figure A.1: The relative phase velocity compared with Stokes’ target solution. The 2DH module (two horizontal space coordinates) we used is typically selected for calculation of short and long period wave disturbance in ports and harbors. It solves the enhanced Boussinesq equations by an implicit finite difference techniques Alternating Direction Implicit with variable defined on a space-staggered rectangular grid. For the more detail on the numerical scheme, see the reference (Madsen and Sørensen, 1992). For more information on MIKE 21 see DHI Software (2005).

143

Appendix B The Elliptic Functions Related to the Cnoidal Waves

The incomplete elliptic integral of the first kind is defined as following,

2 2 2 2 20 0

( , )1 sin (1 )(1 )

xd dvw F m

m v m v

φ θφφ

= = =− − −∫ ∫ , (B.1)

where, φ = am w is called the amplitude of w and x=sinφ , and where here and below 0 1m≤ < . Then, the complete elliptic integral of the first kind K is defined as,

/ 2 1

2 2 2 2 20 0( ) ( , / 2)

1 sin (1 )(1 )

d dvK m F m

m v m v

π θπθ

= = =− − −∫ ∫ . (B.2)

The incomplete elliptic integral of the second kind is defined as following,

2 2 2 2 2

0 0( , ) 1 sin (1 )(1 )

xE m m d v m v dv

φφ φ θ= − = − −∫ ∫ , (B.3)

thus, the complete elliptic integral of the second kind E is defined as,

/ 2 12 2 2 2 2

0 0( ) ( , / 2) 1 sin (1 )(1 )E m E m m d v m v dv

ππ θ θ= = − = − −∫ ∫ (B.4)

The Jacobian elliptic functions can be defined as,

( ) sin sin( )sn w am w xφ= = = (B.5)

cn (w) = cos φ=cos (am w)= 21 x− , (B.6)

dn (w)= 2 21 m sn w− = 2 21 m x− . (B.7) Reference for these functions is made to Milne-Thomson (1965).

144 Appendix B. The Elliptic Functions Related to the Cnoidal Waves

145

Appendix C Stream Function Wave Theory The fully nonlinear steady wave problem on a flat bottom can be solved by stream function theory using a Fourier series expansion of the stream function (Rienecker and Fenton 1981, and Fenton 1988). This method is based on the Laplace equation of the stream function y(x,z) for irrotational flow with boundary conditions. 2 0ψ∇ = (C.1) with horizontal velocity u(x,z) and vertical velocity w(x,z) given by

( , ) ; ( , )=-u x z w x zz x

ψ ψ∂ ∂=∂ ∂

(C.2)

in a frame of reference moving at the phase velocity c. The impermeable bottom condition is ( , ) 0w x h− = (C.3) where, h is the water depth. The kinematic and dynamic free surface boundary conditions imposing zero flux through the free surface and constant pressure at the free surface are expressed respectively as ( , ) at zx t qψ η= = (C.4)

2 21( ) at z

2g u w Rη η+ + = = (C.5)

where h is the surface elevation, q and R are constants. The stream function y(x, z), and velocity components u(x,z) and w(x,z) are expended in Fourier functions as

1

( ) cos( )N

jj

x A jkxη=

=∑ (C.6)

146 Appendix C. Stream Function Wave Theory

01

sinh[ ( )]( , ) cos( )

cosh( )

Nj

j

B jk z hx z B z jkx

jk jkhψ

=

+= +∑ (C.7)

01

cosh[ ( )]( , ) cos( )

cosh( )

N

jj

jk z hu x z B B jkx

jkh=

+= +∑ (C.8)

1

sinh[ ( )]( , ) sin( )

cosh( )

N

jj

jk z hw x z B jkx

jkh=

+=∑ (C.9)

where 1... 0 1..., , N NA B B are constants for a given wave.

backend.orbit.dtu.dki Preface This thesis is submitted as partial fulfillment for the degree of Doctor of Philosophy at the Technical University of Denmark …

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