University of Southern California Dissertations and Theses

The development of a hydraulic-control wave-maker (HCW) for the study of non-linear surface waves

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The development of a hydraulic-control wave-maker (HCW) for the study of non-linear surface waves

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Content THE DEVELOPMENT OF A HYDRAULIC-CONTROL WA VE-MAKER (HCW) FOR THE STUDY OF NON-LINEAR SURFACE WA VES by Haeng Sik Ko A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CIVIL AND ENVIRONMENTAL ENGINEERING) December 2016 Copyright 2016 Haeng Sik Ko To my lovely wife, Hee Sung Mun, and beautiful children, Kaylee and Kaiden ii Acknowledgements I would like to express my deepest appreciation to Dr. Patrick Lynett for his guidance, support, patience and trust throughout my doctoral studies. It has truly been an honor and a pleasure to work with you for past five years. I also would like to thank my committee members, Dr. Jiin-Jen Lee, Dr. Mitul Luhar, Dr. Felipe de Barros and Dr. Adam Fincham for their valuable comments on my works. It has been a great pleasure to share my graduate studies and life with the former and current colleagues in Wave Research Group. Dr. Sangyoung Son, Hoda El Safty, Nikos Kalligeris, Aykut Ayca, Sasan Tavakkol, Luis Montoya and Adam Keen. Most importantly, none of this work would have been possible without unconditional support from my family. I would like to thank my parents and mother-in-law for giving me love, care and encouragement during my study. I also deeply appreciate my beloved daughter, Kaylee who always sings a song for cheering me up as well as my son, Kaiden, who brings my family good luck. Finally, I would like to thank Hee Sung Mun for her support, encouragement, quiet patience and unwavering love. iii Contents List of Tables vii List of Figures viii 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Numerical Simulation for Design of HCW 6 2.1 Introduction of OpenFOAM R . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 The ke Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 V olume of Fluid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Linear Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Solitary Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Stokes Second Order Theory . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Numerical Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.1 Optimized Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.2 Wave Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Small-Scale Prototype 21 3.1 Experimental Design and Methodology . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Calibration between Flow Rate and Motor Speed . . . . . . . . . . . . . . . . . . 21 3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Physical Model with 2 Baffles 28 4.1 Experiment Equipment and Procedures . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.1 General Principle of Operation for Wave Generation . . . . . . . . . . . . 28 4.1.2 Calibration System for Capacity Wave Gauge . . . . . . . . . . . . . . . . 30 4.1.3 Motor Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iv 4.1.4 Straighten Flow in Baffles . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Verification of Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.1 Solitary Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.2 Sinusoidal Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Physical Model with 3 Baffles 41 5.1 Movable Top Baffle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Ramp Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Verification of Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3.1 Solitary Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3.2 Sinusoidal Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3.3 Stokes Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4 Active Wave Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.5 Three Paired Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.5.1 Synchronization of three Paired Cylinders . . . . . . . . . . . . . . . . . . 77 5.5.2 Corrected Calibration between Flow Rate and RPM . . . . . . . . . . . . . 78 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6 Measurement Precision and Uncertainty 82 7 Wave Group Interactions over a Sloping Beach 84 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.2 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.3 Top-hat Spectral Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.4 OpenFOAM with a movable top baffle . . . . . . . . . . . . . . . . . . . . . . . . 92 7.5 COULWA VE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.6 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8 Measurement for the Time Series of Wave Runup 116 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.2 Instrumental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.3 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9 Future Research 149 Reference List 150 Appendix List 156 v A Integration of Horizontal Velocity Profile in 3 Baffles 156 A.0.1 Linear Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A.0.2 Stokes Second Order Wave Theory . . . . . . . . . . . . . . . . . . . . . 158 A.0.3 Solitary Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B Block Diagram of Labview for Controlling a Motor 161 B.0.1 Block Diagram of Labview for Wave Generation: Controlling Three Servo Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 B.0.2 Block Diagram of Labview for Calibration of Wave Gauge: Controlling a Stepper Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 C Coefficients of Second Order Solution 163 vi List of Tables 6.1 Quantifications of precision errors and uncertainty . . . . . . . . . . . . . . . . . . 83 7.1 Wave group conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.1 Comparison of maximum solitary and top-hat spectral wave amplitudes at WG1 . . 126 8.2 Comparison of maximum solitary wave runups between numerical simulations and experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 vii List of Figures 1.1 Major ocean currents of the world (Pidwirny (2006)) . . . . . . . . . . . . . . . . 1 1.2 General schematic of the HCW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Schematic drawing in terms of baffle height, length and position and the number of baffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Comparison of the time series of water surface with different baffle heights (dash line: theory; dash dot line: 0.01m; dot line: 0.02m; solid line: 0.03m) . . . . . . . 12 2.3 Comparison of the time series of water surface with different baffle lengths (dash line: theory; dash dot line: 0.05m; green solid line: 0.1m; dot line: 0.2m; cyan solid line: 0.3m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Comparison of the time series of water surface with different baffle number and position (dash line: theory; dot line: 3 inlets; dash dot line: 5 inlets; solid line: 6 inlets) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Schematic optimized design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.01, kh=0.1) . . . . . . . . . . . . . . . . . . 15 2.7 Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.01, kh=1) . . . . . . . . . . . . . . . . . . . 16 2.8 Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.05, kh=0.1) . . . . . . . . . . . . . . . . . . 16 2.9 Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.05, kh=1) . . . . . . . . . . . . . . . . . . . 17 2.10 Schematic drawing for wave absorption without wave celerity . . . . . . . . . . . 18 2.11 Time series of water surface elevation of solitary wave with a/h=0.05 (left) and 0.3 (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.12 Schematic drawing for wave absorption with wave celerity . . . . . . . . . . . . . 19 2.13 Time series of water surface elevation of solitary wave with of a/h=0.05 (left) and 0.3 (right) with using wave celerity . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1 Schematic design of prototype (bottom) and physical model (top) . . . . . . . . . . 22 3.2 Comparison between flow rate for piston and flow meter . . . . . . . . . . . . . . 23 3.3 The snapshot of sinusoidal wave with a/h=0.3, kh=1 . . . . . . . . . . . . . . . . . 24 3.4 Time series of RPM in sinusoidal wave with a/h=0.3, kh=1 . . . . . . . . . . . . . 24 3.5 The snapshot of sinusoidal wave with a/h=0.3, kh=0.1 . . . . . . . . . . . . . . . . 25 3.6 Time series of RPM in sinusoidal wave with a/h=0.3, kh=0.1 . . . . . . . . . . . . 25 3.7 The snapshot of sinusoidal wave with a/h=0.013, kh=1 using big flume . . . . . . 26 viii 3.8 Time series of RPM in sinusoidal wave with a/h=0.013, kh=1 using big flume . . . 26 4.1 Physical model with 3 baffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Design of 3 baffles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 The calibration system for the capacity wave gauge . . . . . . . . . . . . . . . . . 31 4.4 Comparison between motor synchronization before and after . . . . . . . . . . . . 32 4.5 Snapshots of flow pattern in baffle by sinusoidal wave with kh=1 (top) and kh=0.5 (bottom) (blue arrow: flow direction) . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.6 Snapshots of flow pattern in baffle by sinusoidal wave with kh=0.5 with meshes (blue arrow: flow direction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.7 Time series of RPM (top) and water surface elevation (middle) and comparison between the result and analytical solution (bottom) (a/h=0.08, h=33cm) . . . . . . 36 4.8 Time series of water surface elevation (top), RPM (middle) and enlarged RPM (bottom) (a/h=0.08, kh=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.9 Time series of water surface elevation (top), RPM (middle) and enlarged RPM (bottom) (a/h=0.08, kh=0.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.10 Comparison between the current and a new baffle design (left:current design, right:new design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.1 HCW system with 3 baffles (left) and a movable top baffle (right) . . . . . . . . . . 41 5.2 Snapshots of flow pattern in the movable top baffle by solitary wave without tooth- type meshes (top) and with tooth-type meshes (bottom) . . . . . . . . . . . . . . . 42 5.3 The time-varying volume of top baffle . . . . . . . . . . . . . . . . . . . . . . . . 44 5.4 Increasing signal with ramp period . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.5 Wave flume and instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.6 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.05) . . . . . . . . . . . . . . . . 48 5.7 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.08) . . . . . . . . . . . . . . . . 49 5.8 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.03, kh=1) . . . . . . . . . . . . . 51 5.9 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.05, kh=1) . . . . . . . . . . . . . 52 5.10 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.07, kh=1) . . . . . . . . . . . . . 53 5.11 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.03, kh=0.75) . . . . . . . . . . . 54 5.12 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.05, kh=0.75) . . . . . . . . . . . 55 5.13 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.07, kh=0.75) . . . . . . . . . . . 56 5.14 Reflection coefficient with different kh (a/h=0.05) . . . . . . . . . . . . . . . . . . 58 5.15 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.03, kh=0.5) . . . . . . . . . . . . 59 ix 5.16 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.05, kh=0.5) . . . . . . . . . . . . 60 5.17 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.07, kh=0.5) . . . . . . . . . . . . 61 5.18 Snapshots of flow pattern in the top baffle according to its movement (a/h=0.07, kh=0.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.19 Photos of gaps of the top baffle hinge: overview (left); enlarged view (right) . . . . 62 5.20 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.03, kh=1) . . . . . . . . . . . . . 64 5.21 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.05, kh=1) . . . . . . . . . . . . . 65 5.22 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.07, kh=1) . . . . . . . . . . . . . 66 5.23 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.03, kh=0.75) . . . . . . . . . . . 67 5.24 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.05, kh=0.75) . . . . . . . . . . . 68 5.25 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.07, kh=0.75) . . . . . . . . . . . 69 5.26 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.03, kh=0.5) . . . . . . . . . . . . 70 5.27 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.05, kh=0.5) . . . . . . . . . . . . 71 5.28 Time series of free surface elevation (top) and RPM (bottom). Solid line: labora- tory observations; dashed line: target data (a/h=0.07, kh=0.5) . . . . . . . . . . . . 72 5.29 Wave flume and instrumentation for active wave absorption . . . . . . . . . . . . . 73 5.30 Time series of free surface elevation (top) and RPM (bottom) (a/h=0.05) . . . . . . 74 5.31 Time series of free surface elevation (top) and RPM (bottom) (a/h=0.08) . . . . . . 75 5.32 HCW with 3 baffles and 3 paired cylinders . . . . . . . . . . . . . . . . . . . . . . 76 5.33 Time series of RPM from three paired cylinders. top-hat spectral wave (top); bichromatic wave (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.34 Time series of RPM from three paired cylinders by old calibration results (top); corrected calibration results (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.35 Comparison between target and observed max a/h from WG 1 . . . . . . . . . . . 80 7.1 Classification of the spectrum of ocean waves according to wave period (Munk (1950)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 Wave flume and instrumentation for top-hat spectral waves . . . . . . . . . . . . . 87 7.3 Comparison of target RPMs and monitored RPMs for top-hat spectral waves (case C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.4 Comparison of target RPMs and monitored RPMs for top-hat spectral waves after time modulation(case C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.5 Comparison of the experimental data and the model prediction by OpenFOAM before applying the correct target wave height, Case A (top), B (middle) and C (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 x 7.6 Dynamic mesh motion: mesh at initial time (top); mesh at wave crest (middle); mesh at wave trough (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.7 Time series of free surface elevation and spectrum analysis, case A. Blue: labora- tory observations; red: model predictions by OpenFOAM . . . . . . . . . . . . . . 97 7.8 Time series of free surface elevation and spectrum analysis, case A. Blue: labora- tory observations;red: model predictions by COULWA VE . . . . . . . . . . . . . . 98 7.9 Time series of free surface elevation and spectrum analysis, case B. Blue: labora- tory observations;red: model predictions by OpenFOAM . . . . . . . . . . . . . . 99 7.10 Time series of free surface elevation and spectrum analysis, case B. Blue: labora- tory observations; red: model predictions by COULWA VE . . . . . . . . . . . . . 100 7.11 Time series of free surface elevation and spectrum analysis, case C. Blue: labora- tory observations;red: model predictions by OpenFOAM . . . . . . . . . . . . . . 101 7.12 Time series of free surface elevation and spectrum analysis, case C. Blue: labora- tory observations; red: model predictions by COULWA VE . . . . . . . . . . . . . 102 7.13 Time series of long wave elevation, case A. Solid line: laboratory observations; dotted line: model predictions by OpenFOAM; blue: total wave elevation; green: long wave elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.14 Time series of long wave elevation, case A. Solid line: laboratory observations; dotted line: model predictions by COULWA VE; blue: total wave elevation; green: long wave elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.15 Time series of long wave elevation, case B. Solid line: laboratory observations; dotted line: model predictions by OpenFOAM; blue: total wave elevation; green: long wave elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.16 Time series of long wave elevation, case B. Solid line: laboratory observations; dotted line: model predictions by COULWA VE; blue: total wave elevation; green: long wave elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.17 Time series of long wave elevation, case C. Solid line: laboratory observations; dotted line: model predictions by OpenFOAM; blue: total wave elevation; green: long wave elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.18 Time series of long wave elevation, case C. Solid line: laboratory observations; dotted line: model predictions by COULWA VE; blue: total wave elevation; green: long wave elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.19 Surface elevation amplitude spectra of case A. Blue: laboratory observations; dashed green: model predictions by COULWA VE; dashed-dot red: model pre- diction by OpenFOAM; dotted black: second-order theory at wave-maker position (x= 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.20 Surface elevation amplitude spectra of case B. Blue: laboratory observations; dashed green: model predictions by COULWA VE; dashed-dot red: model prediction by OpenFOAM; dotted black: second-order theory at wave-maker position (x= 0) . . 113 7.21 Surface elevation amplitude spectra of case C. Blue: laboratory observations; dashed green: model predictions by COULWA VE; dashed-dot red: model prediction by OpenFOAM; dotted black: second-order theory at wave-maker position (x= 0) . . 114 8.1 A definition sketch for a solitary wave runup and inundation on a bi-linear sloping beach (m 1 =1/10 and m 2 =1/15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 xi 8.2 Instrumental setup. Top: side view; bottom: top view . . . . . . . . . . . . . . . . 120 8.3 Process of edge detection.a) RGB image; b) grayscale image with mask; c) Edge detection in black & whited; d) edge overlapped RGB . . . . . . . . . . . . . . . . 121 8.4 Comparison of Edge detection without and with image contrast control during wave rundown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.5 Distorted image (top) and undistorted image with DLT transformation (bottom) . . 124 8.6 The time series of wave runups: Distribution of maximum, median, minimum points by Canny edge detection. Solitary wave with a/h=0.07 (top) and case C of top-hat spectral wave (bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.7 Comparisons of solitary wave with a/h=0.12 (top) and Case C of top-hat spectral wave (bottom) surface elevations time series at WG1. Blue: model prediction by COULWA VE; Red: experimental data; green: model prediction by OpenFOAM . . 127 8.8 Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.06 . . . . . . . . . . . 129 8.9 Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.07 . . . . . . . . . . . 130 8.10 Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.08 . . . . . . . . . . . 131 8.11 Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.09 . . . . . . . . . . . 132 8.12 Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.10 . . . . . . . . . . . 133 8.13 Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.11 . . . . . . . . . . . 134 8.14 Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.12 . . . . . . . . . . . 135 8.15 Comparison of maximum solitary wave runup of experiment, COULWA VE and OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.16 Comparison of the time series of top-hat spectral wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: Case A . . . . . . . 138 8.17 Comparison of the time series of top-hat spectral wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: Case B . . . . . . . 139 8.18 Comparison of the time series of top-hat spectral wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: Case C . . . . . . . 140 8.19 Time series of wave runup and space-time evolution of surface elevation of exper- imental data, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.20 Time series of wave runup and space-time evolution of surface elevation of Open- FOAM, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.21 Time series of wave runup and space-time evolution of surface elevation of COUL- WA VE, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 xii 8.22 Time series of low frequency component of wave runup and space-time evolution of low frequency surface elevation of experimental data, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. . . . . . . . . 145 8.23 Time series of low frequency component of wave runup and space-time evolution of low frequency surface elevation of OpenFOAM, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. . . . . . . . . . . . . 146 8.24 Time series of low frequency component of wave runup and space-time evolution of low frequency surface elevation of COULWA VE, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. . . . . . . . . . . . 147 A.1 Definition and notation of linear and second wave . . . . . . . . . . . . . . . . . . 156 A.2 Definition and notation of solitary wave . . . . . . . . . . . . . . . . . . . . . . . 159 B.1 Block Diagram of Labview for controlling 3 servo motors . . . . . . . . . . . . . . 161 B.2 Block Diagram of Labview for controlling a stepper motor . . . . . . . . . . . . . 162 xiii Abstract This study is aimed at developing new experimental equipment which would permit the study of multi-scale and vertically-variable oceanographic flows using a system called Hydraulic-Control Wave-maker (HCW). Both the inlet and outlet flume boundaries are composed of an adjustable set of vertical baffles. Each baffle is connected to an individual flow control system, such that the vertical distribution of flow is entirely controllable. In such a system, a main advantage is that any arbitrary flow can be reasonably created. Firstly, a new method of wave generation by using HCW was tested through numerical analysis. The optimized design of HCW is suggested through sensitivity analyses with respect to baffle length, height, number and position. For verification, the numerical results from the optimized HCW were compared with analytical solutions. In the case of a relative high wave amplitude, the phase lagged behind the analytical solution, but most of the results are in good agreement with the analytical solutions. Furthermore, wave absorption techniques were performed by solitary wave generation. Although the results of wave absorption cannot perfectly remove the reflected wave, the technique may be feasible because the lack of the reflected wave signal is a reasonable indicator. Secondly, a small physical model as a prototype with one baffle was developed and tested. In order to create a wave, two calibrations between the motor speed and the flow rate were carried out, and linear relationships are obtained. Both calibrations show similar linear increase so that mass conservation is approved. By using this relationship, sinusoidal waves were generated. The time series of motor speed profiles are presented and the possibility of a HCW system can be confirmed. Thirdly, HCW with three baffles including a movable top baffle were built. The top baffle can be moved along a surface elevation by using a combination of screw-jack and motor, a kind of xiv linear actuator, similar to the flow control system. The time-varying volume of top baffle has been considered and Ramp period has been introduced for a smooth piston movement. For verification, solitary waves, sinusoidal waves and stokes waves were created. The time series of surface eleva- tions for laboratory observation were compared to the desired target data. The results are in good agreement with the target data in spite of the reflected waves. Yet, the results were overpredicted because the difference of two calibrations through the small prototype HCW was not considered. In cases of periodic relative long waves, sinusoidal and stokes wave with kh=0.5, shows unwanted harmonics are appeared and the results are not overpredicted. This is because the air is sucked into the top baffle through very small gaps of the top baffle hinge and it disturbed flow and mass con- servation of water. An active wave absorption by calculating an arrival time through wave celerity was carried out in the case of solitary wave. The results show the reflection coefficients are low and the active wave absorption would be a feasible way. The enhanced physical model with three paired cylinders has been built for higher wave generations than with three cylinders. By using the enhanced HCW, the corrected calibration was investigated. By the comparison between measured and desired solitary wave amplitude, the corrected calibration was obtained. The generation of long wave by the shoaling and breaking of the propagation of top-hat spectral waves, a transient focused wave group, over composite slopes as one of the applications of HCW. Moreover, the time series of water surface elevation and the amplitude spectra by the laboratory observations were compared with numerical simulations, OpenFOAM and COULWA VE, based on Boussinesq and RANS equations, respectively. The comparisons show that long wave gener- ation in the experiments is in close agreement with numerical simulations. The times series of water surface elevations shows that the short wave groups are transformed to long waves, and their propagation paths can be observed directly. In addition, the comparison of the amplitude spectra presents that the primary and superharmonic wave amplitudes are increased or decreased by wave shoaling or wave breaking, respectively. Long wave components obtained from low-pass filtered surface elevations by time-series measurement obviously describe that the increased amplitudes in shallow water by shoaling and their propagation path. The wave amplitude spectra measured and predicted are compared to the second-order wave theory in association with an interaction of xv wave groups. The comparisons of the theory at initial location and experimental data and numer- ical results at various cross shore locations present that a spatial redistribution of wave energy is obviously described during the shoaling and breaking processes. The wave amplitude spectra with low-frequency increase over nearshore regions because the wave amplitudes with low-frequency become increased in shallow water due to wave shoaling and the energy is transferred from the waves with higher frequency. The measurement for wave runup were performed with images captured by using the one action camera, food colorings, light diffuser materials low costs as well as edge detection function in MATLAB toolbox. In solitary waves with various amplitudes, the time series of inundations and wave runups measured are in close agreement with the predictions by numerical simulations, COULWA VE and OpenFOAM. The predictions of the maximum wave runups are in close agree- ment with the solid line which means perfect agreement with the experiments. In the cases of top-hat spectral waves, the time series of inundations and wave runups for all cases measured and predicted with COULWA VE and OpenFOAM have discrepancies because the wave conditions before wave runup are different as well. However, the tendencies of wave runups and rundowns are so close between the measurements and the predictions. In detail, the sequence of wave runups by the bound long waves can be observed. In addition, the time series of runups with high resolutions including wave runups and rundown with high frequencies can be observed. xvi 1 Introduction “When dealing with water first experiment then use judgement. ” – Leonardo da Vinci, (Price (1978)) 1.1 Motivation 71 percent of the earth’s surface is covered with the ocean. The ocean plays dominant roles in absorbing solar radiation and redistributing heat and salt. In order to perform those important roles, the ocean currents continuously move (Fig. 1.1). Some ocean currents are generated not only by local wind but also by gravity wave. In the thermocline layer in the ocean, internal waves propagate due to density variations. The ocean has a complex combination of wave and flow. Figure 1.1: Major ocean currents of the world (Pidwirny (2006)) 1 Studies of complex oceanographic flows, such as those governed by nonlinear and multi-scale physics, have been done in relevant experimental studies to date. For the study of wavy oceano- graphic flows, most studies have used traditional techniques of wave generation with solid bound- ary movement, a moving wall boundary. The movement is based on the velocity profile under the wave to be created. For a long wave generation, for instance, the vertical wave-maker is commonly driven by a piston. For deep water waves, a hinge of paddle mounted on tank bottom mimics a orbital particle motion decays with water depth. When the wave-maker shape does not perfectly match the vertical kinematic profile of the targeted wave, evanescent modes result. In cases of poor wave-maker wave matching, spurious and undesirable free waves are generated. The wave-maker theories encompass dispersive and shallow water theory, and linear to weakly nonlinear waves. In general, these wave-maker theories are well established (Dalrymple and Dean (1991)). In the field of internal wave laboratory studies, a wide range of generation approaches has been used, and are similar to the free surface wave studies. The most common types are hinged flap and plunger wave-makers. The flap type wave-maker has been studied (Davis and Acrivos (1967); Thorpe (1978); Wallace and Wilkinson (1988)). A plunger type wave-maker, where a solid object vertically oscillates near the interface, also has been researched (Kamachia and Honjia (1988); Ostrovsky and Zaborskikh (1996); Troy and Koseff (2005)). Wave-makers for natural internal wave generation have been introduced, such as forcing a stratified current over a sill (Baines (1995)) and a vertically-segmented, dual-plunge wave-maker (Koop and Redekopp (1981)) and dual-piston wave-maker (Wessels and Hutter (1996)). These wave-makers can successfully generate internal waves by targeting a narrow frequency range. The equipment is designed to isolate a particular element of an interesting physical question; multi- scale and nonlinear wave interactions are often too complex a problem to tackle with these existing laboratory devices. Experimentally, studies with respect to wave-current interaction have been performed by a tra- ditional wave generator coupled with a pump-driven current generator. These studies have had difficulty in generating a steady current and a wave train in tandem. Several authors have consid- ered wave-current interaction in the near bed (Brevik and Bjørn (1979); Brevik (1980); Kemp and 2 Simons (1982); Umeyama (2005); Kemp and Simons (2006)). For distributing adverse current to the direction of wave propagation in whole water depth, Thomas (1981) added a rotataing louvre blade. Swan et al. (2001) used a complicated current-apparatus to generate depth-varying currents and wave, simultaneously. The true complexity of oceanographic flows can only be observed in the field, but, of course, field experiments suffer from the enormity of scales that must be covered. Most field instruments can capture good time resolution data at a limited number of points. To map a flow field, for instance, velocity profiles can be collected along a network of tracks using a bottom-tracking Acoustic Doppler Current Profile (ADCP). The ADCP records a short-term average velocity in several bins below the towing platform. Though each instantaneous profile has high resolution, the data along a track is not synoptic, and the tracks are very sparse when a large region must be mapped, such as an inlet or headland. Turbulence data can be collected at individual points using Acoustic Doppler Velocimeters (ADV), and these can be moored at important locations to get a complete view of the velocity structure of the water column at a point. Yet, to understand the spa- tial transformation of directional wave spectrum, multiple moorings are required at high expense. Moreover, all of these acoustic methods are strictly limited to weakly stratified flows and none give an accurate measure of the turbulent mixing over large scales. Conductivity, Temperature and Depth (CTD) profilers equipped with fluorometers can be used to track a natural or injected dye tracer to better understand mixing (Chadwick and Largier (1999)). However, these results are still limited to non-synoptic profile data. To capture the dynamics of an internal breaking wave, laboratory experiments are still required (Troy and Koseff (2005)). 1.2 Objective This study is aimed at developing new experimental equipment which would permit the study of multi-scale and vertically-variable oceanographic flows using a system called Hydraulic-Control Wave-maker (HCW). An example of three baffles by the HCW is presented as shown in the Fig. 1.2. Both the inlet and outlet flume boundaries are composed of an adjustable set of vertical baffles. 3 Each baffle is connected to an individual flow control system, such that the vertical distribution of flow is entirely controllable. In such a system, any arbitrary flow can be reasonably created, for a long and short wave generation, for instance, each flow control system can make each target flow at each position to mimic the shape of a quadratic and parabolic velocity profile, respectively. If different sets of baffles can be connected to different reservoirs, an internal wave with multi-phase profiles can be created. If, furthermore, pairs of cylinders which work in tandem are created, the problem of a finite input volume can be solved. Thus, the HCW system can be applied to experi- mental studies due to a wide spectrum of waves. In this study, the HCW system is developed, and then a wave generation is verified. Focused wave groups are generated over a bi-linear slope beach and long wave generation by interactions of the wave groups and the process of wave shoaling and breaking are investigated. Figure 1.2: General schematic of the HCW 1.3 Organization In this dissertation, a new experimental device which would permit the study of multi-scale and vertically-variable oceanographic flows using a system called the Hydraulic-Control Wave-maker (HCW) is developed and the performance of HCW system is verified by comparing theoretical 4 solution. Numerical analyses to verify the method of HCW, such as its ability to generate waves are carried out in Chapter 2. For the verification, the time series of wave height and velocity profile are compared with those of a theoretical solution. A wave absorption of solitary wave is performed. A small scale physical model of the HCW as a prototype is built and its operation is detailed in Chapter 3. Calibration between motor speed and flow rate via a cylinder is performed. This calibration is verified by comparing flow data to calculation of cylinder volume and screw-jack movement. Snapshots of generating sinusoidal wave as well as propagating wave to a slope beach are shown. Moreover, histories of actual velocity for generating sinusoidal waves are shown as well. A physical experiment with two baffles is described in detail in Chapter 4. Solitary waves and sinusoidal waves are generated and those results from wave gauges are compared with target data for verification. Chapter 5 describes an enhanced physical model of HCW including three baffles, a movable top baffle and three paired cylinders. In addition, a variety of waves such as solitary waves with various wave amplitudes and sinusoidal waves and Stokes waves with various wave amplitude and length are verified. Chapter 6 presents measurement precision and uncertainty, and investigates how those impact output data. Wave group interactions over a bi-linear slope beach are investigated by generating top-hat spectral waves in Chapter 7. Experimental data are compared to numerical results by OpenFOAM and COULWA VE and second order analytical solutions for verification. Chapter 8 explains a measurement technique for obtaining a time series of shoreline movement by an action camera and shows the time series of shoreline movement and runup by generating solitary waves and top-hat spectral waves. Future research is discussed in Chapter 8. 5 2 Numerical Simulation for Design of HCW 2.1 Introduction of OpenFOAM R The OpenFOAM (Open Source field Operation and Manipulation) is an open-source Compu- tational Fluid Dynamics (CFD) software, and consists of C++ libraries and codes that is used for creating applications. Applications includes solvers or utilities. Solvers are each designed to solve a specific problem in continuum mechanics, and utilities are designed to perform tasks that involve data manipulation (OpenFOAM (2014)). The package distribution enables us to use numerous solvers and utilities and to write our own solver that is suitable for our desired problem, but a solid knowledge of physics and programming is needed. In this study, the interFoam solver is used,which is the solver for multiphase problems with incompressible fluids. 2.1.1 Governing Equations Navier-Stokes equation for an incompressible flow and constant viscosity is: Du Dt ¶u i ¶t + u j ¶u i ¶x j = 1 r ¶ p ¶x i +n ¶ 2 u i ¶x j 2 (2.1) where x i ; j(i: j= 1;2;3) are Cartesian coordinates, u i are the Cartesian components of the velocity, while t represents the time, r is the density of the fluid, p is the pressure, g is the acceleration of gravity, andn is the fluid dynamic viscosity. The continuity equation for an incompressible flow must also be satisfied: ¶u j ¶x j = 0 (2.2) 6 The Reynolds Averaged Navier-Stokes (RANS) equations are time-averaged equations used to analyze turbulent flows. The continuity (Eq. (2.3)) and momentum (Eq. (2.4)) equations for an incompressible fluid can be written, using Einstein summation convention as: ¶u j ¶x j = 0 (2.3) ¶u i ¶t + u j ¶u i ¶x j = 1 r ¶ ¶x j pd i j m ¶u i ¶x j + ¶u j ¶x i +ru 0 i u 0 j (2.4) where u j is the time-averaged velocity andd i j is Kronecker delta function. The relationship of the turbulent-viscosity hypotheses to analyze the stress-rate-of-strain rela- tion for a Newtonian fluid is u 0 i u 0 j =n T ¶u i ¶x j + ¶u j ¶x i 2 3 kd i j (2.5) wheren T is the turbulent viscosity. The turbulent-viscosity hypothesis substituted into the momen- tum of RANS (Eq. (2.4)) is ¶u i ¶t + u j ¶u i ¶x j = ¶ ¶x j n e f f ¶u i ¶x j + ¶u j ¶x i 1 r ¶ ¶x i p+ 2 3 rk (2.6) where n e f f is the effective viscosity, which takes into account the summation of the molecular viscosity (n) and the turbulent viscosity (n t ), and k is turbulent kinetic energy 2.1.2 The ke Model The two-equation k-e turbulence model is one of the most common turbulence models. The equation solves the turbulent kinetic energy (k) and the dissipation rate of the turbulent kinetic energy (e) to represent turbulent properties such as convection and diffusion of turbulent energy. The model is widely used for most types of engineering applications. The modeled transport equations for k ande are: ¶k ¶t + u j ¶k ¶x j = ¶ ¶x j n+ n t s k ¶k ¶x j +n t ¶u i ¶x j + ¶u j ¶x i ¶u i ¶x j e (2.7) 7 ¶e ¶t + u j ¶e ¶x j = ¶ ¶x j n t s e ¶e ¶x j +C e1 e k n t ¶u i ¶x j + ¶u j ¶x i ¶u i ¶x j C e2 e 2 k (2.8) where turbulent viscosity is given byn t = C m k 2 =e, ands k ands are the turbulent Prandtl number for k and e, and C m , C e1 and C e2 are model constants. The standard ke model constants in the model equations are: s k = 1:0;s= 1:3;C m = 0:09;C e1 = 1:44;C e2 = 1:92 (2.9) 2.1.3 Volume of Fluid Method In OpenFOAM, the VOF (V olume of Fluid) method is used for tracking interface movement between air and water. The equation determines volume fraction, but the sharpness interface can possibly become smeared due to false diffusion (Versteeg and Malalasekera (2007)). In Open- FOAM, an extra term called artificial compression is introduced into the phase fraction equation: ¶a ¶t +Ñ(aU)+Ñ(a(1a)U r )= 0 (2.10) where U is the velocity field composed of u, v and w, and al pha is the phase fraction of water and air. Al pha= 1 and al pha= 0 mean that a cell is full of water and air, respectively. The artificial compression velocity in OpenFOAM can be controlled by cAl pha. For no compression velocity, cAl pha set to zero value, and higher values than zero mean that an artificial velocity at the interface is applied. The phase fraction a determines the density of mixture in the Navier-Stokes equations. The equation is as follows: r =ar w +(1a)r a (2.11) wherer w andr a represent the density of water and air, respectively. 8 2.2 Wave Generation In the HCW system, all waves are generated by flow velocity via each baffle. Before generating waves, it is important that a horizontal particle velocity profile of those waves is known. First, horizontal particle velocities are averaged over each baffle height. The averaged velocities are imposed at each baffle as inlet condition. Then, the vertical flow distribution mimics the horizontal particle velocity of wave propagation. 2.2.1 Linear Wave Theory The well-known linear wave theory in engineering applications is Airy wave theory. Nonlinear free surface boundary conditions can be linearized to apply the Taylor series expansion and this is called small amplitude wave theory, because the approximation is valid for small wave steepness (Dalrymple and Dean (1991)). The free surface elevation (h), horizontal particle velocity (u) and vertical particle velocity (w) for the linear wave theory are as follows: h = asin(kxst) (2.12) u= gak s cosh(h+ z) coshkh cos(kxst) (2.13) w=gak sinh(h+ z) coshkh sin(kxst) (2.14) where a is wave amplitude and k= 2p=L and s = 2p=T , and L and T are wave length and wave period, respectively. h is water depth and z is vertical coordinate. The integration of the horizontal velocity profile for 3 baffles in order to generate a linear wave is described in Appendix A.0.1. 9 2.2.2 Solitary Wave Theory The solitary wave consists of a single wave elevation, and theoretically has infinite wave length. The characteristics of a solitary wave are theoretically derived by Russell’s experimental research (Russell (1838, 1845)). Theoretical analyses for a solitary wave over a flat bottom have studied (Boussinesq (1872); McCowan (1891); Munk (1949)). In this study, the free surface elevation (h), horizontal particle velocity (u) and vertical particle velocity (w) for solitary wave theory are adapted from Daily and Stephan (1952). h = asech 2 r 3a 4h 3 (x ct);c 2 = gz(1+ a z ) (2.15) u= r g h h h 2 4h + h 2 3 z 2 2 ¶ 2 h ¶x 2 (2.16) w=z r g h 1 h 2h ¶h ¶x + h 2 3 + z 2 2 ¶ 3 h ¶x 3 (2.17) where c is wave celerity and z is vertical coordinate. The integration of the horizontal velocity profile for 3 baffles in order to generate a linear wave is described in Appendix A.0.3. 2.2.3 Stokes Second Order Theory Stokes theory is a well known nonlinear wave theory, and has various names according to the degree of non-linearity. The fifth order of Stokes theory proposed by Skjelberia and Hendrickson (1961) exists, but the second order of Stokes theory is implemented in this study. The equations of the free surface elevation (h), horizontal particle velocity (u) and vertical particle velocity (w) for Stokes second order wave theory are adapted from Dalrymple and Dean (1991). h = H 2 cos(kxst)+ H 2 k 16 coshkh sinh 3 kh (2+ cosh2kh)cos2(kxst) (2.18) 10 u= gHk 2 coshk(h+ z) coshkh cos(kxst)+ 3H 2 sk 16 cosh2k(h+ z) sinh 4 kh cos2(kxst) (2.19) w= gHk 2 sinhk(h+ z) coshkh sin(kxst)+ 3H 2 sk 16 sinh2k(h+ z) sinh 4 kh sin2(kxst) (2.20) where H = 2a is wave height. The integration of the horizontal velocity profile for 3 baffles in order to generate a nonlinear wave (Stokes second wave theory) is described in Appendix A.0.2. Dalrymple and Dean (1991) have noted that the Stokes expansion is not very good in the second order for high wave in shallow water (shallow water and deep water are defined as h=L< 1=20 and h=L 1=2) and have proposed that the Ursell parameter, U r is subject to: U r = L 2 H h 3 < 8p 2 3 (2.21) for Stokes second order theory to be valid. 2.3 Sensitivity Analysis Numerical analysis by OpenFOAM is implemented to verify the HCW technique. Firstly, sensitivity analysis in terms of baffle vertical height, baffle horizontal length, the number of baffles and baffle position is performed. Fig. 2.1 shows the definition of baffle length and height and the difference of baffle number and position. For all sensitivity analysis, the time series of water surface by numerical simulation are compared with linear wave theory as the analytical solution. Baffle height, vertical length of baffle, increases from 0.01 to 0.03m. Fig. 2.2 presents the sensitivity analysis for vertical length of baffle. The results shows baffle height does not affect the results. For the sensitivity analysis for baffle length, baffle length increases from 0.05m 0.3m. Fig. 2.3 presents the sensitivity analysis with respect to baffle length, horizontal length of baffle. The results shows the longer baffle length is, the longer the phase lag is. 11 Figure 2.1: Schematic drawing in terms of baffle height, length and position and the number of baffles Figure 2.2: Comparison of the time series of water surface with different baffle heights (dash line: theory; dash dot line: 0.01m; dot line: 0.02m; solid line: 0.03m) For an optimized design, baffle length should not be relatively long. For the sensitivity analysis about baffle number and position, baffles are regularly positioned at the inlet boundary in 3 inlets and 5 inlets, and only the number of baffles is different. In the case of 6 inlets, baffles are irregularly 12 Figure 2.3: Comparison of the time series of water surface with different baffle lengths (dash line: theory; dash dot line: 0.05m; green solid line: 0.1m; dot line: 0.2m; cyan solid line: 0.3m) positioned at inlet boundary, and two upper baffles are located between the wave crest and trough of the target wave. As may be intuitively expected, results with increased number of baffles show agreement closer to the analytical solution as seen in the Fig. 2.4. Additionally, when more baffles are located between the wave crest and trough, the better the agreement is. Figure 2.4: Comparison of the time series of water surface with different baffle number and posi- tion (dash line: theory; dot line: 3 inlets; dash dot line: 5 inlets; solid line: 6 inlets) 13 2.4 Numerical Results and Analysis 2.4.1 Optimized Design Figure 2.5: Schematic optimized design The design of an optimized baffle is suggested by the sensitivity analysis as shown in Fig. 2.5. Baffle height is not considered because it does not affect the result. Baffle length is 0.05m to avoid phase lag while waves with various amplitudes are being created. 10 baffles are positioned over the wave amplitude range of a/h=0.05, and 2 baffles are placed under wave trough. To verify this design, a relatively long wave (kh=0.1) and short wave (kh=1) with a/h=0.01 and 0.05 are simulated, and then wave surface elevations and horizontal and vertical velocity profiles near wave crest and trough are compared with an analytical solutions as shown in Fig. 2.6 to 2.9. The results of the short wave are in good agreement with the analytical solution as seen in Fig. 2.6 and 2.7. The results of the long wave show phase differences as shown in Fig. 2.8 and 2.9. Wave crest and trough points in the numerical results lag behind those points in the analytical solution, and the comparison of vertical velocity profile shows almost the opposite direction to analytical solution. This implies that a sinusoidal wave with high wave amplitude will have more phase lag than a 14 wave with small amplitude. In this case, the fluid can be more affected by the baffle boundary than the case of the small wave amplitude because its speed is faster than the case of the small wave amplitude. Especially, the phase lag can obviously be shown in the case of the short wave length as seen in Fig. 2.9. Figure 2.6: Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.01, kh=0.1) 15 Figure 2.7: Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.01, kh=1) Figure 2.8: Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.05, kh=0.1) 16 Figure 2.9: Comparison of the time series of water surface and horizontal and vertical velocity profile with analytical solution (a/h=0.05, kh=1) 17 2.4.2 Wave Absorption Two techniques for wave absorption by the HCW system are discussed in this section. To analyze wave reflection against an outlet boundary, a solitary wave with single wave amplitude is generated from the inlet boundary. Fig. 2.10 illustrates an approach to avoid wave reflection without using wave celerity. First, to absorb the wave along the outgoing boundary, horizontal velocity profiles near the end of the wave flume are extracted. Experimentally, this would be done in real time with an ADV (Acoustic Doppler Velocimeter), and averaged over a short time window. Then, the averaged velocities are applied along the outlet boundary, essentially an experimental radiation boundary condition. Figure 2.10: Schematic drawing for wave absorption without wave celerity The Fig. 2.11 shows the time series of solitary wave height at 10m from the wave-maker when the wave amplitude ratio is 0.05 and 0.3. In this case, the reflected wave, if the back end of the flume was a vertical wall, would arrive at the time series location at around 35 second. The lack of this signal in the time series is a reasonable indicator that this absorption approach may be feasible. The Fig. 2.12 illustrates another method to avoid wave reflection by using wave celerity. First, to absorb the wave along the outgoing boundary, horizontal velocity profiles at any point of the 18 Figure 2.11: Time series of water surface elevation of solitary wave with a/h=0.05 (left) and 0.3 (right) flume are extracted, and the arrival time from extracted point to outlet boundary should be cal- culated by using wave celerity, c (Eq. 2.10). Then, the averaged velocities are applied along the outlet boundary. Figure 2.12: Schematic drawing for wave absorption with wave celerity The Fig. 2.13 shows the time series of solitary wave height at 10m when the wave amplitude ratio is 0.05 and 0.3. In this case, the reflected wave, if the back end of the flume was a vertical wall, would arrive at the time series location at 32 seconds. Although the reflected wave signal is 19 a little more unstable than the prior approach, this method can be allowed to gain more time for extracting data and applying the averaged data than the prior approach. Figure 2.13: Time series of water surface elevation of solitary wave with of a/h=0.05 (left) and 0.3 (right) with using wave celerity 2.5 Conclusions This study is aimed to develop the experimental Hydraulic Control Wave-make (HCW) system for creating various kinds of waves. A new method of wave generation by using HCW was first tested through numerical analysis. The optimized design of HCW is suggested that through sensi- tivity analyses with respect to baffle length, height, number and position. The baffle length should not be relatively long to avoid phase lag. The more is the baffle number between the desired wave amplitude, the more exact wave can be generated. For verification, the numerical results from the optimized HCW were compared with analytical solutions. In the case of the relative high wave amplitude, the phase lagged behind the analytical solution, but most of the results are in good agreement with the analytical solutions. Furthermore, wave absorption techniques were performed by solitary wave generation. Although the results of wave absorption cannot perfectly remove the reflected wave, the technique may be feasible because the lack of the reflected wave signal is a reasonable indicator. 20 3 Small-Scale Prototype 3.1 Experimental Design and Methodology A flow control system is developed by using piston action on a cylinder with a motor and screw- jack as shown in Fig. 3.1. The motor is connected to the screw-jack by shaft coupling and the plate end of screw-jack is connected to the piston. The screw-jack is functioned for changing a rotary motion of the motor to a translation of the piston. Two O-rings on the piston are used for sealing. A small wave flume (150cm of length, 5cm of width and 10cm of height) with an impermeable slope beach is connected to the cylinder (5.25 inch of inner diameter and 12 inch of height) via 1 inch diameter of PVC pipe connection. The wave flume and the cylinder consist of plexiglass. In this system, a servo motor is used because it is excellent in applications requiring high torque at high speed and high dynamic response. For controlling the servo motor, a NI USB 6009 DAQ (Data AcQuisition) system is used, which can allow analog input/output and digital input/output. The RPM of the motor and rotation direction is controlled by an analog signal (0-5V) and digital signal from the DAQ via Labview software, respectively. The block diagram of Labview for controlling a servo-motor is described in Appendix B.0.1. An air-release valve installed on the top side of the cylinder because trapped air in cylinder adds an extra force to the piston and do not allow inflow and outflow of an exact flow rate. 3.2 Calibration between Flow Rate and Motor Speed HCW system is operated to mimic a velocity profile of desired wave. To generate a wave, the flow rate should be known according to motor speed because the flow control system is controlled by the displacement of screw-jack according to motor speed. For solid calibration between the 21 Figure 3.1: Schematic design of prototype (bottom) and physical model (top) motor speed and flow rate, two calibrations are implemented. One measures flow rate according to an inline flow meter, and the other measures the movement of the piston in certain time. The flow rate for the piston is calculated by multiplying the cylinder area and velocity of piston movement. Fig. 3.2 shows a comparison between the flow rate for piston movement and the flow meter. In the plot, red triangular and blue diamond notes the results of flow meter and the calculation by piston movement, respectively. Both results are similar and show a linear increase, satisfying mass conservation. Yet, the results by flow meter are a little higher than the calculation by piston movement after 3000RPM. For convenience, three decimal digits of both slopes between RPM and flow rate are taken to make them a same slope value. However, the calibration affects wave amplitudes in experiment is bigger than in target data if it is applied to a bigger scale experiment. 22 Although the difference between calibrations by the piston movement and by the flow meter is very small in the small scale, it becomes large in a larger scale. To find a corrected calibration between flow rate and RPM, the section 5.5.2 will be described. Figure 3.2: Comparison between flow rate for piston and flow meter 3.3 Experimental Results Using the small physical model, sinusoidal waves with a/h=0.3 and kh=1 and kh=0.1 are cre- ated. Fig. 3.3 presents snapshots of sinusoidal wave propagation on the slope beach while the wave is generated by the HCW small physical model, and Fig. 3.4 shows the times series of motor speed. In these snapshots, it is hard to see exact sinusoidal wave forms because the flume length is too short to avoid the reflected wave. The time series of motor speed shows similar sinusoidal wave forms. This means that the motor is accelerated and decelerated properly to reach the targeted sinusoidal wave form. 23 Figure 3.3: The snapshot of sinusoidal wave with a/h=0.3, kh=1 Figure 3.4: Time series of RPM in sinusoidal wave with a/h=0.3, kh=1 Fig. 3.5 presents snapshots of sinusoidal wave propagation and run-up on the slope beach while the wave is generated by the HCW small physical model, and Fig. 3.6 shows the times series of motor speed. In these snapshots, it is hard to capture the sinusoidal wave form with at least one wave period because the flume length is too short compared to wave length generated. The time series of motor speed is similar the sinusoidal wave form with a relatively long wave length. The possibility of the HCW system can be also confirmed. A problem that air and water are pulled into the baffle is found in the case of the long wave generation. This is because baffle length is short 24 and the baffle connection with the cylinder is higher than water surface elevation. Design of baffle to prevent pulling in air will be shown in Chapter 4.1.1. Figure 3.5: The snapshot of sinusoidal wave with a/h=0.3, kh=0.1 Figure 3.6: Time series of RPM in sinusoidal wave with a/h=0.3, kh=0.1 Although a sinusoidal wave with at most a/h=0.013 is generated by using the small cylinder and the motor, the small HCW system is transferred to a larger flume(40 cm of width, 60 cm of height and 12m of length). Fig. 3.7 and 3.8 show the snapshot and the time series of motor speed for generating sinusoidal wave with a/h=0.013 and kh=1. Although it may be hard to see a sinusoidal 25 wave with such a small wave amplitude in the snapshot, a smoother sinusoidal wave form than the small flume can be observed. The time series of motor speed shows a sinusoidal wave form and the maximum and minimum RPM almost reach the maximum speed of motor, 5500RPM. Figure 3.7: The snapshot of sinusoidal wave with a/h=0.013, kh=1 using big flume Figure 3.8: Time series of RPM in sinusoidal wave with a/h=0.013, kh=1 using big flume 3.4 Conclusions A small physical model with one baffle was developed and tested. In order to create a wave, two calibrations between the motor speed and the flow rate were carried out, and linear relationships 26 are obtained. Both calibrations show similar linear increase so that mass conservation is approved. By using this relationship, sinusoidal waves in the small wave flume were generated and presented in the snapshots. The time series of motor speed profiles are presented and the possibility of a HCW system can be confirmed. 27 4 Physical Model with 2 Baffles 4.1 Experiment Equipment and Procedures This section describes the HCW system in detail, covering the mechanical aspects between piston transitional and motor rotational movement as well as the control procedure of a servo motor by a DAQ (Data AcQuisition) system. The precise calibration system for a wave gauge, which is being developed, is introduced as well. 4.1.1 General Principle of Operation for Wave Generation The large scale physical model with 2 baffles is set up as shown in Fig. 4.1. Each flow control system (the individual cylinders) is developed by using a piston in a cylinder and the same principle of operation is same as the small scale prototype in chapter 3. Yet, most of the components scale up dimension or size. The cylinder has a 15 inch inner diameter and 12 inch height, and the piston diameter fits the inner diameter of the cylinder. To move the big piston with high acceleration, the motor is changed to permit higher torque than for the prototype. However, the screw-jack is the same as for the prototype, so that calibration results between the motor speed and flow rate can be valid for the large physical scale model. Each motor is connected to a screw-jack by shaft coupling and the plate end of the screw-jack is connected to the piston. The screw-jack function is for changing the rotary motion of the motor to the translation of the piston. Two O-rings on the piston are used for sealing. A large wave flume with 1/7 of an impermeable slope beach is connected to the cylinder via 3 inch diameter of PVC pipe connection. If the PVC pipe diameter becomes large, the applied force to the piston is lessened due to decreased head loss. In this system, a servo motor is also used because it is excellent in applications requiring high torque at high speed and high dynamic response. For controlling the servo motor, a NI 9477 32-channel 5 V to 60 V with sinking 28 digital output module and a NI 9264 16-channel 10V with 16-Bit analog output module are used, which can allow analog input/output and digital input/output. Additionally, these digital output and analog output module are slotted into a compact DAQ-9178 chassis with 8 slot USB. The RPM of the motor and rotation direction is controlled by analog signal (-10 to +10V), and on/off and stop are controlled by a digital signal from the DAQ via Labview software, respectively. A block-diagram of Labview for controlling two motors for generating wave is described in Appendix B.0.4. The motor speed is obtained by analog input signal (-10 to +10V), and the signal is collected by NI 9205 32-channel 10 V with 16-bit analog input module. Two air-release valves are installed on the top of each piston. Figure 4.1: Physical model with 3 baffles Fig. 4.2 shows a design of 3 baffles prevent pulling in air if water surface elevation is lower than baffle mouth, as mentioned in Chapter 3. The baffle mouth to the cylinder is lower than baffle 29 outlet to the wave flume so that only water can flow in and out baffle. 5 meshes in each baffle are used for making flow uniform. These will be described in Chapter 4.1.4 in detail. Figure 4.2: Design of 3 baffles 4.1.2 Calibration System for Capacity Wave Gauge The capacitance wave gauges are nearly perfect linear instruments. A calibration of wave height to output voltage can be performed by the variation of output voltage when the wave gauge probe is lowered or raised by a known distance in still water level. The calibration system for the capacitance wave gauge by a stepper motor with a lead screw is built as seen in the Fig. 4.3. The stepper motor with the lead screw is operated as same as a linear actuator, but the device has more competitive price than a linear actuator. The stepper is suitable for linear motion because the 30 motor can be rotated with a regular step. The motor is controlled by a pulse from the digital output module of DAQ via Labview software. A block-diagram is described in Appendix B.0.5 in detail. Figure 4.3: The calibration system for the capacity wave gauge 4.1.3 Motor Synchronization In the HCW system with multiple baffles, it is essential that multiple cylinders should simul- taneously push and pull in sync to create the vertical velocity distribution for generating a desired wave. To operate multiple cylinders at the same time, multiple motors must be synchronized. To analyze motor synchronization, the same signal is sent into 2 motors via drives, and then actual RPMs from two motors are received. The Fig. 4.4 shows the comparison between motor synchro- nization before and after. The plot on the top presents the histories of RPM of two motors that 31 have time lag for operation. To synchronize two motors, a same environment of 2 drives is set, such as motion profile and some parameters to operating motors. In addition, a same length of wires between drives and DAQ system is also connected. The plot on the bottom shows that the two motors can be synchronized. Figure 4.4: Comparison between motor synchronization before and after 4.1.4 Straighten Flow in Baffles If the area of a baffle is the same as an outlet from a cylinder and the piping system is with- out major transitions, uniform flow may be transferred to an outlet of baffle without any change. However, it is hard to build the connection system, in practice. A PVC pipe with 3 inch diameter is connected to both the cylinder and a baffle mouth. An area of a baffle becomes gradually wider so that a velocity from the baffle can be distributed to the outlet of baffle. Yet, the shape creates a disordered flow in the baffle due to area change. A flow pattern in the baffle is visualized by 32 dye injection when sinusoidal wave with short and long wave length (kh=1.0 and kh=0.5) is gen- erated as shown in Fig. 4.5. In the case of the sinusoidal wave with short wave length (kh=1), the flow pattern is straightened, whereas a big eddy is created in the case of it with relative long wave length (kh=0.5). To make flow straight, a plastic mesh is used in the baffle as seen in Fig. 4.6. The mesh opening size is chosen to be as small as possible, without the motor being overloaded. Additionally, the number of meshes is determined such that the eddy disappears as well. Figure 4.5: Snapshots of flow pattern in baffle by sinusoidal wave with kh=1 (top) and kh=0.5 (bottom) (blue arrow: flow direction) 33 Figure 4.6: Snapshots of flow pattern in baffle by sinusoidal wave with kh=0.5 with meshes (blue arrow: flow direction) 34 4.2 Verification of Wave Generation In this section, HCW system with 2 baffles and 2 cylinders with respect to wave generation is verified. The still water depth is 0.33m during the experiment at the baffle position. Two wave gauges are placed over a flat bottom to measure water surface elevation. Data from wave gauges with a sampling frequency of 100Hz are collected and compared to desired target data for verifica- tion. Solitary waves and sinusoidal waves are created and are compared with their target data for verification in next sections. 4.2.1 Solitary Wave Solitary waves with various wave amplitudes are generated to verify capability of solitary wave generation. The 4.7 shows solitary wave with a/h=0.08, the maximum wave amplitude, that the HCW system with 2 baffles and 2 cylinders can create. The times series of water surface elevation from two wave gauges which are positioned at 3.2m and 4.6m from the end of baffle and the histories of actual speed of two motors are presented, as shown in Fig. 4.7. The histories of RPM shows two motors are synchronized and operated to mimic the solitary wave well. The time series of water surface elevations from two wave gauges also show the typical solitary wave foam excepting the data after 11 second by reflected waves. By comparing between the wave amplitude from wave gauges and it of analytical solution, the data from wave gauges (2.80cm) are overestimated than it of desired data (2.64cm). As mentioned in the section 3.2, the difference between calibrations by the flow meter and the piston movement is not taken into account. The section 5.5.2 will show new results of solitary wave generation by using a corrected calibration data. However, the times series of wave amplitude shows solitary wave form is generated well except for an effect of reflected wave after 11 second. 35 Figure 4.7: Time series of RPM (top) and water surface elevation (middle) and comparison between the result and analytical solution (bottom) (a/h=0.08, h=33cm) 36 4.2.2 Sinusoidal Wave Capability of sinusoidal wave generation with various wave amplitude and length has being verified. Fig. 4.8 and 4.9 show the results in terms of the times series of wave amplitude and the histories of motor speed when the relative sinusoidal high waves amplitude (a/h=0.08) with kh=1 and kh=0.5, respectively. Both results presents the wave amplitude in wave crest is higher than it in wave trough as shown in the top plots. The bottom plot shows negative peak motor RPM for piston downward movement is unstable because it means the applied force for piston upward movement is stronger than it for piston downward movement. The reason is not only the cylinder is located higher than still water level as shown in Fig. 4.1 but also piston weight is only added when the piston is moved upward. To resolve this issue, the cylinder is moved to a lower position almost same level as the still water level or a little bit higher than it. Additionally, pipe diameter for connecting between the cylinders and the baffles is increased due to reduce a head loss. Figure 4.8: Time series of water surface elevation (top), RPM (middle) and enlarged RPM (bottom) (a/h=0.08, kh=1) 37 Figure 4.9: Time series of water surface elevation (top), RPM (middle) and enlarged RPM (bottom) (a/h=0.08, kh=0.5) 38 In the case of the relative long wave in Fig. 4.9, a non-linear wave pattern is regularly shown. Some wiggle trends in the peak of wave crest are observed in the time series of wave amplitude in the relative long wave case as seen in the top Fig. 4.9. When water is pulled into cylinder to generate relative wave trough and then water is pushed to the baffle, water can be bumped to baffle wall. To solve the problem, a design of baffle is suggested to prevent that water is bumped as shown in Fig. 4.10. The design is extended to an operation with 3 baffles and 3 cylinders and has a movable top baffle by using a screw-jack and a motor, which is same concept as the flow control system of HCW. A movable top baffle prevents air from sucking into an empty space because the top baffle moves to follow water surface elevation. Moreover, the design is no need to block an above top baffle to avoid an overflow of wave over the top baffle. Figure 4.10: Comparison between the current and a new baffle design (left:current design, right:new design 4.3 Conclusions The scaled-up HCW system with two outflow baffles into the flume is developed. Before generating a wave by the system, the synchronization of multiple motors and the non-disordered flow in baffles are confirmed. For verifying wave generation, a solitary wave is generated and its results with respect to the histories of motor speed and the times series of wave amplitude 39 from the wave gauge is presented and verified. However, the experimental data is overestimated by comparing it to target data because the calibration results from the small prototype are not precisely considered. For verifying a sinusoidal wave, there are still some issues. First, the applied load to motor for the upward piston movement is higher than for the downward piston movement. Second, the non-linear wave pattern appears in the long wave generation because wave is bumped by the baffle wall in the case of long wave generation. For the first problem, the cylinder is moved to a lower position to decrease the hydraulic head and a large PVC pipe is replaced to reduce a head loss in chapter 5. For the last problem, a design of baffle with a movable top baffle is suggested to prevent wave bumping to the baffle wall. 40 5 Physical Model with 3 Baffles Figure 5.1: HCW system with 3 baffles (left) and a movable top baffle (right) Fig. 5.1 shows a physical model with 3 baffles including a movable baffle which compensates the shortcomings of the physical model with two baffles and the extended model. As mentioned in the chapter 4, three cylinders are placed to a lower position than still water depth in the wave flume and outflows via three baffles are provided from three cylinders to create waves. The top baffle moves up and down to match water surface elevation and it is controlled by a coupled mode of screw-jack and a motor. This coupled mode is same as the flow meter in HCW and its function is similar with a linear actuator. According to the movable top baffle, the volume inner top baffle takes into account a flow rate through the top baffle because of the time-varying volume inner top baffle. Next section will be described in detail. To remove disordered flow, filters should be installed in each baffle as described in the section 4.1.4. Tooth-type meshes are installed for the 41 movable top baffle and a flow pattern via the baffle is visualized by using a dye, as seen in Fig. 5.2. Snapshots of flow pattern in the movable top baffle by solitary wave with and without the tooth- type meshes are compared. The flow pattern shows a big eddy without the meshes, whereas the flow is nearly straight with the meshes by splitting the big eddy. In addition, without the meshes, free surface near the top baffle are wiggled because it is disturbed by the big eddy. Figure 5.2: Snapshots of flow pattern in the movable top baffle by solitary wave without tooth-type meshes (top) and with tooth-type meshes (bottom) 42 5.1 Movable Top Baffle To generate waves, velocity profiles are integrated over each baffle and flow rates per unit baffle width for each baffle are obtained. These flow rates are sent from cylinders as inflows of baffles and a velocity distribution from the outflows of baffle same as a desired profile can be created if baffles are fixed. In this study, a movable top baffle are applied to HCW system to remove the shortcomings of the fixed baffle system, such as the wiggle of free surface and the wave overflow over top baffle as described in the section 4.3. A velocity distribution from the outflows cannot mimic a target wave profile because the top baffle has time-varying volume. By introducing the time-varying volume of top baffle, the continuity equation is expressed: Q in = Q out + ¶V ¶t (5.1) where total volume, V = V 0 +DV , and V 0 is a initial volume as water level is still and DV is a time varying volume as water surface elevation is changed. Fig. 5.3 (a)-(c) show the total volume variation as the top baffle are moved to match wave crest, still water level and wave trough, respectively. The top baffle are moved to follow the time series of water surface elevation and the movement of screw-jack is calculated by the calibration between RPM and the movement of screw- jack as described in the section 3.2. 43 Figure 5.3: The time-varying volume of top baffle 44 5.2 Ramp Period If a motor suddenly starts to reach a desired high speed, the motor cannot perfectly reach the desired speed or the motor can be overloaded due to the requirement of high torque. To avoid the drawback, a ramp period is introduced into the beginning of RPM profile. It allows that RPM signal smoothly increase with ramp period, as shown in Fig. 5.4. As a result of an original time-series signal multiplied by a formula with positive ramp, a time-series of signal with smooth increase is obtained. Figure 5.4: Increasing signal with ramp period The functions with ramp period are defined as three times of wave period (T ) is applied to the time series of RPM for three cylinder operation and top baffle movement: New RPM signal (t)= 8 < : RPM signal (t) t 3T for t 3T RPM signal (t) for t > 3T (5.2) Top baffle movement (t)= 8 < : Surface elevation (t) t 3T for t 3T Surface elevation (t) for t > 3T (5.3) 45 5.3 Verification of Wave Generation Figure 5.5: Wave flume and instrumentation In the chapter 4, solitary and sinusoidal wave are created by using HCW with two baffles and cylinders and the time series of solitary wave surface elevation are compared to target data for verification. In this section, wave generation will be verified by using the extended and improved HCW with 3 baffles and a movable top baffle. The still water depth is 0.38m during the experiment at the baffle position. A constant slope of 1:7 composed of a plexiglass is set at downstream and a damping material was placed over the slope to reduce reflected waves. A pipe diameter to connect cylinders to baffles increases to 4 inch larger than the HCW with 2 baffles. Four wave gauges are placed over a flat bottom to measure water surface elevation. Data from wave gauges with a sampling frequency of 100Hz are collected and compared to desired target data for verification. 5.3.1 Solitary Wave Solitary waves with various wave amplitudes are generated to verify solitary wave generation by using HCW with 3 baffles and a movable top baffle. In this section, results of solitary waves with two different wave amplitudes are shown in Fig. 5.6 and 5.7. The fig 5.6 shows the times series of water surface elevation of solitary wave with a/h=0.05 from wave gauges and the histories 46 of actual speed of three motors are presented. The time series of water surface elevations from two wave gauges also show a typical solitary wave shape excepting the data after 11 sec by reflected waves. By comparing between the wave amplitude from wave gauges and the target data, the wave amplitude from wave gauges (2.03cm) are more overestimated than it of desired data (1.90cm). As mentioned in the section 3.2, the difference between calibrations by the flow meter and the piston movement is not taken into account. The section 5.5.2 will show new results of solitary wave generation by using a corrected calibration data. The time series of RPM from a motor to operate a cylinder for outflow via top baffle is out of phase by comparing other RPMs because the outflow via the top baffle includes the effect of the time varying top baffle volume. The Fig. 5.7 shows the times series of water surface elevation with the maximum wave amplitude, a/h=0.08, that the HCW system can generate and the histories of actual speed of two motors are presented. By comparing between the wave amplitude from wave gauges and the desired data, the wave amplitude from wave gauges (3.13cm) are more overestimated than the desired data (3.04cm). The over-prediction is same as the case of wave amplitude of a/h=0.05, but the difference between the data observation and the desired data become increased. This can be expected by the two calibrations between RPM and flow rate by measuring flow meter and screw-jack movement in the Fig 3.2. The difference between calibrations by measuring flow meter and screw-jack movement also become increased according to RPM increase. However, the surface elevations from wave gauges are good agreement with the target surface elevations and the history of motor speeds presents a solitary wave curve well. 47 Figure 5.6: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.05) 48 Figure 5.7: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.08) 49 5.3.2 Sinusoidal Wave To verify generation of sinusoidal waves, sinusoidal waves with various wave amplitudes and various wave lengths are created by HCW with 3 baffles and a movable top baffle. Time series of surface elevations and RPMs with respect to nine cases of sinusoidal waves that include three wave amplitude, a/h=0.03, 0.05 and 0.07, and three wave length, kh=0.5, 0.75 and 1 are presented and are compared the desired data in this section. Theoretically, short waves (deep water waves) and long waves (shallow water waves) are defined as kh less than p and kh larger than p=10, respectively. In this study, the wave lengths for laboratory experiments are all intermediate depth waves because the HCW system has some limitations about a finite cylinder volume and a maximum motor speed. In addition, it is difficult to create high wave amplitudes due to the limitations as well. To solve the limitation, paired cylinders for increasing cylinder volume will be explained in the section 5.5. Fig. 5.8 to 5.13 show that the time series of surface elevations are more overestimated than the desired surface elevations. The reflected wave after approximate 11 second is appeared by compar- ing the surface elevation from the wave gauge 1 and the wave gauge 3. The surface elevations in the relative short wave (kh=1) after around 20 second show phase lags similar with the numerical simulation in Fig. 2.3 because of friction by baffle length, as shown in Fig. 5.10 to 5.12. How- ever, the surface elevations from laboratory observations are closely in agreement with the desired surface elevations. The history of motor speeds for the top baffle movement in Fig. 5.8 to 5.13 shows a different shape with other data because the time-varying top baffle volume is applied in the continuity equation (Eq. 5.1). Yet, the RPM signal shows symmetric sinusoidal wave form to reach a maximum and a minimum RPMs well. 50 Figure 5.8: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.03, kh=1) 51 Figure 5.9: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.05, kh=1) 52 Figure 5.10: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.07, kh=1) 53 Figure 5.11: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.03, kh=0.75) 54 Figure 5.12: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.05, kh=0.75) 55 Figure 5.13: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.07, kh=0.75) 56 To investigate wave reflection, a reflection coefficient (Kr) is calculated by the method of Goda and Suzuki (1976), which uses the time series of surface elevations measured by two fixed wave gauges. The time series of surface elevations can be superposed the surface elevations of incident waves (h I ) and reflected waves (h R ) as follows: h I = a I cos(kxst+e I ) h R = a R cos(kxst+e R ) 9 = ; (5.4) where a is the wave amplitudes, k is the wave number of 2p=L with L being the wave length, s is the angular frequency of 2p=T with T being the wave period, and e is the phase angles. Subscripts I and R represent incident and reflected waves, respectively. The two measured time series of surface elevations as adjacent stations of x 1 and x 2 = x 1 +Dl (Dl is distance between two wave gauges) by using the trigonometric functions can be represented: h 1 =(h I +h R ) x=x 1 = A 1 cosst+ B 1 sinst h 2 =(h I +h R ) x=x 2 = A 2 cosst+ B 2 sinst 9 = ; (5.5) where, A 1 = a I cosf I + a R cosf R B 1 = a I sinf I a R sinf R A 2 = a I cos(kDl+f I )+ a R cos(kDl+f R ) B 2 = a I sin(kDl+f I ) a R sin(kDl+f R ) 9 > > > > > > > = > > > > > > > ; (5.6) f I = kx 1 +e I f R = kx 1 +e R 9 = ; (5.7) The Eq. 5.6 can be solved to find the amplitudes of incident and reflected waves. a I = 1 2jsinkDlj h (A 2 A 1 coskDl B 1 sinkDl) 2 +(B 2 + A 1 sinkDl B 1 coskDl) 2 i 1=2 a R = 1 2jsinkDlj h (A 2 A 1 coskDl+ B 1 sinkDl) 2 +(B 2 A 1 sinkDl B 1 coskDl) 2 i 1=2 9 > = > ; (5.8) Thus, Kr can be calculated: 57 Kr= a R a I (5.9) The reflection analysis by Goda and Suzuki (1976) is constrained to the range of distance between wave gauges: 0:05< Dl L < 0:45 (5.10) Two time series of surface elevation from the wave gauge 3 and the wave gauge 4 are used to yield the reflection coefficients and the distance between the wave gauges 3 and the wave gauge 4 is 60.2cm to satisfy the Eq. 5.10. Sinusoidal waves with a/h=0.05 and kh=0.5, 0.75 and 1 are created and reflections coefficients are calculated. The Fig. 5.14. presents all reflection coefficients (Kr) are higher than 10%. Figure 5.14: Reflection coefficient with different kh (a/h=0.05) 58 Fig. 5.15 to 5.17 present the time series of surface elevations and motor speeds from generating the relative long wave period (kh=0.5). The time series of motor speeds reach the maximum and minimum target RPM. The time series of surface elevations are not overestimated, and the results are different with the previous results. To investigate a reason, flow patterns near baffles are recorded. Fig. 5.18 shows snapshots of flow pattern near baffles at initial, trough, rest and crest positions of the top baffle. Flow patterns at the initial and the trough positions of the top baffle is normal, but air occurs at the rest position of the top baffle, and then it make free surface flow disturbed near the baffle at the crest position of the top baffle. The air is sucked into the top baffle through very small gaps of the top baffle hinge as shown in Fig. 5.19. The air disturbed flow and mass conservation of water so that sinusoidal wave amplitudes are not fully generated and that the unwanted second harmonics are appeared as shown in Fig. 5.15 to 5.17. Figure 5.15: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.03, kh=0.5) 59 Figure 5.16: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.05, kh=0.5) 60 Figure 5.17: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.07, kh=0.5) 61 Figure 5.18: Snapshots of flow pattern in the top baffle according to its movement (a/h=0.07, kh=0.5) Figure 5.19: Photos of gaps of the top baffle hinge: overview (left); enlarged view (right) 62 5.3.3 Stokes Wave In this study, the second order of Stokes wave theory in the section 2.2.3 is used to generate a periodic nonlinear wave. The laboratory observation of Stokes waves with different wave ampli- tudes, a/h=0.03, 0.05 and 0.07 and different wave length, kh=0.5, 0.75 and 1 are obtained and are compared to the desired target data for verification as shown in Fig. 5.20 to 5.28. The time series of motor speeds as feedback data from motors are evaluated to investigate how motors are operated well. The Ursell parameters of all cases satisfy Eq. 2.21 in order that the second order Stokes theory can be valid. Fig. 5.20 to 5.25 show that the time series of surface elevations are more overpredicted than the desired surface elevations, which is similar to the results of sinusoidal waves. The reflected wave after approximate 11 second is appeared by comparing the surface ele- vation from the wave gauge 1 and the wave gauge 3. The surface elevations in the relative short wave (kh=1) after around 18 second show phase lags similar with the numerical simulation in Fig. 2.3. This is also the similar reason to the results of sinusoidal waves by friction by baffle length. In addition to the wave reflection, water surface levels become increased as time goes on. Mass is unbalanced between positive waves and negative waves from sea water level. For this reason, the Stokes wave generations cannot be performed during short time to prevent a piston from reaching a cylinder top. However, the surface elevations from laboratory observations are good in agreement with the desired surface elevations. 63 Figure 5.20: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.03, kh=1) 64 Figure 5.21: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.05, kh=1) 65 Figure 5.22: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.07, kh=1) 66 Figure 5.23: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.03, kh=0.75) 67 Figure 5.24: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.05, kh=0.75) 68 Figure 5.25: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.07, kh=0.75) 69 Fig. 5.26 to 5.28 show the time series of surface elevations and motor speeds from generating the relative long wave period (kh=0.5). In other words, the results are less steep wave amplitude than the previous results. The times series of motor speeds periodically form to mimic periodic waves. The time series of surface elevations are not overestimated, and the results are different with the previous results. Moreover, nonlinear patterns are obviously appeared in the time series of surface elevations. These are caused by air penetration via the gap at the top baffle hinge and disturbance to flow in the top baffle similar to the sinusoidal waves with relative long wave (kh=0.5) as seen in Fig. 5.18 and 5.19. Figure 5.26: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.03, kh=0.5) 70 Figure 5.27: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.05, kh=0.5) 71 Figure 5.28: Time series of free surface elevation (top) and RPM (bottom). Solid line: laboratory observations; dashed line: target data (a/h=0.07, kh=0.5) 72 5.4 Active Wave Absorption Figure 5.29: Wave flume and instrumentation for active wave absorption As discussed in the section 2.4.2, two techniques for wave absorption were carried out by numerical simulations. One is using velocity profiles collected near outlet wall as an outlet bound- ary condition. The other is calculating an arrival time to an outlet boundary wall velocity profiles collected at any point by using wave celerity. Then, the numerical results show that these are fea- sible methods to absorb a solitary wave. In this study, the latter method is performed by using the experimental HCW system. Fig. 5.29 illustrates the wave flume with a vertical wall and set-up of wave gauges for a active wave absorption. A solitary wave are created and absorbed through a same baffle system and experiments are performed at least two times. The first time experiment is for collecting time series of surface elevation at the fixed wave gagues, and then wave celerity and arrival time are calculated by using the Eq. 5.11. In the second experiment, RPM signals for wave generation are multiplied by negative one and added into the time series of RPM signal for wave generation with the arrival time. Solitary wave from is given as a function of distance (x) and time (t) by h = asech 2 q 3a 4h 3 (x ct) c 2 = gz(1+ a h ) 9 = ; (5.11) where a is wave amplitude, c is the wave celerity and h is the water depth. 73 Fig. 5.30 and 5.31 show the time series of surface elevation from four wave gauges and moni- tored motor speeds of three motors by the solitary wave generation with a/h=0.05 and 0.08 and the wave absorption. The monitored motor speeds presents a positive solitary wave curve for genera- tion and a negative solitary wave curve for absorption well. The time series of surface elevations show the solitary wave generation and propagation with almost equal wave amplitudes at all posi- tions of wave gauges, and then the wave absorption. Yet, after the waves are absorbed, the small reflected waves are observed at approximate 20 second. The reflection coefficients of a/h=0.05 and a/h=0.08 are around 10% and 8%, respectively. The method for wave absorption is feasible in an experiment since the reflection values are small. Furthermore, a DAQ system including a real time control system and a instrument for measuring a real time velocity profile would allow an active wave absorber without a pre-experiment for obtaining a velocity profile and calculating an arrival time of wave. Figure 5.30: Time series of free surface elevation (top) and RPM (bottom) (a/h=0.05) 74 Figure 5.31: Time series of free surface elevation (top) and RPM (bottom) (a/h=0.08) 75 5.5 Three Paired Cylinders Figure 5.32: HCW with 3 baffles and 3 paired cylinders Fig. 5.32 shows HCW with three baffles including a movable top baffle connected three paired cylinders to create waves with higher wave amplitudes than the previous cases. Additional cylin- ders for two paired cylinders are placed back of the original cylinders. In addition, all upper plexiglass plates to support the system of motors and screw-jacks are replaced to aluminum plates which have 23 times stronger Young’s Modulus than the plexiglass. While the previous cases were running, some cases required a high torque made the upper plates deflected up and down in spite of reinforced steel frames attached on the plexiglass plate. It resulted in a motor and a screw-jack mis-aligned or a motor overloaded. The synchronization among the motors are connected to each baffle was validated as descried in the section 4.1.3. Similarly, each paired motors in the enhanced system are validated for the synchronization. For one same movement of paired motors, two wires from two drivers with same set-up environment for operating one same movement of paired motors 76 are connected one output of DAQ. The others two paired motors are connected to two outputs in the same manner. Next section will be explained in detail. 5.5.1 Synchronization of three Paired Cylinders Figure 5.33: Time series of RPM from three paired cylinders. top-hat spectral wave (top); bichro- matic wave (bottom) Fig. 5.33 presents the time series of monitored RPMs from three paired motors for generating a top-hat spectral wave and a bi-chromatic wave. RPM 1 and 2, RPM 3 and 4 and RPM 5 and 6 represent 3 paired motor signals and controlling flows via the bottom, middle and top baffle, respectively. Each signals from paired motors show the paired motors equally moves in tandem. The signals for controlling flows via the top baffle look like out of phase because the time-varying top baffle volume is applied. However, the other signals show a top-hat spectral and bichromatic waves are created well. 77 5.5.2 Corrected Calibration between Flow Rate and RPM As described in the section 3.2, two calibrations between RPM and flow rate by measuring a flow meter and a piston movement were carried out for solid verification. Both calibrations show a linear relationship and are in good agreement. However, the results by the flow meter are a little higher than the calculation by piston movement after 3000RPM as shown in Fig. 3.2. By ignoring the difference of two calibrations, three decimal digits of both slopes between RPM and flow rate were taken to perform all previous tests. In all experimental tests, wave amplitudes are bigger than in target data. To find a corrected calibration, solitary wave with a/h=0.1 are generated by each original calibration and the time series of surface elevations are observed as shown in Fig. 5.34. The wave amplitudes by the calibration from the measured piston movement, a=4.2cm, are more overestimated than target data, a=3.8cm, which are similar to the previous results. However, the wave amplitudes by the calibration from the flow meter, 3.8cm, are good agreement with the target data. Therefore, the calibration by measuring a piston movement is less reliable than by a flow meter since the former is measured by an eye observation. Fig. 5.35 shows the comparison between measured and desired maximum wave amplitude ratio at the wave gauge 1 by generating solitary waves with a/h= 0.06 to 0.12. The results are distributed near a solid line which represents perfect agreement. Therefore, the calibration is correct to create any desired waves. 78 Figure 5.34: Time series of RPM from three paired cylinders by old calibration results (top); corrected calibration results (bottom) 79 Figure 5.35: Comparison between target and observed max a/h from WG 1 80 5.6 Conclusions In this study, physical HCW with three baffles including a movable top baffle were built. The top baffle can be moved along a surface elevation by using a combination of screw-jack and motor, a kind of linear actuator, similar to the flow control system. The time-varying volume of top baffle has been considered and Ramp period has been introduced for a smooth piston movement. For verification, solitary waves, sinusoidal waves and stokes waves were created. The time series of surface elevations for laboratory observation were compared to the desired target data. The results are in good agreement with the target data in spite of the reflected waves. Yet, the results were overestimated because the difference of two calibrations through the small prototype HCW was not considered. In cases of periodic relative long waves, sinusoidal and stokes wave with kh=0.5, shows unwanted harmonics are appeared and the results are not overestimated. This is because the air is sucked into the top baffle through very small gaps of the top baffle hinge and it disturbed flow and mass conservation of water. An active wave absorption by calculating an arrival time through wave celerity was carried out in the case of solitary wave. The results show the reflection coefficients are low and the active wave absorption would be a feasible way. The enhanced physical model with three paired cylinders has been built for higher wave generations than with three cylinders. By using the enhanced HCW, corrected calibration was investigated. By the comparison between measured and desired solitary wave amplitude, the corrected calibration was obtained. 81 6 Measurement Precision and Uncertainty Sources of potential errors are investigated before studies of wave group interaction on a slop- ing beach and measurement of wave runup. For the HCW system, the flow control system com- bined with a screw-jack and a motor have a potential error because the screw-jack have that a lifting screw outer diameter and sleeve diameter are not perfectly matched for a smooth movement. The motor is controlled by an input signal via DAQ devices so that the DAQs play a more dominant role than the motor. Moreover, the DAQ for analog output which can control the motor speed directly relates the motor movement. For measuring water surface elevation, wave gauges with a small accuracy have a potential error. In a study of measurement of wave runup, an error by coordinate transformation is calculated by comparing real control points and the predicted control points after DLT (Direct Linear Trans- formation) is applied. Furthermore, a max and a min solitary wave runup of detected edges are obtained, and then an error is calculated. Table 6.1 shows quantifications of precision errors and uncertainty in each source and direc- tional relationship of errors. In the HCW system and wave gauges, vertical and horizontal errors relate a wave height and wave length, respectively. In the wave runup, all errors only relate a cross-shoreline movement. In the HCW system, the errors of screw-jack and DAQ for analog output play a essential role with a piston movement which relates a flow rate in each inlet. The error of screw-jack affects that a vertical movement of the screw which relates the piston movement can more over- or under- estimated than a target movement. Then, a wave height can be higher or lower than a target wave height because a flow rate via each inlet can be over- or under- estimated. The percentage error of DAQ system becomes bigger when it outputs a high voltage for a fast movement of motor. Those errors can potentially make a wave height more over- or under-estimated than a target wave 82 Table 6.1: Quantifications of precision errors and uncertainty Source Bias Uncertainty Direction HCW system: Screw-jack - 0:02cm Vertical HCW system: DAQ-NI 9264 (analog output) 0:05% 0:05% Vertical Wave gauge - 0:1cm Vertical Wave runup: Coordinate transform for solitary wave - 1:73cm - Wave runup: Coordinate transform for top-hat spectrum - 0:51cm - Wave runup: Edge detection filtering for solitary wave - 0:4419cm - height. In addition, the error of wave gauges can also make experimental data about water surface elevation more over- or under-estimated than a target water surface elevation. In wave runup, the uncertainty error of coordinate transform for solitary wave runup is bigger than for top-hat spectrum wave. In the former case, a larger area of interest than in the latter case including an area far from a camera is used. The area includes image data with high as well as low resolutions. The control points in low resolution make the mean error increased. In addition to the error of coordinate transform, the error of edge detection filtering can makes a wave runup over- or under-estimated. 83 7 Wave Group Interactions over a Sloping Beach 7.1 Introduction Surface gravity waves created in offshore represent a group of high and low waves with super- posing individual wave with various frequencies. The gravity waves propagate toward onshore with shallowing water, and eventually shoal in and break on beaches. Since the gravity wave is naturally dispersive, the groups are transient, and individual wave trains are focused in time and space. The focusing of wave energy leads to the formation of extreme, rogue or freak waves (Peregrine et al. (1988); Dean (1990)). Extensive studies have shown analyses of local nonlinear mechanics associated with the focused-transient wave groups in uniform depth (Rapp and Melville (1990); Stansberg (1990); Pierson et al. (1992); Baldock et al. (1996); Johannessen and Swan (2003)). Baldock (2006) investigated that the generation of long waves by shoaling and break- ing of the propagation of transient short wave group over a sloping beach with the high spatial resolution laboratory data. Watson and Peregrine (1992) and Watson et al. (2011) used a focused wave by superimposing a number of solitary waves onto a trough with a sinusoidal wave shape to investigate the generation of long waves during shoaling, breaking and swashing. Furthermore, infragravity waves are appeared by global nonlinear interaction that the mean water level is slowly varied as low-frequency currents and oscillations. In nature, wind and swell waves have typical dominant periods of 1 to 30 second, while infragravity waves have typical dom- inant periods of 80 to 300 second, as shown in Fig. 7.1 (Munk (1950)). The short wave height becomes diminished toward shoreline, while the long wave height grows toward shoreline. Thus, shoreline movement or wave runup is dominated by the long wave. In addition to the long wave 84 Figure 7.1: Classification of the spectrum of ocean waves according to wave period (Munk (1950)) effect, the shoreline motions become increased under an extreme storm condition. These low- frequency waves have been identified by two main generation mechanisms. Longuet-Higgins and Stewart (1962) analytically explained short wave groups in shallower water shoaled and forced infragravity bound waves by introducing radiation stress gradient. The other mechanism con- templates the radiation of free long waves at different depth leads the time-varying break point (Symonds et al. (1982)). Infragravity wave oscillation in the surf zone associated with the breaking of wave groups, called ’surfbeat’, have been studied (Munk (1949); Tucker (1950); Guza and Thornton (1985); Elgar et al. (1992)). The phenomenon strongly influences sediment transport (Goda (1975); Pere- grine (1983); Holman and Bowen (1982); Yu and Mei (2000); Aagaard and Greenwood (2008) and bay resonance (Sand (1982); Harkins and Briggs (1994)). Infragravity wave generation over an slope beach by a time-varying breakpoint has been investigated by Baldock et al. (2000) and Baldock and Huntley (2002) and corroborated by comparing the breakpoint forcing model by Symonds et al. (1982). Schffer (1993) established a theoretical model for infragravity wave taking into account the time-varying break position and a partial transmission of grouping into the surf zone shoaling bound long wave the extended analytical studies of Longuet-Higgins and Stewart (1962) and Symonds et al. (1982). Janssen et al. (2003) investigated the phase lag and short-wave 85 envelope growth with shoaling short-wave group. Battjes et al. (2004) studies the rate of energy transfer from short wave to bound long waves by using this laboratory data. The numerical study of nearshore wave dynamics and long wave transformation has tradi- tionally been approached using Nonlinear Shallow Water (NSW) and Boussinesq-type equations models. Numerical studies based on NSW (Hibberd and Peregrine (1979); Kobayashi et al. (1989); van Leeuwen and Battjes (1990); List (1992); Dongeren et al. (1994) and the Boussinesq equations (M. H. Freilich (1984); Watson et al. (2011); Madsen et al. (1997)) have provided satisfactory results. However, these numerical models require setting both the triggering wave breaking mech- anism and the subsequent wave energy dissipation owing to wave breaking (Torres-Freyermuth et al. (2010)). Numerical models based on Reynolds-Averaged Navier-Stokes (RANS) equation have been developed by using the turbulence closure problem to overcome the limitation of NSW and Boussinesq models. Torres-Freyermuth et al. (2010) analyzed the study of long wave trans- formation on a mildly-sloping beach. Lara et al. (2010) presented infragravity wave induced by a transient-focused wave group over a sloping bottom and investigated the influence that a large flat bottom induced on the propagation pattern of long wave. In this chapter, top-hat spectral waves, a transient focused wave group, are generated over com- posite slopes by using HCW. In addition, the laboratory observations are compared with numerical simulations, OpenFOAM and COULWA VE, based on Boussinesq and RANS equations, respec- tively. Long wave generation in association with an interaction of a transient focused wave group is based on the second-order wave theory (Longuet-Higgins and Stewart (1960)) and the shoal- ing and breaking of the wave group on a composite slope by Fast Fourier Transform (FFT) are investigated. 7.2 Experimental set-up The experiment were performed in the wave flume, 0.4m of width, 0.6cm of height and 12m of length as mentioned in the section 3.3. The still water depth was set at 0.38m same as the top baffle height at a initial position during the experiments. HCW is located next to one side of the flume to 86 generate waves. Two fixed baffles and one top movable top baffle with 0.31m of length are placed at the side of the flume and three pipe lines with 4inch diameter are connected to three paired cylinders. To generate waves, flow rates via three baffles are controlled by piston movements in the cylinders and combinations of a screw-jack and a motor play a key role as a similar line actuator to move pistons. The system is applied to the movable top baffle. A composite beach is placed at the other side of the flume as shown in Fig. 7.2. The beach is made up of a relatively steep slope of 1:10 and relatively mild slope of 1:15 and the beach starts about 4.2m from the baffle. Water surface elevations were at fixed spatial locations were collected and measured from four surface piercing resistance wave gauges with a sampling frequency of 100Hz. Two wave gauges were placed at a horizontal bottom and the other two wave gauges were located at the first slope and the second slope, respectively. The detailed wave flume set-up and wave gauge locations as seen in the Fig. 7.2. Figure 7.2: Wave flume and instrumentation for top-hat spectral waves 7.3 Top-hat Spectral Wave In this study, transient-focused wave groups composed of 50 primary wave components with a same amplitude were generated because the top-hat spectral wave are transformed to long wave and the propagation path can be observed directly. The primary wave components were uniformly spaced over two different frequency bandwidths to obtain ’top-hat’ frequency spectra (Rapp and 87 Melville (1990)). Total wave group amplitude, A, the frequency bandwidth, D f , and the central frequency, f c , are defined as A= H 2 = a n N; D f = f 1 f N ; f c = 1 2 ( f 1 + f N ) (7.1) where a n is the amplitude of the nth frequency component and N is the number of individual wave components. The surface elevation of top-hat spectrum, h, is given by summation of linear wave components as follows: h(x;t)= N å n=1 a n cos(k n xs n t) (7.2) where k n ands n are wave number and wave frequency of the nth wave component, respectively. The six cases with same frequency bandwidths by Lara et al. (2010) were examined as seen in table 7.1. The top-hat spectral waves are periodic with period T 0 = N=D f so that the wave signals are truncated in time to obtain a single wave group signal (Rapp and Melville (1990)). The time is decided by T g = 2=D f , the minimum value of the period of the wave packets at focal point. The practical reason makes that the ’top-hat’ spectral shape is transformed to ’Sombrero (Mexican) hat’ spectral shape. The number of waves in the groups is obtained by N w = 2 f c =D f . The time series of monitored RPM signals gradually become delayed by comparing the target (input) RPM signals as shown in Fig. 7.3. It means that sudden acceleration or deceleration with different direction makes the motors delayed or input data for motor movement with a certain fre- quency become delayed. To solve the delayed signal, the time module factor is multiplied on the original RPM signal time series. Fig. 7.4 shows the comparison of target RPMs and monitored RPMS for case C of top-hat spectral waves after time modulation. In addition, the top-hat spectral waves generated by the experimental HCW were more over-estimated than numerical results by COULWA VE and OpenFOAM as shown in Fig. 7.5. To find a corrected target wave height (H c ), the time series of water surface elevation by numerical simulations are compared to the experi- mental data with maximum wave height. The corrected target wave height is obtained by a simple proportional expression as seen in table 7.1. 88 Table 7.1: Wave group conditions case D f (Hz) f c (Hz) f 1 (Hz) f N (Hz) T g (sec) H t (m) H c (m) A 0.167 0.75 0.833 0.667 12 0.0336 0.0452 B 0.167 0.75 0.833 0.667 12 0.0617 0.0812 C 0.167 0.75 0.833 0.667 12 0.0917 0.1207 Figure 7.3: Comparison of target RPMs and monitored RPMs for top-hat spectral waves (case C) 89 Figure 7.4: Comparison of target RPMs and monitored RPMs for top-hat spectral waves after time modulation(case C) 90 Figure 7.5: Comparison of the experimental data and the model prediction by OpenFOAM before applying the correct target wave height, Case A (top), B (middle) and C (bottom). 91 7.4 OpenFOAM with a movable top baffle The interDyMFoam solver is combination of the interFoam solver and a dynamic mesh motion is used for a simulation of top baffle movement. For the dynamic mesh motion, one of mesh motion methods in OpenFOAM, dynamicMotionSolverFvMesh, is used, and requires a cell motion equa- tion and a diffusivity model. The cell motion equation, displacementLaplacian, is based on the Laplacian of the diffusivity and needs a pointDisplacement file at initial time. For the diffusivity model, inverseDistance is used, and makes that the points after solving the cell motion equation can be moved by the diffusivity of the field based on the inverse distance from boundaries. With the solver, a dynamic mesh motion with 6 DOF (Degrees Of Freedom) can be simulated. In this study, 1 DOF, a rotation on z-axis is only considered to simulate a movable top baffle. The time series of rotations on z-axis are calculated depending on the time series of water surface elevation and obtained as input data for the top baffle movement. In the numerical simulation, the time-varying volume of top baffle same as the experiment in the chapter 5.1 are considered. Fig. 7.6 shows dynamic mesh motions and refinement at initial position and wave crest and trough according to the top baffle movement. 92 Figure 7.6: Dynamic mesh motion: mesh at initial time (top); mesh at wave crest (middle); mesh at wave trough (bottom) 93 7.5 COULWA VE COULWA VE (COrnell University Long and intermediate WA VE modeling package) is based on the expanded Boussinesq equation with some modifications by weakly dispersive terms (Mad- sen and Sorensen (1992)) and highly nonlinear free surface disturbances (Liu (1994) and Wei et al. (1995)). The COULWA VE including a moving boundary treatment has shown to give good predic- tions with respect to wave runup (Lynett et al. (2002)). The continuity and momentum equations of the model are given in dimensional form: h t +Ñ(Hu a ) Ñ H 1 6 (h 2 hh+ h 2 ) 1 2 z 2 a Ñ(Ñ u a ) Ñ H 1 2 (hh h) z a Ñ(Ñ(hu a )) = 0 (7.3) u at + u a Ñu a + gÑh + z 2 a 2 Ñ(Ñ u at )+ z a Ñ(Ñ(hu at )) +[(Ñ(hu a ))Ñ(Ñ(hu a ))Ñ(h(Ñ(hu at )))+(u a Ñz a )Ñ(Ñ(hu a ))] + z a (u a Ñz a )Ñ(Ñ u a )+ z 2 a 2 Ñfu a Ñ(Ñ u a )g +Ñ h 2 2 Ñ u at hu a Ñ(Ñ(hu a ))+h(Ñ(hu a )Ñ u a +Ñ h 2 2 (Ñ u a ) 2 u a Ñ(Ñ u a ) = 0 (7.4) where h is the free surface elevation and h is the local water depth. u a is the reference velocity vector at the elevation z a =0:531h recommended by Nwogu (1993). The first and second terms in the left-hand side of continuity equation (Eq. 7.3) and the first, second and third terms in the left-hand side of momentum equation (Eq. 7.4) are given by the linearized Boussinesq and shallow water equation. Otherwise, the other terms are introduced by the expanded Boussinesq- type derivation. For the generation of top-hat spectral waves, a source location is same as the end 94 of baffle of experiment. The top-hat spectral waves are generated without a sponge layer to predict a bounding long wave motion same as the experiment and OpenFOAM simulation. 7.6 Results and Analysis Fig. 7.7 to 7.12 present the time series of water surface elevation and the wave amplitude spectra by FFT (Fast Fourier Transform) analysis with different wave amplitudes, case A, B and C cross-shore locations, corresponding to four wave gauges. All data are compared to numerical simulations which are performed by OpenFOAM (Fig. 7.7, 7.9 and 7.11) and COULWA VE (Fig. 7.8, 7.10 and 7.12). The overall water surface elevations of the experimental data time series are in good agreement with the numerical simulations. Especially, long waves with small wave amplitude after the propagation of short wave groups are observed well in the experimental and numerical results. However, the time series of water surface elevations at WG 3 and 4 where those are located at nearshore zone of numerical simulations present that wave amplitudes in short wave groups are underpredicted by comparing experimental data because of an inherent numerical dispersion. The OpenFOAM is more underpredicted than COULWA VE because the OpenFOAM considers two phase fluids by VOF scheme. The scheme used difference of density to recognize air and water phase and the velocity of air phase is much bigger than water phase by solving transport equations. This affect unwanted a high velocity in interface region (Afshar (2010)). The wave amplitude spectra show that the initial spectral shape of the primary short wave components in the range 0.7Hz<f<0.8Hz are created. However, the times series of water surface elevations shows that the short wave groups are transformed to long wave, and the propagation path can be observed directly. While the waves are progressing to the bi-linear slope beach at WG1 to WG4), the primary short wave amplitude (0.7Hz<f<0.8Hz) is same in case A or a little decreases in case B, whereas the amplitudes of the short wave superharmonics (1.4Hz<f<1.6Hz) and the bounded long wave subharmonics (f<0.2Hz) increase. However, the spectrum analysis at WG 2 shows the short wave superhamonics more decrease than WG1 because the reflected wave from the slope can affect the wave. The period of wave packets is 12 second and the arrival time from WG2 to 95 WG3 is around 2 second so the wave packets can be affected by the reflected wave while they are propagating at WG2. The overall results well explain how a wave energy can be redistributed during the shoaling process. In case C, the highest wave amplitude, while the waves are progressing to the bi-linear slope beach at WG1 to WG3, the primary short wave amplitude (0.7Hz<f<0.8Hz) decreases, whereas the amplitudes of the short wave superharmonics (1.4Hz<f<1.6Hz) and the bounded long wave subharmonics (f<0.2Hz) increase. The wave amplitude spectra at WG4 show that the short wave superharmonics decreases by comparing WG3 due to wave breaking. The results present that the amplitudes of primary short wave group are transformed on the slope due to shoaling and breaking. 96 Figure 7.7: Time series of free surface elevation and spectrum analysis, case A. Blue: laboratory observations; red: model predictions by OpenFOAM 97 Figure 7.8: Time series of free surface elevation and spectrum analysis, case A. Blue: laboratory observations;red: model predictions by COULWA VE 98 Figure 7.9: Time series of free surface elevation and spectrum analysis, case B. Blue: laboratory observations;red: model predictions by OpenFOAM 99 Figure 7.10: Time series of free surface elevation and spectrum analysis, case B. Blue: laboratory observations; red: model predictions by COULWA VE 100 Figure 7.11: Time series of free surface elevation and spectrum analysis, case C. Blue: laboratory observations;red: model predictions by OpenFOAM 101 Figure 7.12: Time series of free surface elevation and spectrum analysis, case C. Blue: laboratory observations; red: model predictions by COULWA VE 102 Fig. 7.13 to 7.18 show the measured (solid line) and the predicted (dashed line) total wave elevation (blue) and long wave elevation (green) time series at different cross-shore location for case A, B and C. Long-wave components were obtained from low-pass filtered surface elevations by time-series measurement and cut-off frequency of f c < 0:4Hz. In offshore, in the locations of WG1 and WG2, the bound long wave beneath the group is clearly visible and the scale of the long wave is 5 times magnification of the right-hand vertical scale. In the locations of WG3, single large wave in the group makes wave energy focused (Rapp and Melville (1990); Baldock et al. (1996)). The bound long wave and the positive surge in advance of the short wave groups are enhanced by short-wave focusing and shoaling. The negative pulse after the bound long wave is identified as the radiated long wave from the breakpoint. In the locations of WG4, the surging motion is more dominant than bounding wave in numerical simulations. This is because the numerical simulations make less energy dissipated than the experimental results. Moreover, it is observed that the bounded long wave and the positive wave surge in advance the wave group are amplified and reach their maximum wave height in shallow water. 103 Figure 7.13: Time series of long wave elevation, case A. Solid line: laboratory observations; dotted line: model predictions by OpenFOAM; blue: total wave elevation; green: long wave elevation 104 Figure 7.14: Time series of long wave elevation, case A. Solid line: laboratory observations; dotted line: model predictions by COULWA VE; blue: total wave elevation; green: long wave elevation 105 Figure 7.15: Time series of long wave elevation, case B. Solid line: laboratory observations; dotted line: model predictions by OpenFOAM; blue: total wave elevation; green: long wave elevation 106 Figure 7.16: Time series of long wave elevation, case B. Solid line: laboratory observations; dotted line: model predictions by COULWA VE; blue: total wave elevation; green: long wave elevation 107 Figure 7.17: Time series of long wave elevation, case C. Solid line: laboratory observations; dotted line: model predictions by OpenFOAM; blue: total wave elevation; green: long wave elevation 108 Figure 7.18: Time series of long wave elevation, case C. Solid line: laboratory observations; dotted line: model predictions by COULWA VE; blue: total wave elevation; green: long wave elevation 109 Using the solution procedure by Stokes (1847), Longuet-Higgins and Stewart (1960) proposed that the second order solutions of water surface elevation (h) and velocity potential (f) considering the interaction between two progressive waves are: h =h 1 +h 2 + a 1 a 2 2g [Ccos(y 1 y 2 ) Dcos(y 1 +y 2 )] (7.5) f =f 1 +f 2 + Ecosh((k 1 k 2 )(z+ h))sin(y 1 y 2 ) g((k 1 k 2 )sinh(k 1 k 2 )h)(s 1 s 2 ) 2 cosh((k 1 k 2 )h) Fcosh((k 1 + k 2 )(z+ h))sin(y 1 +y 2 ) g((k 1 + k 2 )sinh(k 1 + k 2 )h)(s 1 +s 2 ) 2 cosh((k 1 + k 2 )h) (7.6) where the subscripts denote each progressive wave, andh 1 ,h 2 ,f 1 andf 2 are the first approx- imations of water surface elevation and velocity potential expressed in terms of the phase angle is given by y = kxst, and a, k and s are wave amplitude, the wave number and the wave frequency, respectively. The coefficients of second order solutions (C, D, E and F) defined by Longuet-Higgins and Stewart (1960) are reproduced in Appendix C. The term ofy 1 y 2 is asso- ciation with the global or low frequency interaction, whereas the term of y 1 +y 2 corresponds to the local nonlinear interaction. To apply the equations 7.5 and 7.6 to a focused wave group, Baldock et al. (1996) introduced a summation of the interactions due to each pair of wave components and produced a total second- order solution. The second-order solutions of the water surface elevation and the velocity potential are respectively expressed as h = N å n=1 h(n)+ N å n=1 N å m=n+1 h(n;m) (7.7) f = N å n=1 f(n)+ N å n=1 N å m=n+1 f(n;m) (7.8) whereh n andf n are the first approximations of water surface elevation and velocity potential for the nth wave component, andh ( n;m) andf ( n;m) are the second order interactions between nth and mthe wave components. The terms with respect to the interactions for the paired waves are 110 calculated by using the Eq. 7.7 and 7.8. The analytical solutions will be compared to the laboratory data and the numerical results. Fig. 7.19 to 7.21 show the surface elevation amplitude spectra with different wave ampli- tudes at different cross shore locations for case A, B and C. The laboratory data (solid line) are compared to the numerical results with OpenFOAM (dashed-dot line) and COULWA VE (dashed line) and the second order solution (dotted line) by Longuet-Higgins and Stewart (1960). In addi- tion, the amplitude spectra of the second order solution is predicted at the wave-maker position (x= 0). The comparisons of the theory at initial location and experimental data and numerical results at various cross shore locations present that a spatial redistribution of wave energy is obvi- ously described during the shoaling and breaking processes. The wave amplitude spectra in WG1 of all cases measured and predicted shows the primary short wave amplitude in the frequency range (0.7Hz<f<0.8Hz) and the second-order superharmonics in the frequency range (1.4Hz<f<1.6Hz) are in close agreement with the second-order wave theory by Longuet-Higgins and Stewart (1960). In contrast, the bound long wave subharmonics in the frequency range ( f <0.2Hz) measured and predicted are different with the second-order wave theory. The additional radiation of the energy at shoreward (on sloping beach) for long wave generation is required such as shoaling and break- ing processes (Baldock (2006)). The wave amplitude spectra in the superhamonic band frequency measured and predicted in WG3 and WG4 of case A, B and C are more increased than the second- order theory due to the wave shoaling on the composite sloping beach. In the case C, the wave amplitude spectra in the primary short wave group band frequency measured and predicted in WG4 more decreased than the second-order solution because of wave breaking. The wave ampli- tude spectra with low-frequency increase over nearshore regions because the wave amplitudes with low-frequency become increased in shallow water due to wave shoaling and the energy is trans- ferred from the waves with higher frequency. 111 Figure 7.19: Surface elevation amplitude spectra of case A. Blue: laboratory observations; dashed green: model predictions by COULWA VE; dashed-dot red: model prediction by OpenFOAM; dotted black: second-order theory at wave-maker position (x= 0) 112 Figure 7.20: Surface elevation amplitude spectra of case B. Blue: laboratory observations; dashed green: model predictions by COULWA VE; dashed-dot red: model prediction by OpenFOAM; dotted black: second-order theory at wave-maker position (x= 0) 113 Figure 7.21: Surface elevation amplitude spectra of case C. Blue: laboratory observations; dashed green: model predictions by COULWA VE; dashed-dot red: model prediction by OpenFOAM; dotted black: second-order theory at wave-maker position (x= 0) 114 7.7 Conclusions In this chapter, the generation of long wave by the shoaling and breaking of the propagation of top-hat spectral waves, a transient focused wave group, over composite slopes by using HCW. Moreover, the time series of water surface elevation and the amplitude spectra by the laboratory observations were compared with numerical simulations, OpenFOAM and COULWA VE, based on Boussinesq and RANS equations, respectively. The comparisons show that long wave gener- ation in the experiments is in close agreement with numerical simulations. The times series of water surface elevations shows that the short wave groups are transformed to long waves, and their propagation paths can be observed directly. In addition, the comparison of the amplitude spectra presents that the primary and superharmonic wave amplitudes are increased or decreased by wave shoaling or wave breaking, respectively. Long wave components obtained from low-pass filtered surface elevations by time-series measurement obviously describe that the increased amplitudes in shallow water by shoaling and their propagation path. The wave amplitude spectra measured and predicted were compared to the second-order wave theory in association with an interaction of wave groups (Longuet-Higgins and Stewart (1960)). The comparisons of the theory at initial location and experimental data and numerical results at various cross shore locations present that a spatial redistribution of wave energy is obviously described during the shoaling and breaking pro- cesses. The wave amplitude spectra with low-frequency increase over nearshore regions because the wave amplitudes with low-frequency become increased in shallow water due to wave shoaling and the energy is transferred from the waves with higher frequency. 115 8 Measurement for the Time Series of Wave Runup 8.1 Introduction Wave runup and inundation are defined as the maximum extent of vertical elevation from still water level and the maximum horizontal distance from initial shoreline, as depicted in Fig. 8.1. It is important to evaluate and predict wave runup and inundation on slope because extreme events associated with them cause huge damages in the coastal area. Especially, the study of wave runup by long waves is more essential because wave runup and rundown in swash zone is dominant by long waves. Studies of tsunami inundation or runup, one of extreme events, have been widely performed by solitary wave in an experimental wave flume because the wave is easily generated or reproduced with single parameter, wave height. Synolakis (2006) studied linear and nonlinear theories of solitary wave runup on plane beaches and compared the theories with experimental results. He found that different runup regimes of breaking and non-breaking solitary waves are existent. The maximum runup law of non-breaking solitary wave was presented. Kˆ anoˆ glu and Synolakis (1998) investigated solitary wave runup on piecewise linear topographies. They found asymptotic results with respect to solitary wave interaction with piecewise linear topographies in a counter-intuitive manner. For a composite slope beach, for instance, only a slope of closest to the shoreline affects the runup. Li and Raichlen (2001) derived nonlinear solution from shallow water equation, which includes higher order terms than the solution derived by Synolakis (2006). Slevik et al. (2013) investigated and compared runups of solitary waves on a straight and a composite beach. They 116 thought that a relatively thinner runup above vertex of the second slope is created than the straight slope. Figure 8.1: A definition sketch for a solitary wave runup and inundation on a bi-linear sloping beach (m 1 =1/10 and m 2 =1/15) On the other hands, the wave runup is a summation of wave setup and swash motion. To under- standing wave setup and swash oscillations composed of individual broken waves and infragravity waves is important for the beach environment. Guza and Thornton (1982) first discovered that swash oscillations have two components by spectral analysis. Hunt (1959) proposed a formula to predict wave runup on open-coast beaches. Battjes (1974) rewrote the Hunt formula and sug- gested the dimensionless runup as Iribarren number or surf similarity parameter (x ) depending on an incoming wave condition and a beach slope. It is given by x = R H = tanb p H 0 =L 0 (8.1) where R is wave runup, tanb is a beach slope and H 0 and L 0 are the wave height and wave length in deep water, respectively. If the surf similarity parameter is greater than 1.75, steep slopes or reflective beaches are clas- sified. Otherwise, gentle slopes or dissipative beaches are called (Hunt (2003)). For steep slopes, incoming bores are more dominant than infragravity waves, whereas an effect of infragravity waves is bigger than incoming bores. 117 For dissipative slopes, wave setups become large in proportion to small swash motions since the wave setup is independent of the slope (Kobayashi et al. (1989)). In contrast, swash oscillations are bigger than wave setups on reflective slopes (Battjes (1974)). It is important to measure wave runup in an experimental wave flume, but the measurement is not easy because of the turbulence and aeration of the flow (Schttrumpf and van Gent). A typical measurement of wave runup is using serial wave gauges a step-type array. Yet, the measurement has low resolution. A resistant type runup gauge parallel to a slope is often used, but the runup is underestimated because of a distance between the runup gauge and the slope. Recently a laser scanner has been used to detect water surface and wave runup (Hofland et al. (2015)). Pedersen et al. (2013) captured colored wave runup with a high-speed video camera and measured wave runup with an edge detection method in MATLAB. In the image analysis method, an accuracy of data depends on an image quality. Moreover, when the wave retreats, the colored wave becomes dim so that capturing so that detecting the wave rundown is difficult. In this chapter, the wave runups by solitary waves and top-hat spectral waves are investigated. The solitary wave and the top-hat spectral wave are for the study of tsunami runup and infra- gravity wave runup, respectively. An action camera cheaper than other measuring instruments is used for capturing and detecting the time series of shoreline movement by solitary waves and top- hat spectral waves with an edge detection method in MATLAB. In addition, wave rundowns are captured and detected by controlling an image contrast. The time series of wave runups are com- pared to numerical simulations with OpenFOAM and COULWA VE to investigate an accuracy of the method. Furthermore, the maximum runup of solitary waves and top-hat spectral waves with various amplitudes measured is compared to numerical simulations. To investigate the space-time evolution of wave transformation and runup, the contour plots of surface elevation and time series of wave runup by the laboratory observation and the model predictions are illustrated. In addition, the space-time evolutions of low-frequency component of surface elevation and the time series of wave runup by the laboratory observation and the model predictions to investigate the relationship between low frequency wave and wave runup are presented. 118 8.2 Instrumental Setup Fig. 8.2 shows the instrumental setup for detecting the time series of shoreline movement on x-direction. The rectangular structure built by perforated steel frames is positioned on slope, an area of interest. The frame is for covering the area by the light diffuser as seen in the Fig. 8.2. If an image has glared spots due to a light, it makes the shoreline detection harder. An action camera with FHD resolution (1920x1080) and 30 fps is mounted on the right-hand side of the frame to cover overall the composite slope area. The composite slope is composed of a transparent plexiglass material so white PVC films are attached on the slope. Adhesive rulers are attached on the both sides of the slopes for using control points to transform local coordinate in image to world coordinate. Water is dyed with red food coloring. For all instrument setup, the total cost is around $400. 8.3 Image Processing For the preprocessing, first, the video for monitoring shoreline movement was extracted to the sequence of image frames. The image frames have RGB color as seen in Fig. 8.3 (a). The RGB images were masked except for area of interest to reduce analyzing time and the images were con- verted to grayscale images for using edge detection in the Fig. 8.3 (b). Next, the grayscale images were analyzed with Canny edge function in MATLAB toolbox to detect shoreline movement. The function read the grayscale images to find edges and then outputs white pixel as the detected edge in black & white images in the Fig. 8.3 (c). The canny edge looks for local maximum values of the gradient of the grayscale images to find the edges so the setting appropriate threshold values in the gradient is important. imopen function in MATLAB toolbox carries out morphological opening on a grayscale image with a structuring elements. It makes that a relative small or weak noise in the grayscale images is filtered out before the edge detection. The Fig. 8.3 (d) shows the detected edges are overlapped on the RGB image. The shoreline movement during wave runup can be detected well by the image processing. However, it is difficult to detect the shoreline movement during wave rundown. Water color 119 Figure 8.2: Instrumental setup. Top: side view; bottom: top view becomes faint and the faint water still leave on the composite slope even though wave is run- ning down inner the faint water. To make the wave rundown more obvious, the image contrast in RGB images were controlled. Fig. 8.4 shows the comparison of edge detection without and with image contrast control during wave rundown. The process for coordinate transformation is similar to the process by Kalligeris et al. (2016). First, the lens of the action camera with a fisheye lens for wide angles of view creates more dis- torted image of the FOV (Field Of View) than other cameras. The camera was calibrated with 13 check-board images including different angles of lens by using Caltech image calibration (Bouguet (2015)) and the intrinsic parameters were obtained to remove the lens distortion. Secondly, local coordinates (u,v) in the image were transformed to global coordinates (x,y,z) by using DLT (Direct 120 Figure 8.3: Process of edge detection.a) RGB image; b) grayscale image with mask; c) Edge detection in black & whited; d) edge overlapped RGB Linear Transformation) equation (Holland et al. (1997)). The set of collinearity equations is as follows: u= xL 1 + yL 2 + zL 3 + L 4 xL 9 + yL 10 + zL 11 + 1 ; v= xL 5 + yL 6 + zL 7 + L 8 xL 9 + yL 10 + zL 11 + 1 (8.2) where L 1 , L 2 ,..., L 11 are the DLT parameters. 121 Figure 8.4: Comparison of Edge detection without and with image contrast control during wave rundown Once the DLT coefficients are known, the world coordinates can directly be transformed to local coordinates in images as: 2 4 L 1 L 9 u L 2 L 10 u L 3 L 11 u L 5 L 9 u L 6 L 10 u L 7 L 11 u 3 5 2 6 6 6 4 x y z 3 7 7 7 5 = 2 4 u L 4 v L 8 3 5 (8.3) which gives a direct correspondence from local coordinates in images to world coordinates with z= 0 , provided DLT parameters in Eq. (8.4), are known. 2 4 x y 3 5 = 2 4 L 1 L 9 u L 2 L 10 u L 5 L 9 u L 6 L 10 u 3 5 1 2 4 u L 4 v L 8 3 5 (8.4) 122 To calculate DLT parameters with the camera with ground control points, rewriting Eq. (8.4) in the form Ax= b with x is the vector as unknown DLT parameters. 2 4 x i y i 1 0 0 0 u i x i u i y i 0 0 0 x i y i 1 v i x i v i y i 3 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 L 1 L 2 L 4 L 5 L 6 L 8 L 9 L 10 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 4 u i v i 3 5 (8.5) where (u i ,v i ) and (x i ,y i ) are known image and world coordinates with ground controls points. At least 4 control points are needed to obtain 8 DLT parameters. The more control points are, the more accurate is. 220 ground control points for a larger area of interest (for solitary wave runup) and 164 ground control points for a smaller area of interest (for top-hat spectrum wave runup) obtained by 2 measuring tapes attached on the sides of the flume for pairing the local in image and world coordinate are used. The error e xy is calculated by the mean distance between the real (x, y) and predicted (x p , y p ) coordinates: e xy = 1 N N å i=1 q [x p (i) x(i)] 2 +[y p (i) y(i)] 2 (8.6) where N is the number of ground control points. The mean error value for solitary wave runup is 0.68113 inches and standard deviation is 0.40298, whereas the mean error value for top-hat spectrum wave runup is 0.19994 inches and standard deviation is 0.1103. The error becomes bigger where ground control points are far from the camera because the image resolution becomes low. Fig. 8.5 shows a distorted image and undistorted image with DLT transformation. Once the DLT parameters are obtained, the detected points in local coordinates of images can be transformed to global coordinates by using Eq. (8.4) considering local coordinated in undistorted 123 Figure 8.5: Distorted image (top) and undistorted image with DLT transformation (bottom) image. Although the process to reduce noises and to obtain clear edges was carried out, there are still noises from water drops during wave runup and rundown. It is not a effective way to filter all noises from distributions of detected points. The median value of detected points was chosen with transformed coordinate to plot the time series of wave runup. In addition, the maximum, median and minimum detected point time series were plotted to check abnormal points of median points as shown in the Fig. 8.6. If abnormal points were found, the points were interpolated between neighbor points. Only one case of 10 cases in this chapter, the solitary wave with a/h=0.07 had abnormal points. 124 Figure 8.6: The time series of wave runups: Distribution of maximum, median, minimum points by Canny edge detection. Solitary wave with a/h=0.07 (top) and case C of top-hat spectral wave (bottom) 125 8.4 Results and Analysis For solitary and top-hat spectral wave generations, a source location is same as the end of baffle of experiment. In the case of solitary wave, a sponge layer is located at the opposite of wave propagation, while the top-hat spectral wave is generated without a sponge layer to predict a bounding long wave motion same as the experiment. Fig. 8.7 presents comparisons of solitary wave with a/h=0.12 and top-hat spectral wave with C case surface elevations time series between experimental data and numerical results by COULWA VE and OpenFOAM at WG1. Furthermore, the maximum wave amplitudes of all cases from the experimental data, COULWA VE and Open- FOAM at WG1 are described in table 8.1. The comparisons show that incident waves between the experiment and the numerical simulation before wave runup the sloping beach are closely same. Table 8.1: Comparison of maximum solitary and top-hat spectral wave amplitudes at WG1 case target a/h Experiment (a/h) COULWA VE (a/h) OpenFOAM (a/h) Solitary 0.0600 0.0600 0.0600 0.0595 Solitary 0.0700 0.0699 0.0701 0.0694 Solitary 0.0800 0.0804 0.0802 0.0794 Solitary 0.0900 0.0894 0.0902 0.0893 Solitary 0.1000 0.1006 0.1003 0.0993 Solitary 0.1100 0.1121 0.1103 0.1093 Solitary 0.1200 0.1233 0.1204 0.1193 Top-hat A 0.0452 0.0442 0.0445 0.0445 Top-hat B 0.0812 0.0787 0.0786 0.0770 Top-hat C 0.1207 0.1120 0.1170 0.1184 126 Figure 8.7: Comparisons of solitary wave with a/h=0.12 (top) and Case C of top-hat spectral wave (bottom) surface elevations time series at WG1. Blue: model prediction by COULWA VE; Red: experimental data; green: model prediction by OpenFOAM 127 Fig. 8.8 to 8.14 presents the comparison of the time series of solitary wave inundations and runups with a/h=0.06 to 0.12 measured and predicted with COULWA VE and OpenFOAM. The time series of runup height was obtained by multiplying inundation length and sin(tan 1 (m 2 )), where m 2 is the second slope from offshore. The time series of inundations and wave runups measured are in close agreement with the predictions by numerical simulations, COULWA VE and OpenFOAM. In detailed discrepancies between observations and predictions, the predictions with COULWA VE are faster rundown than the prediction with OpenFOAM and the measurements. This reason is that the predictions with COULWA VE may not be fully dispersive in very shallow region over the composite slope due to the inherent characteristics. On the other hands, the predictions with OpenFOAM with relatively small wave amplitudes (a/h<0.09) are good agreement with the measurements. However, the predictions with OpenFOAM with relatively high wave amplitudes (a/h>0.08) are higher wave runups and more abnormal path of wave runups and rundowns such as near t=13 and 15 seconds than the experiments. The reason results from inherent characteristics of VOF scheme difficult to track an exact free surface elevation in strong interaction area between air and water phase. In the simulation of solitary wave with a/h=0.12, water bubbles over the compos- ite slope are observed during the maximum wave runup and the minimum rundown. The second runups by the reflected solitary wave measured and predicted with OpenFOAM are in close agree- ment with the experiments. Yet, the predictions with COULWA VE are different because a sponge layer is located at the opposite of wave propagation as mentioned earlier. The comparison of max- imum solitary wave runups between numerical simulations and experimental data are described in table 8.2. The maximum experimental runups are expressed as the median values of the detected maximum runups and the errors between maximum and minimum values of the detected maximum runups. Moreover, Fig. 8.15 presents the comparison of maximum runups with the experimental measurements and the model predictions with COULWA VE and OpenFOAM. The predictions are in close agreement with the solid line which means perfect agreement with the experiments. 128 Figure 8.8: Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.06 129 Figure 8.9: Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.07 130 Figure 8.10: Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.08 131 Figure 8.11: Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.09 132 Figure 8.12: Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.10 133 Figure 8.13: Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.11 134 Figure 8.14: Comparison of the time series of solitary wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: a/h=0.12 135 Table 8.2: Comparison of maximum solitary wave runups between numerical simulations and experimental data a/h Experiment (R/h) COULWA VE (R/h) OpenFOAM (R/h) 0.0600 0.2334 0.0142 0.2163 0.2334 0.0700 0.2701 0.0144 0.2533 0.2676 0.0800 0.3013 0.0134 0.2959 0.3031 0.0900 0.3262 0.0120 0.3360 0.3382 0.1000 0.3535 0.0106 0.3410 0.3598 0.1100 0.3799 0.0075 0.3885 0.3937 0.1200 0.3935 0.0093 0.3944 0.4277 Figure 8.15: Comparison of maximum solitary wave runup of experiment, COULWA VE and Open- FOAM 136 Fig. 8.16 to 8.18 presents the comparison of the time series of inundations and runups of top- hat spectral waves (case A, B and C) measured and predicted with COULWA VE and OpenFOAM. The time series of runup height is obtained by multiplying inundation length and sin(tan 1 (m 2 )), where m 2 is the second slope from offshore. The time series of inundations and wave runups for all cases measured and predicted with COULWA VE and OpenFOAM have discrepancies because the wave conditions at WG4 are also different as described in the chapter 7. However, the tendencies of wave runups and rundowns are so close between the measurements and the predictions. In detail, the sequence of wave runups by the bound long waves can be observed. In addition, the time series of runups with high resolutions including wave runups and rundown with high frequencies can be observed. In the case A, the runups of COULWA VE is overpredicted because the comparison of wave amplitude spectra in the Fig. 7.5 and 7.6 shows the COULWA VE is slightly higher than the experiment due to innate characteristic weakly dispersive as seen in Fig. 8.16. The Fig.8.17 presents the runups predicted during peak period (t=12 to 15 second) are underpredicted compared to it measured because the time series of water surface elevations during the period of short wave group predicted are also more underpredicted than the measurement as shown in the Fig. 7.7 and 7.8. In Fig. 8.18, the time series of runups with numerical simulations at 15 second are underpredicted compared to the experiments. The time series of water surface elevations with numerical simulations are underpredicted at the end of wave packet period (t=16 to 17 second) as shown in the Fig. 7.9 and 7.10. 137 Figure 8.16: Comparison of the time series of top-hat spectral wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: Case A 138 Figure 8.17: Comparison of the time series of top-hat spectral wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: Case B 139 Figure 8.18: Comparison of the time series of top-hat spectral wave inundation (top) and runup (bottom) of COULWA VE, OpenFOAM and experimental data: Case C 140 To investigate the space-time evolution of wave transformation and runup, the contour plots of surface elevation and time series of wave runup by the laboratory observation for case A, B and C are illustrated in Fig. 8.19. In spite of the low spatial resolution, the Fig. 8.19 shows that the waves begin to climb up the slope when the transient wave groups approach shoreline and that small long wave crest (light green color) makes the subsequent wave runup. Fig. 8.20 and 8.21 illustrate the contour plots of surface elevation and time series of wave runup for case A, B and C by OpenFOAM and COULWA VE, respectively. These figures with higher spatial resolution than the observation show that the paths of wave groups linearly propagate on constant water depth and then the path propagates with highly non-linear motion on the composite slope. The Fig. 8.22 presents the space-time evolution of low-frequency component ( f c < 0.4) of surface elevation and the time series of wave runup by the laboratory observation for case A, B and C. The figure shows that the bound long wave is amplified by shoaling after passing the first slope and that the subsequent wave runups are dominated by the reflected bound long wave. The Fig. 8.20 and 8.21 illustrate the contour plots of low frequency of surface elevation and time series of wave runup for case A, B and C by OpenFOAM and COULWA VE, respectively. These figures obviously show that the subsequent wave runups are dominated by the reflected bound long wave. The positive surges in advance of bound long wave trough leads a low frequency runup on the composite slope. 141 Figure 8.19: Time series of wave runup and space-time evolution of surface elevation of experi- mental data, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. 142 Figure 8.20: Time series of wave runup and space-time evolution of surface elevation of Open- FOAM, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. 143 Figure 8.21: Time series of wave runup and space-time evolution of surface elevation of COUL- WA VE, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. 144 Figure 8.22: Time series of low frequency component of wave runup and space-time evolution of low frequency surface elevation of experimental data, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. 145 Figure 8.23: Time series of low frequency component of wave runup and space-time evolution of low frequency surface elevation of OpenFOAM, case A (top), B (middle) and C (bottom). Hori- zontal lines indicate the bottom slope changes. 146 Figure 8.24: Time series of low frequency component of wave runup and space-time evolution of low frequency surface elevation of COULWA VE, case A (top), B (middle) and C (bottom). Horizontal lines indicate the bottom slope changes. 147 8.5 Conclusions The measurement for wave runup were performed with images captured by using the one action camera, food colorings, light diffuser materials low costs as well as edge detection function in MATLAB toolbox. In solitary waves with various amplitudes, the time series of inundations and wave runups measured are in close agreement with the predictions by numerical simulations, COULWA VE and OpenFOAM. The predictions of the maximum wave runups are in close agree- ment with the solid line which means perfect agreement with the experiments. In the cases of top-hat spectral waves, the time series of inundations and wave runups for all cases measured and predicted with COULWA VE and OpenFOAM have discrepancies because the wave conditions before wave runup are different as well. However, the tendencies of wave runups and rundowns are so close between the measurements and the predictions. In detail, the sequence of wave runups by the bound long waves can be observed. In addition, the time series of runups with high resolutions including wave runups and rundown with high frequencies can be observed. To investigate the space-time evolution of wave transformation and runup, the contour plots of surface elevation and time series of wave runup by the laboratory observation and the model predictions. The paths of wave group propagating with highly non-linear motion on the composite slope can be presented. The space-time evolution of low-frequency component ( f c < 0.4) of surface elevation and the time series of wave runup by the laboratory observation and the model predictions is illustrated. The bound long wave is amplified by shoaling after passing the first slope and that the subsequent wave runups are dominated by the reflected bound long wave. In addition, the positive surges in advance of bound long wave trough lead a low frequency runup on the composite slope. 148 9 Future Research As mentioned in the chapter 7, the top-hat spectral waves generated by the experimental HCW were much more over-estimated than numerical results by COULWA VE and OpenFOAM. It means that sudden accelerative or decelerative movements with different direction by make target flow rates different with real flow rates to generate waves. It may solve multiplying a scale factor on the input signal. However, the factors can be changeable depending on wave height and frequency. Thus, the relationship of the scale factor between the wave height and frequency will be investi- gated according to performances of numerous cases. The image processing technique for measuring wave runup with low-cost devices (an action camera, a light diffuser, etc.) was developed as described in chapter 8. Viriyakijja and Chinnarasri (2015) carried out measuring wave surface elevation with an CCD camera by using the image processing technique. Similarly, wave surface elevations will be measured by multiple cameras on the side of wave flume to replace traditional wave gauges. 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Journal of Fluid Mechanics, 416:315–348, 2000. 155 Appendix A Integration of Horizontal Velocity Profile in 3 Baffles A.0.1 Linear Wave Figure A.1: Definition and notation of linear and second wave Horizontal particle velocity by linear wave theory is as follow: u = gak s cosh(h+ z) coshkh cos(kxst) (A.1) 156 Integrating the horizontal particle velocity in arbitrary water depth from a to b, flow rate, Q, is as follow: Q = Z b a udz = Z b a gak s cosh(h+ z) coshkh cos(kxst)dz Q = ga s sinhk(h+ z) coshkh cos(kxst) b a (A.2) The flow rates for 3 baffle as shown in the Fig. A.1 are expressed as follow: Q 1 = ga s sinhk(h+ z) coshkh cos(kxst) z 1 z 2 Q 2 = ga s sinhk(h+ z) coshkh cos(kxst) z 2 0 Q 3 = ga s sinhk(h+ z) coshkh cos(kxst) 0 z 3 (A.3) Ifh is below still water level, 0, Q 2 = ga s sinh(h+ z) coshkh cos(kxst) z2 h Q 3 = 0 (A.4) 157 A.0.2 Stokes Second Order Wave Theory Horizontal particle velocity by Stokes second wave theory is as follow: u= gHk 2 coshk(h+ z) coshkh cos(kxst)+ 3H 2 sk 16 cosh2k(h+ z) sinh 4 kh cos2(kxst) (A.5) Integrating the horizontal particle velocity in arbitrary water depth from a to b, flow rate, Q, is as follow: Q = Z b a udz = Z b a gHk 2 coshk(h+ z) coshkh cos(kxst)+ 3H 2 sk 16 cosh2k(h+ z) sinh 4 kh cos2(kxst)dz Q = ga s sinhk(h+ z) coshkh cos(kxst)+ 3H 2 s 32 sinh2k(h+ z) sinh 4 kh cos2(kxst) b a (A.6) The flow rates for 3 baffle as shown in the Fig. A.1 are expressed as follow: Q 1 = ga s sinhk(h+ z) coshkh cos(kxst)+ 3H 2 s 32 sinh2k(h+ z) sinh 4 kh cos2(kxst) z 1 z 2 Q 2 = ga s sinhk(h+ z) coshkh cos(kxst)+ 3H 2 s 32 sinh2k(h+ z) sinh 4 kh cos2(kxst) z 2 0 Q 3 = ga s sinhk(h+ z) coshkh cos(kxst)+ 3H 2 s 32 sinh2k(h+ z) sinh 4 kh cos2(kxst) 0 z 3 (A.7) Ifh is below still water level, 0, Q 2 = ga s sinhk(h+ z) coshkh cos(kxst)+ 3H 2 s 32 sinh2k(h+ z) sinh 4 kh cos2(kxst) z 2 0 Q 3 = 0 (A.8) 158 A.0.3 Solitary Wave Figure A.2: Definition and notation of solitary wave Free surface elevation and horizontal particle velocity by solitary wave theory is as follow: h = asech 2 r 3a 4h 3 (x ct);c 2 = gz(1+ a z ) (A.9) u= r g h h h 2 4h + h 2 3 z 2 2 ¶ 2 h ¶x 2 (A.10) where ¶ 2 h ¶x 2 can be derived as follow: ¶ 2 h ¶x 2 = ¶ ¶x ¶ ¶x asech 2 r 3a 4h 3 (x ct) !! = ¶ ¶x 2a r 3a 4h 3 sech 2 r 3a 4h 3 (x ct)tanh r 3a 4h 3 (x ct) ! = 3a 2 h 3 sech 2 r 3a 4h 3 (x ct)tanh 2 r 3a 4h 3 (x ct) 3a 2h 3 sech 4 r 3a 4h 3 (x ct) = 3a h 3 tanh 2 r 3a 4h 3 (x ct)h 3 2h 3 h 2 (A.11) 159 Integrating the horizontal particle velocity in arbitrary water depth from a to b, flow rate, Q, is as follow: Q = Z b a udz = Z b a r g h h h 2 4h + h 2 3 z 2 2 ¶ 2 h ¶x 2 dz = Z b a r g h " h h 2 4h + h 2 3 z 2 2 3a h 3 tanh 2 r 3a 4h 3 (x ct)h 3 2h 3 h 2 !# dz Q = r g h " h h 2 4h + a h tanh 2 r 3a 4h 3 (x ct)h h 2 2h # z " a 2h 3 tanh 2 r 3a 4h 3 (x ct)h h 2 4h 3 # z 3 b a (A.12) The flow rates for 3 baffle as shown in the Fig. A.2 are expressed as follow: Q 1 = r g h " h h 2 4h + a h tanh 2 r 3a 4h 3 (x ct)h h 2 2h # z " a 2h 3 tanh 2 r 3a 4h 3 (x ct)h h 2 4h 3 # z 3 z 1 z 2 Q 2 = r g h " h h 2 4h + a h tanh 2 r 3a 4h 3 (x ct)h h 2 2h # z " a 2h 3 tanh 2 r 3a 4h 3 (x ct)h h 2 4h 3 # z 3 z 2 0 Q 3 = r g h " h h 2 4h + a h tanh 2 r 3a 4h 3 (x ct)h h 2 2h # z " a 2h 3 tanh 2 r 3a 4h 3 (x ct)h h 2 4h 3 # z 3 0 z 3 (A.13) 160 Appendix B Block Diagram of Labview for Controlling a Motor B.0.1 Block Diagram of Labview for Wave Generation: Controlling Three Servo Motors Figure B.1: Block Diagram of Labview for controlling 3 servo motors 161 B.0.2 Block Diagram of Labview for Calibration of Wave Gauge: Control- ling a Stepper Motor Figure B.2: Block Diagram of Labview for controlling a stepper motor 162 Appendix C Coefficients of Second Order Solution C= [2s 1 s 2 (s 1 s 2 )(1+a 1 a 2 )+s 3 1 (a 2 1 1)s 3 2 (a 2 2 1)](s 1 s2)(a 1 a 2 1) s 2 1 (a 2 1 1) 2s 1 s 2 (a 1 a 2 1)+s 2 2 (a 2 2 1) +(s 2 1 +s 2 2 )s 1 s 2 (a 1 a 2 + 1) (C.1) D= [2s 1 s 2 (s 1 +s 2 )(a 1 a 2 1)+s 3 1 (a 2 1 1)+s 3 2 (a 2 2 1)](s 1 +s2)(a 1 a 2 1) s 2 1 (a 2 1 + 1) 2s 1 s 2 (a 1 a 2 + 1)+s 2 2 (a 2 2 1) (s 2 1 +s 2 2 )+s 1 s 2 (a 1 a 2 1) (C.2) E = 1 2 a 1 a 2 [2s 1 s 2 (s 1 s 2 )(1+a 1 a 2 )+s 3 1 (a 2 1 1)s 3 2 (a 2 2 1)] (C.3) F = 1 2 a 1 a 2 [2s 1 s 2 (s 1 +s 2 )(1a 1 a 2 )s 3 1 (a 2 1 1)s 3 2 (a 2 2 1)] (C.4) wherea coefficients area 1 = coth(k 1 h) anda 2 = coth(k 2 h). 163

Abstract (if available)

Abstract This study is aimed at developing new experimental equipment which would permit the study of multi-scale and vertically-variable oceanographic flows using a system called Hydraulic-Control Wave-maker (HCW). Both the inlet and outlet flume boundaries are composed of an adjustable set of vertical baffles. Each baffle is connected to an individual flow control system, such that the vertical distribution of flow is entirely controllable. In such a system, a main advantage is that any arbitrary flow can be reasonably created. ❧ Firstly, a new method of wave generation by using HCW was tested through numerical analysis. The optimized design of HCW is suggested through sensitivity analyses with respect to baffle length, height, number and position. For verification, the numerical results from the optimized HCW were compared with analytical solutions. In the case of a relative high wave amplitude, the phase lagged behind the analytical solution, but most of the results are in good agreement with the analytical solutions. Furthermore, wave absorption techniques were performed by solitary wave generation. Although the results of wave absorption cannot perfectly remove the reflected wave, the technique may be feasible because the lack of the reflected wave signal is a reasonable indicator. ❧ Secondly, a small physical model as a prototype with one baffle was developed and tested. In order to create a wave, two calibrations between the motor speed and the flow rate were carried out, and linear relationships are obtained. Both calibrations show similar linear increase so that mass conservation is approved. By using this relationship, sinusoidal waves were generated. The time series of motor speed profiles are presented and the possibility of a HCW system can be confirmed. ❧ Thirdly, HCW with three baffles including a movable top baffle were built. The top baffle can be moved along a surface elevation by using a combination of screw-jack and motor, a kind of linear actuator, similar to the flow control system. The time-varying volume of top baffle has been considered and Ramp period has been introduced for a smooth piston movement. For verification, solitary waves, sinusoidal waves and stokes waves were created. The time series of surface elevations for laboratory observation were compared to the desired target data. The results are in good agreement with the target data in spite of the reflected waves. Yet, the results were overpredicted because the difference of two calibrations through the small prototype HCW was not considered. In cases of periodic relative long waves, sinusoidal and stokes wave with kh=0.5, shows unwanted harmonics are appeared and the results are not overpredicted. This is because the air is sucked into the top baffle through very small gaps of the top baffle hinge and it disturbed flow and mass conservation of water. An active wave absorption by calculating an arrival time through wave celerity was carried out in the case of solitary wave. The results show the reflection coefficients are low and the active wave absorption would be a feasible way. The enhanced physical model with three paired cylinders has been built for higher wave generations than with three cylinders. By using the enhanced HCW, the corrected calibration was investigated. By the comparison between measured and desired solitary wave amplitude, the corrected calibration was obtained. ❧ The generation of long wave by the shoaling and breaking of the propagation of top-hat spectral waves, a transient focused wave group, over composite slopes as one of the applications of HCW. Moreover, the time series of water surface elevation and the amplitude spectra by the laboratory observations were compared with numerical simulations, OpenFOAM and COULWAVE, based on Boussinesq and RANS equations, respectively. The comparisons show that long wave generation in the experiments is in close agreement with numerical simulations. The times series of water surface elevations shows that the short wave groups are transformed to long waves, and their propagation paths can be observed directly. In addition, the comparison of the amplitude spectra presents that the primary and superharmonic wave amplitudes are increased or decreased by wave shoaling or wave breaking, respectively. Long wave components obtained from low-pass filtered surface elevations by time-series measurement obviously describe that the increased amplitudes in shallow water by shoaling and their propagation path. The wave amplitude spectra measured and predicted are compared to the second-order wave theory in association with an interaction of wave groups. The comparisons of the theory at initial location and experimental data and numerical results at various cross shore locations present that a spatial redistribution of wave energy is obviously described during the shoaling and breaking processes. The wave amplitude spectra with low-frequency increase over nearshore regions because the wave amplitudes with low-frequency become increased in shallow water due to wave shoaling and the energy is transferred from the waves with higher frequency. ❧ The measurement for wave runup were performed with images captured by using the one action camera, food colorings, light diffuser materials low costs as well as edge detection function in MATLAB toolbox. In solitary waves with various amplitudes, the time series of inundations and wave runups measured are in close agreement with the predictions by numerical simulations, COULWAVE and OpenFOAM. The predictions of the maximum wave runups are in close agreement with the solid line which means perfect agreement with the experiments. In the cases of top-hat spectral waves, the time series of inundations and wave runups for all cases measured and predicted with COULWAVE and OpenFOAM have discrepancies because the wave conditions before wave runup are different as well. However, the tendencies of wave runups and rundowns are so close between the measurements and the predictions. In detail, the sequence of wave runups by the bound long waves can be observed. In addition, the time series of runups with high resolutions including wave runups and rundown with high frequencies can be observed.

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Asset Metadata

Creator Ko, Haeng Sik (author)

Core Title The development of a hydraulic-control wave-maker (HCW) for the study of non-linear surface waves

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 11/09/2016

Defense Date 10/24/2016

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,solitary wave,top-hat spectrum,wave flume,wave run-up,wavemaker

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lynett, Patrick J. (committee chair), Lee, Jiin-Jen (committee member), Luhar, Mitul (committee member)

Creator Email haengsik@usc.edu,kalbread@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-320190

Unique identifier UC11213717

Identifier etd-KoHaengSik-4910.pdf (filename),usctheses-c40-320190 (legacy record id)

Legacy Identifier etd-KoHaengSik-4910.pdf

Dmrecord 320190

Document Type Dissertation

Format application/pdf (imt)

Rights Ko, Haeng Sik

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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