AN ABSTRACT OF THE THESIS OF Robert Edward Jensen for the degree Master of Ocean Engineering (Name) (Degree) in Civil Engineering presented on June 9, 1978 (Major Department) Title: FINITE AMPLITUDE DEEP WATER WAVES: A COMPARISON OF THEORETICAL AND EXPERIMENTAL KINEMATICS AND DYNAMICS Redacted for Privacy Abstract approved: Dr. Charles K. Sollitt Problems associated with quantification of deepwater wave kine- matics and dynamics are examined. Summaries of Stokes' theory and Dean's stream function theory are reported. A computer algorithm is developed to solve the hydrodynamic characteristics for Stokes' fifth andthird order and linear theories. Tests are initiated to verify the results obtained from the computer routine. Experimental data are obtained for several hydrodynamic characteristics in deepwater conditions. Com- parisons between Dean's stream function, Stokes' fifthand third order and linear wave theory, and experimental resultsare made for various deep water conditions. Finite Amplitude Deep Water Waves: A Comparison of Theoretical and Experimental Kinematics and Dynamics by Robert Edward Jensen A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Ocean Engineering Completed June 9, 1978 Commencement June 1979 APPROVED: Redacted for Privacy Professor of Civil Engineering in charge of major Redacted for Privacy Head of D rtment of ivil Engineerin Redacted for Privacy/ Dean of Graduate School Date thesis is presented June 91978 Typed by Pameila Jensen for Robert Edward Jensen ACKNOWLEDGEMENTS This project would not be complete without acknowledging various individuals for their support of this thesis. Special thanks to the Sea Grant College Program for the funding of this project; to Dr. John Nath, Director of the Oregon State University Wave Research Facility for donating the needed time to conduct the experimental program; and to Dr. D.R. Basco, Associate Professor at Texas A&M University for stimulating discussions on the subject of water wave kinematics and dynamics. Two other individuals proven to be indispensable during the experimental work are Terry Dibble and Larry Crawford. Final re- view of the thesis could not have been accomplished without the support of Dr. R.T. Hudspeth, Dr. P.C. Klingeman and Dr. T. Yamamoto.A very special thanks to a professor whom I have grown to admire for his intelligence, guidance and support on this project and throughout the course of my academic career at Oregon State University goes to Dr. C.K. Sollitt. Finally, I would like to express my appreciation to my wife, Pam, not only for typing of the manuscript but also for her patience and understanding. TABLE 02 CONTENTS I. INTRODUCTION 01 1.1 Problem Statement 01 1.2 Previous Theoretical Work 02 1.3 Previous Experimental Work 04 II. THEORY 06 2.1 Boundary Valu2 Problem 06 2.2 Stokes' and Linear Wave Theories 11 2.2.1 Solution tc the Boundary Value Problem 11 2.2.2 Iteration Technique for Solving Definite Equations 17 2.2.3 Calculation of Hydrodynamic Parameters Above the Still Water Level 20 2.3 Dean's Stream Function Theory 27 III. DIGITAL COMPUTATIONS 31 3.1 Stokes' Wave Theory Computer Algorithm 31 3.2 Implementation of Dean's Stream Function 40 IV. EXPERIMENTAL APPARATUS AND PROCEDURES 41 4.1 Wave Research Facility 41 4.2 Kinematic Measurements 41 4.3 Dynamic Measurements 48 V. COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS . 52 5.1 Background Information 52 5.2 Theoretical and Experimental Results 53 VI. CONCLUSIONS 08 6.1 Summary 98 6.2 Evaluation 98 6.3 Future investigations 102 Bibliography 103 Appendix A 106 Appendix B 103 Appendix C 109 Program STOKES 110 Program PERTA 126 Appendix D 141 TABLE OF CONTENTS (Continue6) Appendix E 143 Appendix F 151 Index 161 List of Figures Figure Page 1.1 Approximate Range of E:perimental Work 05 2.1 Definition Sketch 07 2.2 Definition Sketch of the Method Employed in the Taylor 22 Series Expansion 3.1 Flow Diagram of the Program STOKES 32 3.2 Definition Sketch 33 3.3 Relative Operational Range of the STOKES Program 39 4.1 Test Facility 42 4.2 Calibration Curve for the Current Meter 45 4.3 Current Meter and Pressure Transducer Set up, Locking 46 From the Wave Board 5.1 Percent Difference of Theoretical Wave Crest Amplitudes 56 in Comparison to Measured. Results 5.2 Percent Difference of Theoretical Wave Trough Amplitudes 57 in Comparison to Measured Results 5.3 Case 7.5C' Maximum Nondimensional Kinematic and Dynamic 61 Profiles 5.4 Case 7.5B Maximum Nondimensional Kinematic and Dynamic 64 Profiles 5.5 Case 7.5A Maximum Nondimensional Kinematic andDynamic 66 Profiles 5.6 Case 8C' Maximum Nondimensional Kinematic andDynamic 68 Profiles 5.7 Case 8B Maximum Nondimensional Kinematic andDynamic 70 Profiles 5.8 Case 8A Maximum Nondimensional Kinematicand Dynamic 71 Profiles 5.9 Case 9C' Maximum Nondimensional Kinematic andDynamic 73 Profiles 5.10 Case 9B Maximum Nondimensional Kinematic andDynamic 74 Profiles List cf Figures (Continued) Figure Page 5.11 Case 9A Maximum Nondimensional Kinematic and Dynamic 75 Profiles 5.12 Comparison Between Theoretical and Measured Horizontal 77 Accelerations at Two Different Depths 5.13 Case 7.5C' and 7.5B Horizontal Velocity and Dynamic 80 Pressure Profiles Under Wave Trough 5.14 Case 8C' and 8B Horizontal Velocity and Dynamic 81 Pressure Profiles Under Wave Trough 5.15 Case 9C' and 9B Horizontal Velocity and Dynamic 84 Pressure Profiles Under Wave Trough 5.16 Dynamic Breaking Wave Criteria 85 5.17 Case 7.5C' and 7,5B Maximum Nondimensional Horizontal 89 Velocity. Gradient. Profiles 5.18 Case 8C' and 8B Maximum Nondimensional Horizontal 90 Velocity Gradient. Profiles 5.19Case 9C' and 9B Maximum Nondimensional Horizontal 91 Velocity Gradient Profiles 5.20Definition Sketch, the Adjustment in Theoretical 95 Velocity Profiles List of Photographs Photograph Page 4.1 Current Meter 44 4.2 Dynamic Pressure Measurement Apparatus 50 4.3 Pressure Gradient Measurement Apparatus 50 List of Tables Table Page 2.1 Comparison of Hydrodynamic Parameters Above the S.W.L. 25 3.1 Approximate Running Time and Ccst for STOKES 38 5.1 Experimental Wave a)nditions 52-53 5.2 Comparison of Crest and Trough Amplitudes 54-55 5.3 Phase Angle Relationships Between Experimental and 78-79 Theoretical Results 5.4 Wave Reflection Analysis Causing Contamination of 93 Horizontal Velocity Measurements 5.5 Adjustments in the Horizontal Velocity Distribution 96 List of Syrobols A Dean's stream function representation of wave heights which are 0.25 of the theoretical breaking wave height A.. Coefficients associated with the expansion 1J of the velocity potential to the i andjth order of Stokes' theory B Dean's stream function representation of wave heights which are 0.5 of the theoretical breaking wave height, referenced in Chapter 5 and 6 B(t) - B Bernoulli constant B Average value of all Bn's, in the least squares analysis of Dean's stream function theory B.. Coefficients associated with the expansion of 13 the water surface profile to the i and 3nth order of Stokes' theory C Wave celerity CO Linear wave theory wave celerity C. Coefficients associated with the expansion of 1 the dynamic free surface boundary condition to the ith order of Stokes' theory C Classification of wave heights which are ap- proximately equal to 0.75 of the theoretical breaking wave height F Arbitrary function in Section 2.2.3; force elsewhere H Wave height HO Wave height in computer routines Hz Hertz JJ The maximum order of Dean's stream function theory K. Shorthand notation in conjunction with the J coefficients associated with the jth order of the velocity potential for Stokes' theory List of Symbols (Continued) K . Shorthand notation in conjunction with the 3 coefficients associated with the jth order of the water surface profile for Stokes' theory L Results obtained employing linear wave theory, Chapter 5; wavelength elsewhere I, Deep water wavelength NUMETA Number of locations above the S.W.L. to the wave crest where the hydrodynamics are com- puted employing the STOKES program NUMH Number of locations from one wave height below the S.W.L. to the mudline where the hydrody- namics are computed employing the STOKESpro gram NUMH Number of locations from the S.W.L. to one wave height in depth where the hydrodynamics are computed employing the STOKES program NUMX Number of uniform phase angle locations where the hydrodynamics are computed employing the STOKES program P Pressure field P D Dynamic pressure P Total pressure T S Results obtained employing stream function theory S.W.L. Still water level T Wave period a Acceleration a Horizontal acceleration x a Vertical acceleration z c Hyperbolic cosin, Appndix A and B List of Symbol9 (Continued) d Total Derivative e Exponential . th f. Arbitary function to the 3 order f(a,a), fl(a,a), f2(a,0) Arbitrary functions in Section 2,2.2 g Gravitational acceleration h Water depth i Unit vector in the x-- direction j Order of the wave theory k Unit vector in the z-direction m Mass q Velocity vector s Hyperbolic sine, Appendix A and B t Time variable u Horizontal velocity component w Vertical velocity component x Longitudinal coordinate z Depth coordinate 3 Results obtained employing Stokes' third order wave theory 5 Results obtained employing Stokes' fifth order wave theory a Arbitrary constant equal to Kh Arbitrary constant equal to A a Specific weight of water A The change of a particular variable List of Symbols (Continued) Error term Displacement of water particles in the vertical direction Water surface profile 0 Phase angle equal to K(x Ct) K Wave number equal to 27/L Order multiplier PI equal to 3.1415926 p Mass density of water Summation Velocity potential X Stream function coefficients Stream function Vector gradient operator i 3 /3x + 173/3z 3 Partial derivative operator Percent Subscripts I Depth intervals J Order of Stokes' wave theory M Phase angle intervals i Order of Stokes' wave theory j Order of any wave theory max Maximum n Incremental phase angles fl Evaluated at the free surface List of Symbols (Continu,..d) Superscripts K Iteration number The derivative with respect to a or Kh FINITE AMPLITUDE DEEP WATER WAVES: A COMPARISON OF THEORETICAL AND EXPERIMENTAL KINEMATICS AND DYNAMICS I.