
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/UF00075474/00001
Material Information
 Title:
 Water wave interaction with porous structures of irregular cross sections
 Series Title:
 UFLCOELTR
 Creator:
 Gu, Zhihao, 1957 ( Dissertant )
University of Florida  Coastal and Oceanographic Engineering Dept
Dean, Robert G. ( Thesis advisor )
Sheppard, D. Max ( Reviewer )
Kurzweg, Ulrich H. ( Reviewer )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1990
 Copyright Date:
 1990
 Language:
 English
 Physical Description:
 xvii, 201 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Berms ( jstor )
Best fit ( jstor ) Boundary conditions ( jstor ) Breakwaters ( jstor ) Damping ( jstor ) Energy dissipation ( jstor ) Ocean floor ( jstor ) Reflectance ( jstor ) Velocity ( jstor ) Waves ( jstor ) Coastal and Oceanographic Engineering thesis Ph. D Dissertations, Academic  Coastal and Oceanographic Engineering  UF
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Abstract:
 A general unsteady porous flow model is developed based on the assumption that the porous media can be treated as a continuum. The model clearly defines the role of solid and fluid motions and henceforth their interaction. All the important resistant forces are clearly and rigorously defined. The model is applied to the gravity wave field over a porous bed of finite depth. By applying linear wave theory, an analytical solution is obtained, which is applicable to the full range of permeability. The solution yields significantly different results from those of contemporary theory. The solution requires three empirical coefficients, respectively representing linear, nonlinear, and inertial resistance. Laboratory experiments using a standing wave system over a porous seabed were conducted to determine these coefficients and to compare with analytical results. The coefficients related to linear and non.linear resistances were found to be close to those obtained by previous investigators. The virtual mass coefficient was determined to be around 0.46, close to the theoretical value of 0.5 for a sphere. The analytical solution compared well with the experiments. Based on this porous flow model and linear wave theory, two numerical models using boundary integral element method with linear elements are developed for permeable submerged breakwaters and berm breakwaters, respectively. Due to the establishment of a boundary integral expression for wave energy dissipation in a porous domain and the application of the radiation boundary condition on the lateral boundary (ies), the numerical models are highly efficient while maintaining sufficient accuracy. The numerical results show that the wave energy dissipation within a porous domain has a well defined maximum value at certain permeability for a specified wave and geometry condition. The nonlinear effects int he porous flow model are clearly manifested, as all the flow field properties are no longer linearly proportional to the incident wave heights. The numerical results agreed reasonably well with the experimental data on the seaward side. On the leeward of the breakwater, despite the appearance of higher order harmonics, the numerical model produces acceptable results of energy transmission based on energy balance.
 Thesis:
 Thesis (Ph. D.)University of Florida, 1990.
 Bibliography:
 Includes bibliographical references (leaves 197199).
 General Note:
 Typescript.
 General Note:
 Vita.
 Funding:
 This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
 Statement of Responsibility:
 by Zhihao Gu.
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 University of Florida
 Holding Location:
 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact Digital Services (UFDC@uflib.ufl.edu) with any additional information they can provide.
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UFL/COEL/TR083
WATER WAVE INTERACTION WITH POROUS STRUCTURES OF IRREGULAR CROSS SECTIONS
by
Zhihao Gu
Dissertation
1990
WATER WAVE INTERACTION WITH
IRREGULAR CROSS
POROUS STRUCTURES OF SECTIONS
By
Zhihao Gu
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1990
ACKNOWLEDGEMENTS
It is almost impossible to fully express the author's appreciation in words to everyone who has contributed to the completion of this dissertation.
First of all, the author wishes to express his deepest appreciation to his advisor and the chairman of his advisory committee, Dr. Hsiang Wang, for his advice, friendship, encouragement and financial support which enabled the author to come to the United States to pursue his higher education. The four years of graduate work with Dr. Wang have been a challenging and enjoyable experience in the author's life.
The author would also like to thank Dr. Robert G. Dean, Dr. D. Max Sheppard and Dr. Ulrich H. Kurzweg for serving as the members of his doctoral advisory committee; Dr. Daniel M. Hanes for revising the dissertation and attending the final exam. Thanks are also due to all other teaching faculty members in the department during the author's graduate study: Dr. Michel K. Ochi, Dr. Ashish J. Mehta, Dr. Peter Y. Sheng and Dr. James T. Kirby (now at University of Delaware), for their teaching efforts, time and being role models to the author. The author has benefitted substantially from their invaluable experience, knowledge and continuous inspiration throughout the four years' study. The author is especially grateful for the valuable discussions with Dr. Dean and Dr. Kirby on numerous topics directly or indirectly related to this work. The continuous financial support throughout the Ph.D. program by the Coastal and Oceanography Engineering Department is also greatly appreciated.
The author is deeply indebted to his former advisors and supervisors in China, Profs. WenFa Lu, Zhizhuang Xieng and Weicheng Wang. Without their generous support and recommendation, it would have been impossible for the author to study abroad in the first place. The continuous support and understanding by them and all other faculty and staff members in the author's former working unit are highly appreciated.
Very special thanks are given to Prof. Alf Torum, Head of Research in Norwegian Hydrotechnical Laboratory, Trondheim, Norway, and Dr. Hans Dette in LeichtweiB Institut ffir Wasserbau, Technische Universitft Braunschweig, West Germany, for providing computer, office and accommodation facilities while the author was visiting the two institutions in late 1988 with his advisor. The initiation and the provision of technical information by Prof. Torum for the topic of berm breakwater which later became a chapter in this dissertation are gratefully acknowledged.
Thanks are given to Mr. Sidney Schofield, Mr. Jim Joiner and other staff members in the Coastal Laboratory for their help during the experimental phases of the study. The assistance provided by Mr. Subarna Malakar in the use of computer facilities is also appreciated. Thanks are extended to Helen Twedell for the efficiency and courtesy in running the archives and for the parties, cookies and cakes; and to Becky Hudson and all other secretarial personnel in the department for the great job they have done related to this work.
The support of many friends and fellow students is warmly appreciated. Special thanks go to Mrs. Jean Wang for her moral support, constant encouragement and hospitality. Deep appreciation is given to fellow students Rajesh Srinivas and Paul Work for proofreading the manuscripts and to Byunggi Hwang for assisting with rock sorting during the experiments. The friendship shared with Yixin Yan, Steve Peene, Yuming Liu and Richard McMillen and other friends, made the author's stay in Gainesville a delightful period of time.
The author's gratitude towards his family is beyond words. The author would like to thank his best friend and wife, Liqiu (also a Ph.D. student at the time), for her support, both moral and physical, patience and useful technical discussions. She was always the one to count on for assistance on weekend laboratory work, graphics work and more often on the housework beyond her share. Without her support, the road leading to this goal would have been much more tortuous. Finally, this work is dedicated to the author's wonderful mother and father. Their affection and early family education are reflected in between the lines of this work and the author's daily conduct. With all these efforts, thank god it's Phinally Done!
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ............................
LIST OF FIGURES ................................ viii
LIST OF TABLES ................................ xv
ABSTRACT .................................... xvi
CHAPTERS
1 INTRODUCTION ............................... 1
1.1 Problem Statement ............................ 1
1.2 Objectives and Scope ........................... 3
2 LITERATURE REVIEW ........................... 5
2.1 Porous Flow Models ........................... 5
2.2 WavePorous Seabed Interactions .................... 10
2.3 Modeling of Permeable Structures of Irregular Cross Sections .... 13 3 POROUSFLOW MODEL .......................... 20
3.1 The Equation of Motion ......................... 20
3.2 Force Coefficients and Simplifying Assumptions ......... 23 3.3 Relative Importance of The Resistant Forces ............. 28
4 GRAVITY WAVES OVER FINITE POROUS SEA BOTTOMS ..... 31
4.1 Boundary Value Problem ........................ 31
4.2 The Solutions of The Complex Dispersion Equation ......... 35 4.3 Results ........................ .. ......... 41
5 LABORATORY EXPERIMENT FOR POROUS SEABEDS ....... 5o
5.1 Experiment Layout and Test Conditions ................ 50
5.2 Determination of The Empirical Coefficients . 52
5.3 Relative Importance of The Resistances in The Experiment 55 5.4 Comparison of The Experimental Results and The Theoretical Values 55 6 BOUNDARY INTEGRAL ELEMENT METHOD . 64
6.1 Basic Formulation . . . 64
6.2 Local Coordinate System for A Linear Element . 68
6.3 Linear Element and Related Integrations . . 70
6.4 Boundary Conditions . . . 74
7 NUMERICAL MODEL FOR SUBMERGED POROUS BREAKWATERS 76
7.1 Governing Equations . . . 76
7.2 Boundary Conditions . . . 78
7.2.1 Boundary Conditions for The Fluid Domain . 78
7.2.2 Boundary Conditions for The Porous Medium Domain 81
7.3 BIEM Formulations . . . 82
7.3.1 Fluid Domain . . . 82
7.3.2 Porous Medium Domain . . 86
7.3.3 Matching of The Two Domains . . 87
7.4 Linearization of the Nonlinear Porous Flow Model . 88
7.5 Transmission and Reflection Coefficients . . 91
7.6 Total Wave Forces on an Impervious Structure . 92
7.7 General Description of The Computer Program . 94
7.8 Numerical Results . . . 95
8 LABORATORY EXPERIMENTS OF A POROUS SUBMERGED
BREAKWATER . . . . 107
8.1 General Description of The Experiment . . 107
8.2 Wave Transmission and Reflection . . 109
8.3 Pressure Distribution and Wave Envelope Over The Breakwater 121
9 NUMERICAL MODEL FOR BERM BREAKWATERS............ 128
9.1 Mathematical Formulations............................ 128
9.1.1 CFSBC for The Free Surface Inside Porous Medium....... 130 9.1.2 BIEEM Formulations............................ 131
9.2 Linearization..................................... 133
9.3 Numerical Results .. .. .. ... ... ... ... .... ... ....134
10 SUMMARY AND CONCLUSIONS. .. .. ... ... ... .... .....146
10.1lSummnary. .. .. .. ... ... .... .. .... ... ... ... ..146
10.2 Conclusions. .. .. .. ... ... ... ... ... ... .... .....148
10.3 Recommendations for Future Studies...................... 151
APPENDICES
A BOUNDARY INTEGRAL FORMULATION FOR ENERGY DISSIPATION
IN POROUS MEDIA................................... 153
B EXPERIMENTAL DATA FOR POROUS SEABEDS.............. 159
BIBLIOGRAPHY........................................ 197
BIOGRAPHICAL SKETCH................................. 200
LIST OF FIGURES
3.1 Definition sketch for the porous flow model . 21
3.2 Regions with different dominant resistant forces . 30
4.1 Definition Sketch .......................... 32
4.2 Progressive wave case: h, = DS = 5.0 m and h = DW = 2.0
6.0 m. (a) Nondimensional wave number kr/(aCr2/g), (b) Nondimensional wave damping rate ki/(oa2/g) . 45
4.3 Maximum nondimensional damping rate (kg),.a/(a2/g) and its
corresponding permeability parameter R as functions of nondimensional water depth h. (a2/g) . . 46
4.4 Standing wave case. (a) Nondimensional wave frequency or/(L/g) ;
(b) Nondimensional wave damping rate ag/(L/g). . 48
4.5 Solutions based on four porous flow models. (a) Nondimensional
wave frequency u,/(L/g)1, (b) Nondimensional wave damping
rate o, /(L/g)1 ...... .. ................. .... 49
5.1 Experimental setup ......................... 51
5.2 Typical wave data: (a) Averaged nondimensional surface elevation (rt/H), (b) Nondimensional wave heights (H/H1) and the
best fit to the exponential decay function . 54
5.3 The Measurements and the predictions vs. R. for L = 200.0 cm,
ho = 20.0 cm, h = 25.0 cm. (a) Wave frequency ra,, (b) Wave
damping rate ai ........................... 58
5.4 Theoretical values by the present model vs. experimental data
of Table 5.5, Table 5.6 and Table 5.7. (a) Wave frequency o,,
(b) Wave damping rate ai....................... 61
5.5 Theoretical values by the model of Liu and Dalrymple vs. experimental data of Table 5.5, Table 5.6 and Table 5.7. (a) Wave
frequency ar, (b) Wave damping rate ai. . 62
6.1 Auxiliary coordinate system . . 68
7.1 Computational domains. .. .. .. .. ... ... ... .... ..77
7.2 Flow chart of the numerical model for porous submerged breakwaters .. .. .. .. ... ... ... ... ... .... ... ....96
7.3 Wave envelopes for (a) 'Transparent' submerged breakwater; (b)
Impermeable step .. .. .. .. .. ... .... ... ... ... ..98
7.4 Porous submerged breakwater: (a) Wave form and wave envelope; (b) Envelopes of pressure and normal velocity .. .. .. ....99
7.5 Wave field over submerged breakwaters: (a) Impermeable; (b)
Permeable .. .. .. .. ... ... ... ... ... ... ... ..101
7.6 Transmission and reflection coefficients vs. stone size for different wave periods. (a) Transmission coefficient; (b) Reflection
coefficient. .. .. .. .. .. ... ... ... ... ... ... ....103
7.7 Transmission and reflection coefficients vs. R for different wave
periods. (a) Transmission coefficient; (b) Reflection coefficient. 104
7.8 Transmission and reflection coefficients vs. R for different wave
heights. (a) Transmission coefficient; (b) Reflection coefficient. .105
7.9 Wave forces and over turning moment for a impermeable submerged breakwater: (a) Wave forces; (b) Overturning moment. .106 8.1 Experiment layout .. .. .. .. ... ... ... ... ... ....108
8.2 Typical wave record. (a) Partial standing waves on the up wave
side, (b) Transmitted waves on the down wave side. .. .. .. ..111 8.3 The wave spectrum of the transmitted waves .. .. .. .. ....112
8.4 The predicted Kt and K, versus the measured Kt and K,. (a)
Kpvs. Kt,; (b) Kt vs. Ki.... .. .. .. .. .. .. .. ... ...117
8.5 The predicted and measured Kt and K, versus Hj/gT. (a) Kt
and Ktm..; (b) Kp, and Kr.. .. .. .. .. .. .. .... ... ...118
8.6 The predicted and measured Kt and K, versus H,/gT. (a) Ks,,
and Kt m; (b) K7, and K .. .. .. .. .. .. .. .... ... ...119
8.7 The predicted and measured Kt and K7 versus H,/gT. (a) K1,,
and Ktm; (b) Kp, and Krm,.. .. .. .. .. ... ... .... ...120
8.8 Transmitted and reflected wave heights versus the incident wave
heights. (a) Transmitted waves; (b) Reflected waves .. .. .. ...122
8.9 Transmitted and reflected wave heights versus the incident wave
heights. (a) Transmitted waves; (b) Reflected waves .. .. .. ...123
8.10 The envelopes of wave and pressure distribution for T = 0.858
sec.; nonbreaking wave case. (a) Wave envelope; (b) Envelope
of pressure distribution. .. .. ... ... ... ... ... ....126
8.11 The envelopes of wave and pressure distribution for T = 0.858
sec.; breaking wave case. (a) Wave envelope; (b) Envelope of
pressure distribution. .. .. .. .. .... ... ... ... ....127
9.1 Definition sketch for berm breakwaters. .. .. .. ... ... ..129
9.2 Flow chart of the numerical model for berm breakwaters .135
9.3 Berm breakwaters of vertical face: (a) Zero permeability; (b)
Infinite permeability. .. .. .. .. .. ... ... ... .... ...136
9.4 Berm breakwaters of inclined face: (a) Zero permeability; (b)
Infinite permeability. .. .. .. .. .. ... ... ... .... ...137
9.5 The Cross Section of The Berm Breakwater. .. .. ... ....138
9.6 Permeable berm breakwater of model scale with H = 5.0 cm: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity
distribution .. .. .. .. ... ... ... .... ... ... ....140
9.7 Permeable berm breakwater of model scale with H = 10.0 cm:
(a) Wave envelope; (b) Envelopes of pressure and normal velocity
distribution .. .. .. .. ... ... ... ... .... ... ....141
9.8 Permeable berm breakwater of model scale with H = 20.0 cm:
(a) Wave envelope; (b) Envelopes of pressure and normal velocity
distribution .. .. .. .. ... ... ... .... ... ... ....142
9.9 Permeable berm breakwater of prototype scale with H = 2.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions. .. .. .. .. ... ... .... ... ....143
9.10 Permeable berm breakwater of prototype scale with H = 4.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions. .. .. .. .. ... ... .... ... ....144
9.11 Permeable berm breakwater of prototype scale with H = 8.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions. .. .. .. .. ... ... .... ... ....145
A.1 Geometric relations between the vectors .. .. .. ... .....156
B.1 Caseof L =200 cm, h =DW =30cm, h, =DS =2Ocm
and d4o = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (?71H,), (b) Nondimensional wave heights (IT/RI) and
the best fit to the exponential decay function .. .. .. .. .. ...161
B.2 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 0.93 cm. (a) Averaged nondimensional surface elevation (q/H1), (b) Nondimensional wave heights (H/I) and
the best fit to the exponential decay function . 162
B.3 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and ds50 = DD = 1.20 cm. (a) Averaged nondimensional surface elevation (,l/H1), (b) Nondimensional wave heights (H/) and
the best fit to the exponential decay function . 163
B.4 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (q7/H1), (b) Nondimensional wave heights (H/7) and
the best fit to the exponential decay function . 164
B.5 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (q7/HI), (b) Nondimensional wave heights (H/) and
the best fit to the exponential decay function . 165
B.6 Case of L = 200 cm, h = DW = 30 cm, h. = DS = 20 cm
and ds0 = DD = 2.84 cm. (a) Averaged nondimensional surface elevation (t7/Hi), (b) Nondimensional wave heights (H/Rl) and
the best fit to the exponential decay function . 166
B.7 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (ti/Hi), (b) Nondimensional wave heights (H/H) and
the best fit to the exponential decay function . 167
B.8 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (qt/Hi), (b) Nondimensional wave heights (H/) and
the best fit to the exponential decay function . 168
B.9 Case of L = 200 cm, h= DW = 25 cm, h, = DS = 20 cm
and dso = DD = 0.93 cm. (a) Averaged nondimensional surface elevation (t/Hi), (b) Nondimensional wave heights (H/71) and
the best fit to the exponential decay function . 169
B.10 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 1.20 cm. (a) Averaged nondimensional surface elevation (77/Hi), (b) Nondimensional wave heights (H/HIV) and
the best fit to the exponential decay function. . 170
B.11 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (77/Hi), (b) Nondimensional wave heights (H/) and
the best fit to the exponential decay function . 171
B.12 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (t/Hi), (b) Nondimensional wave heights (H/H) and
the best fit to the exponential decay function . 172
B.13 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 2.84 cm. (a) Averaged nondimensional surface elevation (7/Hi), (b) Nondimensional wave heights (H/R1) and
the best fit to the exponential decay function . 173
B.14 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (?/H1), (b) Nondimensional wave heights (H/h) and
the best fit to the exponential decay function . 174
B.15 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and dso = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (7/H1), (b) Nondimensional wave heights (H/I7) and
the best fit to the exponential decay function . 175
B.16 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and dso = DD = 0.93 cm. (a) Averaged nondimensional surface elevation (?/H1), (b) Nondimensional wave heights (IH/Y) and
the best fit to the exponential decay function . 176
B.17 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and ds50 = DD = 1.20 cm. (a) Averaged nondimensional surface elevation (n/H1), (b) Nondimensional wave heights (H/I) and
the best fit to the exponential decay function . 177
B.18 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (n/HI), (b) Nondimensional wave heights (IH/HI) and
the best fit to the exponential decay function . 178
B.19 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and ds50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (q/Hi), (b) Nondimensional wave heights (H/) and
the best fit to the exponential decay function . 179
B.20 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and dso = DD = 2.84 cm. (a) Averaged nondimensional surface elevation (q/H), (b) Nondimensional wave heights (F/R) and
the best fit to the exponential decay function . 180
B.21 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and ds50 = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (n/H1), (b) Nondimensional wave heights (H/I) and
the best fit to the exponential decay function . 181
B.22 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 15 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (77/H1), (b) Nondimensional wave heights (H/H1) and
the best fit to the exponential decay function . 182
B.23 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 15 cm
and ds0 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (7/Hi), (b) Nondimensional wave heights (H/ and
the best fit to the exponential decay function . 183
B.24 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 15 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (i7/Hj), (b) Nondimensional wave heights (H/H) and
the best fit to the exponential decay function . 184
B.25 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 10 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (i7/Hi), (b) Nondimensional wave heights (H/) and
the best fit to the exponential decay function . 185
B.26 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 10 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rl/Hi), (b) Nondimensional wave heights (H/7) and
the best fit to the exponential decay function . 186
B.27 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 10 cm
and dso = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (q/Hi), (b) Nondimensional wave heights (H/H) and
the best fit to the exponential decay function . 187
B.28 Case of L = 225 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (7l/Hi), (b) Nondimensional wave heights (7/Wj) and
the best fit to the exponential decay function . 188
B.29 Case of L =225 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (7/Hi), (b) Nondimensional wave heights (7/ and
the best fit to the exponential decay function . 189
B.30 Case of L = 225 cm, h = DW = 20 cm, h, = DS = 20 cm
and dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (77/H1), (b) Nondimensional wave heights (H/1) and
the best fit to the exponential decay function . 190
B.31 Case of L = 250 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (7/H1), (b) Nondimensional wave heights (H/i and
the best fit to the exponential decay function . 191
B.32 Case of L = 250 cm, h = DW = 25 cm, h, = DS = 20 cm
and dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (ql/H1), (b) Nondimensional wave heights (7/) and
the best fit to the exponential decay function . 192
B.33 Case of L = 250 cm, h = DW = 20 cm, h, = DS = 20 cm
and ds50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (iq/H1), (b) Nondimensional wave heights (H/) and
the best fit to the exponential decay function . 193
B.34 Case of L = 275 cm, h = DW = 30 cm, h, = DS = 20 cm
and ds50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (q/H,), (b) Nondimensional wave heights (H/H) and
the best fit to the exponential decay function . 194
B.35 Case of L = 275 cm, h = DW = 25 cm, h, = DS = 20 cm
and ds50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (/HI1), (b) Nondimensional wave heights (7H/) and
the best fit to the exponential decay function . 195
B.36 Case of L = 275 cm, h = DW = 20 cm, h, = DS = 20 cm
and ds50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (q/H1), (b) Nondimensional wave heights (H/1) and
the best fit to the exponential decay function . 196
LIST OF TABLES
3.1 Illustration of Dominant Force Components Under Coastal Wave
Conditions: [a O(1)rad/sec, V(p/y) 0(101)] ........... 30
4.1 Comparison of The Predictions and The Measurements ..... ..43
5.1 Material Information ...... ........................ 51
5.2 Test Cases ....... .............................. 52
5.3 Measured a, and ai for h, = 20 cm and L = 200 cm....... ...53
5.4 Comparison of The Resistances. h, = 20 cm, L = 200 cm ... 56
5.5 Comparison of Measurements and Predictions for h, = 20 cm,
L = 200 cm...................................... 57
5.6 Comparison of a, and ap for do = 1.48 cm, L = 200 cm..... 59
5.7 Comparison of a,, and ap for d50 = 2.09 cm, h, 20 cm..... 59
5.8 Comparison of a, and ap for do = 0.16 am, h, = 20 cm,
L = 200 cm ..................................... 60
5.9 Comparison of a, and ap for h = dso ................... 63
8.1 Test Results of Nonbreaking Waves .................... 114
8.2 Comparison of Km and Kp ........ ..................... ..115
8.3 Test Results for Breaking Waves ....................... 121
8.4 Normalized Pressure Distribution ..................... 125
B.1 Test Cases ....... .............................. 159
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
WATER WAVE INTERACTION WITH POROUS STRUCTURES OF IRREGULAR CROSS SECTIONS
By
Zhihao Gu
December. 1990
Chairman: Dr. Hsiang Wang
Major Department: Coastal and Oceanographic Engineering
A general unsteady porous flow model is developed based on the assumption that the porous media can be treated as a continuum. The model clearly defines the role of solid. and fluid motions and henceforth their interactions. All the important resistant forces are clearly and rigorously defined. The model is applied to the gravity wave field over a porous bed of finite depth. By applying linear wave theory, an analytical solution is obtained, which is applicable to the full range of permeability. The solution yields significantly different results from those of contemporary theory. The solution requires three empirical coefficients, respectively representing linear, nonlinear and inertial resistance. Laboratory experiments using a standing wave system over a porous seabed were conducted to determine these coefficients and to compare with analytical results. The coefficients related to linear and nonlinear resistances were found to be close to those obtained by previous investigators. The virtual mass coefficient was determined to be around 0.46, close to the theoretical value of 0.5 for a sphere. The analytical solution compared well with the experiments.
Based on this porous flow model and linear wave theory, two numerical models using boundary integral element method with linear elements are developed for
permeable submerged breakwaters and berm breakwaters, respectively. Due to the establishment of a boundary integral expression for wave energy dissipation in a porous domain and the application of the radiation boundary condition on the lateral boundary(ies), the numerical models are highly efficient while maintaining sufficient accuracy. The numerical results show that the wave energy dissipation within a porous domain has a well defined maximum value at certain permeability for a specified wave and geometry condition. The nonlinear effects in the porous flow model are clearly manifested, as all the flow field properties are no longer linearly proportional to the incident wave heights. The numerical results agreed reasonably well with the experimental data on the seaward side. On the leeward of the breakwater, despite the appearance of higher order harmonics, the numerical model produces acceptable results of energy transmission based on energy balance.
Xvii
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Among the existing shore protecting breakwaters, a large number of them are rubblemound structures made of quarry stones and/or artificial blocks. They can be treated as structures of granular materials. A breakwater is called subaerial when its crest is protruding out of the water surface and submerged when its crest is below the water level. Sometimes a breakwater is subaerial at low tide and submerged at high tide. These types of structures are usually termed as low crest breakwaters. When a beach is to be protected from wave erosion, a detached shore parallel submerged breakwater may provide an effective and economic solution. The advantages of submerged breakwaters as compared to subaerial ones are low cost, aesthetics (they do not block the view of the sea) and effectiveness in triggering the early breaking of incident waves, thus reducing the wave energy in the protected area. With the increasing interest in recreational beach protection, where complete wave blockage is not necessary, submerged breakwaters may find more and more applications.
On the other hand, if wave blockage is the main objective such as for harbor and port protection, a subaerial breakwater may be more effective. The traditional design of such a structure is a trapezoidally shaped rubble mound with an inner core covered with one or more thin layers of large blocks to form the armor layer(s) to protect the core. Owing to the demand for deeper water applications, the armor sizes have become larger and larger. This would greatly increase not only the cost but also the structural vulnerability. A relatively new type of structures, called
2
berm breakwater, is attracting more and more attention. For a berm breakwater, the armor layer(s) is replaced by thicker layer(s) of blocks of much smaller sizes. The seaward face of the structure has a berm section instead of the traditional uniform slope. The berm section is intended to enhance the structural stability and to trigger wave breaking.
Wave attenuation behind a porous breakwater is affected by three mechanisms: wave reflection on the seaward face of the structure, waves breaking over the structure and flow percolation inside the porous structure. The former one is conservative and the latter two axe dissipative. The main focus of this study is on the third mechanism, that is the dissipation due to flow percolation. For such a purpose, a numerical solution is sought since an analytical solution for such irregularly shaped structures is nearly impossible. The main advantage of numerical methods is the flexibility of handling complex geometries and boundary conditions. The disadvantage of any numerical approach is the lack of generality for the solution, since it is usually implicit in terms of the variables so that the influences of the parameters can only be examined case by case.
In order to examine the validity of the model and the nature of the dissipative force, the porous flow model is applied to an infinitely long flat seabed of finite thickness subject to wave action. This condition can also be viewed as a submerged breakwater with infinitely long crest. Because of the geometrical simplicity, an analytical solution is attainable. This case is used to compare with the existing solutions proposed by other investigators, to examine the nature of wave attenuation as a function of flow and material properties and, more importantly, to guide the design of an experiment so that the empirical coefficients in the porous model can be determined. Successful determination of these coefficients is crucial to the validity of the model. The experiment which is to be carried out subsequently must demonstrate that the results are stable (or the experiments are repeatable) and that
3
they cover, at least, an adequate range of intended application.
A numerical solution with computer code is then to be developed for arbitrary geometry. Boundary Integral Element Method (BIEM) is proven to be very efficient for boundary value problems with complicated domain geometries. With this method, the solution is expressed in boundary integrals and no interior points have to be involved in the solution procedure. The aim of this study is to develop an efficient computer code based on such a method. Finally, the validity of the solution is to be assessed by a set of experiments.
1.2 Objectives and Scope
Specifically, the objectives of this study are listed as follows:
1. Develop a percolation model suitable for unsteady and turbulent porous flows,
2. Verify the model through an analytical solution and laboratory experiments for the case of a flat porous seabed subject to linear gravity waves,
3. Based upon the porous flow model, develop a numerical solution for submerged and berm breakwaters of arbitrary cross sections using the boundary integral element method.
To achieve these goals, the research is carried out in the following steps:
1. Derive the porous flow model,
2. Examine and interpret the relative importance of the various terms in the model to narrow the scope of the study,
3. Obtain the analytical solution of wave attenuation over a flat porous seabed, compare the solution with the existing ones and examine the wave energy dissipation process,
4. Conduct a laboratory experiment for the flat porous seabed case to determine the empirical coefficients in the porous flow model and to verify the analytical solution, 5. Develop a numerical model using the BIEM method for porous submerged breakwaters,
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6. Verify the numerical model with laboratory experiments of a submerged rubblemound breakwater,
7. Modify the numerical model to represent porous berm breakwaters.
CHAPTER 2
LITERATURE REVIEW
This chapter is a brief literature review of the past efforts made in studying the flow in porous media with and without water waves. The review is separated into three groups: one on porous flow model, one on analytical solutions for wave and porousseabed interactions and the other one on the modeling of interactions between waves and porous structures of irregular cross sections.
Since the porous media encountered in coastal engineering are largely of the granular type, made of sand, gravel, quarry stones or artificial blocks, the deformation of the solid skeleton of the media is usually negligibly small as compared to that of the pore fluid. The literature review deals with granular media only; other types of porous media, such as poroelastic or poroplastic media and so on, are not included.
2.1 Porous Flow Models
A large number of porous flow models have been proposed over the past few decades as an effort to quantitatively model the flows in porous media. Some of the models were based purely on experiments and some of the others on certain theoretical considerations. The derivations of the these models are based primarily on the following three approaches: 1) Simple element approach which views a porous medium as the assembly of simple elementary elements such as a bundle of tubes and so on, 2) Microscopic approach which recognizes the midcrostructure of porous media in derivations. The final formulations of the models usually have to be obtained by taking the spatial average of the microscopic equations, 3) Phenomenological approach which homogenizes the porous media as a continuum and the flow within
6
the medium is assumed to be continuous. The third approach is the most popular for porous flows in granular materials since the microscopic structure is not well understood and the corresponding constitutive equation not well established.
The first porous flow modelDarcy's lawwas proposed by the French engineer Darcy based on his experiments more than a century ago. In his model, the characteristic properties of porous media are lumped into one parameterpermeability coefficientand the model has the form
 Vp= (2.1)
where Vp is the gradient of the pore pressure, it and q are, respectively, the dynamic viscosity and the specific discharge velocity of the pore fluid and K. is called the intrinsic permeability coefficient, which reflects the collective characteristics of porous media, such as the porosity, the roughness of the pore walls, the tortuosity, the connectivity etc.
This model is very simple in form and reasonably accurate for steady porous flows within media of low permeability, generally of the order of Kp = 109 1012 m2, where the flows are normally laminar dominant. However, for porous media with relatively higher permeability, the porous flow is no longer laminar dominant, turbulence plays a larger and larger role with increasing pore size. In such cases, Darcy's model tends to over estimate the discharge velocity for a given pressure gradient. Dupuit and Forchheimer (cited in Madsen, 1974) modified Darcy's law by adding a term quadratic in velocity to take into account the turbulent effects:
 Vp = (a + b I 4I)q" (2.2)
where a = A/KP, and b is coefficient for turbulent resistance defined by Ward (1964) as
b = pC! (2.3)
7
where p is the density of the fluid and C1 is a nondimensional coefficient.
The two coefficients a and b were further expressed in terms of particle diameter by Engelund (1953) and Bear et al. (1968) in the forms (1 n)v (2.4)
a = ao n2d2
1n r
b = bo 1n (2.5)
where ao and b0 are nondimensional coefficients and n is the volumetric porosity of porous media.
The above two models, Eq.(2.1) and (2.2) are both for steady porous flows, although some applications have been made for certain unsteady flows, such as wave induced porous flows in seabeds (Putnam, 1949; Liu, 1973; etc.). These applications were primarily on low permeability media like fine sand. In general, for unsteady flows, the inertial force arises due to the acceleration of the pore fluid and such a force can be quite large and can even be dominant when the permeability is high. To model the unsteady porous flow, a number of models have been suggested.
The first model found in the literature containing the inertial force term was the one by Reid and Kajiura (1957). It is the direct extension of Darcy's model:
 Vp = P8t (2.6)
KP nat
The inertial term in this model is in fact the local acceleration term in NavierStokes equation with the fluid velocity expressed as a specific discharge velocity q It is clear that this term contains only the inertia of the fluid but not the inertia induced by the fluidsolid interaction, or the added mass effect.
The extension of DupuitForchheimer model leads to PolubarinovaKochina's model (Scheidegger, 1960; McCorquodale, 1972). It has the form
 Vp = a+b2 + (2.7)
8
where a, b and c are empirical coefficients.
Murray (1965), when studying the viscous damping of gravity waves over a permeable seabed, proposed
 Vp = !6 +pn (2.8)
where k is the permeability of porous media (defined differently from Kp) and 6 is the spatially averaged microscopic velocity of the pore fluid, related to 4* by 4 = n6Z. Comparing to Eq.(2.7), the coefficient c is equal to p in Murray's model. McCorquodale in 1972 further modified the expression of this coefficient as P (2.9)
n
and his model (McCorquodale, 1972) becomes
 Vp= (a +bT) pa j (2.10)
n at
At about the same time as McCorquodale, Sollitt and Cross (1972) also extended Eq.(2.2) to include the effects of unsteadiness. In their model, the inertia resistance consists of two components, one is the inertia of the pore fluid, and the other one is the inertia induced by the virtual mass effect due to the fluidsolid interactions; the model reads
+ z~pC+ 1n Ca a
Vp = ~ (n + n'P) (2.11)
where C is again the spatially averaged microscopic velocity vector and Ca is the virtual mass coefficient.
The value of the virtual mass coefficient C. in Sollitt and Cross's model was assumed to be zero in their actual computation because it was unknown at the time. It has been so assumed in almost all the later applications of this model (Sulisz, 1985, etc.). When comparing to the experimental data, Sollitt and Cross found that the correlation improved by taking nonzero values for Ca and asserted that "one
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cannot predict the magnitude of this coefficient a priori because the virtual mass of densely packed fractured stone is not known" (Sollitt and Cross, 1972, p1842). They also pointed out that "Evaluation of Ca, however, may serve as a calibrating link between theory and experiment in future studies" (same page).
Hannoura and McCorquodale (1978) made an attempt to determine the value of C0 by laboratory experiment. They measured the instantaneous velocity and the pressure gradient in the experiment, and calibrated the data with their semitheoretical porous flow model:
 Vp = (a + b ap(l+ Co)Lq (2.12)
The results so obtained for C. scattered in a range of 7.5 +5.0.
Dagan in 1979, based on the microscopic approach, arrived at a generalized Darcy's law for nonuniform but steady porous flows (Dagan, 1979):
Vp = V2q (2.13)
where y is a constant coefficient depending only on the geometry of the media and it was defined as
dZL (2.14)
In applying this model to the waveporousseabed interaction, Liu and Dalrymple (1984) added an acceleration term to the Dagan's model and it becomes
 Vp = j ppq+nPq V2 (2.15)
K,, n at K,
It was found, by performing a dimensional analysis (Liu, 1984), that the last term in Dagan's model is a second order quantity in comparison to the other terms. The inertial term in this model is the same as that of Reid and Kajiura and that of Sollitt and Cross where C. is set to be zero.
Based on the phenomenological approach, Barends (1986) added another porous flow model to the list:
 V = C aq gq (2.16)
= at K(q)
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and K(q) was defined by Hannoura and Barends (1981, see Barends, 1986) as /pd, 2.7
K(q) = g (2.17)
with
f Ca (1 n)
Pins
where C is the drag coefficient related to the porous Reynolds number (R =1 q do/v), a and /3 are two constants and d, is the relevant grain size.
With so many models, it is difficult to decide which one is most appropriate for the porous flow problem in this study without further analysis.
2.2 WavePorous Seabed Interactions
The computation of gravity wave attenuation over a rigid porous bottom has been performed by a number of investigators. The classic approaches by Putnam (1949) and Reid and Kajiura (1957) was to assume that the Laplace equation is satisfied in the overlying fluid medium and that the bottom layer can be treated as a continuum following Darcy's law of permeability. In Reid and Kajiura's paper, although the inertial term was included in the porous flow model, they later neglected it and concluded that Darcy's law is adequate for sandy seabeds. Under the assumption of low permeability, Reid and Kajiura found, for infinitely thick seabeds, that
a2 gk,. tanh krh (2.18)
ki = 2(agp/v)k,. (2.19)
2kh + sinh 2krh
for progressive waves, where a is the wave frequency, k, and k are, respectively, the wave number and the damping rate, h is the water depth and v is the kinematic viscosity of the fluid.
Hunt (1959) examined the damping of gravity waves propagating over a permeable surface using Reid and Kajiura's porous flow model and retained the inertial term to the end. The thickness of the permeable beds was infinite and water above
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the permeable surface was assumed viscous as opposed to the inviscid approach by the previous investigators. The tangential velocities at the surface of the permeable bed were set to be zero and the continuity of the normal velocity and the pressure were observed on the same boundary. The resulting dispersion equation was quite lengthy. Under the consideration of sand beds of low permeability, the results for kr and k, are
k = ko + 2k (2.20)
r 2o 2koh + sinh 2koh
2k0 aK
k = 2ko ( + ko ) (2.21)
S 2koh+sinh2koh (2.21)2
with
a' = gko tanh koh (2.22)
Murray (1965) investigated the same problem by the use of stream function and the porous flow model given in Eq.(2.7). Instead of letting the tangential velocities at the interface be zero as done by Hunt, Murray assumed that the tangential velocities were finite and continuous. The first order solution of the dispersion equation for the spatial damping was k + 2koN
S2koh + sinh 2koh (2.23)
where k = ko + k, with ko being the initial wave number for the impervious bottom as defined in Eq.(2.22) and N= naKp
For fine sand, N + 0 and k, = v~ a(2.24) 2kh + sinh 2krh
and the temporal damping for known k in such case is
a = i/uk (2.25)
vAsinh 2kh
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Liu (1973) added an interface laminar boundary layer in his solution for the same problem. In his paper, the water in the fluid domain was regarded as inviscid and the porous flow was assumed to follow Darcy's law. The imaginary part of k, after the boundary layer correction, was given by 2k,+ L)
, 2kh + sinh 2kh (aKF2/v + ,7 a 2 (2.26)
with k, remaining the same as that in Eq.(2.18). It is noted that this is the same expression as that given by Hunt (1959). The correction term of Eq.(2.26) to Eq. (2.19) is found to be insignificant unless Kp is extremely small.
Dean and Dalrymple (1984) later obtained the expressions for shallow water waves over sand beds:
k, =a[1 1 4 h/ 211/2 (2.27)
k, (1 hh/g) K
= ( ) (2.28)
All the solutions above are for infinitely thick seabeds. Liu (1977) solved the problem for a stratified porous bed where each layer has a finite thickness and different permeability. It was found that the wave damping rate was insignificant to the permeability stratification while the wave induced pressure and its gradients are affected significantly by stratification. The porous flow model employed in the analysis also obeys Darcy's law only.
Liu and Dalrymple (1984) further modified the solution, for a bed of finite thickness, by replacing Darcy's law with Dagan's unsteady porous flow model, which partially included the effects of unsteady flows. The dispersion relationship from the homogeneous problemwithout laminar boundary layer correctionswas found as
R gk gk
(R + i)(1 tanhkh) +Rtanhkh, tanhkh(1 kctnhkh) = 0 (2.29)
n a2 72
or equivalently
Cr2 gk tanh kh= tanh kh, (gk r2 tanh kh) (2.30)
Rn
where h, is the seabed thickness and R is the permeability parameter defined as R = crK'
It has been shown that the corrections by the laminar boundary layers were not significant since "the damping is largely due to the energy losses in the porous medium rather than the boundary layer losses" (Liu and Dalrymple, 1984, p47). Therefore, it is reasonable to believe that if an appropriate porous flow model is adopted, the solution for a homogeneous problem, without the correction of laminar boundary layers, will be sufficient for engineering purposes.
Besides the theoretical studies, the interaction of waves and porous seabeds was also investigated experimentally by Savage (1953). The experiment was carried out with progressive waves over sand beds of median diameters, d, = 0.382 cm. and d, = 0.194 cm, respectively. The permeability coefficient of the sand bed was measured as 44.9 x 1010m', and the dimension of the seabed was 0.3 meters thick and 18.3 meters long. The wave heights at the beginning and the end of the sand bed were recorded and adjusted to eliminate the effects of the side friction. Most of the theories listed above were claimed to agree well with the data in this experiment.
2.3 Modeling of Permeable Structures of Irregular Cross Sections
The ability to predict the performance of rubblemound breakwaters under the attacks of ocean waves is critically important in designing such structures. Significant amounts of effort have been devoted to this subject. In the early stage of development, research was basically confined to laboratory experiments and only the empirical formulae extracted from the experiments were available for designs. In recent years, with the continuing efforts by researchers, more and more theoretical models for different types of breakwaters are becoming available. For subaerial
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breakwaters, in addition to a large number of laboratory experiments, a multitude of theoretical models have been developed. The analysis of such structures has been approached both analytically and numerically.
Analytically, Sollitt and Cross (1972) investigated the problem of wave transaission and reflection by a vertical face (rectangular cross section) porous breakwater. The porous flow model used in the formulation was Sollitt and Cross's model although the virtual mass coefficient C. was assumed to be zero in the computation. In their solution, the whole computation domain was divided into three regions, two fluid regions and one porous region. The velocity potential functions in the three regions were assumed to be the summations of the fundamental mode and the evanescent modes according to the wave maker theory. The nonlinear term in the flow model was linearized according to Lorentz's hypothesis and the normal velocities and the pressure were matched at both vertical faces of the breakwater. The transmission and the reflection coefficients for long waves, simplified from the complete solution of complex matrix form, were given by Kb = i (2.31)
1+ n (S if + n2)
K, = (2.32)
Sif +n2i2nab
where S is the coefficient for the inertial resistance of porous media, i is the imaginary unit, f is the linearized coefficient for velocity related resistances, b is the width (or thickness) of the breakwater and all the other symbols have the same meanings as before.
Madsen (1974) later reexamined the same problem beginning directly with the linear long wave theory. He adopted the Dupuit Forchheimer model and introduced the inertial resistance into the momentum equation. The resulting expression of Kt and Kr are almost identical to Eq.(2.31) and (2.32). By carrying out the volume
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integration over the breakwater for the energy dissipation, Madsen obtained an explicit expression for the linearized friction factor f, that is n KpIa. Kpla. 16b 1
f = n[(1 pa) + (, + pa)2 + 16b ai (2.33)
K,,1 2a 2a 37rh
where a and b are the coefficients in DupuitForchheimer's porous flow model. A similar solution for this problem was also given by Scarlatos and Singh (1987). Madsen et al. (1976, 1978) further extended the long wave solution to a trapezoidal porous breakwater with, again, DupuitForchheimen's model. The solution was, however, much more complicated than that for the crib type.
Since the shore protection breakwaters are usually irregular in shape and very often with several layers of stones of different sizes, analytical solutions become impractical. For such complicated geometries, numerical approaches have to be adopted.
The most commonly used numerical schemes are finite element and Boundary Integral Element Methods (BIEM). The first finite element model was developed by McCorquodale (McCorquodale, 1972) using the McCorquodale porous flow model, for computing the wave energy dissipation in rockfalls. In his model, the entire cross section of the structure was divided into small triangleelements and the variation of the physical quantities was interpolated by a time dependent element function: = (1 + 2X + 3y)t + 34 + SX + 6y (2.34)
Since the numerical computation was only carried out in the porous domain, the interaction between the porous domain and the fluid domain was not modeled, although the free surface within the porous region was well predicted.
Due to the large amount of work for data preparation in using a finite element model, the Boundary Integral Element Method (BIEM) became popular since the mid1970s. With BIEM, the discretization is only on the boundaries as opposed to the entire domain with the finite element method. Ijima et al. (1976) applied
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this method to porous breakwaters with constant elements, which assumes that the physical quantities remain constant over each individual element. The whole computation domain was divided into three regions, two fluid regions and one porous region. In the porous domain, Darcy's law was used and in the two fluid regions, two artificially defined vertical boundaries were placed at the offshore and inshore ends. The patterns of the vertical distribution of the velocity potential at these two boundaries was assumed undistorted by the presence of a structure. The reflected and the transmitted potential functions outside the computational domain were given by
,.(X, z) = [ekz'+Ae'z] cosh k(z +h) (2.35)
i())Iv r cosh kh
Ot(x,z) = BOeiA(,+,,)coshk'(z + h) (2.36)
cosh k'h'
where I and 1 are the distances from the left and the right vertical boundaries to the origin, k and k' are the wave numbers in the reflection and the transmission regions, respectively. The reflection and transmission coefficients are then K, =1 Ao 1 (2.37)
Kt =l0Bo 1 (2.38)
The agreement of the transmission coefficient for impermeable floating structures with the experiment data was fairly good, but the comparison of Kt and K, for a slopedface permeable breakwater was not as satisfactory.
Finnigan and Yamamoto (1979) further added the modulated standing wave modes to and Ot, while keeping the constant element and Darcy's law unchanged in their model. The two potential functions were, respectively,
_,k.z_,.cosh k(z + h) +cosk,(z + h) ,(XZ) = [e,+c oecosh kh + Amek+(z) cos k,+h)
m(1 cos2kh
(2.39)
B~'''+ l~hI C cosh kz + h') + 0 M a''cskC~o 'h
t(x,z) = B0eik(z+1)I + B e cos k'(z + h) (2.40)
cosh k'h' m1cos kmlh' with
gk tanh kh = gkm tan kmh = 2 (2.41)
= gk' tan knh = a2 (2.42)
where Ai and Bi are unknown complex coefficients. In their model, the series were truncated at a certain point to make the number of unknown coefficients equal to the number of the elements. Owing to the introduction of the series, the procedure became much more complicated and the computation much more time consuming than that of Ijima et al.
The advantage of a constant element is its simplicity and efficiency. But it has an inherent problemawkward behavior around "corner points". It has been found that significant errors arise around such points even for the simplest casesinusoidal wave propagating in a constant water depth over a impermeable bottom.
Sulisz (1985) employed linear elements in his BIEM model for subaerial porous breakwaters. Eqs.(2.39) and (2.40) were adopted for the lateral boundary conditions. As a result, the formulation became awfully complicated. In modelling the porous media, Sollitt and Cross's model was adopted. However, the inertia term due to virtual mass effect was dropped as the corresponding coefficient C. was not known.
For submerged breakwaters, the modelling is still largely based on laboratory experiments and empirical formulae. For impermeable submerged breakwaters, quite a few empirical formulae for transmission coefficient have been established from experimental data by different authors (see Ba.ba, 1986). For example, the equation given by Goda (1969) for the transmission coefficient due to overtopping is
K= = 0.5[1 sin F (2.43)
2a H,
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with a = 2.2 and /3 = 0.0 0.8 for vertical face breakwaters, where F is the depth of submergence of the breakwater crest and Hi is the incident wave height. Another equation for impermeable submerged breakwaters was given by Seelig (1980): = C(1 F (1 2C)F (2.44)
where R is the wave run up given by Franzius (1965, cited in Baba, 1986) R = HIC(O.123HI)(C2vTH7U+c3) (2.45)
with C, = 1.997, C2 = 0.498 and C3 = 0.185. d is the total water depth and C = 0.51 0,11B (2.46)
h
where B is the crest width and h is the height of the breakwater. Several other empirical equations and methods are also available for designs.
The latest theoretical approach for impermeable submerged breakwaters was given by Kobayashi et al. (1989) by using finite amplitude shallowwater equations. The numerical results were found to agree well with the experimental data by Seelig (1980) for such structures.
However, for permeable structures, no parallel empirical formula could be found in the literature besides the nomograph by Averin and Sidorchuk (1967, cited in Baba, 1986). The research for such structures is still in the stage of physical experiments.
Dick and Brebner (1968) carried out an experiment on solid and permeable breakwaters of vertical faces. In the experiment, it was discovered that, over a certain wave length range, the permeable breakwater was much better than the solid one of the same dimension in terms of wave damping, and that the permeable breakwater had a well defined minimum value for the coefficient of transmission. It was also found that a substantial portion, 30% 60%, of the wave energy of the transmitted waves was transferred to higher frequency components. The equivalent
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wave height for the transmitted waves in his paper was defined as Heq = 2 vf2 jT2 (t t(2.47)
Due to the smallness of the submergence of the breakwater crest, it is felt that the majority of the waves in the experiment were breaking waves.
Dattatri et al. (1978) tested a number of submerged breakwaters of different types, permeable, impermeable, rectangular and trapezoidal. One of the conclusions was that the important parameters affecting the performance of a submerged breakwater are the crest width and the depth of submergence. The transmitted waves were found "to be irregular though periodic". They also reported that porosity and wave steepness did not have significant influence on the transmission coefficient, which contradictory with the observations by Dick et al. (1968).
Seelig (1980) conducted a large number of tests on the cross sections of 17 different breakwaters for both regular and irregular waves. Most of the breakwaters tested were rubblemound porous structures with multilayer designs. Beyond the experiments for impermeable breakwaters by which Eq. (2.44) was obtained, he also tested these permeable structures as submerged breakwaters. Since "no generalized model was available" at the time for such breakwaters, the experimental data with regular waves were compared to Eq.(2.44). It was found that the formula is quite conservative in estimating the transmission coefficient for permeable submerged breakwaters. The phenomenon of wave energy shifting to higher order harmonics in the transmitted waves was again reported without further explanation.
Baba (1986) conducted an experiment on concrete submerged breakwaters and compared the data with the four widely used computational methods for wave transmission coefficient for impermeable submerged breakwaters. He concluded that the formula given by Goda (1969) was the most suitable one in the case of a shore protecting submerged breakwater.
CHAPTER 3
POROUS FLOW MODEL
Examining the existing porous flow models listed in the literature review, it is noted that most of them contain one, two or all of the following components: the linear velocity term, the quadratic velocity term and the term proportional to the acceleration of the fluid. For the first component all the models give the same definition while for the rest two terms, different models have different forms. To be able to model the porous media with confidence, it is necessary to formulate the porous flow from fundamental principles.
3.1 The Equation of Motion
The equation of motion for a solid body placed in an unsteady flow field can be expressed in the following general terms:
FP +FD+FI+FbFF =ma (3.1)
where m = mass of the solid; a = acceleration of the body; Fp = pressure force; FD = velocity related force also known as the drag force; F = accelerationrelated force also known as inertial force; Fb = body force; and Ff = frictional force due to surrounding solids. We now apply this equation to a control volume of a mixture of fluid and solids, as shown in Fig. 3.1, and restrict our discussion to the twodimensional case. The xaxis coincides with the horizontal direction and the zaxis points vertically upward. The spatially averaged equation of motion for the solid skeleton in the xdirection becomes
a dx(1 nAZ)dz + FD. + P' + Fb.z FfZ = p.(1 n) ii,, dxdz (3.2)
where Lp = pressure gradient in the xdirection; nA = averaged area porosity
aw p w + dz p + d
8z Oz
U
P
Bu
udx p + "axdx
Op
p+ dz
Figure 3.1: Definition sketch for the porous flow model
defined as the ratio of Aid to Atotal with Avo being the area of the voids and Atomt being the total control area; n = volumetric porosity defined as the ratio of Void to Vtota~; Tb, is the body force of the solid; p, = density of solid; u, and ti, = velocity and acceleration of the solid. The subscript z denotes the x direction and the overbar denotes spatial average. Since, from this point on, we will be dealing exclusively with spatially averaged values, the overbar will be dropped.
The force balance on the pore fluid can be established in a similar manner:
aP
 dzx na. dz F, FI, + Ffy, = pn dz dz (y, i,)
ax (l.f.
with p being the density of the fluid and Fb&1 being the body force of the fluid in z direction; u and tif equal, respectively, the actual spatially averaged velocity and acceleration of the pore fluid, defined as
u = ua dv (3.4)
od = V old
(3.3)
22
where ua = actual pore velocity. If the fluid and solid mixture is now being treated as a porous medium, and thus a continuum, we require that
1. All the field variables, such as that defined by Eq.(3.4), be independent of the volume of integration.
2. 6 < L, where 5 and L are respectively, the length scales of the pore and the system.
For the wave attenuation problem, Eq.(3.3) is of special interest. We introduce here yet another field variable q representing the apparent spatially averaged fluid velocity, which is related to Uf by
q = 1 uadv "f uadv = n.u (3.5)
here V is the total volume Vtota.
This velocity also known as the discharge velocity, is related to the actual discharge over a unit surface area, or, Q/A. These apparent properties are of final interest in engineering. Since we are now dealing with a continuum, we have Dqq
+ V =t = n" (3.6)
ijt at q*V at
here it has been assumed that the convective acceleration is negligible.
The various force terms are established as follows:
(A). Drag Force
The drag force consists of two terms: that due to laminar skin friction and that due to turbulent form drag; the former is proportional to the velocity and the latter is proportional to the velocity squared. Their functional form, when expressed in terms of the final field variables, may be written as
FD2 = [Ax(qz nu..) + B. q nu, I (q nu.)Idx nA dz (3.7)
The coefficients A, and B. are to be determined later.
(B). Inertia Force
The inertial force can be treated as an added mass effect and expressed in the following form:
Fj = C(6 nt,,)dx nAz dz (3.8)
The coefficient C, will be determined later.
(C). Body Force
In the pore fluid, there is no horizontal body force, and the vertical body force is balanced by the static pressure gradient and, thus, vanishes. In the solid skeleton, the vertical body force is the net weight. If the solid skeleton is subjected to an unsteady vertical motion, this term should be included; otherwise, it can be ignored.
Substitution of Eqs.(3.5) through (3.8) into Eq.(3.3) results in the following differential equation:
aP
  Ax(q< nu,2) + Bz I q nu, I (q. nu,)
ax
+ ( +Cz)(. nik..) (3.9)
This is the basic equation of pore fluid motion. In a general case, it is coupled with Eq.(3.2) and must be solved simultaneously. Only a special case with no movement of the solid skeleton will be analyzed here. For such a case, Eq.(3.9) reduces to
ap = Aqz + Bx, I q I q. + (' + C. )4. (3.10)
axn
The force balance in the zdirection leads to aP
az . A~q + B. I qI q. + (' + C,)5k (3.11)
3.2 Force Coefficients and Simplifying Assumptions
The evaluation of force coefficients involves a great deal of empiricism. However, successful engineering application relies heavily upon successful estimation of these coefficients. The part of the flow resistance which is linear in q clearly will lead to
24
a Darcytype resistance law. The coefficient A is, at least, related to four foundational properties, one of the fluidthe dynamic viscosityand three of the porous structure the porosity, the tortuosity (one that defines effective flow length) and the connectivity (one that defines the manner and number of pore connections). Obviously, the more those factors can be specified explicitly, the more accurately the value of A can be determined. However, it is also generally true that the more factors explicitly introduced the more restrictive the range of application. If none of these four fundamental properties is expressed explicitly, A is simply the inverse of Darcy's hydraulic permeability coefficient. If the dynamic viscosity, tL, is factored out, we have the empirical law:
A=KA (3.12)
Kp
where Kp is known as the intrinsic permeability. A number of investigators including Engelund (1953, cited in Madsen, 1974) and Bear et al. (1968, cited in Madsen, 1974) attempted to relate A to porosity as well and came up with the relationship of the following type:
(1in)3 it
A = a0 n 2 (3.13)
where d, is a characteristic particle size of the pore material. The coefficient a0 obviously still contains the other properties such as tortuosity and connectivity. Since both of these factors have directional preference, a0 should also be a directional property and, in general, is a secondorder tensor. For isotropic material ao becomes a scalar. Engelund (1953, see Madsen, 1974) recommended ao = 780 to 1500 or more (3.14)
with the values increasing with increasing irregularity of the solid particle.
Attempts were also made to sort out the effects of tortuosity and connectivity (Fatt, 1956a,b). The conditions invariably become more restrictive and one is required to specify soil characteristics beyond the normal engineering properties.
25
The characteristics of the coefficient B can be determined by examining the total form drag resistance acting on a unit volume of granular material of characteristic size d,. For this case, we have FDN. = 6, P,CDAP I U1 I ,U, (3.15)
2
where AP is the projected area of individual particles; CD is the form drag coefficient of the individual particles; b is a correction coefficient for CD accounting for the influence of surrounding particles; and m is the total number of particles per unit volume. Since the total projected area should be proportional to (1 n)dx dz/d., we have, after substituting q for uf: FDN. = CD nI q dxdz (3.16)
2 nd.
the constant of proportionality is absorbed in 6. Comparing Eq.(3.16) with the second part of Eq.(3.7), we obtain
2 DnAn2d,
If we let:
b0 = 5 (3.18)
Eq.(3.17) can be expressed as B.n (3.19)
Pboflf2d,
This equation is very similar to that recommended by Engelund (1953), with the exception that nA was replaced by n, the volumetric porosity, in his formula. Again, because b0 has directional preference, it should also be a tensor of second order. Engelund recommended b0 = 1.8 to 3.6 or more (3.20)
for granular material of sandsized particles.
26
The coefficient C. is the overall added mass coefficient for the porous medium. Its characteristics can be determined in much the same manner as those of B,, or
m
Fl. = (3.21)
where C. is the added mass coefficient of each individual particle and V is the volume of each particle. One may then readily obtain, by comparing Eq.(3.21) with Eq.(3.8),
C. = pC.(1n) (3.22)
nAz n
Since C. is related to the volume of the solid skeleton, one does not expect significant directional preference, unless the geometry of the element deviates significantly from a sphere shape. A new coefficient, Cm, can be defined such that C. = p + n = p[n + Co(1 n) (3.23)
nAz nA,
which has the same meaning as the mass coefficient in hydrodynamics.
Clearly the force coefficients in other directions can be defined in a similar fashion. Equations.(3.10) and (3.11) can now be expressed in the following general form:
A4+B q1 + Cm =Vp (3.24)
Here, the carat indicates a second order tensor. This equation can be generalized to include the motion of the solid simply by replacing q with q Uo.
We now proceed to assume that the porous material is isotropic and that all the coefficients reduce to scalar quantities and
nAz = Ax = n
Recall that the convective acceleration in q has been assumed, in Eq.(3.6), to be negligible, i.e.
q= T
27
Substituting these conditions into Eq.(3.24), we obtain
(A + B I q 1)4 + C,,  Vp (3.25)
at
This equation is similar to the wellknown equation of motion in a permeable medium such as given by Reid and Kajiura (1957) and others, with the exception that the added mass effect is now formally introduced.
To seek a solution to this equation, the common approach is to ignore the inertial term and linearize or ignore the nonlinear term. Under such conditions, Eq.(3.25) can be reduced to the Laplace equation by virtue of mass conservation. To retain the inertia term, the most convenient approach is to assume the motion to be oscillatory, as is the case under wave excitation. Now we let
q= qtx, z)e cit (3.26)
where Fis a spatial variable only, and a is a generalized wave frequency, which could be a complex number. Substituting Eq.(3.26) into Eq.(3.25) leads to
 Vp = (A iaCm, + B I q 1)" (3.27)
A new set of nondimensional parameters are defined as follows:
Permeability parameter, R,
R pgp (3.28)
with
K = nd; (3.29)
A a o (1 n)
Inertia parameter, /)
=n + C.(1 n) (3.30)
n2
Volumetric averaged drag coefficient, Cd
Cd = B = b01 n (3.31)
p n~d,
28
Substituting these parameters into the above equation reduces it to the following familiar form:
 Vp = po( 1 ig + Iq I) (3.32)
R GC
This is essentially the same equation used by Sollit and Cross (1972) and others for porous breakwater analysis.
3.3 Relative Importance of The Resistant Forces
Before solving Eq.(3.32) with an overlying wave motion, it is useful to examine the importance of the resistance forces and to determine the nature of the fluid motion in the porous medium. By taking ratios of the three respective resistance forces, three nondimensional parameters can be established:
Inertial Resistance f1 n + C(1 (3.33)
Laminar Resistance f = ao(1 n)3
Turbulent Resistance f, b0
Laminar Resistance fJ = aon(1 n)21 (3.34)
Turbulent Resistance f,, b0(1 n) Rf (3.35)
Inertial Resistance f = Can(1 n + n R, C.
where R, and R are two forms of Reynolds number defined as R_ q Ido (3.36)
and
II d! (3.37)
where v is the kinematic viscosity of the fluid.
Obviously both Reynolds numbers signify. the relative importance of the inertial force to the viscous force. The origins of the inertial forces are, however, different. In Rf, the inertia is of a convective nature and the resistance arises due to change of velocity in space (fore and aft the body) whereas in Ri, the inertia is of a local
29
nature and the resistance arises due to the rate of change of velocity at a specific location. The ratio of Ri and R! is the Strouhal number, which clearly identifies the different origins of the two inertial forces.
In the range of common engineering applications, the magnitudes of various coefficients can be estimated as follows:
n s 0.30.6
ao ~0(1)
bo 0(1)
Therefore, Eqs.(3.33) through (3.35) reduce to
t I ~ 102R1 (3.38)
I_ I ~ 102Rf (3.39)
f I _R (3.40)
When Eqs.(3.38) to (3.40) are plotted on a R1 and Ri plane as given in Fig. 3.2, we identify seven regions where the three resistance forces are of varying degrees of importance. There are three regions where only one resistance force dominates, that is, the dominant force is at least one order of magnitude larger than the other two. There is one region where the three forces are of equal importance. Then, there are three intermediate regions where two out of three forces could be important.
Since both Reynolds numbers are flowrelated parameters, accurate position of a situation within the graph cannot be determined a priori. However, a general guideline can be provided with the aid of the graph or with Eqs.(3.38) through (3.40). We give here an example of practical interest. In coastal waters, we commonly encounter wind waves with frequency in the order of 0(1)(rad/sec) and the nondimensional bottom pressure gradient defined as V(p/y) in the order of O(101).
106 IN Kegion
10 > lo f
10o > lOf,
102 A, f, f, A
1
L Region
101 f I Region
ft > 10f,
0A > lof, > 10f
106R,
106 104 102 1 10 10 106
Figure 3.2: Regions with different dominant resistant forces
Under such conditions, the importance of the three resistance components for bottom materials of various sizes can be assessed. Table 3.1 illustrates the results.
Table 3.1:
Illustration of Dominant Force Components Under Coastal Wave
Conditions: [o O(1)rad/sec, V(p/q) 0(101)
Description Size Disch. Vel. Rf R Dominant
S Range (m/sec) Force
Coarse sand <2 mm. < O(103) < O(1) < 0(1) laminar
or finer
Pebble, laminar
or small 1 cm. O(102) O(102) O(102) turbulence
gravel inertia
Large gravel 10 cm. O(101) O(104) O(104) turbulence
crusted stone inertia
Boulder 0.3 1.0 m O(100) O(106) O(0) turbulence
crusted stone inertia
Artificial turbulence
blocks, > 1.0 m > O(100) > O(106) > O(106) inertia
large rocks III
CHAPTER 4
GRAVITY WAVES OVER FINITE POROUS SEA BOTTOMS
We consider here the case of a small amplitude wave in a fluid of mean depth h above a porous medium of finite thickness h.. The bottom beneath the porous medium is impervious and rigid. The subscript s will be used here to denote variables in the porous bed. The basic approach is to establish governing equations for different zones separately and to then obtain compatible solutions by applying proper matching boundary conditions. In the pure fluid zone, it is common to assume the motion essentially irrotational except near the interface. Most of the investigators neglected the influence of the boundary layer with the exception of Liu (1973) and Liu and Dalrymple (1984) who included in their solutions two laminar boundary layers at the mud line. Since 'the damping is largely due to the energy losses in the porous medium rather than the boundary layer losses (Liu and Dalrymple, 1984) ', the boundary layer effect will be ignored in this study to simplify the mathematics.
4.1 Boundary Value Problem
The governing equation for the velocity potential function in the fluid domain is
v2D= 0 h< z
In the porous medium domain, the linearization of the nonlinear pore pressure equation, Eq.(3.32), yields
 VP. = pufoq h+ h.) <<h (4.2)
where fo is the linearized resistance coefficient.
{z
T 7 AV x 1k1
I I
V {\ [x,
h
Figure 4.1: Definition Sketch If the porous medium is rigid, thus, incompressible, the continuity equation for the discharge velocity, V i = 0, leads to
VP=0 ( A + h < h 4.3)
The boundary conditions of entire system are
r(x, t) = = a ei(ko0 at z = 0 (4.4)
g 8t
. g. 0=0 at z = 0 (4.5)
0*o :; ;*o'o'ono 04 o oooo0o0oo0 0O 00
Figure 4.1: Definition Sketch If the porous medium is rigid, thus, incompressible, the continuity equation for the discharge velocity, V. = 0, leads to
VIP, =0 (h + h.) <h (4.3)
The boundary conditions of entire system are
1 aD ei(kz._at)
,7(x, t) = gat= a eat z=0 (4.4)
a,.1) z=O at z=0 (4.5)
alD
p = P, at z=h (4.6)
at
8_ 1 8P,
az f z at z=h (4.7)
8z pufo 8z
33
BP,
7 = 0 at z = (h + h,) (4.8)
8z
The potential and pore pressure functions are assumed to have the forms
#(z,z,t) = [Acoshk(h + z) + Bsinhk(h + z)ei(kzat) (4.9)
P,(x,z,t) = Dcoshk(h+ h, + z)es(kzat) (4.10)
where A, B and D are unknown complex constants. Equation (4.8) is already satisfied by P, in Eq.(4.10). Introducing Eqs.(4.9) and (4.10) into Eqs.(4.6) and (4.7), A and B can be expressed in terms of D, A = i Dcoshkh, (4.11)
Pa
D
B sinh kh, (4.12)
pefo
4 is then given by
D(Xz, t) = D [icosh coshcoshk(h + z) + 1 sinhkh sinh k(h+ z)]eCi(kzat) (4.13) Pa fo
D can be obtained by the free surface boundary condition given by Eq.(4.4): Spga (4.14)
cosh kh cosh kh.(1 tanh kh tanh kh.) fo
Finally, Eq.(4.5) along with Eq.(4.13) gives the complex wave dispersion equation:
a2 gk tanh kh = tanh kh,(gk oa2 tanh kh) (4.15)
fo
here either a or k, or both, could be complex, as well as the coefficient fo.
In the above equation, when setting Ca = Cf = 0, it becomes the dispersion equation obtained by Liu and Dalrymple (1984). If C1 # 0, as will be the case in this study, the coefficient fo can not be a known value a priori. It becomes another unknown besides a or k, and therefore the procedure of solution will be different from that employed by Liu and Dalrymple. Before solving Eq.(4.15), a few limiting cases are examined here.
34
For the case of infinite seabed thickness and low permeability (laminar skin friction dominates, fo  ), Eq.(4.15) reads a2 gk tanh kh = iR(gk a2 tanlh kh) (4.16)
which is the same as obtained by Reid and Kajiura (1957). If R 0 as with an impervious bed, Eq. (4.16) reduces to the ordinary dispersion relationship for a finite water depth h. On the other hand, if the permeability approaches infinity, we have
R +oo, n 1.0, # 1.0, fo.i
The dispersion relationship expressed in Eq.(4.15) can be shown as 02 = gk tanh k(h + h,) (4.17)
Physically, this is the case of water waves over a finite depth h + ha, or, the solid resistance in layer h, vanishes.
Another limiting case is where the water depth approaches zero and waves now propagate completely inside the porous medium. The dispersion relationship from Eq.(4.15) becomes
a i tanh kh, (4.18)
It can be easily shown that the same dispersion relationship can be obtained by directly solving the problem of linear gravity waves in a porous medium alone. Again, for the case of laminar resistance only, Eq.(4.18) becomes a2 = iRgk tanh kh (4.19)
which states the Darcytype resistance law. On the other hand, when R  co, we obtain from Eq.(4.18)
a2 = gk tanh kh (4.20)
or, as expected, the dispersion relationship in a pure fluid medium.
35
When the virtual mass and the drag coefficients are set to be zero, Eq.(4.15) gives the dispersion equation for the homogeneous solution by Liu and Dalrymple (1984) which is
u2 gk tanh kh= 1 tanh kh,(gk a2 tanh kh) (4.21)
R
4.2 The Solutions of The Complex Dispersion Equation
As mentioned in the previous section, Eq.(4.15) is an equation involving both of the complex unknown variables of either a or k, and the complex coefficient fo. The other necessary equation can be obtained from the linearization process (This step is not necessary if C1 = 0, as in Liu and Dalrymple's solution). The common method of evaluating the linearized resistance coefficient, fo, is to apply the principle of equivalent work. This principle states that the energy dissipation within a volume of porous medium during a time period should be the same when evaluated from the true system or from its equivalent linearized system
(ED)I = (ED)a (4.22)
where ED is the energy dissipation in a controlled volume during one wave period and the subscripts I and n1 refer to linearized and nonlinear systems, respectively.
It can be readily shown (Appendix A) that such energy dissipation (considered as a positive value) can be expressed in the form of a boundary integral ED = d d (4.23)
where ED is a complex energy dissipation function and is the complex energy flux normal to S, with the real parts of them being the corresponding physical quantities. Here S is the closed boundary of the computation domain. The complex energy flux function 7,, can be expressed as (Appendix A)
1 t e 2i t
. w(t + UnP*) (4.24)
2
36
where u, is the normal velocity at the boundary S and p" is the conjugate of pore pressure p, both of them are complex quantities; a, is the wave frequency, areal value, and the subscript r is used to distinguish the complex a.
Physically, there are two classes of problems: standing waves of a specified wave number, and progressive waves of a specified wave frequency. In the former case, k is real, a is complex, and the solution has the following form r7(xt) = ae'i' cos kxe t o, < 0 (4.25)
where ar is the imaginary part of the complex a and a, is the wave frequency. In the latter case, a is real, k is complex and 7 becomes
ti(x,t) = aekze(ktzat) ki > 0 (4.26)
where k, and ki are, respectively, the real and imaginary parts of k with k, being the wave number.
For standing waves, the pore pressure function is
P, = PC'" = Dcoshk(h + h, + z) coskxeitea't (4.27)
Since P, is periodic in x, the boundary curve S for the contour integral in Eq.(4.23) can be chosen as x = 0, z = (h + h,), x = L and z = h with L being the wave length.
As x = 0 and x = L are the antinodes of the standing wave, there is no normal velocity, i.e. u,, = 0, on these two vertical boundaries and also at the impermeable boundary z = (h + h.). Then f rt+r lL( pit
ED jt 0 + wo p) dx dt (4.28)
with
kD
Wo= U, I,=h= k sinh kh, cos kx eait (4.29)
paf
and
f = fo for linearized system (4.30)
1 i )3+ C IwoI
R a
 fA + f2 wO j for nonlinear system (4.31)
with
f =  i = Cd (4.32)
R a
In Eq.(4.31), the magnitude of w0 is approximately taken as
wo  (fo) sinhkh, (4.33)
af
with
(fo) = (4.34)
cosh kh cosh kh,(1 tanh kh tanh kh,) where the decaying wave amplitude ae
With such approximations, f for the nonlinear system is no longer a function of x and t. The approximations made above are not expected to cause significant errors for the final results.
By introducing Eq. (4.29) into Eq.(4.28) and carrying out the contour integration regarding f as a constant, the energy dissipation within that onewavelong portion of a seabed is obtained as
kLTsih oD'_(ff )  [D(f) I
ED = kLT sinh2kh,( T(t) + T2(t)] (4.35)
8p h f f
where Ti (t) and T (t) are the nondimensional functions of t generated by the integration with respect to time.
Substituting the linearized and the nonlinear resistance coefficients given in Eqs.(4.30) and (4.31) for f in Eq.(4.35) respectively, equating the energy dissipations
38
by the two systems according to Eq.(4.22) and assuming that a and k are the same for both systems, we have D(fT) T (t) + 72I D (t) =
10 fo
D'(fA + f2 Iwol)1) 12 D(T = m)
fi f2 Iwo I + D(f + f2 Iwo 1) 2(t) (4.36)
2fl +AIwo I fl+f2 IWoI
Since this equation has to be satisfied at any time instant, we obtain the following equations:
D 2(fo) D 2(f + f2 IO 1) (437)
 (4.37)
fo A + f2 IwoI
I D(fo) j I D(f + f2 Iwo 1)12 (4.38)
O fl + f2 IWOI
Therefore,
fo = fh + f2 IwO 1 (4.39)
Substituting the expressions for fl, f2 and wo in the above equation, it becomes fo = 1 i + Cd I kD(fo) sinh kh, I (4.40)
R C Ia Ufo
It is clear, from the definitions of the parameters, that all the terms in Eq.(4.40) are complex quantities in the standing wave case.
For progressive waves, the pore pressure function P, is given by Eq.(4.10) and the contour S for the integration is the same as that for standing waves. The energy dissipation is
t+T
E = ds dt
t+
= f ( 7 +1 F dz + f n d)dt (4.41)
t =0 ==L =h
By substituting the expression of ', derived from Eq.(4.10) into the above equation, the summation of the first two integrals is found to be O(I ki 1) while the last
39
one is 0(1). If we assume that I ki j< 1, which is generally true in coastal waters (see the results for the progressive wave case), then the energy dissipation becomes ED = t 7,% dx dt (4.42)
Eq.(4.42) will clearly lead to the same relationship as that given by Eq.(4.40) except that the averaged wave amplitude d in D, in this case, is the spatially averaged value of ae kiz over [0, L].
In the progressive wave case, we assume that the wave period is given and that there is no time dependent damping, i.e. a is real. Therefore, the first and the last term on the R.H.S. of Eq.(4.40) are all real numbers. It is then obvious that
(fo)s/3 (4.43)
and
1 Cd I kD(k, fo) I (4.44)
R alfol
where the subscripts i and r are for imaginary and real parts, respectively.
To actually compute fo requires specification of two fundamental quantities of n and d, (see Eq.(4.40) and the definitions for R, Cd and 6). It is often more convenient to specify the permeability parameter R = aK,/v (R' = a, Kp/v for standing waves), as opposed to specifying the actual granular size, d,. Under these circumstances, we could replace Cd with the following expression (Ward, 1964): C, Cf f (4.45)
Cd = a
with
 bo(1 n) (4.46)
6 O =nao(1 n)3
Now C1 is a nondimensional turbulent coefficient. To be consistent with the suggested values of ao and b0 given in Eqs.(3.14) and (3.20), C1 should be in the range of 0.3 1.1 for a porosity of n = 0.4. Equation (4.40) now becomes:
1 i/+ C1 I kD(k, fo) sinh kh (47)
f = a I afo1
40
This is the final form of the equation used for computing fo for both cases.
The dispersion equation, Eq.(4.15), which is coupled with Eq.(4.47), is solved iteratively. The procedures to obtain the solution for the cases of standing waves and progressive waves are different.
a) For standing waves a = oa, + icai is complex, k is real and known. Equation (4.15) can be rewritten as
gk(tanhkh tanhkh,)
a2' = o = Q(fo) + iQ,(fo) (4.48)
1 tanh kh, tanh kh
fo
with Qr(fo) and Qi(fo) being the real and imaginary parts of a2, respectively. The iteration procedures are summarized as follows:
,(n+1) 1 [ VQ(f'n')+Q;)(fO() + Qr(f(n))] (4.49)
 [Q(f))+Q2(fn)) Q,(fo) ] (4.50)
1 C k I r( n), (4.51)f
V") i i(n)f() (4.51)
f 1) 1 if (4.52)
0 R
where the superscript n denotes the level of iteration, and the criterion of convergence is
f a) fo(1 and o") '(n) j< E (4.53)
I f(n) a_, n (n)
with e being a prespecified arbitrarily small number. It is set to be 1.0% in this model.
b) For progressive waves k = k, + iki is complex and a is real and known. In this case, the dispersion equation can be written as
i
F(k, fo) = a2 gk tanh kh + tanh kh,(gk a tanh kh) = 0 (4.54)
and the iteration was carried out as follows:
and the iteration was carried out as follows:
(4.55)
r(k"+), f")) = 0
41
F,(k(n+1),fon)) = 0 (4.56)
f(n) _1 j,3+ 1 C1 I kD(k(n),fo"))l (4.57)
R VraV' lafoCn)I
fo i) (4.58)
R
where F, and F are the real and imaginary parts of F and n indicates the iteration level.
The criterion of convergence for such case is
fn) f O I) and I k() ) ( (4.59)
fofl) k(n)
with e being a prespecified arbitrarily small number. It is again set to be 1.0% in this model.
4.3 Results
The predicted damping rates kip from the solution of Eqs.(4.15) and (4.47) are first compared to the laboratory data ki, by Savage (1953) for progressive waves propagating over a sandy seabed. The experiment was conducted in a wave tank of 29.3 meters (96 ft) long, 0.46 meters (1.5 ft) wide and 0.61 (2 ft) meters deep. The porous seabed was composed of 0.3 meters thick of sand with the medium diameter of 3.82 mm. The water depths are h = 0.229 m. and h = 0.152 m. The data for the water depth of h = 0.102 m was ignored for the reasons given by Liu and Dalrymple (1984). The wave conditions and the comparison of the damping rates are listed in Table 4.1. The parameters used in the solution of Eqs.(4.15) and (4.47) are: n = 0.3, ao = 570, b0 = 2.0 and C. = 0.46 and the average wave height H in the table was calculated according to Ho HLg)
In Ho (1 Ho
HLg
where Ho and HLg are the wave heights measured at two points of Lg apart with Lg = 18.3 m (60 ft). The values computed by Liu and Dalrymple (1984) are also
42
listed. The relative error in the table is defined as A =1kim ,IX100% (4.60)
kim
with ki being either ki or kiLD.
In this case, the errors are of similar magnitude between the present model and the model by Liu and Dalrymple (1984). This is because the experimental values fall in the region where the inertial resistance due to the virtual mass effect and the turbulent resistance, neglected in Liu and Dalrymple's model, are unimportant.
In Fig. 4.2 through Fig. 4.5, the solutions of the complex dispersion equation are illustrated graphically. The case of progressive waves are demonstrated first. The specified conditions for the progressive waves are: H = 1 m, T = 4 seconds, h, = 5 m and n = 0.4. Figure. 4.2 plots the values of k, (wave number) and kic (damping rate) against R (nondimensional permeability parameter) for three different water depths, h = 2, 4, and 6 meters. The equivalent particle sizes for the range of R values are also shown in the figure; they cover a range from 2.3 mmn to 2.3 mx. The thick dlash line is the solution of the dispersion equation given by Liu and Dalrymple (1984, Eq.(4.5)) for the case of h 4 m. The correction term of the laminar boundary layer was not included in this curve since it is negligible in this case. From these results, a number of observations can be made:
1. As expected, the wave number decreases (or wave length increases) monotonically with increasing R, from one limiting value k,,, corresponding to the case of an impervious bottom at depth h, to the other limiting value k,2, corresponding to the case of a water depth equal to h + h, when the lower layer becomes completely porous.
2. Compared with the linear resistance, the nonlinear and the inertial resistances dominate for the complete range of R values displayed. In the region where only linear resistance dominates (Rf < 1, R,~ < 1), the bottom effects are relatively small and often negligible (outside the R range shown).
Table 4.1: Comparison of The Predictions and The Measurements
Run No. H (cm) T (sec.) ki,(m') k1p(m1) A,% kLDF m) ALD% h = 0.229 m
1 6.74 1.27 0.0379 0.0313 17.4 0.0338 10.8
2 4.45 0.0318 0.0337 6.0 0.0338 6.3
3 5.26 0.0303 0.0328 8.3 0.0338 11.6
4 1.93 0.0317 0.0369 16.4 0.0338 6.3
12 6.65 0.0334 0.0313 6.3 0.0338 1.2
13 4.36 0.0283 0.0338 19.4 0.0338 19.4
14 1.90 0.0374 0.0370 1.1 0.0338 9.6
5 6.25 1.00 0.0411 0.0350 14.8 0.0390 5.1
6 4.66 0.0357 0.0372 4.2 0.0390 9.2
7 2.17 0.0397 0.0411 3.5 0.0390 1.8
8 2.08 0.0393 0.0413 5.9 0.0390 0.8
15 5.48 0.0383 0.0360 6.0 0.0390 1.8
16 4.52 0.0397 0.0374 5.8 0.0390 1.8
17 1.93 0.0373 0.0415 11.3 0.0390 4.5
9 5.29 0.80 0.0296 0.0314 6.1 0.0343 15.8
10 3.55 0.0297 0.0334 12.5 0.0343 15.5
11 2.28 0.0335 0.0351 4.8 0.0343 2.4
18 7.05 0.0264 0.0296 12.1 0.0343 29.9
19 4.08 0.0320 0.0328 2.5 0.0343 7.2
20 2.14 0.0305 0.0353 15.7 0.0343 12.5
h = 0.152 m
21 4.00 1.27 0.0552 0.0574 3.9 0.0600 8.7
22 1.83 0.0379 0.0640 68.9 0.0600 58.9
23 3.83 0.0555 0.0579 4.3 0.0600 8.1
24 1.36 1.00 0.0449 0.0770 71.5 0.0731 62.8
25 3.40 0.80 0.0658 0.0700 6.4 0.0767 16.6
ki, the experimental damping rate given by Savage (1953); kip the theoretical values by the present model; k LD the theoretical value by Liu and Dalrymple (1984)
44
3. The wave attenuation, and hence the wave energy dissipation, shows a peak. This peak occurs when the magnitude of the dissipative force (velocity related) equals to that of the inertial force (acceleration related). Depending upon the water depth, this attenuation could be quite pronounced under optimum R values. For instance, for the case h = 4 m, Fig. 4.2 shows the wave height is reduced to about 74% of its original value (or about 46% energy dissipation) over approximately 2 wave lengths.
4. The locations of the peak damping are quite different from that of Liu and Dalrymple (illustrated here for the case of h= 4 m); they occur at a higher permeability. The magnitude of the peak damping is generally smaller than that of Liu and Dalrymple's solution. The values of k,. from the two solutions are also different.
In Fig. 4.3 the maximum damping rate and the corresponding permeability parameter R are plotted as functions of nondimensional water depth. The curves from Liu and Dalrymple (1984) are also plotted for comparison. The trends of
(kj),,'s are similar but the corresponding R behaves quite differently from the two solutions.
The case of a standing wave system is also illustrated here in Fig. 4.4. The behavior is very similar to that of the progressive waves. The grain size in the figure is calculated according to a,. for the case of h = 4.0 m. Finally, the solutions of the dispersion equation for standing waves based on Darcy's model (DARCY: = 0, C1 = 0), Dagan's model (DAGAN: P3 = .1, C1 = 0), DupuitForchheimer's model (D F: = 0 and C1 follows Eq.(4.46) and SollittCross's model (S C: 0 follows Eq.(3.30) with C1 following Eq.(4.46) and n = 0.4, ao = 570, b0 = 3.0 and C. = 0.46) for standing waves are compared in Fig. 4.5. Under the same wave conditions, the differences among the various solutions are seen to be very pronounced. for the permeability range displayed here. When the permeability parameter becomes less
h h ............. h
h
PARTICLE SIZE (CM)
2.32 23.20
= 4.0 m Liu and Dalrymple H = 1.0 m = 2.0 m T = 4.0sec
= 4.0 m h, = 5.0 m
= 6.0 m
I i I I V m
1.0 0.0 1.0 2.0
PERMEABILITY PARAMETER LOG(R)
2.32
23.20
2.0 1.0 0.0 1.0 2.0
PERMEABILITY PARAMETER LOG(R)
232.00
3.0
Figure 4.2: Progressive wave case. (a) Nondimensional wave number k,/(a2/g), (b) Nondimensional wave damping rate ki/(a'/g).
0.23
4
I.8 
1.4 
232.00
2.0
0.23
0.10 0.08 0.06
0.04 0.02
0.00
.... .. . . . . .
. .. .. .. .. .. .. .. .. ..
I m
46
4.0
(LD): Liu and Dalrymple (1984)
3.0
2.0.
R
.0
o
~ (k,) ma:(LD)
0.0
" sR (LD)
1.0
2.0
2.0 1.5 1.0 0.5 0.0 0.5 1.0
Normalized water depth log(h a'2/g) Figure 4.3: Maximum nondimensional damping rate (ks).,=/(a2/g) and its corresponding permeability parameter R as functions of nondimensional water depth h ('/g).
47
than 10'2, all solutions converge to a single curving following Darcy's law. For very highly permeable seabeds, the damping rate based on D F model is very high and tends to increase with the permeability as oppose to approaching to zero according to common sense. Also in the high permeability region, the wave frequency based on Darcy's model is much larger than the correct value (determined by Eq.(4.17)). The reason for such large error is that the force balance of the pore fluid in highly permeable media is now mainly between pressure gradient and the inertia rather than the velocity related frictions.
PARTICLE SIZE (CM)
2.28 22.69
 A = 4.0 m Liu and Dalrymple
h = 2.0 m ............. h = 4.0 m
 h = 6.0 m
225.44
H = 1.0m L = 20.0 m h, = 5.0 m
1.60 1.50
1 .40 I .30 1 20 1.10
2.0 1.0 0.0
PERMEABILITY
i i
1.0 PARAMETER
(a)
2.0 LOG (RI
0.23 2.28 22.69
225.44
0.01
0.0
2.0
1.0 0.0 1.0 2.0 3.0
PERMERBILITT PARAMETER LOG(RI
Figure 4.4: Standing wave case. (a) Nondimensional wave frequency ar,/(L/g)1; (b) Nondimensional wave damping rate aj/(L/g)i.
0.23
 .... ........ .. ...
1.00
0.90
0.06
0.05 0.04
0.03 0.02
.
.
.
.
PARTICLE SIZE (CM)
2.28 22.69
S C ............. ODAGAN
DARCY 0 F
H = 1.0 m
L = 20.0 m
h, = 5.0m

/
/ h=4m
/'
../: ... ... ... ... ... ... ................................ "
......
1.0 0.0
PERMEABILITY
2.28
~I ~ I I 1.0 2.0
PARAMETER LOG(R)
22.69
2.0
1.0 0.0 1.0 2.0
PERMEABILITT PARAMETER LOG(R)
Figure 4.5: Solutions based on four porous flow models. (a) Nondimensional wave frequency o,./(L/g)i, (b) Nondimensional wave damping rate ai/(L/g)i.
0.23
.i.
225.44
1.4 1.3
1.2
2.0
0.23
225.44
0.12
0. 10
0.08 0.06 0.04 0.02
0.00
S 5 C ,............. ORGAN
/' \
.O... ARCY
. \ .... DARCY
/\
./ .. 
I t
.
.
CHAPTER 5
LABORATORY EXPERIMENT FOR POROUS SEABEDS
The experiment was carried out in a wave flume in the Laboratory of Coastal and Oceanographic Engineering Department of University of Florida. The flume is about 15.5 meters long, 0.6 meters wide, 0.9 meters high and equipped with a mechanically driven pistontype wave maker. All the tests were conducted with standing waves.
5.1 Experiment Layout and Test Conditions
Figure 5.1 shows the experiment arrangement. A porous gravel seabed was constructed at the end of the wave flume in the opposite side of the wave maker. A sliding gate was positioned at one wavelength from this end of the tank to trap the standing wave after a sinusoidal wave system was established by the wave maker. The decay of the freely oscillating standing wave was then measured by a capacitance wave gage mounted at the center of the compartment. The damping rate of the porous seabed for each particular test condition was determined by applying a least squares fit to the data according to the following equation:
1 Hi
(o,)i = mCHln ) (5.1)
Ti Hji
where j refers to jth wave. The corresponding wave frequency was obtained by averaging individual waves.
The contribution to the wave damping due to the side walls and the bottom was subtracted from the data according to the following equation
1 ln[1I + e2aieT e2a;.T (5.2)
2T
wave
maker
hs o **S
L/2 L/2
Figure 5.1: Experimental setup Table 5.1: Material Information d5o(cn) f0.72 0.93 1.20 1.48 2.09 2.84 3.74
porosity 0.349 0.349 0.351 0.359 0.369 0.376 0.382
where u, is the actual seabed damping rate, a; is the gross damping rate of the seabedwall system, ai,,, is the damping rate by the side walls and the bottom, and T is the wave period.
The bed material used in the experiment was river gravel of seven sizes, ranging from d50 = 0.72 cm to 3.47 cm, with all sizes having a fairly round shape and smooth surface. The porosity of the material increases slightly with the diameter, as given in Table 5.1.
With the grain sizes being determined by the material selection, the adjustable independent parameters left in the dispersion equation are the water depth h, the seabed thickness h,, and the wave length L. A total of 36 different cases was tested
Table 5.2: Test Cases
d5o (cm) h, (cm) h (cm) L (cm)
0.72 20 20,25,30 200
0.93 20 20,25,30 200
1.20 20 20,25,30 200
1.48 10,15,20 20,25,30 200
2.09 20 20,25,30 200,225,250,275
2.84 20 20,25,30 200
3.74 20 20,25,30 200
and these test conditions are summarized in Table 5.2.
The damping effect due to side walls alone was found to be very small, and an average value of uri,, = 0.01 secI was used for all the corrections.
5.2 Determination of The Empirical Coefficients
The experimental results for the conditions of h, = 20 cm and L = 200 cm are given in Table 5.3 (complete results are given in Appendix B). Each data point represents an averaged value of 10 to 20 tests.
Figure 5.2(a) shows a typical example of measured waves and the exponential fitting to the wave heights. The solid line in Fig. 5.2(a) is the ensemble average whereas the two dash lines are the envelopes of plus and minus one standard deviation from the mean value at each time instant. From the wave form, it is noted that the waves in the experiment were more or less nonlinear waves as oppose to what was assumed. The effect of the nonlinearity to the calculation of the measured damping rate can be minimized by using the ratio of wave heights instead of wave amplitudes. In Fig. 5.2(b), the dashed curve is the exponential decay function with the a, obtained from the data ensemble by the least square analysis.
Based on the measured a, and a, listed in Table 5.3, the empirical coefficients ao, bo and Ca were then determined by multivariate linear regression analysis such
Table 5.3: Measured a, and ai for h, = 20 cm and L = 200 cm
h (cm) dso (cm) R; (cm) Y" (cm) o,(s') a',g(s1) o,(s1)
0.72 11.08 6.49 4.9054 0.0686 0.0570 0.93 10.93 6.35 4.9402 0.0678 0.0562 1.20 10.47 5.52 4.9370 0.0814 0.0694
30.0 1.48 9.03 4.71 4.9620 0.0878 0.0757
2.09 9.75 4.85 4.9673 0.0948 0.0824
2.84 9.80 4.95 5.0111 0.0931 0.0808
3.74 9.80 5.71 5.0341 0.0658 0.0543
0.72 8.70 5.15 4.6218 0.0799 0.0678
0.93 8.51 4.75 4.6856 0.0905 0.0781
1.20 8.00 4.46 4.6793 0.0984 0.0858
25.0 1.48 7.28 3.91 4.7352 0.1162 0.1030
2.09 6.46 3.57 4.7422 0.1105 0.0975
2.84 7.37 4.34 4.7893 0.0940 0.0816
3.74 7.37 4.26 4.7865 0.0813 0.0693
0.72 6.70 4.19 4.2716 0.1184 0.1047
0.93 6.28 3.55 4.3426 0.1334 0.1192
1.20 6.07 3.60 4.3565 0.1320 0.1179
20.0 1.48 4.49 2.57 4.4027 0.1484 0.1337
2.09 5.19 3.12 4.4325 0.1546 0.1396
2.84 5.41 3.33 4.4728 0.1456 0.1311
3.74 5.56 3.58 4.4646 0.1313 0.1173
* H1 is the average value of the first wave heights of the data group and H is the average height of the complete train.
54
0.60
z H = 5.29CM
o
 0.40 L = 200. CM
> .T = 1.27 5
, 0.20
U
U 0.00.
o 0.20
= 0.40
O z
0.60
0.0 4.0 8.0 12.0 16.0 20.0
TIME (SEC)
1.20
ODW= 30.0 CM 1.00 S.. = 15.0CM
00= 1.48CM
LIi
= 0.80 .
2: 0.60 .. ...
C
LIN4
; 0.40 ...... ..
S0.20 .
0.00
0.0 4.0 8.0 12.0 16.0 20.0
TIME (SEC)
Figure 5.2: Typical wave data: (a) Averaged nondimensional surface elevation (7l/Hi), (b) Nondimensional wave heights (1/H) I and the best fit to the exponential decay function.
that the error function defined as
= 1 r p)2 + (Uim iP)2] (5.3)
_1=1 arm im
was minimized, where the subscript m represents the measured values and p denotes the predicted values; M is the number of data points. The best fit was found to exist when:
ao = 570
bo = 3.0
C, = 0.46
The added mass coefficient obtained here, C, = 0.46, is close to the theoretical value of 0.5 for a smooth sphere. The values of a0 and b0, to an extent, reconfirm those given by Engelund (1953) and the others.
5.3 Relative Importance of The Resistances in The Experiment
Introducing a0, bo, C., and the averaged wave height F into Eq.(4.29) for I qI and using Eqs.(3.33) through (3.35), the relative importance of the various resistances of the tested cases can be established precisely. The results are given in Table 5.4. As we can see from these ratios, for the first four grain sizes, all three resistances are about equally important. Whereas for the last two larger diameter materials, the turbulent and the inertial resistances are evidently dominant over the linear resistance, with the inertial force approximately double that of the nonlinear resistance.
5.4 Comparison of The Experimental Results and The Theoretical Values
Table 5.5 shows the comparisons between experimental results and the theoretical values of ai and a,. The relative error, defined as A% =1 a I x100% (5.4)
am
Table 5.4: Comparison of The Resistances. h, = 20 cm, L = 200 cm.
d5o (cm) q I (cm/s) c I R IR, R If,/f, If.if,'f, /fi h = 30.0 cm
0.72 0.57 4.788 37 225 0.93 1.32 1.42
0.93 0.69 4.800 58 379 1.56 2.06 1.32
1.20 0.75 4.819 81 630 2.63 2.88 1.10
1.48 0.78 4.842 104 960 4.18 3.71 0.89
2.09 0.96 4.869 181 1933 8.90 6.48 0.73
2.84 1.09 4.889 281 3577 17.12 10.10 0.59
3.74 1.33 4.900 453 6240 30.90 16.34 0.53
h = 25.0 cm
0.72 0.53 4.526 34 213 0.88 1.21 1.38
0.93 0.62 4.545 52 358 1.48 1.85 1.25
1.20 0.70 4.567 76 597 2.49 2.71 1.09
1.48 0.75 4.596 101 915 3.99 3.60 0.90
2.09 0.84 4.639 159 1833 8.44 5.70 0.67
2.84 1.10 4.657 284 3407 16.31 10.21 0.63
3.74 1.18 4.679 400 5959 29.51 14.43 0.49
h = 20.0 cm
0.72 0.50 4.180 32 196 0.81 1.14 1.41
0.93 0.55 4.206 46 332 1.37 1.64 1.20
1.20 0.66 4.232 72 553 2.31 2.56 1.11
1.48 0.61 4.286 81 849 3.70 2.89 0.78
2.09 0.85 4.322 160 1707 7.86 5.73 0.73
2.84 1.00 4.356 258 3187 15.26 9.27 0.61
3.74 1.15 4.381 393 5579 27.63 14.18 0.51
Table 5.5: Comparison of Measurements and Predictions for h, = 20 cm, L = 200 cm.
d,,) (cm)] 1? Urm( .. o'.()A% (S )1 aip(S ) AjZ
_______ h = 30.0 cm __ _
0.72 0.1791 4.9054 4.7880 2.39 0.0570 0.0541 5.07 0.93 0.3021 4.9402 4.8001 2.84 0.0562 0.0628 11.60 1.20 0.5110 4.9370 4.8187 2.40 0.0694 0.0706 1.66 1.48 0.8448 4.9620 4.8419 2.42 0.0757 .0.0751 0.76
2.09 1.8755 4.9673 4.8685 1.99 0.0824 0.0734 11.00 2.84 3.7429 5.0111 4.8891 2.43 0.0808 0.0661 18.26 3.74 6.9543 5.0341 4.9001 12.66 0.0543 0.0613 12.85
h = 25.0 cm___0.72 0.1687 4.6218 4.5264 2.06 0.0678 0.0709 4.49 0.93 0.2866 4.6856 4.5445 3.01 0.0781 0.0840 7.49 1.20 0.4843 4.6793 4.5666 2.41 0.0858 0.0935 8.97 1.48 0.8095 4.7352 4.5963 2.93 0.1030 0.1000 2.94 2.09 1.7819 4.7422 4.6389 2.18 0.0975 0.0956 1.94 2.84 3.5772 4.7893 4.6566 2.77 0.0816 0.0912 11.76 3.74 6.6123 4.7865 4.6795 12.24 0.0693 0.0781 12.67
h = 20.0 cm______0.72 0.1559 4.2716 4.1799 2.15 0.1047 0.0907 13.34 0.93 0.2656 4.3426 4.2061 3.14 0.1192 0.1109 6.96 1.20 0.4509 4.3565 4.2318 2.86 0.1179 0.1228 4.16 1.48 0.7496 4.4027 4.2857 2.66 0.1337 0.1319 1.32 2.09 1.6656 4.4325 4.3217 2.50 0.1396 0.1308 6.33 2.84 3.3408 4.4728 4.3560 2.61 0.1311 0.1200 8.42 3.74 6.1676 14.4646 14.3805 11.88 0.1173 0.1076 18.24
is uniformly very small for a, (wave frequency). For a, (wave damping), the range
of error is larger, with the maximum being 18.26% of the measured value.
The agreement between the predictions and the measurements shown in the
above table is demonstrated by Fig. 5.3 where the values of a7, and ai in Table 7 for
the case of h = 25 cm are plotted against the permeability parameter.
10.0
1.0
0.120
0. 100 0.080
0.060
0.040 0.020
0.000
1.0
0.5 0.0 0.5
PERMEABILITY PARAMETER LOG(R)
0.5
0.0
0.5
PERMEABILITY PARAMETER LOG()
Figure 5.3: The Measurements and the predictions vs. R. for L = 200.0 cm, h, = 20.0 cm, h = 25.0 cm. (a) Wave frequency at, (b) Wave damping rate oj.
WAVE LENGTH = 200 CM BED THICKNESS = 20 CH WATER DEPTH = 25 CH
a0 & 0
+ +
* PREDICTED
x MEASURED
I
Table 5.6: Comparison of a, and ap for do = 1.48 cm, L = 200 cm.
h (cm) IH (cm) I rm.,(s) I o7.(s1) IA,.% Iaim(s ) Icip(S )A% h. = 15.0 cm
30.0 5.29 4.9268 4.8291 1.98 0.0525 0.0600 14.13 25.0 4.00 4.6715 4.5836 1.88 0.0697 0.0794 13.98 20.0 3.12 4.3173 4.2565 1.41 0.0928 0.1054 13.56 h. = 10.0 cm
30.0 5.97 4.9163 4.8145 2.07 0.0368 0.0416 13.13 25.0 4.61 4.6310 4.5629 1.47 0.0492 0.0551 11.92 20.0 3.48 4.2688 4.2309 0.89 0.0724 0.0729 0.70
Table 5.7: Comparison of 0, and oap for d5o = 2.09 cm, h, = 20 cm.
h (cm) H (cm) I ) crpC5T I A% Uim(s ) I i(s ) IZ%
L = 225.0 cm
30.0 4.42 4.5114 4.4401 1.58 0.0831 0.0766 7.73 25.0 3.91 4.2913 4.1991 2.15 0.0977 0.1006 2.92 20.0 2.96 3.9699 3.8995 1.77 0.1270 0.1286 1.21 L = 250.0 cm
30.0 4.40 4.1524 4.0725 1.92 0.0752 0.0790 5.02 25.0 3.79 3.9119 3.8358 1.94 0.1010 0.1002 0.81 20.0 3.32 3.5862 3.5361 1.40 0.1384 0.1282 7.34 L = 275.0 cm
30.0 4.18 3.8371 3.7592 2.03 0.0783 0.0788 0.68 25.0 3.32 3.6285 3.5339 2.61 0.0993 0.0962 3.15 20.0 2.90 3.3298 3.2497 2.41 0.1264 0.1211 4.16
We now proceed to compare the theoretical computations with those data which
were not included in the calibration. These results are given in Table 5.6 and
Table 5.7. The agreement is just as good, if not better.
In Fig. 5.4, the theoretical values of a,'s and a,'s by the present model are
plotted, respectively, against the measurements for all the experimental data (36
cases). The overall least mean square error defined in Eq.(5.3) for the data of all 36
cases is about 0.008. While the solution of the dispersion equation (Eq.(4.21) based
on the linear porous flow model by Liu and Dalrymple (Liu and Dalrymple, 1984)
Table 5.8: Comparison of a,, and op for ds0 = 0.16 rm, h, = 20 cm, L = 200 cm.
h (cm) R a,.m(s ) arp(S) A,% aim(s5 )
30.0 5.4 x 10. 4.8908 4.7638 2.60 0.0070 4.5 x 10.7
25.0 5.1 x 107 4.6312 4.4957 2.93 0.0090 5.6 x 107
20.0 4.8 x 107 4.2824 4.1428 3.26 0.0122 6.8 x 107
is plotted in Fig. 5.5. In obtaining the solution, the only empirical coefficient ao appeared in their dispersion equation was calibrated to the data in Table 5.3 with the same routine as that for the solution of Eq.(4.15), and the nonlinear regression yielded a0 = 2200. The overall least mean square error for the complete data set (36 cases) is as high as 0.327. It can be seen from the figure that considerable discrepancies occur for the damping rate, and the maximum relative error is about 87% of the experimental values. The prediction of wave frequency by their solution is about the same as that by this study.
A test over a sand bed (d50 = 0.16 mM) was also conducted. Active sand movement was observed along with the rapidly formed ripples. As a result, theoretical predictions are several orders of magnitude smaller than those actually measured (Table 5.8). The basic assumption of no particle movement was violated. However, the predictions of wave frequencies were as good as the coarsegrained cases. The same test, with dye injection into the sand bed, disclosed that the flow inside the sand bed was very slow, and indicated that the assumption of impermeability invoked in the linear wave theory is justified in such situations.
When the thickness of a gravel bed is reduced to one grain diameter, the model also failed to yield good results as shown by the values in Table 5.9. In this case, the effect of the boundary layer at the interface, which was not included in the model, obviously cannot be ignored.
The nature of a turbulent boundary layer over a thicker seabed was qualitatively investigated for the case of h,= 20 cm, ds0 = 1.48 cm, h =20 cm and L =200 cm,
2.0 3.0 4.0 5.0
EXPERIMENTAL FREDUENCT
0.25
0.20
0.15
0. 10 .,
0.05 "
0.00 p
0.00 0.05 0.10 0.15
EXPERIMENTAL DAMPING
0.20 0.25 RATE
Figure 5.4: Theoretical values by the present model vs. experimental data of Table
5.5, Table 5.6 and Table 5.7. (a) Wave frequency at, (b) Wave damping rate a,.
I
5.0 4.0
3.0
1 1 t I 1 6 i
2.0 3.0 4.0 5.0
EXPERIMENTAL FREQUENCY
0.25
0.20
0.15
0.10 0.05
0.00
0.00 0.05 0.10 0.15 0.20
0.25
EXPERIMENTAL DAMPING RATE
Figure 5.5: Theoretical values by the model of Liu and Dalrymple vs. experimental data of Table 5.5, Table 5.6 and Table 5.7. (a) Wave frequency or, (b) Wave damping rate a0.
~1' A.
* .'
* A
.** .
.4
Table 5.9: Comparison of crn. and ap for h, = d5o
d50 (cm) H (cm) crn(S ) Ir.5 ) I r% UI(,(s) rp(s ) TA,% h = 30.0 cm
0.72 6.05 4.9157 4.7666 3.03 0.0024 0.0031 29.66
1.20 5.88 4.9374 4.7723 3.34 0.0040 0.0041 3.04
2.09 5.58 4.9559 4.7827 3.50 0.0075 0.0042 43.69
3.74 5.21 4.9808 4.8002 3.63 0.0097 0.0049 48.99
h = 25.0 cm
0.72 4.64 4.6495 4.4991 3.23 0.0031 0.0041 31.58
1.20 4.53 4.6675 4.5068 3.44 0.0055 0.0056 0.96
2.09 4.28 4.6822 4.5205 3.45 0.0100 0.0056 43.88
3.74 4.02 4.7318 4.5438 3.97 0.0130 0.0064 50.59
h = 20.0 cm
0.72 3.24 4.2936 4.1470 3.41 0.0057 0.0053 7.19
1.20 3.23 4.3134 4.1571 3.62 0.0094 0.0075 20.32
2.09 3.07 4.3339 4.1755 3.66 0.0147 0.0074 49.62
3.74 2.88 4.3950 4.2061 4.30 0.0170 0.0081 52.29
by dye studies. Turbulent diffusion was spotted over virtually the entire porous
domain and up to approximately 1.0 cm above the interface. The effects of turbulent
boundary layers to the wave damping are important in wave interaction with porous
seabeds and more theoretical and quantitative experimental research is needed.
CHAPTER 6
BOUNDARY INTEGRAL ELEMENT METHOD
The Boundary Integral Element Method (BIEM) is an efficient numerical method for conservative systems such as those governed by Laplace equation. It is efficient because the computation is carried out only on the boundaries rather than over the entire domain as it would be in finite difference and finite element methods. Considerable amount of computation time and storage space can be saved without losing accuracy. In addition to the efficiency, the data preparation becomes much simpler as compared to the other two methods. For nonLaplace equations, the application of this method may become difficult and sometimes impossible. By the efforts of many scientists, the scheme has been extended to more and more problems governed by nonLaplace equations (Brebbia, 1987). In this chapter, we restrict ourselves to the two dimensional Laplace equation only.
6.1 Basic Formulation
Let U and V be any two continuous two dimensional functions, twice differentiable in domain D which has a close boundary C. According to the divergence theorem (Loss, 1950; Franklin, 1944, cited in Ligget and Liu, 1983) for continuity in a volume
fD(V ifC6 d (6.1)
where V is the vector operator, V is any differentiable vector and n is the unit outward vector normal to C. If we define that 6 = UVV and then V = VVU in Eq.(6.1), two integral equations can be generated in terms of U and V,
f(VU VV + UV2V)dA = UVV i.s (
C (6.2)
65
(VV VU + VV2U)dA= VVU iids (6.3)
Subtracting (6.3) from (6.2), we have
(UV'V VV'2U)dA = (UVV VVU) d (6.4)
Introducing the expressions
VV au
V n = VU n = n (6.5)
and assuming that both U and V satisfy Lapalce equation, i.e.
V2U V2V = 0 (6.6)
Equation (6.4) becomes
r av au
(U V an )ds = 0 (6.7)
To apply this equation, we choose U as the velocity potential 4 and V as a free space Green's function, G. Both of them satisfy Lapalce equation. Equation (6.7) can then be rewritten, in terms of 1 and G, as 8G(PQ) 8(Q)ld. C[.D(Q) aG(PQ) G(P,Q) ]ds = 0 (6.8)
where P is a point in the domain DnC and Q is a point on C.
One of the free space Green's functions for two dimensional problems is
G(P,Q) = In r (6.9)
where r is the distance between point P and point Q and it can be expressed by r = V(zp ZQ)2 + (zp zq)2 on the z z plane.
Substituting Eq.(6.9) into Eq.(6.8), it becomes #(Q) Br 8(Q)
[ r In r (Q) ]ds = 0 (6.10)
r 5n an
66
Despite the fact that the Green's function has a singularity at r = 0, the contour integral in Eq.(6.10) exists and can be worked out by removing the singular point from the domain.
In general, Eq.(6.10) becomes
C In r( ]d (6.11)
cz(P) TfQ)ar8
after removing the singularity at point P. Here a is 27r if P is an interior point and is equal to the inner angle between the two boundary segments joining at P if it is on the boundary. Generally, point P can be anywhere in the domain DnC, while Q is always on C due to the contour integration. In BIEM, since only the boundary values of D and a are solved, P has to be kept on the boundary C in an
the process of computation. When 4 and an on C are solved, the interior values can be derived with Eq.(6.11) by placing P at the point of interest inside D.
The closed boundary C in Eq.(6.11) is the same as that in Eq.(6.10) except that it does not pass point P as it does in Eq.(6.10). The singular point of the Green's function r = 0 has been excluded by a circular arc of infinitesimal radius from the domain DnC. Thus there is no singularity on the new contour C. Eq.(6.11) is the basic equation for BIEM formulations.
Discretizing the boundary C into N segments, and breaking the contour integration into N parts accordingly, Eq.(6.11) reads N 4D ar, ao N
=I ri5)ds=Ei (6.12)
3=1 arn an1
The curved segment C is usually replaced by a straight line to simplify the integration. This simplification generally does not introduce significant error provided that the segment is small enough.
Before carrying out the integration, the type of element has to be determined. The formulations are different for different types of element. The commonly used elements in two dimensional BIEM problems are: constant element, linear element,
67
quadratic element, special element and so on so forth. The classification of elements is based on the type of the function used to interpolate the values over an element. For example, on a constant element, the physical quantities and their normal derivatives are assumed to have no change over the element; while on a linear element, the quantities and their normal derivatives are interpolated by a linear function between the values at the ends of the element, i.e.
= N1( )DF g(C )4y+1 (6.13)
o (C) = Ng (e)( a ) i+ N2( alp )j+l (6.14)
Cj C ej+i
where N, and N2 are linear functions with the feature of N, (C.) = 1 N1(ej+1) = 0
g2( ) = 0 N2(e,+1) = 1
In the same manner, higher order elements can be defined by replacing N1 and N2 with higher order functions. When the variation of the quantities is known, the variation function can then be chosen as the interpolation function to obtain a more accurate element. Such elements are usually classified as special elements. Generally, it is difficult to judge which element is superior over the others without analyzing the particular problem. As a rule of thumb, the higher the order of an element, the more accurate the approximation would be for the same element size. But for a higher order element, as a tradeoff for precision, the formulation would be much more complicated and tedious, and the computation might be much more time consuming. The constant element works fairly well for problems with smooth boundaries and continuously changing boundary conditions, but could generate considerable errors at 'corner points' where the boundary conditions are not continuous. For boundary with such 'corners', the linear element is usually a better choice.
/
/
pi
Figure 6.1: Auxiliary coordinate system
6.2 Local Coordinate System for A Linear Element
To formulate a problem with linear elements, an auxiliary local coordinate system, as shown in Fig. 6.1, has to be established to facilitate line integrations over the element. The two axis C j7 are perpendicular to each other with one lying on the element and pointing in the direction of integration, from Pj to Pj+1, and the other one being in the same direction as the outward normal vector ii. Based on this auxiliary coordinate system, the integration over each segment can be completed analytically and be expressed in terms of 4., 4j+j and t7, the distances from P, P+i and P to the local origin defined in Fig. 6.1. The values of 4, e,+i and j7i are determined by the global coordinate information of three points.
Since
S+ 171 = ?
(+ AX) + 171 = rt," +i
where
4,,+4 = vi Xi+z) + (i zi+)
then
. "<+ (6.15)
2Ay
+ = fi + A i (6.16)
The value of mi is therefore
r/,I= ,/ C (6.17)
However, the numerical test showed that Eq.(6.17) could lead to significant error of j iri due to runoff errors. A more accurate expression for I 7i j can be obtained by computing the distance from Pi to the straight line PyP+ directly from the global coordinates, i.e.
i ne I + z zi (6.18)
I r/, I +A,
where
A = zj+1 zy zy+1 zy
The sign of ri depends on the relative position of Pi to the boundary line element PjPi+1I If P is in the same side of the element with ii, 7i is positive, otherwise, it is negative. Mathematically, it can be expressed by the following two equations for vertical and nonvertical elements: sign (77i) = (sy Xi) Az +y4 = xy (6.19)
(xj x,) Az sign (7i) = AI (Azio X+,) Az 1 x (6.2019)
Ax Azio
sign (ri) = AxzI x1+1 x (6.20)
where
Ax = xY+ Xi Az = zy+1 zy Az
Azio = (X ) +z z
Therefore
(6.21)
ri = sign (7i) i, I
70
6.3 Linear Element and Related Integrations
By the definition stated in the last section, on a linear element, a quantity and its normal derivative are interpolated by a linear function between the values at the two nodes, P and P+1. For the velocity potential function 4 and its normal
BA
derivative we have, in terms of 9 coordinates,
an
( +(i+ i) + ~ ii+) fi< C i+1 (6.22)
&+1
as as as as
=[()j+l (a)i]C + [+ e(a+1 )
5n M ej(6.23)
ej+1 ej+
It is obvious that
a= ( D) ( e. ) = Il ) ;
= 1 CC+1) =C )+
Substituting Eqs.(6.22) and (6.23) into Eq.(6.12), and noting that
r, = 7, + 2 (6.24)
Bri ar= r (6.25)
an= a = ri
the integration over segment j (between Pj and Pi+1) can be carried out analytically in the local coordinate system (Ligget and Liu, 1983):
le (r In r, )a d
[1 ap 2 (
= Ill + H*2 >+1 Kix( )t K ( (6.26)
where the superscripts and + refer to the positions immediately before and after the nodal point denoted by the corresponding subscript. And
(6.27)
H = I1 + e1+2
71
I = I 22. Ki4 = , + (j+1llj Ki, = Ii2j ( l 2
_ 1 fi+1 r = fi+ fifi ri 8n
i i,7 2+1
2As 17+?+
= 1~j
fJ +, ti
1 ari d = fi+ ri 8n 2Ai fiJ
1 [t an (+1) Yi Vi
(6.32)
 tan'( )]
'i
S 1 f Inri+d2
fi+1 fi fi
 {(,? + C+1)[1n(7 + C+1) 1]
+ +
(,7, + C)[11n(, + I) 1]}
1 f +In ri d(
~ 1nn + e;C 'i7~+~ 2~+ j
1 2 s 2
= ~ {.i(77? + ex)6n(to + ) 2(e6 1 e) 2A(
+2rjj[tanl(e7+l) 7t7(,)]}
th 77
with
(6.28) (6.29) (6.30)
i
2A,
I CJ+ i
e ;dC
r.
(6.31)
d xi
r?
(6.33)
(6.34)
72
for all ij and i#j+l and Ai = fi+x fi If i = j, the above integrals have to be reevaluated to avoid the singularity of the Green's function,
1
I j = lim
1 cI = 1 lin
;j+1 i e0
:i+: 'o
1
= lim
ij+1 ieo
J i+i 1 r d Ji+1 1 ar 0
+ dg = 0 ei+, ri a fI +1 1 B i+, In r d fi+
A(2InA 1)
1 l i + d
= lirm+; In r de
+ 1 ~ 0 44+.
= lnA 1 Similarly, when i = j + 1,
 fiEfz+ a ri
1lim la d( = 0
fi+ ~ iC0 t i r8n
S +1 i eo 1 ri an
1 lim I nried(
j +1 i e
= (12lInAj)
4
(6.35) (6.36)
(6.37)
I1+11 1+11J
(6.38)
(6.39) (6.40)
(6.41)
; 1i ur
++1 i
= InA 1
lnri d(
(6.42)
Summarizing all the If's according to Eq.(6.12) for node points i = 1  N, and assuming that
(6.43) (6.44)
80
at all the nodes, a system of linear equations with unknowns of P and a can be
anobtained: obtained:
N
= L(8n)i j= 1
P4H1= ,N + THil bi, 1 C A?j = Hi... + Hj 
1 ifi=j 4. = 0 if i # j
Li, = K, N + K,1 LiE = K_ i1 + Ke ai j = 2, 3, ..., N
Equation (6.45) can also be expressed by a matrix equation
where
(6.45)
with
(6.46)
(6.47)
(6.48)
= ()
74
where 4' and q,, are the vectors of order N x 1 containing the unknown T and L respectively. The coefficients in R and L are determined solely by the boundary geometries.
6.4 Boundary Conditions
In Eq.(6.45), there are N linear equations and 2N unknowns. The additional N equations necessary for obtaining a unique solution are usually introduced by the application of the boundary conditions. There are essentially three types of boundary conditions for boundary value problems.
I i = T (6.49)
II =  (6.50)
an an
III = k5 (6.51)
n
where D, .'n' f and k are known functions on the boundary.
In water wave problems, the first relation usually means that the pressure distribution is known, and the second describes a given flux distribution on the boundary. They are also referred as 'essential' and 'natural' conditions (Brebbia,1989), respectively. The third one some times takes place when none of the quantities in the first two equations are specified but only the relationship between them can be found. A typical example of this is the free surface boundary condition in linear wave theory. For nonlinear wave problems, the right hand side of the above equations may include other known functions.
As might be noticed in Eq.(6.45) or Eq.(6.48) there are two unknowns for each nodal point. Since we have obtained one equation for every node on the boundary, only one of the three relations needs to be specified at each node, if the assumptions in Eqs.(6.43) and (6.44) are strictly satisfied. However, for some nodal points where the boundary changes directions, ('90) may not be equal to (P)+. For such cases, more than one boundary conditions may be necessary for one node. The treatment
75
of such points will be discussed in the next chapter.
By introducing the boundary conditions at all N boundary nodal points, another system of N equations with the same unknown in Eq.(6.48) is established. The number of unknowns is now equal to the number of the equations. It is usually convenient to eliminate N unknowns with the 2N equations, and the resulted matrix equation can be expressed as
AX=b (6.52)
in which A is a known matrix of order N x N, X is the unknown vector containing 4 or &D or D for some part of the boundary and a for the rest of it, depending on which one is not specified by the boundary condition; and b is a known vector resulted from boundary conditions.
CHAPTER 7
NUMERICAL MODEL FOR SUBMERGED POROUS BREAKWATERS
In this chapter, a numerical model of BIEM for submerged porous breakwaters is developed by using the unsteady porous flow model given in Chapter 3. The basic function of the numerical model is to compute the wave flow field and related quantities such as the wave form, the dynamic pressure and the normal velocity along all the boundaries and so on. With these quantities, the wave transmission and reflection coefficients and wave forces can then be obtained.
7.1 Governing Equations
The computation domain of the problem, as shown in Fig. 7.1, consists of two subdomains, the fluid domain and the domain of the porous medium. In the fluid domain, the water is considered inviscid and incompressible. The flow induced by gravity waves is assumed irrotational. Thus, the governing equation in this domain, for the velocity function 41, is the Laplace equation, vl.(D 0(7.1)
with fluid velocities being defined as
U = (7.2)
ax
W, = aD (7.3)
While in the porous medium domain, the viscosity of the fluid cannot be ignored since the flow is largely within the low Reynolds number region. The flow is
0 0 0 0 0 0 0 0 0 O
Figure 7.1: Computational domains described by the linearized porous flow model, pafof= VP (7.4)
where P is the pore pressure, fo is the linearized resistance coefficient and q is the discharge velocity vector in the porous medium, q = iu + kw with iand j being the unit vector in z and z directions.
Taking curl of Eq.(7.4),
pafoV x f= Vx VP = 0 (7.5)
leads to
Vx= 0 (7.6)
This means that the homogenized porous flow is irrotational, and hence a velocity potential function 0, in the porous domain exists such that
T= V ,
(7.7)
According to Eq.(7.4) and Eq.(7.7)
P
po 0 (7.8)
here p, a and fo are all treated as constants.
Substituting Eq.(7.4) into the continuity condition for the porous flow in terms of the discharge velocity, V. 0 (7.9)
the governing equation for the porous medium domain becomes the Laplace equation in terms of the pore pressure, V2P=0 (7.10)
In this study, the waves are assumed to follow the linear wave theory
= Oeit (7.11)
P = Pe iat (7.12)
It is to be noted that the time function is now expressed as eiat, as opposed to eit used in the seabed problem presented in previous chapters.
7.2 Boundary Conditions The whole computation domain has four types of boundaries, free surface, impermeable bottom, permeable interface of different subdomains and the artificial lateral boundaries. These boundary conditions are discussed here.
7.2.1 Boundary Conditions for The Fluid Domain
1. The free surface
The combined linear free surface boundary condition in the linear wave theory is
a = a (7.13)
wz 9
where z is the vertical coordinate and g is the gravity acceleration.
2. Impervious bottom
The boundary condition on an impermeable bottom is simply nonflux condition: a_ = 0 (7.14)
n
where n is the direction normal to the boundary which can be of any shapesand bar, or soil trench and so on, as long as it can be considered impermeable.
3. The permeable interface
The permeable interface is the common boundary either between the fluid domain and the porous domain, or between two different porous domains. For the sake of simplicity, we consider only one porous subdomain in this chapter. The same formulation can be easily extended to multiporousdomain configuration. The boundary conditions on such a boundary are the continuity of pressure and mass flux. They are
p = P (7.15)
at
or equivalently
iap' p (7.16)
with the linear wave assumption, and .a = 1 ap (7.17)
an, pafo an~i
where n1 is the outward normal of the fluid domain and nn is the outward normal of the porous domain. However, these two notations are only used when confusion could occur, otherwise, n without subscript always denotes the outward normal vector of the domain in discussion.
4. The lateral boundaries
There are two vertical lateral boundaries, one is on the offshore side and one is on the lee side of the porous domain, as shown in Fig. 7.1. In this study, the radiation boundary condition for these two boundaries is adopted. Such a boundary
80
condition assumes that the waves at these two boundaries are purely progressive, and that the decaying standing waves generated by the object inside the domain are negligible at these boundaries. Comparing with the matching boundary condition with the wave maker theory employed by Sulisz (1985) and some other authors, the radiation boundary condition offers significant simplicity in programming and provides sufficient accuracy with much less CPU time. In applying this boundary condition, the two lateral boundaries have to be placed far enough from the structure. Numerical tests show that a distance of about two wave lengths from the toe of the structure is more than adequate for this purpose.
At these two lateral boundaries, the potential functions for the transmitted and reflected waves are assumed to have the forms OrC(,z) = e'k(z+'l)R(z) (7.18)
't(x,z) = ek'('")T(z) (7.19)
where T(z) and R(z) are two unknown functions of z; k and k' are the wave numbers at the two boundaries respectively, and the subscripts t and r here refer to the transmitted and reflected waves, respectively. On the lateral boundary of upwave side, the potential function for the incident wave is known: 01(x, z) = e'Ck(+')A(z) (7.20)
with
A(z) = Hg cosh k(z + h) (7.21)
2a cosh kh
Therefore,
= ki+ 01 at x (7.22)
0= 4 at x=l' (7.23)
where I and are, respectively, the distances from the origin of the coordinates to the offshore and onshore lateral boundaries and 0 is the unknown potential function.
81
On the lateral boundary of transmission (onshore side), x = 1',
(g ik'', = ik'4 (7.24)
an ax ax
in which k' is determined by gk'tanh k'h' = a2 (7.25)
where h' is the water depth at x = 1'.
While on the lateral boundary of reflection (offshore side), x = 1,
,. = 4 01 (7.26)
ao a 8 a4,. ik, iko,
an x ax ax
 ik4' ik(O 0') = 2ik, ikO or
a= 2iko' ikO (7.27)
an
in which k is the incident wave number determined by gk tanh kh = a2 (7.28)
where h is the water depth at x = 1.
7.2.2 Boundary Conditions for The Porous Medium Domain
In this domain, there are two types of boundaries, one is the common boundary with the fluid domain and the other one is the impervious bottom. On the interface, the boundary condition is the same as the one specified by Eq.(7.16) and (7.17).
On the impermeable bottom, the noflux condition, in terms of the pore pressure, is
.p= 0 (7.29)
n=
Again, this boundary can have any shape, it can be the surface of the domain for core materials if it is considered as impermeable, etc.
82
7.3 BIEM Formulations
The formulations in the two domains are slightly different because of the different boundary conditions. In the fluid domain, due to the radiation boundary condition on the lateral boundary of offshore side, the terms containing the incident wave potential are introduced, which will form the RHS vector of the matrix equation. On the free surface, the normal derivative are expressed in terms of owing to the CFSBC. In the porous subdomain, the matrix equation, after applying the noflux condition, has to be manipulated to match with the fluid domain.
7.3.1 Fluid Domain
According to the formulae given in chapter 6, Eq.(6.12) can be expanded, in terms of node values, assuming that 4 = + for all the nodes, as:
jOf = (H,1 + HN)41 + (Ha + H2)02 + (H?2 + Hg3)s...
+ (HIN,1 + H N)ON, + ... + (~U.2 1 + H.)+N2N .
+ H~3..1+ 4 3)ON3 + + (H 1 +X HNb
+ (H, .~ + H ..)ON.. + ... + (H + H1)qN
 KS,+1 Ks,2 Kg+f K320 K,35+ ...
 KFN11, K ,1 N +, ... K~.24_f K +
% ,N3_10; K,, + .. K,?,,,, n K ++,.
 F K 2
 K~NN..1;N. K, N,.4N,,. ... KN1N K.,N;l (7.30)
i = 1, 2 ......, N
where ai is the inner angle of node i and the subscripts 1, NI, N2, N3, Nes and Ne,, refer to the nodes of 'corner points' as shown in Fig. 7.1. The superscripts and + refer to the positions immediately before and after the nodal point in the direction of contour integration, and the subscript n refers to the normal derivative. Here 0is not necessarily equal to 0+ for all the nodes.

Full Text 
165
Figure B.5: Case of L 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d5o = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rÂ¡/Hi), (b)
Nondimensional wave heights (H/Si) and the best fit to the exponential decay
function.
154
Since in this study, we deal exclusively with two dimensional cases, Eq.(A.l)
becomes
(A.4)
To perform the integration in Eq.(A.4) analytically, the quantities F and q
have to be expressible in explicit forms and the integrand has to be analytically
integrable over a specified domain A. Otherwise, a numerical integration over the
entire area needs to be employed, provided that the values of those two quantities
are known everywhere in the area. Although numerical integration of Eq.(A.4)
is applicable for most of the problems, the tedious grid generation and lengthy
numerical computation usually force people to make some simplifying assumptions
or approximations to the integrand to keep it analytically integrable. The accuracy
of the result from such an approach will no doubt be compromised.
However, if the energy dissipation can be expressed as a contour integration
along the boundary of the computation domain, the chance of working it out ana
lytically would be greatly increased. A boundary integral formulation of the energy
dissipation is also essentially important for the solutions of Boundary Integral El
ement Method where the unknowns are solved only on the boundaries. In this
appendix, such an attempt is made by reformulating the expression of energy dis
sipation and the use of Greens formula.
If we define U and W be the discharge velocities (real quantities) in x and z
directions, respectively and eD be the rate of energy dissipation per unit volume (also
a real quantity, considered as a possitive value), the rate of total energy dissipation
in an arbitrarily small cube of dx 1 dz can be expressed, according to Fig.3.1, as
ed dx dz
[(Pr + ^dx)[U + dx) dz + [Pr + ^dz)(W + "dz) dx
ox ox az oz
PrU dz PrW dx)
(A.5)
42
listed. The relative error in the table is defined as
A% = ki 7  X100% (4.60)
k\m
with ki being either kip or knD.
In this case, the errors are of similar magnitude between the present model and
the model by Liu and Dalrymple (1984). This is because the experimental values
fall in the region where the inertial resistance due to the virtual mass effect and the
turbulent resistance, neglected in Liu and Dalrymples model, axe unimportant.
In Fig. 4.2 through Fig. 4.5, the solutions of the complex dispersion equation are
illustrated graphically. The case of progressive waves are demonstrated first. The
specified conditions for the progressive waves are: H = 1 m, T = 4 seconds, h, 5
m and n = 0.4. Figure. 4.2 plots the values of kr (wave number) and kÂ¡ (damping
rate) against R (nondimensional permeability parameter) for three different water
depths, h = 2, 4, and 6 meters. The equivalent particle sizes for the range of R
values are also shown in the figure; they cover a range from 2.3 mm to 2.3 m.
The thick dash line is the solution of the dispersion equation given by Liu and
Dalrymple (1984, Eq.(4.5)) for the case of h = 4 m. The correction term of the
laminar boundary layer was not included in this curve since it is negligible in this
case. From these results, a number of observations can be made:
1. As expected, the wave number decreases (or wave length increases) monoton
ically with increasing R, from one limiting value kTi, corresponding to the case of
an impervious bottom at depth h, to the other limiting value kTÂ¡, corresponding to
the case of a water depth equal to h + ht when the lower layer becomes completely
porous.
2. Compared with the linear resistance, the nonlinear and the inertial resistances
dominate for the complete range of R values displayed. In the region where only
linear resistance dominates (R; < 1, R, < 1), the bottom effects are relatively small
and often negligible (outside the R range shown).
168
Figure B.8: Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm and
d50 = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (t)/Hi), (b)
Nondimensional wave heights (H/Jl[) and the best fit to the exponential decay
function.
97
First, the numerical model is applied to two special structures, one is a trans
parent submerged breakwater with infinite permeability on a flat impermeable
bottom while the other one is an impermeable step with the water depths being
23 cm before the step and 8 cm after it. In the former case, the computations was
performed as if there were a permeable breakwater, even though the transparent
breakwater is equivalent to a fluid breakwater or no breakwater. Figure 7.3 (a)
and (b) show the computational domains, the wave forms and the wave envelopes
for the two cases. In Fig. 7.3 (a), the perfect sinusoidal wave form matches the
theoretical solution for the case, and the waves are completely transmitted without
any reflection as expected. For the later case, the wave form is of a partial standing
wave in the up wave side of the step, and of a monotonic sinusoidal wave in the
downwave side of it. The wave length, at some distance over the step, has been
shortened to the exact value predicted by the dispersion equation for that depth. It
can also be shown that the wave energy is conserved since the equation for energy
flux, 7j JT + ft, is satisfied. Both cases are for conditions at laboratory scale.
Figure 7.4 shows the results for a submerged porous breakwater, for laboratory
conditions again. The crest width of the breakwater is 60.0 cm and it is 8.0 cm
below the mean water level in a water depth of 23.0 cm. The slopes axe 1:1.5 on
both sides. In the top frame, the wave form and the wave envelope directly above
the breakwater are plotted. The lower frame illustrates the pressure and the normal
velocity distribution along the surface of the breakwater. One can see that after the
treatment for the comer nodes, the pressure function, or equivalently the potential
function along the common boundary in the fluid domain, is continuous and well
behaved at all corner points. But the singularities remain for the normal velocity.
In Figure 7.5, two submerged breakwaters, for field conditions, are computed,
one is impermeable, and the other one is made of stones of d, = 0.4 meters. The
dimensions and the wave conditions are the same for both structures. The crest
156
W
U
Figure A.l: Geometric relations between the vectors
continuous on domain A n S, then
IL^Tz)dxi2 = ii,iz+3iz) (A9)
where S is the closed boundary of A.
Let
UP, = g WP, = f (A.10)
Then
ffA[Â§x(UPr) Â£z(WP,)]dxdz = f [(WPr)dx+{UPr)dz] (A.ll)
According to the geometrical relations of the vectors on the boundary shown in
Fig.A.l, the following expression can readily be obtained.
js{P,U~ P,W~)ds = js P,{Usino: + Wcos a)ds = Â£ P,q ds (A.12)
with q = U sin a + W cos a being the discharge velocity normal to the boundary
S.
Therefore, the energy dissipation within the area bounded by 5 during a time
ft+T r rt+T r
eD = J js P,q rids dt = ^ js(Jn),ds
dt
period T is
(A.13)
Then
155
fndU ndW TTdPr rdPT,
* = +
(A.6)
= +<^1
with Pr being the real part of the complex pore pressure function.
To compare with Eq.(A.4), we apply the continuity condition of pore fluid to
Eq.(A.6), the first two terms vanish. For the remaining two terms, by substituting
the porous flow model given by Eq.(A.2) for the derivatives of Pr and keeping only
the dissipative terms, noticing that q iU+jW and qq = U2+W2, one immediately
recognizes that Eq.(A.7) is just an alternate expression of the integrand in Eq.(A.4).
The total energy dissipation within the whole computational area during a time
period of T is then
eD
tp dtdxdz
/t+T red d
Jttei(up) + Tz(wp')]dxdzit (A'8)
where A is the area of computation domain. It could be the cross section area of a
submerged breakwater and so on.
Eq.(A.8) is an equivalent expression to Eq.(A.4) for the energy dissipation in
a porous medium. The only difference between the two is that the nondissipative
resistance is included in Eq.(A.8) but not in Eq.(A.4). From the procedure of
derivation of Eq.(A.8), one can clearly see that the nondissipative resistance should
not be simply left out. However, the value of the energy dissipation by both formulas
should be the same regardless whether this resistance is included or not, simply
because of its nondissipative nature.
By the use of Green formula, the area integration in Eq.(A.8) can be converted
df dg
into a contour integral. The Greens formula states that if /, o, and are
oz ox
40
This is the final form of the equation used for computing /0 for both cases.
The dispersion equation, Eq.(4.15), which is coupled with Eq.(4.47), is solved
iteratively. The procedures to obtain the solution for the cases of standing waves
and progressive waves are different.
a) For standing waves a ar + t<7, is complex, k is real and known. Equation
(4.15) can be rewritten as
a2 =
pfc(tanh kh tanh kht)
Jo
1 tanh kh, tanh kh
Jo
= Qr{fo) + Qi{fo)
(4.48)
with Qr{fo) and <5(/o) being the real and imaginary parts of o2, respectively. The
iteration procedures are summarized as follows:
^ i JqUF)+o/F)+wh )
*Â¡n+,) = I jQHfP) + QKfP) Q'ifP) r}
w i c,
f R ,i3+Vrw laW/^1
= 5*
(4.49)
(4.50)
(4.51)
(4.52)
where the superscript n denotes the level of iteration, and the criterion of conver
gence is
An) Ani)
Jo Jo
ft
(n)
l< e
and
aW a(n1)
oM
l< e
(4.53)
with e being a prespecified sixbitrajily small number. It is set to be 1.0% in this
model.
b) For progressive waves k = kr + is complex and a is real and known. In
this case, the dispersion equation can be written as
F(k, fo) = c2 gk tanhk/i+ j tanh kh,[gk a2 tanh kh) = 0 (4.54)
Jo
and the iteration was carried out as follows:
0
(4.55)
CHAPTER 8
LABORATORY EXPERIMENTS OF A POROUS SUBMERGED
BREAKWATER
In the last two chapters, a numerical model of boundary integral element method
for porous submerged breakwaters has been completed based on the full resistance
model developed in chapter 3. To further verify this numerical model, and at
the same time to examine the applicability of the empirical coefficients ao, b0 and
Ca determined by the seabed experiments described in chapter 5, a laboratory
experiment for a porous submerged breakwater was carried in a wave channel.
8.1 General Description of The Experiment
The experiment was conducted in the Coastal Engineering Laboratory of Coastal
and Oceanographic Engineering Department, University of Florida. The wave tank
is 25 meters long, 0.6 meters wide and 1.7 meters deep with glass walls on both
sides. The wave maker is of piston type furnished with tin absorbing system which
was designed to absorb the wave energy reflected back to the piston. The tank is
also equipped with a motorized rail cart on the top to facilitate wave measurements.
The model of the porous submerged breakwater was of trapezoidal shape and
made of river gravel of <50 = 0.93 cm, the same material used in the seabed exper
iments. The crest height of the model was 15 cm, crest width was 60 cm and the
slope for both sides was 1:1.5. The center of the model was 12 m from the piston
and 13 m to the other end of the tank. In this experiment, only one water depth of
h = 23 cm was tested.
The measurements were concentrated on wave reflection and transmission, in
cluding the cases when waves broke in the region directly above the breakwa
ter crest, the wave envelope over the breakwater and the pressure distribution.
107
196
1.20
<Â£ 1.00
o
UJ
1 0.80
UJ
>
2 0.60
O
UJ
rvi
Z 0.40
0.00
0.0 2.0 4.0 6.0 8.0 10.0 12.0
TIME (SEC)
Figure B.36: Case of L = 275 cm, h = DW = 20 cm, ht = DS = 20 cm and
<50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rz/ifi), (b)
Nondimensional wave heights (if/57) and the best fit to the exponential decay
function.
DW= 20.0 CM
DS= 20.0 CM
DD= 2.09 CM
i 1 1 1 1 1 1 1 1 1 r
35
When the virtual mass and the drag coefficients are set to be zero, Eq.(4.15)
gives the dispersion equation for the homogeneous solution by Liu and Dalrymple
(1984) which is
cr2 gk tanh kh = = tanh kh,(gk a2 tanh kh)
4.2 The Solutions of The Complex Dispersion Equation
(4.21)
As mentioned in the previous section, Eq.(4.15) is an equation involving both
of the complex unknown variables of either a or k, and the complex coefficient
/0. The other necessary equation can be obtained from the linearization process
(This step is not necessary if Cj = 0, as in Liu and Dalrymples solution). The
common method of evaluating the linearized resistance coefficient, /o, is to apply
the principle of equivalent work. This principle states that the energy dissipation
within a volume of porous medium during a time period should be the same when
evaluated from the true system or from its equivalent linearized system
(Ed)i = {Ed)t
(4.22)
where Ed is the energy dissipation in a controlled volume during one wave period
and the subscripts / and nl refer to linearized and nonlinear systems, respectively.
It can be readily shown (Appendix A) that such energy dissipation (considered
as a positive value) can be expressed in the form of a boundary integral
rt+T r
ED = Jt js?ndsdt (4.23)
where Ed is a complex energy dissipation function and Jn is the complex energy flux
normal to S, with the real parts of them being the corresponding physical quantities.
Here S is the closed boundary of the computation domain. The complex energy
flux function 7n can be expressed as (Appendix A)
?n =
e2io,t
+ p')
(4.24)
.50
60.0 40.0 20.0 0.0 20.0 40.0 60.
NORMALIZED SURFACE ELEVATION
en O en o cn o
o o o o o o
o
.50
180
Figure B.20: Case of L 200 cm, h = DW = 20 cm, h, = DS = 20 cm and
Nondimensional wave heights (H/ST) and the best fit to the exponential decay
function.
90
and
(Ed), = 270/0py
For the nonlinear model,
(7.56)
(^n)nl
P{f\ + / I q )
r^rPn
(7.57)
and
(eD)^=y
PnP
ds
'c fi + h iil
Equating Ed(/q)i to En{fi + h \ 9 )ru and take approximately Â¡ q\
the linearized coefficient f0 is then
(7.58)
\Pn/pof0\,
fo =
[ PnP* ds
Jc
L
PnP
(7.59)
ds
c fi + h\ Pn/pcfo
The linearization is in fact an iteration process starting with an approximate
value of f0. The procedures are as follows:
1 Let ft = fu
2 Substitute this approximate /0 into Eq.(7.49) to form A, for Eq.(7.48),
3 Solve Eq.(7.48) for ^ and compute nc using Eq.(7.47) with the obtained ^c,
4 Calculate according to Eq.(7.59) and compare to fo\ if they are close enough,
stop; otherwise
5 /0(2) = fo^ and repeat steps 2, 3, 4 and 5 until
I ft] ft'" l< Â£ (7.60)
where Â£ is a prespecified arbitrarily small quantity.
6Solve Eq.(7.48) with for the final solution .
122
INCIDENT HAVE HEIGHT (CM)
INCIDENT WAVE HEIGHT (CH)
Figure 8.8: Transmitted and reflected wave heights versus the incident wave heights,
(a) Transmitted waves; (b) Reflected waves.
B.12
Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and d50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (tj/Hi), (b) Nondimensional wave heights (H/IT/) and
the best fit to the exponential decay function 172
B.13 Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm
and (so = DD = 2.84 cm. (a) Averaged nondimensional surface
elevation {rj/Hi), (b) Nondimensional wave heights (H/TT/) and
the best fit to the exponential decay function 173
B.14 Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm
and (so = DD = 3.74 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/H7) and
the best fit to the exponential decay function 174
B.15 Case of L 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 0.72 cm. (a) Averaged nondimensional surface
elevation (t]/Hi), (b) Nondimensional wave heights [H/H/) and
the best fit to the exponential decay function 175
B.16 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 0.93 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/Hi) and
the best fit to tne exponential decay function 176
B.17 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and so = DD = 1.20 cm. (a) Averaged nondimensional surface
elevation [r)/HiV (b) Nondimensional wave heights (H/H7) and
the best fit to tne exponential decay function 177
B.18 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/Hi) and
the best fit to the exponential decay function 178
B.19 Case of L = 200 cm, h = DW = 20 cm, ht = DS = 20 cm
and (so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation {rj/Hi), (b) Nondimensional wave heights (H/Hj/) and
the best fit to the exponential decay function 179
B.20 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and so = DD = 2.84 cm. (a) Averaged nondimensional surface
elevation (r}/Hi)t (b) Nondimensional wave heights (H/H7) and
the best fit to tne exponential decay function 180
B.21 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 3.74 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/TT/) and
the best fit to tne exponential decay function 181
Xll
74
where and n are the vectors of order N x 1 containing the unknown $ and
respectively. The coefficients in R and L are determined solely by the boundary
geometries.
6.4 Boundary Conditions
In Eq.(6.45), there are N linear equations and 2N unknowns. The additional
N equations necessary for obtaining a unique solution axe usually introduced by
the application of the boundary conditions. There are essentially three types of
boundary conditions for boundary value problems.
I
II
III
$ = $
Â£$ _~dÂ¥
dn dn
dn
(6.49)
(6.50)
(6.51)
where $, , / and k are known functions on the boundary.
on
In water wave problems, the first relation usually means that the pressure distri
bution is known, and the second describes a given flux distribution on the boundary.
They axe also referred as essential and natural conditions (Brebbia,1989), respec
tively. The third one some times takes place when none of the quantities in the first
two equations are specified but only the relationship between them can be found.
A typical example of this is the free surface boundary condition in linear wave the
ory. For nonlinear wave problems, the right hand side of the above equations may
include other known functions.
As might be noticed in Eq.(6.45) or Eq.(6.48) there are two unknowns for each
nodal point. Since we have obtained one equation for every node on the boundary,
only one of the three relations needs to be specified at each node, if the assumptions
in Eqs.(6.43) and (6.44) are strictly satisfied. However, for some nodal points where
the boundary changes directions, (Â§Â£) may not be equal to (Â£)+. For such cases,
more than one boundary conditions may be necessary for one node. The treatment
NORMAL I ZED WAVE ENVELOPE
141
Figure 9.7: Permeable berm breakwater of model scale with H = 10.0 cm: (a) Wave
envelope; (b) Envelopes of pressure and normal velocity distribution
CHAPTER 6
BOUNDARY INTEGRAL ELEMENT METHOD
The Boundary Integral Element Method (BIEM) is an efficient numerical method
for conservative systems such as those governed by Laplace equation. It is efficient
because the computation is carried out only on the boundaries rather them over the
entire domain as it would be in finite difference and finite element methods. Consid
erable amount of computation time and storage space can be saved without losing
accuracy. In addition to the efficiency, the data preparation becomes much simpler
as compared to the other two methods. For nonLaplace equations, the application
of this method may become difficult and sometimes impossible. By the efforts of
many scientists, the scheme has been extended to more and more problems governed
by nonLaplace equations (Brebbia, 1987). In this chapter, we restrict ourselves to
the two dimensional Laplace equation only.
6.1 Basic Formulation
Let U and V be any two continuous two dimensional functions, twice differen
tiable in domain D which has a close boundary C. According to the divergence
theorem (Loss, 1950; Franklin, 1944, cited in Ligget and Liu, 1983) for continuity
in a volume
J (V v)dA j> v nds (6.1)
where V is the vector operator, v is any differentiable vector and ft is the unit
outward vector normal to C. If we define that v = TJW and then v = WU in
Eq.(6.1), two integral equations can be generated in terms of U and V,
W + UVV)dA = Â£ UVV ads
(6.2)
116
predicted Kr and Kt are plotted against the measured ones. In Fig. 8.5 through
Fig. 8.7, both measured and predicted values forifj and Kr for all nine periods (see
Table 8.2) aire plotted against the nondimensional parameter of H{/gT2.
In the case of nonbreaking waves, the wave energy dissipation is mainly due to
percolation inside the breakwater. As the wave heights increase, breaking occurs
on top of the submerged breakwater because of the change in water depth. The
wave energy is, in such situation, dissipated both by the porous medium and by
breaking.
Due to the wave energy dissipation by breaking, submerged porous breakwaters
axe generally more effective for steeper waves. To evaluate the effectiveness of the
submerged breakwater, the wave transmission and reflection coefficients for breaking
waves were also measured and the test results are given in Table 8.3.
Figures. 8.8 and 8.9 are the plots of the transmitted wave heights versus the
corresponding incident wave heights for the data in Table 8.3. The results for the
corresponding nonbreaking cases listed in Table 8.2 axe also included. From the
figure, one can readily observe that the breaking point for the incident wave height
is about Hi = 4.5 cm. It appears that the transmitted wave heights are controlled
mainly by the water depth over the submerged model (h = 8 cm for this case) after
the breaking point, when the wave length is less than certain value. For longer wave
lengths, the waves seem to be affected less by the submerged bump, and the wave
transmission starts to increase with the increasing incident wave heights.
After wave breaking, the numerical model developed for non breaking wave
apparently over estimates the transmitted wave heights. Comparing with the wave
transmission, the effect of breaking on wave reflection is not as significant. As a
result, the prediction for the reflection coefficients for the breaking waves is roughly
as good as that for nonbreaking waves.
181
Figure B.21: Case of L 200 cm, h = DW = 20 cm, h, = DS = 20 cm and
d5Q = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (rÂ¡Â¡H\), (b)
Nondimensional wave heights (if/Sj) and the best fit to the exponential decay
fraction.
79
2.Impervious bottom
The boundary condition on an impermeable bottom is simply nonflux condition:
dn
= 0
(7.14)
where n is the direction normal to the boundary which can be of any shapesand
bar, or soil trench and so on, as long as it can be considered impermeable.
3.The permeable interface
The permeable interface is the common boundary either between the fluid do
main and the porous domain, or between two different porous domains. For the
sake of simplicity, we consider only one porous subdomain in this chapter. The
same formulation can be easily extended to multiporousdomain configuration.
The boundary conditions on such a boundary are the continuity of pressure and
mass flux. They are
(7.15)
or equivalently
iap4> = p
(7.16)
with the linear wave assumption, and
= (7.17)
dn: pafa dn\i
where nÂ¡ is the outward normal of the fluid domain and nn is the outward normal
of the porous domain. However, these two notations are only used when confusion
could occur, otherwise, n without subscript always denotes the outward normal
vector of the domain in discussion.
4.The lateral boundaries
There are two vertical lateral boundaries, one is on the offshore side and one
is on the lee side of the porous domain, as shown in Fig. 7.1. In this study, the
radiation boundary condition for these two boundaries is adopted. Such a boundary
171
Figure B.ll: Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm and
d50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
CHAPTER 5
LABORATORY EXPERIMENT FOR POROUS SEABEDS
The experiment was carried out in a wave flume in the Laboratory of Coastal
and Oceanographic Engineering Department of University of Florida. The flume
is about 15.5 meters long, 0.6 meters wide, 0.9 meters high and equipped with a
mechanically driven pistontype wave maker. All the tests were conducted with
standing waves.
5.1 Experiment Layout and Test Conditions
Figure 5.1 shows the experiment arrangement. A porous gravel seabed was
constructed at the end of the wave flume in the opposite side of the wave maker. A
sliding gate was positioned at one wavelength from this end of the tank to trap the
standing wave after a sinusoidal wave system was established by the wave maker.
The decay of the freely oscillating standing wave was then measured by a capacitance
wave gage mounted at the center of the compartment. The damping rate of the
porous seabed for each particular test condition was determined by applying a least
squares fit to the data according to the following equation:
.(o<)i = sln(#) (5.1)
Ij Mji
where j refers to jth wave. The corresponding wave frequency was obtained by
averaging individual waves.
The contribution to the wave damping due to the side walls and the bottom
was subtracted from the data according to the following equation
<*i = ~ ln[l + e7o<>7 e7^7} (5.2)
50
83
At a smooth point, where the inner angle a, formed by the two boundary
segments joining at the node is ?r, the normal velocity are continuous across the
node, such that
t = t (731)
But at a comer point where the boundary changes direction, usually
t # t (732)
unless both of them are zero.
This discontinuity in normal velocity introduces one more unknown at a corner
node without introducing the necessary equation. Before applying the boundary
conditions to all the nodes, it is necessary to examine the nature of those corners
and to establish appropriate conditions.
In general, if the normal velocities specified by the boundary conditions across
a corner node satisfy any one of the following conditions, the extra unknown intro
duced by the discontinuity can be eliminated without generating any computation
errors:
1) t = t =
2) 4>~ = ki(f> on one side and = k2 on the other side
with ki and k2 being two nonzero known functions
3) 4>~ = ki on one side and + = 0 on the other side, or vice versa
In Fig. 7.1, the corner points 1, Ni, N, IV3 and the corners on the impervious
bottoms aire such nodes. After introducing the boundary conditions, the only re
maining unknown is , which is continuous anywhere. For Ne4, Nee and the corners
between them on the porous interface, none of the above conditions can be satisfied
and singularities occur. In the present model, for corner points along the porous
interface, ~ + is assumed. This is a reasonable approximation considering the
36
where un is the normal velocity at the boundary S and p* is the conjugate of pore
pressure p, both of them are complex quantities; oT is the wave frequency, areal
value, and the subscript r is used to distinguish the complex a.
Physically, there are two classes of problems: standing waves of a specified wave
number, and progressive waves of a specified wave frequency. In the former case, k
is real, a is complex, and the solution has the following form
rÂ¡{xyt) = areoskxe~,a,t a < 0 (4.25)
where is the imaginary part of the complex a and ar is the wave frequency. In
the latter case, o is real, k is complex and rÂ¡ becomes
(*,!)= k,> 0 (4.26)
where kr and A:, are, respectively, the real and imaginary parts of k with kr being
the wave number.
For standing waves, the pore pressure function is
P, = p = D cosh k(h + h, + z) cos kxec
Since P, is periodic in z, the boundary curve S for the contour integral in
Eq.(4.23) can be chosen as z = 0, z (h + ht), x = L and z h with L being
the wave length.
As z = 0 and x = L are the antinodes of the standing wave, there is no normal
velocity, i.e. u = 0, on these two vertical boundaries and also at the impermeable
boundary z = (h + h,). Then
1 rt+T rL
Ed = ~ / (w0pe~2,errt + w0p') dxdt (4.28)
with
kD
paf
sinh kh, cos kx tait
tt>0 = n *=/ =
(4.29)
CHAPTER 9
NUMERICAL MODEL FOR BERM BREAKWATERS
In this chapter, attention is diverted from submerged breakwaters to subaerial
ones. A subaerial breakwater is a breakwater with its crest protruding out of the
water surface. There are many kinds of subaerial breakwaters classified according
to their designs. Berm breakwater is one of them. In the conventional design
of rubblemound subaerial breakwaters, there are usually one or two thin cover
layers of large blocks, to resist the wave destruction. In a berm breakwater, the
cover layer(s) of large blocks is replaced by a much thicker layer of, or say, a berm
of, much smaller blocks. It has been shown (T0rum et al. 1988, 1989) that this
kind of berm cover, if designed properly, can provide sufficient resistance to the
wave destructions with relatively low cost, as compared to the conventional cover
layer of large blocks, especially at localities where natural large quarry stones are
not available. Even in places where large blocks are available, berm breakwaters
may still be economical because the cost reduces quite drastically with the block
size. New development has shown (Civil Engineering, Feb. 1990) that it would be
more efficient and economical if a submerged breakwater is placed some distance
offshore of a berm breakwater. The combined breakwater system would provide
more energy dissipation at an even lower cost. In this part of the study, the scope
has been limited to single berm breakwater only.
9.1 Mathematical Formulations
Typical berm breakwaters can have either one of the idealized configurations
shown in Fig. 9.1. In the figure, the superscript s refers to the quantities for the
porous subdomain. The core of the berm breakwater can be either permeable or
128
148
A permeable submerged breakwater of trapezoidal shape made of river gravel of
50 = 0.93 was tested in a wave channel for both nonbreaking waves and breaking
waves. The measured transmission and reflection coefficients, pressure distribution
over the breakwater and the wave amplitude modification over the breakwater were
compared well to the corresponding values predicted by the numerical model.
10.2 Conclusions
The analysis and the assessment of the three resistance components in the
porous flow model show that a wide range of conditions in the coastal environ
ment are outside the linear resistance region such that either inertial or turbulent
or both terms could be important. The effects of the nonlinear and the inertial
resistances are pronounced over a wide range of R values, or equivalently a wide
range of particle sizes, both in terms of wave attenuation and altering wave kine
matics. The effects on gravity waves due to the linear resistance are insignificant
whether it dominates the other two components or not. Consequently, a sandy
bottom of coarse sand or finer can be treated as impermeable over a wide range of
environmental conditions in coastal water.
The nature of the solution for the porous seabed problem differs significantly
from that of available solutions. The wave damping, hence the wave attenuation,
first increases with increasing R\ and reaches a peak value. Beyond this point,
the wave damping decreases with increasing permeability. This peak attenuation is
found to occur when the dissipative resistance (velocity related) is equal to the non
dissipative resistance (acceleration related). The magnitude and the correspond
ing permeability for peak damping calculated from this study are quite different
from those by the existing theories. This maximum damping phenomenon opens
an interesting possibility for designing a porous structure with optimum damping
capabilities.
15
integration over the breakwater for the energy dissipation, Madsen obtained an
explicit expression for the linearized friction factor /, that is
J Kpr v 2a
) + \/(! +
; 3;r h1
(2.33)
where a and 6 are the coefficients in DupuitForchheimers porous flow model. A
similar solution for this problem was also given by Scarlatos and Singh (1987).
Madsen et al. (1976, 1978) further extended the long wave solution to a trapezoidal
porous breakwater with, again, DupuitForchheimens model. The solution was,
however, much more complicated than that for the crib type.
Since the shore protection breakwaters are usually irregular in shape and very
often with several layers of stones of different sizes, analytical solutions become
impractical. For such complicated geometries, numerical approaches have to be
adopted.
The most commonly used numerical schemes are finite element and Boundary
Integral Element Methods (BIEM). The first finite element model was developed by
McCorquodale (McCorquodale, 1972) using the McCorquodale porous flow model,
for computing the wave energy dissipation in rockfills. In his model, the entire cross
section of the structure was divided into small triangleelements and the variation
of the physical quantities was interpolated by a time dependent element function:
= (0i + @2% + 02 y)t + 0^+ 0*,x + 0ey (2.34)
Since the numerical computation was only carried out in the porous domain,
the interaction between the porous domain and the fluid domain was not modeled,
although the free surface within the porous region was well predicted.
Due to the large amount of work for data preparation in using a finite element
model, the Boundary Integral Element Method (BIEM) became popular since the
mid1970s. With BIEM, the discretization is only on the boundaries as opposed
to the entire domain with the finite element method. Ijima et al. (1976) applied
185
Figure B.25: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 10 cm and
d5o = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (ij/Hy), (b)
Nondimensional wave heights [H/~H^) and the best fit to the exponential decay
function.
125
Table 8.4: Normalized Pressure Distribution
X
T=0.854
T = 1.12
T = 1.379
T = 0.854
T = 1.12
T = 1.379
46.98
0.260
0.406
0.326
0.247
0.313
0.375
42.02
0.284
0.432
0.366
0.282
0.347
0.403
37.07
0.293
0.432
0.390
0.281
0.371
0.401
32.11
0.343
0.437
0.417
0.316
0.386
0.400
27.44
0.363
0.451
0.418
0.331
0.393
0.414
15.55
0.315
0.365
0.315
0.261
0.262
0.321
5.96
0.269
0.410
0.318
0.228
0.262
0.351
5.96
0.241
0.401
0.318
0.207
0.251
0.330
15.55
0.302
0.419
0.294
0.242
0.262
0.332
27.44
0.253
0.338
0.289
0.201
0.204
0.269
32.11
0.220
0.313
0.236
0.165
0.180
0.245
37.07
0.186
0.263
0.184
0.135
0.144
0.210
42.02
0.177
0.244
0.168
0.127
0.138
0.192
46.98
0.158
0.235
0.162
0.111
0.128
0.185
Hi's and x are in centimeters and Ts are in seconds.
In both cases, the wave setups over the breakwater crest in the experiment are
obvious. The maximum set up for the nonbreaking case in Fig. 8.10 is more than
20% of the wave height at the same point. Similar to that in a flat porous seabed
problem addressed in Chapter 4., the depth increase due to such large wave set up
causes decrease in dynamic pressure at the porous bottom. This may have been the
main reason for the discrepancies in the prediction of pressure magnitude since the
numerical model is not designed to take account moving free surface boundaries.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF FIGURES vui
LIST OF TABLES xv
ABSTRACT xvi
CHAPTERS
1 INTRODUCTION 1
1.1 Problem Statement 1
1.2 Objectives and Scope 3
2 LITERATURE REVIEW 5
2.1 Porous Flow Models 5
2.2 WavePorous Seabed Interactions 10
2.3 Modeling of Permeable Structures of Irregular Cross Sections .... 13
3 POROUS FLOW MODEL 20
3.1 The Equation of Motion 20
3.2 Force Coefficients and Simplifying Assumptions 23
3.3 Relative Importance of The Resistant Forces 28
4 GRAVITY WAVES OVER FINITE POROUS SEA BOTTOMS 31
4.1 Boundary Value Problem 31
4.2 The Solutions of The Complex Dispersion Equation 35
4.3 Results 41
5 LABORATORY EXPERIMENT FOR POROUS SEABEDS 50
5.1 Experiment Layout and Test Conditions 50
v
84
fact that angles on the surface of a porous breakwater made of quarry stones cannot
be sharp enough to cause a sudden change of the normal velocity. At Ncb and Net,
the above approximation cannot be applied since it is known that 4>n = 0 on one
side of the node and n ^ 0 on the other side. The singularities at these two nodes
are avoided by splitting each node into two.
In Eq.(7.30), it can seen that Greens formula was discretized under the as
sumption of cf)~ = <7Â¡>+. If we revoke such an assumption at a singular point and
split the node into two, applying the Greens formula to the two nodes separately,
the original equation for the singular point will be replaced by a pair of equations.
Letting i be either Neb or Nce, the two equations are
Oi4>i
<*Â¡4}
(hi i+aj,)* +...+Hjwtf + nyt+
XIA ~ ~ *>. ~ KhKi ~ (733)
(Hi + + ... + + Hit} + ...
Xim   Klt ... KI.K, (7.34)
where the same Hi/s and KÂ¡/s are used in both equations because the two nodal
points are so close that the distances from a field point to either one of them are
virtually the same. In doing so, two more unknowns, ~ (or +) and ~ (or <Â£+),
are introduced, instead of 4> and 4>n only for the original node. Since am additional
boundary condition is also introduced for each additional node, the system is still
determinant. Numerical tests show that without such treatment, discontinuities
(spikes) may occur in the potential function, , at nodes Ncb, Nee and all the cor
ner nodes on the interface. Applying the boundary conditions given in Eq.(7.13),
Eq.(7.14), Eq.(7.24) and Eq.(7.27) into Eq.(7.30) and setting (j)~ = and 4>~ = +
for all the node except Neb and Ncet results in
Kti
& Ai + HiNtb.xrNti + h}# 4>n. +
i=1
2
berm breakwater, is attracting more and more attention. For a berm breakwater,
the armor layer(s) is replaced by thicker layer(s) of blocks of much smaller sizes.
The seaward face of the structure has a berm section instead of the traditional
uniform slope. The berm section is intended to enhance the structural stability and
to trigger wave breaking.
Wave attenuation behind a porous breakwater is affected by three mechanisms:
wave reflection on the seaward face of the structure, waves breaking over the struc
ture and flow percolation inside the porous structure. The former one is conserva
tive and the latter two are dissipative. The main focus of this study is on the third
mechanism, that is the dissipation due to flow percolation. For such a purpose, a
numerical solution is sought since an analytical solution for such irregularly shaped
structures is nearly impossible. The main advantage of numerical methods is the
flexibility of handling complex geometries and boundary conditions. The disadvan
tage of any numerical approach is the lack of generality for the solution, since it is
usually implicit in terms of the variables so that the influences of the parameters
can only be examined case by case.
In order to examine the validity of the model and the nature of the dissipative
force, the porous flow model is applied to an infinitely long flat seabed of finite
thickness subject to wave action. This condition can also be viewed as a submerged
breakwater with infinitely long crest. Because of the geometrical simplicity, an
analytical solution is attainable. This case is used to compare with the existing
solutions proposed by other investigators, to examine the nature of wave attenua
tion as a function of flow and material properties and, more importantly, to guide
the design of an experiment so that the empirical coefficients in the porous model
can be determined. Successful determination of these coefficients is crucial to the
validity of the model. The experiment which is to be carried out subsequently must
demonstrate that the results are stable (or the experiments are repeatable) and that
161
Figure B.l: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (HÂ¡TTÂ¡) and the best fit to the exponential decay
function.
cannot predict the magnitude of this coefficient a priori because the virtual mass
of densely packed fractured stone is not known (Sollitt and Cross, 1972, pl842).
They also pointed out that Evaluation of Ca, however, may serve as a calibrating
link between theory and experiment in future studies (same page).
Hannoura and McCorquodale (1978) made an attempt to determine the value
of Ca by laboratory experiment. They measured the instantaneous velocity and
the pressure gradient in the experiment, and calibrated the data with their semi
theoretical porous flow model:
Vp=( + 6f) + />(l + C) (2.12)
The results so obtained for Ca scattered in a range of 7.5 ~ +5.0.
Dagan in 1979, based on the microscopic approach, arrived at a generalized
Darcys law for nonuniform but steady porous flows (Dagan, 1979):
_ n 77* _2.
~ Vp= q~ rV q
Tip
(2.13)
where 7 is a constant coefficient depending only on the geometry of the media and
it was defined as
7 =
80
(2.14)
In applying this model to the waveporousseabed interaction, Liu and Dalrym
ple (1984) added an acceleration term to the Dagans model and it becomes
v'5<215>
It was found, by performing a dimensional analysis (Liu, 1984), that the last
term in Dagans model is a second order quantity in comparison to the other terms.
The inertial term in this model is the same as that of Reid and Kajiura and that of
Sollitt and Cross where Ca is set to be zero.
Based on the phenomenological approach, Barends (1986) added another porous
flow model to the list:
r,_ ,Cadq Q x ,
5.2 Determination of The Empirical Coefficients 52
5.3 Relative Importance of The Resistances in The Experiment 55
5.4 Comparison of The Experimental Results and The Theoretical Values 55
6 BOUNDARY INTEGRAL ELEMENT METHOD 64
6.1 Basic Formulation 64
6.2 Local Coordinate System for A Linear Element 68
6.3 Linear Element and Related Integrations 70
6.4 Boundary Conditions 74
7 NUMERICAL MODEL FOR SUBMERGED POROUS BREAKWATERS 76
7.1 Governing Equations 76
7.2 Boundary Conditions 78
7.2.1 Boundary Conditions for The Fluid Domain 78
7.2.2 Boundary Conditions for The Porous Medium Domain .... 81
7.3 BIEM Formulations 82
7.3.1 Fluid Domain 82
7.3.2 Porous Medium Domain 86
7.3.3 Matching of The Two Domains 87
7.4 Linearization of the Nonlinear Porous Flow Model 88
7.5 Transmission and Reflection Coefficients 91
7.6 Total Wave Forces on an Impervious Structure 92
7.7 General Description of The Computer Program 94
7.8 Numerical Results 95
8 LABORATORY EXPERIMENTS OF A POROUS SUBMERGED
BREAKWATER 107
8.1 General Description of The Experiment 107
8.2 Wave Transmission and Reflection 109
8.3 Pressure Distribution and Wave Envelope Over The Breakwater . 121
vi
55
that the error function defined as
M
=, = T7E[(7''
)? + (
m G'
P\ 2
)?i
(5.3)
/=i &rm &im
was minimized, where the subscript m represents the measured values and p denotes
the predicted values; M is the number of data points. The best fit was found to
exist when:
oo = 570
60 = 3.0
Ca = 0.46
The added mass coefficient obtained here, Ca 0.46, is close to the theoretical
value of 0.5 for a smooth sphere. The values of a0 and b0, to an extent, reconfirm
those given by Engelund (1953) and the others.
5.3 Relative Importance of The Resistances in The Experiment
Introducing a0) >o, Ca, and the averaged wave height H into Eq.(4.29) for  qn \
and using Eqs.(3.33) through (3.35), the relative importance of the various resis
tances of the tested cases can be established precisely. The results are given in
Table 5.4. As we cam see from these ratios, for the first four grain sizes, all three
resistances are about equally important. Whereas for the last two larger diameter
materials, the turbulent and the inertial resistances are evidently dominant over the
linear resistance, with the inertial force approximately double that of the nonlinear
resistance.
5.4 Comparison of The Experimental Results and The Theoretical Values
Table 5.5 shows the comparisons between experimental results and the theoret
ical values of aÂ¡ and aT. The relative error, defined as
A% = ZOZ  X100%
(5.4)
21
dw
w + Tdz p +
oz
du
u + dx
ox
dp
w p
Figure 3.1: Definition sketch for the porous flow model
defined as the ratio of AVOid to Atotai with AVOid being the area of the voids and
Atotai being the total control area; n = volumetric porosity defined as the ratio of
Vvoid to Vtotai; Fb, is the body force of the solid; p, = density of solid; u, and ti, =
velocity and acceleration of the solid. The subscript x denotes the x direction and
the overbar denotes spatial average. Since, from this point on, we will be dealing
exclusively with spatially averaged values, the overbar will be dropped.
The force balance on the pore fluid can be established in a similar manner:
dP_
dx
dx n\z dz Fox Fix + Fb/Z = pndxdz (fx ,x)
(3.3)
with p being the density of the fluid and Fb/X being the body force of the fluid in x
direction; u/ and ti/ equal, respectively, the actual spatially averaged velocity and
acceleration of the pore fluid, defined as
uf = 3 f ua dv
Vvoi Jv.oU
(3.4)
174
TIME (SEC)
Figure B.14: Case of L = 200 cm, h DW = 25 cm, h, = DS = 20 cm and
dso = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights [H/Hi) and the best fit to the exponential decay
function.
81
On the lateral boundary of transmission (onshore side), x =
= = ^ = m, = m*
on ox ox
in which k' is determined by
gk' tanh k'h1 cr2
where h' is the water depth at x = l'.
While on the lateral boundary of reflection (offshore side), x = l,
dcf>
dn
<>r ~ I
di d(j>r
dx
dx dx
= ikcfrj tkcfir
(7.24)
(7.25)
(7.26)
or
= ik(f>Â¡ ik(cf> i) = 2ik(f>i ik
d
= 2 ik
dn
in which k is the incident wave number determined by
gritanh kh = o2
(7.27)
(7.28)
where h is the water depth at x = /.
7.2.2 Boundary Conditions for The Porous Medium Domain
In this domain, there are two types of boundaries, one is the common boundary
with the fluid domain and the other one is the impervious bottom. On the interface,
the boundary condition is the same as the one specified by Eq.(7.16) and (7.17).
On the impermeable bottom, the noflux condition, in terms of the pore pres
^ = 0
dn
(7.29)
Again, this boundary can have any shape, it can be the surface of the domain
for core materials if it is considered as impermeable, etc.
60
Table 5.8: Comparison of crm and ov for d50 = 0.16 mm, h, = 20 cm, L = 200 cm.
h (cm)
R
Ms l)
Ar%
0Vm(s i)
M5 l)
30.0
5.4 x KT7
4.8908
4.7638
2.60
0.0070
4.5 x 10"7
25.0
5.1 x 10'7
4.6312
4.4957
2.93
0.0090
5.6 x 10'7
20.0
4.8 x 107
4.2824
4.1428
3.26
0.0122
6.8 x 10"7
is plotted in Fig. 5.5. In obtaining the solution, the only empirical coefficient oo
appeared in their dispersion equation was calibrated to the data in Table 5.3 with
the same routine as that for the solution of Eq.(4.15), and the nonlinear regression
yielded a0 = 2200. The overall least mean square error for the complete data set
(36 cases) is as high as 0.327. It can be seen from the figure that considerable
discrepancies occur for the damping rate, and the maximum relative error is about
87% of the experimental values. The prediction of wave frequency by their solution
is about the same as that by this study.
A test over a sand bed (d50 = 0.16 mm) was also conducted. Active sand move
ment was observed along with the rapidly formed ripples. As a result, theoretical
predictions are several orders of magnitude smaller than those actually measured
(Table 5.8). The basic assumption of no particle movement was violated. How
ever, the predictions of wave frequencies were as good as the coarsegrained cases.
The same test, with dye injection into the sand bed, disclosed that the flow inside
the sand bed was very slow, and indicated that the assumption of impermeability
invoked in the linear wave theory is justified in such situations.
When the thickness of a gravel bed is reduced to one grain diameter, the model
also failed to yield good results as shown by the values in Table 5.9. In this case, the
effect of the boundary layer at the interface, which was not included in the model,
obviously cannot be ignored.
The nature of a turbulent boundary layer over a thicker seabed was qualitatively
investigated for the case of ht= 20 cm, dso = 148 cm, h =20 cm and L =200 cm,
log(J?) and log [(A:,) max l("2/9)\
46
Figure 4.3: Maximum nondimensional damping rate {o* jg) and its corre
sponding permeability parameter R as functions of nondimensional water depth
h{o2/g).
195
Figure B.35: Case of L = 275 cm, h = DW = 25 cm, h, DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/H\) and the best fit to the exponential decay
function.
198
Goda, Y., and Moriya, Y., 1967, Laboratory Investigation on Wave Transmis
sion over Breakwaters, Report of Port and Harbour Research Institute,
Tokyo, No.13.
Hannoura, A.A, and McCorquodale, J.A., 1978, Virtual Mass of Coarse Gran
ular Media, J. of the Waterways Harbors and Coastal Engineering Div.,
ASCE, Vol. 104, 191200.
Hunt, J.N., 1959, On the Damping of gravity Waves Propagated over a Per
meable Surface, J. of Geophysical Res., Vol. 64, No. 4, 437442.
Ijima, T., Chou., C.R., and Yoshida, A., 1976, Method of Analysis for
TwoDimensional Water Wave Problems, Proc. 15th Coastal Eng. Conf.,
ASCE, 27172736.
Johnson, J.W., Hondo, H., and Wallihan, R., 1972, Scale Effects in Wave
Action through Porous Structures, Proc. 10th Coastal Eng. Conf., ASCE,
10221024.
Kim, S.K., Liu, P.LF., and Ligget, J.A., 1983, Boundapr Integral Equa
tion Solutions for Solitary Wave Generation, Propagation and Runup,
Coastal Engineering, Vol. 7, 299317.
Hondo, H., and Toma, S., 1972, Reflection and Transmission for a Porous
Structure, Proc. 13th Coastal Eng. Conf., ASCE, 18471865.
Hondo, H., and Toma, S., 1974, Breaking Wave Transformation by Porous
Breakwaters, Coastal Engineering in Japan, Vol. 17, 8191.
Ligget., J.A., and Liu, P.LF., 1983, The Boundary Integral Equation
Method for Porous Media Flow. George Allen & Unwin Ltd., London.
Liu, P.LF., 1973, Damping of Water Waves over Porous Bed,
J. Hydraulics Div., ASCE, Vol. 99, 22632271.
Liu, P.LF., 1977, On Gravity Waves Propagated over a Layered Permeable
Bed, Coastal Engineering, Vol. 1, 135148.
Liu, P.LF., and Dalrymple, R. A., 1984, Damping of Gravity Water Waves
Due To Percolation, Coastal Engineering, Vol. 8, 3349.
Madsen, O.S., 1974, Wave Transmission through Porous Structures,
J. of the Waterways Harbors and Coastal Engineering Div., ASCE, Vol.
100, WW3, 169188.
Madsen, O.S., and White, S. M., 1976, Wave Transmission through Trape
zoidal Breakwaters, Proc. 15th Coastal Eng. Conf., ASCE, 26622676.
Madsen, O.S., Shusang, P., and Hanson, S.A., 1978, Wave Transmission
Through Trapezoidal Breakwaters, Proc. 16th Coastal Eng. Conf., ASCE,
18271846.
166
Figure B.6: Case of L 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
so = DD = 2.84 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
72
for all i =Â¡ j and i ^ j + 1 and
Ay = 6+1 6
If: = j, the above integrals have to be reevaluated to avoid the singularity of
the Greens function,
1 /Â£/+> 1 <9rt
J}j = iim/y+>^ee = o
6+i 6 Udn
(6.35)
= lim / 7 dÂ£ = 0
6+i 6 <'iy+< r<3n
(6.36)
J?1. =
lim e,+llnr,^d
6+i ~ 0 0 /y+<
= ^(21nAyl)
(6.37)
fy+i
Ijj = 77" i f In r, dÂ£
JJ 6+1 6 'y+
= In Ay 1
(6.38)
Similarly, when i = j + 1,
iy+ij = 7lunf,+ ^Â£dÂ£ = 0
* J 6+i ~ 6 'y r 5n
(6.39)
1 .. /j+~ 1 dr,
Ijh y = 7 r hm/ rdÂ£ = 0
,+J 6+i 6 <,0*'fy r*5rl
(6.40)
(1 2 In Ay)
(6.41)
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149
The experimental results of porous seabed for 36 cases show that the wave
damping rate and the corresponding permeability can be successfully predicted by
the analytical solution. The measurements of wave frequency are also consistent
with the predictions. The added mass coefficient Ca for densely pact gravel is
determined to be equal to 0.46, very close to the theoretical value for a smooth
sphere, as opposed to zero as assumed by many authors. In fact, the least mean
square error of the nonlinear regression process was found to be very sensitive to the
value of Ca. The values of the other two velocity related resistance coefficients, a0,
b0, reconfirm those given by Engelund and others. The comparison of the data to
the predicted values by the analytical solution shows much better agreement than
those of all existing theories.
When the particles of a porous bed are so small that movement occurs at the bed
surface, the experimental data revealed that the analytical solution is not applicable
because of the violation of the noparticlemovement assumption. The data also
indicate that the turbulent boundary layer at the interface cannot be ignored when
it dominates the energy dissipation. This is the case when a bed is as thin as one
particle diameter. However, the wave frequency prediction for both cases remains
almost as accurate as that for the other cases.
For porous structures of irregular shapes, the numerical results show that the
linear element behaves much better than the constant element, especially at corner
points. With the special treatment applied in the numerical model, the singularities
at the corner points for velocity potential function or pressure distribution can be
successfully removed, although the discontinuities in the normal velocities around
such points remain. The radiation boundary condition at the far field in the fluid
domain renders a great deal of simplicity in the numerical formulations, as compared
to the matching boundary condition with the wave maker theory. Generally, as long
as the vertical boundary(ies) is placed far enough from the structure (about two
61
id
o
cc
Li
CD
C3
0
o
EXPERIMENTAL FREQUENCY
EXPERIMENTAL DAMPING RATE
Figure 5.4: Theoretical values by the present model vs. experimental data of Table
5.5, Table 5.6 and Table 5.7. (a) Wave frequency ar, (b) Wave damping rate a.
NORMALIZED WAVE ENVELOPE
144
Figure 9.10: Permeable berm breakwater of prototype scale with H = 4.0 m: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity distributions
95
(7.40). In the porous domain, the matrices C and D are divided into four blocks
and the operation formulated in Eq.(7.44) is carried out. The boundary matching
is accomplished by completing Eq.(7.49) and the resulting matrix equation is then
solved by the IMSL subroutine LEQTlC, a complex matrix solver.
4. Linearization
This is an iteration process which repeats the solution procedure with the con
temporary resistance coefficient /q The starting point of the iteration is Eq.(7.49),
the matching operation where the linearized coefficient /o first appears in the com
putation. The criterion of convergence for the iteration is
I /0 ffi l< (785)
with e being a prespecified arbitrarily small number. It is set to be 1.0% in this
model. Figure 7.2 show the flow chart of the model.
To accommodate further extension, perhaps to higher order stokes waves, the
velocity potential function of the linear incident wave has been nondimensionalized,
in this model, by the factor of Hgl2\fgfcy i.e.,
4>i{x,z) = (786)
Therefore, the resulted vector is also nondimensional.
7.8 Numerical Results
In this section, numerical results for a group of example structures are presented.
The empirical coefficients in the porous flow model in all the computations are taken
to be the values determined in Chapter 5, i.e.,
Oo = 570
o = 3.0
Ca = 0.46
16
this method to porous breakwaters with constant elements, which assumes that
the physical quantities remain constant over each individual element. The whole
computation domain was divided into three regions, two fluid regions and one porous
region. In the porous domain, Darcys law was used and in the two fluid regions,
two artificially defined vertical boundaries were placed at the offshore and inshore
ends. The patterns of the vertical distribution of the velocity potential at these two
boundaries was assumed undistorted by the presence of a structure. The reflected
and the transmitted potential functions outside the computational domain were
given by
r[x,z)
cosh k{z + h)
cosh kh
(2.35)
Mx,z) = cosh *'(*+'l')
cosh k>h' ^2'36)
where l and V axe the distances from the left and the right vertical boundaries to
the origin, k and k' are the wave numbers in the reflection and the transmission
regions, respectively. The reflection and transmission coefficients are then
Kr = 1 Ao
(2.37)
Kt = B0  (2.38)
The agreement of the transmission coefficient for impermeable floating struc
tures with the experiment data was fairly good, but the comparison of Kt and Kr
for a slopedface permeable breakwater was not as satisfactory.
Finnigan and Yamamoto (1979) further added the modulated standing wave
modes to r and 4>t, while keeping the constant element and Darcys law unchanged
in their model. The two potential functions were, respectively,
M*,*) = [*<>+.V*>]C0Sh*[Z,t*) + t h)
cosh kh r'r'*' u h
m=l
cos kmh
(2.39)
118
1.0
o
GO <
r
0
0 o
0.6
0.4
o
T 0.642 SEC.

T = 0.858 SEC.
0.2

T = 0.952 SEC.
a .
EXPERIMENT DATA
0.0
0
" \
.000
1 1 i 1 1 1
0.002 0.004 0.006
1 1 1 l
0.008 0.010 0.
HI/GTk*2
Figure 8.5: The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp and
Ktmi (h) Krp and Krm.
14
breakwaters, in addition to a large number of laboratory experiments, a multitude
of theoretical models have been developed. The analysis of such structures has been
approached both analytically and numerically.
Analytically, Sollitt and Cross (1972) investigated the problem of wave transmis
sion and reflection by a vertical face (rectangular cross section) porous breakwater.
The porous flow model used in the formulation was Sollitt and Crosss model al
though the virtual mass coefficient Ca was assumed to be zero in the computation.
In their solution, the whole computation domain was divided into three regions,
two fluid regions and one porous region. The velocity potential functions in the
three regions were assumed to be the summations of the fundamental mode and
the evanescent modes according to the wave maker theory. The nonlinear term in
the flow model was linearized according to Lorentzs hypothesis and the normal
velocities and the pressure were matched at both vertical faces of the breakwater.
The transmission and the reflection coefficients for long waves, simplified from the
complete solution of complex matrix form, were given by
Kt =
1
(2.31)
1+LZls~if+n2)
Kr =
S if n2
(2.32)
S if + n2 i2n^
gh
where S is the coefficient for the inertial resistance of porous media, i is the imag
inary unit, / is the linearized coefficient for velocity related resistances, b is the
width (or thickness) of the breakwater and all the other symbols have the same
meanings as before.
Madsen (1974) later reexamined the same problem beginning directly with the
linear long wave theory. He adopted the Dupuit Forchheimer model and introduced
the inertial resistance into the momentum equation. The resulting expression of Kt
and Kr are almost identical to Eq.(2.31) and (2.32). By carrying out the volume
85
C (3',:4>: + + H}tNct4>Nc, + Yl P'J^i ~
J=#ek+1
=Net+1
Ni 1
ik K{1(p^ tk1 Y2 7 (iklKfNi_1 KÂ¡Nx)q>nx
y=2 9
a1 Wal a2
T X) 71y^; + ~^,N2iPn2 + ikKl'N2(2jN2 4>n3)
9 j=Ni+1 ^
#sl
+ * IT 7.j(2^/ 4>i) + ikKlNz_x{24>INz 4>N3)
:=n3+i
Net1
+ ^\,Nch^nNeh + Y2 lijPnj + K^Ncl^nN,. (7.35)
=#*+!
where the following notations are introduced
to
= Bfj, + St\, 6Â¡,Â¡ai
(7.36)
to
= Hl_l+HÂ¡jSÂ¡Jai
(7.37)
(7.38)
The matrix form of Eq.(7.35) is, after some manipulations,
iAi a,]{Â£Hbi Bii{^}+b <739)
where A, and B, are known matrices, determined purely by the boundary geometry,
c and 4>nc are, respectively, the vectors of the unknown potential function and its
normal derivative on the interface boundaries, with
<Â¡>C = {^jv,t> 4>Nci+l> M 4>Nu4>Nc,}T
nc = nNci+U finNe.U ^n#e,}
f and 4>nf are the vectors of the unknown potential function and its normal deriva
tive along the boundaries other than the interface boundaries, respectively, b is the
known vector containing 4>i resulting from the radiation boundary condition on the
offshore side lateral boundary. The expression for an element in b can be obtained
from Eq.(7.35):
Nzl
bi = 2ik(KliNj<Â¡>iN2 + Y2 + Ki.Nsi^iNz)
#3 + 1
(7.40)
75
of such points will be discussed in the next chapter.
By introducing the boundary conditions at all N boundary nodal points, another
system of N equations with the same unknown in Eq.(6.48) is established. The
number of unknowns is now equal to the number of the equations. It is usually
convenient to eliminate N unknowns with the 2N equations, and the resulted matrix
equation can be expressed as
AX = b (6.52)
in which A is a known matrix of order N x N, X is the unknown vector containing
<9$ 3$
$ or or $ for some part of the boundary and 5 for the rest of it, depending
on On
on which one is not specified by the boundary condition; and b is a known vector
resulted from boundary conditions.
LIST OF FIGURES
3.1 Definition sketch for the porous flow model 21
3.2 Regions with different dominant resistant forces 30
4.1 Definition Sketch 32
4.2 Progressive wave case: h, = DS = 5.0 m and h = DW = 2.0 
6.0 m. (a) Nondimensional wave number kr/(o2 fg), (b) Nondi
mensional wave damping rate ki/(o2/g) 45
4.3 Maximum nondimensional damping rate (&,)./(cr2/y) and its
corresponding permeability parameter R as functions of nondi
mensional water depth h (cr2 jg) 46
4.4 Standing wave case, (a) Nondimensional wave frequency orl{LÂ¡g)*\
(b) Nondimensional wave damping rate cr,/(L/<7)3 48
4.5 Solutions based on four porous flow models, (a) Nondimensional
wave frequency or/(L/g) 3, (b) Nondimensional wave damping
rate Ci/(L/g)t 49
5.1 Experimental setup 51
5.2 Typical wave data: (a) Averaged nondimensional surface eleva
tion iv/Hi), (b) Nondimensional wave heights (H/~Hi) and the
best fit to the exponential decay function 54
5.3 The Measurements and the predictions vs. R. for L = 200.0 cm,
h, = 20.0 cm, h = 25.0 cm. (a) Wave frequency crr, (b) Wave
damping rate <7, 58
5.4 Theoretical values by the present model vs. experimental data
of Table 5.5, Table 5.6 and Table 5.7. (a) Wave frequency ar,
(b) Wave damping rate <7, 61
5.5 Theoretical values by the model of Liu and Dalrymple vs. ex
perimental data of Table 5.5, Table 5.6 and Table 5.7. (a) Wave
frequency crr, (b) Wave damping rate 62
6.1 Auxiliary coordinate system 68
via
93
the nonflux condition. If we assume that the dimension of the structure is compa
rable with the wave length (usually greater than 0.2 of the wave length), the total
horizontal and vertical forces can be obtained by integrating the pressure over the
exposed surface of the structure.
Since
p = ipafi
and
4> = 4>j)Â£ + (<Â£>+1Â£; + 4>jÂ£j+1)
the total wave force on a single element is
u =
~r~ / [(^j+x ~ })Z + (&+16 + 4>jÂ£j+i)]dÂ£
Llj J ij
1 4>j)ttj+i &) =
(7.75)
7, =
i N^1
Fxr 4 tFxi = po\ 22 {4>i+1 cos(ny, 2)]
}=Ncb
(7.76)
7Z =
%
FZr + iFti = po[ Ys (4j+1 4*3)cos(j, z)}
}=Nci
(7.77)
Fx = 1 7Z I e^t+^
(7.78)
Fr =  Jz  {at+(]
(7.79)
with phase lags of
et = tan
*XT
(7.80)
+ X
e* = tan 
**r
(7.81)
where (n:,x) and (n,,z) are the angles from the normal direction of element j to x
and z axis, respectively.
167
Figure B.7: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (t]/Hi), (b)
Nondimensional wave heights (if/57) and the best fit to the exponential decay
function.
151
breakwater as well as the pressure distribution along the breakwater surface in the
experiment are reasonably well predicted by the numerical model of linear waves,
when no breaking occurs.
Despite the existence of the higher order harmonics which are not predicted
in the numerical model for porous submerged breakwaters, when the concept of
equivalent wave height for transmitted waves is adopted, the agreement between the
experimental and the predicted values for transmission and reflection coefficients is
fairly good considering the complex nature of the problem. Such good agreement
indicates that the wave energy dissipating process within the porous breakwater can
be successfully modeled with the porous flow model using the empirical coefficients
determined in the porous seabed experiments.
The numerical results for porous berm breakwaters show that a significant
amount of the wave energy (as high as 80%) is dissipated within the berm. The
rest of the energy is reflected offshore. Compared to conventional subaerial break
waters with thin cover layers, the energy dissipation due to percolation in berm
breakwaters is certainly much larger.
All the field quantities vary with the incident wave height due to the nonlinearity
in the porous flow model. The percentage of the wave energy dissipation within the
porous berm decreases with increasing incident wave height for the cases computed.
For larger stone size and/or different cross sections, it is possible to make such
dissipation increase for higher incident waves.
10.3 Recommendations for Future Studies
For porous flow model, the inclusion of the deformation of solid skeletons, such
as elastic or plastic deformations, should be of interest to future engineering appli
cations when the materials for porous structures are deformable rather than rigid.
For porous seabed problem, further research work should be directed to develop
turbulent boundary layer solution so that the tangential boundary condition at
138
Figure 9.5: The Cross Section of The Berm Breakwater
The intrinsic permeability for the porous medium made of such stones is about
1.09 x 106 m2. The wave period is T 2.52 s and the wave heights are H =
5.0, 10.0 and 20.0 cm for the three computations, respectively. The permeability
parameter R is 2.48 with oq = 570 and the other two empirical coefficients 60 and
Ca are unchanged in the model (>0=3.0 and Ca = 0.46). In these figures, three
nondimensional quantities the wave envelope (rÂ¡Â¡H), the pressure (p/^/H) and the
normal velocity (unT/H) distribution along the interface, are plotted.
As revealed by shapes of the nondimensional wave envelopes that wave height
reduces inside of the berm and near zero deep into the berm. The dynamic pressure
distribution along the interface increases quite uniformly from the bottom to the
surface as expected. The normal velocity distribution, varies very mildly below
certain depth and then has a rather sharp increase near the surface.
43
Table 4.1: Comparison of The Predictions and The Measurements
Run No.
H (cm)
T (sec.)
kim{m l)
kip(m 1)
Ap%
kiLD{m *)
A id%
h = 0.229 m
1
6.74
1.27
0.0379
0.0313
17.4
0.0338
10.8
2
4.45
0.0318
0.0337
6.0
0.0338
6.3
3
5.26
0.0303
0.0328
8.3
0.0338
11.6
4
1.93
0.0317
0.0369
16.4
0.0338
6.3
12
6.65
0.0334
0.0313
6.3
0.0338
1.2
13
4.36
0.0283
0.0338
19.4
0.0338
19.4
14
1.90
0.0374
0.0370
1.1
0.0338
9.6
5
6.25
1.00
0.0411
0.0350
14.8
0.0390
5.1
6
4.66
0.0357
0.0372
4.2
0.0390
9.2
7
2.17
0.0397
0.0411
3.5
0.0390
1.8
8
2.08
0.0393
0.0413
5.9
0.0390
0.8
15
5.48
0.0383
0.0360
6.0
0.0390
1.8
16
4.52
0.0397
0.0374
5.8
0.0390
1.8
17
1.93
0.0373
0.0415
11.3
0.0390
4.5
9
5.29
0.80
0.0296
0.0314
6.1
0.0343
15.8
10
3.55
0.0297
0.0334
12.5
0.0343
15.5
11
2.28
0.0335
0.0351
4.8
0.0343
2.4
18
7.05
0.0264
0.0296
12.1
0.0343
29.9
19
4.08
0.0320
0.0328
2.5
0.0343
7.2
20
2.14
0.0305
0.0353
15.7
0.0343
12.5
h = 0.152 m
21
4.00
1.27
0.0552
0.0574
3.9
0.0600
8.7
22
1.83
0.0379
0.0640
68.9
0.0600
58.9
23
3.83
0.0555
0.0579
4.3
0.0600
8.1
24
1.36
1.00
0.0449
0.0770
71.5
0.0731
62.8
25
3.40
0.80
0.0658
0.0700
6.4
0.0767
16.6
k{m the experimental damping rate given by Savage (1953);
kip the theoretical values by the present model;
kuD the theoretical value by Liu and Dalrymple (1984)
91
Recalling is that when treating the singularities at JVej and Nce, tf>~ is not nec
essarily equal to cf>+, but are two different unknowns instead. Physically they have
to be equal since the pressure is unique anywhere. This has been confirmed by the
numerical results in the solution for every step of the iteration.
When is known, the rest of the field quantities can be readily obtained, since
p = iop4>
(7.61)
to ,
T] =
9
(7.62)
4>nt = BCtt = ik'lt
o2
(7.63)
4>n, = = 14>.
9
(7.64)
4* nr Brr ik\(f>r
(7.65)
4>nc E c
(7.66)
where the subscripts t, s, r and c refer to the transmission, free surface, reflection
and the common interface boundaries, respectively, Bc,s are the B.C. matrices and
I is an unit matrix.
7.5 Transmission and Reflection Coefficients
When computing the coefficients of transmission and reflection, the velocity
potential functions at both lateral boundaries are assumed to have the forms of
*(*.*) =
cosh kh
+
m=l
k(x+i)Cskm{z + h)
cos kmh
(7.67)
* 2a cosh k'h!
+
f R (,n cosk'mjz + h!)
cos k'mh'
m= 1
(7.68)
11
the permeable surface was assumed viscous as opposed to the inviscid approach by
the previous investigators. The tangential velocities at the surface of the permeable
bed were set to be zero and the continuity of the normal velocity and the pressure
were observed on the same boundary. The resulting dispersion equation was quite
lengthy. Under the consideration of sand beds of low permeability, the results for
kr and /:, are
kr
ko +
nr 2kl
V 2c 2kQh + sinh 2koh
(2.20)
with
ki
2ko (aKp 1 k )
2k0h + sinh 2k0h' u 0 2c
(2.21)
c2 gk0 tanh koh
(2.22)
Murray (1965) investigated the same problem by the use of stream function and
the porous flow model given in Eq.(2.7). Instead of letting the tangential velocities
at the interface be zero as done by Hunt, Murray assumed that the tangential
velocities were finite and continuous. The first order solution of the dispersion
equation for the spatial damping was
V2k2J^+2k0N
kli = 2koh + sinh 2k0h i2*23)
where k = k0 + ki with k0 being the initial wave number for the impervious bottom
as defined in Eq.(2.22) and
N =
noKp
v
For fine sand, JV 0 and
,
* 2krh + sinh 2 krh
and the temporal damping for known k in such case is
yjvok
a =
\/2sinh2A:/i
(2.24)
(2.25)
178
Figure B.18: Case of L = 200 cm, h DW = 20 cm, h, = DS = 20 cm and
d50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/7fÂ¡) and the best fit to the exponential decay
function.
82
7.3 BIEM Formulations
The formulations in the two domains are slightly different because of the dif
ferent boundary conditions. In the fluid domain, due to the radiation boundary
condition on the lateral boundary of offshore side, the terms containing the incident
wave potential are introduced, which will form the RHS vector of the matrix equa
tion. On the free surface, the normal derivative are expressed in terms of 4> owing
to the CFSBC. In the porous subdomain, the matrix equation, after applying the
noflux condition, has to be manipulated to match with the fluid domain.
7.3.1 Fluid Domain
According to the formulae given in chapter 6, Eq.(6.12) can be expanded, in
terms of node values, assuming that 4>~ = + for all the nodes, as:
ou4>i = + + + + +
+ i + HÂ¡Nl)4>Nl + ... + i + + ...
+ [Hlffs_i + H}tNi)s3 + + (S'tV.ii +
+ 1 + + + {Ef/ri + hIn)4>n
 Klfa  Khfa Kfjfa Kl s^3 .
 ~ ^Â¡.Ni^nNi ~ ~ ^i.Ni^nN, ~
 ^i,Nal^nN3 ~ Ki.Ns^nNs ~ ~ ~
~ ^i,Ncl^nN,. ~ Klx"4>nNc. ~ ~ ^i.Nl^nN ~
i = 1,2, ,JV
(7.30)
where a, is the inner angle of node i and the subscripts 1, Ni, JV2, JV3, Neb and Net
refer to the nodes of comer points as shown in Fig. 7.1. The superscripts and +
refer to the positions immediately before and after the nodal point in the direction
of contour integration, and the subscript n refers to the normal derivative. Here 4>~
is not necessarily equal to cf>+ for all the nodes.
26
The coefficient Cx is the overall added mass coefficient for the porous medium.
Its characteristics can be determined in much the same manner as those of Bz, or
m
Fix = T. pCgVyfx (321)
where C is the added mass coefficient of each individual particle and Vp is the
volume of each particle. One may then readily obtain, by comparing Eq.(3.21) with
Eq.(3.8),
Cx = p
Ca{ 1 n)
(3.22)
nAx n
Since Cx is related to the volume of the solid skeleton, one does not expect
significant directional preference, unless the geometry of the element deviates sig
nificantly from a sphere shape. A new coefficient, Cm, can be defined such that
,n + Ca(l n)
riAz
+ Cx=p[
nAxn
(3.23)
which has the same meaning as the mass coefficient in hydrodynamics.
Clearly the force coefficients in other directions can be defined in a similar
fashion. Equations.(3.10) and (3.11) can now be expressed in the following general
form:
q+ B  q  q + Cmq = Vp (3.24)
Here, the carat indicates a second order tensor. This equation cam be generalized
to include the motion of the solid simply by replacing q with q u,.
We now proceed to assume that the porous material is isotropic amd that all
the coefficients reduce to scalar quantities and
nAx = uax = n
Recall that the convective acceleration in q has been assumed, in Eq.(3.6), to be
negligible, i.e.
19
wave height for the transmitted waves in his paper was defined as
Hcq = ^~ j%2{t)dt (2.47)
Due to the smallness of the submergence of the breakwater crest, it is felt that
the majority of the waves in the experiment were breaking waves.
Dattatri et al. (1978) tested a number of submerged breakwaters of different
types, permeable, impermeable, rectangular and trapezoidal. One of the conclusions
was that the important parameters affecting the performance of a submerged break
water are the crest width and the depth of submergence. The transmitted waves
were found to be irregular though periodic. They also reported that porosity and
wave steepness did not have significant influence on the transmission coefficient,
which contradictory with the observations by Dick et al. (1968).
Seelig (1980) conducted a large number of tests on the cross sections of 17
different breakwaters for both regular and irregular waves. Most of the breakwaters
tested were rubblemound porous structures with multilayer designs. Beyond the
experiments for impermeable breakwaters by which Eq.(2.44) was obtained, he also
tested these permeable structures as submerged breakwaters. Since no generalized
model was available at the time for such breakwaters, the experimental data with
regular waves were compared to Eq.(2.44). It was found that the formula is quite
conservative in estimating the transmission coefficient for permeable submerged
breakwaters. The phenomenon of wave energy shifting to higher order harmonics
in the transmitted waves was again reported without further explanation.
Baba (1986) conducted an experiment on concrete submerged breakwaters and
compared the data with the four widely used computational methods for wave
transmission coefficient for impermeable submerged breakwaters. He concluded
that the formula given by Goda (1969) was the most suitable one in the case of a
shore protecting submerged breakwater.
10
and K(q) was defined by Hannoura and Barends (1981, see Baxends, 1986) as
KM=gmÂ¡T\ (2'I7)
with
_ Ca( 1 n)
f ~ Jrf
where C is the drag coefficient related to the porous Reynolds number (R = q 
djv), a and /? are two constants and d, is the relevant grain size.
With so many models, it is difficult to decide which one is most appropriate for
the porous flow problem in this study without further analysis.
2.2 WavePorous Seabed Interactions
The computation of gravity wave attenuation over a rigid porous bottom has
been performed by a number of investigators. The classic approaches by Putnam
(1949) and Reid and Kajiura (1957) was to assume that the Laplace equation is
satisfied in the overlying fluid medium and that the bottom layer can be treated
as a continuum following Darcys law of permeability. In Reid and Kajiuras pa
per, although the inertial term was included in the porous flow model, they later
neglected it and concluded that Darcys law is adequate for sandy seabeds. Under
the assumption of low permeability, Reid and Kajiura found, for infinitely thick
seabeds, that
a1 gkr tanh kTh (2.18)
k. 2(oKp/v)kr
* 2krh + sinh 2 kTh
for progressive waves, where a is the wave frequency, kr and ki are, respectively, the
wave number and the damping rate, h is the water depth and v is the kinematic
viscosity of the fluid.
Hunt (1959) examined the damping of gravity waves propagating over a perme
able surface using Reid and Kajiuras porous flow model and retained the inertial
term to the end. The thickness of the permeable beds was infinite and water above
163
Figure B.3: Case of L 200 cm, h DW = 30 cm, h, DS = 20 cm and
d5o = DD s= 1.20 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H fHÂ¡) and the best fit to the exponential decay
function.
191
Figure B.31: Case of L = 250 cm, h DW = 30 cm, h, = DS = 20 cm and
50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (r]/Hi), (b)
Nondimensional wave heights (H/1TÂ¡) and the best fit to the exponential decay
function.
183
Figure B.23: Case of L = 200 cm, h = DW = 25 cm, h, DS = 15 cm and
so = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rÂ¡/Hi), (b)
Nondimensional wave heights (H/HI) and the best fit to the exponential decay
function.
8
where a, b and c are empirical coefficients.
Murray (1965), when studying the viscous damping of gravity waves over a
permeable seabed, proposed
_ a du ,
 Vp = u + pn (2.8)
where k is the permeability of porous media (defined differently from Kp) and u
is the spatially averaged microscopic velocity of the pore fluid, related to q by
q = nu. Comparing to Eq.(2.7), the coefficient c is equal to p in Murrays model.
McCorquodale in 1972 further modified the expression of this coefficient as
n
and his model (McCorquodale, 1972) becomes
(2.9)
Vp=(a + bg)g + ^ (2.10)
n at
At about the same time as McCorquodale, Sollitt and Cross (1972) also extended
Eq.(2.2) to include the effects of unsteadiness. In their model, the inertia resistance
consists of two components, one is the inertia of the pore fluid, and the other one
is the inertia induced by the virtual mass effect due to the fluidsolid interactions;
the model reads
~ VP = injr + I s l) + Pi1 + (2n)
where u is again the spatially averaged microscopic velocity vector and Ca is the
virtual mass coefficient.
The value of the virtual mass coefficient Ca in Sollitt and Crosss model was
assumed to be zero in their actual computation because it was unknown at the time.
It has been so assumed in almost all the later applications of this model (Sulisz,
1985, etc.). When comparing to the experimental data, Sollitt and Cross found that
the correlation improved by taking nonzero values for Ca and asserted that one
27
Substituting these conditions into Eq.(3.24), we obtain
{A + B  q l)q + cjÂ£ = Vp (3.25)
This equation is similar to the wellknown equation of motion in a permeable
medium such as given by Reid and Kajiura (1957) and others, with the exception
that the added mass effect is now formally introduced.
To seek a solution to this equation, the common approach is to ignore the
inertial term and linearize or ignore the nonlinear term. Under such conditions,
Eq.(3.25) can be reduced to the Laplace equation by virtue of mass conservation.
To retain the inertia term, the most convenient approach is to assume the motion
to be oscillatory, as is the case under wave excitation. Now we let
q = g(z, z)e~%at (3.26)
where g*is a spatial variable only, and a is a generalized wave frequency, which could
be a complex number. Substituting Eq.(3.26) into Eq.(3.25) leads to
 Vp = (A iaCm + B\q  )g (3.27)
A new set of nondimensional parameters are defined as follows:
Permeability parameter, R,
R = Kp
with
Inertia parameter, 0
'
F A a0 (1 n)3
0 = n + C1~n)
n2
Volumetric averaged drag coefficient, C
1 n
nsd,
(3.28)
(3.29)
(3.30)
(3.31)
59
Table 5.6: Comparison of om and ap for d50 = 1.48 cm, L = 200 cm.
h (cm)
H (cm)
<7rm(>S X)
*)
Ar%
OVm(s J)
otAs1)
A,%
h, = 15.0 cm
30.0
5.29
4.9268
4.8291
1.98
0.0525
0.0600
14.13
25.0
4.00
4.6715
4.5836
1.88
0.0697
0.0794
13.98
20.0
3.12
4.3173
4.2565
1.41
0.0928
0.1054
13.56
h, = 10.0 cm
30.0
5.97
4.9163
4.8145
2.07
0.0368
0.0416
13.13
25.0
4.61
4.6310
4.5629
1.47
0.0492
0.0551
11.92
20.0
3.48
4.2688
4.2309
0.89
0.0724
0.0729
0.70
Table 5.7: Comparison of
om and op
for d5C
= 2.09 cm, h, = 20 cm.
h (cm)
H (cm)
0rm{s *)
^rp(51)
A T%
<7,m(s *)
A i%
L = 225.C
cm
30.0
4.42
4.5114
4.4401
1.58
0.0831
0.0766
7.73
25.0
3.91
4.2913
4.1991
2.15
0.0977
0.1006
2.92
20.0
2.96
3.9699
3.8995
1.77
0.1270
0.1286
1.21
L = 250.C
cm
30.0
4.40
4.1524
4.0725
1.92
0.0752
0.0790
5.02
25.0
3.79
3.9119
3.8358
1.94
0.1010
0.1002
0.81
20.0
3.32
3.5862
3.5361
1.40
0.1384
0.1282
7.34
L = 275.C
cm
30.0
4.18
3.8371
3.7592
2.03
0.0783
0.0788
0.68
25.0
3.32
3.6285
3.5339
2.61
0.0993
0.0962
3.15
20.0
2.90
3.3298
3.2497
2.41
0.1264
0.1211
4.16
We now proceed to compare the theoretical computations with those data which
were not included in the calibration. These results are given in Table 5.6 and
Table 5.7. The agreement is just as good, if not better.
In Fig. 5.4, the theoretical values of oys and cr,s by the present model are
plotted, respectively, against the measurements for all the experimental data (36
cases). The overall least mean square error defined in Eq.(5.3) for the data of all 36
cases is about 0.008. While the solution of the dispersion equation (Eq.(4.2l) based
on the linear porous flow model by Liu and Dalrymple (Liu and Dalrymple, 1984)
188
Figure B.28: Case of L = 225 cm, h = DW = 30 cm, h, = DS = 20 cm and
<50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights [H/B7) and the best fit to the exponential decay
function.
CHAPTER 10
SUMMARY AND CONCLUSIONS
10.1 Summary
In this study, a general, unsteady, porous flow model has been developed based
on the assumption that the porous media can be treated as continuum. The model
clearly defines the roles of solid and fluid motions and henceforth their interactions.
When the solids are held stationary, the model reduces to that used by Sollitt and
Cross for analyzing flows in porous media. All the important resistant forceslinear,
nonlinear and inertialare included and more rigorously defined. It is shown that
the relative importance of these resistance components are functions of two nondi
mensional parameters, R/ = Ud,/v and Rt = ad2Ju. General assessment of the
nature of the resistance as well as of the pore flow is possible prior to detailed flow
computations.
The model is applied to a gravity wave field over a porous seabed of finite depth.
The analytical solution is for a linear wave system, but is applicable, theoretically, to
the full range of permeability, from zero to infinity. The inclusion of the inertial and
turbulent resistances results in a complex dispersion relationship that could only be
solved iteratively together with the equation of linearization obtained according to
the Lorentz law of equivalent energy dissipation. The derivation of the linearization
equation was facilitated by the boundary integral expression for such energy dis
sipation. Solutions for two classes of problems of physical significanceprogressive
waves of spatial damping and standing waves of temporal dampingaxe provided.
146
184
Figure B.24: Case of L = 200 cm, h = DW = 20 cm, h, = DS = 15 cm and
<5o = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (ry/jEfj), (b)
Nondimensional wave heights (H/TF[) and the best fit to the exponential decay
function.
152
the interface could be more realistic. Another aspect is to extend the solution to
nonlinear waves. As far as the porous material is concerned, the solution of the
problem with a poroelastic flow model should be useful for modeling and designs
of deformable porous structures, such as breakwaters or seawalls made of discarded
rubber tires.
For modeling of porous structures with irregular cross sections, immediate at
tention should be given to the nonlinear wave approach since the mechanism of
energy transformation to the higher order harmonics observed in the experiment
cannot be explained by the present model. Further effort should also be devoted
to the computation of forces on individual blocks, especially on the blocks at the
surface of a porous structure. This will entail the modeling of the wave force as well
as the interlocking force of the blocks.
For porous submerged breakwaters, the modeling of breaking waves is of prac
tical interest since a large portion of the wave energy is dissipated by breaking and
the majority of large waves will break over such a structure. It is very difficult to
rigorously model a breaking wave due to the complexity involved in representing it
mathematically. One of the possibilities of attacking this problem is the empirical
formulation for wave decay pattern with systematic experiments and incorporate
this pattern into the numerical model.
In reality, the direction of incident waves is not always normal to the long axis
of a shoreparallel breakwater but oblique to it. Such oblique incident wave will
change the governing equation from Laplace equation to homogeneous Helmholtz
equation for the twodimensional approach adopted in this study. The modification
of the BIEM scheme for such change will make the numerical model more general
and more practical. Due to the fact that the actual breakwaters have finite lengths
and varying orientations (perpendicular or inclined to shorelines), three dimensional
modeling should certainly be the ultimate goal.
CHAPTER 2
LITERATURE REVIEW
This chapter is a brief literature review of the past efforts made in studying
the flow in porous media with and without water waves. The review is separated
into three groups: one on porous flow model, one on analytical solutions for wave
and porousseabed interactions and the other one on the modeling of interactions
between waves and porous structures of irregular cross sections.
Since the porous media encountered in coastal engineering are largely of the
granular type, made of sand, gravel, quarry stones or artificial blocks, the deforma
tion of the solid skeleton of the media is usually negligibly small as compared to
that of the pore fluid. The literature review deals with granular media only; other
types of porous media, such as poroelastic or poroplastic media and so on, are not
included.
2.1 Porous Flow Models
A large number of porous flow models have been proposed over the past few
decades as an effort to quantitatively model the flows in porous media. Some of
the models were based purely on experiments and some of the others on certain
theoretical considerations. The derivations of the these models are based primarily
on the following three approaches: 1) Simple element approach which views a porous
medium as the assembly of simple elementary elements such as a bundle of tubes and
so on, 2) Microscopic approach which recognizes the microstructure of porous media
in derivations. The final formulations of the models usually have to be obtained
by taking the spatial average of the microscopic equations, 3) Phenomenological
approach which homogenizes the porous media as a continuum and the flow within
5
and
37
for linearized system
(4.30)
f = fo
f = + Nl
= /i + /2 w0  for nonlinear system (4.31)
with
fi = ^ *0 /* = ^ (432)
XL U
In Eq.(4.31), the magnitude of two is approximately taken as
wo 1=1 v) sinhfch,  (4.33)
0Jo
with
S(/o) = (4.34)
cosh /i cosh fc/i, (1 tanh kh tanh kht)
Jo
where the decaying wave amplitude ae{t has been approximately replaced by a,
which is the averaged wave amplitude in the period of [0,T].
With such approximations, / for the nonlinear system is no longer a function
of x and t. The approximations made above are not expected, to cause significant
errors for the final results.
By introducing Eq.(4.29) into Eq.(4.28) and carrying out the contour integration
regarding / as a constant, the energy dissipation within that onewavelong portion
of a seabed is obtained as
Ed = ^sinh2fch,[^p 71(f) + T2(f)] (4.35)
where Tj(f) and (f) are the nondimensional functions of t generated by the inte
gration with respect to time.
Substituting the linearized and the nonlinear resistance coefficients given in
Eqs.(4.30) and (4.31) for / in Eq.(4.35) respectively, equating the energy dissipations
201
With four years of hard work at college and the thirst for knowledge of science,
he entered graduate school in the same university right after receiving his B.S.
degree in early 1982. His major was offshore structures. After about two years of
course work on structures, he found that he was more interested in water waves
than in solid mechanics. As a result, he chose the problem Wave Forces on Large
Three Dimensional Objectes of Arbitrary Shapes as the topic of his thesis. At the
end of 1984, he graduated with a Master of Science degree and was appointed as a
faculty member in the same department.
Unsatisfied with his knowledge in his field and longing for the advanced tech
nology of the west, he decided to pursue his higher education in the United States.
In early 1986, he was admitted to the Graduate School and, at the same time,
appointed as a research assistant by the Coastal and Oceanographic Engineering
Department of the University of Florida. Since August 1986, he has been working
towards his Ph.D. under the Florida sun.
Gu was married, in 1984, to Liqiu Guan who came to the United States to
join him in April 1987 and later became a Ph.D. student in the Civil Engineering
Department at the same university.
22
where ua = actual pore velocity. If the fluid and solid mixture is now being treated
as a porous medium, and thus a continuum, we require that
1. All the field variables, such as that defined by Eq.(3.4), be independent of
the volume of integration.
2. 6 <Â§; L, where 6 and L are respectively, the length scales of the pore and the
system.
For the wave attenuation problem, Eq.(3.3) is of special interest. We introduce
here yet another field variable q representing the apparent spatially averaged fluid
velocity, which is related to u/ by
q = V L Ua dv = V L udv = n'uf (35)
V J V V J
here V is the total volume Vtotal.
This velocity also known as the discharge velocity, is related to the actual dis
charge over a unit surface area, or, Q/A. These apparent properties are of final
interest in engineering. Since we are now dealing with a continuum, we have
4 =
Dq
Dt
dq
+ q
dt 4
Vq ~
dq
8t
ri'iij
(3.6)
here it has been assumed that the convective acceleration is negligible.
The various force terms are established as follows:
(A). Drag Force
The drag force consists of two terms: that due to laminar skin friction and that
due to turbulent form drag; the former is proportional to the velocity and the latter
is proportional to the velocity squared. Their functional form, when expressed in
terms of the final field variables, may be written as
Fdz = [As{qx nutx) + Bx  q nu,  (q* nutz)]dx nAs dz (3.7)
The coefficients Ax and Bx are to be determined later.
NORMAL I ZED WAVE ENVELOPE NORMALIZED WAVE ENVELOPE
137
Figure 9.4: Berm breakwaters of inclined face: (a) Zero permeability; (b) Infinite
permeability.
APPENDIX A
BOUNDARY INTEGRAL FORMULATION FOR ENERGY DISSIPATION IN
POROUS MEDIA
The conventional definition for the energy dissipation eD within a volume V of
a porous medium during the time period T is (Sollitt et al., 1972, Madsen, 1974,
and Sulisz, 1985)
r r*+T >
eD = J J Fpqdtdv (A.l)
where F, which is a function of spatial coordinates and time, is the dissipative
stress in the medium, q*is the discharge velocity of the porous flow and p is the fluid
density. Here both F and q are real quantities.
In the problems of interaction between water waves and porous media, a non
linear unsteady porous flow model with all the resistance terms has the form
 iVP = (Â£ + ia/3 + % I q 1)5 (A.2)
P KP yjKp
where P is pore pressure function, u is the kinematic viscosity of fluid and Kp is the
intrinsic permeability of the porous medium; a is wave frequency, (3 is the inertial
resistance parameter, Cj is a constant characterizes the nonlinear resistance and q
is the complex vector of discharge velocity in the porous medium, with
q = Re(q)
The dissipative stress in this model is
(A3)
with the inertial term, ipo(3, which is non dissipative, excluded from Eq.(A.2).
153
7.1 Computational domains 77
7.2 Flow chart of the numerical model for porous submerged break
waters 96
7.3 Wave envelopes for (a) Transparent submerged breakwater; (b)
Impermeable step 98
7.4 Porous submerged breakwater: (a) Wave form and wave enve
lope; (b) Envelopes of pressure and normal velocity 99
7.5 Wave field over submerged breakwaters: (a) Impermeable; (b)
Permeable 101
7.6 Transmission and reflection coefficients vs. stone size for differ
ent wave periods, (a) Transmission coefficient; (b) Reflection
coefficient 103
7.7 Transmission and reflection coefficients vs. R for different wave
periods, (a) Transmission coefficient; (b) Reflection coefficient. 104
7.8 Transmission and reflection coefficients vs. R for different wave
heights, (a) Transmission coefficient; (b) Reflection coefficient. 105
7.9 Wave forces and over turning moment for a impermeable sub
merged breakwater: (a) Wave forces; (b) Overturning moment. 106
8.1 Experiment layout 108
8.2 Typical wave record, (a) Partial standing waves on the up wave
side, (b) Transmitted waves on the down wave side Ill
8.3 The wave spectrum of the transmitted waves 112
8.4 The predicted Kt and Kr versus the measured Kt and Kr. (a)
Ktp vs. Ktm', (b) Ktp vs. Ktm 117
8.5 The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp
and Ktm; (b) Krp and Krm 118
8.6 The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp
and Ktm; (b) Krp and K 119
8.7 The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp
and Ktm5 (b) Krp and Krm 120
8.8 Transmitted and reflected wave heights versus the incident wave
heights, (a) Transmitted waves; (b) Reflected waves 122
8.9 Transmitted and reflected wave heights versus the incident wave
heights, (a) Transmitted waves; (b) Reflected waves 123
IX
77
Figure 7.1: Computational domains
described by the linearized porous flow model,
pcf0q = VP (7.4)
where P is the pore pressure, f0 is the linearized resistance coefficient and q is the
discharge velocity vector in the porous medium, q = iu + kw with i and j being the
unit vector in x and z directions.
Taking curl of Eq.(7.4),
pofo V x q = V x VP = 0 (7.5)
leads to
Vx? = 0 (7.6)
This means that the homogenized porous flow is irrotational, and hence a velocity
potential function in the porous domain exists such that
q = V$,
(7.7)
177
Figure B.17: Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm and
dso = DD = 1.20 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights {H/Hi) and the best fit to the exponential decay
function.
53
Table 5.3: Measured oT and for ht = 20 cm and L = 200 cm
h (cm)
d50 (cm)
H\ (cm)
H' (cm)
cr.i(s l)
cr,(s x)
0.72
11.08
6.49
4.9054
0.0686
0.0570
0.93
10.93
6.35
4.9402
0.0678
0.0562
1.20
10.47
5.52
4.9370
0.0814
0.0694
30.0
1.48
9.03
4.71
4.9620
0.0878
0.0757
2.09
9.75
4.85
4.9673
0.0948
0.0824
2.84
9.80
4.95
5.0111
0.0931
0.0808
3.74
9.80
5.71
5.0341
0.0658
0.0543
0.72
8.70
5.15
4.6218
0.0799
0.0678
0.93
8.51
4.75
4.6856
0.0905
0.0781
1.20
8.00
4.46
4.6793
0.0984
0.0858
25.0
1.48
7.28
3.91
4.7352
0.1162
0.1030
2.09
6.46
3.57
4.7422
0.1105
0.0975
2.84
7.37
4.34
4.7893
0.0940
0.0816
3.74
7.37
4.26
4.7865
0.0813
0.0693
0.72
6.70
4.19
4.2716
0.1184
0.1047
0.93
6.28
3.55
4.3426
0.1334
0.1192
1.20
6.07
3.60
4.3565
0.1320
0.1179
20.0
1.48
4.49
2.57
4.4027
0.1484
0.1337
2.09
5.19
3.12
4.4325
0.1546
0.1396
2.84
5.41
3.33
4.4728
0.1456
0.1311
3.74
5.56
3.58
4.4646
0.1313
0.1173
* Hi is the average value of the first wave heights of the data group and H is the
average height of the complete train.
179
Figure B.19: Case of L 200 cm, h DW = 20 cm, h, = DS = 20 cm and
so = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (r\jH\), (b)
Nondimensional wave heights {H/H7) and the best fit to the exponential decay
function.
TRANSMISSION COEFF.
103
STONE SIZE DS (CM)
STONE SIZE OS (CM)
Figure 7.6: Transmission and reflection coefficients vs. stone size for different wave
periods, (a) Transmission coefficient; (b) Reflection coefficient.
24
a Darcytype resistance law. The coefficient A is, at least, related to four founda
tional properties, one of the fluidthe dynamic viscosityand three of the porous
structure the porosity, the tortuosity (one that defines effective flow length) and
the connectivity (one that defines the manner and number of pore connections).
Obviously, the more those factors can be specified explicitly, the more accurately
the value of A can be determined. However, it is also generally true that the more
factors explicitly introduced the more restrictive the range of application. If none of
these four fundamental properties is expressed explicitly, A is simply the inverse of
Darcys hydraulic permeability coefficient. If the dynamic viscosity, n, is factored
out, we have the empirical law:
A = Y (3.12)
where Kp is known as the intrinsic permeability. A number of investigators including
Engelund (1953, cited in Madsen, 1974) and Bear et al. (1968, cited in Madsen,
1974) attempted to relate A to porosity as well and came up with the relationship
of the following type:
A = Co
(1 n)3 /i
n*
(3.13)
where d, is a characteristic particle size of the pore material. The coefficient clq
obviously still contains the other properties such as tortuosity and connectivity.
Since both of these factors have directional preference, ao should also be a directional
property and, in general, is a secondorder tensor. For isotropic material ad becomes
a scalar. Engelund (1953, see Madsen, 1974) recommended
oq = 780 to 1500 or more
(3.14)
with the values increasing with increasing irregularity of the solid particle.
Attempts were also made to sort out the effects of tortuosity and connectiv
ity (Fatt, 1956a,b). The conditions invariably become more restrictive and one is
required to specify soil characteristics beyond the normal engineering properties.
28
Substituting these parameters into the above equation reduces it to the following
familiar form:
Vp = pa(^t/? + ?)g (3.32)
it a
This is essentially the same equation used by Sollit and Cross (1972) and others for
porous breakwater analysis.
3.3 Relative Importance of The Resistant Forces
Before solving Eq.(3.32) with an overlying wave motion, it is useful to examine
the importance of the resistance forces and to determine the nature of the fluid
motion in the porous medium. By taking ratios of the three respective resistance
forces, three nondimensional parameters can be established:
Inertial Resistance /,
Laminar Resistance//
n + C(l n)
ao(ln)
Turbulent Resistance fn
Laminar Resistance //
0
Oon(l n)2
R/
Turbulent Resistance /
Inertial Resistance /,
60(1 n) R/
Can(l n + J) Rt
where R/ and R/ are two forms of Reynolds number defined as
R/ =
gl d,
(3.33)
(3.34)
(3.35)
(3.36)
and
(3.37)
where v is the kinematic viscosity of the fluid.
Obviously both Reynolds numbers signify, the relative importance of the inertial
force to the viscous force. The origins of the inertial forces are, however, different.
In R/, the inertia is of a convective nature and the resistance arises due to change
of velocity in space (fore and aft the body) whereas in R, the inertia is of a local
115
Table 8.2: Comparison of Km and Kp
T
(sec.)
JL
(cm)
w
(cm)
Krm
Krp
Ar%
Ktm
Ktp
At%
2.67
1.87
0.0247
0.0852
71.0
0.6999
0.7989
12.4
0.642
3.38
2.37
0.0296
0.0854
65.4
0.7014
0.7965
11.9
4.07
2.88
0.0268
0.0858
68.8
0.7071
0.7959
11.2
4.28
3.00
0.0355
0.0859
58.7
0.7006
0.7960
12.0
2.22
1.62
0.0564
0.0474
19.1
0.7310
0.7641
4.3
0.858
2.92
2.21
0.0390
0.0509
23.4
0.7561
0.7665
1.4
3.31
2.55
0.0456
0.0525
13.0
0.7709
0.7687
0.3
4.31
3.19
0.0378
0.0555
31.9
0.7395
0.7748
4.6
2.14
1.66
0.0907
0.0712
27.5
0.7761
0.7691
0.9
0.952
2.59
2.03
0.0841
0.0697
20.7
0.7829
0.7721
1.4
3.38
2.65
0.0727
0.0676
7.6
0.7836
0.7784
0.7
3.89
3.15
0.0948
0.0667
42.1
0.8096
0.7818
3.5
1.96
1.47
0.1584
0.1583
0.0
0.7485
0.7673
2.5
1.020
2.47
1.89
0.1553
0.1584
2.0
0.7643
0.7711
0.9
3.38
2.80
0.1378
0.1586
13.2
0.8296
0.7789
6.5
4.29
3.48
0.1220
0.1589
23.2
0.8104
0.7857
3.1
2.06
1.55
0.1655
0.2119
21.9
0.7543
0.7698
2.0
2.29
1.75
0.1779
0.2132
16.6
0.7629
0.7714
1.1
2.42
1.84
0.2135
0.2143
0.4
0.7613
0.7729
1.5
1.120
2.69
2.07
0.1863
0.2158
13.7
0.7698
0.7751
0.7
3.18
2.56
0.1885
0.2183
13.6
0.8055
0.7793
3.4
4.18
3.40
0.2002
0.2219
9.8
0.8144
0.7865
3.5
4.43
3.45
0.1618
0.2228
27.4
0.7786
0.7884
1.2
4.77
3.40
0.1631
0.2243
27.3
0.7126
0.7918
10.0
1.58
1.18
0.2199
0.2434
9.6
0.7478
0.7715
3.1
1.265
2.10
1.67
0.2377
0.2495
4.7
0.7971
0.7751
2.8
3.00
2.43
0.2141
0.2588
17.3
0.8103
0.7828
3.5
3.39
2.76
0.1571
0.2611
39.8
0.8151
0.7851
3.8
1.67
1.28
0.2179
0.2699
19.3
0.7642
0.7952
3.9
2.95
2.40
0.2119
0.2106
0.6
0.8136
0.7879
3.3
1.379
3.09
2.36
0.1623
0.2251
27.9
0.7630
0.7986
4.5
3.38
2.74
0.1728
0.2262
23.6
0.8107
0.7997
1.4
3.89
3.11
0.1931
0.2285
15.5
0.7987
0.8018
0.4
1.81
1.40
0.1989
0.2320
14.3
0.7735
0.8055
4.0
1.453
2.35
1.91
0.1529
0.1791
14.6
0.8135
0.8018
1.5
3.08
2.43
0.1762
0.1861
5.3
0.7887
0.8075
2.3
3.96
3.17
0.1661
0.1920
13.5
0.8013
0.8133
1.5
2.19
1.70
0.0786
0.1985
60.4
0.7770
0.8206
5.3
1.778
2.66
2.03
0.0522
0.0534
2.3
0.7623
0.8595
11.3
3.28
2.63
0.0796
0.0519
53.5
0.8026
0.8634
7.0
3.98
3.25
0.0553
0.0499
10.9
0.8170
0.8686
5.9
6
the medium, is assumed to be continuous. The third approach is the most popular
for porous flows in granular materials since the microscopic structure is not well
understood and the corresponding constitutive equation not well established.
The first porous flow modelDarcys lawwas proposed by the French engineer
Darcy based on his experiments more than a century ago. In his model, the char
acteristic properties of porous media are lumped into one parameterpermeability
coefficientand the model has the form
 Vp = Â£? (2.1)
where Vp is the gradient of the pore pressure, p and q are, respectively, the dy
namic viscosity and the specific discharge velocity of the pore fluid and Kp is called
the intrinsic permeability coefficient, which reflects the collective characteristics of
porous media, such as the porosity, the roughness of the pore walls, the tortuosity,
the connectivity etc.
This model is very simple in form and reasonably accurate for steady porous
flows within media of low permeability, generally of the order of Kp = 109 ~
1012 m2, where the flows are normally laminar dominant. However, for porous
media with relatively higher permeability, the porous flow is no longer laminar dom
inant, turbulence plays a larger and larger role with increasing pore size. In such
cases, Darcys model tends to over estimate the discharge velocity for a given pres
sure gradient. Dupuit and Forchheimer (cited in Madsen, 1974) modified Darcys
law by adding a term quadratic in velocity to take into account the turbulent effects:
 Vp = (o + 6  q I)? (2.2)
where a ii/Kp, and b is coefficient for turbulent resistance defined by Ward (1964)
as
(2.3)
BIOGRAPHICAL SKETCH
Zhihao Gu was bom and raised in Dalian, a beautiful port city in NorthEast
China, on July 4, 1957. About two miles from his home, there was a pocket beach
where he used to go swimming every afternoon in summer. Perhaps it was there
that his bond with the beach and waves started. On weekends, he usually went
fishing with his father in the nearby reservoirs where he enjoyed calm waters and
learned many fishing tricks.
When he was 9, the disastrous cultural revolution began. The quality of edu
cation nosedived and he had to spend a lot of time for the revolution instead of
learning the things he was supposed to learn in school at his age. Under the guid
ance and scrutiny of his father and mother, he managed to complete high school
mathematics and physics by reading at home after school. His hobby at the time
was to play with electronic elements and assemble radios at home.
At the age of 18, right after graduating from high school, he was sent to a farm in
Inner Mongolia along with a group of his schoolmates during the revolution. Their
task was to repair the earth with shovels. It was then that he realized how much
he needed the beach and waves. Fortunately, the sweat of two and onehalf years
harsh labor did not completely wash away his sketchy knowledge attained from
the revolutionary school education and selflearning, which enabled him to drop
shovels and pick up pens again on the campus of Dalian Institute of Technology (a
university in his home town) in February 1978, after passing the first nationwide
college exam following the tenyear revolution. He was assigned to the offshore
engineering section in the hydraulic engineering department.
200
25
The characteristics of the coefficient B can be determined by examining the total
form drag resistance acting on a unit volume of granular material of characteristic
size d,. For this case, we have
m n
FdNx X) 2SCdAp I uÂ¡  u/s (3.15)
where Ap is the projected area of individual particles; Cd is the form drag coefficient
of the individual particles; 6 is a correction coefficient for Cd accounting for the
influence of surrounding paxticles; and m is the total number of particles per unit
volume. Since the total projected area should be proportional to (1 n)dxdz/d
we have, after substituting q for
. J ^
Fdnx = 2^D~^d~ I q I j* dxdz (3.16)
the constant of proportionality is absorbed in 8. Comparing Eq.(3.16) with the
second part of Eq.(3.7), we obtain
1 n
nAsn2d,
If we let:
Eq.(3.17) can be expressed as
(3.17)
(3.18)
S, = pba
1 n
nAxn2d,
(3.19)
This equation is very similar to that recommended by Engelund (1953), with
the exception that nA was replaced by n, the volumetric porosity, in his formula.
Again, because bo has directional preference, it should also be a tensor of second
order. Engelund recommended
60 = 1.8 to 3.6 or more
(3.20)
for granular material of sandsized particles.
NORMALIZED PRESSURE DISTRIBUTION NORMALIZED HAVE ENVELOPE
126
Figure 8.10: The envelopes of wave and pressure distribution for T 0.858 sec.
nonbreaking wave case, (a) Wave envelope; (b) Envelope of pressure distribution
139
Due to the nonlinear effects included in the porous flow model, the variation
patterns of all three nondimensional quantities are nonlinear with wave heights, as
can be detected by comparing the three figures. The gradient of spatial wave damp
ing (as manifested by the decay of wave envelopes) inside the berm becomes steeper
with increasing incident wave heights. The gradient as well as the magnitude of
nondimensional pressure on the interface increases with increasing incident wave
height. Thus the pressure is a nonlinear function of wave height. The nondimen
sional normal velocity decreases with increasing wave height. As a consequence of
nonlinear damping, the reflection coefficient also varies with the changing incident
wave height. The fact that the reflection coefficient increases with incident wave
height indicates that the energy dissipation inside the berm is not proportional to
the square of the wave height, but rather at a lower power.
In Figures 9.9 to 9.11, the numerical results for a berm breakwater in prototype
condition, with the scale ratio of 40:1 to the one shown in Fig. 9.5, are presented.
The water depth, and the wave heights and stone size are 40 times of those model
values and the wave period is Tp = \^40Tm = 15.9 seconds. As a result of scale
change, the reflection coefficients only changed slightly from the values at the model
scale. Comparing the curves for the prototype (Fig. 9.9 through Fig. 9.11) with
those of the model scale (Fig. 9.6 through Fig. 9.8), the scale effect on the three
quantities, rj, p and un is apparently also insignificant. This render some confidence
on the model scale results, at least in the range of values tested here.
Since the experimental data shown in T0rum et al. (1988) were mainly for
random wave tests, no direct comparison has been made since the numerical model
is for regular waves.
182
TIME (SEC)
Figure B.22: Case of L = 200 cm, h = DW = 30 cm, h, DS = 15 cm and
<50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation [rj/Hi), (b)
Nondimensional wave heights (H/TT[) and the best fit to the exponential decay
function.
162
TIME (SEC)
Figure B.2: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 0.93 cm. (a) Averaged nondimensional surface elevation {tÂ¡Â¡Hi), (b)
Nondimensional wave heights [Hand the best fit to the exponential decay
function.
B.22
Case of L = 200 cm, h = DW = 30 cm, h, = DS = 15 cm
and 50 = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (77/if j), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 182
B.23 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 15 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/iM, (b) Nondimensional wave heights (if/Hi) and
the best fit to the exponential decay function 183
B.24 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 15 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (ti/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 184
B.25 Case of L = 200 cm, h DW = 30 cm, h, = DS = 10 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights (H/Ex) and
the best fit to the exponential decay function 185
B.26 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 10 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (tj/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 186
B.27 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 10 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights [H/H7) and
the best fit to tne exponential decay function 187
B.28 Case of L = 225 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights (ifÂ¡Hx) and
the best fit to the exponential decay function 188
B.29 Case o L 225 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation [rj/Hx), (b) Nondimensional wave heights (iffHÂ¡) and
the best fit to tne exponential decay function 189
B.30 Case o L 225 cm, h = DW = 20 cm, h, = DS = 20 cm
and 50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 190
B.31 Case of L = 250 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation [rj/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to the exponential decay function 191
xm
88
unknown potential function on the boundary of the fluid domain is then
A = b
(7.48)
with
A = [Ai Bi A2 + rEBijArxW (749)
Jo
and
(750)
In Eq.(7.48), the number of the unknowns N, which is the total number of nodes
on the boundary of the fluid domain, is equal to the number of the equations. It
can be readily solved by a complex equation solver if the linearized coefficient /o
for the resistances in the porous flow model is given. Unfortunately, it is still an
unknown at this point and has to be determined by the linearization process.
7.4 Linearization of the Nonlinear Porous Flow Model
The principle for the linearization is the equivalent energy dissipation by both
linear and nonlinear systems, i.e.
(Ed)i = (Ev)ni (751)
The conventional definition for the energy dissipation within a control volume
of porous medium during the time period T is (Sollitt et al., 1972, Madsen, 1974,
and Sulisz, 1985)
r ft+T 
Ed = Jv J F pqdt dv (7.52)
where F is the dissipative resistant force per unit volume of the porous medium,
which is a function of the spatial coordinates and the time, with F = Re(afQcj) for
a linearized system and F = Re[a{fl + fi l^l)?] for the nonlinear physical system;
q is the complex discharge velocity of the porous flow and p is the density of the
fluid.
9 NUMERICAL MODEL FOR BERM BREAKWATERS 128
9.1 Mathematical Formulations 128
9.1.1 CFSBC for The Free Surface Inside Porous Medium 130
9.1.2 BIEM Formulations 131
9.2 Linearization 133
9.3 Numerical Results 134
10 SUMMARY AND CONCLUSIONS 146
10.1 Summary 146
10.2 Conclusions 148
10.3 Recommendations for Future Studies 151
APPENDICES
A BOUNDARY INTEGRAL FORMULATION FOR ENERGY DISSIPATION
IN POROUS MEDIA 153
B EXPERIMENTAL DATA FOR POROUS SEABEDS 159
BIBLIOGRAPHY 197
BIOGRAPHICAL SKETCH 200
Vll
then
69
Â£;+i Â£y + Ay
The value of is therefore
k. i=
(6.15)
(6.16)
(6.17)
However, the numerical test showed that Eq.(6.l7) could lead to significant error
of  rÂ¡i  due to runoff errors. A more accurate expression for  rji  can be obtained by
computing the distance from P, to the straight line PP.+1 directly from the global
coordinates, i.e.
 A(x, Xj) +Zj Zj 1
Vi + A2
where
^4 = ~ xi
zi+1 zi
The sign of 77, depends on the relative position of P to the boxindary line element
PjPj+v If P% is in the same side of the element with n, rji is positive, otherwise, it
is negative. Mathematically, it can be expressed by the following two equations for
vertical and nonvertical elements:
(6.18)
where
Therefore
sign (rji)
sign (rji)
(xj X{) Az
I [xj Xi) Az
Ax Az,o
j Ax AziQ I
*y+i x:
xi+\ 7^ xj
A x Xj j Xj
Az = Zj+x Zj
a Az, '
Azo = (x, Xj) + Zj Zi
(6.19)
(6.20)
T]i = sign (rji) j rji j
(6.21)
NORMALIZED WAVE ENVELOPE
145
Figure 9.11: Permeable berm breakwater of prototype scale with H = 8.0 m: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity distributions
41
Fi(4(+1),/<)) = O
,M 1 ... 1 C,  4"1)
lo R ,p+VRV^ k/<"> I
& =
(4.56)
(4.57)
(4.58)
where Er and F{ are the real and imaginary parts of F and n indicates the iteration
level.
The criterion of convergence for such case is
, n) 
Si
jfc(n) _
< e and  ^ l<
(4.59)
with e being a prespecified arbitrarily small number. It is again set to be 1.0% in
this model.
4.3 Results
The predicted damping rates kip from the solution of Eqs.(4.15) and (4.47) are
first compared to the laboratory data kim by Savage (1953) for progressive waves
propagating over a sandy seabed. The experiment was conducted in a wave tank of
29.3 meters (96 ft) long, 0.46 meters (1.5 ft) wide and 0.61 (2 ft) meters deep. The
porous seabed was composed of 0.3 meters thick of sand with the medium diameter
of 3.82 mm. The water depths are h = 0.229 m. and h = 0.152 m. The data
for the water depth of h 0.102 m was ignored for the reasons given by Liu and
Dalrymple (1984). The wave conditions and the comparison of the damping rates
are listed in Table 4.1. The parameters used in the solution of Eqs.(4.15) and (4.47)
are: n 0.3, clq = 570, b0 = 2.0 and C = 0.46 and the average wave height H in
the table was calculated according to
*= H
In
Ho_
HLl
where Hq and Hl3 are the wave heights measured at two points of Lg apart with
Lg 18.3 m (60 ft). The values computed by Liu and Dalrymple (1984) are also
permeable submerged breakwaters and berm breakwaters, respectively. Due to the
establishment of a boundary integral expression for wave energy dissipation in a
porous domain and the application of the radiation boundary condition on the
lateral boundary(ies), the numerical models are highly efficient while maintaining
sufficient accuracy. The numerical results show that the wave energy dissipation
within a porous domain has a well defined maximum value at certain permeability
for a specified wave and geometry condition. The nonlinear effects in the porous
flow model are clearly manifested, as all the flow field properties are no longer
linearly proportional to the incident wave heights. The numerical results agreed
reasonably well with the experimental data on the seaward side. On the leeward of
the breakwater, despite the appearance of higher order harmonics, the numerical
model produces acceptable results of energy transmission based on energy balance.
XVII
12
Liu (1973) added an interface laminar boundary layer in his solution for the
same problem. In his paper, the water in the fluid domain was regarded as inviscid
and the porous flow was assumed to follow Darcys law. The imaginary part of k,
after the boundary layer correction, was given by
ki =
2 kT
2 kTh + sinh 2 krh
(oKp/v \
(2.26)
with kr remaining the same as that in Eq.(2.18). It is noted that this is the same ex
pression as that given by Hunt (1959). The correction term of Eq.(2.26) to Eq.(2.19)
is found to be insignificant unless Kp is extremely small.
Dean and Dalrymple (1984) later obtained the expressions for shallow water
waves over sand beds:
_ (1 a2h/g) oKp
2 h { u ]
All the solutions above are for infinitely thick seabeds. Liu (1977) solved the
problem for a stratified porous bed where each layer has a finite thickness and
different permeability. It was found that the wave damping rate was insignificant
to the permeability stratification while the wave induced pressure and its gradients
are affected significantly by stratification. The porous flow model employed in the
analysis also obeys Darcys law only.
Liu and Dalrymple (1984) further modified the solution, for a bed of finite
thickness, by replacing Darcys law with Dagans unsteady porous flow model, which
partially included the effects of unsteady flows. The dispersion relationship from
the homogeneous problemwithout laminar boundary layer correctionswas found
as
( + t)(l ~ tanh/i) + R tanh kh, tanh/:h(l
n a1
~ ctnh kh) = 0
(2.29)
158
not a function of time, and the complex energy dissipation is reduced to
Ed~ j> unp* ds
(A.21)
39
one is O(l). If we assume that  fc, c 1, which is generally true in coastal waters
(see the results for the progressive wave case), then the energy dissipation becomes
Ed = J Jndxdt (442)
Eq.(4.42) will clearly lead to the same relationship as that given by Eq.(4.40)
except that the averaged wave amplitude a in D, in this case, is the spatially
averaged value of ae~k'x over [0,1/].
In the progressive wave case, we assume that the wave period is given and that
there is no time dependent damping, i.e. a is real. Therefore, the first and the last
term on the R.H.S. of Eq.(4.40) are all real numbers. It is then obvious that
(/o)i = 0 (4.43)
and
,,, 1 Cd\kD(kJa)
(/o)r R+ <7 I oh I
(4.44)
where the subscripts i and r are for imaginary and real parts, respectively.
To actually compute /0 requires specification of two fundamental quantities of
n and d, (see Eq.(4.40) and the definitions for R, C and /?). It is often more
convenient to specify the permeability parameter R = oKpÂ¡u (R,. arKPlu for
standing waves), as opposed to specifying the actual granular size, d,. Under these
circumstances, we could replace Cd with the following expression (Ward, 1964):
fa
Cf Cf
d~ VSVl/
with
C7 =
&o(l n)
(4.45)
(4.46)
 n)s
Now Cj is a nondimensional turbulent coefficient. To be consistent with the
suggested values of a0 and b0 given in Eqs.(3.14) and (3.20), Cf should be in the
range of 0.3 ~ 1.1 for a porosity of n = 0.4. Equation (4.40) now becomes:
1 .a Cf \ kD{k, fo) sinh kh [
R yjRov  crfo 
(4.47)
102
interacting with waves.
To demonstrate the influence of incident wave heights, the same breakwater
configuration is computed for the case of T 1.2 seconds and H = 2.0 ~ 8.0 cm.
The transmission and reflection coefficients are plotted against R in Fig. 7.8. The
permeability corresponding to the minimum transmission is shifted to larger values
for higher incident waves, while the value of minimum transmission remains more or
less the same. It is seen that when the permeability (or equivalently the stone size)
is greater than a certain value, say R = 10.0 (log R = 1.0), the wave transmission
decreases with increasing wave height, while if 12 is less than, say 1.0 in this case,
the wave transmission increases with increasing wave height. This means that larger
stones are more efficient for storm protection.
Figure 7.9 is an example of the total wave forces on an impermeable submerged
breakwater sitting on an impervious bottom. The breakwater has the same cross
section dimensions as the one depicted in Fig. 7.4(a). In Fig. 7.7, the horizontal and
the vertical wave forces and the overturning moment are all nondimensional values.
The nondimensional factors are 7Hs for the forces and 7H3L for the moment. As
far as the stability of the breakwater is concerned, the most dangerous point is when
Fz is close to its positive maxima, the overturning moment is close to its counter
clockwise maximum value and Fx is negative. In such a situation, the structure
tends to overturn around the toe at the seaward side. In Fig. 7.9, the vertical wave
force Ft is seen much larger than Fz and the overturning moment is almost in phase
with Fx. This is because of the assumption of impermeable sea bottom. In reality,
a seabed is usually more or less porous, the vertical seepage force on the bottom of
the impermeable breakwater is generally out of phase with Fs in certain degrees,
and the summation of the two will make the total vertical wave force smaller and
more reasonable. To correctly estimate the seepage wave force is another interesting
problem to both coastal and ocean engineers.
REFLECTION COEFF. TRANSMISSION COEFF.
105
1.00
0.95
0.90
0.85
0.80
0.75
0.70
5.0 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0
PERMEABILITY PRRRMETER LOG (R)
T 1 1 1 ii ill t i i I i I
PERMEABILITY PARAMETER LOG (R)
Figure 7.8: Transmission and reflection coefficients vs. R for different wave heights
(a) Transmission coefficient; (b) Reflection coefficient.
LIST OF TABLES
3.1 Illustration of Dominant Force Components Under Coastal Wave
Conditions: [a ~ 0(l)rad/sec,V(pf^) ~ O(101)] 30
4.1 Comparison of The Predictions and The Measurements 43
5.1 Material Information 51
5.2 Test Cases 52
5.3 Measured ar and cr, for h, = 20 cm and L = 200 cm 53
5.4 Comparison of The Resistances, k, = 20 cm, L = 200 cm. ... 56
5.5 Comparison of Measurements and Predictions for h, = 20 cm,
L = 200 cm 57
5.6 Comparison of om and ap for d^o = 1.48 cm, L = 200 cm 59
5.7 Comparison of am and cp for d50 = 2.09 cm, h, = 20 cm 59
5.8 Comparison of om and op for d50 = 0.16 mm, h, = 20 cm,
L = 200 cm 60
5.9 Comparison of am and cp for h, = 63
8.1 Test Results of Nonbreaking Waves 114
8.2 Comparison of Km and Kp 115
8.3 Test Results for Breaking Waves 121
8.4 Normalized Pressure Distribution 125
B.l Test Cases 159
xv
COMPUTED DAMPING RATE COMPUTEO FREQUENCY
62
EXPERIMENTAL FREQUENCY
0.25
0.20
0. 15
0. 10
0.05
0.00
0.00 0.05 0.10 0.15 0.20 0.25
EXPERIMENTAL DAMPING RATE
Figure 5.5: Theoretical values by the model of Liu and Dairymple vs. experimental
data of Table 5.5, Table 5.6 and Table 5.7. (a) Wave frequency ar, (b) Wave damping
rate cr,.
132
+ f(Ki,NcmPNcm+ lijPi + Ki.N.flPN.f) (99)
9 J=ATm + l
N.f1
Writing this equation into a matrix form, it reads
Cll C12 C13
C21 c22 C23
C31 C32 C33 _
Pc
P
PÂ¡>
D11 Dn
D2i D22
fr}
(9.10)
where pc and pnc are the vectors of pressure function and its normal derivative
along the common boundary referred by the subscript c. Similarly, the subscripts
s and b refer to the corresponding vectors on the free surface and the impermeable
boundaries, respectively.
After moving p, on the right hand side to the left Eq.(9.10) can be further
simplified to
' r* r* 1 ~ / tv \
Pnc (9.11)
Cu C12
C21 C22
Pc
Pci
Du
D2i
with
Cu Cu
C12
C21
= [ C12 + D12 CIS ]
C22
C21 j
C31 J
C22 + D22 C23
C32 C33
and
={p;) <9i2>
Splitting Eq.(9.11) into two matrix equations and eliminating one of the vectors
pc or p,j at a time, we obtain the following equations,
Pnc = Ei pc (9.13)
P,i = E2pe (9.14)
with
Ei = (DiiCi2C221D2i)1(CiiCa2C221C2i) (9.15)
E2 = (C22D2iDri1Ci2)1(C2iD2iDr11C1i) (9.16)
193
Figure B.33: Case of L = 250 cm, h = DW = 20 cm, ht DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (t]/Hi), (b)
Nondimensional wave heights [H/li\) and the best fit to the exponential decay
function.
4
6. Verify the numerical model with laboratory experiments of a submerged rubble
mound breakwater,
7. Modify the numerical model to represent porous berm breakwaters.
65
jjyv VU + VV2U)dA = jcVVU fids
Subtracting (6.3) from (6.2), we have
[ {UV2V W2U)dA = {UVV VVU) fids
Jd jc
Introducing the expressions
dV dU
Wn = ^ VZ7 n =
on on
and assuming that both U and V satisfy Lapalce equation, i.e.
V2U = V2F = 0
(6.3)
(6.4)
(6.5)
(6.6)
Equation (6.4) becomes
L^v^ds <67>
To apply this equation, we choose U as the velocity potential $ and V as a free
space Greens function, G. Both of them satisfy Lapalce equation. Equation (6.7)
can then be rewritten, in terms of $ and G, as
/C1W)  g(p, I** = 0 (6s)
where P is a point in the domain DnC and Q is a point on C.
One of the free space Greens functions for two dimensional problems is
G(P,
where r is the distance between point P and point Q and it can be expressed by
r = \]\xP xQy + (zP zQ)2
on the x z plane.
Substituting Eq.(6.9) into Eq.(6.8), it becomes
(6.10)
29
nature and the resistance arises due to the rate of change of velocity at a specific
location. The ratio of R, and R/ is the Strouhal number, which clearly identifies
the different origins of the two inertial forces.
In the range of common engineering applications, the magnitudes of various
coefficients can be estimated as follows:
n 0.3 ~ 0.6
Ca ~ 0(1)
do ~ O(103)
bo ~ 0(1)
Therefore, Eqs.(3.33) through (3.35) reduce to
 10"2R,
n
fi
k
ft
k
fi
io_2r f
R
(3.38)
(3.39)
(3.40)
When Eqs.(3.38) to (3.40) are plotted on a R/ and R, plane as given in Fig. 3.2,
we identify seven regions where the three resistance forces are of varying degrees of
importance. There are three regions where only one resistance force dominates, that
is, the dominant force is at least one order of magnitude larger than the other two.
There is one region where the three forces are of equal importance. Then, there are
three intermediate regions where two out of three forces could be important.
Since both Reynolds numbers are flowrelated parameters, accurate position of
a situation within the graph cannot be determined a priori. However, a general
guideline can be provided with the aid of the graph or with Eqs.(3.38) through
(3.40). We give here an example of practiced interest. In coastal waters, we com
monly encounter wind waves with frequency in the order of 0(l)(rad/sec) and the
non dimensional bottom pressure gradient defined as V (p/7) in the order of 0(1O1).
44
3. The wave attenuation, and hence the wave energy dissipation, shows a peak.
This peak occurs when the magnitude of the dissipative force (velocity related)
equals to that of the inertial force (acceleration related). Depending upon the water
depth, this attenuation could be quite pronounced under optimum R values. For
instance, for the case h = 4 m, Fig. 4.2 shows the wave height is reduced to about
74% of its original value (or about 46% energy dissipation) over approximately 2
wave lengths.
4. The locations of the peak damping are quite different from that of Liu
and Dalrymple (illustrated here for the case of h= 4 m); they occur at a higher
permeability. The magnitude of the peak damping is generally smaller than that
of Liu and Dalrymples solution. The values of kr from the two solutions are also
different.
In Fig. 4.3 the maximum damping rate and the corresponding permeability
parameter R are plotted as functions of nondimensional water depth. The curves
from Liu and Dalrymple (1984) are also plotted for comparison. The trends of
(&)maxs are similar but the corresponding R behaves quite differently from the two
solutions.
The case of a standing wave system is also illustrated here in Fig. 4.4. The
behavior is very similar to that of the progressive waves. The grain size in the figure
is calculated according to aT for the case of h = 4.0 m. Finally, the solutions of the
dispersion equation for standing waves based on Darcys model (DARCY: 0 = 0, Cj
= 0), Dagans model (DAGAN: 0 = CÂ¡ = 0), DupuitForchheimers model (D F:
0 = 0 and Cf follows Eq.(4.46) and SollittCrosss model (S C: 0 follows Eq.(3.30)
with Cj following Eq.(4.46) and n = 0.4, Oq = 570, 60 = 3.0 and Ca = 0.46) for
standing waves are compared in Fig. 4.5. Under the same wave conditions, the
differences among the various solutions are seen to be very pronounced, for the
permeability range displayed here. When the permeability parameter becomes less
89
To carry out the integration in Eq.(7.52), a new grid system over the entire
porous domain has to be introduced, which is obviously a set back for BIEM which
deals with only the boundary values. However, by using an alternate definition for
the energy dissipation and applying Greens formula, the averaged energy dissipa
tion can be expressed by a boundary integral (Appendix A). If we further define a
complex energy dissipation function with the real part being the physical quantity,
it has the form
Ed =
unp* ds
(7.53)
where S is the closed boundary of the porous domain, p* is the conjugate of the
complex pressure function and un is the complex normal discharge velocity on the
boundary. Introducing the noflux boundary condition on the impervious bottom,
Eq.(7.53) becomes
Ed
unp* ds
(7.54)
for both linear and nonlinear systems. Where C is the common boundary. The
sign is used here because Ed is considered as a positive value.
The physical explanation of the expression is that the energy dissipation inside
the porous domain in one wave period T is equal to the net energy flux into the
domain in the same time period. It is equivalent to the conventional volumetric
integral form defined in Eq.(7.52). By expressing the energy dissipation in the
boundary integral form, the advantage of BIEM can be well explored.
The energy dissipation by the two systems can be obtained by substituting the
expressions for the normal velocities from the linearized and the nonlinear porous
flow models, respectively.
For the linearized model,
1 dp
pa fo dn
Mi =
(7.55)
119
1 O
0.8
...
. i V ,l
0.6
0.4
T = 1.020 SEC.
T = 1.120 SEC.
0.2
T = 1.265 SEC.
o A
EXPERIMENT DATA
0.0
0
1
.000
i i i i i i
0.00! 0.002 0.003
i 1 i 1
0.003 0.004 0.
HI /GT**2
Figure 8.6: The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp and
Ktm\ (b) Krp 3nd Krm
80
condition assumes that the waves at these two boundaries are purely progressive,
and that the decaying standing waves generated by the object inside the domain are
negligible at these boundaries. Comparing with the matching boundary condition
with the wave maker theory employed by Sulisz (1985) and some other authors,
the radiation boundary condition offers significant simplicity in programming and
provides sufficient accuracy with much less CPU time. In applying this boundary
condition, the two lateral boundaries have to be placed far enough from the struc
ture. Numerical tests show that a distance of about two wave lengths from the toe
of the structure is more than adequate for this purpose.
At these two lateral boundaries, the potential functions for the transmitted and
reflected waves are assumed to have the forms
&(*,*) = e,kWR(z)
(7.18)
*(*,*) = e'i'')j(2)
(7.19)
where T(z) and R(z) are two unknown functions of z\ k and k' are the wave numbers
at the two boundaries respectively, and the subscripts t and r here refer to the
transmitted and reflected waves, respectively. On the lateral boundary of upwave
side, the potential function for the incident wave is known:
4>i{x,z) =
(7.20)
with
Therefore,
, Hgcoshk(z + h)
2 a cosh kh
(7.21)
] + 4>r at 2= l
4> = t at x = l'
(7.22)
(7.23)
where l and l' are, respectively, the distances from the origin of the coordinates to
the offshore and onshore lateral boundaries and if> is the unknown potential function.
WAVE DAMPING RATE
49
PARTICLE SIZE (CM)
0.23 2.28 22.69 225.44
PERMEABILITT PARAMETER LOG (R)
(a)
0.23 2.28 22.69 225.44
(b)
Figure 4.5: Solutions based on four porous flow models, (a) Nondimensional wave
frequency or/{LÂ¡g)h, (b) Nondimensional wave damping rate
30
R/
Under such conditions, the importance of the three resistance components for bot
tom materials of various sizes can be assessed. Table 3.1 illustrates the results.
Table 3.1: Illustration of Dominant Force Components Under Coastal Wave
Conditions: [a ~ 0(l)rad/sec, V(p/7) ~ O(l01)]
Description
Size
Range
Disch. Vel.
(m/sec)
R/
R,
Dominant
Force
Coarse sand
or finer
< 2 mm.
< 0(KT3)
< O(l)
laminar
Pebble,
or small
gravel
1 cm.
O(102)
o(io2)
O(l02)
laminar
turbulence
inertia
Large gravel
crusted stone
10 cm.
O(K)1)
O(104)
O(104)
turbulence
inertia
Boulder
crusted stone
0.3 1.0 m
0(10)
O(10)
0(10)
turbulence
inertia
Artificial
blocks,
large rocks
> 1.0 m
> 0(10)
> O(l06)
> o(io6)
turbulence
inertia
UFL/COEL/TR083
WATER WAVE INTERACTION WITH POROUS
STRUCTURES OF IRREGULAR CROSS SECTIONS
by
Zhihao Gu
Dissertation
1990
xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0007547400001datestamp 20090220setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Water wave interaction with porous structures of irregular cross sectionsUFLCOELTR dc:creator Gu, Zhihao, 1957University of Florida  Coastal and Oceanographic Engineering Deptdc:subject Coastal and Oceanographic Engineering thesis Ph. DDissertations, Academic  Coastal and Oceanographic Engineering  UFdc:description b Statement of Responsibility by Zhihao GuThesis Thesis (Ph. D.)University of Florida, 1990.Bibliography Includes bibliographical references (leaves 197199).Typescript.Vita.Funding This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.dc:date 1990dc:type Bookdc:format xvii, 201 leaves : ill. ; 29 cm.dc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00075474&v=0000124160971 (oclc)dc:source University of Floridadc:language Englishdc:rights All rights reserved, Board of Trustees of the University of Florida
169
Figure B.9: Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
dso = DD = 0.93 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights {H/H7) and the best fit to the exponential decay
function.
The authors gratitude towards his family is beyond words. The author would
like to thank his best friend and wife, Liqiu (also a Ph.D. student at the time), for
her support, both moral and physical, patience and useful technical discussions. She
was always the one to count on for assistance on weekend laboratory work, graphics
work and more often on the housework beyond her share. Without her support, the
road leading to this goal would have been much more tortuous. Finally, this work
is dedicated to the authors wonderful mother and father. Their affection and early
family education axe reflected in between the lines of this work and the authors
daily conduct. With all these efforts, thank god its Phinally Done!
iv
47
than 102, all solutions converge to a single curving following Darcys law. For very
highly permeable seabeds, the damping rate based on D F model is very high and
tends to increase with the permeability as oppose to approaching to zero according
to common sense. Also in the high permeability region, the wave frequency based
on Darcys model is much larger than the correct value (determined by Eq.(4.17)).
The reason for such large error is that the force balance of the pore fluid in highly
permeable media is now mainly between pressure gradient and the inertia rather
than the velocity related frictions.
7
where p is the density of the fluid and CÂ¡ is a nondimensional coefficient.
The two coefficients a and b were further expressed in terms of particle diameter
by Engelund (1953) and Bear et al. (1968) in the forms
(1 n)sv
a Cq
n2d2
(2.4)
b = b0
1 n
n2d,
(2.5)
where
porous media.
The above two models, Eq.(2.l) and (2.2) are both for steady porous flows,
although some applications have been made for certain unsteady flows, such as wave
induced porous flows in seabeds (Putnam, 1949; Liu, 1973; etc.). These applications
were primarily on low permeability media like fine sand. In general, for unsteady
flows, the inertial force arises due to the acceleration of the pore fluid and such a
force can be quite large and can even be dominant when the permeability is high.
To model the unsteady porous flow, a number of models have been suggested.
The first model found in the literature containing the inertial force term was
the one by Reid and Kajiura (1957). It is the direct extension of Darcys model:
T7 Pd$
P Kp,+ nSt
(2.6)
The inertial term in this model is in fact the local acceleration term in NavierStokes
equation with the fluid velocity expressed as a specific discharge velocity q. It is
clear that this term contains only the inertia of the fluid but not the inertia induced
by the fluidsolid interaction, or the added mass effect.
The extension of DupuitForchheimer model leads to PolubarinovaKochinas
model (Scheidegger, 1960; McCorquodale, 1972). It has the form
 Vp = aq + bq + 
(2.7)
The author is deeply indebted to his former advisors and supervisors in China,
Profs. WenFa Lu, Zhizhuang Xieng and Weicheng Wang. Without their generous
support and recommendation, it would have been impossible for the author to study
abroad in the first place. The continuous support and understanding by them and
all other faculty and staff members in the authors former working unit are highly
appreciated.
Very special thanks are given to Prof. Alf Tprum, Head of Research in Nor
wegian Hydrotechnical Laboratory, Trondheim, Norway, and Dr. Hans Dette in
Leichtweifi Institut fur Wasserbau, Technische Universitat Braunschweig, West Ger
many, for providing computer, office and accommodation facilities while the author
was visiting the two institutions in late 1988 with his advisor. The initiation and the
provision of technical information by Prof. Tprum for the topic of berm breakwater
which later became a chapter in this dissertation are gratefully acknowledged.
Thanks are given to Mr. Sidney Schofield, Mr. Jim Joiner and other staff
members in the Coastal Laboratory for their help during the experimental phases
of the study. The assistance provided by Mr. Subama Malakar in the use of
computer facilities is also appreciated. Thanks are extended to Helen Twedell for
the efficiency and courtesy in running the archives and for the parties, cookies and
cakes; and to Becky Hudson and all other secretarial personnel in the department
for the great job they have done related to this work.
The support of many friends and fellow students is warmly appreciated. Special
thanks go to Mrs. Jean Wang for her moral support, constant encouragement and
hospitality. Deep appreciation is given to fellow students Rajesh Srinivas and Paul
Work for proofreading the manuscripts and to Byunggi Hwang for assisting with
rock sorting during the experiments. The friendship shared with Yixin Yan, Steve
Peene, Yuming Liu and Richard McMillen and other friends, made the authors stay
in Gainesville a delightful period of time.
in
130
9.1.1 CFSBC for The Free Surface Inside Porous Medium
On the porous free surface boundary, the dynamic boundary condition cam
simply be taken as the pressure at z 0:
P P9V.
or
and the linearized kinematic boundary condition is
drÂ¡,
dt
w
(9.1)
(9.2)
with r]t denoting the surface elevation inside the porous domain.
According to the linearized porous flow model defined in Eq.(4.2), the vertical
velocity w in Eq.(9.2) can be expressed in terms of the pore pressure P as
1 dP
pufo dz
(9.3)
therefore
drÂ¡Â¡ 1 dP
dt pofo dz
Differentiating Eq.(9.1) with respect to t and substituting
following equation can be obtained,
dr?,
at
(9.4)
into Eq.(9.4), the
dP ofpdP
dz g dt
If
P = ptiot
(9.5)
(9.6)
we have
dp .a2f0
 p
dz g
(9.7)
It is clear that when the porous medium becomes completely permeable, fo * t
and p = ipcr<Â£, and Eq.(9.7) is the CFSBC in a fluid domain.
NORMALIZED WAVE ENVELOPE
142
Figure 9.8: Permeable berm breakwater of model scale with H 20.0 cm: (a) Wave
envelope; (b) Envelopes of pressure and normal velocity distribution
170
o
<\J
DW= 25. 0 CM
I
o
o
1H
* DS= 20.0 CM
j 00= 1.20 CM
E Hi
o
00
o
1
1 i
>
cr
* 1
3 0.60 .
4
o
LU
rvi
ft....
J 0.40 .
cr
*it
NORM
o
ro
o
i
* 4 f
o
o
o
j
0.0 M.O 8.0 12.0 16.0 20.0
TIME (SEC)
Figure B.10: Case of L 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 1.20 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
71
Zb
 I1 f/12.
'j ^j>j
(6.28)
zb
= ^5 + 6+iiS
(6.29)
Kh
 j? f.j
(6.30)
1 [*+l ju_ [!>+'
6+164 r{dn* 2A,4 r?
>2L
2Ay 1 r?? + e? J
(6.31)
1 [ti+1 1 dr, T7, f/+ a:
ti+itjhi ri dn 2Ay 4 r?
T"[tan 1(~~) tan
Ay *7 *7
(6.32)
f. j I**' Inuidi
6'+i ~ 6 4
4Ay
{tf + ej+1)Mvi + th) 1]
iv? + $){Hvl + 62) ~ i]}
(6.33)
~ y J*1+llnridt
6+i ~ 6 4
2Ay
{6+i M*?, + 6?+i) 6 in(*7i + 6) 2(6+i 6)
+2*7i[tan 1{^)tan 1(^)]}
Vi Vi
(6.34)
108
Cart
T 1
0
Gauge 1 Wave Direction 
Have
Gauge 2
if
0 0 0 0 0 0 0 0
X 0.0.0.0.0.0.0.0.0. X
k
Figure 8.1: Experiment layout
Figure 8.1 shows the experiment layout.
The waves between the model and the wave maker were partially standing waves
due to the presence of the breakwater in the middle of the tank. The incident
and the reflected wave heights were determined from the maxima and the minima
of the envelope of the partially standing waves. The envelope was measured by
moving the rail cart slowly with a constant speed along the wave tank, from its
original position towards the wave maker during each data acquisition period. The
wave height modulation over the distance traveled by the cart was then recorded.
Similarly, the wave envelope over the submerged model was measured by moving
the cart from the toe on one side to the toe of the other side, during each sampling
time.
The pressure distribution along the surface of the breakwater was measured
by two underwater pressure transducers. The distance between the two pressure
ACKNOWLEDGEMENTS
It is almost impossible to fully express the authors appreciation in words to
everyone who has contributed to the completion of this dissertation.
First of all, the author wishes to express his deepest appreciation to his advisor
and the chairman of his advisory committee, Dr. Hsiang Wang, for his advice,
friendship, encouragement and financial support which enabled the author to come
to the United States to pursue his higher education. The four years of graduate work
with Dr. Wang have been a challenging and enjoyable experience in the authors
life.
The author would also like to thank Dr. Robert G. Dean, Dr. D. Max Sheppard
and Dr. Ulrich H. Kurzweg for serving as the members of his doctoral advisory
committee; Dr. Daniel M. Hanes for revising the dissertation and attending the final
exam. Thanks are also due to all other teaching faculty members in the department
during the authors graduate study: Dr. Michel K. Ochi, Dr. Ashish J. Mehta,
Dr. Peter Y. Sheng and Dr. James T. Kirby (now at University of Delaware), for
their teaching efforts, time and being role models to the author. The author has
benefitted substantially from their invaluable experience, knowledge and continuous
inspiration throughout the four years study. The author is especially grateful for
the valuable discussions with Dr. Dean and Dr. Kirby on numerous topics directly
or indirectly related to this work. The continuous financial support throughout the
Ph.D. program by the Coastal and Oceanography Engineering Department is also
greatly appreciated.
ii
CHAPTER 4
GRAVITY WAVES OVER FINITE POROUS SEA BOTTOMS
We consider here the case of a small amplitude wave in a fluid of mean depth
h above a porous medium of finite thickness h,. The bottom beneath the porous
medium is impervious and rigid. The subscript s will be used here to denote vari
ables in the porous bed. The basic approach is to establish governing equations
for different zones separately and to then obtain compatible solutions by applying
proper matching boundary conditions. In the pure fluid zone, it is common to
assume the motion essentially irrotational except near the interface. Most of the
investigators neglected the influence of the boundary layer with the exception of Liu
(1973) and Liu and Dalrymple (1984) who included in their solutions two laminar
boundary layers at the mud line. Since the damping is largely due to the energy
losses in the porous medium rather than the boundary layer losses (Liu and Dal
rymple, 1984), the boundary layer effect will be ignored in this study to simplify
the mathematics.
4.1 Boundary Value Problem
The governing equation for the velocity potential function in the fluid domain
is
V2$ = 0 h < z < 0 (4.1)
In the porous medium domain, the linearization of the nonlinear pore pressure
equation, Eq.(3.32), yields
 VP, = pafoq (h + h,) < z < h (4.2)
where /0 is the linearized resistance coefficient.
31
92
with
gk tanh kh = gkm tan kmh = a1 (7.69)
gk' tanh k'h' = gk'm tan k'mh' = a2 (7.70)
where AÂ¡ and i?, axe complex coefficients. After is obtained, all As and i?,s can
readily be determined by applying the orthogonality of the hyperbolic and cosine
functions. The only coefficients which affect the transmission and reflection are
Ao and jE?0 since all the other terms are standing wave modes. The reflection and
transmission coefficients are
o
11
(7.71)
Kt = \B0\
(7.72)
Kr =
Kt =
(7.73)
(7.74)
Letting x = l in Eq.(7.67) and x = V in Eq.(7.68), integrating 4>{l,z) coshfc(z+
h) and '(l',z) cosh k'(z + h') over [/i,0] and [h',0] respectively, the reflection and
transmission coefficients are found to be
rO
cos kh (l,z) cosh k[z + h) dz
11
f0 1 I
/ cosh2 k(z + h) dz
J h
cos k'h' J
[ cosh2 k'(z + h') dz
Jh'
where <Â£(l,z) and 4>'(l',z) aie known in a discretized basis from the solution of
Eq.(7.48).
7.6 Total Wave Forces on an Impervious Structure
When a submerged structure is impermeable, such as a breakwater made of
concrete instead of quarry stones, the total wave forces and the overturning moment
will be one of the major factors to be concerned in design.
Such an impermeable structure is merely a special case of d, = 0 in the model
presented before. The boundary condition on the surface of the structure is now
38
by the two systems according to Eq.(4.22) and assuming that o and k axe the same
for both systems, we have
fo fo
(4.36)
Since this equation has to be satisfied at any time instant, we obtain the follow
ing equations:
D2(fo) = D2(fx + f2\wo\)
fo fl + fl I ^0 
(4.37)
1 D(fo) i2
fo
 P(/l + fj >0 I) l2
fl + fl I Wo I
(4.38)
Therefore,
fo fi + fi  Wo 
(4.39)
Substituting the expressions for fi, fi and too in the above equation, it becomes
Cd  kD(f0) sinh khs 
a  <7/01
fo=RiP +
(4.40)
It is clear, from the definitions of the parameters, that all the terms in Eq.(4.40)
Eire complex quantities in the standing wave case.
For progressive waves, the pore pressure function P, is given by Eq.(4.10) and
the contour S for the integration is the same as that for standing waves. The energy
dissipation is
By substituting the expression of Jn derived from Eq.(4.10) into the above equa
tion, the summation of the first two integrals is found to be 0( i,1) while the last
23
(B). Inertia Force
The inertial force can be treated as an added mass effect and expressed in the
following form:
Fix = CX(q, nutx)dxnAx dz (3.8)
The coefficient Cx will be determined later.
(C). Body Force
In the pore fluid, there is no horizontal body force, and the vertical body force
is balanced by the static pressure gradient and, thus, vanishes. In the solid skeleton,
the vertical body force is the net weight. If the solid skeleton is subjected to an
unsteady vertical motion, this term should be included; otherwise, it can be ignored.
Substitution of Eqs.(3.5) through (3.8) into Eq.(3.3) results in the following
differential equation:
Bx  q nu,  (qz nu$x)
( + C'zXq* niits) (3.9)
n
This is the basic equation of pore fluid motion. In a general case, it is coupled
with Eq.(3.2) and must be solved simultaneously. Only a special case with no
movement of the solid skeleton will be analyzed here. For such a case, Eq.(3.9)
reduces to
^ = A.q, + B,qq* + (Â£ + CI)q, (3.10)
The force b'alance in the zdirection leads to
+ Bz i 9 I 9* + (Â£ + C*)4* 311)
3.2 Force Coefficients and Simplifying Assumptions
dP
dx
= Az(qxnu,x) +
+
The evaluation of force coefficients involves a great deal of empiricism. However,
successful engineering application relies heavily upon successful estimation of these
coefficients. The part of the flow resistance which is linear in q clearly will lead to
32
<1
>
k Z
Lv
r / V o
i
i
i
h
'44t''84w'44''i"i'S'4'4'4w4'4'i
Â¡Â¡Â¡Porous seabed*Â¡*Â¡*Â¡*Â¡*Â¡*Â¡Â¡*Â¡Â¡*Â¡*Â¡*Â¡*Â¡*
oVAVoVoVoVoVoVoVoVoVoVo0
iWÂ¡iKW
, 0 0 0 0 0 Â£
;h,
5 S .0,0.0 00,
0 0 0 0 0 c
VoVoVoVc
jVoVoVoV
Figure 4.1: Definition Sketch
If the porous medium is rigid, thus, incompressible, the continuity equation for
the discharge velocity, V q = 0, leads to
V2P, = 0
(h + h,) < h
(4.3)
The boundary conditions of entire system are
T](x,t) = ^ = aei^x~(,t'> at 2 = 0
g at
(4.4)
a2$
dt2 + 9 dz~
at 2 = 0
(4.5)
a*
dz
3$
P dt ~F
1 dP,
pafo dz
at 2 = h
at 2 = h
(4.6)
(4.7)
133
Although Eqs.(9.1l), (9.13) and (9.15) look exactly like their counterparts in
Chapter 7, the contents are different. In this case they are all complex matrices
instead of real ones because the CSFBC on the porous free surface inside the berm
contains the complex number /0.
The matching equation in matrix form can be obtained by substituting the
continuities in pressure and the mass flux along the interface, expressed by Eq.(7.45)
and (7.46), into Eq.(9.13):
*nC = ~jVl4>c (9.17)
The matching of the two regions is carried out by introducing Eq.(9.l7) into the
matrix equation for the fluid domain, and the resulting matrix equation is
A = b (9.18)
with
A = [Aj. Bi A2 + yEx B2]atxW
Jo
(9.19)
and
(9.20)
where the notations are the same as those defined in Chapter 7.
After Eq.(9.18) is solved, the surface fluctuation inside the berm can be obtained
from Eq.(9.1) and (9.14).
9.2 Linearization
Similar to the case for submerged breakwaters, Eq.(9.18) is solvable only when
the linearized resistance coefficient /o is known. The technique of finding such
a coefficient is again iteration. However, due to the presence of the porous free
surface, the procedure of iteration has to be altered. For submerged breakwaters,
the starting point for the iteration loop is the matching, right before the formation
of the final matrix equation. But for berm breakwaters, the starting point of the
iteration loop goes back up to the formation of the matrices for the porous domain.
73
= In Ay 1
Summarizing all the Jys according to Eq.(6.12) for node points t = 1
assuming that
= $+
O'
at all the nodes, a system of linear equations with unknowns of $ and
obtained:
dn
where
N N
y=i y=i
= if?* + *ti<*
= Bfji + BliStjOi
j = 2,3, ...,N
with
. 1 if t = j
,J \ 0 if 5 j
Am
L
+ Kl
Ki+Kij
j = 2,3,
Equation (6.45) can also be expressed by a matrix equation
(6.42)
N, and
(6.43)
(6.44)
can be
(6.45)
(6.46)
(6.47)
R 14>n
(6.48)
157
Here (Jn)r, a real quantity, is the energy flux normal to the boundary S.
The physical explanation of the above equation is that the energy dissipation
in the time period T is equal to the net energy flux across the boundary S into the
enclosed area in the same period.
Up to this point, we have been dealing with real quantities only. In many cases,
it is often found convenient to use complex variables with the real parts of them
being the corresponding physical quantities.
Let
Pr = Re[P{x,z,t)] = Re[p{x,z)e ,at]
q n = Re[un(x,z)e~,at]
(A.14)
(A.1S)
where both p and u are complex variables, it is not difficult to
following identity is true,
prove that the
PTq = Re\^(unpe~ilot + unp*)]
(A.16)
with p* being the conjugate of p.
Therefore, a complex energy flux function can be defined as
= ^(nP e_2,vt + u p*)
(A.17)
and the corresponding complex energy dissipation function is then
(A.18)
such that
IT.), = Rt{T.)
tu Rc[Ed)
(A.19)
(A.20)
If the time period T is chosen as the wave period, the integration of the first
term of the integrand in Eq.(A.18) with respect to time will vanish when p u is
57
Table 5.5: Comparison of Measurements and Predictions for h, = 20 cm, L = 200
cm.
50 (Cm)
R
Orm{is *)
<7rp(s *)
Ar%
<7.m(s l)
A i%
h 30.0 cm
0.72
0.1791
4.9054
4.7880
2.39
0.0570
0.0541
5.07
0.93
0.3021
4.9402
4.8001
2.84
0.0562
0.0628
11.60
1.20
0.5110
4.9370
4.8187
2.40
0.0694
0.0706
1.66
1.48
0.8448
4.9620
4.8419
2.42
0.0757
0.0751
0.76
2.09
1.8755
4.9673
4.8685
1.99
0.0824
0.0734
11.00
2.84
3.7429
5.0111
4.8891
2.43
0.0808
0.0661
18.26
3.74
6.9543
5.0341
4.9001
2.66
0.0543
0.0613
12.85
h 25.0 cm
0.72
0.1687
4.6218
4.5264
2.06
0.0678
0.0709
4.49
0.93
0.2866
4.6856
4.5445
3.01
0.0781
0.0840
7.49
1.20
0.4843
4.6793
4.5666
2.41
0.0858
0.0935
8.97
1.48
0.8095
4.7352
4.5963
2.93
0.1030
0.1000
2.94
2.09
1.7819
4.7422
4.6389
2.18
0.0975
0.0956
1.94
2.84
3.5772
4.7893
4.6566
2.77
0.0816
0.0912
11.76
3.74
6.6123
4.7865
4.6795
2.24
0.0693
0.0781
12.67
h = 20.0 cm
0.72
0.1559
4.2716
4.1799
2.15
0.1047
0.0907
13.34
0.93
0.2656
4.3426
4.2061
3.14
0.1192
0.1109
6.96
1.20
0.4509
4.3565
4.2318
2.86
0.1179
0.1228
4.16
1.48
0.7496
4.4027
4.2857
2.66
0.1337
0.1319
1.32
2.09
1.6656
4.4325
4.3217
2.50
0.1396
0.1308
6.33
2.84
3.3408
4.4728
4.3560
2.61
0.1311
0.1200
8.42
3.74
6.1676
4.4646
4.3805
1.88
0.1173
0.1076
8.24
is uniformly very small for or (wave frequency). For cq (wave damping), the range
of error is larger, with the maximum being 18.26% of the measured value.
The agreement between the predictions and the measurements shown in the
above table is demonstrated by Fig. 5.3 where the values of or and a, in Table 7 for
the case of h = 25 cm are plotted against the permeability parameter.
109
gauges was about 12 cm and the locations of the gauges can be found in Fig. 8.1. The
pore pressures at 14 points along the surface of the physical model were measured
subsequently with two points at a time. The wave conditions were kept as identical
as possible for different runs and the pressure readings were later normalized by the
incident wave height for each test.
8.2 Wave Transmission and Reflection
The wave transmission and reflection coefficients are two very important pa
rameters in assessing the performance of a breakwater. In the experiment, the
measurements of these two coefficients were carried out for 9 wave periods ranging
from T 0.642 seconds to T = 1.778 seconds with several different wave heights
for each wave period. Both nonbreaking and breaking waves were tested. Here
breaking waves refer to those incident waves break over the submerged breakwater
model.
The transmitted waves were directly measured by the fixed wave gauge be
hind the model whereas the reflected waves were indirectly measured by the gauge
mounted on the rail cart. Figure 8.2(a) shows a typical data series recorded by
the moving gauge. The incident and reflected wave heights were separated by the
following relationship for partially standing waves,
S{ = TÂ¡max TJmin (8.l)
Hr fjmax T] min (8.2)
In most cases, at least two pairs of quasinodes and quasiantinodes were recorded,
and the averaged values of the two were used for and rj^ to reduce possible
errors.
Figure 8.2(b) shows the record of the corresponding transmitted waves. It is seen
from the figure that the transmitted waves are not monochromatic for a sinusoidal
incident wave. They are, instead, the superposition of waves of fundamental
176
TIME (SEC)
Figure B.16: Case of L = 200 cm, h DW 20 cm, h, DS = 20 cm and
d50 = DD = 0.93 cm. (a) Averaged nondimensional surface elevation (tj/H\), (b)
Nondimensional wave heights (H/H7) and the best fit to the exponential decay
function.
94
The overturning moment about the center of a structure at the mudline is
N"~1 i 1
M0 = //{cOs(y,z)[2J + Aycos(y,x)]cOs(J,z)[x;AycOs(y,2)]} (7.82)
jN,d
M0 =  M0  e><+Â£> (7.83)
em = tan"1^ (7.84)
Mr
7.7 General Description of The Computer Program
The computer program is written in FORTRAN 77 computer language and com
pleted on a VAX 8352. It has 12 subroutines and a main assembling program. The
functions of the subroutines ranging from element generation to the computation
of Kr and Kt.
The model consists of four major parts:
1. Element generation
This part enables users to input the geometry data of a computation bound
ary by breaking it into straight lines or circular arcs, if any, and specify only the
positional information of each segment along with the number of elements to be dis
cretized on this segment. The preparation for input data is therefore very simple.
For example, to compute a trapezoidal submerged breakwater, the input data for
the geometry is less than ten lines with no more than five values in each line.
2. Matrix formation
This part computes the coefficients H and K for both fluid and porous medium
domains according to the formulae given in Chapter 6, with element information
generated in the first part. Since there is no boundary condition involved, it is a
generic subroutine and can be used as the basic subroutine in any BIEM program
with linear elements.
3. Matrix assembling and boundary matching
In this part, the boundary conditions are introduced for both domains. In the
fluid domain, the matrices A,, B, and b are formed according to Eqs.(7.35) through
CHAPTER 3
POROUS FLOW MODEL
Examining the existing porous flow models listed in the literature review, it
is noted that most of them contain one, two or all of the following components:
the linear velocity term, the quadratic velocity term and the term proportional to
the acceleration of the fluid. For the first component all the models give the same
definition while for the rest two terms, different models have different forms. To be
able to model the porous media with confidence, it is necessary to formulate the
porous flow from fundamental principles.
3.1 The Equation of Motion
The equation of motion for a solid body placed in an unsteady flow field can be
expressed in the following general terms:
Fp + Fd + Fj f Ft, Ff = ma (31)
where m = mass of the solid; a acceleration of the body; Fp = pressure force;
Fd = velocity related force also known as the drag force; F/ = accelerationrelated
force also known as inertial force; Fb = body force; and Fj = frictional force due to
surrounding solids. We now apply this equation to a control volume of a mixture
of fluid and solids, as shown in Fig. 3.1, and restrict our discussion to the two
dimensional case. The xaxis coincides with the horizontal direction and the zaxis
points vertically upward. The spatially averaged equation of motion for the solid
skeleton in the xdirection becomes
dP _____
 dx(l nAx)dz + Fdx + F/s + Fbtx FJz = p,( 1 n) ,x dxdz (3.2)
ox
where = pressure gradient in the xdirection; = averaged area porosity
20
B.32 Case of L 250 cm, h DW = 25 cm, h, = DS = 20 cm
and d50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (77/ffi), (b) Nondimensional wave heights (H/Hi) and
the best fit to tne exponential decay function 192
B.33 Case of L = 250 cm, h = DW = 20 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (r?/iM, (b) Nondimensional wave heights (H/Hi) and
the best fit to tne exponential decay function 193
B.34 Case of L = 275 cm, h = DW = 30 cm, h, = DS = 20 cm
and Â£50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (r\/HiJ, (b) Nondimensional wave heights (if/i7) and
the best fit to tne exponential decay function 194
B.35 Case of L 275 cm, h = DW = 25 cm, h, = DS = 20 cm
and Â£50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (ri/Hi), (b) Nondimensional wave heights (if/Hi) and
the best fit to the exponential decay function 195
B.36 Case of L = 275 cm, h = DW = 20 cm, hs = DS = 20 cm
and (so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (77/ffi), (b) Nondimensional wave heights (if /H/) and
the best fit to the exponential decay function 196
xiv
172
Figure B.12: Case of L 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation [ijjHi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
68
Pi
\
Figure 6.1: Auxiliary coordinate system
6.2 Local Coordinate System for A Linear Element
To formulate a problem with linear elements, an auxiliary local coordinate sys
tem, as shown in Fig. 6.1, has to be established to facilitate line integrations over
the element. The two axis Â£ rj are perpendicular to each other with one lying on
the element and pointing in the direction of integration, from PÂ¡ to Pj+n and the
other one being in the same direction as the outward normal vector . Based on this
auxiliary coordinate system, the integration over each segment can be completed
analytically and be expressed in terms of Â£y, Â£J+i and r?,, the distances from Py,
Pi* i and Pi to the local origin defined in Fig. 6.1. The values of Â£y, Â£y+1 and 77, are
determined by the global coordinate information of three points.
Since
where
= y/(z. Xj)2 + (2, ~ Zj)2
54
TIME (SEC)
Figure 5.2: Typical wave data: (a) Averaged nondimensional surface elevation
{t)/Hi), (b) Nondimensional wave heights (H/H7) and the best fit to the expo
nential decay function.
123
INCIDENT HAVE HEIGHT (CH)
INCIDENT HAVE HEIGHT (CM)
Figure 8.9: Transmitted and reflected wave heights versus the incident wave heights,
(a) Transmitted waves; (b) Reflected waves.
134
For example, the iteration starts from Eq.(9.9) or Eq.(9.10) due to the presence of
fo in the formulae. The remaining procedures and the convergence criterion are
still the same. Figure 9.2 shows the flow chart of the computer program for berm
breakwaters.
9.3 Numerical Results
The numerical model for berm breakwaters has been applied to several different
cross sections with impermeable cores.
The first step in the computation was to verify the model with some special cases
for which theoretical solutions are available. In Fig. 9.3(a) and (b), the dotted curves
are the wave envelopes, computed by the numerical model for the two porous berm
breakwaters with vertical front face, in a water depth of 0.5 m. The permeability
of the berm breakwater in (a) is zero, i.e., equivalent to a concrete wall on the
front face, and is infinity for the one in (b), i.e., equivalent to the case of no berm
but a concrete wall on the back side. The perfect standing wave envelopes and
the complete reflections demonstrate that the model is well behaved and reliable
for the two extreme cases. Figures 9.4(a) and (b) show similar cases with the
berm of a parallelogram cross section. The reflection is, again, complete for both
cases and the wave envelopes are of perfect standing wave shapes except near the
reflective surfaces. The nondimensional wave runups in this case are greater than
those for the case of vertical face, as expected. The wave envelopes have been
nondimensionalized by the incident wave height.
Figure 9.5 shows the cross section of a berm breakwater designed and tested in
Norwegian Hydrotechnical Laboratory (T0rum el at., 1988, 1989) for the extension
of the fishing port in Arviksand, Norway. The cross section is dimensioned in model
scale which is 1/40 of the prototype.
Figures 9.6 to 9.8 are the numerical results for the berm breakwater sketched
in Fig. 9.5, with the stone size of dso = 2.9 cm and the porosity of 0.4 (assumed).
96
Figure 7.2: Flow chart of the numerical model for porous submerged breakwaters
175
Figure B.15: Case of L = 200 cm, h = DW = 20 cm, h, DS = 20 cm and
d50 = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
63
Table 5.9: Comparison of om and ap for h, = d50
d50 (cm)
H (cm)
^rm(>S *)
X)
Ar%
X)
A,%
h 30.0 cm
0.72
6.05
4.9157
4.7666
3.03
0.0024
0.0031
29.66
1.20
5.88
4.9374
4.7723
3.34
0.0040
0.0041
3.04
2.09
5.58
4.9559
4.7827
3.50
0.0075
0.0042
43.69
3.74
5.21
4.9808
4.8002
3.63
0.0097
0.0049
48.99
h = 25.0 cm
0.72
4.64
4.6495
4.4991
3.23
0.0031
0.0041
31.58
1.20
4.53
4.6675
4.5068
3.44
0.0055
0.0056
0.96
2.09
4.28
4.6822
4.5205
3.45
0.0100
0.0056
43.88
3.74
4.02
4.7318
4.5438
3.97
0.0130
0.0064
50.59
h = 20.0 cm
0.72
3.24
4.2936
4.1470
3.41
0.0057
0.0053
7.19
1.20
3.23
4.3134
4.1571
3.62
0.0094
0.0075
20.32
2.09
3.07
4.3339
4.1755
3.66
0.0147
0.0074
49.62
3.74
2.88
4.3950
4.2061
4.30
0.0170
0.0081
52.29
by dye studies. Turbulent diffusion was spotted over virtually the entire porous
domain and up to approximately 1.0 cm above the interface. The effects of turbulent
boundary layers to the wave damping are important in wave interaction with porous
seabeds and more theoretical and quantitative experimental research is needed.
WAVE FREQUENCY (1/S)
58
10.0
8.0
6.0
4.0
2.0
0.0
WRVE LENGTH = 200 CM
BED THICKNESS = 20 CM
HATER DEPTH = 25 CM
& a 6 6 6 *
t 1 1 1 1 : r
1.0 0.5 0.0 0.5 1.0
PERMEABILITY PARAMETER LOG (R)
cx
cc
cr
Q
a:
3
0.120
0.100 .
. $ .
0.080 .
0.060 .
O
+
O + 4
+
O PREDICTED
0
+
0.040 .
0.020 .
x MEASURED
0.000 .
i
1.0
1 1 i 1 1
0.5 0.0 0.5
PERMEABILITY PARAMETER LOG (R)
i
1.
Figure 5.3: The Measurements and the predictions vs. R. for L = 200.0 cm,
h, = 20.0 cm, h = 25.0 cm. (a) Wave frequency ar, (b) Wave damping rate er,.
120
i .0
O
CD (
J 1 1
...a. .
* /o O
I
CO
O
O
JZ
L
CD T = 1.379 SEC.
a T = 1.453 SEC.
0.2 .
o T = 1.778 SEC.
o a o EXPERIMENT DATA
O
o
0.000 0.000 0.001 0.001 0.002 0.002 0.003
HI/GThk2
Figure 8.7: The predicted and measured Kt and Kr versus HjgT2. (a) Ktp and
Ktm\ (h) Krp and Krm.
UFL/COEL/TR083
WATER WAVE INTERACTION WITH POROUS
STRUCTURES OF IRREGULAR CROSS SECTIONS
by
Zhihao Gu
Dissertation
1990
WATER WAVE INTERACTION WITH POROUS STRUCTURES OF
IRREGULAR CROSS SECTIONS
By
Zhihao Gu
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1990
ACKNOWLEDGEMENTS
It is almost impossible to fully express the authors appreciation in words to
everyone who has contributed to the completion of this dissertation.
First of all, the author wishes to express his deepest appreciation to his advisor
and the chairman of his advisory committee, Dr. Hsiang Wang, for his advice,
friendship, encouragement and financial support which enabled the author to come
to the United States to pursue his higher education. The four years of graduate work
with Dr. Wang have been a challenging and enjoyable experience in the authors
life.
The author would also like to thank Dr. Robert G. Dean, Dr. D. Max Sheppard
and Dr. Ulrich H. Kurzweg for serving as the members of his doctoral advisory
committee; Dr. Daniel M. Hanes for revising the dissertation and attending the final
exam. Thanks are also due to all other teaching faculty members in the department
during the authors graduate study: Dr. Michel K. Ochi, Dr. Ashish J. Mehta,
Dr. Peter Y. Sheng and Dr. James T. Kirby (now at University of Delaware), for
their teaching efforts, time and being role models to the author. The author has
benefitted substantially from their invaluable experience, knowledge and continuous
inspiration throughout the four years study. The author is especially grateful for
the valuable discussions with Dr. Dean and Dr. Kirby on numerous topics directly
or indirectly related to this work. The continuous financial support throughout the
Ph.D. program by the Coastal and Oceanography Engineering Department is also
greatly appreciated.
ii
The author is deeply indebted to his former advisors and supervisors in China,
Profs. WenFa Lu, Zhizhuang Xieng and Weicheng Wang. Without their generous
support and recommendation, it would have been impossible for the author to study
abroad in the first place. The continuous support and understanding by them and
all other faculty and staff members in the authors former working unit are highly
appreciated.
Very special thanks are given to Prof. Alf Tprum, Head of Research in Nor
wegian Hydrotechnical Laboratory, Trondheim, Norway, and Dr. Hans Dette in
Leichtweifi Institut fur Wasserbau, Technische Universitat Braunschweig, West Ger
many, for providing computer, office and accommodation facilities while the author
was visiting the two institutions in late 1988 with his advisor. The initiation and the
provision of technical information by Prof. Tprum for the topic of berm breakwater
which later became a chapter in this dissertation are gratefully acknowledged.
Thanks are given to Mr. Sidney Schofield, Mr. Jim Joiner and other staff
members in the Coastal Laboratory for their help during the experimental phases
of the study. The assistance provided by Mr. Subama Malakar in the use of
computer facilities is also appreciated. Thanks are extended to Helen Twedell for
the efficiency and courtesy in running the archives and for the parties, cookies and
cakes; and to Becky Hudson and all other secretarial personnel in the department
for the great job they have done related to this work.
The support of many friends and fellow students is warmly appreciated. Special
thanks go to Mrs. Jean Wang for her moral support, constant encouragement and
hospitality. Deep appreciation is given to fellow students Rajesh Srinivas and Paul
Work for proofreading the manuscripts and to Byunggi Hwang for assisting with
rock sorting during the experiments. The friendship shared with Yixin Yan, Steve
Peene, Yuming Liu and Richard McMillen and other friends, made the authors stay
in Gainesville a delightful period of time.
in
The authors gratitude towards his family is beyond words. The author would
like to thank his best friend and wife, Liqiu (also a Ph.D. student at the time), for
her support, both moral and physical, patience and useful technical discussions. She
was always the one to count on for assistance on weekend laboratory work, graphics
work and more often on the housework beyond her share. Without her support, the
road leading to this goal would have been much more tortuous. Finally, this work
is dedicated to the authors wonderful mother and father. Their affection and early
family education axe reflected in between the lines of this work and the authors
daily conduct. With all these efforts, thank god its Phinally Done!
iv
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF FIGURES vui
LIST OF TABLES xv
ABSTRACT xvi
CHAPTERS
1 INTRODUCTION 1
1.1 Problem Statement 1
1.2 Objectives and Scope 3
2 LITERATURE REVIEW 5
2.1 Porous Flow Models 5
2.2 WavePorous Seabed Interactions 10
2.3 Modeling of Permeable Structures of Irregular Cross Sections .... 13
3 POROUS FLOW MODEL 20
3.1 The Equation of Motion 20
3.2 Force Coefficients and Simplifying Assumptions 23
3.3 Relative Importance of The Resistant Forces 28
4 GRAVITY WAVES OVER FINITE POROUS SEA BOTTOMS 31
4.1 Boundary Value Problem 31
4.2 The Solutions of The Complex Dispersion Equation 35
4.3 Results 41
5 LABORATORY EXPERIMENT FOR POROUS SEABEDS 50
5.1 Experiment Layout and Test Conditions 50
v
5.2 Determination of The Empirical Coefficients 52
5.3 Relative Importance of The Resistances in The Experiment 55
5.4 Comparison of The Experimental Results and The Theoretical Values 55
6 BOUNDARY INTEGRAL ELEMENT METHOD 64
6.1 Basic Formulation 64
6.2 Local Coordinate System for A Linear Element 68
6.3 Linear Element and Related Integrations 70
6.4 Boundary Conditions 74
7 NUMERICAL MODEL FOR SUBMERGED POROUS BREAKWATERS 76
7.1 Governing Equations 76
7.2 Boundary Conditions 78
7.2.1 Boundary Conditions for The Fluid Domain 78
7.2.2 Boundary Conditions for The Porous Medium Domain .... 81
7.3 BIEM Formulations 82
7.3.1 Fluid Domain 82
7.3.2 Porous Medium Domain 86
7.3.3 Matching of The Two Domains 87
7.4 Linearization of the Nonlinear Porous Flow Model 88
7.5 Transmission and Reflection Coefficients 91
7.6 Total Wave Forces on an Impervious Structure 92
7.7 General Description of The Computer Program 94
7.8 Numerical Results 95
8 LABORATORY EXPERIMENTS OF A POROUS SUBMERGED
BREAKWATER 107
8.1 General Description of The Experiment 107
8.2 Wave Transmission and Reflection 109
8.3 Pressure Distribution and Wave Envelope Over The Breakwater . 121
vi
9 NUMERICAL MODEL FOR BERM BREAKWATERS 128
9.1 Mathematical Formulations 128
9.1.1 CFSBC for The Free Surface Inside Porous Medium 130
9.1.2 BIEM Formulations 131
9.2 Linearization 133
9.3 Numerical Results 134
10 SUMMARY AND CONCLUSIONS 146
10.1 Summary 146
10.2 Conclusions 148
10.3 Recommendations for Future Studies 151
APPENDICES
A BOUNDARY INTEGRAL FORMULATION FOR ENERGY DISSIPATION
IN POROUS MEDIA 153
B EXPERIMENTAL DATA FOR POROUS SEABEDS 159
BIBLIOGRAPHY 197
BIOGRAPHICAL SKETCH 200
Vll
LIST OF FIGURES
3.1 Definition sketch for the porous flow model 21
3.2 Regions with different dominant resistant forces 30
4.1 Definition Sketch 32
4.2 Progressive wave case: h, = DS = 5.0 m and h = DW = 2.0 
6.0 m. (a) Nondimensional wave number kr/(o2 fg), (b) Nondi
mensional wave damping rate ki/(o2/g) 45
4.3 Maximum nondimensional damping rate (&,)./(cr2/y) and its
corresponding permeability parameter R as functions of nondi
mensional water depth h (cr2 jg) 46
4.4 Standing wave case, (a) Nondimensional wave frequency orl{LÂ¡g)*\
(b) Nondimensional wave damping rate cr,/(L/<7)3 48
4.5 Solutions based on four porous flow models, (a) Nondimensional
wave frequency or/(L/g) 3, (b) Nondimensional wave damping
rate Ci/(L/g)t 49
5.1 Experimental setup 51
5.2 Typical wave data: (a) Averaged nondimensional surface eleva
tion iv/Hi), (b) Nondimensional wave heights (H/~Hi) and the
best fit to the exponential decay function 54
5.3 The Measurements and the predictions vs. R. for L = 200.0 cm,
h, = 20.0 cm, h = 25.0 cm. (a) Wave frequency crr, (b) Wave
damping rate <7, 58
5.4 Theoretical values by the present model vs. experimental data
of Table 5.5, Table 5.6 and Table 5.7. (a) Wave frequency ar,
(b) Wave damping rate <7, 61
5.5 Theoretical values by the model of Liu and Dalrymple vs. ex
perimental data of Table 5.5, Table 5.6 and Table 5.7. (a) Wave
frequency crr, (b) Wave damping rate 62
6.1 Auxiliary coordinate system 68
via
7.1 Computational domains 77
7.2 Flow chart of the numerical model for porous submerged break
waters 96
7.3 Wave envelopes for (a) Transparent submerged breakwater; (b)
Impermeable step 98
7.4 Porous submerged breakwater: (a) Wave form and wave enve
lope; (b) Envelopes of pressure and normal velocity 99
7.5 Wave field over submerged breakwaters: (a) Impermeable; (b)
Permeable 101
7.6 Transmission and reflection coefficients vs. stone size for differ
ent wave periods, (a) Transmission coefficient; (b) Reflection
coefficient 103
7.7 Transmission and reflection coefficients vs. R for different wave
periods, (a) Transmission coefficient; (b) Reflection coefficient. 104
7.8 Transmission and reflection coefficients vs. R for different wave
heights, (a) Transmission coefficient; (b) Reflection coefficient. 105
7.9 Wave forces and over turning moment for a impermeable sub
merged breakwater: (a) Wave forces; (b) Overturning moment. 106
8.1 Experiment layout 108
8.2 Typical wave record, (a) Partial standing waves on the up wave
side, (b) Transmitted waves on the down wave side Ill
8.3 The wave spectrum of the transmitted waves 112
8.4 The predicted Kt and Kr versus the measured Kt and Kr. (a)
Ktp vs. Ktm', (b) Ktp vs. Ktm 117
8.5 The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp
and Ktm; (b) Krp and Krm 118
8.6 The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp
and Ktm; (b) Krp and K 119
8.7 The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp
and Ktm5 (b) Krp and Krm 120
8.8 Transmitted and reflected wave heights versus the incident wave
heights, (a) Transmitted waves; (b) Reflected waves 122
8.9 Transmitted and reflected wave heights versus the incident wave
heights, (a) Transmitted waves; (b) Reflected waves 123
IX
8.10 The envelopes of wave and pressure distribution for T 0.858
sec.; nonbreaking wave case, (a) Wave envelope; (b) Envelope
of pressure distribution 126
8.11 The envelopes of wave and pressure distribution for T 0.858
sec.; breaking wave case, (a) Wave envelope; (b) Envelope of
pressure distribution 127
9.1 Definition sketch for berm breakwaters 129
9.2 Flow chart of the numerical model for berm breakwaters .... 135
9.3 Berm breakwaters of vertical face: (a) Zero permeability; (b)
Infinite permeability 136
9.4 Berm breakwaters of inclined face: (a) Zero permeability; (b)
Infinite permeability 137
9.5 The Cross Section of The Berm Breakwater 138
9.6 Permeable berm breakwater of model scale with H = 5.0 cm: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity
distribution 140
9.7 Permeable berm breakwater of model scale with H = 10.0 cm:
(a) Wave envelope; (b) Envelopes of pressure and normal velocity
distribution 141
9.8 Permeable berm breakwater of model scale with H = 20.0 cm:
(a) Wave envelope; (b) Envelopes of pressure and normal velocity
distribution 142
9.9 Permeable berm breakwater of prototype scale with H = 2.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions 143
9.10 Permeable berm breakwater of prototype scale with H = 4.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions 144
9.11 Permeable berm breakwater of prototype scale with H = 8.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions 145
A.l Geometric relations between the vectors 156
B.l Case of L = 200 cm, h = DW = 30 cm, k, = DS = 20 cm
and d50 = DD = 0.72 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/Hi) and
the best fit to the exponential decay function 161
x
B.2 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and d50 = DD = 0.93 cm. (a) Averaged nondimensional surface
elevation (r?/ifi), (b) Nondimensional wave heights (H/Hi) and
the best fit to the exponential decay function 162
B.3 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 1.20 cm. (a) Averaged nondimensional surface
elevation (rj/HA, (b) Nondimensional wave heights (H/HA and
the best fit to the exponential decay function 163
B.4 Case of L = 200 cm, h DW = 30 cm, h, = DS = 20 cm
and d60 = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (tj/HA, (b) Nondimensional wave heights (H/HA and
the best fit to the exponential decay function 164
B.5 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (r]/HA> (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 165
B.6 Case of L 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.84 cm. (a) Averaged nondimensional surface
elevation (rÂ¡/HA, (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 166
B.7 Case of L = 200 cm, h = DW = 30 cm, h, DS = 20 cm
and so = DD = 3.74 cm. (a) Averaged nondimensional surface
elevation (r)/H\), (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 167
B.8 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 0.72 cm. (a) Averaged nondimensional surface
elevation (rÂ¡/HA, (b) Nondimensional wave heights (H/HA and
the best fit to the exponential decay function 168
B.9 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 0.93 cm. (a) Averaged nondimensional surface
elevation (rj/HA, (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 169
B.10 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 1.20 cm. (a) Averaged nondimensional surface
elevation (rÂ¡/H\), (b) Nondimensional wave heights (H /HA and
the best fit to tne exponential decay function 170
B.ll Case of L = 200 cm, h DW = 25 cm, h, = DS = 20 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (t)/HA (b) Nondimensional wave heights [H/HA and
the best fit to the exponential decay function 171
B.12
Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and d50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (tj/Hi), (b) Nondimensional wave heights (H/IT/) and
the best fit to the exponential decay function 172
B.13 Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm
and (so = DD = 2.84 cm. (a) Averaged nondimensional surface
elevation {rj/Hi), (b) Nondimensional wave heights (H/TT/) and
the best fit to the exponential decay function 173
B.14 Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm
and (so = DD = 3.74 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/H7) and
the best fit to the exponential decay function 174
B.15 Case of L 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 0.72 cm. (a) Averaged nondimensional surface
elevation (t]/Hi), (b) Nondimensional wave heights [H/H/) and
the best fit to the exponential decay function 175
B.16 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 0.93 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/Hi) and
the best fit to tne exponential decay function 176
B.17 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and so = DD = 1.20 cm. (a) Averaged nondimensional surface
elevation [r)/HiV (b) Nondimensional wave heights (H/H7) and
the best fit to tne exponential decay function 177
B.18 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/Hi) and
the best fit to the exponential decay function 178
B.19 Case of L = 200 cm, h = DW = 20 cm, ht = DS = 20 cm
and (so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation {rj/Hi), (b) Nondimensional wave heights (H/Hj/) and
the best fit to the exponential decay function 179
B.20 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and so = DD = 2.84 cm. (a) Averaged nondimensional surface
elevation (r}/Hi)t (b) Nondimensional wave heights (H/H7) and
the best fit to tne exponential decay function 180
B.21 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm
and (so = DD = 3.74 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/TT/) and
the best fit to tne exponential decay function 181
Xll
B.22
Case of L = 200 cm, h = DW = 30 cm, h, = DS = 15 cm
and 50 = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (77/if j), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 182
B.23 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 15 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/iM, (b) Nondimensional wave heights (if/Hi) and
the best fit to the exponential decay function 183
B.24 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 15 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (ti/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 184
B.25 Case of L = 200 cm, h DW = 30 cm, h, = DS = 10 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights (H/Ex) and
the best fit to the exponential decay function 185
B.26 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 10 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (tj/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 186
B.27 Case of L = 200 cm, h = DW = 20 cm, h, = DS = 10 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights [H/H7) and
the best fit to tne exponential decay function 187
B.28 Case of L = 225 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights (ifÂ¡Hx) and
the best fit to the exponential decay function 188
B.29 Case o L 225 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation [rj/Hx), (b) Nondimensional wave heights (iffHÂ¡) and
the best fit to tne exponential decay function 189
B.30 Case o L 225 cm, h = DW = 20 cm, h, = DS = 20 cm
and 50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (rj/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to tne exponential decay function 190
B.31 Case of L = 250 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation [rj/Hx), (b) Nondimensional wave heights (if/Hi) and
the best fit to the exponential decay function 191
xm
B.32 Case of L 250 cm, h DW = 25 cm, h, = DS = 20 cm
and d50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (77/ffi), (b) Nondimensional wave heights (H/Hi) and
the best fit to tne exponential decay function 192
B.33 Case of L = 250 cm, h = DW = 20 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (r?/iM, (b) Nondimensional wave heights (H/Hi) and
the best fit to tne exponential decay function 193
B.34 Case of L = 275 cm, h = DW = 30 cm, h, = DS = 20 cm
and Â£50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (r\/HiJ, (b) Nondimensional wave heights (if/i7) and
the best fit to tne exponential decay function 194
B.35 Case of L 275 cm, h = DW = 25 cm, h, = DS = 20 cm
and Â£50 = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (ri/Hi), (b) Nondimensional wave heights (if/Hi) and
the best fit to the exponential decay function 195
B.36 Case of L = 275 cm, h = DW = 20 cm, hs = DS = 20 cm
and (so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (77/ffi), (b) Nondimensional wave heights (if /H/) and
the best fit to the exponential decay function 196
xiv
LIST OF TABLES
3.1 Illustration of Dominant Force Components Under Coastal Wave
Conditions: [a ~ 0(l)rad/sec,V(pf^) ~ O(101)] 30
4.1 Comparison of The Predictions and The Measurements 43
5.1 Material Information 51
5.2 Test Cases 52
5.3 Measured ar and cr, for h, = 20 cm and L = 200 cm 53
5.4 Comparison of The Resistances, k, = 20 cm, L = 200 cm. ... 56
5.5 Comparison of Measurements and Predictions for h, = 20 cm,
L = 200 cm 57
5.6 Comparison of om and ap for d^o = 1.48 cm, L = 200 cm 59
5.7 Comparison of am and cp for d50 = 2.09 cm, h, = 20 cm 59
5.8 Comparison of om and op for d50 = 0.16 mm, h, = 20 cm,
L = 200 cm 60
5.9 Comparison of am and cp for h, = 63
8.1 Test Results of Nonbreaking Waves 114
8.2 Comparison of Km and Kp 115
8.3 Test Results for Breaking Waves 121
8.4 Normalized Pressure Distribution 125
B.l Test Cases 159
xv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
WATER WAVE INTERACTION WITH POROUS STRUCTURES OF
IRREGULAR CROSS SECTIONS
By
Zhihao Gu
December. 1990
Chairman: Dr. Hsiang Wang
Major Department: Coastal and Oceanographic Engineering
A general unsteady porous flow model is developed based on the assumption
that the porous media can be treated as a continuum. The model clearly defines
the role of solid, and fluid motions and henceforth their interactions. All the im
portant resistant forces are clearly and rigorously defined. The model is applied to
the gravity wave field over a porous bed of finite depth. By applying linear wave
theory, an analytical solution is obtained, which is applicable to the full range of
permeability. The solution yields significantly different results from those of con
temporary theory. The solution requires three empirical coefficients, respectively
representing linear, nonlinear and inertial resistance. Laboratory experiments using
a standing wave system over a porous seabed were conducted to determine these
coefficients and to compare with analytical results. The coefficients related to lin
ear and nonlinear resistances were found to be close to those obtained by previous
investigators. The virtual mass coefficient was determined to be around 0.46, close
to the theoretical value of 0.5 for a sphere. The analytical solution compared well
with the experiments.
Based on this porous flow model and linear wave theory, two numerical models
using boundary integral element method with linear elements are developed for
xvi
permeable submerged breakwaters and berm breakwaters, respectively. Due to the
establishment of a boundary integral expression for wave energy dissipation in a
porous domain and the application of the radiation boundary condition on the
lateral boundary(ies), the numerical models are highly efficient while maintaining
sufficient accuracy. The numerical results show that the wave energy dissipation
within a porous domain has a well defined maximum value at certain permeability
for a specified wave and geometry condition. The nonlinear effects in the porous
flow model are clearly manifested, as all the flow field properties are no longer
linearly proportional to the incident wave heights. The numerical results agreed
reasonably well with the experimental data on the seaward side. On the leeward of
the breakwater, despite the appearance of higher order harmonics, the numerical
model produces acceptable results of energy transmission based on energy balance.
XVII
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Among the existing shore protecting breakwaters, a large number of them are
rubblemound structures made of quarry stones and/or artificial blocks. They can
be treated as structures of granular materials. A breakwater is called subaerial
when its crest is protruding out of the water surface and submerged when its crest
is below the water level. Sometimes a breakwater is subaerial at low tide and
submerged at high tide. These types of structures are usually termed as low crest
breakwaters. When a beach is to be protected from wave erosion, a detached shore
parallel submerged breakwater may provide an effective and economic solution. The
advantages of submerged breakwaters as compared to subaerial ones are low cost,
aesthetics (they do not block the view of the sea) and effectiveness in triggering the
early breaking of incident waves, thus reducing the wave energy in the protected
area. With the increasing interest in recreational beach protection, where complete
wave blockage is not necessary, submerged breakwaters may find more and more
applications.
On the other hand, if wave blockage is the main objective such as for harbor
and port protection, a subaerial breakwater may be more effective. The traditional
design of such a structure is a trapezoidally shaped rubble mound with an inner
core covered with one or more thin layers of large blocks to form the armor layer (s)
to protect the core. Owing to the demand for deeper water applications, the armor
sizes have become larger and larger. This would greatly increase not only the cost
but also the structural vulnerability. A relatively new type of structures, called
1
2
berm breakwater, is attracting more and more attention. For a berm breakwater,
the armor layer(s) is replaced by thicker layer(s) of blocks of much smaller sizes.
The seaward face of the structure has a berm section instead of the traditional
uniform slope. The berm section is intended to enhance the structural stability and
to trigger wave breaking.
Wave attenuation behind a porous breakwater is affected by three mechanisms:
wave reflection on the seaward face of the structure, waves breaking over the struc
ture and flow percolation inside the porous structure. The former one is conserva
tive and the latter two are dissipative. The main focus of this study is on the third
mechanism, that is the dissipation due to flow percolation. For such a purpose, a
numerical solution is sought since an analytical solution for such irregularly shaped
structures is nearly impossible. The main advantage of numerical methods is the
flexibility of handling complex geometries and boundary conditions. The disadvan
tage of any numerical approach is the lack of generality for the solution, since it is
usually implicit in terms of the variables so that the influences of the parameters
can only be examined case by case.
In order to examine the validity of the model and the nature of the dissipative
force, the porous flow model is applied to an infinitely long flat seabed of finite
thickness subject to wave action. This condition can also be viewed as a submerged
breakwater with infinitely long crest. Because of the geometrical simplicity, an
analytical solution is attainable. This case is used to compare with the existing
solutions proposed by other investigators, to examine the nature of wave attenua
tion as a function of flow and material properties and, more importantly, to guide
the design of an experiment so that the empirical coefficients in the porous model
can be determined. Successful determination of these coefficients is crucial to the
validity of the model. The experiment which is to be carried out subsequently must
demonstrate that the results are stable (or the experiments are repeatable) and that
3
they cover, at least, an adequate range of intended application.
A numerical solution with computer code is then to be developed for arbitrary
geometry. Boundary Integral Element Method (BIEM) is proven to be very effi
cient for boundary value problems with complicated domain geometries. With this
method, the solution is expressed in boundary integrals and no interior points have
to be involved in the solution procedure. The aim of this study is to develop an
efficient computer code based on such a method. Finally, the validity of the solution
is to be assessed by a set of experiments.
1.2 Objectives and Scope
Specifically, the objectives of this study are listed as follows:
1. Develop a percolation model suitable for unsteady and turbulent porous flows,
2. Verify the model through an analytical solution and laboratory experiments for
the case of a flat porous seabed subject to linear gravity waves,
3. Based upon the porous flow model, develop a numerical solution for submerged
and berm breakwaters of arbitrary cross sections using the boundary integral ele
ment method.
To achieve these goals, the research is carried out in the following steps:
1. Derive the porous flow model,
2. Examine and interpret the relative importance of the various terms in the model
to narrow the scope of the study,
3. Obtain the analytical solution of wave attenuation over a flat porous seabed,
compare the solution with the existing ones and examine the wave energy dissipa
tion process,
4. Conduct a laboratory experiment for the flat porous seabed case to determine the
empirical coefficients in the porous flow model and to verify the analytical solution,
5. Develop a numerical model using the BIEM method for porous submerged break
waters,
4
6. Verify the numerical model with laboratory experiments of a submerged rubble
mound breakwater,
7. Modify the numerical model to represent porous berm breakwaters.
CHAPTER 2
LITERATURE REVIEW
This chapter is a brief literature review of the past efforts made in studying
the flow in porous media with and without water waves. The review is separated
into three groups: one on porous flow model, one on analytical solutions for wave
and porousseabed interactions and the other one on the modeling of interactions
between waves and porous structures of irregular cross sections.
Since the porous media encountered in coastal engineering are largely of the
granular type, made of sand, gravel, quarry stones or artificial blocks, the deforma
tion of the solid skeleton of the media is usually negligibly small as compared to
that of the pore fluid. The literature review deals with granular media only; other
types of porous media, such as poroelastic or poroplastic media and so on, are not
included.
2.1 Porous Flow Models
A large number of porous flow models have been proposed over the past few
decades as an effort to quantitatively model the flows in porous media. Some of
the models were based purely on experiments and some of the others on certain
theoretical considerations. The derivations of the these models are based primarily
on the following three approaches: 1) Simple element approach which views a porous
medium as the assembly of simple elementary elements such as a bundle of tubes and
so on, 2) Microscopic approach which recognizes the microstructure of porous media
in derivations. The final formulations of the models usually have to be obtained
by taking the spatial average of the microscopic equations, 3) Phenomenological
approach which homogenizes the porous media as a continuum and the flow within
5
6
the medium, is assumed to be continuous. The third approach is the most popular
for porous flows in granular materials since the microscopic structure is not well
understood and the corresponding constitutive equation not well established.
The first porous flow modelDarcys lawwas proposed by the French engineer
Darcy based on his experiments more than a century ago. In his model, the char
acteristic properties of porous media are lumped into one parameterpermeability
coefficientand the model has the form
 Vp = Â£? (2.1)
where Vp is the gradient of the pore pressure, p and q are, respectively, the dy
namic viscosity and the specific discharge velocity of the pore fluid and Kp is called
the intrinsic permeability coefficient, which reflects the collective characteristics of
porous media, such as the porosity, the roughness of the pore walls, the tortuosity,
the connectivity etc.
This model is very simple in form and reasonably accurate for steady porous
flows within media of low permeability, generally of the order of Kp = 109 ~
1012 m2, where the flows are normally laminar dominant. However, for porous
media with relatively higher permeability, the porous flow is no longer laminar dom
inant, turbulence plays a larger and larger role with increasing pore size. In such
cases, Darcys model tends to over estimate the discharge velocity for a given pres
sure gradient. Dupuit and Forchheimer (cited in Madsen, 1974) modified Darcys
law by adding a term quadratic in velocity to take into account the turbulent effects:
 Vp = (o + 6  q I)? (2.2)
where a ii/Kp, and b is coefficient for turbulent resistance defined by Ward (1964)
as
(2.3)
7
where p is the density of the fluid and CÂ¡ is a nondimensional coefficient.
The two coefficients a and b were further expressed in terms of particle diameter
by Engelund (1953) and Bear et al. (1968) in the forms
(1 n)sv
a Cq
n2d2
(2.4)
b = b0
1 n
n2d,
(2.5)
where
porous media.
The above two models, Eq.(2.l) and (2.2) are both for steady porous flows,
although some applications have been made for certain unsteady flows, such as wave
induced porous flows in seabeds (Putnam, 1949; Liu, 1973; etc.). These applications
were primarily on low permeability media like fine sand. In general, for unsteady
flows, the inertial force arises due to the acceleration of the pore fluid and such a
force can be quite large and can even be dominant when the permeability is high.
To model the unsteady porous flow, a number of models have been suggested.
The first model found in the literature containing the inertial force term was
the one by Reid and Kajiura (1957). It is the direct extension of Darcys model:
T7 Pd$
P Kp,+ nSt
(2.6)
The inertial term in this model is in fact the local acceleration term in NavierStokes
equation with the fluid velocity expressed as a specific discharge velocity q. It is
clear that this term contains only the inertia of the fluid but not the inertia induced
by the fluidsolid interaction, or the added mass effect.
The extension of DupuitForchheimer model leads to PolubarinovaKochinas
model (Scheidegger, 1960; McCorquodale, 1972). It has the form
 Vp = aq + bq + 
(2.7)
8
where a, b and c are empirical coefficients.
Murray (1965), when studying the viscous damping of gravity waves over a
permeable seabed, proposed
_ a du ,
 Vp = u + pn (2.8)
where k is the permeability of porous media (defined differently from Kp) and u
is the spatially averaged microscopic velocity of the pore fluid, related to q by
q = nu. Comparing to Eq.(2.7), the coefficient c is equal to p in Murrays model.
McCorquodale in 1972 further modified the expression of this coefficient as
n
and his model (McCorquodale, 1972) becomes
(2.9)
Vp=(a + bg)g + ^ (2.10)
n at
At about the same time as McCorquodale, Sollitt and Cross (1972) also extended
Eq.(2.2) to include the effects of unsteadiness. In their model, the inertia resistance
consists of two components, one is the inertia of the pore fluid, and the other one
is the inertia induced by the virtual mass effect due to the fluidsolid interactions;
the model reads
~ VP = injr + I s l) + Pi1 + (2n)
where u is again the spatially averaged microscopic velocity vector and Ca is the
virtual mass coefficient.
The value of the virtual mass coefficient Ca in Sollitt and Crosss model was
assumed to be zero in their actual computation because it was unknown at the time.
It has been so assumed in almost all the later applications of this model (Sulisz,
1985, etc.). When comparing to the experimental data, Sollitt and Cross found that
the correlation improved by taking nonzero values for Ca and asserted that one
cannot predict the magnitude of this coefficient a priori because the virtual mass
of densely packed fractured stone is not known (Sollitt and Cross, 1972, pl842).
They also pointed out that Evaluation of Ca, however, may serve as a calibrating
link between theory and experiment in future studies (same page).
Hannoura and McCorquodale (1978) made an attempt to determine the value
of Ca by laboratory experiment. They measured the instantaneous velocity and
the pressure gradient in the experiment, and calibrated the data with their semi
theoretical porous flow model:
Vp=( + 6f) + />(l + C) (2.12)
The results so obtained for Ca scattered in a range of 7.5 ~ +5.0.
Dagan in 1979, based on the microscopic approach, arrived at a generalized
Darcys law for nonuniform but steady porous flows (Dagan, 1979):
_ n 77* _2.
~ Vp= q~ rV q
Tip
(2.13)
where 7 is a constant coefficient depending only on the geometry of the media and
it was defined as
7 =
80
(2.14)
In applying this model to the waveporousseabed interaction, Liu and Dalrym
ple (1984) added an acceleration term to the Dagans model and it becomes
v'5<215>
It was found, by performing a dimensional analysis (Liu, 1984), that the last
term in Dagans model is a second order quantity in comparison to the other terms.
The inertial term in this model is the same as that of Reid and Kajiura and that of
Sollitt and Cross where Ca is set to be zero.
Based on the phenomenological approach, Barends (1986) added another porous
flow model to the list:
r,_ ,Cadq Q x ,
10
and K(q) was defined by Hannoura and Barends (1981, see Baxends, 1986) as
KM=gmÂ¡T\ (2'I7)
with
_ Ca( 1 n)
f ~ Jrf
where C is the drag coefficient related to the porous Reynolds number (R = q 
djv), a and /? are two constants and d, is the relevant grain size.
With so many models, it is difficult to decide which one is most appropriate for
the porous flow problem in this study without further analysis.
2.2 WavePorous Seabed Interactions
The computation of gravity wave attenuation over a rigid porous bottom has
been performed by a number of investigators. The classic approaches by Putnam
(1949) and Reid and Kajiura (1957) was to assume that the Laplace equation is
satisfied in the overlying fluid medium and that the bottom layer can be treated
as a continuum following Darcys law of permeability. In Reid and Kajiuras pa
per, although the inertial term was included in the porous flow model, they later
neglected it and concluded that Darcys law is adequate for sandy seabeds. Under
the assumption of low permeability, Reid and Kajiura found, for infinitely thick
seabeds, that
a1 gkr tanh kTh (2.18)
k. 2(oKp/v)kr
* 2krh + sinh 2 kTh
for progressive waves, where a is the wave frequency, kr and ki are, respectively, the
wave number and the damping rate, h is the water depth and v is the kinematic
viscosity of the fluid.
Hunt (1959) examined the damping of gravity waves propagating over a perme
able surface using Reid and Kajiuras porous flow model and retained the inertial
term to the end. The thickness of the permeable beds was infinite and water above
11
the permeable surface was assumed viscous as opposed to the inviscid approach by
the previous investigators. The tangential velocities at the surface of the permeable
bed were set to be zero and the continuity of the normal velocity and the pressure
were observed on the same boundary. The resulting dispersion equation was quite
lengthy. Under the consideration of sand beds of low permeability, the results for
kr and /:, are
kr
ko +
nr 2kl
V 2c 2kQh + sinh 2koh
(2.20)
with
ki
2ko (aKp 1 k )
2k0h + sinh 2k0h' u 0 2c
(2.21)
c2 gk0 tanh koh
(2.22)
Murray (1965) investigated the same problem by the use of stream function and
the porous flow model given in Eq.(2.7). Instead of letting the tangential velocities
at the interface be zero as done by Hunt, Murray assumed that the tangential
velocities were finite and continuous. The first order solution of the dispersion
equation for the spatial damping was
V2k2J^+2k0N
kli = 2koh + sinh 2k0h i2*23)
where k = k0 + ki with k0 being the initial wave number for the impervious bottom
as defined in Eq.(2.22) and
N =
noKp
v
For fine sand, JV 0 and
,
* 2krh + sinh 2 krh
and the temporal damping for known k in such case is
yjvok
a =
\/2sinh2A:/i
(2.24)
(2.25)
12
Liu (1973) added an interface laminar boundary layer in his solution for the
same problem. In his paper, the water in the fluid domain was regarded as inviscid
and the porous flow was assumed to follow Darcys law. The imaginary part of k,
after the boundary layer correction, was given by
ki =
2 kT
2 kTh + sinh 2 krh
(oKp/v \
(2.26)
with kr remaining the same as that in Eq.(2.18). It is noted that this is the same ex
pression as that given by Hunt (1959). The correction term of Eq.(2.26) to Eq.(2.19)
is found to be insignificant unless Kp is extremely small.
Dean and Dalrymple (1984) later obtained the expressions for shallow water
waves over sand beds:
_ (1 a2h/g) oKp
2 h { u ]
All the solutions above are for infinitely thick seabeds. Liu (1977) solved the
problem for a stratified porous bed where each layer has a finite thickness and
different permeability. It was found that the wave damping rate was insignificant
to the permeability stratification while the wave induced pressure and its gradients
are affected significantly by stratification. The porous flow model employed in the
analysis also obeys Darcys law only.
Liu and Dalrymple (1984) further modified the solution, for a bed of finite
thickness, by replacing Darcys law with Dagans unsteady porous flow model, which
partially included the effects of unsteady flows. The dispersion relationship from
the homogeneous problemwithout laminar boundary layer correctionswas found
as
( + t)(l ~ tanh/i) + R tanh kh, tanh/:h(l
n a1
~ ctnh kh) = 0
(2.29)
13
or equivalently
a2 gktsuikh yy tanh kh,{gk o2tanhkh)
(2.30)
R 1 n
where h, is the seabed thickness and R is the permeability parameter defined as
V
It has been shown that the corrections by the laminar boundary layers were
not significant since the damping is largely due to the energy losses in the porous
medium rather than the boundary layer losses (Liu and Dalrymple, 1984, p47).
Therefore, it is reasonable to believe that if an appropriate porous flow model is
adopted, the solution for a homogeneous problem, without the correction of laminar
boundary layers, will be sufficient for engineering purposes.
Besides the theoretical studies, the interaction of waves and porous seabeds was
also investigated experimentally by Savage (1953). The experiment was carried out
with progressive waves over sand beds of median diameters, d, = 0.382 cm and d, =
0.194 cm, respectively. The permeability coefficient of the sand bed was measured
as 44.9 x 10~10m2, and the dimension of the seabed was 0.3 meters thick and 18.3
meters long. The wave heights at the beginning and the end of the sand bed were
recorded and adjusted to eliminate the effects of the side friction. Most of the
theories listed above were claimed to agree well with the data in this experiment.
2.3 Modeling of Permeable Structures of Irregular Cross Sections
The ability to predict the performance of rubblemound breakwaters under the
attacks of ocean waves is critically important in designing such structures. Signif
icant amounts of effort have been devoted to this subject. In the early stage of
development, research was basically confined to laboratory experiments and only
the empirical formulae extracted from the experiments were available for designs.
In recent years, with the continuing efforts by researchers, more and more theoreti
cal models for different types of breakwaters are becoming available. For subaerial
14
breakwaters, in addition to a large number of laboratory experiments, a multitude
of theoretical models have been developed. The analysis of such structures has been
approached both analytically and numerically.
Analytically, Sollitt and Cross (1972) investigated the problem of wave transmis
sion and reflection by a vertical face (rectangular cross section) porous breakwater.
The porous flow model used in the formulation was Sollitt and Crosss model al
though the virtual mass coefficient Ca was assumed to be zero in the computation.
In their solution, the whole computation domain was divided into three regions,
two fluid regions and one porous region. The velocity potential functions in the
three regions were assumed to be the summations of the fundamental mode and
the evanescent modes according to the wave maker theory. The nonlinear term in
the flow model was linearized according to Lorentzs hypothesis and the normal
velocities and the pressure were matched at both vertical faces of the breakwater.
The transmission and the reflection coefficients for long waves, simplified from the
complete solution of complex matrix form, were given by
Kt =
1
(2.31)
1+LZls~if+n2)
Kr =
S if n2
(2.32)
S if + n2 i2n^
gh
where S is the coefficient for the inertial resistance of porous media, i is the imag
inary unit, / is the linearized coefficient for velocity related resistances, b is the
width (or thickness) of the breakwater and all the other symbols have the same
meanings as before.
Madsen (1974) later reexamined the same problem beginning directly with the
linear long wave theory. He adopted the Dupuit Forchheimer model and introduced
the inertial resistance into the momentum equation. The resulting expression of Kt
and Kr are almost identical to Eq.(2.31) and (2.32). By carrying out the volume
15
integration over the breakwater for the energy dissipation, Madsen obtained an
explicit expression for the linearized friction factor /, that is
J Kpr v 2a
) + \/(! +
; 3;r h1
(2.33)
where a and 6 are the coefficients in DupuitForchheimers porous flow model. A
similar solution for this problem was also given by Scarlatos and Singh (1987).
Madsen et al. (1976, 1978) further extended the long wave solution to a trapezoidal
porous breakwater with, again, DupuitForchheimens model. The solution was,
however, much more complicated than that for the crib type.
Since the shore protection breakwaters are usually irregular in shape and very
often with several layers of stones of different sizes, analytical solutions become
impractical. For such complicated geometries, numerical approaches have to be
adopted.
The most commonly used numerical schemes are finite element and Boundary
Integral Element Methods (BIEM). The first finite element model was developed by
McCorquodale (McCorquodale, 1972) using the McCorquodale porous flow model,
for computing the wave energy dissipation in rockfills. In his model, the entire cross
section of the structure was divided into small triangleelements and the variation
of the physical quantities was interpolated by a time dependent element function:
= (0i + @2% + 02 y)t + 0^+ 0*,x + 0ey (2.34)
Since the numerical computation was only carried out in the porous domain,
the interaction between the porous domain and the fluid domain was not modeled,
although the free surface within the porous region was well predicted.
Due to the large amount of work for data preparation in using a finite element
model, the Boundary Integral Element Method (BIEM) became popular since the
mid1970s. With BIEM, the discretization is only on the boundaries as opposed
to the entire domain with the finite element method. Ijima et al. (1976) applied
16
this method to porous breakwaters with constant elements, which assumes that
the physical quantities remain constant over each individual element. The whole
computation domain was divided into three regions, two fluid regions and one porous
region. In the porous domain, Darcys law was used and in the two fluid regions,
two artificially defined vertical boundaries were placed at the offshore and inshore
ends. The patterns of the vertical distribution of the velocity potential at these two
boundaries was assumed undistorted by the presence of a structure. The reflected
and the transmitted potential functions outside the computational domain were
given by
r[x,z)
cosh k{z + h)
cosh kh
(2.35)
Mx,z) = cosh *'(*+'l')
cosh k>h' ^2'36)
where l and V axe the distances from the left and the right vertical boundaries to
the origin, k and k' are the wave numbers in the reflection and the transmission
regions, respectively. The reflection and transmission coefficients are then
Kr = 1 Ao
(2.37)
Kt = B0  (2.38)
The agreement of the transmission coefficient for impermeable floating struc
tures with the experiment data was fairly good, but the comparison of Kt and Kr
for a slopedface permeable breakwater was not as satisfactory.
Finnigan and Yamamoto (1979) further added the modulated standing wave
modes to r and 4>t, while keeping the constant element and Darcys law unchanged
in their model. The two potential functions were, respectively,
M*,*) = [*<>+.V*>]C0Sh*[Z,t*) + t h)
cosh kh r'r'*' u h
m=l
cos kmh
(2.39)
17
cosh k'(z + h!) ^ D _JfcLfiC_ncos A^fc + h*)
cosh 'A'
+ E
m=l
cos /i'
rrx
with
gk tanh kh = gkm tan kmh a2
gk' tanh k'h' = gk'm tan k'mh' = a
(2.40)
(2.41)
(2.42)
where A, and I?, are unknown complex coefficients. In their model, the series were
truncated at a certain point to make the number of unknown coefficients equal to
the number of the elements. Owing to the introduction of the series, the procedure
became much more complicated and the computation much more time consuming
than that of Ijima et al.
The advantage of a constant element is its simplicity and efficiency. But it has
an inherent problemawkward behavior around corner points. It has been found
that significant errors arise around such points even for the simplest casesinusoidal
wave propagating in a constant water depth over a impermeable bottom.
Sulisz (1985) employed linear elements in his BIEM model for subaerial porous
breakwaters. Eqs.(2.39) and (2.40) were adopted for the lateral boundary condi
tions. As a result, the formulation became awfully complicated. In modelling the
porous media, Sollitt and Crosss model was adopted. However, the inertia term
due to virtual mass effect was dropped as the corresponding coefficient Ca was not
known.
For submerged breakwaters, the modelling is still largely based on laboratory ex
periments and empirical formulae. For impermeable submerged breakwaters, quite
a few empirical formulae for transmission coefficient have been established from ex
perimental data by different authors (see Baba, 1986). For example, the equation
given by Goda (1969) for the transmission coefficient due to overtopping is
K, = 0.5[1sin ;Â£(/? A)]
(2.43)
18
with a = 2.2 and (3 0.0 ~ 0.8 for vertical face breakwaters, where F is the depth
of submergence of the breakwater crest and Hi is the incident wave height. Another
equation for impermeable submerged breakwaters was given by Seelig (1980):
Jf, = C(l)(l2C) (2.44)
where R is the wave run up given by Franzius (1965, cited in Baba, 1986)
R ii/C1(0.123^)(C3'/Fi7S+C3> (2.45)
Hi
with Ci = 1.997, C2 = 0.498 and C3 = 0.185. d is the total water depth and
C = 0.51 (2.46)
h
where B is the crest width and h is the height of the breakwater. Several other
empirical equations and methods are also available for designs.
The latest theoretical approach for impermeable submerged breakwaters was
given by Kobayashi et al. (1989) by using finite amplitude shallowwater equations.
The numerical results were found to agree well with the experimental data by Seelig
(1980) for such structures.
However, for permeable structures, no parallel empirical formula could be found
in the literature besides the nomograph by Averin and Sidorchuk (1967, cited in
Baba, 1986). The research for such structures is still in the stage of physical exper
iments.
Dick and Brebner (1968) carried out an experiment on solid and permeable
breakwaters of vertical faces. In the experiment, it was discovered that, over a
certain wave length range, the permeable breakwater was much better than the
solid one of the same dimension in terms of wave damping, and that the permeable
breakwater had a well defined minimum value for the coefficient of transmission. It
was also found that a substantial portion, 30% ~ 60%, of the wave energy of the
transmitted waves was transferred to higher frequency components. The equivalent
19
wave height for the transmitted waves in his paper was defined as
Hcq = ^~ j%2{t)dt (2.47)
Due to the smallness of the submergence of the breakwater crest, it is felt that
the majority of the waves in the experiment were breaking waves.
Dattatri et al. (1978) tested a number of submerged breakwaters of different
types, permeable, impermeable, rectangular and trapezoidal. One of the conclusions
was that the important parameters affecting the performance of a submerged break
water are the crest width and the depth of submergence. The transmitted waves
were found to be irregular though periodic. They also reported that porosity and
wave steepness did not have significant influence on the transmission coefficient,
which contradictory with the observations by Dick et al. (1968).
Seelig (1980) conducted a large number of tests on the cross sections of 17
different breakwaters for both regular and irregular waves. Most of the breakwaters
tested were rubblemound porous structures with multilayer designs. Beyond the
experiments for impermeable breakwaters by which Eq.(2.44) was obtained, he also
tested these permeable structures as submerged breakwaters. Since no generalized
model was available at the time for such breakwaters, the experimental data with
regular waves were compared to Eq.(2.44). It was found that the formula is quite
conservative in estimating the transmission coefficient for permeable submerged
breakwaters. The phenomenon of wave energy shifting to higher order harmonics
in the transmitted waves was again reported without further explanation.
Baba (1986) conducted an experiment on concrete submerged breakwaters and
compared the data with the four widely used computational methods for wave
transmission coefficient for impermeable submerged breakwaters. He concluded
that the formula given by Goda (1969) was the most suitable one in the case of a
shore protecting submerged breakwater.
CHAPTER 3
POROUS FLOW MODEL
Examining the existing porous flow models listed in the literature review, it
is noted that most of them contain one, two or all of the following components:
the linear velocity term, the quadratic velocity term and the term proportional to
the acceleration of the fluid. For the first component all the models give the same
definition while for the rest two terms, different models have different forms. To be
able to model the porous media with confidence, it is necessary to formulate the
porous flow from fundamental principles.
3.1 The Equation of Motion
The equation of motion for a solid body placed in an unsteady flow field can be
expressed in the following general terms:
Fp + Fd + Fj f Ft, Ff = ma (31)
where m = mass of the solid; a acceleration of the body; Fp = pressure force;
Fd = velocity related force also known as the drag force; F/ = accelerationrelated
force also known as inertial force; Fb = body force; and Fj = frictional force due to
surrounding solids. We now apply this equation to a control volume of a mixture
of fluid and solids, as shown in Fig. 3.1, and restrict our discussion to the two
dimensional case. The xaxis coincides with the horizontal direction and the zaxis
points vertically upward. The spatially averaged equation of motion for the solid
skeleton in the xdirection becomes
dP _____
 dx(l nAx)dz + Fdx + F/s + Fbtx FJz = p,( 1 n) ,x dxdz (3.2)
ox
where = pressure gradient in the xdirection; = averaged area porosity
20
21
dw
w + Tdz p +
oz
du
u + dx
ox
dp
w p
Figure 3.1: Definition sketch for the porous flow model
defined as the ratio of AVOid to Atotai with AVOid being the area of the voids and
Atotai being the total control area; n = volumetric porosity defined as the ratio of
Vvoid to Vtotai; Fb, is the body force of the solid; p, = density of solid; u, and ti, =
velocity and acceleration of the solid. The subscript x denotes the x direction and
the overbar denotes spatial average. Since, from this point on, we will be dealing
exclusively with spatially averaged values, the overbar will be dropped.
The force balance on the pore fluid can be established in a similar manner:
dP_
dx
dx n\z dz Fox Fix + Fb/Z = pndxdz (fx ,x)
(3.3)
with p being the density of the fluid and Fb/X being the body force of the fluid in x
direction; u/ and ti/ equal, respectively, the actual spatially averaged velocity and
acceleration of the pore fluid, defined as
uf = 3 f ua dv
Vvoi Jv.oU
(3.4)
22
where ua = actual pore velocity. If the fluid and solid mixture is now being treated
as a porous medium, and thus a continuum, we require that
1. All the field variables, such as that defined by Eq.(3.4), be independent of
the volume of integration.
2. 6 <Â§; L, where 6 and L are respectively, the length scales of the pore and the
system.
For the wave attenuation problem, Eq.(3.3) is of special interest. We introduce
here yet another field variable q representing the apparent spatially averaged fluid
velocity, which is related to u/ by
q = V L Ua dv = V L udv = n'uf (35)
V J V V J
here V is the total volume Vtotal.
This velocity also known as the discharge velocity, is related to the actual dis
charge over a unit surface area, or, Q/A. These apparent properties are of final
interest in engineering. Since we are now dealing with a continuum, we have
4 =
Dq
Dt
dq
+ q
dt 4
Vq ~
dq
8t
ri'iij
(3.6)
here it has been assumed that the convective acceleration is negligible.
The various force terms are established as follows:
(A). Drag Force
The drag force consists of two terms: that due to laminar skin friction and that
due to turbulent form drag; the former is proportional to the velocity and the latter
is proportional to the velocity squared. Their functional form, when expressed in
terms of the final field variables, may be written as
Fdz = [As{qx nutx) + Bx  q nu,  (q* nutz)]dx nAs dz (3.7)
The coefficients Ax and Bx are to be determined later.
23
(B). Inertia Force
The inertial force can be treated as an added mass effect and expressed in the
following form:
Fix = CX(q, nutx)dxnAx dz (3.8)
The coefficient Cx will be determined later.
(C). Body Force
In the pore fluid, there is no horizontal body force, and the vertical body force
is balanced by the static pressure gradient and, thus, vanishes. In the solid skeleton,
the vertical body force is the net weight. If the solid skeleton is subjected to an
unsteady vertical motion, this term should be included; otherwise, it can be ignored.
Substitution of Eqs.(3.5) through (3.8) into Eq.(3.3) results in the following
differential equation:
Bx  q nu,  (qz nu$x)
( + C'zXq* niits) (3.9)
n
This is the basic equation of pore fluid motion. In a general case, it is coupled
with Eq.(3.2) and must be solved simultaneously. Only a special case with no
movement of the solid skeleton will be analyzed here. For such a case, Eq.(3.9)
reduces to
^ = A.q, + B,qq* + (Â£ + CI)q, (3.10)
The force b'alance in the zdirection leads to
+ Bz i 9 I 9* + (Â£ + C*)4* 311)
3.2 Force Coefficients and Simplifying Assumptions
dP
dx
= Az(qxnu,x) +
+
The evaluation of force coefficients involves a great deal of empiricism. However,
successful engineering application relies heavily upon successful estimation of these
coefficients. The part of the flow resistance which is linear in q clearly will lead to
24
a Darcytype resistance law. The coefficient A is, at least, related to four founda
tional properties, one of the fluidthe dynamic viscosityand three of the porous
structure the porosity, the tortuosity (one that defines effective flow length) and
the connectivity (one that defines the manner and number of pore connections).
Obviously, the more those factors can be specified explicitly, the more accurately
the value of A can be determined. However, it is also generally true that the more
factors explicitly introduced the more restrictive the range of application. If none of
these four fundamental properties is expressed explicitly, A is simply the inverse of
Darcys hydraulic permeability coefficient. If the dynamic viscosity, n, is factored
out, we have the empirical law:
A = Y (3.12)
where Kp is known as the intrinsic permeability. A number of investigators including
Engelund (1953, cited in Madsen, 1974) and Bear et al. (1968, cited in Madsen,
1974) attempted to relate A to porosity as well and came up with the relationship
of the following type:
A = Co
(1 n)3 /i
n*
(3.13)
where d, is a characteristic particle size of the pore material. The coefficient clq
obviously still contains the other properties such as tortuosity and connectivity.
Since both of these factors have directional preference, ao should also be a directional
property and, in general, is a secondorder tensor. For isotropic material ad becomes
a scalar. Engelund (1953, see Madsen, 1974) recommended
oq = 780 to 1500 or more
(3.14)
with the values increasing with increasing irregularity of the solid particle.
Attempts were also made to sort out the effects of tortuosity and connectiv
ity (Fatt, 1956a,b). The conditions invariably become more restrictive and one is
required to specify soil characteristics beyond the normal engineering properties.
25
The characteristics of the coefficient B can be determined by examining the total
form drag resistance acting on a unit volume of granular material of characteristic
size d,. For this case, we have
m n
FdNx X) 2SCdAp I uÂ¡  u/s (3.15)
where Ap is the projected area of individual particles; Cd is the form drag coefficient
of the individual particles; 6 is a correction coefficient for Cd accounting for the
influence of surrounding paxticles; and m is the total number of particles per unit
volume. Since the total projected area should be proportional to (1 n)dxdz/d
we have, after substituting q for
. J ^
Fdnx = 2^D~^d~ I q I j* dxdz (3.16)
the constant of proportionality is absorbed in 8. Comparing Eq.(3.16) with the
second part of Eq.(3.7), we obtain
1 n
nAsn2d,
If we let:
Eq.(3.17) can be expressed as
(3.17)
(3.18)
S, = pba
1 n
nAxn2d,
(3.19)
This equation is very similar to that recommended by Engelund (1953), with
the exception that nA was replaced by n, the volumetric porosity, in his formula.
Again, because bo has directional preference, it should also be a tensor of second
order. Engelund recommended
60 = 1.8 to 3.6 or more
(3.20)
for granular material of sandsized particles.
26
The coefficient Cx is the overall added mass coefficient for the porous medium.
Its characteristics can be determined in much the same manner as those of Bz, or
m
Fix = T. pCgVyfx (321)
where C is the added mass coefficient of each individual particle and Vp is the
volume of each particle. One may then readily obtain, by comparing Eq.(3.21) with
Eq.(3.8),
Cx = p
Ca{ 1 n)
(3.22)
nAx n
Since Cx is related to the volume of the solid skeleton, one does not expect
significant directional preference, unless the geometry of the element deviates sig
nificantly from a sphere shape. A new coefficient, Cm, can be defined such that
,n + Ca(l n)
riAz
+ Cx=p[
nAxn
(3.23)
which has the same meaning as the mass coefficient in hydrodynamics.
Clearly the force coefficients in other directions can be defined in a similar
fashion. Equations.(3.10) and (3.11) can now be expressed in the following general
form:
q+ B  q  q + Cmq = Vp (3.24)
Here, the carat indicates a second order tensor. This equation cam be generalized
to include the motion of the solid simply by replacing q with q u,.
We now proceed to assume that the porous material is isotropic amd that all
the coefficients reduce to scalar quantities and
nAx = uax = n
Recall that the convective acceleration in q has been assumed, in Eq.(3.6), to be
negligible, i.e.
27
Substituting these conditions into Eq.(3.24), we obtain
{A + B  q l)q + cjÂ£ = Vp (3.25)
This equation is similar to the wellknown equation of motion in a permeable
medium such as given by Reid and Kajiura (1957) and others, with the exception
that the added mass effect is now formally introduced.
To seek a solution to this equation, the common approach is to ignore the
inertial term and linearize or ignore the nonlinear term. Under such conditions,
Eq.(3.25) can be reduced to the Laplace equation by virtue of mass conservation.
To retain the inertia term, the most convenient approach is to assume the motion
to be oscillatory, as is the case under wave excitation. Now we let
q = g(z, z)e~%at (3.26)
where g*is a spatial variable only, and a is a generalized wave frequency, which could
be a complex number. Substituting Eq.(3.26) into Eq.(3.25) leads to
 Vp = (A iaCm + B\q  )g (3.27)
A new set of nondimensional parameters are defined as follows:
Permeability parameter, R,
R = Kp
with
Inertia parameter, 0
'
F A a0 (1 n)3
0 = n + C1~n)
n2
Volumetric averaged drag coefficient, C
1 n
nsd,
(3.28)
(3.29)
(3.30)
(3.31)
28
Substituting these parameters into the above equation reduces it to the following
familiar form:
Vp = pa(^t/? + ?)g (3.32)
it a
This is essentially the same equation used by Sollit and Cross (1972) and others for
porous breakwater analysis.
3.3 Relative Importance of The Resistant Forces
Before solving Eq.(3.32) with an overlying wave motion, it is useful to examine
the importance of the resistance forces and to determine the nature of the fluid
motion in the porous medium. By taking ratios of the three respective resistance
forces, three nondimensional parameters can be established:
Inertial Resistance /,
Laminar Resistance//
n + C(l n)
ao(ln)
Turbulent Resistance fn
Laminar Resistance //
0
Oon(l n)2
R/
Turbulent Resistance /
Inertial Resistance /,
60(1 n) R/
Can(l n + J) Rt
where R/ and R/ are two forms of Reynolds number defined as
R/ =
gl d,
(3.33)
(3.34)
(3.35)
(3.36)
and
(3.37)
where v is the kinematic viscosity of the fluid.
Obviously both Reynolds numbers signify, the relative importance of the inertial
force to the viscous force. The origins of the inertial forces are, however, different.
In R/, the inertia is of a convective nature and the resistance arises due to change
of velocity in space (fore and aft the body) whereas in R, the inertia is of a local
29
nature and the resistance arises due to the rate of change of velocity at a specific
location. The ratio of R, and R/ is the Strouhal number, which clearly identifies
the different origins of the two inertial forces.
In the range of common engineering applications, the magnitudes of various
coefficients can be estimated as follows:
n 0.3 ~ 0.6
Ca ~ 0(1)
do ~ O(103)
bo ~ 0(1)
Therefore, Eqs.(3.33) through (3.35) reduce to
 10"2R,
n
fi
k
ft
k
fi
io_2r f
R
(3.38)
(3.39)
(3.40)
When Eqs.(3.38) to (3.40) are plotted on a R/ and R, plane as given in Fig. 3.2,
we identify seven regions where the three resistance forces are of varying degrees of
importance. There are three regions where only one resistance force dominates, that
is, the dominant force is at least one order of magnitude larger than the other two.
There is one region where the three forces are of equal importance. Then, there are
three intermediate regions where two out of three forces could be important.
Since both Reynolds numbers are flowrelated parameters, accurate position of
a situation within the graph cannot be determined a priori. However, a general
guideline can be provided with the aid of the graph or with Eqs.(3.38) through
(3.40). We give here an example of practiced interest. In coastal waters, we com
monly encounter wind waves with frequency in the order of 0(l)(rad/sec) and the
non dimensional bottom pressure gradient defined as V (p/7) in the order of 0(1O1).
30
R/
Under such conditions, the importance of the three resistance components for bot
tom materials of various sizes can be assessed. Table 3.1 illustrates the results.
Table 3.1: Illustration of Dominant Force Components Under Coastal Wave
Conditions: [a ~ 0(l)rad/sec, V(p/7) ~ O(l01)]
Description
Size
Range
Disch. Vel.
(m/sec)
R/
R,
Dominant
Force
Coarse sand
or finer
< 2 mm.
< 0(KT3)
< O(l)
laminar
Pebble,
or small
gravel
1 cm.
O(102)
o(io2)
O(l02)
laminar
turbulence
inertia
Large gravel
crusted stone
10 cm.
O(K)1)
O(104)
O(104)
turbulence
inertia
Boulder
crusted stone
0.3 1.0 m
0(10)
O(10)
0(10)
turbulence
inertia
Artificial
blocks,
large rocks
> 1.0 m
> 0(10)
> O(l06)
> o(io6)
turbulence
inertia
CHAPTER 4
GRAVITY WAVES OVER FINITE POROUS SEA BOTTOMS
We consider here the case of a small amplitude wave in a fluid of mean depth
h above a porous medium of finite thickness h,. The bottom beneath the porous
medium is impervious and rigid. The subscript s will be used here to denote vari
ables in the porous bed. The basic approach is to establish governing equations
for different zones separately and to then obtain compatible solutions by applying
proper matching boundary conditions. In the pure fluid zone, it is common to
assume the motion essentially irrotational except near the interface. Most of the
investigators neglected the influence of the boundary layer with the exception of Liu
(1973) and Liu and Dalrymple (1984) who included in their solutions two laminar
boundary layers at the mud line. Since the damping is largely due to the energy
losses in the porous medium rather than the boundary layer losses (Liu and Dal
rymple, 1984), the boundary layer effect will be ignored in this study to simplify
the mathematics.
4.1 Boundary Value Problem
The governing equation for the velocity potential function in the fluid domain
is
V2$ = 0 h < z < 0 (4.1)
In the porous medium domain, the linearization of the nonlinear pore pressure
equation, Eq.(3.32), yields
 VP, = pafoq (h + h,) < z < h (4.2)
where /0 is the linearized resistance coefficient.
31
32
<1
>
k Z
Lv
r / V o
i
i
i
h
'44t''84w'44''i"i'S'4'4'4w4'4'i
Â¡Â¡Â¡Porous seabed*Â¡*Â¡*Â¡*Â¡*Â¡*Â¡Â¡*Â¡Â¡*Â¡*Â¡*Â¡*Â¡*
oVAVoVoVoVoVoVoVoVoVoVo0
iWÂ¡iKW
, 0 0 0 0 0 Â£
;h,
5 S .0,0.0 00,
0 0 0 0 0 c
VoVoVoVc
jVoVoVoV
Figure 4.1: Definition Sketch
If the porous medium is rigid, thus, incompressible, the continuity equation for
the discharge velocity, V q = 0, leads to
V2P, = 0
(h + h,) < h
(4.3)
The boundary conditions of entire system are
T](x,t) = ^ = aei^x~(,t'> at 2 = 0
g at
(4.4)
a2$
dt2 + 9 dz~
at 2 = 0
(4.5)
a*
dz
3$
P dt ~F
1 dP,
pafo dz
at 2 = h
at 2 = h
(4.6)
(4.7)
33
 = 0 at z (h + h,) (4.8)
02
The potential and pore pressure functions axe assumed to have the forms
$(x,z,t) = [>lcosh/:(h + z) + Bsmhk[h + z)\e^kx~at^ (4.9)
P,(x,z,t) = D cosh k(h + h, + z)e^kx~ot^ (4.10)
where A, B and D are unknown complex constants. Equation (4.8) is already
satisfied by P, in Eq.(4.10). Introducing Eqs.(4.9) and (4.10) into Eqs.(4.6) and
(4.7), A and B can be expressed in terms of D,
t cosh kha
pa
(4.11)
sinh kh,
pofo
(4.12)
$ is then given by
[it 1 1 *
pa
D can be obtained by the free surface boundary condition given by Eq.(4.4):
$(x, z, t) = [: cosh kh, cosh k(h + z) + sinh kh, sinh k[h + z) ]c^fcx at^ (4.13)
pa Jo
D =
pga
cosh kh cosh kh,{l ttinh kh tanh kht)
Jo
(4.14)
Finally, Eq.(4.5) along with Eq.(4.13) gives the complex wave dispersion equa
tion:
a2 gk tanh kh = ~r tanh kht(gk a2 tanh kh) (4.15)
Jo
here either a or k, or both, could be complex, as well as the coefficient f0.
In the above equation, when setting Ca = CÂ¡ = 0, it becomes the dispersion
equation obtained by Liu and Dalrymple (1984). If C/ 0, as will be the case in
this study, the coefficient /0 can not be a known value a priori. It becomes another
unknown besides a or k, and therefore the procedure of solution will be different
from that employed by Liu and Dalrymple. Before solving Eq.(4.15), a few limiting
cases are examined here.
34
For the case of infinite seabed thickness and low permeability (laminar skin
friction dominates, /0 * ^), Eq.(4.15) reads
a2 gk tanh kh iR(gk a2 tanh kh) (416)
which is the same as obtained by Reid and Kajiura (1957). If R 0 as with an
impervious bed, Eq.(4.16) reduces to the ordinary dispersion relationship for a finite
water depth h. On the other hand, if the permeability approaches infinity, we have
R * oo, n >1.0, /3 1.0, /o + t
The dispersion relationship expressed in Eq.(4.15) can be shown as
a2 = gfctanh k(h + h,) (417)
Physically, this is the case of water waves over a finite depth h + hti or, the solid
resistance in layer h, vanishes.
Another limiting case is where the water depth approaches zero and waves now
propagate completely inside the porous medium. The dispersion relationship from
Eq.(4.15) becomes
ofc
a2 t^jtanh kh, (418)
Jo
It can be easily shown that the same dispersion relationship cam be obtained
by directly solving the problem of linear gravity waves in a porous medium alone.
Again, for the case of laminar resistance only, Eq.(4.18) becomes
a2 = iRgk tanh kh, (419)
which states the Darcytype resistance law. On the other hand, when R * oo, we
obtain from Eq.(4.18)
cr2 = gk tanh. kh (420)
or, as expected, the dispersion relationship in a pure fluid medium.
35
When the virtual mass and the drag coefficients are set to be zero, Eq.(4.15)
gives the dispersion equation for the homogeneous solution by Liu and Dalrymple
(1984) which is
cr2 gk tanh kh = = tanh kh,(gk a2 tanh kh)
4.2 The Solutions of The Complex Dispersion Equation
(4.21)
As mentioned in the previous section, Eq.(4.15) is an equation involving both
of the complex unknown variables of either a or k, and the complex coefficient
/0. The other necessary equation can be obtained from the linearization process
(This step is not necessary if Cj = 0, as in Liu and Dalrymples solution). The
common method of evaluating the linearized resistance coefficient, /o, is to apply
the principle of equivalent work. This principle states that the energy dissipation
within a volume of porous medium during a time period should be the same when
evaluated from the true system or from its equivalent linearized system
(Ed)i = {Ed)t
(4.22)
where Ed is the energy dissipation in a controlled volume during one wave period
and the subscripts / and nl refer to linearized and nonlinear systems, respectively.
It can be readily shown (Appendix A) that such energy dissipation (considered
as a positive value) can be expressed in the form of a boundary integral
rt+T r
ED = Jt js?ndsdt (4.23)
where Ed is a complex energy dissipation function and Jn is the complex energy flux
normal to S, with the real parts of them being the corresponding physical quantities.
Here S is the closed boundary of the computation domain. The complex energy
flux function 7n can be expressed as (Appendix A)
?n =
e2io,t
+ p')
(4.24)
36
where un is the normal velocity at the boundary S and p* is the conjugate of pore
pressure p, both of them are complex quantities; oT is the wave frequency, areal
value, and the subscript r is used to distinguish the complex a.
Physically, there are two classes of problems: standing waves of a specified wave
number, and progressive waves of a specified wave frequency. In the former case, k
is real, a is complex, and the solution has the following form
rÂ¡{xyt) = areoskxe~,a,t a < 0 (4.25)
where is the imaginary part of the complex a and ar is the wave frequency. In
the latter case, o is real, k is complex and rÂ¡ becomes
(*,!)= k,> 0 (4.26)
where kr and A:, are, respectively, the real and imaginary parts of k with kr being
the wave number.
For standing waves, the pore pressure function is
P, = p = D cosh k(h + h, + z) cos kxec
Since P, is periodic in z, the boundary curve S for the contour integral in
Eq.(4.23) can be chosen as z = 0, z (h + ht), x = L and z h with L being
the wave length.
As z = 0 and x = L are the antinodes of the standing wave, there is no normal
velocity, i.e. u = 0, on these two vertical boundaries and also at the impermeable
boundary z = (h + h,). Then
1 rt+T rL
Ed = ~ / (w0pe~2,errt + w0p') dxdt (4.28)
with
kD
paf
sinh kh, cos kx tait
tt>0 = n *=/ =
(4.29)
and
37
for linearized system
(4.30)
f = fo
f = + Nl
= /i + /2 w0  for nonlinear system (4.31)
with
fi = ^ *0 /* = ^ (432)
XL U
In Eq.(4.31), the magnitude of two is approximately taken as
wo 1=1 v) sinhfch,  (4.33)
0Jo
with
S(/o) = (4.34)
cosh /i cosh fc/i, (1 tanh kh tanh kht)
Jo
where the decaying wave amplitude ae{t has been approximately replaced by a,
which is the averaged wave amplitude in the period of [0,T].
With such approximations, / for the nonlinear system is no longer a function
of x and t. The approximations made above are not expected, to cause significant
errors for the final results.
By introducing Eq.(4.29) into Eq.(4.28) and carrying out the contour integration
regarding / as a constant, the energy dissipation within that onewavelong portion
of a seabed is obtained as
Ed = ^sinh2fch,[^p 71(f) + T2(f)] (4.35)
where Tj(f) and (f) are the nondimensional functions of t generated by the inte
gration with respect to time.
Substituting the linearized and the nonlinear resistance coefficients given in
Eqs.(4.30) and (4.31) for / in Eq.(4.35) respectively, equating the energy dissipations
38
by the two systems according to Eq.(4.22) and assuming that o and k axe the same
for both systems, we have
fo fo
(4.36)
Since this equation has to be satisfied at any time instant, we obtain the follow
ing equations:
D2(fo) = D2(fx + f2\wo\)
fo fl + fl I ^0 
(4.37)
1 D(fo) i2
fo
 P(/l + fj >0 I) l2
fl + fl I Wo I
(4.38)
Therefore,
fo fi + fi  Wo 
(4.39)
Substituting the expressions for fi, fi and too in the above equation, it becomes
Cd  kD(f0) sinh khs 
a  <7/01
fo=RiP +
(4.40)
It is clear, from the definitions of the parameters, that all the terms in Eq.(4.40)
Eire complex quantities in the standing wave case.
For progressive waves, the pore pressure function P, is given by Eq.(4.10) and
the contour S for the integration is the same as that for standing waves. The energy
dissipation is
By substituting the expression of Jn derived from Eq.(4.10) into the above equa
tion, the summation of the first two integrals is found to be 0( i,1) while the last
39
one is O(l). If we assume that  fc, c 1, which is generally true in coastal waters
(see the results for the progressive wave case), then the energy dissipation becomes
Ed = J Jndxdt (442)
Eq.(4.42) will clearly lead to the same relationship as that given by Eq.(4.40)
except that the averaged wave amplitude a in D, in this case, is the spatially
averaged value of ae~k'x over [0,1/].
In the progressive wave case, we assume that the wave period is given and that
there is no time dependent damping, i.e. a is real. Therefore, the first and the last
term on the R.H.S. of Eq.(4.40) are all real numbers. It is then obvious that
(/o)i = 0 (4.43)
and
,,, 1 Cd\kD(kJa)
(/o)r R+ <7 I oh I
(4.44)
where the subscripts i and r are for imaginary and real parts, respectively.
To actually compute /0 requires specification of two fundamental quantities of
n and d, (see Eq.(4.40) and the definitions for R, C and /?). It is often more
convenient to specify the permeability parameter R = oKpÂ¡u (R,. arKPlu for
standing waves), as opposed to specifying the actual granular size, d,. Under these
circumstances, we could replace Cd with the following expression (Ward, 1964):
fa
Cf Cf
d~ VSVl/
with
C7 =
&o(l n)
(4.45)
(4.46)
 n)s
Now Cj is a nondimensional turbulent coefficient. To be consistent with the
suggested values of a0 and b0 given in Eqs.(3.14) and (3.20), Cf should be in the
range of 0.3 ~ 1.1 for a porosity of n = 0.4. Equation (4.40) now becomes:
1 .a Cf \ kD{k, fo) sinh kh [
R yjRov  crfo 
(4.47)
40
This is the final form of the equation used for computing /0 for both cases.
The dispersion equation, Eq.(4.15), which is coupled with Eq.(4.47), is solved
iteratively. The procedures to obtain the solution for the cases of standing waves
and progressive waves are different.
a) For standing waves a ar + t<7, is complex, k is real and known. Equation
(4.15) can be rewritten as
a2 =
pfc(tanh kh tanh kht)
Jo
1 tanh kh, tanh kh
Jo
= Qr{fo) + Qi{fo)
(4.48)
with Qr{fo) and <5(/o) being the real and imaginary parts of o2, respectively. The
iteration procedures are summarized as follows:
^ i JqUF)+o/F)+wh )
*Â¡n+,) = I jQHfP) + QKfP) Q'ifP) r}
w i c,
f R ,i3+Vrw laW/^1
= 5*
(4.49)
(4.50)
(4.51)
(4.52)
where the superscript n denotes the level of iteration, and the criterion of conver
gence is
An) Ani)
Jo Jo
ft
(n)
l< e
and
aW a(n1)
oM
l< e
(4.53)
with e being a prespecified sixbitrajily small number. It is set to be 1.0% in this
model.
b) For progressive waves k = kr + is complex and a is real and known. In
this case, the dispersion equation can be written as
F(k, fo) = c2 gk tanhk/i+ j tanh kh,[gk a2 tanh kh) = 0 (4.54)
Jo
and the iteration was carried out as follows:
0
(4.55)
41
Fi(4(+1),/<)) = O
,M 1 ... 1 C,  4"1)
lo R ,p+VRV^ k/<"> I
& =
(4.56)
(4.57)
(4.58)
where Er and F{ are the real and imaginary parts of F and n indicates the iteration
level.
The criterion of convergence for such case is
, n) 
Si
jfc(n) _
< e and  ^ l<
(4.59)
with e being a prespecified arbitrarily small number. It is again set to be 1.0% in
this model.
4.3 Results
The predicted damping rates kip from the solution of Eqs.(4.15) and (4.47) are
first compared to the laboratory data kim by Savage (1953) for progressive waves
propagating over a sandy seabed. The experiment was conducted in a wave tank of
29.3 meters (96 ft) long, 0.46 meters (1.5 ft) wide and 0.61 (2 ft) meters deep. The
porous seabed was composed of 0.3 meters thick of sand with the medium diameter
of 3.82 mm. The water depths are h = 0.229 m. and h = 0.152 m. The data
for the water depth of h 0.102 m was ignored for the reasons given by Liu and
Dalrymple (1984). The wave conditions and the comparison of the damping rates
are listed in Table 4.1. The parameters used in the solution of Eqs.(4.15) and (4.47)
are: n 0.3, clq = 570, b0 = 2.0 and C = 0.46 and the average wave height H in
the table was calculated according to
*= H
In
Ho_
HLl
where Hq and Hl3 are the wave heights measured at two points of Lg apart with
Lg 18.3 m (60 ft). The values computed by Liu and Dalrymple (1984) are also
42
listed. The relative error in the table is defined as
A% = ki 7  X100% (4.60)
k\m
with ki being either kip or knD.
In this case, the errors are of similar magnitude between the present model and
the model by Liu and Dalrymple (1984). This is because the experimental values
fall in the region where the inertial resistance due to the virtual mass effect and the
turbulent resistance, neglected in Liu and Dalrymples model, axe unimportant.
In Fig. 4.2 through Fig. 4.5, the solutions of the complex dispersion equation are
illustrated graphically. The case of progressive waves are demonstrated first. The
specified conditions for the progressive waves are: H = 1 m, T = 4 seconds, h, 5
m and n = 0.4. Figure. 4.2 plots the values of kr (wave number) and kÂ¡ (damping
rate) against R (nondimensional permeability parameter) for three different water
depths, h = 2, 4, and 6 meters. The equivalent particle sizes for the range of R
values are also shown in the figure; they cover a range from 2.3 mm to 2.3 m.
The thick dash line is the solution of the dispersion equation given by Liu and
Dalrymple (1984, Eq.(4.5)) for the case of h = 4 m. The correction term of the
laminar boundary layer was not included in this curve since it is negligible in this
case. From these results, a number of observations can be made:
1. As expected, the wave number decreases (or wave length increases) monoton
ically with increasing R, from one limiting value kTi, corresponding to the case of
an impervious bottom at depth h, to the other limiting value kTÂ¡, corresponding to
the case of a water depth equal to h + ht when the lower layer becomes completely
porous.
2. Compared with the linear resistance, the nonlinear and the inertial resistances
dominate for the complete range of R values displayed. In the region where only
linear resistance dominates (R; < 1, R, < 1), the bottom effects are relatively small
and often negligible (outside the R range shown).
43
Table 4.1: Comparison of The Predictions and The Measurements
Run No.
H (cm)
T (sec.)
kim{m l)
kip(m 1)
Ap%
kiLD{m *)
A id%
h = 0.229 m
1
6.74
1.27
0.0379
0.0313
17.4
0.0338
10.8
2
4.45
0.0318
0.0337
6.0
0.0338
6.3
3
5.26
0.0303
0.0328
8.3
0.0338
11.6
4
1.93
0.0317
0.0369
16.4
0.0338
6.3
12
6.65
0.0334
0.0313
6.3
0.0338
1.2
13
4.36
0.0283
0.0338
19.4
0.0338
19.4
14
1.90
0.0374
0.0370
1.1
0.0338
9.6
5
6.25
1.00
0.0411
0.0350
14.8
0.0390
5.1
6
4.66
0.0357
0.0372
4.2
0.0390
9.2
7
2.17
0.0397
0.0411
3.5
0.0390
1.8
8
2.08
0.0393
0.0413
5.9
0.0390
0.8
15
5.48
0.0383
0.0360
6.0
0.0390
1.8
16
4.52
0.0397
0.0374
5.8
0.0390
1.8
17
1.93
0.0373
0.0415
11.3
0.0390
4.5
9
5.29
0.80
0.0296
0.0314
6.1
0.0343
15.8
10
3.55
0.0297
0.0334
12.5
0.0343
15.5
11
2.28
0.0335
0.0351
4.8
0.0343
2.4
18
7.05
0.0264
0.0296
12.1
0.0343
29.9
19
4.08
0.0320
0.0328
2.5
0.0343
7.2
20
2.14
0.0305
0.0353
15.7
0.0343
12.5
h = 0.152 m
21
4.00
1.27
0.0552
0.0574
3.9
0.0600
8.7
22
1.83
0.0379
0.0640
68.9
0.0600
58.9
23
3.83
0.0555
0.0579
4.3
0.0600
8.1
24
1.36
1.00
0.0449
0.0770
71.5
0.0731
62.8
25
3.40
0.80
0.0658
0.0700
6.4
0.0767
16.6
k{m the experimental damping rate given by Savage (1953);
kip the theoretical values by the present model;
kuD the theoretical value by Liu and Dalrymple (1984)
44
3. The wave attenuation, and hence the wave energy dissipation, shows a peak.
This peak occurs when the magnitude of the dissipative force (velocity related)
equals to that of the inertial force (acceleration related). Depending upon the water
depth, this attenuation could be quite pronounced under optimum R values. For
instance, for the case h = 4 m, Fig. 4.2 shows the wave height is reduced to about
74% of its original value (or about 46% energy dissipation) over approximately 2
wave lengths.
4. The locations of the peak damping are quite different from that of Liu
and Dalrymple (illustrated here for the case of h= 4 m); they occur at a higher
permeability. The magnitude of the peak damping is generally smaller than that
of Liu and Dalrymples solution. The values of kr from the two solutions are also
different.
In Fig. 4.3 the maximum damping rate and the corresponding permeability
parameter R are plotted as functions of nondimensional water depth. The curves
from Liu and Dalrymple (1984) are also plotted for comparison. The trends of
(&)maxs are similar but the corresponding R behaves quite differently from the two
solutions.
The case of a standing wave system is also illustrated here in Fig. 4.4. The
behavior is very similar to that of the progressive waves. The grain size in the figure
is calculated according to aT for the case of h = 4.0 m. Finally, the solutions of the
dispersion equation for standing waves based on Darcys model (DARCY: 0 = 0, Cj
= 0), Dagans model (DAGAN: 0 = CÂ¡ = 0), DupuitForchheimers model (D F:
0 = 0 and Cf follows Eq.(4.46) and SollittCrosss model (S C: 0 follows Eq.(3.30)
with Cj following Eq.(4.46) and n = 0.4, Oq = 570, 60 = 3.0 and Ca = 0.46) for
standing waves are compared in Fig. 4.5. Under the same wave conditions, the
differences among the various solutions are seen to be very pronounced, for the
permeability range displayed here. When the permeability parameter becomes less
HAVE DAHPINC RATE WAVE NUMBER
45
PARTICLE SIZE (CM)
0.23 2.32 23.20 232.00
0.23 2.32 23.20 232.00
(b)
Figure 4.2: Progressive wave case, (a) Nondimensional wave number kr/(c*/g), (b)
Nondimensional wave damping rate fc,/(<72/^).
log(J?) and log [(A:,) max l("2/9)\
46
Figure 4.3: Maximum nondimensional damping rate {o* jg) and its corre
sponding permeability parameter R as functions of nondimensional water depth
h{o2/g).
47
than 102, all solutions converge to a single curving following Darcys law. For very
highly permeable seabeds, the damping rate based on D F model is very high and
tends to increase with the permeability as oppose to approaching to zero according
to common sense. Also in the high permeability region, the wave frequency based
on Darcys model is much larger than the correct value (determined by Eq.(4.17)).
The reason for such large error is that the force balance of the pore fluid in highly
permeable media is now mainly between pressure gradient and the inertia rather
than the velocity related frictions.
WOVE DAMPING RATE WAVE FREQUENCY
48
PARTICLE SIZE (CM)
(a)
0.23 2.28 22.69 225.MM
Figure 4.4: Standing wave case, (a) Nondimensional wave frequency crr/(L/g); (b)
Nondimensional wave damping rate Ci/(L/g)t.
WAVE DAMPING RATE
49
PARTICLE SIZE (CM)
0.23 2.28 22.69 225.44
PERMEABILITT PARAMETER LOG (R)
(a)
0.23 2.28 22.69 225.44
(b)
Figure 4.5: Solutions based on four porous flow models, (a) Nondimensional wave
frequency or/{LÂ¡g)h, (b) Nondimensional wave damping rate
CHAPTER 5
LABORATORY EXPERIMENT FOR POROUS SEABEDS
The experiment was carried out in a wave flume in the Laboratory of Coastal
and Oceanographic Engineering Department of University of Florida. The flume
is about 15.5 meters long, 0.6 meters wide, 0.9 meters high and equipped with a
mechanically driven pistontype wave maker. All the tests were conducted with
standing waves.
5.1 Experiment Layout and Test Conditions
Figure 5.1 shows the experiment arrangement. A porous gravel seabed was
constructed at the end of the wave flume in the opposite side of the wave maker. A
sliding gate was positioned at one wavelength from this end of the tank to trap the
standing wave after a sinusoidal wave system was established by the wave maker.
The decay of the freely oscillating standing wave was then measured by a capacitance
wave gage mounted at the center of the compartment. The damping rate of the
porous seabed for each particular test condition was determined by applying a least
squares fit to the data according to the following equation:
.(o<)i = sln(#) (5.1)
Ij Mji
where j refers to jth wave. The corresponding wave frequency was obtained by
averaging individual waves.
The contribution to the wave damping due to the side walls and the bottom
was subtracted from the data according to the following equation
<*i = ~ ln[l + e7o<>7 e7^7} (5.2)
50
51
h
h,
wave
maker
Figure 5.1: Experimental setup
Table 5.1: Material Information
so (cm)
0.72
0.93
1.20
1.48
2.09
2.84
3.74
porosity
0.349
0.349
0.351
0.359
0.369
0.376
0.382
where a, is the actual seabed damping rate, cr,^ is the gross damping rate of the
seabedwall system, aiw is the damping rate by the side walls and the bottom, and
T is the wave period.
The bed material used in the experiment was river gravel of seven sizes, ranging
from so = 0.72 cm to 3.47 cm, with all sizes having a fairly round shape and smooth
surface. The porosity of the material increases slightly with the diameter, as given
in Table 5.1.
With the grain sizes being determined by the material selection, the adjustable
independent parameters left in the dispersion equation are the water depth h, the
seabed thickness h and the wave length L. A total of 36 different cases was tested
52
Table 5.2: Test Cases
50 (cm)
h, (cm)
h (cm)
L (cm)
0.72
20
20,25,30
200
0.93
20
20,25,30
200
1.20
20
20,25,30
200
1.48
10,15,20
20,25,30
200
2.09
20
20,25,30
200,225,250,275
2.84
20
20,25,30
200
3.74
20
20,25,30
200
and these test conditions are summarized in Table 5.2.
The damping effect due to side walls alone was found to be very small, and an
average value of oltu = 0.01 sec1 was used for all the corrections.
5.2 Determination of The Empirical Coefficients
The experimental results for the conditions of ht = 20 cm and L 200 cm
are given in Table 5.3 (complete results are given in Appendix B). Each data point
represents an averaged value of 10 to 20 tests.
Figure 5.2(a) shows a typical example of measured waves and the exponential
fitting to the wave heights. The solid line in Fig. 5.2(a) is the ensemble average
whereas the two dash lines are the envelopes of plus and minus one standard devi
ation from the mean value at each time instant. From the wave form, it is noted
that the waves in the experiment were more or less nonlinear waves as oppose to
what was assumed. The effect of the nonlinearity to the calculation of the measured
damping rate can be minimized by using the ratio of wave heights instead of wave
amplitudes. In Fig. 5.2(b), the dashed curve is the exponential decay function with
the a, obtained from the data ensemble by the least square analysis.
Based on the measured cT and a, listed in Table 5.3, the empirical coefficients
do, b0 and Ca were then determined by multivariate linear regression analysis such
53
Table 5.3: Measured oT and for ht = 20 cm and L = 200 cm
h (cm)
d50 (cm)
H\ (cm)
H' (cm)
cr.i(s l)
cr,(s x)
0.72
11.08
6.49
4.9054
0.0686
0.0570
0.93
10.93
6.35
4.9402
0.0678
0.0562
1.20
10.47
5.52
4.9370
0.0814
0.0694
30.0
1.48
9.03
4.71
4.9620
0.0878
0.0757
2.09
9.75
4.85
4.9673
0.0948
0.0824
2.84
9.80
4.95
5.0111
0.0931
0.0808
3.74
9.80
5.71
5.0341
0.0658
0.0543
0.72
8.70
5.15
4.6218
0.0799
0.0678
0.93
8.51
4.75
4.6856
0.0905
0.0781
1.20
8.00
4.46
4.6793
0.0984
0.0858
25.0
1.48
7.28
3.91
4.7352
0.1162
0.1030
2.09
6.46
3.57
4.7422
0.1105
0.0975
2.84
7.37
4.34
4.7893
0.0940
0.0816
3.74
7.37
4.26
4.7865
0.0813
0.0693
0.72
6.70
4.19
4.2716
0.1184
0.1047
0.93
6.28
3.55
4.3426
0.1334
0.1192
1.20
6.07
3.60
4.3565
0.1320
0.1179
20.0
1.48
4.49
2.57
4.4027
0.1484
0.1337
2.09
5.19
3.12
4.4325
0.1546
0.1396
2.84
5.41
3.33
4.4728
0.1456
0.1311
3.74
5.56
3.58
4.4646
0.1313
0.1173
* Hi is the average value of the first wave heights of the data group and H is the
average height of the complete train.
54
TIME (SEC)
Figure 5.2: Typical wave data: (a) Averaged nondimensional surface elevation
{t)/Hi), (b) Nondimensional wave heights (H/H7) and the best fit to the expo
nential decay function.
55
that the error function defined as
M
=, = T7E[(7''
)? + (
m G'
P\ 2
)?i
(5.3)
/=i &rm &im
was minimized, where the subscript m represents the measured values and p denotes
the predicted values; M is the number of data points. The best fit was found to
exist when:
oo = 570
60 = 3.0
Ca = 0.46
The added mass coefficient obtained here, Ca 0.46, is close to the theoretical
value of 0.5 for a smooth sphere. The values of a0 and b0, to an extent, reconfirm
those given by Engelund (1953) and the others.
5.3 Relative Importance of The Resistances in The Experiment
Introducing a0) >o, Ca, and the averaged wave height H into Eq.(4.29) for  qn \
and using Eqs.(3.33) through (3.35), the relative importance of the various resis
tances of the tested cases can be established precisely. The results are given in
Table 5.4. As we cam see from these ratios, for the first four grain sizes, all three
resistances are about equally important. Whereas for the last two larger diameter
materials, the turbulent and the inertial resistances are evidently dominant over the
linear resistance, with the inertial force approximately double that of the nonlinear
resistance.
5.4 Comparison of The Experimental Results and The Theoretical Values
Table 5.5 shows the comparisons between experimental results and the theoret
ical values of aÂ¡ and aT. The relative error, defined as
A% = ZOZ  X100%
(5.4)
Table 5.4: Comparison of The Resistances, h, = 20 cm, L = 200 cm.
d50 (cm)
1 7 1 (cm/s)
a 
R/
R,
fi/fi
fn/fl
fn/fi
h =
30.0 cm
0.72
0.57
4.788
37
225
0.93
1.32
1.42
0.93
0.69
4.800
58
379
1.56
2.06
1.32
1.20
0.75
4.819
81
630
2.63
2.88
1.10
1.48
0.78
4.842
104
960
4.18
3.71
0.89
2.09
0.96
4.869
181
1933
8.90
6.48
0.73
2.84
1.09
4.889
281
3577
17.12
10.10
0.59
3.74
1.33
4.900
453
6240
30.90
16.34
0.53
h =
25.0 cm
0.72
0.53
4.526
34
213
0.88
1.21
1.38
0.93
0.62
4.545
52
358
1.48
1.85
1.25
1.20
0.70
4.567
76
597
2.49
2.71
1.09
1.48
0.75
4.596
101
915
3.99
3.60
0.90
2.09
0.84
4.639
159
1833
8.44
5.70
0.67
2.84
1.10
4.657
284
3407
16.31
10.21
0.63
3.74
1.18
4.679
400
5959
29.51
14.43
0.49
h =
20.0 cm
0.72
0.50
4.180
32
196
0.81
1.14
1.41
0.93
0.55
4.206
46
332
1.37
1.64
1.20
1.20
0.66
4.232
72
553
2.31
2.56
1.11
1.48
0.61
4.286
81
849
3.70
2.89
0.78
2.09
0.85
4.322
160
1707
7.86
5.73
0.73
2.84
1.00
4.356
258
3187
15.26
9.27
0.61
3.74
1.15
4.381
393
5579
27.63
14.18
0.51
57
Table 5.5: Comparison of Measurements and Predictions for h, = 20 cm, L = 200
cm.
50 (Cm)
R
Orm{is *)
<7rp(s *)
Ar%
<7.m(s l)
A i%
h 30.0 cm
0.72
0.1791
4.9054
4.7880
2.39
0.0570
0.0541
5.07
0.93
0.3021
4.9402
4.8001
2.84
0.0562
0.0628
11.60
1.20
0.5110
4.9370
4.8187
2.40
0.0694
0.0706
1.66
1.48
0.8448
4.9620
4.8419
2.42
0.0757
0.0751
0.76
2.09
1.8755
4.9673
4.8685
1.99
0.0824
0.0734
11.00
2.84
3.7429
5.0111
4.8891
2.43
0.0808
0.0661
18.26
3.74
6.9543
5.0341
4.9001
2.66
0.0543
0.0613
12.85
h 25.0 cm
0.72
0.1687
4.6218
4.5264
2.06
0.0678
0.0709
4.49
0.93
0.2866
4.6856
4.5445
3.01
0.0781
0.0840
7.49
1.20
0.4843
4.6793
4.5666
2.41
0.0858
0.0935
8.97
1.48
0.8095
4.7352
4.5963
2.93
0.1030
0.1000
2.94
2.09
1.7819
4.7422
4.6389
2.18
0.0975
0.0956
1.94
2.84
3.5772
4.7893
4.6566
2.77
0.0816
0.0912
11.76
3.74
6.6123
4.7865
4.6795
2.24
0.0693
0.0781
12.67
h = 20.0 cm
0.72
0.1559
4.2716
4.1799
2.15
0.1047
0.0907
13.34
0.93
0.2656
4.3426
4.2061
3.14
0.1192
0.1109
6.96
1.20
0.4509
4.3565
4.2318
2.86
0.1179
0.1228
4.16
1.48
0.7496
4.4027
4.2857
2.66
0.1337
0.1319
1.32
2.09
1.6656
4.4325
4.3217
2.50
0.1396
0.1308
6.33
2.84
3.3408
4.4728
4.3560
2.61
0.1311
0.1200
8.42
3.74
6.1676
4.4646
4.3805
1.88
0.1173
0.1076
8.24
is uniformly very small for or (wave frequency). For cq (wave damping), the range
of error is larger, with the maximum being 18.26% of the measured value.
The agreement between the predictions and the measurements shown in the
above table is demonstrated by Fig. 5.3 where the values of or and a, in Table 7 for
the case of h = 25 cm are plotted against the permeability parameter.
WAVE FREQUENCY (1/S)
58
10.0
8.0
6.0
4.0
2.0
0.0
WRVE LENGTH = 200 CM
BED THICKNESS = 20 CM
HATER DEPTH = 25 CM
& a 6 6 6 *
t 1 1 1 1 : r
1.0 0.5 0.0 0.5 1.0
PERMEABILITY PARAMETER LOG (R)
cx
cc
cr
Q
a:
3
0.120
0.100 .
. $ .
0.080 .
0.060 .
O
+
O + 4
+
O PREDICTED
0
+
0.040 .
0.020 .
x MEASURED
0.000 .
i
1.0
1 1 i 1 1
0.5 0.0 0.5
PERMEABILITY PARAMETER LOG (R)
i
1.
Figure 5.3: The Measurements and the predictions vs. R. for L = 200.0 cm,
h, = 20.0 cm, h = 25.0 cm. (a) Wave frequency ar, (b) Wave damping rate er,.
59
Table 5.6: Comparison of om and ap for d50 = 1.48 cm, L = 200 cm.
h (cm)
H (cm)
<7rm(>S X)
*)
Ar%
OVm(s J)
otAs1)
A,%
h, = 15.0 cm
30.0
5.29
4.9268
4.8291
1.98
0.0525
0.0600
14.13
25.0
4.00
4.6715
4.5836
1.88
0.0697
0.0794
13.98
20.0
3.12
4.3173
4.2565
1.41
0.0928
0.1054
13.56
h, = 10.0 cm
30.0
5.97
4.9163
4.8145
2.07
0.0368
0.0416
13.13
25.0
4.61
4.6310
4.5629
1.47
0.0492
0.0551
11.92
20.0
3.48
4.2688
4.2309
0.89
0.0724
0.0729
0.70
Table 5.7: Comparison of
om and op
for d5C
= 2.09 cm, h, = 20 cm.
h (cm)
H (cm)
0rm{s *)
^rp(51)
A T%
<7,m(s *)
A i%
L = 225.C
cm
30.0
4.42
4.5114
4.4401
1.58
0.0831
0.0766
7.73
25.0
3.91
4.2913
4.1991
2.15
0.0977
0.1006
2.92
20.0
2.96
3.9699
3.8995
1.77
0.1270
0.1286
1.21
L = 250.C
cm
30.0
4.40
4.1524
4.0725
1.92
0.0752
0.0790
5.02
25.0
3.79
3.9119
3.8358
1.94
0.1010
0.1002
0.81
20.0
3.32
3.5862
3.5361
1.40
0.1384
0.1282
7.34
L = 275.C
cm
30.0
4.18
3.8371
3.7592
2.03
0.0783
0.0788
0.68
25.0
3.32
3.6285
3.5339
2.61
0.0993
0.0962
3.15
20.0
2.90
3.3298
3.2497
2.41
0.1264
0.1211
4.16
We now proceed to compare the theoretical computations with those data which
were not included in the calibration. These results are given in Table 5.6 and
Table 5.7. The agreement is just as good, if not better.
In Fig. 5.4, the theoretical values of oys and cr,s by the present model are
plotted, respectively, against the measurements for all the experimental data (36
cases). The overall least mean square error defined in Eq.(5.3) for the data of all 36
cases is about 0.008. While the solution of the dispersion equation (Eq.(4.2l) based
on the linear porous flow model by Liu and Dalrymple (Liu and Dalrymple, 1984)
60
Table 5.8: Comparison of crm and ov for d50 = 0.16 mm, h, = 20 cm, L = 200 cm.
h (cm)
R
Ms l)
Ar%
0Vm(s i)
M5 l)
30.0
5.4 x KT7
4.8908
4.7638
2.60
0.0070
4.5 x 10"7
25.0
5.1 x 10'7
4.6312
4.4957
2.93
0.0090
5.6 x 10'7
20.0
4.8 x 107
4.2824
4.1428
3.26
0.0122
6.8 x 10"7
is plotted in Fig. 5.5. In obtaining the solution, the only empirical coefficient oo
appeared in their dispersion equation was calibrated to the data in Table 5.3 with
the same routine as that for the solution of Eq.(4.15), and the nonlinear regression
yielded a0 = 2200. The overall least mean square error for the complete data set
(36 cases) is as high as 0.327. It can be seen from the figure that considerable
discrepancies occur for the damping rate, and the maximum relative error is about
87% of the experimental values. The prediction of wave frequency by their solution
is about the same as that by this study.
A test over a sand bed (d50 = 0.16 mm) was also conducted. Active sand move
ment was observed along with the rapidly formed ripples. As a result, theoretical
predictions are several orders of magnitude smaller than those actually measured
(Table 5.8). The basic assumption of no particle movement was violated. How
ever, the predictions of wave frequencies were as good as the coarsegrained cases.
The same test, with dye injection into the sand bed, disclosed that the flow inside
the sand bed was very slow, and indicated that the assumption of impermeability
invoked in the linear wave theory is justified in such situations.
When the thickness of a gravel bed is reduced to one grain diameter, the model
also failed to yield good results as shown by the values in Table 5.9. In this case, the
effect of the boundary layer at the interface, which was not included in the model,
obviously cannot be ignored.
The nature of a turbulent boundary layer over a thicker seabed was qualitatively
investigated for the case of ht= 20 cm, dso = 148 cm, h =20 cm and L =200 cm,
61
id
o
cc
Li
CD
C3
0
o
EXPERIMENTAL FREQUENCY
EXPERIMENTAL DAMPING RATE
Figure 5.4: Theoretical values by the present model vs. experimental data of Table
5.5, Table 5.6 and Table 5.7. (a) Wave frequency ar, (b) Wave damping rate a.
COMPUTED DAMPING RATE COMPUTEO FREQUENCY
62
EXPERIMENTAL FREQUENCY
0.25
0.20
0. 15
0. 10
0.05
0.00
0.00 0.05 0.10 0.15 0.20 0.25
EXPERIMENTAL DAMPING RATE
Figure 5.5: Theoretical values by the model of Liu and Dairymple vs. experimental
data of Table 5.5, Table 5.6 and Table 5.7. (a) Wave frequency ar, (b) Wave damping
rate cr,.
63
Table 5.9: Comparison of om and ap for h, = d50
d50 (cm)
H (cm)
^rm(>S *)
X)
Ar%
X)
A,%
h 30.0 cm
0.72
6.05
4.9157
4.7666
3.03
0.0024
0.0031
29.66
1.20
5.88
4.9374
4.7723
3.34
0.0040
0.0041
3.04
2.09
5.58
4.9559
4.7827
3.50
0.0075
0.0042
43.69
3.74
5.21
4.9808
4.8002
3.63
0.0097
0.0049
48.99
h = 25.0 cm
0.72
4.64
4.6495
4.4991
3.23
0.0031
0.0041
31.58
1.20
4.53
4.6675
4.5068
3.44
0.0055
0.0056
0.96
2.09
4.28
4.6822
4.5205
3.45
0.0100
0.0056
43.88
3.74
4.02
4.7318
4.5438
3.97
0.0130
0.0064
50.59
h = 20.0 cm
0.72
3.24
4.2936
4.1470
3.41
0.0057
0.0053
7.19
1.20
3.23
4.3134
4.1571
3.62
0.0094
0.0075
20.32
2.09
3.07
4.3339
4.1755
3.66
0.0147
0.0074
49.62
3.74
2.88
4.3950
4.2061
4.30
0.0170
0.0081
52.29
by dye studies. Turbulent diffusion was spotted over virtually the entire porous
domain and up to approximately 1.0 cm above the interface. The effects of turbulent
boundary layers to the wave damping are important in wave interaction with porous
seabeds and more theoretical and quantitative experimental research is needed.
CHAPTER 6
BOUNDARY INTEGRAL ELEMENT METHOD
The Boundary Integral Element Method (BIEM) is an efficient numerical method
for conservative systems such as those governed by Laplace equation. It is efficient
because the computation is carried out only on the boundaries rather them over the
entire domain as it would be in finite difference and finite element methods. Consid
erable amount of computation time and storage space can be saved without losing
accuracy. In addition to the efficiency, the data preparation becomes much simpler
as compared to the other two methods. For nonLaplace equations, the application
of this method may become difficult and sometimes impossible. By the efforts of
many scientists, the scheme has been extended to more and more problems governed
by nonLaplace equations (Brebbia, 1987). In this chapter, we restrict ourselves to
the two dimensional Laplace equation only.
6.1 Basic Formulation
Let U and V be any two continuous two dimensional functions, twice differen
tiable in domain D which has a close boundary C. According to the divergence
theorem (Loss, 1950; Franklin, 1944, cited in Ligget and Liu, 1983) for continuity
in a volume
J (V v)dA j> v nds (6.1)
where V is the vector operator, v is any differentiable vector and ft is the unit
outward vector normal to C. If we define that v = TJW and then v = WU in
Eq.(6.1), two integral equations can be generated in terms of U and V,
W + UVV)dA = Â£ UVV ads
(6.2)
65
jjyv VU + VV2U)dA = jcVVU fids
Subtracting (6.3) from (6.2), we have
[ {UV2V W2U)dA = {UVV VVU) fids
Jd jc
Introducing the expressions
dV dU
Wn = ^ VZ7 n =
on on
and assuming that both U and V satisfy Lapalce equation, i.e.
V2U = V2F = 0
(6.3)
(6.4)
(6.5)
(6.6)
Equation (6.4) becomes
L^v^ds <67>
To apply this equation, we choose U as the velocity potential $ and V as a free
space Greens function, G. Both of them satisfy Lapalce equation. Equation (6.7)
can then be rewritten, in terms of $ and G, as
/C1W)  g(p, I** = 0 (6s)
where P is a point in the domain DnC and Q is a point on C.
One of the free space Greens functions for two dimensional problems is
G(P,
where r is the distance between point P and point Q and it can be expressed by
r = \]\xP xQy + (zP zQ)2
on the x z plane.
Substituting Eq.(6.9) into Eq.(6.8), it becomes
(6.10)
66
Despite the fact that the Greens function has a singularity at r = 0, the contour
integral in Eq.(6.10) exists and can be worked out by removing the singular point
from the domain.
In general, Eq.(6.10) becomes
(6.11)
after removing the singularity at point P. Here a is 2n if P is an interior point
and is equal to the inner angle between the two boundary segments joining at P if
it is on the boundary. Generally, point P can be anywhere in the domain DnC,
while Q is always on C due to the contour integration. In BIEM, since only the
3$
boundary values of $ and are solved, P has to be kept on the boundary C in
an
3$
the process of computation. When $ and r on C are solved, the interior values
on
can be derived with Eq.(6.1l) by placing P at the point of interest inside D.
The closed boundary C in Eq.(6.1l) is the same as that in Eq.(6.10) except that
it does not pass point P as it does in Eq.(6.10). The singular point of the Greens
function r = 0 has been excluded by a circular arc of infinitesimal radius from the
domain DnC. Thus there is no singularity on the new contour C. Eq.(6.1l) is the
basic equation for BIEM formulations.
Discretizing the boundary C into N segments, and breaking the contour inte
gration into N parts accordingly, Eq.(6.1l) reads
*< Inrfl)fa=Js ^ (6,12)
The curved segment Cj is usually replaced by a straight line to simplify the inte
gration. This simplification generally does not introduce significant error provided
that the segment is small enough.
Before carrying out the integration, the type of element has to be determined.
The formulations are different for different types of element. The commonly used
elements in two dimensional BIEM problems aire: constant element, linear element,
67
quadratic element, special element and so on so forth. The classification of elements
is based on the type of the function used to interpolate the values over an element.
For example, on a constant element, the physical quantities and their normal deriva
tives axe assumed to have no change over the element; while on a linear element,
the quantities and their normal derivatives are interpolated by a linear function
between the values at the ends of the element, i.e.
= JVi(0*; + ^(Â£)*/+i (6.13)
= )
< e < Â£i+l
where JVj and Nz are linear functions with the feature of
Nd(j) = 1 JVaiij+i) = 0
= 0 N2((+Â¡) = 1
In the same manner, higher order elements can be defined by replacing Ni and
Nz with higher order functions. When the variation of the quantities is known,
the variation function can then be chosen as the interpolation function to obtain
a more accurate element. Such elements are usually classified as special elements.
Generally, it is difficult to judge which element is superior over the others with
out analyzing the particular problem. As a rule of thumb, the higher the order of
an element, the more accurate the approximation would be for the same element
size. But for a higher order element, as a tradeoff for precision, the formulation
would be much more complicated and tedious, and the computation might be much
more time consuming. The constant element works fairly well for problems with
smooth boundaries and continuously changing boundary conditions, but could gen
erate considerable errors at corner points where the boundary conditions are not
continuous. For boundary with such corners, the linear element is usually a better
choice.
d$
68
Pi
\
Figure 6.1: Auxiliary coordinate system
6.2 Local Coordinate System for A Linear Element
To formulate a problem with linear elements, an auxiliary local coordinate sys
tem, as shown in Fig. 6.1, has to be established to facilitate line integrations over
the element. The two axis Â£ rj are perpendicular to each other with one lying on
the element and pointing in the direction of integration, from PÂ¡ to Pj+n and the
other one being in the same direction as the outward normal vector . Based on this
auxiliary coordinate system, the integration over each segment can be completed
analytically and be expressed in terms of Â£y, Â£J+i and r?,, the distances from Py,
Pi* i and Pi to the local origin defined in Fig. 6.1. The values of Â£y, Â£y+1 and 77, are
determined by the global coordinate information of three points.
Since
where
= y/(z. Xj)2 + (2, ~ Zj)2
then
69
Â£;+i Â£y + Ay
The value of is therefore
k. i=
(6.15)
(6.16)
(6.17)
However, the numerical test showed that Eq.(6.l7) could lead to significant error
of  rÂ¡i  due to runoff errors. A more accurate expression for  rji  can be obtained by
computing the distance from P, to the straight line PP.+1 directly from the global
coordinates, i.e.
 A(x, Xj) +Zj Zj 1
Vi + A2
where
^4 = ~ xi
zi+1 zi
The sign of 77, depends on the relative position of P to the boxindary line element
PjPj+v If P% is in the same side of the element with n, rji is positive, otherwise, it
is negative. Mathematically, it can be expressed by the following two equations for
vertical and nonvertical elements:
(6.18)
where
Therefore
sign (rji)
sign (rji)
(xj X{) Az
I [xj Xi) Az
Ax Az,o
j Ax AziQ I
*y+i x:
xi+\ 7^ xj
A x Xj j Xj
Az = Zj+x Zj
a Az, '
Azo = (x, Xj) + Zj Zi
(6.19)
(6.20)
T]i = sign (rji) j rji j
(6.21)
70
6.3 Linear Element and Related Integrations
By the definition stated in the last section, on a linear element, a quantity and
its normal derivative axe interpolated by a linear function between the values at
the two nodes, Pj and Ph i. For the velocity potential function $ and its normal
3$
derivative we have, in terms of Â£ rj coordinates,
on
4>() = (*> y)t + (fe+i*f &*<+> Â£,
Vj'+l V)
It is obvious that
..3$. ,5$. .3$. ,3$. i
i(^;W (ggjjf +
ei+. &
, < < {,
$(Â£,) = <Â¡>i;
3$ 3$
(6.22)
(6.23)
*(ii+i) = *i+1; ()(fi+i) = (Â§Â£),
Substituting Eqs.(6.22) and (6.23) into Eq.(6.12), and noting that
r, = 1
JVi+2
(6.24)
5rf =
dn
dr{ r?i
3t7, r,
(6.25)
the integration over segment j (between Pj and PJ+1) cam be carried out analytically
in the local coordinate system (Ligget and Liu, 1983):
3r, 3$,
_e //+1 3r, 3$,,.
 4 4^nr'^)di
= Hi ft + Hi $;+1  Hi(i )7+1
(6.26)
where the superscripts ~ and + refer to the positions immediately before and after
the nodal point denoted by the corresponding subscript. And
Hh = /g + feJS
(6.27)
71
Zb
 I1 f/12.
'j ^j>j
(6.28)
zb
= ^5 + 6+iiS
(6.29)
Kh
 j? f.j
(6.30)
1 [*+l ju_ [!>+'
6+164 r{dn* 2A,4 r?
>2L
2Ay 1 r?? + e? J
(6.31)
1 [ti+1 1 dr, T7, f/+ a:
ti+itjhi ri dn 2Ay 4 r?
T"[tan 1(~~) tan
Ay *7 *7
(6.32)
f. j I**' Inuidi
6'+i ~ 6 4
4Ay
{tf + ej+1)Mvi + th) 1]
iv? + $){Hvl + 62) ~ i]}
(6.33)
~ y J*1+llnridt
6+i ~ 6 4
2Ay
{6+i M*?, + 6?+i) 6 in(*7i + 6) 2(6+i 6)
+2*7i[tan 1{^)tan 1(^)]}
Vi Vi
(6.34)
72
for all i =Â¡ j and i ^ j + 1 and
Ay = 6+1 6
If: = j, the above integrals have to be reevaluated to avoid the singularity of
the Greens function,
1 /Â£/+> 1 <9rt
J}j = iim/y+>^ee = o
6+i 6 Udn
(6.35)
= lim / 7 dÂ£ = 0
6+i 6 <'iy+< r<3n
(6.36)
J?1. =
lim e,+llnr,^d
6+i ~ 0 0 /y+<
= ^(21nAyl)
(6.37)
fy+i
Ijj = 77" i f In r, dÂ£
JJ 6+1 6 'y+
= In Ay 1
(6.38)
Similarly, when i = j + 1,
iy+ij = 7lunf,+ ^Â£dÂ£ = 0
* J 6+i ~ 6 'y r 5n
(6.39)
1 .. /j+~ 1 dr,
Ijh y = 7 r hm/ rdÂ£ = 0
,+J 6+i 6 <,0*'fy r*5rl
(6.40)
(1 2 In Ay)
(6.41)
73
= In Ay 1
Summarizing all the Jys according to Eq.(6.12) for node points t = 1
assuming that
= $+
O'
at all the nodes, a system of linear equations with unknowns of $ and
obtained:
dn
where
N N
y=i y=i
= if?* + *ti<*
= Bfji + BliStjOi
j = 2,3, ...,N
with
. 1 if t = j
,J \ 0 if 5 j
Am
L
+ Kl
Ki+Kij
j = 2,3,
Equation (6.45) can also be expressed by a matrix equation
(6.42)
N, and
(6.43)
(6.44)
can be
(6.45)
(6.46)
(6.47)
R 14>n
(6.48)
74
where and n are the vectors of order N x 1 containing the unknown $ and
respectively. The coefficients in R and L are determined solely by the boundary
geometries.
6.4 Boundary Conditions
In Eq.(6.45), there are N linear equations and 2N unknowns. The additional
N equations necessary for obtaining a unique solution axe usually introduced by
the application of the boundary conditions. There are essentially three types of
boundary conditions for boundary value problems.
I
II
III
$ = $
Â£$ _~dÂ¥
dn dn
dn
(6.49)
(6.50)
(6.51)
where $, , / and k are known functions on the boundary.
on
In water wave problems, the first relation usually means that the pressure distri
bution is known, and the second describes a given flux distribution on the boundary.
They axe also referred as essential and natural conditions (Brebbia,1989), respec
tively. The third one some times takes place when none of the quantities in the first
two equations are specified but only the relationship between them can be found.
A typical example of this is the free surface boundary condition in linear wave the
ory. For nonlinear wave problems, the right hand side of the above equations may
include other known functions.
As might be noticed in Eq.(6.45) or Eq.(6.48) there are two unknowns for each
nodal point. Since we have obtained one equation for every node on the boundary,
only one of the three relations needs to be specified at each node, if the assumptions
in Eqs.(6.43) and (6.44) are strictly satisfied. However, for some nodal points where
the boundary changes directions, (Â§Â£) may not be equal to (Â£)+. For such cases,
more than one boundary conditions may be necessary for one node. The treatment
75
of such points will be discussed in the next chapter.
By introducing the boundary conditions at all N boundary nodal points, another
system of N equations with the same unknown in Eq.(6.48) is established. The
number of unknowns is now equal to the number of the equations. It is usually
convenient to eliminate N unknowns with the 2N equations, and the resulted matrix
equation can be expressed as
AX = b (6.52)
in which A is a known matrix of order N x N, X is the unknown vector containing
<9$ 3$
$ or or $ for some part of the boundary and 5 for the rest of it, depending
on On
on which one is not specified by the boundary condition; and b is a known vector
resulted from boundary conditions.
CHAPTER 7
NUMERICAL MODEL FOR SUBMERGED POROUS BREAKWATERS
In this chapter, a numerical model of BIEM for submerged porous breakwaters
is developed by using the unsteady porous flow model given in Chapter 3. The
basic function of the numerical model is to compute the wave flow field and related
quantities such as the wave form, the dynamic pressure and the normal velocity
along all the boundaries and so on. With these quantities, the wave transmission
and reflection coefficients and wave forces can then be obtained.
7.1 Governing Equations
The computation domain of the problem, as shown in Fig. 7.1, consists of two
subdomains, the fluid domain and the domain of the porous medium. In the fluid
domain, the water is considered inviscid and incompressible. The flow induced by
gravity waves is assumed irrotational. Thus, the governing equation in this domain,
for the velocity function $, is the Laplace equation,
V2$ = 0 (7.1)
with fluid velocities being defined as
w =
Â£Â£
dz
(7.3)
While in the porous medium domain, the viscosity of the fluid cannot be ig
nored since the flow is largely within the low Reynolds number region. The flow is
76
77
Figure 7.1: Computational domains
described by the linearized porous flow model,
pcf0q = VP (7.4)
where P is the pore pressure, f0 is the linearized resistance coefficient and q is the
discharge velocity vector in the porous medium, q = iu + kw with i and j being the
unit vector in x and z directions.
Taking curl of Eq.(7.4),
pofo V x q = V x VP = 0 (7.5)
leads to
Vx? = 0 (7.6)
This means that the homogenized porous flow is irrotational, and hence a velocity
potential function in the porous domain exists such that
q = V$,
(7.7)
According to Eq.(7.4) and Eq.(7.7)
P
Pfo
(7.8)
here p, cr and /0 are all treated as constants.
Substituting Eq.(7.4) into the continuity condition for the porous flow in terms
of the discharge velocity,
Vg = 0 (7.9)
the governing equation for the porous medium domain becomes the Laplace equation
in terms of the pore pressure,
V2P = 0 (7.10)
In this study, the waves are assumed to follow the linear wave theory
$ = eiot (7.11)
P = peiot (7.12)
It is to be noted that the time function is now expressed as e'at, as opposed to e~'at
used in the seabed problem presented in previous chapters.
7.2 Boundary Conditions
The whole computation domain has four types of boundaries, free surface, im
permeable bottom, permeable interface of different subdomains and the artificial
lateral boundaries. These boundary conditions are discussed here.
7.2.1 Boundary Conditions for The Fluid Domain
1. The free surface
The combined linear free surface boundary condition in the linear wave theory
s.d,
dz gV
where z is the vertical coordinate and g is the gravity acceleration.
(7.13)
79
2.Impervious bottom
The boundary condition on an impermeable bottom is simply nonflux condition:
dn
= 0
(7.14)
where n is the direction normal to the boundary which can be of any shapesand
bar, or soil trench and so on, as long as it can be considered impermeable.
3.The permeable interface
The permeable interface is the common boundary either between the fluid do
main and the porous domain, or between two different porous domains. For the
sake of simplicity, we consider only one porous subdomain in this chapter. The
same formulation can be easily extended to multiporousdomain configuration.
The boundary conditions on such a boundary are the continuity of pressure and
mass flux. They are
(7.15)
or equivalently
iap4> = p
(7.16)
with the linear wave assumption, and
= (7.17)
dn: pafa dn\i
where nÂ¡ is the outward normal of the fluid domain and nn is the outward normal
of the porous domain. However, these two notations are only used when confusion
could occur, otherwise, n without subscript always denotes the outward normal
vector of the domain in discussion.
4.The lateral boundaries
There are two vertical lateral boundaries, one is on the offshore side and one
is on the lee side of the porous domain, as shown in Fig. 7.1. In this study, the
radiation boundary condition for these two boundaries is adopted. Such a boundary
80
condition assumes that the waves at these two boundaries are purely progressive,
and that the decaying standing waves generated by the object inside the domain are
negligible at these boundaries. Comparing with the matching boundary condition
with the wave maker theory employed by Sulisz (1985) and some other authors,
the radiation boundary condition offers significant simplicity in programming and
provides sufficient accuracy with much less CPU time. In applying this boundary
condition, the two lateral boundaries have to be placed far enough from the struc
ture. Numerical tests show that a distance of about two wave lengths from the toe
of the structure is more than adequate for this purpose.
At these two lateral boundaries, the potential functions for the transmitted and
reflected waves are assumed to have the forms
&(*,*) = e,kWR(z)
(7.18)
*(*,*) = e'i'')j(2)
(7.19)
where T(z) and R(z) are two unknown functions of z\ k and k' are the wave numbers
at the two boundaries respectively, and the subscripts t and r here refer to the
transmitted and reflected waves, respectively. On the lateral boundary of upwave
side, the potential function for the incident wave is known:
4>i{x,z) =
(7.20)
with
Therefore,
, Hgcoshk(z + h)
2 a cosh kh
(7.21)
] + 4>r at 2= l
4> = t at x = l'
(7.22)
(7.23)
where l and l' are, respectively, the distances from the origin of the coordinates to
the offshore and onshore lateral boundaries and if> is the unknown potential function.
81
On the lateral boundary of transmission (onshore side), x =
= = ^ = m, = m*
on ox ox
in which k' is determined by
gk' tanh k'h1 cr2
where h' is the water depth at x = l'.
While on the lateral boundary of reflection (offshore side), x = l,
dcf>
dn
<>r ~ I
di d(j>r
dx
dx dx
= ikcfrj tkcfir
(7.24)
(7.25)
(7.26)
or
= ik(f>Â¡ ik(cf> i) = 2ik(f>i ik
d
= 2 ik
dn
in which k is the incident wave number determined by
gritanh kh = o2
(7.27)
(7.28)
where h is the water depth at x = /.
7.2.2 Boundary Conditions for The Porous Medium Domain
In this domain, there are two types of boundaries, one is the common boundary
with the fluid domain and the other one is the impervious bottom. On the interface,
the boundary condition is the same as the one specified by Eq.(7.16) and (7.17).
On the impermeable bottom, the noflux condition, in terms of the pore pres
^ = 0
dn
(7.29)
Again, this boundary can have any shape, it can be the surface of the domain
for core materials if it is considered as impermeable, etc.
82
7.3 BIEM Formulations
The formulations in the two domains are slightly different because of the dif
ferent boundary conditions. In the fluid domain, due to the radiation boundary
condition on the lateral boundary of offshore side, the terms containing the incident
wave potential are introduced, which will form the RHS vector of the matrix equa
tion. On the free surface, the normal derivative are expressed in terms of 4> owing
to the CFSBC. In the porous subdomain, the matrix equation, after applying the
noflux condition, has to be manipulated to match with the fluid domain.
7.3.1 Fluid Domain
According to the formulae given in chapter 6, Eq.(6.12) can be expanded, in
terms of node values, assuming that 4>~ = + for all the nodes, as:
ou4>i = + + + + +
+ i + HÂ¡Nl)4>Nl + ... + i + + ...
+ [Hlffs_i + H}tNi)s3 + + (S'tV.ii +
+ 1 + + + {Ef/ri + hIn)4>n
 Klfa  Khfa Kfjfa Kl s^3 .
 ~ ^Â¡.Ni^nNi ~ ~ ^i.Ni^nN, ~
 ^i,Nal^nN3 ~ Ki.Ns^nNs ~ ~ ~
~ ^i,Ncl^nN,. ~ Klx"4>nNc. ~ ~ ^i.Nl^nN ~
i = 1,2, ,JV
(7.30)
where a, is the inner angle of node i and the subscripts 1, Ni, JV2, JV3, Neb and Net
refer to the nodes of comer points as shown in Fig. 7.1. The superscripts and +
refer to the positions immediately before and after the nodal point in the direction
of contour integration, and the subscript n refers to the normal derivative. Here 4>~
is not necessarily equal to cf>+ for all the nodes.
83
At a smooth point, where the inner angle a, formed by the two boundary
segments joining at the node is ?r, the normal velocity are continuous across the
node, such that
t = t (731)
But at a comer point where the boundary changes direction, usually
t # t (732)
unless both of them are zero.
This discontinuity in normal velocity introduces one more unknown at a corner
node without introducing the necessary equation. Before applying the boundary
conditions to all the nodes, it is necessary to examine the nature of those corners
and to establish appropriate conditions.
In general, if the normal velocities specified by the boundary conditions across
a corner node satisfy any one of the following conditions, the extra unknown intro
duced by the discontinuity can be eliminated without generating any computation
errors:
1) t = t =
2) 4>~ = ki(f> on one side and = k2 on the other side
with ki and k2 being two nonzero known functions
3) 4>~ = ki on one side and + = 0 on the other side, or vice versa
In Fig. 7.1, the corner points 1, Ni, N, IV3 and the corners on the impervious
bottoms aire such nodes. After introducing the boundary conditions, the only re
maining unknown is , which is continuous anywhere. For Ne4, Nee and the corners
between them on the porous interface, none of the above conditions can be satisfied
and singularities occur. In the present model, for corner points along the porous
interface, ~ + is assumed. This is a reasonable approximation considering the
84
fact that angles on the surface of a porous breakwater made of quarry stones cannot
be sharp enough to cause a sudden change of the normal velocity. At Ncb and Net,
the above approximation cannot be applied since it is known that 4>n = 0 on one
side of the node and n ^ 0 on the other side. The singularities at these two nodes
are avoided by splitting each node into two.
In Eq.(7.30), it can seen that Greens formula was discretized under the as
sumption of cf)~ = <7Â¡>+. If we revoke such an assumption at a singular point and
split the node into two, applying the Greens formula to the two nodes separately,
the original equation for the singular point will be replaced by a pair of equations.
Letting i be either Neb or Nce, the two equations are
Oi4>i
<*Â¡4}
(hi i+aj,)* +...+Hjwtf + nyt+
XIA ~ ~ *>. ~ KhKi ~ (733)
(Hi + + ... + + Hit} + ...
Xim   Klt ... KI.K, (7.34)
where the same Hi/s and KÂ¡/s are used in both equations because the two nodal
points are so close that the distances from a field point to either one of them are
virtually the same. In doing so, two more unknowns, ~ (or +) and ~ (or <Â£+),
are introduced, instead of 4> and 4>n only for the original node. Since am additional
boundary condition is also introduced for each additional node, the system is still
determinant. Numerical tests show that without such treatment, discontinuities
(spikes) may occur in the potential function, , at nodes Ncb, Nee and all the cor
ner nodes on the interface. Applying the boundary conditions given in Eq.(7.13),
Eq.(7.14), Eq.(7.24) and Eq.(7.27) into Eq.(7.30) and setting (j)~ = and 4>~ = +
for all the node except Neb and Ncet results in
Kti
& Ai + HiNtb.xrNti + h}# 4>n. +
i=1
85
C (3',:4>: + + H}tNct4>Nc, + Yl P'J^i ~
J=#ek+1
=Net+1
Ni 1
ik K{1(p^ tk1 Y2 7 (iklKfNi_1 KÂ¡Nx)q>nx
y=2 9
a1 Wal a2
T X) 71y^; + ~^,N2iPn2 + ikKl'N2(2jN2 4>n3)
9 j=Ni+1 ^
#sl
+ * IT 7.j(2^/ 4>i) + ikKlNz_x{24>INz 4>N3)
:=n3+i
Net1
+ ^\,Nch^nNeh + Y2 lijPnj + K^Ncl^nN,. (7.35)
=#*+!
where the following notations are introduced
to
= Bfj, + St\, 6Â¡,Â¡ai
(7.36)
to
= Hl_l+HÂ¡jSÂ¡Jai
(7.37)
(7.38)
The matrix form of Eq.(7.35) is, after some manipulations,
iAi a,]{Â£Hbi Bii{^}+b <739)
where A, and B, are known matrices, determined purely by the boundary geometry,
c and 4>nc are, respectively, the vectors of the unknown potential function and its
normal derivative on the interface boundaries, with
<Â¡>C = {^jv,t> 4>Nci+l> M 4>Nu4>Nc,}T
nc = nNci+U finNe.U ^n#e,}
f and 4>nf are the vectors of the unknown potential function and its normal deriva
tive along the boundaries other than the interface boundaries, respectively, b is the
known vector containing 4>i resulting from the radiation boundary condition on the
offshore side lateral boundary. The expression for an element in b can be obtained
from Eq.(7.35):
Nzl
bi = 2ik(KliNj<Â¡>iN2 + Y2 + Ki.Nsi^iNz)
#3 + 1
(7.40)
86
* = 1,2,
N
Equation (7.39) is not solvable at this point since the number of unknowns is
still greater than the number of equations. The additional required equations arise
from the interfacing with the porous subdomain.
7.3.2 Porous Medium Domain
In this subdomain, the integration is in the opposite direction to the one in the
fluid domain, therefore, on the common boundary of the two domains, the contour
integrations in both domains are in the same direction as shown in Fig. 7.1. It is so
arranged as to facilitate matching with the boundary conditions along the interface.
Similar to the fluid domain, the expanded form of Eq.(6.12) for the porous
subdomain is
+ E + r
Ncrn*hi
= urti + Eli + K&.p~. (7.41)
y=2
after applying the noflux boundary condition on the impervious bottom. Where
the superscript s refers to the quantities in the porous subdomain, N* is the total
number of nodes on the entire closed boundary of this domain and Ncm is the node
number on the common boundary with the fluid domain. In this equation, p~ is
assumed to be equal to p+ for all the points except node 1 and Nem
In matrix form, Eq.(7.41) reads,
Cn
C21
(7.42)
where pe and pnc are the vectors of pressure function and its normal derivative
along the common boundary and p and pnj are the corresponding vectors on the
87
impervious bottom
of the subdomain with
Pc =
{pt,P2,~;PNeml,pNcJT
Pnc =
{PnliPn2i ) PnNcml, Pntfcm}T
Pi =
{PNcm > PNcm + U > Pni,P }T
PnA =
i.PnNcm'PnN'm + 1) > PnN'l j Pnl}
Expanding Eq.(7.42) and solving pne in terms of pc by eliminating p6,
a rela
tionship between p,
ac SJld pc
can be established
Pnc = Epc
(7.43)
with
E
= (Du 
Cl2 C22 D2i) 1 (CU C12 C^J1 C2i)
(7.44)
The boundary conditions at the interface are the continuity statements for the
pressure and mass flux, given by Eq.(7.16) and (7.17). In the matrix form, they are
Pc = ipoc (7.45)
P ne = pefotnc (7.46)
They are also called compatibility equations (Ligget and Liu, 1983).
Substitution of these two equations into Eq.(7.43) yields
4>nC = yEc (7.47)
This is the additional matrix equation needed by Eq.(7.39) in order to have a
unique solution. The final combined matrix equation for such a solution in is
obtained by matching the two domains along the common boundary.
7.3.3 Matching of The Two Domains
The matching of the two domains along the common boundary can be accom
plished by introducing Eq.(7.47) into Eq.(7.39). The resulting equation for the
88
unknown potential function on the boundary of the fluid domain is then
A = b
(7.48)
with
A = [Ai Bi A2 + rEBijArxW (749)
Jo
and
(750)
In Eq.(7.48), the number of the unknowns N, which is the total number of nodes
on the boundary of the fluid domain, is equal to the number of the equations. It
can be readily solved by a complex equation solver if the linearized coefficient /o
for the resistances in the porous flow model is given. Unfortunately, it is still an
unknown at this point and has to be determined by the linearization process.
7.4 Linearization of the Nonlinear Porous Flow Model
The principle for the linearization is the equivalent energy dissipation by both
linear and nonlinear systems, i.e.
(Ed)i = (Ev)ni (751)
The conventional definition for the energy dissipation within a control volume
of porous medium during the time period T is (Sollitt et al., 1972, Madsen, 1974,
and Sulisz, 1985)
r ft+T 
Ed = Jv J F pqdt dv (7.52)
where F is the dissipative resistant force per unit volume of the porous medium,
which is a function of the spatial coordinates and the time, with F = Re(afQcj) for
a linearized system and F = Re[a{fl + fi l^l)?] for the nonlinear physical system;
q is the complex discharge velocity of the porous flow and p is the density of the
fluid.
89
To carry out the integration in Eq.(7.52), a new grid system over the entire
porous domain has to be introduced, which is obviously a set back for BIEM which
deals with only the boundary values. However, by using an alternate definition for
the energy dissipation and applying Greens formula, the averaged energy dissipa
tion can be expressed by a boundary integral (Appendix A). If we further define a
complex energy dissipation function with the real part being the physical quantity,
it has the form
Ed =
unp* ds
(7.53)
where S is the closed boundary of the porous domain, p* is the conjugate of the
complex pressure function and un is the complex normal discharge velocity on the
boundary. Introducing the noflux boundary condition on the impervious bottom,
Eq.(7.53) becomes
Ed
unp* ds
(7.54)
for both linear and nonlinear systems. Where C is the common boundary. The
sign is used here because Ed is considered as a positive value.
The physical explanation of the expression is that the energy dissipation inside
the porous domain in one wave period T is equal to the net energy flux into the
domain in the same time period. It is equivalent to the conventional volumetric
integral form defined in Eq.(7.52). By expressing the energy dissipation in the
boundary integral form, the advantage of BIEM can be well explored.
The energy dissipation by the two systems can be obtained by substituting the
expressions for the normal velocities from the linearized and the nonlinear porous
flow models, respectively.
For the linearized model,
1 dp
pa fo dn
Mi =
(7.55)
90
and
(Ed), = 270/0py
For the nonlinear model,
(7.56)
(^n)nl
P{f\ + / I q )
r^rPn
(7.57)
and
(eD)^=y
PnP
ds
'c fi + h iil
Equating Ed(/q)i to En{fi + h \ 9 )ru and take approximately Â¡ q\
the linearized coefficient f0 is then
(7.58)
\Pn/pof0\,
fo =
[ PnP* ds
Jc
L
PnP
(7.59)
ds
c fi + h\ Pn/pcfo
The linearization is in fact an iteration process starting with an approximate
value of f0. The procedures are as follows:
1 Let ft = fu
2 Substitute this approximate /0 into Eq.(7.49) to form A, for Eq.(7.48),
3 Solve Eq.(7.48) for ^ and compute nc using Eq.(7.47) with the obtained ^c,
4 Calculate according to Eq.(7.59) and compare to fo\ if they are close enough,
stop; otherwise
5 /0(2) = fo^ and repeat steps 2, 3, 4 and 5 until
I ft] ft'" l< Â£ (7.60)
where Â£ is a prespecified arbitrarily small quantity.
6Solve Eq.(7.48) with for the final solution .
91
Recalling is that when treating the singularities at JVej and Nce, tf>~ is not nec
essarily equal to cf>+, but are two different unknowns instead. Physically they have
to be equal since the pressure is unique anywhere. This has been confirmed by the
numerical results in the solution for every step of the iteration.
When is known, the rest of the field quantities can be readily obtained, since
p = iop4>
(7.61)
to ,
T] =
9
(7.62)
4>nt = BCtt = ik'lt
o2
(7.63)
4>n, = = 14>.
9
(7.64)
4* nr Brr ik\(f>r
(7.65)
4>nc E c
(7.66)
where the subscripts t, s, r and c refer to the transmission, free surface, reflection
and the common interface boundaries, respectively, Bc,s are the B.C. matrices and
I is an unit matrix.
7.5 Transmission and Reflection Coefficients
When computing the coefficients of transmission and reflection, the velocity
potential functions at both lateral boundaries are assumed to have the forms of
*(*.*) =
cosh kh
+
m=l
k(x+i)Cskm{z + h)
cos kmh
(7.67)
* 2a cosh k'h!
+
f R (,n cosk'mjz + h!)
cos k'mh'
m= 1
(7.68)
92
with
gk tanh kh = gkm tan kmh = a1 (7.69)
gk' tanh k'h' = gk'm tan k'mh' = a2 (7.70)
where AÂ¡ and i?, axe complex coefficients. After is obtained, all As and i?,s can
readily be determined by applying the orthogonality of the hyperbolic and cosine
functions. The only coefficients which affect the transmission and reflection are
Ao and jE?0 since all the other terms are standing wave modes. The reflection and
transmission coefficients are
o
11
(7.71)
Kt = \B0\
(7.72)
Kr =
Kt =
(7.73)
(7.74)
Letting x = l in Eq.(7.67) and x = V in Eq.(7.68), integrating 4>{l,z) coshfc(z+
h) and '(l',z) cosh k'(z + h') over [/i,0] and [h',0] respectively, the reflection and
transmission coefficients are found to be
rO
cos kh (l,z) cosh k[z + h) dz
11
f0 1 I
/ cosh2 k(z + h) dz
J h
cos k'h' J
[ cosh2 k'(z + h') dz
Jh'
where <Â£(l,z) and 4>'(l',z) aie known in a discretized basis from the solution of
Eq.(7.48).
7.6 Total Wave Forces on an Impervious Structure
When a submerged structure is impermeable, such as a breakwater made of
concrete instead of quarry stones, the total wave forces and the overturning moment
will be one of the major factors to be concerned in design.
Such an impermeable structure is merely a special case of d, = 0 in the model
presented before. The boundary condition on the surface of the structure is now
93
the nonflux condition. If we assume that the dimension of the structure is compa
rable with the wave length (usually greater than 0.2 of the wave length), the total
horizontal and vertical forces can be obtained by integrating the pressure over the
exposed surface of the structure.
Since
p = ipafi
and
4> = 4>j)Â£ + (<Â£>+1Â£; + 4>jÂ£j+1)
the total wave force on a single element is
u =
~r~ / [(^j+x ~ })Z + (&+16 + 4>jÂ£j+i)]dÂ£
Llj J ij
1 4>j)ttj+i &) =
(7.75)
7, =
i N^1
Fxr 4 tFxi = po\ 22 {4>i+1 cos(ny, 2)]
}=Ncb
(7.76)
7Z =
%
FZr + iFti = po[ Ys (4j+1 4*3)cos(j, z)}
}=Nci
(7.77)
Fx = 1 7Z I e^t+^
(7.78)
Fr =  Jz  {at+(]
(7.79)
with phase lags of
et = tan
*XT
(7.80)
+ X
e* = tan 
**r
(7.81)
where (n:,x) and (n,,z) are the angles from the normal direction of element j to x
and z axis, respectively.
94
The overturning moment about the center of a structure at the mudline is
N"~1 i 1
M0 = //{cOs(y,z)[2J + Aycos(y,x)]cOs(J,z)[x;AycOs(y,2)]} (7.82)
jN,d
M0 =  M0  e><+Â£> (7.83)
em = tan"1^ (7.84)
Mr
7.7 General Description of The Computer Program
The computer program is written in FORTRAN 77 computer language and com
pleted on a VAX 8352. It has 12 subroutines and a main assembling program. The
functions of the subroutines ranging from element generation to the computation
of Kr and Kt.
The model consists of four major parts:
1. Element generation
This part enables users to input the geometry data of a computation bound
ary by breaking it into straight lines or circular arcs, if any, and specify only the
positional information of each segment along with the number of elements to be dis
cretized on this segment. The preparation for input data is therefore very simple.
For example, to compute a trapezoidal submerged breakwater, the input data for
the geometry is less than ten lines with no more than five values in each line.
2. Matrix formation
This part computes the coefficients H and K for both fluid and porous medium
domains according to the formulae given in Chapter 6, with element information
generated in the first part. Since there is no boundary condition involved, it is a
generic subroutine and can be used as the basic subroutine in any BIEM program
with linear elements.
3. Matrix assembling and boundary matching
In this part, the boundary conditions are introduced for both domains. In the
fluid domain, the matrices A,, B, and b are formed according to Eqs.(7.35) through
95
(7.40). In the porous domain, the matrices C and D are divided into four blocks
and the operation formulated in Eq.(7.44) is carried out. The boundary matching
is accomplished by completing Eq.(7.49) and the resulting matrix equation is then
solved by the IMSL subroutine LEQTlC, a complex matrix solver.
4. Linearization
This is an iteration process which repeats the solution procedure with the con
temporary resistance coefficient /q The starting point of the iteration is Eq.(7.49),
the matching operation where the linearized coefficient /o first appears in the com
putation. The criterion of convergence for the iteration is
I /0 ffi l< (785)
with e being a prespecified arbitrarily small number. It is set to be 1.0% in this
model. Figure 7.2 show the flow chart of the model.
To accommodate further extension, perhaps to higher order stokes waves, the
velocity potential function of the linear incident wave has been nondimensionalized,
in this model, by the factor of Hgl2\fgfcy i.e.,
4>i{x,z) = (786)
Therefore, the resulted vector is also nondimensional.
7.8 Numerical Results
In this section, numerical results for a group of example structures are presented.
The empirical coefficients in the porous flow model in all the computations are taken
to be the values determined in Chapter 5, i.e.,
Oo = 570
o = 3.0
Ca = 0.46
96
Figure 7.2: Flow chart of the numerical model for porous submerged breakwaters
97
First, the numerical model is applied to two special structures, one is a trans
parent submerged breakwater with infinite permeability on a flat impermeable
bottom while the other one is an impermeable step with the water depths being
23 cm before the step and 8 cm after it. In the former case, the computations was
performed as if there were a permeable breakwater, even though the transparent
breakwater is equivalent to a fluid breakwater or no breakwater. Figure 7.3 (a)
and (b) show the computational domains, the wave forms and the wave envelopes
for the two cases. In Fig. 7.3 (a), the perfect sinusoidal wave form matches the
theoretical solution for the case, and the waves are completely transmitted without
any reflection as expected. For the later case, the wave form is of a partial standing
wave in the up wave side of the step, and of a monotonic sinusoidal wave in the
downwave side of it. The wave length, at some distance over the step, has been
shortened to the exact value predicted by the dispersion equation for that depth. It
can also be shown that the wave energy is conserved since the equation for energy
flux, 7j JT + ft, is satisfied. Both cases are for conditions at laboratory scale.
Figure 7.4 shows the results for a submerged porous breakwater, for laboratory
conditions again. The crest width of the breakwater is 60.0 cm and it is 8.0 cm
below the mean water level in a water depth of 23.0 cm. The slopes axe 1:1.5 on
both sides. In the top frame, the wave form and the wave envelope directly above
the breakwater are plotted. The lower frame illustrates the pressure and the normal
velocity distribution along the surface of the breakwater. One can see that after the
treatment for the comer nodes, the pressure function, or equivalently the potential
function along the common boundary in the fluid domain, is continuous and well
behaved at all corner points. But the singularities remain for the normal velocity.
In Figure 7.5, two submerged breakwaters, for field conditions, are computed,
one is impermeable, and the other one is made of stones of d, = 0.4 meters. The
dimensions and the wave conditions are the same for both structures. The crest
NORMALIZED SURFACE ELEVATION NORMALIZED SURFACE ELEVATION
98
Figure 7.3: Wave envelopes for (a) Transparent submerged breakwater; (b) Im
permeable step.
NORMALIZED PRES. RND VEL.DIST. NORMALIZED WAVE ENVELOPE
99
Figure 7.4: Porous submerged breakwater: (a) Wave form and wave envelope; (b)
Envelopes of pressure and normal velocity.
100
of the breakwater is 12 meters wide with 1.6 meter submergence in a water of
4.6 meters deep. The slopes are again 1:1.5 on both sides. Comparing the wave
envelope for the impervious breakwater to the one for the porous breakwater, it
is obvious that they are similar in form, but the transmitted wave height by the
permeable breakwater is less than that by the concrete one as expected. It is noticed
that the energy is conserved for the impervious structure, i.e., K\ + Kjj = 1, while
some energy dissipation occurred in the porous breakwater. This dissipation is
1.0 K\ K\ = 29% of the total wave energy. The wave length of the transmitted
waves restored to the same value as that for the incident wave after being disturbed
by the breakwater.
Figures. 7.6 (a) and (b) illustrate the transmission and reflection coefficients
for submerged porous breakwater, which has the same configuration as the one in
Fig. 7.4, as a function of stone size for four different wave periods. The wave height
was kept constant, at H = 4 cm throughout the computation. In the top frame,
the transmission coefficient is shown to have a minimum value for each wave period
for a particular stone size (around d, = 1.0 to 2.0 cm). Beyond this point, the wave
transmission does not decrease with increasing stone size any more, but starts to
increase. This indicates that a larger stone size is not always better in terms of wave
damping. Therefore, there exists a optimum stone size with which maximum wave
energy dissipation and minimum wave transmission can be achieved. The reflection
coefficient is not significantly affected by the stone size as shown in Fig. 7.6, although
it decreases with increasing stone size for most of the cases.
Another phenomenon to be noted is that neither the transmission coefficient nor
the reflection coefficient changes monotonically with changing wave period. When
the same set of data is plotted against the permeability parameter R in Fig. 7.7,
it clearly shows that the lowest values for Kt and the highest values for Kr are
achieved at T = 1.2 seconds. This means that the breakwater is selective when
.50
60.0 40.0 20.0 0.0 20.0 40.0 60.
NORMALIZED SURFACE ELEVATION
en O en o cn o
o o o o o o
o
.50
102
interacting with waves.
To demonstrate the influence of incident wave heights, the same breakwater
configuration is computed for the case of T 1.2 seconds and H = 2.0 ~ 8.0 cm.
The transmission and reflection coefficients are plotted against R in Fig. 7.8. The
permeability corresponding to the minimum transmission is shifted to larger values
for higher incident waves, while the value of minimum transmission remains more or
less the same. It is seen that when the permeability (or equivalently the stone size)
is greater than a certain value, say R = 10.0 (log R = 1.0), the wave transmission
decreases with increasing wave height, while if 12 is less than, say 1.0 in this case,
the wave transmission increases with increasing wave height. This means that larger
stones are more efficient for storm protection.
Figure 7.9 is an example of the total wave forces on an impermeable submerged
breakwater sitting on an impervious bottom. The breakwater has the same cross
section dimensions as the one depicted in Fig. 7.4(a). In Fig. 7.7, the horizontal and
the vertical wave forces and the overturning moment are all nondimensional values.
The nondimensional factors are 7Hs for the forces and 7H3L for the moment. As
far as the stability of the breakwater is concerned, the most dangerous point is when
Fz is close to its positive maxima, the overturning moment is close to its counter
clockwise maximum value and Fx is negative. In such a situation, the structure
tends to overturn around the toe at the seaward side. In Fig. 7.9, the vertical wave
force Ft is seen much larger than Fz and the overturning moment is almost in phase
with Fx. This is because of the assumption of impermeable sea bottom. In reality,
a seabed is usually more or less porous, the vertical seepage force on the bottom of
the impermeable breakwater is generally out of phase with Fs in certain degrees,
and the summation of the two will make the total vertical wave force smaller and
more reasonable. To correctly estimate the seepage wave force is another interesting
problem to both coastal and ocean engineers.
TRANSMISSION COEFF.
103
STONE SIZE DS (CM)
STONE SIZE OS (CM)
Figure 7.6: Transmission and reflection coefficients vs. stone size for different wave
periods, (a) Transmission coefficient; (b) Reflection coefficient.
TRANSMISSION COEFF.
104
Figure 7.7: Transmission and reflection coefficients vs. R for different wave periods
(a) Transmission coefficient; (b) Reflection coefficient.
REFLECTION COEFF. TRANSMISSION COEFF.
105
1.00
0.95
0.90
0.85
0.80
0.75
0.70
5.0 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0
PERMEABILITY PRRRMETER LOG (R)
T 1 1 1 ii ill t i i I i I
PERMEABILITY PARAMETER LOG (R)
Figure 7.8: Transmission and reflection coefficients vs. R for different wave heights
(a) Transmission coefficient; (b) Reflection coefficient.
106
WT (DEGREE)
Figure 7.9: Wave forces and over turning moment for a impermeable submerged
breakwater: (a) Wave forces; (b) Overturning moment.
CHAPTER 8
LABORATORY EXPERIMENTS OF A POROUS SUBMERGED
BREAKWATER
In the last two chapters, a numerical model of boundary integral element method
for porous submerged breakwaters has been completed based on the full resistance
model developed in chapter 3. To further verify this numerical model, and at
the same time to examine the applicability of the empirical coefficients ao, b0 and
Ca determined by the seabed experiments described in chapter 5, a laboratory
experiment for a porous submerged breakwater was carried in a wave channel.
8.1 General Description of The Experiment
The experiment was conducted in the Coastal Engineering Laboratory of Coastal
and Oceanographic Engineering Department, University of Florida. The wave tank
is 25 meters long, 0.6 meters wide and 1.7 meters deep with glass walls on both
sides. The wave maker is of piston type furnished with tin absorbing system which
was designed to absorb the wave energy reflected back to the piston. The tank is
also equipped with a motorized rail cart on the top to facilitate wave measurements.
The model of the porous submerged breakwater was of trapezoidal shape and
made of river gravel of <50 = 0.93 cm, the same material used in the seabed exper
iments. The crest height of the model was 15 cm, crest width was 60 cm and the
slope for both sides was 1:1.5. The center of the model was 12 m from the piston
and 13 m to the other end of the tank. In this experiment, only one water depth of
h = 23 cm was tested.
The measurements were concentrated on wave reflection and transmission, in
cluding the cases when waves broke in the region directly above the breakwa
ter crest, the wave envelope over the breakwater and the pressure distribution.
107
108
Cart
T 1
0
Gauge 1 Wave Direction 
Have
Gauge 2
if
0 0 0 0 0 0 0 0
X 0.0.0.0.0.0.0.0.0. X
k
Figure 8.1: Experiment layout
Figure 8.1 shows the experiment layout.
The waves between the model and the wave maker were partially standing waves
due to the presence of the breakwater in the middle of the tank. The incident
and the reflected wave heights were determined from the maxima and the minima
of the envelope of the partially standing waves. The envelope was measured by
moving the rail cart slowly with a constant speed along the wave tank, from its
original position towards the wave maker during each data acquisition period. The
wave height modulation over the distance traveled by the cart was then recorded.
Similarly, the wave envelope over the submerged model was measured by moving
the cart from the toe on one side to the toe of the other side, during each sampling
time.
The pressure distribution along the surface of the breakwater was measured
by two underwater pressure transducers. The distance between the two pressure
109
gauges was about 12 cm and the locations of the gauges can be found in Fig. 8.1. The
pore pressures at 14 points along the surface of the physical model were measured
subsequently with two points at a time. The wave conditions were kept as identical
as possible for different runs and the pressure readings were later normalized by the
incident wave height for each test.
8.2 Wave Transmission and Reflection
The wave transmission and reflection coefficients are two very important pa
rameters in assessing the performance of a breakwater. In the experiment, the
measurements of these two coefficients were carried out for 9 wave periods ranging
from T 0.642 seconds to T = 1.778 seconds with several different wave heights
for each wave period. Both nonbreaking and breaking waves were tested. Here
breaking waves refer to those incident waves break over the submerged breakwater
model.
The transmitted waves were directly measured by the fixed wave gauge be
hind the model whereas the reflected waves were indirectly measured by the gauge
mounted on the rail cart. Figure 8.2(a) shows a typical data series recorded by
the moving gauge. The incident and reflected wave heights were separated by the
following relationship for partially standing waves,
S{ = TÂ¡max TJmin (8.l)
Hr fjmax T] min (8.2)
In most cases, at least two pairs of quasinodes and quasiantinodes were recorded,
and the averaged values of the two were used for and rj^ to reduce possible
errors.
Figure 8.2(b) shows the record of the corresponding transmitted waves. It is seen
from the figure that the transmitted waves are not monochromatic for a sinusoidal
incident wave. They are, instead, the superposition of waves of fundamental
110
frequency, and the waves of higher frequencies. This phenomenon has been reported
by a number of investigators in the past (e.g., Seelig, 1980), but the physical mech
anism regarding the generation of higher harmonics has not been clearly described.
In this study, spectral analysis was performed on the transmitted wave data.
The spectrum of the transmitted waves revealed that the high frequency waves are
just the higher harmonics of the fundamental wave, as shown in Fig. 8.3. This
implies that due to the interaction with the submerged breakwater, part of the
energy of the fundamental wave is transferred to the waves of higher harmonics. It
was observed in the experiment that such energy transfer occurs around the lee side
end of the crest.
The test results for the incident and reflected wave heights of nonbreaking
waves are listed in Table 8.1. Also listed in the table are the corresponding wave
heights of the higher harmonics up to the third order. Generally, the wave energy of
the first three modes possess more than 98% of the total transmitted wave energy.
It is interesting to note that some of the second harmonics possess more energy
than those of the fundamentals.
The numerical model is based upon the energy balance of the fundamental
waves. In order to be consistent with the concept of energy transmission and to
compare the measured transmission coefficients with the predicted values, an equiv
alent height for a transmitted wave was adopted for the experimental data by adding
the wave energy flux of the higher harmonics to that of the fundamental wave, i.e.
4 = = jr(EtC) (8.3)
1=1
with
Cgi = Crii 
2kih .
sinh 2 kih
and
of = gk{ tanh kih
TRANSMI TED MOVES (CM) INCID. AND REFL. MOVES (CM)
111
TIME (SEC)
TIME (SEC)
Figure 8.2: Typical wave record, (a) Partial standing waves on the up wave side,
(b) Transmitted waves on the down wave side.
SPECTRUM (CMmm2SEC)
112
5.0
4.5 .
4.0 .
3.5 .
3.0 .
2.5 .
2.0
1.5
1.0
0.5
0.0
0.0 0.5 1.0 1.5
FI =
0.
78
1/SEC.
F 2 
1.
56
1/SEC.
F 3 =
2.
34
1/SEC.
HT =
2.
86
CM
T
2.0 2.5 3.0
FREQUENCY (1/SEC)
Figure 8.3: The wave spectrum of the transmitted
3.5 4.0
waves
113
where N is the total number of harmonics, C* is the group velocity of the tth order
harmonics (i = 1 refers to the fundamental wave) and k{ is the corresponding wave
number. With the definition of
, pgHf
8
the equivalent transmitted wave height is then
(HtU =
N n
\ ni
(8.4)
where (#<), is the wave height of the tth order harmonic wave, which could be
determined by the corresponding spike area of the spectrum diagram. The reflection
and transmission coefficients for the experiment data axe then defined as
Kr = H (8.5)
K, = (8.6)
Table 8.2 shows the comparison of the experimental and the predicted trans
mission and reflection coefficients. The relative error A in the table is defined as
A% = _gpl x ioo% (8.7)
Kp
where the subscripts m and p refer to the measured and the predicted values and K
can be either Kr or Kt.
As shown in the table, the agreement between the predicted and the measured
transmission coefficients is reasonably good with the maximum relative error being
about 12.4%. The relative errors for the computed and the experimental reflection
coefficients appear to be fairly large (e.g., Ama* = 71%), even though the absolute
errors axe generally not as large as those for the transmission coefficients. The
majority of the large relative errors occurs when the absolute values of Kr are small.
The comparison is also demonstrated in the following two figures. In Fig. 8.4, the
Table 8.1: Test Results of Nonbreaking Waves
T (sec.)
Hi (cm)
Hr (cm)
Ha (cm)
Ht2 (cm)
Hti (cm)
2.67
0.07
1.87
0.18
0.03
0.642
3.38
0.10
2.37
0.18
0.04
4.07
0.11
2.88
0.19
0.03
4.28
0.15
3.00
0.18
0.06
2.22
0.13
1.54
0.58
0.08
0.858
2.92
0.11
2.03
0.97
0.18
3.31
0.15
2.33
1.16
0.31
4.31
0.16
2.72
1.85
0.61
2.14
0.19
1.44
0.92
0.13
0.952
2.59
0.22
1.69
1.26
0.17
3.38
0.25
2.05
1.80
0.66
3.89
0.37
2.40
2.28
0.34
1.96
0.31
1.32
0.72
0.14
1.020
2.47
0.38
1.61
1.08
0.23
3.38
0.47
2.17
1.95
0.20
4.29
0.52
2.54
2.56
0.64
2.06
0.34
1.35
0.83
0.12
2.29
0.41
1.50
0.96
0.12
2.42
0.52
1.47
1.19
0.24
1.120
2.69
0.50
1.68
1.31
0.14
3.18
0.60
1.75
1.96
0.69
4.18
0.84
2.30
2.67
0.63
4.43
0.72
2.06
3.02
0.20
4.77
0.78
2.18
2.78
0.69
1.58
0.35
1.14
0.30
0.16
1.265
2.10
0.50
1.54
0.57
0.38
3.00
0.64
2.07
1.08
0.94
3.39
0.53
2.34
1.21
1.12
1.67
0.37
1.21
0.34
0.30
2.95
0.63
2.01
0.87
1.08
1.379
3.09
0.50
2.02
0.86
1.16
3.38
0.58
2.28
1.05
1.34
3.89
0.75
2.48
1.30
1.64
1.81
0.36
1.28
0.52
0.34
1.453
2.35
0.36
1.65
0.84
0.60
3.08
0.54
1.98
1.11
1.10
3.96
0.66
2.40
1.66
1.54
2.19
0.17
1.53
0.66
0.43
1.778
2.66
0.14
1.75
0.87
0.63
3.28
0.26
2.15
1.27
0.97
3.98
0.22
2.41
1.69
1.58
115
Table 8.2: Comparison of Km and Kp
T
(sec.)
JL
(cm)
w
(cm)
Krm
Krp
Ar%
Ktm
Ktp
At%
2.67
1.87
0.0247
0.0852
71.0
0.6999
0.7989
12.4
0.642
3.38
2.37
0.0296
0.0854
65.4
0.7014
0.7965
11.9
4.07
2.88
0.0268
0.0858
68.8
0.7071
0.7959
11.2
4.28
3.00
0.0355
0.0859
58.7
0.7006
0.7960
12.0
2.22
1.62
0.0564
0.0474
19.1
0.7310
0.7641
4.3
0.858
2.92
2.21
0.0390
0.0509
23.4
0.7561
0.7665
1.4
3.31
2.55
0.0456
0.0525
13.0
0.7709
0.7687
0.3
4.31
3.19
0.0378
0.0555
31.9
0.7395
0.7748
4.6
2.14
1.66
0.0907
0.0712
27.5
0.7761
0.7691
0.9
0.952
2.59
2.03
0.0841
0.0697
20.7
0.7829
0.7721
1.4
3.38
2.65
0.0727
0.0676
7.6
0.7836
0.7784
0.7
3.89
3.15
0.0948
0.0667
42.1
0.8096
0.7818
3.5
1.96
1.47
0.1584
0.1583
0.0
0.7485
0.7673
2.5
1.020
2.47
1.89
0.1553
0.1584
2.0
0.7643
0.7711
0.9
3.38
2.80
0.1378
0.1586
13.2
0.8296
0.7789
6.5
4.29
3.48
0.1220
0.1589
23.2
0.8104
0.7857
3.1
2.06
1.55
0.1655
0.2119
21.9
0.7543
0.7698
2.0
2.29
1.75
0.1779
0.2132
16.6
0.7629
0.7714
1.1
2.42
1.84
0.2135
0.2143
0.4
0.7613
0.7729
1.5
1.120
2.69
2.07
0.1863
0.2158
13.7
0.7698
0.7751
0.7
3.18
2.56
0.1885
0.2183
13.6
0.8055
0.7793
3.4
4.18
3.40
0.2002
0.2219
9.8
0.8144
0.7865
3.5
4.43
3.45
0.1618
0.2228
27.4
0.7786
0.7884
1.2
4.77
3.40
0.1631
0.2243
27.3
0.7126
0.7918
10.0
1.58
1.18
0.2199
0.2434
9.6
0.7478
0.7715
3.1
1.265
2.10
1.67
0.2377
0.2495
4.7
0.7971
0.7751
2.8
3.00
2.43
0.2141
0.2588
17.3
0.8103
0.7828
3.5
3.39
2.76
0.1571
0.2611
39.8
0.8151
0.7851
3.8
1.67
1.28
0.2179
0.2699
19.3
0.7642
0.7952
3.9
2.95
2.40
0.2119
0.2106
0.6
0.8136
0.7879
3.3
1.379
3.09
2.36
0.1623
0.2251
27.9
0.7630
0.7986
4.5
3.38
2.74
0.1728
0.2262
23.6
0.8107
0.7997
1.4
3.89
3.11
0.1931
0.2285
15.5
0.7987
0.8018
0.4
1.81
1.40
0.1989
0.2320
14.3
0.7735
0.8055
4.0
1.453
2.35
1.91
0.1529
0.1791
14.6
0.8135
0.8018
1.5
3.08
2.43
0.1762
0.1861
5.3
0.7887
0.8075
2.3
3.96
3.17
0.1661
0.1920
13.5
0.8013
0.8133
1.5
2.19
1.70
0.0786
0.1985
60.4
0.7770
0.8206
5.3
1.778
2.66
2.03
0.0522
0.0534
2.3
0.7623
0.8595
11.3
3.28
2.63
0.0796
0.0519
53.5
0.8026
0.8634
7.0
3.98
3.25
0.0553
0.0499
10.9
0.8170
0.8686
5.9
116
predicted Kr and Kt are plotted against the measured ones. In Fig. 8.5 through
Fig. 8.7, both measured and predicted values forifj and Kr for all nine periods (see
Table 8.2) aire plotted against the nondimensional parameter of H{/gT2.
In the case of nonbreaking waves, the wave energy dissipation is mainly due to
percolation inside the breakwater. As the wave heights increase, breaking occurs
on top of the submerged breakwater because of the change in water depth. The
wave energy is, in such situation, dissipated both by the porous medium and by
breaking.
Due to the wave energy dissipation by breaking, submerged porous breakwaters
axe generally more effective for steeper waves. To evaluate the effectiveness of the
submerged breakwater, the wave transmission and reflection coefficients for breaking
waves were also measured and the test results are given in Table 8.3.
Figures. 8.8 and 8.9 are the plots of the transmitted wave heights versus the
corresponding incident wave heights for the data in Table 8.3. The results for the
corresponding nonbreaking cases listed in Table 8.2 axe also included. From the
figure, one can readily observe that the breaking point for the incident wave height
is about Hi = 4.5 cm. It appears that the transmitted wave heights are controlled
mainly by the water depth over the submerged model (h = 8 cm for this case) after
the breaking point, when the wave length is less than certain value. For longer wave
lengths, the waves seem to be affected less by the submerged bump, and the wave
transmission starts to increase with the increasing incident wave heights.
After wave breaking, the numerical model developed for non breaking wave
apparently over estimates the transmitted wave heights. Comparing with the wave
transmission, the effect of breaking on wave reflection is not as significant. As a
result, the prediction for the reflection coefficients for the breaking waves is roughly
as good as that for nonbreaking waves.
117
Figure 8.4: The predicted Kt and Kr versus the measured Kt and Kr. (a) Ktp vs.
Ktm\ (h) Ktp VS. Ktm*
118
1.0
o
GO <
r
0
0 o
0.6
0.4
o
T 0.642 SEC.

T = 0.858 SEC.
0.2

T = 0.952 SEC.
a .
EXPERIMENT DATA
0.0
0
" \
.000
1 1 i 1 1 1
0.002 0.004 0.006
1 1 1 l
0.008 0.010 0.
HI/GTk*2
Figure 8.5: The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp and
Ktmi (h) Krp and Krm.
119
1 O
0.8
...
. i V ,l
0.6
0.4
T = 1.020 SEC.
T = 1.120 SEC.
0.2
T = 1.265 SEC.
o A
EXPERIMENT DATA
0.0
0
1
.000
i i i i i i
0.00! 0.002 0.003
i 1 i 1
0.003 0.004 0.
HI /GT**2
Figure 8.6: The predicted and measured Kt and Kr versus Hi/gT2. (a) Ktp and
Ktm\ (b) Krp 3nd Krm
120
i .0
O
CD (
J 1 1
...a. .
* /o O
I
CO
O
O
JZ
L
CD T = 1.379 SEC.
a T = 1.453 SEC.
0.2 .
o T = 1.778 SEC.
o a o EXPERIMENT DATA
O
o
0.000 0.000 0.001 0.001 0.002 0.002 0.003
HI/GThk2
Figure 8.7: The predicted and measured Kt and Kr versus HjgT2. (a) Ktp and
Ktm\ (h) Krp and Krm.
121
Table 8.3: Test Results for Breaking Waves
T (sec.)
Hi (cm)
(Ht)tq( cm)
Hr{ cm)
Kt
Kr
5.66
3.53
0.26
0.6233
0.0468
0.642
6.99
3.02
0.33
0.4320
0.0469
7.03
2.98
0.31
0.4240
0.0448
5.68
3.40
0.27
0.5981
0.0478
0.858
7.76
3.44
0.44
0.4431
0.0564
8.40
3.28
1.02
0.3903
0.1218
9.15
3.43
1.01
0.3748
0.1102
6.37
3.40
0.77
0.5339
0.1212
1.020
7.96
3.53
0.83
0.4434
0.1042
9.38
3.89
0.91
0.4149
0.0975
10.47
3.82
1.47
0.3648
0.1402
5.19
3.52
0.76
0.6780
0.1470
6.18
3.45
1.07
0.5585
0.1727
1.120
6.27
3.55
1.05
0.5665
0.1672
7.74
3.96
1.18
0.5118
0.1520
7.83
3.91
1.18
0.4996
0.1504
9.00
3.99
1.20
0.4435
0.1329
4.75
3.59
0.93
0.7552
0.1946
1.379
6.60
3.87
1.21
0.5866
0.1830
7.53
4.30
1.22
0.5713
0.1623
9.02
4.29
1.59
0.4755
0.1763
6.01
4.06
0.41
0.6760
0.0679
1.778
7.48
4.49
0.88
0.6003
0.1183
8.02
4.62
1.11
0.5757
0.1387
8.3 Pressure Distribution and Wave Envelope Over The Breakwater
In the numerical model of BIEM, the unknown pressure function is solved along
the boundaries, and the kinematic properties such as the velocities and accelerations
in the flow field can then be computed according to Greens formula as stated in
chapter 6. It is therefore very important for a numerical model of BIEM to be able to
predict the pressure distribution accurately. In this section, the predicted pressure
distributions as well as the wave envelopes over the breakwater are compared with
those measured from the experiment.
122
INCIDENT HAVE HEIGHT (CM)
INCIDENT WAVE HEIGHT (CH)
Figure 8.8: Transmitted and reflected wave heights versus the incident wave heights,
(a) Transmitted waves; (b) Reflected waves.
123
INCIDENT HAVE HEIGHT (CH)
INCIDENT HAVE HEIGHT (CM)
Figure 8.9: Transmitted and reflected wave heights versus the incident wave heights,
(a) Transmitted waves; (b) Reflected waves.
124
The pressure and the wave height distributions were measured both for non
breaking wave and breaking wave conditions of three wave periods. Table 8.4 shows
the averaged nondimensional pressure fluctuations and the locations of measure
ments. The normalized pressure head in the table was calculated according to
N
y \Pytak Ptrough)j
p tl
2N^H
(8.8)
where N is the number of waves recorded, H is the incident wave height and 7 is
the specific weight of water.
For comparison between theoretical and experimental results, the normalized
wave envelopes in the region above the crest and the pressure distributions on the
breakwater surface are plotted in Figs. 8.10 and 8.11 for nonbreaking and break
ing cases respectively, for T 0.858 seconds. It is seen from Fig. 8.10 that in
the nonbreaking case, although the measured wave envelope above the breakwa
ter crest is shifted slightly upward, the numerical model is able to predict, with
sufficient accuracy, both the variation patterns and the magnitudes of the wave
heights (the distances between the two envelope profiles). Fair good agreement
was also found in pressure distribution for nonbreaking waves (Fig. 8.10 b), espe
cially the variation pattern. The magnitude of the measured pressure distribution
is consistently smaller than that of the predicted. In the case of breaking waves
(Fig. 8.11), greater discrepancies (between the predicted and the measured values)
were expected because the nonbreaking assumption in the numerical model had
been violated. However, the prediction for both wave envelope and pressure distri
bution before the breaking point turned out to agree fairly well with those measured.
The good prediction of the flow field for the upper front portion of the breakwater,
which was found in the experiment to be most susceptible to damage, suggests that
the numerical model could be applied to breaking wave cases for the purpose of
extreme force prediction.
125
Table 8.4: Normalized Pressure Distribution
X
T=0.854
T = 1.12
T = 1.379
T = 0.854
T = 1.12
T = 1.379
46.98
0.260
0.406
0.326
0.247
0.313
0.375
42.02
0.284
0.432
0.366
0.282
0.347
0.403
37.07
0.293
0.432
0.390
0.281
0.371
0.401
32.11
0.343
0.437
0.417
0.316
0.386
0.400
27.44
0.363
0.451
0.418
0.331
0.393
0.414
15.55
0.315
0.365
0.315
0.261
0.262
0.321
5.96
0.269
0.410
0.318
0.228
0.262
0.351
5.96
0.241
0.401
0.318
0.207
0.251
0.330
15.55
0.302
0.419
0.294
0.242
0.262
0.332
27.44
0.253
0.338
0.289
0.201
0.204
0.269
32.11
0.220
0.313
0.236
0.165
0.180
0.245
37.07
0.186
0.263
0.184
0.135
0.144
0.210
42.02
0.177
0.244
0.168
0.127
0.138
0.192
46.98
0.158
0.235
0.162
0.111
0.128
0.185
Hi's and x are in centimeters and Ts are in seconds.
In both cases, the wave setups over the breakwater crest in the experiment are
obvious. The maximum set up for the nonbreaking case in Fig. 8.10 is more than
20% of the wave height at the same point. Similar to that in a flat porous seabed
problem addressed in Chapter 4., the depth increase due to such large wave set up
causes decrease in dynamic pressure at the porous bottom. This may have been the
main reason for the discrepancies in the prediction of pressure magnitude since the
numerical model is not designed to take account moving free surface boundaries.
NORMALIZED PRESSURE DISTRIBUTION NORMALIZED HAVE ENVELOPE
126
Figure 8.10: The envelopes of wave and pressure distribution for T 0.858 sec.
nonbreaking wave case, (a) Wave envelope; (b) Envelope of pressure distribution
NORMALIZED PRESSURE DISTRIBUTION NORMALIZED WAVE ENVELOPE
127
1.50
0.50
0.00
1.50
BRK
I*.
BI EM MODEL H= 6.70 CM
pT o EXPERIMENT T = 0.858 S
*
50.0 30.0 10.0 10.0
X (CM)
30.0
50.0
Figure 8.11: The envelopes of wave and pressure distribution for T = 0.858 sec
breaking wave case, (a) Wave envelope; (b) Envelope of pressure distribution
CHAPTER 9
NUMERICAL MODEL FOR BERM BREAKWATERS
In this chapter, attention is diverted from submerged breakwaters to subaerial
ones. A subaerial breakwater is a breakwater with its crest protruding out of the
water surface. There are many kinds of subaerial breakwaters classified according
to their designs. Berm breakwater is one of them. In the conventional design
of rubblemound subaerial breakwaters, there are usually one or two thin cover
layers of large blocks, to resist the wave destruction. In a berm breakwater, the
cover layer(s) of large blocks is replaced by a much thicker layer of, or say, a berm
of, much smaller blocks. It has been shown (T0rum et al. 1988, 1989) that this
kind of berm cover, if designed properly, can provide sufficient resistance to the
wave destructions with relatively low cost, as compared to the conventional cover
layer of large blocks, especially at localities where natural large quarry stones are
not available. Even in places where large blocks are available, berm breakwaters
may still be economical because the cost reduces quite drastically with the block
size. New development has shown (Civil Engineering, Feb. 1990) that it would be
more efficient and economical if a submerged breakwater is placed some distance
offshore of a berm breakwater. The combined breakwater system would provide
more energy dissipation at an even lower cost. In this part of the study, the scope
has been limited to single berm breakwater only.
9.1 Mathematical Formulations
Typical berm breakwaters can have either one of the idealized configurations
shown in Fig. 9.1. In the figure, the superscript s refers to the quantities for the
porous subdomain. The core of the berm breakwater can be either permeable or
128
129
Figure 9.1: Definition sketch for berm breakwaters
impermeable. In this study, we consider only the case of impervious core, which
is usually the case in present practice (T0rum et al. 1988). There are again two
types of domains, the fluid domain and the porous domain. In the fluid domain,
the governing equation and the boundary conditions are essentially the same as
those described in Chapter 7 for submerged breakwaters. In the porous domain,
the governing equation, the impermeable and the interface boundary conditions
are also the same as those for the porous regions of submerged breakwaters. The
special aspect for the computation of berm breakwaters (or for regular subaerial
breakwaters) is the treatment of the free surface inside the porous medium. The
boundary condition for this boundary is different from the CFSBC for the fluid
domain.
130
9.1.1 CFSBC for The Free Surface Inside Porous Medium
On the porous free surface boundary, the dynamic boundary condition cam
simply be taken as the pressure at z 0:
P P9V.
or
and the linearized kinematic boundary condition is
drÂ¡,
dt
w
(9.1)
(9.2)
with r]t denoting the surface elevation inside the porous domain.
According to the linearized porous flow model defined in Eq.(4.2), the vertical
velocity w in Eq.(9.2) can be expressed in terms of the pore pressure P as
1 dP
pufo dz
(9.3)
therefore
drÂ¡Â¡ 1 dP
dt pofo dz
Differentiating Eq.(9.1) with respect to t and substituting
following equation can be obtained,
dr?,
at
(9.4)
into Eq.(9.4), the
dP ofpdP
dz g dt
If
P = ptiot
(9.5)
(9.6)
we have
dp .a2f0
 p
dz g
(9.7)
It is clear that when the porous medium becomes completely permeable, fo * t
and p = ipcr<Â£, and Eq.(9.7) is the CFSBC in a fluid domain.
131
9.1.2 BIEM Formulations
In the fluid domain, the BIEM formulation is basically the same as that given
in Chapter 7 for submerged breakwaters and the matrix equation for this domain
is identical to Eq.(7.39). It will not be repeated in this chapter. For the porous
domain, because of the existence of the CFSBC, the formulation is quite different.
Parallel to Eq.(7.41), the Greens formula given in Eq.(6.10) is discretized for the
berm breakwater of the first configuration shown in Fig. 9.1(a), as
Hlipt + E PhPi+H!k.iPk.+
3=2
+ f pjPj+Hfr"_Â¡PN+
j=Ncm +1
HÂ¡,N,fPN$/ + E fljPi + BfalPl
3=N.,+1
Ncml
= Ki\ Pnl + E 'lijPni + Ki,NemlPnNcm
3=2
N.fl
b ^i,Ncm PnNcm *b E Pni "b 1 PnN.f
+ E 1Â¡jP*Â¡ + K&:iP* (98)
i=K.t+1
where the superscript refers to the porous subdomain, Ncm is the last node number
of the common boundary and N,Â¡ is the last node number of the porous free surface.
Introducing the nonflux boundary condition on the surface of the impermeable
core and the CFSBC given by Eq.(9.7) into the above equation, we get
Ncm * 1
HP + E 0jp + h&ip? +
3=2
Hlkm PNcrn + E Pi + Hikf ~ 1 P +
n!k, + s' fljPi + Biji: 1P1
y=Ar./+i
Ncm 1
= KlPn1+ E ^jPni+KlkmlPnNc
3=2
132
+ f(Ki,NcmPNcm+ lijPi + Ki.N.flPN.f) (99)
9 J=ATm + l
N.f1
Writing this equation into a matrix form, it reads
Cll C12 C13
C21 c22 C23
C31 C32 C33 _
Pc
P
PÂ¡>
D11 Dn
D2i D22
fr}
(9.10)
where pc and pnc are the vectors of pressure function and its normal derivative
along the common boundary referred by the subscript c. Similarly, the subscripts
s and b refer to the corresponding vectors on the free surface and the impermeable
boundaries, respectively.
After moving p, on the right hand side to the left Eq.(9.10) can be further
simplified to
' r* r* 1 ~ / tv \
Pnc (9.11)
Cu C12
C21 C22
Pc
Pci
Du
D2i
with
Cu Cu
C12
C21
= [ C12 + D12 CIS ]
C22
C21 j
C31 J
C22 + D22 C23
C32 C33
and
={p;) <9i2>
Splitting Eq.(9.11) into two matrix equations and eliminating one of the vectors
pc or p,j at a time, we obtain the following equations,
Pnc = Ei pc (9.13)
P,i = E2pe (9.14)
with
Ei = (DiiCi2C221D2i)1(CiiCa2C221C2i) (9.15)
E2 = (C22D2iDri1Ci2)1(C2iD2iDr11C1i) (9.16)
133
Although Eqs.(9.1l), (9.13) and (9.15) look exactly like their counterparts in
Chapter 7, the contents are different. In this case they are all complex matrices
instead of real ones because the CSFBC on the porous free surface inside the berm
contains the complex number /0.
The matching equation in matrix form can be obtained by substituting the
continuities in pressure and the mass flux along the interface, expressed by Eq.(7.45)
and (7.46), into Eq.(9.13):
*nC = ~jVl4>c (9.17)
The matching of the two regions is carried out by introducing Eq.(9.l7) into the
matrix equation for the fluid domain, and the resulting matrix equation is
A = b (9.18)
with
A = [Aj. Bi A2 + yEx B2]atxW
Jo
(9.19)
and
(9.20)
where the notations are the same as those defined in Chapter 7.
After Eq.(9.18) is solved, the surface fluctuation inside the berm can be obtained
from Eq.(9.1) and (9.14).
9.2 Linearization
Similar to the case for submerged breakwaters, Eq.(9.18) is solvable only when
the linearized resistance coefficient /o is known. The technique of finding such
a coefficient is again iteration. However, due to the presence of the porous free
surface, the procedure of iteration has to be altered. For submerged breakwaters,
the starting point for the iteration loop is the matching, right before the formation
of the final matrix equation. But for berm breakwaters, the starting point of the
iteration loop goes back up to the formation of the matrices for the porous domain.
134
For example, the iteration starts from Eq.(9.9) or Eq.(9.10) due to the presence of
fo in the formulae. The remaining procedures and the convergence criterion are
still the same. Figure 9.2 shows the flow chart of the computer program for berm
breakwaters.
9.3 Numerical Results
The numerical model for berm breakwaters has been applied to several different
cross sections with impermeable cores.
The first step in the computation was to verify the model with some special cases
for which theoretical solutions are available. In Fig. 9.3(a) and (b), the dotted curves
are the wave envelopes, computed by the numerical model for the two porous berm
breakwaters with vertical front face, in a water depth of 0.5 m. The permeability
of the berm breakwater in (a) is zero, i.e., equivalent to a concrete wall on the
front face, and is infinity for the one in (b), i.e., equivalent to the case of no berm
but a concrete wall on the back side. The perfect standing wave envelopes and
the complete reflections demonstrate that the model is well behaved and reliable
for the two extreme cases. Figures 9.4(a) and (b) show similar cases with the
berm of a parallelogram cross section. The reflection is, again, complete for both
cases and the wave envelopes are of perfect standing wave shapes except near the
reflective surfaces. The nondimensional wave runups in this case are greater than
those for the case of vertical face, as expected. The wave envelopes have been
nondimensionalized by the incident wave height.
Figure 9.5 shows the cross section of a berm breakwater designed and tested in
Norwegian Hydrotechnical Laboratory (T0rum el at., 1988, 1989) for the extension
of the fishing port in Arviksand, Norway. The cross section is dimensioned in model
scale which is 1/40 of the prototype.
Figures 9.6 to 9.8 are the numerical results for the berm breakwater sketched
in Fig. 9.5, with the stone size of dso = 2.9 cm and the porosity of 0.4 (assumed).
135
Figure 9.2: Flow chart of the numerical model for berm breakwaters
NORMALIZED WAVE ENVELOPE NORMALIZED WAVE ENVELOPE
136
X (M)
3.00
2.00
1.00
0.00
1.00
2.00
3.00
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5

***'*
H= 15.00 CM
T = 2.52 S
" *
KR 1.001
BERM
R*
X (M)
Figure 9.3: Berm breakwaters of vertical face: (a) Zero permeability; (b) Infinite
permeability.
NORMAL I ZED WAVE ENVELOPE NORMALIZED WAVE ENVELOPE
137
Figure 9.4: Berm breakwaters of inclined face: (a) Zero permeability; (b) Infinite
permeability.
138
Figure 9.5: The Cross Section of The Berm Breakwater
The intrinsic permeability for the porous medium made of such stones is about
1.09 x 106 m2. The wave period is T 2.52 s and the wave heights are H =
5.0, 10.0 and 20.0 cm for the three computations, respectively. The permeability
parameter R is 2.48 with oq = 570 and the other two empirical coefficients 60 and
Ca are unchanged in the model (>0=3.0 and Ca = 0.46). In these figures, three
nondimensional quantities the wave envelope (rÂ¡Â¡H), the pressure (p/^/H) and the
normal velocity (unT/H) distribution along the interface, are plotted.
As revealed by shapes of the nondimensional wave envelopes that wave height
reduces inside of the berm and near zero deep into the berm. The dynamic pressure
distribution along the interface increases quite uniformly from the bottom to the
surface as expected. The normal velocity distribution, varies very mildly below
certain depth and then has a rather sharp increase near the surface.
139
Due to the nonlinear effects included in the porous flow model, the variation
patterns of all three nondimensional quantities are nonlinear with wave heights, as
can be detected by comparing the three figures. The gradient of spatial wave damp
ing (as manifested by the decay of wave envelopes) inside the berm becomes steeper
with increasing incident wave heights. The gradient as well as the magnitude of
nondimensional pressure on the interface increases with increasing incident wave
height. Thus the pressure is a nonlinear function of wave height. The nondimen
sional normal velocity decreases with increasing wave height. As a consequence of
nonlinear damping, the reflection coefficient also varies with the changing incident
wave height. The fact that the reflection coefficient increases with incident wave
height indicates that the energy dissipation inside the berm is not proportional to
the square of the wave height, but rather at a lower power.
In Figures 9.9 to 9.11, the numerical results for a berm breakwater in prototype
condition, with the scale ratio of 40:1 to the one shown in Fig. 9.5, are presented.
The water depth, and the wave heights and stone size are 40 times of those model
values and the wave period is Tp = \^40Tm = 15.9 seconds. As a result of scale
change, the reflection coefficients only changed slightly from the values at the model
scale. Comparing the curves for the prototype (Fig. 9.9 through Fig. 9.11) with
those of the model scale (Fig. 9.6 through Fig. 9.8), the scale effect on the three
quantities, rj, p and un is apparently also insignificant. This render some confidence
on the model scale results, at least in the range of values tested here.
Since the experimental data shown in T0rum et al. (1988) were mainly for
random wave tests, no direct comparison has been made since the numerical model
is for regular waves.
NORMALIZED WAVE ENVELOPE
140
Figure 9.6: Permeable berm breakwater of model scale with H = 5.0 cm: (a) Wave
envelope; (b) Envelopes of pressure and normal velocity distribution
NORMAL I ZED WAVE ENVELOPE
141
Figure 9.7: Permeable berm breakwater of model scale with H = 10.0 cm: (a) Wave
envelope; (b) Envelopes of pressure and normal velocity distribution
NORMALIZED WAVE ENVELOPE
142
Figure 9.8: Permeable berm breakwater of model scale with H 20.0 cm: (a) Wave
envelope; (b) Envelopes of pressure and normal velocity distribution
NORMALIZED WAVE ENVELOPE
143
Figure 9.9: Permeable berm breakwater of prototype scale with H = 2.0 m: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity distributions
NORMALIZED WAVE ENVELOPE
144
Figure 9.10: Permeable berm breakwater of prototype scale with H = 4.0 m: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity distributions
NORMALIZED WAVE ENVELOPE
145
Figure 9.11: Permeable berm breakwater of prototype scale with H = 8.0 m: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity distributions
CHAPTER 10
SUMMARY AND CONCLUSIONS
10.1 Summary
In this study, a general, unsteady, porous flow model has been developed based
on the assumption that the porous media can be treated as continuum. The model
clearly defines the roles of solid and fluid motions and henceforth their interactions.
When the solids are held stationary, the model reduces to that used by Sollitt and
Cross for analyzing flows in porous media. All the important resistant forceslinear,
nonlinear and inertialare included and more rigorously defined. It is shown that
the relative importance of these resistance components are functions of two nondi
mensional parameters, R/ = Ud,/v and Rt = ad2Ju. General assessment of the
nature of the resistance as well as of the pore flow is possible prior to detailed flow
computations.
The model is applied to a gravity wave field over a porous seabed of finite depth.
The analytical solution is for a linear wave system, but is applicable, theoretically, to
the full range of permeability, from zero to infinity. The inclusion of the inertial and
turbulent resistances results in a complex dispersion relationship that could only be
solved iteratively together with the equation of linearization obtained according to
the Lorentz law of equivalent energy dissipation. The derivation of the linearization
equation was facilitated by the boundary integral expression for such energy dis
sipation. Solutions for two classes of problems of physical significanceprogressive
waves of spatial damping and standing waves of temporal dampingaxe provided.
146
147
A series of systematic physical experiments was conducted in a wave channel
with standing waves on the porous seabeds of river gravel of seven sizes. The
empirical coefficients in the analytical solution were determined for experimental
data by nonlinear regression.
The porous flow model with coefficients determined from the seabed experiment
was further applied to the computation of porous structures of irregular cross sec
tions. Two numerical models using boundary integral element method had been
developed for such structures. One is for porous submerged breakwaters and the
other one is for porous berm breakwaters. In these two numerical models, the
boundary element used in the numerical formulations are linear elements. The
far field boundary condition in the fluid domain is the radiation condition, which
assumes that the waves at such a boundary are purely progressive waves. The lin
earization of the nonlinear porous flow model is carried out by an iteration process
in which the energy dissipation within a porous breakwater was computed by using
the boundary integral expression instead of the conventional volumetric integration.
The autoelement generation subroutine in the numerical models reduces the data
preparation for input data files to a minimum and, henceforth, making the use of
the programs an easy task.
The main function of the numerical models is to compute the wave field around
and inside the porous structures of arbitrary cross sections. With the numerical
model for submerged breakwaters, the wave transmission and reflection coefficients,
dynamic pressure distribution along the boundaries and the total wave forces on a
impermeable submerged breakwater and so on can be directly obtained. In addition
to all these quantities, the numerical model for berm breakwaters also gives the
water surface fluctuation inside porous berm. Both models are programmed with
linear gravity waves.
148
A permeable submerged breakwater of trapezoidal shape made of river gravel of
50 = 0.93 was tested in a wave channel for both nonbreaking waves and breaking
waves. The measured transmission and reflection coefficients, pressure distribution
over the breakwater and the wave amplitude modification over the breakwater were
compared well to the corresponding values predicted by the numerical model.
10.2 Conclusions
The analysis and the assessment of the three resistance components in the
porous flow model show that a wide range of conditions in the coastal environ
ment are outside the linear resistance region such that either inertial or turbulent
or both terms could be important. The effects of the nonlinear and the inertial
resistances are pronounced over a wide range of R values, or equivalently a wide
range of particle sizes, both in terms of wave attenuation and altering wave kine
matics. The effects on gravity waves due to the linear resistance are insignificant
whether it dominates the other two components or not. Consequently, a sandy
bottom of coarse sand or finer can be treated as impermeable over a wide range of
environmental conditions in coastal water.
The nature of the solution for the porous seabed problem differs significantly
from that of available solutions. The wave damping, hence the wave attenuation,
first increases with increasing R\ and reaches a peak value. Beyond this point,
the wave damping decreases with increasing permeability. This peak attenuation is
found to occur when the dissipative resistance (velocity related) is equal to the non
dissipative resistance (acceleration related). The magnitude and the correspond
ing permeability for peak damping calculated from this study are quite different
from those by the existing theories. This maximum damping phenomenon opens
an interesting possibility for designing a porous structure with optimum damping
capabilities.
149
The experimental results of porous seabed for 36 cases show that the wave
damping rate and the corresponding permeability can be successfully predicted by
the analytical solution. The measurements of wave frequency are also consistent
with the predictions. The added mass coefficient Ca for densely pact gravel is
determined to be equal to 0.46, very close to the theoretical value for a smooth
sphere, as opposed to zero as assumed by many authors. In fact, the least mean
square error of the nonlinear regression process was found to be very sensitive to the
value of Ca. The values of the other two velocity related resistance coefficients, a0,
b0, reconfirm those given by Engelund and others. The comparison of the data to
the predicted values by the analytical solution shows much better agreement than
those of all existing theories.
When the particles of a porous bed are so small that movement occurs at the bed
surface, the experimental data revealed that the analytical solution is not applicable
because of the violation of the noparticlemovement assumption. The data also
indicate that the turbulent boundary layer at the interface cannot be ignored when
it dominates the energy dissipation. This is the case when a bed is as thin as one
particle diameter. However, the wave frequency prediction for both cases remains
almost as accurate as that for the other cases.
For porous structures of irregular shapes, the numerical results show that the
linear element behaves much better than the constant element, especially at corner
points. With the special treatment applied in the numerical model, the singularities
at the corner points for velocity potential function or pressure distribution can be
successfully removed, although the discontinuities in the normal velocities around
such points remain. The radiation boundary condition at the far field in the fluid
domain renders a great deal of simplicity in the numerical formulations, as compared
to the matching boundary condition with the wave maker theory. Generally, as long
as the vertical boundary(ies) is placed far enough from the structure (about two
150
wave lengths from the toe of a structure), the accuracy of the numerical results can
be assured to satisfy engineering applications.
As in agreement with the solution for porous seabeds, the transmission coeffi
cient of porous submerged breakwaters computed by the numerical model shows a
well defined minimum value when it is plotted against the permeability parameter
R (or stone size). This means there exists a optimum stone size for the maximum
wave energy dissipation for a specified breakwater geometry and. wave condition.
This suggests that a breakwater can be tuned to achieve maximum effect.
Due to the inclusion of nonlinear effects in the porous flow model, the transmis
sion coefficient is a function of the incident wave height instead of independent to
it as in a linear model. The optimum stone size (or equivalently the permeability)
corresponding to the minimum wave transmission, as mentioned in the previous
paragraph, increases with increasing incident wave height, while the magnitude of
the peak dissipation remains more or less the same when wave height changes. In
a certain range of incident wave heights, when the stone size becomes greater than
the optimum size for the highest wave, the transmission coefficient decreases with
increasing incident wave height, while the same coefficient increases with the wave
height for a stone size smaller than the optimum size for the smallest wave height.
Thus, stones of larger size axe more effective for storm protection
In the laboratory experiment for the porous submerged breakwater, the phe
nomenon of wave energy shifting to higher frequencies was observed and the spectral
analysis of the wave records for transmitted waves clearly show that the high fre
quency waves in the lee side of the breakwater are the higher order harmonics of
the incident wave. The observation reveals that such higher order waves are not
significant in the offshore side or above the breakwater, but become stronger and
start to have phase lag with the fundamental wave from somewhere around the toe
of the structure in the lee side. Due to this reason, the wave envelope over the
151
breakwater as well as the pressure distribution along the breakwater surface in the
experiment are reasonably well predicted by the numerical model of linear waves,
when no breaking occurs.
Despite the existence of the higher order harmonics which are not predicted
in the numerical model for porous submerged breakwaters, when the concept of
equivalent wave height for transmitted waves is adopted, the agreement between the
experimental and the predicted values for transmission and reflection coefficients is
fairly good considering the complex nature of the problem. Such good agreement
indicates that the wave energy dissipating process within the porous breakwater can
be successfully modeled with the porous flow model using the empirical coefficients
determined in the porous seabed experiments.
The numerical results for porous berm breakwaters show that a significant
amount of the wave energy (as high as 80%) is dissipated within the berm. The
rest of the energy is reflected offshore. Compared to conventional subaerial break
waters with thin cover layers, the energy dissipation due to percolation in berm
breakwaters is certainly much larger.
All the field quantities vary with the incident wave height due to the nonlinearity
in the porous flow model. The percentage of the wave energy dissipation within the
porous berm decreases with increasing incident wave height for the cases computed.
For larger stone size and/or different cross sections, it is possible to make such
dissipation increase for higher incident waves.
10.3 Recommendations for Future Studies
For porous flow model, the inclusion of the deformation of solid skeletons, such
as elastic or plastic deformations, should be of interest to future engineering appli
cations when the materials for porous structures are deformable rather than rigid.
For porous seabed problem, further research work should be directed to develop
turbulent boundary layer solution so that the tangential boundary condition at
152
the interface could be more realistic. Another aspect is to extend the solution to
nonlinear waves. As far as the porous material is concerned, the solution of the
problem with a poroelastic flow model should be useful for modeling and designs
of deformable porous structures, such as breakwaters or seawalls made of discarded
rubber tires.
For modeling of porous structures with irregular cross sections, immediate at
tention should be given to the nonlinear wave approach since the mechanism of
energy transformation to the higher order harmonics observed in the experiment
cannot be explained by the present model. Further effort should also be devoted
to the computation of forces on individual blocks, especially on the blocks at the
surface of a porous structure. This will entail the modeling of the wave force as well
as the interlocking force of the blocks.
For porous submerged breakwaters, the modeling of breaking waves is of prac
tical interest since a large portion of the wave energy is dissipated by breaking and
the majority of large waves will break over such a structure. It is very difficult to
rigorously model a breaking wave due to the complexity involved in representing it
mathematically. One of the possibilities of attacking this problem is the empirical
formulation for wave decay pattern with systematic experiments and incorporate
this pattern into the numerical model.
In reality, the direction of incident waves is not always normal to the long axis
of a shoreparallel breakwater but oblique to it. Such oblique incident wave will
change the governing equation from Laplace equation to homogeneous Helmholtz
equation for the twodimensional approach adopted in this study. The modification
of the BIEM scheme for such change will make the numerical model more general
and more practical. Due to the fact that the actual breakwaters have finite lengths
and varying orientations (perpendicular or inclined to shorelines), three dimensional
modeling should certainly be the ultimate goal.
APPENDIX A
BOUNDARY INTEGRAL FORMULATION FOR ENERGY DISSIPATION IN
POROUS MEDIA
The conventional definition for the energy dissipation eD within a volume V of
a porous medium during the time period T is (Sollitt et al., 1972, Madsen, 1974,
and Sulisz, 1985)
r r*+T >
eD = J J Fpqdtdv (A.l)
where F, which is a function of spatial coordinates and time, is the dissipative
stress in the medium, q*is the discharge velocity of the porous flow and p is the fluid
density. Here both F and q are real quantities.
In the problems of interaction between water waves and porous media, a non
linear unsteady porous flow model with all the resistance terms has the form
 iVP = (Â£ + ia/3 + % I q 1)5 (A.2)
P KP yjKp
where P is pore pressure function, u is the kinematic viscosity of fluid and Kp is the
intrinsic permeability of the porous medium; a is wave frequency, (3 is the inertial
resistance parameter, Cj is a constant characterizes the nonlinear resistance and q
is the complex vector of discharge velocity in the porous medium, with
q = Re(q)
The dissipative stress in this model is
(A3)
with the inertial term, ipo(3, which is non dissipative, excluded from Eq.(A.2).
153
154
Since in this study, we deal exclusively with two dimensional cases, Eq.(A.l)
becomes
(A.4)
To perform the integration in Eq.(A.4) analytically, the quantities F and q
have to be expressible in explicit forms and the integrand has to be analytically
integrable over a specified domain A. Otherwise, a numerical integration over the
entire area needs to be employed, provided that the values of those two quantities
are known everywhere in the area. Although numerical integration of Eq.(A.4)
is applicable for most of the problems, the tedious grid generation and lengthy
numerical computation usually force people to make some simplifying assumptions
or approximations to the integrand to keep it analytically integrable. The accuracy
of the result from such an approach will no doubt be compromised.
However, if the energy dissipation can be expressed as a contour integration
along the boundary of the computation domain, the chance of working it out ana
lytically would be greatly increased. A boundary integral formulation of the energy
dissipation is also essentially important for the solutions of Boundary Integral El
ement Method where the unknowns are solved only on the boundaries. In this
appendix, such an attempt is made by reformulating the expression of energy dis
sipation and the use of Greens formula.
If we define U and W be the discharge velocities (real quantities) in x and z
directions, respectively and eD be the rate of energy dissipation per unit volume (also
a real quantity, considered as a possitive value), the rate of total energy dissipation
in an arbitrarily small cube of dx 1 dz can be expressed, according to Fig.3.1, as
ed dx dz
[(Pr + ^dx)[U + dx) dz + [Pr + ^dz)(W + "dz) dx
ox ox az oz
PrU dz PrW dx)
(A.5)
Then
155
fndU ndW TTdPr rdPT,
* = +
(A.6)
= +<^1
with Pr being the real part of the complex pore pressure function.
To compare with Eq.(A.4), we apply the continuity condition of pore fluid to
Eq.(A.6), the first two terms vanish. For the remaining two terms, by substituting
the porous flow model given by Eq.(A.2) for the derivatives of Pr and keeping only
the dissipative terms, noticing that q iU+jW and qq = U2+W2, one immediately
recognizes that Eq.(A.7) is just an alternate expression of the integrand in Eq.(A.4).
The total energy dissipation within the whole computational area during a time
period of T is then
eD
tp dtdxdz
/t+T red d
Jttei(up) + Tz(wp')]dxdzit (A'8)
where A is the area of computation domain. It could be the cross section area of a
submerged breakwater and so on.
Eq.(A.8) is an equivalent expression to Eq.(A.4) for the energy dissipation in
a porous medium. The only difference between the two is that the nondissipative
resistance is included in Eq.(A.8) but not in Eq.(A.4). From the procedure of
derivation of Eq.(A.8), one can clearly see that the nondissipative resistance should
not be simply left out. However, the value of the energy dissipation by both formulas
should be the same regardless whether this resistance is included or not, simply
because of its nondissipative nature.
By the use of Green formula, the area integration in Eq.(A.8) can be converted
df dg
into a contour integral. The Greens formula states that if /, o, and are
oz ox
156
W
U
Figure A.l: Geometric relations between the vectors
continuous on domain A n S, then
IL^Tz)dxi2 = ii,iz+3iz) (A9)
where S is the closed boundary of A.
Let
UP, = g WP, = f (A.10)
Then
ffA[Â§x(UPr) Â£z(WP,)]dxdz = f [(WPr)dx+{UPr)dz] (A.ll)
According to the geometrical relations of the vectors on the boundary shown in
Fig.A.l, the following expression can readily be obtained.
js{P,U~ P,W~)ds = js P,{Usino: + Wcos a)ds = Â£ P,q ds (A.12)
with q = U sin a + W cos a being the discharge velocity normal to the boundary
S.
Therefore, the energy dissipation within the area bounded by 5 during a time
ft+T r rt+T r
eD = J js P,q rids dt = ^ js(Jn),ds
dt
period T is
(A.13)
157
Here (Jn)r, a real quantity, is the energy flux normal to the boundary S.
The physical explanation of the above equation is that the energy dissipation
in the time period T is equal to the net energy flux across the boundary S into the
enclosed area in the same period.
Up to this point, we have been dealing with real quantities only. In many cases,
it is often found convenient to use complex variables with the real parts of them
being the corresponding physical quantities.
Let
Pr = Re[P{x,z,t)] = Re[p{x,z)e ,at]
q n = Re[un(x,z)e~,at]
(A.14)
(A.1S)
where both p and u are complex variables, it is not difficult to
following identity is true,
prove that the
PTq = Re\^(unpe~ilot + unp*)]
(A.16)
with p* being the conjugate of p.
Therefore, a complex energy flux function can be defined as
= ^(nP e_2,vt + u p*)
(A.17)
and the corresponding complex energy dissipation function is then
(A.18)
such that
IT.), = Rt{T.)
tu Rc[Ed)
(A.19)
(A.20)
If the time period T is chosen as the wave period, the integration of the first
term of the integrand in Eq.(A.18) with respect to time will vanish when p u is
158
not a function of time, and the complex energy dissipation is reduced to
Ed~ j> unp* ds
(A.21)
APPENDIX B
EXPERIMENTAL DATA FOR POROUS SEABEDS
In this appendix, the complete experimental data from the tests on porous
seabeds made of river gravel are given in graphical forms. The 36 test cases are
listed in Table B.l.
In the following figures, the averaged surface fluctuation and the wave height
attenuation for each case are plotted. For each case, the experiment was repeated 10
to 20 times and the surface fluctuations were normalized by the first wave height of
each data record. In top frames, the averaged wave forms, represented by solid lines,
were obtained by averaging all the normalized wave records for that case. The dash
lines are the envelopes of plus and minus one standard deviation from the averaged
value at each time instant. In lower frames, the dots represent the wave heights
deduced from each normalized record of surface elevation and the dashed curves are
the exponential decay function with the a,s obtained from the data ensemble by
the least square analysis.
In the figures, H denote the averaged wave height if, T is the averaged wave
Table B.l: Test Cases
d60 (cm)
h$ (cm)
h (cm)
L (cm)
0.72
20
20,25,30
200
0.93
20
30,25,20
200
1.20
20
30,25,20
200
1.48
20,15,10
30,25,20
200
2.09
20
30,25,20
200,225,250,275
2.84
20
30,25,20
200
3.74
20
30,25,20
200
159
160
period T and L is the wave length; DS is the bed thickness h, and DW is the water
depth h\ DDS is the particle diameter d50.
161
Figure B.l: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (HÂ¡TTÂ¡) and the best fit to the exponential decay
function.
162
TIME (SEC)
Figure B.2: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 0.93 cm. (a) Averaged nondimensional surface elevation {tÂ¡Â¡Hi), (b)
Nondimensional wave heights [Hand the best fit to the exponential decay
function.
163
Figure B.3: Case of L 200 cm, h DW = 30 cm, h, DS = 20 cm and
d5o = DD s= 1.20 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H fHÂ¡) and the best fit to the exponential decay
function.
164
Figure B.4: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
165
Figure B.5: Case of L 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d5o = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rÂ¡/Hi), (b)
Nondimensional wave heights (H/Si) and the best fit to the exponential decay
function.
166
Figure B.6: Case of L 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
so = DD = 2.84 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
167
Figure B.7: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (t]/Hi), (b)
Nondimensional wave heights (if/57) and the best fit to the exponential decay
function.
168
Figure B.8: Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm and
d50 = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (t)/Hi), (b)
Nondimensional wave heights (H/Jl[) and the best fit to the exponential decay
function.
169
Figure B.9: Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
dso = DD = 0.93 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights {H/H7) and the best fit to the exponential decay
function.
170
o
<\J
DW= 25. 0 CM
I
o
o
1H
* DS= 20.0 CM
j 00= 1.20 CM
E Hi
o
00
o
1
1 i
>
cr
* 1
3 0.60 .
4
o
LU
rvi
ft....
J 0.40 .
cr
*it
NORM
o
ro
o
i
* 4 f
o
o
o
j
0.0 M.O 8.0 12.0 16.0 20.0
TIME (SEC)
Figure B.10: Case of L 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 1.20 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
171
Figure B.ll: Case of L = 200 cm, h = DW = 25 cm, h, DS = 20 cm and
d50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
172
Figure B.12: Case of L 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation [ijjHi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
173
Figure B.13: Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 2.84 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (HÂ¡TT\) and the best fit to the exponential decay
function.
174
TIME (SEC)
Figure B.14: Case of L = 200 cm, h DW = 25 cm, h, = DS = 20 cm and
dso = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights [H/Hi) and the best fit to the exponential decay
function.
175
Figure B.15: Case of L = 200 cm, h = DW = 20 cm, h, DS = 20 cm and
d50 = DD = 0.72 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
176
TIME (SEC)
Figure B.16: Case of L = 200 cm, h DW 20 cm, h, DS = 20 cm and
d50 = DD = 0.93 cm. (a) Averaged nondimensional surface elevation (tj/H\), (b)
Nondimensional wave heights (H/H7) and the best fit to the exponential decay
function.
177
Figure B.17: Case of L = 200 cm, h = DW = 20 cm, h, = DS = 20 cm and
dso = DD = 1.20 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights {H/Hi) and the best fit to the exponential decay
function.
178
Figure B.18: Case of L = 200 cm, h DW = 20 cm, h, = DS = 20 cm and
d50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/7fÂ¡) and the best fit to the exponential decay
function.
179
Figure B.19: Case of L 200 cm, h DW = 20 cm, h, = DS = 20 cm and
so = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (r\jH\), (b)
Nondimensional wave heights {H/H7) and the best fit to the exponential decay
function.
180
Figure B.20: Case of L 200 cm, h = DW = 20 cm, h, = DS = 20 cm and
Nondimensional wave heights (H/ST) and the best fit to the exponential decay
function.
181
Figure B.21: Case of L 200 cm, h = DW = 20 cm, h, = DS = 20 cm and
d5Q = DD = 3.74 cm. (a) Averaged nondimensional surface elevation (rÂ¡Â¡H\), (b)
Nondimensional wave heights (if/Sj) and the best fit to the exponential decay
fraction.
182
TIME (SEC)
Figure B.22: Case of L = 200 cm, h = DW = 30 cm, h, DS = 15 cm and
<50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation [rj/Hi), (b)
Nondimensional wave heights (H/TT[) and the best fit to the exponential decay
function.
183
Figure B.23: Case of L = 200 cm, h = DW = 25 cm, h, DS = 15 cm and
so = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rÂ¡/Hi), (b)
Nondimensional wave heights (H/HI) and the best fit to the exponential decay
function.
184
Figure B.24: Case of L = 200 cm, h = DW = 20 cm, h, = DS = 15 cm and
<5o = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (ry/jEfj), (b)
Nondimensional wave heights (H/TF[) and the best fit to the exponential decay
function.
185
Figure B.25: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 10 cm and
d5o = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (ij/Hy), (b)
Nondimensional wave heights [H/~H^) and the best fit to the exponential decay
function.
186
Figure B.26: Case o L 200 cm, h = DW = 25 cm, h, = DS = 10 cm and
so = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights [HÂ¡TT?) and the best fit to the exponential decay
function.
187
Figure B.27: Case of L 200 cm, h DW = 20 cm, h, = DS = 10 cm and
so = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/H^) and the best fit to the exponential decay
function.
188
Figure B.28: Case of L = 225 cm, h = DW = 30 cm, h, = DS = 20 cm and
<50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights [H/B7) and the best fit to the exponential decay
function.
189
Figure B.29: Case of L = 225 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights {H/B7) and the best fit to the exponential decay
function.
190
1.20
Â£ 1.00
o
LU
x 0.80
UJ
>
3 0.60
O
UJ
rvi
j 0.40
cc
I 0.20
0.00
0.0 2.0 4.0 6.0 8.0 10.0 12.0
TIME ISEC)
Figure B.30: Case of L = 225 cm, h = DW = 20 cm, h, = DS = 20 cm and
dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights [H/TTÂ¡) and the best fit to the exponential decay
function.
DW= 20.0 CM
DS= 20.0 CM
DD= 2.09 CM
i 1 1 1 1 1 1 1 1 1 r
191
Figure B.31: Case of L = 250 cm, h DW = 30 cm, h, = DS = 20 cm and
50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (r]/Hi), (b)
Nondimensional wave heights (H/1TÂ¡) and the best fit to the exponential decay
function.
192
Figure B.32: Case of L = 250 cm, h = DW = 25 cm, h, = DS = 20 cm and
<5o = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/H^) and the best fit to the exponential decay
function.
193
Figure B.33: Case of L = 250 cm, h = DW = 20 cm, ht DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (t]/Hi), (b)
Nondimensional wave heights [H/li\) and the best fit to the exponential decay
function.
194
Figure B.34: Case of L 275 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights {H/H7) and the best fit to the exponential decay
function.
195
Figure B.35: Case of L = 275 cm, h = DW = 25 cm, h, DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/H\) and the best fit to the exponential decay
function.
196
1.20
<Â£ 1.00
o
UJ
1 0.80
UJ
>
2 0.60
O
UJ
rvi
Z 0.40
0.00
0.0 2.0 4.0 6.0 8.0 10.0 12.0
TIME (SEC)
Figure B.36: Case of L = 275 cm, h = DW = 20 cm, ht = DS = 20 cm and
<50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rz/ifi), (b)
Nondimensional wave heights (if/57) and the best fit to the exponential decay
function.
DW= 20.0 CM
DS= 20.0 CM
DD= 2.09 CM
i 1 1 1 1 1 1 1 1 1 r
BIBLIOGRAPHY
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NearSurface Breakwaters, Proc. 20th Coastal Eng. Conf., ASCE, 1729
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Developments in Breakwaters, Thomas Telford Ltd, London.
Brebbia, C. A. and Dominguez, J., 1989, Boundary Elements An
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Dagan, G., 1979, The Generalization of Darcys Law for Nonuniform Flows,
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Dick, T.M., and Brebner, A., 1968, Solid and Permeable Submerged Break
waters, Proc. 11th Coastal Eng. Conf., ASCE, 11411158.
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Characteristics, Pet. Trans. AIME, 207, pl44.
Fatt, I., 1956b, The Network Model of Porous Media: II. Capillary Pressure
Characteristics, Pet. Trans. AIME, 216, p449.
Finnigan, T.D., and Yamamoto, T., 1979, Analysis of SemiSubmerged Porous
Breakwaters, ASCE Proc. Civil Engineering in the Oceans, Vol. 1, 380
397.
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Breakwaters, Report of Port and Harbour Research Institute, Tokyo, Vol.
8, No.3, 18pp.
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198
Goda, Y., and Moriya, Y., 1967, Laboratory Investigation on Wave Transmis
sion over Breakwaters, Report of Port and Harbour Research Institute,
Tokyo, No.13.
Hannoura, A.A, and McCorquodale, J.A., 1978, Virtual Mass of Coarse Gran
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ASCE, Vol. 104, 191200.
Hunt, J.N., 1959, On the Damping of gravity Waves Propagated over a Per
meable Surface, J. of Geophysical Res., Vol. 64, No. 4, 437442.
Ijima, T., Chou., C.R., and Yoshida, A., 1976, Method of Analysis for
TwoDimensional Water Wave Problems, Proc. 15th Coastal Eng. Conf.,
ASCE, 27172736.
Johnson, J.W., Hondo, H., and Wallihan, R., 1972, Scale Effects in Wave
Action through Porous Structures, Proc. 10th Coastal Eng. Conf., ASCE,
10221024.
Kim, S.K., Liu, P.LF., and Ligget, J.A., 1983, Boundapr Integral Equa
tion Solutions for Solitary Wave Generation, Propagation and Runup,
Coastal Engineering, Vol. 7, 299317.
Hondo, H., and Toma, S., 1972, Reflection and Transmission for a Porous
Structure, Proc. 13th Coastal Eng. Conf., ASCE, 18471865.
Hondo, H., and Toma, S., 1974, Breaking Wave Transformation by Porous
Breakwaters, Coastal Engineering in Japan, Vol. 17, 8191.
Ligget., J.A., and Liu, P.LF., 1983, The Boundary Integral Equation
Method for Porous Media Flow. George Allen & Unwin Ltd., London.
Liu, P.LF., 1973, Damping of Water Waves over Porous Bed,
J. Hydraulics Div., ASCE, Vol. 99, 22632271.
Liu, P.LF., 1977, On Gravity Waves Propagated over a Layered Permeable
Bed, Coastal Engineering, Vol. 1, 135148.
Liu, P.LF., and Dalrymple, R. A., 1984, Damping of Gravity Water Waves
Due To Percolation, Coastal Engineering, Vol. 8, 3349.
Madsen, O.S., 1974, Wave Transmission through Porous Structures,
J. of the Waterways Harbors and Coastal Engineering Div., ASCE, Vol.
100, WW3, 169188.
Madsen, O.S., and White, S. M., 1976, Wave Transmission through Trape
zoidal Breakwaters, Proc. 15th Coastal Eng. Conf., ASCE, 26622676.
Madsen, O.S., Shusang, P., and Hanson, S.A., 1978, Wave Transmission
Through Trapezoidal Breakwaters, Proc. 16th Coastal Eng. Conf., ASCE,
18271846.
199
McCorquodale, J.A., 1970, A Variational Approach to NonDarcy Flow,
J. of the Hydraulics Div., ASCE, Vol. 96, HY11, 22652278.
McCorquodale, J.A., 1972, Wave Dissipation in Rockfill,
Proc. 13th Coastal Eng. Conf., ASCE, 18851900.
Murray, J.D., 1965, Viscous Damping of Gravity Waves over a Permeable
Bed, J. of Geophysical Res., Vol. 30, No. 10, 23252331.
Putnam, J.A., 1949, Loss of Wave Energy Due to Percolation in a Permeable
Sea Bottom, Trans., American Geophysical Union, Vol. 30, No. 3, 349356.
Raichlen, F. and Lee, JJ., 1978, An Inclined Plate Wave Generator,
Proc. 16th Coastal Eng. Conf., ASCE, 388399.
Reid R.O., and Kajiura, K., 1957, On the Damping of Gravity Waves over
a Permeable Sea Bed, Trans., American Geophysical Union, Vol. 38, No.
5, 662666.
Savage, R.P., 1953, Laboratory Study of Wave Energy Losses by Bottom
Friction and Percolation, TM No. 31, Beach Erosion Board, Department
of the Army.
Scarlatos, P.D., and Singh, V.P., 1987, Long Wave Transmission through
Porous Breakwaters, Coastal Engineering Vol. 11, 141157.
Scheidegger, A.E., ed., 1960, The Physics of Flow Through Porous Media, 2nd
ed., University of Toronto Press, Toronto, Canada.
Seelig, W.N., 1980, TwoDimensional Tests of Wave Transmission and Reflec
tion Characteristics of Laboratory Breakwaters, Technical Report 801,
U.S. Army, Corps of Engineers, CERC., Ft. Belvoir, Virginia.
Sollitt, C.K., and Cross, R.H., 1972, Wave Transmission through Permeable
Breakwaters, Proc. 13th Coastal Eng. Conf., ASCE, 18271846.
Sulisz, W., 1985, Wave Reflection and Transmission at Permeable Breakwaters
of Arbitrary Cross Section, Coastal Engineering Vol. 9, 371386.
T0rum, A., Naess, S., Instanes A. and Void S., 1988, On Berm Breakwaters,
Proc. 21th Coastal Eng. Conf., ASCE, 19972012.
Wang, H. and Gu, Z., 1988, "Gravity Waves over Porous Bottom,
Proc. 2nd Intern. Symp. on Wave Res, and Coastal Eng., Hannover Ger
many.
Ward, J.C., 1964, Turbulent Flow in Porous Media, J. Hydraulics Div.,
ASCE, Vol. 90, No. HY 5, 112.
Wilson, W.W., and Cross, R.H., 1972, Scale Effects in RubbleMound Break
waters, Proc. 13th Coastal Eng. Conf., ASCE, 18731883.
BIOGRAPHICAL SKETCH
Zhihao Gu was bom and raised in Dalian, a beautiful port city in NorthEast
China, on July 4, 1957. About two miles from his home, there was a pocket beach
where he used to go swimming every afternoon in summer. Perhaps it was there
that his bond with the beach and waves started. On weekends, he usually went
fishing with his father in the nearby reservoirs where he enjoyed calm waters and
learned many fishing tricks.
When he was 9, the disastrous cultural revolution began. The quality of edu
cation nosedived and he had to spend a lot of time for the revolution instead of
learning the things he was supposed to learn in school at his age. Under the guid
ance and scrutiny of his father and mother, he managed to complete high school
mathematics and physics by reading at home after school. His hobby at the time
was to play with electronic elements and assemble radios at home.
At the age of 18, right after graduating from high school, he was sent to a farm in
Inner Mongolia along with a group of his schoolmates during the revolution. Their
task was to repair the earth with shovels. It was then that he realized how much
he needed the beach and waves. Fortunately, the sweat of two and onehalf years
harsh labor did not completely wash away his sketchy knowledge attained from
the revolutionary school education and selflearning, which enabled him to drop
shovels and pick up pens again on the campus of Dalian Institute of Technology (a
university in his home town) in February 1978, after passing the first nationwide
college exam following the tenyear revolution. He was assigned to the offshore
engineering section in the hydraulic engineering department.
200
201
With four years of hard work at college and the thirst for knowledge of science,
he entered graduate school in the same university right after receiving his B.S.
degree in early 1982. His major was offshore structures. After about two years of
course work on structures, he found that he was more interested in water waves
than in solid mechanics. As a result, he chose the problem Wave Forces on Large
Three Dimensional Objectes of Arbitrary Shapes as the topic of his thesis. At the
end of 1984, he graduated with a Master of Science degree and was appointed as a
faculty member in the same department.
Unsatisfied with his knowledge in his field and longing for the advanced tech
nology of the west, he decided to pursue his higher education in the United States.
In early 1986, he was admitted to the Graduate School and, at the same time,
appointed as a research assistant by the Coastal and Oceanographic Engineering
Department of the University of Florida. Since August 1986, he has been working
towards his Ph.D. under the Florida sun.
Gu was married, in 1984, to Liqiu Guan who came to the United States to
join him in April 1987 and later became a Ph.D. student in the Civil Engineering
Department at the same university.
33
 = 0 at z (h + h,) (4.8)
02
The potential and pore pressure functions axe assumed to have the forms
$(x,z,t) = [>lcosh/:(h + z) + Bsmhk[h + z)\e^kx~at^ (4.9)
P,(x,z,t) = D cosh k(h + h, + z)e^kx~ot^ (4.10)
where A, B and D are unknown complex constants. Equation (4.8) is already
satisfied by P, in Eq.(4.10). Introducing Eqs.(4.9) and (4.10) into Eqs.(4.6) and
(4.7), A and B can be expressed in terms of D,
t cosh kha
pa
(4.11)
sinh kh,
pofo
(4.12)
$ is then given by
[it 1 1 *
pa
D can be obtained by the free surface boundary condition given by Eq.(4.4):
$(x, z, t) = [: cosh kh, cosh k(h + z) + sinh kh, sinh k[h + z) ]c^fcx at^ (4.13)
pa Jo
D =
pga
cosh kh cosh kh,{l ttinh kh tanh kht)
Jo
(4.14)
Finally, Eq.(4.5) along with Eq.(4.13) gives the complex wave dispersion equa
tion:
a2 gk tanh kh = ~r tanh kht(gk a2 tanh kh) (4.15)
Jo
here either a or k, or both, could be complex, as well as the coefficient f0.
In the above equation, when setting Ca = CÂ¡ = 0, it becomes the dispersion
equation obtained by Liu and Dalrymple (1984). If C/ 0, as will be the case in
this study, the coefficient /0 can not be a known value a priori. It becomes another
unknown besides a or k, and therefore the procedure of solution will be different
from that employed by Liu and Dalrymple. Before solving Eq.(4.15), a few limiting
cases are examined here.
100
of the breakwater is 12 meters wide with 1.6 meter submergence in a water of
4.6 meters deep. The slopes are again 1:1.5 on both sides. Comparing the wave
envelope for the impervious breakwater to the one for the porous breakwater, it
is obvious that they are similar in form, but the transmitted wave height by the
permeable breakwater is less than that by the concrete one as expected. It is noticed
that the energy is conserved for the impervious structure, i.e., K\ + Kjj = 1, while
some energy dissipation occurred in the porous breakwater. This dissipation is
1.0 K\ K\ = 29% of the total wave energy. The wave length of the transmitted
waves restored to the same value as that for the incident wave after being disturbed
by the breakwater.
Figures. 7.6 (a) and (b) illustrate the transmission and reflection coefficients
for submerged porous breakwater, which has the same configuration as the one in
Fig. 7.4, as a function of stone size for four different wave periods. The wave height
was kept constant, at H = 4 cm throughout the computation. In the top frame,
the transmission coefficient is shown to have a minimum value for each wave period
for a particular stone size (around d, = 1.0 to 2.0 cm). Beyond this point, the wave
transmission does not decrease with increasing stone size any more, but starts to
increase. This indicates that a larger stone size is not always better in terms of wave
damping. Therefore, there exists a optimum stone size with which maximum wave
energy dissipation and minimum wave transmission can be achieved. The reflection
coefficient is not significantly affected by the stone size as shown in Fig. 7.6, although
it decreases with increasing stone size for most of the cases.
Another phenomenon to be noted is that neither the transmission coefficient nor
the reflection coefficient changes monotonically with changing wave period. When
the same set of data is plotted against the permeability parameter R in Fig. 7.7,
it clearly shows that the lowest values for Kt and the highest values for Kr are
achieved at T = 1.2 seconds. This means that the breakwater is selective when
NORMALIZED PRESSURE DISTRIBUTION NORMALIZED WAVE ENVELOPE
127
1.50
0.50
0.00
1.50
BRK
I*.
BI EM MODEL H= 6.70 CM
pT o EXPERIMENT T = 0.858 S
*
50.0 30.0 10.0 10.0
X (CM)
30.0
50.0
Figure 8.11: The envelopes of wave and pressure distribution for T = 0.858 sec
breaking wave case, (a) Wave envelope; (b) Envelope of pressure distribution
NORMALIZED WAVE ENVELOPE
143
Figure 9.9: Permeable berm breakwater of prototype scale with H = 2.0 m: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity distributions
124
The pressure and the wave height distributions were measured both for non
breaking wave and breaking wave conditions of three wave periods. Table 8.4 shows
the averaged nondimensional pressure fluctuations and the locations of measure
ments. The normalized pressure head in the table was calculated according to
N
y \Pytak Ptrough)j
p tl
2N^H
(8.8)
where N is the number of waves recorded, H is the incident wave height and 7 is
the specific weight of water.
For comparison between theoretical and experimental results, the normalized
wave envelopes in the region above the crest and the pressure distributions on the
breakwater surface are plotted in Figs. 8.10 and 8.11 for nonbreaking and break
ing cases respectively, for T 0.858 seconds. It is seen from Fig. 8.10 that in
the nonbreaking case, although the measured wave envelope above the breakwa
ter crest is shifted slightly upward, the numerical model is able to predict, with
sufficient accuracy, both the variation patterns and the magnitudes of the wave
heights (the distances between the two envelope profiles). Fair good agreement
was also found in pressure distribution for nonbreaking waves (Fig. 8.10 b), espe
cially the variation pattern. The magnitude of the measured pressure distribution
is consistently smaller than that of the predicted. In the case of breaking waves
(Fig. 8.11), greater discrepancies (between the predicted and the measured values)
were expected because the nonbreaking assumption in the numerical model had
been violated. However, the prediction for both wave envelope and pressure distri
bution before the breaking point turned out to agree fairly well with those measured.
The good prediction of the flow field for the upper front portion of the breakwater,
which was found in the experiment to be most susceptible to damage, suggests that
the numerical model could be applied to breaking wave cases for the purpose of
extreme force prediction.
129
Figure 9.1: Definition sketch for berm breakwaters
impermeable. In this study, we consider only the case of impervious core, which
is usually the case in present practice (T0rum et al. 1988). There are again two
types of domains, the fluid domain and the porous domain. In the fluid domain,
the governing equation and the boundary conditions are essentially the same as
those described in Chapter 7 for submerged breakwaters. In the porous domain,
the governing equation, the impermeable and the interface boundary conditions
are also the same as those for the porous regions of submerged breakwaters. The
special aspect for the computation of berm breakwaters (or for regular subaerial
breakwaters) is the treatment of the free surface inside the porous medium. The
boundary condition for this boundary is different from the CFSBC for the fluid
domain.
According to Eq.(7.4) and Eq.(7.7)
P
Pfo
(7.8)
here p, cr and /0 are all treated as constants.
Substituting Eq.(7.4) into the continuity condition for the porous flow in terms
of the discharge velocity,
Vg = 0 (7.9)
the governing equation for the porous medium domain becomes the Laplace equation
in terms of the pore pressure,
V2P = 0 (7.10)
In this study, the waves are assumed to follow the linear wave theory
$ = eiot (7.11)
P = peiot (7.12)
It is to be noted that the time function is now expressed as e'at, as opposed to e~'at
used in the seabed problem presented in previous chapters.
7.2 Boundary Conditions
The whole computation domain has four types of boundaries, free surface, im
permeable bottom, permeable interface of different subdomains and the artificial
lateral boundaries. These boundary conditions are discussed here.
7.2.1 Boundary Conditions for The Fluid Domain
1. The free surface
The combined linear free surface boundary condition in the linear wave theory
s.d,
dz gV
where z is the vertical coordinate and g is the gravity acceleration.
(7.13)
192
Figure B.32: Case of L = 250 cm, h = DW = 25 cm, h, = DS = 20 cm and
<5o = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights (H/H^) and the best fit to the exponential decay
function.
189
Figure B.29: Case of L = 225 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights {H/B7) and the best fit to the exponential decay
function.
164
Figure B.4: Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/Hi) and the best fit to the exponential decay
function.
135
Figure 9.2: Flow chart of the numerical model for berm breakwaters
WOVE DAMPING RATE WAVE FREQUENCY
48
PARTICLE SIZE (CM)
(a)
0.23 2.28 22.69 225.MM
Figure 4.4: Standing wave case, (a) Nondimensional wave frequency crr/(L/g); (b)
Nondimensional wave damping rate Ci/(L/g)t.
70
6.3 Linear Element and Related Integrations
By the definition stated in the last section, on a linear element, a quantity and
its normal derivative axe interpolated by a linear function between the values at
the two nodes, Pj and Ph i. For the velocity potential function $ and its normal
3$
derivative we have, in terms of Â£ rj coordinates,
on
4>() = (*> y)t + (fe+i*f &*<+> Â£,
Vj'+l V)
It is obvious that
..3$. ,5$. .3$. ,3$. i
i(^;W (ggjjf +
ei+. &
, < < {,
$(Â£,) = <Â¡>i;
3$ 3$
(6.22)
(6.23)
*(ii+i) = *i+1; ()(fi+i) = (Â§Â£),
Substituting Eqs.(6.22) and (6.23) into Eq.(6.12), and noting that
r, = 1
JVi+2
(6.24)
5rf =
dn
dr{ r?i
3t7, r,
(6.25)
the integration over segment j (between Pj and PJ+1) cam be carried out analytically
in the local coordinate system (Ligget and Liu, 1983):
3r, 3$,
_e //+1 3r, 3$,,.
 4 4^nr'^)di
= Hi ft + Hi $;+1  Hi(i )7+1
(6.26)
where the superscripts ~ and + refer to the positions immediately before and after
the nodal point denoted by the corresponding subscript. And
Hh = /g + feJS
(6.27)
86
* = 1,2,
N
Equation (7.39) is not solvable at this point since the number of unknowns is
still greater than the number of equations. The additional required equations arise
from the interfacing with the porous subdomain.
7.3.2 Porous Medium Domain
In this subdomain, the integration is in the opposite direction to the one in the
fluid domain, therefore, on the common boundary of the two domains, the contour
integrations in both domains are in the same direction as shown in Fig. 7.1. It is so
arranged as to facilitate matching with the boundary conditions along the interface.
Similar to the fluid domain, the expanded form of Eq.(6.12) for the porous
subdomain is
+ E + r
Ncrn*hi
= urti + Eli + K&.p~. (7.41)
y=2
after applying the noflux boundary condition on the impervious bottom. Where
the superscript s refers to the quantities in the porous subdomain, N* is the total
number of nodes on the entire closed boundary of this domain and Ncm is the node
number on the common boundary with the fluid domain. In this equation, p~ is
assumed to be equal to p+ for all the points except node 1 and Nem
In matrix form, Eq.(7.41) reads,
Cn
C21
(7.42)
where pe and pnc are the vectors of pressure function and its normal derivative
along the common boundary and p and pnj are the corresponding vectors on the
NORMALIZED PRES. RND VEL.DIST. NORMALIZED WAVE ENVELOPE
99
Figure 7.4: Porous submerged breakwater: (a) Wave form and wave envelope; (b)
Envelopes of pressure and normal velocity.
13
or equivalently
a2 gktsuikh yy tanh kh,{gk o2tanhkh)
(2.30)
R 1 n
where h, is the seabed thickness and R is the permeability parameter defined as
V
It has been shown that the corrections by the laminar boundary layers were
not significant since the damping is largely due to the energy losses in the porous
medium rather than the boundary layer losses (Liu and Dalrymple, 1984, p47).
Therefore, it is reasonable to believe that if an appropriate porous flow model is
adopted, the solution for a homogeneous problem, without the correction of laminar
boundary layers, will be sufficient for engineering purposes.
Besides the theoretical studies, the interaction of waves and porous seabeds was
also investigated experimentally by Savage (1953). The experiment was carried out
with progressive waves over sand beds of median diameters, d, = 0.382 cm and d, =
0.194 cm, respectively. The permeability coefficient of the sand bed was measured
as 44.9 x 10~10m2, and the dimension of the seabed was 0.3 meters thick and 18.3
meters long. The wave heights at the beginning and the end of the sand bed were
recorded and adjusted to eliminate the effects of the side friction. Most of the
theories listed above were claimed to agree well with the data in this experiment.
2.3 Modeling of Permeable Structures of Irregular Cross Sections
The ability to predict the performance of rubblemound breakwaters under the
attacks of ocean waves is critically important in designing such structures. Signif
icant amounts of effort have been devoted to this subject. In the early stage of
development, research was basically confined to laboratory experiments and only
the empirical formulae extracted from the experiments were available for designs.
In recent years, with the continuing efforts by researchers, more and more theoreti
cal models for different types of breakwaters are becoming available. For subaerial
NORMALIZED SURFACE ELEVATION NORMALIZED SURFACE ELEVATION
98
Figure 7.3: Wave envelopes for (a) Transparent submerged breakwater; (b) Im
permeable step.
TRANSMISSION COEFF.
104
Figure 7.7: Transmission and reflection coefficients vs. R for different wave periods
(a) Transmission coefficient; (b) Reflection coefficient.
160
period T and L is the wave length; DS is the bed thickness h, and DW is the water
depth h\ DDS is the particle diameter d50.
147
A series of systematic physical experiments was conducted in a wave channel
with standing waves on the porous seabeds of river gravel of seven sizes. The
empirical coefficients in the analytical solution were determined for experimental
data by nonlinear regression.
The porous flow model with coefficients determined from the seabed experiment
was further applied to the computation of porous structures of irregular cross sec
tions. Two numerical models using boundary integral element method had been
developed for such structures. One is for porous submerged breakwaters and the
other one is for porous berm breakwaters. In these two numerical models, the
boundary element used in the numerical formulations are linear elements. The
far field boundary condition in the fluid domain is the radiation condition, which
assumes that the waves at such a boundary are purely progressive waves. The lin
earization of the nonlinear porous flow model is carried out by an iteration process
in which the energy dissipation within a porous breakwater was computed by using
the boundary integral expression instead of the conventional volumetric integration.
The autoelement generation subroutine in the numerical models reduces the data
preparation for input data files to a minimum and, henceforth, making the use of
the programs an easy task.
The main function of the numerical models is to compute the wave field around
and inside the porous structures of arbitrary cross sections. With the numerical
model for submerged breakwaters, the wave transmission and reflection coefficients,
dynamic pressure distribution along the boundaries and the total wave forces on a
impermeable submerged breakwater and so on can be directly obtained. In addition
to all these quantities, the numerical model for berm breakwaters also gives the
water surface fluctuation inside porous berm. Both models are programmed with
linear gravity waves.
SPECTRUM (CMmm2SEC)
112
5.0
4.5 .
4.0 .
3.5 .
3.0 .
2.5 .
2.0
1.5
1.0
0.5
0.0
0.0 0.5 1.0 1.5
FI =
0.
78
1/SEC.
F 2 
1.
56
1/SEC.
F 3 =
2.
34
1/SEC.
HT =
2.
86
CM
T
2.0 2.5 3.0
FREQUENCY (1/SEC)
Figure 8.3: The wave spectrum of the transmitted
3.5 4.0
waves
186
Figure B.26: Case o L 200 cm, h = DW = 25 cm, h, = DS = 10 cm and
so = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights [HÂ¡TT?) and the best fit to the exponential decay
function.
BIBLIOGRAPHY
Adams, C.B., and Sonu, C.J., 1986, Wave Transmission across Submerged
NearSurface Breakwaters, Proc. 20th Coastal Eng. Conf., ASCE, 1729
1737.
Baba, M., 1986, Computation of Wave Transmission over a Shore Protecting
Submerged Breakwater, Ocean Engineering, Vol. 13, No. 3, 227237.
Barends, F.B.J., 1986, Geotechnical Aspects of RubbleMound Breakwaters,
Developments in Breakwaters, Thomas Telford Ltd, London.
Brebbia, C. A. and Dominguez, J., 1989, Boundary Elements An
Introductory Course, Computational Mechanics Publications, Great
Britain.
Dagan, G., 1979, The Generalization of Darcys Law for Nonuniform Flows,
Water Resources Research. Vol. 15, No. 1, 17.
Dattatri, J., Raman, H., and Shankar, N.J., 1978, Performance Characteristics
of Submerged Breakwaters, Proc. 16th Coastal Eng. Conf., ASCE, 2153
2171.
Dean, R. G., and Dalrymple, R.A., 1984, Water Wave Mechanics for Engineers
and Scientists. PrenticeHall, Englewood Cliffs, New Jersey.
Delmonte, R.C., 1972, Scale Effects of Wave Transmission through Permeable
Structures, Proc. 13th Coastal Eng. Conf., ASCE, 18671871.
Dick, T.M., and Brebner, A., 1968, Solid and Permeable Submerged Break
waters, Proc. 11th Coastal Eng. Conf., ASCE, 11411158.
Fatt, I., 1956a, The Network Model of Porous Media: I. Capillary Pressure
Characteristics, Pet. Trans. AIME, 207, pl44.
Fatt, I., 1956b, The Network Model of Porous Media: II. Capillary Pressure
Characteristics, Pet. Trans. AIME, 216, p449.
Finnigan, T.D., and Yamamoto, T., 1979, Analysis of SemiSubmerged Porous
Breakwaters, ASCE Proc. Civil Engineering in the Oceans, Vol. 1, 380
397.
Goda, Y., 1969, Reanalysis of Laboratory Data on Wave Transmission over
Breakwaters, Report of Port and Harbour Research Institute, Tokyo, Vol.
8, No.3, 18pp.
197
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
WATER WAVE INTERACTION WITH POROUS STRUCTURES OF
IRREGULAR CROSS SECTIONS
By
Zhihao Gu
December. 1990
Chairman: Dr. Hsiang Wang
Major Department: Coastal and Oceanographic Engineering
A general unsteady porous flow model is developed based on the assumption
that the porous media can be treated as a continuum. The model clearly defines
the role of solid, and fluid motions and henceforth their interactions. All the im
portant resistant forces are clearly and rigorously defined. The model is applied to
the gravity wave field over a porous bed of finite depth. By applying linear wave
theory, an analytical solution is obtained, which is applicable to the full range of
permeability. The solution yields significantly different results from those of con
temporary theory. The solution requires three empirical coefficients, respectively
representing linear, nonlinear and inertial resistance. Laboratory experiments using
a standing wave system over a porous seabed were conducted to determine these
coefficients and to compare with analytical results. The coefficients related to lin
ear and nonlinear resistances were found to be close to those obtained by previous
investigators. The virtual mass coefficient was determined to be around 0.46, close
to the theoretical value of 0.5 for a sphere. The analytical solution compared well
with the experiments.
Based on this porous flow model and linear wave theory, two numerical models
using boundary integral element method with linear elements are developed for
xvi
150
wave lengths from the toe of a structure), the accuracy of the numerical results can
be assured to satisfy engineering applications.
As in agreement with the solution for porous seabeds, the transmission coeffi
cient of porous submerged breakwaters computed by the numerical model shows a
well defined minimum value when it is plotted against the permeability parameter
R (or stone size). This means there exists a optimum stone size for the maximum
wave energy dissipation for a specified breakwater geometry and. wave condition.
This suggests that a breakwater can be tuned to achieve maximum effect.
Due to the inclusion of nonlinear effects in the porous flow model, the transmis
sion coefficient is a function of the incident wave height instead of independent to
it as in a linear model. The optimum stone size (or equivalently the permeability)
corresponding to the minimum wave transmission, as mentioned in the previous
paragraph, increases with increasing incident wave height, while the magnitude of
the peak dissipation remains more or less the same when wave height changes. In
a certain range of incident wave heights, when the stone size becomes greater than
the optimum size for the highest wave, the transmission coefficient decreases with
increasing incident wave height, while the same coefficient increases with the wave
height for a stone size smaller than the optimum size for the smallest wave height.
Thus, stones of larger size axe more effective for storm protection
In the laboratory experiment for the porous submerged breakwater, the phe
nomenon of wave energy shifting to higher frequencies was observed and the spectral
analysis of the wave records for transmitted waves clearly show that the high fre
quency waves in the lee side of the breakwater are the higher order harmonics of
the incident wave. The observation reveals that such higher order waves are not
significant in the offshore side or above the breakwater, but become stronger and
start to have phase lag with the fundamental wave from somewhere around the toe
of the structure in the lee side. Due to this reason, the wave envelope over the
WATER WAVE INTERACTION WITH POROUS STRUCTURES OF
IRREGULAR CROSS SECTIONS
By
Zhihao Gu
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1990
NORMALIZED WAVE ENVELOPE
140
Figure 9.6: Permeable berm breakwater of model scale with H = 5.0 cm: (a) Wave
envelope; (b) Envelopes of pressure and normal velocity distribution
HAVE DAHPINC RATE WAVE NUMBER
45
PARTICLE SIZE (CM)
0.23 2.32 23.20 232.00
0.23 2.32 23.20 232.00
(b)
Figure 4.2: Progressive wave case, (a) Nondimensional wave number kr/(c*/g), (b)
Nondimensional wave damping rate fc,/(<72/^).
TRANSMI TED MOVES (CM) INCID. AND REFL. MOVES (CM)
111
TIME (SEC)
TIME (SEC)
Figure 8.2: Typical wave record, (a) Partial standing waves on the up wave side,
(b) Transmitted waves on the down wave side.
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Among the existing shore protecting breakwaters, a large number of them are
rubblemound structures made of quarry stones and/or artificial blocks. They can
be treated as structures of granular materials. A breakwater is called subaerial
when its crest is protruding out of the water surface and submerged when its crest
is below the water level. Sometimes a breakwater is subaerial at low tide and
submerged at high tide. These types of structures are usually termed as low crest
breakwaters. When a beach is to be protected from wave erosion, a detached shore
parallel submerged breakwater may provide an effective and economic solution. The
advantages of submerged breakwaters as compared to subaerial ones are low cost,
aesthetics (they do not block the view of the sea) and effectiveness in triggering the
early breaking of incident waves, thus reducing the wave energy in the protected
area. With the increasing interest in recreational beach protection, where complete
wave blockage is not necessary, submerged breakwaters may find more and more
applications.
On the other hand, if wave blockage is the main objective such as for harbor
and port protection, a subaerial breakwater may be more effective. The traditional
design of such a structure is a trapezoidally shaped rubble mound with an inner
core covered with one or more thin layers of large blocks to form the armor layer (s)
to protect the core. Owing to the demand for deeper water applications, the armor
sizes have become larger and larger. This would greatly increase not only the cost
but also the structural vulnerability. A relatively new type of structures, called
1
8.10 The envelopes of wave and pressure distribution for T 0.858
sec.; nonbreaking wave case, (a) Wave envelope; (b) Envelope
of pressure distribution 126
8.11 The envelopes of wave and pressure distribution for T 0.858
sec.; breaking wave case, (a) Wave envelope; (b) Envelope of
pressure distribution 127
9.1 Definition sketch for berm breakwaters 129
9.2 Flow chart of the numerical model for berm breakwaters .... 135
9.3 Berm breakwaters of vertical face: (a) Zero permeability; (b)
Infinite permeability 136
9.4 Berm breakwaters of inclined face: (a) Zero permeability; (b)
Infinite permeability 137
9.5 The Cross Section of The Berm Breakwater 138
9.6 Permeable berm breakwater of model scale with H = 5.0 cm: (a)
Wave envelope; (b) Envelopes of pressure and normal velocity
distribution 140
9.7 Permeable berm breakwater of model scale with H = 10.0 cm:
(a) Wave envelope; (b) Envelopes of pressure and normal velocity
distribution 141
9.8 Permeable berm breakwater of model scale with H = 20.0 cm:
(a) Wave envelope; (b) Envelopes of pressure and normal velocity
distribution 142
9.9 Permeable berm breakwater of prototype scale with H = 2.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions 143
9.10 Permeable berm breakwater of prototype scale with H = 4.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions 144
9.11 Permeable berm breakwater of prototype scale with H = 8.0
m: (a) Wave envelope; (b) Envelopes of pressure and normal
velocity distributions 145
A.l Geometric relations between the vectors 156
B.l Case of L = 200 cm, h = DW = 30 cm, k, = DS = 20 cm
and d50 = DD = 0.72 cm. (a) Averaged nondimensional surface
elevation (rj/Hi), (b) Nondimensional wave heights (H/Hi) and
the best fit to the exponential decay function 161
x
3
they cover, at least, an adequate range of intended application.
A numerical solution with computer code is then to be developed for arbitrary
geometry. Boundary Integral Element Method (BIEM) is proven to be very effi
cient for boundary value problems with complicated domain geometries. With this
method, the solution is expressed in boundary integrals and no interior points have
to be involved in the solution procedure. The aim of this study is to develop an
efficient computer code based on such a method. Finally, the validity of the solution
is to be assessed by a set of experiments.
1.2 Objectives and Scope
Specifically, the objectives of this study are listed as follows:
1. Develop a percolation model suitable for unsteady and turbulent porous flows,
2. Verify the model through an analytical solution and laboratory experiments for
the case of a flat porous seabed subject to linear gravity waves,
3. Based upon the porous flow model, develop a numerical solution for submerged
and berm breakwaters of arbitrary cross sections using the boundary integral ele
ment method.
To achieve these goals, the research is carried out in the following steps:
1. Derive the porous flow model,
2. Examine and interpret the relative importance of the various terms in the model
to narrow the scope of the study,
3. Obtain the analytical solution of wave attenuation over a flat porous seabed,
compare the solution with the existing ones and examine the wave energy dissipa
tion process,
4. Conduct a laboratory experiment for the flat porous seabed case to determine the
empirical coefficients in the porous flow model and to verify the analytical solution,
5. Develop a numerical model using the BIEM method for porous submerged break
waters,
199
McCorquodale, J.A., 1970, A Variational Approach to NonDarcy Flow,
J. of the Hydraulics Div., ASCE, Vol. 96, HY11, 22652278.
McCorquodale, J.A., 1972, Wave Dissipation in Rockfill,
Proc. 13th Coastal Eng. Conf., ASCE, 18851900.
Murray, J.D., 1965, Viscous Damping of Gravity Waves over a Permeable
Bed, J. of Geophysical Res., Vol. 30, No. 10, 23252331.
Putnam, J.A., 1949, Loss of Wave Energy Due to Percolation in a Permeable
Sea Bottom, Trans., American Geophysical Union, Vol. 30, No. 3, 349356.
Raichlen, F. and Lee, JJ., 1978, An Inclined Plate Wave Generator,
Proc. 16th Coastal Eng. Conf., ASCE, 388399.
Reid R.O., and Kajiura, K., 1957, On the Damping of Gravity Waves over
a Permeable Sea Bed, Trans., American Geophysical Union, Vol. 38, No.
5, 662666.
Savage, R.P., 1953, Laboratory Study of Wave Energy Losses by Bottom
Friction and Percolation, TM No. 31, Beach Erosion Board, Department
of the Army.
Scarlatos, P.D., and Singh, V.P., 1987, Long Wave Transmission through
Porous Breakwaters, Coastal Engineering Vol. 11, 141157.
Scheidegger, A.E., ed., 1960, The Physics of Flow Through Porous Media, 2nd
ed., University of Toronto Press, Toronto, Canada.
Seelig, W.N., 1980, TwoDimensional Tests of Wave Transmission and Reflec
tion Characteristics of Laboratory Breakwaters, Technical Report 801,
U.S. Army, Corps of Engineers, CERC., Ft. Belvoir, Virginia.
Sollitt, C.K., and Cross, R.H., 1972, Wave Transmission through Permeable
Breakwaters, Proc. 13th Coastal Eng. Conf., ASCE, 18271846.
Sulisz, W., 1985, Wave Reflection and Transmission at Permeable Breakwaters
of Arbitrary Cross Section, Coastal Engineering Vol. 9, 371386.
T0rum, A., Naess, S., Instanes A. and Void S., 1988, On Berm Breakwaters,
Proc. 21th Coastal Eng. Conf., ASCE, 19972012.
Wang, H. and Gu, Z., 1988, "Gravity Waves over Porous Bottom,
Proc. 2nd Intern. Symp. on Wave Res, and Coastal Eng., Hannover Ger
many.
Ward, J.C., 1964, Turbulent Flow in Porous Media, J. Hydraulics Div.,
ASCE, Vol. 90, No. HY 5, 112.
Wilson, W.W., and Cross, R.H., 1972, Scale Effects in RubbleMound Break
waters, Proc. 13th Coastal Eng. Conf., ASCE, 18731883.
113
where N is the total number of harmonics, C* is the group velocity of the tth order
harmonics (i = 1 refers to the fundamental wave) and k{ is the corresponding wave
number. With the definition of
, pgHf
8
the equivalent transmitted wave height is then
(HtU =
N n
\ ni
(8.4)
where (#<), is the wave height of the tth order harmonic wave, which could be
determined by the corresponding spike area of the spectrum diagram. The reflection
and transmission coefficients for the experiment data axe then defined as
Kr = H (8.5)
K, = (8.6)
Table 8.2 shows the comparison of the experimental and the predicted trans
mission and reflection coefficients. The relative error A in the table is defined as
A% = _gpl x ioo% (8.7)
Kp
where the subscripts m and p refer to the measured and the predicted values and K
can be either Kr or Kt.
As shown in the table, the agreement between the predicted and the measured
transmission coefficients is reasonably good with the maximum relative error being
about 12.4%. The relative errors for the computed and the experimental reflection
coefficients appear to be fairly large (e.g., Ama* = 71%), even though the absolute
errors axe generally not as large as those for the transmission coefficients. The
majority of the large relative errors occurs when the absolute values of Kr are small.
The comparison is also demonstrated in the following two figures. In Fig. 8.4, the
190
1.20
Â£ 1.00
o
LU
x 0.80
UJ
>
3 0.60
O
UJ
rvi
j 0.40
cc
I 0.20
0.00
0.0 2.0 4.0 6.0 8.0 10.0 12.0
TIME ISEC)
Figure B.30: Case of L = 225 cm, h = DW = 20 cm, h, = DS = 20 cm and
dso = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (tj/Hi), (b)
Nondimensional wave heights [H/TTÂ¡) and the best fit to the exponential decay
function.
DW= 20.0 CM
DS= 20.0 CM
DD= 2.09 CM
i 1 1 1 1 1 1 1 1 1 r
106
WT (DEGREE)
Figure 7.9: Wave forces and over turning moment for a impermeable submerged
breakwater: (a) Wave forces; (b) Overturning moment.
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B.2 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and d50 = DD = 0.93 cm. (a) Averaged nondimensional surface
elevation (r?/ifi), (b) Nondimensional wave heights (H/Hi) and
the best fit to the exponential decay function 162
B.3 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and dso = DD = 1.20 cm. (a) Averaged nondimensional surface
elevation (rj/HA, (b) Nondimensional wave heights (H/HA and
the best fit to the exponential decay function 163
B.4 Case of L = 200 cm, h DW = 30 cm, h, = DS = 20 cm
and d60 = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (tj/HA, (b) Nondimensional wave heights (H/HA and
the best fit to the exponential decay function 164
B.5 Case of L = 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.09 cm. (a) Averaged nondimensional surface
elevation (r]/HA> (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 165
B.6 Case of L 200 cm, h = DW = 30 cm, h, = DS = 20 cm
and so = DD = 2.84 cm. (a) Averaged nondimensional surface
elevation (rÂ¡/HA, (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 166
B.7 Case of L = 200 cm, h = DW = 30 cm, h, DS = 20 cm
and so = DD = 3.74 cm. (a) Averaged nondimensional surface
elevation (r)/H\), (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 167
B.8 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 0.72 cm. (a) Averaged nondimensional surface
elevation (rÂ¡/HA, (b) Nondimensional wave heights (H/HA and
the best fit to the exponential decay function 168
B.9 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 0.93 cm. (a) Averaged nondimensional surface
elevation (rj/HA, (b) Nondimensional wave heights (H/HA and
the best fit to tne exponential decay function 169
B.10 Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm
and so = DD = 1.20 cm. (a) Averaged nondimensional surface
elevation (rÂ¡/H\), (b) Nondimensional wave heights (H /HA and
the best fit to tne exponential decay function 170
B.ll Case of L = 200 cm, h DW = 25 cm, h, = DS = 20 cm
and so = DD = 1.48 cm. (a) Averaged nondimensional surface
elevation (t)/HA (b) Nondimensional wave heights [H/HA and
the best fit to the exponential decay function 171
Table 8.1: Test Results of Nonbreaking Waves
T (sec.)
Hi (cm)
Hr (cm)
Ha (cm)
Ht2 (cm)
Hti (cm)
2.67
0.07
1.87
0.18
0.03
0.642
3.38
0.10
2.37
0.18
0.04
4.07
0.11
2.88
0.19
0.03
4.28
0.15
3.00
0.18
0.06
2.22
0.13
1.54
0.58
0.08
0.858
2.92
0.11
2.03
0.97
0.18
3.31
0.15
2.33
1.16
0.31
4.31
0.16
2.72
1.85
0.61
2.14
0.19
1.44
0.92
0.13
0.952
2.59
0.22
1.69
1.26
0.17
3.38
0.25
2.05
1.80
0.66
3.89
0.37
2.40
2.28
0.34
1.96
0.31
1.32
0.72
0.14
1.020
2.47
0.38
1.61
1.08
0.23
3.38
0.47
2.17
1.95
0.20
4.29
0.52
2.54
2.56
0.64
2.06
0.34
1.35
0.83
0.12
2.29
0.41
1.50
0.96
0.12
2.42
0.52
1.47
1.19
0.24
1.120
2.69
0.50
1.68
1.31
0.14
3.18
0.60
1.75
1.96
0.69
4.18
0.84
2.30
2.67
0.63
4.43
0.72
2.06
3.02
0.20
4.77
0.78
2.18
2.78
0.69
1.58
0.35
1.14
0.30
0.16
1.265
2.10
0.50
1.54
0.57
0.38
3.00
0.64
2.07
1.08
0.94
3.39
0.53
2.34
1.21
1.12
1.67
0.37
1.21
0.34
0.30
2.95
0.63
2.01
0.87
1.08
1.379
3.09
0.50
2.02
0.86
1.16
3.38
0.58
2.28
1.05
1.34
3.89
0.75
2.48
1.30
1.64
1.81
0.36
1.28
0.52
0.34
1.453
2.35
0.36
1.65
0.84
0.60
3.08
0.54
1.98
1.11
1.10
3.96
0.66
2.40
1.66
1.54
2.19
0.17
1.53
0.66
0.43
1.778
2.66
0.14
1.75
0.87
0.63
3.28
0.26
2.15
1.27
0.97
3.98
0.22
2.41
1.69
1.58
CHAPTER 7
NUMERICAL MODEL FOR SUBMERGED POROUS BREAKWATERS
In this chapter, a numerical model of BIEM for submerged porous breakwaters
is developed by using the unsteady porous flow model given in Chapter 3. The
basic function of the numerical model is to compute the wave flow field and related
quantities such as the wave form, the dynamic pressure and the normal velocity
along all the boundaries and so on. With these quantities, the wave transmission
and reflection coefficients and wave forces can then be obtained.
7.1 Governing Equations
The computation domain of the problem, as shown in Fig. 7.1, consists of two
subdomains, the fluid domain and the domain of the porous medium. In the fluid
domain, the water is considered inviscid and incompressible. The flow induced by
gravity waves is assumed irrotational. Thus, the governing equation in this domain,
for the velocity function $, is the Laplace equation,
V2$ = 0 (7.1)
with fluid velocities being defined as
w =
Â£Â£
dz
(7.3)
While in the porous medium domain, the viscosity of the fluid cannot be ig
nored since the flow is largely within the low Reynolds number region. The flow is
76
87
impervious bottom
of the subdomain with
Pc =
{pt,P2,~;PNeml,pNcJT
Pnc =
{PnliPn2i ) PnNcml, Pntfcm}T
Pi =
{PNcm > PNcm + U > Pni,P }T
PnA =
i.PnNcm'PnN'm + 1) > PnN'l j Pnl}
Expanding Eq.(7.42) and solving pne in terms of pc by eliminating p6,
a rela
tionship between p,
ac SJld pc
can be established
Pnc = Epc
(7.43)
with
E
= (Du 
Cl2 C22 D2i) 1 (CU C12 C^J1 C2i)
(7.44)
The boundary conditions at the interface are the continuity statements for the
pressure and mass flux, given by Eq.(7.16) and (7.17). In the matrix form, they are
Pc = ipoc (7.45)
P ne = pefotnc (7.46)
They are also called compatibility equations (Ligget and Liu, 1983).
Substitution of these two equations into Eq.(7.43) yields
4>nC = yEc (7.47)
This is the additional matrix equation needed by Eq.(7.39) in order to have a
unique solution. The final combined matrix equation for such a solution in is
obtained by matching the two domains along the common boundary.
7.3.3 Matching of The Two Domains
The matching of the two domains along the common boundary can be accom
plished by introducing Eq.(7.47) into Eq.(7.39). The resulting equation for the
121
Table 8.3: Test Results for Breaking Waves
T (sec.)
Hi (cm)
(Ht)tq( cm)
Hr{ cm)
Kt
Kr
5.66
3.53
0.26
0.6233
0.0468
0.642
6.99
3.02
0.33
0.4320
0.0469
7.03
2.98
0.31
0.4240
0.0448
5.68
3.40
0.27
0.5981
0.0478
0.858
7.76
3.44
0.44
0.4431
0.0564
8.40
3.28
1.02
0.3903
0.1218
9.15
3.43
1.01
0.3748
0.1102
6.37
3.40
0.77
0.5339
0.1212
1.020
7.96
3.53
0.83
0.4434
0.1042
9.38
3.89
0.91
0.4149
0.0975
10.47
3.82
1.47
0.3648
0.1402
5.19
3.52
0.76
0.6780
0.1470
6.18
3.45
1.07
0.5585
0.1727
1.120
6.27
3.55
1.05
0.5665
0.1672
7.74
3.96
1.18
0.5118
0.1520
7.83
3.91
1.18
0.4996
0.1504
9.00
3.99
1.20
0.4435
0.1329
4.75
3.59
0.93
0.7552
0.1946
1.379
6.60
3.87
1.21
0.5866
0.1830
7.53
4.30
1.22
0.5713
0.1623
9.02
4.29
1.59
0.4755
0.1763
6.01
4.06
0.41
0.6760
0.0679
1.778
7.48
4.49
0.88
0.6003
0.1183
8.02
4.62
1.11
0.5757
0.1387
8.3 Pressure Distribution and Wave Envelope Over The Breakwater
In the numerical model of BIEM, the unknown pressure function is solved along
the boundaries, and the kinematic properties such as the velocities and accelerations
in the flow field can then be computed according to Greens formula as stated in
chapter 6. It is therefore very important for a numerical model of BIEM to be able to
predict the pressure distribution accurately. In this section, the predicted pressure
distributions as well as the wave envelopes over the breakwater are compared with
those measured from the experiment.
18
with a = 2.2 and (3 0.0 ~ 0.8 for vertical face breakwaters, where F is the depth
of submergence of the breakwater crest and Hi is the incident wave height. Another
equation for impermeable submerged breakwaters was given by Seelig (1980):
Jf, = C(l)(l2C) (2.44)
where R is the wave run up given by Franzius (1965, cited in Baba, 1986)
R ii/C1(0.123^)(C3'/Fi7S+C3> (2.45)
Hi
with Ci = 1.997, C2 = 0.498 and C3 = 0.185. d is the total water depth and
C = 0.51 (2.46)
h
where B is the crest width and h is the height of the breakwater. Several other
empirical equations and methods are also available for designs.
The latest theoretical approach for impermeable submerged breakwaters was
given by Kobayashi et al. (1989) by using finite amplitude shallowwater equations.
The numerical results were found to agree well with the experimental data by Seelig
(1980) for such structures.
However, for permeable structures, no parallel empirical formula could be found
in the literature besides the nomograph by Averin and Sidorchuk (1967, cited in
Baba, 1986). The research for such structures is still in the stage of physical exper
iments.
Dick and Brebner (1968) carried out an experiment on solid and permeable
breakwaters of vertical faces. In the experiment, it was discovered that, over a
certain wave length range, the permeable breakwater was much better than the
solid one of the same dimension in terms of wave damping, and that the permeable
breakwater had a well defined minimum value for the coefficient of transmission. It
was also found that a substantial portion, 30% ~ 60%, of the wave energy of the
transmitted waves was transferred to higher frequency components. The equivalent
110
frequency, and the waves of higher frequencies. This phenomenon has been reported
by a number of investigators in the past (e.g., Seelig, 1980), but the physical mech
anism regarding the generation of higher harmonics has not been clearly described.
In this study, spectral analysis was performed on the transmitted wave data.
The spectrum of the transmitted waves revealed that the high frequency waves are
just the higher harmonics of the fundamental wave, as shown in Fig. 8.3. This
implies that due to the interaction with the submerged breakwater, part of the
energy of the fundamental wave is transferred to the waves of higher harmonics. It
was observed in the experiment that such energy transfer occurs around the lee side
end of the crest.
The test results for the incident and reflected wave heights of nonbreaking
waves are listed in Table 8.1. Also listed in the table are the corresponding wave
heights of the higher harmonics up to the third order. Generally, the wave energy of
the first three modes possess more than 98% of the total transmitted wave energy.
It is interesting to note that some of the second harmonics possess more energy
than those of the fundamentals.
The numerical model is based upon the energy balance of the fundamental
waves. In order to be consistent with the concept of energy transmission and to
compare the measured transmission coefficients with the predicted values, an equiv
alent height for a transmitted wave was adopted for the experimental data by adding
the wave energy flux of the higher harmonics to that of the fundamental wave, i.e.
4 = = jr(EtC) (8.3)
1=1
with
Cgi = Crii 
2kih .
sinh 2 kih
and
of = gk{ tanh kih
67
quadratic element, special element and so on so forth. The classification of elements
is based on the type of the function used to interpolate the values over an element.
For example, on a constant element, the physical quantities and their normal deriva
tives axe assumed to have no change over the element; while on a linear element,
the quantities and their normal derivatives are interpolated by a linear function
between the values at the ends of the element, i.e.
= JVi(0*; + ^(Â£)*/+i (6.13)
= )
< e < Â£i+l
where JVj and Nz are linear functions with the feature of
Nd(j) = 1 JVaiij+i) = 0
= 0 N2((+Â¡) = 1
In the same manner, higher order elements can be defined by replacing Ni and
Nz with higher order functions. When the variation of the quantities is known,
the variation function can then be chosen as the interpolation function to obtain
a more accurate element. Such elements are usually classified as special elements.
Generally, it is difficult to judge which element is superior over the others with
out analyzing the particular problem. As a rule of thumb, the higher the order of
an element, the more accurate the approximation would be for the same element
size. But for a higher order element, as a tradeoff for precision, the formulation
would be much more complicated and tedious, and the computation might be much
more time consuming. The constant element works fairly well for problems with
smooth boundaries and continuously changing boundary conditions, but could gen
erate considerable errors at corner points where the boundary conditions are not
continuous. For boundary with such corners, the linear element is usually a better
choice.
d$
52
Table 5.2: Test Cases
50 (cm)
h, (cm)
h (cm)
L (cm)
0.72
20
20,25,30
200
0.93
20
20,25,30
200
1.20
20
20,25,30
200
1.48
10,15,20
20,25,30
200
2.09
20
20,25,30
200,225,250,275
2.84
20
20,25,30
200
3.74
20
20,25,30
200
and these test conditions are summarized in Table 5.2.
The damping effect due to side walls alone was found to be very small, and an
average value of oltu = 0.01 sec1 was used for all the corrections.
5.2 Determination of The Empirical Coefficients
The experimental results for the conditions of ht = 20 cm and L 200 cm
are given in Table 5.3 (complete results are given in Appendix B). Each data point
represents an averaged value of 10 to 20 tests.
Figure 5.2(a) shows a typical example of measured waves and the exponential
fitting to the wave heights. The solid line in Fig. 5.2(a) is the ensemble average
whereas the two dash lines are the envelopes of plus and minus one standard devi
ation from the mean value at each time instant. From the wave form, it is noted
that the waves in the experiment were more or less nonlinear waves as oppose to
what was assumed. The effect of the nonlinearity to the calculation of the measured
damping rate can be minimized by using the ratio of wave heights instead of wave
amplitudes. In Fig. 5.2(b), the dashed curve is the exponential decay function with
the a, obtained from the data ensemble by the least square analysis.
Based on the measured cT and a, listed in Table 5.3, the empirical coefficients
do, b0 and Ca were then determined by multivariate linear regression analysis such
194
Figure B.34: Case of L 275 cm, h = DW = 30 cm, h, = DS = 20 cm and
d50 = DD = 2.09 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights {H/H7) and the best fit to the exponential decay
function.
117
Figure 8.4: The predicted Kt and Kr versus the measured Kt and Kr. (a) Ktp vs.
Ktm\ (h) Ktp VS. Ktm*
66
Despite the fact that the Greens function has a singularity at r = 0, the contour
integral in Eq.(6.10) exists and can be worked out by removing the singular point
from the domain.
In general, Eq.(6.10) becomes
(6.11)
after removing the singularity at point P. Here a is 2n if P is an interior point
and is equal to the inner angle between the two boundary segments joining at P if
it is on the boundary. Generally, point P can be anywhere in the domain DnC,
while Q is always on C due to the contour integration. In BIEM, since only the
3$
boundary values of $ and are solved, P has to be kept on the boundary C in
an
3$
the process of computation. When $ and r on C are solved, the interior values
on
can be derived with Eq.(6.1l) by placing P at the point of interest inside D.
The closed boundary C in Eq.(6.1l) is the same as that in Eq.(6.10) except that
it does not pass point P as it does in Eq.(6.10). The singular point of the Greens
function r = 0 has been excluded by a circular arc of infinitesimal radius from the
domain DnC. Thus there is no singularity on the new contour C. Eq.(6.1l) is the
basic equation for BIEM formulations.
Discretizing the boundary C into N segments, and breaking the contour inte
gration into N parts accordingly, Eq.(6.1l) reads
*< Inrfl)fa=Js ^ (6,12)
The curved segment Cj is usually replaced by a straight line to simplify the inte
gration. This simplification generally does not introduce significant error provided
that the segment is small enough.
Before carrying out the integration, the type of element has to be determined.
The formulations are different for different types of element. The commonly used
elements in two dimensional BIEM problems aire: constant element, linear element,
51
h
h,
wave
maker
Figure 5.1: Experimental setup
Table 5.1: Material Information
so (cm)
0.72
0.93
1.20
1.48
2.09
2.84
3.74
porosity
0.349
0.349
0.351
0.359
0.369
0.376
0.382
where a, is the actual seabed damping rate, cr,^ is the gross damping rate of the
seabedwall system, aiw is the damping rate by the side walls and the bottom, and
T is the wave period.
The bed material used in the experiment was river gravel of seven sizes, ranging
from so = 0.72 cm to 3.47 cm, with all sizes having a fairly round shape and smooth
surface. The porosity of the material increases slightly with the diameter, as given
in Table 5.1.
With the grain sizes being determined by the material selection, the adjustable
independent parameters left in the dispersion equation are the water depth h, the
seabed thickness h and the wave length L. A total of 36 different cases was tested
17
cosh k'(z + h!) ^ D _JfcLfiC_ncos A^fc + h*)
cosh 'A'
+ E
m=l
cos /i'
rrx
with
gk tanh kh = gkm tan kmh a2
gk' tanh k'h' = gk'm tan k'mh' = a
(2.40)
(2.41)
(2.42)
where A, and I?, are unknown complex coefficients. In their model, the series were
truncated at a certain point to make the number of unknown coefficients equal to
the number of the elements. Owing to the introduction of the series, the procedure
became much more complicated and the computation much more time consuming
than that of Ijima et al.
The advantage of a constant element is its simplicity and efficiency. But it has
an inherent problemawkward behavior around corner points. It has been found
that significant errors arise around such points even for the simplest casesinusoidal
wave propagating in a constant water depth over a impermeable bottom.
Sulisz (1985) employed linear elements in his BIEM model for subaerial porous
breakwaters. Eqs.(2.39) and (2.40) were adopted for the lateral boundary condi
tions. As a result, the formulation became awfully complicated. In modelling the
porous media, Sollitt and Crosss model was adopted. However, the inertia term
due to virtual mass effect was dropped as the corresponding coefficient Ca was not
known.
For submerged breakwaters, the modelling is still largely based on laboratory ex
periments and empirical formulae. For impermeable submerged breakwaters, quite
a few empirical formulae for transmission coefficient have been established from ex
perimental data by different authors (see Baba, 1986). For example, the equation
given by Goda (1969) for the transmission coefficient due to overtopping is
K, = 0.5[1sin ;Â£(/? A)]
(2.43)
NORMALIZED WAVE ENVELOPE NORMALIZED WAVE ENVELOPE
136
X (M)
3.00
2.00
1.00
0.00
1.00
2.00
3.00
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5

***'*
H= 15.00 CM
T = 2.52 S
" *
KR 1.001
BERM
R*
X (M)
Figure 9.3: Berm breakwaters of vertical face: (a) Zero permeability; (b) Infinite
permeability.
APPENDIX B
EXPERIMENTAL DATA FOR POROUS SEABEDS
In this appendix, the complete experimental data from the tests on porous
seabeds made of river gravel are given in graphical forms. The 36 test cases are
listed in Table B.l.
In the following figures, the averaged surface fluctuation and the wave height
attenuation for each case are plotted. For each case, the experiment was repeated 10
to 20 times and the surface fluctuations were normalized by the first wave height of
each data record. In top frames, the averaged wave forms, represented by solid lines,
were obtained by averaging all the normalized wave records for that case. The dash
lines are the envelopes of plus and minus one standard deviation from the averaged
value at each time instant. In lower frames, the dots represent the wave heights
deduced from each normalized record of surface elevation and the dashed curves are
the exponential decay function with the a,s obtained from the data ensemble by
the least square analysis.
In the figures, H denote the averaged wave height if, T is the averaged wave
Table B.l: Test Cases
d60 (cm)
h$ (cm)
h (cm)
L (cm)
0.72
20
20,25,30
200
0.93
20
30,25,20
200
1.20
20
30,25,20
200
1.48
20,15,10
30,25,20
200
2.09
20
30,25,20
200,225,250,275
2.84
20
30,25,20
200
3.74
20
30,25,20
200
159
131
9.1.2 BIEM Formulations
In the fluid domain, the BIEM formulation is basically the same as that given
in Chapter 7 for submerged breakwaters and the matrix equation for this domain
is identical to Eq.(7.39). It will not be repeated in this chapter. For the porous
domain, because of the existence of the CFSBC, the formulation is quite different.
Parallel to Eq.(7.41), the Greens formula given in Eq.(6.10) is discretized for the
berm breakwater of the first configuration shown in Fig. 9.1(a), as
Hlipt + E PhPi+H!k.iPk.+
3=2
+ f pjPj+Hfr"_Â¡PN+
j=Ncm +1
HÂ¡,N,fPN$/ + E fljPi + BfalPl
3=N.,+1
Ncml
= Ki\ Pnl + E 'lijPni + Ki,NemlPnNcm
3=2
N.fl
b ^i,Ncm PnNcm *b E Pni "b 1 PnN.f
+ E 1Â¡jP*Â¡ + K&:iP* (98)
i=K.t+1
where the superscript refers to the porous subdomain, Ncm is the last node number
of the common boundary and N,Â¡ is the last node number of the porous free surface.
Introducing the nonflux boundary condition on the surface of the impermeable
core and the CFSBC given by Eq.(9.7) into the above equation, we get
Ncm * 1
HP + E 0jp + h&ip? +
3=2
Hlkm PNcrn + E Pi + Hikf ~ 1 P +
n!k, + s' fljPi + Biji: 1P1
y=Ar./+i
Ncm 1
= KlPn1+ E ^jPni+KlkmlPnNc
3=2
Table 5.4: Comparison of The Resistances, h, = 20 cm, L = 200 cm.
d50 (cm)
1 7 1 (cm/s)
a 
R/
R,
fi/fi
fn/fl
fn/fi
h =
30.0 cm
0.72
0.57
4.788
37
225
0.93
1.32
1.42
0.93
0.69
4.800
58
379
1.56
2.06
1.32
1.20
0.75
4.819
81
630
2.63
2.88
1.10
1.48
0.78
4.842
104
960
4.18
3.71
0.89
2.09
0.96
4.869
181
1933
8.90
6.48
0.73
2.84
1.09
4.889
281
3577
17.12
10.10
0.59
3.74
1.33
4.900
453
6240
30.90
16.34
0.53
h =
25.0 cm
0.72
0.53
4.526
34
213
0.88
1.21
1.38
0.93
0.62
4.545
52
358
1.48
1.85
1.25
1.20
0.70
4.567
76
597
2.49
2.71
1.09
1.48
0.75
4.596
101
915
3.99
3.60
0.90
2.09
0.84
4.639
159
1833
8.44
5.70
0.67
2.84
1.10
4.657
284
3407
16.31
10.21
0.63
3.74
1.18
4.679
400
5959
29.51
14.43
0.49
h =
20.0 cm
0.72
0.50
4.180
32
196
0.81
1.14
1.41
0.93
0.55
4.206
46
332
1.37
1.64
1.20
1.20
0.66
4.232
72
553
2.31
2.56
1.11
1.48
0.61
4.286
81
849
3.70
2.89
0.78
2.09
0.85
4.322
160
1707
7.86
5.73
0.73
2.84
1.00
4.356
258
3187
15.26
9.27
0.61
3.74
1.15
4.381
393
5579
27.63
14.18
0.51
187
Figure B.27: Case of L 200 cm, h DW = 20 cm, h, = DS = 10 cm and
so = DD = 1.48 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (H/H^) and the best fit to the exponential decay
function.
34
For the case of infinite seabed thickness and low permeability (laminar skin
friction dominates, /0 * ^), Eq.(4.15) reads
a2 gk tanh kh iR(gk a2 tanh kh) (416)
which is the same as obtained by Reid and Kajiura (1957). If R 0 as with an
impervious bed, Eq.(4.16) reduces to the ordinary dispersion relationship for a finite
water depth h. On the other hand, if the permeability approaches infinity, we have
R * oo, n >1.0, /3 1.0, /o + t
The dispersion relationship expressed in Eq.(4.15) can be shown as
a2 = gfctanh k(h + h,) (417)
Physically, this is the case of water waves over a finite depth h + hti or, the solid
resistance in layer h, vanishes.
Another limiting case is where the water depth approaches zero and waves now
propagate completely inside the porous medium. The dispersion relationship from
Eq.(4.15) becomes
ofc
a2 t^jtanh kh, (418)
Jo
It can be easily shown that the same dispersion relationship cam be obtained
by directly solving the problem of linear gravity waves in a porous medium alone.
Again, for the case of laminar resistance only, Eq.(4.18) becomes
a2 = iRgk tanh kh, (419)
which states the Darcytype resistance law. On the other hand, when R * oo, we
obtain from Eq.(4.18)
cr2 = gk tanh. kh (420)
or, as expected, the dispersion relationship in a pure fluid medium.
173
Figure B.13: Case of L = 200 cm, h = DW = 25 cm, h, = DS = 20 cm and
d50 = DD = 2.84 cm. (a) Averaged nondimensional surface elevation (rj/Hi), (b)
Nondimensional wave heights (HÂ¡TT\) and the best fit to the exponential decay
function.

