Citation
Structure-induced sediment scour potential near a rectangular structure due to waves

Material Information

Title:
Structure-induced sediment scour potential near a rectangular structure due to waves
Series Title:
UFLCOEL
Creator:
Karunamuni, Anura J., 1954- ( Dissertant )
Sheppard, D. Max ( Thesis advisor )
University of Florida -- Coastal and Oceanographic Engineering Dept
Place of Publication:
Gainesville, Fla.
Publisher:
Coastal & Oceanographic Engineering Dept., University of Florida
Publication Date:
Copyright Date:
1991
Language:
English
Physical Description:
xvi, 120 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Sediment transport ( lcsh )
Scour (Hydraulic engineering) ( lcsh )
Water waves ( lcsh )
Coastal and Oceanographic Engineering thesis M.S ( local )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF ( local )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The problem of structure-induced sediment scour has been a subject of increasing importance in numerous branches of civil engineering. The problem considered here is the potential for sediment scour in the vicinity of a rectangular, partially submerged fixed structure separated from the bottom by a gap and exposed to two-dimensional monochromatic waves. A potential flow solution for the flow field in the vicinity of the structure is obtained using linear wave theory. In order to simplify the solution method, the flow field is divided into three regions; flow upstream of the structure, flow below the structure and flow downstream of the structure. A dimensional analysis of the problem was carried out in order to obtain the pertinent dimensionless groups. The solution procedure includes solving Laplace’s equation and applying the standard bottom and free surface boundary conditions together with the continuity of pressure and velocity conditions at the interregional boundaries. Satisfaction of the boundary conditions results in a system of simultaneous algebraic equations with complex coefficients. This set of equations is solved numerically. Wave reflection and transmission coefficients were computed as part of this work and compared with the results of other theoretical studies. The ratio of maximum bottom velocities under the structure to the maximum velocity under the incident wave was computer for a range of structure parameters and wave conditions. Laboratory experiments were conducted where incident, reflected and transmitted wave heights along with flow velocities beneath he structure were measured and the results compared with the theoretically predicted values. The results give an indication of the sediment scour potential as a function of the structure and wave parameters.
Thesis:
Thesis (M.S.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaf 89).
General Note:
"UFL/COEL-91/007."
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 Anura J. Karunamuni.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
24654452 ( OCLC )

Full Text
UFL/COEL-91/007

STRUCTURE-INDUCED SEDIMENT SCOUR POTENTIAL NEAR A RECTANGULAR STRUCTURE DUE TO WAVES
by
Anura J. Karunamuni
Thesis

1991




STRUCTURE-INDUCED SEDIMENT SCOUR POTENTIAL NEAR A
RECTANGULAR STRUCTURE DUE TO WAVES
By
ANURA J. KARUNAMUNI

A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA

1991




To my parents




ACKNOWLEDGEMENTS

To begin, I wish to express my sincere appreciation and gratitude to the supervisory committee chairman, Prof. D. Max Sheppard, for his continuous support and guidance in and out of the academic framework. I truly enjoyed working under his patient leadership.
A special debt of gratitude is owed to Dr. Alan Wm. Niedoroda for his contributions and guidance throughout. Further, I wish to thank Prof. Ashish J. Mehta for his sponsorship at the beginning of my studies and advisement thereafter. Appreciation is also extended to Prof. Robert G. Dean for many helpful discussions at the latter part of this study.
Many thanks go to Sidney Schofield, Jim Joiner and the other Coastal Engineering Laboratory staff for their friendship and cooperation. I would also like to take this opportunity to express gratitude to all my past teachers who contributed in one way or another to my achievement of this educational goal.
I am also grateful to the Bureau of Coastal Engineering and Regulation of the Division of Beaches and Shores of the Florida Department of Natural Resources for providing partial support for this research.
Finally, I am most grateful to my wife Lalanie, for her patience, love and sacrifice during the course of this work.

iii




TABLE OF CONTENTS

ACKNOWLEDGEMENTS LIST OF FIGURES ...... LIST OF TABLES ....... LIST OF SYMBOLS .... ABSTRACT .........
CHAPTERS
1 INTMCT O N Tr

2 BACKGRO 3 PROBLEM
3.1 Proble 3.2 Assum 3.3 Metho
3.3.1
3.3.2
3.3.3
3.3.4
3.3.5
3.3.6
3.3.7
4 PROBLEM
4.1 Interre

. .
UND ...........................
STATEMENT AND SOLUTION METHOD n Statement ......................
options ...........................
d of Solution ..................
Input parameters ..................
Output parameters ...............
Governing differential equation and boundary Boundary conditions for region 1 ....... Boundary conditions for region 2 ....... Boundary conditions for region 3 . . . Solution procedure . . . . . . ..
SOLUTION . . . . . . . . . .
gional Boundary Conditions . . . . .

4.2 The Eigenvalue Solution for the Flow Field .

iv

iii vii
x
xi xv

conditions

19




4.2.1 Use of first interregional boundary condition . . . . . 1

4.2.2
4.2.3

Use of second interregional boundary condition Us oftrd ;-".rr iaL bondr n diin

4.2.4 Use of fourth int 4.3 Summary of Equations

erregional bou

5 EXPERIMENTAL SET-UP AND PROCE
5.1 Test Facility . . . . . .
5.2 Experimental Procedure . . . 6 RESULTS . . . . . . . . .
7 CONCLUSIONS AND FUTURE WORK
7.1 Conclusions . . . . . . .
7.2 Recommendations for Future Work BIBLIOGRAPHY . . . . . . . .
APPENDICES
A VELOCITY POTENTIALS . . . .
A.1 Region 1 .
A.1.1 Incident wave . . . . .
A.1.2 Reflected wave . . . .
A.1.3 Resultant wave . . . .
A.2 Region 2 . . . . . . . .
A.2.1 Governing differential equation A.2.2 Solution . . . . . .
A.3 Region 3 . . . . . .
B DIMENSIONAL ANALYSIS . . . .

and

boundary

C EXAMPLE SOLUTION

D EXPERIMENTAL DATA REDUCTION TECHNIQUE

. . 23
. . 26
. . 29
. . 31
. . 36
. . 36
. . 37
. . 39
. . 84
. . 84
. . 87
. . 89

.. ........ 90
. . . . . 90
. . . . . 90
. . . . . 91
. . . . . 92
. . . . . 93
conditions . 93 . . . . . 93
. . . . . 99
. . . . . 102
. . . . . 106
. . . . . 112

D.1 Wave Data

v

112

19

ndary condition . . . . . .
DURE . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .




D.2 Velocity Data . . . . . . . . . . . . . . . 113
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . 121

Vi




LIST OF FIGURES

3.1 Definition sketch of the flow field in the vicinity of a rectangular
structure subjected to waves . . . . . . . . . . 16
5.1 Schematic diagram of experimental set-up . . . . . . 38
6.1 Transmission coefficient versus dimensionless structure length (for
deep water waves). . . . . . . . . . . . . . 42
6.2 Transmission coefficient versus dimensionless structure length (for
intermediate depth water waves). . . . . . . . . . 43
6.3 Transmission coefficient versus dimensionless structure length (for
shallow water waves) . . . . . . . . . . . . 44
6.4 Transmission coefficient versus ratio of water depth to structure
draft (for deep water waves) . . . . . . . . . ... 45
6.5 Transmission coefficient versus ratio of water depth to structure
draft (for intermediate depth water waves). . . . . . . 46
6.6 Transmission coefficient versus ratio of water depth to structure
draft (for shallow water waves). . . . . . . . . . 47
6.7 Reflection coefficient versus dimensionless structure length (for
deep water waves). . . . . . . . . . . . . . 49
6.8 Reflection coefficient versus dimensionless structure length (for intermediate depth water waves). . . . . . . . . . 50
6.9 Reflection coefficient versus dimensionless structure length (for
shallow water waves).. . . . . . . . . . . . . 51
6.10 Reflection coefficient versus ratio of water depth to structure draft
(for deep water waves). . . . . . . . . . . . . 52
6.11 Reflection coefficient versus ratio of water depth to structure draft
(for intermediate depth water waves). . . . . . . . . 53
6.12 Reflection coefficient versus ratio of water depth to structure draft
(for shallow water waves). . . . . . . . . . . . 54

vii




6.13 Ratio of maximum bottom velocity at the center of the structure
to the maximum incident wave bottom velocity(r5) versus dimensionless structure length. . . . . . . . . . . . 57
6.14 Ratio of maximum bottom velocity at the center of the structure
to the maximum incident wave bottom velocity(r5) versus dimensionless structure length (h/d = 1.05) . . . . . . . . 59
6.15 Ratio of maximum bottom velocity at the center of the structure
to the maximum incident wave bottom velocity(r5) versus dimensionless structure length (h/d = 1.3).. . . . . . . . . 60
6.16 The effect of the structure length on the maximum bottom velocity
along the bed . 60
6.17 The effect of the gap below the structure on the maximum bottom
velocity along the bed. . . . . . . . . . . . . 61
6.18 The effect of different types of incident waves (deep and intermediate depth water waves) on the maximum bottom velocity along
the bed. 62
6.19 The maximum horizontal velocity profile for regions 1, 2 and 3 for
the structure length shown in fig. 6.16 (1=4.0 ft). . . . . 63
6.20 The maximum horizontal velocity profile for regions 1, 2 and 3 for
the structure length shown in fig. 6.16 (1=74.7 ft) . . . . 64
6.21 The maximum horizontal velocity profile for regions 1, 2 and 3
for the ratio of water depth to structure draft shown in fig. 6.17
(h/d = 2.0). 65
6.22 The maximum horizontal velocity profile for regions 1, 2 and 3 for
the deep water wave shown in fig. 6.18 (T = 1.0 sec and h/gT2
0.01). 66
6.23 Comparison of theoretical and experimental results for run numbers 2, 4, 6 and 8 (see table 6.4 and 6.5). . . . . . . . 71
6.24 Comparison of theoretical and experimental results for run numbers 9, 11, 13 and 15 (see table 6.6 and 6.7). . . . ... . 72
6.25 Comparison of theoretical and experimental results for run numbers 10, 12, 14 and 16 (see table 6.6 and 6.7). . . . . . 73
6.26 Comparison of theoretical and experimental results for run numbers 1, 3, 5 and 7 (see table 6.4 and 6.5). . . . . . . . 74
6.27 Comparison of theoretical and experimental results for run numbers 2, 4, 6 and 8 (see table 6.4 and 6.5). . . . . . . . 75
6.28 Comparison of theoretical and experimental results for run numbers 9, 11, 13 and 15 (see table 6.6 and 6.7). . . . . . . 76

viii




6.29 Comparison of theoretical and experimental results for run numbers 10, 12, 14 and 16 (see table 6.6 and 6.7). . . . . . 77
6.30 Comparison of theoretical and experimental results for run numbers 1, 3, 5 and 7 (see table 6.4 and 6.5). . . . . . . . 78
6.31 Comparison of theoretical and experimental results for run numbers 1, 2, 6, 9, 10, 13 and 14 (see table 6.4 through 6.7). . . 79
6.32 Comparison of theoretical and experimental results for run numbers 3, 4, 8, 11, 12, 15 and 16 (see table 6.4 through 6.7). . . 80
6.33 Comparison of theoretical transmission coefficient with results from
Steimer and Black. . . 81
D.1 Measured wave height envelope (incident and reflected wave) for
region 1 .. 115
D.2 Measured wave height envelope (transmitted wave) for region 3 116 D.3 Measured velocity time history for the flow in region 2 . . . 117 D.4 Wave energy density function of the wave height for the flow in
region 1 .. 118
D.5 Wave energy density function of the wave height of the transmitted
wave 119
D.6 Wave energy density function of the velocity under the structure 120

ix




LIST OF TABLES

6.1 Summary of figure contents regarding transmission coefficients 41 6.2 Summary of figure contents regarding reflection coefficient . . 48
6.3 Summary of figure contents regarding maximum velocity along the
bed ...... ... .................................. 55
6.4 Experiment set-up number 1 . . . . . . . . . . 67
6.5 Experiment set-up number 2 . . . . . . . . . . 68
6.6 Experiment set-up number 3 . . . . . . . . . . 69
6.7 Experiment set-up number 4 . . . . . . . . . . 70
6.8 Macagno's experimental conditions . . . . . . . . 82
6.9 Comparison of present theory with other studies for experimental
conditions described in table 6.8 . . . . . . . . . 83

x




LIST OF SYMBOLS

A Complex matrix of dimension (4 + 4N) by (4 + 4N) B Column vector of length (4 + 4N) B(,) Velocity potential coefficients of reflected wave
in evanescent mode (ft2/sec) Co A constant (ft2/sec) C(,) Velocity potential coefficients of transmitted wave
in evanescent mode (ft2/sec) C(t) Bernoulli term (ft2/sec2) d Structure draft below the still water level (ft) g Acceleration of gravity (ft/sec2) h Water depth (ft) H;.C Incident wave height (ft) H.ax Maximum height of the wave height envelope for
region 1 (in)
Hmin Minimum height of the wave height envelope for
region 1 (in)
H., Reflected wave height (ft) Ht Transmitted wave height (ft) kinc Wave number of incident wave (ft-1) k, Wave number of reflected wave in progressive mode (f-1)

xi




kq,() Wave number of reflected wave in evanescent mode (ft-') k,(,) Wave number of transmitted wave in evanescent mode (ft-1) kt Wave number of transmitted wave in progressive mode (ft-1) k(m) Wave number for the flow in region 2 (ft-1)
1 Structure half length (ft) m, n mth and nth eigenmodes N Number of eigenmodes p Pressure, p(x, z,t), (lbf/f2) P1A p(-l, z, t) for h < z < -d (1b/ft2)
(in region 1)
P2A p(-l, z, t) for h < z < -d (1b/ft2)
(in region 2)
P(m) Coefficients in the expression for 4 in region 2 (ft2/sec)
- Coefficients in the expression for T Wave period (sec) T(t) Time dependent component of the solution for 4 in region 2 U Uniform horizontal velocity for the flow in region 2 (ft/sec) U1 Horizontal velocity for the flow in region 1, U1(x, z, t) (ft/sec) UlA U1(-l, z, t) for h < z < -d (ft/sec)
(in region 1)
U2 Horizontal velocity for the flow in region 2 U2(x, z, t) (fi/sec) U2A U2(-l, z, t) for h < z < -d (fi/sec)
(in region 2)

xii




U2B U2(l, z,t) for h < z < -d (ft/sec)
(in region 2)
U3 Horizontal velocity for the flow in region 3, U3(x, z, t) (ft/sec) U3B U3(l, z, t) for h < z < -d (fi/sec)
(in region 3)
Ubj The magnitude of the maximum bottom velocity
due to incident waves (ft/sec)
Uc- Critical bottom velocity (velocity that initiates sediment
motion) (ft/sec)
Uic Horizontal velocity due to incident wave, Uic(x, z, t) (ft/sec) UncA U;,c(-l, z, t) for h < z < -d (ft/sec)
(in region 1)
U1. Horizontal velocity due to reflected wave U,.(x, z, t) (ft/sec) U- Horizontal velocity of transmitted wave, Ut(x, z, t) (ft/sec) w Vertical velocity component in the positive z direction (ft/sec) x Horizontal coordinate measured from the center of the
structure in the direction of incident wave propogation (ft) X(x) x-dependent component of the solution for 4 in region 2 X Column vector of length (4 + 4n) z Vertical coordinate measured upward from the
still water level (ft)
Z(z) z-dependent component of the solution for D in region 2 ( Velocity potential (ft2) 4) Velocity potential for the flow in region 1, 4D (x, z, ) (ft2/seC)

xiii




1 = 41(-l, z, t) for h < z < -d (f12/sec) (in region 1)
02 Velocity potential for the flow in region 2, D2(x, z, t) (ft2/sec) 02,1 Velocity potential for the flow in region 2 (ft2/sec)
(for real roots of #)
02,2 Velocity potential for the flow in region 2 (ft2/sec)
(for imaginary roots of P)
02A 02(-l, z,t) for h < z < -d (ft2/sec)
(in region 2)
02B 02(l, z, t) for h < z < -d (ft2/sec)
(in region 2)
3 Velocity potential for the flow in region 3, 03(x, z, t) (ft2/sec) 3B 4)3(l, z, t) for h < z < -d (ft2/sec) (in region 3)
O Velocity potential of incident wave, 0; c(X, Z, t) (ft2/sec) OJ incA = (-l, z,t) for h < z < -d (ft2/sec)
(in region 1)
- Velocity potential of reflected wave (ft2)
- Velocity potential of transmitted wave (ft2) # A complex constant w Wave frequency (sec-1) r7 Free surface elevation measured from still water level (ft) IC A constant

xiv




Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
STRUCTURE-INDUCED SEDIMENT SCOUR POTENTIAL NEAR A
RECTANGULAR STRUCTURE DUE TO WAVES By
ANURAJ.KARUNAMUNI
May 1991
Chairman: Prof. D. Max Sheppard
Major Department: Coastal and Oceanographic Engineering
The problem of structure-induced sediment scour has been a subject of increasing importance in numerous branches of civil engineering. The problem considered here is the potential for sediment scour in the vicinity of a rectangular, partially submerged fixed structure separated from the bottom by a gap and exposed to two-dimensional monochromatic waves.
A potential flow solution for the flow field in the vicinity of the structure is obtained using linear wave theory. In order to simplify the solution method, the flow field is divided into three regions; flow upstream of the structure, flow below the structure and flow downstream of the structure. A dimensional analysis of the problem was carried out in order to obtain the pertinent dimensionless groups.
The solution procedure includes solving Laplace's equation and applying the standard bottom and free surface boundary conditions together with the continuity of pressure and velocity conditions at the interregional boundaries. Satisfaction of the boundary conditions results in a system of simultaneous algebraic equations with complex coefficients. This set of equations is solved numerically.

xv




Wave reflection and transmission coefficients were computed as part of this work and compared with the results of other theoretical studies. The ratio of maximum bottom velocities under the structure to the maximum velocity under the incident wave was computed for a range of structure parameters and wave conditions.
Laboratory experiments were conducted where incident, reflected and transmitted wave heights along with flow velocities beneath the structure were measured and the results compared with the theoretically predicted values. The results give an indication of the sediment scour potential as a function of the structure and wave parameters.

xvi




CHAPTER 1
INTRODUCTION
Sediment scour in the vicinity of structure foundations is a significant factor in the design of submerged or partially submerged structures, particularly when the sea bed is composed of fine cohesionless sediments. This phenomena occurs in numerous branches of civil engineering. Three major areas are discussed below.
In river engineering, local scour around bridge piers has been a problem for many years. Scour holes around bridge piers created by the flowing water is a major cause of bridge pier foundation failure. As a result, many investigators have worked on this subject and produced a number of empirical scour prediction formulas.
In offshore engineering, structure-induced scour is a subject of increasing importance with the expanding oil and gas exploration activities in offshore waters where more and more fixed and gravity based structures are being placed on the sea bed. Here, the problem areas include scour and sedimentation processes around pipelines, risers and platform legs.
In the discipline of coastal engineering, structure-induced scour problems have escalated with the ever increasing development of coastlines and the associated large numbers of man-made structures of various shapes and sizes. Even though these structures are normally above the water level, they can be submerged by storm surge (meteorological tides) during severe storms and hurricanes causing the water level to rise and flow through and around these structures. In the coastal environment both currents and waves can exist simultaneously.
In many structure-induced sediment scour problems the concern is strictly for the integrity of the structure causing the scour. For coastal structures in low lying

1




2

coastal areas, the volume of sediment removed as a result of scour may be of equal importance due to the potential for loss of lives and property in the vicinity of the structure.
When a structure is placed in a flow field, it will alter the flow in the vicinity of the structure. The local flow field around even a simple structure element, such as a circular pile, is extremely complex and has so far been impossible to analyze analytically for situations of practical interest due to the presence of flow separation, vortex shedding, wave breaking, etc. This flow modification generally results in increased bottom shear stress in the local flow field and thus sediment scour or erosion.
The net effect is
1. Weakening of the structure foundation, and
2. The loss of sediment from the vicinity of the structure and possibly from the
system (i.e., the sediment may be transported offshore or to an adjoining bay).
The extent of the scour depends upon the shape and size of the structure, its orientation relative to the flow, its location relative to the bottom, the nature of the incoming flow (e.g., water depth, wave period, wave height, current velocity etc.) and sediment parameters (e.g., sediment size and distribution, sediment density, cohesiveness, etc.).
The process of structure-induced scour is somewhat different for waves than for steady currents. Since much of the work on local scour has been directly or indirectly associated with bridge piers, currents have received most of the attention. The present understanding is that 1) in general, waves alone cause less scour than currents alone for comparable reference velocities (Eadie and Herbich 1987), 2) equilibrium scour depths due to waves and currents together are only slightly greater than those due to currents alone, but the equilibrium scour depth is reached much faster than for currents alone. There are many situations, however, where the currents are small




3
and waves dominate the scour mechanism. Scour phenomena due to waves can be described as follows.
The processes through which water waves move cohesionless bottom sediment are complex, but it is generally believed that the sediment transport is related to the magnitude of the shear stress exerted on the bed by the water motion Oscillatory fluid motion associated with surface gravity waves exerts shear stresses on the bottom that are often several times larger than shear stresses produced by unidirectional currents of the same magnitude. Thus, the importance of wave motion in initiating sediment movement and suspension in a coastal environment is apparent. Shear stresses produced by wave motion may put sediment into suspension where it can be transported by currents otherwise too small to initiate sediment motion, (Ippen 1966 and Nielsen 1979).
The addition of currents to waves generally accelerates the rate of scour and indeed, most circumstances of practical significance will involve a combination of waves and currents.
Substantial research has been conducted on local scour of noncohesive sediment near fixed, bottom penetrating structures subjected to steady currents. Less effort has been devoted to the study of local scour due to wave action. In recent years, investigations of scour near pipelines (which are somewhat different from fixed bottom-founded structures) have been initiated, but again the emphasis has been on currents and not waves.
Many of the structural elements used in coastal construction are similar to those used in bridge and offshore construction. For example, support piles for individual dwellings and large multi-unit structures are similar to some bridge piers and nearshore jacket-type oil platforms. Some structure shapes, however, are more unique to coastal construction. The situation of interest in this thesis is a large, fixed, water surface penetrating, flat bottom rectangular structure that is separated from a cohe-




4
sionless sediment bottom and subjected to monochromatic waves (figure 3.1).
The objective of this study was to determine the conditions (i.e., structure dimensions and wave parameters) under which structure-induced scour is likely to occur. The analysis is strictly hydrodynamic, uses potential flow theory and linear wave approximations, and thus is subject to these limitations.




CHAPTER2
BACKGROUND
Several wave-structure interaction investigators have dealt with scattering and radiation of water waves, reflection and transmission coefficients and energy loss of surface waves due to submerged and partially submerged structures. Most of this work was done in relation to breakwaters, vertical barriers, underground installations, etc. A summary of related work is given below. This work was primarily aimed at the hydrodynamic aspects of wave-structure interaction problems. The fluid-structure-soil interaction problems have received less attention.
Most of the work discussed below was for breakwaters and their effectiveness in attenuating waves. Consequently, these investigators were not interested in and did not report near bottom velocities or the effect of structure and wave parameters on the bottom velocities. Note that for a given wave, there is a direct correlation between the near bottom velocity and the bottom shear stress.
Black and Mei (1970). Black investigated the far field effects of waves incident on rectangular and cylindrical structures. He used a variational technique to solve the problem. After solving for the scattered wave potential, transmission and reflection coefficients were computed. Black compared his theoretical results with experimental data.
Tuck (1971), Porter (1972) and Evans (1970). Tuck investigated the transmission of waves through a small gap in a thin vertical wall. First, he constructed the inner and outer approximations to the flow using the method of matched asymptotic expansions and then deduced transmission coefficients as a function of the gap dimension. Here, the width of the gap is small compared to the depth of the gap and the

5




6
water depth. One drawback of this solution is that, for combinations of a deep gap, deep water and narrow structure width, it requires an inordinate number of modes to solve the problem.
Porter (1972) later solved essentially the same problem, but allowed for a variable gap width to depth ratio rather than a deep gap only. Both Porter and Tuck referenced the earlier work by Evans (1970) on reflection and transmission of waves past a submerged plate.
Ippen (1966). Ippen (1966) compiled the theoretical results of
o a thin fixed barrier extending through the free surface downward to some variable depth, and
9 a rigid fixed surface dock with variable draft starting with zero.
Further, Ippen (1966) presented experimental data for reflection coefficients for a fixed rectangular semi-submerged structure.
Sumer et al. (1989). Sumer et al. presented results of an investigation on scour around vertical piles and pipelines exposed to waves. Pipelines have some similarities to the problem considered in this thesis, because of the gap below the structure in both cases. But, it would be difficult to apply the present theory to the latter case due to the complexity of the boundary conditions.
Steimer (1977). Steimer modeled the problem of a fixed rectangular structure of unit width due to simple harmonic normally incident waves. Solutions were obtained for a range of structure dimensions and incident wave parameters. The solution also includes the effects of expansion and contraction losses and allows for the inclusion of a porous media under the structure. The solution was obtained by analytically solving Laplace's equation within the flow field with matching pressure and horizontal velocity components at vertical interfacial boundaries. The results presented include design curves for reflection and transmission coefficients, horizontal and vertical force




7
components for various structure configurations and wave parameters. The work by Steimer is closely related to parts of the present study; hence, some of the results produced here are compared with those of Steimer's.




CHAPTER3
PROBLEM STATEMENT AND SOLUTION METHOD
3.1 Problem Statement
The problem considered in this thesis is that of structure-induced sediment scour potential near a rectangular solid body displaced upward from the bottom as shown in figure 3.1. The water motion is due to linear, monochromatic, progressive surface waves normally incident on the structure as indicated in figure 3.1.
The rectangular structure is considered to be rigid with a flat bottom parallel to the sea floor. In order to evaluate the scour potential underneath the structure, the bottom velocity in the vicinity of the structure must be determined.
The assumptions are that 1) when the maximum bottom velocity under the structure is greater than the maximum bottom velocity under the incident wave and 2) the velocity is greater than the critical bottom velocity needed for incipient motion, there will be scour. This presumes that there are currents present to carry the sediment to an area of reduced bottom shear stress.
The unsteady potential flow problem will be formulated and solved in terms of velocity potentials. Once the velocity potential is'known, the velocity at any point in the flow field can be computed, along with the reflected and transmitted wave parameters.
The idealizations or assumptions used in formulating the model are listed below.
3.2 Assumptions
1. Two-dimensional simple harmonic linear waves.
2. Incompressible, irrotational and inviscid flow.

8




9

3. Wave *amplitude at the structure is less than the draft of the structure.
4. Fixed impermeable bed.
5. Rigid rectangular structure with a flat bottom parallel to bed.
3.3 Method of Solution For a given situation, the wave parameters and structure dimensions are known. In order to find the bottom velocities, reflection and transmission coefficients, the velocity potentials for all three regions must be determined. This is achieved by solving Laplace's equation and satisfying the relevant boundary conditions. This will entail solving a system of linear algebraic equations. The solution procedure is discussed below.
3.3.1 Input parameters
The following quantities are known for a given situation; hence, are considered as input parameters to the set of simultaneous governing equations.
" Water depth, h,
" Incident wave period, T, and height, Hi,, and
* Structure horizontal length, 2 1, and draft, d.
3.3.2 Output parameters
The parameters shown below are obtained by solving the set of simultaneous equations.
9 Flow below structure: Co, U, P(m) and Q(m)
(in region 2)
where




10
Co A constant in the expression for 4 in region 2 (ft2/sec),
U Uniform horizontal velocity for the flow in region 2 (ft/sec),
P(m) Coefficients in the expression for 4 in region 2 (ft2/sec) and
Q (,) Coefficients in the expression for D in region 2 (ft2/sec).
* Reflected wave: H, and B(,)
(in region 1)
where
H, Reflected wave height (ft), and
B(,) Velocity potential coefficients for reflected wave
in evanescent mode (ft2/sec).
* Transmitted wave: Ht and C(n)
(in region 3)
where
Ht Transmitted wave height (ft), and
C(n) Velocity potential coefficients for transmitted wave
in evanescent mode (ft2/sec).
Once these quantities are determined the velocity potentials for the entire flow field can be computed. Hence, the bottom velocities, reflection and transmission coefficients, etc. can be computed.
3.3.3 Governing differential equation and boundary conditions
The governing differential equation for this flow is the continuity equation for twodimensional incompressible, irrotational and inviscid flow which can be expressed as a2 ( 24
2 4 = + _- = 0.(3.1) where 4 is the velocity potential.
In order to solve the boundary value problem discussed below, it is necessary to divide the flow field into three regions as shown in figure 3.1.




11

These regions are defined as follows.
region 1: -h This linear second order partial differential equation in x and z must be satisfied throughout the flow.
The boundary conditions for the three regions defined above and shown in figure
3.1 are given below.
3.3.4 Boundary conditions for region 1
1. The rigid, impermeable bottom requires that the normal velocity be zero on
this boundary. That is,
W 0 at z = -h (3.3) 9z
where w is the vertical component of the velocity.
2. At the free surface two boundary conditions must be satisfied.
a) The dynamic free surface boundary condition (DFSBC),
1 a41
r7 = Z=O and (3.4) g 8t
b) The kinematic free surface boundary condition (KFSBC),
& D i ~ a 77 (3 .5 ) where 41 is the velocity potential in region 1 and 77 is the free surface position.
3. The Summerfeld radiation boundary condition, which states that a wave at
infinity behaves like an outgoing spherical wave from an oscillatory point source,
must be satisfied as x approaches -oo.




12
4. The kinematic boundary condition at x = -l for h < z < 0 is (0 for -d < z < 0 and U1(-, zi) =(3.6) U2A for -h < z < -d where U, is the horizontal component of velocity in region n and
U2A = U2(-l,z,t) for -h < z < -d.
3.3.5 Boundary conditions for region 2
1. The rigid, impermeable bottom requires that the normal velocity be zero on
this boundary. That is,
W = = 0 at z = -h. (3.7) 9z
2. Similarly, the rigid, impermeable bottom face of the structure requires that the
velocity normal to the bottom face be zero on this surface; i.e.,
W 0 at z = -d. (3.8) 8z
3. The pressure must be continuous over the boundary dividing regions 1 and 2,
i.e.,
p1(-l, z, t) = p2(-l, z, t) for h < z < -d (3.9) where p, is the pressure in region n.
4. The pressure must be continuous over the boundary dividing regions 2 and 3,
i.e.,
P2(l, z, t) = p3(l, z, t) for h < z < -d. (3.10)
3.3.6 Boundary conditions for region 3
1. The rigid, impermeable bottom requires that the normal velocity be zero on
this boundary; i.e.,




13

W 0 at z = -h. (3.11) az
2. At the free surface, two boundary conditions must be satisfied.
a) The dynamic free surface boundary condition (DFSBC),
1 a43
= 4 a O and. (3.12) b) The kinematic free surface boundary condition (KFSBC),
_ (977
0 3- = (3.13) where '3 is the velocity potential in region 3 and 77 is the free surface position.
3. The Summerfeld radiation boundary condition, which states that a wave at
infinity behaves like an outgoing spherical wave from an oscillatory point source,
must be satisfied as x approaches oo.
4. The kinematic boundary condition at x = 1 for h < z < 0 is
0 for-d U3(l, z,i) = (3.14) U2B for -h < z < -d,
where U2B = U2(l,z,t) for -h < z < -d.
3.3.7 Solution procedure
In order to solve the above boundary value problem, an expression for the velocity potential that satisfies the governing differential equation and the corresponding boundary conditions must be found for each of three regions. Convergence of these different velocity potential expressions for regions 1, 2 and 3 at interregional boundaries will ensure a continuous solution over the entire flow field.
Region 2 does not have a free surface. Laplace's equation can be solved in this region using separation of variables techniques. An expression for the velocity potential that satisfies boundary conditions 1 and 2 in region 2 must be found. This results




14
in an expression, 12, with (2+2N) unknowns, namely, Co, U P(m) and Q(m); where N = 1, 2, 3, ... is the number of eigenmodes and m = 1, 2, 3, .. ., N is the mth eigenmode.
One approach that can be used to obtain solutions for the velocity potential in regions 1 and 3 makes use of "wave maker theory". This will allow the interregional boundary conditions to be satisfied.
Using the above approach, a velocity potential, 4, representing the reflected wave, satisfying boundary conditions 1 through 3 for region 1 can be found (see e.g., Dean and Dalrymple 1984). This expression for the velocity potential will have (1+N) unknowns. These unknowns are the propagating mode wave height, H,., and the coefficients, B., in the evanescent modes.
Similarly, a velocity potential, 4D, representing the transmitted wave, satisfying boundary conditions 1 through 3 for region 3 can be found. This velocity potential, 4), will also have (1+N) unknowns which are the transmitted mode wave height, Ht and the coefficients, C(n), in the evanescent modes.
The above described solution technique is given in Appendix A.
The remaining boundary conditions that must be satisfied are
1. Velocity continuity at x = -1.
(4th boundary condition for region 1)
2. Velocity continuity at x = L
(4th boundary condition for region 3)
3. Pressure continuity at x = -1.
(3rd boundary condition for region 2)
4. Pressure continuity at x = 1.
(4th boundary condition for region 2)




15
The velocity potentials produced by this procedure (given in Appendix A) along with the above four velocity and pressure continuity boundary conditions, result in (4+4N) depth dependent algebraic equations with (4+4N) unknowns. These equations are solved numerically. Details of this solution procedure is given in Appendix C. Increasing the number of modes will increase the number of unknowns as well as the number of equations. For example, two eigenmodes will have (4+4*2) unknowns and 12 equations, while 20 eigenmodes will have (4+4*20) unknowns and 84 equations. An example solution for 3 eigenmodes is given at the end of Appendix C.
Once the velocity potential is known, the velocity field, transmission and reflection coefficients, etc. can be computed.




DFSBC: Dynaic Free Surface Boundary Condition KFSBC: Kinematic Free Surface Boundary Condition PLBC : Periodic Lateral Boundary Condition

incident and Reflected Wave
DFSBC
KFSBC
I t
PLBC
Region 1
V201=0

lo lo
x~B:

Region 2
V2q,2=0

Transmitted Wave
DFSBC
KFSBC
=0
Region 3
V21,3=0

-I0

,)D 0

Mean Water Level PLBC-

A B
Figure 3.1: Definition sketch of the flow field in the vicinity of a rectangular structure subjected to waves




CHAPTER 4
PROBLEM SOLUTION
Expressions for velocity potentials in regions 1 and 3 that satisfy the bottom, free surface and periodic lateral boundary conditions and in region 2 that satisfy the boundary conditions at the structure bottom and water bottom, are given in Appendix A. The expression for velocity potential in region 3 can be found in a number of references (e.g., Dean and Dalrymple 1984) but is included in Appendix A for completeness.
These expressions for velocity potential were formulated so as to allow satisfaction of the boundary conditions at the regional boundaries (referred to here as interregional boundaries). Expressions for velocity potential for the 3 regions are given below. For region 1,
.g Hi,,, cosh ki,,c(h + z )
41(x, z, t) = w-2 cosh ki,(h exp i{ kic(x + 1) wt} w 2 cosh knch
+ .gH, coshk,.(h + z) expi{ -k(+l)-4 w 2 cosh k,.h
N
+ E B(1) exp { k,(j)(x + l)} cos kq()(h + z) exp (-iwt). (4.1) j=1
For region 2,
2(x, z, t) = Co + Ux + E [P(1) exp {k(j)(x l)} j=1
+ Q(j) exp {-k(j)(x + l)}] cos k()(z + h)} exp(-iwi). (4.2)

17




18

For region 3,
.g Hi cosh kt(h + z)
)3(Xw 2 cosh kh exp i{ kt(x 1) wt}
N
+ E C(j) exp {k,(I)(x l)} cos k,()(h + z) exp (-iwi). (4.3) j=1
Horizontal velocities and velocity potentials in regions 1, 2 and 3 at interregional boundaries were expressed in terms of (4+4N) unknowns in Appendix A. In this Chapter, the boundary conditions will be applied to the velocity potential solutions to produce (4+4N) algebraic equations.
4.1 Interregional Boundary Conditions
1. Continuity of velocity at x = -1:
0 for -d < z < 0 and U2A for -h < z < -d. This satisfies the zero velocity boundary condition normal to the rigid, impermeable vertical face of the structure at x = -l and -d < z < 0 and the continuity
of velocity at x = -1, -h < z < -d .
2. Continuity of velocity at x = I:
0 for -d < z < 0 and U3(l, z, t) = (4.5) U2B for -h < z < -d .
This satisfies the zero velocity boundary condition normal to the rigid, impermeable vertical face of the structure at x = 1 and -d < z < 0 and the continuity
of velocity at x = l, -h < z < -d.
3. Continuity of pressure at x = -1:
P1A = P2A for h < z < -d. (4.6)




19
By using the unsteady form of the Bernoulli equation,
- +p 2 1 + + gz= C(t), (4.7) These pressures, PIA and P2A, can be found in terms of velocity potentials.
Hence, the continuity in pressure reduces to the continuity in velocity potential, 1PA = 2A for h < z < -d. (4.8)
4. Continuity of pressures at x = I :
Using the same approach as above, this boundary condition reduces to
03B = (2B for h < z < -d. (4.9) The unknown coefficients in the expressions for 01, (2 and 03 (equation 4.1 through
4.3) will be chosen such that the above four boundary conditions are satisfied.
4.2 The Eigenvalue Solution for the Flow Field
4.2.1 Use of first interregional boundary condition
Velocity in region 1 is due to incident and reflected waves. Since the governing differential equation is linear, the principle of superposition can be applied. Therefore, for x < -1,
U = Uinc + U,. for h < z < 0. (4.10) At x = -1, the reflected wave velocity will be
(U,)_= UIA Ui.cA for h < z < 0. (4.11)
Multiplying the above equation 4.11 by cosh kr(h + z) and integrating w.r.t. z over the interval -h to 0 and making use of the orthogonality conditions results in,




20

0
(U),_1 cosh k,(h + z) dz
0 0(4.12) = U1 cosh k,(h + z) dz- j UincA cosh k,(h + z) dz.
The first integral on the right hand side (RHS) can be expanded as follows:
f0UI cosh k,(h + z) dz
d (4.13) = U1A cosh k,(h + z) dz+ U1A cosh k,(h + z) dz.
--h -d
The boundary condition at x = -l (equation 4.4) is
U1A = U2A for h < z < -d and U1A = 0 for d < z < 0.
If the first integral on the RHS of equation 4.12 is replaced by the expression given in equation 4.13 and use of the above velocity boundary condition, at x = -1, is made, equation 4.12 reduces to
0
1- (U,.),__, cosh k,(h + z) dz
(4.14)
= U2A cosh k,(h + z) dz- UincA cosh k,(h + z) dz.




21
After substituting expressions for the velocities (U,) =-I, U2 and (U~ncA) from Appendix A, equation 4.14 will take the form jo gk, H, cosh2 k,.(h + z) dz
-h Lo 2 cosh k,.h
0 N
- L B(j)kq(j) cos k,()(h + z) cosh k,(h + z) dz
-hj=1
U cosh k,(h + z) dz
_h (4.15)
- E k(m) [P(m) exp {-2k(m)l} Q(m)
cos k(m)(h + z) cosh k,.(h + z) dz
[0 gkin Hin cosh kinc(h + z) cosh k,(h + z) dz.
-h W 2 cosh kinch It can be shown that the functions cos k,(j)(h + z) cosh k,.(h + z) form an orthogonal set, (see e.g., Churchill (1941)), over the interval z = -h to z = 0. Thus,
0
Jh cos k(j)(h + z) cosh k,.(h + z) dz = 0. (4.16) Equation 4.15 thus reduces to
gk, H, 0 cosh2 k,(h + z) dz
w 2 f-h cosh k,.h
+U cosh k,(h + z) dz
-h
N
+ Z k(m) [P(m) exp {2k(m) Q(m) (4.17)
m=1
-d
J-dh cos k(m)(h + z) cosh k,-(h + z) dz
gkin Hn 1 h0
- cosh kh -h cosh kinc(h + z) cosh k,(h + z) dz. This is the first of the (4+4N) required equations. Next, multiply equation 4.11 by cos kq(n)(h + z) and integrate w.r.t z over the interval




22
-h to 0 and make use of the second orthogonality condition.
0gkr H cosh k,(h + z)
cosh kh+ cos k,(n)(h + z) dz
-h w 2 cosh k,-h
0 N
E B(j) kq(j) cos kq(j)(h + z) cos kq(n)(h + z) dz h =1
- dU cosh k, ()( h + z) dz (4.18)
-d N
- d 1k(m) [P(m) exp {-2k(m)l} Q(mn)]
cos k(m)(h + z) cos kq(n)(h + z) dz
J k gn Hinc cosh kine(h + z) cos kq(-)(h + z) dz.
-h w 2 cosh kinco It can be shown that the functions cosh k,(h + z) cos kq(n)(h + z) form an orthogonal set over the interval z = -h to z = 0. Thus,
/0
-fJ cosh kr(h + z) cos kq(n)(h + z) dz 0 (4.19) and
f0
- cosh kinc(h + z) cos kq(n)(h + z) dz = 0. (4.20) Likewise, the functions cos kq()(h + z) cos kq(n)(h + z) form an orthogonal set over the interval z = 0 to -h. Thus,
-h cs k(j)(h + z) cos k,(n)(h + z) dz = 0,' for j 5 n. (4.21)




23

For j = n, equation 4.18 reduces to
-B(n)kq(n) Cos2 kq(n)(h + z) dz
+U] cosh kq(n)(h + z) dz
N
+ E k(m) [P(m) exp { -2k(m)l} Q(m) (4.22) M=1
cos k(m)( h + z) cosh kq(n)( h + z) dz =0.
Note that equation 4.22 represents N of the (4+4N) required equations.
4.2.2 Use of second interregional boundary condition
Velocity in region 3 will be due to transmitted waves only. Hence, for x > 1,
U3 = Ut for -h < z < 0. (4.23) at x = l, the transmitted wave velocity will be
(U ) 1 = U3B for h < z < 0. (4.24) Multiplying equation 4.24 by cosh kt(h + z) and integrating w.r.t z over the interval
-h to 0 and making use of the orthogonality condition results in
0 0
-h (Ut),, cosh kt(h + z) dz = h U3B cosh kt(h + z) dz. (4.25) The integral on the RHS can be expanded as follows,
hU3B cosh kt(h + z) dz
(4.26)
= U3B cosh kt(h + z) dz+ U3B cosh kt(h+ z) dz. The boundary condition at x = I (equation 4.5) is U3B = U2B in h < z < -d and U3B = 0 in -d



24
Replacing the RHS of equation 4.25 by the expression in equation 4.26 and making use of the above boundary condition at x = 1, reduces equation 4.25 to
h U)=, cosh kt(h + z) dz = U2B cosh kt(h + z) dz. (4.27) After substituting expressions for the velocities at x = 1, (Ut),= and U2B from Appendix A, equation 4.27 will take the form:
[0 gkt Htcosh2 kt(h+z) dz
-h w 2 cosh kth
0 N
+10 Ct)k() cos k8()(h + z) cosh kt(h + z) dz
- -d dz(4.28) = -dU cosh kt (h + z) dz (.8
-h
_-d N
- j-d k(m) [P(m) Q(m) exp {-2k(m)
cos k(m)(h + z) cosh kt(h + z) dz.
As before in equation 4.16, the orthogonality of the function will result in the second integral on the LHS being 0; i.e.,
0
-J cos k,()(h + z) cosh k(h + z) dz = 0. (4.29) Hence, equation 4.28 reduces to gk Ht 1 f0h k hd g k-t t 1 0cosh 2 kt(h + z) dz w 2 coshkthh -h c z +U] cosh kt(h+ z) dz
N
+ >3 k(m) [P(m) Q(m) exp I-2k(m)] (4.30) =1
-d cos k(m)(h + z) cosh kt(h + z) dz =0.




25
Equation 4.30 represents one of the (4+4N) required equations. Multiplying equation 4.24 by cos k,(,)(h + z) and integrating w.r.t z over the interval -h to 0 and making use of the second orthogonality condition, results in
[0 gkt Ht cosh kt(h + z) cos k,(n)(h + z) dz
-h w 2 cosh kth
0 N
+f EC(j k,(j) cos k8()(h + z) cos k,(n)(h + z) dz
U-d cosh k,(,)(h + z) dz (4.31)
I-d N
-h E Zk(m) [P(m) Q(m) exp {-2k(m) cos k(m)(h + z) cosh k,(,)(h + z) dz .
As before, the orthogonality condition reduces the first integral on the LHS to 0; i.e.,
0
- cosh kt(h + z) cos k.,()(h + z) dz = 0 (4.32) Also the orthogonality condition yields
0
J cos k,(j)(h + z) cos k.,(,)(h + z) dz = 0, for j n (4.33) For j = n, equation 4.31 reduces to
0
C(n)ks(n)J_ cos2 k,(,)(h + z) dz
+U] cosh k,5()(h + z) dz
+ E k(m) [P(m) Q(m) exp {2k(m) (4.34) M=1
d
-h Cos k(m)(h + z) cosh k.,(n)(h + z) dz S0.
Equation 4.34 represents N equations of the (4+4N) required equations.




26
4.2.3 Use of third interregional boundary condition
The velocity potential in region 1 will be due to the velocity potentials of the incident and the reflected waves. For x < -1, the principle of superposition leads to 41=, +nc + for -h < z < 0 (4.35) At x = -1, the velocity potential in the region 1 will be
4)1A = 4incA + (4,),=-_ for h < z < 0 (4.36) But, the 3rd boundary condition (equation 4.8) is
I1A = 2A for -h 02A = OIincA + (rx1in h -h to -d, results in
L-d
h2A cos king(h + z) dz
(4.39)
=-d (DincA cos k(.,)(h + z) dz+ d(,),=_ cos k(n)(h + z) dz .
_h -h where n is an integer and
k(-) = hn- d (4.40) Details of this proof is given in Appendix A. After substituting expressions for the velocity potentials for (0)i, I2A and OincA from Appendix A, equation 4.39 will




27

take the form
-d d
J-h Cocos k(,)(h + z) dz-1 Ulcos k()(h + z) dz
-d N
+ E~ [P(m) exp {-2k(m)l} + Q(n)]
-hm=1
cos k(m)(h + z) cos k(n)( h + z) dz
-d igkc Hi, cosh ki e(h + z) (4.41)
-h W 2 cosh kinch cos k()(h + z) dz d ik. H, cosh k,(h + z) +Jf W 29 CO'r cos k(n)(h +z) dz
-h w 2 cosh k,-h
-d N
+ E B(m) cos k,(m) (h + z) cos k(,) (h + z) dz .
Again, it can be shown that the functions cos k(n)(h + z) form an orthoganal set over the interval z = -h to z = -d, thus,
-hdcos k(n)(h + z) dz =0 (4.42) Orthogonality also yields
L os k(n)(h + z) cos k(m)(h + z) dz = 0, for n m (4.43) For n = m, equation 4.41 reduces to
d
[P(n) exp f-2kn)l + Q(n)] cos2 k(n)(h + z) dz
igk2 coshkh h dcosh kr(h + z) cos k(n)(h + z) dz
N ,-d (4.44)
- B(m) cos kq(m)(h + z) cos k(m)(h + z) dz
CgsM=1ic1 -C
w 2in Hicos 1 -- ] cosh kin(h + z) cos k(m)(h + z) dz .




28
Equation 4.44 represents N equations of the (4+4N) required equations. Next, integrate equation 4.38 w.r.t z over the interval -h to -d and make use of the second orthogonality condition,
-d -d -d
12A dz= j hincA dz+ (4,)XI dz (4.45) After substituting expressions for the velocity potentials at x = -1, ( ., 4)2A and bincA (from Appendix A) equation 4.45 takes the form
d d
khCo dz- Ul dz
f-d N
+d [P(m) exp {-2k(m)l} + Q(m)] cos k(m)(h + z) dz M=1
-d igki,,e Hinc cosh kinc(h + z) dz 4.46)
-h w 2 cosh kinch
+L-d igk, H,. cosh k,(h + z) dz
-h w 2 cosh k,-d N
+ Bm) cos k,(m)(h+ z) dz .
M=1
As before, the orthogonality condition reduces the second integral on the LHS to 0,
-d
f-h cos (m)(h + z) dz = 0 (4.47) Thus, equation 4.46 reduces to
(h d) {Co Ul}
igkH 1cosh k,(h + z) dz w 2 cosh k,.h ih o N f-d (4.48)
- B(m) cos kq(m)(h + z) dz M=1
= igkin Hinc 1 -d cosh kinc(h + z) dz .
w 2 coshkinch h Equation 4.48 represents one of the (4+4N) required equations.




29
4.2.4 Use of fourth interregional boundary condition
The velocity potential in region 3 is due to the transmitted wave only. Therefore, for x > 1, the velocity potential is 4)3 = 4 for h < z < 0 (4.49) At x = 1, the above equation reduces to 4D3 = ("Dt),_ for h < z < 0 (4.50) The fourth boundary condition (equation 4.9) is DB = D2B for -h -h to -d and making use of the orthogonality condition results in
-d -d
L2B cos k(,,)(h + z) dz= cos k(,,)(h + z) dz (4.53) After substituting expressions for the velocity potentials (4Q),=,, and D2B from Appendix A, equation 4.53 will take the form
l-d -d
-h Co cos k(,)(h + z) dz+ Ul cos k(,)(h + z) dz
- d N
+ E [P(j) + Q(i) exp {-2kj)l}]
-hj=1
cos k(j )(h + z) cos k(,)(h + z) dz (4.54)
*-d igk Ht cosh kt(h + z) cos k(n)(h + z) dz
-h w 2 cosh kth
- d N
+ 1:C(,,) cos k.,(, ( h + z) cos k(,, ( h + z ) d z .




30

Due to the orthogonality condition presented in equation 4.42, the first and second integrals on the LHS become 0. Also, using the orthogonality condition given in equation 4.43, for j = n, equation 4.54 reduces to
P(n) + Q(.) exp {-2k(,,)l}] J cos2 k(,)(h + z) dz
igkt Ht 1_ fd cosh kt(h + z) cos k(,)(h + z) dz
w 2 coshkth Jh
- C(m) cos ks(m)(h + z) cos k(,)(h + z) dz
=0.
Equation 4.55 represents N equations of the (4+4N) required equations. Integrating equation 4.52 w.r.t z over the interval -h to -d and making use of the second orthogonality condition, results in
j-d -d
-h 2B dz= t)= dz (4.56) After substituting expressions for the velocity potentials ((Pt),=, and 42B from Appendix A, equation 4.56 becomes
-d -d
COdz+ Ul dz
--h --h
+ -d N
+ -E mi [P(m) + Q(m) exp {-2k(n)l}] cos k(m)(h + z) dz
(4.57)
= -d igkt Ht cosh kt(h + z) dz
-h w 2 cosh kth
-d N
+ C C(m) cos ks(m)(h + z) dz.
+ -hM=1
Due to the orthogonality condition presented in equation 4.47, the second integral on the LHS of equation 4.57 becomes 0.




31

Hence, equation 4.57 reduces to (h d) {Co + Ul} igkt Ht 1 -d w 2t c1sh -dh cosh kt(h + z) dz W 2 cosh kth h N (4.58)
- C(m) -h cos ks(m)(h + z) dz M=1
=0 .
Equation 4.58 represents one of the (4+4N) required equations.
4.3 Summary of Equations The unknown variables to be determined are Flow below structure: Co, U, P(,) and Q(n)
* Reflected wave: H,. and B,)
" Transmitted wave: Ht and C(n).
The above variables are defined in the list of symbols and are contained in the expressions for velocities and velocity potentials in Appendix A.
The (4+4N) simultaneous equations are summerized below. Note that n=1, 2, 3, . ,N.
gk, H, O cosh2 k,(h + z) w 2 J-h coshhk,.h
+U cosh k,(h + z) dz
N
+ E k(m) [P(m) exp {-2k(m)l} Q(.)] (4.59)
M=1
d
] cos k(m)(h + z) cosh k,.(h + z) dz
- os1kinc h 0 cosh kinc(h + z) cosh k,(h + z) dz .
W 2 cosh kinch J-h




32

-B(n)k(n)j cos2 kq(n)(h + z) dz
+. -d
+U cosh kq(n)(h + z) dz
+ E k(m) [P(m) exp {-2k(m)l} Q()(4.60) M=l
-d
-h cos k(m)(h + z) cosh kq(n)(h + z) dz =0.
gktsHt 1 -0cosh2kt(h+z)d w 2 coshkthJh\Z
d
+U coshkt(h+z) dz
+ E k(m) [P(m) Q(n) exp {2k(m) (4.61)
- cos k(m)(h + z) cosh kt(h + z) dz =0.
C(n)ks()J Cos2 k,(n)(h + z) dz
+U] cosh k,(n)(h + z) dz
+ E k(m) [P(m) Q(m) exp {-2k(m) (4.62) M=l
-d
-h cos k(m)(h + z) cosh k,(n)( h + z) dz =0.




33
i-d
[P(n) exp {-2k(n)l} + Q()] -h cos2 k(,)(h + z) dz
wk 2H k1h cosh k,(h + z) cos k()(h + z) dz igkr2 cosh k~h
N (4.63)
- B( COs kq(m)(h + z) cos k(n)(h + z) dz igkn c s 1 cosh ki,,,(h + z) cos k(n)(h + z) dz w 2 cosh kic cosh
(h d) {Co Ul}
igkH s k d cosh k,(h + z) dz
w 2 co-shkkhh
N ,-d (4.64)
- B(m) cos k(m)(h + z) dz
M=1
igkincH 1 cosh kin(h + z) dz .
w 2 cosh kin osh
[P(n) + Q(,) exp {-2k(n)l}] j-d cos2 k(n)(h + z) dz
sigk Ht 1 h d cosh kt (h + z) cos k(n)(h + z) dz
w 2 coshTh Jh (4.65)
N -d
- ME C(m) J cos ksm)( h + z) cos k(n)(h + z) dz

= 0 .




34

(h d) {Co + Ul}
igk, He 1 -d
w 2t coh -d I coshkt(h+ z)dz w 2 cosh kth h
N _d (4.66)
- C(m)I cos ks(m)(h + z) dz M=1
=0.
These equations can be rearranged and expressed in matrix form; i.e., AX = B. (4.67)
where
A is a known complex matrix of dimension (4+4N) by (4+4N), B is a known complex column matrix of length (4+4N), and X is an unknown complex column matrix of length(4+4N).
This system of complex linear equations is solved using an IMSL (International Mathematics and Statistics Library) subroutine called LEQ2C: Linear equation solution Complex matrix High accuracy solution. This subroutine calls several other subroutines within the IMSL package. The method of solution used by this program is as follows.
For a given N by N complex matrix, LEQ2C factors the matrix A into the L U decomposition of a rowwise permutation of A and/or solves the system of equations AX = B. LEQ2C calls LEQT1C to factor the matrix A and solve the system of equations AX = B. LEQ2C computes the residuals and uses iterative improvement to obtain an accurate solution.
The computation is carried out using single precision accuracy. The double precision accuracy solution did not produce any significant changes for eigenmodes up to 20.




35
The limiting case of a structure with a gap approaching zero was considered to test the validity of the solution. This represents the case of a perfect reflection from a vertical barrier. In this solution incident wave energy is totally reflected hence zero energy is transmitted. Solutions for several other cases were compared with the results of Steimer (1978).
An example solution for three eigenmodes is given in Appendix C.




CHAPTER5
EXPERIMENTAL SET-UP AND PROCEDURE
5.1 Test Facility
A series of experiments were designed and conducted to verify the validity of theoretical results. Model structure dimensions and wave parameters for experiments were chosen so that experimental data would cover a reasonable range of theoretical data.
The experiments were conducted at the Coastal Engineering Laboratory in the "Internal Wave Tank" (23" width x 4' height x 78' length). A diagram of the experimental set-up and the data acquisition system is shown in figure 5.1. Two-dimensional monochromatic surface waves were generated by a piston type wave maker at the front end of the tank as shown in figure 6.18. The aft end of the wave tank has a 1:20 constant slope beach covered with 'horse hair' to dissipate transmission wave energy and minimize reflection.
Two model structures (I = 0.5 ft. and 1.5 ft.) were used. These are of plywood construction (5/16 in. thick) covered by three coats of fiberglass resin. The model structure was firmly fixed to the wave tank to avoid vibration or movement as a result of wave loading. 'weather stripping' was used between the model structure and the glass sides of the tank to seal the gaps between the structure and the sides of the tank.
The instrumentation includes a Marsh McBirney Model 523 current meter for measuring velocity below the center of the model structure; a traversing capacitor wave gage in front of the structure to measure the incident and reflected wave heights, and a stationary capacitor wave gage behind the model to measure the transmitted

36




37
wave height. The wave gages were placed approximately three times the water depth away from the front and aft faces of the structure to avoid transient wave height regions.
Two water depths were used with each of the two model structures and 32 runs were conducted. The data collection frequency was 20 Hz and the duration of each run was 60 seconds.
5.2 Experimental Procedure
The model structure was placed approximately 34 feet away from the wave generator. The gap between the lower horizontal surface of the model structure and the wave tank bottom was adjusted to a known value.
The traversing capacitor wave gage was used to measure incident and reflected wave heights, the stationary capacitor wave gage was used to measure transmitted wave height and the Marsh Mcbirney current meter was used to measure velocity beneath the model structure. All these instruments were connected to a data logger.
Data recording was initiated when the first reflected wave from the structure reached the traversing wave gage and ceased before waves reflected from the wave generator reached the traversing wave gage. During the data recording period, the traversing wave gage always travelled away from the model structure and towards the wave generator. The traversing speed of the wave gage (approximately 80 percent of the wave propagation speed) was chosen to obtain a proper envelope of wave heights.
Later, the time history of recorded data was analyzed using Fast Fourier Transformation techniques to obtain wave height envelopes and wave energy density functions. An example of these plots for Run 4B is given in the Appendix D.




48' 6

5 34'10"
-I ( TIT

Traversing Capacitor Wave Gage

d
h f_

I

Mean Water
L _____________________________ I

Rixed Capaciter
Wave Gage
Marsh McBirney Current Meter

Mean Water
Level
S1:20 Sloping Bottom

Figure 5.1: Schematic diagram of experimental set-up

Piston Type Wave Maker

S i




CHAPTER 6
RESULTS
A dimensional analysis was conducted to obtain the important dimensionless groups for this problem. This analysis is presented in Appendix B. By presenting the results in dimensionless form their range of application is greatly enhanced.
Reflection coefficients (the ratio of reflected wave height to incident wave height) and transmission coefficients (the ratio of transmitted wave height to incident wave height) predicted by the theory are plotted against dimensionless structure length (k;.cl) and the ratio of water depth to structure draft (h/d) while holding other relevant dimensionless parameters constant.
The dimensionless group, 7rs (the ratio of maximum bottom velocity at the center of the structure to maximum bottom velocity due to incident wave), is a good indicator of the potential for structure-induced scour. Figures 6.13 to 6.15 show 7r5 vs. dimensionless structure length (k;.cl) for different water depths to draft ratios (h/d). When 7r5 exceeds unity, a potential for structure-induced scour exists. For a given wave climate, where the maximum incident wave bottom velocity has reached its threshold value (velocity that initiates sediment motion), these plots will predict a range of structure parameters that will cause scouring underneath the structure. In section 6.1.3, figures 6.16 through 6.21 represent conditions where the maximum bottom velocity due to an incident wave maximum bottom velocity (U;) is at the threshold value (Uc) for a selected sea bed of sediment diameter of 0.25 mm and a mass density of 165 ib,m/ft.

39




40
The dependence of the velocity ratio, irs, on other relevant dimensionless parameters is also shown in section 6.1.3. These will assist in the understanding of scour potential due to the variation of wave parameters and structure dimensions. Maximum horizontal velocity profiles for regions 1, 2 and 3 for the stated conditions shown in figures 6.16 through 6.18 are shown in figures 6.19 through 6.22.




41

Region 3
Table 6.1: Brief description of figures on transmission coefficient Figure no. Description of the figure fig. 6.1 Demonstrates the effect of structure length on transmission
coefficient for different gaps below the structure
(for deep water waves).
fig. 6.2 Demonstrates the effect of structure length on transmission
coefficient for different gaps below the structure
(for intermediate depth water waves).
fig. 6.3 Demonstrates the effect of structure length on transmission
coefficient for different gaps below the structure
(for shallow water waves).
fig. 6.4 Demonstrates the effect of gap below the structure on
transmission coefficient for different structural lengths
(for deep water waves).
fig. 6.5 Demonstrates the effect of gap below the structure on
transmission coefficient for different structural lengths
(for intermediate depth water waves).
fig. 6.6 Demonstrates the effect of gap below the structure on
transmission coefficient for different structural lengths
(for shallow water waves).




42
0.0 1.0 2.0 3.0 4.0 5.0
1.0 I I 1.0
0.8 -0.8
h
-- = 0.08 (deep water waves) gT2
h
3.0, 4.5, 6.0, 8.0, 12.0 and 20.0
0.6 .0.6
0
h
Increasing
S0.11 -0.'4
0.2 -.
0.0 0.0
0 .0 1 .0 2. 0 3. 0 4 .0 5.0O Dimensionless structure length (k;,cl) Figure 6.1: Transmission coefficient versus dimensionless structure length (for deep
water waves).




43

0.0 1.0 2.0 3.0 4.0 5.0
1.OL I. I . 1.0

0.8

- = 0.02 (intermediate depth water waves) 9T2
h
-= 3.0, 4.5, 6.0, 8.0, 12.0 and 20.0

0.8

.6 0.6
h
Increasing
-0.4
). 2 0.2
0.0 -7----. I0.0
0.0 1.0 2.0 3.0 4.0 5.0
Dimensionless structure length (k;.cl)
Figure 6.2: Transmission coefficient versus dimensionless structure length (for intermediate depth water waves).

E
0
0
r.
If)




44

0.0 1.0 2.0 3.0 41.0 5.0 l.O L . I . I .

0.8

h
- = 0.003 (shallow water waves)
9T2
h
-= 3.0, 4.5, 6.0, 8.0, 12.0 and 20.0

0.8

0.6 -0.6
h
Increasing
d
0.4 -0. 4
0.2 0.2
0.0 ,0.0
0.0 1.0 2.0 3.0 L4.0 5.0 Dimensionless structure length (kincl) Figure 6.3: Transmission coefficient versus dimensionless structure length (for shallow water waves).

LE
0
0
W) U2




45

0.0 4.0 8.0 12.0 16.0 20.0
1 .0 , , 1 .0
0.8 -0.8
C 0.6 -0.6
0
o Increasing k nel S 0.4 .0.4
0.2 0.2
0.0 0.0
0.0 4.0 8.0 12.0 16.0 20.0 Ratio of water depth to structure draft (h/d) Figure 6.4: Transmission coefficient versus ratio of water depth to structure draft
(for deep water waves).
h
g-2 0.08 (deep water waves)
k;.,l = 0.08, 0.32, 0.64, 1.28, 2.50 and 6.32




46
0.0 4.0 8.0 12.0 16.0 20.0
1.0 1.
0.8 -0.8
92 0.02 (intermediate depth water waves)
7 0. 6 kiel= 0.08, 0.32, 0.64, 1.28, 2.50 and 6.32 0. 6
0
0.4
0.2 Increasing k el 0.2
0.0 0.0
0.0 4.0 8.0 12.0 16.0 20.0 Ratio of water depth to structure draft (h/d) Figure 6.5: Transmission coefficient versus ratio of water depth to structure draft
(for intermediate depth water waves).




47

4.0

8.0

12.0

16.0

20.0
ml 0f

h
-, = 0.003 (shallow water waves) k,,l = 0.08, 0.32, 0.64, 1.28, 2.50 and 6.32
Increasing kindcl

0.8 0.6
0.4 0.2

10.0

01 .4.0 8.0 12.0 16.0 20.0
Ratio of water depth to structure draft (h/d)
Figure 6.6: Transmission coefficient versus ratio of water depth to structure draft (for shallow water waves).

1.0

0.8
0.6
0
0.4
.2
0.2

0. 01

0.0

, ,, ,

I Who...&MMUMA-

.




48

Region 1
Table 6.2: Brief description of figures on reflection coefficient Figure no. Description of the figure fig. 6.7 Demonstrates the effect of structure length on reflection
coefficient for different gaps below the structure
(for deep water waves).
fig. 6.8 Demonstrates the effect of structure length on reflection
coefficient for different gaps below the structure
(for intermediate depth water waves).
fig. 6.9 Demonstrates the effect of structure length on reflection
coefficient for different gaps below the structure
(for shallow water waves).
fig. 6.10 Demonstrates the effect of gap below the structure on
reflection coefficient for different structural lengths
(for deep water waves).
fig. 6.11 Demonstrates the effect of gap below the structure on reflection coefficient for different structural lengths
(for intermediate depth water waves).
fig. 6.12 Demonstrates the effect of gap below the structure on reflection coefficient for different structural lengths
(for shallow water waves).




49

0.
1.0 0.8 + 0.6
0

0 1.0 2.0 3.0 4.0
h
Increasing h
d
h
-- = 0.08 (deep water waves)
9T 2
h
- = 3.0, 4.5, 6.0, 8.0, 12.0 and 20.0

5.0
1.0 0.8 0.6
.0.4

-0.2

0.0 1.0 2.0 3.0 4.0 5.0 Dimensionless structure length (k;,.l)
Figure 6.7: Reflection coefficient versus dimensionless structure length (for deep water waves).

0.2




50
0.0 1.0 2.0 3.0 4.0 5.0
1.0 1.0
0.8 0.8
h
Increasing
d
0.6 -0.6
0
0.41 0.4
h
- 0.02 (intermediate depth water waves)
h
= 3.0, 4.5, 6.0, 8.0, 12.0 and 20.0
0.2 -0.2
0.0............................0.0
0.0 1 0 2. 0 3.0 4.0 5.0 Dimensionless structure length (k;.cl)
Figure 6.8: Reflection coefficient versus dimensionless structure length (for intermediate depth water waves).




51

0 1.0 2.0 3.0 .41.0 5.0

0.
1.0
0.8
*0.6
0
t= 0.4
W)

h
Increasing
h
-- = 0.003 (shallow water waves)
h
= 3.0, 4.5, 6.0, 8.0, 12.0 and 20.0

-0.2

0 .0 1 1 1 1 1 1- I I IU U
0.0 1.0 2.0 3.0 4.0 5.0
Dimensionless structure length (k;,cl)
Figure 6.9: Reflection coefficient versus dimensionless structure length (for shallow water waves).

0.8 0.6
-0.4

0.2




52

4.0

8.0

12.0

16.0

20.0

1.0 0.8 0..6
0. 4 0.2

|0.0

0.0 4.0 8.0 12.0 16.0 20.0
Ratio of water depth to structure draft (h/d)
Figure 6.10: Reflection coefficient versus ratio of water depth to structure draft (for deep water waves).

0.0

Increasing kinci .
'= 0.08 (deep water waves) ,I = 0.08, 0.32, 0.64, 1.28, 2.50 and 6.32

0.01

I i

0.8 0.6 0.4
0.2

h
T k;.

, ,

11 0




53
0.0 4.0 8.0 12.0 16.0 20.0
1.0 1.0
0.8 .
0.8 h = 0.02 (intermediate depth water waves) -.
kincl = 0.08, 0.32, 0.64, 1.28, 2.50 and 6.32
.0.6 -0.6
Increasing kindl
0
U
=a 0. 4 -0.4
0.2 0.2
0.0 0.0
0.0 4.0 8.0 12.0 16.0 20.0
Ratio of water depth to structure draft (h/d) Figure 6.11: Reflection coefficient versus ratio of water depth to structure draft (for
intermediate depth water waves).




54

0.0 4.0 8.0 12.0 16.0 20.0
1.0 1.0
0.8 0.8 = 0.003 (shallow water waves)
9T2
= 0.08, 0.32, 0.64, 1.28, 2.50 and 6.32
0.6 0.6
0 Increasing kied
0.4 ..0.4y
0.2 --0.2
0 .0 .4. 0 8'.0 12.0 16.0 2.
Ratio of water depth to structure draft (h/d)
Figure 6.12: Reflection coefficient versus ratio of water depth to structure draft (for shallow water waves).




55

Region 2
Table 6.3: Brief description of figures on maximum velocity along the bed Figure no. Description of the figure fig. 6.13 Demonstrates the effect of different incident waves (deep,
intermediate depth or shallow water wave) on the
maximum bottom velocity at the center of the structure
(for very small gap, h/d = 1.05).
fig. 6.14 Demonstrates the effect of different incident waves (deep,
intermediate depth or shallow water wave) on the
maximum bottom velocity at the center of the structure
(for small gap, h/d = 1.30).
fig. 6.15 Demonstrates the effect of different incident waves (deep,
intermediate depth or shallow water wave) on the
maximum bottom velocity at the center of the structure
(for moderate gap, h/d = 2.00).
fig. 6.16 Demonstrates the effect of structure length on the
maximum bottom velocity along the bed.
fig. 6.17 Demonstrates the effect of gap below the structure on the maximum bottom velocity along the bed.
fig. 6.18 Demonstrates the effect of different incident waves (deep, Intermediate depth or shallow water wave) on the
maximum bottom velocity along the bed.




56
Figure no.] Description of the figure fig. 6.19 The variation of the maximum'horizontal velocity
profile in the regions 1, 2 and 3 due to the presence of
a small structure (length = 4.0 ft).
fig. 6.20 The variation of the maximum horizontal velocity
profile in the regions 1, 2 and 3 due to the presence of
a large structure (length = 74.7 ft)
(In this case, wave length = structure length).
fig. 6.21 The variation of the maximum horizontal velocity
profile in the regions 1, 2 and 3 due to a moderate gap
below the structure (h/d = 2.0).
fig. 6.22 The variation of the maximum horizontal velocity
profile in the regions 1, 2 and 3 due to a deep water incident
wave (h/gT2 = 0.10).




57

0.0 0.5 1.0 1.5 2.0
12 .0 .. . . 12 .0
h
--- -T = 0.08 (deep water waves) T2
80 - 0.04 (int.depth water waves) 8. 0 gT2
8.0 h 8
----= 0.02 (int.depth water waves) gT2
h
.....-..-..-. -- 0.003 (shallow water waves)
9T2
h
1.05
S4.0 -4.0
0.0 0.5 1.0 1.5 2.0 Dimensionless structure length (ki, l)
Figure 6.13: Ratio of maximum bottom velocity at the center of the structure to the maximum incident wave bottom velocity(7r5) versus dimensionless structure length.




58
0.0 0.5 1.0 1.5 2.0
12.0 12.0
h
- - 0.08 (deep water waves)
------ -- = 0.04 (int.depth water waves) 9T 2
h
- = 0.02 (int.depth water waves) 9T2z
h
- -..= 0.003 (shallow water waves) 9T2
& h
- 1.3
E 4.0 4.0
0.0 0.0
0.0 0.5 1.0 1.5 2.0 Dimensionless structure length (ki,,l)
Figure 6.14: Ratio of maximum bottom velocity at the center of the structure to the maximum incident wave bottom velocity(7rs) versus dimensionless structure length.




0

59

0.5

0.0
12.0
8.0
-.
C
4. .

0.0 1
0.

h
h.=
gT2
h
gT2
h
9T2
h
- =

1.0

1.5

0.08 (deep water waves)
0.04 (int.depth water waves) 0.02 (int.depth water waves) 0.003 (shallow water waves)
2.0

.0
12. 0

- ---- - -

0.5

1.0

1.5

2

8.0
1.0 0.0 .0

Dimensionless structure length (k;.cl)

Figure 6.15: Ratio of maximum bottom velocity at the center of the structure to the maximum incident wave bottom velocity(7rs) versus dimensionless structure length.

' ' '

2


.




Note: The horizontal scale is normalized by different structure lengths
Water depth = 8.0 ft Wave period = 5.0 sec Wave height = 0.5 ft Wave length = 74.7 ft 0
It3
-= 0.01
TF2
1.05 0 d w U, = U = 0.43 ft/sec Structure length 1 = 2.0 ft .------ Structure length 2 = 74.7 ft
;.- .. ......

-16 -15 -14 -13 -12

I I I I I

-11 -10 -9 -8 -7 -6

-5 -4 --3 -2 -1

-5.0

-4.0
-3.0
.2.0
-1.

0

n~n

1 2 3 4

Dimensionless Horizontal Coordinate (-) Figure 6.16: The effect of the structure length on the maximum bottom velocity along the bed.

0




I I
-16 -15 -14 -13

Water depth = 8.0 ft Wave period = 5.0 sec Wave height = 0.5 ft Structure length = 2.0 ft
--=0.01 c T2 00
U, = Ubic = 0.43 ft/sec h -
2.00
d 2

5.0

-3.0
.2.0
-1.0

An

-12 -1 -10 -9 -8 -7 -6 -21 2 Dimensionless Horizontal Coordinate (-) Figure 6.17: The effect of the gap below the structure on the maximum bottom velocity along the bed

I-'

3 4




Water depth = 8.0 ft Wave height = 0.5 ft
Structure length = 2.0 ft
h 0
-= 1.05
.d
Wave period = 5.0 sec
h
_- = 0.01, U, = U = 0.43 ft/sec
gT2
Wave period = 1.6 sec
h
-----. -- = 0.10
72
I .... .

-16 -15 -14 -13 -12 -It -10 -9 -8 -7 -6 -5 -4
Dimensionless

.5.0
.4.0
..3.0
1.0 n .........

-3 -2 -1 0 1
X
Horizontal Coordinate()

Figure 6.18: The effect of different types of incident waves (deep and intermediate depth water waves) on the maximum bottom velocity along the bed.

2 3 4




I I I
-16 -1S -14 -1

T-.---1----------- I ~1 I I I I
I I

2
4J 4,
'.4 4,

.0.0

3 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
Dimensionless Horizontal Coordinate (X) Figure 6.19: The maximum horizontal velocity profile for regions 1, 2 and 3 for the structure length shown in fig. 6.16 (1=4.0 ft).

Water depth = 8.0 ft Wave period = 5.0 sec Wave height = 0.5 ft Structure length = 2.0 ft
h
T2 0.01
= 1.05
Uc =Ube = 0.43 ft/sec

C~3

4

3




I I S III I I -A Q

-16 -15 -14 -13 -12

-11 -10 -9 -8 -7 -G -5 -4
Dimensionless

42
04

-3 -2
Horizontal

-0.0

-1 0 1 Coordinate (,)

Figure 6.20: The maximum horizontal velocity profile for regions 1, 2 and 3 for the structure length shown in fig. 6.16 (1=74.7 ft).

-4

Water depth = 8.0 ft Wave period = 5.0 sec Wave height = 0.5 ft Structure length = 74.7 ft
= 0.01
- = 1.05
d
Uc = Ubi, = 0.43 ft/sec

2 3

4

0
ktbl




-16 -15 -14

-4

2
4,
'.4
4,

.0.0

I I IIn- J

a-I I I I I I I I I I I
-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
Dimensionless Horizontal Coordinate ( )
Figure 6.21: The maximum horizontal velocity profile for regions 1, 2 and 3 for the
ratio of water depth to structure draft shown in fig. 6.17 (h/d = 2.0).

Water depth = 8.0 ft Wave period = 5.0 sec Wave height = 0.5 ft Structure length = 2.0 ft
h
T2 = 0.01
d- 2.0
U= Ubi = 0.43 ft/sec

-9

--4

0)i

3

4

I




0.0
Water depth = 8.0 ft Wave period = 1.6 sec --2.0 Wave height = 0.5 ft Structure length = 2.0 ft
h
-_= 0.10
T2
= 1.05
-6.0
-16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Dimensionless Horizontal Coordinate ( ) Figure 6.22: The maximum horizontal velocity profile for regions 1, 2 and 3 for the deep water wave shown in fig. 6.18 (T = 1.0 sec and h/gT2 = 0.01)




67
Experimental results
Table 6.4: Experiment set-up number 1 Water depth (h)= 12.0 inches
Structure length (2 l)=12.0 inches
Experiment Incident w. height Wave period Draft kil
I d I 7r
no. H, (in.) T (sec.) d (in.)
1A 0.96 0.88 6.00 2.0 0.04 0.85 1B 0.92 0.88 6.00 2.0 0.04 0.85 2A 1.23 1.25 6.00 2.0 0.02 0.51 2B 1.88 1.25 6.00 2.0 0.02 0.51
3A 1.91 0.88 9.23 1.3 0.04 0.85 3B 2.15 0.88 9.23 1.3 0.04 0.85
4A 0.90 1.25 9.23 1.3 0.02 0.51 4B 1.92 1.25 9.23 1.3 0.02 0.51




68

Table 6.5: Experiment set-up number 2 Water depth (h)= 12.0 inches
Structure length (2 l)=36.0 inches
Experiment Incident w. height Wave period Draft T2 kic1
no. Hinc (in.) T (sec.) d (in.)
5A 0.67 0.88 6.00 2.0 0.04 2.53 5B 1.48 0.88 6.00 2.0 0.04 2.53 6A 0.98 1.25 6.00 2.0 0.02 1.53 6B 3.17 1.25 6.00 2.0 0.02 1.53
7A 1.01 0.88 9.23 1.3 0.04 2.53 7B 1.64 0.88 9.23 1.3 0.04 2.53 8A 1.00 1.25 9.23 1.3 0.02 1.53 8B 3.17 1.25 9.23 1.3 0.02 1.53




69

Table 6.6: Experiment set-up number 3 Water depth (h)= 18.0 inches
Structure length (2 l)=12.0 inches
Experiment Incident w. height Wave period Draft h T k;,cl no. H;nc (in.) T (sec.) d (in.)
9A 1.87 1.08 9.00 2.0 0.04 0.56 9B 3.58 1.08 9.00 2.0 0.04 0.56 10A 1.30 1.53 9.00 2.0 0.02 0.34 10B 2.30 1.53 9.00 2.0 0.02 0.34 11A 1.53 1.08 13.85 1.3 0.04 0.56 11B 3.23 1.08 13.85 1.3 0.04 0.56 12A 1.25 1.53 13.85 1.3 0.02 0.34 12B 2.61 1.53 13.85 1.3 0.02 0.34




70

Table 6.7: Experiment set-up number 4 Water depth (h)= 18.0 inches
Structure length (2 l)=36.0 inches
Experiment Incident w. height Wave period Draft h hT2 k;,c I Id 9T2
no. H ac (in.) T (sec.) d (in.)
13A 0.95 1.08 9.00 2.0 0.04 1.68 13B 2.45 1.08 9.00 2.0 0.04 1.68 14A 1.98 1.53 9.00 2.0 0.02 1.02 14B 4.04 1.53 9.00 2.0 0.02 1.02 15A 1.11 1.08 13.85 1.3 0.04 1.68 15B 2.15 1.08 13.85 1.3 0.04 1.68 16A 1.87 1.53 13.85 1.3 0.02 1.02 16B 4.24 1.53 13.85 1.3 0.02 1.02




71

Comparison of results
Comparison of theoretical and experimental results
0.0 1.0 2.0 3.0 ,4.0 5.0
1.0 1.0
theoretical, kincl =0.51 C experimental, k;,l =0.51
----....... theoretical, kinl =1.53
08 experimental, kgcl =1.53 .0.8
h
-=0.02
cT2
E 0.6 -0.6
a)
0
0
U,
0 ...(-D- 0 4
0.2 0.2
0. 0 0.0
0.0 1.0 2.0 3.0 4.0 5.0
Ratio of water depth to structure draft (h/d)
Figure 6.23: Comparison of theoretical and experimental results for run numbers
2, 4, 6 and 8 (see table 6.4 and 6.5).




72

1.0

2.0

3.0

4.0

5

,
1.0 2.0 3.0 '4.0 Ratio of water depth to structure draft (h/d)

0.0

5.0

Figure 6.24: Comparison of theoretical and experimental results for run numbers 9, 11, 13 and 15 (see table 6.6 and 6.7). .

1.0 0.8 0.6 0.4

theoretical, k .cl =0.56
C experimental, k1scl =0.56
-.---------. theoretical, .k ,l =1.68
experimental, k .c, =1.68
h
=0.04

. , i ,

0.2

0.0
0.0

.0
1.0 0.8 0.6
-0.4
0.2
10.0

..(D-




0.0 1.0 2.0 3.0 4.0 5.0
1.0 1.0 theoretical, k e =0.34
0 experimental, kjI =0.34
----..------ theoretical, k ,,l =1.02
experimental, k I =1.02
0.8 =0.02 -0.8
0.6 0.6
0
0.4 --'-0.4
0.2 -0.2
0.
0.0 ,,,
0.0 1.0 2.0 3.0 4.0 5.0
Ratio of water depth to structure draft (h/d)
Figure 6.25: Comparison of theoretical and experimental results for run numbers
10, 12, 14 and 16 (see table 6.6 and 6.7).




74

1.0

2.0

3.0

4.0

1.0' 2.0 3.0 4.0 Ratio of water depth to structure draft (h/d)

Figure 6.26: 1, 3, 5 and 7

Comparison of theoretical and experimental results for run numbers (see table 6.4 and 6.5).

0.0

5

1.0

0.8
.6
v
0
0
U2
0.LI
0.2

theoretical, k nel =0.85
C experimental, k sel =0.85
-----------. theoretical, k .cl =2.53
a experimental, k nel =2.53
h
-T =0.04

, i . i .

0.0 1
0.

0

. 0

5

.0
1.0 0.8 0.6
0.4 0 .2

, , , ,

............-- -- -




75

1.0

2.0

3.0

.........A.............
.............
.A .......
0M
(D 0D

1.0 2.0 3.0 q.0 Ratio of water depth to structure draft (h/d)

Figure 6.27: 2, 4, 6 and 8

Comparison of theoretical and experimental results for run numbers (see table 6.4 and 6.5).

. . . . .

1.0 0.8
-0.6
-0
U .4

0.0

5

.

..

.

.0
1.0 0.8 0.6
0 .4 0.2

- theoretical, k;.cl =0.51
CD experimental, kje, =0.51
------------ theoretical, k;,cl =1.53
S experimental, ki,cl =1.53
h
OP =0.02

L

0.2

0 0 .....
0.0

..... 0
5.0




0.0
1.0 0.8
- 0.6
0
0.2

1.0
Ratio of water

5

2.0 3.0 4.0 depth to structure draft (h/d)

10.0 .0

Figure 6.28: Comparison of theoretical and experimental results 9, 11, 13 and 15 (see table 6.6 and 6.7).

for run numbers

76

1.0

2.0

3.0

4.0

5

.I.

theoretical, k;nel =0.56
CD experimental, k;nd =0.56
----. --- theoretical, k;,l =1.68
experimental, k;,c1 =1.68
h
-- =0.04-

0.01
0.

0

.............................
CD
... .......................................... ..... *" .............

.

.0
1.0 0.8 0.6
0.
.0.2




77

0.0 1.0 2.0 3.0 4.0 5.0
1.0 . 1 .0
...... .....
0.8 0.8
C.))
.2 0 .6 -0. 6
0
theoretical. kincl =0.34 0.'4
o experimental, kisel =0.34
------------ theoretical, ki,,l =1.02
. experimental, kicl =1.02
h
o- =0.02
0.2 10.2
0 .0, ,
0.0 1.0 2.0 3.0 4.0 5.0
Ratio of water depth to structure draft (h/d)
Figure 6.29: Comparison of theoretical and experimental results for run numbers
10, 12, 14 and 16 (see table 6.6 and 6.7).-




78

0.0
1.0 0.8 0.6
0
0.4 0.2

0.01
0.

0

Figure 6.30: 1, 3, 5 and 7

1.0

2.0

3.0

4.0

5

7

1.0 2.0 3.0 4.0 Ratio of water depth to structure draft (h/d)

10.0 .0

Comparison of theoretical and experimental results for run numbers (see table 6.4 and 6.5).

theoretical, kil =0.85
0 experimental, kind =0.85
------------ theoretical, k cj =2.53
a experimental, k1 cl =2.53
h
-~ =0.04

5

. ...... .................................................................
CD
-------------- -------------

, ,

.0
1.0 0.8 0.6 0.4
0.2




79

0.0 0.5 1.0 1.5 2.0
6.0 . I 6.0
S------ theoretical, =0.04
h
experimental, =0.04 gT2
. theoretical, =0.02 gT2
h
C experimental, =0.02
=2.0
2.0 -2.0
0.0 10.0
0.0 0.5 1.0 1.5 2.0 Dimensionless structure length (k acl)
Figure 6.31: Comparison of theoretical and experimental results for run numbers
1, 2, 6, 9, 10, 13 and 14 (see table 6.4 through 6.7).




80
0.0 0.5 1.0 1.5 2.0
6.0 ---- ---6.0
theoretical, =0.04 PT2
4. 0 -& experimental, =0.04 -4.0
gT2
- experimental, h =0.02 h -h
-=1.3
S2.0 -2.0
-At
0.0 0.0
0.0 0.5 1.0 1.5 2.0 Dimensionless structure length (k .cl)
Figure 6.32: Comparison of theoretical and experimental results for run numbers
3, 4, 8, 11, 12, 15 and 16 (see table 6.4 through 6.7).




81
Comparison of results with the other researchers
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
1.0 1.,,1 0
).8 0.8
9 'Present theory ...--------- Steimer's theory
0.6 Black's theory -0.6
0
0.4 -0.4
0.2 .0.2
0.00.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Dimensionless structure draft (kincd)
Figure 6.33: Comparison of theoretical transmission coefficient with results from
Steimer and Black




82
As reported by Steimer (1977), Macagno conducted a series of experiments to measure transmission coefficient on a problem that is similar to described in this thesis. These experimental conditions are listed in the following table.
Later, several other authors theoretically computed transmission coefficients for for above coditions listed in the table 6.8. Following table compares the results predicted by the present theory over other available theoretical and experimental values.

Table 6.8: Macagno's experimental conditions Run no. Depth Draft Length of struc. W. period
h (cm.) d (cm.) 2! (cm.) T (sec.)
EXP 1 30.0 15.0 175.0 1.0 EXP 2 30.0 15.0 175.0 1.4 EXP 3 30.0 15.0 175.0 1.8 EXP 4 30.0 21.5 175.0 1.0 EXP 5 30.0 4.4 58.3 1.3 EXP 6 30.0 4.4 58.3 1.0




83

Table 6.9: Comparison of present theory with other studies for experimental conditions described in table 6.8
Present Theory by Experiment by Run no. Theory Steimer Ito Ursell Macagno Macagno
EXP 1 0.073 0.073 0.076 0.080 0.098 0.064 EXP 2 0.152 0.151 0.159 0.980 0.173 0.130 EXP 3 0.221 0.218 0.229 1.000 0.239 0.190 EXP 4 0.036 0.037 0.039 0.480 0.054 0.039 EXP 5 0.631 0.628 0.638 1.000 0.635 0.480 EXP 6 0.459 0.460 0.459 0.790 0.484 0.370