Wave Transmission through Multi-layered Wave Screenscollectionscanada.gc.ca/obj/s4/f2/dsk1/tape3/PQDD_0016/MQ54487.pdfwave screens within 3% of the actual value for a single screen

Wave Transmission through

Multi-layered Wave Screens

by

Gordon Grant Thomson

A thesis submitted to the

Department of Civil Engineering

in conformity with the requirernents for the degree of

Master of Science (Engineering)

Queen's University Kingston, Ontario, Canada

March, 2000

Copyright O Gordon Grant Thomson, 2000

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Abstract

A wave screen is a porous vertical wall, usually constructed using rectangular slats

oriented in either a horizontal or vertical direction and attached to vertical piles or

suppoa structures. A wave screen breakwater can be composed of a series of screens

attached to the support system. They have several advantages over conventional

mbblemound breakwaters incIuding a smalIer footprint, allowing increased circdation

within the sheltered zone, less environmental impact and Iower cost. However, there are

few practical design guidelines for predicting their performance over a wide range of

design variables such as wave height, wave period and water depth.

Hydraulic mode1 tests were undertaken in a two-dimensional wave flurne at the Queen's

University Coastd Engineering Research Laboratory in Kingston, Ontario. The tests

investigated the impact of slat orientation, screen porosity and screen spacing on the

transmission coefficient for single, double and triple screen systems. The screens were

subjected to irregular wave conditions of varying wave height, wave period and water

depth.

The results showed that wave screens could reduce wave transmission by up to 60% for a

single screen and 80% for a double screen. Wave transmission was found to be a

function of the number of screens used, screen porosity and orientation, wave steepness,

relative depth, d / g ~ 2 and dimensionless gap space. Empirical equations were developed

for single and double screen systems and were able to predict the performance of the

wave screens within 3% of the actual value for a single screen system and 9% for a

double screen system. Prediction of wave transmission through a triple screen system

was shown to be possible by combining a single screen equation with a double screen

equation. Given the compIex nature of the wave transmission process, these equations

should be used as an initial estirnate of wave transmission and physical mode1 tests are

recomrnended for confirmation of final design parameters.

Acknowledgements

I would like to express my deep gratitude to Dr. Kevin Hall for his professional and

financial support throughout this project. His boundless enthusiasm for coastai

en-gineering is contagious and served as a source of inspiration when there was doubt.

Many thanks are due to Stuart Seabrook for his endless patience for answering endless

questions and making the time at the coastai lab a tme joy.

1 am also grateful to rny coueagues, Mohamed Dabees for his advice during late nights in

the offrce, Kevin Brown and Paul Tschirky for acting as sounding boards, and everyone

else sharing Room 342 for creating a working environment unlikely to be equaled.

1 would iike to thank Neil Porter for his technical support and Tiago Pereira Caldas for

his assistance in making the wave screens.

1 have to express my thanks to my family for ttieir unswerving belief and support.

Finally, the Mysliveceks, for whom 1 cannot begin to express my deep gratitude, and

especially Paula for her support that wiU never be forgotten.

Table of Contents

Abs tract

Acknowledgennents

Table of Contents

List of Tables

List of Figures

List of Symbols

2 Literature Review

2.1 Basic Eqrtations

2.2 Influence of Physical Variables

2.2.1 Single Screen System

2.2.2 MultipleScreen System

2.3 muence of Wave Climate

2.4 Wave Pressure and Force

3 Experimental Equipment and Procedure

Introduction

Wave Flurne

Wave Generator

Wave Generation

Wave Probes

Data Acquisition and Analysis

Wave Screen Construction

Dimensional Ana lysis

4 Parametric Analysis

4.1 Introduction

4.2 rnfluence of Wave Period

4.2.1 Single Screen

4.2.2 Double Screens

4.2.3 Triple Screens

4.3 Influence of Wave Height

4.3.1 Single Screen



4.4 Influence of Depth

4.4.1 Single Screen


4.5 Influence of W m e Steepness

4.5.1 Single Screen



4.6 Influence of Porosiv

4.6.1 Single Screen



4.7 influence of Screen On'enration

4-7.1 Single Screen



4.8 Influence of Spacing between Screens

4.8 1 Double Screens


4.9 Influence of Relative Depth

4.9- L Single Screen

4.9-2 Double Screens

4.10 influence of ug'15

4.10.1 Single Screen

4.10.2 Double Screen

5.1 Introduction

5.2 Boundary Condition

5.3 Equation Variables and Form

5.4 Data Manipulation

5.5 Interpretatiun of Systat Results

5.6 Single Screen Equations

5.6.1 Single Horizontal Screen

5.6.2 Single Vertical Screen

5.6.3 Cornparison of Horizontal and Vertical Screen Orientations

5.6.4 Evaluation of Screen 30b

5.6.5 Evaluation of Screen 50

5.7 Double Screen Equations

5.7.1 H-H Screen Orientation

5.7.2 H-V Screen Orientation

5.7.3 V-H Screen Orientation

5.7.4 V-V Screen Orientation

vii

5.7.5 Combined H-H and H-V Screen Systerns

5.7.6 Combined V-H and V-V Screen Systems

5.7.7 Other Data Set Combinations 108

5.7.8 Prediction of Double Screen Performance using Single Screen Equations 109

5.8 Triple Screen Equation 110

5.9 Cornparison with Other Theones 113

6 Conclusions and Recommendations 119

6.1 Conclusions I l 9

6.2 Recommendations 122

List of References 124

Appendix A Pictures of constmcted wave screens 127

Appendix B Figures showing Kt vs Wave Period for screen systems differentiated by wave height

Appendix C Figures showing Kt vs Wave Height for screen systems differentiated by wave penod 143

Appendix D Figures showing Kt vs Wave Height for screen systems differentiated by water depth 155

Appendix E Figures showing Kt vs Wave Steepness for screen systems differentiated by wave period 160

Appendix F Figures showing Kt vs Wave Steepness for screen systems differentiated by screen porosity 172

Appendix G Figures showing Kt vs Wave Steepness for screen systems differentiated by screen orientation 177

. . . vlll

Appendix H Figures showing Kt vs Wave Steepness for screen systems differentiated by gap size

Appendix 1 Figures showing Kt vs Relative Depth for screen systems differentiated by depth

Appendix J Figures showing Kt vs cU~T' for screen systems differentiated by depth

end& K Sensitivity Analysis graphs of Wave Steepness vs R~ for approximating zero

Appendix L Graphs showing actual versus predicted Kt using two single prediction equations to mode1 a double screen system

Appendix M Graphs showing actual versus predicted Kt using three single prediction equations, a single and a double equation and a double then single equation to mode1 a triple screen system 201

Vita

List of Tables

Table 3-1

Table 3-2

Table 5-1

Table 5-2

Table 5-3

TabIe 5-4

Table 5-5

Table 5-6

Table 5-7

Table 5-8

Table 5-9

Table 5-10

TabIe showing target wave train characteristics

Sumrnary of wave screen porosities and slat sizes

Table showing contribution of variables to Kt for horizontal screen equation

Table comparing predicted and actual Kt for data points withheld from the single horizontal screen statistical analysis

Table comparing predicted and actual Kt for data points withheld f'rom the single vertical screen statistical analysis

Table showing contribution of variables to Kt for H-H screen equation

Table comparing predicted and actual Kt for data points withheld from the H-H screen statistical analysis

Table showing contribution of variables to Kt for H-V screen equation

Table comparing predicted and actual Kt for data points witliheld fiom the H-V screen statistical analysis

Table showing contribution of variables to Kt for V-H screen equation

Table cornparing predicted and actual Kt for data points withheld from the V-H screen statistical analysis

Table showing contribution of variables to Kt for V-V screen equation

Table 5-11 Table comparing predicted and actual Kt for data points withheld fiom the V-V screen statistical analysis

Table 5-12 Table cornparing predicted and actual Kt for data points withheld fiom the combined H-H and H-V statistical analysis

Table 5-13 Table comparing predicted and actual Kt for data points withheId from the combined V-H and V-V statistical analysis using the natural log equation

Table 6-1 Table summarizing: variable range

Figure 1-1 Diagram showing a single and a double wave screen breakwater (Allsop, 1995) 3

Figure 2-1 Notation for wave screen porosity 9

Figure 2-2 Kt vs Wall Element Ratio taken fiom Gmne and Kohlhase (1974) 9

Figure 2-3 Schematic for nomenclature of wave screen height 14

Figure 2-4 Actual vs predicted Kt for a double screen system (Kondo, 1979) 17

Figure 2-5 Kt vs dIg~' for one and two rows of piles 21

Figure 2-6 Graphic cornparisons of large and srnall wave steepnesses (Bergmann and Oumeraci, 1998)

Figure 2-7 Notation for wave approach

Figure 3-1 Wave flume schematic (not to scale)

Figure 3-2 Wave probe schematic

Figure 3-3 Schematic showing horizontal and vertical orientations and screen spacing 37

Figure 3-4 A picture of an installed wave screen with a vertical orientation 40

Figure 4-1 Kt vs Wave Penod for double screen system differentiated by wave height

Figure 4-2 Kt vs Wave Period for triple screen system differentiated by wave height

Figure 4-3 Kt vs Wave Height for double screen system differentiated by wave period

Figure 4-4 Kt vs Wave Height for triple screen system differentiated by wave period

Figure 4-5 Kt vs Wave Height for single screen system differentiated by depth 50

Figure 4-6 Kt vs Wave Height for a double screen system differentiated by depth 51

Figure 4-7 Kt vs Wave Steepness for a single screen system differentiated by wave period

Figure 4-8 Kt vs Wave Steepness for a double screen system differentiated by wave period

Figure 4-9 Kt vs Wave Steepness for a triple screen system differentiated by wave period

Figure 4-10 Kt vs Wave Steepness for a single screen system differentiated by screen porosity

Figure 4-11 Kt vs Wave Steepness for a double screen system differentiated by screen porosity

Figure 4-12 Kt vs Wave Steepness for a triple screen system differentiated by screen porosity

Figure 4-13 Kt vs Wave Steepness for a single screen system dserentiated by orientation

Figure 4-14 Kt vs Wave Steepness for a double screen system differentiated by orientation

Figure 4-15 Kt vs Wave Steepness for a triple screen system differentiated by orientation

Figure 4-16 Kt vs Wave Steepness for a double screen system differentiated by gap size

Figure 4-17 Kt vs Wave Steepness for a triple screen system differentiated by gap size

Figure 4-18 Kt vs Relative Depth for a single screen system differentiated by depth

Figure 4-19 Kt vs Relative Depth for a double screen system differentiated by depth

Figure 4-20 Kt vs dgT2 for a single screen system differentiated by depth

Figure 4-21 Kt vs dgT2 for a double screen system differentiated by depth

Figure 5-1 Sensitivity analysis graph of wave steepness vs lS2 for approximating zero 74

Figure 5-2 Sample Systat output for a single, horizontally oriented screen 77

Figure 5-3 Graph of residuais after regression analysis for a single, horizontally onented screen 80

Figures 5-4a & 5-4b Graphs of actual vs predicted Kt values for a single horizontal screen using both the naturai log and power equations

Figures 5-5a & 5-Sb Graphs of predicted vs actual Kt values for a single vertical screen using both the natural log and power equations

Figure 5-6 Graph comparing horizontal and vertical natural log equations

Figure 5-7 Graph directly comparing horizontal and predicted Kt using the natural log equation

Figures 5-Sa & 5-8b Graphs of Actuai vs Predicted Kt for Screen 30b using the natural log and power equations

Figures 5-9a & 5-9b Graphs of Actual vs Predicted Kt for Screen 50 using the natural log and power equations

Figures 5-10a & 5-lob Graphs of actual vs predicted Kt values for an H-H screen system using both the natural log and power equations

Figures 5-Lla & 5-llb Graphs of predicted vs actual Kt values for an H-V screen systern using both the natural log and power equations

Figures 5-12a & 5-12b Graphs of predicted vs actual Kt values for an V-H screen system using both the natural log m d power equations

Figures 5-13a & 5-13b Graphs of predicted vs actual Kt vaIues for an V-V screen system using both the natural log and power equations

Figures 5-14a & 5-14b Graphs of actual vs predicted Kt values for H-H and H-V screen systems using the natural log and power equations

Figures 5-15a & 5-15b Graphs of actuai vs predicted Kt values for V-H and V-V screen systems using the natural log and power equations

Figure 5-16 Graph showing actual vs predicted Kt values for prediction of a double screen system using the single screen equation twice

Figure 5-17 Acniai vs predicted Kt when using the single screen equations to predict Kt through a triple screen 111

Figure 5-18 Actual vs predicted Kt using a single then double screen equation to predict Kt through a triple screen 112

Figure 5-19 Actual vs predicted results using a double then single screen equation to predict Kt for a triple screen system 113

Figure 5-20 Actual vs Predicted Kt using Hartmann's Equation for a single screen 114

Figure 5-21 Actual vs Predicted Kt using Mei's equation for a single screen 115

xiv

List of Svmbols

Variable Definition

Empirical shape coefficient

Slat width

Contraction coefficient

Depth of water

Distance kom water surface to structure bottom

Height of submerged structure from bottom

Dimensionless depth

Energy density

Dissipated wave energy

Incident wave energy

Reflected wave energy

Transmitted wave energy

Centre to centre distance of hvo slats

Force on wall element

Reduced dynamic force

Gravitational constant

Dimensionless gap width

Horizontal slat orientation

Wave height

Incident wave height

Transmitted wave height

Relative depth

Wave steepness

Refiection coefficient

Transmission coefficient

Energy loss coefficient

wave number

Wavelength in local depth

Deepwater waveIength

Screen porosity

Hydrostatic pressure

Dynamic pressure due to incident wave

Dynamic pressure due to reflected wave

Dynamic pressure due to transrnitted wave

Gap between two slats

Wave penod

Slat thickness

Velocity of transrnitted wave at given depth

Verticai slat orientation

Wall element ratio

Incident wave angle

Fluid viscosity

Density of fluid

xvi

One of the primary goals of coastal engineers when constructing harbours or marinas is to

reduce the wave energy entering the sheltered area, while stU maintaining water

circulation within the harbour. Vertical permeable walls, more commonly cailed wave

screens, are an appealing option for achieving this goal. However, there is Little research

upon which an engineer can base an initial design of a wave screen systern. This thesis

presents the results of a two-dimensional flume study undertaken to determine empirical

equations describing wave transmission through various wave screen systems.

There are many factors that have to be considered during the design of breakwaters. The

type and size of a protecting structure and its footprint are important considerations for

aesthetic and environmental reasons. The location of any structure with respect to the

depth of water and wave breaking zone can drastically increase the cost. FinalIy, the

construction of a barrier can have environmental repercussions due to disrupted water

circulation patterns and sedirnent transport, which affects aquatic and nearshore habitat.

Conventional rubblernound breakwaters and berm breakwaters are conimonly used to

create protected berthing areas for ships, They are simply a large voIume of rock

material that is piled up and projects above the water surface to minimize the

transmission and overtopping of wave energy.

Rubblemound breakwaters do an excellent job of creating a sheltered area behind them as

they stop the majority of the incident wave energy fiom passing through them. They also

reduce reflection within the harbour through dissipation of energy in the voids at the back

of the structure. However, they have a very large footprint and c m be very expensive.

These drawbacks are dramatically increased as the depth of water increases. The size of

the footprint is especially problematic in environmentally sensitive regions or in areas of

heavy ship tr&c with limited navigation space. Due to their impermeable nature, these

breakwaters can also reduce water circulation and water quality within the harbour, again

affecting the environment. Rubblemound breakwaters c m be aesthetically unappealing.

Submerged breakwaters are becomuig more popular due to these latter two reasons. In

certain cases, various types of vertical breakwaters may be preferable.

Vertical caisson breakwaters are usually monolithic gravity structures consisting of a

concrete box (caisson) which is filled with sand. They do not aUow any transmission (if

not overtopped) and have an advantage over rubblemound and berm breakwaters because

of their srnaller footprint. The main problem with vertical caisson breakwaters is that

most of the wave energy is reflected which can result in standing waves that are twice as

high as the incident waves, creating a dangerous environment for ships. They do not

dissipate wave energy within the harbour either and again Iimit water circulation and

quality .

Another option that rninimizes footprint size is a pile breakwater. A pile breakwater

consists of several rows of piles driven into the substrate. Pile breakwaters do not occupy

a large footprint but the;. cm be quite expensive due to the specialized equipment needed

for pile driving from a floating platform. Furthemore, several rows of piles are often

needed to achieve satisfactory dissipation, adding to the cost It is also difncult to drive

the piles sufficiently close to each other to effectively reduce transmission. The cost of

pile driving also Limits the depth of water in which a large nurnber of piles can be placed,

for the deeper the water, the deeper the piles have to be driven into the substrate to resist

the increased wave forces. Lastly, piles are not very effective against waves with large

penods. However, to their advantage, pile breakwaters only reduce water circulation and

sediment transport and do not halt it altogether, uniike the previously discussed

structures.

A wave screen is a permeable wall made of closely spaced elements such as steel,

concrete or h b e r planks, which are hung off of piles. A diagram is given in figurel-1

showing piles being used to suspend both a single and double wave screen system.

Figure 1-1 Diagram showing a single and a double wave screen breakwater (AUsop, 1995)

A wave screen breakwater can have multiple layers of screens, with each screen having a

different porosity. The larger porosity screen is generally placed on the seaward side of

the structure as larger porosities decrease reflection. The second screen generdy has a

srnalier porosity than the first screen in order to minimize transmission. An impermeable

waIl c m be included within the system to eliminate ail transrnission but this increases

reflection. The screens can extend through the fidl dtpth of water or have ody a limited

depth of projection into the water. The slats can be aligned at any angle but are usuaUy

horizontal or vertical for ease of construction. The versatility of wave screens is one of

their greates t features.

Wave screens combine the small footprint nature of a vertical caisson with the partial

transmission characteristics of a pile breakwater. They foilow the same concept as a pile

breakwater but require far fewer piles and are easier to construct. Allowing transmission

through the breakwater reduces the reflection of energy from the fiont face of the screen.

Dissipation within the system reduces the energy passing into the harbour. Furthemore,

wave energy within the harbour can be dissipated as it interacts with the wave screen

from the harbour side, Wave screens still allow water circulation and sediment transport

to occur so that natural processes can stilI occur, although at reduced rates.

At present, not much is known about the performance characteristics of wave screens. In

general, site specific hydraulic mode1 studies must be conducted to quant@ the effect of

different screen configurations to maximize their usefuhess. Very few guidelines are

available to act as a starting point.

Thus, the goals of this research were:

1. to investigate the influences of wave height, wave period and water depth on the

wave transmission coefficient, Kt;

2. to investigate the influences of screen porosity, orientation and spacing (gap)

between multiple screens on Kt, and;

3. to develop practical empirical equations to predict Kt through single, double and

triple screen systems.

A two-dimensional hydraulic mode1 study was performed to fulfil these goals. The study

incorporated combinations of screen orientations, porosities, and gap sizes for single,

double and triple screen systems. Tests were completed for a variety of wave heights,

wave periods and water depths using irregular wave conditions. Screen slat size was

sumrnarily examined but was omitted from the focus of the research because it would

have greatly decreased the extent to which other variables could have been investigated.

Furthemore, it was decided to center this research on transmission through a full depth,

emergent screen system.

A dimensional analysis was undertaken to determine dimensionless relationships for the

physical processes that occurred.

A parametric analysis was performed to identify trends within the data and the influence

of individuai parameters. Combining the parametric and dimensional analyses

highlighted which dimensiodess variables should be included in the statistical analysis.

A multiple linear regression analysis was then used to develop a series of empincal

equations that cm predict Kt for various screen systerns.

In this thesis, a literature review in chapter 2 outiines what information is available

regarding previous research on wave screens. The equipment and procedures used for

testing are discussed in chapter 3. This is followed by the parametric analysis, chapter 4,

which qualitatively discusses the infIuence of individual parameters on wave

transmission. Chapter 5 contains the results of a statistical anaiysis used to develop the

empirical equations. Conclusions and recommendations are presented in chapter 6.

2 Literature Review

2.1 Basic Equations

This thesis examines the transmission of wave ener,c.y through a wave screen. The most

common rnethod of quan-ing wave transmission is with the transmission coefficient,

Kt, where Kt is defined as a ratio of the transmitted wave height to the incident wave

height.

where HI = incident wave height

HT = transrnitted wave height

The primary purpose of a wave screen is to reduce the wave height entering a sheltered

area while minimizing the size of the reflected waves. An d-encompassing approach to

exarnining the effectiveness of wave screens is to look at the amount of wave energy they

reflect, transmit or dissipate. The wave energy density can be calcutated using:

where E = energy density

p = density of the fluid

g = gravitational constant

H = wave height

The general energy balance can then be stated as follows:

where Et = incident wave energy

ER = reflected wave energy

ED = dissipated wave energy

ET = transrnitted wave energy

This chapter discusses previous studies that have documented the performance of wave

screens by studying the transmission coefficient.

2.2 Influence of PhysicaI Variables

2.2.1 Single Screen System

The first variable considered when building a wave screen is the porosity, P, cf the

screen. The porosity is defined as the ratio of the gap between slats (s) to the centre to

centre distance of adjacent slats (e). The complement of porosity is the wall element

ratio, W, which is the ratio of the impermeable screen area to the total screen area.

Figure 2- 1 graphically describes this.

Wave Direction

I - P = s/e W = b/e

Screen axis

IndMdual Screen Slat r-@-

Figure 2-1 Notation for wave screen porosiw

Grune and Kohlhase (1974) showed that increasing the wall element ratio (decreasing

porosity) decreases transmission while increasing reflection. They tested screens with a

wail element rztio ranging fiom 0.25 to 0.6 while varying wave steepness (defined as

wave height divided by waveiength, WL) from 0.025 to 0.067. The results c m b e seen in

figure 2-2 and show that screen porosity is one of the key variables to be studied.

O ~ ~ r r s n r a l V r HAYASHI c3f

Figure 2-2 Kt vs Wall Element Ratio taken from Gmne and Kohlhase (1974)

Slat shape is another variable to be considered when designîng a wave screen. Generally,

two basic options are used; a rectangular or a circular cross-section. Wiegel(1961) and

Hayashi et al (1966, 1968) provided results for transmission through vertical screens with

circiùar cross-sections (in essence a pile breakwater). Their results are included in figure

2-2 and also illustrate that wave transmission is a function of wave steepness and wall

element ratio (or porosity). Wiegel developed an equation for wave transmission through

a pile breakwater that was based solely on the wail element ratio. However, this equation

is not applicable for screens with slats of rectangular cross-sections. The advantages of

wave screen breakwaters over pile breakwaters have already been noted in chapter 1.

Wiegel(1961) also developed an equation for vertical slotted wave barriers with

rectangular slats. He estimated the transmission coefficient by assuming that the

transmitted wave energy flux over the width (s+b) was equal to the portion of the incident

wave energy flux passing through the gap width S. He then postulated equation

Cs-41 -

This approach underestimates wave transmission because it ignores the effect of wave

reflection. Reflection must be accounted for since the presence of reflected waves

increases the pressure gradient across the screen causing increased iransmission.

Hartmann (1969) denved a similar formula for a wave dissipator composed of wire mesh

screens by means of the energy trânsferabiIity method. He obtained equation [2-51, but

this generalized equation contains only the wall element ratio, W, which again restricts its

predictive abilities, because it ignores the influence of wave reflection and other variables

on Kt.

Mei (1983) examined scattering by a slotted or perforated breakwater and derived a

quasi-theoretical formula based on wave steepness, porosity and a contraction coefficient,

Cc, as given below.

where Cc = 0.6 + 0 . 4 ~ ~

KI, = ((~/C~P)-I)'

= deepwater wavelength

This equation has a theoretical basis (Daily and Harleman, 1973) but relies on an

empirical contraction coefficient, Cc, for a sharp edged orifice. Cc can Vary between 0.6

and 1 depending on the nature of the slat edge. The difference in Kt when using either

value is no more than about 0.2 (Bennett et al, 1992). The possible effect of slat

thickness on Cc was not discussed in any of the three papers.

Gmne and Kohlhase (1974) showed that slat thickness has a rninor impact on Kt, stating

that as the slat thickness was increased, Kt decreased. They tested six rectangular slat

shapes with b/t ratios varying f'rom 0.66 to 10. Each test series was comprised of 12 runs,

each with a different wave condition where wave height was varied between 4 and 14cm

and the wave period between 0.7 and 1.7 seconds. They found the influence of thickness

was proportional to the wall fiction, so as porosity was decreased this trend strengthened.

However, they concluded that the effect of slat thickness on Kt was minimal.

Gardner et ai (1986) agreed that slat thickness had limited impact on Kt. They performed

two-dimensional model tests in a 45m long flume at a 1: 15 scale. They tested two

screens, one with twice the slat thickness of the other (30cm vs 15cm) while varying the

wave period between 3 and 12 seconds and found that the screens gave similar hydraulic

performances- Based on a review of previous research, it was decided not to investigate

screen slat thickness during this thesis and focus on other variables such as screen

porosity, orientation and gap space.

Muramaki et al (1986) performed a hydraulic model investigation of irregular slat shapes

in a two-dimensional flume. They investigated seven shapes including a rectangular slat

shape. Water depth and wave period were kept constant at 15cm and 0.75 seconds

respectively, while wave steepness was varied from 0.01 to 0.05. They used a very small

width/thickness ratio (b't) of 0.25 for the rectangdar slats used during their tests but they

still found that, of aU the shapes tested, the rectangular section had the smallest Kt but the

highest reflection coefficient, Kr. A screen porosity of 12% was used. In protoîype, most

screens are composed of rectangular slats for practical reasons and since this shape also

appeared to give the lowest values of Kt in previous research, rectangular slats were used

throughout this experimental program.

Surprisingly little information was available comparing vertical slat orientation versus

horizontal slat orientation. Gardner and Townend (1988) claimed that there was no

obvious hydraulic advantage between the two orientations and that the choice shodd be

made on sû-ucîural grounds. It was decided to make orientation a focal point of the

current research because of the lack of available design criteria regarding this variable.

No information could be found regarding slat orientation being at an angle, other than

90°, to the horizontai.

Another physical variable for single wave screens was the height or depth to which they

extend above or below the water surface. Figure 2-3 graphically descnbes the

nomenclature for wave screen height. A full depth, emergent screen allows no

overtopping and extends the full depth, d. A partial depth, emergent screen ailows no

overtopping but only extends a distance db from the water surface. Common reasons for

partial depth screens to be utilized are the cost considerations and enhanced bottom

circulation. A submerged screen extends from the bottom to a height, d,. Submerged

screens are generally used for aesthetic reasons.

Full Depth Partial Depth Submergent Emergent bergent

Figure 2-3 Schernatic for nomenclature of wave screen height

Wave overtopping can result in an increase in Kt as well as increasing the complexity of

an empincal transmission prediction (Goda, 1985). An ernergent screen, which lirnits

overtopping, is thus preferable for reducing Kt.

Isaacson et al (1998) and Gardner et ai (1986) showed experimentally that partial depth

screens have a greater Kt than full depth screens, as is to be expected. Isaacson et ai used

a 20m long flume to perform a two-dimensional hydraulic test using ddd values of 0.5

and I with screen porosities of 5, 10, 20,30,40 and 50%. Each slat was 2.0cm wide and

1.3cm thick. The water depth was held constant at 45cm, while the wave period was

varied from 0.6 to 1.4 seconds. Wave height was altered so that wave steepness was heId

constant at 0.07. They also examined Kt for wave periods of 1.0 and 1.6 seconds and

varied wave height to give wave steepness values of 0.02,0.04,0.07 and 0.09. The

findings showed that at a porosity of 10 and 2096, Kt values for a fulI depth screen were

about 0.6 and 0.65 respectively, but 0.76 and 0.75 for a screen extending to only half the

depth. Isaacson et al also found that Kt decreased with an increase in wave steepness.

14

Knebel and Bollmam (1996) compared three theoretical methods (power transmission

theory, eigenfunction expansion and the modified power transmission theory) for

predicting wave transmission through a vertical wave b d e r and the findings of Isaacson

et al (1998) matched those of Knebel and Bollmann. The power transmission theory is

based on the presumption that wave motion behind the wall is related to wave power

transmission below the wali. The modified power transmission theory uses the same

approach, but the effects of wave reflection are considered. The eigenfunction expansion

method involves solving for the velocity potentials on both sides of the wall and then

matching them at the location of the wall. Kriebel compared these models with tests

conducted in a 37m long, 2.4m wide, and 1.5m deep wave tank. He used four ddd

values, 0.4,0.5,0.6 and 0.7, during the 80 tests run. The wave periods were varied fkom

0.9 to 2.5 seconds with wave heights ranging from 2.5cm to 23cm resulting in wave

steepness values between 0.01 and 0.06. Both the theory and experimental data showed

that Kt decreased with an increase in ddd.

Clauss and Habel (1999) investigated submerged wave screens and found that they allow

greate transrnission than a full depth screen. They used screens with 5, 11,20 and 27%

porosities with dJd ranging from 0.3 to 1.2. They were tested for wave periods ranging

from 1 to 12 seconds and wave heights ranging from 0.1 to 1.5m in an 80m long, 4m

wide two-dimensional wave flume. Results showed that at a dJd value of 0.4, screen

porosity makes little difference as a 5% porosity screen allows a Kt of 0.95 while a 50%

porosity screen allows a Kt of 0.98. At a dJd value of 1 however, a 5% porosity screen

allows a Kt of only 0.08 (minor overtopping) while the 50% porosity screen still had a Kt

of 0.92.

It was decided to focus on full depth, emergent screens. There were several reasons for

choosing this particular type of wave screen. Firsf of the three types of wave screens

described here, the full depth ernergent screen allows the least energy transmission.

Second, it is the most commonly used style of wave screen, maînly because of its lower

Kt relative to the other two styles. Third, the bottom of a partiai depth screen or the top

of a submerged screen would have had to be supported to avoid rocking. A support

system would have influenced Kt whereas, with the full depth screen, the screen could be

fastened into a railing on the floor so that the support structure had a negligible impact.

Fourth, a square screen would reduce the number of screens that had to be consîructed

because one screen could be tested in both a horizontal and verticai orientation. Since the

fiume was 1. lm wide, but the highest practicd water depth was 0.9rn, an emergent screen

was the most convenient.

2.2.2 Multiple Screen S ystem

A multiple screen system is obviously more cornplex than a single screen system because

the reflected wave from the second screen interacts with the incident wave that has just

passed through the first screen. The reflected wave also changes the pressure gradient

behind the first screen, which directly impacts wave transmission through that screen.

AU previous research has indicated that multiple screens perform better than a single

screen. However, there are very few equations developed for energy transmission

through a double screen system- Kondo (1979) presented a complex series of theoretical

equations to predict Kt for screens with circular holes. The equations form the bais for

an iterative procedure based on gap space, porosity, hole diameter, wave height and wave

period. Kondo benchmarked the results against two-dimensional tests performed by

other researchers. Figure 2-4 gives a cornparison of Kondo's theory and the experimental

results. The tests were performed in a two-dimensional flume. Holes were drilled in a

steel plate that was 0.6cm thick. For each hole size (2.0cm and 1-2cm) there was a 19,20

and 34% porosity plate. The equation and actual Kt results differ by as much as 0.2.

Thus, a prirnary goal of this research was to develop a double and triple screen empirical

Kt prediction eqiation that closely matched the actual Kt.

' 0.41 Poroua va- Expet . Theory

RELATIVE WIDTH; G/L

Figure 2-4 Actual vs predicted Kt for a double screen system (Kondo, 1979)

While a multi-layer system cannot be regarded as two individual screen systems, the

same physical variables, such as porosity, height, and thickness, stiLl apply for each

screen. Jarnieson and Mansard (1987) performed hydraulic tests in a 30.5111 long, l m

17

wide and 2.3m deep wave Bume to investigate minimizing reflection in a wave basin,

using multiple wave screens. The wave screens were perforated, expanded metal sheets,

ranging in thickness from 1.2 to 3.2mm, and had a louvered design. The porosity of the

sheets ranged from 5 to 40%. They showed that the spacing of the screens required to

achieve high levels of wave energy dissipation is related to the horizontal displacements

of water particle motion. So, wider spacing is required for higher wave heights and

longer periods. They found that optimum spacing is associated with a progressive

decrease in spacing of the sheets towards the rear of the absorber. They further noted that

it was important for the waves to have sufficient space for energy dissipation by

turbulence before the next screen is encountered, to avoid excessive reflection.

Generally, the screen closest to the incoming wave has the Iargest porosity with each

subsequent screen having a progressively smaller porosity, as this reduces both the

reflection and wave force for each screen.

Most previous works have described the spacing between multiple screens in terms of the

dimensionless parameter, GfL, the gap divided by the local wavelength of the incident

wave. Kondo (1979) used this term as shown in figure 2-4. This graph shows no distinct

trend in Kt when G L is increased. Cox et al (1998) conducted two-dimensional tests in a

32m long, lm wide and 1.2m deep wave flume with screen porosity ranging from 10 to

30% and wave steepness from 0.02 to 0.10, for partial depth screens. They presented

analysis showing that GiL was an important factor in reducing Kt as they varied G/L

from 0.1 to 0.3. However, Cox et al found that increasing the depth of the screen was

more influentid than increasing the gap between screens. Neither Jarnieson and

Mansard, Kondo nor Cox et al gave a relationship for Kt with respect to G L G/L was

included in this research because so iittle information was available-

2.3 Influence of Wave Climate

Wave height, wave period and water depth can provide a lirnited definition of the wave

clirnate. Wave height can affect Kt since the larger the wave, the larger the area of the

screen that is coatacted. The larger contact area results in greater friction and thus

dissipation, so bigger waves are expected to have a lower Kt. However, wave height can

be very srnall compared to the water depth, and since wave energy is distributed over the

full depth, a relatively small amount of the total wave energy is within the water surface

fluctuation range. Thus, while wave height is an important factor affecting Kt, it is not

the only one.

Wave period is directly linked to wave celerity, thus a large penod wave has a

correspondingly large celerity. This resuIts in higher water velocities passing through the

screen and higher associated friction and turbulence. However, it is the combination of

wave penod and wave height that wili determine the actual water velocity and pressure

build up on one side of the screen. These parameters can be combined through wave

steepness, HL. Al1 previous work reviewed discussed wave height and wave period in

terms of wave steepness, not as independent variables.

Water depth was found to have a negligible effect by Hartmann (1969). He used the

term, d/L, to explain the influence of depth. This implies that bottom friction does not

have a signincant influence on Kt, making depth unimportant. Jamieson and Mansard

(1987) investigated depth, varying it between 1 and 2m, but found little change in Kt.

Grune and Kohlhase (1974) conducted al1 their tests at a constant depth of 35crn, as did

Isaacson et al (1998) who maintained the depth at 45cm during his tests. Gardner et al

(1986) did not mention at what depth they performed their tests. It was decided to

examine relative depth, Wd, in the parametric analysis and see if it should be included as

a variable in the statistical analysis.

Herbich (1990) noted that wave transmission for monochromatic waves decreased as

d / g ~ 2 was increased for pile breakwaters. Herbich analyzed data collected by Tmitt and

Herbich (1986) which were based on tests conducted in a two-dimensional, 37m long,

0.6m wide and 0.6m deep wave flume. Wave penod was varied between 0.5 and 2-0

seconds for single and double piie breakwaters. Tests were run at depths of 4 l , 5 l and

61cm As can be clearly seen in figure 2-5, cU~T' had a si@~cant influence on Kt. It

was thus decided to include d/gg~' in the list of variables investigated in the present

research.

PILE DIAMETER: 1-3116 INCH b - =0.1 0

A 1 ROW OF PILES h 2 ROWS OF PILES

1 ROW A

Figure 2-5 Kt vs d / g ~ 2 for one and two rows of piles

Wave steepness appears to be one of the dominant wave climate variables when

examining transmission through a wave screen. It incorporates wave height, wave period

and water depth in a dimensionless fonn. Steeper waves create a larger pressure

differentid across the screen compared to less steep waves. This in turn creates larger

water velocities passing through the gaps in the screen. Since energy loss due to flow

separation is proportional to the square of the flow velocity through the opening (Kakuno

et al, 1992), larger velocities result in higher flow resistance and turbulence. The net

result is that there is greater energy dissipation and thus a lower Kt with steeper waves.

Figure 2-6 graphicaliy illustrates this point.

( large veiocities

Figure 2-6 Graphic cornparisons of large and smali wave steepnesses (Bergmann and Oumeraci, 1998)

f i e b e l (1992) stated that higher wave steepnesses resulted in a lower value of Kt. He

also noted that for low wave steepness waves in deepwater, Kt may stiU be as high as 0.7

or 0.8 for wail porosities of only 0.15. Isaacson et al (1998) also stated that increased

wave steepness Leads to a reduction in Kt. Al1 previous research points to wave steepness

being an influentid variable on Kt and so it was included in the current analysis.

An increase in the incident wave angle has been shown, by Grune and Kohihase (1974),

to reduce Kt but increase Kr. Figure 2-7 shows the notation they used to describe the

incident angle wave approach to a wave screen.

direction of wave advance O 0 a

wave crests CJ screen

/ O

/ O

Figure 2-7 Notation for wave approach

Gmne and Kohlhase exarnined four incident wave angles during their tests: O", 4S0, 67.5'

and 90". From the results, they developed the semi-empuical equation given below.

where Ktp = transmission coefficient for any incident wave direction p

Kto = transmission coefficient for the wave direction p=OO

a = empirical shape coefficient of the wall element

It should be noted that even with an incident wave angle of 90°, Kt is about half that of an

incident wave angle of O". Grune and Kohlhase explained that this was due to

diffraction. The angle of wave incidence could not be incorporated into the experirnental

work described in this thesis since the tests were conducted in a two-dimensiond wave

flume.

2.4 Wave Pressure and Force

The design and construction of a functiond wave screen must include consideration of

wave pressure and the associated force on the screen. While this thesis only investigates

the prediction of Kt, a discussion of the forces and pressures on a screen was deemed

important,

The normal methods of predicting pressure @oth dynamic and hydrostatic) on a vertical

wall cannot be used to predict the pressure on a wave screen. Since there is water on the

leeside of the screen, the hydrostatic pressure acting on a wave screen is lower than that

for a retaining waLl. Hydrostatic pressure is present because the water levels on either

side of the screen are diEerent due to differences in phase and wave height between the

incident and transmitted waves. The equation for hydrostatic pressure on a wave screen

is given in equation [2-81.

where ph = hydrostatic pressure

g = gravitational constant

Ad = instantaneous difference in depth between the front and back of the screen

p = density of fluid

Wave screens are subjected to dynamic pressures due to the water particle velocities of

the waves. Kriebel(1992) developed an expression for the dynamic pressure drop across

a vertical wave screen.

where Cc = contraction coefficient = 0.6 + 0 . 4 ~ ~

d = water depth

k = wave number = 2TdL

pi = dynamic pressure due to incident wave = '/i p g Hi Z,

p, = dynamic pressure due to reflected wave = '/i p g H, Zp

p, = dynamic pressure due to transmitted wave = H p g H, Z,

ut = velocity of transmitted wave at given depth

Z, = cosh(kh+kd) / cosh (kh)

This equation was based upon the assumption that wave transmission and wave forces are

deterrnined by the horizontal fluid velocities through the breakwater gaps. The maximum

pressure gradient can then be associated with the maximum fluid velocities, and these

occur at the wave crest and trough. Integrating this expression from the seafloor to the

still water surface (because equation [2-91 estimates the pressure drop across the wall in a

thui horizontal layer) yields the total force on one wall element as shown in equation

[2- 1 O].

F = p g (H,/k) tanh (kd) (1-Kt) (stb)

where b = width of slat

F = total force on one vertical wall element

s = space between slats

So, the force on a wall depends on wall porosity and wave steepness because these affect

Kt. Equation [2-101 can be reduced to the full depth, vertical wall equation (Kriebel et al,

1998). This can be clearly seen as the (1-Kt) term disappears, as does the (s+b) tem, for

a full depth vertical wall, leaving:

f iebe l also found that the maximum force on a wave screen can be reduced to the

standing wave solution for a solid wall.

For waves approaching at an angle, the United States Army Corps of Engineers (1984)

States the force rnay be reduced by:

where F' = reduced dynamic force

p = incident wave angle

However, equation [2-121 should only be applied to dynamic wave force components of

breaking or broken waves and should not be applied to the hydrostatic cornponent.

The bulk of the experimental work previously undertaken has been completed using

regular waves and lïmited test conditions.

Grune and Kohlhase (1974) showed that increasing the wall element ratio (decreasing

porosity) decreases transmission while increasing reflection. Slat shape has a rninor

impact on wave transmission as show by Wiegel(1961), Hayashi et al (1966, 1968) and

Muramaki et al (1986). Slat thickness also has a limited impact on Kt as shown by Grune

and Kohlbase (1974) and Gardner et al (1986, 1988). Gardner et al (1986), Cox et al

(1998) and Isaacson et al (1998) showed that the greater the depth to which the screen

extends into the water the less the wave energy transmission. Clauss and Habel (1999)

fuaher showed that submerged screens allow greater h.ansrnission than emergent screens.

Multiple screen systems were found to be more effective at reducing Kt than single

screen systems (Kondo (1979) and Jamieson and Mansard (1987)). Very little data was

available regarding the effect of the gap space on Kt.

AU previous work accounted for the wave height and wave period variables through wave

steepness. It was found that as wave steepness increases, Kt decreases. Depth was found

to little influence on Kt (Hartmann (19691, Jamieson and Mansard (1987), Tnritt and

Herbich (1986)).

3 Experirnental Eaui~ment and Procedure

3.1 Introduction

This chapter describes the test procedure and equipment used to physically mode1 and

analyze wave transmission through wave screen breakwaters. The research was

conducted at the Queen's University Coastal Engineering Research Laboratory,

Department of Civil Engineering, Queen's University, Kingston, Ontario, during the

period spanning April1999 to September 1999.

3.2 Wave Flume

The wave screen experiments were perfomed in a 2m wide, 45m long wave flume. The

total depth of the flume was 1.2m, but the water depth was varied between 0.7m and

0.9m during the tests. The flume had 6 large observation windows measuring 1.2m high

by 0.8m wide in the test section. Figure 3-1 shows the flume layout.

At the end of the flume an absorbing beach was constructed consisting of a plywood

ramp at a slope of approximately 1: 10. The plywood ramp was covered with a thin layer

of small Stone and then two layers of rubberized horsehair matting which helped to

absorb wave energy and reduce reflection. The average reflection was found to be Iess

than 5% of the incident wave energy and was neglected during analysis.

The flume contained a divider that bisected the flume dong its length and split the test

area in half. This reduced the required width of the wave screens. The divider extended

from approxhately 15m in front of the test area to the end of the flume. It was 6mm

(1/4') thick and caused negligible distortion to the incident wave. The divider also helped

to reduce secondary wave reflections off of the wave paddle, which could have

infiuenced the results.

1 Side View 1 Probe Rack 1 Probe Rack 2

7 -

Wave Generator / Observation 1: 10 Beach Wave Screen ' Windows

1 Plan View

Divider 1 T

l . lm 00.0 0 -0 a O

L t L T d

Observation Windows

Figure 3-1 Wave flurne schematic (not to scale)

3.3 Wave Generator

The wave generator was a Kempf and Remmer bottom-hinged, flapper-type wave paddle

and was capable of creating both regular and irregular waves. The piston had a working

stroke of lOcm resulting in 24' of motion of the paddle. The paddIe followed command

30

signds in the fiequency range of 0.1 Hertz to 3.0 Hertz. Calibration of the paddle

showed that the padde was capable of creating waves ranging fiom 2.5cm to lOcm with

a frequency range of 0.4Hertz to lSHertz, corresponding to periods of 2.5 to 0.7 seconds-

3.4 Wave Generation

Wave û-ains were generated for 33 different wave height, wave penod and water depth

combinations. A list of target wave characteristics is given in table 3-1. OnIy irregular

waves were generated, as there was concern about resonance occumng within the flume

with regular waves.

Wave signal generation was accomplished using GEDAP Version 2.0 Time Control

Table 3-1 Table showing target wave train characteristics

Wave Generation Package (Miles, 1990). The package consisted of three modules. The

Depth (cm)

70

80

90

fust module, called PARSPEC, generated the wave spectmm. This spectrum was based

upon a target significant wave height, depth, spectrum type, spectrum peak frequency and

Wave Period (s) 1

1-5 2

-75 1

1.25 1.5 1.75 2

2.25 1

1-5 2

Wave Height (cm) 395

3 ,5 ,7 3,597

3 39597 5 ,9

3-5, 7 - 9 5

395,799 5

3959 7 3 ,5 ,7 3,597

peakedness factor. A JONSWAP spectrum was used with a peakedness value, y, of 3.3

and a Phillips constant, a, of 0.0081. The next module, IREG-WAVESYNJKG

synthesized the wave train for this spectrum using an inverse Fourier tramform. A

sequence of 200 waves was used to determine the cycle duration of the synthesized

signal. The final stage was to generate a voltage signal for the paddle. A wave machine

calibration file, WMCAL, was used to enter the specific characteristics of the paddle that

the module RWREP2 used to determine the voltage signal sent to the paddle.

Calibration of the paddle involved measuring the stroke, or displacement, of the paddle

when subjected to a standard calibration signal of + 5 volts. GEDAP determined the

stroke and stroke rate (voltage and s1ew rate) required to generate the desired wave train

based on this data as well as the depth of water at the paddle and the gap between the

bottom of the paddle and the floor. A full description of the GEDAP system and

appropriate calibration procedures is documented in Pelletier (1990).

The waves initially generated by the paddle did not exactly match the desired wave

characteristics. To account for this discrepancy, each signal was altered using an

amplification factor. The arnplification factor was continually modified until the actual

wave height and wave period recorded in the flume were within 5% of the target

spectrum, which was deemed acceptabIe for the purposes of this research. While all

discussions regarding wave conditions refer to the target conditions presented in table 3-1

the analysis was performed using the actual data acquired during individual tests.

3.5 Wave Probes

Each water surface time series was rneasured using ten capacitance-type wave probes.

The wave probe consisted of a 30cm (12") bras bow with a thinly insulated wire

stretched between its two ends. Figure 3-2 shows a schematic of a wave probe. The

water closed the circuit between the wire wrap, so as the water level changed the

capacitance within the wire wrap also varied. A water-resistant transducer box converted

this resistance reading into a current that was proportional to the resistance. The current

was then converted to a voltage signal and amplified. The data was then coliected by an

analog/di,oital converter using LabTech Notebook control software (Laboratory

Technologies Corporation, 1991). Finally, the voltage signal was converted to a water

level by cross-referencing a calibration N e using linear interpolation. The probes were

calibrated daily at 3 different water levels to account for variations in temperature or

water level changes. The probes responded linearly with errors less than 1.5% (Weigert

and Edwards, 1981) and a correlation coefficient of 0.995 and greater was deemed

acceptable for calibration. The resolution of the probe was better than lmrn, which was

less than 4% of the smallest incident wave height tested and 1% of the highest wave

tes ted.

Probe Bow 0 0 4

Probe Transducar Box

Iasulatuig Grornet

insuiated Wire

Water Level

Bras Reference Post

Figure 3-2 Wave probe schernatic

The probes were placed in two arrays of five probes each, one array set in fiont of the

screen and the other array set behind the screen. The probes within each array were

spaced 18cm, 40cm, 66cm, and 125cm away from the f ~ s t probe in each array. This is

the standard probe spacing required by GEDAP to separate incident and reflected wave

trains. This separation was cornpleted using GEDAP's refiection analysis which

performed a least squares analysis (Mansard and Funke, 1980, 1987) using data from

three of the five probes. A cross-spectra between probes one and two and probes one and

three was obtained using the Welch Method, which was then used to separate the incident

and reflected spectra.

3.6 Data Acquisition and Analysis

Data acquisition was controlled with the software package LabTech Notebook. This was

a real time controi @TC) program that provided an interface between the user and the

instrumentation, via a Metrabyte DAS 8 analog to digital converter. The sampling rate

was set at 20Hz while the sampling period was based on the duration of 200 waves. The

discretized water levels for each probe were output as an ASCII file.

The data was analyzed using the National Research Council of Canada's GEDAP Version

6.0 software package, but the ASCII file first had to be converted to a GEDAP

compatible format. This simply involved removing the end of file marker that LabTech

Notebook generated in the last Line of the ASCII file.

A spectral analysis for each wave probe record was undertaken using GEDAP. The

incident and reflected wave spectra were then separated as discussed in section 3.5. Thus

the first probe rack, located in front of the wave screen, measured the incident wave

conditions, HI and Ti, while the second rack, situated behind the wave screen, was used to

determine the transmitted wave conditions, HT and TT. It should be noted here that a

spectral analysis yields a zero moment wave height, not the significant wave height.

The transmission coefficient, Kt, is defmed as:

where HI = incident zero-moment wave height

HT = transmitted zero-moment wave height

The deepwater wavelength was then calculated using equation D-21.

where g = gravitational constant

= deepwater wavelength

T = peak incident wave period

This equation cannot be used for waves in intermediate depth water (0.05 < d/Lo < 0.5).

In this case, the local wavelength was approximated using Hunt's formula (Hunt, 1979).

Hunt's was used as it calculates Iocal wavelength with an accuracy of El%. Hunt's

formula is defined as:

and

where d = depth

L = local wavelength

3.7 Wave Screen Construction

The wave screens were constmcted fiom wood (#1 grade-pine)- Six screens were

constructed, with 4 different porosities, 20%, 30%, 40% and 50%. Porosity is defined as

the percent open area divided by the total area of the screen. Each screen measured

1.13m (44%') by 1.13m (44%') as a square screen allowed a screen to be used in both a

horizontal and vertical orientation, which is graphically depicted in figure 3-3.

Horizontal

Figure 3-3 Schematic showing horizontal and vertical orientations and screen spacing

Three different slat sizes were used, Table 3-2 summarizes the porosities and slat size

used for the six screens.

Table 3-2 Surnrnary of wave screen porosities and slat sizes

Porosity 20% 30%

30% (a) 30% (b)

40% 50%

Slat size (cm) 2.5 x 1.3 2.5 x 1.3 2.5 x 1.3 1.7 x 1.7 2.5 x 1.3 2.5 x 1.7

Slat spacing (cm) 0.64 1.1 1.1 0.73 1.7 2.5 .

Three screens were made with 30% porosity. Two screens (30 and 30a) were identical so

that two similar screens could be tested in combination. The third screen (30b) was made

with a M e r e n t slat size so that the effect of slat size could be evaluated at a constant

porosity.

A brief description of the wave screen coding system is given here. The screen setup is

first described by porosity: 20, 30,30b, 40 and 50- The merence between 30 and 30b is

described above. The orientation of the screen is then given as either vertical, V, or

horizontal, H. If there are two screens, then the size of the gap (in cm) between the

screens is located between the description of the two screens. The screen closest to the

incoming wave is listed first. So 40H : 59.5 : 30V denotes a two screen system with a

40% screen oriented with horizontal slats located closest to the wave machine. There is

then a 59.5cm gap followed by the 30% screen oriented in a vertical direction.

The 50 screen was the fxst screen constructed. The slats were cut from 3.8cm x 2Scm x

305cm, #2 grade-pine. There were only two plywood fiarnirig pieces each 6.4cm wide

and 1.3cm thick. While the paint did protect the wood for the limited time the screen was

in the water, there was concem that the plywood might swell excessively causing

buckling of the slats. Thus, the fiaming pieces for the other five screens were made of

pine. Furthermore, since the framing pieces for the 50 screen were over 11% of the width

of the fiurne, it was decided to reduce the frarning piece width for the remaining screens

to decrease their intrusion. The Iast alteration for the construction of the remaining

screens was to increase the number of frarning pieces to four, to reduce flexing of the

screens. Each framing piece was less thick (1.3cm) than those for the 50 screen (19cm),

although the overail thickaess was greater (2.6cm compared to 1 -9cm).

For the 20,30, and 40 screens, the slats were cut fiom 1.3cm x 15.2cm x 243.8cm pine

boards. Each slat was predrilled at both ends to prevent splitting when they were screwed

into fiamhg pieces. Due to flexing and buckling concems, four framing pieces were

used to form the supporting structure for the slats in each screen. The fiaming pieces

were 3.8cm wide. This was less than 7% of the total width of the flume and was ignored

when analyzing the results. The slats were sandwiched between the two sets of fiaming

pieces discussed earlier. The slats were both glued and screwed into the frame. Finally,

the screens were painted with marine paint to prolong their life. Pictures of these screens

can be seen in Appendix A. The 30b screen was prepared sirnilarly to the previous ones

except the slats were cut from 1.7cm x 15.2cm x 243.8cm pine boards.

The screens were fixed at the top and floor of the flume. Brackets, consisting of two

2.5cm L-shaped steel rails, were screwed into the concrete fioor. The bottoms of the

screens were then wedged into these brackets, using wooden wedges to stop shifting if

necessary. The top of the screen was clamped to 2.5cm thick plywood using 16cm "C"

clamps. The plywood was, in tum, h n l y fastened to the fiume waU and divider. Five

brackets were prepared with gaps of 40.7cm, 59.5cm, 49.8cm, and 25.Scm starting at the

first bracket. Figure 3-4 shows a vertical wave screen fastened in place in the wave

flume.

Figure 3-4 A picture of an installe.' ~ 2 ~ e screen with a vertical orientation

3.8 Dimensional Analysis

In order to develop a design relationship for Kt in terms of the physical processes

occurrïng at the wave screen, a dimensional analysis was performed. A dimensional

analysis considers al i of the variables affecting the transmission process and detemines

the physically relevant forms of these variables. In general, the transrnitted wave height,

HT> can be expressed as a function of a number of independent variables as follows:

where p = fluid viscosity

Q = density

However, it has already been stated that slat thickness, t, will only be investigated

qualitatively, and not quantitatively, and will therefore not be evaluated. Furthemore,

this research is being limited to a two-dimensional investigation so the incident wave

angle, p, cannot be evaluated either. Lastly, fluid viscosity and density wiil not be altered

during the test, as only water wiii be used as a fluid, and thus the influence of these two

variables wili not be investigated.

Kt has been previously defined as the ratio of transrnitted wave height, HT, to incident

wave height, HI. Therefore, HI is an obvious choice as a repeater variable using

conventional dimensional andysis techniques. The choice of the other two repeaters

should be based on making a i l the variables dimensioniess. There are several advantages

of a dimensionles s relationship over its dimensional counterpart according to Y alin

(197 l), including the dimensionless relationship having three fewer variables, the

numericd value not depending on a system of units and it is a correct version of a natural

law. The other two repeaters must contain a mass term and a time term. It was thus

decided to use wave period, T, and fluid viscosity, p, as repeaters. The terms in equation

[3-51 c m be made dimensionless by multiplying andor dividing by the repeater variables

leading to equations [3-6 1 and [3-71.

These dimensionless variables can now be combined to form comrnonly used variables.

So X7 can be divided by X3 to give (de) which defines porosity. X2 can be inverted to

give relative depth (Wd) as c m & to give wave steepness (HL). It is also possible to

create variables that are independent of wave height. & can be divided by X6 to give

G/L, which now relates gap width to wavelength instead of wave height. Finally, the

variable d / g ~ 2 can be created by dividing X by X5.

Therefore, equation [3-71 c m be rewritten as:

4.1 Introduction

A parametric analysis was used to study the effect of independent variables on the

transmission coefficient. This preliminary examination was stnctly quaiitative and only

involved describing general trends without placing values to them. A more quantitative

analysis is found in the next chapter, which descnbes the regïession analyses performed

to produce equations associated with the trends identified in this chapter.

This examination was perfonned by plotting the dependent variable, Kt, against a specific

independent variabIe, where di the other variables were held constant. This allowed the

effect of that particdar variable to be assessed.

The analysis discussed in this chapter is based on figures presented in appendices B

through J where the independent variables are: wave period, wave height, depth, screen

porosity, orientation, and gap size; and the dimensionless variables: wave steepness,

wave heighddepth, and gap size/wavelength. Trends visible when comparing different

graphs were not noted however, until the section regarding the particular variable was

discussed.

4.2 Influence of Wave Period

4.2.1 Single Screen

The single screen plots of Kt versus wave penod, shown in figures B 1 through B 10,

displayed a large scatter. There were few clear trends present with respect to penod. The

lack of a defining trend suggests that period had little influence on Kt or its effect is

highly variable. The literature review points to the latter. Striations between wave height

groupings (isolines of wave height stacked above each other) showed that wave height

could be an important factor afîecting Kt.


A trend between Kt and wave period was observed with the double screens (a slight

convex shape of results with the apex around Ms), as can be seen in figures B 11 to B48.

This suggests that period did have an effect on Kt so a statistical analysis was performed

to see if the influence of wave period was statistically significant (see chapter 5).

While the decrease in Kt afier the 2.5s peak is not large, it is consistent for each wave

height. About 80% of the double screen results displayed this trend. A sample using the

40H:59.5:30H plot is given below in figure 4-1. Notably, any screen system that had a

20% porosity screen at the back defied this trend and gave very scattered results as can be

seen in figures B 11 through B 14 and B29 through B32. Distinct striations within the data

set showed that wave height had an influence on Kt. The Iower wave heights had higher

Kt values.

There are several possible explanations for this trend. The most likely is that Kt is

dependent on the size of the slats and slat spacing within the screen- A 1.5s wave rnay

have been "tuned" to the particdar slat size and thus passed more easiiy through the

screen whereas waves with a different wave penod experienced greater interference.

This trend may be linked through wave steepness which will be discussed more

thouroughly in section 4.5. Tests with a different size slat (screen 30b) did show that slat

size makes a substantial difference to Kt (figures B5 and B6), but the results were so

scattered that the two test series were diff~cult to compare. Overall, the results fiom

screen 30b were inconclusive. Another possible cause is resonance within the fiume,

though this is unlikely because of the sloped beach which rninimized reflection.

0.00 025 0.50 0.75 1.00 1.25 1.50 1.75 2.00 225 2.5(

Wave Period (s)

Figure 4-1 Kt vs Wave Penod for double screen system differentiated by wave height


The relationship between Kt and wave period for triple screen systems also showed a

convex shape very similar to that found for the double screen systems, with the peak of

the c u v e at 1.5s. The 3cm-0.75s wave foUowed the general curved shape more closely

than with the double screen system. The triple screen graphs are shown in figures B49 to

Bol, wkde a sample is given in figure 4-2 below- Striations between wave height data

sets c m be clearly seen in thïs figure, suggesting wave height had a strong influence on

Kt.

ODO 025 050 0.75 1.00 125 150 1.75 2.00 225 2 3

Wave Period (s)

Figure 4-2 Kt vs Wave Period for triple screen system differentiated by wave height

4.3 Influence of Wave Height

4.3.1 SingleScreen

The variation of Kt with wave height for single screen results was notable; some plots

were scattered, some well grouped; some sloped downwards, others upwards while yet

others were horizontal; some were sûiated, sorne were not- This suggested that while

wave height was an important factor affecting Kt, no strong trend could be identified

when wave height was the lone parameter. Al1 the plots showing the variation of Kt with

wave height for the single screen systems are shown in appendix C, figures Cl through

Clo.


The double screen systems showed a distinct trend ihstrating that as wave height

increased, Kt decreased. This is consistent with the trends noted in section 4.2. Figures

CI 1 through C48 show the variation of Kt with wave height for double screen systems. It

was decided to include wave height in the statistical analysis so as to develop a

quantitative evaluation for this relationship.

Some striation was visible for the data sets separated by wave period. Figure 4-3, on the

following page, is a representative sample of the double screen results. Note that the 1.5

second data set had the highest average Kt, which corresponded to the apex of the convex

curve for the penod graph. It was found that if the second screen had a 30% porosity,

those waves with periods less than 1.5 seconds resulted in lower Kt values. However, if

the second screen had a 20% porosity, it was found that wave periods greater than 1.5

seconds gave the lowest average Kt.

J

0.00 1 1 0.û 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 IO.(

Wsive Height (cm)

Figure 4-3 Kt vs Wave Height for double screen system differentiated by wave period


The triple screen system foilowed the trernds described above for the double screen

system. There was a distinct decrease in Kt with increasing wave height, with some

differentiation by wave period, showing that penod and wave height are important factors

affecting Kt. Figure 4-4 shows an exampue of the variation of Kt with wave height while

the remaining plots are included in appendix C, figures C49 to C6 1.

Wave He ight (cm)

Figure 4-4 Kt vs Wave Height for triple screen system differentiated by wave penod

4.4 Infïuence of Depth

4.4.1 Single Screen

There are distinct variations ia Kt with depth for the three different depths tested, as

shown in appendix D. Tests at a depth of 0.7m had the lowest values of Kt while those at

0.9m deep had the highest average value of Kt. There was considerable overlap between

the results, especially for deptfis of 0.8m and 0.9m. Friction frorn the sides and bottom of

the flume may have influenced the results, having a greater effect at Iower depths and

thus a greater influence on o v e r d Kt. Figure 4-5 gives an example of the variation in

results between depths while the complete set of single screen graphs is contained in

appendix D, figures D 1 to D IO.

The 30% screen had the smallest Kt range, varying from 0.6 to 0-8, while the dtered 30%

screen had the largest range, varying from 0.4 to 0.9. Most results exhibited a slight

decrease in Kt with an increase in wave height, as mentioned earlier in section 4.3.

20 Horizontal 1

0.0 1 I

0.0 1.0 2.0 3.0 4 8 5.0 6.0 7.0 813 9.0 10.1

Wave Height (cm)

Figure 4-5 Kt vs Wave Height for single screen system differentiated by depth

4-4.2 Double Screens

The double screen results were found to give differing resuIts compared with the single

screen results, as the 0.7m depth tests generaily had an equal or higher Kt value than the

0.8m results for all screen configurations. This may be due to the waves becoming more

agitated between the two screens allowing steeper waves at Iarger depths and thus

transmitting less. This served to cancel the bottom and side effects prominent in the

single screen analysis. Again a statistical analysis needed to be performed to distinguish

if this contradiction was statisticaIly significant or not. Fiame 4-6 shows a cornparison

between 0.7m and 0.8m only as there are no results for 0.9m depth for either a double or

triple screen system. The remaining graphs in this particular analysis set are figures Dl 1

to D24.

Wave Height (cm)

Figure 4-6 Kt vs Wave Height for a double screen system differentiated by depth

4.5 Muence of Wave Steepness

4.5.1 Single Screen

The literature review, presented in chapter 2, suggested that wave steepness was an

important factor affecting Kt. Steepness is defined as wave heightlwavelength and

intrinsically incorporates wave height, wave period and water depth. It proved to be a

good indicator, as there was a very defined trend showing that as steepness increased, Kt

decreased. This was apparent for all wave periods, as steeper waves encountered more of

the screen surface in a smaüer space of time resulting in greater reflection and thus lower

transmission. Furthemore, when steeper waves pass through the screen they create

Larger vortices on the back side due to a larger elevation head ciifference than flatter

waves, resulting in greater energy dissipation.

Including so many variables (wave height, wave period and water depth) withïn the one

dimensionless variable resulted in a more linear trend. Figure 4-7 illustrates this

tendency. The orientation of the screen also affècted the magnitude of Kt, as can be seen

in appendix E (figures E l to EIO), but the influence of screen orientation will be covered

more fblly in section 4.7.1.

20 Horizontal 1 .O

0.0 1 I 0.00 0.01 0.02 0.03 0.04 0.05 O.O(

Wave Steepness

Figure 4-7 Kt vs Wave Steepness for a single screen system differentiated by wave period

4.52 Double Screens

A sùnilar &end to that described in section 4.5.1 was found for the double screen system.

Figure 4-8 demonstrates that waves with a 2 second period had a higher Kt than those

with a 1 second period, which can be related to a lower wave steepness. Waves with a

penod of 1.5 seconds did not folIow the trend of lower Kt values, as noted in section

The orientation and spacing between screens altered the magnitude of Kt but did not

change the trend of decreasing Kt with increasing wave steepness. Appendix E contains

a l l of the double screen graphs (figures E l 1 through E48) showing wave steepness versus

Kt.

I

0.00 0.01 0.02 0.03 0.04 0.05 o.oe

Wave Steepness

Figure 4-8 Kt vs Wave Steepness for a double screen system differentiated by wave period


Similar to the double and single screens, the triple screen system had a decreased Kt with

increased wave steepness. There was some striation noticeable between 1, 1.5 and 2

second waves showing that Kt was dependent upon wave period, as was discussed in

section 4.2.3. Period is intrinsicdy incorporated within the wave steepness variable, as

smaller wave penods produce waves with smaller wavelengths and smaller wavelengths

produce steeper waves. Figure 4-9 is a representative sample of ail the triple screen

graphs given in appendix E, figures E49 through E6 1.

0.00 0.01 0.02 0.03 0.04 0.05 0.Ot

Wave Steepness

Figure 4-9 Kt vs Wave Steepness for a triple screen system differentiated by wave period

4.6 Influence of Porosity

It was readily apparent that porosity had a substantial influence on Kt. Figure 4-10 shows

distinct striations between the three screens of diffenng porosities, with the lower

porosity screens resulting in lower Kt values. Some overlap was prevalent between the

30% and 40% porosity screens. This banding and slight overlap was present for all

orientations at depths of 0.8m and 0.9m, as c m be seen in the remaining graphs in

appendix F, figures F1 through F6. However, at 0.7m depth the distinction between the 3

different screen porosities was less clear. The orientation of the screens affected the

magnitude of Kt but not the general trend. It should be noted that only the screens with

the same slat size were compared in this analysis.

Single Screen - Horizontal Depth = 0.8m

1 . 0 ,

0.0 4 1 OaO 0.01 0.02 0.03 0.04 0.05 0.Oc

Wave Steepness

Figure 4-10 Kt vs Wave Steepness for a single screen system differentiated by screen porosity


As expected, the screens with lower total porosity combinations aüowed less

transmission. It was found that 30-20 had the greatest wave height reduction followed by

40-20,30-30, 30-40 and finally 40-30. The 40-20 system was slightly superior to the 30-

30 system because steeper waves hpacted the 20% screen than the second 30% screen

aüowing less transmission. The Kt rnaopitude difference was not large because the first

30% screen allowed Iess transmission than the 40% but the net Kt result favoured the 40-

20 system. These results also showed that the order of the screens is important (40-30 vs

30-40). Figure 4-1 1 shows an example of how Kt varied for different screen systerns

while sirnilar graphs c m be found in appendix F, figures F7 to F20.

Horizontal : 40.7 : Horizontal Depth = 0.8m

1.0,

, , , 1 X 40%- 30%

0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.Ot

Wave Steepness

Figure 4-11 Kt vs Wave Steepness for a double screen system differentiated by screen porosity


No triple screen porosity variations were tested as there were too many combinations and

permutations. While a 40-30-20 screen system was the only one tested, the orientations

and gaps between successive screens were varied.

The force of a wave on a screen is inversely proportional to its porosity, the larger the

porosity the smaller the force. As such, it is generally appropriate to place the larger

porosity screens closer to the incoming waves to minimize the force exerted on the screen

and thus reduce the required strength of the screen. Each screen reduces the amount of

wave energy hitting the following screen, and thus a smaller porosity screen cm be used.

This also results in a reduced reflection coefficient.

Interestingly, two double screen systems (30-20 and 40-20) had a lower average Kt than

the triple screen system for sirnilarly onentated screens. Figure 4-12 displays these

trends for horizontally onentated screen systems. As discussed in section 4.5, the steeper

the wave, the higher the dissipation the wave expenenced on passing through the screen.

However, the triple screen system gradually reduced the wave steepness whereas the

double screen system allowed steeper waves to impact the 20% screen of the double

screen system. The steeper waves were not as easily transmitted through the 20% screen

in the double system and it is believed that this accounted for the lower Kt.


Wave Steepness

Figure 4-12 Kt vs Wave Steepness for a triple screen system differentiated by screen porosiîy

4.7 Muence of Screen Orientation

4.7.1 Single Screen

Screens positioned with their slats horizontally aligned were more effective at reducing

Kt than screens with a vertical slat orientation. Kt can be related to the disruption of the

waveform as it passes through the screen. A horizontal slat disrupts this waveform more

than a vertical one, causing more energy to be dissipated and reflected, resulting in less

transmission. The extent of this reduction due to slat orientation was not consistent and a

statistical analysis was performed to try and distinguish between the two orientations.

Figure 4-13 provides an example of the effect of orientation on Kt. Note that this graph

also shows that Kt decreases with increasing wave steepness. Appendix G, figures G1

through G12, contains the remaining plots for the various screen configurations.

Screen 30 Depth=O.gm

ID

0.00 OD1 0D2 0D3 0.04 0.05 0.06

Wave Steepness

Figure 4-13 Kt vs Wave Steepness for a single screen system differentiated by orientation


It was found that a system of two horizontal screens (H-H) had the best performance in

reducing transmission, followed by H-V. The other orientations, V-V and V-H, did not

reduce Kt to the same magnitude as either H-H or H-V. However, it was only a minor

difference in Kt, as shown by the tight clustering of results in figure 4-14. This figure

shows the small advantage H-H has over the other orientations as weU as the clustering of

results for constant depth. Again a statistical anaiysis was performed to try and quanti@

this effect, and is discussed in section 5.7.8. Structural design and cost may be a more

important factor in deterrnining orientation in a real-life application.

Screen 40 : 99.5 : Screen 30 Depth=O.irm

1 .O

0-1 H;+;I 013 3

0.00 0.01 0.02 0.03 0.04 0.05 0.01

i Wave Steepness

Figure 4-14 Kt vs Wave Steepness for a double screen system differentiated by orientation


The orientation of the screens in a triple screen system had no significant influence on Kt

for a given wave period, wave height or depth. The range in variation of Kt for a given

wave train was about 0.05 between the best and worst possible setup. This differed from

the conclusions drawn in the double screen analysis. It was deduced that three screens, in

any orientation, sufficiently disrupted the waveform so that orientation no longer made a

substantial ciifference to Kt, whereas in the two screen system, screen orientation still had

a discernable effect.

While orientation differences in the triple screen system appeared to have a limited effect,

a statistical analysis was still performed to differentiate between the systems. An initial

assessrnent suggested that H-H-V and V-V-H were preferred over other orientations for

reducing Kt while V-V-V was the least preferable. Figure 4-15 highlights the narrow

range of results that the different orientations produced.

x v-v-v O-l w

, 0.0 0.00 0.0 1 0.02 0.03 0.04 0.05

Wave Steepness

Figure 4-15 Kt vs Wave Steepness for a triple screen system differentiated by orientation

4.8 Influence of Spacing between Screens


Some striation between the data sets was visible s ignaing that the spacing between the

screens, or gap, was an important factor affecting Kt, as can be noted in figure 4-16 or

appendix H, figures HI through HlO. No strong trend regarding the influence of the

magnitude of the gap on Kt was apparent though. A screen system with a gap of 99Scm

generally had the lowest Kt while a system with a gap of 25.5crn had the highest Kt-

40 Horuontal : Gap : 30 Horizontal Depth=0.8m

I

' 0.00 0131 0.02 003 0 .O4 OR!

Wave Steepness

Figure 4-16 Kt vs Wave Steepness for a double screen system differentiated by gap size

An effort was made to make gap a dimensionIess variable by dividing by wavelength.

However, a plot of Kt versus G/L did not prove usefil as the gap size dominated,

grouping the data sets vertically. The parametric analysis showed that a statistical

analysis should be performed to determine the effect of gap.


Gap size had a lirnited effect on Kt for the triple screen systems. The system with the

smaller gap size (49.8 : 25.5) appeared to be marginally more effective than the system

with the larger gaps (59.5 : 49.8). A statisticd analysis was perfomed to quanti@ this

effect and the results are given in section 5.7. The sample plot given in figure 4-17 is

typical of al1 the plots for the triple screens. This graph highlights the decrease in Kt with

an increase in steepness and shows that steepness has a Iarger impact on Kt than gap size.

40H : Gap : 30H : Gap : 20H 1 1.0,

f 0.01 0.02 0.03 0.04 0.0:

I Wave Steepness

Figure 4-17 Kt vs Wave Steepness for a triple screen system differentiated by gap size

4.9 Muence of Relative Depth

4.9.1 Single Screen

Relative depth is defined here as the incident wave height divided by depth (HI/D). A

trend of decreased Kt with increased H m was noted when Kt was plotted against relative

depth. This identifies the same trends as the plots of Kt versus incident wave height,

showing that as wave height increases, Kt decreases. Striation showed that depth played

a role in affecting Kt but suggested that wave height was a more important factor. Figure

4-18 shows the increased Kt with increased Hr/D trend and accompanying striation. The

full set of plots can be found in appendix 1, figures 11 to 110.

20 Horizontal

1

Incident Wave Height / Depth

Figure 4-18 Kt vs Relative Depth for a single screen system differentiated by depth


There was a weak trend showing a decrease in Kt with an increase in relative depth.

There was not as much visible striation as with the plot of Kt versus incident wave height.

The results were fairly scattered if a 20% porosity screen was in the second location but

somewhat tighter if a 30% screen followed the f i s t screen. This was probably caused

because the 20% screen caused greater reflection than the 30% screen, which in tum

caused greater variabiliw in the wave steepness and thus Kt. Figure 4-19 shows a set of

typical results when the second screen has a 30% porosity. Figures Il 1 through 124

display Kt verses Hr/D results for a l i the single screen tests. The screen orientation was

difficult to compare in this preliminary analysis because different orientations were

placed on separate g-raphs.

Incident Wave Height 1 Depth

Figure 4-19 Kt vs Relative Depth for a double screen system differentiated by depth

4.10.1 Single Screen

The dimensiodess parameter d / g ~ 2 was investigated to incorporate the wave penod more

directly within an empirical equation. The trend of decreasing depth causing decreasing

Kt within the results showed that depth played a minor role in determining Kt. Only a

weak trend could be identifed that a change in dg~' caused a change in Kt. Figure 4-20

shows a weak colrelation that as dlg~' increases Kt decreases. The r ema idg graphs,

included in appendix J (figures J I to JlO), follow this trend as well.

20 Horizontal 1 1.0,

Figure 4-20 Kt vs d / g ~ 2 for a single screen system differentiated by depth

4.10.2 Double Screen

The results for the double screen mirrored those in section 4.4.2, except Kt was higher for

a depth of 0.7m than for a depth of 0.8rn, for alI orientations. No tests were run at 0.9m

depth. However, these results match with the results reported in section 4.4. Figure 4-21

shows how scattered the results were (the remaining graphs can be found in figures J11 to

524). Only weak trends of decreasing Kt with increasing d / g ~ 2 were visible from these

plots.

Figure 4-21 Kt vs dg~' for a double screen system merentiated by depth

Since depth appeared to play a rninor role, based on this analysis and previous fiterature,

only one term including depth needed to be included in the statistical analysis. Both

options (H[/d and dIgT2) are commoniy used dimensioniess parameters. The fiterature

review suggested that dIgT2 is a more influentid variable than Hr/d however, so this was

the variable chosen. A discussion on the relative contributions of HI/d and d / g ~ 2 can be

found in section 5.6.1 that confïrrns the validity of this choice.

A parametric analysis was performed to quaiitatively determine the effect of individual

parameters on Kt, The following generd observations were made.

No clear trend was present with respect to wave period.

Kt was found to decrease with an increase in wave height.

Kt was found to decrease with a decrease in depth.

Kt was found to decrease with an increase in wave steepness.

Kt was found to decrease with a decrease in screen porosity.

Screen orientation did have an innuence on Kt; horizontal screens having a lower

Kt than vertical screens.

No distinct trend could be noted with an increase in gap space between two

screens,

Kt was found to decrease with an increase in reIative depth (Wd).

Kt was found to decrease with an increase in d / g ~ 2 .

Kt was found to decrease with an increase in the number of screens placed in

series.

5.1 Introduction

This chapter discusses the statistical analysis perforrned on the test results. The goal of

the statistical anaiysis was to develop a series of empirical equations that predict the

transmission through one, two and three wave screens in a variety of orientations, subject

to a variety of input conditions (wave height, wave period and depth). The statistical

analysis was perforrned using SYSTAT 6.0 (SPSS, 1998). SYSTAT is a commercial

statistical software package specifïcally catered towards scientific research.

Section 5.2 of this chapter explains the boundary condition that was applied to each data

set in order to more accurately predict Kt at smail value of wave steepnesses. Section 5.3

discusses the choice of the fom of the equation and variables that were incorporated into

the mode1 equations. The manipulation of the data is descnbed in section 5.4, including

the removal of outliers, withdrawal of data points for later validation and application of

Systat. An interpretation of the Systat output is included in section 5.5. The coefficients

for the single and double screen equations are presented and statistical findings discussed

in sections 5.6 and 5.7 respectiveIy. A bnef mention of the limitations associated with

the triple screen system follows this in section 5.8. A cornparison is made with other

theones in section 5.9. Section 5.10 presents a summary of the entire chapter.

5.2 Boundary Condition

Several variables were identined as being significant folIowing completion of the

parametric analysis (chapter 4). It was proposed that wave steepness was the

predominant dimensioniess variable as it included wave height, wave period and water

depth. Thus, Kt and wave steepness were used as axes when fitting a trend h e through

the data points. However, these trends did not accurately define a realistic scenario at

=O, as when extended, they intercepted the y-axis at a Kt value less than one. This

implied that at a negligible wave steepness, there was still s i b d c a n t dissipation. A

boundary condition was applied to all data sets to remedy this physical inconsistency.

A boundary condition of cornplete transmission can be applied to cases where the period

of the wave is very long. Long period waves, such as tides, have correspondingly long

wavelengths, so steepness can be considered to approach zero. The screen should have

no impact on these waves. Thus, a boundary condition of zero steepness (-0) and

cornplete transmission (Kt=l) was acceptable based on this premise. The other situation

represented by this boundary condition is for a very small wave height. In this case, the

size of the gap within the wave screen will be much larger than the wave and,

consequently, wiU have little effect on the wave, again resulting in complete

transmission. Thus, this imposed boundary condition is applicable for al1 situations

having the wave steepness approach zero.

5.3 Equation Variables and Form

The first step in developing an equation to predict Kt was to select which variables to

include in the equation. The parametric analysis showed that wave steepness and screen

porosity were important factors affecthg Kt. Since Kt is dimensionless, it is desirable for

the components within the equation to be dimensionless. Porosity and wave steepness

are already dimensionless. To incorporate a dimensionless wave period and depth into

the equation ( d g S ) was included. Finally, to non-dimensionalize screen spacing, the

gap was divided by wavelength to form a dimensionless gap variable (GL) for the

multiple screen analyses.

The form of the equation was the next consideration. The simplest equation is linear

which has the basic fonn:

where: CI,C2,C3, etc are fitted parameters

X,,Xb, etc are dimensionless variables

However, the pararnetric analysis results clearly revealed that the variation of Kt with the

dependent variables was not a simple linear mode1 (see section 4.1 1). Statisticdy, the

Pearson correlation coefficient (a rneasure of the strength of the linear relationship

between two variables) also showed that the dimensionless variables were not linearly

related to Kt, confirming the initial observations.

71

Higher order equations were considered and they had to sat ise several criteria. A

poiynomid, exponential, natural logarithmic and power form were aU investigated and

their suitability was judged by the following criteria:

1) the magnitude of the squared multiple correlation coefficient, R', which is a

general measure of how well the equation fit the data.

2) the ability of the equation to satisQ the boundary condition. This key factor has

been discussed in section 5.2.

3) the ability of the equation to incorporate al1 the desired variables while

minimizing the number of statistically fitted parameters.

4) the ability of the equation to Lirnit values withh its approximation to O I Kt I 1.

5) residuals should be normally distributed showing that there is no bias in the

equation or that s i b d c a n t trends have been missed.

Both the polynomial and exponential equation foms were rejected because they couId

not meet the second requirement. A polynomid equation such as:

has a constant Cl, which may not equal one. As a result, when the X, and XI, terms go to

zero, the predicted curve intercepts the y-axis at CI, inftïnging on the boundary condition.

Furthemore, a polynomial equation needs to have many more fitted parameters which

rnakes it unattractive.

The exponential equation has fewer fitted parameters than the polynomial equation as c m

be seen in equation [S-31.

However, it breaches the boundary condition as it can cross the y-axis at a value not equd

to one. Thus, the exponential form was rejected.

The n a W log equation appears to have a sirnilar problem to the polynomial and

exponential equation forms. However, the naturd log curve does not support negative

(XJ or (Xb) values as it is asymptotic to the y-axis. Exarnining the general nahml log

equation shows this.

This results in the curve simulating the boundary condition once CI is negative and

(X,)(Xb) is less than one. The problem is that the equation only sirnulates the boundary

condition but is not exact because ln(0) is infinite. This leads to the predicted curve being

asymptotic to the y-axis so critenon 4 is broken as Kt >1, when (X,)(Xb) approaches zero.

The power equation has the identical problem.

Based on the desired criteria, the natural log and power equations would be the most

preferable once the boundary condition (0,l) problem was resolved. To solve this

problem, an approximation for wave steepness close to zero had to be used. The zero

value boundary condition was added to the data sets and a sensitivity analysis was

performed using various values approximating zero ranging fiom IX~O-' to 1x10-". The

goal of this analysis was to find a value close to zero that minimized roundoff and

numericd errors in order to maxirnize R~. A graph of this sensitivity analysis is given in

figure 5-1 and shows that 1x10~ gave the highest R~ value so this was used to represent a

wave steepness equal to O. This value was consistent for all plots as can be seen in

appendix K, figures K 1 to K6. Limits had to be applied to uie range of all the equations

because this approximation was made. The equations should not be used below a wave

steepness of 1x10-~ or outside of the 95% confidence lùnit.

/ Sensitivity Anal ysis of FI2 for WL clos t o 1 1 .- Zero and K t 4 for 20H 1

Value of Zero I Figure 5-1 Sensitivity analysis graph of wave steepness vs R~ for approximating zero

The natural log and power trends had similar shapes when fit through individual data sets.

While the nanual log curve had larger R~ values on average (in the order of 0.1 higher

than that for the power curve), it was not deemed significant enough to neglect the power

cuve relationship. Therefore, analysis was performed for both types of cwes .

It was decided to f o m separate equations based on the screen orientation rather than add

an additional variable. It was possible to use an effects coding statistical anaiysis to

differentiate between the two orientations. Effect coding is a statistical method whereby

an integer is applied to a group subset so that SYSTAT can differentiate between separate

data groups. This would have proved to be very cornplicated for the double screen

orientations as there would have been a code for each individual screen orientation and

gap width combination. Thus, it was decided to simply develop separate equations.

5.4 Data Manipulation

The first step in developing an equation to predict Kt was to remove any obvious outliers

fiom the data sets. Any Kt value greater than one was obviously an outüer as the

transmitted wave height cannot be greater than the incident wave height. There were

three instances of this happening, which resulted f?om sampling errors.

Single test results were removed randomly fiom the data set. These results were used

later to assess the validity of the prediction equation. This will be discussed M e r in

section 5.6.1.

A multiple Linear regression was performed on the remaining results within the data sets

to find the fitted parameters for the natural logarithm and power cuves. The natural

logarithm coefficients were developed for the single screen system by taking the natural

logarithrn of screen p o r o s i ~ (Pl), wave steepness (WL) and d / g ~ 2 and regressing these

variables against Kt. For the double screen systern, the natural log of the porosity of the

second screen (P2) and dimensionless gap (GIL) were added to the variables analyzed for

the single screen equation. The power analysis was identical to the natural analysis,

except that the variables were regressed against the log of Kt.

A regression analysis was not possible for the triple screen system because the screen

porosity and order were the sarne for aIl the tests. This would have excluded the three

porosity variables fiom the analysis and these variables have a significant influence on

Kt. Future research should investigate the triple screen system in more detail.

Systat v8.0 used complete regression estimation for this analysis. This estimate method

inputs all the independent variables in a single step. Systat also sets a tolerance limit,

which prevents the entry of a variable that is highly correlated with the independent

variable. A tolerance of 1 xlo-'' was used even though a tolerance of 0.01 did not affect

the results. Above a tolerance of O. 1, the program recornmended that wave steepness and

G L be excluded from the modeI. It was known that wave steepness and G L were

correlated since they both shared wavelength as their denominator, so setting a high

tolerance limit was impractical and it was lcft at the default setting.

5.5 Interpretation of Systat Results

Systat retunied the results from the multiple linear regression, which included the fitted

parameters, R', the standard error and any possible outliers. A sample output is shown in

figure 5-2.

4 case(s1 deleted due to missing data.

Dep Var: KT N: 95 Multiple R: 0.882 Squared multiple R: 0,777

Adjusted squared multiple R: 0.767 Standard error of estimate: 0.064

Effect Coef f i c i ~ r Std Error Std Coef Tolerance t P(2 Tail)

CONSTANT 0.681 0,047 0-000 14,630 0.000 LNPl O. 198 O. 023 O. 433 0.998 8.694 O. O00 LNSTEEPNESS -0.063 O. 024 -0.767 0.029 -2.636 O. 010 LNHD -0.022 0.012 -0.313 0.078 -1-756 0 - 082 LNDGT2 O. 023 O. 017 0 - 3 1 9 0.045 1.361 0.177

Analysis of Variance

Source Sum-of-Squares df Mean-Square F-ratio P

Regression 1.280 4 0.320 78.521 O. 000 Residual 0 -367 90 0.004

9C * * W m N G f f f

Case 68 is an outlier (Studentized Residual = -3.448

Durbin-Watson D Statistic 0.880 First Order Autocorrelation 0.553

Figure 5-2 Sample Systat output for a single, horizontally oriented screen

The important statistical data is bnefly described here. As stated earlier, the squared

multiple R value, or R ~ , is an indicator as to how well the equation fits the data. An R'

above 0.7 was considered to be the lowest acceptable lùnit for the purposes of this thesis.

The standard error of estimate is a measure of how variable the estimates are expected to

be if data was continually sarnpled from the population using a least squares analysis

equation. Mathematicaily, it is the square root of the residual mean-square.

The coefficient column returns the fitted parameter for each of the variables while the

standard error column gave the expected error in the particuiar variable. The standard

coefficient (or beta coefficient) nonnalized the variable coefficient and shows which

variables had a greater influence on Kt (Kleinbaum & Kupper, 1978). Although

standardized regression coefficients eliminated the problem of scale-dependence, they

should not be used to judge the relative importance if the variables are highly correlated

(Schulman, 1992). Tolerance was a rneasure of high correlation arnong the independent

variables and is mathematically represented by (1-R'); one minus the squared multiple

correlation between the predictor and other predictors in the mode1 (the dependent

variable is not used). The 't' test evaluates the significance of the slope, which is

equivalent to testing the significance of the correIation between the variable and Kt. The

errors in predicting Kt from each variable were based on a normal distribution but since

the standard errors of the regression coefficients were estimated from the data, a 't'

distribution had to be used. The two tail value for probability, shown as "P (2 tail)"

represented the area under the theoretical 't' probability curve. This corresponded to

coefficient estimates whose absolute values were more extreme, by extending the range

of the ones found during testing. The smaller the p-value the more certain that a

prediction of Kt from the variable was possible. So in the example case shown in figure

5-2,ln(P) and ln(HL) give confidence that these two variables gave a good prediction of

Kt. The p-value increased for the other variables as they were trying to account for the

residuals that enlarged the uncertainty of their prediction.

Analysis of Variance, fiequently abbreviated anova, is a technique used to determine

whether there is a probable qualitative relahonship between the variables. The sum-of-

squares is a sum of total error between the predicted value and actual value. Obviously,

the lower the sum-of-squares the better the model fits all the data. The next column in

the Systat output dispIays the number of degrees of fieedom, df. The mean-square is a

ratio of the sum-of-squares and the degree of fieedom and is the same as the sarnple

variance- The "F-ratio" is an indication whether the slope of the model is zero, similar to

the 't' test but for the complete equation. The F is large when the independent variables

help tu explain the variation in the dependent variable. The last column in the anova

section gives the 'P' value. The closer this value is to zero the more likely the slope of the

regression line is non-zero signaling that a relationship between the variables and the

dependent is highly probable.

The outliers are based on the point being M e r than three standard deviations away

from the mean. The Studentized residual divides the residual by the standard error. The

higher the Studentized residual the higher the chance that it is an outiier. Such data

points may have been more than three standard deviations away from the mean but may

not be true outliers. Such points were first checked against any inconsistencies during

testing such as calibration problems or deviation f?om target spectrum. They were then

plotted with the other test results in that series to see if they significantly broke the trend

within the series. If either of these had occurred then the data points were rernoved and

another regression performed.

If the data point did not break the trend within the series but was part of a compIete test

series, which challenged the prediction line, then the point was still removed and another

regression performed. This was continued until there were no statisticaUy declared

outliers. The outliers highlighted in this process were then studied for trends, such as all

being in one data senes. This gave an indication as to which data series was skewing the

prediction equation coefficients.

The plot of residuals against predicted values gave a fair indication of which situation

was encountered with the outliers. A single point away from the prediction line

suggested a true outlier where an inconsistency had developed. Several points offset

£rom the prediction line, Like in figure 5-3, suggested a data series was at fault and the

outlier was not a typically defined outlier,

Plot of Residuals against Predicted Values

Figure 5-3 Graph of residuals afier regression analysis for a single, honzontaily oriented screen

The Durbin-Watson D statistic is specifically designed to detect patterns within the

residuals by checking for adjacent residuals having sirnilar signs. This is mainly used

80

when analyzing time dependent variables and was ignored dunng this thesis. The first

order autocorrelation is simila. to the Durbin-Watson D and was also ignored.

5.6 Single Screen Equations

Four equations were developed to predict wave transmission through a single screen; two

each for the horizontally and verticdy oriented slats.

5.6.1 Single Horizontal Screen

The naturai log equation related to a single horizontal wave screen is given in equation

[S-61.

The power equation is given in equation [5-71.

These equations should only be used when the screen porosity is between 0.2 and 0.4,

wave steepness between 0.005 to 0.5 and d / g ~ 2 between 0.145 and 0.16.

The natural log equation had a better R' value than the power equation (0.84 versus 0.75).

The standard error of estimate was higher for the naturai log equation than for the power

equation (0.05 versus 0.04). Ninety-three data points were analyzed during the statisticd

analysis. The most influential variables were the screen porosity and wave steepness

based on the standard coefficient and tolerance results. While the literature cautions

against using the standard coefficient as a measure of influence, these two variables were

not correlated (the tolerance of porosity was 0.999) and had significant standard

coefficient values (0.48 and - 1.3 respectively). Wave steepness had a standard coefficient

value, double that of d / g ~ 2 (-0.60). The standard coefficient can be misleading when the

individual parameters have a low tolerance as was the case with wave steepness and

each a tolerance value of 0.044. in this case, the variable with the larger standard

coefficient (wave steepness) is deemed to be more influential than the one with the

srnalier standard coefficient (dlg~'). Wave steepness also had a lower p-value (0.00) than

d / g ~ 2 (0.027) signaling that this variable gives a more certain prediction.

Examining the contribution of each term within the natural log equation also shows that

wave steepness is the most influentid variable, foliowed by the screen porosity, as shown

in table 5-1. This agrees with the statistical output fiom Systat. A similar analysis was

perfomed with relative depth, Wd, included in the equation. It was found that H/d did

not change the R~ value. It was further found that the contribution of the Wd term to the

total Kt in the natural log equation was, at most, 0.04. Thus, this term was omitted when

performing the statistical analyses to develop the Kt prediction equations.

Table 5-1 Table showing contribution of variables to Kt for horizontal screen equation

Systat returned only one outlier during the k s t round of analysis for both the natural log

and power regressions. 136iwd was identified as an outlier during the natural log

analysis while 206jyd was identified in the power analysis. It was interesting that they

were separate data points with different wave characteristics but shared a similar wave

steepness. Both outliers were accepted as the value of Kt was noticeably lower than the

other data points within their respective series. During the second round of analysis,

these points were again highlighted as outliers, so 206jyd was now identified dunng the

natural log regression and vice versa. Since outliers should match irrespective of the

equation form, the final results omitted both of these points from the regression analysis.

However, continued regression while removing stated outliers did produce a trend. Only

data points within the 20H and 40H groups at a 0.7m depth were returned as outliers.

This shows that the equation did not predict Kt as well for these particular ranges as for

others, but this could be due to the low number of data po-hts within these groupings.

Variable

Porosity H/L d / g ~ 2

A plot of the actual Kt values versus the predicted Kt values using the naturd log

equation is given in figure 5-4a. There is very Little clifference between this figure and

figure 5-4b, which gives the actual Kt versus the predicted Kt using the power equation.

The natural logarithrn curve and power curve can, on average, predict Kt within 2% or an

absolute percent magnitude of 8%. It also highlights the fact that the prediction curves

83

Range of variable

0.2 + 0.4 0.0048 + 0.448 0.016 + 0.145

Range of variable ccritribution to Kt -0.332 + -0.189 0.078 + 0.518

-0.128 + -0.000

tended to overestimate Kt values for 20H and 40H. The average corresponding

underestimate (because the regression line tried to minimize error) was much smaller

than the average overestimate but more data points were underestimated,

Actual Kt vs Predcted Ktfor a Single Horizontal Screen uslng the Naturai Log Equatlor,

1.0 =/'-=-l

Actual Kt vs Predicted Kt fora Single Horizontal Screen using the Power muation

0.0 0.1 0 2 03 O A O S Ob 0.7 O B 0.9 1.0 1.1

Predicted Kt

Figures 5-4a & 5-4b Graphs of actual vs predicted Kt values for a single horizontal screen using both the natural log and power equations

The natural log equation was then used to predict Kt for the four data points that were

randomly removed h m the complete data set pnor to the statistical analysis. A

summary of the results is shown in table 5-2 and clearly demonstrates that the mode1

works well. It under predicted al1 of the Kt values in this case, but by no more than 6%

and in the best case it was only a negligible 1% difference. The average absolute

prediction error was 3%-

Table 5-2 Table comparing predicted and actual Kt for data points withheld fkom the single horizontal screen statistical analysis

Screen 1 Test Number 1 Predicted Kt 1 Actual Kt 1 % Difference

5.6.2 Single Vertical Screen

20H 30H 30H 40H

Equations developed to represent transmission through single vertical screens had lower

R~ values than the horizontal screen equations (0.77 for the natural log and 0.72 for the

power equation). The standard error was again higher for the natural log equation (0.066)

compared to the power equation (0.045). These equations are given as [5-81 and [5-91

respectively.

These equations should only be used when the variables are within the variable range

given in table 5- 1.

217jxe 7 2 k ~ d 82jwe 1461ve

The residual plot was checked for both forms of equations and there were no visible

outliers for the natural Iog analysis. However, there was one definite outlier identified in

Average absolute prediction error, %

0.63 0.66 0.70 0.68

3

0.64 0.66 0.74 0.72

-2 - 1 -6 -5

the power analysis and this was removed. The only adequate explanation that could be

given for this ciifference was that 238iyd was a relatively flat wave (HL = 0.007) and the

two curves have noticeably different shapes when H L < 0.01, so an outlier would be

apparent for one but not the other.

Cornparison of actual Kt and predicted Kt in figure 5-Sa shows that the natural log

equation for the single vertical screen acted comparably to the equation for the horizontal

screen. Most data points have an actual Kt slightly higher than was predicted by the

equation. This was again offset by a few points that had a lower actual Kt than what was

predicted. However, these points were further away from the 45" line that marked a

perfect estimate.

Actwl Kt K Predlcted Kt for Single Vertical Screen using Natural Log Equatfon

Actual Kt vs Predicted Kt for Single Vertical Screen using the Power eqation

i I

r ozon

JO" 1 i AOH

OB 0.1 0 2 0.3 OA 0 5 Ob 0.7 04 0.9 Id 1.1

Redlcted Kt

Figures 5-Sa & 54b Graphs of predicted vs actud Kt values for a single vertical screen using both the natural log and power equations

As with the horizontal screen data, four data points were removed randomly fiorn the

initial input data to be used to ven@ the model. The sumrnary of the verification results

is shown in table 5-3. These results appear to support the equation except for the third

data point, 171kwd. The prediction for this point is almost 40% away from the actual Kt.

However, this point was removed and pIaced on the validation list prior to the outlier

analysis. After the validation test, the data point was checked and it was found that, had

it not been randomly chosen as a validation point, it would have been removed from the

data set because it was an outlier.

Table 5-3 Table comparing predicted and actual Kt for data points withheld fkorn the single vertical screen statistical analysis

The average absolute prediction error was 3%.

% Difference -2

Screen 20V 30V 40V 40V

5.6.3 Comparison of Horizontal and Vertical Screen Orientations

The goal of this thesis was to provide an equation that accurately predicts Kt through a

Test Number 247jwe 561ye

l7lkwd 196iyf

permeable wave screen. ïhe equations presented so far have been developed by

Average absolute prediction error, % (excluding outlier)

statisticaLly examining the horizontal and vertical data sets independently. An attempt

Predicted Kt 0.62 0.69 0.77 0.87

3

was made to reduce the number of equations by combining the data sets of the horizontal

Actual Kt 0.64

and vertical screen orientations.

0.68 0.55 0.83

When the two data sets were combined, the R' value for the best fit equation was reduced

1 40 5

to 0-74 using a natural log equation form and 0.69 for the power equation form. These

are lower than the R~ values discussed in sections 5.6.1 and 5.6.2. Furthemore, the

87

standard error of estimate and mean square residud are both higher for the combined

equations, thus it was deterrnined that it was necessary to have two separate sets of

equations in order to correctly represent screen orientation.

A comparison of horizontal and vertical screen orientation was performed. This

comparison examined the magnitude of Kt for a given screen setup and wave climate.

Also, an attempt was made to quanti@ generd trends noted in the pararneûic analysis,

such as the effects of increased wave period, wave height, depth and porosiq on Kt and

whether a horizontal or vertical orientation was more greatly effected by these variables.

It was noted in the parametric analysis, that the horizontal screens appeared to be more

effective at reducing Kt than the vertical screens. Figure 5-6 provides a comparison of

equations [5-61 and [S-81. It shows that the natural log equation predicts a lower Kt using

the horizontaily oriented screen equation than the verticaliy oriented screen one by about

8% on average for a 30% porosity screen, 1s wave period and a depth of 0.8m. As wave

steepness increases, the influence of the horizontal screen becornes slightly more

pronounced. At a wave steepness of 0.01, the difference between the predicted Kt for a

horizontal and a vertical screen is 7%, which increases to 9% at a steepness of 0.06.

1 M u n l L o g C u m f o r P ~ T = l s , d d . ~ f o r

1 Horizontal and Vertical Screen ûrientatlons

on O 091 0.02 00J O M 0.05 OaC

Wave Steepness

Figure 5-6 Graph comparing horizontal and ver t id natural log equations

Figure 5-7 gives a direct cornparison of the predicted Kt for a horizontal screen and the

predicted Kt for a vertical screen, using equations [5-61 and [S-81 respectively. Figure

5-7 highlights the fact that the predictions for the horizontal and vertical screens are

sirnilar, although not identical, and that the horizontal screen was better at reducing Kt

than the vertical screen for the particular setup used in this thesis.

Predtded Horizontal vs Predlctad Vertical Kt

! udng a Natural Log Equation for Pdû%. T=ls

and d=0.8rn

Figure 5-7 Graph Oirectly cornparhg horizontal and predicted Kt using the natural log equation

The difference between the horizontal and vertical prediction was reduced M e r as the

wave period was increased. At a period of 2 seconds, the average diKerence was only

1%. At periods larger than 2 seconds, the vertical screen was projected to have a Iower

Kt than the horizontal screen. Examining the effect of wave period on an individual

screen showed that longer penod waves had lower Kt values than shorter period waves.

5.6.4 Evaluation of Screen 30b

Sections 5.6.1 and 5.6.2 presented empincal equations to predict Kt based on results of

tests using screens with a widWthickness @/t) ratio of 2. Screen 30b, however, had a b/t

ratio of 1 as square shaped slats were used. Equations 15-61 through [5-91 which were

developed for a b/t of 2, were used to predict Kt for screen 30b. Figure 5-8 shows the

cornparison of the actual versus the predicted Kt.

Figures 5-Sa & 5-Sb Graphs of Actual vs Predicted Kt for Screen 30b using the naturai log and power equations

Acîual vs Predicted Kt for Screen 30b uçing the Power Equation

I

OD 0.1 0 2 03 Od 08 O b 0.7 O8 0.9 10

Redlded Kt

It can be seen that the horizontal orientation equation did not provide an accurate

prediction of Kt as the horizontal results are scattered and the vertical results are

I i Actual vs Predicted Kt for Screen 30b using

the Natural Log Equation

0 0 0.1 02 03 04 05 Ob 03' OB 0.9 1 0

Redided Kt

overestimated, The average absolute difference between the actud Kt and predicted Kt

was 18% using either the natural log and power equations. The vertical orientation

equation provided a better prediction of Kt for the 30b screen in a vertical orientation

than the horizontal equation did for the 30b screen in a horizontal orientation. The

absolute average difference between the actual and predicted Kt was only 6% for the

vertical equation. Further research needs to be performed, extending the current b/t range

tested, to confidently use these equations for a b/t ratio other than 2.

5.6.5 Evaluation of Screen 50

Screen 50 had a different b/t ratio (1 -45) compared to the four screens used to develop the

equations in sections 5.6.1 and 5.6.2. 1t also used larger fiaming pieces and had a larger

porosity than the other screens. However, the actuai and predicted Kt values were still

comparable when using equations [S-61 through [5-91 to predict Kt. The absolute

percentage difference was 9% for the horizontal screen and 10% for the vertical screen.

Figure 5-9 shows a graph of the actual verses predicted Kt values and that the equation

generally overestimates Kt for screen 50. Further tests should be done to verify the

prediction equations past a 40% porosity screen and with different b/t ratios.

Actual vs Predicted K i for Screen 50 using the Naîural Log Equation

I D r

Ackral vs Predicted Kt for Screen 50 using the Power Equation

1 A

Homontal - o Vertical

OD 0.1 03 03 04 O 5 Ob 0.7 0 8 0.9 10

Rmdicted Kt

Figures 5-9a & 5-9b Graphs of Actual vs Predicted Kt for Screen 50 using the natural log and power equations

5.7 Double Screen Equations

The parametric analysis showed that results for the double screen tests were grouped by

screen orientation. This suggested that an equation would have to be developed for each

of the four screen orientations, as one equation might not satisfactorily predict Kt for all

scenarios. Two additional ternis were included for the double screen equations; a

porosity term for the second screen, P2, and a dimensionless gap term, G/L. The

generalized form of the natural log equation is given in equation [5-i0] while the

generalized form of the power equation is given in equation [5-111.

5.7.1 H-H Screen Orientation

The full natural log and power equations, resulting fkom the regression analysis, are given

below as equations [5-121 and [5-131.

As with the single screens, the natural log c w e provided a superior prediction curve than

the power one. The R~ values for the nanird log and power curve prediction equation

were 0.83 and 0.74 respectively. The standard error of estimate was 0.058 and 0.05 1

respectively. There were 18 test series run for the H-H orientation, totaling 248

individual tests. After removing ten points for later verification, and satisQing the

boundary condition, 256 data points were used for the statistical analysis. The statisticd

analysis suggests, based on the standard coefficient values, that the porosity of the second

screen had a greater influence on Kt than the porosity of the first screen (their standard

coefficients being 0.36 and 0.25 respectively). The standard coefficient also implies that

wave steepness (-1.1), d / g ~ 2 (0.82) and dimensionless gap width (-0.51) had a greater

influence on Kt than the porosity of either screen. However, the low tolerance values of

these parameters (0.06, 0.03 and 0.07 respectively) highlighted that they were al1 related

and the only firrn co~clusion that could be drawn was that the overall wave climate had a

larger impact on Kt than screen porosity.

Comparing the relative contribution of each variable within the H-H equation agrees with

the assessment of the statistical output, as shown in table 5-4. Wave steepness has the

highest contribution range and can have the largest ef6ect on the Kt value (adds 0.59).

G L provides the least contribution to the overail Kt value (adding at most 0.11). It can

also be seen that the porosity of the second screen (P2) affects the overall value of Kt

more than the porosity of the fist screen (PL). P2 will reduce the Kt value by at least

0.30 while Pl will reduce the o v e r d Kt by only 0.29 at most. It should be stressed that

these equations should only be used within the variable range gïven in table 5-4.

Table 5-4 Table showing contribution of variables to Kt for H-H screen equation

No outliers were present within the data set. Figure 4-8 shows this as no points lie far

fiom the 45" line representing a perfect model.

Range of variable contribution to Kt -0.287 + -0.218 -0.396 + -0.296 0.587 + 0.088 -0.289 + -0.135 O. 1 1 + -0.02

Variable

PL P2 H/L d / g ~ 2 G/L

Range of variable

0.3 + 0.4 0.2 + 0.3

0.0048 + 0.448 0.016 + 0.145 0.043 + 1.71

Actual Kt vs Predkted Kt for an K-ii Screen System using Natural Log Ejquatim

02 / i 0.t 1 on I On 0.1 03 03 0 4 05 Ob 0.7 OB 0.9 IO 1.1

Redtcted Kt

Actual Kt vs Predkted Kt for an WH SCteen System using Power Equaîkm

l n 0.9

Ob

Figures 5-10a & 5-lob Graphs of actual vs predicted Kt values for an H-H screen system using both the natural log and power equations

The natural log curve was then used to predict Kt values for ten data points that were

removed prior to the statistical anaiysis. These results were not as good as the single

horizontal screen predictions, but it was anticipated that when a second screen was

utilized for greater energy dissipation, wave reflection and agitation would increase,

scattering the results more widely than for a single screen. The larger scatter resulted in a

poorer prediction. Table 5-5 compares the predicted and actual Kt value, for the data

points randomly removed before analysis, and shows the maximum discrepancy between

the predicted and actual Kt was 15% while the absolute average was 8%.

Table 5-5 Table cornparing predicted and actual Kt for data points withheld fiom the H-H screen statistical analysis

I Screen Test Nurnber 6451we 1899jxe 1737kye 1663lve 1798kwe 1933jye 409jve 667iyd 90 1 iue 945jye

Predicted Kt 0.4 1 0.6 1 0.57 0.50 0.52 0.67 0.52 0.7 1 0.63

Average absolute prediction error, %

5.7.2 H-V Screen Orientation

The same number of experimental mns was conducted using the H-H and H-V screen

orientations. The full natural log equation, [5-141, continues to provide a better estimate

of Kt than the power equation, [S-151, shown by the higher R~ value of 0.8 1 as compared

to 0.71. The standard error is slightly high for the natural log equation though, 0.058

versus 0.049 for the power equation.

Again it was found that the second screen had a greater influence on Kt than the first

screen, judging by the standard coefficient (0.30 and 0.18 respectively). Since the

tolerance was very high (0.96) and the two screens are known to be independent

variables, this assessrnent was deemed accurate. Wave steepness was suggested as the

most infiuential parameter with a standard coefficient of - 1.1 but, as discussed previously,

the rernaining variables are highly correlated which can influence the accuracy of the

standard coefficient. Table 5-6 gives a suxnmary of the variable contribution to the total

Kt for the H-V equation. It also gives the applicable range for use of these equations.

Again it cm be seen that wave steepness is the most influential variable while G/L is the

leas t influential.

Table 5-6 Table showing contribution of variables to Kt for H-V screen equation

Range of variable Range of variable 1 contribution to Kt

The initiai statistical analysis suggested three possible outliers. While these points had

large residuds, they were not removed from the data set as they were all from the same

test set (40H-60-20V) and were closely grouped. The large nurnber of data points within

the whole test range reduced the leverage of these three outliers and did not cause the R'

value to be much Iower than that for the H-H system. However, figure 5-1 1 demonstrates

that there were several data points that skewed the relationship. As a result, the mode1

under predicted the true Kt by a small margin for most points, but over predicted Kt more

97

substantially for a few others. There was one data set in particdar, 40H-60-20V, for

which the regression equation over predicted Kt, as is highlighted in figure 5- 1 1.

Accual Kt vs Predkted Kt for an WV Screen System uslng the Natural Log Eiquation

Actual Kt vs Preùicted Kt for an K V Screen System wlng the Pawer Equation

ID

Figures 5-lla & 5-l lb Graphs of predicted vs actual Kt values for an H-V screen system using both the natural log and power equations

Two data points fiom the 40H-60-20V were randomly removed before the statistical

analysis for the purpose of benchmarking. The transmission coefficient for these two

data points were si+cantly over predicted as can be seen in table 5-7. The 40H-60-

20V test set should be repeated to see whether this group was an anomaly or a true result.

There is no justification for rernoving the group completely from the analysis at this time.

Seven other validation points, shown in table 5-7, had a predicted and actual Kt within

10% of each other, two being perfect predictions. There was one other point that was

over predicted by 16% and this was explained as natural variation.

Table 5-7 Table comparing predicted and actual Kt for data points withheld fiom the -

H-V screen statisticai analysis

5.7.3 V-H Screen Orientation

Screen 30H-60-20V

30AH-25-30V 30AH-99-30V 30H4 1 -4OV 40H-60-20V 40H-60-20V 4OH-26-30V 4OH-60-30V 40H-15 1-30V 40H- 15 1 -30V

The foilowing naturd log and power equations were developed for the V-H orientation.

A greater number of tests were perfomed using the H-H and H-V orientations than V-H

or V-V. These latter two orientations were omitted fiom several test series to allow a

wider range of gap spaces to be tested. WhiIe this should have statisticaLly weakened the

results there was littie ciifference in the R~ values of the four orientations. Again, the

Test Number 538iue

Average absolute prediction error % (excluding outliers)

1871jze 1838jze 1990kue 51'7kye 5131we 1 134kue 2050iwe 1029Lwd 1045Qe

Predicted Kt 0.5 1 0.64 0.60 0.59 0.54 0.49 0.59 0.69 0.57 0.60

0.60 0.6 1 0.50 0.30 0.38 0.61 0.68 0.62 0.60

Actuai Kt 0.47

% Difference 9

naturd log mode1 had a higher lX2 value than the powet curve, 0.79 to 0.69. The standard

error of estimate 0.085 for equation [5-161 and 0.078 for equation [S-171.

The analysis of the standard coefficient, tolerance values and variable contribution

continued to show that wave steepness was the most influential variable, while G/L was

the least influential. Table 5-8 sumrnarizes the results for the variable contributions

towards Kt for the V-H screen system, as well as the range of variables to which the

equations can be applied.

Table 5-8 Table showing contribution of variables to Kt for V-H screen equation

Variable

No points lay outside the three standard deviation definition of an outfier. However, a

trend was noted within the 30V-60-20H result, as shown in figure 5-12. The Kt for tests

614iye, 615jye, 616kye, 6171ye and 6l8jze were al1 over predicted by the regression

equatioii by about 70%. The other data points within the group are unrernarkabIe which

hints that experimvntal error could have been the cause. Furthermore, these were the last

5 tests of the day, but there is no conclusive proof that these are experimental errors. It

would have been desirable to repeat these tests, but this was not possible due to the

dismantling of equiprnent.

Range of variable

0.3 + 0.4 0.2 -+ 0.3

0.0048 + 0.448 0.0 16 + O. 145 0.043 + 1.7 1

Range of variable contribution to Kt -0.242 + -0.184 -0.583 + -0.436 0.790 + 0.1 19

-0.409 + -0.191 0.10 + -0.02

Actuol Kt vs Predkted Kt for a V-H Screen System osing the Naturat Log Equation

10

O-' y on O D 0.1 02 0 3 Od O S Ob 0.7 0.8 0.9 ID 1.1

Predicted Kt

Actual Ktvs Predicted Kt for a V-H Screen System using the Power Equatkm

OB t

-- O 0 0.1 0 2 0J 0 4 05 Ob 0.7 0.8 0.9 1 0 1.1

Predictsd Kt

Figures 5-12a & 5-l2b Graphs of predicted vs actual Kt values for an V-H screen system using both the natural log and power equations

Test 6 18jze was not actuaUy plotted in figure 5- 12 because it was removed pior to the

statistical analysis. Table 5-9 contains all the points that were withheld from the

statistical analysis but test 618jze stands out because the predicted value is almost double

the actual Kt. The three other verification points closely match predicted and actuai Kt

magnitudes, showing that the mode1 does work for the majority of the data set.

Table 5-9 Table comparing predicted and actual Kt for data points withheld from the V-H screen statistical analysis

Screen 1 Test Number 1 Predicted Kt 1 Achial Kt 1 % Difference 30V-60-20H

30AV-50-30H 40V-60-20H

40V- 15 1 -30H

6 18jze 1691ite 439iue 96 liwd

Average absolute prediction error % (excluding outlier)

0.48 0.53 0.54 0.70

6

0.26 0.53 0.62 0.72

83 O

- 14 3

5.7.4 V-V Screen Orientation

The natural log equation is given in equation [5-181 while the power equation is given in

equation 15-191. The R' values were 0.80 and 0.70 respectively while the standard errors

of estimate were 0.08 1 and 0.073 respectively.

The V-V orientation produced the largest average Kt of all the screen orientations. The

P l standard coefficient was lowest for the V-V orientation at 0.1 1. In cornparison, the

next lowest standard coefficient for this pararneter was 0.16 for the V-H orientation. The

V-V orientation dso had the second lowest standard coefficient, 0.29, for the second

screen porosity coefficient. These arguments quantitatively demonstrate that this

orientation was the poorest solution for reducing Kt within the test range.

The variable contribution analysis shown in table 5-10 continues to demonstrate that

wave steepness is the most influential variable while the porosity of the second screen is

the second most influential. The variable range for these equations is identical to those

given previously.

Table 5-10 Table showing contribution of variables to Kt for V-V screen equation

Variable

Pl P2 H/L d / g ~ Z G/L

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

The V-V test series also shows that a high R' term can be accornpanied by substantial

differences between the predicted and actual Kt values. While no outliers were recorded

in this analysis, figure 5-13 graphically shows that there is still a fair amount of scatter.

Most of the points lie within 10% of the actual Kt but it is interesting to note that the

boundary condition of perfect transmission is poorly modeled. The reason for this is the

ballooning of errors when wave steepness approaches zero.

Range of variable

0.3 + 0.4 0.2 + 0.3

0.0048 + 0.448 0.016 + O. 145 0.043 + 1.71

Actual Kt vs Predicted Kt for a V-V Screen System using the Natural Log Equation

Range of variable contribution to Kt -0.158 + -0-120 -0.620 + -0.464 0.780 + 0.1 17

-0.455 4 -0.212 0.13 + -0.02

on J On 0.1 OZ 03 OA 05 0.6 0.7 0.8 0.9 ID 1.1

Predlcted Kt

Actual Kt vs Predicted Kt for a V-V Screen System using the Power Equation

OD OD 0.1 02 03 OA as oh 0.7 oa 0.9 in 1.1

Predided Kt

Figures 5-13a & 5-13b Graphs of predicted vs actual Kt values for an V-V screen system using both the natural Log and power equations

The four points removed to validate the mode1 show how rnisleading the R~ value can be.

The mode1 gives a general equation that passes through the points but individual data

points can still be over or under predicted by as much as 34%, as shown in table 5-1 1.

Table 5-11 Table comparing predicted and actual Kt for withheld data points within the V-V screen statistical analysis

1 Screen 1 Test Number 1 Predicted Kt 1 Actud Kt 1 % Difference 1

5.7.5 Combined H-H and H-V Screen S ys tems

3OV-59-20V 3OV-59-20V 4OV-59-20V 4OV-4 1-3 OV

The differences observed between the performance of the four double screen orientations

were relatively smail. Thus it was decided to analyze the data in groups in order to

reduce the number of equations required to predict Kt.

The H-H and H-V data sets were analyzed as one group and had an R~ value which lay

between the R~ value of the separately analyzed H-H and H-V for both the natural log

and power equations. The values were 0.82 and 0.72 respectively. The standard errors of

estimate was similar (0.058 and 0.050) but the standard error of each variable was lower

with the combined data, as was the residual mean-square. The F-ratio was higher when

the two sets were grouped. It was thus deemed preferable to formulate one equation for

the H-K and H-V screen systems and the equations are given below. The combined data

set had 509 data points were used in the regression analysis.

562iud 570ite 4851ye 733iyd

Average absolute prediction error %

0.49 0.45 0.44 0.75

17

0.54 0.59 0.33 0.73

9 -23 34 2

There were three suggested outliers but these were all from the same test set, 40H-60-

2OV. As there is no information that gives cause to remove them, they were included in

the analysis, though the tests should be repeated in the future to ensure that they are not

tme outliers. Figure 5-14 shows a cornparison of measured and predicted Kt for the

complete data set while highlighting the 40H-60-20V group.

Actual Kt vs Preâkted Kt for a WH and K V Screen Systems ushg the Naturat Log 6quaUoci

ActuaI Kt vr Predicted Kt for WH and K V Screen Systems using Uie Power GquaUon

Figures 5-14a & 5-14b Graphs of actual vs predicted Kt values for H-H and H-V screen systems using the natural log and power equations

Before completing the regression analysis for the combined data set, 20 points were

removed for validation purposes. The actual and predicted Kt values for these points are

given in table 5-12. The average percentage difference between the predicted and actual

Kt was 3%. The average absolute difference was 8%. This showed that the prediction

mode1 was accurate and could be used instead of separate equations for H-H and H-V.

Table 5-12 Table comparing predicted and actud Kt for withheld data points from the combined H-H and H-V statistical anaiysis

Screen 30H-60-20H

30AH-25-30H 30-1-30H 30AEf-50-30H 3 OAH-99-3 OH 30H-60-40H 40H-60-20H 40H-4 1 -3OH 40H-99-30H

4OH- 15 1-30H 30H-60-20V

3OAH-25-30V 3OAH-99-30V 30H-41-40V 40H-60-20V 40H-26-30V 40H-60-30V 4OH-15 1-30V 40H-15 1-30V

Test Number 645LWe 1899JXe 1737KYe 1663LVe 1798KWe 1933JYe 409JVe 667iYd 901iUe 945Ne 538iUe 187 1 JZe 1838JZe 1990KUe 513LWe 1 l34KUe 2050iWe 1029KWd 1045KYe

Average absolute nrt

0.35 0.63 0.57 0.56 0.6 1 0.59 0.46 0.72 0.60 0.63 0.47 0.60 0.6 1 0.50 0.3 8 0.61 0.68 0.62 0.60

iction error %

% Difference 21 -2 2 -9

-13 13 15 -2 6 1 5 6 -3 16 26 -2 2

-10 O 8

5.7.6 Combined V-H and V-V Screen Systems

The second grouping analyzed was the combination of the V-H and V-V resdts, that is

the data for multiple screens having a vertical screen as the initial screen. The R~ value

was as high for the combined data as for the individual data sets, 0.79 for the natural log

equation and 0.69 for the power equation. The standard error of estimate (0.082 and

0.074 respectively) and residud mean-square were similar while the standard error of

each individual coefficient was lower. This indicates that this equation is statistically

comparable to either the V-H or V-V equations presented earlier. The prediction

equations for the combined data are presented on the following page.

The statistical analysis did not indicate any outliers or points having a large leverage

amidst the 185 data points included in the analysis. Figure 5- 15 shows the distribution of

points around the perfect prediction mode1 line.

Adual Kt vs Predided Kt for V-H and V-V Screen Sys tems uslng the Natural Log 6quatlon

, ln i -

oa I OD 0.1 0 2 03 04 OS Ob 0.7 08 0.9 ID 1.1

Predicted Kt

Actual Kt vs Predicted Kt for V-H and V-V ~creen- Systems using the Ehtural Log Equation

Figures 5-15a & 5-15b Graphs of actual vs predicted Kt values for V-H and V-V screen systems using the natural log and power equations

This figure shows that there is similar scatter to figures 5-12 and 5-13. Combining the

data was thus justifiable and reduced the nurnber of equations needed to predict Kt.

Table 5-13 confirmed this as the absolute average error for the randomly removed data

points was only 7% compared to 5% and 11 % when V-H and V-V were analyzed

separately.

Table 5-13 Table comparing predicted and actual Kt for data points withheld fiom the cornbined V-H and V-V statistical analysis using the natural log equation

Screen 30AV-50-30H

5.7.7 Other Data Set Combinations

40V-60-20H 40V-151-30H 30V-59-20V 30V-59-20V 4OV-59-20V 40V-41-30V

Several other data set combinations were examined. The most obvious combination was

Test Number 1691iTe

to group al1 the double screen data sets and analyze them together. This had a lirnited

Average absolute prediction error % (excluding outliers)

439iUe 961iWd 562iUd 57OiTe 485LYe 733iYd

degree of success. The R' value of 0.80 was lower than the grouped H-H, H-V and V-H,

Predicted Kt 0.55

V-V combinations. This analysis did not return any outiiers, which showed that the

0.53 0.69 0.49 0.45 0.45 0.75

standard deviation was much larger than the previous analyses, increasing the uncertainty

Actuai Kt 0.53 0.62 0.72 0.54 0.59 0.33 0-73

of the prediction. Both the individual and grouped equations developed earlier in the

% Difference 3

chapter are better modeIs so the lone equation to predict ai l double screen systems was

rejected.

A grouping of data based on both screens in the series having identical orientations was

then investigated, so that the H-H and V-V data was cornbined. The natural log R~ value

was Iower for this analysis, 0.80, than when the data sets were separated. The sarne is

true for the power equation, where the R' value was 0.70. This grouping naturaily lead to

comparing screen systems where each screen in the series had a different orientations. So

the H-V and V-H data sets were merged and anaiyzed. The R~ values for the natural log

and power equations were 0.79 and 0.69 respectively, which was again lower than the

individual anaiyses. The standard error of estimate and residual mean-square for the

merged data was the average of the descriptors when the data was separate. The

variability caused by the grouping sequence, and the questionable method of combining

the data in these particular combinations caused the H-H, V-V and H-V, V-H grouped

data equations to be rejected as a prediction tool.

5.7.8 Prediction of Double Screen Performance using Single Screen Equations

An attempt was made to see if the single screen equations could adequately mode1 a

double screen system or if there was increased dissipation or transmission due to

reflection and wave-wave interaction between the two screens. This involved applying

the single screen equation in series to allow for different screens systems. For exarnple,

modelling an H-V screen system would require using the horizontal screen equation with

the incident wave height to find the transrnitted wave height through the first screen.

This transrnitted wave height would then act as the incident wave height for the vertical

screen equation to calculate the transmission through the second screen.

As can be noted in figure 5-16, the predicted Kt was much iower than the actual Kt using

this method. Five other figures are included in Appendix L for the following data test

series: 30H:60:20H, 30H:SO:30V, 4OH:ZS: 3OH, 40H:40:30H, 40H: 1 OO:3OH and

40H:150:30H. These remaining figures show that this method does not always under

predict Kt. Figures L1 and L4 over predict Kt by an average 29% and 15% respectively.

Figure L2 shows a good estimate while the other series, L3, L5 and L6 are ali under

estimated by the extrapolation. The R~ value for 30H:60:20H was 0.35. These figures

and low R~ value illustrate that this method is not reliable and that the wave interaction

between the two screens caused either an increase in dissipation or transmission. It was

concluded that the double screen equations provide a better prediction of the performance

of a double screen system than a combination of single screen equations.

Actual vs Predicted Kt uslng Single Scrssn Quation Twlœ for JOtt5952ûH

0.7

Figure 5-16 Graph showing acîual vs predicted Kt values for prediction of a double screen system using the single screen equation twice

5.8 Triple Screen Equation

It was not possible to develop an empirical triple screen equation because there were not

enough screens with differing porosity and similar slat size to develop an equation. The

slat size had to be the same for each screen to eliminate this variable from the equation.

This did not aLlow the porosity to be changed as only 4 screens had the same slat size and

two of these were identical screens. The goal was thus to try and develop a triple screen

equation by combining the single and double screen equations. There are three options

for predicting the triple screen Kt; use the single screen equation three times, use the

single screen equation followed by the double screen equation or use the double screen

equation followed by the single screen equation.

The extrapolation of the single screen equation to predict a three screen system was

similar to the extrapolation for predicting the double screen Kt. The input wave condition

for the first screen was the incident wave height and penod while the reduced wave

height was then used as the input wave height for the second and third screens. As c m be

seen in figure 5- 17, this method over predicts Kt by an average of 23%. The

correspondhg R~ value using this method was 0.33. Appendix M contains 5 other graphs

confirming this trend (M2 to M6). Kt is over predicted because the method of using three

single screens to calculate Kt ignores the effect of reflection and wave-wave interaction.

Reflection between screens would cause an increase in wave steepness and thus an

increase in energy dissipation. This method does not accurately predict Kt and was

rejected.

Aaual vcr Predlcted Kt uslng Single Screen Quatlons for 4ûHSCMCIHSOMH

Figure 5-17 Actual vs predicted Kt when using the single screen equations to predict Kt through a triple screen

The second option for trying to predict Kt through a triple screen system is to combine

the single and double screen equations. The single screen equation was applied to the

incident wave conditions. The transmitted wave conditions were then entered as the

incident wave conditions for the appropriate double screen equation. This method

worked well as can be seen in figure 5-18 and appendix M, figures M7 to M18. The R'

value of 0.67 confirmed this. The average absolute difference between the actual and

predicted Kt was 5%. However, the average difference was only 0.5% showing that this

method neither over or under predicts Kt. This method can be used to predict Kt for a

triple screen system. Further experiments should still be performed to confirm this, using

different screen porosities. A larger number of results can also be used to empirically

derive an equation that predicts a triple screen systern for comparison with this method.

Actual vs Predicted Kt using Single then üouble Screen 6quation for

40tt6030H50MH

R

09 0.1 02 O 3 Od 05 Ob 0.7 O 8 0.9 1 1

Predicted Kt

Figure 5-18 Actual vs predicted Kt using a single then double screen equation to predict Kt through a triple screen

Using a double screen equation first followed by a single screen equation proved futile, as

displayed in figure 5-19. This method yielded an R~ value of only 0.37. The remaining

graphs are shown in appendix M, figures Ml9 through M30. The average over prediction

of Kt was 17% and furthemore, not a single point was under predicted. Thus this

method was rejected for predicting Kt for a triple screen system. It is interesting that the

order in which the single and double screen equations are applied to the incident wave

conditions can alter the result.

I Acturl vs Predicted Kt using Double than Single Sueen Quation for

40HSOSût-kSûMH

OD 0.1 0 2 0 2 0d 05 Ob 0.7 OB 0.9 11

Predictsd Kt

Figure 5-19 Actual vs predicted results using a double then single screen equation to predict Kt for a triple screen system

5.9 Cornparison with Other Theories

Hartmann (1969) presented the following equation for predicting Kt based on the wall

elernent ratio.

where W = wall element ratio @/e)

As can be seen clearly in figure 5-20, this expression cannot be used to predict Kt

measured in these experirnents for a single vertical screen. This was expected as the

equation does not consider any variable except porosity. This results in the perfectly

vertical lines that can be noted in the figure.

Cornparison of Hartmann's Equation (1 969) and Ejquatlon [S-61 for a 20H Screen

OD 0.1 02 03 Od 0.5 Ob 0.7 0 8 0.9 ID Predlded Kt

Figure 5-20 Actual vs Predicted Kt using Hartmann's Equation for a single screen

Hartmann's equation under predicts Kt by an average of 56%, whereas

presented in this thesis, over predicts Kt by an average of 2%.

Mei (1983) developed the foilowing equation.

equation

where Cc = 0.6 + 0 . 4 ~ ~

IQ = deepwater wavelength

P = porosity

Mei's equation included deepwater wavelength and porosity. However, it stiiI did not

give an accurate prediction of Kt for a single screen, as can be seen from figure 5-21,

because it was theoretically devised and did not include the effects of reflection,

Camparison of Mei's Equation and Equation [Se] 1 for a 20H screen

:; o Më

1 Eadm (5-61 0.0 0.0 0.1 0 2 0.3 0.4 05 04 O.? OB 0.9 1 0 1.1

Predicted Kt

Figure 5-21 Actual vs Predicted Kt using Mei's equation for a single screen

Mei's prediction was superior to Hartmann's but it stiU over predicted Kt by 19% on

average compared to 2% for equation [5-61.

5.10 Summary

Sixteen empirïcal equations were derived for predicting Kt through single and double

screen systems, two for each orientation (H, V, H-H, H-V, V-H, V-V) and two for each

set of grouped data (H-H, H-V and V-H, V-V). A natural log and power form equation

was developed for each system. The naturd log form always had a higher R~ value than

the power form. A boundary condition had to be applied to these equations so that at

negligible wave steepness there was complete transmission. These equations should

therefore not be used for wave steepnesses less than 1x10~~.

The single screen equation incorporated three variables: screen porosity, wave steepness

and It was found that wave clirnate had a greater effect on Kt than screen porosity

but, because the wave climate variables were interrelated, it was not possible to quantify

this effect. The best equation for predicting Kt for a single horizontal screen is equation

[5-61 and is summarized directly below. The R~ value is 0.84 while the standard error is

0.052.

The best equation for predicting Kt through a single vertical screen is equation [5-81. The

R~ value is 0.77 while the standard error is 0.066.

These equations should only be used when the screen porosity is between 20 and 40%,

wave steepness is between 0.005 and 0.5 and d / g ~ 2 is between 0.016 and 0.145.

These equations were developed using a b/t ratio of 2. The difference between the actual

and predicted Kt (using equation [S-61 or [5-81) for screen 30b (b/t ratio of 1) was 18%

for the horizontal orientation and 6% for the vertical orientation.

The double screen equations incorporated five variables: porosity of the first screen,

porosity of the second screen, wave steepness, d / g ~ 2 and dimensioniess gap space. Wave

period and height were key factors affecting Kt. The second screen had a greater impact

on Kt than the first screen. Depth and gap space had a negligible effect on Kt. Grouping

the individual data sets by the orientation of the first screen reduced the number of

equations necessary to derive an estimate of Kt.

A double screen with either an H-H or H-V slat orientation equation is best predicted

using equation 15-20]. The R~ value is 0.82 while the standard error is 0.058.

Finally, a screen system with either a V-H or V-V slat orientation should be predicted

using equation [5-221. The R~ value is 0.79 while the standard error is 0.082.

Using the single screen equation twice to predict the behaviour of a double screen system

did not provide an accurate prediction tool because of increased surface agitation between

the two screens that the single screen equation did not incorporate during its formulation.

Triple screen equations could not be produced because there was insufficient variation of

screen porosity. However, combining a single screen equation foiiowed by a double

screen equation produced an accurate prediction of Kt through a triple screen systern.

The equations developed in this thesis predict Kt better than those developed by

Hartmann or Mei.

6 Conclusions and Recommendations

6.1 Conclusions

Wave screens are a compelling breakwater option due to their srnall footprïnt, aesthetic

value and ability to allow water circulation within a harbour. However, an engineer has

little practical basis for the initial design of a wave screen breakwater.

Based on an extensive set of two-dimensionai hyclraulic mode1 tests, designed to

investigate transmission of wave energy through full depth, emergent wave screens, the

following general observations were made with respect to the effect of several variables

on the transmission coefficient, Kt.

1. Kt was found to decrease with increasing wave steepness.

2. Kt was found to decrease with an increase in wave height.

3. Kt was found to decrease with a decrease in depth.

4. Kt was found to decrease with an decrease in screen porosity.

5. Kt was found to decrease with an increase in the number of screens placed in

series.

6. No distinct trend could be noted with an increase in gap space between screens.

Wave steepness was found to have the most significant influence on Kt. Depth had a

very limited influence on Kt in intermediate or deep water (d/L > 0.05).

A systern utilizing two screens in series was evaluated and it was found that the porosity

of the f is t screen was the determinhg factor in the grouping of double screen

orientations but the second screen had a greater iduence on the overall Kt. It was found

that H-H and H-V screens had similar performance characteristics, wMe V-H and V-V

had comparable Kt values. It was therefore concluded that only the orientation of the

first screen is important. The orientation of the second screen was found to have no

significant influence on the total Kt.

A series of empirical design equations were developed for single and double screen

systerns based on a detailed statistical analysis. Four variables were incorporated into the

single screen equation, including the porosity of the screen (Pl), wave steepness (Hn)

and dlg~ ' . Two additional variables, porosity of the second screen (P2) and

dixnensionless gap width (G/L), were included in the double screen system equations.

Natural log and power equations were developed for each screen orientation combination.

The nahual log equation always had a higher correlation coefficient (an average of 0.1

higher) than the corresponding power equation. Zt was found that accurate predictions of

Kt were possible using the following four equations for the single or double screen

system in various orientations.

For a single horizontal screen (lX2 equals 0.84, standard error of 0.052):

For a single vertical screen (lX2 equals 0.77, standard error of 0.066):

For double screen with either an H-H or H-V slat orientation (R' equals 0.82, standard

error of 0.058):

For a double screen system with either a V-H or V-V slat orientation (R' equals 0.79,

standard error of 0.082):

These equations are only applicable when the variables lie within the range given in table

6-1.

Table 6-1 Table summarizing variable range

An equation for predicting total Kt was not developed for a triple screen system.

However, it was possible to predict Kt (within an average of 1% of the actual value) for a

triple screen system by using a combination of the appropriate single screen equation

followed by the double screen equation.

Equations 16-11 to 16-41 are based on screens having a slat width to thickness ratio @/t) of

2. These equations did not accurately predict Kt for screen 30b (b/t ratio of 1) resulting

in an average absolute error of 18% for the horizontal orientation. Furthemore, these

equations were developed fiom tests conducted in intermediate and deepwater only.

Double Screen Range 0.3 -+ 0.4 0.2 + 0.3

0.0048 -+ 0.448 0.016 + 0.145 0.043 + 1.7 1

Variable Porosiw of 1" screen Porosity of 2nd screen H/L cI/~T' G/L

6.2 Recomrnendations

Single Screen Range 0.2 + 0.4

- 0.0048 + 0.448 0.016 + 0.145

-

The following recommendations for future work and are put forth on the bais of the

present findings. The equations presented could be further vefiied and expanded by

conducting additional tests in a two dimensional flume investigating:

1. Screen porosities lower than 20%.

2. Triple screen systems with different gap spaces and porosities.

3. b/t ratios of slats at values other than 2.

4. Screens made of different materials to investigate influence of flexùig.

Associated work in this area would also help deveiop practicai tools on which engineers

could base an initial design of a wave screen. Recornrnended work includes:

1. Three-dimensional tests of wave screens to investigate the effects of incident

wave angle and diffraction on Kt.

2. The development of equations that focus on reflection and energy dissipation.

Reflection is often a concern and major design criterion for engineers, especidy

in areas of ship traffic. Maximizing energy dissipation May dso be of g e a t

importance, for both reflection and transmission purposes.

3. The investigation of partial depth, ernergent screens and submerged screens.

Research has been performed in these areas but no useful, ernpirical equations

have been presented.

Alisop, N.W.H., (1995), "Vertical W d s and Breakwaters: Optimisation to Improve Vessel Safety and Wave Disturbance by Reducing Wave Reflections", Wave Forces on Inched and Vertical Wall Structures, ASCE, pp 232-257.

Bennett, G.S ., McIver, P. and Srnailman, J.V., (1992), "A Mathematical Mode1 of a Slotted Wave Screen Breakwater", International Journal for Coastal, Harbour and Offshore Engineering, Vol 18, Nos. 3,4, pp 23 1-249.

Bergmann, H. and Oumeraci, H., (1998), "Wave Pressure Distribution on Permeable Vertical Walls", Proc., 26" Coastal Engineering Conference, ASCE, Vol 2, pp 2042- 2055.

Clauss, G.F. and Habel, R., (L999), "Hydrodynamic Charac teris tics of Underwater Filter Systems for Coastal Protection", Canadian Coastal Conference 1999, pp 139-154.

Cox, R.J., Horton, P.R. and Bettington, S.H., (1998), "Double Walled, Low Reflection Wave Barriers", Proc., 26" Coastal Engineering Conference, ASCE, Vol 2, pp 2221- 2234.

Daily, J. W. and Harleman, D.R.F., (1 973), "Fiuid Dynamics", Addison-Wesley Publishing Co., pp 317-320.

Gardner, J.D., Townend, I.H. and Fleming, C.A., (1986), "The Design of a Slotted Vertical Screen Breakwater", Proc., 20" Coastal Engineering Conference, ASCE, Vol 3, Ch 138, pp 1881-1893.

Gardner, J.D. and Townend, I.H., (1 988), "Slotted Vertical Screen Breakwaters ", Design of Breakwaters, Proc. Breakwaters '88 Conference, Thomas Teiford Ltd., Ch 15, pp 283- 298.

Goda, Y., (1985), "Random Seas and Design of Maritime Structures", University of Tokyo Press, Ch 3, pp 100-106.

Grune, 3. and Kohlhase, S., (1974), "Wave Transmission Through Vertical Slotted Walls", Proc., 14'~ Coastal Engineering Conference, ASCE, Vol 3., Ch 11 1, pp 1906- 1923.

Hartmann, (1969), "Das Stabgitter in Instationarer Stromungsbewegung", Mitt. des Instituts fur Wasserbau und Wasserwirtschaft, TU Berlin, Heft 69.

Hayashi, T. et al, (1966), "Hydraulic Research on the Closely Spaced Pile Breakwater", Coastal Engineering in Japan, Vol 9.

Hayashi, T., et al, (1968), "Closely Spaced Pile Breakwater as a Protection Structure Against Beach Erosion", Coastal Engineering in Japan, Vol 2.

Herbich, J.B., (1990), "Pile and Offshore Breakwaters", Handbook of Coastai and Ocean Engineering, Gulf Publishing Co., Vol 1, Ch 19, pp 895-920

Hunt, J.N., (1979), "Direct Solution of Wave Dispersion Equation", Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol 105, WW4, pp 457-459.

Isaacson, M., Premasiri. S., and Yang, G., (1998), "Wave Interactions with Vertical Slotted Barrier", Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE, Vol 124, NO. 3, pp 118-126.

Jarnieson, W.W., and Mansard, E.P.D., (1987), "An Efficient Upright Wave Absorber", Coastd Hydrodynamics, ASCE, pp 124-1 39.

Kakuno, S., Oda, K. and Liu, P.L.F., (1992), "Scattering of Water Waves by Vertical Cylinders with a Back Wall", Proc., 23rd Coastal Engineering Conference, ASCE, Vol 2, Ch 95, pp 1258-1271.

Kleinbaum, D.G., and Kupper, L.L., (1978), "Applied Regression Analysis and Other Multivariable Methods", PWS Publishers, pp 170.

Kondo, H., (1979), "Anaiysis of Breakwaters Having Two Porous Wdls", Coastal Structures 79, ASCE, pp 962-977.

Kriebel, D.L. (1992), "Vertical Wave Barriers: Wave Transmission and Wave Forces", Proc., 23rd Coastal Engineering Conference, ASCE, Vol 2, Ch 100, pp 13 13- 1326.

Knebel, D.L. and Bolimann, C.A., (1996), "Wave Transmission Past Vertical Wave Barriers", Proc., 2~~ Coastd Engineering Conference, ASCE, Vol 2, Ch 19 1, pp 2470- 2483.

Kriebel, D.L., Sollitt, C. and Gerken, W., (1998), "Wave Forces on a Vertical Wave B&ern, Proc., 26' Coastal Engineering Conference, ASCE, Vol 2, pp 2069-208 1.

Laboratory Technologies Corporation, (199 l), "Labtech Notebook, Technical Software

Laboratory Technologies Corporation, (199 l), "Labtech Notebook Reference Manual", Laboratory Technologies Corporation.

Mansard, E.P.D., and Funke, E.R., (1980), "The Measurement of Incident and Reflected Wave Spectra using a Least Squares Method", hoc., 17" Coastal Engineering Conference, ASCE, pp 154- 172

Mansard, E.P.D. and Funke, E.R., (1987), "On the Reflection Analysis of Zrregular Waves", Technical Report No.TR-HY-017, National Research Council of Canada, Ottawa.

Mei, C.C., (1983), "The Applied Dynamics of Ocean Surface Waves", John Wiley and Sons, New York, pp 253-268.

Miles, M.D., (1990), "User Guide for GEDAP Version 2.0 Wave Generation Software", Technical Memorandum LM-HY-034, National Research Councii of Canada, Ottawa.

Muramaki, H., Hosoi, Y. and Goda, Y., (1986), "Analysis of Permeable Breakwaters", Proc., 2 0 ~ Coastai Engineering Conference, ASCE, Vol 3, Ch 155, pp 2104-2 1 18.

Pelletier, D., (1990), "Real Tirne Control Systern Users Manual", Technical Report TR- HY-035, National Research Council of Canada, Ottawa.

Schulman, R.S ., (1992), "S tatistics in Plain English with Cornputer Applications", Van Nostrand Reinhold, pp 408.

SPSS Inc., (1998), "Systat V8.0", (Technical Software).

Truitt, C.L. and Herbich, J.B., (1986), "Transmission of Random Waves Through Pile Breakwaters", Proc., 2 0 ~ Coastal Engineering Conference, ASCE, Vol 3, Ch 169, pp 2303-23 13.

United States Army Corps of Engineers, (1984), "Shore Protection Manual", 4b Ed., US Government Printing Office, Vol 2, pp 7-198.

Weigert, A.W. and Edwards, E.P., (198 l), "Capacitative Water Level Gauges for Hydraulic Models: Laboratory Memorandum", National Research Council of Canada, pp 20.

Wiegel, R.L., (1961), "Closely Spaced Piles as a Breakwater", Dock and Harbour Authority, Vol 41, No 49 1.

Yalin, M.S ., (197 1)- "Theory of Hydraulic Models", Macmillan Press, pp 1-33.

Pictures of constmcted wave screens

Figure A Screen 20

Figure A Screen 30

Fig . .. -..-.+ .. - . . . . . . - ... ... ..7 . . . ...- ... ' - - -

Screen 30A

Figure A 4 Screen 30B

Figure A 5 Screen 40

Figure A 6 Screen 50

Figures showing Kt vs Wave Period for screen systerns differentiated by wave height

20 Horizontal

1

::I==j:i 0-10 x w

0.00 0.00 0.25 050 0.75 1.00 1.25 1150 1.75 200 225 25(1

Period (s)

30 Horizontal

on=5an

OZü f 6 H=7m I l

0-10 XH=9ani

0.00 0.00 025 050 0.75 1.00 125 150 50.75 200

Period (s)

30b Horizontal 1.00 , 1

O.W! . . . 0.00 025 0 3 0.75 1.m 1.25 1 3 1-75 200 225 2 3

Period (s)

0.00 ! I 0.00 025 050 0.75 1.00 125 150 1.75 200 22!ï 25(

Period (s)

30 Vertical 1.00 , I

0.00 ! 0.00 025 0- 0.75 1.00 125 1.50 1.75 200 225 W

Period (s)

30b Vertical

0.00 025 050 0.75 1.00 125 1 5 0 1.75 200 225 2%

Period (s)

Figures B I to 86 Kt vs Period for screen system differentiated by wave height

40 Horizontal

9 t

Period (s)

50 Horizontal 1

o . , C . . i 0.00 O25 0.50 0.75 1.00 125 1.50 1.75 200 225 2M

Period (s)

1 o m !- . . i l 0.00 025 0.50 0.75 1.00 125 150 1.75 200 225 2 5 C

Period (SI

40 Vertical / i.m,

0.00 U*-I . 0.00 025 050 0.75 1.00 125 1.50 1.75 2 0 0 225 W

Period (s)

50 Vertical

0-1° x H=9cm 1 1

0.00 ! I 0.00 025 0 s 0.75 1.00 125 150 1.75 ZOO 2.25 250

Period (s)

Periad (s)

Figures 87 to 812 Kt vs Period for screen system differentiated by wave height

1

,

0.00 025 0.50 0.75 1.00 1 2 5 1.50 1-75 2-00 225 W

Period (SI

o . m U x ~ - l . c I 0.00 025 0.50 0.75 1.00 1 2 5 150 1.75 200 225 2bC

Period (s)

Period (s)

0.00 025 0.50 0.75 1.00 125 1.50 1.75 200 225 ZSO~

Period (s) 1

Period (s) 1

0.00 025 050 0.75 1.00 125 1.50 1.75 2.00 225 250

Perlod (s) 1

Perioci (s) I

Period (s) f

Period (s)

0.00 025 O S 0.75 1.00 125 150 1.75 200 225 W

Period (s)

0.00 025 O S 0.75 1.00 125 150 1.75 200 225 250

Period (s)

Period (s)

Figures 81 9 to 824 Kt vs Period for screen system differentiated by wave height 1 8 1 9 1 8201

' o s { A 1 I

Period (s)

0.00 025 050 0.75 1.00 125 150 1.75 200 225 250'

Period (s) 1

0.60

S o m

0.40

0.30

0.30 O Hdcnl

Oz(- PH-

- Pm 1 . . . r . . . I I

0.00 XH- 0.00 025 050 0.75 1.00 1.25 150 1.75 200 225 25C

Period (SI

0.00 025 050 0.75 1.00 1.25 1.50 1.75 200 225 2 5 C

Period (s)

0.00 025 050 0.75 1.00 125 1.50 1-75 2.00 225 250

Period (s) Period (s)

0.00 0.00 025 050 0.75 1 . 0 1.25 150 1.75 200 225 250

Period (s)

Period (s)

A H=7an

. . . , , . , ,

0.00 025 O S 0.75 1.00 125 150 1.75 200 225 250

Period (s)

Oa Ha- ' le--+

Period (s)

Period (s)

Period (s)

Figures B31 to 836 Kt vs Period for screen system differentiated by wave height

0.10 -- O- A -7m

0.001 T

0.00 0 2 5 050 0.75 1.00 125 1 3 135 200 225 W

Period (s)

Period (s)

0.60 - .

Q o m -

Period (s)

Period (s)

0.00 025 050 0.75 1.00 1 2 5 1.50 1.75 200 2.25 250

Period (s)

1 '=A P o r i f A i

0.00 025 O 0 5 0.75 1.00 125 150 1.75 200 225 25

Period (s)

Figures 837 to B42 Kt vs Period for screen system differentiated by wave height

Period (s)

Period (s)

0.00 025 050 0.75 1.00 125 150 1.75 200 225 W

Period (s)

0.00 025 0.50 0.75 1.00 725 1 5 0 1.75 200 225 250'

Period (s) 1

1 Period (SI

Period (s)

Figures 843 to B48 Kt vs Period for screen systern differentiated by wave height

0.00 025 O50 0.75 1.00 125 150 1.75 200 225 2 5 0

Period (s)

:: - . ' a , ,

A x

1 Period (s)

0.00 025 0.50 0.75 1.00 125 150 1-75 200 225 250

Period (s)

Period (s) 1

Period (s)

Figures B49 to 854 Kt vs Period for Screen System differentiated by Wave Height

Period (s)

0.00 025 050 0.75 1.00 125 1 5 0 1.75 200 225 W

Period (s)

0.00 025 O 5 0 0.75 1.00 1 2 5 150 1-75 200 225 W

Period (s)

PerÏod (s)

0.00 025 0.50 0.75 1.00 1 .2 s 1-50 1.75 2.00 225 2.54

Period (SI

0.00 0.25 0.50 0.75 1.00 125 150 1.75 200 225 W

Period (s)

Figures B55 to 660 Kt vs Period for Screen System differentiated by Wave Height rn

Period (s)

Figure 661 Kt vs Period for Screen System differentiated by Wave Height

Figures showing Kt vs Wave Height for screen systems differentiated by wave period

20 Horizontal 1 .O0 ! I

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Wave Height (cm)

30 Horizontal 1.00

1

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0/

Wave Height (cm) 1

30b Horizontal 1.00 , I

Wave Height (cm) 1

20 Vertical 1.00 7

I 1 /

0.0 1.0 2 0 3.0 4.0 5.0 &O 7.0 8.0

Wave Height (cm) 9.0 10.0/

I

30 Vertical 1 1.001 1

1 Wave Ueight (cm) 1 30b Vertical

1 .00 I

i 1

Wave Height (cm)

Figures Cl to C6 Kt vs Wave Height for screen system differentiated by wave period pJ

40 Horizontal 1 .m

O

O90

- 0.00 I

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.C

Wave Height (cm)

50 Horizontal

I

I OBI-

#& X X 1 0.70.. f P

Lb 0

I 0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Wave Height (cm)

Wave Height (cm)

40 Vertical 1.00 ,

I

0.00 ! 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 lOA

Wave Height (cm)

50 Vertical 1.00 I

0.00 i . I 0.0 1.0 20 3.0 4.3 5.0 6.0 7.0 6.0 9.0 IO.(

Wave Height (cm)

Wave Height (cm)

Figures C7 to Cl 2 Kt vs Wave Height for screen system differeniiated by wave penod

C 11 C 12

, ! 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

z0.50 - nX fa I

X

+T=1.75s 1: [;-,Tl 0.00

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 102

Wave Height (cm)

oa

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 a 0 9.0 l0.c

Wave Height (cm)

XT=l.!jS

o.m!l~~d , , , . , 1 ' 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

1 1 I

Figures Cl 3 to Cl 8 Kt vs Wave Height for screen system differentiated by wave period

0-10

* h l *

+T=1.756

O T ~ O c t

XT=225a

020 1

0.00 4 XT=-

0.0 1.0 2.0 3.0 4.0 5.0 6-0 7.0 8.0 9.0 10.0

Wave Height (cm)

-

--

O.W+ 3 7

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 81) Q.0 10.1

Wave Height (cm)

0.10

+T=1.759

OT=20a

T-

--

Wave Height (cm) I l

I

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

0.10 -

3

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm) I

Wave Height (cm)

0.00 i 1

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1O.i

Wave Heiaht (cm)

-

Wave Hetght (cm) f

O Teos

XT-

figures Cl9 to C24 Kt vs Wave Height for screen system differentiated by wave period

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 6.0 9.0 10.0

Wave Height (cm)

0.00

Wave Height (cm)

l

?

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 6.0 9.0 10.

Wave Height (cm)

0.30

XT=225s 0.00

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Wave Height (cm) 10iOl

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 101

Wave Height (cm)

- --

Figures C25 to C30 Kt vs Wave Height for screen system differentiated by wave period

+T=1.7!3 0.10 .- oTao5

0.00 1 . . . .

o T 4 . 7 5 ~

a T=l .Os

AT=ISS

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10J

Wave Height (cm)

O

Wave Height (cm)

0.00 I I 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 101

Wave Height (cm)

Wave Height (cm) J

4 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm) f

0.10 OT=20s

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 IO.(

Wave Height (cm)

o . 0 0 1 l x ~ - I . . I 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 IO.(

Wave Height (cm)

Figures C31 to C36 Kt vs Wave Height for screen systern differentiated by wave period

1 Wave Height (cm) 1 l

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.

Wave Height (cm)

I Wave Height (cm)

1 loT=2.01 1 0.00 i

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

0.70 0 0.60 i d - % 3 4 Y

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.C

Wave Helsht (cm)

020 #:zn:I +T=1.7Ss

0.10 OT=2oS

0.00 0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.1

Wave Height (cm)

Wave Heig ht (cm)

1 Wave Height (cm)

a-O 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1o.c

Wave Height (cm)

1 O." :

I Wave Helght (cm)

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 IO.(

Wave Height (cm)

Wave Height (cm)

Figures C43 to C48 Kt v s Wave Height for s c r e e n systern differentiated by wave penod

0.00 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7-0 8.0 9.0 10.0

Wave Height (cm)

0 .0 ! IxT-.J . 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1O.C

Wave Height (cm)

Wave Height (cm)

Wave Height (cm) 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.01

l I

Y

OT=l.Os 030 - ( L T = ~ W

Wave Height (cm)

Wave Height (cm)


I X T I Z Z ~ ~ 1 * I 0.0 1.0 2 0 3.0 4.0 5.0 6.0 711 a 0 9.0 1O.C

Wave Height (cm)

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.C

Wave Height (cm)

1 ~ ~ ~ 2 2 5 5 ] I 0.00

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.C

Wave Height (cm)

,, 1 X T = Z ~ S ~ 1 0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 4.0 10.C

Wave Height (cm)

Wave Height (cm)

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)


1 Wave Height (cm)

Figure C61 Kt vs Wave Height for screen system diHerentiated by wave period

Figures showing Kt vs Wave Height for screen systems differentiated by water depth

20 Horizontal

Wave Height (cm) 1

30 Horizontal 1 .O0 1

I 0.0 1.0 20 8 0 4.0 5.0 6.0 7.0 8.0 9.0 10.i

Wave Height (cm)

30b Horizontal 1.00 , 1

I o m J . . . 1

I 0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.1

Wave Hsight (cm)

20 Vertical 1.00 -

1 1

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

30 Vertical 1 .O0

I I

i 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 a o 9.0 10.0

Wave Height (cm)

30b Vertical 1 .O0

t i i

Wave Height (cm)

Figures Dl to D6 Kt vs Wave Height for screen systern differentiated by depth

40 Horizontal 1.00 -.

O

0.90 -

! A DepW.9m 0.00 i . ,

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1O.C

Wave Height (cm)

0.00 1 :O 1 0 & 40 & 7:O & 910 1;.

[ Wave Height (cm)

Wave Height (cm)

I D O...

.O- 1; CT O - Y LI u

40 Vertical 1 .00 1

l l

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 IO.[

Wave Height (cm)

50 Vertical 1.00

0.0 t.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

Wave Height (cm)

Figures D7 to D l 2 Kt vs Wave Height for screen system differentiated by depth

0.00 . 0.0 1.0 2 0 3.0 4.0 5-0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

1 Wave Heiaht (cm1 1

Wave Height (cm)

"0????0 5:. 6. 7:O 9:O

Wave Height (cm)

O20 -

0.00 1 0.0 la 2 0 3.0 4.0 5.0 6.0 7.0 a.o 9.0 1o.a

Wave Height (cm)

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.1

Wave Height (cm)

Figures Dl 9 to 024 Kt vs Wave Height for screen system differentiated by depth

Appendix E

Figures showing Kt vs Wave Steepness for screen systems differentiated by wave penod

20 Horizontal 1.00

0.00 1 1 0.w 0.01 0.02 0.03 0.04 0.05 0.M

- - Steepness

30 Horizontal 1 1 I

1 Stee pness

30b Horizontal 1.00 1-, 1

0 . L 0.- o.".- J Steepness

20 Vertical 1.m , 1

Posa 0.40 - OT=O.75s

0.w ! -

1 0.00 0.01 0.02 0.03 0.04 0.05 0.M

Steepness

30b Vertical 1.w I

i I

Steepness

Figures El to €6 Kt vs Steepness for screen system differentiated by wave period 1 E 1 1 E 2 1

40 Horizontal t.00

O

40 Vertical

1

O20 -- XT=lSs 1 +T=l.75s

0.10 -- ,OT&Os

0.00 4 I 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Steepness / O-"o.; 0,l o . 0 O 0:

i Steepness

50 Horizontal

I

1 Pori Ili

1 0.00 0.01 0.02 0.03 0.04 0.05 0.m

Steepness

A 0.70 f - '+~=1.75s - X 1 0.60 - 0 Te.&

7 - -

XT=22!3 1 ~0.50 - a-

I

0.00 0.01 0.02 0.03 0.04 0.05 0.W l Steepness

50 Vertical

I

0.00 i I

0.00 0.01 0.02 0.03 0.04 0.05 0.w

f Steepness

t 0.00 0.01 0.02 0.03 0.04 0.05 0.M

Stee pness

Figures R to El 2 Kt vs Steepness for screen system differentiated by wave period

E l 1 E l 2

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Steepness

0.10 ,OT*Os

om / XT=ZZ%

0.00 0.01 0.02 0.03 0.04 0.05 0.M

Steepness

Stee pness

Steepness

Steepness l

steepness 1

Figures El 3 to El 8 Kt vs Steepness for screen system differentiated by wave penod

-.-

0.60 - O + x *~?h-- A

Zoso -n

Steepness

0.00 0.01 o . 0.03 0.04 0.05 0.M

Steepness

L X T = l b +T=c1.75s

0.10 -- O T d . 0 ~

0.00 1 XT- 0.00 0.01 0.02 0.03 0.00 0.05 0.Q

Steepness

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Steepness

-.. -

0.60 - O Ar O *

O 2 A

ZOSO -t n

0.00 UXT- l 1 0.W 0.01 0.02 0.03 0.04 0.05 0.M

Steepness

steepness I

020 20-

i

0.10 -

0.00 i 0.00 0.01 0.02 0.03 0.w 0.05 0.a

Steepness

Steepness

Steepness i

I Steepness

Figures €25 to E30 Kt vs Steepness for screen syçtem differentiated by wave period

40V : 59.5 : 20H 1.00 -.

O 0.m -

osa. X , Xt'1.a

I Steepness

XT4.58 - +TOI .75s

I 0.10 -- OT=20r

0.00, XT-?= 2 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Steepness

Steepness

Steepness

Stee pness

Figures E31 to E36 Kt vs Steepness for screen system differentiated by wave period

f l

Osa 1

Steepness

Steepness

o = ~ ~ ; ~ ~ 0.10 O T=2.09

0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.M

Steepness

0.1 0 t

0.00 0.00 0.01 o.m 0.03 o.os o.as o.a

Steepness

[ 0.10 l OT&O?r 1

Steepness

O T e o s

0.00 0.01 0.02 0.03 0.04 0.0s 0.a

Steepness

Figures E37 to €42 Kt vs Steepness for screen system differentiated by wave perÏod

0.00 0.L 0 O O O o., o.Li

Steepness

0.00 XT=2259 O P 0.0 ' O O 0 o., o.Li

Steepness

0.00 4 1 0.00 0.01 0.02 0.03 0.04 0.05 0.a

Steepness


0m XT=15S

t T ~ l . 7 5 ~ 0.10 -- OTeOs

0.00 * Xr-

0.00 0.01 0.02 0.03 O . 0.05 0.a

Steepness

0.00 0.01 0.02 0.03 0.04 0.05

Steepness

0.00 0.01 0.02 0.03 0 . 0.05 0.a

Steepness 0.00 0.01 0.02 0.03 0.04 0.05 0.a

Steepness


;z-j:i::i 1 0.00 XT-

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Steepness

0.00 0.01 0.02 0.03 0.w 0.05 0.06

Steepness

Steepness

Steepness

I

I 0.00 0.01 0.02 0.03 0.04 0.05 O.@

Steepness

l 0.00 0.01 0.02 0.03 0.04 0.05 O M

Steepness

Figures ES5 to E60 Kt vs Steepness for screen system differentiated by wave period

I Steepness

Figure E61 Kt vs Steepness for screen system differentiated by wave period

Figures showing Kt vs Wave Steepness for screen systems differentiated by screen

porosity


lm A

0.90 Y

0.00 0.01 0.02 0.03 0.04 0.05 Wave Steepness


1.00

I I

1 I

0.00 ' Wave Steepness

Single Screen - Vertical ûepth = 0.8m

1 .O0

0.00 ! 0.00 0.01 0.02 0.03 0.04 0.05

Wave Steapness

Single Screen - Horizontal Depîh = 0.8m

1 .O0 1

0.00 A i l o.^ .L Al Wave Steepness I

Single Screen - Vertical Depth = 0.7m

0.00 0.01 0.02 0.03 0.04 0.05 0.a

Wave Steepness

Single Screen - Vertical Depth = 0.9m

0.00 J I

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

Figures FI to F6 Kt vs Period for screen system differentiated by porosity


0.m 0.01 0.02 0.03 0.04 0.w 0.06

Wave Steepness

- -


1.00 1 i

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

Horizontal : 255 : Vertical Depth = 0.8m

0.00 0.01 0.02 0.03 0.04 0.05 0.M

Wave Steepness

1

Horizontal : 40.7 : Vertical Depth = 0.8m

0.00 0.01 0.02 0.03 0.04 0.05

Wave Steeuness

Horizontal : 595 : Horizontal Depth = 0.7m

1 .O0

0.00 J f 0.00 0.01 0.02 0.03 0.04 0.05 0.a

Wave Steepness


I

0.00 0.01 0.02 0.03 0.M 0.05 0.M

Wave Steepness

Figures F7 to FI 2 Kt vs Period for screen system differentiated by porosity

Horizontal : 595 : Vertical Horizontal : 59.5 : Vertical Depth = 0.8m

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

1 Vertical : 59.5 : Horizontal üepth = 0.7m

l 1.00 - 1

090 - + 1 0.80 1

Wave Steepness

1 Vertical : 59.5 : Vettlcal ûepth = 0.7m I l m .

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

Wave Steepness

Vertical : 59.5 : ~orizon&l Depth = 0.8m

1 0.00 0.01 o.oz 0.03 O.CM oar 0 . s

Wave Steepness I

I f

Vertical : 59.5 : Vertical Depth = 0.8m

! l

0.00 rn

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Stee~ness

Figures FI 3 to FI 8 Kt vs Period for screen system differentiated by porosity

Horizontal : 99.5 : Horizontal Deptti = 0.8m

1 .O0 1

0.00 0.01 0.02 OB3 0.04 0.05 0.M

Wave Steepness

Horizontal : 995 : Vertical Depth = 0.8m

I 0.00 0.01 0.M 0.03 0.04 0.05 0.a I 0 m 4 Wave Steepness

Figures FI 9 to F20 Kt vs Period for screen system differentiated by porosity 1 F 1 9 ( F201

Figures showing Kt vs Wave Steepness for screen systems differentiated by screen

orientation

- - -

Screen 20 Dspth=O.?m

1 .O0

0.00 4- 0.00 0.01 0.02 0.03 O W 0.05

Wave Stee~ness

Screen 30 Depth=O.7m

1.00 7 1

0.00 0.01 0.02 0.03 0.04 0.05 O M

Wave Steepness

Screen 20 Depth4.8m / 1.00,

1 I 1 1 i 0.; 0.01 0.02 0, 0.04 0.05 0.061

Wave Steepness I

1 - - -

Screen 30

I

O Horizontal 0.10 -

! 1 overrid [

0.00 . I

I 0.00 0.01 0.02 0.03 0.w 0.05 0.a

Wave Steepness

Screen 30 Depth=O.Sm

1.00 , I /

1 / /OWO<IIOOW

-1 ci venicai

0.00 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

Screen 30b 1 Depth=O.irrn

las i i

0.00 0.01 0.02 0.03 O.@ 0.05

Wave Steepness

Figures G1 to G6 Kt vs Period for screen system differentiated by screen orientation m

Screen 30b DepthdJJrn

1 .Ca I

1 / Screen 40

DepthdJJrn 1 .W

O 1

l 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness I

t

0.00 0.01 0.02 0.03 0.04 0.05 0.N

Wave Steepness

Screen 50 DepthdJm 1 1-00 , i

O20

0.00 r 0.00 0.01 0.M 0.03 0.04 0.05 0.M

Wave Steepness

0.00 0.01 0.02 0.03 0.04 0.05 0.Q

Wave Steepness

Screen 40 Depth=O.Sm

11)0 I 1 --- P a a n a O i

0.80 n -b 4p

O O O 0.70 O

I

Wave Steepness

Screen 50 Depth=0.8m

1-00 T 1

0.00 O.Ot 0.02 0.03 0.04 0.05 0.02

Wave Stee~ness

Figures G7 to G12 Kt vs Period for screen system differentiated by screen orientation

G 11 G 12

Screen 30 : 59.5 : Screen 20 üepth4.7m I lm/

0.1 0

0.m 0.00 0-01 0.02 0.03 0.04 0.05 0.M

Wave Stee~ness

Screen 30 : 25.5 : Screen 30 Depth=0.8rn

1-00 T I I

I 1

0.00 0.01 0.02 0.03 0.04 0.05 0.m Wave Steepness

Screen 30 : 49.8 : Screen 30 Depth=O.am

1.00 , I 1

030

0.00

Wave Steepness

Screen 30 : 59.5 : Screen 20 Depth=0.8m

lm, 1

0.10

[ X V - H 0.00 4

0.00 0.01 0.02 0.03 0.04 0.05 0.R

Wave Steepness

l Screen 30 : 40.7 : Screen 30 Depth=O.lm

I I

0.00

Wave Steepness


1 .00 1

0.00 4 1 0.00 0.01 0.02 0- 0.04 0.05 0.0

Wave Steepness

Figures G13 to G18 Kt vs Period for screen sysfem differentiated by screen orientation

Screen 30 : 40.7 : Screen 40 0 020 on-n .1° P OH-V

Wave Steepness

0.00 0.00 0.01 0.M 0.03 0.04 0.05 0.06

Wave Steepness

Screen 40 : 59.5 : Screen 20 Depth=O.irm

1.00 -

Screen 40 : 25.5 : Screen 30 Deptha.8m

1 .00

0.90 -

0.00 ! 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

X

Screen 30 : 59.5 : Screen 40 Depth=O.lrn

1 .O0 1

0.00 0.01 o.m 0.03 0.04 o.os 0.o~

Wave Stee~ness


1 .O0 O

0.10

0.00 . 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

Screen 40 : 40.7 : Screen 30 üepthd.7rn

0.00 0.01 0.02 0.03 0.04 0.05 0.06

Wave Steepness

Figures G19 to G24 Kt vs Period for screen system differentiated by screen orientation

Screen 40 : 40.7 : Screen 30 Deptk0.8m

1 .00 1

0.00 f 0.00 0+01 0.02 0.03 0.04 0.05 0.üi

Wave Steepness

Screen 40 : 99.5 : Screen 30 üepth=O.fm

1 .O0 1

0.00 0.01 o.m 0.113 0.04 o.os o.as Wave Steepness

Screen 40 : 150.5 : Screen 30 DepthdIJm

1 .O0 I

0.00 0.01 o . 0.03 0.04 O.os 0.m

Wave Steepness

Screen 40 : 59.5 : Screen 30 Depth;;O.Bm

1 .O0 I

0.00 ! 1 t

0.00 o.or 0-02 0.03 0.04 o.os o.a Wave Steetpness


'ao 1 1

I

0.00 m l 0.00 0.01 0.02 0.03 0.09 0.05 0.06

Wave Steepness

- - - - -


0.00 0.01 0.02 0.03 0.09 0.05 0.0

Wave Steapness

Figures G25 to G30 Kt vs Period for screen systern differentiated by screen orientation

Figures showing Kt vs Wave Steepness for screen systems differentiated by gap size

30 Horizontal : Gap : 30 Horizontal Depthdi.8m

1 .DO 1

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

30 Horizontal : Gap : 40 Horizontal Depth=0.8m

1

0.00 ! f 0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.C

Wave Height (cm)


1 -00 1

0.00s . I 0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

30 Horizontal : Gap : 30 Vertical Depthd.8rn

1-03 , 1

I

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm)

30 Horizontal : Gap : 40 Vertical Depth=0.8m

1.w , I

0.00 4 . I 0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 9.0 101

Wave Height (cm)


1 .O0 I

0.M 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 S.0 IO.(

Wave Height (cm)

Figures Hl to H5 Kt vs Period for screen system differentiated by gap size

f 40 Horizontal : Gap : 30 Vertical 40 Horizontal : Gap : 30 Vertical DepthdI.7m

1-00 I

0.00 J . , -I 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm) / 0.0 1.0 2 0 3.0 4.0 50 6.0 7-0 110 9.0 10.C

Wave Height (cm)

40 Vertical : Gap : 30 Horizontal DepthS.7 m

1 .00 I 1 i

40 Vertical : Gap : 30 Vertical DepthdIJm , i

0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm) 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Wave Height (cm) I l

4ûH : Gap : 30H : Gap : 20H l 40H : Gap : 30H : Gap : 20V

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 0.0 9.0 10.0

Wave Height (cm) 0.0 1.0 2 0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 IO.(

Wave Height (cm)

Figures H7 to H l 1 Kt vs Period for screen system differentiated by gap size 1 H 7 1 H 8 1

40V : Gap : 30H : Gap : 20H 1.00

XGap = 5 9 5 : 49.8

I 0.007 . . . . ,

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 0.0 9.0 ion Wave Height (cm)

40V : Gap : 30V : Gap : 20H

0.0 1.0 20 3.0 4.0 5.0 6.0 7.0 0.0 9.0

Wave Height (cm)

40V : Gap : 30H : Gap : 20V lm, I

Wave Height (cm)

Figures Hl 3 to Hl 5 Kt vs Period for screen system differentiated by gap size

Figures showing Kt vs Relative Depth for screen systems differentiated by depth

20 Horizontal 1 .O0 I

lncident Wave Height I Depth

30 Horizontal 1

Incident Wave Height 1 Depth 1

30b Horizontal 1.00 1 1

0.00 1 I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

lncident Wave Height 1 Depth

Figures II to 16


30 Vertical 1


30b Vertical

0.00 4 I 0.00 0.02 0.04 0.08 0.08 0.10 0.12 0.v


Kt vs Relative Depth for screen systern differentiated by depth

40 Horizontal 1.00 -

O

0.90 -.

Incident Wave HeigM / Depth 1

50 Horizontal

0.00 1 I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0-14


0a2O

0.00 4 4

0.00 0.02 O . 0.m 0.08 0.10 0.12 0.14


40 vertical I


50 Vertical

. 0.00 0.02 0.04 0.08 0.08 0.10 0.12 0.1,


Incident Wave Height I Depth

Figures 17 to 11 2 Kt vs Relative Depth for screen system differentiated by depth

0.W 0.M 0.04 0.06 0.08 0.10 0.12 0.1


0.00 0.02 0.04 0.06 0.08 0.10 0.12

lncident Wave Height / Depth

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

lncident Wave Height / Oepth


Incident Wave HeIght I Depth 1

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14


Figures 11 3 to 11 8 Kt vs Relative Depth for screen system differentiated by depth

0.00 0.M 0.09 0.06 0.08 0.10 0.12 0.14

lncldent Wave Height 1 Depth

0.00 0.02 0.04 0.06 0.08 0-10 0.12

lncident Wave Helght 1 Depth

0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.b


020 J

lncident Wave Heiaht 1 Denth

0.10

020

0.00 o.m 0.04 o.os 0.00 0.10 0.12 0-14

1 lncfdent Wave HeIght 1 Depth

Figures 119 to 124 Kt vs Relative Depth for screen system differentiated by depth

1 23 1 24

0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.18


.- oDep--7"

O DepUlrO.8m

Appendix J

Fi,gues showing Kt vs d /g~ ' for screen systems differentiated by depth

1 20 Horizontal 1

] O." 1 f i

30 Horizontal

l-O0 I f

30b Horizontal

Figures J I to J6

20 Vertical 1.00 1

30b Vertical

Kt vs d/gTz for screen systern differentiated by depth

J 5

50 Horizontal l 1 1

50 Vertical I l I

Figures J7 to J 12 Kt vs digT2 for screen systern differentiated by depth m

Figures J I 3 to 51 8 Kt vs d/gT2 for screen system differentiated by depth

o... ", z0.50

O 0 o n

- - O

Figures J I 9 to J24 Kt vs d/gT2 for screen system differentiated by depth

Sensitivity Analysis graphs of Wave Steepness vs R~ for approximating zero

Sensitivity Analysis of R* for Hn close to Zero and K k l for 20H

1 ,

02 . . + Power l Natural Log

Sensitivity Analysis of R2 for H L close to Zero and K t 4 for 40H

1 [

1 1 i Natural L O ~ J

Sensitivity Analysis of R2 for M close to Zero for 40H: 59.5 :30H

1

02 I 4 Power 1

O 1 [ i Natural Log /

Sensitivity Analysis of for H L close to Zero and K k l for 3OH

' m m . . . . . 0.8' . . . 2 . " L

Sensitivity Anaiysis of FI2 for HR close to Zero and K k l for 30H: 49.8 :30H i

1 I I

. 1 Power 1 . NaturaI Log

Sensltivity Analysis of RZ for M close to Zero for 4 0 ~ : 59.5 :30H: 49.8 :20H

I

Figures K I to K6 Sensitivity Analysis of R* for H/L close to Zero

Appendix L

Graphs showing actual versus predicted Kt using two single prediction equations to

mode1 a double screen system

Actual vs Pledicted Kt using Single Screen EquaUon Twice for 30H:49.8:30V

Actual vs Predfcted Kt using Single Screen 1

0.0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Predicted Kt

‘

0.0 0.1 0 2 0.3 0.4 0.5 0.6 0.7 0.0 0.9 1.0

Predicted Kt

Actual vs Predicted Kt using Single Screen Equation Twice for 40H:25530H

1 l

Actual vs Predicted Kt uslng Single Screen Equation Twlce for 4CiH:40.7:30H

0.0 0.1 02 0.3 0.4 O 5 0.6 0.7 0.8 0.9 1.t

Predided Kt Predicted Kt

Actual vs Predicted Kt using Single Screen Equation Twice for 40H:1505330H

Actual vs Predicted Kt using Single Screen Equation Twice for #H:99.5:30H

0.0 0.1 02 03 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Predicted Kt

Figures LI to L6 Single screen prediction equation for double screen system

Graphs showing actual versus predicted Kt using three single prediction equations, a

single and a double equation and a double then single equation to mode1 a triple screen

sys tem

Actual vs Predicthd Kt uslng Single Scraen Equations for 40H:60:30H:5020H

0.0

0.7

0.0 0.1 0 2 0.3 0.4 0.5 0.6 0.7 0.8 0.8 1.0

Predicted Kt

Actual vs Predicted Kt using Single Screen Equations for 40H:50:30H:25:20H

Predicted Kt

Actual vs Predicted Kt using Single Scieen Equations for WH:50:30V:25:20H

0.0 ' ! . I 0.0 0.1 0 2 0.3 0.4 OS 0.6 0.7 0.8 0.9 l.C

Predicted Kt

Actual w Predicted Kt uslng Single Screen Equations for 40H:60:30H:50:20V

Actual vs Predicted Kt using Single Screen Equations for 40H:50:30H:25:20V

0.0 0.1 0 2 0 3 0.4 05 0.6 0.7 0.8 0.9 11

Predicted Kt

Actual vs Predicted Ut using Single Screen Equations for 40H:60:30V:50:20V

Predicted Kt

Figures Ml to M6 Single screen equations combined to predict Kt for triple screen system

Actual vs Predicted Kt using Single then Double Screen Equation for 4ûH:60:30H:5&20H

0.0 0.0 0.1 0 2 0.3 0.4 OS 0.6 0.7 0.8 0.9 1.C

Predicted Kt

Double Screen Equations for 40H:50:30H:25:20H

Actual vs Predicted Kt uslng Single Vien Double Screen Equations for

4ûH:50:30V:25:20H

Predicted Kt 1

Actual vs Predicted Kt using Single then Double Screen Equations for

40H:W:30H:5020V

I 0.0 0.1 0 2 0.3 0.4 0 5 0.6 0.7 0.8 0.9 1.0

Predicted K t


40H:50:30H:25:20V 1 .O A

Predicted Kt

1

Actual vs Predicted Kt using Single Vien Double Screen Equations for

40H:60:30V:50:20V

0.0 0.1 0 2 0 3 0.4 0 6 0.6 0-7 0.8 OS 1.0

Figures M 7 to M l 2 Single then double screen equations combined to predict Kt for triple screen system

203

ActuaI vs Predlded Kt using Single then Double Screen Equations for

40V:60-30H:SOSOH

0.8

O. 1

0.0 T?z5EE2 0.0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Predlcted Kt


40V:50:30H:25:20H 1.0 ,

Actual vs Predlcted Kt uslng Single then DoubleScreen Equations for 4üV:60:30V:50:20H

1 .O A

Actual vs Prdlcted Kt using Single then Double Screen EquatIons for

40V.60:30H-50:20V 1.0 A

Double Screen Equations for 4ûV:50:30H:25:20V

0.9

0.0 0.1 02 0.3 0.4 0 5 0.6 0 3 0.8 0.9 1.0

Predicted Kt

Actual vs Predicted Kt uslng S1ng:e then Double Screen EquatIons for 1

Figures Ml3 to Ml8 Single then double screen equations combined to predict Kt for triple screen system

Actuat vs Predicted Kt using Double then Single Screen Equation for 40H:W:30H:5020H I

I 0.0 0.1 0 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Predicted Kt

Actual vs Predicted Kt using Double then 1 Single Screen Equations for 40H:50:30H:25:20H

0.1

0.0 Il: 1 0.0 0.1 O 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.C

1 Predicted Kt

Actual vs Predicted Kt using Double Vien Singie Screen Equations for 40H:50:30V:25:20H

1.0 A

Predicted Kt

Actual vs Predicted Kt using Double then 1 Single S c m n Equations for 4OH:W:3On:50;20V 1

0.0 I

0 i2 0 O O 0:0 017 018 019 l!O 1 Predicted Kt

Actual vs Predicted Kt using Double then Single Screen Equations for 40H:50:30H:25:20V

1 .O /l

Actuai vs Predicted Kt using Double then Single Screen Equations for 40H:60:30V:50:20V

1 .O

0.9 ' t

Figures Ml9 to M24 Double then single screen equations combined to predict Kt for triple screen system

Actual vs Predicted Kt uslng Single then

l

J

Double Scnien ~ ~ u a t i o n s for 4WS0:30H:5020H

1.0 A

i O-' r/ . 0.0

0.0 0.1 0 2 0 3 0.4 05 0.6 0.1 0.8 0 3 1.0

Predicted Kt

Actual vs P d i c t e d Kt using Single then Double Screen Equations for

40V:50:30H:25:20H 1 .O

O-' r 0.0 : l l 0.0 0.1 0.2 03 0.4 05 0.6 0.7 0.8 0.9 1.0

Predicted Kt

Actual vs Predicted Kt using Single then DoubleScreen Equatlons for 40V:60:30V:50:20H

0.8

Actuat vs Predicted Kt uslng Single then Double Screen Equatlons for

40V:60:30H:50:20V

Predicted Kt 1


40V:50:30H:25:20V

I 1.0 A

Preciicteci ~t I

r I Actuai vs Predicted Kt uslng Single then

Double Screen Equations for

1.0, 40V:60:30V:50:20V

Figures M25 to M30 Double then single screen equations combined to predict Kt for triple screen system

Documents

Wave Transmission through Multi-layered Wave Screenscollectionscanada.gc.ca/obj/s4/f2/dsk1/tape3/PQDD_0016/MQ54487.pdfwave screens within 3% of the actual value for a single screen