Simulation of velocity field in front of breakwater and wave breaking impact

DEPARTMENT OF CIVIL ENGINEERING YOKOHAMA NATIONAL UNIVERSITY SIMULATION OF VELOCITY FIELD IN FRONT OF BREAKWATER AND WAVE BREAKING IMPACT By Nguyen Danh Thao A dissertation submitt

DEPARTMENT OF CIVIL ENGINEERING

YOKOHAMA NATIONAL UNIVERSITY

SIMULATION OF VELOCITY FIELD

IN FRONT OF BREAKWATER AND

WAVE BREAKING IMPACT

By

Nguyen Danh Thao

A dissertation submitted in partial fulfillment of the requirements for the

degree of Doctor of Engineering

Academic Advisor: Professor TOMOYA SHIBAYAMA, D.Eng

September, 2007

ABSTRACT

A three-dimensional breaking wave model has been developed to investigate the spatial and temporal variation of fluid motions under breaking waves A three-dimensional Large Eddy Simulation (LES) is used to solve the governing equations for the flow parameters Fluid motions in the area near the breakwater and its turbulences as well as wave overtopping on an inclined breakwater are investigated Through the model, the spatial and temporal variations of the hydrodynamic problems of breaking waves on a breakwater are examined

Breaking waves on vertical or sloping coastal structures produce impulse pressures that are high in magnitude and short in duration, compared with pressures exerted by non-breaking waves In recent years, model and prototype tests have been conducted to determine the history and spatial distribution of impulse pressures on smooth seawalls Improved awareness of wave impulse pressure induced failures has focused attention on the need to include dynamic responses to wave impact in the analysis of loadings to maritime structures

The present study is concerned with a theoretical approach, which is based on the momentum impulse, to simulate the impact pressures on a vertical wall The theoretical impact pressures are determined using varied velocity before impact process The velocity field in the vicinity of the coastal structure is calculated by LES model including turbulence The necessary coefficient to evaluate impulse pressures is tentatively obtained through a comparison between the laboratory tests and the numerical results The value of the impact pressure rise time is determined through the model The model has been also applied to examine the impact pressure of solitary wave The computational results of the application using the coefficient obtained show nonlinear time histories of pressure compared to the simplified linear time histories assumed by other researchers These results are reasonable agreements with the laboratory data

*

λ

ACKNOWLEDGMENTS

First of all, the author would like to express the deepest gratitude to his academic advisor, Dr Tomoya Shibayama, Professor, for his tireless advises, numerous suggestions, and patient encouragement which have been invaluable throughout all stages of the research program

The author also wishes to express his sincere appreciation to Dr Jun Sasaki for his kindness and impressive comments and suggestions over the course of this study Sincere thanks are also due to Mr Hiroshi Takagi, Research Associate for his encouragement, and valuable suggestions and discussions

Acknowledgment is gratefully extended to Prof Tatsuya Tsubaki, Prof Fumihiko Nakamura and Dr Toshiyuki Okamura, Associate Professor for their helpful comments, suggestions and serving as members of the examination committee

The author would like to extend a deep sense of gratitude to all of his teachers who directly or indirectly have given lectures and encouraged him during the time in research program

Acknowledgement is gratefully extended to Mr Miguel Esteban, a Doctoral Candidate in the laboratory for joint experiment and generous assistance in doing laboratory tests which have been very useful for this study

The author is grateful to all members in the Coastal Engineering Laboratory for friendly learning environment and all his friends who helped him in many respects

Grateful acknowledgement is conveyed to the Ministry of Education, Science and Culture of Japan for providing the financial support and Yokohama National University for providing necessary facilities for the completion of this study

Finally, the author would like to dedicate this work to his family for their spiritual support during his study in Japan

CONTENTS

ABSTRACT ii

ACKNOWLEDGEMENTS iii

LIST OF FIGURES vii

LIST OF TABLES x

LIST OF SYMBOLS xi

CHAPTER 1 INTRODUCTION 1

1.1 Significance of the Study 1

1.2 Objectives and Scope of the Study 3

CHAPTER 2 LITERATURE REVIEWS 4

2.1 Numerical Studies on Breaking Waves 4

2.1.1 The Direct Numerical Simulation (DNS) 5

2.1.2 The Large Eddy Simulation (LES) 6

2.2 Wave Forces on Vertical Breakwater 7

CHAPTER 3 THREE-DIMENSIONAL LARGE EDDY SIMULATION OF

BREAKING WAVE 16

3.1 Governing Equations and Sub-grid Scale Model 16

3.1.1 The Direct Numerical Simulation (DNS) 16

3.1.2 Large Eddy Simulation (LES) 17

3.1.3 The Sub-grid Scale Model 19

3.1.4 The Simplified Governing Equations 20

3.2 Boundary Conditions 21

3.3 Computational Domain and Staggered Grid System 26

3.4 Method of Solution and Numerical Procedures 27

3.4.1 Free Surface Computation 28

3.4.2 Pressure Computation 29

3.4.3 Velocity Computation 33

3.5 Dummy Arrays 37

3.6 Numerical Results of Large Eddy Simulation 39

3.6.1 Numerical Set-up and Incident Wave Conditions 39

3.6.2 LES of Breaking Wave Motions on Breakwater 39

3.6.3 Eddy Vortices around a Slope-Type Breakwater 45

CHAPTER 4 BREAKING WAVE IMPACT PRESSURE ON VERTICAL BREAKWATER 48

4.1 General Description of Breaking Waves Impact on a Vertical Breakwater 48

4.1.1 Probability of Occurrence of Wave Impact 48

4.1.2 Classification of the Wave Impact Pressures 51

4.1.3 Characteristics of Impact Pressures under Various Collision Conditions 52

4.1.4 Remained Problems 53

4.2 Theoretical Analysis 54

4.2.1 Effect of the Shape of Breaking Wave 54

4.2.2 Breaking Wave Velocity 55

4.2.3 Time Interval of Wave Impact 56

4.2.4 The Role of Entrapped Air 57

4.3 The Impulse-Momentum Model 58

4.3.1 Assumptions from Previous Research need to be Improved 58

4.3.2 Formulations 59

4.3.3 Empirical parameter λ 62

4.4 Simulation Results 63

4.4.1 Rise time of Impact Force History 64

4.4.2 Wave Profiles at the Breakwater 64

4.4.3 Breaking Wave Impact Pressures 66

4.4.4 Pressure Distributions at the Breakwater 69

4.4.5 Characteristics of Breaking Wave Impacts 72

4.5 Applications of the Model – Solitary Wave 74

4.5.1 Introduction 74

4.5.2 Boundary Conditions and Formulations 74

4.5.3 Experiments on Solitary Wave Impact 75

4.5.4 Verification of Numerical Results with Experimental data 82

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 84

5.1 Conclusions 84

5.2 Recommendations 86

REFERENCES 87

APPENDIX A CUPIC-INTERPOLATED PSEUDOPARTICLE (CIP) METHOD 93

B ADDITIONAL IMAGES OF SOLITARY WAVE EXPERIMENTS 97

List of Figures

2.1 Hiroi and Sainflou wave pressure diagram 8

2.2 Minikin wave pressure diagram 10

2.3 Wave pressure distribution (Goda, 2000) 11

3.1 The time dependence of a velocity component at a point 18

3.2 Boundary conditions 21

3.3 Waves at the offshore boundary 22

3.4 Marking of cells at free surface boundary 25

3.5 Irregular Star Method 26

3.6 Computational domain and the 3-D grid system 26

3.7 Staggered grid system 27

3.8 Dummy arrays in computational domain 37

3.9 Flow diagram of computational algorithm 38

3.10 Flow velocity vectors of wave breaking in a cross-shore vertical section (Case 1, y=60cm) 40

3.11 LES velocity field on a horizontal section (Case 1, z=44cm) 41

3.12 Wave overtopping on an inclined breakwater (Case 2, cross-shore vertical section)43 3.13 Wave overtopping on an inclined breakwater (Case 2, horizontal section, z=22cm)44 3.14 A lateral cross section in front of the breakwater (Case 2, x=160cm) 45

3.15 A lateral cross section just behind the breakwater (Case 2, x=60cm) 45

3.16 Vorticity plot for a cross-shore vertical section (Case 2, y=30cm) 46

3.17 Vorticity plot for a horizontal section (Case 2, z=22cm) 46

3.18 Vorticity plot for a lateral cross section in front of the breakwater (Case 2, x=160cm) 47

4.1 PROVERBS parameter map 50

4.2 Profiles of a breaking wave colliding with a vertical wall 55

4.3 A typical diagram of an impact pressure history 60

4.4 Impact pressure as a function of (simplified from Blackmore and Hewson, 1984) 62

2 b TC ρ 4.5 Detail of wave profiles close to the wall (case 1 – slight early breaking) 65

4.6 Detail of wave profiles close to the wall (case 2 – perfect breaking) 66

4.7 Time histories of impact pressures on the wall (case 1 – slight early breaking) 67

4.8 Time histories of impact pressures on the wall (case 2 – perfect breaking) 68

4.9 Time histories of impact pressures on the wall (case 3 – late breaking) 68

4.10 Time histories of impact pressures on the wall (case 4 – perfect breaking) 69

4.11 Vertical distributions of impact pressures on the wall (case 1 – slight early breaking) 70

4.12 Vertical distributions of impact pressures on the wall (case 2 – perfect breaking) 71 4.13 Vertical distributions of impact pressures on the wall (case 4 – perfect breaking) 71 4.14 Relationship between maximum impact pressures p m and rise time t r 73

4.15 Boundary conditions of solitary wave 75

4.16 Experiment set-up in the wave flume (unit: cm) 76

4.17 Model caisson and arrangement of measuring devices 77

4.18 Model caisson 77

4.19 Snapshots of the experiment set-up 78

4.20 Typical free surface time series (C3) 79

4.21 Comparison of measured and computed maximum impact pressures 80

4.22 Comparison of measured and computed maximum impact pressures 81

4.23 Comparison of time history of impact pressures 83

B.1 Sequential snapshots of the solitary wave impact (Case 2, t=0s-0.20s) 97

B.2 Sequential snapshots of the solitary wave impact (Case 2, t=0.27s-0.60s) 98

B.3 Sequential snapshots of the solitary wave impact (Case 3, t=0s-0.33s) 99

B.4 Sequential snapshots of the solitary wave impact (Case 3, t=0.40s-0.73s) 100

List of Tables

2.1 Overview of design methods for impact wave loading 13

3.1 Numerical set-up and incident wave conditions 39

4.1 The danger of impulsive breaking wave pressure (Goda, 2000) 49

4.2 Review of rise time t r 57

4.3 Numerical results of rise time 64

4.4 Solitary wave conditions 78

4.5 Initial conditions and computed pressures 82

H wave height at breaking point

i horizontal coordinate (offshore-onshore)

j horizontal coordinate (lateral)

k vertical coordinate

b

L breaking wave length

v

l virtual length of the water mass involved the impact

n current time step

m iteration step

u mean (spatially averaged) velocity in the i th component

v component of velocity in y direction

Vol volume of water

w component of velocity in z direction

x horizontal axis from offshore to onshore

y horizontal axis in lateral direction

z vertical axis (upwards positive)

ν molecular viscosity of fluid

t

ν eddy viscosity

ρ density of fluid

ij

τ sub-grid scale Reynolds stress

Δ spatial length scale

Chapter 1

INTRODUCTION

1.1 Significance of the Study

It is known that turbulence generated by breaking waves has important effects on most of the fluid dynamical processes within the surf zone such as wave transformation, generation of near-shore currents, diffusion of materials, and sediment transport Under the effect of breaker-generated turbulence, the surf zone is an area where the hydrodynamics are very complex with extremely complicated fluid motion The turbulent flow field has been the subject of intensive experimental study Most investigations have been carried out

in the laboratory, where the well controlled conditions make a systematic study possible, but field measurements have also been made of turbulence in breaking waves

Due to development of more advanced measuring technology and more advanced numerical models that make use of the faster computers now available, the hydrodynamics can be studied in much more detail than earlier Although a great number of researches have been carried out on the hydrodynamics, the turbulence in the surf zone is still one of the most challenging natural problems to any researcher in hydrodynamics or fluid mechanics

In the past, the primary approach to studying turbulent flows was experimental Overall parameters such as the time-averaged drag or heat transfer are relatively easy to measure but as the sophistication of engineering devices increases, the levels of detail and accuracy required also increase, as does cost and the expense and difficulty of making measurements However, the major difficulty is that turbulent flows contain variations on a much wider range of length and time scales than laminar flows So, even though they are

similar to the laminar flow equations, the equations describing turbulent flows are usually much more difficult and expensive to solve

The most accurate approach to turbulence simulation is to solve the Navier-Stokes equations directly without averaging or approximation It is also the simplest approach from the conceptual point of view In such simulation, all of the motions contained in the flow are resolved This approach is called Direct Numerical Simulation (DNS) Although the Direct Numerical Simulation is a good method for understanding turbulence, the applicable phenomena are limited because the scale of the turbulence is confined to the grid scale used in the calculation Then, DNS requires the use of very smooth meshes to resolve turbulence

In order to solve this problem, Large Eddy Simulation (LES), which is another way

of simulating turbulence in wave breaking, with turbulence model for the subgrid turbulence only, have been carried out due to its good evaluating anisotropic turbulence The use of LES models gives very detailed information about the flow structures A number of characteristic coherent turbulent flow structures, such as horizontal eddies, obliquely descending eddies, and longitudinal eddies, have been recognized

Many researches have assumed breaking wave motion to be two-dimensional in the wave direction and have tried to evaluate this phenomenon by a two-dimensional breaking wave model due to the difficulty of estimating this complex three-dimensional phenomenon However, wave breaking changes the two-dimensional potential fluid motion into three-dimensional rotational motion because of the transition to turbulence With this phenomenon, two-dimensional modeling cannot be simulated exactly fluid motion since the stretching of eddies, which is characteristic for true turbulence, is not simulated at all For this reason, it is better to simulate three-dimensional turbulence flow

Another very important thing in coastal engineering field is the wave forces acting on the coastal structures, in which the wave pressure is the most important force for the design

of maritime structures A number of formulae for wave pressure have been proposed because of its practical important Many researchers have treated the same problem with irrotational theory and achieved satisfactory results

While much research has been carried out on the hydrodynamics for breaking waves and the force generated by them, research on wave impacts is still at a preliminary stage

However, wave impacts on vertical breakwaters are among the most severe and dangerous loads this type of structure can suffer This impact pressure can be very high comparing with hydrostatic pressure corresponding to the wave height, though its duration is very short Many design methods are based on experimental research or empirical models To

be simulated the impact pressure, a number of assumption have been made through the process of getting results Due to the lack of series of parameter, most studies on breaking wave impact pressure so far have considered only on the maximum pressure These methods, generally, can not simulate reasonably the mechanism of wave impact, especially the time history of impact pressure The combination between LES and impulse momentum model is the good solution to solve this problem

1.2 Objectives and Scope of the Study

The present study aims at simulating the very complex surf zone dynamics using numerical techniques to investigate three-dimensional turbulence and fluid motion due to wave breaking A three-dimensional Large Eddy Simulation (LES) will be used to solve the governing equations for the flow parameters The scopes of the study are described as follows:

1 A three-dimensional breaking wave model using Large Eddy Simulation (LES) will

be developed and applied to investigate the spatial and temporal variation of fluid motions under breaking waves in front of and around a breakwater

2 A theoretical method based on the momentum impulse, using varied velocity just in front of the breakwater throughout the impact process, will be applied to compute the breaking wave impact pressures on a vertical breakwater

Chapter 2

LITERATURE REVIEWS

This review of literature will focus on researches in the past which are considered to

be most close-related to the following two problems:

1 The numerical simulation of breaking waves in time and space domains

2 Wave forces on vertical breakwater

2.1 Numerical Studies on Breaking Waves

It is known that turbulence generated by breaking waves has important effects on most of the fluid dynamical processes within the surf zone such as wave transformation, generation of near-shore currents, diffusion of materials, and sediment transport Under the effect of breaker-generated turbulence, the surf zone is an area where the hydrodynamics are very complex with extremely complicated fluid motion The turbulent flow field has been the subject of intensive experimental study Most investigations have been carried out

in the laboratory, where the well controlled conditions make a systematic study possible, but field measurements have also been made of turbulence in breaking waves

Fluid motion under progressive water waves can be assumed as two-dimensional along the wave direction However, wave breaking changes the two-dimensional potential fluid motion into three-dimensional rotational motion because of the transition to turbulence Two-dimensional turbulent flows occur only with certain specific conditions and are rarely observed Many researches have assumed breaking wave motion to be two-dimensional in the wave direction and have tried to evaluate this phenomenon by a two-dimensional breaking wave model due to the difficulty of estimating this complex three-dimensional phenomenon

2.1.1 The Direct Numerical Simulation (DNS)

The most direct way to investigate the flow in the surf zone numerically is to solve the basic equations for Newtonian fluids, namely the Navier-Stokes equations, since this approach can provide time-dependant detail of the flow characteristics It is possible to calculate the complex flow to give detailed information about the velocities, motion properties and others This direct solution of the dynamic equations is called Direct Numerical Simulation (DNS)

DSN means a complete three-dimensional and time-dependent solution of the Navier-Stokes and continuity equations The value of such simulations is obvious, in principle, numerically-accurate solution to the turbulence problem As the flow in the surf zone is turbulent and has significantly effective boundary layers, a more accurate formulation has to be based on the Navier-Stokes equation for incompressible viscous fluid For a free surface domain, the Navier-Stokes equations and additional continuity condition can be written in tensor notation as

i j i i

i j

i j i

x x

u x

P f

x

u u t

u

∂

∂

∂+

P, total pressure and time

ν molecular viscosity of fluid

The main thrust of present day research in computational fluid mechanics in turbulent flows is through the time-averaged Navier-Stokes equations These equations are also referred to as the Reynolds equations of motion The Reynolds equations are derived by decomposing the dependent variables in the turbulent Navier-Stokes equations into time mean and fluctuating components and then time averaging the entire equation

Duy and Shibayama (1997) carried out a detailed investigation and evaluation of breaking wave kinematics and turbulence in the surf zone, using time-averaged technique

to solve averaged forms of Navier-Stokes equations and show that the algebra Reynolds stress model can give good results in the inner surf zone Lin and Liu (1998), Bradford (2000) also got very good results using this technique Although DNS is a good method for understanding turbulence, the applicable phenomena are limited because the scale of the turbulence is confined to the grid scale used in the calculation In this case, DNS requires the use of very smooth meshes to resolve turbulence

2.1.2 The Large Eddy Simulation (LES)

Despite a great deal of effort and advancement in turbulence modeling for the past century, difficulties still remain in geometrically and physically complicated flow fields One approach to solve this problem is called Large Eddy Simulation (LES), which simulates only the larger-scale eddy motions which are resolvable by the numerical methods, while the effect of smaller-scale eddy motions is simulated indirectly The fundamental premise of this approach is that the large-scale eddies are affected by boundary conditions, causing most of the Reynolds stress and must be computed The small-scale turbulences are weaker, therefore less critical since they contribute less to the Reynolds stresses The use of LES models gives very detailed information about the flow structures with good evaluating anisotropic turbulence, and it is possible to achieve much higher Reynolds numbers than with DNS

There are two major steps involved in the LES analysis: filtering and sub-grid scale modeling Traditionally, filtering is carried out using the box function, Gaussian function,

or Fourier cutoff function Sub-grid modeling includes eddy viscosity model, structure function model, dynamic model, scale similarity model, and mixed model, among others The large scale motions are generally much more energetic than the small scale ones; their size and strength make them by far the most effective transporters of the conserved properties The small scales are usually much weaker and provide little transport of these properties A simulation which treats the large eddies more exactly than the small ones may make sense; large eddy simulation is just such an approach Large eddy simulations are three dimensional, time dependent and expensive but much less costly than DNS of the same flow In general, since it is more accurate, DNS is the preferred method whenever it

is feasible For flows in which the Reynolds number is too high or the geometry is too complex, the using of LES is more feasible than DNS

Wijayaratna et al (2000) presented a numerical simulation of wave transformation, breaking and run-up on sloping beds He used a full three-dimensional Navier–Stokes solver to study the three-dimensional turbulent flow structures in the surf and swash zones Watanabe and Saeki (1999), Christensen and Deigaard (2001), Suzuki et al (2004), and Christensen (2006) also found similar results from three-dimensional simulations of breaking waves Okayasu et al (2005) carried out laboratory experiment and three-dimensional LES to survey wave overtopping on gentle slope seawalls Thao et al (2006) investigated breaking wave motions around a slope-type breakwater using LES and got reasonable results although they have not directly verified the model with experimental data

2.2 Wave Forces on Vertical Breakwater

The wave force is the most important force for the design of maritime structures In parallel with the history of the construction of vertical breakwater, the formulas of the wave pressure exerted on upright sections have long history of development In an analysis

of wave forces acting on vertical breakwaters, a distinction is usually made between the effects of non-breaking and breaking waves In general, because of the effects of breaking waves, the slope of the sea bottom, wave interference with the rubble mound in composite breakwaters, overtopping, and other factors, the determination of the wave pressure on vertical breakwaters is very difficult and in most cases is not straightforward This demonstrates the reason that in most practical cases, important breakwaters are designed based on the results obtained from laboratory model tests or on the basis of empirical formulas from the modal tests A number of formulae for wave pressure have been proposed because of its practical important

The wave pressure on vertical breakwater has been, until recently, evaluated using the significant wave height, which probably produces a smaller value of wave pressure comparing with that produced by the maximum wave At present, since some important breakwater had been destroyed by big waves, the maximum wave height is normally used

to calculate the wave pressure

In the conventional design standard of Japan, both Hiroi’s formula for breaking waves and Sainflou’s formula for clapotis was used Breaking or non-breaking waves depended on the relation between the water depth in front of the structure and the incident wave height Hiroi (1919) considered wave pressure as similar to the pressure of a water jet and derived the formula for wave pressure

)

(d

)(H1/3

p Sainflou (1928) used the rotational wave theory as a base to derive the expressions of pressure distribution along

a vertical wall This method simplified wave pressure theory and provided wave pressure distributions at the wave crest and wave trough Accordingly, wave pressure at a wave crest is determined by the following equations (see Figure 2.1)

Hiroi (1919) Sainflou (1928)

Figure 2.1 Hiroi and Sainflou wave pressure diagram

0

0 2

1 ( )

h h H

h H gh p

p

++

++

)/2cosh(

2

L h

gH p

2 '

2

L h

gH p

In Eq (2.3) through (2.7), H is the height of the original free wave (in water depth ), is the height of the mean level of the clapotis above the still water level; and

wave length

Since the resulting pressure distribution may be reasonably approximated by a straight line, the Sainflou’s method was easy to apply Experimental observations by Rundgren (1958) have showed that Sainflou’s method overestimates the non-breaking wave force for steep waves The higher-order theory by Miche (1944), as modified by Rundgren (1958), to consider the wave reflection coefficient of the structure, gives the good results of wave forces on vertical walls for steep waves, while Sainflou’s method gives better results for long waves of low steepness

Waves breaking directly against vertical or sloping faced coastal structures produce impact pressures high in magnitude and short in duration, compared with pressures exerted

by non-breaking waves Wave impacts on vertical or sloping breakwaters are one of the most severe and dangerous loads that maritime structures can suffer The pressures measured are much greater than those expected from the parameters associated with the incident wave (wave height H, water depth h, gravity g and water density ρ ) This impulse breaking wave pressure may easily exceed 10ρg(h+H) (Goda, 2000) However, in the past, the short duration wave impact pressures have often been neglected although the pressures can be very high In recent years, model and prototype tests have been conducted

to determine the history and spatial distribution of impulse pressures on smooth seawalls to improve the awareness of wave impact pressure

Bagnold (1939) found that impact pressures occur at the instant that the vertical front face of a breaking wave hits a wall and only when a plunging wave entraps a cushion of air against the wall Impact wave pressure is one of the most important problems in the design

of a vertical breakwater, and therefore, it effects on breakwater performance must be evaluated thoroughly Based on Bagnold’s laboratory data (Bagnold, 1939), Minikin (1950) proposed a formula for breaking wave pressure According to Minikin, the maximum dynamic pressure from a breaking wave is assumed to act at the still water level (SWL) and could be determined from the formula

)(

H w

where is the maximum dynamic pressure, is the breaker height, is the wave length in water depth , is the depth at the toe of the wall, and is the depth one wave length in front of the wall

H P

Figure 2.2 Minikin wave pressure diagram Ito (1966), based on hydraulic model experiments, proposed a single formula covering both breaking and non-breaking wave pressures, including the effect of the

d P

SWL

b

H

Dynamic Component Hydrostatic Component

d

d

Combined Total

)2

d

( H b

d

presence of a rubber mound foundation After that, Tanimoto (1976) modified the formula

to account for the effect of oblique wave approach

Based on the results from model tests and using empirical methods, Goda (2000) proposed a comprehensive formula for calculation of wave forces acting on vertical breakwaters It was later modified to account for the effect of the oblique waves and was used successfully for the design of numerous vertical breakwaters built in Japan In the formula, the wave pressure acting along the vertical wall is assumed to have a trapezoidal distribution both above and below the SWL as shown in Figure 2.3

Figure 2.3 Wave pressure distribution (Goda, 2000) The theoretical elevation at which the wave pressure could be exerted , and the representative wave pressure intensities , , in generalized form, are obtained from the following:

max 2

2 1

1 2

L h

p p

π

1 3

3 p

in which

2 1

)/4sinh(

/42

16

=

L h

L h

2,3

min

H

d d

H h

d h b

11

'1

3

L h h

is the smaller of and b ; is water depth at the location at a distance

seaward of the breakwater

{a, b

The above pressure intensities are assumed not to change even if wave overtopping takes place The value of coefficients α1, α2, α3 are obtained from the laboratory tests Takahashi et al (1994) modified Goda formula for pressure calculation and proposed a set

of formulas to estimate the intensity of impulsive pressure by modifying impulsive pressure coefficients, which are derived, based on caisson sliding experiments

Many researchers have treated the same problem to evaluate the mechanism of impact pressure and achieved satisfactory results Table 2.1 shows the overview of design methods for impact wave loading Most of the methods are based on the experimental results and empirical formulas

Based on experimental results, Weggel and Maxwell (1970) observed and classified impact pressures into two general types, according to the relative area over which the impact pressure acts simultaneously when waves break against coastal structures: significant impact pressures and ordinary impact pressures Both types were of very short duration and their over-all significance might depend on their effect on the materials of which the structure is built as well as wave steepness

Table 2.1 Overview of design methods for impact wave loading

Quasi-Static Waves

Sainflou 1928 yes yes, but

difficult Vertical wall, no berm Miche-Rundgen 1944

Design curves from SPM,

1984 Goda

1985 yes yes Most-widely used design

method

Impact Waves

Minikin 1963 yes yes Sometimes incorrect

dimensions

Blacmore & Hewson 1984 yes yes

Partenscky 1988 yes unknown Air content of wave needed

Takahashi et al 1994 yes yes Extension of Goda model

Allsop et al 1996 no yes

Walkden et al

1996 no yes Relation of forces and rise

time Oumeraci &

McConnell 1998 no yes Amendment of O&K, 1997

Hull & Müller 1998 yes yes Amendment of O&K, 1997

Vicinanza 1998 yes yes Amendment of O&K, 1997

Broken Waves

SPM 1984 yes yes Vertical wall only

Camfield 1991 yes yes Amendment of SPM, 1984

Bradbury & Allsop 1988 yes yes Crown walls

Martín et al 1997 yes yes Crown walls

CEM 2006 yes yes Vertical wall only

Hattori et al (1994) conducted laboratory tests and found that the highest short duration pressure is observed when a vertical wave front strikes the wall while trapping a

small amount of air in the form of either bubbles or a thin lens-shaped pocket The larger

the amount of the entrapped air at impact of the plunging breakers, the lower the

magnitude and the longer the rise or compression time of the impact pressures Through a series of experiments, he found that the impact pressures acting on structures are generated

by all types of breaking waves, from plunging to collapsing breakers in shallow water It is, therefore, generally accepted that the impact process depends considerably on the kinetic properties of breaking waves during the impingement onto the structures

Kirkgör (1991, 1995), Wood et al (2000), Bullock et al (2004), Wolters & Müller (2004), Wienke and Oumeraci (2005) also carried out laboratory tests to investigate the mechanism of impact pressures From these experiments, it has become clear that impact pressures occur regularly when the wave front is approximately parallel to the vertical wall

Allsop et al (1996) have investigated a large data set to predict horizontal wave forces on vertical breakwaters The relative wave height H si d has been found to most significantly influence the wave forces non-dimensionalized by the water depth over the berm His formula provides a quick estimation of the expected horizontal force to the structure but cannot predict the length of its duration

On the other hand, theoretical analyses and numerical solutions of impact pressure have generally been based on the impulse momentum Numerical computations by Peregrine (2003), Kirkgör and Mamak (2004) gave reasonable results of the wave impact pressures on vertical seawalls if the impact pressure rise time is known However, it is very difficult to model breaking wave impact theoretically since the impact mechanism is nonlinear and unsteady Theoretical or numerical attempts to predict impact pressure have failed to give reasonable answers regarding the following three problems:

1 The time integral of pressure through the impact is still not determined Rise time

or duration time should be used for the calculation of the time integral?

2 During the time of impact, the impulse velocity will be change Therefore, the assumption of using only maximum velocity at breaking point to calculate impact pressure will give imprecise results

3 The rise time is unknown The impact pressure rise times used in the numerical model were only selected from experiment impact pressure data

Improved awareness of wave impact induced failures has focused attention on the need to include dynamic responses to wave impact in the analysis of loadings to maritime structures (Kirkgöz and Mamak, 2004) Until now, theoretical attempts to predict wave impact pressure have generally failed to give reasonable answers for practical purposes, and it still remains as one of the most challenging topics in coastal engineering

3.1 Governing Equation and Sub-grid Scale Model

3.1.1 The Direct Numerical Simulation (DNS)

The most accurate approach to turbulence simulation is to solve the Navier-Stokes equations without averaging or approximation It is also the simplest approach from the conceptual point of view In such simulations, all of the motions contained in the flow are resolved The computed flow field obtained is equivalent to a single realization of a flow or

a short-duration laboratory experiment; as noted above, this approach is called direct numerical simulation (DSN)

Turbulent flows in continuum regime are governed by the Navier-Stokes equations, which are derived based on the universal laws of conservation For a free surface flow, the Navier-Stokes equations for incompressible viscous fluid are used as the basic governing equations

i j i

g x x

u x

p x

u u t

i g

3.1.2 The Large Eddy Simulation (LES)

For the direct numerical simulations, Eq (3.1) and (3.2) are used as the governing equations Even though great progresses have been made in the field of computer science nowadays, the storage capacity and speed of present-day computers are still not sufficient

to allow a numerical solution of the turbulent Navier-Stokes equations The main reason is that the time and space scales of the turbulent motion are so small that the number of grid points required and the small size of the required time steps put the practical computation

of turbulent flows by numerical methods outside the realm of possibility for present computer Although estimates from various sources differ on the required mesh spacing, a common estimate is that at least 10 grid points would be required to adequately resolve the motion of a turbulent eddy The scale of the smallest eddies are typically times the size of the flow domain for flow along a solid surface It means that the size of the computational domain should be large enough in order to simulate a turbulent eddy, in according with so many grid points Moreover, for three-dimensional domains and larger two-dimensional set-up, the grid points must be in very large number This requirement can not be satisfied with present-day computers’ capacity

of the smaller scale turbulences, i.e smaller than the grid scale, are simulated indirectly by taking into account a sub-grid scale (SGS) model

Figure 3.1 The time dependence of a velocity component at a point

j i i

j

i j i

g x

x

u x

p x

u u t

τ is the sub-grid scale (SGS) Reynolds stress:

( i j i j)

j i

i u u

The models used to approximate the SGS Reynolds stress (3.5) are called sub-grid scale (SGS) or subfilter-scale models, which can account the effect of small-scale turbulences, the length scale of which are smaller than the grid scale

The sub-grid scale Reynolds stress contains local averages of the small scale field so models for it should be based on the local velocity field or, perhaps, on the past history of the local fluid The later can be accomplished by using a model that solves partial

differential equations to obtain the parameters needed to determine the SGS Reynolds stress

3.1.3 The Sub-grid Scale Model

The sub-grid scale Reynolds stress contains local averages of the small scale field, so the model for it should be based on the local velocity field or on the past history of the local fluid in order to obtain a closure So far, several models for determining the SGS Reynolds stress have been proposed, such as Smagorinsky model (Smagorinsky, 1963), dynamic model (Germano et al., 1991), k-equation model (see Christensen, 2006), deconvolution model (Domaradzki and Saiki, 1997) Smagorinski (1963) was the first to postulate a model for the SGS stresses At present, Smagorinsky model is the most widely used model, due to two main reasons: the model yields sufficient diffusion and dissipation

to stabilize the numerical computation, and low-order statistics of the larger eddies are usually insensitive to the SGS motions In the model, τ is modeled by using an isotropic ijSGS eddy viscosity to present the momentum exchange

ij t

ij ν S

where νt is the eddy viscosity in the sub-grid scale (Smagorinski eddy viscosity) and

is the deformation tensor

∂

∂

=

i j j

i ij

x

u x

u S

C is a model parameter to be determined Theories provide estimates of the

parameter Most of these methods apply only to isotropic turbulence for which they all agree that C s ≈0.2 Unfortunately, C s is not constant; it may be a function of Reynolds

number and/or other non-dimensional parameters and may take different values in different flows

In Smagorinsky sub-grid scale model, is considered as the Smagorinsky coefficient Applications of the Smagorinsky closure model have typically been made with values of the Smagorinsky constant in the order of 0.07 to 0.21 (Schumann, 1987) However, is used for the present simulations since this value is typically used for similar studies (e.g Christensen and Deigaard, 2001)

s C

1.0

∂

∂Δ

j j

i s

j ij

u x

u x

u x

u C

x

12

12

2 2

However, Eq (3.11) is very complicated in numerical formulations, especially in boundary conditions of three-dimensional models Therefore, an approximate expression is used

∂

∂Δ

i j j

i s

ij

u x

u x

x

u x

u C

12

12

2 2

This can be further simplified by using the continuity condition (3.5)

j i i

j j

i s

ij

u x

u x

u C

2 2 2

∂

∂Δ

=

∂

∂

3.1.4 The Simplified Governing Equations

Replace sub-grid scale model into Eq (3.4) The axis , and are replaced with

1

x x2 x3

x , and In addition, the velocity components , and will be referred as , and The governing equations for three-dimensional model using LES can be rewritten as follows:

( )θ

νρ

sin

22

1

1

2

2 2

2 2

2 2

2

2 2

2 2 2

g

z

u y

u x

u x

w z

u x

v y

u x

u C

z

u y

u x

u x

p z

u w y

u v x

u u

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂Δ

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂Δ

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂

2

2 2

2 2

2 2

2

2 2

2 2 2

22

1

1

z

v y

v x

v y

w z

v y

v y

u x

v C

z

v y

v x

v y

p z

v w y

v v x

v u

t

v

s

νρ

(3.15)

( )θ

νρ

cos

22

1

1

2

2 2

2 2

2 2

2

2 2

2 2 2

g

z

w y

w x

w z

w z

v y

w z

u x

w C

z

w y

w x

w z

p z

w w y

w v x

w u

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂Δ

∂

∂+

∂

∂+

∂

∂+

∂

∂+

v x

u

(3.17)

where u , v and w are the mean (spatially averaged) velocity in x , and component,

respectively To have simpler notations, in the following section,

u , v, w and p will be replaced with their simpler counterparts u , v , and Therefore, the meaning of all u ,

, and in those sections are actually

For the 3-D, the boundary conditions as in Figure 3.2 are specified at the following boundaries:

Offshore Boundary

The boundary conditions for the incoming wave at the offshore boundary are obtained by applying 3rd order cnoidal wave theory or 5th order Stokes wave theory, depending on the calculated Ursell parameter U r (Nishimura et al., 1977)

3 2

The reflected wave from the domain inside, if existing, will be treated to pass freely

and undisturbed through the open offshore boundary In the model, velocity in x-direction

and free surface are generated at the offshore boundary using a wave theory Pressure and the other velocity components are not generated but assumed to be equal to the same value

at the immediate interior neighbor

Incident

Figure 3.3 Waves at the offshore boundary

The actual water surface at each time level t is computed as the sum of the incident

and reflected waves

()()

ζi t− incident wave component at grid point 2 at time t−τ

where the incident wave at grid point 2 can be computed from the generated incident wave

at the offshore boundary:

2()

Bottom Boundary

Non-slip condition is applied at the bottom boundary, where the velocity components

on this boundary are set to be zero and the pressure is approximated to be hydrostatic

A boundary condition for the pressure at the bottom boundary is obtained by simplifying the Eq (3.16) using the zero velocity condition on the boundary

ν

11

2

2 2

2

2

g z

w z

v z

u C

z

w z

∂

∂Δ

Neumann boundary condition for the pressure is derived in y-direction using zero

velocity condition on the side of the walls

∂

∂Δ

2

2

2

11

y

v y

w y

u C

y

v y

p

s

ν

Free Surface Boundary

Density Function method (Watanabe and Saeki, 1997) is used to detect the free surface In the computational domain, each cell is categorized as being a surface cell, full cell or empty cell The density function f indicates 1 at any grid cell fully occupied by fluid and 0 if it contains no fluid The exact value of , therefore, would represent the fractional volume of the cell occupied by fluid In fact, in the region of breaking waves, a sharp definition of free surface is very difficult Water particles escape from the main flow, mix with air and form a region in without clear water-air boundary To model this region accurately, the presence of air bubbles must be take into account This model can be done

by combining a module of the air region with Navier-Stokes equations for compressible and another module of the incompressible flow equations using an interface model However, this method is very complex and requires very high computer capacities For simplicity, the free surface is assumed to be located on a surface where is equal to 0.5 When is greater or equal to 0.5, the cell is considered to be a water cell while it is an air cell otherwise (see Figure 3.4) The status of each cell is checked and corrected at each time step by using the following equation:

f

f f

=

∂

∂+

∂

∂+

∂

∂+

∂

∂

z

f w y

f v x

f u

f

Pressure of the grid cells in the free surface is not calculated in the same way with the fully filled inner grid cells which use the Poisson equation The Irregular Star Method (Chan and Street, 1970) is used to solve the pressure at the grid cells neighboring the free surface

Pressure on the boundary cell at free surface is now interpolated as

++

+

++

++

++

=

γβαη

ηηη

ηη

ηηηη

η

ηηηηη

ηη

ηηηηηηηηηηηη

ηηηηηη

,,

6 5 6 5

5 6 6 5

4 2 4 2

4 2 2 4 3 1 3 1

3 1 1 3

4 3 2 1 6 5 3 1 6 5 4 2

6 5 4 3 2 1

f p p

p p

p p

where f(α,β,γ) is function of velocity and will be expressed in following sections

Free Surface

Free Surface

Figure 3.5 Irregular Star Method

3.3 Computational Domain and Staggered Grid System

Both sloping and horizontal-vertical grid systems are used with x along the wave

propagation direction, y in the horizontal transversal direction and z in a vertically upward direction For three-dimensional simulations, the computational domain of the grid system is illustrated as in Figure 3.6

Figure 3.6 Computational domain and the 3-D grid system

The simplified governing equations and corresponding boundary conditions can be solved numerically by finite difference method Numerical calculations are carried out in a staggered grid system due to its higher accuracy The solution of LES equations in the computational domain includes the computations of pressure, volume of fluid function and velocities In the staggered grid system, the pressure and the volume of fluid function are defined at the center of the cell, while the velocities are at the center of the surfaces of the cell A single computational cell is illustrated in Figure 3.7

Figure 3.7 Staggered grid system

3.4 Method of Solution and Numerical Procedures

As mentioned above, the numerical model consists of a number of elements which are combined with a formulation of the boundary conditions for simulation of breaking waves General sequence of solution algorithm is:

(1) Update free surface using information from previous step;

(2) Compute the pressure for the new free surface; and (3) Update velocities for new free surface and pressure field