Sand Foundation Instability due to Wave-Seabed-Structure Dynamic
Interaction
NAKAMURA Tomoaki
Sand Foundation Instability due to Wave-Seabed-Structure Dynamic
Interaction
NAKAMURA Tomoaki supervised by Prof. MIZUTANI NorimiCoastal
& Ocean Engineering Laboratory Department of Civil Engineering
Nagoya University JAPAN
NAKAMURA TomoakiCoastal & Ocean Engineering Laboratory,
Department of Civil Engineering, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya 464-8603, JAPAN Web Site: http://tnakamura.net/
E-mail: [email protected]
i
AcknowledgmentsThe author would like to express his deepest
appreciation and gratitude to his supervisor Prof. MIZUTANI Norimi,
Nagoya University, for his constant and invaluable guidance during
his bachelor, master and Ph.D. courses. His board knowledge, sharp
methodical observations and subjective discussions have navigated
the author through dicult pathways. The author will remain ever
deeply grateful to him for giving this opportunity to undertake
undergraduate and graduate studies under such a kind supervisor.
Special gratitude also goes to this dissertation committee, which
consisted of Prof. MIZUTANI Norimi, Nagoya University (Chair);
Prof. TSUJIMOTO Tetsuro, Nagoya University; Prof. NODA Toshihiro,
Nagoya University; Assoc. Prof. KAWASAKI Koji, Nagoya University;
and Assoc. Prof. KITANO Toshikazu, Nagoya Institute of Technology,
for their careful review, valuable suggestions, helpful comments,
constructive criticisms, and highly professional but personal touch
on this work. The author wishes to express his sincere gratitude to
Prof. IWATA Koichiro, Chubu University, for his helpful technical
guidance especially during the early stage of this dissertation.
Sincere thanks are also due to Assoc. Prof. HUR Dong-Soo,
Gyeongsang National University, Korea, for his unfailing support in
various aspects of this work. The author believes that this work
could not be done without the help of the former and present
members of the Coastal & Ocean Engineering Laboratory, Nagoya
University, in particular Assist. Prof. LEE Kwang-Ho, Nagoya
University, for valuable discussions and helpful comments, and Mr.
KURAMITSU Yasuki, NTT West Co., for his kind assistance in
conducting the hydraulic model experiments and numerical
simulations. The latter part of this dissertation was partly
supported by the Grant-in-Aid for JSPS (Japan Society for the
Promotion of Science) Fellows No. 19623 (Representative: NAKAMURA
Tomoaki) oered by the Ministry of Education, Culture, Sports,
Science and Technology, Japan. The author is grateful for the
nancial support. Last but not least, the author uses this
opportunity to express his deepest respect to his father Mr.
NAKAMURA Fumio and his mother Ms. NAKAMURA Hiroko for their
permanent inspiration, encouragement and support in all aspects of
his life.
iii
Table of ContentsAcknowledgments Chapter 1 Introduction 1.1
Motivation . . . . 1.2 Literature review 1.3 Study objectives . 1.4
Contents . . . . . References . . . . . . . i 1 1 4 6 7 8 11 11 12
12 15 17 17 19 20 21 21 24 24 27 27 27 28 34 35 36 39 40 42 46
46
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Chapter 2 Numerical Model 2.1 General . . . . . . . . . . . . .
. . . . . . . . . 2.2 VOF-based numerical submodel for a wave eld .
2.2.1 Governing equations . . . . . . . . . . . . . 2.2.2
Computational schemes . . . . . . . . . . . . 2.3 FEM-based
numerical submodel for a sand bed . 2.3.1 Governing equations . . .
. . . . . . . . . . 2.3.2 Computational schemes . . . . . . . . . .
. . 2.4 Coupling technique between the submodels . . . 2.5 Remarks
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Chapter 3 Wave-Induced Sand Leakage Phenomena 3.1 General . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Sand
leakage from behind a rubble mound breakwater . . . . . 3.2.1
Hydraulic model experiments . . . . . . . . . . . . . . . . 3.2.1.1
Dimensional analysis . . . . . . . . . . . . . . . . . . 3.2.1.2
Experimental setups and conditions . . . . . . . . . . . 3.2.2
Experimental results and discussions . . . . . . . . . . . .
3.2.2.1 Wave eld around the upright rubble mound breakwater 3.2.2.2
Sand leakage from behind the rubble mound breakwater 3.2.3
Numerical results and discussions . . . . . . . . . . . . . 3.2.3.1
Applicability of the numerical simulation . . . . . . . . 3.2.3.2
Backlling sand leakage mechanism . . . . . . . . . . 3.3 Sand
leakage through a gap under a revetment . . . . . . . . . 3.3.1
Hydraulic model experiments . . . . . . . . . . . . . . . .
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iv 3.3.1.1 Dimensional analysis . . . . . . . . . . 3.3.1.2
Experimental setup and conditions . . . 3.3.2 Experimental results
and discussions . . . . 3.3.3 Numerical results and discussions . .
. . . 3.3.3.1 Applicability of the numerical simulation 3.3.3.2
Backlling sand leakage mechanism . . 3.4 Remarks . . . . . . . . .
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Table of Contents . . . . . . . . . . . . . . . . . . . . . . .
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46 47 48 49 50 52 55 56 58 58 60 60 61 62 63 64 64 68 72 75 78 78
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Chapter 4 Tsunami-Induced Local Scouring Phenomena 4.1 General .
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Hydraulic model experiments . . . . . . . . . . . . . . . . . .
4.2.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . .
4.2.2 Experimental setups and conditions . . . . . . . . . . . . .
4.2.2.1 Solitary wave . . . . . . . . . . . . . . . . . . . . . .
4.2.2.2 Isolated long wave . . . . . . . . . . . . . . . . . . . .
4.3 Experimental results and discussions . . . . . . . . . . . . .
. 4.3.1 Tsunami scour and its time development . . . . . . . . . .
4.3.2 Final scour depth . . . . . . . . . . . . . . . . . . . . . .
4.4 Numerical results and discussions . . . . . . . . . . . . . . .
4.4.1 Validation of the numerical simulation . . . . . . . . . . .
4.4.2 Velocity and stress elds and their eects on tsunami scour
4.4.2.1 Velocity eects on tsunami scour . . . . . . . . . . . .
4.4.2.2 Stress eects on tsunami scour . . . . . . . . . . . . . 4.5
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5 Time-Domain Analysis on Tsunami-Induced Topographic
Change 5.1 General . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 5.2 Sediment transport model . . . . . . . . .
. . . . . . . . . . . . . . . . 5.2.1 Governing equations and
computational schemes . . . . . . . . . . 5.2.2 Sediment transport
formula with eective stress . . . . . . . . . . . 5.3 Results and
discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 6 Conclusions 99 References . . . . . . . . . . . . . .
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104
1
Chapter 1
Introduction1.1 MotivationOur mother planet Earth is a watery
planet. Approximately 70.8 % of the surface is covered by water,
most of which consists of ocean with several principal oceans and
smaller seas. The ocean plays an important role in keeping the
global climate balance and in maintaining a wide variety of
ecosystems. The ocean also provides a large number of treasured
places for relaxation and healing. The best places of them are
shallow water regions, especially coastal beaches, in which many
people gather and spend precious time together to enjoy their
benets. In recent years, suitable beaches, however, have been
decreasing because of land reclamation, coastal erosion and water
pollution. Figure 1.1 shows annual changes in the land reclamation
area from 1950 to 2004 in Japan. As indicated in Fig. 1.1, the land
reclamation area gradually declined from 1975 to 1990, and then the
reclamation area hovers around 10 km2 per year through the 1990s
and 2000s. The solid line of Fig. 1.1 also shows that the total
reclamation area is steadily increasing even today. Furthermore,
the coastal erosion area was approximately 16 km2 per year on an
average from 1978 to 1992 (Ministry of Land, Infrastructure and
Transport, 2006). Both the land reclamation and costal erosion
therefore cause the substantial loss of natural beaches in Japan.
The Japanese Ministry of Land, Infrastructure and Transport
authorized a few dozens of coastal community zones (CCZ) in Japan,
which were mostly built on erodingTotal Reclamation Area [km ]120
1200Reclamation Area [km ]2
Reclamation Area [km ]
2
100
Total Reclamation Area [km ]
2
1000
80
800
60
600
40
400
20
200
0
0
0-1 95 5 56 -19 65
70
5
0
19 85
19 90
95
19 7
19 8
19
19
19 5
19
Year
Fig. 1.1 Annual changes in the land reclamation area in Japan
(Geographical Survey Institute, 2004).
20
00
2
2
Chapter 1 Introduction
(a)
(b)
Photo 1.1 Aerial photos of the coastal community zones (CCZ)
authorized by the Japanese Ministry of Land, Infrastructure and
Transport: (a) the Kozakai beach, Toyama; and (b) the Shiroya
beach, Aichi (National CCZ Maintenance and Promotion Conference,
2007).
coasts to compensate for unsuitable and lost beaches. Aerial
photos of the CCZs are exemplied in Photo 1.1. The Kozakai and
Shiroya beaches in Photo 1.1 were designated as the CCZs in 1988
and 1990, respectively. Because of the predominance of beach
erosion trends, preventing coastal re-erosion obviously requires
the installation of maritime structures at the CCZs. However, awed
designs and faulty constructions occasionally led to crucial
coastal disasters such as beach subsidence reclaimed behind the
structures, and then people playing at healing places were
ironically exposed to unexpected risk. Examples of actual accidents
will be introduced in the following chapter. Besides such
accidents, there exist several wave-induced coastal disasters,
namely, sliding, settlement and overturning of coastal structures
and failure of their foundation, which closely relate to not only
acting wave proles but also the stability of the foundation. Many
researchers have focused on investigating wave-structure
interactions, while inadequate studies have so far been devoted to
detailed dynamics of the foundation, in particular various eects of
stress and strain elds within the foundation. As indicated in the
review paper of Sumer et al. (2001), it is still unclear whether
wave-induced pore water pressure gradients close to critical, i.e.,
nearly zero eective stresses called liquefaction, have an eect on
dynamic behavior of the foundation such as local scour and sediment
transport phenomena. Jeng (2003) also suggested that the link
between scour and liquefaction is a challenging and interesting
task. Coastal structures at the CCZs are generally designed to
protect articial beaches against ordinary wind waves, which have so
far been concentrated in many technical papers. However, special
attention to rare events such as tsunami waves has gradually been
growing up particularly since the 2004 Indian Ocean Tsunami. It is
well known that the Sumatra-Andaman earthquake occurred with an
epicenter under the Indian Ocean near the west coast of Sumatra,
Indonesia at 7:58 a.m. local time (0:58 a.m. UTC) on December 26,
2004. A series of devastating tsunamis triggered by the earthquake
unfortunately struck the coastal areas of India, Indonesia, Sri
Lanka, Thailand and other countries along the Indian Ocean, killing
a surprising number of people in many countries including South
1.1 Motivation
3
(a)
(b)
(c)
Photo 1.2 Photos of destroyed structures due to the Indian Ocean
Tsunami : (a) a damaged house due to both tsunami waves and
tsunami-borne debris in Banda Aceh, Indonesia; (b) a boat lifted on
the top of damaged houses in Banda Aceh, Indonesia; and (c) exposed
bridge piers of a road in Lhoknga, Indonesia (Earthquake
Engineering Research Institute, 2005).
Africa over 8,000 km away from the epicenter. Moreover, the
tsunami waves as well as tsunami-borne drifting objects caused the
destruction of many structures. Photo 1.2 illustrates destroyed
structures due to the Indian Ocean Tsunami. Photos 1.2(a) and
1.2(b) show that drifting debris such as lumbers and vessels led to
catastrophic damages of houses in Banda Aceh, Indonesia, and Photo
1.2(c) shows exposed bridge piers of a road washed away by the
runup tsunami waves in Lhoknga, Indonesia. The tsunami waves also
caused large-scale sediment transport, resulting in substantial
erosion and scour around a large number of structures. Actual
examples of tsunami-induced local scour will be explained at the
beginning of Chapter 4. Such severe wave action due to tsunami
waves is expected to cause considerable uctuations of stress and
strain elds within a foundation, and it is,
4
Chapter 1 Introduction hence, of primary importance that the
author focuses on wave-induced responses of the foundation in
investigating interactions between tsunami waves and the
foundation.
1.2 Literature reviewSince Yamamoto (1977) and Yamamoto et al.
(1978), many researchers have, analytically and numerically,
investigated interactions between wind waves and foundations of
coastal structures. In this section, the author introduces detailed
reviews of principal studies especially on numerical investigations
into the wave-foundation interactions. Yamamoto (1977) and Yamamoto
et al. (1978) obtained exact closed-form analytical solutions for
compressible pore water and homogeneous isotropic seabeds of nite
and innite thickness on the basis of the consolidation equations
(Biot, 1941). They consequently conrmed that the derived pore water
pressure inside coarse and ne sand seabeds with a small amount of
air bubbles showed a good agreement with experimental one, and
revealed that seabed response strongly depended on the stiness
ratio and permeability of the seabeds. However, they assumed that
wave-induced water pressure and sand displacement on the seabeds
were periodic in time and space, and also omitted ow velocity
across the wave-seabed interface. Furthermore, the adopted
consolidation equations included neither the acceleration of sand
particles nor that of pore water. Since exact solutions, e.g.,
Yamamoto (1977) and Yamamoto et al. (1978), were extremely
complicated in practical problems, Mei and Foda (1981) proposed an
approximate analytical solution based on the boundary layer
approximation, in which a sand skeleton and pore water inside a
seabed were assumed to move together according to the laws of
classical elasticity for a single phase, and then the boundary
layer correction based on the consolidation equations was needed
inside a thin boundary layer near the surface of the seabed. Sakai
et al. (1990) investigated applicability of the exact solution of
Yamamoto et al. (1978), the approximate solution of Mei and Foda
(1981) and another analytical solution based on the seepage ow
analysis (Finn et al., 1983). They, as a result, revealed that the
applicability was sensitive to a ratio of the shear modulus of a
seabed to the eective bulk modulus of pore water and a
nondimensional parameter proportional to the hydraulic conductivity
and shear modulus of the seabed, and also found a better
applicability of the approximate solution as well as the exact
solution for the larger former parameter and the smaller latter
parameter. The approximate solution, however, had the same
drawbacks as the exact solutions of Yamamoto (1977) and Yamamoto et
al. (1978). To overcome the aforementioned drawbacks, Jeng et al.
(2001) adopted the potential theory for incompressible irrotational
ow, the poro-elastic theory of Mei (1989) with the accelerations of
sand particles and pore water, and boundary conditions including
the continuity of water pressure and ow velocity on the wave-seabed
interface, and derived a closed-form analytical dynamic solution
for water waves and a semi-innite homogeneous isotropic seabed.
Their paper demonstrated that the accelerations of sand particles
and pore water signicantly increased vertical sand displacement and
pore water pressure in comparison with the previous quasi-static
solution, and pore water displacement was obtainable only with the
dynamic solution. However, the dynamic solution was inapplicable to
strong nonlinear wave elds such as wave breaking because of the
potential theory
1.2 Literature review for wave elds. After that, Jeng and Cha
(2003) obtained another analytical dynamic solution for a
homogeneous isotropic seabed of nite thickness based on the
complete form of the Biot equations, although this dynamic solution
excluded ow velocity across the wave-seabed interface unlike the
previous dynamic solution of Jeng et al. (2001). Analytical
approaches above have a limited applicability in investigating the
wavefoundation interactions particularly with complex geometry.
Mase et al. (1989) developed a nite element model based on the
consolidation equations (Biot, 1941) with Hookes law for an
isotropic elastic seabed to investigate pore water pressure around
a composite-type breakwater and uplift force on the bottom of a
caisson. They conrmed in preliminary calculations that computed
sand displacement and pore water pressure inside a homogeneous
isotropic seabed of nite thickness were in an excellent agreement
with analytical ones (Yamamoto, 1977); however, they applied
analytical water pressure based on the small amplitude wave theory
to the wave-seabed interface, and also modeled caisson-seabed
interactions with not a nite element approach but an analytical
one. Moreover, the use of the consolidation equations and Hookes
law caused the same problems as the analytical solutions of
Yamamoto (1977) and Yamamoto et al. (1978). Mase et al. (1989) also
ignored uplift force on the bottom of the caisson in calculating
the caisson-seabed interactions; hence, Mase et al. (1991) improved
the nite element model of Mase et al. (1989), assuming that a
concrete caisson was composed of a high stiness poro-elastic
material with very small hydraulic conductivity. Since the above
models could not reproduce residual pore water pressure and its
accumulation because of the linear constitutive equation, Kuwahara
and Ohmaki (1992) proposed another nite element model on the basis
of the u-p approximation of the Biot equations and an
elasto-plastic constitutive equation of Akai et al. (1988).
However, Kuwahara and Ohmaki (1992) also employed the same
analytical water pressure as the models of Mase et al. (1989,
1991). Park et al. (1996) developed a nite element model for not
only a seabed but also water waves to investigate
wave-seabed-structure interactions. They applied the potential
theory to incompressible irrotational ow, and the u-w form of the
Biot equations to homogeneous isotropic seabeds of nite and innite
thickness, and also incorporated boundary conditions including
continuity of water pressure and ow velocity on the wave-seabed
interface. Here, the u-w form of the Biot equations is identical to
the complete form of the Biot equations for compressible pore
water, which form includes the accelerations of both sand
displacement and pore water. Although computed results accurately
predicted analytical ones (Yamamoto, 1981), this model was
inapplicable to nonlinear wave elds because of a linear
frequency-domain calculation. To eliminate this drawback, Jiang et
al. (2000) built a VOF-FEM (Volume Of Fluid-Finite Element Method)
model on the basis of the CADMAS-SURF (Super Roller Flume for
Computer Aided Design of MAritime Structure; CDIT, 2001), the u-w
form of the Biot equations and a Voigt-type viscoelastic
constitutive equation. To incorporate continuity of water pressure,
ow velocity and normal and shear stresses on the wave-seabed
interface, they adopted the following quasitwo-way coupling method
between the VOF and FEM models: velocity and pressure elds for
water waves were rstly computed with the VOF model; the FEM model
was secondly solved with computed water pressure on the wave-seabed
interface; and the VOF model at
5
6
Chapter 1 Introduction the next step was nally calculated with
computed ow velocity across the wave-seabed interface, and this
process was repeated until the nal step of the computation.
However, the time-domain calculation of the u-w form of the Biot
equations reproduced inappropriate second compressible waves
propagating inside pore water because pore water pressure was
indirectly computed in the u-w form of the Biot equations, as
indicated by Takahashi et al. (2002). For this reason, Takahashi et
al. (2002) proposed an improved VOF-FEM model called CADMAS
GEO-SURF with the u-p approximation of the Biot equations instead
of the u-w form of the Biot equations, although this improved model
was inapplicable to typical elasto-plastic materials because of the
viscoelastic constitutive equation. Takayama et al. (2005), then,
investigated pore water pressure within a seabed foundation around
a composite-type breakwater with the CADMAS GEO-SURF; however, they
did not use the most important process of the quasi-two-way
coupling method, i.e., the feedback calculation from the FEM model
to the VOF model. Mizutani et al. (1998) and Mostafa et al. (1999)
developed a combined BEM-FEM model based on the boundary element
method (BEM) for water waves and the FEM for pore water. The
BEM-FEM model employed the potential theory for incompressible
irrotational ow and modied Navier-Stokes equations (McCorquodale et
al., 1978) for incompressible rotational viscous ow inside porous
media. They also developed another poro-elastic FEM model for a
homogeneous isotropic seabed based on the Biot consolidation
equations and a one-way coupling technique with the BEM-FEM model.
However, these two models included no feedback calculation from the
poro-elastic FEM model to the BEM-FEM model because of the
computational cost, and hence they calculated velocity and pressure
elds inside the porous media twice per computational step with each
model. In addition, many other drawbacks resulted from the use of
the potential theory, consolidation equations and linear
constitutive equation. Using the same coupling method as the models
of Mizutani et al. (1998) and Mostafa et al. (1999), Hur et al.
(2007) proposed a newly-developed numerical model with the VOF
method for incompressible air-water two-phase ow and the FEM for a
homogeneous isotropic elastic seabed. In the FEM, they adopted not
the u-w form of the Biot equations but the u-p approximation of the
Biot equations to accurately compute wave-induced pore water
pressure inside the seabed after Takahashi et al. (2002); however,
they left several problems, e.g., the use of the linear
constitutive equation. Moreover, the model of Hur et al. (2007)
also had another problem of inapplicability to three-dimensional
phenomena, which was a common fatal drawback to all analytical and
numerical approaches explained above. Further review will be
discussed at the beginning of the relevant chapters.
1.3 Study objectivesThe purpose of this study is to
experimentally and numerically investigate wave-induced macroscopic
states of sand particles under uctuating stress and strain
conditions inside a seabed. The specic targets of the present
research are as follows: Sand leakage from behind a rubble mound
breakwater and vertical seawall; and Local scour around a
land-based square structure due to a runup tsunami wave.
1.4 Contents In each phenomenon, wave-induced large stresses are
expected to be locally concentrated near the structure, and then
the author expects to conrm obvious eects of stress and strain elds
within the seabed; hence, the author here focuses on the
aforementioned two separate phenomena connected with wave-seabed
interactions, i.e., the sand leakage and local scour. To achieve
numerical investigations of the targets stated above, the author
adopts a numerical simulation applicable to the following
wave-seabed interactions: Wave eld inside and outside porous media
including a seabed; and Wave-induced pore pressure, stress and
strain elds inside the seabed. In this paper, the author develops a
three-dimensional numerical simulation model capable of
investigating the wave-seabed interactions, which consists of two
numerical submodels: one is a VOF-based submodel; and another is an
FEM-based submodel. The VOF-based submodel is a numerical wave tank
with a non-reective wave generation technique to model
three-dimensional wave elds. Assuming incompressible air-water
two-phase ow for water waves, the numerical submodel employs
governing equations based on the dynamic two-parameter mixed model
(DTM) of the LES (Large Eddy Simulation), which equations include
laminar and turbulent resistance forces due to porous media and
surface tension force based on the CSF (Continuum Surface Force)
model. Furthermore, the author introduces the MARS (Multi-interface
Advection and Reconstruction Solver) to track air-water interfaces,
which is one of the extensions of the original VOF method. On the
other hand, the author builds the FEM-based submodel based on the
u-p approximation of the Biot equations and Hookes law for
no-tension isotropic elastic materials to compute sand skeleton
displacement and pore water pressure inside the seabed. In this
submodel, the author incorporates boundary conditions including
continuity of both water pressure and ow velocity on the
wave-seabed interface after Mizutani et al. (1998) and Mostafa et
al. (1999), and dierentiates the governing and constitutive
equations with the Galerkin method of the FEM. As explained later,
the author evaluates stress and strain elds with the present
submodel. The author nally mentions excluded topics beyond the
scope of this study. Severe wave action causes the movement of sand
particles especially around structures. It seems probable that
wave-induced stress inside a seabed aects not only the initial
stage of the movement but also moving individual sand particles.
However, the author here treats not the microscopic state of the
seabed, i.e., the movement of individual sand particles, but the
macroscopic one, as mentioned above. The author therefore does not
develop a numerical model for tracking individual sand particles
coupled with stress and strain elds inside the seabed.
7
1.4 ContentsThe contents of this dissertation are summarized
below. In Chapter 2, the author explains governing equations and
computational schemes of each submodel in the newly-developed
three-dimensional numerical simulation, and also describes a
coupling technique between both numerical submodels. As
mentioned
8
Chapter 1 Introduction above, the formulation of the VOF-based
submodel is based on the DTM, whose governing equations include
laminar and turbulent resistance forces due to porous media and
surface tension force based on the CSF model. The formulation of
the FEM-based submodel is based on the u-p approximation of the
Biot equations and Hookes law for no-tension isotropic elastic
materials, which are dierentiated with the Galerkin method. In
Chapter 3, the author performs experimental and numerical
investigations of sand leakage phenomena from behind coastal
structures. In this work, the author treats two types of coastal
structure: one is a rubble mound breakwater in the absence of lter
layers and geotextile sheets against sand leakage; and the other is
a vertical seawall in the absence of wave absorption and foot
protection blocks. Focusing on ow velocity and eective stress, the
author investigates their eects on leakage mechanisms of backlling
materials. The author nally demonstrates eciency of lter layers
against sand leakage from behind the rubble mound breakwater. In
Chapter 4, the author investigates local scour around a land-based
structure with a square cross-section due to a runup tsunami with a
series of hydraulic model experiments and numerical simulations.
The author here adopts a solitary wave and isolated long wave in
modeling an acting runup tsunami wave. The hydraulic model
experiments reveal the time development of local scour around the
seaward corner of the structure with a miniature video camera
installed inside the structure, and also the inuences of tsunami
wave proles on nal scouring form in the vicinity of the structure.
In the numerical simulations, the author pays special attention to
ow velocity and eective stress around the surface of a sand
foundation, and investigates detailed mechanisms of tsunami-induced
local scour. In Chapter 5, the author proposes a new sediment
transport formula coupled with eective stress inside a sand
foundation, and incorporates the new formula into an
alreadyproposed sediment transport model. The author, then, applies
the new model to sediment transport simulations on tsunami-induced
local scour around the land-based square structure discussed above,
and nally veries the applicability of the present proposed model.
In Chapter 6, the author presents an overview of main conclusions
as well as perspectives and recommendations for future
extensions.
References[1] Akai, K., Adachi, T. and Oka, F. (1988): A cyclic
elasto-plastic constitutive model for sand, Proc., Int. Workshop on
Constitutive Equations for Granular Non-Cohesive Soils, Cleveland,
Ohio, pp. 101-114. [2] Biot, M. A. (1941): General theory of
three-dimensional consolidation, J. Appl. Phys., Vol. 12, pp.
155-164. [3] Coastal Development Institute of Technology (CDIT)
(2001): Research and Development of Numerical Wave Channel
(CADMAS-SURF), CDIT Library, No. 12, 296 p. (in Japanese). [4]
Earthquake Engineering Research Institute (2005): The Great Sumatra
Earthquake and Indian Ocean Tsunami of December 26, 2004, Internet
Resource, http://www. eeri.org/lfe/clearinghouse/sumatra
tsunami/presentation/Tsunami FINAL 4-19-05 novideo website.ppt.
References [5] Finn, W. D. L., Siddharthan, R. and Martin, G. R.
(1983): Response of seaoor to ocean waves, J. Geotech. Eng., ASCE,
Vol. 109, No. 4, pp. 556-572. [6] Geographical Survey Institute
(2004): Changes in the Reclamation Area from 1950 to 2004, Internet
Resource, http://www.gsi.go.jp/WNEW/PRESS-RELEASE/2004/ 0209d.htm
(in Japanese). [7] Hur, D.-S., Nakamura, T. and Mizutani, N.
(2007): Sand suction mechanism in articial beach composed of rubble
mound breakwater and reclaimed sand area, Ocean Eng., Elsevier,
Vol. 34, No. 8-9, pp. 1104-1119. [8] Jeng, D. S. (2003):
Wave-induced sea oor dynamics, Appl. Mech. Rev., Vol. 56, No. 4,
pp. 407-429. [9] Jeng, D. S., Barry, D. A. and Li, L. (2001): Water
wave-driven seepage in marine sediments, Advances in Water
Resource, Elsevier, Vol. 24, No. 1, pp. 1-10. [10] Jeng, D. S. and
Cha, D. H. (2003): Eects of dynamic soil behavior and wave
nonlinearity on the wave-induced pore pressure and eective stresses
in porous seabed, Ocean Eng., Elsevier, Vol. 30, No. 16, pp.
2065-2089. [11] Jiang, Q., Takahashi, S., Muranishi, Y. and Isobe,
M. (2000): A VOF-FEM model for the interaction among waves, soils
and structures, Proc., Coastal Eng., JSCE, Vol. 47, pp. 51-55 (in
Japanese). [12] Kuwahara, H. and Ohmaki, M. (1992): Wave-induced
elasto-plastic seabed behavior around a composite breakwater,
Proc., Coastal Eng., JSCE, Vol. 39, pp. 861-865 (in Japanese). [13]
Mase, H., Kawasako, I. and Sakai, T. (1991): Study on wave-induced
foundation response of a composite breakwater, Proc., Coastal Eng.,
JSCE, Vol. 38, pp. 129133 (in Japanese). [14] Mase, H., Sakai, T.,
Nishimura, Y. and Maeno, Y. (1989): Analysis of wave-induced uplift
force acting on caisson and pore-water pressure around breakwater
by using poro-elastic theory, J. Japan Soc. Civil Eng., JSCE, No.
411, II-12, pp. 9-17 (in Japanese). [15] McCorquodale, J. A.,
Hannoura, A. and Nasser, M. S. (1978): Hydraulic conductivity of
rockll, J. Hydr. Res., Vol. 16, No. 2, pp. 123-137. [16] Mei, C. C.
(1989): Applied Dynamics of Ocean Surface Waves, World Scientic,
Singapore, 740 p. [17] Mei, C. C. and Foda, M. A. (1981):
Wave-induced responses in a uid-lled poroelastic solid with a free
surface a boundary layer theory, Geophys. J. R. astr. Soc., Vol.
66, pp. 597-631. [18] Ministry of Land, Infrastructure and
Transport (2006): An approach to the Coastal Erosion Measures,
Internet Resource, http://www.mlit.go.jp/river/kaigan
dukuri/gijutsu-kondan/shinshoku/giron.pdf (in Japanese). [19]
Mizutani, N., Mostafa, A. M. and Iwata, K. (1998): Nonlinear
regular wave, submerged breakwater and seabed dynamic interaction,
Coastal Eng., Elsevier, Vol. 33, pp. 177-202. [20] Mostafa, A. M.,
Mizutani, N. and Iwata, K. (1999): Nonlinear wave, composite
breakwater, and seabed dynamic interaction, J. Waterway Port
Coastal Ocean Eng.,
9
10
Chapter 1 Introduction ASCE, Vol. 125, No. 2, pp. 88-97.
National CCZ Maintenance and Promotion Conference (2007): Coastal
Community Zone Website, Internet Resource, http://www.ccz.jp/,
Retrieved on Sep. 1, 2007 (in Japanese). Park, W.-S., Takahashi,
S., Suzuki, K. and Kang, Y.-K. (1996): Finite element analysis on
wave-seabed-structure interactions, Proc., Coastal Eng., JSCE, Vol.
43, pp. 1036-1040 (in Japanese). Sakai, T., Hatanaka, K. and Mase,
H. (1990): Applicability of solutions for transient wave-induced
porewater pressures in seabed and liquefaction conditions of
seabed, J. Japan Soc. Civil Eng., JSCE, No. 417, II-13, pp. 41-49
(in Japanese). Sumer, B. M., Whitehouse, R. J. S. and Trum, A.
(2001): Scour around coastal structures: a summary of recent
research, Coastal Eng., Elsevier, Vol. 44, No. 2, pp. 153-190.
Takahashi, S., Suzuki, K., Muranishi, Y. and Isobe, M. (2002): U-
form VOF-FEM program simulating wave-soil interaction:
CADMAS-GEO-SURF, Proc., Coastal Eng., JSCE, Vol. 49, pp. 881-885
(in Japanese). Takayama, T., Yasuda, T., Tsujio, D., Taniguchi, S.
and Mizutani, M. (2005): Field observations for the response
properties of pore water pressures in the seabed beneath a
composite breakwater covered with concrete blocks, Ann. J. Coastal
Eng., JSCE, Vol. 52, pp. 846-850 (in Japanese). Yamamoto, T.
(1977): Wave induced instability in seabeds, Proc., Coastal
Sediments 77, ASCE, Charleston, South Carolina, pp. 898-913.
Yamamoto, T. (1981): Wave-induced pore pressures and eective
stresses in inhomogeneous seabed foundations, Ocean Eng., Elsevier,
Vol. 8, No. 1, pp. 1-16. Yamamoto, T., Koning, H. L., Sellmeijer,
H. and Hijum, E. V. (1978): On the response of a poro-elastic bed
to water waves, J. Fluid Mech., Vol. 87, No. 1, pp. 193-206.
[21]
[22]
[23]
[24]
[25]
[26]
[27] [28] [29]
11
Chapter 2
Numerical Model2.1 GeneralIn general, numerical simulations have
great advantages over hydraulic model experiments for no measured
experimental data because of their laborious or impossible
measurement, e.g., detailed ow velocity elds, sand particle
displacement and dynamic stress inside impermeable and permeable
coastal structures. Furthermore, the numerical simulations also
have superiority from the standpoint of safety, time eciency and
cost eectiveness. Wave-induced sand foundation dynamics is one of
the key factors in investigating the behavior of sand leakage and
local scouring phenomena; however, eective stresses within the
foundation cannot be directly measured in hydraulic model
experiments, and it is therefore extremely important whether a
numerical simulation enables us to accurately compute
wave-foundation interactions. Numerous analytical and numerical
models have so far been proposed to investigate the wave-seabed
interaction; however, most of the existing models such as Yamamoto
(1977) employ analytical wave pressure distributions on the surface
of the seabed, and also assume no ow velocity across the
wave-seabed interface. Park et al. (1996) investigated
wave-seabed-structure interactions with a developed nite element
model (FEM) based on the potential theory and the u-w form of the
Biot equations with continuity of water pressure and ow velocity at
the wave-seabed interface, although this model could not analyze
nonlinear wave elds such as wave breaking because of a linear
frequency-domain calculation. Mizutani et al. (1998) and Mostafa et
al. (1999) developed combined BEMFEM and poro-elastic FEM models
for simulating nonlinear interactions among waves, seabed and
composite/submerged breakwater, which had the capability to compute
seabed response due to nonlinear waves. In the poro-elastic FEM
model, the accelerations of sand particles and pore water were,
however, eliminated because of the Biot poro-elastic consolidation
equations, and hence it is possible that wave-induced eective
stresses are inaccurate under nonlinear wave conditions (Jeng and
Cha, 2003). Jiang et al. (2000) performed numerical simulations on
wave-seabed-structure interactions, e.g., seabed behavior due to
wave breaking and wave-foundation interactions around a
composite-type breakwater, using a VOF-FEM model based on the
CADMAS-SURF (Super Roller Flume for Computer Aided Design of
MAritime Structure; CDIT, 2001) and the u-w form of the Biot
equations. As mentioned below, Takahashi et al. (2002), then,
adopted the u-p approximation of the Biot equations instead of the
u-w form of the Biot equations for more
12
Chapter 2 Numerical Model accurate calculation of pore water
pressures around the surface of the seabed, and nally proposed a
modied VOF-FEM model called CADMAS GEO-SURF. Hur et al. (2007)
developed a new numerical simulation model capable of wave-seabed
interactions, which was composed of two numerical submodels: one
was a numerical wave tank based on the volume of uid (VOF) method
for computing velocity and pressure elds inside airwater two-phase
ow; and the other was a wave-induced soil-water coupled FEM based
on the u-p approximation of the Biot equations for calculating
eective stresses and pore water pressures inside a sand foundation.
Following Mizutani et al. (1998) and Mostafa et al. (1999), they
applied a mixed boundary condition to the coupling scheme between
the VOF-based and FEM-based submodels unlike the CADMAS GEO-SURF,
which employed a rst-type (Dirichlet) boundary condition as the
coupling method between the VOF and FEM submodels (Takahashi et
al., 2002). Important advances in analytical and numerical
simulation models for the wave-seabed interaction system were
summarized in Jengs review paper (2003). In this paper, the author
improved the numerical model of Hur et al. (2007) to compute
wave-seabed interactions, especially ow velocity and eective stress
elds around the surface of a sand foundation in the vicinity of
coastal structures. The author here provides a detailed explanation
of governing equations and computational schemes of each submodel,
and then briey explains a coupling technique between both numerical
submodels proposed by Mizutani et al. (1998) and Mostafa et al.
(1999).
2.2 VOF-based numerical submodel for a wave eld2.2.1 Governing
equations Golshani et al. (2003) developed a three-dimensional
fully nonlinear numerical model based on the VOF method (Hirt and
Nichols, 1981) to investigate wave-induced ow elds inside and
around a vertical permeable structure. In their model, equations of
motion explicitly included laminar and turbulent resistance forces
with the size eects of porous media (Mizutani et al., 1996) to deal
with the interaction between water waves and the porous media.
They, however, eliminated molecular viscosity terms instead of the
introduction of the laminar resistance force, and their model was
also inapplicable to air-water two-phase ow, which is greatly
important in a complicated wave eld, e.g., air bubble entrainment
due to wave breaking. Hur et al. (2007), then, added the molecular
viscosity terms omitted by Golshani et al. (2003) to improve the
computational accuracy of molecular viscosity regions such as
near-wall zones, and also incorporated computational schemes
applicable to air-water two-phase ow. In this study, the author
additionally improved the numerical model of Hur et al. (2007). The
author rst added surface tension eects based on the CSF (Continuum
Surface Force) model (Brackbill et al., 1992), and then introduced
the large eddy simulation (LES) based on the dynamic two-parameter
mixed model (DTM; Salvetti and Banerjee, 1995) for modeling eddy
viscosity eects. This simulation thus employed the following
governing equations, i.e., a continuity equation (2.1), modied
Navier-Stokes equations (2.2) and an advection equation (2.3) of
the VOF function F (0 F 1), which represents the volume fraction of
water in a numerical mesh:
2.2 VOF-based numerical submodel for a wave eld ( ) mv j x j ( =
q ,
13
(2.1)
( ) ) 1 m vi vi v j 1 + CA + m t x j ) fis ( 1 p gi + + i j + 2
Di j Ri + Qi i j v j , = xi x j ( ) mv j F (mF) + = Fq , t x j
(2.2)
(2.3)
[ ] where vi = [u, v, w]T is the seepage velocity vector; p is
the pressure; xi = x, y, z T is the position vector; t is the time;
gi = gi3 is the gravitational acceleration vector, in which g is
the gravitational acceleration and i j is the Kronecker delta; = Fw
+ (1 F) a is the uid density, in which w and a are the densities of
water and air, respectively; = Fw + (1 F) a is the uid kinematic
molecular viscosity, in which w and a are the kinematic molecular
viscosities of water and air, respectively; m is the porosity; q =
q (y, z; t) /x s is the wave generation source, in which q (y, z;
t) is the source density at a source position (x = x s ) and x s is
the x-directional mesh width at the source position (x = x s ); C A
is the added mass coecient; fis is the surface tension force based
on )the ( CSF model; i j is the turbulent stress based on the DTM;
Di j = vi /x j + v j /xi /2 is the strain rate tensor; Ri is the
resistance force vector due to porous media proposed by Mizutani et
al. (1996); Qi is the wave source vector; i j = i3 j3 is the
dissipation factor matrix, in which is the dissipation factor which
is equal to zero except for added dissipation zones (Hinatsu,
1992); the superscript T represents a transposed matrix; and the
subscripts i and j are governed by the Einstein summation
convention. The author varied the porosity m in space to model
impermeable structures such as a revetment (m = 0.0), permeable
structures such as a reclaimed beach (0.0 < m < 1.0), and the
other void regions (m = 1.0). The surface tension force fis , the
resistance force vector Ri and the wave source vector Qi are
formulated as follows: fis = Ri = F , xi (2.4) (2.5)
12C D2 (1 m) C D1 (1 m) vi + vi v j v j , 2 2md50 md50 ( ) q 2 q
Qi = vi , m 3 xi m
(2.6)
where is the surface tension coecient; is the local surface
curvature; = (w + a ) /2 is the uid density at the air-water
interface; C D2 and C D1 are the laminar and turbulent resistance
coecients, respectively; and d50 is the median grain size of porous
media. In the present paper, the author used the following physical
constants (see Table 2.1):
14
Chapter 2 Numerical ModelTable 2.1 Physical constants of
gravity, water and air.
g [m/s2 ] w [kg/m3 ] a [kg/m3 ] w [m2 /s] a [m2 /s] [N/m]
9.81 9.97 102 1.18 8.93 107 1.54 105 7.20 102
the gravitational acceleration g = 9.81 m/s2 ; the water density
w = 9.97 102 kg/m3 ; the air density a = 1.18 kg/m3 ; the water
kinematic molecular viscosity w = 8.93 107 m2 /s; the air kinematic
molecular viscosity a = 1.54 105 m2 /s; and the surface tension
coecient = 7.20 102 N/m, in which the author adopted the values of
w , a , w , a and at 25.0 C and 1.01 105 Pa (National Astronomical
Observatory of Japan, 2003). The author also set the added mass
coecient C A = 0.04, the turbulent resistant coecient C D1 = 0.45
and the laminar resistant coecient C D2 = 25.0 after Fig. 2.1
(Mizutani et al., 1996), although the author decided the values of
C A and C D2 with preliminary tests in Chapter 3.2.3. As for the
wave generation technique, the author applied the fth-order Stokes
wave theory (Horikawa, 1988) to the computation of the source
density q for periodic waves, the third-order solitary wave theory
(Fenton, 1972) for a solitary wave, and the Airy wave theory for an
isolated long wave. This technique required the x-directional mesh
width at the source position x s to apply the non-reective wave
generator in the boundary element method (Ohyama and Nadaoka, 1991)
to the nite element method. Further details of the wave generation
technique are available in Kawasaki (1999) and Hur and Mizutani
(2003). The direct numerical simulation of turbulent ows is
restricted to low Reynolds numbers because of the need to resolve
all spatial scales of turbulence (Salvetti and Banerjee, 1995). In
the large eddy simulation (LES), the large-scale eld greater than
the gridscale (GS) is directly calculated, while only the
small-scale eld called the subgrid-scale (SGS) is modeled with the
pioneering Smagorinsky model (Smagorinsky, 1963), dynamic
Smagorinsky model (DSM; Germano et al., 1991), dynamic mixed model
(DMM; Zang et al., 1993) or DTM. The author here utilized the DTM,
in which the SGS stress i j is assumed to be proportional to the
modied Leonard stress Lim = vi v j vi v j and the strain j rate
tensor Di j as follows (Morinishi and Vasilyev, 2001): aj = C L
Lima CS |D| Di j , j i CL =a Lia Hi a Mk Mk Lia Mi j Hk Mk j j j a
Hi a Hi a Mk Mk Hi a Mi j Hk Mk j j j a a a Lia Mi j Hk Hk Lia Hi a
Hk Mk j j j a Hi a Hi a Mk Mk Hi a Mi j Hk Mk j j j
(2.7) , (2.8)
CS =
,
(2.9)
where |D| is the absolute value of the strain rate tensor Di j ;
Li j = vi v j vi v j is the Germano identity; Hi j = vi v j vi v j
; Mi j = 2 |D|Di j |D| Di j ; = /, in which and are
2.2 VOF-based numerical submodel for a wave eld
15
(a)
(b)
(c)Fig. 2.1 Experimental results of (a) the inertia force
coecient C M = 1 +C A ; (b) the turbulent resistant coecient C D1 ;
and (c) the laminar resistant coecient C D2 against the
Keulegan-Carpenter number KC (Mizutani et al., 1996). Averaging
measured values of each coecient, they estimated C M = 0.96 (C A =
0.04) and C D1 = 0.45 for all KC numbers and C D2 = 25.0 for KC
< 10.0.
the lter widths of the grid and test scales, respectively; the
superscript a represents the anisotropic part of a tensor, e.g., aj
= i j i j kk /3; and the subscripts i, j, k and are i governed by
the Einstein summation convention. As indicated in Eqs. (2.8) and
(2.9), the coecients C L and CS can be dynamically computed with
resolved GS velocities vi . The input parameter in the DTM is only
, and the author adopted = 2.0 after Germano et al. (1991). For
details of the DTM, see Salvetti and Banerjee (1995) and Morinishi
and Vasilyev (2001). 2.2.2 Computational schemes In this submodel,
the author utilized the SMAC (Simplied MAC) method (Amsden and
Harlow, 1970) for coupling velocities vi and pressures p in Eqs.
(2.1) and (2.2). For accurate and stable calculation of the time
integration of Eq. (2.2), especially the linear resistance force
term, this simulation employed the second-order Crank-Nicolson
and
16
Chapter 2 Numerical Model third-order Adams-Bashforth schemes
instead of the standard rst-order forward dierence scheme, and thus
the dierence equations lead to vip = [ vn i { 1 pn tn+1/2 1 12C D2
n (1 m) n + n gi vi 2 1 + C A (1 m) /m xi 2 md50 ( + An +
0Crank-Nicolson scheme ( )2 )}]/ tn+1/2 tn+1/2 n Bn , A1 + An 2
2
6
(2.10)
Third-order Adams-Bashforth scheme
vn+1 = vip i
1 tn+1/2 n+1/2 / n B, n 1 + C A (1 m) /m xi
(2.11)
where vip is the predicted seepage velocity vector; the
superscript n is the time step number; tn+1/2 is the time increment
between the n-th and (n + 1)-th time steps; n+1/2 = pn+1 pn is the
pressure increment at the (n + 1/2)-th time step, which is computed
with the following Poisson equation: ( ) (1 n+1/2 / ) mvip /xi q
n+1 m ; (2.12) Bn = n 1 + C (1 m) /m xi xi tn+1/2 A and the
time-dependent parameters An , An , An and Bn are expressed as 0 1
2 An 0 = ) ( vn vn i j x j ) fis n ( n i + n + i j + 2 n Dnj x j C
D1 (1 m) n n n vi v j v j + Qn nj vn , i j i 2md50
(2.13)
{ ( ) ( )2 An = tn3/2 tn3/2 + 2tn1/2 An tn3/2 + tn1/2 An1 1 0 0
( )2 }/{ ( )} + tn1/2 An2 tn3/2 tn1/2 tn3/2 + tn1/2 , 0 { ( ) } An
= 2 tn3/2 An tn3/2 + tn1/2 An1 + tn1/2 An2 2 0 0 0 /{ ( )} tn3/2
tn1/2 tn3/2 + tn1/2 , (2.15) (2.16) (2.14)
Bn = 1 +
tn+1/2 1 12C D2 n (1 m) , 2 1 + C A (1 m) /m 2
md50Crank-Nicolson scheme
in which the author applied the third-order TVD (Total Variation
Diminishing) scheme of Osher and Chakravarthy (1984) and
Chakravarthy and Osher (1985) to the convective terms of Eq.
(2.13), and the second-order central dierence scheme to the other
terms of Eqs. (2.10) to (2.16). Figure 2.2 shows the performance of
the third-order TVD scheme
2.3 FEM-based numerical submodel for a sand bed1.5 1.5
17
1.0
1.0
0.5
0.5
y
0.0
y0.0 -0.5Initial profile 3rd-order TVD 1st-order upwind
3rd-order upwind Initial profile 3rd-order TVD 1st-order upwind
3rd-order upwind
-0.5
-1.0 -0.5
0.0
0.5
(a)
x
1.0
1.5
-1.0 -0.5
0.0
0.5
(b)
x
1.0
1.5
Fig. 2.2 Performance of the third-order TVD scheme after 100 and
200 time steps for the Courant number 0.1: (a) a rectangular wave;
and (b) a sinusoidal wave.
as well as the rst-order and third-order upwind dierence schemes
for simple rectangular and sinusoidal waves. A box lter in physical
space was used as both the grid and test lters of the LES, and the
lter width was dened as = xi in each direction, where xi is the
mesh width in the i-th direction. For the linear solver of the
Poisson equation (2.12), the author adopted the MICCG (Modied
Incomplete Cholesky Conjugate Gradient) method. For the accurate
tracking of air-water interfaces, the advection equation (2.3) was
computed with the MARS (Multi-interface Advection and
Reconstruction Solver) of Kunugi (2000), one of the PLICs
(Piecewise Linear Interface Calculation) such as pioneering PLIC
(Youngs, 1982) and TELLURIDE (Rider and Kothe, 1998). In the MARS,
the air-water interface in each numerical mesh is described as an
inclined plane, which can be easily determined with the neighboring
VOF functions F. Detailed explanations of the MARS are available in
Kunugi (2000).
2.3 FEM-based numerical submodel for a sand bed2.3.1 Governing
equations Assuming that the Lagrangian coordinate system transfers
with sand displacement, one can derive a momentum conservation
equation for total system , and mass and momentum conservation
equations for compressible pore water (Zienkiewicz and Shiomi,
1984). This complete form is generally called the u-w form of the
Biot equations. Jiang et al. (2000) developed a VOF-FEM coupled
model with this form of the Biot equations; however, Takahashi et
al. (2002) revealed that the second (slow) compressible wave
propagating in pore water could not be accurately computed in the
time-domain calculation of the u-w form of the Biot equations, and
instead recommended the use of the u-p approximation of the Biot
equations, which is derived from the complete form of the Biot
equations on the assumption that the acceleration of relative water
displacement is negligible. Zienkiewicz and Shiomi (1984) also
indicated that this approximation is valid for most frequencies
slower than earthquake analysis. For this reason, Hur et al. (2007)
developed a soil-water coupled two-dimensional FEM with the u-p
approximation of the Biot equations for computing wave-induced sand
displacement and pore water pressure inside
18
Chapter 2 Numerical Model a sand foundation. However, they
adopted Hookes law for isotropic elastic materials as a
constitutive equation, and hence there still remain some drawbacks.
In the present paper, the author expanded the two-dimensional FEM
(Hur et al., 2007) into a three-dimensional one, and utilized
Hookes law for isotropic elastic materials nonresistant to tensile
force, so-called no-tension isotropic elastic materials. The author
notes that liquefaction of a sand foundation strongly depends on
its stress-strain history, modeling technique and evaluation index;
however, the author here adopted the simple constitutive equation
to clarify the mechanism of the sand leakage and local scouring
phenomena. This simulation thus employed the following governing
equations: i = ji, j p,i gi , u { } ) kk m ks ( + p+ w ui p,i w g =
0, t Kw w g ,i (2.17) (2.18)
where ui is the displacement vector of a sand skeleton; p is the
pore water pressure; [ ] xi = x, y, z T is the position vector; t
is the time; = (1 m) s + mw is the sand density, in which m is the
porosity and s and w are the densities of sand particles and water,
respectively; j is the eective stress; gi = gi3 is the
gravitational acceleration i vector, in which g is the
gravitational acceleration and i j is the Kronecker delta; i j is
the strain tensor; Kw is the bulk modulus of water; k s is the
hydraulic conductivity; the superscript dot means the time
derivative; the subscript , j denotes /x j ; the superscript T
represents a transposed matrix; and the subscripts i, j and k are
governed by the Einstein summation convention. In soil mechanics,
positive j and i j are generally dened as i compression and
contraction, respectively, and the author hence followed this
denition. The constitutive equation adopted in this simulation is j
= kk i j + 2i j , i where and are Lam constants, which are
expressed as e = E , (1 + ) (1 2) E = G, 2 (1 + ) (2.20) (2.21)
(2.19)
=
in which E is the modulus of elasticity; G is the shear modulus
of elasticity; and is the Poisson ratio. In this work, the author
utilized the following physical constants: the gravitational
acceleration g = 9.81 m/s2 ; the water density w = 9.97 102 kg/m3 ;
and the bulk modulus of water Kw = 2.20 109 N/m2 . There exist
several studies in which an apparent Kw was employed to represent
highly compressible pore water containing a small percentage of air
(e.g., Yamamoto, 1977; Mizutani et al., 1998; and Jeng and Cha,
2003); however, the author here adopted the value of Kw for pure
water at 20.0 C and 1.01 105 Pa (National Astronomical Observatory
of Japan, 2003) because of not only laborious measurement of the
dissolved air percentage in hydraulic model experiments and
2.3 FEM-based numerical submodel for a sand bed eld surveys but
also the too technical approach to the apparent Kw . As for the
calculation of the hydraulic conductivity k s , the author used the
Kozeny-Carman equation (Bear, 1972) expressed as 2 1 m3 gd50 .
(2.22) ks = 180 (1 m)2 w 2.3.2 Computational schemes As mentioned
in the previous section, the author utilized the nonlinear
constitutive equation, and the author hence developed the numerical
submodel with the FEM based on the following incremental form of
the nite element equation: ][ ] [ ] ][ ] [ UU [ UU ][ ] [ dU dF dU
K K UP M 0 dU 0 0 = , + + CPU CPP dP dQ MPU 0 dP dP 0 K PP (2.23)
where dU and dP are the incremental forms of the global sand
displacement vector U and the global pore water pressure vector P,
respectively; and the coecient matrices MUU , MPU , CPU , CPP , K
UU , K UP , K PP , dF and dQ are formulated as UU M = NT N dv,
(2.24) MPU = C CPU
19
GT T
ks N dv, g
(2.25) (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32)
= = =
N mT B dv,
PP
N
T
m N dv, Kw
K
UU
BT DB dv,
K K
UP
=
BT mN dv,
PP
=
GT
ks G dv, w g
dF = dQ =
NT d t ds,t
q
N dq ds,
T
in which N and N are the shape functions of dU and dP,
respectively; D is the stress-strain matrix; B = LN is the
displacement-strain matrix; G = L N; d t is the incremental form of
on the traction boundary t ; dq is the incremental form of relative
external force vector t
20
Chapter 2 Numerical Model water discharge q = vn un in the
outward normal direction on the natural boundary q (Takahashi et
al., 2002), in which un and vn are the velocities of the sand
skeleton and pore water in the outward normal direction on q ,
respectively; m is the vector notation of the Kronecker delta i j ;
and L and L are the matrix and vector notations of the Laplace
operator , respectively. As explained in the next section, the
author computed vn with the seepage velocity vector vi of the
VOF-based submodel. The previous two-dimensional submodel (Hur et
al., 2007) employed the four-node linear isoparametric
quadrilateral elements for both shape functions N and N; however,
Sandhu et al. (1977) and Arai et al. (1983) revealed from numerical
experiments that the more stable calculation of the spatial
discretization resulted from the higher-order shape function N of
dU compared with N of dP, and hence the author applied the
twenty-node quadratic and eight-node linear isoparametric brick
elements to N and N, respectively. This simulation also utilized
the Newmark and Crank-Nicolson schemes to discretize the time
integrals of dU and dP, respectively. Judging from the
computational stability of preliminary simulations, the author set
the parameters of the Newmark scheme at N = 0.3 and N = 0.6. In
addition, the author adopted the modied Newton-Raphson scheme as
the iterative computational technique to induce no tensile normal
stresses in each direction. In the Newmark , Crank-Nicolson and
modied Newton-Raphson schemes, this submodel employed the linear
solver called the CGSTAB (Conjugate Gradient STABilized)
method.
2.4 Coupling technique between the submodelsParticularly for
high permeability of the seabed, it is of great importance to
impose continuity conditions of not only water pressure but also ow
velocity on the surface of the seabed. As discussed later, it is
possible that severe wave action causes large ow velocity across
the wave-seabed interface. An actual example will be explained with
Figure 3.24 in Chapter 3.3.3.2. In this study, the aforementioned
two submodels, i.e., the VOF-based submodel for a wave eld
(hereafter referred to as VOF) and the FEM-based submodel for a
sand foundation (hereafter referred to as FEM), were therefore
coupled with the following coupling technique proposed by Mizutani
et al. (1998): 1. Using the VOF, velocities vi and pressures p were
computed in the whole numerical domain, including inside the sand
foundation. 2. Discharges q and pressures p on the surface of the
sand foundation were interpolated with the above computed vi and p,
which discharges q and pressures p were used as the boundary
conditions of the FEM at the following step. 3. Using the FEM with
the above computed boundary conditions, sand displacement ui and
pressure p were computed only inside the sand foundation, following
which stresses j and strains i j were determined. i This process
was repeated until the nal step of the computation. Through this
process, the author had the time series of the velocities vi and
pressures p in the whole numerical domain, and also the stresses j
and strains i j within the sand foundation. The author notes i that
no feedback calculation from the FEM to the VOF was carried out
following Mizutani et al. (1998) and Maeno and Fujita (2001)
because wave-induced sand displacement was
2.5 Remarks adequately small, that is, the maximum order of the
sand displacement was 107 m under all conditions presented in the
following chapters. Details of this coupling technique are
available in Mizutani et al. (1998) and Mostafa et al. (1999).
21
2.5 RemarksWave-induced sand foundation dynamics is one of the
key factors in investigating the behavior of sand leakage and local
scouring phenomena. In this chapter, the author therefore developed
a numerical simulation model with a coupling technique proposed by
Mizutani et al. (1998) to improve the computational accuracy of
wave-foundation interactions, which was composed of two numerical
submodels: one was a numerical wave tank based on the VOF method
for computing velocity and pressure elds inside air-water two-phase
ow; and the other was a wave-induced soil-water coupled FEM based
on the u-p approximation of the Biot equations for calculating
eective stresses and pore water pressures inside a sand foundation.
The details of each submodel are summarized as follows: 1. The
VOF-based submodel was a large eddy simulation based on the dynamic
twoparameter mixed model (Salvetti and Banerjee, 1995) with the
laminar and turbulent resistance forces due to porous media
(Mizutani et al., 1996), the surface tension force based on the CSF
model (Brackbill et al., 1992) and the wave generation source of
Kawasaki (1999). This submodel employed the SMAC method (Amsden and
Harlow, 1970) and MARS (Kunugi, 2000) with the third-order TVD,
secondorder Crank-Nicolson and third-order Adams-Bashforth schemes
as the spatial and temporal discretization of the governing
equations. 2. The FEM-based submodel was based on the u-p
approximation of the Biot equations and Hookes law for no-tension
isotropic elastic materials. This submodel employed the Newmark and
Crank-Nicolson schemes to discretize the time integrals of sand
displacement and pore water pressure, respectively. For stable
calculation of the spatial discretization, this submodel also
applied the twenty-node quadratic and eight-node linear
isoparametric brick elements to the shape functions of sand
displacement and pore water pressure, respectively.
References[1] Amsden, A. A. and Harlow, F. H. (1970): A simplied
MAC technique for incompressible uid ow calculation, J. Comp.
Phys., Vol. 6, pp. 322-325. [2] Arai, K., Watanabe, T. and Tagyo,
K. (1983): Comparison of numerical techniques for multi-dimensional
consolidation problem, J. Japanese Soc. Soil Mech. Foundation
Mech., Vol. 23, No. 3, pp. 189-195 (in Japanese). [3] Bear, J.
(1972): Dynamics of Fluids in Porous Media, American Elsevier Pub.
Co., New York, 764 p. [4] Brackbill, J. U., Kothe, D. B. and
Zemach, C. (1992): A continuum method for modeling surface tension,
J. Comp. Phys., Vol. 100, pp. 335-354. [5] Chakravarthy, S. R. and
Osher, S. (1985): A new class of high accuracy TVD schemes for
hyperbolic conservation law, AIAA Paper, 85-0363.
22
Chapter 2 Numerical Model [6] Coastal Development Institute of
Technology (CDIT) (2001): Research and Development of Numerical
Wave Channel (CADMAS-SURF), CDIT Library, No. 12, 296 p. (in
Japanese). [7] Fenton, J. (1972): A ninth-order solution for the
solitary wave, J. Comp. Phys., Vol. 53, pp. 257-271. [8] Germano,
M., Piomelli, U., Moin, P. and Cabot, W. H. (1991): A dynamic
subgridscale eddy viscosity model, Phys. Fluids A, Vol. 3, No. 7,
pp. 1760-1765. [9] Golshani, A., Mizutani, N., Hur, D.-S. and
Shimizu, H. (2003): Three-dimensional analysis on nonlinear
interaction between water waves and vertical permeable breakwater,
Coastal Eng. J., JSCE, Vol. 45, No. 1, pp. 1-28. [10] Hinatsu, M.
(1992): Numerical simulation of unsteady viscous nonlinear waves
using moving grid system tted on a free surface, J. Kansai Soc.
Naval Architects, Vol. 217, pp. 1-11. [11] Hirt, C. W. and Nichols,
B. D. (1981): Volume of uid (VOF) method for dynamics of free
boundaries, J. Comp. Phys., Vol. 39, pp. 201-225. [12] Horikawa, K.
(1988): Nearshore Dynamics and Coastal Processes, Univ. of Tokyo
Press, 522 p. [13] Hur, D.-S. and Mizutani, N. (2003): Numerical
estimation of the wave forces acting on a three-dimensional body on
submerged breakwater, Coastal Eng., Elsevier, Vol. 47, pp. 329-345.
[14] Hur, D.-S., Nakamura, T. and Mizutani, N. (2007): Sand suction
mechanism in articial beach composed of rubble mound breakwater and
reclaimed sand area, Ocean Eng., Elsevier, Vol. 34, No. 8-9, pp.
1104-1119. [15] Jeng, D. S. (2003): Wave-induced sea oor dynamics,
Appl. Mech. Rev., Vol. 56, No. 4, pp. 407-429. [16] Jeng, D. S. and
Cha, D. H. (2003): Eects of dynamic soil behavior and wave
nonlinearity on the wave-induced pore pressure and eective stresses
in porous seabed, Ocean Eng., Elsevier, Vol. 30, pp. 2065-2089.
[17] Jiang, Q., Takahashi, S., Muranishi, Y. and Isobe, M. (2000):
A VOF-FEM model for the interaction among waves, soils and
structures, Proc., Coastal Eng., JSCE, Vol. 47, pp. 51-55 (in
Japanese). [18] Kawasaki, K. (1999): Numerical simulation of
breaking and post-breaking wave deformation process around a
submerged breakwater, Coastal Eng. J., JSCE, Vol. 41, No. 3-4, pp.
201-223. [19] Kunugi, T. (2000): MARS for multiphase calculation,
CFD J., Vol. 9, No. 1, IX-563. [20] Maeno, S. and Fujita, S.
(2001): VOF-FEM analysis of dynamic seabed behavior around
revetment under wave motion, Proc., Coastal Eng., JSCE, Vol. 48,
pp. 971975 (in Japanese). [21] Mizutani, N., McDougal, W. G. and
Mostafa, A. M. (1996): BEM-FEM combined analysis of nonlinear
interaction between wave and submerged breakwater, Proc., 25th Int.
Conf. Coastal Eng., ASCE, Orlando, Florida, pp. 2377-2390. [22]
Mizutani, N., Mostafa, A. M. and Iwata, K. (1998): Nonlinear
regular wave, submerged breakwater and seabed dynamic interaction,
Coastal Eng., Elsevier, Vol. 33,
References pp. 177-202. Morinishi, Y. and Vasilyev, O. V.
(2001): A recommended modication to the dynamic two-parameter mixed
subgrid scale model for large eddy simulation of wall bounded
turbulent ow, Phys. Fluids, Vol. 13, No. 11, pp. 3400-3410.
Mostafa, A. M., Mizutani, N. and Iwata, K. (1999): Nonlinear wave,
composite breakwater, and seabed dynamic interaction, J. Waterway
Port Coastal Ocean Eng., ASCE, Vol. 125, No. 2, pp. 88-97. National
Astronomical Observatory of Japan (2003): Chronological Scientic
Tables, Maruzen Co., Ltd., Tokyo, 945 p. (in Japanese). Ohyama, T.
and Nadaoka, K. (1991): Development of a numerical wave tank for
analysis of nonlinear and irregular wave eld, Fluid Dynamics R.,
Vol. 8, pp. 231251. Osher, S. and Chakravarthy, S. (1984): Very
high order accurate TVD schemes, ICASE Rep., No. 84-44, NASA
Langley Research Center, Virginia, 64 p. Park, W.-S., Takahashi,
S., Suzuki, K. and Kang, Y.-K. (1996): Finite element analysis on
wave-seabed-structure interactions, Proc., Coastal Eng., JSCE, Vol.
43, pp. 1036-1040 (in Japanese). Rider, W. J. and Kothe, D. B.
(1998): Reconstruction volume tracking, J. Comp. Phys., Vol. 141,
pp. 112-152. Salvetti, M. V. and Banerjee, S. (1995): A priori
tests of a new dynamic subgrid-scale model for nite dierence
large-eddy simulations, Phys. Fluids, Vol. 7, No. 11, pp.
2831-2847. Sandhu, R. S., Liu, H. and Singh, K. J. (1977):
Numerical performance of some nite element schemes for analysis of
seepage in porous elastic media, Int. J. Num. Anal. Methods
Geomech., Vol. 1, pp. 177-194. Smagorinsky, J. (1963): General
circulation experiments with the primitive equations, Mon. Weath.
Rev., Vol. 91, No. 3, pp. 99-164. Takahashi, S., Suzuki, K.,
Muranishi, Y. and Isobe, M. (2002): U- form VOF-FEM program
simulating wave-soil interaction: CADMAS-GEO-SURF, Proc., Coastal
Eng., JSCE, Vol. 49, pp. 881-885 (in Japanese). Yamamoto, T.
(1977): Wave induced instability in seabeds, Proc., Coastal
Sediments 77, ASCE, New York, pp. 898-913. Youngs, D. L. (1982):
Time dependent multimaterial ow with large uid distortion,
Numerical Methods for Fluid Dynamics, ed. Morton, K. M. and Baines,
M. J., Academic Press, 517 p. Zang, Y., Street, R. L. and Kose, J.
R. (1993): A dynamic mixed subgrid-scale model and its application
to turbulent recirculating ows, Phys. Fluids A, Vol. 5, No. 12, pp.
3186-3196. Zienkiewicz, O. C. and Shiomi, T. (1984): Dynamic
behaviour of saturated porous media; the generalized Biot
formulation and its numerical solution, Int. J. Num. Anal. Methods
Geomech., Vol. 8, pp. 71-96.
23
[23]
[24]
[25] [26]
[27] [28]
[29] [30]
[31]
[32] [33]
[34] [35]
[36]
[37]
24
Chapter 3
Wave-Induced Sand Leakage Phenomena3.1 GeneralSuccessive severe
wave action due to heavy storms may lead to coastal disasters such
as sliding, settlement and overturning of coastal structures and
subsidence of their reclaimed areas. Photo 3.1 shows an actual
example of reclaimed land subsidence behind a caissontype seawall.
On December 30, 2001, a huge cave-in suddenly appeared on an
articial reclaimed beach just behind the seawall at the Ohkura
beach, Hyogo, Japan, unfortunately causing a tragic accident
resulting in the death of a little girl of four years old. After
this accident, an investigation by the Japanese Ministry of Land,
Infrastructure and Transport revealed many caves and cave-ins at
several articial beaches. Photo 3.2 shows another example of
cave-ins behind a rubble mound breakwater at the Shiroya beach,
Aichi, Japan. In addition to these accidents, it is well known that
similar phenomena occurred on roads built behind a seawall. In the
early morning on April 7, 2000, a certain Japanese national road
subsided behind a vertical seawall in the Chita peninsula, Aichi,
Japan, as shown in Photo 3.3. Bierawski et al. (2002) and Bierawski
and Maeno (2006) also reported similar accidents to Photo 3.3. Once
this type of accident occurs near roads, it is possible that trac
passing on the roads is obstructed for a long period; however, few
studies attempted to clarify subsidence mechanisms of a reclaimed
area behind coastal structures. In general, subsidence of a
reclaimed beach results from tow scouring and back-
Caisson
CaissonCave-in(a)
Cave-in
(b)
Photo 3.1 Cave-in on an articial beach reclaimed behind a
caisson-type seawall at the Ohkura beach, Hyogo, Japan (JSCE
Coastal Engineering Committee, 2002).
3.1 General
25
wate reak B
r
Cave-in
Cave-in(a) (b)Photo 3.2 Cave-ins on an articial beach reclaimed
behind a rubble mound breakwater at the Shiroya beach, Aichi,
Japan.
v Re
etm
t en
Re vet me ntCave-in
Cave-in(b)
(a)
Revetment
Gap Ocean(c)Photo 3.3 Subsidence of a Japanese national road
built behind a vertical seawall at the low tide in the Chita
peninsula, Aichi, Japan (provided by Aichi Prefectural
Government).
lling sand leakage; hence, the installation of protecting
facilities such as lter layers, geotextile sheets and wave
absorption and foot protection blocks prevents the aforementioned
accidents. However, it is known that there are some cases in which
no installation of these facilities caused subsidence behind
coastal structures. Takahashi et al. (1996) investigated subsidence
mechanisms of a reclaimed area behind a caisson-type seawall with
damaged protecting facilities using a series of hydraulic model
experiments. They consequently revealed that if there was a hole on
geotextile sheets against backlling sand leakage, especially a
larger hole located around the still water level, backlling sand in
the
26
Chapter 3 Wave-Induced Sand Leakage Phenomena vicinity of the
hole loosened due to the uprush and leaked due to the following
backwash, resulting in caves and cave-ins on the reclaimed area.
Maeno et al. (2002) conducted an investigation on dynamic behavior
of the sandy seabed around a sheet-pile seawall with small-scale
laboratory tests and a numerical simulation combining the volume of
uid (VOF) method with the nite element method (FEM). Using a
stability analysis of the seabed in front of the seawall, they
found that the propagation of wave troughs with large incident wave
height caused an increase in negative pressure on the seabed,
resulting in a rise in the sand leakage risk. Bierawski et al.
(2002) investigated the motion of backlling sand grains around a
sheet-pile seawall using hydraulic model experiments and a
two-dimensional numerical model with the distinct element method
(DEM) and the FEM. They, as a result, conrmed the applicability of
the DEM-FEM model, and revealed that the intensity of sand leakage
depended mainly on the seawall height, the backlling sand
characteristics, the foundation depth and the wave pressure acting
on the seabed in front of the seawall. Previous studies including
above treated subsidence phenomena behind a caisson-type seawall, a
sheet-pile seawall, a coastal dike and a gentle slope revetment;
however, no studies have so far been intended for a rubble mound
breakwater such as Photo 3.2 and a vertical revetment such as Photo
3.3. Furthermore, although Maeno et al. (2002) investigated
stability of the seabed in front of the seawall, there is little
research on a critical condition of backlling sand leakage and its
mechanisms. For clarifying sand leakage mechanisms from behind
coastal structures, waveinduced backlling sand behavior is of
primary importance. Since Yamamoto (1977), a number of researchers
focused on the wave-seabed interaction with a series of Biot
equations (Biot, 1941). Using a computational technique based on
the FEM, Mizutani et al. (1998), Mostafa et al. (1999), Jeng et al.
(2001) and many other papers revealed waveinduced seabed response
around a submerged rubble mound breakwater and a caissontype
breakwater. Further detailed explanations are available in Jengs
review paper (2003). However, little attention has been paid to
wave-seabed interaction for investigating leakage phenomena of
backlling materials reclaimed behind coastal structures. In this
chapter, the author investigates sand leakage phenomena from behind
coastal structures with a series of hydraulic model experiments and
the numerical simulation developed in the previous chapter. The
author here adopts two types of seawall: one is a rubble mound
breakwater in the absence of lter layers and geotextile sheets such
as Photo 3.2; and the other is a vertical revetment in the absence
of wave absorption and foot protection blocks such as Photo 3.3. In
the hydraulic model experiments, the author tries to reproduce
subsidence of a reclaimed area due to backlling sand leakage from
behind the seawall, and examines sand leakage conditions with
nondimensional parameters of incident waves, the seawall and the
reclaimed area. In the numerical simulation, the author rst
compares numerical results with experimental data to conrm the
applicability of the developed numerical model. The author, then,
focuses on ow velocities around the reclaimed beach and eective
stresses inside the backlling sand, and investigates their eects on
sand leakage phenomena in the rubble mound breakwater and the
vertical revetment. Moreover, the author demonstrates eciency of
lter layers against backlling sand leakage from behind the rubble
mound breakwater.
3.2 Sand leakage from behind a rubble mound breakwaterRubble
Mound Breakwater
27
Wave
Reclaimed Beach
:7
33.0
Wave Generator
1
Impermeable Bed
Impermeable Wall Unit: cm
Fig. 3.1 Schematic gure of a wave ume for hydraulic model
experiments on backlling sand leakage from behind a rubble mound
breakwater.
3.2
Sand leakage from behind a rubble mound breakwater
3.2.1 Hydraulic model experiments Hydraulic model experiments
were conducted on the basis of the Froude similarity with the
length ratio of 1/20 using a wave ume of 30.0 m in length, 0.7 m in
width and 0.9 m in depth at the Department of Civil Engineering,
Nagoya University. As shown in Fig. 3.1, a ap-type wave generator
was equipped at one side of the ume, and an impermeable rigid bed
and wall were installed at the other side of the wave ume. On the
impermeable rigid bed, the author placed a rubble mound breakwater
and a reclaimed beach composed of gravel and sand, respectively. In
this section, the author adopted two types of rubble mound
breakwater: one was an inclined-type breakwater modeled on the
Shiroya beach, Aichi, Japan (Photo 3.2); and the other was an
upright-type breakwater. 3.2.1.1 Dimensional analysis Backlling
sand leakage from behind inclined and upright rubble mound
breakwaters is described by the following physical parameters: f
(Hi , T, h, g, w , w , B, hr , s s , s , r , mr , D50 , , h s , s ,
m s , d50 ) = 0, (3.1)
where Hi is the incident wave height; T is the wave period; h is
the still water depth; g is the gravitational acceleration; w is
the water density; w is the molecular viscosity of water; B is the
breakwater width at the still water level; hr is the breakwater
height; s s and s are the seaward and landward slopes of the
breakwater, respectively; r is the density of the gravel; mr is the
porosity of the breakwater; D50 is the median diameter of the
gravel; is the beach length at the still water level; h s is the
beach height; s is the density of the sand particles; m s is the
porosity of the backlling sand; and d50 is the median diameter of
the sand particles. Applying the Buckingham theorem, the
nondimensional parameters are Hi h D50 gh B hr r D50 h s s d50 = 0,
(3.2) , , , s s , s , , mr , , , , , ms , f , , L L w L h w B L h w
D50 where L gT 2 is the wavelength; and w = w /w is the kinematic
molecular viscosity of water. As mentioned below, w , hr , r , mr ,
s and m s were constant in this study; hence, the
28
Chapter 3 Wave-Induced Sand Leakage PhenomenaRubble Mound
Breakwater Wave Impermeable Wall
W1
W2 W3 W4
W5
W6
W7 W8
45.0
2
h
1:
B
hs
Reclaimed Beach140.0
450.0 50.0 50.0 30.0 60.0
25.0 60.0 30.0 70.0
W1-W5: Wave Gage, W6-W8: Groundwater Gage
Unit: cm
Fig. 3.2 Schematic gure of an experimental setup for
investigating backlling sand leakage from behind an inclined rubble
mound breakwater and the positions of wave and groundwater
gages.
Wave Reclaimed Beach
Rubble Mound Breakwater GravelPhoto 3.4 Initial condition of the
hydraulic model experiments for investigating backlling sand
leakage from behind the inclined rubble mound breakwater. Before
generating incident waves, the backlling sand owed into the onshore
side of the inclined rubble mound breakwater.
Photo 3.5 Groundwater gage.
author neglected the nondimensional parameters D50 gh/w , hr /h,
r /w , mr , s /w and m s in Eq. (3.2). In addition, the author
adopted the Ursell parameter Ur = Hi L2 /h3 instead of the relative
water depth h/L. As a result, the backlling sand leakage from
behind the rubble mound breakwater is described by the following
nondimensional parameters: ( ) B D50 h s d50 Hi f , Ur, , s s , s ,
, , , = 0. (3.3) L L B L h D50 3.2.1.2 Experimental setups and
conditions 3.2.1.2.1 Inclined-type rubble mound breakwater
Hydraulic model experiments were carried out for investigating
backlling sand leakage from behind an inclined rubble mound
breakwater modeled on the actual breakwater at the Shiroya beach,
Aichi, Japan (Photo 3.2). As shown in Fig. 3.2, the author set the
inclined rubble mound breakwater (hr = 45.0 cm, s s = 1/2, s = 1/2)
with gravel (D50 = 2.0 cm) and the reclaimed beach with sand (d50 =
0.010 cm) on the impermeable rigid bed. In this kind of experiment,
more attention should be paid to install the rubble mound
breakwater and the reclaimed beach; hence, the author adopted the
following procedure: the author rst placed the rubble mound
breakwater on the impermeable rigid bed, and then the author poured
the water into the wave ume until the still water depth was
about
3.2 Sand leakage from behind a rubble mound breakwaterTable 3.1
Experimental conditions for investigating backlling sand leakage
from behind the inclined rubble mound breakwater.
29
Case Case 01 Case 02 Case 03 Case 04 Case 05 Case 06 Case 07
Case 08 Case 09 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15
Case 16 Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23
Case 24 Case 25 Case 26 Case 27 Case 28
Hi [cm] 3.9 4.0 3.9 6.8 6.9 7.3 6.9 7.1 7.2 7.0 6.8 3.8 3.8 4.0
3.7 4.1 1.9 1.9 1.9 1.8 2.0 10.4 4.0 3.9 4.9 6.5 5.7 7.5
T [s] 0.9 1.3 1.7 0.9 1.3 1.7 0.9 1.3 1.7 1.1 1.5 0.9 1.1 1.3
1.5 1.7 0.9 1.1 1.3 1.5 1.7 1.0 0.7 0.8 0.8 0.8 0.9 1.0
h [cm] 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0
35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0 35.0
35.0 35.0 35.0 35.0
B [cm] 85.0 85.0 85.0 85.0 85.0 85.0 85.0 85.0 85.0 85.0 85.0
65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0 65.0
65.0 65.0 65.0 65.0
h s [cm] 40.0 40.0 40.0 40.0 40.0 40.0 45.0 45.0 45.0 45.0 45.0
45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0 45.0
45.0 45.0 45.0 45.0
D50 [cm] 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
d50 [cm] 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010
0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010
0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010
20 cm. Next, the author set the reclaimed beach between the
breakwater and the wall, and the author nally poured the water
again. The sand, as a result, owed into the onshore side of the
breakwater, as shown in Photo 3.4. In this paper, the author
generated incident waves under this initial condition. For
measuring water surface uctuations at the oshore side of the
breakwater, the author utilized ve capacitance-type wave gages
(KENEK: CHT6-40). In addition, the author installed three
groundwater gages (KENEK: CHT6-43-SQ) inside the breakwater and the
reclaimed beach, which gages were newly-developed equipments for
measuring groundwater table uctuations. If a capacitance-type wave
gage is surrounded with an open-bottomed impermeable pipe in order
not to touch a capacitance line to gravel and sand, the gage
measures not groundwater table uctuation inside the porous media
but pore water pressure at the bottom of the pipe; hence, the
author covered the capacitance line with adequately ne wire gauze
(mesh size: 75 m), as shown in Photo 3.5. In preliminary tests, the
author compared output data between a groundwater gage and a wave
gage with periodic waves. The author, as a result, found that the
delay time of the output data of the groundwater gage was 50 ms or
less; however, the gauze around the gage had little eect on
measuring data since the delay time was negligible in comparison
with the wave
30
Chapter 3 Wave-Induced Sand Leakage PhenomenaRubble Mound
Breakwater Wave Impermeable Wall
W1
W2 W3 W4 W5
W6 W7
W8
W920.0
30.0
P1
P2 P3 P4 P5 P6 P7 P8 P9 P10Gauze Reclaimed Beach
45.0
860.0 50.0 22.5 22.5 22.5 22.5 15.0 15.0 15.0 15.0 30.0
60.0
W1-W3: Wave Gage, W4-W9: Groundwater Gage, P1-P10: Pressure Gage
Unit: cmFig. 3.3 Schematic gure of an experimental setup for
measuring water surface uctuations and pore water pressures around
an upright rubble mound breakwater and the positions of wave,
groundwater and pore water pressure gages. Table 3.2 Experimental
conditions for measuring water surface uctuations and pore water
pressures around the upright rubble mound breakwater.
Case Case 01 Case 02 Case 03 Case 04 Case 05 Case 06 Case 07
Case 08 Case 09 Case 10
Hi [cm] 1.9 1.9 1.8 1.8 1.7 4.8 4.7 4.6 4.6 4.5
T [s] 0.9 1.1 1.3 1.5 1.7 0.9 1.1 1.3 1.5 1.7
h [cm] 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0
B [cm] 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0
D50 [cm] 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0
d50 [cm] 0.045 0.045 0.045 0.045 0.045 0.045 0.045 0.045 0.045
0.045
periods. Figure 3.2 shows the positions of the gages. The author
also recorded the leakage process of the backlling sand with a
digital video camera (SONY: DCR-PC110). Incident waves for the
inclined rubble mound breakwater were regular ones. The author
changed the incident wave height Hi with the range of 1.8-10.4 cm
and the wave period T with the range of 0.7-1.7 s. In addition, the
author set the still water depth h at 30.0 and 35.0 cm and the
beach height h s at 40.0 and 45.0 cm. All experimental conditions
are listed in Table 3.1. In each experimental case, the wave
generator worked for 60 minutes. 3.2.1.2.2 Upright-type rubble
mound breakwater As mentioned in the following section, the author
conrmed that the backlling sand leakage depended partly on the
relative breakwater width B/L for the inclined rubble mound
breakwater. It is, however, very dicult to discuss inuences of the
wavelength L on the sand leakage since the breakwater width at the
still water level B is highly dependent on the still water depth h.
Hydraulic model experiments for an upright rubble mound breakwater
were therefore conducted to investigate eects of the wavelength L
more clearly. Furthermore, the author measured groundwater table
uctuations and pore water pressures to discuss wave elds inside the
porous media due to incident wave action. In this section,
3.2 Sand leakage from behind a rubble mound breakwaterRubble
Mound Breakwater Wave Impermeable Wall
31
W1
W2
W3
B 30.0
45.0
Reclaimed Beach1000.0 150.0
W1-W2: Wave Gage, W3: Groundwater Gage
Unit: cm
Fig. 3.4 Schematic gure of an experimental setup for
investigating backlling sand leakage from behind an upright rubble
mound breakwater and the positions of wave and groundwater
gages.
Reclaimed Beach
Rubble MoundWave
Breakwater
GravelPhoto 3.6 Initial condition of the hydraulic model
experiments for investigating backlling sand leakage from behind
the upright rubble mound breakwater. Before generating incident
waves, the backlling sand owed into the onshore side of the upright
rubble mound breakwater.
the author carried out two kinds of experiment: one was for the
measurement of groundwater table uctuations and pore water
pressures inside the rubble mound breakwater and the reclaimed
area; and the other was for the investigation of the backlling sand
leakage. Measurement of groundwater table uctuations and pore water
pressures The author rst performed hydraulic model experiments for
measuring groundwater table uctuations and pore water pressures
inside the breakwater and the reclaimed beach. As shown in Fig.
3.3, the author installed an upright rubble mound breakwater (hr =
45.0 cm and B = 90.0 cm) with gravel of D50 = 3.0 cm and a
reclaimed beach (h s = 45.0 cm and = 150.0 cm) with sand of d50 =
0.045 cm. To prevent the backlling sand from owing into the
breakwater, the author set ne wire gauze (mesh size: 75 m) on the
oshore side of the reclaimed beach. The author measured water
surface uctuations and pore water pressures with three
capacitance-type wave gages (KENEK: CHT6-40), three groundwater
gages (KENEK: CHT6-43-SQ) and ve pore water pressure gages (KYOWA:
BP-500GRS), respectively. Generated incident waves were regular,
and the experimental conditions are summarized in Table 3.2. In
this study, the author generated the waves for about one
minute.
32
Chapter 3 Wave-Induced Sand Leakage Phenomena
Table 3.3 Experimental conditions for investigating backlling
sand leakage from behind the upright rubble mound breakwater.
Case Case 01 Case 02 Case 03 Case 04 Case 05 Case 06 Case 07
Case 08 Case 09 Case 10 Case 11 Case 12 Case 13 Case 14 Case 15
Case 16 Case 17 Case 18 Case 19 Case 20 Case 21 Case 22 Case 23
Case 24 Case 25 Case 26 Case 27 Case 28 Case 29 Case 30 Case 31
Case 32 Case 33 Case 34 Case 35 Case 36 Case 37 Case 38 Case 39
Case 40 Case 41 Case 42 Case 43 Case 44 Case 45 Case 46
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