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UFL/COEL-TR/112
IMPROVED CROSS-SHORE SEDIMENT TRANSPORT
RELATIONSHIPS AND MODELS
by
Jie Zheng
Dissertation
1996
IMPROVED CROSS-SHORE SEDIMENT
TRANSPORT RELATIONSHIPS AND MODELS
By
JIE ZHENG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1996
ACKNOWLEDGMENTS
I would like to express my deepest gratitude and appreciation to my advisor and the
chairman of my supervisory committee Professor Robert G. Dean, for his constructive
direction, enthusiasm, advice and unflagging support. His zeal and love for coastal
engineering have inspired me throughout this four-year study, which has been a challenging,
joyful and unforgettable experience in my life.
My thanks also go to committee members, Professors Daniel M. Hanes, Robert J.
Thieke, Hsiang Wang and Ulrich H. Kurzweg, for their valuable advice, suggestions,
discussions, comments and patience in reviewing this dissertation.
Appreciation is extended to all other faculty members and staff in the department for
supplying various components of knowledge and help during this study. Special thanks go to
Becky Hudson, Sandra Bivins, Lucy Hamm, Cynthia Vey, Helen Twedell, John Davis and Dr.
Li-Hwa Lin for their help and kindness in the completion of this study. Gratitude is due to my
friends in the department, Xu Wang, Yigong Li, Linda Charles, Emre Otay, Albert Browder,
Michael Dombrowski, Taerim Kim, Susan Harr, Michael Krecic, Michael Bootcheck, and
Paul Devine, for their friendship and assistance.
My final acknowledgment is reserved for those whom I owe the most, my husband
Jian Liu, for his love, support, encouragement and patience through these years, and my
parents, who instilled in me the values of life that have made me to go this far, and have
supported me all my life.
TABLE OF CONTENTS
Pages
ACKNOWLEDGMENTS ............................................... ii
ABSTRACT .................................................... vi
CHAPTERS
1 INTRODUCTION ......................................... 1
1.1 M otivation ......................................... 1
1.2 Nearshore Sediment Transport .......................... 2
1.2.1 Transport Processes and Related Problems ........... 2
1.2.2 Bedload and Suspended Load ..................... 5
1.3 Review of Cross-Shore Sediment Transport Relationships and Models
................................................ 7
1.3.1 G general ...................................... 7
1.3.2 "Closed Loop" Model ........................... 8
1.3.3 "Open Loop" Model ........................... 11
1.4 Scope of Study ..................................... 15
2 SHEAR STRESSES ACTING IN THE NEARSHORE ............ 20
2.1 Introduction ....................................... 20
2.2 Critical Shear Stresses ................................ 22
2.3 G ravity ........................................... 23
2.4 Shear Stress Due to Nonlinear Waves .................... 24
2.5 Boundary Rotation Flow Related Shear Stress ............. 25
2.6 Shear Stress Related to Undertow ....................... 33
2.6.1 Linear Shear Stress Relationship .................. 34
2.6.2 Quadratic Shear Stress Relationship ............... 37
2.7 Comparison of Magnitude of Shear Stress Terms ........... 48
3 DEVELOPMENT OF CROSS-SHORE TRANSPORT MODEL CROSS
........................ ............. ................. 53
3.1 Equilibrium Beach Profile ............................ 53
3.2 Scale Analysis ...................................... 57
3.3 Discussion of Transport Relationship ...................... 62
3.4 M odeling Process ................................... 66
3.4.1 Numerical Method ............................. 66
3.4.2 Wave Run-up and Set-up ....................... 69
3.4.3 Dune, Shoreline and Offshore Slopes ............... 70
3.4.4 Random Wave Generation ....................... 72
4 CALIBRATION OF CROSS MODEL WITH LABORATORY
EXPERIM ENTS ........................................ 79
4.1 Introduction ....................................... 79
4.2 Saville's Experiments ................................ 80
4.2.1 General Description ............................ 80
4.2.2 Calibration ................................... 81
4.3 German "Large Wave Flume" ......................... 86
4.3.1 G general ..................................... 86
4.3.2 Experiments With Constant Waves ................ 90
4.3.3 Experiment With Irregular Waves ................. 95
4.4 SUPERTANK Experiments .......................... 99
4.5 Results of Laboratory Calibration and Comparison ......... 101
5 EVALUATION OF CROSS WITH LABORATORY EXPERIMENTS 106
5.1 Application of CROSS to Laboratory Experiments ......... 106
5.1.1 Saville's Experiments ......................... 107
5.1.2 German "Large Wave Flume" ................... 112
5.1.3 SUPERTANK Experiments ..................... 118
5.2 Results of Evaluation ............................... 120
6 APPLICATION OF CROSS FOR NOVEMBER 1991 AND JANUARY
1992 STORM EROSION AT OCEAN CITY, MARYLAND ....... 124
6.1 Background ...................................... 124
6.2 Brief Description of Three Commonly Used Models ........ 126
6.2.1 CCCLModel ............................... 127
6.2.2 EDUNE Model .............................. 127
6.2.3 SBEACH Model .............................. 128
6.3 Storm and Beach Profile Characteristics ................. 132
6.3.1 Storm Assessment ............................ 132
6.3.2 Beach Profile Characteristics .................... 136
6.4 Comparisons of CROSS Model With Three Commonly Used Models
................................. ...... ....... 140
6.4.1 Input Case Design ............................ 140
6.4.1.1 CROSS M odel ......................... 141
6.4.1.2 CCCL M odel .......................... 141
6.4.1.3 EDUNE Model ........................ 142
6.4.1.4 SBEACH Model ....................... 143
6.4.2 Numerical Results and Comparisons .............. 143
6.4.3 Average Profile Change at Different Elevation Contours 168
6.5 Sensitivity Study of CROSS Model ..................... 170
6.5.1 Transport Coefficient .......................... 170
6.5.2 Active W ater Depth ........................... 174
6.5.3 Storm Surge ................................ 184
6.5.4 W ave Height ................................ 185
6.6 Summ ary ........................................ 197
7 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ..... 200
7.1 Summary ........................................ 200
7.2 Conclusions ...................................... 201
7.3 Discussion and Recommendations ...................... 204
REFERENCES ................................................... 206
BIOGRAPHICAL SKETCH .......................................... 213
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
IMPROVED CROSS-SHORE SEDIMENT
TRANSPORT RELATIONSHIPS AND MODELS
By
Jie Zheng
August 1996
Chairperson: Robert G. Dean
Major Department: Coastal and Oceanographic Engineering
A modified nonlinear cross-shore sediment transport relationship is developed based
on equilibrium beach profile concepts and dimensional scaling relationships. This nonlinear
relationship provides a reasonable explanation for the significantly different time scales of
beach evolution evident in various laboratory experiments. To predict a beach profile
response under wave action, a finite difference method is applied to solve the sediment
transport and continuity equations numerically. The proposed nonlinear model called
"CROSS" is calibrated and compared with the commonly employed linear transport
relationship using laboratory data. A total of seven large scale wave tank experiments from
three different facilities are examined. The results demonstrate that the nonlinear transport
model provides overall better predictions than the linear transport equations. The CROSS
model and the three other commonly used models are applied to predict beach erosion at
Ocean City, Maryland, during the November 11, 1991, and January 4, 1992, storms. Seven
survey lines are available for comparison with the numerical simulations. Two versions (2.0
and 3.0) of SBEACH are applied. Among the four models, the CCCL model is the only one
overpredicting average dune erosion and the other three models have different degrees of
underprediction with SBEACH version 2.0 underpredicting most. Overall, CROSS, EDUNE
and SBEACH version 3.0 present reasonable predictions for both dune erosion and the entire
profiles. The sensitivity of CROSS to the transport coefficient, active water depth, storm
surge levels and the storm wave heights are studied for the storm erosion at Ocean City. It
appears that CROSS is very insensitive to the transport coefficient. The subaqueous part of
a profile is quite sensitive to the wave height and the subaerial part is less affected. The
CROSS model provides better predictions with the ratio of active water depth to incoming
wave height of 1 than with the ratio of 1.28, and the 20% increased storm surge yields a
better simulation.
CHAPTER 1
INTRODUCTION
1.1 Motivation
The beaches are the transition zone between land and sea. They protect the shore from
damage by coastal storms and hurricanes and provide valuable recreational resources. Their
effectiveness as natural barriers depends on their size and shape, sediment characteristics and
the severity of storms. Under the seasonal action of waves, currents and winds, beaches often
erode during winters and accrete during summers. During storm surges the sea level rises
considerably above the normal high water level. As a result the storm driven waves reach the
front of the dunes and beach erosion occurs. The eroded sand is moved in an offshore
direction and a new beach profile is developed. In fact, the process of storm erosion can be
considered as the continuous adjustment of the beach profile to the changing hydrodynamic
and meteorological conditions during the storm.
In coastal engineering, accurate estimates of beach profile evolution in response to
storm waves and high water levels and the adjustment of beach fill to long term wave action
are particularly important for a variety of regulatory and design purposes. The time scale
associated with storm induced beach erosion is on the order of hours to days and depends on
the duration, water level and wave conditions of the storm, while the time scale of beach fill
adjustment is from months to years and depends on the fill material and the wave climate at
2
the site. Due to the complexities of surf zone hydrodynamics, sediment characteristics, and
wave and water level conditions, an analytical treatment of beach profile change is difficult
and requires recourse to numerical modeling, which requires an understanding of sediment
transport. The objective of this study is to provide a better understanding of cross-shore
sediment transport relationships and develop a numerical model to represent beach profile
changes associated with variable waves and tides.
1.2 Nearshore Sediment Transport
1.2.1 Transport Processes and Related Problems
Sediment transport at a point in the nearshore zone can be considered in terms of
cross-shore and longshore components. It appears that under a number of coastal engineering
scenarios of interest, the transport is dominated by either the cross-shore or longshore
component and this, in part, has led to a history of separate investigative programs for each
of these two components. Longshore sediment transport figures prominently in situations
involving loss of sediment supply, such as damming of rivers, and impoundment at structures
and inlets. In these cases, longshore sediment transport is the major process governing
nearshore topographic change. The cross-shore component determines profile evolution
primarily for beaches far away from structures, inlets and river mouths, and under the case
of increased water levels, storms and beach nourishment. In contrast to longshore sediment
transport modeling, which has been studied for about five decades, a focus on cross-shore
sediment transport modeling is relatively recent (about 20 years) and the uncertainty in
predicting effects of all variables thus may be considerably greater.
3
Cross-shore sediment transport encompasses both offshore transport such as occurs
during storms and onshore transport which dominates during mild wave conditions. Transport
in these two directions appears to occur in significantly distinct modes and with remarkably
disparate time scales. The offshore transport moves sand from the dune to the offshore bar
in hours to days during a storm, whereas the onshore transport may take months to years to
move sand from the offshore bar to the dry beach. The difficulties in predicting transport in
the two directions also differ substantially. Offshore transport is simpler and the more
predictable of the two with transport more or less in phase over the entire active profile. This
is fortunate for coastal engineers since there is considerably greater engineering concern and
interest in offshore transport due to the potential for damage to structures and loss of land.
Cross-shore sediment transport is relevant to a number of coastal engineering
problems including (a) beach and dune response to storms, (b) the equilibration of a beach
project that is placed at a slope steeper than equilibrium, (c) "profile nourishment" in which
the sand is placed in the nearshore with the expectation that it will move landward nourishing
the beach (this involves the more difficult problem of onshore transport), (d) shoreline
response to sea level rise, (e) seasonal changes of shoreline positions, (f) overwash, the
process of landward transport due to overtopping of the normal land mass due to high tides
and waves, (g) scour immediately seaward of shore parallel structures (e.g. seawalls), and (h)
cross-shore and longshore coupled transport around shore normal structures (e.g. jetties and
groins) in which the steeper and milder slopes on the updrift and downdrift sides induce
seaward and landward components, respectively. These problems are schematized in Figure
1.1.
Pre-Slorm Profile
Storm Water Level \ Immediately
S-- ..Z.- .- -.. /Post-Nourished
S.ZPoat-Nourlished
,, _Normal Water Level ..
Post-Storm .' ,-Pre-Nourlshment '-
Profile Equllibrate' h'~ '---
a) Response to Storms b) Beach Nourishment Evolution
Shoreline Response
Increased Sea Level
S.... Initial Sea Level
"^g~ a~rnShoreward?
"Profile" Nourishment
c) Profile Nourishment Evolution d) Profile Response to Sea Level Rise
"Summer" Profile
Post-Storm Profile with
Washover Deposits
St By Norm Water L evel
,. Be Normal Water Level
"Winter" Profile .-- --
'.'..- Pre-Storm Profile *--.-
e) Seasonal Profile Changes f) Effects of Storm-Induced Overwash
Seawallt
Storm Water Lev.
A$
Storm-induced Scour
Adjacent to Seawall
g) Storm-Induced Scour Adjacent to Seawalls h) Flow of Sand Around Structures
Figure 1.1 Problems and processes in which cross-shore sediment transport is relevant (after
Dean, draft of Chapter II of Coastal Engineering Manual).
1.2.2 Bedload and Suspended Load
A complete understanding of cross-shore sediment transport is complicated by the
contribution of both bed and suspended load transport components. Bagnold (1956) defined
bedload as the part of the total load which is supported by intergranular forces, and the
suspended load as the part supported by fluid drag. Usually particles entrained in the bedload
exhibit motions of rolling, sliding and sometimes jumping in the bed layer, whereas particles
entrained in the suspended load are suspended in the water column. Turbulence is the most
important factor in the suspension of sediment. Due to gravity, particles tend to settle to the
bottom; however, irregular motions of the fluid introduced by turbulence tend to lift particles
from the bed and support them in suspension. For the cross-shore sediment transport, bedload
is dominant outside the surf zone because of the relative mild fluid conditions. However,
owing to the highly turbulent flow field caused by wave breaking, suspended load prevails
inside the surf zone.
Generally bedload is related to the shear stress acting on the bottom. Several
researchers have proposed expressions for the transport rate in an oscillatory flow. Bailard
and Inman (1981) suggested that bedload transport rate was proportional to the 1.5 power
of bottom shear stress. Trowbridge and Young (1989) presented a linear relationship between
the transport rate and the bottom shear stress. There are several causes of shear stresses
acting on the bottom in the nearshore active zone: (1) shear stress induced by gravity, (2)
shear stress due to nonlinear waves, (3) shear stress caused by return flow, and (4) shear
stress related to boundary rotation flow. The magnitudes of these stresses can be remarkably
6
different inside and outside surf zone. A detail discussion of these shear terms will be
presented in Chapter 2.
Suspended sediment transport in the cross-shore direction under wave action primally
results from the contributions due to the oscillatory first-order velocity (intermittent
transport) and the mean cross-shore current. Dean (1973) developed a heuristic model for
cross-shore transport in the surf zone based on the intermittent suspension of sand grains by
wave breaking and the eventual settling of sand to the bottom. A wave breaking in the surf
zone lifts sand from the bottom up into the water column to a particular elevation. The time
required for the sand to fall back to the bottom is determined by the ratio of the elevation to
the sediment falling velocity. If the fall time is less than half the wave period, the sand grain
should move onshore. Alternatively if the fall time is greater, the sand particle would be
carried offshore. The elevation is dependent on the wave height, as it seems reasonable that
larger waves would lift sand to a higher elevation than smaller waves. Therefore Dean
assumed that the elevation is proportional to the breaking wave height. The direction of
intermittent transport was determined by the so-called "fall velocity parameter," which is a
non-dimensional parameter of breaking wave height divided by wave period and sediment fall
velocity. For most ocean wave cases, the mean cross-shore current is caused by the mean
mass return flow in the offshore direction. As a result the suspended transport contributed
from the mean flow is always in the offshore direction no matter how high the sediment
suspended. Because the cross-shore velocity profiles inside the surf zone are different from
those outside the surf zone, the contributions of the mean flow to suspended sediment
transport inside the surf zone are different from those outside the surf zone. Under unbroken
7
wave conditions since the upper part of water column has greater return flow velocity but
very little suspended sediment, the suspended transport due to the mean flow is almost
negligible. By contrast, inside the surf zone the lower part of water column has larger return
flow velocities and high sediment concentration, therefore the suspended sediment transport
caused by mean flow could be significant.
1.3 Review of Cross-Shore Sediment Transport Relationships and Models
1.3.1 General
Some recent interesting reviews of cross-shore sediment transport models have been
reported by Broker Hedegaard et al. (1992), Steetzel (1993), and Schoonees and Theron
(1995). Different methods and viewpoints have been applied. Hedegaard et al. made an inter-
comparison of six coastal profile models against measured profile evolutions from a large
wave flume (under dune erosion conditions). Steetzel classified transport formulae as
empirical transport, shear stress related transport, energy dissipation related transport, energy
dissipation related suspension and velocity related transport. Schoonees and Theron
summarized ten internationally developed transport models according to their theoretical
basis and morphological verification data.
In the present review, cross-shore sediment transport models are broadly classified
into two groups: "open loop" and "closed loop" models. An "open loop" model is not
constrained a priori to the final profile and the sediment transport is determined by sediment
concentration and fluid motions. "Closed loop" sediment transport models are based on
equilibrium beach profile concepts (Bruun 1954 and Dean 1977) and assume that a profile will
eventually achieve equilibrium if exposed to the same conditions for a long time. Cross-shore
8
transport is caused by deviations of a beach profile from equilibrium. Methodologically "open
loop" models are microscale approaches based on detailed physics of the process, while
"closed loop" models are more macroscale and utilize conservation laws and heuristic
arguments that provide reasonable solutions. Owing to the complexities of highly turbulent
flow inside the surf zone, there is considerable uncertainty in the detailed physics of wave
breaking processes and related sediment transport. Therefore the macroscale approach is a
powerful tool to predict beach profile change at the present time.
1.3.2 "Closed Loop" Models
The earliest relationship between increased water level and profile response was
developed by Bruun (1962), which does not require any specific form of the equilibrium
profile, but only requires that the form be known. The response is considered in terms of the
horizontal recession of the profile and sea level rise. Two requirements must be satisfied by
the new profile. First, the profile shape does not change with respect to the sea level.
Secondly, the sand volume in the profile must be conserved. One restriction of the Bruun
model is that the sea level rise should be much less than the berm height, since the berm height
relative to the rising sea level is considered fixed. Edelman (1972) removed the limitation of
relative small changes in water level by accounting for the progressive decrease in relative
dune elevation. Since the two above assume that a beach profile maintains its equilibrium form
immediately in response to the water level rise, no sediment transport relationship is required.
However, it is not suitable to apply such static models (no lag) to short-term profile changes.
Following the idea of Bakker (1968), Swart (1974 a, b) schematized the beach profile
into 4 zones referred to as backshore, onshore profile, offshore profile and transition area and
9
developed a time-dependent cross-shore model. With L, and L2 representing the schematized
lengths of onshore and offshore profiles, respectively, the transport equation is written as
Q = q [W-(L2-Li),] (1.1)
where Q is the time dependent cross-shore sediment transport rate, q is a constant for a
specific set of boundary conditions, and W and (L2-LI)t are equilibrium and time dependent
values of (L2-L,), respectively.
Chiu and Dean (1984,1986) developed the CCCL model, which is still used by the
State of Florida to establish the location of the State Coastal Construction Control Line
associated with the erosion limits of a 100 year return period storm event. Under erosive
water level and wave conditions, the time dependent beach recession, R(t), at each depth
contour is given by
R(t) = R(1 -e -Kt) (1.2)
in whichR_ is the equilibrium recession for the particular condition at that time and K is a
decay parameter. Strictly speaking, this is a beach profile change model since there is no
sediment transport relationship involved. Compared with Bruun and Edelman models, CCCL
is time dependent and can be applied to short-term profile change caused by storms.
Two wave energy dissipation related transport models are EDUNE (Kriebel 1982,
1990, Kriebel and Dean 1985) and SBEACH (Larson and Kraus 1989, Larson et al. 1989).
According to equilibrium beach profile concepts, offshore transport continues until the wave
energy dissipation per unit volume of water becomes uniform over the entire surf zone. Based
10
on this consideration, EDUNE expresses the local cross-shore sediment transport rate, Q, in
the surf zone as
Q = K(D D) (1.3)
in which K, is a dimensional transport coefficient, D and D. are the actual and equilibrium
wave energy dissipation per unit volume of water, respectively, and D. depends on the sand
size. Eq. (1.3) was first proposed by Moore (1982) and later modified by Kriebel (1982) and
Kriebel and Dean (1985). In SBEACH, the sediment transport rate across the nearshore
active zone is calculated by different relationships in four distinct zones identified as pre-
breaking zone (extending offshore from the breaking point), transition zone (from the break
point to the plunge point), broken wave zone (the plunge point to the seaward limit of the
swash zone), and swash zone (from the seaward swash zone to the run-up limit). A transport
relationship similar to Eq. (1.3) is applied in the broken wave zone with a term added to
account for the effect of local slope and written as
E dh E dh
Q = Ks(D -D, + -), D>D,
= O, D
K, dx
where K, and e are dimensional coefficients and dh/dx is the local profile slope. The transport
rates in the other three ranges are established empirically in accordance with Saville's large
wave tank experiments (Kraus 1988).
1.3.3 "Open Loop" Models
"Open loop" models usually relate cross-shore sediment transport to the detailed
physics of the flow field such as sediment concentration, fluid velocity, and bottom shear
stress. Three different types of formulae will be briefly reviewed here.
(1) Sediment transport rate based on the product of flow velocity and sediment concentration.
The Dally model (Dally 1980, Dally and Dean 1984) presented the net time-averaged
flux of suspended sediment past a section in the nearshore zone as
Q = f u(z) c(z) dz (1.5)
depth
where u(z) and c(z) are averaged horizontal velocity and sediment concentration at level z,
respectively. The flow regime is divided into an upper layer where only mean flow is
considered and a lower layer where both mean flow and orbital velocity are taken into
account. In the lower layer, the fall time of a sand particle is determined by its elevation and
fall velocity. According to linear wave theory, the particle will traverse a horizontal
displacement during the fall time. Thus, the average horizontal velocity of the particle during
one wave cycle due to the orbital motion is equal to the ratio of the horizontal displacement
to the fall time. The interface between the layers is determined by the elevation at which a
particle will fall back to the bottom in one wave period.
LITCROSS (Broker Hedegaard et al. 1992) is a coastal profile model developed by
the Danish Hydraulic Institute. The net sediment transport is calculated as bed and suspended
load. The instantaneous bed load and nearbed boundary condition for the vertical distribution
12
of suspended sediment are determined as functions of the instantaneous shear stress
(Engelund and Fredsoe 1976). The total suspended sediment transport over a wave period
is found from
Q= f fcudzdt+ f u,dz (1.6)
period depth depth
where the velocity, u, and the concentration, c, in the first integral are functions of time and
elevation, u, is Lagrangian drift, E denotes the time averaged concentration, and T is the
wave period. The time varying vertical distribution of suspended concentrations is calculated
by the vertical diffusion equation according to Deigaard et al. (1986).
Steetzel (1993) also presented his cross-shore transport model as the integral of the
product of concentration and velocity over time and depth. The sediment concentration is
solved from a one-dimensional, non-stationary convection-diffusion equation. The reference
concentration at the bed is assumed to be related to the kind of breaking and the dissipation
of the turbulent kinetic energy. The cross-shore transport rate is calculated in two parts, that
is, below and above the trough level. Above the trough level, concentration is a constant and
given by that at the mean water level.
(2) Sediment transport rate related to velocity powers
A representation of this type of model was proposed by Bailard (Bailard and Inman
1981, Bailard 1981, 1982) based on the concepts of stream transport developed by Bagnold
(1963, 1966). Bagnold assumed that the sediment was transported in two distinct modes,
each differing by way of the support of the sediment grains. Sediment transport as bedload
13
is supported by the bed via grain to grain interactions, while sediment transport as suspended
load is supported by the stream fluid via turbulent diffusion. In both modes, energy is
expended by the stream in transporting the sediment load. Bagnold, comparing the stream to
a machine, defined the sediment transport efficiency as the ratio of the rate of energy
expended by sediment load to the total rate of energy produced by the stream. In essence, he
assumed that the instantaneous sediment transport rate was directively proportional and
reacted immediately to the instantaneous energy dissipation rate per unit bed area.
Bailard (1981) refined Bagnold's work for the cross-shore sediment transport problem
and expressed the total load time averaged sediment transport equation as
Q = pcf n2 > t anf 3> pc u3u> tanp < I j 5> (1.7)
Stan4 tan I W W I.
where p is the density of water, cf is drag coefficient of the bed, EB and Es are bed and
suspended load efficiency factors, ( is the internal angle of friction for the sediment, tanp is
the bottom slope, iu is the instantaneous velocity vector, i is a unit vector in onshore
direction, W is the sediment fall velocity, and the < >means time average over a wave period.
This formula has been widely applied by others such as Stive (Stive 1986, Roelvink and Stive
1989) and Nairn (1990). Three coastal profile models have been developed based on Bailard
transport relationship. They are UNIBEST of Delft Hydraulics, NPM of Hydraulic Reseach,
and SEDITEL of Laboratoire National d'Hydraulique.
There are other velocity power related sediment transport relationships. One example
was given by Shibayama and Horikawa (1982). The transport rate accounts for bedload and
14
suspended load. Both bed and suspended loads are proportional to the velocity to the sixth
power.
(3) Sediment transport determined by the product of velocity and bottom shear stress
Watanabe (1988) presented a sediment transport rate equation related to local wave-
current conditions and bottom shear stress. The total transport rate is the sum of contributions
from mean current, Qc, and waves, Q,, which are expressed, respectively, by
Qc = Ac(t-tc) p pg(1.8)
Q, = AwF ,(z-Zc)u pg
where Ac and A, are nondimensional coefficients for mean current and wave induced
transport, respectively, U is the current velocity vector, ib is the amplitude of the near-
bottom wave orbital velocity vector, T is the maximum value of the bottom shear stress in a
wave-current coexisting system, T; is the critical shear stress, F D is a direction function for
wave-induced net transport, and g is the gravitational acceleration. The coastal profile model
WATAN3 developed by University of Liverpool is based on the Watanabe transport
equations.
There are some relationships in addition to the above three types such as the one
developed by Nishimura and Sunamura (1986), in which the transport rate is related
empirically to the Ursell number representing the skewness of water particle velocity profile
and the Hallermeier parameter indicating the intensity of sediment movement.
Most "open loop" profile change models include modeling of hydrodynamic processes
to different degrees. The sediment transport calculation is related to the properties of
15
sediment and flow field. A interesting inter-comparison of five coastal profile change models
developed in Europe was presented by Broker Hedegaard et al. (1992). The five models are
LITCROSS of Danish Hydraulic Institute, UNIBEST of Delft Hydraulics, NPM of Hydraulic
Research, WATAN3 of University of Liverpool, and SEDITEL of Laboratory National
d'Hydraulique. The associated transport relationships have been described above. The models
were tested against measured profile evolution from a large wave flume. The initial profile is
a planar beach with a slope of 1:20. The experiment was carried out under regular waves with
a wave height of 1.5 m and period of 6 s. The velocities and the transport rates calculated
across the planar profile are shown in Figures 1.2 and 1.3. The measured and calculated
profiles after 4.3 hours are compared in Figure 1.4. It is noticed that the all five models
present quite good fits for the profile although the calculated velocity and transport rates are
markedly different. It seems that considerable uncertainty in the understanding of surf zone
hydrodynamics and sediment transport process still exists.
1.4 Scope of Study
A major objective of this dissertation is to improve "closed-loop" cross-shore
sediment transport relationships. In the following studies, a modified transport equation is
proposed according to dimensional arguments. A cross-shore profile change model called
CROSS is developed based on the modified transport equation and the conservation of sand.
Three sets of different laboratory experiments are applied to the calibration of CROSS.
Finally, the model is tested and compared with other three commonly used "closed loop"
models against data obtained from field storm erosion.
I U L ~
I I I I I
i I
I I
I I
o 1i 2 30 1 10 0S90 160Cr) X
DHI, Litcross
l, ------- .
1.0
i 44 .. 4.1 ` -- .12 as J. 1 -
LNH, Seditel
I's
IAj1-.-1-
izil' Iz f~1I F
I
I
HR, NPM
.- -- I4
I
a -U,.
10 46. 4 4 42 .0 0. 0.4 aL keI 1.
1-14
DH. Unibest
-- W* -- -- IY
. L1
I -I- -. .....I" /..l I I I
It -
S .1. 48 4* 4. 4.2 4.o 1 O. as as 1.*
tlioo
S- Le $ad
1 Vertial Curenl
IO Componren at 52 Mi
Figure 1.2 Velocity components calculated by NPM, LITCROSS, UNIBEST and SEDITEL
and 2-D current field calculated by SEDITEL (after Broker Hedegaard et al., 1992).
2
0 4.1 -4. 4.4 41s 4. .I 1.4 0. as 1.
lnot*
S;;
I
17
HR, NPM
z
0- SWI.
0 . .
50 100 m
DHI, Ltcross DH, Unibest
s M 0 0 o I" -li e Is so a t s i i B oW I
LNH, Seditel UL, Woian3
4 -----------. .... '. -- -- ----
A io 4U 1 4U a i l ie 1 14 I is s U 40 i i o0 SO IN 1 10 I
Figure 1.3 Cross-shore sediment transport calculated by NPM, UNIBEST, LITCROSS,
SEDITEL and WATAN3 (after Broker Hedegaard et al., 1992).
To better understand the surf zone sediment transport process, Chapter 2 commences
with a qualitative description of the time-averaged bottom shear stresses acting in the
nearshore. The total shear stress is composed of four components. They are shear stress due
to gravity, shear stress induced by non-linear waves, shear stress related to the boundary
rotation flow, and shear stress caused by the mean mass return flow.
Chapter 3 is devoted to the development of CROSS based on the concepts of
equilibrium beach profiles. The relationship between the transport rate and the deviation of
the actual wave energy dissipation per unit volume from equilibrium is discussed according
(m)
*o -NPM
------------------ -----
0_ UN1BEST TC
o.0
": .... 5- -- .... -
o woo -.0 -0-00 7,S0.00 W.oot(m)
4.o
10- ----- ---- -0- EXPE00 -o-TAL (ES,
-10.00 t0.0 J.OO 5o0 7000 0.00 (m)
0.0* ---- ----- -----l----- ^^^lCY*LLE-I
-O100 00 00 OO 00 OCt
Cm)
.to SEDTEL
4.0
so
o WATAN 3
MMERICAL ROSIJU
C -XPCERU dNUtIo6L uSL
------- ----- -- ------,__ -------
-10.00 10.0 30.0 000 70.00 0.00 (m)
REGULAR WAVES
H 1.5m
T 6 s8
d k 0.33mm
Initial slopes: lower beach 1:20
upper beach 1:40
Figure 1.4 Comparison of measured and calculated coastal profiles after 4.3 hours of
exposure (after Broker Hedegaard et al. 1992).
to dimensional considerations. The profile change is determined by both the transport and the
continuity equations. The two equations are solved by the finite difference method with a
staggered grid. A detailed numerical process is presented.
The calibrations of CROSS are described in Chapter 4. Three sets of large wave tank
data are utilized, including three cases in the large German wave flume in Hanover (Dette and
Uliczka 1987), three cases from Saville's large wave tank experiments (Kraus 1988), and one
case from the SUPERTANK experiments at Oregon State University (Kraus and Smith
1994). Among the seven experiments, one of the German set was carried out with a Jonswap-
Spectrum of random waves, the Supertank case was run with the combination of random
19
and regular waves, and the others are conducted under regular waves. The best-fit transport
coefficient is determined as the value providing the least squares error for the fitting of eroded
volume in each experiment. Based on the calibration results, a transport coefficient is
recommended for the applications of CROSS according to the average of the best-fit values
in the seven experiments.
In Chapter 5, the CROSS model with the recommended transport coefficient is
evaluated by the same experiments used in Chapter 4 for the calibration of CROSS. The
predictions of CROSS with the recommended transport coefficient are compared with those
with the best-fit values for each experiment. The comparisons include the time history of
eroded volume and the profile evolutions.
The applications of CROSS for the November 1991 and January 1992 storm erosion
at Ocean City, Maryland, are presented in Chapter 6. The predictions of CROSS are
compared with three other models commonly used in the United States. The three models are
CCCL (Chiu and Dean 1984,1986), EDUNE (Kriebel and Dean 1985, Kriebel 1989,1990),
and SBEACH (Larson and Kraus 1989, Larson et al. 1989). A sensitivity study of CROSS
is also included in this chapter to test the sensitivity of the model to changes wave and surge
conditions during storms.
Finally, Chapter 7 presents a summary, documents conclusions of the study, and
proposes some recommendations for future study.
CHAPTER 2
SHEAR STRESSES ACTING IN THE NEARSHORE
2.1 Introduction
A requirement for a proper understanding of coastal sediment transport is a
consideration of forces which act in the nearshore active zone. There are three kinds of
forces governing the behavior of sediment particles whether they are resting on the bed or
suspended in the water column. These are gravity forces, intergranular forces related to
continuous sediment contact, and the wave and current induced fluid forces. The magnitudes
of wave induced forces can by remarkably different inside and outside the surf zone. Due to
the strong and large scale turbulent flow caused by wave breaking, sediment is typically highly
mobilized inside the surf zone. Changes in beach profiles result from intermittent suspension,
turbulence, bottom shear stress and gravity. Outside the surf zone, the wave energy
dissipation and the turbulence occur mainly within the near-bed wave boundary layer. Within
this zone, the bottom shear stress related bedload sediment transport becomes the primary
factor in controlling profile change.
Under equilibrium conditions, the forces acting on the sediment particles are in
dynamic balance, and, although there is motion of individual particles under even low wave
activity, the profile remains more or less static. Cross-shore sediment transport occurs when
the nearshore hydrodynamic conditions change thereby modifying one or more of the forces
21
resulting in an imbalance and causing a sediment transport gradient and profile change.
Established terminology is that onshore and offshore directed forces are referred to as
"constructive" and "destructive," respectively. It is evident that some forces could behave as
constructive under certain conditions and destructive under others. As the result of the
combined constructive or destructive forces, sediment is transported onshore or offshore,
respectively. For suspended particles, the movement is determined by gravity, drag, lift, added
mass force, etc. acting on the particle. For particles resting on the bed, the bottom shear stress
is the dominant force. When the bottom shear stress is less than the critical shear stress
(friction), particles are under static conditions and no motion occurs. As the bottom shear
stress overcomes friction, particles will be moved in the shear stress direction and bedload
transport commences. Generally, the bottom shear stresses in the cross-shore direction
include contributions from gravity, waves, currents, bottom boundary layer rotation flow, and
undertow due to the mass and momentum flux. As waves and currents vary with time, the
shear stresses also change temporally. In principle, the sediment transport processes are
related directly to the instantaneous shear stresses. However, as a first step, only mean shear
stresses averaged over one wave period will be discussed here.
In this chapter, the critical shear stress is discussed first. When the total bottom shear
stress is less then the critical shear stress, the intergranular friction will keep sand particles
from moving. Under wave conditions, the total bottom shear stress is due to the superposition
of four components. These are shear stress induced by gravity, shear stress due to nonlinear
waves, shear stress related to the boundary layer rotation flow, and shear stress caused by the
mass return flow. Their magnitudes depend on the wave breaking, wind speed, sediment size
22
and bed slope. To provide a better understand of these forces, a detailed example will be
presented to quantify the directions and magnitudes of the four stress terms both inside and
outside the surf zone.
2.2 Critical Shear Stresses
The intergranular forces acting on a sediment particle are the normal and frictional
forces resulting from the surrounding sediment. For grains at rest, the static angle of repose,
(k, is determined by the friction coefficient, i.e. the ratio between the maximum sustainable
shear stress (i.e. critical shear stress), tr and the effective normal stress, oe (see Figure 2.1).
For mostly sandy materials, the static angle of repose is between 26 and 34 degrees (Nielsen,
1992). For shearing (moving) materials of near maximum concentration, the friction
coefficient is only slightly less than the static. Hence the dynamic angle of repose, (d is of
a similar magnitude to the static angle. Hance and Inman (1985) suggested a typical value of
31 degrees for beach sand.
A different parameter, the Shields parameter, describing the frictional shear stress was
suggested by Shields in a study of the incipient motion in steady flow as
= (2.1)
p(s-1)gD
where 0, represents the Shields parameter, p is mass density of water, s is relative density of
sediment, and D is the sediment diameter. The critical Shields parameter, 0,cr, which
corresponds to the critical shear stress, tr,, is the effective Shields parameter at which
sediment movement starts. For sand grains quiescent under static water, the critical Shields
parameter is related to the angle of repose by the relationship, 0,cr = sin (4. Under waves and
/; Mg Figure 2.1. The friction
M Pi coefficient, (W/o),
equals tank,, where (,
f is the static angle of
a = Mg cosp Mg repose.
currents, this parameter becomes much less because the fluid lift force and near-bottom
turbulence decreases the effective normal stress. Although some laboratory studies (Natarajan
1969, and Hammond & Collins 1979) and field studies (Amos et al. 1988) have been
conducted, it is impossible to define a sharp criterion for the initiation of motion under
combined wave and current flows. Based on their observations, Amos et al. recommended
0.04 as the critical Shields parameter. Therefore, for the case of bottom shear stress less than
0.04 times the submerged weight of the sand, sediment will remain static on the bottom due
to friction effects.
2.3 Gravity
When the sand bed is horizontal, gravity acts normally to the bottom and has no
contribution in the shear direction. For a monotonic beach profile, gravity becomes an
obvious destructive force which acts downslope, i.e. seaward. However, for the case of a
barred profile, gravity can act in the shoreward direction over a portion of the profile. Overall,
gravity tends to "smooth" any irregularities that occur in the profile. If gravity were the only
force acting, the only possible equilibrium profile would be horizontal and sandy beaches as
24
we know them would not exist. For particles in the sand bed, the tangent component of
gravity force per unit bottom area at a layer of one grain thickness will be determined by the
component in the beach slope direction as
Tb = apg(s-1)dsinp (2.2)
Where a is the sediment concentration in the sand bed, d is the mean diameter of the
sediment, and p is the beach slope.
2.4 Shear Stress Due to Nonlinear Waves
As noted above, the constructive forces are those that tend to cause onshore sediment
transport. For classic nonlinear wave theories (Stokes, Cnoidal, Solitary, Stream Function
etc.), the wave crests are higher and of less duration than the troughs. This feature is most
pronounced just outside the breaking point. The oscillatory water particle velocities are
usually expressed as a sum of phase-locked sinusoids such as for the Stokes and Stream
Function wave theories. Although the time averaged water particle velocity is zero, the
average bottom shear stress, tb is directed onshore and expressed as
b=P IVbl (2.3)
Where fis Darcy-Weisbach friction coefficient and vb is the instantaneous wave induced water
particle velocity at the bottom. Dean (1987) developed the average bottom shear stress based
on the Stream Function nonlinear wave theory and presented the results in the non-
dimensional form shown in Figure 2.2.
2 MIH/H b =1.u--
0.751
2
z5
j 37
,S, 2 -
5 (1) Hb =Breaking Wave Height
(2) La =2-
2
2 5 10'2 2 5 10 2 5 100 2
RELATIVE DEPTH (h/Lo)
Figure 2.2 Isolines of non-dimensional average bottom shear stress,
Tb, versus relative depth and wave steepness (Dean 1987). Note the
bottom shear stresses are directed shoreward.
2.5 Boundary Rotation Flow Related Shear Stress
The second constructive force originates from the net mean onshore directed
streaming velocity induced by wave motion in the bottom boundary layer. This streaming
motion was first observed by Bagnold (1940) in the laboratory and has been studied
quantitatively by Longuet-Higgins (1953). The mechanism producing a mean bottom shear
stress is reviewed as follows"'. To satisfy the no-slip bottom boundary condition, the
[n The following development is a modification of the Longuet-Higgins approach due to
James Kirby (personal comminution).
26
particles of fluid in contact with the boundary must have the same velocity as the boundary
itself. But quite near the bottom the fluid is in motion with velocities comparable to that in
the interior potential flow of the fluid. This implies that there is strong vorticity in the
neighborhood of the bottom. This vorticity modifies the vertical velocity, which is correlated
with the streamwise velocity at the outer edge of the boundary layer, so that there is a change
in the mean flux of streamwise momentum across the top of boundary layer. This change in
the mean momentum flux must be balanced by a mean bottom shear stress.
The case considered here is that of periodic linear waves propagating shoreward
above a gently sloping sand bottom with straight and parallel depth contours. The equations
governing the water waves flow are the continuity equation and the Navier-Stokes equations
av 8w
-+_ = 0
ay az
av 8v2 vw 1 aP 2 v 2v
+ -- + -- = + e (-- (2.4)
t ay 9z p y y2 z2 (4)
8w + vw aw2 1 1P +(a2w 82w
at ay 8z p az ay2 az2
where the y axis is positive offshore and coincides with the mean water surface, the z axis is
perpendicular to the water surface and positive upward, (v,w) are the corresponding velocity
components in (y,z) directions, P is the water pressure, and e is the kinematic eddy viscosity.
It is convenient to divide the flow into two parts with potential and rotational components,
a(I
v = v+vr y+v
(2.5)
S r r
W vt + JZ + W
27
In which, the subscripts p and r represent the potential and rotational components
respectively, and D is the velocity potential for the interior wave flow. The no-slip boundary
condition requires the velocity (v,w) to be equal to zero at the bottom. It is assumed that the
conditions at the free surface are unaffected by the viscous flow since the boundary layer is
assumed thin. This requires
v- -- or v-0 as (h +z)-.o
ay (2.6)
w- -- or w,-0 as (h +z) -
az
where h is the mean water depth. The first order potential and viscous velocity components
can be solved from the linearized Euler and approximate boundary layer equations,
V2D =0
avr a2vr
at az2 (2.7)
wr 2Wr
8t z 2
where V is the Laplace operator. For waves propagating onshore, the solutions of Eqs. (2.7)
are of the forms
1l = 0.5Hei(kyOt)
S= (z) ei(ky+ot)
Vr= r(Z) e i(ky+ot) (2.8)
Wr = r() ei(ky+ot)
28
in which i is the imaginary number ( T/v ), rI is the instantaneous water elevation, H is the
wave height, k is the wave number, and a is the wave angular frequency. Solving Eqs. (2.7)
with the boundary condition (2.6) gives
(z) = A coshk(h+z)+B sinhk(h+z)
h +z
,(z) = Cexp[-(l+i) (2.9)
(z) = D exp[-(l+i) ]
where 6 = i/2e/ represents the boundary layer thickness and is very small compared with the
water depth, and A, B, C and D are constants to be determined. By substituting (2.9) into the
continuity equation and the no-slip bottom boundary condition, the constants B, C and D can
be replaced by the constant A. The solution becomes
0(z) = A [coshk(h+z) kb sinhk(h+z)]
2
=ikA exp[-( l +i)- -
V,(z) = ikAexp[-(l+i) ] (2.10)
*,(z) = -i k28A exp[-(1+i) h+z
2 8
in which, A is related to the wave height through the linearized free surface dynamic and
kinematic boundary conditions as
H = i A [coshkh -ik sinhkh]
2 g 2 (2.11)
H kA 1-i ( )
H = -k [sinhkh- k-bcoshkh]
2 ia 2
29
Eliminating H and A in Eqs. (2.11) results in the dispersion relationship
2 [1 --k tanhkh] = gktanhkh [1 --i k6cothkh] (2.12)
2 2
As viscous effects approach zero, (2.12) clearly approaches the classic wave dispersion
relationship, 02 = gk tanhkh, i.e., Dean and Dalrymple (1991). In most cases, k8 << 1, so that
the deviations to a and k determined by (2.12) are small.
Eq. (2.12) generally predicts complex o and k combinations, which correspond to the
wave energy decay temporally and spatially. The two simplest cases are as follows:
(1) o is real and k=k, +iki. Time periodic waves decay with propagation distance.
(2) k is real and a = or + io,. Spatial periodic waves decay in time.
The first case is more generally applicable to ocean waves, while the second case may be used
to describe cases such as wave decay in basins, etc. For the problems concerned, we assume
k1/k,l<< and klh < 1 and analyze the first alternative further in detail. Substituting k = kr + iki
into Eq. (2.12), retaining only the first order terms, and separating the real and imaginary
parts yield
kr6 k6
o2 [1 rtanhkrh] = gk [tanhkrh- ]
2 2
Skr8 gkrkh gkr2 (2.13)
o2 tanhkh = +- +g ki tanhkrh
2 cosh2krh 2
Assuming kr = k +Ak, where k is determined by 02 = gk tanhkh, we obtain
k26
kr =k+
2kh + sinh2kh (2
(2.14)
k28
k. = k2
2kh + sinh2kh
Thus waves of fixed frequency are shortened by friction, and their phase speed is reduced.
The constant A can be solved from (2.11) as
Ho k6 .n
A = -i Ho [1+-cothkhexp(-i-)] (2.15)
2ksinhkh 42 4
Combing Eqs. (2.8), (2.10) and (1.15) together, the real parts of potential and rotational
velocities are presented as
) H coshk(h +z) sin Hocosh kz sn
2ksinhkh 2/2 sinh2kh 4
Hocoshk(h+z) os Hok6coshkz cos(-)
v,(z) = -cos cos ( ) (2.16)
2 sinhkh 2r2 sinh2kh 4
SHasinhk(h+z) sin Haksinhkz sin(-
w,(z) sin sin(--)
2 sinhkh 22 sinh2kh 4
and
-r Ho H/o kicosh kh Q
r(z) = e s cos(-6-) + H -- coskh Cos -
2sinh kh 2v sinh2 kh 4(2.17)
(2.17)
w(z) = -e- Hk sin(O---)
2v2 sinhkh 4
31
where 6 = ky +ot and h+z Since 6/h is very small, the higher order terms are
neglected.
We now consider the streamwise mean momentum flux in the bottom boundary layer
and the possibility of mean flows. Neglecting the effects of wave height damping and wave
length shortening caused by bottom friction, averaging the second equation of (2.4) over a
wave period yields a second order mean flow in the boundary layer
avw a2pr
e (2.18)
az az 2
where the over bar means the time average over a wave period. The mean shear stress caused
by the boundary viscous flow is obtained by integrating (2.18) from an arbitrary depth, z, to
the surface
t(z) = pe pvw|-pvwL. (2.19)
az
where the velocities v and w include both the potential and rotational parts. Applying Eqs.
(2.16) and (2.17) into the term vw yields
vw = vpwp +vpw,+v,Wp+Vrwr (2.20)
with the terms in the right hand side are expanded as
H2 2k6
VpWp z 16 sinh2kh
VpWr z = -_ H2 2k86 coshk(h+z) .
vw | = e sin-(-+-)
8 V/2sinh2kh 4
VrWp Iz = -e- H2 2 sinhk(h+z) sin (2.21)
8 sinh2kh
H2 2k6 k
-e [sinhk(h+z) coshkh sin ( +-) + sinh kz sin( -c)]
8 /2sinh3kh 4 4
VrWr Iz = e-2 H2a2k6
16 sinh2kh
Therefore the mean shear stress caused by the boundary layer rotation flow is
z(z) = pH2a2 inhk(h+z) sin +k6 [sinhk(h +z) sin +cosh k(h+z) cos -9 e -(2.22)
8 sinh2kh tanhkh 2
which decays exponentially outside the boundary layer. The corresponding bottom shear
stress is given by (2.22) with z=-h as
pH202k6
b = H2 (2.23)
16 sinh2kh
For shallow water conditions, kh <<1, substituting (2.22) into (2.19) and integrating from the
bottom to an arbitrary depth, z, with the no-slip bottom boundary condition, the time mean
flow in the boundary layer is given as
S=- H2ok [3 -2(+3+Z)cosh e- -2(h+z)sin e-+e-2+ ] (2.24)
16 sinh2kh h h
33
which is usually referred to as the "streaming velocity" and is directed in the onshore
direction. It is seen that just outside the boundary layer, this time averaged rotational related
velocity is independent of the value of the viscosity and is given by
3H2ok
VrL = (2.25)
16 sinh2kh
which is 1.5 times the average of the return flow due to the mass transport.
2.6 Shear Stress Related to Undertow
The seaward return flow of wave mass transport induces a seaward stress on the
bottom sediment particles. In a real beach, a net onshore or offshore mass flux may exist in
the presence of three dimensional circulations, therefore an additional shoreward or seaward
bottom shear stress can result from the non-zero net onshore or offshore mass flux. It is well
known that associated with the wave propagation toward shore is a shoreward linear
momentum flux (Longuet-Higgins & Stewart 1964) and wave energy damping. Outside the
surf zone, wave energy damps slowly with distance due to dissipation in the bottom boundary
layer. Inside the surf zone, the waves decay relatively rapidly due to turbulent dissipation
caused by wave breaking. As a result of wave energy damping, the momentum is transferred
to the water column causing a shoreward directed thrust and thus a wave induced set-up, the
gradient of which is proportional to the gradient of wave energy dissipation. This momentum
flux is distributed over depth as shown as Figure 2.3. In shallow water, linear wave theory
predicts that one third of the momentum flux originates between the trough and the crest
levels and has its centroid at the mean water level and that the remaining two thirds, which
Figure 2.3 Distribution of the onshore component of momentum flux over
depth.
originates between the bottom and the mean water level, is uniforrply distributed over this
dimension and thus has its centroid at the mid-depth of the water column. Because of the
momentum contribution within the free surface, wave energy dissipation by breaking induces
an equivalent shear force on the water surface which will be quantified later. This surface
stress causes a seaward bottom shear stress, which does not exist outside the breaking zone
and will become very significant within the breaking zone. The magnitude of the induced
bottom shear stress is dependent on the rate of wave energy dissipation and the relationship
of the velocity gradient and shear stress. Over the water column, this effect of surface shear
stress caused by the momentum transfer must be balanced by the bottom shear stress and the
pressure gradient due to the slope of the water surface.
2.6.1 Linear Shear Stress Relationship
The linear Boussineq shear stress relationship with eddy viscosity, e, is applied as
= pe- (2.26)
az
35
Considering linear wave theory, the momentum equation for cross-shore flow is given by the
second equation of (2.4). Assuming shallow water waves, kh<
period and integrating from the free surface, z = 0, to an arbitrary elevation, z, yield
( E E pgz (T (z)) = 0 (2.27)
ay 2 h ay y
where E = pgH2 is wave energy density per unit surface area, and i is mean water level.
8
The first two terms of Eq. (2.27) represent the so-called radiation stress components with the
first being the contribution between trough and crest and the second below the mean water
surface. The third term is the effect of the slope of the mean water level. The -r in the fourth
term is the wind shear stress on the free surface. Applying Eq. (2.26) into (2.27) and
integrating from the bottom, z= -h, to an arbitrary depth, z, with the no-slip bottom
boundary condition yield
p+(z) = (h+z) + g2- +h a (2.28)
T y 2 y h ay
Integrating the above equation over the entire water depth results in the volumetric flux,
which is equal to the sum of the net volumetric flux, qet, and the wave mass transport flux,
, where c is the wave celerity. The slope of the mean water surface is therefore
pc
determined as
a8 -I -3 p E (2729)
ay pgh 2 4 Sy h pc qne
36
Substituting (2.29) into (2.28), we obtain the final expression for the velocity
h -aE 3 Z)2 z 3 (E +
()=- 2 -+ 1 -2
peC ay) 8 h 2h 8 2h pc J h
(2.30)
The corresponding bottom shear stress is given as
Pb 1 aE3pe E +
r P e- Z=-h + +
8 z 12 4 ay h2 pc
(2.31)
The first term on the above right hand side is the contribution of the surface wind. Onshore
winds cause a shoreward directed surface flow and a seaward directed bottom flow as shown
in Figure 2.4. Of course, offshore winds would cause a shoreward directed flow near bottom.
Thus landward and seaward directed winds result in destructive (offshore) and constructive
(onshore) forces, respectively. The second term is related to the part of momentum flux
originating between the wave trough and crest. Since the gradient of momentum flux is due
*- Wind Stress
-~ -- -
N
Shear Stress *-.
Acting on Bottom
Figure 2.4 Bottom stresses caused by surface winds.
37
to the rate of wave energy dissipation, this term is vary important for breaking waves but
almost negligible outside surf zone. The third term results from the mass flux. The seaward
return flow of the wave mass flux induces a seaward bottom shear stress. A net offshore or
onshore volumetric flux may exist when the return flow is enhanced or weakened due to
gradients in the longshore current. In storm events in which there is overtopping of the barrier
island, a portion or all of the potential return flow due to mass transport can be relieved. A
net onshore volumetric flux induced by the weakened return flow causes an onshore directed
bottom shear stress, while a net offshore volumetric flux acts oppositely and results in a
offshore directed bottom shear stress.
2.6.2 Quadratic Shear Stress Relationship
As waves propagate into shallow water, the wave related flow becomes more
turbulent. After wave breaking, the turbulence near the free surface is much stronger than that
near the bottom. Figures 2.5 and 2.6 present variations of mean turbulent kinetic energy, k,
from experiments of Ting and Kirby (1994), and Stive and Wind (1982) for spilling and
plunging breakers, respectively. In the figures, h/hb represents the ratio of local water depth
to the breaking water depth, and z-_ denotes the relative elevation. It appears that the
h
turbulent level decays in both types of wave breakers. The assumption of constant eddy
viscosity is not suitable anymore. Following Prandtl and Von Karman, the shear stress under
conditions of turbulent flow is related to the velocity shear as follows
av avy
az= pl2 I (2.32)
az aa
where I is the mixing length for an open channel flow. The form presented by Reid (1957)
-0.2
S(a) 0
S-0.4 0 ,
:0 +0 *
0 X
-1-
No -
-0.8 *
0 I I ,.
0.02 0.03 0.04 0.05 0.08 0.07 0.08
(k/gh)1
6)
-0.42
-0.4 0
-0.68
-0.8- o x +
-1
0.03 0.04 0.05 0.08 0.07 0.08 0.09
(Ilgh) I
Figure 2.5 Variation of turbulent kinetic energy with depth in spilling breakers. (a) Ting
and Kirby (1994); h/hb = 0.879 (o), 0.809 (*), 0.744 (+), 0.668 (*), 0.563 (x). (b) Test
1 of Stive and Wind (1982); h/hb = 0.882 (o), 0.765 (+), 0.647 (*), 0.529 (x).
is applied here
Ko
S=K- (zo+h+z)(z -z) (2.33)
h
where zo and z, are roughness lengths at the bottom and the free surface, respectively. Given
a surface stress and nearshore beach profile, the momentum equation (2.27) is solved by
E
equating return flow to the sum of Stoke's drift, -, and net volumetric flux, qner
pc
A comparison of cross-shore velocity variation with depth associated with the linear
and quadratic shear stress relationships is presented in Figures 2.7 and 2.8 for breaking and
0
(a)
S -0.5 +
a o a
-1
0 0.01 0.02 0.03 0.04 0.05 0.08 0.07 0.08 0.09 0.
0
(b) 0
0.3 -0.5
1 -I
s0.01 0.02 0.03 0.04 0.05 0.06 0.07
(T/gh)
Figure 2.6 Variation of turbulent kinetic energy with depth in plunging breakers. (a)
Ting and Kirby (1994); h/hb = 0.929 (+), 0.857 (.), 0.773 (o), 0.675 (x), 0.584 (.).
(b) Test 2 of Stive and Wind (1982); surface, 0.905 (+), 0.709 (), 0.544 (o), 0.381 (x),
0.143 (-).
non-breaking waves, respectively. Bottom streaming is not included in this model. In the
figures, the following wave conditions have been considered: water depth h = 1 m, beach
slope = 0.02, wave height H = 0.78 m, and wave period T = 6 s. In the quadratic relationship,
the bottom roughness is 1 cm for both breaking and non-breaking waves. Since breaking
waves cause a very rough water surface, the surface roughness is taken as half the wave
height for breaking waves and 1 cm for non-breaking waves, respectively. Generally, the
eddy viscosity in the linear stress relationship is difficult to determine and may vary with the
water and wave conditions. For the comparisons presented here, the eddy viscosities are given
S0.4 2...... 0.4 0. ...... .. ..
0 0 0L
-20 0 20 0 0.02 0.04 0 0.2 0.4
stress [N/m^2] eddy viscosity [mA2/s] velocity [m/s]
Figure 2.7 Comparisons of the linear (dashed line) and the quadratic (solid line)shear stress relationships for
breaking waves. Note: the net seaward transport is equal to the shoreward mass transport.
0.5
stress [N/m^2]
0 0.001 0.002 0 0.2
eddy viscosity [mA2/s] velocity [m/s]
Figure 2.8 Comparisons of the linear (dashed line) and the quadratic (solid line)shear stress relationships for
non-breaking waves. Note: the net seaward transport is equal to the shoreward mass transport.
42
by the average values of the quadratic shear stress relationship as 1.23x102 m2/s and
1.22x103 m2/s for the breaking and non-breaking waves, respectively. As shown in Figure
2.7, the magnitudes of velocity predicted by two methods are about the same order, but the
profiles are quite different. The position for the maximum velocity is at an elevation of 50%
and 25% above the bottom in the linear and quadratic relationships, respectively. In both
models, the shear stress varies linearly with the elevation and the surface value is equal to one
half the local wave energy gradient due to the wave breaking; but the bottom shear stress
predicted by the linear model is about three times large as that predicted by the quadratic
model. As shown in Eqs. (2.30) and (2.31) for the linear stress model, the velocity is inversely
proportional to the eddy viscosity and the bottom stress varies directly with eddy viscosity.
It is difficult to determine a proper value of eddy viscosity to satisfy both the velocity
distribution and the bottom shear stress under such conditions. For the non-breaking wave
case shown in Figure 2.8, the two methods yield much closer results. Due to the turbulence
caused by wave breaking, the eddy viscosity is much larger inside the surf zone. Since the
velocity profile is independent of eddy viscosity in the linear model for non-breaking waves,
a better fitting between the two methods for the shear stress can be achieved by decreasing
the eddy viscosity in the linear model. In the following, the predictions of the two methods
are compared with measurements from the literature.
Hansen and Svendsen (1984) measured the velocity distributions for undertow on a
planar bed with a slope of 1:34.3. The incident wave height was 12 cm in the uniform depth
portion of tank with a water depth of 36 cm, and a wave period of 2 seconds. Breaking
occurs at an approximate depth of 20 cm. The velocity distributions were measured in one
43
position outside the surf zone with a water depth of 22.5 cm and four different positions
inside the surf zone with water depths of 7.6 cm, 9.5 cm, 12 cm and 14.9 cm, respectively.
Figure 2.9 shows the comparison between the measured and the predicted results at
a water depth of 22.5 cm for non-breaking waves. The relative bottom and surface
roughnesses are 0.01 for the quadratic stress model. The eddy viscosity in the linear stress
model is taken as 0.609 cm2/s based on the average of the quadratic model. It appears that
the linear shear stress relationship presents a better velocity agreement than the quadratic.
However the difficulty in determine an approximate eddy viscosity in the linear stress model
should be noticed. Under non-breaking wave condition without winds, the velocity is
independent of eddy viscosity, but the shear stress is linearly proportional to it.
The velocity profiles of Hansen and Svendsen at the four positions inside the surf zone
are compared with the model predictions in Figure 2.10. The h/hb values shown in the figure
are the ratios of the local water depth to the depth at the initial breaking point. The relative
bottom roughnesses, zo/h, are 0.01 for all cases. Due to the wave breaking effects, the surface
roughness is given as half of the local wave height for each position. It appears that the linear
shear stress relationship is not suitable inside the surf zone anymore. The measured velocities
and these predicted by the quadratic shear stress relationship are in quite good agreement
except the case with 14.9 cm of water depth, which is a position very close to the breaking
point with a quite uniform velocity distribution.
Figures 2.11 and 2.12 compare the shear stresses, eddy viscosities and the velocities
computed from the quadratic stress model with those measured in Case 2 of Okayasu et al
(1988). The experiments were carried out with a bed slope of 1:20, incident wave height of
5 0 0.05 0 0.5 1
stress [N/m"2] eddy viscosity [cmA2/s]
0 5
velocity [cm/s]
Figure 2.9 Comparisons of the measurements (circles, Hansen and Svendsen 1984) with the predictions of
the linear (dashed line) and the quadratic (solid line)shear stress relationships for non-breaking waves at water
depth of 22.5 cm.
(b) h=12.0cm, h/hb=0.594
0
> 0.6
a)
.0)
*0.4
(
velocity [cm/s]
(c) h=9.5cm, h/hb=0.469
velocity [cm/s]
(d) h=7.6cm, h/hb=0.378
-5 0 5 10 15
velocity [cm/s]
0 5 10 15
velocity [cm/s]
Figure 2.10 Comparisons of the measured velocities (circles, Hansen and
Svendsen 1984) with the predictions of the linear shear stress model (dashed
lines) and the quadratic shear stress model (solid lines). Four different
positions are presented.
(a) h=14.9cm, h/hb=0.738
C a : 0
S0.4 .-- .. 0.4 0*. 0.4 .......... "I*:''-
02 0.2 ( 0.2
0L "- 01---.1 0
-2 0 2 0 5 10 -20 0 20
stress [N/mA2] eddy viscosity [cmA2/s] velocity [cm/s]
Figure 2.11 Comparisons of the measurements (circles, Okayasu et al 1988) with the predictions of the
quadratic (solid line)shear stress relationships for breaking waves at water depth of 6.5 cm and wave height
of 5.1 cm.
stress [N/mA2]
0 2 4 -10 0
eddy viscosity [cm"2/s] velocity [cm/s]
Figure 2.12 Comparisons of the measurements (circles, Okayasu et al 1988) with the predictions of the
quadratic (solid line)shear stress relationships for breaking waves at water depth of 3.0 cm and wave height
of 2.34 cm.
48
5.63 cm and period of 2 seconds at the uniform depth portion of the tank with a water depth
of 40 cm. The measurements inside the surf zone at two different locations are presented here.
In the experimental data, the Reynold stresses were calculated by Okayasu from the measured
turbulent fluctuating velocity; the mean shear stresses shown in the figures was obtained by
averaging the Reynold stresses over one wave period, and the mean eddy viscosity was
calculated from the mean shear stress and the gradient of the mean velocity. As expected, the
shear stresses varied quite linearly from the bottom to the surface. Since waves broke at the
surface, the eddy viscosity in the upper part of the water was greater than that in the lower
part. The turbulent level of flow decreased from the water surface to the bottom. In both
locations, the measurements and the predictions of the quadratic stress model agree well for
the shear stresses and the velocities. The predictions for the eddy viscosity are acceptable.
2.7 Comparison of Magnitude of Shear Stress Terms
In the field, all the bottom shear stress components mentioned in sections 2.3, 2.4, 2.5
and 2.6 exist however their relative magnitudes differed with some terms being much more
important than others. The momentum flux related shear stresses are remarkably different for
breaking and non-breaking waves. To provide an estimate of the relative magnitudes of these
shear stresses, the following conditions have been considered:
Water depth, h = 1 m, and density, p = 1025 kg/m3.
Beach slope of 0.02.
Wave height, H = 0.78 m, and period T = 6 s.
Mean sand diameter, D = 0.2 mm, and relative density, s = 2.65.
Wind speed at 10 m above the water level, uo = 20 m/s.
49
Because of the relative strong turbulence levels, the eddy viscosities are larger for breaking
waves than those for non-breaking waves. Under the water and wave conditions listed above,
the shear stresses will be calculated for both breaking and non-breaking wave cases.
(1) Critical shear stress
Generally, the critical shear stress inside the surf zone is much smaller than that
outside the surf zone due to the turbulence caused by wave breaking. Since very few studies
have been conducted under breaking wave conditions, the results from Amos et al. (1988) are
applied to determine the critical shear stress outside the surf zone as
T = 0.04p(s-l)gD = 0.13 N/m2 (2.34)
Inside the surf zone, the critical shear stress is considered as zero.
(2) Gravitation contribution
In Eq. (2.2), the sediment concentration, a, in the equation is assumed to be 0.66,
therefore the gravity induced shear stress is 0.04 N/m2.
(3) Shear stress due to non-linear waves
For the wave conditions given above, the wave length in the deep water, Lo, is 56.2
meters. The corresponding wave steepness, H/Lo, and the relative depth, h/L o, are 0.0139
and 0.0178, respectively. From Figure 2.2, the non-dimensional average bottom shear stress
is 5.4. Considering a Darcy-Weisbach friction coefficient of 0.08, the shear stress will be
given by Eq.(2.3) as 0.94 N/m2 directed shoreward.
(4) Shear stress due to streaming velocity
The shear stress due to the boundary layer streaming velocity is presented by Eq.
(2.23). The negative sign in the equation means the shear stress is acting in the onshore
direction. The bottom eddy viscosities computed by the quadratic shear stress relationship are
applied here for the calculations. For non-breaking wave conditions, the bottom eddy
viscosity is 1.66x104 rd/s, the boundary layer thickness, 6, is calculated as 1.86 cm, and
therefore the shear stress is 2.17 N/m2. Under breaking waves, the bottom eddy viscosity is
6.69x104 m2/s, the corresponding boundary layer thickness and shear stress are calculated as
3.74 cm and 4.36 N/m2, respectively.
(5) Shear stress due to the wave mass flux
The case considered here assumes no net mass flux in the cross-shore direction, which
means the seaward return flow balances the wave mass transport. The bottom shear stresses
are computed by the quadratic stress model as 0.64 N/m2 and 1.38 N/m2 for the non-breaking
and breaking wave conditions, respectively. The difference between the two conditions is
caused by the different eddy viscosities, which result from the different turbulence levels.
(6) Shear stress due to momentum flux
As seen from Eq.(2.31), the contribution of momentum flux is related to the gradient
of wave energy. Outside the surf zone, the change of wave energy can be neglected, therefore
the shear stress due to the momentum flux is almost zero. Inside the surf zone, there is a
substantial wave energy dissipation due to wave breaking. The shear stress caused by the
momentum flux becomes a very important term. For the wave and water conditions listed
51
above, the bottom stress computed by the quadratic stress model is 4.2 N/m2, which is the
total value presented in Figure 2.7 reduced by the contribution due to the mass flux.
(7) Shear stress due to winds
Winds blowing over water cause a wind shear stress and a slope of water surface. As
a result, a surface flow in the wind direction and a bottom flow in the opposite direction are
generated. The wind shear stress is expressed by a friction factor, f, (assumed as 0.005 here),
and the the wind speed at 10 m elevation
S= fu (2.35)
where Pa is the density of air and equal to 1.20 kg/m3 under the temperature of 20 oC. For the
given wind speed of 20 m/s, the surface stress is calculated as 1.20 N/m2. In the quadratic
stress model, the magnitude of bottom shear stress induced by this wind depends on the wind
direction and wave breaking. An onshore directed wind will cause an additional offshore
directed bottom stress of 0.549 and 0.255 N/m2 for the non-breaking and breaking wave
conditions, respectively. An offshore directed wind will induce an onshore directed bottom
stress of 0.31 and 0.26 N/m2 for the non-breaking and breaking wave conditions, respectively.
A summary of all shear stress terms discussed above is presented in Table 2.1. The
stresses due to the momentum flux transport and streaming velocity are the two dominant
terms. The critical shear stress is quite small compared with other terms. Because winds are
the only source of the surface shear stress outside the surf zone, they have the most significant
effects on bottom stress in this region.
Table 2.1 Constructive and destructive bottom shear stresses.
Magnitude N/m2
Type Description of Force Outside Inside
Surf Zone Surf Zone
Critical Shear Stress Friction 0.13 0
Nonlinear Wave 0.94 0.94
Constructive Streaming Velocity 2.17 4.36
Wind* in Offshore Direction 0.31 0.26
Gravity 0.04 0.04
Mass Flux 0.64 1.38
Destructive
Momentum Flux 0 4.16
Wind* in Onshore Direction 0.55 0.26
* Wind speed at 10 m above the water level, u0o= 20 m/s.
CHAPTER 3
DEVELOPMENT OF CROSS-SHORE TRANSPORT MODEL CROSS
The "closed loop" cross-shore sediment transport relationships, which are studied in
this chapter, are based on wave energy dissipation and implicitly "capture" the dominant
destructive shear stress mechanism (i.e. wave momentum flux) discussed in Chapter 2. Since
these models do not represent any constructive mechanisms, they would not be expected to
perform well for beach accretion. Based on equilibrium beach profile concepts (Dean 1977),
sediment transport is caused by deviations of a beach from its equilibrium form. According
to scale analysis, the linear transport relationship proposed by Moore (1982) should be
modified and a non-linear model is developed. In the non-linear model, the transport rate is
proportional to the cube of the difference of the local wave energy dissipation per unit volume
from the equilibrium value at each location across the surf zone. It appears that the non-
linearity of this transport relationship can reasonably explain the time scale difference in the
beach profile evolution in different experiments. A finite difference method is applied to solve
the transport and the continuity equations numerically. Beach profile response at different
times is determined by the numerical solutions.
3.1 Equilibrium Beach Profile
An equilibrium beach profile represents a dynamic balance of destructive and
constructive forces acting on the beach. A change of these two competing types of forces will
54
result in a disequilibrium. Considering wave energy dissipation per unit water volume to
represent the dominant destructive force, Dean (1977) has proposed that a sediment of a
given size will be stable in the presence of a particular level of wave energy dissipation per
unit volume, D.. This can be expressed in terms of wave energy conservation as
1 d(Eg)
D- (3.1)
h dy
in which y is the shore-normal coordinate directed offshore, E is the wave energy, and Cg is
the wave group velocity. As a first approximation, D. is assumed to be a function only of
sediment size (Moore, 1982), or equivalent sediment fall velocity (Dean, 1987). With linear
wave theory and shallow water assumption, Eq. (3.1) can be integrated to
24D 2
h = ypg23= Ay3 (3.2)
5pg /iK2)
where K is the ratio of breaking wave height to water depth, and A is defined as a profile scale
parameter. The relationship between A values and the median sediment size is shown in Table
3.1 (Dean and Dalrymple 1996). Since the profile scale parameter, A, is only a function of
sediment size, wave conditions are not included in the above relationship.
The two-thirds power law profile formula of Eq. (3.2) was first empirically identified
by Bruun (1954) as an appropriate representation of the profiles in a field study of beach
profiles at Monterey Bay, CA and along the Denmark coast. Dean (1977) examined 502
beach profiles from the east coast and Gulf shoreline of the United States by a least square
55
fit to each profile with a generalized power law profile, h = Ay m. It was found that the
average value of the exponent was 0.66, in very good accord with the result derived in Eq.
(3.2). Hughes and Chiu (1978) carried out a study of beach profiles and associated sediment
characteristics at different locations in the state of Florida and Lake Michigan and found that
Eq. (3.2) described the beach profiles reasonably well.
Table 3.1 Summary of Recommended A values, units of A parameter are min (after
Dean and Dalrymple 1996).
D 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
(mm)
0.1 0.063 0.067 0.071 0.076 0.080 0.084 0.087 0.090 0.094 0.097
0.2 0.100 0.103 0.106 0.109 0.112 0.115 0.117 0.119 0.121 0.123
0.3 0.125 0.127 0.129 0.131 0.133 0.135 0.137 0.139 0.141 0.143
0.4 0.145 0.147 0.148 0.150 0.151 0.153 0.155 0.156 0.158 0.159
0.5 0.161 0.162 0.163 0.165 0.166 0.167 0.168 0.169 0.171 0.172
0.6 0.173 0.174 0.175 0.177 0.178 0.179 0.180 0.181 0.183 0.184
0.7 0.185 0.186 0.187 0.188 0.189 0.190 0.190 0.191 0.192 0.193
0.8 0.194 0.195 0.196 0.196 0.197 0.198 0.199 0.200 0.200 0.201
0.9 0.202 0.203 0.204 0.204 0.205 0.206 0.207 0.208 0.208 0.209
1.0 0.210 0.211 0.212 0.212 0.213 0.214 0.215 0.216 0.216 0.217
Two disadvantages of Eq. (3.2) are the infinite beach slope at the water line and the
monotonic form of the profile. The first shortcoming is overcome by including gravity as a
significant destructive force when a profile becomes steep. In this case, Eq. (3.2) is modified
to include the beach face slope, mo,
h h 3/2
y = O- + (3.3)
mo A
Figure 3.1 Definition sketch for Inman's curve fitting. Crosses denote the origin for bar-berm
(dotted) and shorerise (dashed) curves, with y coordinates, y,, Y2, and vertical coordinates,
hi, h2 (from Inman et al. 1993).
Larson (1988) modified Dean's approach and developed an equation of similar form by
replacing the spilling breaker assumption with the wave breaking model of Dally et al. (1985)
and retaining the requirement of uniform wave energy dissipation per unit volume.
Inman et al. (1993) developed an equilibrium beach profile that treats a profile as two
parts, the inner (bar-berm) and the outer (shorerise) portions. The two portions are matched
at the breakpoint-bar and fit by curves of the form
h = By m (3.4)
The coordinates of Inman's curve fitting are shown in Figure 3.1 with the subscripts 1 and
2 corresponding to bar-berm and shorerise curves respectively. In addition to B and m in Eq.
(3.4), the origins of two curves must be determined from the profile data. In total, seven
57
variables, yl, z1, B, m1, y2, B2,and m2 are required to fit a profile. This method is
diagnostic and generally useful for a beach with measured data available. Comparatively, the
method described by Eq. (3.2) is prognostic and needs only a description of sediment size.
3.2 Scale Analysis
A beach which is steeper than equilibrium has a smaller volume of water over which
the incident wave energy is dissipated. This causes the energy dissipation per unit volume to
be greater than the equilibrium value. As a result, the total destructive forces are greater than
the constructive forces. The profile will respond to the imbalance of forces through
redistribution of the sediments. Over time, sand will be carried from onshore to offshore and
deposited near the breakpoint, resulting in a milder profile which approaches the condition
of uniform wave energy dissipation per unit volume. Similar to this process, for a beach with
a milder slope than equilibrium, sediment will be moved from offshore to onshore.
Based on these concepts, Kriebel and Dean (1985) proposed that the cross-shore
sediment transport rate per unit beach width, Q, could be approximated according to the
deviation of actual wave energy dissipation per unit volume from the equilibrium at a
particular location across surf zone as
Q = K(D -D,) (3.5)
in which, D represents the actual total destructive force,
5 tg3/2
D= 5pg32 Kh /- (3.6)
24 ay
58
The transport parameter, K, was considered as a dimensional constant. The following scaling
relationship is established from Eq. (3.5)
[K(D -D,)]odel
Q [KD = = Kr(D-D,)r (3.7)
[ K(D -D ,) ]prototype
where the subscript r denotes scale ratios. For an undistorted model with same density fluid,
according to the definition of D, the disequilibrium scale (D-D.),can be expressed in terms
of the length scale, L,
(D-D.), = (3.8)
On the other hand, according to the Froude relationship the time scale, Tr, is expressed as
Tr = Vr (3.9)
Following this relationship, the cross-shore sediment transport per unit beach width should
be scaled as
L2
Qr Lr2 (3.10)
This equation provides a basis for evaluation of transport models. If K is independent of the
length scale, Eq. (3.7) does not provide a valid scaling of transport.
Based on field observations, Dean (1973) hypothesized that sediment was suspended
during the wave crest phase position and if the fall time were less or greater than one half
59
wave period, the net transport would be landward or seaward, resulting in a bar formation for
the latter case. This mechanism would exist in the region of wave breaking. Considering the
height of suspended sediment to be proportional to the breaking wave height, Hb, resulted
Hb
in identification so-called fall velocity parameter, -. By examining small scale wave data
wT
for which only deep water reference wave heights, Ho, were available, the following
relationship for net seaward sediment transport and bar formation was found
Ho
> 0.85 (3.11)
wT
Later, Kriebel et al. (1986) examined only prototype and large scale laboratory data and found
a constant of 2.8 instead of 0.85 in Eq. (3.5). Kraus (1988) examined only large tank data and
proposed the following relationship for bar formation
H Ho
o 0.0007 (3.12)
LO [wT)
H0
Several model studies have confirmed that -- is a valid modeling parameter such that if this
wT
parameter is the same in model and prototype, they are scaled versions of each other and the
fall velocity is scaled by the length scale as w, = JL-. Two examples from Hughes et al. (1990)
and Kriebel et al. (1986) are presented in Figures 3.2 and 3.3, respectively.
It is of interest to develop and test a transport model which can ensure convergence
to the target (equilibrium) beach profile and also satisfy the scale relationship given by
Eq.(3.10). One approach is to consider the following form for the sediment transport
60
370 WAVES Proto oshed
S~-~S Ois ff a 0.484 a
.10 0 10 '20 30 40 50
RANGE (m)
L_.. i"* i i ioi c |odei Solid
S15so WAVES Prot Dshd
~.lS iFP R, 0.444 a1
-*1 0 10 20 30 40 SO
RANGE (m)
Figure 3.2 Comparison of beach profiles from medium and large scale wave tank, Scaling
according to (Ho/wT),=1 (from Hughes et al. 1990). Note concrete bore underlying initial
sloping sand beach.
Q = K(D-D,) ID-D,In-1 (3.13)
Which results in the scale relationship
Qr =K(D-D .)rD-D I (3.14)
Substituting Eqs. (3.8) and (3.10) to Eq. (3.14) yields
n 3
KL = L 2 (3.15)
If K is only a function of the fall velocity parameter, Kr equals unity and n=3 is determined
such that both the scaling relationship and convergence to the equilibrium profile can be
satisfied. Otherwise, if Kr is related to the length scale as Kr = Lm, the relationship, m+- = ,
2 2
Me- MdW 1G.T7a,.LTgo
Praftye H*L6a. TP5A*
Figure 3.3 Comparison of beach profile from medium and large scale wave
tank, Scaling according to (Hd/wT)r=1 (from Kriebel et al. 1986).
S500. -- Dune without foreshore
--. Case 300
400.
S300. O
200.
100.
5. 10. 15. 20. 25. 30. 35. 40. 45. 50.
Time (hours)
(a) Eroded volumes versus time for the German "dune without foreshore" case and
Savill's Case 300.
7
~ L,-LO --;-,..-*, ,,,---.-o,, --.",*,,...
L|L,.OIT2.,Se44l,,O.,S'^066SiH
SI RE MARKS: W.6.S07
aSSss saw*
s0o 0ooo 505o 2Sooo 1 0oo
Time (hours)
(b) (LI L2) versus time for Swart's flume B Case with 86 = 0.15m and 6 2 = 0.10m
Figure 3.4 Time history of beach evolution in three different experiments.
must hold to satisfy Eq. (3.10). In this case, there is an infinite number of combinations of m
and n would satisfy the scaling requirement.
3.3 Discussion of Transport Relationship
Sediment transport in the nearshore region is a complicated process. Under different
conditions, some beaches reach equilibrium very fast, but others may vary slowly. The time
scale of beach evolution may vary from tens to thousands of hours for different experiments.
Figure 3.4 presents the results from the German "dune without foreshore" case (Dette and
Uliczka 1987), Saville's Case 300 (Kraus 1988), and Flume B case in Swart's experiments
63
(Swart 1974a). In these tests, the eroded volume at any time is determined as the cumulative
volume of material displaced between initial and current profiles, and L, and I2 are
schematized profile lengths at water depth 6, and (68 +86), respectively. Comparing these
three cases in Figure 3.4, a question arises as to the causes of the greatly different profile
response times.
In an attempt to understand the causes of the different time scale, we examine the
following equation
dx =K(x-x,) Ix-x, n-1 (3.16)
dt
where x. is the equilibrium value of x. Eq. (3.16) is reminiscent of the transport relationship
discussed above. Non-dimensionalizing with x '= and t'= t yields
x. Kx-1
dx- (x -1) x'- 11 n-1 (3.17)
dt'
with the initial condition x'(t'=0) =x0, the normalized solution of Eq. (3.17) is given by
x/- 1 _t'
x e for n=
x0-1
x'- 1 1 for n*(3.18)
for nr l
0 [(n-1)|xo-ll" 1|- +1 -1
Figure 3.5 presents the results of versus t' for n=l, 2 and 3, and x'=2 and 10
xo-1
respectively. It appears that the time scale of the linear system (n=l) is independent of the
initial conditions and the two lines in Figure 3.5 (a) are coincident. However, for the nonlinear
(a) n = 1
(b) n = 2
2 4 6 8
(c) n = 3
2 4 6 8
normalized time, t'
Figure 3.5 The solution of Eq. (3.16).
1
S0.8
- 0.6
a 0.4
a 0.2
1-
~0.8
.0.6
i5i 0.4
o
S0.2
0
0
1
^ 0.8
N I
,0.6
~S 0.4
" 0.2
0
65
systems, the initial conditions affect the time scale by the factor (n-1) |x -l|n-1. As n
increases, this factor becomes more and more significant.
As demonstrated in the solution of Eq. (3.16), a nonlinear transport equation provides
an explanation for the range of time scales observed in laboratory experiments of beach profile
evolution. In such nonlinear systems, the greater variation of initial condition from
equilibrium corresponds to smaller time scales of profile response. Returning to the question
raised about Figure 3.4, it appears the time scale differences in the three cases may be caused
by the different initial conditions and non-linearity of the transport relationship. Among three
cases, the "dune without foreshore" had a sand size of 0.33 mm and a initial slope of 1:4 with
a wave height of 1.5 m and period of 6 s, and its initial condition is farthest from its
equilibrium and the beach takes the shortest time to approach relative equilibrium. For Case
300, the sand size was 0.22 mm and the initial profile slope was 1:15 with a wave height of
1.68 m and period of 11.33 s. Swart's flume B Case had a sand size of 0.17 mm and an initial
slope of 1:10 with a wave height of 0.07 m and period of 1.04 s. Its initial condition is
hypothesized closest to the equilibrium and takes the longest time to arrive at relative
equilibrium. In the next chapter, the transport relationship, Eq. (3.13) with n=l and n=3, is
applied to three sets of large tank experiment data. Time dependent profile response is then
determined by the numerical solution of the transport equation and continuity equation, which
is
ay- -_ Q (3.19)
at ah
MSL
hi.1
yi hn = A(yYmsi)23
---Y4.1 --
Figure 3.6 Grid with h and t as the independent variables and y dependent.
3.4 Modeling Process
3.4.1 Numerical Method
The sediment transport and continuity equations are solved by a finite difference
method. The computational physical domain with the staggered grid is presented in Figure
3.6. The cells are finite increments of the depth variable, h. Thus the depth is the independent
variable and the dependent variable is the distance, y, which varies with time. For each time
step, the transport and continuity equations are solved simultaneously. After applying Eq.
(3.6) as the formula of the actual wave energy dissipation per unit water column, the finite
difference form of the transport equation (3.13) at the (m+6)th time step, t=(m+6)-At,
becomes
P5 32k 2(hi3" -hi) 325 pg k 2(hn/-hi )
Qm+ = K 24 6 -D *24 +6 + -D (3.20)
e tm+6 m+6 m+6 m+6 *
Yi -Yi-1 Yi -Yi-1
where the superscript m+6 denotes the values at (m+8)th time step, and 6 is a value between
67
0 and 1. After rewriting the distance variable at the time, t=(m+l)-At, as yml =yim+Ayim+l
Eq. (3.20) becomes
Qm+1
-pg 32k 2(h3/2 -h.
24
m m
yi -yi-I
-D*
(3.21)
A m+ A m+1
For a small time increment At, the value e is much less than 1. Therefore
m m
Yi -yi-1
Eq. (3.21) is linearized as
5 p2-h) 5 2 (h 3/2_h3 n
QimK1 24 32 (l-) -D,.24 2 (1-e) -D (3.22)
Yi -yi-1 yi -Yi-1
When n is an odd number, the absolute sign in the equation can be dropped. Neglecting higher
order terms, the linearized finite difference transport equation is expressed as
M A-, i mAl
Ai*Ayi1m+ + Bi, Qm. + CiiAy = D
5P 3sk2(h3/ -hi n-I 5 P 32k2(h -hi3)
Ai*=nK 24 -D, ) *
Yi -Yi-1 Yi -Yi-1
B' = 1
C = -Ai
[5 3/22(h 3/2h 3/2 n
K 24
D* = i -- -D
yi -Yi-I
(3.23)
68
For the grid shown in Figure 3.6, the discrete form of the continuity equation (3.19) is
1 1
m+-- m+-
m+1 2 2
yi Qi+l Q (3.24)
At Ah
1+- 1
Recalling Qi 2 IQim + Qi,) Eq. (3.24) is written as
2
Aiimtl +BiAyimt + C-m+l = Di (3.25)
AQ,1 ++BAym+l+ C, =D,
A At
A---
S 2Ah
B.= 1
Ci = -A.
D = At (Q m Qin)
2Ah il-I
Eqs (3.23) and (3.25) are tridiagonal equations and can be solved simultaneously by the
standard double sweep method. At each computation time, the active profile domain is
determined by the instantaneous incoming wave and water level conditions. Assuming this
active domain is the region of hib h< hhie, the boundary conditions, Qib = Qie = 0 can be
applied, which leaves 2x(ie-ib) variables as unknowns. The continuity equation (3.25) is
solved from i=ib to i=ie-l, and the transport equation (3.23) is solved from i=ib+l to
i=ie. They provide total 2x(ie-ib) equations. Starting with boundary condition Fi = Qie and
E i= 0, four new sets of variables are calculated as
A. Di-CiFi1 Ai Di*-Ci*Fi
E= --, F.= E. = Fi = D (3.26)
Bi+CiEi, Bi+CiEi* Bi* +Ci* Ei Bi +C* E
i i +1 +1Ci
69
These coefficients are determined sequentially from i = ie to i = ib. Therefore the current
transport rates and the profile changes are determined as
Ayi = EiQ +Fi ib
(3.27)
Qi = Ei* Ay+Fi* ib+1li ie-1 (
3.4.2 Wave Run-up and Set-up
The active profile considered in the numerical model is from the wave run-up limit to
the wave breaking point. Outside this active region, the net sediment transport does not result
in profile changes. At each time step, the water level is determined by the sum of storm surge,
tide and wave set-up, and the elevation contours used in the transport equation are modified
according to the change of water level.
The wave induced set-up in the surf zone is calculated according to the balance of
pressure gradient and radiation stress. Inside the surf zone, based on the linear wave theory,
the set-up, l, is given (Dean and Dalrymple, 1991) as
3K12/
(h) = + (hb-h) (3.28)
1+3K2/8
where K is the ratio of breaking wave height to the breaking water depth, and rb and hb are
the set-down and the water depth at the breaking point, respectively. For shallow water, the
value of tb is given by Longuet-Higgins and Stewart (1964) as
k2hb
ib- -- (3.29)
16
70
In the numerical model, storm surge and tide are applied to the whole surf zone uniformly,
but the set-up is added according to Eq. (3.28) and varies with water depth. At the shoreline,
the set-up, %0 calculated from Eq. (3.28) is
e -0.0625 K2
-0 i- hb (3.30)
1-e
3K2/8
where e= According to linear wave theory, we have K = 0.78, and = 0.18 hb is
1+3K2/8
determined.
In the study of seawalls and breakwaters design, Hunt (1958) proposed wave run-up,
R, as a function of bed slope, wave height, and wave steepness, which can be expressed as
R =FHb (3.31)
where R is the run-up height measured vertically upward from storm water level, FR is a non-
dimensional coefficient and is approximately 1, m is the average bed slope from the run-up
limit to the breaking point, and Lo is the deep water wave length. Since the wave run-up
calculated by Eq.(3.31) is measured from the storm water level, after including wave set-up
in water level change, the wave run-up is established as the value calculated from Eq. (3.31)
by subtracting the value of wave set-up given in Eq. (3.30).
3.4.3 Dune. Shoreline and Offshore Slopes
In the numerical model, three characteristic slopes must be specified. As shown in
Figure 3.7, they are dune slope, md, shoreline slope, m and offshore slope, mff. The dune
Initial Profile
Dune Scarp
/IM
Run-up (m, / S.W.L.
Final Profile Offshore slope (m,,)
Figure 3.7 Definition sketch for dune, shoreline, and offshore slopes.
slope, which is defined as the averaged dune scarp slope after erosion, is the maximum slope
that the profile is allowed to achieve. If the profile reaches an angle steeper than the dune
slope at the end of any time step, avalanching occurs and the profile is smoothed to the dune
slope.
The shoreline slope, which is the anticipated profile slope between the shoreline and
the run-up limit, controls the profile evolution between the shoreline and the run-up limit. In
this region, the profile change is caused by the uprush and backrush, and the transport
equation discussed in section 3.2 cannot be applied. After each time step, the actual slope is
compared with the shoreline slope. If the actual profile is steeper than the given shoreline
slope at a point, the following adjustment is made for the distance variables of two adjacent
cells
Ayi =-0.5 Yi-Yi-1 1h/ -e
(3.32)
Ayi- = 0.5 Yyi- hi-h 1 -e K)
where the subscript i and i-1 denote the indexes of cells, m, is the shoreline slope, and Ky is
a decay parameter and set somewhat arbitrarily to 10 hr- here, which represents a "folding
time" of 0.1 hours.
Sand conservation across the profile is ensured by the continuity equation, so that the
volume eroded from the beach face must be deposited offshore. Since the profile change ends
at the wave breaking point, there is an unrealistic sudden vertical change in the profile shape.
To overcome this problem, the offshore slope is introduced to control the slope at the end of
the deposited volume. Under most cases, the value of this slope is between 0.1 to 0.2.
3.4.4 Random Wave Generation
Under random wave conditions, the joint distribution of wave periods and heights
developed by Longuet-Higgins (1983) is applied to generate wave height and period series
in the time domain. It is convenient to non-dimensionalize wave height and period with the
quantities related to the spectral density, E(o). The average frequency is defined as
J = mlImo (3.33)
where mn denotes the nth moment of the spectral density,
m= on E(o)do (3.34)
0
73
A spectral width parameter v can then be defined in terms of the variance of E(o) about the
mean as
2 f (o-)2E(o) do mm2
C22 M 2
v_=_=__ _-_ _-- 1 ( 3 .3 5 )
o 0mo m
For a narrow-banded spectral random wave system, v is much less than 1. Under severe wave
conditions, v will increase. In practice, v values are between 0 to 0.6 for ocean waves. After
normalizing the wave height, H, and the wave period, T, by
H a
R 2 = (3.36)
the joint probability density function, p, is expressed as
2 -R2 +( 11 )212
p(R,-) = L(v) 2 -- e R (3.37)
where L(v) = 1+- is a normalization factor introduced to take account of the
fact that only positive values of I are considered.
The marginal probability density of wave height R is found by integrating p(R,') with
respect to 0 < T < c, that is
p(R) = L(v) [1 +erf(R/v)]Re -R2 (3.38)
R/v
where erf(Rv)= f e -t2dt is the error function. Eq. (3.38) states that the density of R
So
nu=0
0.8 ---.. nu= 0.2
Snu =0.4
0.6 ........... nu = 0.6
o0 nuO\.
C0.4
0.2-
0 0.5 1 1.5 2 2.5 3
non-dimensional height
Figure 3.8 The probability density of wave height R when v = 0.2, 0.4 and 0.6
compared with the Rayleigh distribution (v = 0).
is almost Rayleigh distributed, but must be corrected by the factor L(v) [1 +erf(Rlv)]. For
2
large waves and small v values, the correction is exponentially small. Some examples are
presented in Figure 3.8. For the current application, the Rayleigh distribution is used as an
approximation of Eq. (3.38). Therefore, the cumulative probability of the wave height
becomes
P(R) = 1-e-R (3.39)
and the root mean square wave height, I-I, is related to the spectral density as Hrm, = 2 /2mo.
The distribution of wave period at fixed values of the wave height R is found by
dividing p(R,t) in Eq.(3.37) by p(R) in Eq. (38), that is
2 R 1 2R2
p('c/R) = ex 1-1 v (3.40)
rv[l+erf(R/v)] T2 T V2
Integrating p(r/R) with respect to t results in the cumulative probability
1 +tanh 1.2R (
To simulate an irregular wave condition, a random number P, uniformly distributed
from 0 to 1 is generated. A random non-dimensional wave height, R, is obtained from Eq.
(3.39)
R) = -ln(1-P) (3.43)
To obtain the corresponding wave period, a value of v is assumed according to the severity
of wave conditions. A second random number P2 with uniform density distribution over (0,1),
is generated. Substituting P2 and R(1 into Eq. (3.42) yields
is generated. Substituting P, and R i into Eq. (3.42) yields
0.4
0.2
0 0.5 1 1.5 2 2.5 3 3.5 4
x
Figure 3.9 Comparison of the error function erf(x) with tanh(1.2x).
1.2RI
P2 1+tanh
1 v In I v (3.44)
2.4R +tanh 1.2R,
2-P2 1 +tanh--1
Given a root mean square wave height, H-, and an average wave period, T, the dimensional
wave height and period are
H, =H R, T, = T (3.45)
The random wave heights and periods generated numerically are compared with Longuet-
Higgins's joint density distribution in Figure 3.10 and Figure 3.11 for narrow band (v = 0.1)
and wide band (v = 0.5) spectra, respectively. It appears that the numerical results agree with
Longuet-Higgins theory very well.
(a) Scatter diagram of numerical results
0 0.5 1 1.5 2 2.5
non-dimensional period
Contours of L-H Eq (dashed) and numerical
results (solid)
O
0 0.5 1 1.5 2 2.5
non-dimensional period
Figure 3.10 Comparisons of the numerical results with the density
function (Eq. (3.37)) of Longuet-Higgins for v = 0.1.
2.5
I 2
0
o
"5 1.5
E
o
0
0.5
(a) Scatter diagram of numerical results
3
2.5
.C
S2
c
O
, 1.5-
e-e
1 '
0.5 .
0 0.5
0 0.5
1 1.5 2 2.5 3
non-dimensional period
(b) Contours of L-H Eq (dashed) and numerical
3 1 1i
results (solid)
1 1.5 2
non-dimensional period
Figure 3.11 Comparisons of the numerical results with the density
function (Eq. (3.37)) of Longuet-Higgins for v = 0.5.
CHAPTER 4
CALIBRATION OF CROSS MODEL WITH LABORATORY EXPERIMENTS
4.1 Introduction
The cross-shore sediment relationships discussed in Chapter 3 have two coefficients
requiring quantification. They are the transport coefficient, K and the exponent, n. Based on
the scaling analysis, n=3 was proposed in the last chapter. In this section, the proposed
nonlinear transport model with n=3 will be compared with the linear transport relationship,
n=l, for seven laboratory experiments. Calibrations of the two models are accomplished by
a series of simulations in which a trial K value is used to simulate profile development for each
experiment. The eroded volume at any time is defined as the cumulative volume of material
eroded between the initial and current profiles. The errors between the predicted and
observed eroded volume are obtained at the times for which profile information is available.
The best-fit K value is determined as the value yielding the overall least square error. The
calibrations utilize 3 sets of different laboratory data, involving a total of seven large scale
wave tank experiments. Among these, five cases were carried out with monochromatic wave
conditions, and the other two were conducted under random waves. According to the wave
tank facilities, the seven experiments are divided into three groups: Saville's experiments
which were operated by the Beach Erosion Board in the "Large Wave Tank" in 1956-1957
and 1962 (Saville 1957, Saville and Watts 1969, and Kraus 1988); the German "Large Wave
80
Flume" experiments, which were carried out in Hannover in 1987 (Dette and Uliczka 1987,
Dette and Rahlf 1992, and Dette et al. 1992); and SUPERTANK laboratory data collection
project, which was conducted at O.H. Hinsdale Wave Research Laboratory, Oregon State
University, in 1991 (Kraus and Smith 1994, and Smith and Kraus 1995).
4.2 Saville's Experiments
4.2.1 General Description
Saville's experiments (Saville and Watts 1969, and Kraus 1988) described here were
conducted during 1956-1957 and 1962 in the "Large Wave Tank" located at Dalecarlia
Reservation in Washington, DC. The concrete tank was 194 m long, 4.6 m wide, and 6.1 m
deep. Prior to the start of most new experiments, sand was emplaced to form a 1 on 15 slope.
A fine sand of median diameter 0.22 mm was used in the experiment performed during 1956-
1957, and a coarse sand of median diameter 0.40 mm was used in the experiment performed
in 1962. Since beach erosion is the process of interest in this study, only the set with a sand
size of 0.22 mm will be discussed here.
Seven experiments were conducted under constant wave conditions with sand size of
0.22 mm. They were referred as Case 100, 200, 300, 400, 500, 600, and 700. The
corresponding fall velocity for the sand at a temperature of 20 C is 3 cm/s and the profile
scale parameter, A, is 0.106 m1i3. Since the profile eroded back to the end of the tank during
the experiment in Case 100, this case will not be discussed. The wave and water level
conditions for the other six cases are listed in Table 4.1. The initial and the final profiles for
each of these experiments are presented in Figure 4.1 in the order of the fall velocity
parameter, H/wT. Since the wave heights were small in Case 600 and Case 200, there was
81
Table 4.1 Wave height, period, and water depth in horizontal section of the tank.
Case No. Wave Height (m) Wave Period (s) Water Depth (m) H/wT
200 0.55 11.33 4.57 1.62
300 1.68 11.33 4.27 4.94
400 1.62 5.60 4.42 9.64
500 1.52 3.75 4.57 13.51
600 0.61 16.00 4.57 1.27
700 1.62 16.00 4.11 3.38
almost no beach erosion. In the other four cases, beaches eroded and offshore bars formed
near the break points. It appears that as the fall velocity parameter increases the offshore bar
becomes more and more significant. In Case 500, the height of the offshore bar became about
1.5 m. According to observations recorded during the experiment, a second wave breaking
position occurred in this case. The time histories of observed eroded volume for the four
erosion cases are shown in Figure 4.2. It appears that the variation of eroded volume with
time is not consistent in Case 700. Although constant wave conditions were run during the
experiment, the eroded volume reached a maximum at 30 hours instead of increasing
monotonically. Therefore, for the calibration of CROSS model, only Case 300, Case 400 and
Case 500 will be used.
4.2.2 Calibration
The sediment relationship, Eq. (3.13) with both n=l and n=3, are applied. The active
water depth is determined as the breaking water depth, which is 1.28 times of the breaking
wave height according to McCowan (1894) theory. Based on the wave heights listed in Table
Case 600 H/wT = 1.27
20 .. ... .. ... .. ... . . . . .. . . .
-2. .. ... .. ... ...
-4
-6
-40 -20 0 20 40 60 80
distance [m]
Case 700 H/wT = 3.38
2 ... .... ... ....... ....... ....... ......
o .......... ............. ....... .......
- 4 ..... ....... ...... ... ."' ......
-2
-4
-6
-40 -20 0 20 40 60 80
distance [m]
Case 400 H/wT = 9.64
: : : 1 :
.. . .. :. . ... .. ... . .. :. . . .. .
...... .......
: : :
.... .
; .-
-Vi,
-40 -20 0 20 40 60 80
distance [m]
Case 200 H/wT = 1.62
2 ...... .... .. ... .. .. ..... .. ... ... .... .
0 ........ ...... '" ...... .... ............ .......
0
-2
-4
-6
-40 -20 0 20 40 60 80
distance [m]
Case 300 H/wT = 4.94
0 ...... ... ...... ....... ........ ...... .......
2
0
-2 ... .
-4
-6
-40 -20 0 20 40 60 80
distance [m]
Case 500 H/wT = 13.51
2 ...... ..... ....... ....... ....... .. .......
0 ..... .. ... ....... .............. .......
-2 ...... ....... ......... .... .. ... ......
-4 .....
-6
-40 -20 0 20 40 60 80
distance [m]
Figure 4.1 The initial (dashed) and the finial (solid) profiles of Cases 200, 300, 400, 500, 600,
and 700 in Saville's experiments.
^
Case 700 H/wT = 3.38
30
2 5 ........ ... .. 0 ......... ..... .......
200 0 0
2 0 ......... ....... ........ ......... ...........
15 0......
5 ..... ........ ........ ......... ....
0 20 40 60 80 100
time [hr]
Case 400 H/wT = 9.64
30
25 ........................ 0 ..
0
20 ........ .. ... .......... ............
0
10 0
10 5 -0 ....... ....................... ..........
5 0 .. ....................... ..........
10 20
time [hr]
30 40
Case 300 H/wT = 4.94
30
0
25 0.........--- ............
10
50
0 10 20 30 40 5(
time [hr]
Case 500 H/wT= 13.51
O n ________________________-- T-m -- -
0 20 40 60
time [hr]
80 100
Figure 4.2 The time histories of the observed eroded volume for Cases 700, 300, 400 and
500.
4.1, the active water depths are 2.15 m, 2.08 m and 1.95 m for Case 300, Case 400 and Case
500, respectively. The eroded volume error is defined as
Err -1 [(Vol(t -VOlm(t]2
n j=
(4.1)
where the Vol, and Volm are the predicted and measured eroded volumes, respectively, and
ti denotes the time at which the measurements are available. In each case, the dune, shoreline
and offshore slopes used in the numerical model are determined according to the observed
......... ......... o .. .: ... ........ .......
0 0
..... .. ....... ........ .......... .......
0
. ... ............. ...... ... .......
)
I
, A
I I I I
84
Table 4.2 The numerical model inputs and best-fit results for Saville's experiments.
Case Slope Best-fit K Error* [m2]
No. dune shoreline offshore n=3 [m8s2/N3] n=l [m4/N] n=3 n=l
300 0.50 0.15 0.20 8.55x10-o1 3.45x10-6 2.08 3.37
400 1.00 0.17 0.15 7.97x10-'0 3.71x10-6 1.78 2.40
500 0.50 0.13 0.30 5.77x10-'0 1.57x10-6 3.99 1.88
Error represents the root mean square between predicted and observed eroded volume
as shown in Eq. (4.1).
7
-6-
E5-
2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
K [m^8*s^2/N^3] 10-9
Figure 4.3 Root mean square error of eroded volume as a function of transport coefficient
K for the non-linear model (n=3) simulation of Case 300.
final profile. The values of these slopes are shown in Table 4.2. The calibration process for
the non-linear transport model simulation of Case 300 is shown in Figure 4.3. Based on the
results, a best-fit K value of 8.55x10-10 is adopted which gives the overall minimum error for
the eroded volume. The best-fit K values and the corresponding errors of eroded volume for
the three cases are presented in Table 4.2. The best-fit K values vary from 5.77x10'1 to
8.55x101-, a factor of 1.48 for n=3. The corresponding factor for the best-fit K values for
n=l is 2.36. Comparisons of predicted and measured eroded volumes are shown in Figure 4.4
4U i I i I I I I
01 -Case 300 ..--
E o
S20 0o -
20 0 measured
(D
- 10 -- predicted by n=3
S- - .. predicted by n=1
0 5 10 15 20 25 30 35 40 45 50
time [hr]
An.
0)
E
:20
-ol
10
0-
a)
5 10 15 20
time [hr]
25 30 35 40
10 20 30 40 50 60 70 80 90 100
time [hr]
Figure 4.4 Comparisons of prediction with best-fit K value to observed eroded volumes
for Cases 300, 400 and 500 in Saville's experiments.
Case 400 ,.---
0 measured
- predicted by n=3
- - predicted by n=1
E
af20
E
",010
a)
-0
0o
a)
1r V4 i I -- I
I i
86
for the predictions of both the linear (n=l) and the non-linear (n=3) sediment transport
relationships. It appears that the non-linear transport relationship with n=3 provides a better
fit for Case 300 and Case 400, while the linear transport relationship, n=l, predicts better
results for Case 500. As discussed before, Case 500 had a very significant offshore bar
(Figure 4.1) formed during beach erosion. Waves first broke at the offshore bar, then
reshaped in the offshore trough area and broke again in the nearshore region. It is not
appropriate to calculate wave energy dissipation per unit volume according to the offshore
incident wave height. The vortex caused by the first wave breaking also played an important
role in the sediment transport in the area shoreward of the bar.
The beach profile evolutions predicted by the non-linear (n=3) and the linear (n=l)
transport relationship are compared in Figures 4.5, 4.6 and 4.7 for Cases 300, 400, and 500,
respectively. In each case, the profiles at four different times are compared. Among three
cases, the proposed non-linear relationship presents overall better agreement for the profile
evolution in Case 300 and Case 400. Owing to the significant offshore bar in Case 500, the
numerical models which predict a smooth monotonic profile form have some difficulties in
representing the profile around the offshore trough and bar area.
4.3 German "Large Wave Flume"
4.3.1 General
The German "large wave flume" is 324 meters long, 7 meters deep and 5 meters wide.
Three series of experiments were carried out in the flume (Dette and Uliczka 1987, Dette and
Rahlf 1992, and Dette et al. 1992). Two of these experiments had the same constant wave
conditions and different initial profiles. Regular waves with a wave height of 1.5 meters and
E
0
a ,
aM -4
-6
2
C0
(1)
Z -4
-6
-20
-20
20 40
offshore distance [m]
Figure 4.5 Case 300 from Saville's experiments. Comparisons of predicted to observed
profiles at different times for the best-fit K values for each transport relationships.
profiles at 5 hours
observed '
......... predicted by n=1
predicted by n=3
-20 0 20 40 60 8C
profiles at 10 hours
-20 0 20 40 60 8C
profiles at 30 hours
-20 0 20 40 60 8C
.profiles at 50 hours
| N.
2
0
S-2
z -4
_C;
2 profiles at 5 hours
0 -
a -2 -. initial profile. .
) -- observed
5 -4 ... .... predicted by n=1
- predicted by n=3
--6 I I I I
-20 0 20 40 60 80
2- profiles at 10 hours
0 -2-
S-4
-6 I I I I
-20 0 20 40 60 80
2- profiles at 30 hours
S. 0
-0 --
U-4
-6 II
-20 0 20 40 60 80
2 profiles at 40 hours
) -4
-6 I
-20 0 20 40 60 80
offshore distance [m]
Figure 4.6 Case 400 from Saville's experiments. Comparisons of predicted to observed
profiles at different times for the best-fit K values for each transport relationships.
2 profiles at 10 hours
-2
0
-2 -- initial profile
observed
-4 ......... predicted by n=1
- predicted by n=3
-6 I I I
-20 0 20 40 60 8C
2 profiles at 30 hours
0 -
-2 -
-4
-6 i I I I
-20 0 20 40 60 8C
2- profiles at 60 hours
0 -
-2
-4
-6 I
S
-2-
-4- -
_fi
-20
-20
20 40
offshore distance [m]
Figure 4.7 Case 500 from Saville's experiments. Comparisons of predicted to observed
profiles at different times for the best-fit K values for each transport relationships.
E
0
*6--
o
a)
profiles at 100 hours
A
90
a period of 6 seconds were generated in a water depth of 5 meters. The sand used for both
experiments had a mean diameter of 0.33 mm, which corresponding to a fall velocity of 5
cm/s at a temperature of 20 C and a profile scale parameter, A, of 0.131 mi"3. Two initial
profiles, termed "dune without foreshore" and "dune with foreshore", were used. The "dune
without foreshore" had a dune crest of 2 meters above still water level and a seaward slope
of 1:4 down to the channel floor. The "dune with foreshore" had a slope of 1:4 from the dune
crest of 2 meters above still water level to 1 meter water depth followed by a slope of 1:20
down to the channel floor. The third experiment, a "dune without foreshore" was conducted
with the same sediment as the other two but irregular waves. A JONSWAP-Spectrum was
applied to generate random waves with a significant wave height of 1.5 meters and peak
spectral wave period of 6 seconds in a water depth of 5 meters.
4.3.2 Experiments With Constant Waves
Two tests with the same constant wave conditions and different initial profiles were
carried out in the German "Large Wave Flume" in 1986. They are denoted by their initial
profiles as the "dune without foreshore" and the "dune with foreshore". During the
experiments, a test was stopped generally whenever the wave height in the horizontal bottom
portion (water depth of 5 meters) of the flume exceeded 20% of the originally generated
wave height so that the breaking point could stay in its original position. After motions in the
tank subsided, the tests were resumed.
Based on McCowan (1894) theory, waves break when the ratio of water depth to
wave height becomes 1.28. The active water depth, which is the water depth at the breaking
point, is determined as 1.92 meters for the numerical model according to the incident wave
91
Table 4.3 The numerical model inputs and best-fit results for the two constant wave cases
in German "large wave flume'.
Slope Best-fit K Error* [m2]
Case n=3 n=1
dune shoreline offshore [s/N3] [m4/N] n=3 n=l
Dune without 3.00 0.20 0.20 7.64x10'0 2.03x105- 0.74 8.08
foreshore
Dune with 3.00 0.20 0.162 1.03x109 8.13x10-6 0.46 1.11
foreshore
Error represents the root mean square between predicted and observed eroded volume
as shown in Eq. (4.1).
height of 1.5 meters. The input of the dune, shoreline and offshore slopes in the models are
adopted based on the finial profiles in the two tests and are presented in Table 4.3. The best-
fit K values and the corresponding errors of eroded volume resulting from the calibrations are
also shown in Table 4.3. For the two cases, the non-linear transport relationship (n=3)
produces much less error for the eroded volume than the linear transport relationship (n=l),
especially in the case "dune without foreshore". Of relevance is that the best-fit K values for
the non-linear transport equation (n=3) in the calibrations of the two cases here are close to
those in Saville's experiments with a difference of less than a factor of 2. However the
difference in the best-fit K value for the linear transport equation (n=l) is about a factor of
10 in Saville's experiments and German tests. The predicted eroded volumes as a function of
time are compared with measurements in Figure 4.8 for the two cases. The predictions from
the non-linear transport model (n=3) agree very well with the measured eroded volume over
the whole test time for both cases with the same wave and sediment conditions but different
initial profiles. This result supports the discussion in Chapter 3 that the non-linear model has
60 --
Dune without foreshore ..-
a 40
E
0
20 0, measured
--- predicted by n=3
/ -. predicted by n=1
0 1 2 3 4 5 6 7
time [hr]
15 -
S Dune with foreshore o
a 10 -
E
5 0 measured
S- predicted by n=3
2 - -. predicted by n=1
0 I I I I I
0 1 2 3 4 5 6 7
time [hr]
Figure 4.8 Comparisons of predictions with best-fit K values to observed eroded volumes
for the two tests with constant waves in German "large wave flume".
capabilities to handle different initial conditions, whereas the linear transport relationship
(n=l) cannot represent well the difference in time scale of profile evolution caused by the
different initial conditions. Figure 4.8 clearly shows that the linear transport relationship has
a difficulty in simulating the rapid response in the case of "dune without foreshore", although
it could provide an acceptable predictions for the case "dune with foreshore".
Comparisons of the profiles predicted by the non-linear and the linear transport
models and the measurements are presented in Figures 4.9 and 4.10 for the two cases,
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