UFL/COEL95/007
MOVABLEBED MODELLING LAW OF BEACH
PROFILE RESPONSE
by
Xu Wang
Thesis
1995
MOVABLEBED MODELLING LAW OF BEACH PROFILE RESPONSE
By
Xu Wang
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
1995
ACKNOWLEGEMENTS
I would like to express my sincere appreciation and gratitude to Dr. Hsiang
Wang, my advisor, who has given me extremely helpful guidance and his insight into
the problems made this research study presented in this thesis possible.
I also wish to thank my committee members, Dr. R.G.Dean and Dr. Michel K.
Ochi, for their guidance, suggestions during the course of the study and for the review
of the manuscript at the end.
Special thanks go to Dr. LiHwa Lin who gave me helpful advice and suggestions.
And the thanks go to Jim Joiner and Sidney for the help received during the labora
tory work.
TABLE OF CONTENTS
ACKNOWLEDGEMENT .
LIST OF FIGURES ................................
LIST OF TABLES .................................
ABSTRACT .. ....... .. .. ... .. ......... .... .... .
CHAPTERS
1 INTRODUCTION ...............................
2 LITERATURE REVIEW ...........................
3 APPROACH TO PHYSICAL MOVABLEBED MODELING LAW . .
3.1 Principles of Similarity ..........................
3.1.1 Dynamic Similarity ................. .......
3.1.2 Similarity By Dimensional Analysis ..............
3.1.3 Similarity By nondimensionalizing the governing Equations
3.2 Physical Modeling Laws Of Beach Response .............
4 LABORATORY EXPERIMENTS .................
4.1 Test Facilities .... .... ... ........... .. ..
4.1.1 AirSea Wave Tank ...................
4.1.2 Wave Flume Facility .................
4.1.3 Wave Basin Facility ..................
4.2 Initial Beach Profile Design .................
4.2.1 Experimental Procedures ...............
4.2.2 Test Conditions .....................
4.3 Test Results and Data Analysis ...............
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
Profile Classifications of the Test Results .....
Volumetric Changes along the Profiles ......
Beach Erosion ........... ......
Profile Evolution and Bar Migration .. .....
Eqilibrium Beach Profile .............
5 EVALUATION OF SCALING LAWS ..........
5.1 Methodology and Evaluation Criteria ........
5.1.1 Geometrical Scaling and Equilibrium Profile
5.1.2 Wave Height and Wave Period Scaling ..
5.1.3 Evaluation of Morphological Time Scale .
. ii
xi
16
17
17
20
21
21
1 111
5.1.4 Summary of TwoDimensional Test Results . . . ... 92
5.1.5 Test Results from ThreeDimensional Basin . . . ... 92
5.2 Test Results from Undistorted Model . . . . . . . ... 103
6 SUMMARY AND CONCLUSION ...................... 111
APPENDICES
A BEACH PROFILE EVOLUTION AND SEDIMENT TRANSPORT RATE 116
BIBLIOGRAPHY .................... ............. 142
BIOGRAPHICAL SKETCH .......................... 145
LIST OF FIGURES
4.1 Initial Beach Profile In Prototype . . . . . . . . 32
4.2 Cumulative Sand Size Distribution . . . . . . . . 33
4.3 Fall Velocity Of Spherical Grains As A Function Of Size, Rouse
(1937) . . . . . . . . . . . . ........ . . 34
4.4 Designed Initial Beach Profile in Model with D50 = 0.20mm . 36
4.5 Example of Beach Profile with Diffused Bar, Dso = 0.09mm . 45
4.6 Definition Of Net Sediment Transport Rate Across The Beach Profile 47
4.7 Dune Erosion Evolution With Elapsed Time in ASW(1) . . 49
4.8 Dune Erosion Evolution With Elapsed Time in ASW(2) . . 50
4.9 Dune Erosion Evolution With Elapsed Time in WF ...... ..51
4.10 Dune Erosion Evolution With Elapsed Time in WB ...... ..52
4.11 Definition of Longshore Bar With Respect to Initial Profile . 53
4.12 Horizontal Movement Of Bar Crest With Horizontal Scale Equal
to 20 In ASW ............................ 56
4.13 Horizontal Movement Of Bar Crest With Horizontal Scale Equal
to 20 In W F ................... .......... 57
4.14 Horizontal Movement Of Bar Mass Center With Horizontal Scale
Equal to 20 In ASW ...................... .. .. 58
4.15 Horizontal Movement Of Bar Mass Center With Horizontal Scale
Equal to 20 In W F .......................... 59
4.16 Growth of Bar Volume With Horizontal Scale Equal to 20 In ASW 60
4.17 Growth of Bar Volume With Horizontal Scale Equal to 20 In WF 61
4.18 Horizontal Movement Of Bar Crest With Horizontal Scale Equal
to 30. ............................... ... . .. 62
4.19 Horizontal Movement Of Bar Mass Center With Horizontal Scale
Equal to 30. ............................ .. 62
4.20 Growth Of Bar Volume With Horizontal Scale Equal to 30. . 63
4.21 Horizontal Movement Of Bar Crest With Horizontal Scale Equal
to 40. ..................... .......... .. 63
4.22 Horizontal Movement Of Bar Mass Center With Horizontal Scale
Equal to 40. ................. ......... 64
4.23 Growth Of Bar Volume With Horizontal Scale Equal to 40. . 64
4.24 Horizontal Movement Of Bar Crest With D50 = 0.09mm ..... ..65
4.25 Horizontal Movement Of Bar Mass Center With D0o = 0.09mm 65
4.26 Growth Of Bar Volume With D50 = 0.09mm . . . . ... 66
4.27 Beach Profile Evolution And Sediment Transport Rate . . 67
5.1 Definition of The Dune and Bar Regions . . . . . ... 69
5.2 Equilibrium Profile Comparison Between Model Test Results and
M odeling Laws ............................ 72
5.3 ASW Model Tests and Prototype Final Profile Comparison (1) 75
5.4 ASW Model Tests and Prototype Final Profile Comparison (2) 76
5.5 ASW Model Tests and Prototype Final Profile Comparison (3) 77
5.6 WF Model Tests and Prototype Final Profile Comparison . .. 78
5.7 Dune Volume Erosion Error Criterion . . . . . ..... 79
5.8 The RMS Error of The Bar Profile Criterion . . . .... 80
5.9 Bar Volume Error Criterion ..................... 81
5.10 Bar Location Error Criterion . . . . . . ...... . 82
5.11 Summary of ASW Test Results Based On Dune Profile Parameter 85
5.12 Summary of ASW Test Results Based On Bar Profile Parameter 86
5.13 Summary of WF Test Results Based On Dune Profile Parameter 88
5.14 Summary of WF Test Results Based On Bar Profile Parameter .89
5.15 Dune Profile Evolution Scaling from ASW results . . ... 90
5.16 Dune Profile Evolution Scaling From WF results . . . ... 91
5.17 Morphological time scaling of Dune Profile RMS Value from ASW
Tests . . .. . . . . . . . . . . .. . . 93
5.18 Morphological time scaling of Dune Profile RMS Value from WF
Tests . . . . . . . . . . . . . . ..... 94
5.19 Morphological time scaling of Bar Crest Location from ASW Tests 95
5.20 Morphological time scaling of Bar Mass Center Location from
ASW Tests ................... ......... 96
5.21 Morpholigical Time Scaling of Bar Profile RMS Value from ASW
Tests .................... ............ 97
5.22 Morphological time scaling of Bar Volume from ASW Tests . 98
5.23 Morpholigical Time Scalings from WF Tests . . . . ... 99
5.24 Initial Profiles Comparison Between 5 Sections for Case 3 . .. 101
5.25 Final Profile Comparison Between 5 Sections for Case 3 . . 102
5.26 Comparison of Wave Basin Model Test No.3 with Prototype . 104
5.27 Morphological Time Scaling from Wave Basin Test No.1 against
Prototype .............................. 105
5.28 Morphological Time Scaling from Wave Basin Test No.2 against
Prototype ... .. . .. . .. .. . . . .. .. 106
5.29 Morphological Time Scaling from Wave Basin Test No.3 against
Prototype .............................. 107
5.30 Dune Erosion Volume Comparison With Prototype . . ... 108
5.31 Dune Profile RMS Comparison With Prototype . . . ... 109
5.32 Summary of The Undistorted Model Test Results (shore profile) 110
5.33 Summary of The Undistorted Model Test Results (bar profile) .110
6.1 Wave Height Scale as A function of k Value . . . . ... 114
A.1 Beach Profile Evolution and Sediment Transport Pattern of Test
No.1 .................................. 117
A.2 Beach Profile Evolution and Sediment Transport Pattern of Test
No.2 ................................... 118
A.3
A.4
A.5
A.6
A.7
A.8
A.9
A.10 Beach Profile Evolution and Sediment Transport Pattern of Test
M 1in
Beach Profile
No.ll ...
Beach Profile
No.12 ....
Beach Profile
No.13 ....
Beach Profile
No.14 ....
Beach Profile
No.16 ....
Beach Profile
No.17 ....
Beach Profile
No.18 ....
Beach Profile
No.19 ....
Beach Profile
No.20 ....
Beach Profile
No.21 ....
Evo
Evo
Evo
Evo
Evo
Evo
Evo
Evo
Evo
Evo
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
.............lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
lution and Sediment Transport Pattern of Test
Beach Profile Evolution and Sediment Transport Pattern of Test
N o.3 . . . . . . . . . . . . . . . . ..
Beach Profile Evolution and Sediment Transport Pattern of Test
N o.4 . . . . . . . . . . . . . . . . ..
Beach Profile Evolution and Sediment Transport Pattern of Test
N o.5 . . . . . . . . . . . . . . . . ..
Beach Profile Evolution and Sediment Transport Pattern of Test
N o.6 . . . . . . . . . . . . . . . . ..
Beach Profile Evolution and Sediment Transport Pattern of Test
N o.7 . . . . . . . . . . . . . . . . ..
Beach Profile Evolution and Sediment Transport Pattern of Test
N o.8 . . . . . . . . . . . . . . . . ..
Beach Profile Evolution and Sediment Transport Pattern of Test
N o.9 . . . . . . . . . . . . . . . . ..
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
A.11
A.12
A.13
A.14
A.15
A.16
A.17
A.18
A.19
A.20
oL .i.V . .
A.21
A.22
A.23
A.24
A.25
Beach
No.22
Beach
No.23
Beach
No.24
Beach
No.25
Beach
No.26
Profile Evolution
Profile Evolution
Profile Evolution
Profile Evolution
Profile Evolution
Profile Evolution
Profile Evolution
. . . . . .
and Sediment
and Sediment
and Sediment
and Sediment
and Sediment
. . . ..
Transport
Transport
Transport
Transport
Transport
. . o . .
Pattern of Test
Pattern of Test
Pattern of Test
Pattern of Test
Pattern of Test
. . . ..
LIST OF TABLES
3.1 Summary of Fall Speed Distorted Model Laws ....... 22
3.2 Summary of Wave Breaking Index (,yb) .......... 27
4.1 Scale Ratios and Physical Dimensions in ASW Tests .. 35
4.2 Summary of Criteria Governing Beach Classification . 39
4.3 Summary of Test Conditions in ASW ............ 40
4.4 Summary of Beach Profile Classification Parameter for
ASW .................... ............ 41
4.5 Summary of Test Conditions in the WF ......... 42
4.6 Beach Profile Classification Parameter for WF ...... 43
4.7 Test Conditions in ThreeDimensional Wave Basin . .. 43
5.1 Four Fall Speed Distorted Model Laws ........... 70
5.2 Comparison of Model Performances ............. 83
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
MOVABLEBED MODELLING LAW OF BEACH PROFILE RESPONSE
By
Xu Wang
May 1995
Chairman: Dr. Hsiang Wang
Major Department: Coastal and Oceanographic Engineering
Beach profiles in nature occur in a variety of forms and proportions. Storm condi
tions generally cause beach profiles to evolve to milder slopes, with resulting shoreline
recession and a portion of the eroded material usually stored in an offshore bar. Beach
profile response to storm is essential for defining erosion hazard zones associated with
particular storm characteristics. Lack of full understanding of sediment transport
mechanics, the modeling of coastal areas using mobile bed models, has in the past
been an art rather than a science. A number of authors have attempted to derive
formal scaling laws resulting in many varying modeling formulae to choose from.
This study summarizes the insights gained during a yearlong study of the scaling
of mobile bed coastal models. The historical background is reviewed and a modified
modeling law based on Wang's comprehensive work (1990) has been derived with the
assumption that the suspended sediment transport is the dominant mode under storm
wave condition and the breaking index can be preserved. This, in essence, adds to the
constraint that wave height be properly modeled in accordance to the preservation
breaking index instead of as simple vertical geometrical scale. It is further proposed
that the breaking index scale is a function of beach slope such that N, = (N6/Nx)k N.
Four existing scaling laws including the proposed one were tested in the laboratory
xi
using 2D tanks and threedimensional basin. German large wave tank experiment
results were taken as the prototype data. The beach profile was divided into two
regions, the dune region and the barprofile region. Separate criteria were developed
to evaluate the modeling laws.
It is concluded that the modeling law presented in this study has the better overall
evaluation against other existing modeling laws; however, further research is needed
for the verification purpose of beach dune erosion and bar profile evolution modeling
law and the morphological time scaling. Again, one may state that mobile bed models
are workable, but because of the large number of substantial discrepancies between
model and prototype, physical models will remain as works of art rather than precise
scientific instruments.
CHAPTER 1
INTRODUCTION
Beach and dune erosion as well as the related beach profile changes that occur
under storm waves and high water levels are of basic interest in coastal engineering.
Numerous attempts using various techniques have been made to better understand
and predict beach and dune responses under storm wave attack. Numerical simulation
and laboratory experiments are two of the most commonly employed techniques.
In general, numerical models no doubt have the advantage for their quick and
neat answers and usually cost much less than physical models. However, the process
of beach and dune erosion is difficult to formulate owing to the inherently complicated
nature of sediment fluid interaction and the highly nonlinear unsteady and nonuniform
flow condition inside the surf zone. The time varying and irregular bottom changes
further complicate the problem. Therefore, current workable numerical models can
be considered as the first generation as all of them deal with simple macroscale gross
effects such as the total erosional volume and the final shape of the profile. Even
in these terms, the predictive capability already exceeds our fundamental knowledge
employed to build these models. For instance, the swash transport mechanism, the
beach slope effect and the bar formation are already built in some of the models,
mainly based on conjectures with no credible fundamental knowledge. Further model
improvement requires better formulas based on improved fundamental understanding.
Physical models, on the other hand, can be used to reproduce the natural con
dition without a priori knowledge on the basic mechanisms. They also offer the
opportunity of improving our understanding and provide useful data for numerical
models. One of the major difficulties in physical modeling is the problem of the scale
2
effects, particularly when sediment transport is involved. Physical models which in
tend to simulate sediment transport and the associated morphological changes are
also known as movablebed models. The subject of model scaling is a difficult one,
as mentioned. Numerous papers have been written proposing various similitude re
lationships. At present there is no general solution that is also practical. Specific
modeling laws are usually only applicable to certain restricted conditions.
The beach response physical modeling has also been studied by a number of in
vestigators. Currently, all the proposed modeling laws for beach response are for
twodimensional application and most of them are based on the main assumption,
explicitly or implicitly, that within the surf zone suspended sediment is the dominant
mode of transport. These modeling laws can further be classified as distorted and
undistorted depending upon whether the horizontal and vertical geometrical scales
have the same ratio. The general opinion is that the undistorted model is preferred
over distorted model. However, distorted models are more flexible in accommodating
wider ranges of variables, both in physical dimensions and other physical parameters,
such as wave heights, wave lengths and sand sizes. A common weakness of the exist
ing profile modeling laws, whether distorted or undistorted, appears to be the lack of
sufficient information to adequately verify the morphological time scale, i.e., the mod
eling of beach evolution process as a function of time. This is an important class of
problem, particularly if one is to extend the physical modeling to threedimensional
applications such as studying the morphological processes of shoreline changes or
nearshore shoaling phenomena.
This study is aimed at evaluating and improving the scaling laws for distorted
models guided by the modeling theory and through a series of laboratory experiments
carried out at different physical scales. The study is restricted to twodimensional ap
plication although experiments were also carried out in a threedimensional in addition
to the traditional wave tank experiments.
3
Chapter 2 provides a general review on the development of physical modeling laws
concerning coastal sediment transport and that are specifically related to beach profile
response modeling. Chapter 3 reviews the basic similitude theorem and summarizes
existing beach profile modeling laws and their origins. A modified modeling law
based on the work of Wang et al. (1990) was also derived. These modeling laws are
evaluated by the results obtained in the subsequent laboratory model tests. Chapter
4 describes the various types of experiments being carried out both in 2D wave tanks
and in the 3D wave basin. Chapter 5 is the heart of this thesis as it deals with the
laboratory data analysis and the evaluation of various modeling laws based on the
laboratory data. Finally, summary and conclusion of the study is given in Chapter 6.
CHAPTER 2
LITERATURE REVIEW
Physical modeling is essentially a means of replacing the analytical integration of
the differential equations governing a physical process including the often complicated
initial and boundary conditions. It is, thereby, considered an important and useful
engineering tool. Since physical models are usually conducted at reduced scales,
the relationship between the model and the prototype must be clearly established
for model design and for extrapolating laboratory results to prototype scales. The
literature on the subject of scaling law is vast. This chapter is intended to provide a
brief account on the development of movable bed scaling laws for coastal engineering
applications, especially those relevant to beach profile modeling.
To the present knowledge, the first known scalemodel experiments were con
ducted by an English engineer (John Smeaton) during the period 175253, to deter
mine the performance of water wheels and windmills (Hudson et al., 1979); a French
professor (Ferdinand Reech) in 1852, was the first to express what is now known as
the Froude criterion of similitude (Rouse and Ince, 1957). The earliest movable bed
model was conducted in 1875 by a Frenchman (Louis J. Fargue) and in 1885 by Os
borne Reynolds in England. Natural sand was used as bed material and time scale
was taken into consideration.
L.P. VernonHarcourt, who continued the work of Reynolds, investigated scale
models using bed materials of sand as well as lighter weight sediments such as charcoal
and pumice. And the principle of movable bed model verification, which means that
the model is thought to be verified if it can reproduce all the features of bottom
evolution observed in the past, has been proven to be the basic guideline and widely
5
applied ever since, though there has been no rigorous proof of its sufficiency.
The extremely small scale (horizontal length scale up to 40,000) models with large
distortions (up to 100) were used in the early British model studies, which later on
tended to use larger scales with smaller distortions so as to reduce the scale effect for
more reliable results. Also after the work of Reynolds, movable bed model technology
was gradually employed to solving practical engineering problems.
After World War II, experience accumulated over the years through the extensive
movable bed scale model studies made by various laboratories all over the world,
especially in Europe, the United States, Japan, has advanced the state of the art.
And, movablemodel studies of coastal engineering problems were routinely conducted
in connection with almost all coastal and other hydraulic engineering projects.
Meyer(1936) first conducted the laboratory model of beach profile evolution to
investigate scaling effects in movable bed experiments, and derived an empirical rela
tionship between beach slope and wave steepness.
Water(1939) worked on the characteristic response of the beach profile to wave
action and classified profiles as ordinary or storm type and led to the conclusion that
wave steepness can be used to determine the type of beach profile that developed
under certain wave conditions. The process of sediment sorting along the profile was
demonstrated in the experiments in which the coarser material remained near the
plunge point and finer material moved offshore.
Bagnold (1940) used a wave tank to model the interaction of the waves and
beach and presented a thorough discussion of the basic physics of the phenomenon
as well as a detailed discussion on the application of using a hydraulic model study
for the analysis of the process. Rather coarse sand (0.57.0 mm) and waves varied
in amplitude between 30 centimeters and 5 centimeters were used to conduct the
smallscale laboratory experiments and it was found that the foreshore slope was
independent of the wave height and mainly a function of grain size.
6
By reviewing the literature, one may conclude that the earlier works in 1950s and
1960's were mostly related to beach profile classification and that this topic is well
studied and basically understood.
Up to present time, numerous work of the physical processes involved in beach
erosion by wave action (Keulegan, 1945, 1948; Bruun, 1954; Saville, 1957; Eagle
son and Dean, 1961;; Collins, 1963; Vanoni, 1964; Eagleson, 1957, 1959, 1965 and
Galvin,1967; Sumnamura and Horikawa, 1975; Wang, Dalrymple and Shiau, 1975;
Dean, 1976, 1977; Inman and Bailard, 1982; Vellinga, 1982, 1986; Kriebel, Dally, and
Dean, 1987; etc) would lead us to some conclusions.
Wave breaking generates turbulent motion and provides the necessary mecha
nism for suspending and keeping sediment in suspension, thus mobilizing the grains
for transport by mean currents. Although profile change is highly stochastic on a
microscale involving turbulence, movement of individual and collective grains, and
various types of organized flows, if viewed on a macroscale, changes in the profile are
surprisingly regular and consistent with respect to large features such as bars and
berms. And the existence of an equilibrium profile was proven to be a valid concept
under laboratory conditions.
For the laboratory scale modeling law for movablebed models, a number of simil
itude relations have been developed, each having its own practical assumptions and
constraints, and some of them are completely empirical and some completely mathe
matical, while power laws are the most common type of function employed. But the
exact dynamic similitude of the dominant physical processes in two regimes (existing
a transition from one basic regime of boundary flow to another as sediment motion
outside the surf zone is compared to sediment motion in the surf zone) simultane
ously i.e., using the same model fluid and the same model laws for reproducing the
waves, currents, and bottom material for both regimes) is not possible. So far, the
attempts to establish the correct scaling law are not really successful due to the lack
7
of an understanding of the basic mechanism of coastal sediment transport limited
choices of material that can be used in the laboratory resulting in the limited sediment
sizes, specific weights, viscosity, and the lack of quality prototype data for verification
purposes, etc.
Goddet and Jaffry (1960) derived the basic relations between horizontal scale,
vertical scale, sediment diameter scale and relative specific weight scale based on the
sediment motion due to combined action of wind waves and unidirectional currents.
ND = N 7/2ons/5 (2.1)
N. = N3/2023/5 (2.2)
n = Ns/N, (2.3)
where No is the scale of the sediment size, N6 is the geometrical vertical scale, and fS
is the distortion, the ratio of the geometrical horizontal scale (N\) and vertical scale.
The scale relationships proposed by Goddet and Jaffry have not achieved popular
use, and were only proposed as a first step towards a reasonable solution.
Yalin (1963) examined model scale selection for sediment transport involving wind
waves and tidal currents. The phenomenon was considered to be a function of seven
characteristic parameters
p, ,78, D, T, b, Ub (2.4)
where ub and Ub are typical wave and tide horizontal velocities, at the bottom (im
mediately above an oscillatory boundary layer) and T is the wave period. These
parameters were combined to give a dimensionless expression for twophase motion
in the vicinity of the bed. By assuming that the wave period T does not influence
the phenomenon of net sediment transport, and preserving the dimensionless change
in bed level and by application of small amplitude wave theory, the relationships as:
ND = 3/4/2
(2.5)
8
N, N = 1 (2.6)
were obtained from the dimensional analysis.
Fan and Le Mehaute (1969) preserved the characteristics of sediment transport,
i.e., densimetric Froude Number Fr., and Reynolds Number for initiation of sediment
motion Re. and used equilibrium beach profile concept.
N.,N = 1 (2.7)
N,, = N'JN3/2 (2.8)
or
ND = N N1/ 1
The authors reviewed most aspects of coastal mobile bed model technology and con
cluded the following:
a. Similarity of bottom evolution is a basic requirement for a mobile bed model,
which is only possible for a flow regime where the boundary layer is turbulent.
b. A coastal mobile bed model must be geometrically distorted. Geometric
distortion should be defined by the ratio of the equilibrium beach slopes as measured
in model and prototype.
c. Similarity of wave action is only required to the extent that
NH = NL = N6 (2.10)
where L is the short wave length. This prohibits wave height distortion.
Noda (1971) listed seven similitude requirements for profile modeling under equi
librium conditions based on the results of a number of previous investigations, in
cluding Fan and Le Mehaute (1969) and Yalin (1963), etc. So far, this is the most
comprehensive one in beach profile scale modeling. These eight criteria are as follows:
a. Coastal fluid phenomena are generally dominated by gravitational and inertial
forces and hence the Froude number, i.e., u/(gd)1/2 is an important parameter. The
requirement that NF, = 1 yields
N, = /N6 (2.11)
where N, is the velocity scale.
b. The preservation of the densimetric Froude number, u/(gy'D)1/2 based on the
grain size and bed shear velocity gives
NyND = N (2.12)
where u. is the bed shear velocity, and 7' is the relative specific weight of the material
= (7' Yf)/I (2.13)
here 7, is the specific weight of the sediment, and 7y is the specific weight of the fluid.
c. The requirement of identical grain diameter Reynolds number, i.e., u.D/v
produces
NuND = 1 (2.14)
d. Bed shear velocity for steady conditions in turbulent flow is proportional to
free stream velocity:
u. oc fl/2u (2.15)
f. The scale ratio for the friction factor : 7 related to unidirectional uniform flow.
C = (!)1/2 (2.16)
N = (2.17)
7 N\
and
N = N (2.18)
g. Kinematic Similarity:
u z
(2.19)
w y
h. Fall velocity in stokes range:
wo (2.20)
18v
These are all supposed to be pertinent to beach processes, and there are certainly
a many of combinations and Noda derived several of the possible scalemodel laws
from various combinations of seven similitude conditions but failed to sort out the
proper ones only based on a limited experiment data.
Based on the assumed equilibrium conditions and limited experiments data sets,
Noda (1972) derived a completely empirical modeling law.
NDN;84 = N.55 (2.21)
N = N32N0386 (2.22)
Two of the four basic parameters ND, N,, N6 can be chosen freely and the
other two are automatically constrained. Also, the limiting conditions that if Ny =
ND = 1, then, NA = N6 = 1 as found experimentally are also satisfied.
This set of modeling laws has been derived from twodimensional laboratory beach
profile data and thus the extrapolation to threedimensional models is still in question.
The wavelength is scaled according to the vertical scale to preserve the refraction
pattern, but the number of waves in the model will be smaller in proportion to
the distortion when scaled to prototype. Diffraction and reflection processes are
not preserved. Also, mass transport, sediment concentration, and material porosity
phenomena are not scaled. The region of interest was the beach profile in the breaker
zone where the boundary conditions belong to turbulent flow regime. No wind effects
have been accounted for.
11
Dean (1973) reported that the most promising parameter used for the predic
tion of equilibrium beach slopes and the onshore or offshore sediment transport is
the dimensionless fall velocity (H/WT), as this parameter tells whether a sediment
particle thrown into suspension by the passage of a wave will settle to the bed during
the time that the water particle motion is shoreward or seaward, resulting in onshore
or offshore movement of the particle. This parameter actually is the fraction of the
drag force in vertical (gravity) and the drag force horizontal (orbital motion of the
wave), or the time taken for a sediment particle to fall a distance equal to the wave
height.
Dean (1985) reviewed previous movablebed modeling criteria and considered the
dominant physical mechanisms involved in surf zone sediment transport. He argued
that the Shield's criterion is not necessary in the surf zone as turbulence, not bed
shear, is the dominant cause of sediment mobilization. He argued again that sediment
fall path, i.e., the fall speed parameter between the prototype and undistorted model
should be preserved. Hydrodynamics scaled according to Froude similarity and the
undistorted model are large enough to preclude significant viscous, surface tension,
and cohesive sediment effects so that the character of the wave breaking is properly
simulated. The scale relationships were proposed as
NF, = 1 (2.23)
NT = Nt = (Ns)1/2 (2.24)
H = (H) (2.25)
WT 'WT
N6 = (Nw)2/3(NA)2/3
(2.26)
NA = Ns (2.27)
where Fr is the Froude number, W is the particle fall velocity, T the wave period.
Dean's empirical relation was established through synthesizing a large number of field
profiles.
Kamphuis (1974) listed four different dimensionless parameters as requirements
for complete similarity. He proposed a set of four different modeling laws preserving
one or more nondimensional parameters but not all of them and suitable for a specific
range of environmental conditions without comparison with laboratory data. And he
also did not specify the mode of transport assuming that the sediment transport under
two dimensional wave action is related to the four dimensionless parameters and the
asymmetry of the wave motion. These criteria for dynamic similarity in modeling
movable bed can be summarized as
N,, ND
NR,= 1 (2.28)
N,N
NF. 1 (2.29)
NyND
N = 1 (2.30)
N.
N = 1 (2.31)
ND
where D is the particle size, p is the fluid density, p, is the particle density and 7, is
the underwater specific weight, i.e., (p, p)g, and as is the wave orbital amplitude
at the bottom.
The scale effects were also discussed based on the four dimensionless parameters:
a. The Reynolds number scale effect will only be felt at low flows and flow reversals
and expected to be small in the short wave models.
13
b. Incorrect scaling of the densimetric Froude number (Shields parameter) will
result in incorrect sediment transport and an undeterminate time scale.
c. The scale effect resulting from the violation of the Np,/p = 1 is not very serious
if mass movement rather than dynamics of individual particles is the interest.
d. The geometric link criterion is normally violated which means that sediment
transport is not only incorrect but varies throughout the model with depth.
Vellinga (1978) and Graaff (1977) derived a model law for dune erosion by em
pirical correlations of tests done at different scale dimensions in the Delft Hydraulic
Laboratory in the Netherlands when compared to a single prototype condition. The
results of model tests on dune erosion with very fine sand support the validity of the
dimensionless fall velocity parameter H/TW for small scale modeling of beach pro
cesses. If this parameter cannot be satisfied in the model, a profile distortion based
on kinematical similarity NA/Ns = N,/Nw = (N6/Ny)"a with a = 0.5, gives good
results for the finer sands. For coarser sands values of a ranging from 0.5 to 0.3 were
found. The morphological time scale equal to the hydrodynamical time scale and
equals to square root of geometrical vertical scale. But Hughes's attempts (1983) to
verify the Delft modeling laws using the Hurricane Eloise data proved unsuccessful.
Graaff (1977) and Vellinga (1982) conducted a comprehensive laboratory study
by using different scales in attempting to duplicate the beach and dune erosion of the
Dutch's coast. The law requires that the wave steepness (Ho/Lo) and Froude number
be preserved and takes the morphological time scale into consideration.
Saville (1980) conducted a series of tests that the fall velocity of sediment was
scaled correctly and compared the results of these smallscale tests with prototype
model tests, finding that profile similarity was best in the surf zone and on the beach
face, where setting velocity might be expected to be a major parameter affecting the
modeling.
Hughes (1983) presented a midscale modeling law based on consideration of the
14
inertial forces, represented by the turbulent shear stress, and the gravity force in the
nearly horizontal direction of the principal flow, which results in a dynamic scaling
relationship for a distorted model. A great number of experiments using both regular
and irregular wave trains verified the modeling law by reasonable reproduction of the
dune erosion which occurred during prototype event. The scale relationships preserve
the dimensionless fall velocity parameter (H/TW) and Froude number.
Wang (1990) derived a relationship based on different argument from the inspec
tion of the basic governing equation instead of from dimensional analysis of physical
quantities. Based on the twodimensional sediment conservation equation and preser
vation of the surf zone parameter (Tanp//H_/Lo) with Tan# the beach slope and
the number of incoming waves per unit time, a set of modeling relationships had been
derived as:
N\
NT = Nt = (2.32)
1N/2
NH = N6 (2.33)
N, = NsNy/2 (2.34)
Ns = (NsNw)2/5N /5 (2.35)
In reviewing these modeling laws a number of issues appear to be unsettled.
One important question is the morphological time scale. While most of the profile
modeling laws appear to be able to reproduce the final profile between model and
the targeted prototype, the morphological time scale usually was not well established
owing largely to insufficient timeseries data of both prototype and scaled models.
Another important parameter which needs attention is the scaling of wave height NH
in distorted models. So far no matter which type of transport mechanism the modeling
laws are based on, i.e., bed shearstress dominated transport or turbulence dominated
transport, although the latter was preferred by more investigators, wave height scale
is always treat as the geometric vertical scale, which might not be automatically the
15
case. Wang (1985) argued that in movable bed experiment the water depth at any
location is the original depth (which is a geometrical scale) minus (or plus) the sand
accretion (or erosion). Such erosion or accretion depth is not a simple vertical scale
and, therefore, should not be scaled as such. From this point of view, since wave
height also varies with water depth inside the surf zone but not necessarily linearly,
then wave height scale might also not be the same as the geometric vertical scale if the
model is distorted. The apparent question is why wave height should not be scaled
according to horizontal scale so the wave steepness is not distorted. The question of
wave height scaling in distorted model must be further explored in the laboratory.
Finally, a question can be raised on the criteria of modeling law verification. So far,
the performance of modeling is judged by properties in the vicinity of waterline such
as the volume of erosion and profile shape in that region. These quantities are of
obvious importance in engineering application but may or may not be sufficient for
modeling law verification purpose.
In closing this chapter it is reiterated here which has been observed by many
authors that the modeling of the coastal problems remains more of an art than a
science at this stage. There is general guideline but no general solution. Most of all
the model laws are tailored to fit certain restricted circumstances.
CHAPTER 3
APPROACH TO PHYSICAL MOVABLEBED MODELING LAW
A number of different modelling laws for beach profile response modelling have
been proposed in the past; most of them are empirical and can only be applied with
certain restricted conditions and there is no clear indication as to which is more
preferable than the others.
The laws of hydraulic similitude, which should conform to the principles of fluid
mechanics, define the requirements necessary to ensure correspondence between flow
conditions of a scale model and its prototype. These requirements can be established
on the basis of either dynamical considerations, dimensional analysis or the equations
governing the process. However, full correspondence between the model and the
prototype is difficult, if not impossible, for most of the hydraulic problems including
coastal processes. This is because it is usually not possible to obtain a model fluid that
has the required viscosity, surface tension, elasticity and other physical properties to
meet the exact similitude requirement unless the linear scale is such that the model
is as large, or nearly as large, as the prototype. It is known that complete similitude
is not practical and in most cases not necessary. Then, it is extremely important
to select and preserve the parameters that have the dominant effects on the process
being examined.
Different investigators based on different interpretations of the coastal processes,
may choose to preserve different physical parameters, thus, arrived at different simil
itude relationships.
In this chapter, the basic principles on similitude is briefly reviewed first. Then
follow the work by Wang, et al. (1990) but with different consideration on wave
17
height scaling based on a general wave breaking criterion, a modified beach response
modeling law is derived.
3.1 Principles of Similarity
As stated earlier the requirements for similarity between hydraulic scale model and its
prototype can be established on the bases of dynamical considerations, dimensional
analysis or the equations governing the process. These three methods are briefly
discussed here.
3.1.1 Dynamic Similarity
Similarity between the model and the prototype consists of three categories: ge
ometric similarity, kinematic similarity and dynamic similarity. Geometric similarity
means the model and the prototype have similar geometrical and boundary shapes.
A basic geometrical scale is then defined as the ratio of linear dimension between the
model and the prototype:
N = L (3.1)
L,m
where L is the linear dimension and the subscripts m and p refer to model and
prototype, respectively. N\ is called the length scale which dictates the size of the
model. If all the linear dimensions are preserved with the same N\ the model is known
as undistorted. The model could also be distorted if different parts of the model have
different length scales. They are known as distorted models. A common case is a
model with different horizontal and vertical length scales.
Kinematic similarity indicates a similarity of motion between model and proto
type. Kinematic similarity of two systems is obtained if homologous particles are at
homologous points at homologous times (American Society of Civil Engineers, 1942).
The time intervals in the two systems must have a constant ratio,
NT = T (3.2)
Tm
Where NT is time scale.
In geometrically similar models, kinematic similarity is assured when there is
dynamic similarity. Following Newton's second law of motion, dynamic similarity can
be achieved when the ratio of inertial forces (Fi) between model and prototype equals
the vector sums of the ratios of active forces, which are recognized as gravitational
forces (Fg), viscous forces (F,) elastic forces (Fe), surface tension forces (F,t), and
pressure forces (Fr,) in general fluid mechanics problems, i.e.,
(Fi) (F + F + F, + F + F) (33)
(F,)m (F, +F,+ F,, + F, + Fp)
A more restrictive requirement that guarantees Eq.(3.3) to be true is that the ratios
of each and every force be equal, or,
(Fi), (F)p (FA)P (Ft)p (F,)p (Fp)p (3.4)
(Fi)m (Fg)m (F,)m (Ft)m (Fe)m (Fp)m
The above equation can also be written as,
S= (Fg)p/(Fi)p (F,)/(Fi)p (Ft)p/(Fi)p (F.)p/(Fi)p (Fp_ )(Fi)p
(Fg)m/(F)m (F,)~ m(Fj) (F t),l(Fi) (F,)m/(F)m (F,)lI(Fi)m
(3.5)
The above equation consists of five nondimensional force ratios. If both model
and prototype are under the same atmospheric pressure condition as often to be the
case, the nondimensional pressure force ratio will be preserved if other force ratios
are preserved. Therefore, full dynamic similarity, in general, require the other four
force ratios all to be unity. Except with a one to one scale model, it is generally
not possible to achieve such full dynamic similarity. Therefore, it is important to
examine which forces are dominant in the prototype that are to be preserved in the
model. The others which contribute little to the phenomenon under consideration are
then ignored in the model simulation. Familiar examples are the Froude similitude
which preserves only the nondimensional gravitational force ratio and the Reynolds
similitude that preserves only the nondimensional viscous force ratio.
19
These forces can be expressed in basic physical quantities of length (L), mass (M),
gravitational accelation (g), density (p), dynamic viscosity (y/), modulus of elasticity
(E), surface tension (a), and pressure (p), that is,
Fi = mass x acceleration = (pL3)(V2/L) (3.6)
Fg = mass x gravitationalacceleration = pL3g (3.7)
velocity
F, = viscosity x i area = tVL (3.8)
distance
Fst = unitsurfacetension x length = aL (3.9)
F, = modulusofelasticity x area = EL2 (3.10)
Fp = unitpressure x area = pL2 (3.11)
Based on these expressions, the ratio of gravitational force to inertial force can be
readily obtained as 1, the inverse of which is known as the Froude number. The
preservation of gravitational force is equivalent to preserve the Froude number be
tween the model and the prototype. Similarly, the ratio of viscous force to inertial
force given by is known as Reynolds number. The modeling law that preserves
the Reynolds number is called Reynolds similitude.
Other similitude creteria can be derived if one of the other nondimensional force
ratios is preserved. When surface tension force predominates, the force ratio is given
by Weber's number, We = v Similarly, the elastic force to inertial force ratio is
known as Mach or Cauchy number,Ma = v
Usually, for problems in coastal engineering field where gravity wave plays an
important role, Froude similitude needs to be preserved and every effort should be
made in the model design to either compensate or minimize the effects of viscous
force.
20
3.1.2 Similarity By Dimensional Analysis
The methods of dimensional analysis were developed primarily by Rayleigh (1899),
Buckingham (1914), and Bidgman (1922). The best method of analyzing fluidflow
problems is by direct mathematical solution. However, many physical problems are
complex and direct mathematical solutions are not possible. For such cases, labora
tory model experiments are one of a few available alternatives. Dimensional analysis
is a means to organize test parameters and scale the test results without the knowl
edge of the governing equations. Buckingham's theorem, generally known as the 7r
theorem, is used quite commonly.
Any correct mathematical equation which governs the physical process must be
dimensionally homogeneous and each term in the equation must contain identical
powers of each of the fundamental dimensions when the terms are reduced to basic
dimensions of mass, length, and time (M,L,T) or force, length, and time (F,L,T).
If n variables are connected by an unknown dimensionally homogeneous equation,
the equation can be expressed in the form of a relationship among nk dimensionless
products, where k is the number of fundamental dimension in the problem and nk
is the number of products in a complete set of dimensionless products (7r terms) of
the variables, and each 7r term will have k+1 variables of which one must be changed
from term to term. The general function can be written in the form
A1 = f(A2,A3, A4,...,A,) (3.12)
which can also be written
f'(Ai, A2, A, ..., A)=0 (3.13)
or
f"(7 72, a, .., ),nk)=0 O(3.14)
And the form of the function and values of the constant, for different types of flow
21
conditions, must be determined by analytical reasoning, experiment, or a combination
of reasoning and experiment.
3.1.3 Similarity By nondimensionalizing the governing Equations
The mathematical equations that govern a phenomenon may give more insight
into the laws of similarity than the use of dimensional analysis of the variables influ
encing or being suspected of influencing the phenomenon. If a phenomenon can be
described with sufficient accuracy by differential equations, the equations, after being
converted to dimensionless form, provide the basis of determining transfer parameters
between model and prototype. This is simply an extension of the dynamic similitude
techniques described earlier. There are several ways to convert the differential equa
tions to dimensionless forms. For the details the reader may refer to Langhaar
(1951), Duncan (1953), Keulegan (1966), and Young (1971).
3.2 Physical Modeling Laws Of Beach Response
A number of modeling laws have been proposed for beach response modeling.
A summary can be found in Wang et.al (1990) and Hughes (1993). In both texts,
it was pointed out that the scale criteria dependent on fall velocity appeared to be
most promising. Table 3.1 summarizes some of the modeling laws given in the texts
mentioned above. Of these formulas, the Dean's criterion is gemotrically undistorted
whereas all the others permit geometrical distortions. It was pointed out by both
authors that all these criteria converge to the same undistorted condition.
Although an undistorted model is preferred, it lacks flexibility and limits the range
of tests that can be performed either due to facility limitation or material limitation.
Therefore, in this study, the main concern is the development of criteria that permit
scale distortion.
Hughes (1993) made a general assessment on the tests of distorted modeling cri
teria and remarked that the body of experimental evidence presently supports either
Table 3.1: Summary of Fall Speed Distorted Model Laws
Geometric Hydrodynamic Morphological
Author Distortion Time Scale Time Scale
Le Mehaute (1970)) = (N )1/2 N =
Vellinga (1982) = ( N)o.28 NT = N =
Hughes (1983)) = ( 6 = Nt N
Wang, et.al (1990)) = (N', )/4 NT = Ntm =
Wang,et.al (1994) = ( N6 )1/4 T = Nt =
Vellinga's relationships or Wang, et.al.'s guidance. The geometrical relationships of
these two set of criteria are very similar but the time scale is different. Vellinga's
approach is largely empirical based on dimensional analysis of physical quantities.
Wang (1990) took a slightly different approach by the inspection of the basic govern
ing equation to deal with a restricted case, here the two dimensional beach profile
changes under the influence of wave action. Since their approach is based on the
actual sediment transport equation it offers the major advantage that the modeling
laws can be rationally modified to accommodate different hypotheses. These model
ing laws, in turn, can be used to explain the physical process that is being modelled,
not simply producing match scales between model and prototype. Therefore, the
approach by Wang et.al is adopted here with a brief derivation of their results then
followed by a proposed variation.
23
The basic equation, which balancing the spatial change of sediment transport rate
and the temporal change of beach profile, is the twodimensional sediment conserva
tion equation:
Oh = q (3.15)
9t Ox
where h is the bottom elevation, q is the volumetric sediment transport rate in
the direction x. Nondimensionalize the equation:
Oh q 9,t,, O
h qt (3.16)
where the overbar refers to nondimensional quantities and q,, t,, 6, A represent
the reference values of sediment transport rate, the morphorogical time scale, vertical
and horizontal geometrical scale respectively.
To maintain similitude between the model and prototype requires
N 1Nt (3.17)
NsNA
where N refers to the ratio of prototype to model.
It is assumed that suspended load transport mode predominates the sediment
transport inside the surf zone which is approximated by the following equation,
q, = hVc (3.18)
where h is depth, V is mean transport velocity and c is mean sediment concentration.
The suspended sediemnt concentration is then assumed to be directly proportional
to the ratio of stirring power due to turbulence and the settling power due to gravity
and can be expressed as (Hattori and Karvamata, 1980):
pu' u'
coc (3.19)
(pS p)W SW
24
where u' is the turbulent intensity, W is the particle settling velocity and S is the
submerged specific weight.
The ratio of turbulent velocity and wave induced velocity is a function of surf
zone parameter as suggested by Thorton (1978), i.e.,
U/
= (f) (3.20)
U
The surf zone parameter is defined as Tan/3/Hb/Lo with Tan/ the beach slope,
Hb the breaking wave height, and Lo the deep water wave length.
Physically, this equation states that if the surf zone property is similar, the turbu
lent intensity should be proportional to the mean velocity scale provided the surf zone
parameter is preserved. Since in a wave field u is proportional to H/T, combining
Eqs. (1.17), (1.18) and (1.19) with Eq. (1.16) gives the following scaling law,
NvNf(~)NtNH = 1 (3.21)
NANwNTNs
where the subscripts correspond to various physical quantities given earlier. It should
be noted here Nv is the scale ratio of the mean transport velocity which must not be
confused with the waveinduced particle velocity, u. This mean net transport velocity
is assumed to be proportional to wave celerity inside the surf zone. Therefore, Nv
is scaled as (NN6)1/2
By requiring Nf(( ) = 1, i.e.,
(g1/2T Tanf/HO/2), = (g1/2T Tanp/Ho1/2) (3.22)
the hydrodynamic time scale, or here the wave period scale, is obtained as,
N NH/2 (323)
NT = N /2 (3.23)
N9112N1
In the modeling law proposed by Wang et.al. (1990), it was stated that the wave
height is treated as a vertical parameter which leads to the following hydrodynmaic
time scale,
NT = N (3.24)
NT r/2 1/2
25
From here after, the Ng term will be dropped from all the formulas since all of
the coastal models are carried out in the same gravity field. Wang et.al further
examined two possibilities for geometric modeling. The first case was by letting the
morphological time scale to be the same as the distorted hydrodynmaic time scale
which is equivalent to preserve the number of waves in determining morpholical time
scale. The second alternative is to preserve the sediment particle fall trajectory. The
first possibility leads to the following pair of morpholical time scale and geometrical
scaling laws,
NT = Nt = 2
Ns = ( )2/3N3N2 2/3 (3.25)
The second case offers the following scaling equations,
NA
NT = N 2
1/2
Nv
N6 = (N)2/5 /5N4/5 (3.26)
Therefore, geometric distortion is permissible by both scaling laws. Wang et.al. indi
cated that the second pair is perferred by comparing with experimental results. The
distorted modeling law offers a major advantage over undistorted modeling laws in
that it greatly relaxes the requirement of laboratory facilities.
It has been noted that in this modeling law, the wave height is treated as a vertical
scale. This assumption is reexamined here. Clearly, in modeling beach profile change,
wave height inside the surf zone should be similar between model and prototype. By
treating wave height as a vertical geometrical scale in essence implicitly assumes that
wave height is proportional to the local water depth, i.e., H = 7sh with the Yb a
constant value. This relationship was originally proposed by McCowan (1984) as a
breaking criterion and has been widely used. To maintain wave height similitude for
26
the region to be simulated, here the surf zone, however, also requires that the same
criterion be true for the entire surf zone. This may or may not be case. therefore, a
more general wave height scaling law is proposed here with the following form,
NH = NyN6 (3.27)
with N, being the scale ratio of the breaking index, Yb. This is the major difference
from the one proposed by Wang, et.al. Substituting the above equation into Eqs.
(3.21) and (3.23) give the modified modeling laws,
NT = N',2 (3.28)
and
= [NT NsNw NA 2/3
NlN, (3.29)
A guideline on determining N, is proposed here by examining the functional
form of 7y as developed by various investigators. A summary of different empirical
formulas for 7b is given in table 3.2 (from Wang, 1990). From this Table it can be
seen that the breaking index 7b could be affected by beach slope and deepwater wave
steepness. Inside surf zone, the effect of deepwater wave steepness is likely to be
minimal. Therefore, a general power law functional form of N, is proposed as,
N y= [N ] (3.30)
From examining the equations given in Table 3.2, one may conclude that the value
of k is likely in the range from 0 to 1. In the case, k=0, the proposed modeling law
reduces to that of Wang's. On the other extreme if k=l, or N, is linearly proportional
to the local beach slope, then,
N, = (3.31)
and Eqs. (3.28) and (3.29) become
NT = N/2
(3.32)
Table 3.2: Summary of Wave Breaking Index (7b)
Author Yb note
McCowan
(1894)
Munk
(1949)
Galvin
(1968)
Collins and
Weir(1969)
Komar and
Gaughan(1972)
Weggel
(1972)
Singamsetti
and Wind(1980)
Sunamura
(1980)
Moore
(1982)
Larson and
Kraus(1989)
Smith and
Kraus(1990)
Wu
(1990)
Hansen
(1990)
Kampuis
(1991)
Y7 = 0.78
7b = 10/3( )1/3
Yb = 1.087mi
'Yb = 1 Mm<0
'Yb = 1.406.85mm < 0
7% = 0.72 + 5.5m
7b = 1/0.56() 1/5
Yb = b(m) a(m)
a(m) = 43.8(1.0 e19)
b(m) = 1.56(1.0 + e19.")1
7 fb = 0.568mo.l107( o0.237
'Yb ]1/6
= 1.l[H7 )11]/6
tb = b(m) a(m( 4/5
a and b same as Weggel
m 10.21
Yb = 1.14[ ]0.21
yb = b(m) a(m)(V)
y7 = 0.85 + 0.351og(Ho )
Yb = 1.25mO.2
7b = 0.56e3.5"
solitary
solitary
laboratory
linear
linear
laboratory
laboartory
laboratory
hybrid
1
0.007 < Ha < 0.0921
Lo 
laboratory
laboratory
laboratory
and
N6 = (~t)2/ NsNW)2/5NA5 (3.33)
Again, by using the two different hypotheses proposed by Wang on morphological
time scaling, two different scaling laws can be obtained. First, by assuming that the
number of incoming waves per unit time is preserved for the similitude of erosion rate,
we obtain,
N, = NT = Ng/2
Ns = (NsNw)2/5NA/5 (3.34)
Second, if one perserves the fallen particle trajectory, the following modeling laws are
arrived at,
NT = N/2
Nt = N1/2
Ns = (NsNw)/3N5/6 (3.35)
In summary, a new set of profile modeling law have been derived by using the same
approach of Wang et.al. This new modeling law contains an additional scale ratio of
wave height. It seems that Wang et.al's law represents one limiting condition of the
proposed law. The other limiting condition is also derived here and the modeling law
is also included in the Table 3.1.
CHAPTER 4
LABORATORY EXPERIMENTS
4.1 Test Facilities
Laboratory experiments were carried out in three different facilities all located
in the Department of Coastal and Oceanographic Engineering, University of Florida.
The main series of experiment was conducted in what is referred to as the airsea wave
tank (ASW) and a limited number of cases were repeated in a different wave tank,
hereto referred to as the wave flume (WF). Finally, a set of experiment was carried in
a threedimensional basin (WB) with varying input wave angles. The results under
normal wave incident angle were also reported in this thesis for comparison. Brief
descriptions of each facility is givn here.
4.1.1 AirSea Wave Tank
The ASW tank is 1.8meter wide and 45.7meter long. The wave generator section
occupies 3.4 meter and the wave absorbing beaches is 5.8 meter long. The remaining
36.6 meters are divided into two bays, each 0.86meter wide and 1.9 meters deep. The
maximum allowable water depth is about 0.9 meters. The wave tank can be filled
from a well at the rate of 500 gallons per minute.
Wave generation in the AirSea tank is controlled electronically, hydraulically
driven wave paddle measuring 1.8meter wide and 1.2meter high. The wave generator
bulkhead is mounted on acarriage and is driven by two hydraulic rams governed
by hydraulic servovalves. The system provides independent control of the top and
bottom rams in such a way that the bulkhead can move either as a piston on the
carriage or as a paddle. Any combination of piston and paddle motion is possible.
30
For the beach response modelling law verification tests, the piston type is used.
A moving instrument carriage was mounted on the railing system traversing on
top of the tank. The rails for the carriage was installed with a horizontal tolerance of
0.03 mm, providing an leveled reference platform from which precision beach profile
measurement can be performed. The carriage also serves to carry various sensing
instruments. The carriage drive train is powered by a 1/2HP electric motor capable
of moving the carriage at variable speed between 0 to 6.1 meters per second. A hand
held remote control unit is used to start or stop the drive motor. A handoperated
optic probe is used in this series of tests to measure beach profile elevations.
4.1.2 Wave Flume Facility
A limited number of experiments were repeated in the Wave Flume which is
considerably smaller than ASW. It is 0.6meter wide, 0.9meter high and 15.5meter
long with one side of the wall made of glass panel and the other side of steel. The
tank is equipped with a piston type wave maker driven by a mechanically controlled
electric motor. A manually operated ponit gage was installed for profile measurement.
In order to avoid penetrating into the sand at the point of contact the gage was
constructed of light aluminium and the tip was modified by replacing the point with
a small rectangular base which rests flat on the sand.
4.1.3 Wave Basin Facility
Threedimensional model tests were carried out in the wave basin of the Coastal
and Oceanographic Engineering Department, University of Florida. The dimension
of the basin is approximately 28 m x 28 m x 1 m. The basin is equipped with
a snaketype wavemaker, which consists of 88 independent wavepaddles of 24 cm
width each. By adjusting the phase of each individual paddle motion, it can generate
waves of various oblique angles. Also the amplitude can be adjusted individually to
meet different needs with wave heights ranging from 1 centimeter to 15 centimeter
and wave periods from 0.89 to 1.89 seconds.
31
4.2 Initial Beach Profile Design
In this laboratory study, a test series obtained from the Grosser Wellenkanal
(GWK) facility in Hannover, Federal Republic of Germany (Dette and Uliczka,1987)
was selected as the prototypescale target condition. This was chosen on the basis that
GWK is among the largest tank facilities in the world and the experiments carried
out thereout can be considered as near prototype scale. The experiments were also
well documented with rather comprehensive time histories of profile evolution and
the associated wave transformation data. The GWK experiments, hereto referred as
prototype, used sand with a median diameter of 0.33 mm molded to a composite slope
of 1 on 4 on the upper portion of the slope and 1 on 20 on the lower portion of the
slope. This initial prototype profile is shown in figure 4.1. The tests were conducted
at water depth of 5.0 meters under the input wave condition of 1.5 meters wave height
and 6.0 seconds wave period.
In the ASW tank model tests, two types of sediment were used, one of natural fine
sand with median diameter (Ds0) of 0.20 mm, hereon referred to as natural sand and
the other a well sorted very fine quartz sand, hereon referred to as fine quartz sand,
with Dso equal to 0.09 mm. The former was used in the bulk of the experiments. The
latter was used only in one series of tests at undistorted model scale to see whether
very fine sand can be used in the model as this is one option to increase the geometrical
scale ratio between prototype and model without resorting to distorted models. The
size distributions of the two sand sizes obtained by sieve analysis are given in Figure
4.2. The sediment fall velocities corresponding to D50 = 0.09 mm and D50 = 0.2 mm
are, respectively, 0.675 cm/sec and 1.90 cm/sec at a water temperature 200C (see
Figure 4.3 after Rouse, 1937).
The modeling laws listed in Table 3.1 do not dictate the same geometrical distor
tion ratio. The geometrical distortions based on LeMehaute's and Hughes' laws are
identical whereas the geometrical distortions based on Vellinga's, Wang's the newly
INITIAL PROFILE
Horizontal Scale=1:1 Vertical Scale=l:l
Slope 1=4:1 Slope 2=20:1
Length in m
25.0
7.1 S.W.L
0.0 12.0 24.0 106.0
/ N
Figure 4.1: Initial Beach Profile In Prototype
33
SAND SIZE DISTRIBUTION
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.00
0.60 0.70 0.80 0.90
Figure 4.2: Cumulative Sand Size Distribution
proposed one are nearly the same. A choice has to be made here for the laboratory
initial profile design. The geometrical distortion based on the latter group, or the
Wang's modeling law is adopted. This is because the existing field and laboratory
data seemed to have better agreement using either the Vallinga's or the Wang's cri
terion (Hughes, 1993), which are nearly the same in terms of geometrical distortion.
It should also be pointed out here that the geometrical distortion criteria of all the
modeling laws listed do no actually differ significantly from each other and that the
final test profile is not particularly sensitive to the initial profile.
The geometrical distortion law proposed by Wang is,
Ns = N 45,2/5
6" ;1A NW
(4.1)
The initial test profile is shaped according to the above equation based on the
prototype profile dimensions. Three different horizontal scales Nx = 20, 30 and 40
were selected in the test. The relevant geometrical scale ratios and some of the
0.10 0.20 0.30 0.40 0.50
SAND SIZE, D50 (mm)
Figure 4.3: Fall Velocity Of Spherical Grains As A Function Of Size, Rouse (1937)
Table 4.1: Scale Ratios and Physical Dimensions in ASW Tests
Median Sand Size D50 0.20 mm 0.09 mm
Median Fall Velocity W, 1.9 cm/s 0.675 cm/s
Fall Velocity Scale Nw 2.08 6.62
Horizontal Length Scale N, 20.0 30.0 40.0 20.0 30.0 40.0
Vertical Length Scale N6 14.46 20.00 25.18 20.00 30.00 40.00
Distortion (N : Nx) 1:1.38 1:1.5 1:1.59 1:1 1:1 1:1
Water Depth 34.6cm 25.0cm 19.9cm 25.0cm 16.7cm 12.5cm
Corresponding "A" Parameter 0.092 0.065
physical dimensions at different scale models are listed in Table 4.1. The physical
dimensions of these initial profiles are also shown in Figure 4.4 for the natural sand
models. As can be seen from Table 4.1, the fine quartz sand models are all close to
undistorted. Or, in other words, to achieve undistorted modeling very fine material
has to be used. To avoid too shallow water due to the increased scale ratios, shift
the whole profile to maintaining the water level around 40.0 centimeters but keep the
shape of the profile unchanged as designed.
The WF tank tests were design to repeat part of the experiment carried out in
the ASW, namely only the cases with 20:1 horizontal scale. A planar beach of initial
slope 1:2.9 at upper segment and 1:14.46 at the lower segment was formed such as
given in Figure 4.4. The beach material used here was wellsorted fine quartz sand
with a mean diameter of 0.21 mm (close to that used in the ASW) with a sorting
coefficient of 0.58. The corresponding fall velocity is approximately 2.0 cm/sec. The
INITIAL
Horizontal aoalesl:40
Slope 12.6:1
PROFILE
Vertical eale= 1:25.2
Slope 212.6:1
Figure 4.4: Designed Initial Beach Profile in Model with Dso = 0.20mm
Unit In om
I
37
grain size distributions range between 0.1 mm and 0.5 mm. The water depth is 35.3
cm. A total 10 experiments was conducted with test conditions summarized in the
table 4.5.
Finally, in the 3D wave basin tests, the sediment used in the model was also fine
quartz sand with median diameter of 0.20 mm with size distribution given in Figure
4.2. The corresponding fall velocity is 1.9 cm/sec at water temperature 200C; this
yields a N,=2.0 which is the same as the natural sand experiment in the ASW tank.
The horizontal scale was again selected at 20 and the corresponding vertical scale is
14.46 according to the modeling law.
The beach built in the basin has a total length of 19 m with a width of 14 m and
contained 125 tons of quartz sand. Both ends of the beach were confined by rigid
block walls. The entire beach was molded into initial shape by wooden templates cut
into the desired shape.
4.2.1 Experimental Procedures
It is desirable to have set a standard procedure before conducting the experiments
so that the results from the three different setups can be consistent and to reduce the
operating error as much as possible. Hopefully the procedure will also be helpful for
future experiments of a similar nature. This procedures are outlined in the following:
Step 1. Mark the design profile and water level on the outside of the tank glass
wall. Close the drainage valves and fill airsea tank with water to the required depth
and adjust the wave generating machine to the design test conditions including wave
height and wave period .
Step 2. While filling the tank with water, a wooden board cut to just slightly less
than the tank width is used to mold the profile from offshore to beach face according
to the profile line. The uniformity of the profile across the width of the tank is checked
with a leveler.
Step 3. Survey the initial profile after the beach attends saturation at the test
water level. Record the water level.
Step 4. The beach is first stablized by a series of small waves. The wave height
is gradually increased to the designated test height.
Step 5. In most of the twodimensional tests, beach profiles are surveyed only
along the centerline of the tank at time intervals of 5 minutes, 10 minutes, 20 minutes,
40 minutes, 80 minutes. In the threedimensional wave basin experiments, 5 profiles
were regularly surveyed at any time interval.
A number of experimental errors are evident. The seaward position of closure
where sand transport stopped sometimes was difficult to define as a verneer of sand
was often transported beyond the end of the initial profile and deposited as a thin
layer over the horizontal floor of the tank. Also, profile variations across the tank
were common. Uniform sand compaction was also difficult to achieve. These errors
together with survey inaccuracy could lead to erroreous results that did not conserve
the total volume of material. In general, this errors was small. Nevertheless, caution
must be exercised in accepting the data.
4.2.2 Test Conditions
The test conditions were determined by the requirement that the beach should
be erosional under the given wave input. Various criteria have been proposed in the
past, some of them are given in Table 4.2.
From this table it can be seen that three nondimensional parameters are often
used: the wave steepness parameter, Ho/Lo, the relative fall velocity parameter,
7rW/gT and the surf zone parameter, tan/l H/o/Lo.
The test conditions for the twodimensional and threedimensional experiments
as well as the values of these three key paramters were given in Tables 4.3 to 4.7.
The ASW tank tests were the most comprehensive conisting of 26 cases at 3 dif
ferent horizontal geometrical scales whereas 10 tests were performed in WF with a
horizontal scale of 20. In the threedimensional basin, 4 sets of experiments were
Table 4.2: Summary of Criteria Governing Beach Classification
Authors Parameters Erosional Criterion
Johnson, 1952 Ho HI >0.0250.030
Lo Lo
Dean, 1973 ;. _w > 1.7w
Lo 9T Lo 9T
Sunamura and &; Tan/; = C(Tan)027( )067
Horikawa, 1974
Wang and Hb r H > 0.5(')1/3
agqTTan, gT VWTTan, gT
Yang, 1980
Hattori and OTanp; Tan/ > 0.5w
Kawamata, 1980
Kraus, 1991 (Ho/Lo)/(i )3/2 (Ho/Lo)/( T)3/2 < 184
Table 4.3: Summary of Test Conditions in ASW
Test Wave Wave Water Grain Horizontal Vertical
No. period height depth size scale scale
(sec) (cm) (cm) (mm)
1 1.00 11.50 52.0
2 1.14 10.50 34.6
3 1.20 11.25 52.0
4 1.20 12.75 52.0
5 1.33 10.00 52.0
6 1.33 11.00 35.3
7 1.33 11.25 35.3 0.20 20.0 14.46
8 1.33 12.00 52.0
9 1.33 12.75 52.0
10 1.33 13.00 35.3
11 1.33 17.50 52.0
12 1.45 10.50 52.0
13 1.45 13.50 52.0
14 1.45 18.00 52.0
16 1.33 10.00 40.0
17 1.15 9.50 40.0 0.20 30.0 20.0
18 1.00 9.50 40.0
19 0.80 5.50 40.0
20 1.00 9.00 40.0 0.20 40.0 25.2
21 1.15 9.50 40.0
22 1.34 7.50 40.0 20.0 20.0
23 1.34 7.50 40.0 20.0 20.0
24 1.10 5.00 40.0 0.09 30.0 30.0
25 1.10 5.00 40.0 30.0 30.0
26 1.054 3.75 40.0 40.0 40.0
Table 4.4: Summary of Beach Profile Classification Parameter for ASW
Test Wave Wave Wave Wave Relative Fall Surf Zone Kraus
No. period height Length Steepness Velocity Parameter Parameter
(sec) (m) (m) (H_) W T )/()3/2
(=) /oHLo /T L.
(GWK) 6.00 1.50 56.18 0.0267 0.00239 0.3060 228
1 1.00 0.115 1.56 0.0737 0.00689 0.2547 129
2 1.14 0.105 2.03 0.0517 0.00604 0.3041 110
3 1.20 0.1125 2.25 0.0500 0.00574 0.3093 115
4 1.20 0.1275 2.25 0.0567 0.00574 0.2904 130
5 1.33 0.10 2.76 0.0362 0.00518 0.3635 97
6 1.33 0.11 2.76 0.0399 0.00518 0.3462 107
7 1.33 0.1125 2.76 0.0408 0.00518 0.3424 109
8 1.33 0.12 2.76 0.0435 0.00518 0.3316 117
9 1.33 0.1275 2.76 0.0462 0.00518 0.3217 124
10 1.33 0.13 2.76 0.0471 0.00518 0.3187 126
11 1.33 0.175 2.76 0.0634 0.00518 0.2747 170
12 1.45 0.105 3.28 0.0320 0.00475 0.3866 97
13 1.45 0.135 3.28 0.0411 0.00475 0.3411 126
14 1.45 0.18 3.28 0.0549 0.00475 0.2952 168
16 1.33 0.10 2.76 0.0362 0.00518 0.3942 97
17 1.15 0.095 2.06 0.0461 0.00600 0.3493 99
18 1.00 0.095 1.56 0.0609 0.00689 0.3039 120
19 0.80 0.055 1.00 0.0550 0.00861 0.3384 69
20 1.00 0.09 1.56 0.0577 0.00689 0.3304 101
21 1.15 0.095 2.06 0.0461 0.00600 0.3696 99
22 1.34 0.075 2.81 0.0267 0.00161 0.3060 413
23 1.34 0.075 2.81 0.0267 0.00161 0.3060 413
24 1.10 0.050 1.89 0.0265 0.00197 0.3071 303
25 1.054 0.0375 1.73 0.0217 0.00205 0.3394 233
26 1.10 0.050 1.89 0.0265 0.00197 0.3071 303
Table 4.5: Summary of Test Conditions in the WF
Test Wave Wave Water Grain Horizontal Vertical
No. period height depth size scale scale
(sec) (cm) (cm) (mm)
1 1.30 11.50
2 1.30 11.50
3 1.30 12.00
4 1.30 12.50
5 1.30 12.50 35.3 0.21 20.0 14.46
6 1.30 13.00
7 1.30 13.00
8 1.60 15.00
9 1.30 16.50
10 1.65 16.50
carried.
All the test conditions listed above should, in theory, satisfy the erosional condi
tions given in Table 4.2, however, some cases dropped to the region of the accretional
profile according to one or two parameters, especially for the undistorted model tests
the profiles suppose to be accretional according to the Kraus parameter.
4.3 Test Results and Data Analysis
The experimental data consisted of a set of measured profiles for the three test
series. This data set is too voluminous to be included in this thesis. However, they
are archived on computer disks in the Department of Coastal and Oceanographic
Enginnering, University of Florida. In this thesis only reduced information related to
the specific study subject is reported.
Data anaylsis was performed for two objectives: a primary objective of establish
ing the modeling law as stated in the "Introduction" and a secondary objective of
documenting profile evolution process under laboratory condition. To fulfill the first
Table 4.6: Beach Profile Classification Parameter for WF
Table 4.7: Test Conditions in ThreeDimensional Wave Basin
Test No. Wave Height Wave angle Wave Period Vertical Scale
(cm) (sec)
1 10.5 normal 1.14 17.05
2 10.5 normal 1.14 14.16
3 12.5 normal 1.33 14.16
4 12.5 oblique 1.33 14.16
Test Wave Wave Wave Wave Relative Fall Surf Zone Kraus
No. period height Length Steepness Velocity Parameter Parameter
(sec) (m) (m) (& ([ )/()32
LoT TLo Lo 9T
1 1.30 0.13 2.63 0.0494 0.00468 0.311 154
2 1.30 0.11 2.63 0.0418 0.00468 0.338 131
3 1.30 0.115 2.63 0.0418 0.00468 0.338 131
4 1.30 0.12 2.63 0.0456 0.00468 0.324 142
5 1.30 0.10 2.63 0.0380 0.00468 0.354 119
6 1.30 0.135 2.63 0.0513 0.00468 0.305 160
7 1.30 0.105 2.63 0.0399 0.00468 0.346 125
8 1.30 0.1125 2.63 0.0428 0.00468 0.337 134
9 1.30 0.165 2.63 0.0627 0.00468 0.276 196
10 1.65 0.165 4.24 0.0389 0.00370 0.350 173
44
objective, various quantities that can be used for comparisons between prototype
and model are computed; such quantities include the profiles themselves, the volume
of erosion and the bar characteristics. The procedures of computing these quantities
and the results are given here. The subject of modeling law will be treated separately
in the next chapter. The secondary objective, however, is more descriptive and will
be discussed here as an integral part of the data analysis process.
4.3.1 Profile Classifications of the Test Results
As stated earlier that all the test conditions were selected on the permise that
the beach should be erosional. An erosional beach generally is also associated with
the development of a bartype of profile. This has been the case for practically all
the cases tested. Berm and foreshore erosion occurred in every cases. The develop
ment of breaking bar also occurred, but in certain cases the breaking bar is not as
prominent but rather diffused; an example is shown in Figure 4.5. In the very fine
sand experiments (D50=0.09 mm), no prominent breaking bar was formed in any of
the test case.
4.3.2 Volumetric Changes along the Profiles
The basic equation to compute the volumetric change along the profile is the
twodimensional sediment conservation given by
ah 8q
=  (4.2)
&t ax
where h is the elevation of the profile at a given point x and time t; x is onoff shore
direction pointing offshore and zaxis orients upward from an origin located at the
still water shoreline; q is the timeaveraged volumetric sediment transport rate per
unit length of shoreline.
Integrate equation (4.2) from the landward reference, xo, to any other position,
x, gives the net volume change per unit time:
45
2 
.......... INITIAL PROFILE
1  After 80min
      
S
5
10 0 10 20 30 40 50 60 70 80 90 100
OFFSHORE DISTANCE (m)
Figure 4.5: Example of Beach Profile with Diffused Bar, Dso = 0.09mm
SOqz = (4.3)
axo J at
Let q(xo) = 0 at landward closure we have:
q(x) d (4.4)
with
oh [z(x,t2) z(x,ti)] (45)
5t t2 t1
The quantities on LHS are measured. Thus, the spatial variation of transport
rate between any two time levels can be computed from Equation 4.4 and 4.5. In
theory, this transport rate should become zero at the offshore closure depth. Figure
4.6 illustrates a typical spatial variation of the transport rate for the erosional beach
and bartype profile. In the computation transport to the offshore direction is defined
as positive.
46
The cumulative volumatric difference per unit width along the profile between
two type steps defined as VD(volumetric difference) is simply the integration of Ah
along the profile:
V.D. = Ahdx (4.6)
C0
Ah= z(x,t2) z(x,tl) (4.7)
If the profile is truly two dimensional and the sediment bulk density is unchanged
VD should be zero when the integration is carried out to the closure depth to conserve
the mass. In the experiments, nonzero values were often obtained owing to the
combined effects sediment losses to the offshore region and the three dimensional
tank effect.
The beach profiles evolution at different elasped time, the computed transport
rate and V.D. values are presented graphically in Appendix A.
4.3.3 Beach Erosion
There are a number of ways to define beach erosion. The two common ones are
shoreline retreat and volumetric sand loss. The former is clearly more direct and
the latter may be or more engineering concern. In the experiments, the shoreline is
not always easy to define in the final profile as the profile may undulate near mean
water level thus resulting in multiple intersects. To compute volumetric loss, one
must define an offshore cutoff point which also is not necessarily easy. In the present
experiment setups, the initial profile was composite and has a natural break. This
artificial point is chosen as the offshore limit in the volumetric integration based on
Equation 4.6. This volume is defined as the dune volume erosion and the results are
presented in Figure 4.7 to 4.10. As can be seen, most of the erosion took place in
the early period and the erosion rate decreased rapidly on later times. However, not
all cases had reached nonerosional state at the end of the tests (in most cases 80
Final
Initial
Final
Initial
Final
+q +q +q
Offshore Offshore
Onshore x x nshore
q q q
Figure 4.6: Definition Of Net Sediment Transport Rate Across The Beach Profile
48
minutes test time). In fact, the general trend as evidenced from the figures appeared
to indicate otherwise.
4.3.4 Profile Evolution and Bar Migration
As mentioned earlier, most of the profiles evolved into bartype with erosion at
near the shore face and dune region. Initially the material eroded from the beach face
was deposited just seaward of the plunging point to form a breaking bar. A trough
would develop immediately landward of the breaking bar. As the bar grew, waves
started to break farther offshore, causing the location of the bar also to move offshore.
The volume of the bar would also grow as the bar elevation is mainly controlled by
the water depth. The material supply was taken from the region inshore of the
bar. Secondary bar(s) could also be developed shoreward of the breaking bar. The
breaking bar developed rather quickly in the early stage of the test. But the process
of bar migration was more gradual and the rate of offshore migration also slowed
down as time progressed. Analysis was performed here to quantify the process of bar
development and to examine whether a stable profile had been reached at the end of
the test.
First a definition of a bar feature is needed. In the laboratory a convienent
reference is the initial profile and a bar can be defined as material accumulated above
the initial profile. The limits of the bar can be expressed by the intersects with the
initial profile. A number of indices can be defined to charecterize the bar such as the
bar volume, the bar crest location and the mass center. These definions are given in
Figure 4.11. A number of difficulties were encountered in handling the actual test
profiles. The most frequent problem was that there existed no clear intersecting point
at the seaward end, and, at other occasions, the bar appeared reached beyond the toe
of the beach profile. The same problem was also encountered at the landward end
but of fewer cases. For these cases, the intersects were determined subjectively.
Each profile survey was, thus, visually examined for bar feature. The following
AIRSEA TANK MODEL TEST
Horizontal Scale 20, Vertical Scale 14.46, D50=0.20 mm
0
" Test No.l
4  Test No.2
S '" Test No.3
S  Test No.4
.. .. .. .Test No.5
8
S=. Test No.6
0 .. .. . . . . .
9) 12 
2 
V 16
20
0 10 20 30 40 50 60 70
0
0  
Test No.8
\ 4 Test No.9
\  Test No.10
(  Test No.11
........ Test No.12
B 'Test No.13
 Test No.14
S12
0
" ~ ~'~..... 
V 16 ..
2 0 .j^Z z ,
0 10 20 30 40 50 60 70
Elapsed Time (min)
Figure 4.7: Dune Erosion Evolution With Elapsed Time in ASW(1)
80
50
AIRSEA TANK MODEL TEST
Horizontal Scale 30, Vertical Scale 20, D50=0.20 mm
10 20 30 40 50 60 70 80
Undistorted Model With Median Sand Size D50 = 0.09mm
10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.8: Dune Erosion Evolution With Elapsed Time in ASW(2)
0
i
o
o 4
0
m 12
0
Qa 16
20
51
Tilting Flume MODEL TEST
Horizontal Scale 20, Vertical Scale 14.46, D50=0.21 mm
9
5 Test No.1
 Test No.2
1             Test No.3
.  Test No.4
7 3 ....... Test No.5
S Test No.6
" Test No.7
..11 '.. Test No.8
   Test No.9
o 5 ."'
4 19
23

1 27
31
35 I
0 10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.9: Dune Erosion Evolution With Elapsed Time in WF
Wave Basin MODEL TEST
Horizontal Scale 20, Vertical Scale 14.46, D50=0.20 mm
2
 Test No.2 (A) Test No.1 (A)
 (B)  (B)
?6 \'  (c) (c)
\ \,  (D) (D)
1   (E) . (E)
10
22 i1___________
22 I
10 20 30 40 50 60 70 80
2
2 Test No.3 (A)
".  (B)
S~~ (D)
0 ~ ....
0 14
18
22  i  1  1   i  i  1  i 
10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.10: Dune Erosion Evolution With Elapsed Time in WB
30.0
" 0.0
z
0
P10.0
, 20.0
Wd
0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0 550.0 600.0
OFFSHORE DISTANCE (cm)
Figure 4.11: Definition of Longshore Bar With Respect to Initial Profile
properties are then calculated for every identified breaking offshore bar from the
profiles: bar volume = Vb, location of bar center of mass = xcm, location of bar crest
=Xc.
The results of these computed values are given in Figure 4.12 to 4.23 for the
natural fine sand (D50 = 0.20mm). As can be seen from the above figures, breaking
bar development can be broken down into two distinctive stages: the formation stage
and movement stage. During the formation stage the breaking bar grew rapidly as
can be seen from the growth curves of both bar crest and bar volume. The bar quickly
reached a stable condition usually within the first 10 to 20 minutes of the test. Once
the bar reached this mature stage it remained rather stable and only moved at a very
slow rate in the offshore while bar volume also grew. This can be witnessed from the
distinct breaks of the curves shown in the above figures. In most of cases, the bar
and the profile appeared to gradually reach a stationary condition near the end of the
54
tests. There were cases, however, the bar seemed to extend beyond the end of the
initial profile and an offshore intersecting point could not be established; consequently
, the bar volume could not be clearly defined. For these cases, a stationary condition
may or may not have been reached. It is also observed here that as indicators of
bar development, the bar crest and the bar mass center, particularly the bar crest,
appeared to be more stable than bar volume. Part of reason may be that the bar
crest is mainly controlled by the wave breaking point whereas the bar volume could
be rather diffused at times for lack of clear boundaries. As a consequence, the bar
crest usually approached to a stationary position in the test whereas the volume still
was undergone changes.
Similar anayalsis was performed for the very fine quartz sand (Dso = 0.09 mm)tests.
The results were shown in Figure 4.24 to 4.26. In most of these cases, a clean bar
profile such as evidenced in the natural sand tests was not obtained. The bar(s), if
any, was rather diffused. This was likely due to the fact that the very fine material
which when suspended by the wave motion settled over a wider aera.
4.3.5 Eqilibrium Beach Profile
Equilibrium beach profile is an idealized stable beach profile under a given stable
wave condition. The permise is that under a constant wave condition, the profile will
eventually evolve into a shape such that the net sediment transport at any point along
the profile is zero, or in other words, the destructive and constructive forces reached
local balance. This idea is widely accepted and applied in coastal engineering In the
field, since the enviornmntal forces change constantly, the equilibrium profile, if exists,
should also be "dynamic" that might change its shape or shift its location from time
to time and are, therefore, more difficult to define or verify. In theory, the idea of
equilibrium profile can best be tested in a laboratory enviornment under controlled
test conditions. Therefore, although testing the concept of equilibrium profile is not
the main objective of this thesis analysis was performed here to contribute to this
55
topic. Basically, there are two issues; whether equilibrium condition is attainable or
has been attained in the tests and, if it is, what is the final shape?
Based on the analysis given above, one can clearly identify that the timeaveraged
sediment transport rate along the profile deceased monotonically with increasing time.
This is illustrated here by the test case 8 which was tested for an extended period
up to 240 minutes. The profile change as well as transport rate are shown in Figure
4.27.
For most of test cases, the transport rate quickly approached to negilible value
after 40 to 80 minutes run time. This can be seen from the plots in A.1 to A.26.
Another stable feature is the bar profile which was generally achieved before the end
of 80 minutes test time after which the breaking bar maintained its shape with little
or no movement.
The quantity that appeared to have not reached stable condition at the end of
the tests were the dune erosion volume which still showed a trend of increasing. This
suggested that even when the profile appeared to have reached a stable form it may
shift as a whole.
As to the shape of the equilibrium profile, the results will be presented in chapter
56
AIRSEA TANK MODEL TEST
Horizontal Scale 20, Vertical Scale 14.46, D50=0.198mm
Test No.l
Test No.2
 Test No.3
 Test No.4
........ Test No.5
Test No.6
 Test No.7
10 20 30 40 50 60 70 80

III
10 20 30 40 50 60 70 80
I
10 20 30 40 50
Elapsed Time (min)
60 70 80
Figure 4.12: Horizontal Movement Of Bar Crest With Horizontal Scale Equal to 20.
Test No.8
Test No.9
 Test No.10
 Test No.11
........ Test No.12
Test NoJ .
...__ , .*""est No. 14 ....
S_ 14
I. ..
SI 
TILTING FLUME MODEL TEST
Horizontal Scale 20, Vertical Scale 14.46, D50=0.21mm
100.
S90.
80.
. 70.
S60.
0 50.
,j 40.
) 30.
. 20.
. 10.
0.
100.
90. 
80. Te
70.  Te
60 Tea
. ........ Tee
50. 
40. 
.  
10.
10.
o. 1 11i 
0. 10. 20. 30. 40. 50.
Elapsed Time (min)
60. 70. 80.
Figure 4.13: Horizontal Movement Of Bar Crest With Horizontal Scale Equal to 20
In WF
 Test No.1
 Test No.2
Test No.3
 Test No.4
........ Test No.5
  ._...........
  
100
] 90
f 80
0
O
S70
0
0 60
0 50
d 40
30
Ue
96 20
S10
M 0
10 20 30 40 50
Elapsed Time (min)
60 70 80
Figure 4.14: Horizontal Movement Of Bar Mass Center With Horizontal Scale Equal
to 20 In ASW
AIRSEA TANK MODEL TEST
Horizontal Scale 20, Vertical Scale 14.46, D50=0.20 mm
 Test No.l
Test No.2
 Test No.3
 Test No.4
........ Test No.5
Test No.6
 Test No.7
     
     .....

D 10 20 30 40 50 60 70 80
Test No.8
Test No.9
 Test No.10
 Test No.ll
 Test No.13
' Tj._.l4 
,.............. ................ ..............................................
TILTING FLUME
Horizontal Scale 20, Vertical
MODEL TEST
Scale 14.46, D50=0.21mm
100.
90.
80.
70.
60.
50.
40.
30.
20.
10.
0.
100.
90.
80.
70.
60.
50.
40.
30.
20.
10.
0.
0. 10. 20. 30. 40. 50. 60. 70. 80.
Elapsed Time (min)
Figure 4.15: Horizontal Movement Of Bar Mass Center With Horizontal Scale Equal
to 20 In WF
Test No.1
Test No.2
 Test No.3
 Test No.4
........ Test No.5
       
              
AIRSEA TANK
Horizontal Scale 20, Vertical
MODEL TEST
Scale 14.46, D50=0.20 mm
10 20 30 40 50 60 70 80
10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.16: Growth of Bar Volume With Horizontal Scale Equal to 20 In ASW
i
N
C03
' I
0
>
Cd
M
i
N
C)
.**
0
S.
E
'3
>
4
0)
ffl
61
TILTING FLUME MODEL TEST
Horizontal Scale 20, Vertical Scale 14.46, D50=0.21mnm
 Test No.1
Test No.2
 Test No.3
 Test No.4
........ Test No.5
 
   
0. 10. 20. 30. 40. 50. 60. 70. 80.
Elapsed Time (min)
Figure 4.17: Growth of Bar Volume With Horizontal Scale Equal to 20 In WF
62
AIRSEA TANK MODEL TEST
Horizontal Scale 30, Vertical Scale 20, D50=0.20 mm
100
90 Test No.16
 Test No.17
S80  Test No.18
0 70
S60 
S40 
10
0 10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.18: Horizontal Movement Of Bar Crest With Horizontal Scale Equal to 30.
0
Figure
to 30.
10 20 30 40 50 60 70 80
Elapsed Time (min)
4.19: Horizontal Movement Of Bar Mass Center With Horizontal Scale Equal
63
AIRSEA TANK MODEL TEST
Horizontal Scale 30, Vertical Scale 20, D50=0.20 mm
0 10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.20: Growth Of Bar Volume With Horizontal Scale Equal to 30.
Horizontal Scale 40, Vertical Scale 25.2, D50=0.20 mm
S80
0 70
*P"
*j
d 60
0
0
 50
30
m 40
Q)
U. 30
S20
10
0
10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.21: Horizontal Movement Of Bar Crest With Horizontal Scale Equal to 40.
64
AIRSEA TANK MODEL TEST
Horizontal Scale 40, Vertical Scale 25.2, D50=0.20 mm
0 10 20 30 40 50 60 70 80
Elapsed Time (min)
"Figure 4.22: Horizontal Movement Of Bar Mass Center With Horizontal Scale Equal
to 40.
40
35 Test No.19
0 10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.23: Growth Of Bar Volume With Horizontal Scale Equal to 40.
200
180
S160
0 140
g 120
0
S100
s 80
ao
) 60
O 40
20
0
Figure 4.24: Horizontal Movement Of Bar Crest With Dso = 0.09mm
0 10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.25: Horizontal Movement Of Bar Mass Center With Dso = 0.09mm
AIRSEA TANK MODEL TEST
Undistorted Model With Median Sand Size D50 = 0.09mm
Test No.22(Horizontal Scale 20)
Test No.23(Horizontal Scale 20)
. Test No.24(Horizontal Scale 30)
 Test No.25(Horizontal Scale 40)
 Test No.26(Horizontal Scale 30)
 
 
10 20 30 40 50 60 70 80
Elapsed Time (min)
200
180
g 160
0
 140
Cd
0 120
S100
S80
60
d 40
S20
m n
66
AIRSEA TANK MODEL TEST
Undistorted Model With Median Sand Size D50 = 0.09mm
40
35  Test No.22(Horizontal Scale 20)
Test No.23(Horizontal Scale 20)
S. Test No.24(Horizontal Scale 30)
30 Test No.25(Horizontal Scale 40)
C  Test No.26(Horizontal Scale 30)
* 25
20
15
*"^ _   
0 10 "...
10
0 10 20 30 40 50 60 70 80
Elapsed Time (min)
Figure 4.26: Growth Of Bar Volume With Dso = 0.09mm
67
AIRSEA TANK TEST NO.15 (EROSION) X=1:20 6=1:14.46
(H=11.5cm, T=1.33sec, D=52.0cm, SCALE=9.0)
200.
400.
200. 400.
OFFSHORE DISTANCE (cm)
600.
200.
Figure 4.27: Beach Profile Evolution And Sediment Transport Rate
S.W.L.
240.mim
\ 2O n ^*a
V.D. .0049(m3/m)
:_U.UU0
V.D. .0274(m3/m)
0.000    
0.005
0.005,
V.D. 0230(m3/m)
  =  =
600.
I I
CHAPTER 5
EVALUATION OF SCALING LAWS
5.1 Methodology and Evaluation Criteria
As stated earlier a specific set of test results from GWK is treated here as the pro
totype conditions. All the test results were compared with the GWK data set as
basis for evaluating the scaling laws. Basically four different scaling laws from Table
3.1 were selected for the purpose; they are, by first author's name, vallinga, Hughes,
Wang 1 and Wang 2. They practically cover the existing scope of the profile modeling
law. These four scaling laws are repeated here in Table 5.1.
To facilitate quantitative evaluation certain criteria are employed here. The entire
beach profile is first divided into two parts: from the shoreline to the end of the first
slope of the profile is defined as the dune profile region and from the beginning of the
second slope of the profile to the end of the toe is defined as the bar profile region
such as given in Figure 5.1. Six different quantities were used for evaluations, two
related to the dune region and four related to the bar region; they are:
A. Dune Region:
The RMS value of the dune region profile.
The dune erosion volme.
B. Bar Region:
The RMS value of the bar region profile.
The location of bar center of mass.
The location of bar crest.
The volume of bar.
Bar Profile Region
SWL
S    
 Initial Profile
 Storm Profile
A SCHEMATIC OF BAR AND SHORE REGION
Figure 5.1: Definition of The Dune and Bar Regions
Shore Region
Table 5.1: Four Fall Speed Distorted Model Laws
Geometric Hydrodynamic Morphological
Author Distortion Time Scale Time Scale
Vellinga (1982) N = ( )0.28 NT = m Nt,. =
N6 NVN, Nfg I
Hughes (1983)) & = (~ NT =1 Ntm =
Wang, et.al (1990)) A = () NN )1/4 NT = m
Wang,et.al (1994) N= ( N )1 NT = Nt = Nf
All the above quantities were evaluated relative to the reference prototype quan
tites in terms of error such that the set with the least error represents the best fit.
Detailed formulas for computations will be presented later in the appropriate sec
tions.
Geometrical scaling was evaluated first based on the RMS error of the profiles
in the two regions. Wave height and wave period scalings were then evaluated by
comparing the scaled final profiles of all the test cases with the prototype final profile.
Finally, the bestfit data sets from each horizontalscale group, i.e., N\ =20, 30 and
40, were selected to evaluate the morphological time scale.
5.1.1 Geometrical Scaling and Equilibrium Profile
The gerometrical scale was evaluated by comparing the RMS (root mean square)
error of the final test profiles with the prototype. The RMS error is defined as
c = [i(hph)
i=1
(5.1)
where hP and hM are profile elevation of prototype and model, respectively. There
are basically only two geometrical scaling laws in the 4 modeling laws listed in Ta
ble 5.1 as the Vellinga's and the two Wang's laws are nearly the same. These two
geometrical scaling laws are,
Ns = N/3 (5.2)
and
Ns = Ng45 (5.3)
The data were scattered but collective the latter scaling law appeared to perform
better. Figure 5.2 shows the comparisons of all the final profiles for the cases of N\
=20 with the prototype based on the scaling Equation 5.3.
From this figure one may also argue that the gross feature of the final profiles
may be expressed by a simple form which could be defined as the equilibrium profile.
A number of equilibrium profile forms have been proposed, notably by Bruun (1954),
Dean (1977), Vellinga(1982) and Wang (1990). They are all in the following general
form:
SW
h = a( )"m (5.4)
where h is the water depth, x is distance from shoreline, S is the submerged specific
weight, W is the sediment fall velocity and a is nondimensional coefficient. The value
of a(s) is also known as the scale parameter. Moore (1982) analyzed 500 or so
beach profiles and proposed A to be a function of particle size. Later, Dean, Vellinga
and Wang all suggested that particle fall velocity is a more appropriate parameter
instead of the partial size. Vellinga (1982) proposed A to be equal to 0.39W0o' from
analyzing field profile along the Dutch coast. Dean (1985) reanalyzed the values
complied by Moore (1982) and gave the best fit as A = 0.51W0.44; this formula was
also found suitable for GWK data by Wang (1990). As to the value m, Bruun and
WAVE TANK MODEL TESTS EQUILIBRIUM PROFILES
10 0 10 20 30 40 50 60
OFFSHORE DISTANCE (m)
70 80 90 100
Figure 5.2: Equilibrium Profile Comparison Between Model Test Results and Model
ing Laws
73
Dean suggested to be 2/3, Vellinga used 0.78 and Wang came up with 4/5. Equation
5.4 was also plotted in Figure 5.2 using two different values of n = 2/3 and 4/5. The
A parameter is equal to 0.131 for both cases based on the fall velocity equation given
above. It appeared that value of m=4/5 gave a better fit for the specific data set in
the inshore segment of the profile. It should be noted here that, to a certain degree,
irrespective to which value of m one selects, the equilbrium curve can also be adjusted
to fit any specific data set by adjusting the value A.
5.1.2 Wave Height and Wave Period Scaling
Based on the analysis given in the previous section, the basic goemetrical scaling
by Wang is selected for the subsequent analysis of wave height, wave period and mor
phological time scalings. In this section, the scalings of the former two are addressed
and the latter will be addressed in the next section.
As mentioned earlier the analysis of wave height and wave period is based upon
final profiles. Figure 5.3 to figure 5.5 plots the scaled final profile of each test case for
the ASW test series against the protype. The results of the 10 WF tests are presented
in Figure 5.6
74
To establish the best scaling relationships from the data, the four of the criteria
given earlier were employed. The procedures are the same for analyzing each criterion.
It is explained here with the dune volume computation. The dune volume error is
defined as
Vm VP
vM (5.5)
VP
where VP is the volume deficit (eroded volume) between the final and the initial
profile in the prototype and Vm is scaledup volume deficit in the model. These values
are plotted against model wave heights, for different model wave periods, as shown in
Figure 5.7. These constitute a series of curves. The zero crossing(s) should represent
the bestfit model wave heights) and wave periodss. From them the bestfit scaling
ratios can be obtained. A rather clear trend between the dune volume error and wave
height can be detected. This trend is almost linear revealing a expected relationship
that increasing wave height resulted in increased erosional volume. The effect of wave
period on volumetric erosion is less clear revealing a rather weak relationship between
dune volumetric erosion and wave period. From this figure the best fit values of wave
height and wave period are determined to be 13.5 cm and 1.33 sec, respectively. The
corresponding scaling ratios based on this creterion are 11.1 and 4.5 for wave height
and wave period.
The above procedure was also applied to determine the bestfit modeling scales
for the other three criteria. Figure 5.8 to 5.10 show the plots of those other three
parameters: the RMS error of the bar profile, the bar volume and the bar mass
center. From these figures the bestfit scaling ratios based on different creteria are
determined. The results are summarized in Table 5.2.
5.1.3 Evaluation of Morphological Time Scale
To evaluate the morphological time scale one requires a comparison of the model
data with the prototype at each time step. This would be an unwiedy task if all the
AIRSEA TANK MODEL TEST AND GERMAN DATA COMPARISON
3.
2.
o0.
S4.
3.
2.
o6.
3.
E2.
r4.
0. 10. 20. 30. 40. 50. 60.
70. 80. 90. 100.
OFFSHORE DISTANCE (m)
Figure 5.3: ASW Model Tests and Prototype Final Profile Comparison (1)
1.
Z1.
0
2.
~. 3.
M4.
5.
6. 
10.
AIRSEA TANK MODEL TEST AND GERMAN DATA COMPARISON
0. 10. 20. 30. 40. 50. 60. 70. 80.
OFFSHORE DISTANCE (m)
Figure 5.4: ASW Model Tests and Prototype Final Profile Comparison (2)
3.
2.
1.
0.
2.
>3.
S4.
5.
6.
3.
2.
1.
0.
z1.
0
E 2.
4.
5.
6.
i o.
Z1.
0
E2.
13.
~4.
5.
6.
10.
90. 100.
77
AIRSEA TANK MODEL TEST AND GERMAN DATA COMPARISON
1. 
1. 
0
,2.
4.
6. 
10.
10.
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
OFFSHORE DISTANCE (m)
Figure 5.5: ASW Model Tests and Prototype Final Profile Comparison (3)
3.
2.
1.
0.
ZI o.
0
E2.
> 3.
r 4.
5.
6.
3.
2.
1.
0.
'go.
z1.
0
a2.
> 3.
~4.
5.
6.
3.
2.
100.
78
TILTING FLUME MODEL TEST AND GERLMAN DATA COMPARISON
Z1.
0
> 3.
S 4.
5.
6.
3.
2.
0.
z1.
0
.2.
>3.
4.
5.
6.
3.
2.
1. 
24.
10. 0.
1 3
10. 20. 30. 40. 50. 60. 70. 80. 90.
OFFSHORE DISTANCE (m)
Figure 5.6: WF Model Tests and Prototype Final Profile Comparison
A: Vellinga (1982)
B: Hughes (1983)
C: Wang, et al.(1990)
D: Wang, et al.(present)
T=1.20(S)
T=1.T=.30(S)
T=1.33(S)
K. T=1.45(S)
a T=1.00(S)
+ T=1.14(S)
* T=1.65(S)
2
0
o
w
0
+3 ~+
0 ................................... ..................
9 1
Wave Height, H (cm) 19
PLOT OF DUNE VOLUME ERROR (Nx =20, N =14.5 )
Figure 5.7: Dune Volume Erosion Error Criterion
A: Vellinga (1982)
B: Hughes (1983)
C: Wang, et al.(1990)
D: Wang, et al.(present)
*T=1.20(S)
* T=1.30(S)
 T=1.33(S)
 T=1.45(S)
o T=1.00(S)
+ T=1.14(S)
* T=1.65(S)
Wave Height, H (cm) 19
PLOT OF RMS ERROR OF BAR PROFILE ( NX=20, N =14.5)
Figure 5.8: The RMS Error of The Bar Profile Criterion
xA
A: Vellinga (1982)
B: Hughes (1983)
C: Wang, et al.(1990)
D: Wang, et al.(present)
0
0
0
~a.
0
1
 T=1.20(S)
 T=1.30(S)
 T=1.33(S)
x T=1.45(S)
o T=1.00(S)
+ T=1.14(S)
* T=1.65(S)
Wave Height, H (cm)
PLOT OF BAR VOLUME ERROR ( NX=20, N =14.5 )
Figure 5.9: Bar Volume Error Criterion
A: Vellinga (1982)
B: Hughes (1983)
C: Wang, et al.(1990)
D: Wang, et al.(present)
.0.3
0
C.)
03
xT=1.20(S)
 T=1.30(S)
 T=1.33(S)
 T=1.45(S)
e T=1.00(S)
+ T=1.14(S)
* T=1.65(S)
9 Wave Height, H (cm) 19
PLOT OF BAR LOCATION CENTER ERROR ( N =20, N,=14.5)
Figure 5.10: Bar Location Error Criterion
Table 5.2: Comparison of Model Performances
Ex.1: Given NA = 20, Nw = 2.
Author Ns NH NT Nt
Vellinga (1982) 14.5 14.5 3.8 3.8
Hughes (1983) 14.5 14.5 5.3 5.3
Wang, et al.(1990) 14.5 14.5 5.3 3.8
Wang(present) 14.5 10.5 4.5 4.5
Experimental data N6 NH NT Nt
Bar location 14.5 11.5 4.5 4.5
Bar volume change 14.5 11.3 4.5 4.5
Dune volume change 14.5 11.1 4.5 4.5
Ex.2: Given NA = 40, Nw = 2.
Author N6 NH NT Nt
Vellinga (1982) 25.2 25.2 5.0 5.0
Hughes (1983) 25.2 25.2 7.8 7.8
Wang, et al.(1990) 25.2 25.2 7.8 5.0
Wang(present) 25.2 15.9 6.3 6.3
Experimental data Ns NH NT Nt
Bar location 25.2 16.7 6.0 6.3
Bar volume change 25.2 16.7 6.0 6.3
Dune volume change 25.2 16.7 6.0 6.3
84
data sets were used. In here, the bestfit data set from the three different horizontal
scale groups (NA = 20, 30, and 40) were selected from the ASW and WF tests results.
The morphorlogical scale was then examined from this data set. The selection
of the bestfit data set was guided by the results of the previous sections. The error
quantities established above are now plotted for each test case.
Figure 5.11 shows the results of dune region error parameters (dune erosion vol
ume and dune profile RMS) for the cases in the ASW test.
Figure 5.12 gives the results of bar region error parameters for the ASW tests.
From these two figures, the bestfit data sets selected (judged to have the smallest
overall error) are No.8 for NA = 20, No.18 for NA = 30 and No.20 for N\ = 40.
Similarly, Figures 5.13 and 5.12 give the dune and bar parameters results for the WF
tests. From here, No.9 is selected as a compromise for the best overall fit. These data
sets are then used to establish the bestfit morphorlogical scale. In theory, this can
be accomplished by the same procedure as described in the previous section except
that the comparisons are to be made at each time step for the three selected cases.
In practice, this is rather tidious and there is not sufficient data points at each time
step. An alternative procedure is employed.
There are three different morphological time scaling laws guided by the mor
phological time scales given in Table 5.1. Wang(1990) suggested that the particle
trajectory should be preserved which leads to Nt = 'N. Both Vellinga (1982) and
Hughes (1986) modeling laws give Nt = NA/I/NE. The study referred to as Wang 2
suggests that both motion time scale and morphological time scale are governed by
the scaling law, or, Nt = VNA.
Now, the time elaspsed changes of the 6 quantities, i.e., the dune erosion volume,
bar volume the RMS values of profiles, etc., are plotted in accordance with the three
morphological scaling laws and are compared with the GWK results. The GWK data
contained 21 different elapsed time at 5, 13, 20, 27, 34, 48, 62, 76, 90, 111, 132, 162,
AIRSEA TANK DISTORTED MODEL TESTS
(Shore and Dune Profile)
80
() ERROR OF VOLUME EROSION
A7 RMS VALUE OF DUNE PROFILE (*25)
70 WAVE PERIOD SCALE
 WAVE HEIGHT SCALE
65
60
55
NX=20 30 40
50 o ,
45 N =14.5 20 25.2
40
35 oo
30
2 5 ,
20 /
5 z
15 ........
0 (
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
TEST CASE NUMBER
Figure 5.11: Summary of ASW Test Results Based On Dune Profile Parameter
AIRSEA TANK DISTORTED MODEL TESTS
(Nearshore and Bar Profile)
0 ERROR OF BAR LOCATION(center of mass)
A RMS ERROR OF PROFILE (*50)
WAVE PERIOD SCALE
 WAVE HEIGHT SCALE
NX=20 A 30 40
N6 =14.5
25.2
A A
30 A '
25 A A
A/ A
20 A \
5 0, 0  / 
10 S
(D
m rN n
(D
 o ~g5^
. .
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
TEST CASE NUMBER
Figure 5.12: Summary of ASW Test Results Based On Bar Profile Parameter
87
176, 190, 204, 217, 231, 245, 259, 273 minute, respectively. From these comparisons,
the bestfit morphorlogical time scale is obtained.
Dune Region
There are two criteria in the dune region, the dune volume erosion and the profile
RMS value.
For dune erosion, a number of observations can be made. From the ASW results
(Figure 5.15) all three scaling laws over predict the total erosional volume and the
larger the scale ratio, or the smaller the model, the larger the overprediction. The
over prediction largely occurred in the initial stage (0 to 100 minutes prototype time).
Also, it appears that none of the tests including the prototype has reached equilibrium
at the end of the runs.
The results from WF test are shown in Figure 5.16. All modeling laws also over
predict the total erosional volume.
Therefore, these results suggest that the bestfit morphological time scale ratio
has to be smaller than any of the 3 modeling laws.
The comparisons of profile RMS error are given in Figure 5.17 for ASW cases
and in Figure 5.18 for the WF case. For this criterion all 3 modeling laws appeared
to perform reasonably well as the scaled values all custered in a narrow range around
the prototype results. Overall the Wang 2 model may be slightly better.
Bar Profile Region
Four different criteria have been developed for this region, the profile RMS value,
the bar volume, the bar mass center and the bar crest location. The results of these
four criteria are plotted in Figure 5.19 to 5.22 for the ASW tests and Figure 5.23 for
the WF tests.
From these figures a number of observations are made. The bar mass center is
the most stable and consistent criterion. All three modeling laws appeared to give
reasonable results for all three data sets. The bar crest location is next consistent
