• TABLE OF CONTENTS
HIDE
 Front Cover
 Report documentation page
 Title Page
 Table of Contents
 List of Figures
 List of Tables
 Abstract
 Introduction
 Background
 Methodology
 Results
 Summary and conclusions
 Appendix I: References
 Appendix II: Notation






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 91/013
Title: Beach nourishment performance predictions
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 Material Information
Title: Beach nourishment performance predictions
Series Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 91/013
Physical Description: Book
Language: English
Creator: Dean, Robert G.
Publisher: Coastal and Oceanographic Engineering Department, University of Florida
Publication Date: 1991
 Subjects
Subject: Beach nourishment
 Notes
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
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Table of Contents
    Front Cover
        Front Cover
    Report documentation page
        Unnumbered ( 2 )
        Unnumbered ( 3 )
    Title Page
        Page 1
    Table of Contents
        Page 2
    List of Figures
        Page 3
        Page 4
    List of Tables
        Page 5
    Abstract
        Page 6
        Page 7
    Introduction
        Page 7
        Page 8
    Background
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 22
    Methodology
        Page 14
        Page 15
        Page 16
        Page 13
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
    Results
        Page 23
        Page 24
        Page 25
        Page 26
        Page 22
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
    Summary and conclusions
        Page 42
        Page 43
        Page 44
        Page 41
    Appendix I: References
        Page 45
        Page 46
    Appendix II: Notation
        Page 47
Full Text



UFL/COEL-91/013


BEACH NOURISHMENT PERFORMANCE
PREDICTIONS






by




R. G. Dean
and
Chul-Hee Yoo


October 1991






REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient's Accessioon o.


4. Title oad Subtitle 5. Report Date
October, 1991
BEACH NOURISHMENT PERFORMANCE PREDICTIONS 6.

7. Author(s) S. Performing Organizatioo Report No.
R. G. Dean
Chul-Hee Yoo UFL/COEL-91/013
9. Performing Organization Name and Address 10. project/lask/Work Unit No.
Coastal and Oceanographic Engineering Department
University of Florida 11. contract or crant No.
336 Weil Hall
Gainesville, FL 32611 13. Type of ~port
12. Sponsoring Organizaton Name and Address
Miscellaneous


14.
15. Supplementary Notes


16. Abstract


A simple method is developed for representing wave refraction and shoaling in the vicinity of a
beach nourishment project. The method applies for the case of a one-line model of shoreline evolution in
which the active profile is displaced seaward or landward without change of form. The model can include
the presence of shore-perpendicular structures and background erosion. Background erosion rates are
formulated in terms of cross-shore and longshore transport. An underlying concept of the method is that
in cases where large perturbations, such as nourishment projects, are placed in the natural system, the
system erodes on two time scales with the shorter time scale associated with the planform perturbation. It
is recommended that shoreline modeling be carried out by conducting an ad hoc transformation in which
the pre-project contours are represented as straight and parallel.

The simple method is compared to a one-line model which includes a more detailed grid-based refrac-
tion and shoaling algorithm. For all cases tested, the simple method of representing refraction and shoaling
results in shoreline evolution in good correspondence with the detailed method. The models are used to
illustrate the effects of several features of beach nourishment projects that are of engineering interest. For
a long uninterrupted shoreline, which has been nourished with the same material as the native, and in
the absence of structures and background erosion, initially symmetric nourished planforms remain nearly
symmetric as they evolve, even under oblique wave attack. This is interpreted as due to the small aspect
ratio (additional beach width to length) of the nourishment project. For no background erosion or retention

17. Originator's Key Words 18. Availability Statement
Beach nourishment
Sediment transport
Shoreline evolution
Shoreline models

19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Pale 21. No. of Page 22. Price
Unclassified Unclassified 47



















structures, the proportion of sand remaining in the project area over a twenty year period is illustrated
for projects of various lengths and various effective wave heights. The effects of shore-perpendicular struc-
tures placed at the ends of the project with and without background erosion are illustrated. For normally
incident waves and no background erosion, the project with or without retention structures causes only
accretion to the adjacent shorelines. However, in the presence of background erosion or oblique waves,
retention structures can cause localized erosion to the adjacent shorelines.

Finally, the effects of nourishing with material more and less transportable than the native are illus-
trated. If less transportable, the nourishment tends to "armor" the project area and oblique waves can
cause localized downdrift erosion. It is concluded that the simple method can provide reliable predictions
of the evolution of nourishment projects and is thus useful in conducting benefit/cost analyses.








UFL/COEL-91/013


BEACH NOURISHMENT PERFORMANCE PREDICTIONS

by

R. G. Dean and Chul-Hee Yoo


October, 1991








TABLE OF CONTENTS


LIST OF FIGURES


LIST OF TABLES

Abstract

INTRODUCTION

BACKGROUND
Governing Equations ................................

METHODOLOGY
Procedures Common to Both Methods ......................
Grid System and Transformation of Initial Geometry ...........
Background Erosion .............................
Numerical Solution of Governing Equations ...............
Procedures Which Differ for the Two Methods Wave Refraction and Shoaling
Simplified Wave Refraction and Shoaling .................
Detailed Refraction and Shoaling ......................
Effective Wave Height and Period .........................


RESULTS

SUMMARY AND CONCLUSIONS

Appendices
I. REFERENCES ...........
II. NOTATION ............








LIST OF FIGURES


FIGURE PAGE

1 Definition Sketch .................... ........... 11
2 Definition Sketch for Numerical Model ..................... 14
3 Recommended Ad Hoc Transformation for Modelling Coastal Systems in
which Large Perturbations are to be Introduced . . ... 16
4 Definition Sketch for Effect of Beach Nourishment on Contours . 18
5 Example 1. Comparison of Beach Nourishment Evolution for Simple and
Detailed Methods of Wave Refraction and Shoaling, Normal Wave Inci-
dence, Ho = 0.6 m, T = 6.0 sec, co = 900. No Background Erosion 24
6 Example 2. Comparison of Planform Evolution Obtained by Two Methods
for 200 Oblique Waves, Ho = 0.6 m, T = 6.0 sec, ao = 700. No Background
Erosion . . . . . . . . .. 25
7 Example 3a. Planform Evolution by Detailed Method for Deep Water
Wave Directions, oa = 700, 800, 900. Ho = 0.6 m, T = 6.0 sec. No
Background Erosion ................... .......... 27
8 Example 3b. Planform Evolution by Simple Method for Deep Water Wave
Directions, ao = 700, 800, 900. Ho = 0.6 m, T = 6.0 sec. No Background
Erosion ................. ................. 28
9 Example 4. Even and Odd Components of Shoreline Position After 10
Years for Deep Water Wave Directions, ao = 700 and 800. Results Ob-
tained by Detailed Method ....................... 29
10 Example 5. Illustration of Wave Height Effect on Rate of Planform Evo-
lution. Results Based on Simple Method, ao = 900, T = 6.0 sec. Results
Shown for 1, 3, 5 and 10 Years. No Background Erosion . ... 31
11 Example 6. Effects of Various Project Lengths and Wave Heights on
Project Longevity ................... .. .......... 32
12 Example 7a. Effects on Planform Evolution of Two Shore-Normal Reten-
tion Structures of Length Equal to One-Half the Initial Project Width.
Normal Wave Incidence, No Background Erosion. Ho = 0.6 m, T = 6.0
sec . . . . . . . . . 34








13 Example 7b. Effects on Planform Evolution of Two Shore-Normal Reten-
tion Structures of Length Equal to One-Half the Initial Project Width.
Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr,
Zero Background Transport at Project Centerline. Ho = 0.6 m, T = 6.0
sec. . . . . . . . . . 35
14 Example 7c. Effects on Planform Evolution of Two Shore-Normal Reten-
tion Structures of Length Equal to One-Half the Initial Project Width.
Normal Wave Incidence. Uniform Background Erosion Rate at 0.5 m/yr,
Zero Background Transport Located 4,500 m to Left of Left Structure. Ho
= 0.6 m, T = 6.0 sec. ........................... .. 36
15 Plot of K vs D (Modified from Dean, et al., 1982) . . ... 38
16 Example 8a. Planform Evolution for Nourishment Sand Less Transportable
than the Native (KF = 0.693, KN = 0.77). Note Centroid of Planform
Migrates Updrift Variation of Surface Layer K Values with Time at
Locations A, B, C, D and E are presented in Figure 18. Wave and other
Project Conditions Presented in Table I. . . . ... 39
17 Example 8b. Planform Evolution for Nourishment Sand More Trans-
portable than the Native (KF = 0.847, KN = 0.77). Note Centroid of
Planform Migrates Downdrift Variation of Surface Layer K Values with
Time at Locations A, B, C, D and E are presented in Figure 19. Wave
and other Project Conditions Presented in Table I. . .... 40
18 Example 8c. Variation of Surface Layer Longshore Transport Coefficient
K with Time at the Five Locations Shown in Figure 16. Case of KF =
0.693, KN = 0.77, Ymi = 2.0 m. ...................... 42
19 Example 8d. Variation of Surface Layer Longshore Transport Coefficient
K with Time at the Five Locations Shown in Figure 17. Case of KF =
0.847, KN = 0.77, Ymi = 2.0 m. ...................... 43








LIST OF TABLES


1 Characteristics of Examples Presented ................. 23


TABLE


PAGE








BEACH NOURISHMENT PERFORMANCE PREDICTIONS

R. G. Dean and Chul-Hee Yoo




Abstract


A simple method is developed for representing wave refraction and shoaling in the vicinity of

a beach nourishment project. The method applies for the case of a one-line model of shoreline

evolution in which the active profile is displaced seaward or landward without change of form.

The model can include the presence of shore-perpendicular structures and background erosion.

Background erosion rates are formulated in terms of cross-shore and longshore transport. An

underlying concept of the method is that in cases where large perturbations, such as nourishment

projects, are placed in the natural system, the system erodes on two time scales with the shorter

time scale associated with the planform perturbation. It is recommended that shoreline modeling

be carried out by conducting an ad hoc transformation in which the pre-project contours are

represented as straight and parallel.

The simple method is compared to a one-line model which includes a more detailed grid-based

refraction and shoaling algorithm. For all cases tested, the simple method of representing refraction

and shoaling results in shoreline evolution in good correspondence with the detailed method. The

models are used to illustrate the effects of several features of beach nourishment projects that are

of engineering interest. For a long uninterrupted shoreline, which has been nourished with the

same material as the native, and in the absence of structures and background erosion, initially

symmetric nourished planforms remain nearly symmetric as they evolve, even under oblique wave

attack. This is interpreted as due to the small aspect ratio (additional beach width to length)

of the nourishment project. For no background erosion or retention structures, the proportion of

sand remaining in the project area over a twenty year period is illustrated for projects of various

lengths and various effective wave heights. The effects of shore-perpendicular structures placed at

the ends of the project with and without background erosion are illustrated. For normally incident








waves and no background erosion, the project with or without retention structures causes only

accretion to the adjacent shorelines. However, in the presence of background erosion or oblique

waves, retention structures can cause localized erosion to the adjacent shorelines.

Finally, the effects of nourishing with material more and less transportable than the native are

illustrated. If less transportable, the nourishment tends to "armor" the project area and oblique

waves can cause localized downdrift erosion. It is concluded that the simple method can provide

reliable predictions of the evolution of nourishment projects and is thus useful in conducting

benefit/cost analyses.


INTRODUCTION


Beach nourishment, the placement of large quantities of sand in the nearshore region to advance

the shoreline seaward, is being applied increasingly as a method of erosion control. Advantages

over other methods include maintaining a near-natural wide beach for storm protection and recre-

ational purposes. In some localities the wide beach also provides suitable nesting areas for several

endangered species of sea turtles. Sand placed in this manner along a long, uninterrupted shore-

line represents a perturbation which, over time, tends to be smoothed out by longshore sediment

transport. Additionally, if placed at an initial profile that is steeper than "equilibrium", offshore

transport will occur with an associated narrowing of the dry beach. Realistic prediction of the

performance of beach nourishment projects includes analysis of the cross-shore and longshore

transport processes and is of considerable significance to a rational evaluation of the economic

benefits of the project. Unfortunately, at present there are few beach nourishment projects that

have been documented adequately to allow detailed and quantitative evaluation of calculation

procedures.

Methods have been proposed by a number of investigators to predict the performance of nour-

ishment projects. A beach nourishment manual has been developed based on experience in the

Netherlands (Pilarczyk and Overeem, 1987) which, in addition to performance, addresses dredge

equipment, manner of placement, environmental effects, etc. Krumbein and James (1965), James








waves and no background erosion, the project with or without retention structures causes only

accretion to the adjacent shorelines. However, in the presence of background erosion or oblique

waves, retention structures can cause localized erosion to the adjacent shorelines.

Finally, the effects of nourishing with material more and less transportable than the native are

illustrated. If less transportable, the nourishment tends to "armor" the project area and oblique

waves can cause localized downdrift erosion. It is concluded that the simple method can provide

reliable predictions of the evolution of nourishment projects and is thus useful in conducting

benefit/cost analyses.


INTRODUCTION


Beach nourishment, the placement of large quantities of sand in the nearshore region to advance

the shoreline seaward, is being applied increasingly as a method of erosion control. Advantages

over other methods include maintaining a near-natural wide beach for storm protection and recre-

ational purposes. In some localities the wide beach also provides suitable nesting areas for several

endangered species of sea turtles. Sand placed in this manner along a long, uninterrupted shore-

line represents a perturbation which, over time, tends to be smoothed out by longshore sediment

transport. Additionally, if placed at an initial profile that is steeper than "equilibrium", offshore

transport will occur with an associated narrowing of the dry beach. Realistic prediction of the

performance of beach nourishment projects includes analysis of the cross-shore and longshore

transport processes and is of considerable significance to a rational evaluation of the economic

benefits of the project. Unfortunately, at present there are few beach nourishment projects that

have been documented adequately to allow detailed and quantitative evaluation of calculation

procedures.

Methods have been proposed by a number of investigators to predict the performance of nour-

ishment projects. A beach nourishment manual has been developed based on experience in the

Netherlands (Pilarczyk and Overeem, 1987) which, in addition to performance, addresses dredge

equipment, manner of placement, environmental effects, etc. Krumbein and James (1965), James








(1974), and Dean (1974) have proposed ad hoc but quantitative methods of assessing the quality

of material. Application of the James method is described in the Shore Protection Manual (1984).

In general, these methods attempt to establish the required volume of borrow material which is

equivalent to one unit volume of native material. However, other parameters critical to project

performance are not addressed by these procedures.

Pilkey and his co-workers (Pilkey and Clayton, 1989; Leonard, Clayton and Pilkey, 1990;

Leonard, Dixon and Pilkey, 1990) have conducted "broad brush" analyses of the performances of

many projects along the East and Gulf coasts of the United States. Attempts have been made

to define project performance in terms of the actual versus design lifetimes of the projects. Also

performance was plotted versus project length, sediment grain size, etc. In general, they found

little or no correlation between expected and actual performances. Rather, the following equation

was proposed for the East Coast of the United States to represent the required renourishment

volumes, -VR, conducted at intervals of n (in years) for an initial restoration volume, -V, and a

project life, N (in years).

-R = N (1)
n
It was recommended that n = 9 for Florida, n = 3 for New Jersey, and n = 5 for the remaining

portions of the East coast. Eq. (1) simply states that it will be necessary to renourish with an

amount equal to the initial restoration volume every n years. The rationale for Eq. (1) is not

apparent from sediment transport considerations. The studies by Pilkey and his co-workers do

not recognize explicitly the effects of background erosion, sediment quality or project length on

project performance.

Dean (1983) reviewed available methods of predicting beach nourishment performance and

showed that based on the Pelnard Considere (1956) solution and in the absence of background

erosion, the time, tp, required for a project to lose p percent of the material placed is

-tp a 2 (2)

in which i is the project length and Hb is the height of the breaking waves which mobilize the

sediment causing the spreading out beyond the project limits. Recently Dean (1988b) has shown








that the proportionality factors for 50% "loss" are approximately, for the units as shown


t50% = K 2 (3)
H5/2

in which t50% is in years and K" = 0.172 for i in km and Hb in meters, and K" = 8.7 for I in

miles and Hb in feet. It is emphasized that the material "lost" from the project area is transported

alongshore to project adjacent areas and continues to provide benefits there (Dean, 1988a).

BACKGROUND


The bases for predicting the performance of beach nourishment projects are the equations of

continuity and transport. In general, these may be used to develop a one-line model in which

only one contour (usually the mean water line) is used to represent shoreline changes (e.g. Bakker,

1968; LeMehaute and Soldate, 1977; Perlin, 1978; Walton and Chiu, 1979; Perlin and Dean, 1985;

Hanson, 1989; Hanson and Kraus, 1989). Bakker (1968) has developed a two-line model which,

for example, allows for profile steepening and flattening updrift and downdrift of a structure,

respectively. Hanson and Kraus (1987) have compared the results of one-line numerical models to

analytical solutions applicable for several beach nourishment initial planforms. An n-line model

has been developed by Perlin and Dean (1985) in which an arbitrary number (n) of contour lines

is used to represent the beach profile. A restriction of the multi-line models developed to date

is that the profiles represented must be monotonic. An alternative formulation method would be

to represent the topography by a grid system as is commonly done in hydrodynamic modeling

thereby eliminating the monotonic requirement. For models which represent the profile by more

than one contour, it is necessary to specify a relationship for cross-shore sediment transport.

Such models have been developed by Kriebel (1982) and Kriebel and Dean (1985) for profiles

which must vary monotonically and Larson (1988) and Larson and Kraus (1989) for models not

requiring monotonocity.








Governing Equations


The equation of sediment conservation can be expressed for three dimensions as

Oh Oq Oy (4)
t= V- = x + y (4)
-+-
81 Ox ay

in which h is the water depth relative to a fixed datum, t is time, V is the horizontal vec-

tor differential operator (V () = + j q is the horizontal sediment transport vector

(q=i q,+ j q,) as presented in Figure 1, and i and j are the unit vectors in the x and y

directions, respectively. Integrating Eq. (4) in the cross-shore direction from a landward location,

yi, where the cross-shore transport (qy) is zero to a seaward location, y2, where qy is similarly

zero, yields
a v2 9 vY2
I j hdy I2 qdy = 0 (5)

The first term is recognized as minus the time rate of change of volume of sand,-V and the second

integral is the total longshore sediment transport, Q. Making these substitutions

&V aQ
+ = 0 (6)
Ot Ox

In one-dimensional model formulations, it is usually assumed that accretion or erosion of a profile

is associated with a seaward or landward displacement respectively of the profile without change

of form. The vertical extent of this change is from some depth, h., of limiting sediment motion

up to the berm elevation, B. Thus the change, Ay, in any contour associated with a change in

volume, AV, is

Ay =AN (7)
h. + B

which when substituted in Eq. (6) yields the one-dimensional equation for conservation of sand,

ay 1 8Q
y+ 1 0 (8)
Ot + (h + B) Ox

The one-dimensional equation of sediment transport can be expressed as (Komar and Inman,

1970)

I = KPt, (9)



















Depth
Contours


Crest


. h*( (x,y)

Depth of Limiting Motion


Figure 1. Definition Sketch.








in which I is the immersed weight sediment transport rate, Pt, is the longshore energy flux factor

and K is a non-dimensional sediment transport proportionality factor. These two quantities can

be expanded

I = Qpg(s- 1)(1-p) (10)



Pt, = EbCG sin Ob cos b (11)

in which s is the ratio of mass densities of sediment to water (r 2.60), g = gravitational acceleration,

p = in-place sediment porosity (taken here as 0.35), E = wave energy density, CG = wave group

velocity, 0 = angle between the wave crests and the bottom contours, and the subscript "b" denotes

that the subscripted variable is to be evaluated at the breaker position. Based on small-amplitude

shallow water wave theory and assuming that the breaking wave height Hb is proportional to the

breaking water depth, hb (Hb = Khb)

H2
Eb = pg9- (12)



CGb = Cb = = = gHb/ (13)

in which K is a proportionality constant (; 0.78) and Cb = wave celerity at breaking. Substituting

Eqs. (10), (11), (12) and (13) into Eq. (9),
K Hs5/2gi/2
Q = K ( _-1)(- sin Ob cos Ob (14)
8 (s 1)(1 p)
Eqs. (8) and (14) form the basis for a one-dimensional numerical model. Pelnard-Considere

(1956) has shown that by linearizing Eq. (14) with respect to perturbations in the predominant

shoreline alignment and combining the result with Eq. (8), the one-line model is transformed into

the heat conduction equation
ay y
S= G (15)
Tt Ox2
in which G is the longshoree diffusivity", defined as

K H 5/2)(1 -)
G 8(-)(-p)(= (16)
8(s 1)(1 p)(h* + B)








In the finite difference solution of Eq. (15), it can be shown that a critical time increment,

Act, exists which, if exceeded, will cause the numerical solution to become unstable,

1 A2(17)
At- 2 G (

in which Ax is the alongshore grid spacing. By inspecting Eqs. (16) and (17) it can be seen

that the smaller the grid spacing and the larger the wave height, the smaller the allowable time

increment.

For multi-line models, it is necessary to specify relationships for the distributed longshore

(q.) and cross-shore (qy) sediment transport. Kriebel (1982) and Kriebel and Dean (1985) have

proposed the following for the cross-shore transport rate

qy = K'(V ?D.) (18)

in which D and 9. are the actual and equilibrium values of wave energy dissipation per unit water

volume and K' is a proportionality factor.

METHODOLOGY


Only one-line models are discussed here. Two models using quite different methodologies for

representing wave refraction and shoaling will be presented, applied and the results compared. In

the first model, refraction and shoaling will be represented by a very simple one-step procedure

whereas in the other a detailed grid-based solution is used.

Procedures Common to Both Methods

Grid System and Transformation of Initial Geometry The continuity and transport

equations (Eqs. (8) and (14)) were solved using an explicit method with the grid system shown

in Figure 2. The shoreline displacements are maintained fixed while the transport is computed

and in the second part of the same time step the transport is held constant while the shoreline

displacements are computed. Thus although the primitive equations are employed, the limiting

time step specified by Eq. (17) is still valid. In areas of special interest where greater resolution is

required, smaller grid elements, (Ax), can be used in the grid system presented in Figure 2.








and the effective value of C02/C. (or equivalently, wave period) to be used in Eq. (29) is the

denominator of Eq. (35) raised to the 2.4 power.


RESULTS


Several examples were investigated to demonstrate the effects of various design options and to

establish, by comparison with the detailed procedure, the relative validity of the simple procedure.

In the examples the background erosion will be taken as zero unless otherwise stated. For those

examples including effects of background erosion, only longshore transport contributions will be

represented. Unless noted otherwise, the grid spacing, (Ax), and project length for all examples

were 150 m and 6,000 m, respectively. At the lateral boundary conditions, the shoreline position

changed according to the background erosion rate and the domain was selected to be sufficiently

large that the perturbation effects at the ends of the domain were insignificant. Other character-

istics of these examples are presented in Table I.


Example 1 Initially Rectangular Planform, i = 6000 m, Y = 30 m, Ho = 0.60 m,

T = 6.0s, ao = Po = 90, h*, = 5.5 m, B = 2.5 m.

The results of this example comparing the two methods is presented in Figure 5 for 1, 3, 5

and 10 years after nourishment. The simple and detailed procedures yield very similar results

which is somewhat surprising in view of the extremely simple algorithm employed to represent

wave refraction and shoaling. Again the interpretation is that the aspect ratio of the nourishment

project (Y/i) is quite small.


Example 2 Initially Rectangular Planform, = 6000 m, Y = 30 m, Ho = 0.6 m,

T = 6s, o = 700, o = 900, h. = 5.5 m, B = 2.5 m.

Conditions for this example are the same as for Example 1, except the deep water wave di-

rection is 200 oblique to the shoreline. Reference to Figure 6 demonstrates that the two methods

are in quite good agreement with surprisingly little planform asymmetry due to the oblique wave

direction. This asymmetry is examined in greater detail in the next two examples.


























Q.





\ i-i-;
Note:
Ob=1P,-ab
Yi



Ax+ y i+


x


Figure 2. Definition Sketch for Numerical Model.








In conducting numerical modeling of generally straight shorelines where a large perturba-

tion has been introduced, it is recommended, in general, that the original shoreline and offshore

bathymetry be represented by straight and parallel contours. This is equivalent to the ad hoc

transformation shown in Figure 3. The rationale for the transformation is that, with the exception

of the long-term background erosion that will be discussed subsequently, the nearshore system has

approached a near-equilibrium, the details of which present modeling techniques cannot represent

adequately. The equilibrium may depend on rather subtle and perhaps nonlinear three-dimensional

wave transformation (refraction, diffraction and shoaling) over minor bathymetric features. Usu-

ally, if a model is applied to a shoreline situation such as Figure 3a, it is found that the shoreline

and offshore contours will approach unrealistically straight alignments and/or the changes will

occur over time scales that are much shorter than actual. Thus numerical modeling is much more

effective in cases where substantial perturbations (human-induced or natural) have placed the sys-

tem out of balance. The transformation presented in Figure 3 recognizes that the changes caused

by the perturbation introduced will be of a much shorter time scale than those associated with

the pre-existing non-equilibrium causative features. Subsequent to the modeling, the results can

be transformed back to the actual system.

Background Erosion Long-term background shoreline changes must be accounted for ap-

propriately if the nourishment evolution is to be predicted with good accuracy. This is especially

the case if stabilizing structures are to be installed. Long-term background changes could be

caused by divergences in longshore or cross-shore transport (cf Eq. (4)). Thus, the general treat-

ment considers the long-term background changes composed of a longshore transport component,

YB,L, and a cross-shore transport component, YB,C

8yB 9YB,L 9YB,C (19)
at at+ at19)

and the background change, -y, is expressed as background longshore transport, according to

Eq. (8),

,() YBL at









Shoreline
/r- Contours


Ad Hoc Transformation

b


Shoreline
/Contours


a) Initial Actual Shoreline b) Initial Shoreline and
and Contours Contours to be Modeled.

Figure 3. Recommended Ad Hoc Transformation for Modelling Coastal Systems in which Large
Perturbations are to be Introduced.








In the finite difference solution of Eq. (15), it can be shown that a critical time increment,

Act, exists which, if exceeded, will cause the numerical solution to become unstable,

1 A2(17)
At- 2 G (

in which Ax is the alongshore grid spacing. By inspecting Eqs. (16) and (17) it can be seen

that the smaller the grid spacing and the larger the wave height, the smaller the allowable time

increment.

For multi-line models, it is necessary to specify relationships for the distributed longshore

(q.) and cross-shore (qy) sediment transport. Kriebel (1982) and Kriebel and Dean (1985) have

proposed the following for the cross-shore transport rate

qy = K'(V ?D.) (18)

in which D and 9. are the actual and equilibrium values of wave energy dissipation per unit water

volume and K' is a proportionality factor.

METHODOLOGY


Only one-line models are discussed here. Two models using quite different methodologies for

representing wave refraction and shoaling will be presented, applied and the results compared. In

the first model, refraction and shoaling will be represented by a very simple one-step procedure

whereas in the other a detailed grid-based solution is used.

Procedures Common to Both Methods

Grid System and Transformation of Initial Geometry The continuity and transport

equations (Eqs. (8) and (14)) were solved using an explicit method with the grid system shown

in Figure 2. The shoreline displacements are maintained fixed while the transport is computed

and in the second part of the same time step the transport is held constant while the shoreline

displacements are computed. Thus although the primitive equations are employed, the limiting

time step specified by Eq. (17) is still valid. In areas of special interest where greater resolution is

required, smaller grid elements, (Ax), can be used in the grid system presented in Figure 2.








in which QB,L(XR) denotes the background transport at some reference location, XR, to be based

on local knowledge or other calculation procedures not discussed here.

Numerical Solution of Governing Equations The azimuth, Pi, of the shoreline normal

at the nth time level, is established to represent the value at the grid line associated with Qg (see

Figure 2)

i" = ~ + tan-1 ( 1 (21)
2 Zi+l Xi/
The background transport, QB,L, (Eq. (20)), is added to the longshore transport resulting from

the planform anomaly to yield the total transport, Qn,

Q = Q1 + QB,L, (22)

Finally, the shoreline position is updated from the nth to the (n + 1)th time level

y?"1 = y" + z(h + B)( T QT+) + t (23)

Procedures Which Differ for the Two Methods Wave Refraction and Shoaling

Simplified Wave Refraction and Shoaling We start by showing several results from lin-

ear wave theory using the wave and contour directions shown in Figure 4. The difference Ap,

between the nourished contour orientation, s,(h < h.), and the deep water contour orientation,

0o, (h > h.), can be considered small since the ratio of nourished beach width, Y, to length, ,

is generally on the order of 0.02 at most. Conservation of wave energy flux from deep water to a

water depth, h,+, is

EoCGo cos(flo ao) = E+ CG.+ cos(/3o a.+) = E.CG. cos(Po a.) (24)

in which the subscripts "+" and "*" indicate conditions just seaward and landward of the depth

transition respectively (cf. Fig. 4). The energy flux just landward of the transition may be equated

to that at breaking by

E.CG. cos(po a,) = EbCGb cos(3o ab) (25)

Since 3,(x) = 3o + Ap(x)

EC cos(, ) = EoCGo cos(13o Qo) EbCGb sin(o, ab) sin() (26)
EbCGb cosi b) (26)
cos(Ap)








North


h=h*


Y

Shoreline












i.





**.
*
AX


-Contours -




x)
Waves


Deep Water
Contour
N
ao

y ep
V 0-


A
_t


a) Planform Showing Perturbed Contours to Depth h,


b) Profile Through A-A

Figure 4. Definition Sketch for Effect of Beach Nourishment on Contours.
18








Utilizing shallow water linear wave theory and the wave height breaking proportionality factor,

neglecting terms modified by sin(A3) sin(3, ab) and approximating cos(A/) by unity (At is

small),
S2H CG0, coS(PO 0) 0.2
cb = [ ^cos(f3,-a,) J (27)
Cb 2 COS(p, b)

which will be useful later. Applying Snell's Law in a similar manner across the transition and to

breaking,
sin(-, a.+) sin(p, a.) sin(s, ab)
(28)
C*+ C. Cb
and a,+ can be determined by applying Snell's law from deep water to h,.+ Eqs. (26), (27) and

(28) will be used to express the transport relationship (Eq. (14)) primarily in terms of deep water
wave conditions. With a limited amount of algebra, we obtain

KH2.4C 2 0.4 COS1.2(PO )
Q = 0 0sin(P, a, ) (29)
8(s 1)(1 p)C.o.4 a29

in which terms modified by sin(Af) sin( ab) have been neglected and cos(Af) has been ap-

proximated by unity. It is interesting to note that if Eq. (29) is linearized in the form of Eq. (15),

the appropriate value of longshore diffusivity, G, in terms of deep water wave conditions can be

expressed as

G" -KHI 2Co.4 C.20.4 cos1.2(po a0) cos 2(o a.) (
8(s 1)(1 p)C.no.4(h, + B) cos(o a) a.(30)

Eq. (29) is the relationship for the longshore sediment transport rate that completes the simple

model development.

Detailed Refraction and Shoaling The detailed method of solution differs from the sim-

ple method as refraction and shoaling are carried out by the detailed procedure. This method as

first presented by Noda (1972), and employed by Perlin and Dean (1983) and Dalrymple (1988),

is based on the irrotationality of wave number and conservation of wave energy. The reader is

referred to the papers noted above for further detail.


V x = 0 (31)








in which k is vector wave number. Eq. (31) can be expanded to


(k sin 0) = (k cos 0) (32)

In the present model, a two-dimensional grid is established and Eq. (32) is expressed in finite

difference form as
S[r(k cos O)_l,-_- + (1 2r)(k cos 0)i,j+1

1 = cos-1 +r(k cos O)i+1,+x (33)

2_ "((k sin O)ij+l (k sin 0)-, j)

in which r is a smoothing factor taken as 0.25 in application here and all k and 0 values on the

right hand side are understood to be at the (n + 1) time step.

The above equation is solved by considering the wave direction, 0, at the offshore boundary as

known and the wave directions along the lateral boundaries given by Snell's Law. These conditions

along the lateral boundaries should not affect the interior solution if the lateral boundaries are

sufficiently distant from the area of interest.

With the refraction solution for a given time step, the conservation of wave energy equation is

solved to determine the wave height field, Hi,. The equation for conservation of wave energy (Eq.

(24)), is expressed in finite difference form as

r(H2CGsin 8)i-_,j+l + (1 2r)(H2CG sin )i,j+ 1/2

S (C sin +r(H2CG sin O)i+,j+1 (34)
i (CG sin 0)jj

+A- ((H2CG cos 0)i+,,j (H2CG cos 0)i,_,)
As with wave refraction, in solving Eq. (34), the wave height along the outer boundary is considered

as known and along the lateral boundaries, it is assumed that a = 0. Eq. (34) is iterated until

convergence of the solution is obtained to within an arbitrary pre-defined limit.

The wave height calculations are carried landward to a location where the inequality Hf[, >

Khj occurs. The transport, Q', is then calculated by transforming the wave conditions at i,j'- 1

to the breaking location much as was done for the simple method in transforming wave conditions

from deep water to the breaking point.








This concludes description of the simple and detailed procedures for calculating shoreline evo-

lution following a nourishment project. While the required computer times are not large for either

procedure, the times for the simple procedure were approximately 1/250 of those for the more

detailed procedure.

Effective Wave Height and Period

Although the wave height and period change continuously with time, it is evident that in the

absence of littoral barriers, there is an effective constant wave height and period that will produce

the same spreading out of the beach nourishment material as the actual time-varying values.

In the following development, a Rayleigh distribution f(H) for wave heights will be assumed

2f(H) = -(H/Hm (35)
f()- H2rms

in which Hrm, is the root-mean-square wave height. Referring to the sediment transport equation

(Eq. (29)) expressed in terms of deep water conditions, for a given sea state characterized by the

significant deep water wave height H,, the effective deep water wave height, Ho.,,, is given by

Hoeff = [m H2.4p(H)dH] (36)

Evaluating Eq. (36) numerically and using the approximate Rayleigh distribution relationship

between significant and root-mean-square wave heights (H, VI2Hms), it can be shown that

Ho,,f = KrmsHrms = KHs (37)

where Krms = 1.04 and K, = 0.735. Thus the long-term effective wave height Ho.ff at a particular

location is

Ho.f = E (K, HsW)24 (38)

in which H,, is the significant wave height of the nth record in a series of N records encompassing

the time period of interest. Examining Eq. (29), it is clear that a somewhat more appropriate but

U1.
more cumbersome value of effective wave height, H'1 ,is given by



eff = (39)
[1 N GoE n 2.
N n=1 C*n









TABLE I. CHARACTERISTICS OF EXAMPLES PRESENTED*


Wave Characteristics Results Special
Example Ho T ao in Characteristics Purpose of Example
(m) (sec) (0) Figure(s)


0.6




0.6


0.6



0.6




0.4,0.8

Variable



0.6





0.6


900




700


700,800,900



700,800




900

900



900





700


* Unless Stated Otherwise, the Nourished Length, of Shoreline is 6,000 m.


5




6


7,8



9




10

11



12,13,14





16,17


None




None


None



None




None

Various Wave
Heights and
Project Lengths

Two Project
Retention Structures
Present, With and
Without Background
Erosion

Nourishment with
Sands of Different
Transport
Characteristics


Compare Two
(Simple and Detailed)
Methods for Normal
Wave Incidence

Compare Two Methods
for Oblique Waves

Compare Two Methods
for Range of Wave
Directions

Detailed Examination
of Antisymmetric
Component of
Response

Effect of Wave Height

Effects of Wave Heights
and Project Lengths on
Project Longevity

Illustrate Effects of
Retention Structures
for Various Background
Transport Conditions


Effect on Evolution
Patterns













50.0 I1-- I I

... SIMPLE METHOD
DETAILED METHOD




Initial Planform
30.0 ---
S 1 year

S//:. ...... 3 years
y -m) 5 years

10 years

10.0 -








- 10 .0 I I I I I I I I i I
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.

x(m)

Figure 5. Example 1. Comparison of Beach Nourishment Evolution for Simple and Detailed
Methods of Wave Refraction and Shoaling, Normal Wave Incidence, Ho = 0.6 m, T = 6.0 sec, co
= 900. No Background Erosion.


I












50.0


30.0 ,
1 year
3 years

y (m) -5 years
S10 years


10.0







-10.0 I I I i I I I I
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.

x(m)

Figure 6. Example 2. Comparison of Planform Evolution Obtained by Two Methods for 20
Oblique Waves, Ho = 0.6 m, T = 6.0 sec, ao = 70. No Background Erosion.










Example 3 Same As Example 2, Except Response to Different Wave Directions

Figures 7 and 8 present shoreline planforms at 1, 3, 5 and 10 years after nourishment for wave

directions of 70, 800 and 900, i.e. 200, 100 and 0 obliquity to the general shoreline alignment.

It is seen that the results obtained by the two methods are quite similar and that the effects of

wave direction are relatively small. Although slight, the major effect seems to be that for the

more oblique angles, there is less wave energy flux toward the shoreline resulting in less longshore

sediment transport. A small asymmetry is evident.



Example 4 Same as Example 2, Except Detailed Examination of Planform Asym-

metry for Oblique Wave Directions

The purpose of this example is to examine the asymmetry resulting from an oblique wave acting

on a nourished planform which is initially symmetric. Figure 9 presents the results ten years after

nourishment for deep water wave directions, ao, of 700 and 800, i.e. 200 and 100 obliquity to the

pre-nourished shoreline alignment. In order to quantify the antisymmetry, the planforms were

separated into even and odd components, ye(x') and yo(x') where x' is the longshore coordinate

with origin at the project centerline. Denoting yT(x') as the total shoreline displacement, it can

be shown that the even and odd components are determined as

1
ye(x') = [yr(x') + YT(-x')] (40)



o(x') = [YT(') yr(-x')] (41)

The maximum displacement associated with the odd component is 1 m. The reason that the odd

components are so small is that the aspect ration (Y/g) of the nourished planform is so small, in

this case 30 m/6000 m = 1:200. Thus the asymmetries are small and are primarily due to the

nonlinearities in the sin 20 term of the transport equation.








and the effective value of C02/C. (or equivalently, wave period) to be used in Eq. (29) is the

denominator of Eq. (35) raised to the 2.4 power.


RESULTS


Several examples were investigated to demonstrate the effects of various design options and to

establish, by comparison with the detailed procedure, the relative validity of the simple procedure.

In the examples the background erosion will be taken as zero unless otherwise stated. For those

examples including effects of background erosion, only longshore transport contributions will be

represented. Unless noted otherwise, the grid spacing, (Ax), and project length for all examples

were 150 m and 6,000 m, respectively. At the lateral boundary conditions, the shoreline position

changed according to the background erosion rate and the domain was selected to be sufficiently

large that the perturbation effects at the ends of the domain were insignificant. Other character-

istics of these examples are presented in Table I.


Example 1 Initially Rectangular Planform, i = 6000 m, Y = 30 m, Ho = 0.60 m,

T = 6.0s, ao = Po = 90, h*, = 5.5 m, B = 2.5 m.

The results of this example comparing the two methods is presented in Figure 5 for 1, 3, 5

and 10 years after nourishment. The simple and detailed procedures yield very similar results

which is somewhat surprising in view of the extremely simple algorithm employed to represent

wave refraction and shoaling. Again the interpretation is that the aspect ratio of the nourishment

project (Y/i) is quite small.


Example 2 Initially Rectangular Planform, = 6000 m, Y = 30 m, Ho = 0.6 m,

T = 6s, o = 700, o = 900, h. = 5.5 m, B = 2.5 m.

Conditions for this example are the same as for Example 1, except the deep water wave di-

rection is 200 oblique to the shoreline. Reference to Figure 6 demonstrates that the two methods

are in quite good agreement with surprisingly little planform asymmetry due to the oblique wave

direction. This asymmetry is examined in greater detail in the next two examples.













50.0


30.0 ....... -- -
1 year

S-// 3 years

y m) /( m -5 years
1. 10 years


10.0





-10.0 ----' ---------'-----------------------'-

-10.0
0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.
x(m)

Figure 7. Example 3a. Planform Evolution by Detailed Method for Deep Water Wave Direc-
tions, ao = 70, 800, 900. Ho = 0.6 m, T = 6.0 sec. No Background Erosion.









































0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.

x(m)

Figure 8. Example 3b. Planform Evolution by Simple Method for Deep Water Wave Direc-
tions, ao = 70, 800, 90. Ho = 0.6 m, T = 6.0 sec. No Background Erosion.


50.0


30.0



y(m)



10.0








-10.0












50.0


30.0 ---------------- -------------------- .



ym ) Even Components
Y( i
/.0 \


10.0 -/ Odd Components








-10.0
_- __K_- '- .





-10 .0 I I I i I I I i I i I I i I

0. 1800. 3600. 5400. 7200. 9000. 10800. 12600. 14400. 16200. 18000.

x(m)
Figure 9. Example 4. Even and Odd Components of Shoreline Position After 10 Years for
Deep Water Wave Directions, ao = 70 and 80. Results Obtained by Detailed Method.


1








Example 5 Effect of Wave Height

Figure 10 presents the planforms calculated by the simple method at 1, 3, 5 and 10 years after

nourishment and for 0.4, and 0.8 m wave heights and normal wave incidence. Of particular interest

is the major role of wave height in causing spreading out of the nourished planform. This effect is

also evident from Eq. (2) which indicates that the spreading out is proportional to H'/2.



Example 6 Effects of Various Wave Heights and Project Lengths

Earlier examples have demonstrated (for homogeneous sediment conditions) the relative in-

significance of wave direction. The simple model was exercised to demonstrate the effect on the

fraction of sand remaining in the project area for a wide range of wave heights and project lengths.

These results are presented in Figure 11 where the proportion remaining is shown over a 20 year

time period for nine combinations of wave heights and project lengths. For a small wave height

and long project (Ho = 0.30 m, i = 24,000 m), it is seen that at the end of a 20 year period,

over 95% of the material remains. By contrast, for a four-fold larger wave height and a length

one-sixteenth of the former, the sand remaining after one year is less than 40% of that placed.

This illustrates the significance of project length and wave height. It has been noted earlier that

the longevity of a project varies with the square of the project length and inversely with the wave

height to the 2.5 power. It is emphasized that the results presented in Figure 11 do not include

effects of background erosion which can be of considerable magnitude, especially over long time

periods.



Example 7 Effects of Retention Structures

Several examples will be presented illustrating the effects of retention structures, with and

without background erosion. The algorithm for the transport boundary condition at a structure

is fairly complex and will not be described fully here. In general the algorithm is consistent with

Eqs. (21), (22), (23) and (29) with transport determination based on relative positions of structure

tip and the shorelines on the two adjacent grid cells and the background erosion transport rate

components. Diffraction was not included in the examples presented here.












50.0


30.0







10.0 -
S /// \






-10.0 -
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)

Figure 10. Example 5. Illustration of Wave Height Effect on Rate of Planform Evolution.
Results Based on Simple Method, so = 900, T = 6.0 sec. Results Shown for 1, 3, 5 and 10 Years.
No Background Erosion.











1.0- .._ ---- --
S" -- Ho=0.3 m, 2= 24000 m
ft 0 8... H,=0.6 m, -24000 m

Z 0.8 \! -- H=1.2 m, = 24000 m

Sl Ho=0.3 m, j= 6000 m
LI > %

DLu 0.6 \ --- .
S, Ho=0.6 m, = 6000 m


^O \ru .. "". ^ -
0.4 ". Ho=1.2 m, .= 6000 m


o Ho=0.9m, 9= 1500

0Q Ho=0.6m, A=1500
0.
a-
CL 1Ho=1.2m, J= 1500m

0 2 4 6 8 10 12 14 16 18 20

YEARS AFTER NOURISHMENT

Figure 11. Example 6. Effects of Various Project Lengths and Wave Heights on Project Longevity.








Example 7a and Figure 12 present results for no background erosion and normal wave incidence.

Structures one-half the length of the initial project width (30 m) are present at the two ends of

the project. Here, the structure length should be interpreted as the "effective" structure length.

Consistent with intuition, it is seen that the shorelines immediately adjacent to the project advance

as sand from the project area "spills" around the structures; however as the project recedes

approaching the structure length, less and less sand is transported to the project adjacent areas and

the planform evolves toward one of shoreline segments straight and parallel to the incoming waves

with the beach width within the project area equal to the lengths of the stabilizing structures.

Example 7b and Figure 13 present the results for the same conditions as Example 7a, except

there is a uniform background erosion of 0.5 m/year and the reference background erosion is taken

as zero at the project centerline. The transient results are qualitatively similar to those of Exam-

ple 7a. However, because of the background erosion, an equilibrium planform exists only within

the confines of the stabilization structures; within the project area, the equilibrium planform is

concave outward and symmetric about the project centerline. For equilibrium conditions within

the project area, the background erosion transport is exactly balanced locally by the planform

orientation transport due to the waves. This accounts for the character of the planform in Fig-

ure 13. At a great distance from the project area, the shoreline retreats at the background rate

of 0.5 m/year; however, immediately adjacent to the retention structures, the erosion rate would

be greater to compensate for the effect of the reduced erosion (at later times) within the project

area.

The final example (7c) that will be presented illustrating the effects of structures is the same as

example 7b, except now the zero reference background transport is located 4,500 m to the left of

the left structure in Figure 14. It is seen that updrift of the left structure, transport is toward the

structure and the shoreline accretes there. Inside the two structures, sand is transported initially

in both directions, but somewhat later the positive transport prevails and sand is carried past

only the right hand structure. Eventually, as before, the planform within the structures will be

aligned for equilibrium and both the downdrift and updrift shorelines will erode. For zero reference












50 .0 I I I I I -ii






Initial Planform
30.0 .......
1 year

S3 years
ym 5 years

10 years

100 Structure Structure

// /



-10.0~~~~------ ----------------------------- ---

-10.0 I I i I I I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)

Figure 12. Example 7a. Effects on Planform Evolution of Two Shore-Normal Retention
Structures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence, No
Background Erosion. Ho = 0.6 m, T = 6.0 sec.












50.0


30.0 -------------
1 year

/ .- \ 3 years
y(m),/
Uy( 5 years
-- -- 10 years

10.0 Structure Structure








-10.0
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.

x(m)

Figure 13. Example 7b. Effects on Planform Evolution of Two Shore-Normal Retention Struc-
tures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence. Uniform
Background Erosion Rate at 0.5 m/yr, Zero Background Transport at Project Centerline. Ho =
0.6 m, T = 6.0 sec.












50.0


30.0 .-----..- ....-------
1 year

//--- 3 years

m5 years

10 years
10.0 -
Structure Structure


----- ---r-u-c-t-ur--



- 1 0 0 I I I I I I I g I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)

Figure 14. Example 7c. Effects on Planform Evolution of Two Shore-Normal Retention Struc-
tures of Length Equal to One-Half the Initial Project Width. Normal Wave Incidence. Uniform
Background Erosion Rate at 0.5 m/yr, Zero Background Transport Located 4,500 m to Left of
Left Structure. Ho = 0.6 m, T = 6.0 sec.








transport located much farther to the left than for the example shown here, sand would bypass the

updrift structure, flow past the infrastructure segment and onto the downdrift shoreline. However,

both the updrift and downdrift shorelines at great distances from the project would continue to

erode at the background rate and only the updrift shoreline immediately adjacent to the project

area would experience a net accretion.



Example 8 Effect of Different Transport Characteristics for Native and Nourishment

Sands

Dean, et al. (1982) have examined the sediment transport factors, K (Eq. (9)), determined

from various field programs and have proposed the dependency of K on sand diameter, D, as

shown in Figure 15. Intuition suggests that the effects of nourishing with KF 0 KN could have a

significant effect on the adjacent shorelines. The subscripts "F" and "N" denote fill and native,

respectively. This effect will be greater for the case of oblique waves.

Figures 16 and 17 present the case of waves at a 200 obliquity acting on nourishment projects

with KFIKN = 0.9 and 1.1, respectively. It is seen from Figure 16 that for the case in which

the nourishment material is less transportable than the native, the nourishment planform acts as

an erodible barrier with an associated accretion and erosion on the updrift and downdrift sides of

the barrier, respectively. For the case in which the sand is more transportable than the native,

Figure 17, the pattern is qualitatively a mirror image of that noted.

As the project evolves, the calculation procedure requires determination of the degree to which

the sand exposed to the waves is of nourishment and native character. This was accomplished by

the following algorithm. A mixed layer of minimum thickness, Ymix, was assumed. If, during a

time increment, erosion occurred such that the mixed layer was less than Ymix thick, the remaining

material within the mixed layer was mixed with the underlying sand to re-establish a thickness

of Ymix. The character (i.e. K) of the mixed layer was calculated. If deposition occurred, the

mixed layer thickness and character were determined based on the thickness and character at

the previous time step and the character of the material in the littoral stream. Obviously, it was
























2.0 1 1 1 1
2.0
I-





LL
ZLU \

1.o \ -






0 0.5 1.0
DIAMETER, D (mm)

Figure 15. Plot of K vs D (Modified from Dean, et al., 1982).












50.0


30. 0--
Initial Planform

S1 year

/^ \\ 3 years

/ _/_ \ l 5 years

/0/ / 10 years







-10.0 I I I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 16. Example 8a. Planform Evolution for Nourishment Sand Less Transportable than
the Native (KF = 0.693, KN = 0.77). Note Centroid of Planform Migrates Updrift. Variation
of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 18.
Wave and other Project Conditions Presented in Table I.











50.0


30. ------0 ..-
----- Initial Planform

I II


// / | -5 years
/i / '\\ \
10.0 / \\\ 10 years






-10.0
- 10 .0 I I I I I I I I I I I
0. 3000. 6000. 9000. 12000. 15000. 18000. 21000. 24000. 27000. 30000.
x(m)
Figure 17. Example 8b. Planform Evolution for Nourishment Sand More Transportable than
the Native (Kp = 0.847, KN = 0:77). Note Centroid of Planform Migrates Downdrift. Variation
of Surface Layer K Values with Time at Locations A, B, C, D and E are presented in Figure 19.
Wave and other Project Conditions Presented in Table I.








necessary to calculate the character of the sand flowing into and out of a cell; this was accomplished

by starting with the updrift grid line as a boundary condition and, on cells where erosion occurred,

modifying the magnitude and character of the material in the littoral stream. In the nourished

area, the K value was set equal to KN when the nourished thickness reached zero. Figures 18

and 19 show the composite surface layer K values at the five locations indicated in Figures 16

and 17, respectively. It is noted that the centroids of the planform anomalies migrate updrift and

downdrift for KF < KN and KF > KN, respectively and that this profile migrational signature

could possibly be used in conjunction with a field monitoring program to establish the relative

sediment transport coefficients, KF and KN.

SUMMARY AND CONCLUSIONS


Two numerical methods have been presented for calculating shoreline evolution subsequent to

a beach nourishment project. These methods fall within the class of one-line shoreline models

in which the active vertical portion of the profile is represented by only one contour line. Both

models are capable of representing shore-perpendicular structures such as groins that might be

used to stabilize beach nourishment projects. One model is quite simple, with refraction and

shoaling from deep water to breaking conditions occurring in a closed form. In the second method,

refraction and wave shoaling are carried out on a two- dimensional grid using conditions of wave

number irrotationality and conservation of wave energy, respectively. The simple model requires

approximately 1/250 of the computer time of the more detailed model.

In applying the two methods, it is recommended that the actual pre-project bathymetry be

replaced by contour lines which are straight and parallel with an alignment of the general shoreline.

This recognizes implicitly the shorter time scales associated with the project evolution as compared

to that of the original bathymetric and shoreline disequilibrium. Pre-project background erosion is

interpreted as due to cross-shore and longshore transport components and is assumed to continue

unchanged after project construction.















0.80 I i I I






..--T
0.8 -------- ---i--------------------- -------------i-----i-- -- ]--|- -|-







Co



0 / -------- AT
9/ I-

c 0.70 /
bO ( --'- -- - ---- ---
I-







-- T E

0.60
S--- T C


-- AlT E

0.60 --I---I I I i I --I------ I I I I--

0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Years After Nourishment

Figure 18. Example 8c. Variation of Surface Layer Longshore Transport Coefficient K with
Time at the Five Locations Shown in Figure 16. Case of KF = 0.693, KN = 0.77, Ymix = 2.0 m.















0n n


I I I I I I I I


U)
c.
0

o 8





C I
0 -- --
, 0.8 -


0)




O
--. A T C




AT E
0)






0.70 I- T I

0 .7 0 -- ^ } -
0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Years After Nourishment


Figure 19. Example 8d. Variation of Surface Layer Longshore Transport Coefficient K with
Time at the Five Locations Shown in Figure 17. Case of KF = 0.847, KN = 0.77, Ymi, = 2.0 m.


1 I I 1 _








Comparison of results of applying the two models has established that they yield essentially

the same evolution. In the absence of littoral barriers, the planform asymmetry due to oblique

waves is shown to be small and is interpreted to be a result of the small aspect ratio (width to

length) of the nourished project.

The interaction of nourishment retention structures with background transport is examined.

Such structures can cause a wide range of effects within and adjacent to the nourished area

depending on the characteristics of the background transport.

The effects of nourishing with sediment of transport characteristics different than the native

are investigated. If the nourishment material is transported less readily, the nourishment project

acts as an erodible barrier causing accretion and erosion on the updrift and downdrift sides of

the project, respectively. For cases in which the nourishment material is transported more read-

ily than the native, the shorelines updrift and downdrift of the project both accrete with the

greater accretion occurring on the downdrift side. For nourishment materials which are less and

more transportable than the native, the centroids of the planform anomalies migrate updrift and

downdrift respectively with time.

The results presented herein predict much greater longevity than those by Pilkey and co-

workers although it is cautioned that most of examples do not include background erosion which

can be a significant site-specific factor.

It is concluded that for the range of cases and conditions examined, the simple model yields

results as valid as the more detailed model which requires much greater computer time. Finally,

models such as those developed herein can be effective in rational benefit/cost analyses of potential

beach nourishment projects.








necessary to calculate the character of the sand flowing into and out of a cell; this was accomplished

by starting with the updrift grid line as a boundary condition and, on cells where erosion occurred,

modifying the magnitude and character of the material in the littoral stream. In the nourished

area, the K value was set equal to KN when the nourished thickness reached zero. Figures 18

and 19 show the composite surface layer K values at the five locations indicated in Figures 16

and 17, respectively. It is noted that the centroids of the planform anomalies migrate updrift and

downdrift for KF < KN and KF > KN, respectively and that this profile migrational signature

could possibly be used in conjunction with a field monitoring program to establish the relative

sediment transport coefficients, KF and KN.

SUMMARY AND CONCLUSIONS


Two numerical methods have been presented for calculating shoreline evolution subsequent to

a beach nourishment project. These methods fall within the class of one-line shoreline models

in which the active vertical portion of the profile is represented by only one contour line. Both

models are capable of representing shore-perpendicular structures such as groins that might be

used to stabilize beach nourishment projects. One model is quite simple, with refraction and

shoaling from deep water to breaking conditions occurring in a closed form. In the second method,

refraction and wave shoaling are carried out on a two- dimensional grid using conditions of wave

number irrotationality and conservation of wave energy, respectively. The simple model requires

approximately 1/250 of the computer time of the more detailed model.

In applying the two methods, it is recommended that the actual pre-project bathymetry be

replaced by contour lines which are straight and parallel with an alignment of the general shoreline.

This recognizes implicitly the shorter time scales associated with the project evolution as compared

to that of the original bathymetric and shoreline disequilibrium. Pre-project background erosion is

interpreted as due to cross-shore and longshore transport components and is assumed to continue

unchanged after project construction.








Appendix I


REFERENCES

Bakker, W.T. (1968) "The Dynamics of a Coast With a Groyne System", Proceedings, Eleventh
International Conference on Coastal Engineering, ASCE, pp. 492-517.
Dalrymple, R.A. (1988) "Model for Refraction of Water Waves", Journal of Waterway, Port,
Coastal and Ocean Engineering, Vol. 114, No. 4, July, pp. 423-435.
Dean, R.G. (1974) "Compatibility of Borrow Material for Beach Fills", Chapter 77, Proceedings,
Fourteenth International Conference on Coastal Engineering, Copenhagen, Denmark, pp.
1319-1333.
Dean, R.G. (1983) "Principles of Beach Nourishment", In: Komar, P.D. (Ed.), CRC Handbook
of Coastal Processes and Erosion, Boca Raton: CRC Press, pp. 217-232.
Dean, R.G. (1988a) "Realistic Economic Benefits from Beach Nourishment", Chapter 116, Pro-
ceedings, Twenty-First International Coastal Engineering Conference, Malaga, Spain, pp.
1558-1572.
Dean, R.G. (1988b) "Engineering Design Principles", Short Course on Principles and Applications
of Beach Nourishment, Gainesville, FL: Florida Shore and Beach Preservation Association,
42 p.
Dean, R.G., E.P. Berek, C.G. Gable and R.J. Seymour (1982) "Longshore Transport Determined
by an Efficient Trap", Chapter 60, Proceedings, Eighteenth International Conference on
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Hanson, H. (1989) "Genesis A Generalized Shoreline Change Numerical Model", Journal of
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Hanson, H. and N. Kraus (1989) "Genesis: Generalized Model for Simulating Shoreline Change",
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Kriebel, D.L. and R.G. Dean (1985) "Numerical Simulation of Time-Dependent Beach and Dune
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Research Center, Washington, D.C.








Larson, M. (1988) "Quantification of Beach Profile Change", Report No. 1008, Department of
Water Resources Engineering, Lund University, Lund, Sweden.
Larson, M. and N.C. Kraus (1989) "SBEACH: Numerical Model for Simulating Storm-Induced
Beach Change Report 1: Empirical Foundation and Model Development", Technical Re-
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Vicksburg, Mississippi.
Leonard, L.A., K.L. Dixon and O.H. Pilkey (1990) "A Comparison of Beach Replenishment on
the U.S. Atlantic, Pacific and Gulf Coasts", Special Issue No. 6 Artificial Beaches, Journal
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LeMehaute, B. and M. Soldate (1977) "Mathematical Modeling of Shoreline Evolution", U.S.
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Perlin, M. (1978) "A Numerical Model to Predict Beach Planforms in the Vicinity of Littoral
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on Coastal Sediments '79, p. 809-837.








APPENDIX II. NOTATION
The following symbols are used in this paper:

b = subscript denoting breaking;
B = beach berm height;
C = wave celerity;
CG = group velocity;
D = characteristic sediment size;
V = wave energy dissipation per unit volume;
E = wave energy density;
f = probability distribution;
g = gravitational constant;
G = longshore diffusivity;
h = water depth;
h. = depth of closure;
h.+ = depth of original profile, immediately adjacent to filled profile, see Figure 4b;
H = wave height;
I = immersed weight sediment transport rate;
k = wave number;
K = sediment transport proportionality factor;
K' = proportionality factor for cross-shore sediment transport;
K" = proportionality factor for project longevity;
L = nourishment project length;
n = renourishment interval;
N = project life;
p = in place sediment porosity;
Pes = longshore energy flux factor;
s = relative sediment density, ps/p;
t = time;
T = wave period;
qx,y = sediment transport distribution in x, y directions;
Q = total longshore sediment transport;
-V = volume;
x = longshore coordinate;
y = cross-shore coordinate, position seaward;
Y = initial nourished beach width;
a = wave direction, relative to north;
P = shoreline or contour orientation, relative to north;
0 = wave crest orientation, relative to bottom contours;
K = breaking wave height proportionality factor;
pI = measure of ambient shoreline orientation;
p = water mass density;
p, = sediment mass density;
r = smoothing factor; and
V = horizontal vector differential operator.




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