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Friction in hurricane-induced flooding

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Title:
Friction in hurricane-induced flooding
Creator:
Wang, Shang-Yih, 1952-
Publication Date:
Language:
English
Physical Description:
xvi, 165 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Buildings ( jstor )
Canopy ( jstor )
Dehydration ( jstor )
Drag coefficient ( jstor )
Floods ( jstor )
Friction factor ( jstor )
Shear stress ( jstor )
Surface water ( jstor )
Velocity ( jstor )
Water depth ( jstor )
Flood forecasting ( lcsh )
Floods ( lcsh )
Hurricanes ( lcsh )
City of Gainesville ( local )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1983.
Bibliography:
Includes bibliographical references (leaves 161-164).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Shang-Yih Wang.

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
11479424 ( OCLC )
ocm11479424
00458303 ( ALEPH )

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Full Text











FRICTION IN HURRICANE-INDUCED FLOODING


By

SHANG-YIH WANG
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1983




FRICTION IN HURRICANE-INDUCED FLOODING
By
SHANG-YIH WANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1983


ACKNOWLEDGEMENTS
The author wishes to express his sincerest gratitude to the
chairman of his supervisory committee, Dr. B. A. Christensen, for all
his expert guidance, the tremendous benefit of his professional
competence and vast practical experience throughout this study. He
also wishes to thank Dr. T. Y. Chiu for his advice, understanding,
encouragement and support during the author's six years of graduate
study at the University of Florida. Without their untiring patience
and help this dissertation would not have been possible.
Thanks are also due to Dr. D. P. Spangler, Dr. B. A. Benedict,
Dr. T. G. Curtis and Dr. H. Rubin for serving on the author's super
visory committee and for their consulting and assistance.
Appreciation is extented to Drs. A. J. Mehta, D. L. Harris,
F. Morris and P. Nielsen for their suggestions and providing reference,
which contributed greatly to this study.
The author is indebted to Mr. E. Dobson for his technical assis
tance, Ms. L. Pieter for her drafting and Ms. D. Butler for her typing.
Special thanks are due to Mr. E. Hayter for his help in the preparation
of this dissertation.
Finally, the author wishes to thank his wife, Fu-Mei, whose partici
pation in every phase of this study has made these years more joyful.


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF SYMBOLS x
ABSTRACT xv
CHAPTER
I. INTRODUCTION 1
Storm Surge Prediction 4
Objectives of Present Work 6
II. FIELD EXPLORATION OF PHYSICAL ENVIRONMENT 7
Mangrove Areas 7
General View of Mangroves in Florida 7
Sampling of Mangroves 8
Developed Areas 15
High-Rise Building Areas 15
Medium-Rise Building Areas 17
Residential Areas 21
III. THEORETICAL BACKGROUND AND DEVELOPMENT 22
Hydrodynamic Equations for Storm Surges 22
Wind Shear Stress 23
Wind Velocity Profile in Vertical 25
Proposed Wind Shear Stress on Obstructed Areas 27
Bed Shear Stress 31
Post Approach, Friction Factor for Surges in
Unobstructed Areas 33
Proposed Approach, Friction Factor for Surge
in Obstructed Areas 38
iii


Page
IV.EXPERIMENTAL VERIFICATION OF FRICTION FACTORMODEL
LAWS 43
Distorted Model for Buildings 45
Undistorted Model for Mangrove Stems and Roots 49
Distorted Model for Canopy 50
V.MODEL DESIGN 54
Recirculating Flume 54
Instrumentation 57
Velocity Meter 57
Data Acquisition System (DAS) 59
Depth-Measuring Device 59
Selection of Model Scales 63
Mangroves 63
Buildings 64
Model Setup 66
Mangrove Stems and Roots 66
Canopy of Red Mangroves 69
Buildings 72
VI.EXPERIMENTAL TEST SERIES 77
Experimental Procedure 77
Calibration of Velocity Meter 77
Measurements of Mean Velocities 77
Measurements of Water Depths 79
Experimental Runs 81
Mangrove Areas 81
Building Areas 82
VII.PRESENTATION AND ANALYSIS OF DATA 84
Mangrove Areas 84
Building Areas 95
Determination of Drag Coefficient 110
Drag Coefficient-Building Density Relation 110
Drag Coefficient-Disposition Parameter Relation 113
i v


Page
VIII. DISCUSSIONS AND CONCLUSIONS 119
Mangrove Areas 120
Developed Areas 121
Ocean Bottom 123
Forested Areas 127
Grassy Areas 128
Conclusions 129
APPENDICES
A. FIELD RECORDED DATA FOR MANGROVES 132
B. COMPUTER PROGRAM LISTINGS 145
C. TABLES OF EXPERIMENTAL DATA 151
BIBLIOGRAPHY 161
BIOGRAPHICAL SKETCH 165
v


LIST OF TABLES
Table Page
1. Average Parameters of Sampling Mangroves 14
2. Average Characteristics of Canopy 16
3. Average Parameters of High-Rise Buildings 19
4. Average Parameters of Medium-Rise Buildings 20
5. Scale Selection for Canopy 64
6. Average Parameters of Prototype and Model for
Building Areas 65
7. Statistical Values of Experimental Results for
Mangrove Areas 94
8. Statistical Values of Experimental Results for
Building Areas 109
9. Relations Between Disposition Parameters and
Drag Coefficients 115
10. Typical Values for Mangrove Areas 120
11. Bed Friction Characteristics of Three Entrances 126
A1. Parameters of Sampling Red Mangroves 141
A2. Parameters of Sampling Black Mangroves 142
A3. Characteristics of Canopy of Red Mangroves 143
Cl. Experimental Data for Red Mangroves (Without Canopy) 151
C2. Experimental Data for Red Mangroves (with Canopy) 152
C3. Experimental Data for Black Mangroves 153
C4. Experimental Data for Building Areas 154


LIST OF FIGURES
Figure Page
1. Prop Roots of Red Mangroves 9
2. Air Roots of Black Mangroves in 1 Foot Square Areas 9
3. Red Mangrove Area (Sampling Area #4) 11
4. Black Mangrove Area (Sampling Area #7) 11
5. Field Data Record for Red Mangroves 12
6. Field Data Record for Black Mangroves 13
7. Measurement of Density of Canopy 16
8. Section View of Survey Area (Red Mangroves) 16
9. Top View of Building Shapes on Coastal Areas 18
10. Wind Stress Coefficient over Sea Surface 26
11. Plan View for Wind Stress over an Obstructed Area 28
12. Elevation View for Wind Stress over an Obstructed Area 28
13. Distribution of Horizontal Apparent Shear Stress
and of its Drag, Inertial and Viscous Components 39
14. Plan of Flume 55
15. Cross Section of Flume 55
16. Novonic-Nixon Velocity Meter 58
17. Input Box of Data Acquisition System 60
18. HP 9825A Programmable Calculator 60
19. Setup of Water Depth Measuring Device 62
20. Point Gage and Tube 62
vii


Figure Page
21. Model Setup for Red Mangroves 67
22. Model Setup for Black Mangroves 68
23. Stems and Roots of Red Mangroves 70
24. Stems and Roots of Black Mangroves 70
25. Overview of Setup for Mangroves 71
26. Setup of Model Equivalent of Canopy of Red Mangroves 71
27. Distribution of Leaf Stripes in the Model 73
28. Building Patterns Designed for the Tests 74
29. Relation Between U and U 80
n w
30. Designed Building Patterns in the Tests 83
31. Relation Between f' and R' for Red Mangrove Areas
(Without Canopy) 85
32. Relation Between Cn and R' for Red Mangrove Areas
(Without Canopy) 86
33. Relation Between f' and Water Depth d for Red
Mangrove Areas e (Without Canopy) 87
34. Relation Between f and R for Red Mangrove Areas
(with Canopy) e e 88
35. Relation Between Cn and R1 for Red Mangrove Areas
(with Canopy) 89
36. Relation Between f' and Water Depth d for Red Mangrove
Areas (with Canopye) 90
37. Relation Between f and R' for Black Mangrove Areas 91
e e
38. Relation Between Cp and R^ for Black Mangrove Areas 92
39. Relation Between f1 and Water Depth d for Black
Mangrove Areas e 93
40. Relation Between f and R' for Building Areas 94
e e


Figure Page
41. Relation Between Cn and R' for Building Areas 102
D e 3
42. Relation Between f1 and Water Depth d for Building
Areas e 108
43. Relation Between Cn and Density m for High-Rise
Building Areas 111
44. Relation Between Cn and Density m for Medium-Rise
Building and Residential Areas 112
45. Position Spacings. Definition Sketch 114
46. Relation Between Cn and S,/D in Aligned and Staggered
Patterns 116
47. Relation Between and S^/D 118
Al. Field Recorded Data for Red Mangroves (Area #1) 132
A2. Field Recorded Data for Red Mangroves (Area #2) 133
A3. Field Recorded Data for Red Mangroves (Area #3) 134
A4. Field Recorded Data for Red Mangroves (Area #4) 135
A5. Field Recorded Data for Red Mangroves (Area #6) 136
A6. Field Recorded Data for Black Mangroves (Area #8) 137
A7. Field Recorded Data for Black Mangroves (Area #9) 138
A8. Field Recorded Data for Black Mangroves (Area #10) 139
A9. Field Recorded Data for Black Mangroves (Area #11) 140
i x


LIST OF SYMBOLS
area
leaf area for prototype and model, respectively
horizontal width
drag coefficient
skin friction coefficient
average diameter of obstruction
vertical depth for prototype and model, respectively
water depth
water depth at section 1 and section 2, respectively
average water depth of d-j and d^
diameter of pipe
drag force
elastic force
skin friction
gravitational force
inertial force
Froude number =/ U / gd
d d
surface tension force
viscous force
Darcy-Weisbach friction factor based on diameter of the pipe
friction factor based on hydraulic radius


equivalent friction factor
total friction factor
gravitational acceleration
vertical depth above the vertex of Thomson weir
protruding height of obstructions above water surface
indicial functional parameter
wind stress coefficient
drag force scale
skin force scale
gravity force scale
inertial force scale
shear force scale
equivalent sand roughness
apparent roughness
horizontal length
leaf length for prototype and model, respectively
density = no. of obstruction elements/area
indicating the subscripted parameters for model and prototype,
respectively
total number of obstruction elements
vertical length scale
horizontal length scale
force scale
time scale
Manning's n
wetted perimeter of flow cross-section
pressure


Ps pressure on the water surface
Q discharge from the Thomson weir
q q discharge per unit width
x y
R hydraulic radius = A/P
R-|,R2 hydraulic radius at section 1 and section 2, respectively
Rfl average hydraulic radius of R-j and R^
R^ reduction factor for wind stress
Rq Reynolds number based on depth = U d /v
R' Reynolds number based on hydraulic radius = UR /v
6 ad
R* wall Reynolds number = u^.k/v
R v Reynolds number = U^x/v
C )A
r radius of the pipe
S slope of energy grade line = aH/L
S corner to corner distance between the roughness elements in
adjacent transverse raws
S£ longitudinal spacing between two successive roughness elements
S^. laternal spacing between two roughness elements
s free surface displacement from mean sea level
S. Dev. standard deviation
t time variable
U spatial mean flow velocity
U-j, U2 spatial mean flow velocity at section 1 and 2, respectively
Ua average spatial mean velocity of U-| and U2
U U spatial mean flow velocity in x and y directions, respectively
x y
Uoo free stream velocity
u,v,w instantaneous components of the water velocity in the x, y, z
coordinate directions, respectively


u
u1 V'
uf,t
W(z)
w
m
x, y, z
z
0
0
Y
6
AH
^1*^2
e
K
u
V
e
P
pa
7b
Tbx Tby
time-mean velocity in the direction of flow
turbulent velocity fluctuations in the x and z directions,
respectively
friction velocity based on bottom friction
friction velocity based on total friction
time-mean wind velocity at the elevation z above water surface
critical wind velocity
time-mean wind velocity at the elevation 10 meters above water
surface
leaf width for prototype and model, respectively
Cartesian coordinate directions
dynamic roughness
shape factor
specific weight of water = pg
displacement thickness
energy loss per unit weight of fluid
fractions of distances nid and n2d from bottom to the total
depth d, respectively
latitude
Von Karman's constant
molecular viscosity of water
kinematic viscosity of water
percentage of the measured area occupied by obstructions
water density
air density
spatial mean bottom shear stress = tq
bottom shear stress in the x and y directions, respectively
xi i i


Tj hydrodynamic drag
viscous stress
wind shear stress on open area
t wind shear stress on obstructed area
so
Reynolds stress
obstruction correction factor
10 the earth's angular velocity
to* wind friction velocity = /t./p
S d
V2 Laplacian operator
xiv


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
FRICTION IN HURRICANE-INDUCED FLOODING
By
Shang-Yih Wang
December 1983
Chairman: B. A. Christensen
Cochairman: T. Y. Chiu
Major Department: Civil Engineering
With the increasing development of coastal areas, it is necessary
to have a sound method for predicting hurricane-induced flooding in
these areas, especially for studies such as the coastal construction
set-back line, flood insurance rate-making and county land use planning.
The purpose of this study is to develop the capability of describing
the friction factor in coastal areas for improved representation in
numerical models of storm surges.
Five types of areas are considered: A, ocean bottom with bed-
forms and some vegetation; B, mangrove fringe and areas; C, grassy
areas; D, forested areas; and E, developed areas. The friction factors,
which incorporate both the bottom friction coefficient and drag
coefficient due to the submerged parts of obstructions were verified by
conducting laboratory experiments for mangrove and developed areas,
xv


using the typical distribution found in each of these coastal areas.
Analysis of the experimental data revealed that the drag coef
ficient for each case is invariant with the Reynolds number in the most
possible flooding flow ranges, but that it is related strongly to the
density and distribution of the roughness elements. Formulae
expressing these relations were derived for the evaluation of the
friction factor for different coastal areas. In addition, it is found
that the drag coefficient for a staggered disposition is about two to
three times larger than that for an aligned disposition under the same
density for all building areas. A relationship between the drag
coefficient and the disposition parameter of the evenly distributed
roughness elements was developed. The principal reduction of the wind
stress due to wind drag forces on the parts of the obstructions,
including-buildings and vegetation, above the water surface during
overland flooding was determined. Accounting for this reduction of the
wind stress provides a realistic view of wind generation forces in
coastal areas.
Finally, the formulae of the friction factor for the ocean bottom,
forested areas and grassy areas are presented by adopting results from
previous investigations and discussed with the results of the current
study.
xvi


CHAPTER I
INTRODUCTION
The rapid growth of population and industry in very low coastal
areas in recent years has resulted in increased concern and attention
to the potential hazard to these areas from tropical storms and hur
ricanes. A severe tropical storm is called a hurricane when the maximum
sustained wind speeds reach 75 mph or 65 knots (U.S. Army, Corps of
Engineers, 1977). During a hurricane, the wind-driven storm waves are
superimposed on the storm surge, which is the rise above normal water
level due to the action of storm, and sometimes the low coastal areas
are flooded. The worst natural disaster in the history of the United
States came as the result of a hurricane which struck Galveston, Texas,
in 1900. The storm, which hit the Texas coast on September 8, with
winds of 125 mph caused a storm surge 15 feet in height above the usual
two-foot tidal range. The fifteen-foot surge, accompanied by wave
action, demolished the city and caused more than 5,000 deaths (Bascom,
1980). Weather warnings were ineffectual. The people of Galveston,
unprepared for a storm of such intensity, were helpless in the face of
the hurricane. But the hurricane is no longer the unheralded killer it
once was. The years of progress in weather forecasting and wave research
have now made it possible to predict such surges. Hurricane Donna, for
example, which crossed the Florida Keys and then moved northeastward
across the state of Florida from near Fort Myers to Daytona Beach on
September 9-10, 1960, is thought to have been the most destructive
1


2
storm ever experienced in Florida. Fortunately, she was detected in
advance. Thus, even though this hurricane caused an estimated $300
million in damage, only 13 fatalities occurred. In 1961, Hurricane
Carla struck along the Gulf coast of Florida. The area was evacuated
before its arrival, and there was no loss of life.
From the facts mentioned above, it is clear that the great value
of the modern storm warning service is in its reducing the loss of life.
However, it is also apparent that the potential damage to property and
structures has increased dramatically with the rapid development of
coastal areas, if no concession is made to the storms. Therefore, the
future development plans of these coastal counties must take into
consideration this threat to life and property. As a result, the 1971
session of the Florida State Legislature passed a law (Chapter 16,053,
Florida Statutes), requiring the Department of Natural Resources to
establish a coastal construction set-back line (SBL) along Florida's
sandy beaches fronting the Atlantic Ocean and the Gulf of Mexico.
Based upon comprehensive engineering studies and topographical surveys,
such a line, where deemed necessary, is intended to protect upland
properties and control beach erosion. Basically, construction and
excavation seaward of the SBL is prohibited, though a provision for
variances is included in the law (Chiu, 1981).
In 1973 the Congress of the United States enacted the Flood Disaster
Protection Act (Public Law No. 93-234, 87 Statutes 983) which greatly
expanded the available limits of federal flood insurance coverage. The
act also imposed new requirements on property owners and communities
desiring to participate in the National Flood Insurance Program (NFIP)
(Chiu et al., 1979).


3
The Flood Disaster Protection Act of 1973 requires the Department
of Housing and Urban Development (HUD) to notify those communities that
have been designated as flood hazard areas. Such areas are defined as
having a one percent annual chance of flooding at any location within
the areas. Such a community must either make prompt application for
participation in the flood insurance program or must satisfy the Secretary
of HUD that the area is no longer flood prone. Participation in the
program is mandatory (as of July 1,1975) or the community would be denied
both federally related financing and most mortgage money.
Individuals and businesses located in identified areas of special
flood hazard are required to purchase flood insurance as a prerequisite
for receiving any type of federally insured or regulated financial
assistance for acquisition or construction purposes. Effective July 1,
1975, such assistance to individuals and businesses was predicated on
the adoption of effective land use and land management controls by the
community.
Federally subsidized insurance for flood hazard is authorized only
within communities where future development is controlled through adequate
flood plain management. Management may include a comprehensive program
of corrective and preventative measures for reducing flood damage, such
as land use controls, emergency preparedness plans and flood control works.
Participating communities may be suspended from the program for failure
to adopt or to enforce land use regulation (National Flood Insurers
Association, 1974).


4
Storm Surge Prediction
To implement either the coastal construction setback line or the
flood insurance program the flood elevation has to be determined on the
basis of different time intervals. Accumulation of data over many years
in areas of the Old World, such as regions near the North Sea, has led
to relatively accurate empirical techniques of storm surge prediction
for these locations. However, these empirical methods are not applicable
to other locations. In general, not enough storm surge observations are
available in the New World to make accurate prediction of the 100 year
storm surge. Therefore, the general practice has been to use hypothetical
design storms, and to estimate the storm-induced surge by numerical
models, since it is difficult to represent some of the storm-surge
generating processes (such as the direct wind effects and Coriolis
effects) in physical laboratory models. With the use of digital com
puters, numerical models have been able to analytically describe storm
surges to much greater detail than was ever possible with the other

methods. As a result, many numerical models for the prediction of surges
have been proposed to investigate practical cases including irregular
coastlines, irregular bathymetry, islands and arbitrary wind stress
patterns (Mungall and Mattews, 1970; Reid and Bodine, 1968; Platzman,
1958; Platzman and Rao, 1963). Moreover, Pearce (1972) and Reid and
Bodine (1968) developed their models to evaluate the inland extent of
flooding by using a moveable boundary. For different emphases on off
shore and nearshore areas, varied grid systems are used for most finite-
difference models. An orthogonal curvilinear coordinate system with
telescoping computing cells has also been introduced by Wanstrath (1978)


5
to solve the flooding problems of Louisiana. Regardless of the
purposes and differences in approaches of all these numerical storm
surge models, the Navier-Stokes and continuity equations which incor
porate terms accounting for wind stress, bottom friction, inertia,
Coriolis effect, pressure distribution, and other physical parameters
are solved numerically in space and time to determine localized surge
hydrographs. In order to obtain a more realistic and accurate predic
tion of storm surge, these physical parameters should be carefully
determined and incorporated into the numerical models. Modern
achievements in meterology and oceanography have led to an increased
understanding of hurricane to the extent that a model hurricane can be
characterized to a satisfactory degree by certain parameters. A list
of these variables includes central pressure deficit, radius to maximum
winds, speed of hurricane system translation, hurricane direction and
landfall location (or some other descriptor of hurricane track).
Surprisingly little work has been done in measuring another important
term-bottom friction. As suggested by Pearce (1972), future work on
actual hurricane surges and currents is especially needed for improved
representations of bottom friction that would be achieved with a better
understanding of the dissipation mechanism (i.e. friction) during a
hurricane.
Objectives of Present Work
Based on the necessities for a more accurate prediction of
hurricane-induced flooding in coastal areas (especially for studies
like the construction set-back line, flood insurance rate determination


6
and county land use plan), this study will develop a method of
describing the friction factor in coastal areas for improved repre
sentation of numerical storm-surge models. Special emphasis will be
placed on the friction characteristics of mangroves and buildings which
are the two most important causes of frictional resistance in vegetated
and developed land areas, respectively. The effect of these two
roughnesses in reducing overall wind stress on the water is also
introduced. The friction factors for other roughnesses such as the
ocean bottom, the forested and the grassy areas are determined by the
results from previous investigations and present study and are discussed
in the last chapter.


CHAPTER II
FIELD EXPLORATION OF PHYSICAL ENVIRONMENT
As stated in Chapter I, mangroves and buildings are the two major
flow retarding objects and are therefore being investigated in this
study. However, information on the density, dimensions and typical
distributions of these two forms of roughness in the coastal areas is
very scarce. Thus, field trips were taken to a mangrove area and
developed areas in southern Florida in order to collect the most
representative data for use in the model tests.
Mangrove Areas
General View of Mangroves in Florida
Mangrove is a kind of salt-resistant plant that usually grows
densely on sub-tropic shorelines around the world. This special
feature may be an inherent gift from nature in that the mangroves
enable exposed shorelines to resist severe attacks of hurricanes.
Basically, there are three species of mangroves, the red mangrove
(Rhizophora mangle), the black mangrove (Aricennia nitida) and the
white mangrove (Laguncularia racemosa). Each of these three species
occupies a distinct zone within the forest, depending on the degree
of salinity and length of inundation that each species can tolerate.
Red mangroves usually are found at the outer or seaward zone. They
are distinctive in appearance, with arching prop roots that project
7


8
from the trunk or branches down into the water (Figure 1). The root and
trunk systems of red mangroves, which spreads in shallow offshore areas
and onshore areas serve as a soil producer and stabilizer as well as a
storm buffer. In their role as buffers against storm winds and tides,
they prevent devastation of the coastline (Lugo et al., 1974). The
middle zone, at slightly higher elevations, is dominated by the black
mangroves in association with salt marsh plants. This zone is usually
submerged at high tide, but is otherwise exposed. The roots produce
pneumatophores (fingerlike extensions above the soil surface),as shown
in Figure 2. Black mangroves may also be found in pure stands in
shallow basins where sea water remains standing between tides. The
heat from the sun evaporates some of the water, leaving slightly
concentrated salt water behind. The black mangroves are also important
for shoreline stabilization as they present a secondary defense behind
the red mangroves. The white mangrove, which can be found in the most
landward zone that is affected only by the highest spring and storm
tides will not be discussed here since it is not as important in
defending against storm flooding and its usual appearance may be
categorized into buttonwood or other common types of vegetation.
Sampling of Mangroves
Five mangrove forest types Fringe, Riverine, Basin, Overwash and
Dwarf Forests-have been found by Snedaker and Pool (1973) in southern
Florida, with distinctive differences in structure. The pattern is
strongly related to the action of water, both the freguency and the
amount of tidal flushing and freshwater runoff from the upland. The
coastal fringe forest including red and black mangroves, which are the


9
FIGURE 1: Prop Roots of Red Mangroves
FIGURE 2: Air Roots of Black Mangroves in 1 Foot Square Area


10
most important species, was investigated in San Carlos Bay on the
southwest coast of Florida.
Eleven sampling areas which included six red mangrove fringe areas
and five black mangrove areas were selected at random. Each sampling
area was framed by survey poles to form a 12 by 12 foot square area in
which the locations and dimensions of mangrove trunks, roots, and
canopies were recorded. Figures 1 through 4 show some of the features
of both types of mangroves in the surveyed area. Figures 5 and 6 show
two examples of data recorded from red and black mangrove sampling
areas, respectively. Data for the other nine sampling areas are shown
in Appendix A. From these data it is clear that the density and
dimensions of the trunk and root systems of mangroves are quite random;
therefore, averaged characteristics are chosen to describe these
samples as shown in Table 1. Red mangroves in the surveyed areas
extend from the low tidal water line to about 50 feet inland which is
the same distance Veri et al., (1975) recommended for fringe
mangroves in order to form a protective buffer zone. Thus, this value of
50 feet can be considered to be a standard distribution distance for
red mangroves and is used in the present study.
Although the average height of the lower edge of canopy was found
to be about 8 feet above the ground for red mangroves, the canopy along
the water edge was found to generally have a distribution from the
water surface to a few feet high. This feature may be important in
resisting storm surges. Therefore, a detailed measurement of canopy
distribution was done at a later time in Sarasota, Florida. Figure 7
depicts the measuring of the density of leaves by counting the number
3
of leaves in a unit volume (1 ft.). Dimensions of leaves were also


11
FIGURE 3: Red Mangrove Area (Sampling Area #4)
FIGURE 4: Black Mangrove Area (Sampling Area #7)


12
AREA #5
FIGURE 5: Field Data Record for Red Mangroves


13
AREA #7
O Stem
FIGURE 6: Field Data Record for Black Mangroves


14
TABLE 1: Average Parameters of Sampling Mangroves
Average Parameters
Red
Mangroves
Black
Mangroves
Main-
Stem
no.
(12 ft.)2
4
12
diameter
6.0 in.
3.1 In.
height
10.0 ft.
11.2 ft.
% occupied
0.84 %
0.49%
Sub-
Stem
no.
(12 ft.)2
13
diameter
2.0 in.
height
18.0 in.
% occupied
0.19 %
Canopy
height
8.0 ft.
9.4 ft
Root
no.
(12 ft.)2
81
10,800
diameter
1.0 in.
0.25 in
height
18.0 in.
6.0 in.
% occupied
0.25 %
2.39%


15
measured and recorded. Totally six sampling areas along the coastal
fringe were randomly selected. As shown in Figure 8, the survey area
contains three sections in which each section covers a distance of five
feet. The data collected are shown in Appendix A. Table 2 lists the
average densities and dimensions of leaves obtained from these six
sampling areas.
Developed Areas
In developed areas, buildings constitute the principal roughness
elements which would significantly affect the apparent bottom shear
stress as well as the wind shear stress during a storm induced flood.
Buildings are not, in general, arranged in a uniform manner but are
strongly dependent on the environment where they are located. A common
feature found in the coastal counties of Florida, especially in Broward
and Dade counties, is that high-rise buildings are predominant along the
beaches while residential houses are predominant a few miles inland
from the coastline. Three areas, a high-rise building area, medium-
rise building area and residential area, are defined to represent a
developed area in this study.
High-Rise Building Area
Aerial photographs of Broward and Dade counties, Florida, made
by the State Topographic Office, Florida Department of Transportation
in 1980 were used to analyze the dimensions and densities of buildings
in the coastal areas. Dade county is divided into 113 ranges while
Broward county is divided into 128 ranges in the aerial photographs.


16
FIGURE 7: Measurement of Density of Canopy
TABLE 2: Average Characteristics of Canopy
^Sections
Parameters
#1
n
#3
... no. of leaves\
Density! = )
ftJ
10
5
2
Leaf Size
2" x 1"
3" x 1.5"
4" x 1.75"


17
Each range has a length of 1,000 feet approximately parallel to the
shoreline and is marked by monuments both in the field and on the
aerial photographs. Typical shapes and orientation of buildings
found in these two counties are shown in Figure 9. Category (a) is
the most common type found (more than 50 percent) which may be dictated
by the high cost of land per foot along seashore, and is chosen to
present all the buildings in the study.
High-rise buildings are defined as buildings having a surface area
2
larger than 10,000 ft An estimation of the dimensions and densities
of these high-rise buildings from Broward and Dade Counties are 1isted in
Table 3. High rises and hotels/motels are predominant in the area.
Medium-Rise Building Area
Madium-rise buildings cover all buildings which do not belong to
either the high-rise or residential types. They can include two and
more story semidetached houses, row houses, garden apartments and other
buildings which are lower than ten stories. The surface area occupied
2 2
by medium-rise buildings is defined from 2,400 ft to 10,000 ft in
this study. An investigation of buildings in this category was also
made from aerial photographs of Broward and Dade Counties. Table 4
shows the average densities and dimensions of buildings from existing
field data.
The values obtained from these two counties at least present some
general views of buildings in highly developed areas despite their
irregularities in distributions found in the field. To apply these


18
Sea Side
Sea Side
(b)
FIGURE 9: Top View of Building Shapes on Coastal Areas


19
TABLE 3: Average Parameters of High-Rise Buildings
Range
No.
Buildings
County
length
(ft.)
width
(ft.)
density
( ^ )
1000'x500'
% of
land occupied
by buildings
42
280
80
10
45
Broward
43
290
60
4
14
Broward
45
290
110
4
26
Broward
50
140
140
10
39
Broward
54
300
100
3
18
Broward
72
200
70
8
22
Broward
82
200
50
19
38
Broward
83
200
180
5
36
Broward
84
200
230
5
46
Broward
118
140
90
8
20
Broward
119
150
70
12
25
Broward
121
230
90
6
25
Broward
8
240
200
3
29
Dade
11
300
50
8
24
Dade
12
320
70
7
31
Dade
14
240
50
9
22
Dade
15
230
50
8
18
Dade
17
230
60
9
25
Dade
18
250
50
10
25
Dade
19
200
125
5
25
Dade
36
220
65
4
11
Dade
42
190
65
7
17
Dade
43
270
200
6
65
Dade
44
200
200
5
40
Dade
48
120
200
6
29
Dade
52
220
210
4
37
Dade
56
190
160
9
55
Dade
Mean
224
112
7
30
S.Dev.
53
62
3
13


20
TABLE 4: Average Parameters of Medium-Rise Buildings
Range
No.
Buildings
County
length
(ft.)
width
(ft.)
density
l no> )
U000'x500' 1
% of
land occupied
by buildings
26
65
65
23
19
Broward
36
120
40
30
29
Broward
37
100
35
33
23
Broward
46
150
40
27
32
Broward
49
80
60
31
30
Broward
51
100
40
22
18
Broward
52
120
40
22
21
Broward
62
80
30
33
16
Broward
64
100
40
20
16
Broward
66
70
40
43
24
Broward
67
80
70
14
16
Broward
101
60
50
35
21
Broward
109
70
70
18
18
Broward
110
70
40
24
13
Broward
111
70
70
25
25
Broward
116
100
70
17
24
Broward
1
120
75
14
25
Dade
2
90
60
13
14
Dade
3
80
50
15
12
Dade
4
75
75
11
12
Dade
5
100
65
15
20
Dade
33
160
49
24
38
Dade
35
125
35
32
28
Dade
68
130
40
21
22
Dade
69
130
40
24
25
Dade
70
125
40
28
28
Dade
Mean
99
51
24
22
S.Dev.
28
15
8
7


21
data in the prediction of storm surge, it is recommended that the County
Land Use Plan Map published by each county be used so that the most
realistic results can be expected.
Residential Area
Residential houses are usually located behind the commercial areas
?
and have a surface area less than 4,000 ft A typical density value
of detached, one story houses is given as six units per acre (43,560 sq.
ft.) (DeChiara and Callender, 1980). Density ranges in residential areas
can also be found in the Land Use Map of each county which categorizes
these single family houses in the density range of 0-8 units per acre
(Reynolds, Smith and Hills, 1972). The significant difference between
residential, medium-rise building areas and high-rise building areas is
that the former two areas usually have a matrix type distribution while
the latter one has only one or two rows distributed in the coastal
fringe area. The importance of this variation in the building distribu
tion will be shown later in the discussion of modeling studies.
Dimensions of the typical residential house are chosen as 30 feet by 62
feet, 1,860 sq. ft., which are convenient for the model tests and also
realistic for most single family houses.


CHAPTER III
THEORETICAL BACKGROUND AND DEVELOPMENT
Hydrodynamic Equations for Storm Surges
The equations governing incompressible fluid flows are the
Navier-Stokes equations of motion and the equation of continuity.
In the case of storm surges these equations may be written:
3qx
3 t
2o)(sin0)qy
d_ 3Ps
p 3 x
gd H + p (Tsx -Tbx>
(1)
2o)(sine)qx
d_ 3Ps
p 3y
gd |S- + 1
a 9y p
(2)
3qx + 3^y_ + 3S
3 x 3 y 3t
0
(3)
where t is time, u is the angular velocity of the earth, e is the lati
tude, p is the pressure, g is the acceleration of gravity, p is the water
density, s is the free surface displacement from mean sea level, the
subscripts s and b indicate that the subscripted quantities are to be
evaluated at the surface and bed, respectively, d is the total depth, qx
and q^ represent the time mean transport component, i.e. discharge per
unit width, in x and y directions, respectively, i.e.
d(x,y,t) = h(x,y) + s(x,y,t)
qx(x,y,t)
fs(x,y,t)
u(x,y,t) dz
-h(x,y)
(4)
(5)
22


23
qy(x,y,t)
rs(x,y,t)
v(x,y,t) dz
-h(x,y)
(6)
in which h = water depth referenced to mean sea level, u and v are the
instantaneous components of water velocity in the x and y coordinate
directions, respectively.
Expressions for the wind shear stress, and bed shear stress,
x^, for coastal areas in tropical storm induced flooding are presented
in the following sections.
Wind Shear Stress
In general, the wind stress (t ) on a water surface may be expressed
in terms of the mean wind speed (W-^) at anemometer level (10 meters above
water surface), the air density (p ) and a wind-stress coefficient (K),
a
as
T
s
K W
10
(7)
The problem of evaluating the wind stress is therefore reduced to
estimating the wind-stress coefficient, K, at different wind speeds, if
the reference wind speed and air density are known. Numerous studies
have found the quadratic wind speeds relation to be appropriate for a
wide range of wind speeds (Wilson, 1960). A wind-stress relation more
physically satisfying the quadratic law correlation was developed by
Keulegan (1951) and Van Dorn (1953) in the low winds range (<15 ms"^).
The Keulegan-Van Dorn relation for x$ is given as
T
s
Pa [K^o K2(W10 w/]
(8)


24
where K-| and are the constants and Wc is critical wind speed.
Although there are uncertainties in applying the Keulegan-Van Dorn
relation to hurricane winds, it has been applied widely in hurricane-
induced surge cases. To eliminate this deficit, the wind-stress
relation has to be extended to higher wind ranges. Whitaker, Reid and
Vastano (1975) investigated the wind-stress coefficient at hurricane wind
speeds using a numerical simulation of dynamical water changes in Lake
Okeechobee, Florida. Results of their numerical experiments showed
that the Keulegan-Van Dorn wind-stress relation was superior to the more
commonly used quadratic relation for wind speeds in the range of 20 to 40
meters per second. The relation they found for the wind stress ts is
given by:
t = p[0.0000026 + (1.0 ^.)2 x 0.0000030] W?n (9)
s wio 1u
where W^q and 7.0 are in meters per second and p is the water density.
Unfortunately, though this result was verified by a simulation of the
surge associated with a hurricane which occurred in October, 1950, it
still has some deficiencies such as the limited range of applicability
(Whitaker et al., 1975).
Recent studies of the wind-stress coefficient over the sea surface
have produced more complete and perhaps more accurate results with the
refinement of measurements and analysis techniques. Garratt (1977)
reviewed and averaged 17 selected sets of data and proposed an empirical
expression for 'light' winds:
K = (0.75 + 0.67 W1Q) x 10"3
(10)


25
Wu (1980) suggested a similar result for the wind-stress coef
ficient from 33 averaged data sets under 'light* winds
K = (0.8 + 0.065 W1Q) x 10"3 (11)
Furthermore, Wu (1982) compiled and averaged all available data for
'strong' winds. The data were obtained from independent investigations
either cited or reported in the following sources: Wu (1969), Kondo
(1975), Garratt (1977), Smith (1980), Wu (1980), and Large and Pond
(1981). All the data sets selected were obtained under nearly neutral
conditions of atmospheric surface layer. Additional factors which
affect the wind-stress coefficient, such as rainfall and sea spray, are
neglected due to their minor importance compared to the major factor of
wind speeds. As a result, the empirical formula proposed, given by
equation (11), for 'light' winds appears to be applicable even in
'strong' winds. Light and strong winds are defined as those less than
and greater than 15 meters per second, respectively. The averaged data
obtained from those sources and the formula proposed, equation (11), are
shown in Figure 10.
Wind Velocity Profile in Vertical
A vertical profile of wind velocities, usually expressed by the
following logarithmic law, is regarded by meteorologists as a superior
representation of strong winds in the lower atmosphere (Tennekes, 1973):
W(z) = 1, In (f-) (12)
0
where W(z) is the wind velocity at a height z above mean sea level, is
the von Karman's turbulence constant, z is the dynamic roughness of
1 /2
the logarithmic velocity profile, and w* = (t /pa) = friction velocity.


FIGURE 10:
Wind Stress Coefficient over Sea Surface


27
The wind velocity profile given by equation (12) is well defined
except that it fails next to the bed where z approaches zero, and the
wind velocity W(z) approaches minus infinity. This discrepancy can be
corrected by using a modified mixing length approach as proposed by
Christensen (1971) for the flow of water over a rough bed, resulting in
modified logarithmic law for wind velocity profile is given in the form
= 2.5 In (^- + 1) (13)
* zo
By substituting equation (7) into equation (12), an equation for
determining the dynamic roughness, zq, is obtained
4ft = /K (2.5) In (j- + 1) (14)
w10 o
Applying the boundary condition
W(10) = W1Q at z = 10m
The dynamic roughness, zq, in all the wind speeds is found to be a
function of wind-stress coefficient, or on the wind velocity, i.e.,
z
0
10
e'2.5 ,
(15)
Proposed Wind Shear Stress on Obstructed Areas
Consider an obstruction which has an effective width, D, and
protruding height, h, as shown in Figures 11 and 12. The wind drag
force on such an obstruction can be expressed as


28
I
T
D
-L
PLAN VIEW
FIGURE 11: Plan View for Wind Stress over an Obstructred Area
ELEVATION VIEW
Protruding
Obstruction
Wind
=>
FIGURE 12: Elevation View for Wind Stress over an Obstructed Area


29
F
D
CPpaD
2
h
0
W2(z) dz
06)
where CQ is the drag coefficient. Substituting equation (14) into
equation (16) gives
(2.5)
F0 "
CDpa
DK
10
[In { + 1 )]2 dz
o
or
Fd = 3.13 Cdp3D W20 K {(h + zQ)1n(^-+ l)[ln(^- + 1) 2] + 2h } (17)
o o
Recall the equation for wind stress on an open water surface, i.e.,
K a *4
.(7)
This wind stress acts on the water surface and causes a rise of the
elevation of water surface which is called wind setup. The wind energy
is being transformed from the wind field to the water flow by the wind
shear. When the same wind field moves from the open water area to the
obstructed area, the wind setup will be reduced. This is due to the
extra form drag (FQ) acting on the protruding obstruction that can be
contributed to the wind setup per unit area. As a result, this reduced
wind stress causing a wind setup on an obstructed area may be expressed
as:
t = R t = R, K p W._
so d s d pa 10
, 0 < Rj < 1
(18)
in which Rd is a reduction factor which represents the ratio of the wind
stress on the obstructed area to that on an open area under the same
wind condition.


30
Since the reduction of wind stress on an obstructed area is due to
the extra form drag, or wind energy loss caused by the obstruction, the
reduction factor may also be defined as the ratio of the total loss of
wind energy per unit length on an open area to that on an obstructed
area with dimensions 1*1, i.e., (Gee and Jenson, 1974)
2
T. X l
Rd = K 2 (19)
[(1 mgD)xs + mFp] x C
2
in which m = density = N/£ N = total number of obstructions, 8 = a
shape factor defined as the horizontal cross-sectional area of average
obstruction element at surface level divided by D. Substituting
equations (7) and (17) into equation (19) gives
K
W
10
(1 m8D2)KpaW^0
+ 3.13 (mCDPaDW2Q) K{(h+zo)ln(^-+l)[ln(^-+l)-2]+2h}
o o
or
Rd =
1
(l-m6D2) + 3.13 (mDCD) {(h+ZQ) ln(^-+l) [ln(^-+ 1 )-2] + 2h}
o o
(20)
_ o
where K = (0.8 + 0.65 W^q) x 10 W^q is in meters per second.
Based on the result presented in equations (18) and (20), the
reduction factor R^ can be determined and incorporated into the storm
surge model to produce more realistic results for the wind stress on
water in flooding areas. The drag coefficient in equation (20) for
vegetations and buildings will be determined and discussed in Chapter
VII and VIII.


31
Bed Shear Stress
A space averaged bed shear stress, 7b, usually can be expressed as
V i-e
Tb = To = yRS = yR T- (21)
in which tq =average bed shear stress along the wetted perimeter;
y = pg = unit weight of water; R = A/P = hydraulic radius = depth in
sheet flow; A = cross-section area; P = wetted perimeter of flow
cross-section; S = slope of the energy grade line; AH = energy loss
per unit weight of fluid over a bed length of L. Primarily developed
for flow in pipes, the energy loss term, aH, is defined by the Darcy-
Weisbach formula as
iH f 55 T (22)
3 0
where f = friction factor based on depth; U = spatial mean flow
velocity; dQ = diameter of the pipe. Since dQ = 4R, the above equation
/
may be written for an arbitrary cross section as
(23)
where f = friction factor based on hydraulic radius.
Incorporating equation (21) into equation (23) in the x and y
directions, respectively, gives the following quadratic forms for the bed
shear stresses:
fp|U|U
bx
f'p|U|U
by
2
2
(24)


32
or in terms of volume transport
_ f,phlqx flp|qlqy
Tbx 2d2 Tby 2d2
(25)
where f = 4f'; U = /U 2 + U 2; q = /q 2 + q 2; U and U = spatial
x y x y x y
mean flow velocity in x and y directions, respectively.
It is assumed that these steady state relationships for the two
shear stress components are valid for storm surge propagation, which is
generally considered to be quasi-steady, i.e., the velocity variation
with time or the temporal acceleration is very small. The quadratic
Darcy-Weisbach form of bed shear stress is the best formula available
to account for the effect of bottom friction.
The friction factor in the Darcy-Weisbach formula, f, has been
studied by many investigators in both pipe flows and open channel
flows. From the abundant experimental data, numerous empirical
formulae have been established to express the relationship between the
friction factor and the dependent parameters, such as bed roughness,
Reynolds number, Froude number and Strouhal number. For example, the
well-known Stanton diagram (1914), Moody diagram (1944) and many others
(to be discussed in the next section) have enabled determination of the
friction factor in varied flow conditions.
In general, a surge could be expected to travel over five different
terrains (Christensen and Walton, 1980):
A. Ocean (river) bottom with flow induced bed form and completely
submerged vegetation,
B. Mangrove fringes and areas,
C. Forested areas and cypress swamps,
D. Grassy areas, and


33
E. Developed areas.
Each of these five categories has unique roughness characteristics.
However, in evaluating the friction factor in hurricane-induced flooding,
these five terrains can be divided into two major categories,
unobstructed and obstructed areas, based on their distinct functions
to retard flow.
Post Approach, Friction Factor for Surges in Unobstructed Areas
Unobstructed areas include the ocean (river) bottom and grassy
areas, the latter of which are assumed to be completely submerged in
water during floods. Friction factors in these kinds of areas can be
determined from the results of previous research which will be discussed
below and used as basis for the present work. Overland flooding in this
study is considered to be turbulent and in the hydraulically rough
range, i.e., the wall Reynolds number is in excess of about 70.
The effect of wall roughness on turbulent flow in pipes has been
studied during the last century by many investigators. An important
result obtained by Nikuradse (1933) in steady flow using six different
values of the relative roughness k/r with Reynolds numbers ranging from
4 6
R = Ud /v = 10 to 10 has been widely used in flow fields and will be
e o
applied in this study (k is the equivalent sand roughness, r is the
radius of pipe; U is the average velocity, v is the kinematic viscosity).
Nikuradse divided flow conditions into three ranges, smooth flow range I
(u^k/v < 4), transition flow range II (5 <_ ufk/v £ 68), and rough
turbulent flow range III (ufk/v > 68) in which u^k/v = R* = wall Reynolds
number, u^ = friction velocity = /x/p.


34
In range III (rough turbulent flow) the thickness of the viscous
sublayer <5 is negligible compared to the equivalent sand roughness, k,
and the friction factor is independent of the Reynolds number.
The distribution of the time-mean velocity obtained using Prandtls'
mixing length approach in combination with Nikuradse's experimental
results is given by the general expression
jj- = 8.48 + 2.5 In | = 2.5 In
(26)
in which u = time-mean velocity in the direction of flow at a distance z
from the theoretical bed. Theoretical bed is defined as the plane
located such that the volume of grains above the plane equals to the
volume of pores below the plane but above the center of grains.
The classic velocity profile given by equation (26) is well defined
at moderate to large distances from the bed and for roughnesses much
smaller than the depth. However, it falls next to the bed whereas z
approaches zero, the time-mean velocity U approaches minus infinity.
This is especially true in flows where the roughness is not significantly
smaller than the depth. Because of the above-mentioned discrepancies,
Christensen (1971) introduced a new law for the velocity profile by using
a modified mixing length approach over a rough bed in the rough range
= 8.48 + 2.5 In
uf
|+ 0.0338) = 2.5 In ( ?973z + 1)
(27)
The form of this equation is the same as that of classic equation (26)
except for the +1 term in the argument of the logarithmic function which


35
makes the time-mean velocity u equal to zero at the theoretical bed. As
the distance from the bed increases to more than a few times k, very
little difference exists between these two velocity profiles.
For practical purposes, the time-mean velocity profile is trans
formed to a depth averaged velocity profile using the fact that the mean
velocity (depth averaged), U, occurs theoretically at a distance
z = 0.368d from the bed also for the modified logarithmic vertical
velocity profile, where d is the water depth, and d/k is larger than 1.
It shall be noted that the k value used here is the equivalent roughness
height for bottom friction only. Therefore, at z = 0.368d equation (27)
yields
= 2.5 ln[%^ (0.368d) + 1]
U K
or
= 2.5 In [10.94 £ + 1]
U K
(28)
where U = time and depth averaged velocity.
The friction factor may in general be related to the velocity
profile by introducing the Darcy-Weisbach formula into the definition
of the friction velocity, i.e., uf = /tq/p = /gRS, leading to the result
(29)
Solving equation (29) for f and introducing equation (28) gives the
following expression for the friction factor
0.32
f
(30)


36
This depth dependent friction factor is proposed for areas where the
surge moves over bottoms at moderate depths. Another equation for f
obtained from Nikuradse's experimental result (1933) for rough
turbulent flow in circular pipes, is given by
= 1.171 + log £ (31)
4/Tr K
It seems quite clear from equation (30) and equation (31) that
determination of friction factor in unobstructed areas is just a matter
of finding the value of the equivalent sand roughness k. This k value
can be related to Manning's n by using a Strickler-type formula
(Henderson, 1966, Christensen, 1978) in metric units
1 8.25 /q
(32)
given k in meters, or in the English units
1.486 8.25 /q
(33)
with k in feet. Values for n may be determined from various sources
such as textbooks by Henderson (1966) and Chow (1959), charts and graphs
by the Soil Conservation Service (1954), and photographs of a number of
typical channels by the U. S. Geological Survey (Barnes, 1967). Other
specific studies, for instance, the experiments conducted by Palmer
(1946), also provide valuable information on the flow of water through
various grass and leguminous covers. Based on the theoretical velocity


37
distribution in rough channels, the value of Manning's n can also be
determined by analytical methods such as that presented by Boyer (1954).
It should be noted, however, that these values of n from previous
sources may not be applied to every case under natural conditions. It
should be also careful in selecting the values of n, since a small error
on n will be amplified substantially on k by using the Strickler-type
formula. Therefore, a method to determine the k-value from the vertical
velocity distribution in turbulent flow over rough surface is recom
mended (Christensen, 1978). Let Hi be the velocity at m depth, that is,
at a distance md from the bottom of a wide rough channel, where d is the
depth of flow. By equation (26), the velocity may be expressed as
Ui 29.7nid
= 2.5 In £ (34)
Similarly, let u2 be the velocity at r\2 depth; then
u2 29.7r)2d
= 2.5 In u
(35)
Subtracting equation (34) by equation (35) and solving for uf,
uf =
U2 Ux
*12
2.5 In
fii
(36)
Introducing equation (36) into equation (35) and solving for k,
29.7md
(37)
n2
u2- ux


38
Proposed Friction Factors for Surges in Obstructed Areas
Obstructions in these areas are defined as roughness elements with
significant heights which either protrude through the water layer or
consist of relatively rigid elements with heights that are sufficient to
cause form drag that are much larger than surface friction on the same
area. The two major forms of obstructed areas, mangroves and buildings,
to be discussed in this study are often higher than the storm surge
level so that the influence of hydrodynamic drag on the individual
elements should be taken into consideration together with other factors
of resistances in overland floods. The theoretical analysis presented
here is based on the assumption of steady or quasi-steady flow in the
rough flow range.
Consider a design flow that passes over an obstructed area whose
bottom is horizontal. The total averaged shear stress, x in the
direction of flow may be written as equation (21), i.e.,
- dc d aH
t0 yRS = yR
For the steady state case, assume that the viscous stresses, turbulent
stresses and hydrodynamic drag acting on the obstructions contribute
independently to the flow system without any interaction among them.
Following Prandtl's assumption (Schlichting, 1979), that the shear stress
in the x-direction is constant and equal to the wall shear stress at
all distance z from the wall, an equilibrium equation is given in the
following and shown in Figure 13.


t
FIGURE 13: Distribution of Horizontal Apparent Shear Stress and of its Drag,
Inertial and Viscous Components, (Christensen, 1976).


40
du
YRS = ^ (l-e) -
pu'v'
(l-e) + mCDdDY S|
(to}
Viscous
Stress
(Tt)
Turbulent
Stress
(Td>
Hydrodynamic
Drag
(38)
where e = fraction of total area occupied by obstructions, u', v' are
the turbulent velocity fluctuations in the x and z directions,
respectively, m = number of obstructions per unit bed area, CQ = average
drag coefficient, D = average diameter of the obstructions in the pro
jected plane normal to the flow.
For fully developed turbulence, the viscous term is negligible
compared to the turbulent term and may be omitted. Consequently
pgRS = -pu1v'
(1-e) +
mCDdD |
(39)
The turbulent stress, x^, which expresses the rate of flow of x momentum
in the z direction, was first derived by 0. Reynolds from the equation
of motion in fluid dynamics, and is termed the Reynolds stress or the
inertia stress. The Reynolds stress on the right hand side of equation
(39) must be
1
e
(40)


where
1
41
= (
Id
T
1 e
V2
)
, 0 < $ < 1
(41)
may be defined as the obstruction correction factor, which directly
reflects the reduction of the Reynolds stress pu1v1 due to the
presence of the obstructions. If the drag is equal to zero, implying
that no obstruction exists (e = 0), the obstruction correction factor
will of course become equal to 1. The corresponding friction velocity
depends on the bottom friction only. It is defined from equation (40),
\'2
)
uft *
(42)
where uft is the friction velocity based on the total bed shear stress
V i>e uft = /T7p*
Similarly, introducing f and fj. as the friction factors of bottom
friction and total friction including bottom friction and drag acting on
the obstructions, respectively. Therefore, these two friction factors
can be related to the mean velocity as shown in equation (29),
U 2 1^2
= (7.) for the obstruction-free area
Uf T
(43)
and
U
uft
for the total obstruction area
(44)
Eliminating U from the two equations above, relation between f and fj.
is given as
f fi *2
(45)


42
Substituting equations (40), (42) and (44) into equation (39), the
equilibrium equation becomes
yR
A = ¡L
2 (1-e) + mCDdDY ^
or
AH = £. f ,.,2
YR y = f f'U" (1-0 + mCpdDy ^
The head loss, AH, then may be written as
4K= f' <-*> +raCDd0f?s-
and
2 L
AH = [f (1-e) + mCDdD]
(46)
(47)
(48)
which is the form of the Darcy-Weisbach formula shown in equation (23).
An equivalent friction factor, f^, which includes the effects of bottom
friction and form drag based on equation (48) is introduced
f* = f (1-e) + mCpdD
(49)
where, according to equation (30),
0.32
[In Therefore, the equivalent friction factor, f^, must be equal to the
total friction factor, f|, according to the above definitions, as can
be proved easily by substituting equation (45) into equation (49).
An apparent roughness height ka which represents the sum of bottom
roughness and rigid drag element can then be expressed and calculated
from equation (31), i.e.,
-7= 1.171 + eog if-
4/f7 a


CHAPTER IV
EXPERIMENTAL VERIFICATION OF FRICTION FACTORSMODEL LAWS
The formula proposed in Chapter III for the friction factor
(equation (49)) has left some unsolved questions: What is the drag
coefficient (C^) for the kinds of obstructions being studied? What
kind of relationship do drag coefficient and density of obstruction
have? Does the friction factor depend on unspecified flow conditions
like the Reynolds number? All these problems may not be satisfactorily
answered without experimental verification. Therefore, laboratory
measurements were carried out in this study to verify the analytical
results and to build a data base for further development.
The requirements of similarity between hydraulic scale-models and
their prototypes are found by the application of several relationships
generally known as the laws of hydraulic similitude, i.e., geometrical,
kinematic and dynamic similitude. These laws, which are based on the
principles of fluid mechanics, define the requirements necessary to
ensure correspondence between model and prototype.
Complete similarity between model and prototype requires that the
system in question be geometrically, kinematically and dynamically
similar. Geometric similarity implies that the ratio of all corres
ponding lengths in the two systems must be the same, kinematic
similarity exists if all kinematic quantities in the model, such.as
velocity, is similar to the corresponding quantities in the prototype,
43


44
and dynamic similarity requires that two systems with geometrically
similar boundaries have the same ratios of all forces acting on
corresponding fluid element of mass. Following the basic dynamic law
of Newton, which states that force is equal to rate of momentum,
dynamic similarity is achieved when the ratio of inertial forces in
the two systems equals that of the vector sum of the various active
forces, which include gravitational forces, viscous forces, elastic
forces and surface tension forces in fluid-motion phenomena. In other
words, the ratios of each and every force must be the same, as given
in the equation form
V (F) (Fv) (F.) (Fs)
P a P P- P. P
(50)
where subscripts p and m refer to prototype and model, relatively.
Since it is almost always impossible to obtain exact dynamic
similitude, it becomes necessary to examine the flow situation being
modeled to determine which forces contribute little or nothing to the
phenomenon. These forces can then be safely neglected with the goal of
reducing the flow to an interplay of two major forces from which the
pertinent similitude criterion may be analytically developed (Rouse,
1950).
For a model of hurricane-induced flooding of coastal areas,
elastic forces and surface tension forces are sufficiently small and
can be neglected. The condition for dynamic similitude reduces to
equating the ratio of inertial forces to the ratio of either gravity


45
forces or viscous forces. Viscous forces are only considered in the
model of canopies whose surface friction effects are investigated. In
models for measuring form drags in turbulent flow with high Reynolds
numbers (inertia force/viscous force), viscous forces are small
compared to the major forces due to the turbulent fluctuations and this
can be neglected in this instance. Since the vertical dimension scale
(involving flow depth) cannot follow the horizontal dimension scale
in building models as the flow depth would be much too small for
measurements to be made, or the viscous force would become important
and cannot be neglected for the small flow depth if the same fluid is
used for the prototype and the model. Therefore, a model with a
different vertical dimension scale than horizontal dimension scale is
used to keep the Reynolds numbers in the turbulent flow range. For
simplicity, such distorted models will be introduced first since
undistorted models with the same length scale in both the vertical and
horizontal dimensions can be regarded as a special case of the former.
Distorted Model for Buildings
The fundamental model scale ratio may be written as:
L B
Length Scale (horizontal): (51)
Lm Bm
D
Depth Scale (vertical): Nd = D^ (52)
T
Time Scale:
(53)


46
Force Scale:
(54)
where L = horizontal length, B = horizontal width, D = vertical depth,
T = time and F = force.
Following the development of Christensen and Snyder (1975), the
force scale for the gravity component in the nearly horizontal direction
of the principal flow may be written as
unit sine
weight volume of slope
i ^ i r~>
K =
g
L B D (D /L p g 2
P P P P P = (_£) (_£.) N N
Pmgm CCtTU V V Vd
P 9
P^P
m m m
m' nr
m
Jm
(55)
where p is the fluid density, g is the gravitational acceleration and
D/L is the bed slope ar the slope of the energy grade line.
In a unidirectional flow the inertial force can be expressed as a
horizontal, or nearly horizontal area multiplied by the Reynolds shear
stress, which is proportional to the fluid density and the time mean
value of the product of a vertical velocity fluctuation and the
corresponding velocity fluctuation in the direction of the time mean
flow. Consequently, the inertial force is
F.. = puv1 (area) (56)
and the inertial force scale can be written as


47
area
D_ r
Pp
pnW
m m
Pn N! N .
m
N,
(57)
In order to have dynamic similarity between model and prototype,
Kgravity should be equal to Kn-nertial i-e,} equation (55) should be the
same as equation (57). This condition is expressed by
or
(58)
where gp is assumed equal to g^. The ratio between a gravitational
force and an inertial force is commonly known as the Froude number, and
the resulting time scale (equation (58)) is the similarity criterion of
the Froude law for distorted models.
The scale ratios of the drag coefficient and friction factor in
distorted Froude models have to be determined before experimental data
can be interpreted correctly. The drag force proposed for the present
study is given by equation (39)
^d = Td* A = m^gd^P jr- A (59)
The drag force scale in the flow direction can then be written as


48
1
L B
JL£
1
L B
m m
L 2
)(Cn)ABmpJr) L B
DmmmmT mm
m
or
. p
K, *
d ,r \ vp "d l
(Cpj m
m
(60)
To satisfy the ratio of the force scale in the Froude model law, Kd must
be equal to K^, and by substituting the time scale, Nt = N£/(Nd^2, into
equation (60) gives
(CD> p m
(61)
Shear forces generally may be expressed by the Darcy-Weisbach form
F
s
(62)
where f is the friction factor and can be substituted by the equivalent
friction factor, T, or bottom friction factor, f', later for the present
use. The shear force scale is given as
or
Ks \
m pm t
(63)


49
Dynamic similarity requires that = K^. Substituting the time
scale, equation (58) into the required equality, i.e.,
£><>¡ (i)2. p
m pm
(64)
gives the expression for the scale ratio of the friction factor
f N.
_B. = A
fm
(65)
Comparing equation (61) with equation (65) it is noted that the drag
coefficient is the same in the model as in the prototype, however, the
friction factor of the prototype should be modified by an inverse
distortion ratio, N^/N^, in the Froude distorted model. The distortion
ratio usually is defined as
D =
r Nd
(66)
Undistorted Model for Mangrove Stems and Roots
Due to the fact that the dimensions of mangrove stems and roots
are one to two orders of magnitude less than the water depth, an
undistorted model can be used in this part of the study. All the
methodology applied in the previous section for a distorted model is
also applicable for this analysis. In the case of an undistorted
model where = N^, equation (58) reduces to
Nt (67)


50
which is the time scale for an undistorted Froude model. Both the
dimensionless coefficients and f are the same in the prototype and
model, and the distortion ratio becomes unified in this instance.
Distorted Model for Canopy
Before the model law for canopies is derived, it is necessary to
determine what kind of boundary layer forms over the surface of a
mangrove leaf. A prototype red mangrove canopy was tested in the
hydraulic laboratory flume and it was quite apparent that all the leaves
bent in the direction of flow even at a flow velocity less than 10
cm/sec. Such high flexibility makes the leaves more resistant to a
storm attack. As a result, leaves offer only skin friction and no form
drag to resist the flow.
The surface of a leaf is assumed to be smooth in this study so
that theoretical and empirical results on the behaviour of a boundary
layer on a smooth flat plate can be applied. In general, the point
of instability on a flat plate at zero incidence to the flow is
determined by the critical Reynolds number
U x
(R
e.x^crit
'crit
(68)
in which U is the free stream velocity and x is the distance from the
leading edge of the plate measured along the plate. An analytical
stability criterion developed by R. Jordinson, based on W. Tollmien's
theory, is given by
U 6
/
(69)


51
where = displacement thickness and
(Schlichting, 1979)
(70)
Combining the last two equations give (R ) = 9.1 x 104.
c jA Ci I U
In reality, the position of the point of transition from laminar
to turbulent flow will depend on the intensity of the turbulence in
the external flow field. This has been investigated experimentally by
J. M. Burgers, B. G. Van der Hagge Zijnen and M. Hansen in 1924. These
measurements led to the result that the critical Reynolds number was
contained in the range
U x
()crit = 3-5 x 105 t0 5 x 105 (Schlichting, 1979) (71)
Similar experiments done by Schubauer and Skramstad in 1947 also yielded
results which indicated that the critical value of R is in a range
G jX
from 9.5 x 105 to 3 x 106 depending on the relative intensity of the
free-stream turbulence, (l/U^Ku'u'/sj/2 (Hinze, 1975). Therefore, the
minimum value of (R ) .. is chosen as 3.5 x 105 for the present study.
cjX crit
In the prototype, the maximum value of x is the largest leaf length
and was found to be 5 inches; the highest flow velocity is assumed to be
10 ft/sec, which results in a maximum value of (R ) .. of about
c)A Li I l
2.98 x 105, which is still lower than but near the minimum value
(R ) ... This shows that a turbulent boundary layer has very little
g ,x cn l
chance to be formed over such a short length, and that a laminar
boundary layer should prevail over the entire leaf area.


52
Skin friction can be expressed in terms of a dimensionless skin
friction coefficient, C^., times the stagnation pressure, pU2/2, and area
of the plate, A, as follows
Ff = CfP I A (72)
in which, for a laminar boundary layer,
C
f
1.328
(73)
where R = lk/v denotes the Reynolds number formed by the product of the
Xj *
plate length and the free-stream velocity (Schlichting, 1979). The skin
friction scale in the flow direction may be written as
K
f
LnV1/2
(-P P)
TnV
JEL£
f III III \
m m
m
L 2
(/) A
T^~
(j51) A
P
m
(74)
where z and A are the leaf length and leaf area, respectively. Following
the undistorted Froude law, = (N^2 p and v are the same in proto
type as in the model since the same water properties are assumed in the
two systems. Equation (74) is then reduced to
3A -V2 A
V m m
(75)


53
For dynamic similarity, Kp has to be set equal to K i.e.
% £ -V2 A 3
= <"*>
m m
(76)
giving the length scale
£_ An2
f(Ni)
m m
(77)
It is obvious that the dimensions of a leaf need to be distorted
according to the scale ratio shown in equation (77), which is the result
of inclusion of viscous effects on a leaf surface in a Froude law
controlled flow model.


CHAPTER V
MODEL DESIGN
Recirculating Flume
The present model tests were conducted in the hydraulic laboratory
flume of the Civil Engineering Department at the University of Florida.
Figures 14 and 15 show the primary elements of the flume geometry.
The main channel is 120 feet (36.58 meters) long, 8 feet (2.44 meters)
wide, and 2.7 feet (0.81 meter) deep. A false-bottomed section 20 feet
(6.1 meters) in length and 13.4 inches (34 cm) deep is located at the
longitudinal center of the flume. Centered in the false-bottomed area,
observation windows cover a length of 12 feet (3.66 meters) and are 2
feet high (starting at the bed level). The 74 kW (100 HP) flume pump
has a maximum discharge of 40 cfs (1.1 m /sec). Between the pump and
the overflow weir are two sets of 8 inch long, 2 inch diameter poly
vinyl chloride pipes arranged in a honeycomb fashion. Two more sets
of these pipes, which act as flow straighteners, are located just
beyond the outlet weir. By adjusting two gate valves at the main
delivery pipe and return pipe, the flow rate and depth over the
Thomson V-notch weir can be regulated. A Poncelet rectangular weir
is also available for high discharges. A motor-driven sluice gate
at the downstream end on the main channel serves to regulate the water
depth in the main flume and to moderately regulate the discharge.
54


FIGURE 14: Plan of Flume
f
Power Source for Trolley
Rail for Movable Trolley
Return f
Channel
38
(l.llm)
iv > \
Main
Channel
i*r-
8'
(2.44m)
T
28"(0.8lm) .
(1.22m)
FIGURE 15: Cross Section of Flume


56
A movable trolley which spans the entire width of the flume and
which has a maximum towing speed of 2 feet per second provides the
work-deck for calibrating velocity meters as well as collecting data.
To determine the drag coefficient and equivalent friction factor
for a given roughness in the rough turbulent flow, the energy loss, aH,
has to be measured (cf. equation (47)). According to the principle of
conservation of energy, the total energy head at the upstream section 1
should be equal to the total energy head at the downstream section 2
plus the two sections, i.e.,
2 2
u, u
dl + 2g = d2 + 2?
+ aH
or AH = (d1 This equation is applied to the present study in which the channel bed
is horizontal and a value of unity is assumed for the energy coefficient
(Henderson, 1966). Therefore, the energy loss aH due to the friction in
turbulent flow can be measured by knowing the water depths and mean
velocities at the two sections. Relating the measured results of energy
loss to the Darcy-Weisbach equation
AH
,, Hi
fe 2g Ra
(79)
1 ^1 ^2
in which Ua = 2 (U-j + U2) and = 2- the equivalent friction
factor f' can be determined for the designed roughness elements. To
determine the water depths and velocities, some instruments are employed
for this study and described in the following section.


57
Instrumentation
Velocity Meter
A Novonic-Nixon type velocity meter was employed for all velocity
measurements. The probe consists of a measuring head supported by a
thin shaft 18 inches long with an electrical lead connection. The head
consists of a five blade, impeller mounted on a stainless steel spindle,
terminating in conical pivots (Figure 16). These pivots run in jewels
mounted in a sheathed frame. The impeller is 1 cm in diameter, machined
from solid PVC and balanced. An insulated gold wire within the shaft
support terminates 0.1mm from each rotor tip. As the rotor is rotated
by the motion of a conductive fluid, the small clearance between the
blades and the shaft slightly varies the impedance between the shaft
and the gold wire. This impedance variation modulates a 15KHz carrier
signal, which in turn is used to detect rotor rotations.
The range of this velocity meter is from 2.5 to 150 cms-1 (0.08 to
4.92 fps) with an advertised accuracy of + 1 % of true velocity. Its
operating temperature is from 0 to 50C (32 to 122F) with an operating
medium of water or other fluids having similar conductive properties.
The shaft of the current meter was clamped to the rack of a point gage.
The point gage bracket was then bolted to the trolley carriage so that
the instrument could be easily removed from its bracket with no deviation
in the vertical setting. Also all the accuracy and ease of a point gage
and vernier is accrued.


58
O I 2 cm
> I I
FIGURE 16: Novonic Nixon Velocity Meter


59
Data Acquisition System (DAS)
The data acquisition system is composed of two pieces of equipment:
an input box and an HP 9825A desk-top programmable calculator (Figure 17
and 18). The input box, which is specially designed for coupling with
the HP 9825A, has connectors for fifteen thermistors, ten Cushing
electromagnetic current meters, two Ott velocity meters and two Novonic-
Nixon velocity meters. It contains the electronic circuitry which takes
the raw transmission from the measuring devices and converts it into
usable signals tor the programmable calculator. An electronic timer
which registers six counts per second is also contained in the input box.
The HP 9825A interfaces with the input box to provide program
control and data storage capabilities. The calculator has a 32-character
LED display, 16-character thermal strip printer, and a typewriter-!ike
keyboard with upper and lower-case alphnumerics. A tape cartridge with
the capacity of 250,000 bytes is used with this calculator to store and
access the programs. Based on the manual of the HP 9825A and the
instructions provided by Morris (1979), programs designed to calibrate
the Novonic-Nixon meter, measure the velocities and perform linear
regression are listed in Appendix B. Through the DAS a substantial
amount of time usually used in experiments and data reduction was saved
and the accuracy of results was greatly enhanced.
Depth-Measuring Device
Determination of the flowing water depth by measuring the difference
of water surface elevations is the most important part, except for the
measurement of the flow velocity, of the laboratory experiments. However,


60
FIGURE 17: Input Box of Data Acquisition System
FIGURE 18: HP 9825A Programmable Calculator


61
the measurement of water surface elevations is not easy due to the
rough water surface of turbulent flow. In addition, an accuracy of one
millimeter or better is needed for the depth measurement, since the
difference of water surface elevations at two sections is less than one
centimeter in many test cases. Therefore, a stable and sensitive depth
measuring device is required for the present study. Figure 19 shows the
schematic diagram of the device designed, in which the hoses connected to
the tube, which have a diameter of 0.2 inch, were extended to the desired
cross-sections in the flume. A 10 inch long, 0.15 inch diameter glass
tube was attached to the end of each hose and positioned perpendicular
to the water surface. In high velocity flows some weights were added to
the 0.15 inch diameter tube in order to maintain its vertical position.
The diameter of the tube is 2 inches which is large enough to allow the
point gage to be able to contact the plane water surface without the
influence of surface tension on the side wall of the tube. The point
gage was attached to the top of the tube, and the still water level is
indicated when a white ball on the gage appears, which indicates that
the sharp tip of the gage is touching the water surface (Figure 20).
A manual hand operated vacuum pump was used to help initiate a siphon
between the water in the flume and in the tube at the beginning of each
test and to pump air bubbles out of the hoses periodically during
the test.


62
FIGURE 19: Setup of Water Depth Measuring Device
FIGURE 20: Point Gage and Tube


63
Selection of Model Scales
In the last chapter the scale-model relationships based upon the
Froude law were derived. The scale to which the model should be
constructed depends on the following factors: the size of the flume
(length, width and depth), the discharge capacity of pump, the accuracy
of instrumentation and the dimensions of the prototype. According to
these factors, the vertical length scale of 1:10 (or = 10) is selected
through the entire study for both distorted and undistorted models.
Mangroves
The dimensions of all stems and roots, including height and
diameter of the prototype, are reduced to 1/10 for the model based on
the undistorted Froude law. However, for the canopy some distortion
of scale between prototype and model is required according to equation
(77) in the Chapter IV, i.e.,
it.
A*
m
<\>
in which N = Nd = 10, i is the length of a leaf and A is the surface
area of the leaf. For simplicity, the elliptic shape of mangrove leaves
are approximated by a rectangular area with a length n and a width w.
Equation (77) is then reduced to
(80)
By choosing 1:10 for the length scale for this study, the width scale of


64
the leaf becomes
w 7/
= 10/4= 56 (81)
m
Therefore, the dimensions of the leaves used in the model can be
estimated from the derived relations and are shown in Table 5, which
are based upon the prototype data listed in Table 2.
TABLE 5: Scale Selection for canopy
Section
No.
(1)
(2)
(3)
prototype
length (£)
2.00
3.00
4.00
(in)
width (w)
1.00
1.50
1.75
Model
length (£)
0.20
0.30
0.40
(in)
width (w)
0.018
0.027
0.031
p' m
10
10
10
w /w
p m
56
56
56
Buildings
This part includes the three previously discussed kinds of
buildings: high-rise, medium-rise and residential buildings. Table 6
shows a sumnary of the average parameters for these three categories.
In searching for material to be used in constructing the buildings in
the model, it was found that the ratio of length and width of a standard
block was very close to that of the prototype. Another advantage in
using concrete blocks is that they are easy to set up, since each


TABLE 6: Average Parameters of Prototype and Model for Building Areas
PROTOTYPE
MODEL (Nd =
10)
Type
of
Average
Dimension
(ft)
Approximate
Dimension
(ft)
Density
%
Dimension
(in)
Density
Buildings
1ength
width
1ength
width
no.
no.
Deve-
1 oped
1ength
width
no.
no.
l000'x500'
acre
8'x8'
8'x2,871
High-Rise
224
112
225
109
7.19
0.63
36
174
15.50
7.50
28
10
Medium-Rise
99
51
103
' 50
23.62
2.06
24
80
15.50
7.50
20
7
Residential
62
30
68.87
6.00
26
48
15.50
7.50
20
7
Note: 1 acre = 43,560 ft^


66
concrete block is heavy enough to withstand all the flow velocities in
this study. Therefore it was not required to anchor them to the flume
bottom. Scales of the model are then determined from the horizontal
dimension of a concrete block (15.5 in x 7.5 in) and the dimensions of
prototype buildings, as shown in Table 6. It is noted that in order to
scale the prototype buildings into the 8 foot wide flume, it was
necessary to use a distorted model.
Model Setup
Mangrove Stems and Roots
Based on the average parameters obtained from the prototype,
patterns of red and black mangroves were designed and shown in Figures
21 and 22. These two patterns were the best arrangements that could be
achieved in the modeling in order to insure that the stems and roots
were distributed evenly and yet still maintained their own natural
characteristics in dispositions. For example, the prop roots were
arranged in a hexagon pattern, which was found to be the most common
disposition found in natural. The staggered pattern used for the stems
of black mangroves and the root system of red mangroves was considered
to be the best regular pattern to simulate the fully random distribution
found in the prototype. The legends listed in Figures 21 and 22 were
the actual dimensions used in model setup.
The stems of red and black mangroves were simulated by dowels of
the specified diameters and heights. The substems and prop roots were
simulated by galvanized nails with the caps removed. The air roots of
black mangroves would be very hard to model on a one by one basis due to


67
o
o o
0+0
o o
o
O o
0 + 0
o o
o
o O
o + o
o
o
o o
o+o
o O
o
o
o o
0*0
o
o
O o
0+0
o o
o
o o
0+0
o O
o
o O
0+0
o o
o
0
o
o
O O
0+0
o o
o
o o
0+0
o o
o
o
o o
0 0
o
o
15
h3ui
^r?ir
o + o
o o
O Main-Stern
Sub-Stem
0 Prop Root
FIGURE 21: Model Setup for Red Mangroves


68
o Stem
Air Root
FIGURE 22: Model Setup for Black Mangroves


69
their high density and small dimensions. Therefore, a manufactured nylon
door mat whose strings have the same height (0.6 inch) and the same
thickness (0.025 inch) as the design model dimensions of air roots was
used. Density of the strings is 44 per inch square area which is 91 l
of the average design density (48 per inch square area). The only
deficiency in using this mat is that the strings are blade-shaped, which
may cause a higher resistance to the flow than cylindrical air root.
However, considering the advantages of using the mat, this deficiency is
considered to be insignificant.
These dowels, nails and mats were fixed on three 8 feet by 4 feet
marine plywood sheets, which were coated with latex paint to prevent
swelling (Figures 23 and 24). The plywood sheets were secured to the
false bottom by a row of concrete blocks and by a 24 feet long L shaped
steel beam attached to two sides of the plywood sheets (Figure 25).
The row of concrete blocks stacked 15 inches high was placed in the main
flume, starting from the last flow straightener and extending a distance
of 80 feet. Thus only half of the flume width was used in these
experiments.
Canopy of Red Mangroves
During the second part of the experiments a canopy was constructed
in the red mangrove area. Strips of galvanized metal plates were used
to simulate 'leaf strips'. This assumes that the leaves are closely
connected to each other. Strips with three different widths, 0.2, 0.3
and 0.4 inch, represent three different sizes of leaves, as shown in
Table 5. Each stripe has a height of 9.6 inches, which covers 539, 360
and 308 leaves for section #1, #2 and #3, respectively. Stripe numbers


FIGURE 23: Stems and Roots of Red Mangroves
FIGURE 24: Stems and Roots of Black Mangroves


71
FIGURE 25: Overview of Setup for Mangroves
FIGURE 26: Setup of Model Equivalent of Canopy of Red Mangroves


72
for each section, which is 5 feet long, 34.2 feet wide and 8 feet high
in the prototype, can be calculated from the densities measured (cf.
Table 2). As a result, stripe numbers needed in the model for section
#1 to #3 are found to be 25, 20 and 10, respectively. These stripes
were also arranged in a staggered pattern, as shown in Figure 27.
Figure 26 shows the setup of the stripes in which the stripes are sus
pended from the top of the supporters and fixed to the plywood bottom.
Buildings
In this part of the experiments, the whole width of the flume was
used. As mentioned in the last section, the concrete blocks with
dimensions 7.5 inches x 7.5 inches x 15.5 inches were used to simulate
the buildings for the three different types of developed areas. Figure
28 shows 21 patterns to be tested in which no. 1 to 13 were designed to
simulate high-rise building areas, while no. 14 to 21 were for medium-
rise building and residential areas. As can be seen in these patterns,
both the aligned and staggered dispositions were included for each
density of the buildings. The design densities are started from low to
high and will at least cover the average densities obtained from the
prototype for the three developed areas (cf. Table 6). No extra work
was needed to anchor these concrete blocks except to move them into the
desired positions, since each concrete block weighs about 38.5 pounds
and two layers of blocks are steady enough to withstand all the flows
used in this study.


SECTION # I SECTION# 2 SECTION # 3
73
X 0.2" x 9.6"
0.3" x 9.6"
A 0.4" x 9.6"
FIGURE 27
Distribution of Leaf Stripes in the Model
OPEN AREA 4*ROOT AND STEM AREA-*)


74
FLOW

(2)

(3)

(4)

FIGURE 28: Building Patterns Designed for the Tests


75
uu
IT
nn
M
FLOW
V
(9)
(10)
U
U
_n
u u
n n -
(ID
(12)
(13)
8'
i l


8'


11


(14)
(15)
FIGURE 28: CONTINUED


76
FLOW
i
,




0


00
0
(16)
(17)




-
0 0 0 D
0 0 0 0 0
0 D 0 0 0
0 0 0 0 0
(18)
(19)




D 0 0 0
DOOODO
D 0 0 D 0
OOOODO
(20) (21)
FIGURE 28: CONTINUED


CHAPTER VI
EXPERIMENTAL TEST SERIES
Experimental Procedure
Calibration of Velocity Meter
After the construction of the apparatus, the first step in the
experimental procedure was to calibrate the velocity meter. Mounted
with its normal support on the carriage, the Nixon meter was pulled
through still water at constant velocity over a distance of 20 to 40
feet with the trolley. By operating the specific keys on the HP 9825A
calculator to execute program statements which read initial and final
values of propeller revolutions and time, the average frequency of the
current meter, the true velocity and the percent error of the calibra
tion curve were then computed and printed out. If the absolute error
was greater than 5 % the instrument was recalibrated.
The meter was checked in the range of 5 to 60 cm/sec and no less
than 20 points were used to determine a linear least square fit of
frequency versus velocity. Appendix C contains a complete program
listing for the HP 9825A.
Measurements of Mean Velocities
The velocity and depth obtained in this study were measured in the
center line of the test sections where the influence of sidewall was not
felt. The experimental run begins when the main pump is started. It
77


78
usually takes about twenty minutes for flow to reach steady state for
each set of discharge values. After the flow became stabilized,
velocities were taken at one section 30 feet downstream of the last
flow straighteners. Nine points on a vertical at the relative depths
of: z/d = 0.1504, 0.1881, 0.2352, 0.2492, 0.3679, 0.4601, 0.5754,
0.7197, 0.9000, were sampled to best describe the vertical velocity
profile (Christensen, 1978). The velocity at each depth was then
determined by the velocity program (Appendix B) from the calibration
formula and was printed out for immediate checking. Each velocity
obtained is on average velocity over a time span of 30 seconds which is
the maximum time interval that can be used with the HP 9825A. Plotting
the vertically distributed velocities on graph paper and integrating
over the water depth yields the discharge per unit width /udy. The
spatial mean velocity for each run then was obtained from the value of
unit-width area divided by the water depth. Even though this method is
time consuming for the large number of runs, it is still the best way
to determine the mean velocity for the mangrove part of the experiments
in which the test channel occupies one half of the main channel.
For the building part of the experiments, in which the entire main
channel was used, a discharge formula for the Thomson weir derived by
the hydraulic laboratory of Civil Engineering Department, University of
Florida, was applied to determine the mean velocities, i.e.,
2.514 ,
Q = 2.840 H (82)
where Q is the discharge from the Thomson weir in cubic feet per second,
and H is the vertical distance in feet between the elevation of the


79
lowest part of the notch or the vertex and the elevation of the weir
pond. Eighteen runs with mean velocities from 17 cm/sec to 53 cm/sec
were tested by both methods to determine the accuracy of the weir
formula; the results are shown in Figure 29. It is apparent that the
mean velocities obtained using the Nixon meter in the center line of the
flume Un is slightly larger than that given by weir formula Uw, but is
within a limit of 5 %. This small error is considered to be insigni
ficant and may be compensated for by the advantages of using the weir
formula. For instance, the fluctuating water level above the weir vertex
due to the instability of the pump was often observed, therefore, the
mean velocity obtained from an average value of H over a longer period
of time should be more representative than that measured by the Nixon
meter over a 30 second period.
Measurements of Water Depths
For each run three water depths were measured by using the device
shown in Figure 19. Two water depths were taken at the two sections
which covered the roughness area and one was taken at the section where
velocities were measured out of the roughness area, located 30 feet
downstream of the last flow straighteners. Since the water head losses
between the two sections in the model tests were in the range from less
than 1 to a few centimeters, the water depths were measured to an
accuracy of one hundredth of a centimeter for a precise and reliable
result. To implement this fine measurement, all the siphon hoses used
in the tests were kept free of air bubbles and the well graduated
electronic point gages were used. Before each run the still water


80
FIGURE 29: Relation Between U and U
n w


81
depth was measured and its scale reading for water surface elevation on
the point gage was recorded. The same reading was performed for each
section after the flow became stable. From the difference of these two
readings for water surface elevations and the initial still water depth
the flowing water depth can be calculated.
In general, it takes about 5 to 20 minutes for a new water level in
the tube (cf. Figure 19) to reach its equilibrium state, which can be
observed by moving the vernier on the point gage to see whether any
change in the water level is detected. This water depth measuring
device worked very well through the entire experiment and provided
consistent and reliable data.
Experimental Runs
Mangrove Areas
The total model lengths of the red and black mangrove areas were 5
and 15 feet, respectively (Figure 25). For the red mangrove area the
water depths were measured at the two ends of the 5 feet long area. For
the black mangrove area, the first section was chosen 4 feet from the
front end, and the second section was located 3.5 feet from the rear end
of the black mangrove region so that the influences, including the
disturbance caused by the red mangroves in the front, and the depth drop
due to the end of the plywood sheets in the rear could be eliminated.
Therefore, a total length of 7.5 feet centered in the middle section of
the black mangrove area was used to measure the energy loss.
During the first part of experiments, 7 runs were conducted for the
air roots of the black mangrove area to determine its apparent roughness


82
height and friction factor. In the second part of the experiments, 38
runs were performed for the red mangrove areas (without canopy) and
black mangrove areas by adjusting the discharge value and changing the
still water depths so that the flow Reynolds number (R^ = UgR^/ v)
covered a range from 20,000 to 55,000 while the Froude numbers varied
from 0.14 to 0.44. An additional 32 runs were conducted for the red
mangroves with canopy at the later stage to determine the importance of
a canopy in reducing the flow energy.
Building Areas
At least 10 runs were conducted for each of the 21 patterns shown
in Figure 28. These runs for each pattern were controlled by adjusting
the flumes discharge valve and the tail gate so that they covered a
range of Reynolds number (R^) from 20,000 to 70,000, while the Froude
number varied from 0.1 to 0.5. Figure 30 shows 20 pictures of the
designed patterns in which pattern No. 9 is not included due to the
faulty picture. The results obtained for medium-rise building areas
can be converted using appropriate scaling factors to use in residential
areas since these two areas are presumed to have the same relative
distributions and have only dimensional differences.


83
FIGURE 30: Designed Building Patterns in the Tests


Full Text
FRICTION IN HURRICANE-INDUCED FLOODING
By
SHANG-YIH WANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
• UNIVERSITY OF FLORIDA
1983

ACKNOWLEDGEMENTS
The author wishes to express his sincerest gratitude to the
chairman of his supervisory committee, Dr. B. A. Christensen, for all
his expert guidance, the tremendous benefit of his professional
competence and vast practical experience throughout this study. He
also wishes to thank Dr. T. Y. Chiu for his advice, understanding,
encouragement and support during the author's six years of graduate
study at the University of Florida. Without their untiring patience
and help this dissertation would not have been possible.
Thanks are also due to Dr. D. P. Spangler, Dr. B. A. Benedict,
Dr. T. G. Curtis and Dr. H. Rubin for serving on the author's super¬
visory committee and for their consulting and assistance.
Appreciation is extented to Drs. A. J. Mehta, D. L. Harris,
F. Morris and P. Nielsen for their suggestions and providing reference,
which contributed greatly to this study.
The author is indebted to Mr. E. Dobson for his technical assis¬
tance, Ms. L. Pieter for her drafting and Ms. D. Butler for her typing.
Special thanks are due to Mr. E. Hayter for his help in the preparation
of this dissertation.
Finally, the author wishes to thank his wife, Fu-Mei, whose partici¬
pation in every phase of this study has made these years more joyful.

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF SYMBOLS x
ABSTRACT xv
CHAPTER
I. INTRODUCTION 1
Storm Surge Prediction 4
Objectives of Present Work 6
II. FIELD EXPLORATION OF PHYSICAL ENVIRONMENT 7
Mangrove Areas 7
General View of Mangroves in Florida 7
Sampling of Mangroves 8
Developed Areas 15
High-Rise Building Areas 15
Medium-Rise Building Areas 17
Residential Areas 21
III. THEORETICAL BACKGROUND AND DEVELOPMENT 22
Hydrodynamic Equations for Storm Surges 22
Wind Shear Stress 23
Wind Velocity Profile in Vertical 25
Proposed Wind Shear Stress on Obstructed Areas 27
Bed Shear Stress 31
Post Approach, Friction Factor for Surges in
Unobstructed Areas 33
Proposed Approach, Friction Factor for Surge
in Obstructed Areas 38
iii

Page
IV.EXPERIMENTAL VERIFICATION OF FRICTION FACTOR—MODEL
LAWS 43
Distorted Model for Buildings 45
Undistorted Model for Mangrove Stems and Roots 49
Distorted Model for Canopy 50
V.MODEL DESIGN 54
Recirculating Flume 54
Instrumentation 57
Velocity Meter 57
Data Acquisition System (DAS) 59
Depth-Measuring Device 59
Selection of Model Scales 63
Mangroves 63
Buildings 64
Model Setup 66
Mangrove Stems and Roots 66
Canopy of Red Mangroves 69
Buildings 72
VI.EXPERIMENTAL TEST SERIES 77
Experimental Procedure 77
Calibration of Velocity Meter 77
Measurements of Mean Velocities 77
Measurements of Water Depths 79
Experimental Runs 81
Mangrove Areas 81
Building Areas 82
VII.PRESENTATION AND ANALYSIS OF DATA 84
Mangrove Areas 84
Building Areas 95
Determination of Drag Coefficient 110
Drag Coefficient-Building Density Relation 110
Drag Coefficient-Disposition Parameter Relation 113
i v

Page
VIII. DISCUSSIONS AND CONCLUSIONS 119
Mangrove Areas 120
Developed Areas 121
Ocean Bottom 123
Forested Areas 127
Grassy Areas 128
Conclusions 129
APPENDICES
A. FIELD RECORDED DATA FOR MANGROVES 132
B. COMPUTER PROGRAM LISTINGS 145
C. TABLES OF EXPERIMENTAL DATA 151
BIBLIOGRAPHY 161
BIOGRAPHICAL SKETCH 165
v

LIST OF TABLES
Table Page
1. Average Parameters of Sampling Mangroves 14
2. Average Characteristics of Canopy 16
3. Average Parameters of High-Rise Buildings 19
4. Average Parameters of Medium-Rise Buildings 20
5. Scale Selection for Canopy 64
6. Average Parameters of Prototype and Model for
Building Areas 65
7. Statistical Values of Experimental Results for
Mangrove Areas 94
8. Statistical Values of Experimental Results for
Building Areas 109
9. Relations Between Disposition Parameters and
Drag Coefficients 115
10. Typical Values for Mangrove Areas 120
11. Bed Friction Characteristics of Three Entrances 126
A1. Parameters of Sampling Red Mangroves 141
A2. Parameters of Sampling Black Mangroves 142
A3. Characteristics of Canopy of Red Mangroves 143
Cl. Experimental Data for Red Mangroves (Without Canopy) 151
C2. Experimental Data for Red Mangroves (with Canopy) 152
C3. Experimental Data for Black Mangroves 153
C4. Experimental Data for Building Areas 154

LIST OF FIGURES
Figure Page
1. Prop Roots of Red Mangroves 9
2. Air Roots of Black Mangroves in 1 Foot Square Areas 9
3. Red Mangrove Area (Sampling Area #4) 11
4. Black Mangrove Area (Sampling Area #7) 11
5. Field Data Record for Red Mangroves 12
6. Field Data Record for Black Mangroves 13
7. Measurement of Density of Canopy 16
8. Section View of Survey Area (Red Mangroves) 16
9. Top View of Building Shapes on Coastal Areas 18
10. Wind Stress Coefficient over Sea Surface 26
11. Plan View for Wind Stress over an Obstructed Area 28
12. Elevation View for Wind Stress over an Obstructed Area 28
13. Distribution of Horizontal Apparent Shear Stress
and of its Drag, Inertial and Viscous Components 39
14. Plan of Flume 55
15. Cross Section of Flume 55
16. Novonic-Nixon Velocity Meter 58
17. Input Box of Data Acquisition System 60
18. HP 9825A Programmable Calculator 60
19. Setup of Water Depth Measuring Device 62
20. Point Gage and Tube 62
vii

Figure Page
21. Model Setup for Red Mangroves 67
22. Model Setup for Black Mangroves 68
23. Stems and Roots of Red Mangroves 70
24. Stems and Roots of Black Mangroves 70
25. Overview of Setup for Mangroves 7T
26. Setup of Model Equivalent of Canopy of Red Mangroves 71
27. Distribution of Leaf Stripes in the Model 73
28. Building Patterns Designed for the Tests 74
29. Relation Between U and U, 80
n w
30. Designed Building Patterns in the Tests 83
31. Relation Between f' and R' for Red Mangrove Areas
(Without Canopy) 85
32. Relation Between Cn and R' for Red Mangrove Areas
(Without Canopy) 86
33. Relation Between f' and Water Depth d for Red
Mangrove Areas e (Without Canopy) 87
34. Relation Between f and R‘ for Red Mangrove Areas
(with Canopy) e e 88
35. Relation Between Cn and R1 for Red Mangrove Areas
(with Canopy) 89
36. Relation Between f' and Water Depth d for Red Mangrove
Areas (with Canopye) 90
37. Relation Between f and R' for Black Mangrove Areas 91
e e
38. Relation Between CQ and R^ for Black Mangrove Areas 92
39. Relation Between f1 and Water Depth d for Black
Mangrove Areas e 93
40. Relation Between f and R' for Building Areas 94
e e

Figure Page
41. Relation Between Cn and R' for Building Areas 102
D e 3
42. Relation Between f1 and Water Depth d for Building
Areas e 108
43. Relation Between Cn and Density m for High-Rise
Building Areas 111
44. Relation Between Cn and Density m for Medium-Rise
Building and Residential Areas 112
45. Position Spacings. Definition Sketch 114
46. Relation Between Cn and S,/D in Aligned and Staggered
Patterns 116
47. Relation Between and S^/D 118
Al. Field Recorded Data for Red Mangroves (Area #1) 132
A2. Field Recorded Data for Red Mangroves (Area #2) 133
A3. Field Recorded Data for Red Mangroves (Area #3) 134
A4. Field Recorded Data for Red Mangroves (Area #4) 135
A5. Field Recorded Data for Red Mangroves (Area #6) 136
A6. Field Recorded Data for Black Mangroves (Area #8) 137
A7. Field Recorded Data for Black Mangroves (Area #9) 138
A8. Field Recorded Data for Black Mangroves (Area #10) 139
A9. Field Recorded Data for Black Mangroves (Area #11) 140
i x

LIST OF SYMBOLS
area
leaf area for prototype and model, respectively
horizontal width
drag coefficient
skin friction coefficient
average diameter of obstruction
vertical depth for prototype and model, respectively
water depth
water depth at section 1 and section 2, respectively
average water depth of d-j and d2
diameter of pipe
drag force
elastic force
skin friction
gravitational force
inertial force
Froude number =/ U / gd
d d
surface tension force
viscous force
Darcy-Weisbach friction factor based on diameter of the pipe
friction factor based on hydraulic radius

equivalent friction factor
total friction factor
gravitational acceleration
vertical depth above the vertex of Thomson weir
protruding height of obstructions above water surface
indicial functional parameter
wind stress coefficient
drag force scale
skin force scale
gravity force scale
inertial force scale
shear force scale
equivalent sand roughness
apparent roughness
horizontal length
leaf length for prototype and model, respectively
density = no. of obstruction elements/area
indicating the subscripted parameters for model and prototype,
respectively
total number of obstruction elements
vertical length scale
horizontal length scale
force scale
time scale
Manning's n
wetted perimeter of flow cross-section
pressure

Ps pressure on the water surface
Q discharge from the Thomson weir
q , q discharge per unit width
x y
R hydraulic radius = A/P
R-|,R2 hydraulic radius at section 1 and section 2, respectively
Rfl average hydraulic radius of R-j and R^
R^ reduction factor for wind stress
Rq Reynolds number based on depth = U d /v
R' Reynolds number based on hydraulic radius = UR /v
6 ad
R* wall Reynolds number = u^.k/v
R v Reynolds number = U„x/v
C )A
r radius of the pipe
S slope of energy grade line = aH/L
S , corner to corner distance between the roughness elements in
adjacent transverse raws
S£ longitudinal spacing between two successive roughness elements
S^. laternal spacing between two roughness elements
s free surface displacement from mean sea level
S. Dev. standard deviation
t time variable
U spatial mean flow velocity
U-j, U2 spatial mean flow velocity at section 1 and 2, respectively
Ua average spatial mean velocity of U-| and U2
U , U spatial mean flow velocity in x and y directions, respectively
x y
Uoo free stream velocity
u,v,w instantaneous components of the water velocity in the x, y, z
coordinate directions, respectively

u
u1 , V'
uf,t
W(z)
w
m
x, y, z
z
0
8
Y
6
AH
Til >fl2
e
K
u
V
e
P
pa
7b
Tbx’ Tby
time-mean velocity in the direction of flow
turbulent velocity fluctuations in the x and z directions,
respectively
friction velocity based on bottom friction
friction velocity based on total friction
time-mean wind velocity at the elevation z above water surface
critical wind velocity
time-mean wind velocity at the elevation 10 meters above water
surface
leaf width for prototype and model, respectively
Cartesian coordinate directions
dynamic roughness
shape factor
specific weight of water = pg
displacement thickness
energy loss per unit weight of fluid
fractions of distances nid and n2d from bottom to the total
depth d, respectively
latitude
Von Karman's constant
molecular viscosity of water
kinematic viscosity of water
percentage of the measured area occupied by obstructions
water density
air density
spatial mean bottom shear stress = tq
bottom shear stress in the x and y directions, respectively
xi i i

Tj hydrodynamic drag
viscous stress
wind shear stress on open area
t wind shear stress on obstructed area
so
Tj. Reynolds stress
obstruction correction factor
10 the earth's angular velocity
to* wind friction velocity = /t./p
S 3
V2 Laplacian operator
xiv

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
FRICTION IN HURRICANE-INDUCED FLOODING
By
Shang-Yih Wang
December 1983
Chairman: B. A. Christensen
Cochairman: T. Y. Chiu
Major Department: Civil Engineering
With the increasing development of coastal areas, it is necessary
to have a sound method for predicting hurricane-induced flooding in
these areas, especially for studies such as the coastal construction
set-back line, flood insurance rate-making and county land use planning.
The purpose of this study is to develop the capability of describing
the friction factor in coastal areas for improved representation in
numerical models of storm surges.
Five types of areas are considered: A, ocean bottom with bed-
forms and some vegetation; B, mangrove fringe and areas; C, grassy
areas; D, forested areas; and E, developed areas. The friction factors,
which incorporate both the bottom friction coefficient and drag
coefficient due to the submerged parts of obstructions were verified by
conducting laboratory experiments for mangrove and developed areas,
xv

using the typical distribution found in each of these coastal areas.
Analysis of the experimental data revealed that the drag coef¬
ficient for each case is invariant with the Reynolds number in the most
possible flooding flow ranges, but that it is related strongly to the
density and distribution of the roughness elements. Formulae
expressing these relations were derived for the evaluation of the
friction factor for different coastal areas. In addition, it is found
that the drag coefficient for a staggered disposition is about two to
three times larger than that for an aligned disposition under the same
density for all building areas. A relationship between the drag
coefficient and the disposition parameter of the evenly distributed
roughness elements was developed. The principal reduction of the wind
stress due to wind drag forces on the parts of the obstructions,
including-buildings and vegetation, above the water surface during
overland flooding was determined. Accounting for this reduction of the
wind stress provides a realistic view of wind generation forces in
coastal areas.
Finally, the formulae of the friction factor for the ocean bottom,
forested areas and grassy areas are presented by adopting results from
previous investigations and discussed with the results of the current
study.
xvi

CHAPTER I
INTRODUCTION
The rapid growth of population and industry in very low coastal
areas in recent years has resulted in increased concern and attention
to the potential hazard to these areas from tropical storms and hur¬
ricanes. A severe tropical storm is called a hurricane when the maximum
sustained wind speeds reach 75 mph or 65 knots (U.S. Army, Corps of
Engineers, 1977). During a hurricane, the wind-driven storm waves are
superimposed on the storm surge, which is the rise above normal water
level due to the action of storm, and sometimes the low coastal areas
are flooded. The worst natural disaster in the history of the United
States came as the result of a hurricane which struck Galveston, Texas,
in 1900. The storm, which hit the Texas coast on September 8, with
winds of 125 mph caused a storm surge 15 feet in height above the usual
two-foot tidal range. The fifteen-foot surge, accompanied by wave
action, demolished the city and caused more than 5,000 deaths (Bascom,
1980). Weather warnings were ineffectual. The people of Galveston,
unprepared for a storm of such intensity, were helpless in the face of
the hurricane. But the hurricane is no longer the unheralded killer it
once was. The years of progress in weather forecasting and wave research
have now made it possible to predict such surges. Hurricane Donna, for
example, which crossed the Florida Keys and then moved northeastward
across the state of Florida from near Fort Myers to Daytona Beach on
September 9-10, 1960, is thought to have been the most destructive
1

2
storm ever experienced in Florida. Fortunately, she was detected in
advance. Thus, even though this hurricane caused an estimated $300
million in damage, only 13 fatalities occurred. In 1961, Hurricane
Carla struck along the Gulf coast of Florida. The area was evacuated
before its arrival, and there was no loss of life.
From the facts mentioned above, it is clear that the great value
of the modern storm warning service is in its reducing the loss of life.
However, it is also apparent that the potential damage to property and
structures has increased dramatically with the rapid development of
coastal areas, if no concession is made to the storms. Therefore, the
future development plans of these coastal counties must take into
consideration this threat to life and property. As a result, the 1971
session of the Florida State Legislature passed a law (Chapter 16,053,
Florida Statutes), requiring the Department of Natural Resources to
establish a coastal construction set-back line (SBL) along Florida's
sandy beaches fronting the Atlantic Ocean and the Gulf of Mexico.
Based upon comprehensive engineering studies and topographical surveys,
such a line, where deemed necessary, is intended to protect upland
properties and control beach erosion. Basically, construction and
excavation seaward of the SBL is prohibited, though a provision for
variances is included in the law (Chiu, 1981).
In 1973 the Congress of the United States enacted the Flood Disaster
Protection Act (Public Law No. 93-234, 87 Statutes 983) which greatly
expanded the available limits of federal flood insurance coverage. The
act also imposed new requirements on property owners and communities
desiring to participate in the National Flood Insurance Program (NFIP)
(Chiu et al., 1979).

3
The Flood Disaster Protection Act of 1973 requires the Department
of Housing and Urban Development (HUD) to notify those communities that
have been designated as flood hazard areas. Such areas are defined as
having a one percent annual chance of flooding at any location within
the areas. Such a community must either make prompt application for
participation in the flood insurance program or must satisfy the Secretary
of HUD that the area is no longer flood prone. Participation in the
program is mandatory (as of July 1,1975) or the community would be denied
both federally related financing and most mortgage money.
Individuals and businesses located in identified areas of special
flood hazard are required to purchase flood insurance as a prerequisite
for receiving any type of federally insured or regulated financial
assistance for acquisition or construction purposes. Effective July 1,
1975, such assistance to individuals and businesses was predicated on
the adoption of effective land use and land management controls by the
community.
Federally subsidized insurance for flood hazard is authorized only
within communities where future development is controlled through adequate
flood plain management. Management may include a comprehensive program
of corrective and preventative measures for reducing flood damage, such
as land use controls, emergency preparedness plans and flood control works.
Participating communities may be suspended from the program for failure
to adopt or to enforce land use regulation (National Flood Insurers
Association, 1974).

4
Storm Surge Prediction
To implement either the coastal construction setback line or the
flood insurance program the flood elevation has to be determined on the
basis of different time intervals. Accumulation of data over many years
in areas of the Old World, such as regions near the North Sea, has led
to relatively accurate empirical techniques of storm surge prediction
for these locations. However, these empirical methods are not applicable
to other locations. In general, not enough storm surge observations are
available in the New World to make accurate prediction of the 100 year
storm surge. Therefore, the general practice has been to use hypothetical
design storms, and to estimate the storm-induced surge by numerical
models, since it is difficult to represent some of the storm-surge¬
generating processes (such as the direct wind effects and Coriolis
effects) in physical laboratory models. With the use of digital com¬
puters, numerical models have been able to analytically describe storm
surges to much greater detail than was ever possible with the other
methods. As a result, many numerical models for the prediction of surges
have been proposed to investigate practical cases including irregular
coastlines, irregular bathymetry, islands and arbitrary wind stress
patterns (Mungall and Mattews, 1970; Reid and Bodine, 1968; Platzman,
1958; Platzman and Rao, 1963). Moreover, Pearce (1972) and Reid and
Bodine (1968) developed their models to evaluate the inland extent of
flooding by using a moveable boundary. For different emphases on off¬
shore and nearshore areas, varied grid systems are used for most finite-
difference models. An orthogonal curvilinear coordinate system with
telescoping computing cells has also been introduced by Wanstrath (1978)

5
to solve the flooding problems of Louisiana. Regardless of the
purposes and differences in approaches of all these numerical storm
surge models, the Navier-Stokes and continuity equations which incor¬
porate terms accounting for wind stress, bottom friction, inertia,
Coriolis effect, pressure distribution, and other physical parameters
are solved numerically in space and time to determine localized surge
hydrographs. In order to obtain a more realistic and accurate predic¬
tion of storm surge, these physical parameters should be carefully
determined and incorporated into the numerical models. Modern
achievements in meterology and oceanography have led to an increased
understanding of hurricane to the extent that a model hurricane can be
characterized to a satisfactory degree by certain parameters. A list
of these variables includes central pressure deficit, radius to maximum
winds, speed of hurricane system translation, hurricane direction and
landfall location (or some other descriptor of hurricane track).
Surprisingly little work has been done in measuring another important
term-bottom friction. As suggested by Pearce (1972), future work on
actual hurricane surges and currents is especially needed for improved
representations of bottom friction that would be achieved with a better
understanding of the dissipation mechanism (i.e. friction) during a
hurricane.
Objectives of Present Work
Based on the necessities for a more accurate prediction of
hurricane-induced flooding in coastal areas (especially for studies
like the construction set-back line, flood insurance rate determination

6
and county land use plan), this study will develop a method of
describing the friction factor in coastal areas for improved repre¬
sentation of numerical storm-surge models. Special emphasis will be
placed on the friction characteristics of mangroves and buildings which
are the two most important causes of frictional resistance in vegetated
and developed land areas, respectively. The effect of these two
roughnesses in reducing overall wind stress on the water is also
introduced. The friction factors for other roughnesses such as the
ocean bottom, the forested and the grassy areas are determined by the
results from previous investigations and present study and are discussed
in the last chapter.

CHAPTER II
FIELD EXPLORATION OF PHYSICAL ENVIRONMENT
As stated in Chapter I, mangroves and buildings are the two major
flow retarding objects and are therefore being investigated in this
study. However, information on the density, dimensions and typical
distributions of these two forms of roughness in the coastal areas is
very scarce. Thus, field trips were taken to a mangrove area and
developed areas in southern Florida in order to collect the most
representative data for use in the model tests.
Mangrove Areas
General View of Mangroves in Florida
Mangrove is a kind of salt-resistant plant that usually grows
densely on sub-tropic shorelines around the world. This special
feature may be an inherent gift from nature in that the mangroves
enable exposed shorelines to resist severe attacks of hurricanes.
Basically, there are three species of mangroves, the red mangrove
(Rhizophora mangle), the black mangrove (Aricennia nitida) and the
white mangrove (Laguncularia racemosa). Each of these three species
occupies a distinct zone within the forest, depending on the degree
of salinity and length of inundation that each species can tolerate.
Red mangroves usually are found at the outer or seaward zone. They
are distinctive in appearance, with arching prop roots that project
7

8
from the trunk or branches down into the water (Figure 1). The root and
trunk systems of red mangroves, which spreads in shallow offshore areas
and onshore areas serve as a soil producer and stabilizer as well as a
storm buffer. In their role as buffers against storm winds and tides,
they prevent devastation of the coastline (Lugo et al., 1974). The
middle zone, at slightly higher elevations, is dominated by the black
mangroves in association with salt marsh plants. This zone is usually
submerged at high tide, but is otherwise exposed. The roots produce
pneumatophores (fingerlike extensions above the soil surface),as shown
in Figure 2. Black mangroves may also be found in pure stands in
shallow basins where sea water remains standing between tides. The
heat from the sun evaporates some of the water, leaving slightly
concentrated salt water behind. The black mangroves are also important
for shoreline stabilization as they present a secondary defense behind
the red mangroves. The white mangrove, which can be found in the most
landward zone that is affected only by the highest spring and storm
tides will not be discussed here since it is not as important in
defending against storm flooding and its usual appearance may be
categorized into buttonwood or other common types of vegetation.
Sampling of Mangroves
Five mangrove forest types—■ Fringe, Riverine, Basin, Overwash and
Dwarf Forests-have been found by Snedaker and Pool (1973) in southern
Florida, with distinctive differences in structure. The pattern is
strongly related to the action of water, both the freguency and the
amount of tidal flushing and freshwater runoff from the upland. The
coastal fringe forest including red and black mangroves, which are the

9
FIGURE 1: Prop Roots of Red Mangroves
FIGURE 2: Air Roots of Black Mangroves in 1 Foot Square Area

10
most important species, was investigated in San Carlos Bay on the
southwest coast of Florida.
Eleven sampling areas which included six red mangrove fringe areas
and five black mangrove areas were selected at random. Each sampling
area was framed by survey poles to form a 12 by 12 foot square area in
which the locations and dimensions of mangrove trunks, roots, and
canopies were recorded. Figures 1 through 4 show some of the features
of both types of mangroves in the surveyed area. Figures 5 and 6 show
two examples of data recorded from red and black mangrove sampling
areas, respectively. Data for the other nine sampling areas are shown
in Appendix A. From these data it is clear that the density and
dimensions of the trunk and root systems of mangroves are quite random;
therefore, averaged characteristics are chosen to describe these
samples as shown in Table 1. Red mangroves in the surveyed areas
extend from the low tidal water line to about 50 feet inland which is
the same distance Veri et al., (1975) recommended for fringe
mangroves in order to form a protective buffer zone. Thus, this value of
50 feet can be considered to be a standard distribution distance for
red mangroves and is used in the present study.
Although the average height of the lower edge of canopy was found
to be about 8 feet above the ground for red mangroves, the canopy along
the water edge was found to generally have a distribution from the
water surface to a few feet high. This feature may be important in
resisting storm surges. Therefore, a detailed measurement of canopy
distribution was done at a later time in Sarasota, Florida. Figure 7
depicts the measuring of the density of leaves by counting the number
3
of leaves in a unit volume (1 ft.). Dimensions of leaves were also

11
FIGURE 3: Red Mangrove Area (Sampling Area #4)
FIGURE 4: Black Mangrove Area (Sampling Area #7)

12
AREA #5
FIGURE 5: Field Data Record for Red Mangroves

13
AREA #7
O Stem
FIGURE 6: Field Data Record for Black Mangroves

14
TABLE 1: Average Parameters of Sampling Mangroves
Average Parameters
Red
Mangroves
Black
Mangroves
Main-
Stem
no.
(12 ft.)2
4
12
diameter
6.0 in.
3.1 In.
height
10.0 ft.
11.2 ft.
% occupied
0.84 %
0.49%
Sub-
Stem
no.
(12 ft.)2
13
diameter
2.0 in.
height
18.0 in.
% occupied
0.19 %
Canopy
height
8.0 ft.
9.4 ft
Root
no.
(12 ft.)2
81
10,800
diameter
1.0 in.
0.25 in
height
18.0 in.
6.0 in.
% occupied
0.25 %
2.39%

15
measured and recorded. Totally six sampling areas along the coastal
fringe were randomly selected. As shown in Figure 8, the survey area
contains three sections in which each section covers a distance of five
feet. The data collected are shown in Appendix A. Table 2 lists the
average densities and dimensions of leaves obtained from these six
sampling areas.
Developed Areas
In developed areas, buildings constitute the principal roughness
elements which would significantly affect the apparent bottom shear
stress as well as the wind shear stress during a storm induced flood.
Buildings are not, in general, arranged in a uniform manner but are
strongly dependent on the environment where they are located. A common
feature found in the coastal counties of Florida, especially in Broward
and Dade counties, is that high-rise buildings are predominant along the
beaches while residential houses are predominant a few miles inland
from the coastline. Three areas, a high-rise building area, medium-
rise building area and residential area, are defined to represent a
developed area in this study.
High-Rise Building Area
Aerial photographs of Broward and Dade counties, Florida, made
by the State Topographic Office, Florida Department of Transportation
in 1980 were used to analyze the dimensions and densities of buildings
in the coastal areas. Dade county is divided into 113 ranges while
Broward county is divided into 128 ranges in the aerial photographs.

16
FIGURE 7: Measurement of Density of Canopy
TABLE 2: Average Characteristics of Canopy
—^Sections
Parameters
#1
n
#3
„ .... , no. of leaves\
Density! = )
ftJ
10
5
2
Leaf Size
2" x 1”
3" x 1.5"
4" x 1.75"

17
Each range has a length of 1,000 feet approximately parallel to the
shoreline and is marked by monuments both in the field and on the
aerial photographs. Typical shapes and orientation of buildings
found in these two counties are shown in Figure 9. Category (a) is
the most common type found (more than 50 percent) which may be dictated
by the high cost of land per foot along seashore, and is chosen to
present all the buildings in the study.
High-rise buildings are defined as buildings having a surface area
2
larger than 10,000 ft . An estimation of the dimensions and densities
of these high-rise buildings from Broward and Dade Counties are 1isted in
Table 3. High rises and hotels/motels are predominant in the area.
Medium-Rise Building Area
Madium-rise buildings cover all buildings which do not belong to
either the high-rise or residential types. They can include two and
more story semidetached houses, row houses, garden apartments and other
buildings which are lower than ten stories. The surface area occupied
2 2
by medium-rise buildings is defined from 2,400 ft to 10,000 ft in
this study. An investigation of buildings in this category was also
made from aerial photographs of Broward and Dade Counties. Table 4
shows the average densities and dimensions of buildings from existing
field data.
The values obtained from these two counties at least present some
general views of buildings in highly developed areas despite their
irregularities in distributions found in the field. To apply these

18
Sea Side
Sea Side
(b)
FIGURE 9: Top View of Building Shapes on Coastal Areas

19
TABLE 3: Average Parameters of High-Rise Buildings
Range
No.
Buildings
County
length
(ft.)
width
(ft.)
density
( ^ )
1000'x500'
% of
land occupied
by buildings
42
280
80
10
45
Broward
43
290
60
4
14
Broward
45
290
110
4
26
Broward
50
140
140
10
39
Broward
54
300
100
3
18
Broward
72
200
70
8
22
Broward
82
200
50
19
38
Broward
83
200
180
5
36
Broward
84
200
230
5
46
Broward
118
140
90
8
20
Broward
119
150
70
12
25
Broward
121
230
90
6
25
Broward
8
240
200
3
29
Dade
11
300
50
8
24
Dade
12
320
70
7
31
Dade
14
240
50
9
22
Dade
15
230
50
8
18
Dade
17
230
60
9
25
Dade
18
250
50
10
25
Dade
19
200
125
5
25
Dade
36
220
65
4
11
Dade
42
190
65
7
17
Dade
43
270
200
6
65
Dade
44
200
200
5
40
Dade
48
120
200
6
29
Dade
52
220
210
4
37
Dade
56
190
160
9
55
Dade
Mean
224
112
7
30
S.Dev.
53
62
3
13

20
TABLE 4: Average Parameters of Medium-Rise Buildings
Range
No.
Buildings
County
length
(ft.)
width
(ft.)
density
( no- )
U000'x500' 1
% of
land occupied
by buildings
26
65
65
23
19
Broward
36
120
40
30
29
Broward
37
100
35
33
23
Broward
46
150
40
27
32
Broward
49
80
60
31
30
Broward
51
100
40
22
18
Broward
52
120
40
22
21
Broward
62
80
30
33
16
Broward
64
100
40
20
16
Broward
66
70
40
43
24
Broward
67
80
70
14
16
Broward
101
60
50
35
21
Broward
109
70
70
18
18
Broward
110
70
40
24
13
Broward
111
70
70
25
25
Broward
116
100
70
17
24
Broward
1
120
75
14
25
Dade
2
90
60
13
14
Dade
3
80
50
15
12
Dade
4
75
75
11
12
Dade
5
100
65
15
20
Dade
33
160
49
24
38
Dade
35
125
35
32
28
Dade
68
130
40
21
22
Dade
69
130
40
24
25
Dade
70
125
40
28
28
Dade
Mean
99
51
24
22
S.Dev.
28
15
8
7

21
data in the prediction of storm surge, it is recommended that the County
Land Use Plan Map published by each county be used so that the most
realistic results can be expected.
Residential Area
Residential houses are usually located behind the commercial areas
2
and have a surface area less than 4,000 ft . A typical density value
of detached, one story houses is given as six units per acre (43,560 sq.
ft.) (DeChiara and Callender, 1980). Density ranges in residential areas
can also be found in the Land Use Map of each county which categorizes
these single family houses in the density range of 0-8 units per acre
(Reynolds, Smith and Hills, 1972). The significant difference between
residential, medium-rise building areas and high-rise building areas is
that the former two areas usually have a matrix type distribution while
the latter one has only one or two rows distributed in the coastal
fringe area. The importance of this variation in the building distribu¬
tion will be shown later in the discussion of modeling studies.
Dimensions of the typical residential house are chosen as 30 feet by 62
feet, 1,860 sq. ft., which are convenient for the model tests and also
realistic for most single family houses.

CHAPTER III
THEORETICAL BACKGROUND AND DEVELOPMENT
Hydrodynamic Equations for Storm Surges
The equations governing incompressible fluid flows are the
Navier-Stokes equations of motion and the equation of continuity.
In the case of storm surges these equations may be written:
3qx
3 t
2o)(sin0)qy
d_ 3Ps
p 3 x
9“ H + 7 (Tsx •Tbx>
(1)
2ü)(sine)qx
d_ 3Ps
p 3y
gd |S- + 1
a 3y p
(2)
3qx + 3^y_ + 3S
3 x 3 y 3t
0
(3)
where t is time, to is the angular velocity of the earth, e is the lati¬
tude, p is the pressure, g is the acceleration of gravity, p is the water
density, s is the free surface displacement from mean sea level, the
subscripts s and b indicate that the subscripted quantities are to be
evaluated at the surface and bed, respectively, d is the total depth, qx
and q^ represent the time mean transport component, i.e. discharge per
unit width, in x and y directions, respectively, i.e.
d(x,y,t) = h(x,y) + s(x,y,t)
qx(x,y,t)
fs(x,y,t)
u(x,y,t) dz
-h(x,y)
(4)
(5)
22

23
qy(x,y,t)
rs(x,y,t)
v(x,y,t) dz
-h(x,y)
(6)
in which h = water depth referenced to mean sea level, u and v are the
instantaneous components of water velocity in the x and y coordinate
directions, respectively.
Expressions for the wind shear stress, and bed shear stress,
x^, for coastal areas in tropical storm induced flooding are presented
in the following sections.
Wind Shear Stress
In general, the wind stress (t ) on a water surface may be expressed
in terms of the mean wind speed (W-^) at anemometer level (10 meters above
water surface), the air density (p ) and a wind-stress coefficient (K),
a
as
T
s
K W
10
(7)
The problem of evaluating the wind stress is therefore reduced to
estimating the wind-stress coefficient, K, at different wind speeds, if
the reference wind speed and air density are known. Numerous studies
have found the quadratic wind speeds relation to be appropriate for a
wide range of wind speeds (Wilson, 1960). A wind-stress relation more
physically satisfying the quadratic law correlation was developed by
Keulegan (1951) and Van Dorn (1953) in the low winds range (<15 ms"^).
The Keulegan-Van Dorn relation for x$ is given as
T
s
Pa [K^o ♦ K2(W10 - w/]
(8)

24
where K-| and are the constants and Wc is critical wind speed.
Although there are uncertainties in applying the Keulegan-Van Dorn
relation to hurricane winds, it has been applied widely in hurricane-
induced surge cases. To eliminate this deficit, the wind-stress
relation has to be extended to higher wind ranges. Whitaker, Reid and
Vastano (1975) investigated the wind-stress coefficient at hurricane wind
speeds using a numerical simulation of dynamical water changes in Lake
Okeechobee, Florida. Results of their numerical experiments showed
that the Keulegan-Van Dorn wind-stress relation was superior to the more
commonly used quadratic relation for wind speeds in the range of 20 to 40
meters per second. The relation they found for the wind stress ts is
given by:
T = p[0.0000026 + (1.0 - ^.)2 x 0.0000030] W2 (9)
s wio 1u
where W^q and 7.0 are in meters per second and p is the water density.
Unfortunately, though this result was verified by a simulation of the
surge associated with a hurricane which occurred in October, 1950, it
still has some deficiencies such as the limited range of applicability
(Whitaker et al., 1975).
Recent studies of the wind-stress coefficient over the sea surface
have produced more complete and perhaps more accurate results with the
refinement of measurements and analysis techniques. Garratt (1977)
reviewed and averaged 17 selected sets of data and proposed an empirical
expression for 'light' winds:
K = (0.75 + 0.67 W1Q) x 10"3
(10)

25
Wu (1980) suggested a similar result for the wind-stress coef¬
ficient from 33 averaged data sets under 'light* winds
K = (0.8 + 0.065 W1Q) x 10"3 (11)
Furthermore, Wu (1982) compiled and averaged all available data for
'strong' winds. The data were obtained from independent investigations
either cited or reported in the following sources: Wu (1969), Kondo
(1975), Garratt (1977), Smith (1980), Wu (1980), and Large and Pond
(1981). All the data sets selected were obtained under nearly neutral
conditions of atmospheric surface layer. Additional factors which
affect the wind-stress coefficient, such as rainfall and sea spray, are
neglected due to their minor importance compared to the major factor of
wind speeds. As a result, the empirical formula proposed, given by
equation (11), for 'light' winds appears to be applicable even in
'strong' winds. Light and strong winds are defined as those less than
and greater than 15 meters per second, respectively. The averaged data
obtained from those sources and the formula proposed, equation (11), are
shown in Figure 10.
Wind Velocity Profile in Vertical
A vertical profile of wind velocities, usually expressed by the
following logarithmic law, is regarded by meteorologists as a superior
representation of strong winds in the lower atmosphere (Tennekes, 1973):
W(z) = 1», In (£-) (12)
0
where W(z) is the wind velocity at a height z above mean sea level, is
the von Karman's turbulence constant, z is the dynamic roughness of
1 /2
the logarithmic velocity profile, and w* = (Ts/Pa) = friction velocity.

FIGURE 10:
Wind Stress Coefficient over Sea Surface

27
The wind velocity profile given by equation (12) is well defined
except that it fails next to the bed where z approaches zero, and the
wind velocity W(z) approaches minus infinity. This discrepancy can be
corrected by using a modified mixing length approach as proposed by
Christensen (1971) for the flow of water over a rough bed, resulting in
modified logarithmic law for wind velocity profile is given in the form
= 2.5 In (^- + 1) (13)
“* zo
By substituting equation (7) into equation (12), an equation for
determining the dynamic roughness, zq, is obtained
4ft = /K (2.5) In (y- + 1) (14)
w10 o
Applying the boundary condition
W(10) = W1Q at z = 10m
The dynamic roughness, z , in all the wind speeds is found to be a
function of wind-stress coefficient, or on the wind velocity, i.e.,
z
0
10
e'2.5 . ,
(15)
Proposed Wind Shear Stress on Obstructed Areas
Consider an obstruction which has an effective width, D, and
protruding height, h, as shown in Figures 11 and 12. The wind drag
force on such an obstruction can be expressed as

28
I
T
D
Jl
PLAN VIEW
FIGURE 11: Plan View for Wind Stress over an Obstructred Area
ELEVATION VIEW
Protruding
Obstruction
Wind
=>
FIGURE 12: Elevation View for Wind Stress over an Obstructed Area

29
F
D
CPpaD
2
•h
â– 0
W2(z) dz
06)
where CQ is the drag coefficient. Substituting equation (14) into
equation (16) gives
(2.5)‘
F0 "
CDpa
DK
10
[In + 1 )]2 dz
o
or
Fd = 3.13 CDPaD W20 K {(h + zQ)1n(^-+ l)[ln(^- + 1) - 2] + 2h } (17)
o o
Recall the equation for wind stress on an open water surface, i.e.,
■ K “a *4
.(7)
This wind stress acts on the water surface and causes a rise of the
elevation of water surface which is called wind setup. The wind energy
is being transformed from the wind field to the water flow by the wind
shear. When the same wind field moves from the open water area to the
obstructed area, the wind setup will be reduced. This is due to the
extra form drag (FQ) acting on the protruding obstruction that can be
contributed to the wind setup per unit area. As a result, this reduced
wind stress causing a wind setup on an obstructed area may be expressed
as:
t = R , t = R, K p W._
so d s d pa 10
, 0 < Rj < 1
(18)
in which Rd is a reduction factor which represents the ratio of the wind
stress on the obstructed area to that on an open area under the same
wind condition.

30
Since the reduction of wind stress on an obstructed area is due to
the extra form drag, or wind energy loss caused by the obstruction, the
reduction factor may also be defined as the ratio of the total loss of
wind energy per unit length on an open area to that on an obstructed
area with dimensions 1*1, i.e., (Gee and Jenson, 1974)
2
T. * l
Rd = K 2 (19)
[(1 - mgD¿)xs + mFp] x C
2
in which m = density = N/£ , N = total number of obstructions, 8 = a
shape factor defined as the horizontal cross-sectional area of average
obstruction element at surface level divided by D. Substituting
equations (7) and (17) into equation (19) gives
K
W
10
(1 - m8D2)KpaW^0
+ 3.13 (mCDPaDW2Q) K{(h+zo)ln(^-+l)[ln(^-+l)-2]+2h}
o o
or
Rd =
1
(l-m6D2) + 3.13 (mDCD) {(h+ZQ) ln(^-+l) [ln(^-+l)-2] + 2h}
o o
(20)
_ o
where K = (0.8 + 0.65 W^q) x 10 . W^q is in meters per second.
Based on the result presented in equations (18) and (20), the
reduction factor R^ can be determined and incorporated into the storm
surge model to produce more realistic results for the wind stress on
water in flooding areas. The drag coefficient in equation (20) for
vegetations and buildings will be determined and discussed in Chapter
VII and VIII.

31
Bed Shear Stress
A space averaged bed shear stress, 7b, usually can be expressed as
V i-e”
Tb = To = yRS = yR T- (21)
in which tq =average bed shear stress along the wetted perimeter;
y = pg = unit weight of water; R = A/P = hydraulic radius = depth in
sheet flow; A = cross-section area; P = wetted perimeter of flow
cross-section; S = slope of the energy grade line; AH = energy loss
per unit weight of fluid over a bed length of L. Primarily developed
for flow in pipes, the energy loss term, aH, is defined by the Darcy-
Weisbach formula as
iH ■ f ' | • T (22)
3 0
where f = friction factor based on depth; U = spatial mean flow
velocity; dQ = diameter of the pipe. Since dQ = 4R, the above equation
/
may be written for an arbitrary cross section as
= <23>
where f = friction factor based on hydraulic radius.
Incorporating equation (21) into equation (23) in the x and y
directions, respectively, gives the following quadratic forms for the bed
shear stresses:
fp|U|U
bx
f'p|U|U
by
2
2
(24)

32
or in terms of volume transport
f'p|q|qx _ f'p|q|qy
Tbx = -^~ ; Tby= 2d2
(25)
where f = 4f'; U = /U 2 + U 2; q = /q 2 + q 2; U and U = spatial
x y x y x y
mean flow velocity in x and y directions, respectively.
It is assumed that these steady state relationships for the two
shear stress components are valid for storm surge propagation, which is
generally considered to be quasi-steady, i.e., the velocity variation
with time or the temporal acceleration is very small. The quadratic
Darcy-Weisbach form of bed shear stress is the best formula available
to account for the effect of bottom friction.
The friction factor in the Darcy-Weisbach formula, f, has been
studied by many investigators in both pipe flows and open channel
flows. From the abundant experimental data, numerous empirical
formulae have been established to express the relationship between the
friction factor and the dependent parameters, such as bed roughness,
Reynolds number, Froude number and Strouhal number. For example, the
well-known Stanton diagram (1914), Moody diagram (1944) and many others
(to be discussed in the next section) have enabled determination of the
friction factor in varied flow conditions.
In general, a surge could be expected to travel over five different
terrains (Christensen and Walton, 1980):
A. Ocean (river) bottom with flow induced bed form and completely
submerged vegetation,
B. Mangrove fringes and areas,
C. Forested areas and cypress swamps,
D. Grassy areas, and

33
E. Developed areas.
Each of these five categories has unique roughness characteristics.
However, in evaluating the friction factor in hurricane-induced flooding,
these five terrains can be divided into two major categories,
unobstructed and obstructed areas, based on their distinct functions
to retard flow.
Post Approach, Friction Factor for Surges in Unobstructed Areas
Unobstructed areas include the ocean (river) bottom and grassy
areas, the latter of which are assumed to be completely submerged in
water during floods. Friction factors in these kinds of areas can be
determined from the results of previous research which will be discussed
below and used as basis for the present work. Overland flooding in this
study is considered to be turbulent and in the hydraulically rough
range, i.e., the wall Reynolds number is in excess of about 70.
The effect of wall roughness on turbulent flow in pipes has been
studied during the last century by many investigators. An important
result obtained by Nikuradse (1933) in steady flow using six different
values of the relative roughness k/r with Reynolds numbers ranging from
4 6
R = Ud /v = 10 to 10 has been widely used in flow fields and will be
e o
applied in this study (k is the equivalent sand roughness, r is the
radius of pipe; U is the average velocity, v is the kinematic viscosity).
Nikuradse divided flow conditions into three ranges, smooth flow range I
(u^k/v < 4), transition flow range II (5 <_ ufk/v £ 68), and rough
turbulent flow range III (ufk/v > 68) in which ufk/v = R* = wall Reynolds
number, u^ = friction velocity = /x/p.

34
In range III (rough turbulent flow) the thickness of the viscous
sublayer <5 is negligible compared to the equivalent sand roughness, k,
and the friction factor is independent of the Reynolds number.
The distribution of the time-mean velocity obtained using Prandtls'
mixing length approach in combination with Nikuradse's experimental
results is given by the general expression
jj- = 8.48 + 2.5 In | = 2.5 In ^iZl
(26)
in which u = time-mean velocity in the direction of flow at a distance z
from the theoretical bed. Theoretical bed is defined as the plane
located such that the volume of grains above the plane equals to the
volume of pores below the plane but above the center of grains.
The classic velocity profile given by equation (26) is well defined
at moderate to large distances from the bed and for roughnesses much
smaller than the depth. However, it falls next to the bed whereas z
approaches zero, the time-mean velocity U approaches minus infinity.
This is especially true in flows where the roughness is not significantly
smaller than the depth. Because of the above-mentioned discrepancies,
Christensen (1971) introduced a new law for the velocity profile by using
a modified mixing length approach over a rough bed in the rough range
= 8.48 + 2.5 In
uf
£ + 0.0338) = 2.5 In ( ?9¿73z + 1)
(27)
The form of this equation is the same as that of classic equation (26)
except for the +1 term in the argument of the logarithmic function which

35
makes the time-mean velocity u equal to zero at the theoretical bed. As
the distance from the bed increases to more than a few times k, very
little difference exists between these two velocity profiles.
For practical purposes, the time-mean velocity profile is trans¬
formed to a depth averaged velocity profile using the fact that the mean
velocity (depth averaged), U, occurs theoretically at a distance
z = 0.368d from the bed also for the modified logarithmic vertical
velocity profile, where d is the water depth, and d/k is larger than 1.
It shall be noted that the k value used here is the equivalent roughness
height for bottom friction only. Therefore, at z = 0.368d equation (27)
yields
— = 2.5 ln[%^ (0.368d) + 1]
U K
or
— = 2.5 In [10.94 £ + 1]
U K
(28)
where U = time and depth averaged velocity.
The friction factor may in general be related to the velocity
profile by introducing the Darcy-Weisbach formula into the definition
of the friction velocity, i.e., uf = /tq/p = /gRS, leading to the result
(29)
Solving equation (29) for f and introducing equation (28) gives the
following expression for the friction factor
0.32
f
(30)

36
This depth dependent friction factor is proposed for areas where the
surge moves over bottoms at moderate depths. Another equation for f
obtained from Nikuradse's experimental result (1933) for rough
turbulent flow in circular pipes, is given by
— = 1.171 + log £ (31)
4/Tr K
It seems quite clear from equation (30) and equation (31) that
determination of friction factor in unobstructed areas is just a matter
of finding the value of the equivalent sand roughness k. This k value
can be related to Manning's n by using a Strickler-type formula
(Henderson, 1966, Christensen, 1978) in metric units
1 _ 8.25 /q
(32)
given k in meters, or in the English units
1.486 8.25/q
(33)
with k in feet. Values for n may be determined from various sources
such as textbooks by Henderson (1966) and Chow (1959), charts and graphs
by the Soil Conservation Service (1954), and photographs of a number of
typical channels by the U. S. Geological Survey (Barnes, 1967). Other
specific studies, for instance, the experiments conducted by Palmer
(1946), also provide valuable information on the flow of water through
various grass and leguminous covers. Based on the theoretical velocity

37
distribution in rough channels, the value of Manning's n can also be
determined by analytical methods such as that presented by Boyer (1954).
It should be noted, however, that these values of n from previous
sources may not be applied to every case under natural conditions. It
should be also careful in selecting the values of n, since a small error
on n will be amplified substantially on k by using the Strickler-type
formula. Therefore, a method to determine the k-value from the vertical
velocity distribution in turbulent flow over rough surface is recom¬
mended (Christensen, 1978). Let Hi be the velocity at m depth, that is,
at a distance md from the bottom of a wide rough channel, where d is the
depth of flow. By equation (26), the velocity may be expressed as
Ui 29.7nid
— = 2.5 In £ (34)
Similarly, let u2 be the velocity at r\2 depth; then
u2 29.7r)2d
— = 2.5 In —u
(35)
Subtracting equation (34) by equation (35) and solving for uf,
uf =
U2 - Ux
*12
2.5 In —
fii
(36)
Introducing equation (36) into equation (35) and solving for k,
29.7md
(37)
n2
u2- ux

38
Proposed Friction Factors for Surges in Obstructed Areas
Obstructions in these areas are defined as roughness elements with
significant heights which either protrude through the water layer or
consist of relatively rigid elements with heights that are sufficient to
cause form drag that are much larger than surface friction on the same
area. The two major forms of obstructed areas, mangroves and buildings,
to be discussed in this study are often higher than the storm surge
level so that the influence of hydrodynamic drag on the individual
elements should be taken into consideration together with other factors
of resistances in overland floods. The theoretical analysis presented
here is based on the assumption of steady or quasi-steady flow in the
rough flow range.
Consider a design flow that passes over an obstructed area whose
bottom is horizontal. The total averaged shear stress, x , in the
direction of flow may be written as equation (21), i.e.,
- dc d aH
t0 - yRS = yR —
For the steady state case, assume that the viscous stresses, turbulent
stresses and hydrodynamic drag acting on the obstructions contribute
independently to the flow system without any interaction among them.
Following Prandtl's assumption (Schlichting, 1979), that the shear stress
in the x-direction is constant and equal to the wall shear stress at
all distance z from the wall, an equilibrium equation is given in the
following and shown in Figure 13.

t
FIGURE 13: Distribution of Horizontal Apparent Shear Stress and of its Drag,
Inertial and Viscous Components, (Christensen, 1976).

40
du
YRS = ^ (l-e) -
pu'v'
(l-e) + mCDdDY S|
(to}
<*£>
Viscous
Stress
(Tt)
Turbulent
Stress
(Td>
Hydrodynamic
Drag
(38)
where e = fraction of total area occupied by obstructions, u', v' are
the turbulent velocity fluctuations in the x and z directions,
respectively, m = number of obstructions per unit bed area, CQ = average
drag coefficient, D = average diameter of the obstructions in the pro¬
jected plane normal to the flow.
For fully developed turbulence, the viscous term is negligible
compared to the turbulent term and may be omitted. Consequently
pgRS =
-pu1v1
(1-e) +
mCDdD |
(39)
The turbulent stress, x^, which expresses the rate of flow of x momentum
in the z direction, was first derived by 0. Reynolds from the equation
of motion in fluid dynamics, and is termed the Reynolds stress or the
inertia stress. The Reynolds stress on the right hand side of equation
(39) must be
1
e
(40)

where
1
41
♦ = (
Id
T
1 - e
V2
)
, 0 < $ < 1
(41)
may be defined as the obstruction correction factor, which directly
reflects the reduction of the Reynolds stress - pu'v‘ due to the
presence of the obstructions. If the drag is equal to zero, implying
that no obstruction exists (e = 0), the obstruction correction factor
will of course become equal to 1. The corresponding friction velocity
depends on the bottom friction only. It is defined from equation (40),
\'2
) *
uft *
(42)
where uft is the friction velocity based on the total bed shear stress
V i>e” uft = /T7P'
Similarly, introducing f and fj. as the friction factors of bottom
friction and total friction including bottom friction and drag acting on
the obstructions, respectively. Therefore, these two friction factors
can be related to the mean velocity as shown in equation (29),
U 2 1^2
— = (7.) for the obstruction-free area
Uf T
(43)
and
U
uft
for the total obstruction area
(44)
Eliminating U from the two equations above, relation between f and fj.
is given as
f - • *2
(45)

42
Substituting equations (40), (42) and (44) into equation (39), the
equilibrium equation becomes
yR
Aü = ¡L
2 (1-e) + mCpdDy ^
or
AH = £. f ,.,2
YR y = f f'U" (1-0 + mCpdDy ^
The head loss, AH, then may be written as
4H= f i Í ('-*> +raCDdDi?S-
and
Ü2 L
AH = [f (1-0 + mCDdD]
(46)
(47)
(48)
which is the form of the Darcy-Weisbach formula shown in equation (23).
An equivalent friction factor, f^, which includes the effects of bottom
friction and form drag based on equation (48) is introduced
f' = f (1-e) + mCpdD
(49)
where, according to equation (30),
0.32
[In + l)]2
Therefore, the equivalent friction factor, f^, must be equal to the
total friction factor, f|, according to the above definitions, as can
be proved easily by substituting equation (45) into equation (49).
An apparent roughness height ka which represents the sum of bottom
roughness and rigid drag element can then be expressed and calculated
from equation (31), i.e.,
-7= - 1.171 + eog if-
4/f7 a

CHAPTER IV
EXPERIMENTAL VERIFICATION OF FRICTION FACTORS—MODEL LAWS
The formula proposed in Chapter III for the friction factor
(equation (49)) has left some unsolved questions: What is the drag
coefficient (C^) for the kinds of obstructions being studied? What
kind of relationship do drag coefficient and density of obstruction
have? Does the friction factor depend on unspecified flow conditions
like the Reynolds number? All these problems may not be satisfactorily
answered without experimental verification. Therefore, laboratory
measurements were carried out in this study to verify the analytical
results and to build a data base for further development.
The requirements of similarity between hydraulic scale-models and
their prototypes are found by the application of several relationships
generally known as the laws of hydraulic similitude, i.e., geometrical,
kinematic and dynamic similitude. These laws, which are based on the
principles of fluid mechanics, define the requirements necessary to
ensure correspondence between model and prototype.
Complete similarity between model and prototype requires that the
system in question be geometrically, kinematically and dynamically
similar. Geometric similarity implies that the ratio of all corres¬
ponding lengths in the two systems must be the same, kinematic
similarity exists if all kinematic quantities in the model, such.as
velocity, is similar to the corresponding quantities in the prototype,
43

44
and dynamic similarity requires that two systems with geometrically
similar boundaries have the same ratios of all forces acting on
corresponding fluid element of mass. Following the basic dynamic law
of Newton, which states that force is equal to rate of momentum,
dynamic similarity is achieved when the ratio of inertial forces in
the two systems equals that of the vector sum of the various active
forces, which include gravitational forces, viscous forces, elastic
forces and surface tension forces in fluid-motion phenomena. In other
words, the ratios of each and every force must be the same, as given
in the equation form
«V (F) (Fv) (F.) (Fs)
P _ a P . P- P. P
(50)
where subscripts p and m refer to prototype and model, relatively.
Since it is almost always impossible to obtain exact dynamic
similitude, it becomes necessary to examine the flow situation being
modeled to determine which forces contribute little or nothing to the
phenomenon. These forces can then be safely neglected with the goal of
reducing the flow to an interplay of two major forces from which the
pertinent similitude criterion may be analytically developed (Rouse,
1950).
For a model of hurricane-induced flooding of coastal areas,
elastic forces and surface tension forces are sufficiently small and
can be neglected. The condition for dynamic similitude reduces to
equating the ratio of inertial forces to the ratio of either gravity

45
forces or viscous forces. Viscous forces are only considered in the
model of canopies whose surface friction effects are investigated. In
models for measuring form drags in turbulent flow with high Reynolds
numbers (inertia force/viscous force), viscous forces are small
compared to the major forces due to the turbulent fluctuations and this
can be neglected in this instance. Since the vertical dimension scale
(involving flow depth) cannot follow the horizontal dimension scale
in building models as the flow depth would be much too small for
measurements to be made, or the viscous force would become important
and cannot be neglected for the small flow depth if the same fluid is
used for the prototype and the model. Therefore, a model with a
different vertical dimension scale than horizontal dimension scale is
used to keep the Reynolds numbers in the turbulent flow range. For
simplicity, such distorted models will be introduced first since
undistorted models with the same length scale in both the vertical and
horizontal dimensions can be regarded as a special case of the former.
Distorted Model for Buildings
The fundamental model scale ratio may be written as:
L B
Length Scale (horizontal): (51)
Lm Bm
D
Depth Scale (vertical): Nd = D^ (52)
T
Time Scale:
(53)

46
Force Scale:
(54)
where L = horizontal length, B = horizontal width, D = vertical depth,
T = time and F = force.
Following the development of Christensen and Snyder (1975), the
force scale for the gravity component in the nearly horizontal direction
of the principal flow may be written as
unit sine
weight volume of slope
I ( I ‘ >
K =
g
L B D (D /L p g 2
P P P P P = (_£) (IE.) N N
Pmgm CCtTÃœU V V Vd
p q
P^P
m m m
m' nr
m
Jm
(55)
where p is the fluid density, g is the gravitational acceleration and
D/L is the bed slope ar the slope of the energy grade line.
In a unidirectional flow the inertial force can be expressed as a
horizontal, or nearly horizontal area multiplied by the Reynolds shear
stress, which is proportional to the fluid density and the time mean
value of the product of a vertical velocity fluctuation and the
corresponding velocity fluctuation in the direction of the time mean
flow. Consequently, the inertial force is
F.. = pu‘v1 • (area) , (56)
and the inertial force scale can be written as

47
area
D_ r
Pp pn(r)(r1)(LmBm)
m m
Pn N!
m
N,
(57)
In order to have dynamic similarity between model and prototype,
Kgravity should be equal to Kn-nertial * i-6*» equation (55) should be the
same as equation (57). This condition is expressed by
or
(58)
where gp is assumed equal to g^. The ratio between a gravitational
force and an inertial force is commonly known as the Froude number, and
the resulting time scale (equation (58)) is the similarity criterion of
the Froude law for distorted models.
The scale ratios of the drag coefficient and friction factor in
distorted Froude models have to be determined before experimental data
can be interpreted correctly. The drag force proposed for the present
study is given by equation (39)
^d = Td* A = m^gd^P jr- * A (59)
The drag force scale in the flow direction can then be written as

48
1
L B
JL£
1
L B
m m
• L 2
)(Cn)mDmBmpJr) L B
DmmmmT mm
m
or
. p
K, *
«A
d ,r \ vp ' "d l
(Cpj m
m
(60)
To satisfy the ratio of the force scale in the Froude model law, Kd must
be equal to K^, and by substituting the time scale, Nt = N£/(Nd^2, into
equation (60) gives
(CD> - p m
(61)
Shear forces generally may be expressed by the Darcy-Weisbach form
F
s
(62)
where f is the friction factor and can be substituted by the equivalent
friction factor, T, or bottom friction factor, f', later for the present
use. The shear force scale is given as
or
Ks ■ <#<£> <
m pm t
(63)

49
Dynamic similarity requires that = K^. Substituting the time
scale, equation (58) into the required equality, i.e.,
(i)2. (j(^)NA2
p
m pm
(64)
gives the expression for the scale ratio of the friction factor
f N.
_B. = -A
fm
(65)
Comparing equation (61) with equation (65) it is noted that the drag
coefficient is the same in the model as in the prototype, however, the
friction factor of the prototype should be modified by an inverse
distortion ratio, N^/N^, in the Froude distorted model. The distortion
ratio usually is defined as
D =
r Nd
(66)
Undistorted Model for Mangrove Stems and Roots
Due to the fact that the dimensions of mangrove stems and roots
are one to two orders of magnitude less than the water depth, an
undistorted model can be used in this part of the study. All the
methodology applied in the previous section for a distorted model is
also applicable for this analysis. In the case of an undistorted
model where N = N^, equation (58) reduces to
Nt â–  (67)

50
which is the time scale for an undistorted Froude model. Both the
dimensionless coefficients and f are the same in the prototype and
model, and the distortion ratio becomes unified in this instance.
Distorted Model for Canopy
Before the model law for canopies is derived, it is necessary to
determine what kind of boundary layer forms over the surface of a
mangrove leaf. A prototype red mangrove canopy was tested in the
hydraulic laboratory flume and it was quite apparent that all the leaves
bent in the direction of flow even at a flow velocity less than 10
cm/sec. Such high flexibility makes the leaves more resistant to a
storm attack. As a result, leaves offer only skin friction and no form
drag to resist the flow.
The surface of a leaf is assumed to be smooth in this study so
that theoretical and empirical results on the behaviour of a boundary
layer on a smooth flat plate can be applied. In general, the point
of instability on a flat plate at zero incidence to the flow is
determined by the critical Reynolds number
U x
(R
e.x^crit
'crit
(68)
in which U is the free stream velocity and x is the distance from the
leading edge of the plate measured along the plate. An analytical
stability criterion developed by R. Jordinson, based on W. Tollmien's
theory, is given by
U 6
, co
(69)

51
where ó = displacement thickness and
6 = 1.7208 (
co
(Schlichting, 1979)
(70)
Combining the last two equations give (R ) .. = 9.1 x 104.
c )A Li I L
In reality, the position of the point of transition from laminar
to turbulent flow will depend on the intensity of the turbulence in
the external flow field. This has been investigated experimentally by
J. M. Burgers, B. G. Van der Hagge Zijnen and M. Hansen in 1924. These
measurements led to the result that the critical Reynolds number was
contained in the range
U x
(—)crit = 3-5 x 105 t0 5 x 105 (Schlichting, 1979) (71)
Similar experiments done by Schubauer and Skramstad in 1947 also yielded
results which indicated that the critical value of R„ is in a range
G jX
from 9.5 x 105 to 3 x 106 depending on the relative intensity of the
free-stream turbulence, (l/U^Jiu'u'/sj^2 (Hinze, 1975). Therefore, the
minimum value of (R ) .. is chosen as 3.5 x 105 for the present study.
C J X Cllt
In the prototype, the maximum value of x is the largest leaf length
and was found to be 5 inches; the highest flow velocity is assumed to be
10 ft/sec, which results in a maximum value of (R ) ,. of about
c )a Li I L
2.98 x 105, which is still lower than but near the minimum value
(R ) ... This shows that a turbulent boundary layer has very little
g ,x cn L
chance to be formed over such a short length, and that a laminar
boundary layer should prevail over the entire leaf area.

52
Skin friction can be expressed in terms of a dimensionless skin
friction coefficient, C^., times the stagnation pressure, pU2/2, and area
of the plate, A, as follows
Ff = CfP I A (72)
in which, for a laminar boundary layer,
C
f
1.328
(73)
where R = lk/v denotes the Reynolds number formed by the product of the
Xj *
plate length and the free-stream velocity (Schlichting, 1979). The skin
friction scale in the flow direction may be written as
K
f
LnV1/2
(-P P)
TnV
-P-P -
f III III \
m m
m
L 2
(/) A
T^~
(j51) A
P
m
(74)
where z and A are the leaf length and leaf area, respectively. Following
the undistorted Froude law, = (N^2 , p and v are the same in proto¬
type as in the model since the same water properties are assumed in the
two systems. Equation (74) is then reduced to
3A -V2 A
V m m
(75)

53
For dynamic similarity, Kp has to be set equal to K , i.e.
% i -V2 An 3

m m
(76)
giving the length scale
An An2
f(Ni)
m m
(77)
It is obvious that the dimensions of a leaf need to be distorted
according to the scale ratio shown in equation (77), which is the result
of inclusion of viscous effects on a leaf surface in a Froude law
controlled flow model.

CHAPTER V
MODEL DESIGN
Recirculating Flume
The present model tests were conducted in the hydraulic laboratory
flume of the Civil Engineering Department at the University of Florida.
Figures 14 and 15 show the primary elements of the flume geometry.
The main channel is 120 feet (36.58 meters) long, 8 feet (2.44 meters)
wide, and 2.7 feet (0.81 meter) deep. A false-bottomed section 20 feet
(6.1 meters) in length and 13.4 inches (34 cm) deep is located at the
longitudinal center of the flume. Centered in the false-bottomed area,
observation windows cover a length of 12 feet (3.66 meters) and are 2
feet high (starting at the bed level). The 74 kW (100 HP) flume pump
has a maximum discharge of 40 cfs (1.1 m /sec). Between the pump and
the overflow weir are two sets of 8 inch long, 2 inch diameter poly¬
vinyl chloride pipes arranged in a honeycomb fashion. Two more sets
of these pipes, which act as flow straighteners, are located just
beyond the outlet weir. By adjusting two gate valves at the main
delivery pipe and return pipe, the flow rate and depth over the
Thomson V-notch weir can be regulated. A Poncelet rectangular weir
is also available for high discharges. A motor-driven sluice gate
at the downstream end on the main channel serves to regulate the water
depth in the main flume and to moderately regulate the discharge.
54

FIGURE 14: Plan of Flume
f
Power Source for Trolley
Rail for Movable Trolley
l
Return f
Channel
38
(l.llm)
iv > \
Main
Channel
I—*—r-
8'
(2.44m)
T
2‘8"(0.8lm) .
(1.22m)
FIGURE 15: Cross Section of Flume

56
A movable trolley which spans the entire width of the flume and
which has a maximum towing speed of 2 feet per second provides the
work-deck for calibrating velocity meters as well as collecting data.
To determine the drag coefficient and equivalent friction factor
for a given roughness in the rough turbulent flow, the energy loss, aH,
has to be measured (cf. equation (47)). According to the principle of
conservation of energy, the total energy head at the upstream section 1
should be equal to the total energy head at the downstream section 2
plus the two sections, i.e.,
2 2
u, u
dl + 2g = d2 + 2?
+ aH
or AH = (d1 - This equation is applied to the present study in which the channel bed
is horizontal and a value of unity is assumed for the energy coefficient
(Henderson, 1966). Therefore, the energy loss aH due to the friction in
turbulent flow can be measured by knowing the water depths and mean
velocities at the two sections. Relating the measured results of energy
loss to the Darcy-Weisbach equation
AH
,, Hi
fe 2g Ra
(79)
1 ^1 ^2
in which = 2 (U-j + U2) and = 2—- , the equivalent friction
factor f' can be determined for the designed roughness elements. To
determine the water depths and velocities, some instruments are employed
for this study and described in the following section.

57
Instrumentation
Velocity Meter
A Novonic-Nixon type velocity meter was employed for all velocity
measurements. The probe consists of a measuring head supported by a
thin shaft 18 inches long with an electrical lead connection. The head
consists of a five blade, impeller mounted on a stainless steel spindle,
terminating in conical pivots (Figure 16). These pivots run in jewels
mounted in a sheathed frame. The impeller is 1 cm in diameter, machined
from solid PVC and balanced. An insulated gold wire within the shaft
support terminates 0.1mm from each rotor tip. As the rotor is rotated
by the motion of a conductive fluid, the small clearance between the
blades and the shaft slightly varies the impedance between the shaft
and the gold wire. This impedance variation modulates a 15KHz carrier
signal, which in turn is used to detect rotor rotations.
The range of this velocity meter is from 2.5 to 150 cms-1 (0.08 to
4.92 fps) with an advertised accuracy of + 1 % of true velocity. Its
operating temperature is from 0° to 50°C (32° to 122°F) with an operating
medium of water or other fluids having similar conductive properties.
The shaft of the current meter was clamped to the rack of a point gage.
The point gage bracket was then bolted to the trolley carriage so that
the instrument could be easily removed from its bracket with no deviation
in the vertical setting. Also all the accuracy and ease of a point gage
and vernier is accrued.

58
O I 2 cm
> I I
FIGURE 16: Novonic - Nixon Velocity Meter

59
Data Acquisition System (DAS)
The data acquisition system is composed of two pieces of equipment:
an input box and an HP 9825A desk-top programmable calculator (Figure 17
and 18). The input box, which is specially designed for coupling with
the HP 9825A, has connectors for fifteen thermistors, ten Cushing
electromagnetic current meters, two Ott velocity meters and two Novonic-
Nixon velocity meters. It contains the electronic circuitry which takes
the raw transmission from the measuring devices and converts it into
usable signals tor the programmable calculator. An electronic timer
which registers six counts per second is also contained in the input box.
The HP 9825A interfaces with the input box to provide program
control and data storage capabilities. The calculator has a 32-character
LED display, 16-character thermal strip printer, and a typewriter-!ike
keyboard with upper and lower-case alphnumerics. A tape cartridge with
the capacity of 250,000 bytes is used with this calculator to store and
access the programs. Based on the manual of the HP 9825A and the
instructions provided by Morris (1979), programs designed to calibrate
the Novonic-Nixon meter, measure the velocities and perform linear
regression are listed in Appendix B. Through the DAS a substantial
amount of time usually used in experiments and data reduction was saved
and the accuracy of results was greatly enhanced.
Depth-Measuring Device
Determination of the flowing water depth by measuring the difference
of water surface elevations is the most important part, except for the
measurement of the flow velocity, of the laboratory experiments. However,

60
FIGURE 17: Input Box of Data Acquisition System
FIGURE 18: HP 9825A Programmable Calculator

61
the measurement of water surface elevations is not easy due to the
rough water surface of turbulent flow. In addition, an accuracy of one
millimeter or better is needed for the depth measurement, since the
difference of water surface elevations at two sections is less than one
centimeter in many test cases. Therefore, a stable and sensitive depth¬
measuring device is required for the present study. Figure 19 shows the
schematic diagram of the device designed, in which the hoses connected to
the tube, which have a diameter of 0.2 inch, were extended to the desired
cross-sections in the flume. A 10 inch long, 0.15 inch diameter glass
tube was attached to the end of each hose and positioned perpendicular
to the water surface. In high velocity flows some weights were added to
the 0.15 inch diameter tube in order to maintain its vertical position.
The diameter of the tube is 2 inches which is large enough to allow the
point gage to be able to contact the plane water surface without the
influence of surface tension on the side wall of the tube. The point
gage was attached to the top of the tube, and the still water level is
indicated when a white ball on the gage appears, which indicates that
the sharp tip of the gage is touching the water surface (Figure 20).
A manual hand operated vacuum pump was used to help initiate a siphon
between the water in the flume and in the tube at the beginning of each
test and to pump air bubbles out of the hoses periodically during
the test.

62
FIGURE 19: Setup of Water Depth Measuring Device
FIGURE 20: Point Gage and Tube

63
Selection of Model Scales
In the last chapter the scale-model relationships based upon the
Froude law were derived. The scale to which the model should be
constructed depends on the following factors: the size of the flume
(length, width and depth), the discharge capacity of pump, the accuracy
of instrumentation and the dimensions of the prototype. According to
these factors, the vertical length scale of 1:10 (or = 10) is selected
through the entire study for both distorted and undistorted models.
Mangroves
The dimensions of all stems and roots, including height and
diameter of the prototype, are reduced to 1/10 for the model based on
the undistorted Froude law. However, for the canopy some distortion
of scale between prototype and model is required according to equation
(77) in the Chapter IV, i.e.,
£ An
m m
<\>
in which N = Nd = 10, £ is the length of a leaf and A is the surface
area of the leaf. For simplicity, the elliptic shape of mangrove leaves
are approximated by a rectangular area with a length £ and a width w.
Equation (77) is then reduced to
(80)
By choosing 1:10 for the length scale for this study, the width scale of

64
the leaf becomes
w 7/
= 10/4= 56 , (81)
m
Therefore, the dimensions of the leaves used in the model can be
estimated from the derived relations and are shown in Table 5, which
are based upon the prototype data listed in Table 2.
TABLE 5: Scale Selection for canopy
Section
No.
(1)
(2)
(3)
prototype
length (£)
2.00
3.00
4.00
(in)
width (w)
1.00
1.50
1.75
Model
length (£)
0.20
0.30
0.40
(in)
width (w)
0.018
0.027
0.031
p' m
10
10
10
w /w
p m
56
56
56
Buildings
This part includes the three previously discussed kinds of
buildings: high-rise, medium-rise and residential buildings. Table 6
shows a sumnary of the average parameters for these three categories.
In searching for material to be used in constructing the buildings in
the model, it was found that the ratio of length and width of a standard
block was very close to that of the prototype. Another advantage in
using concrete blocks is that they are easy to set up, since each

TABLE 6: Average Parameters of Prototype and Model for Building Areas
PROTOTYPE
MODEL (Nd =
10)
Type
of
Average
Dimension
(ft)
Approximate
Dimension
(ft)
Density
%
Dimension
(in)
Density
Buildings
1ength
width
1ength
width
no.
no.
Deve-
1 oped
1ength
width
no.
no.
l000'x500'
acre
8'x8'
8'x2,871
High-Rise
224
112
225
109
7.19
0.63
36
174
15.50
7.50
28
10
Medium-Rise
99
51
103
' 50
23.62
2.06
24
80
15.50
7.50
20
7
Residential
62
30
68.87
6.00
26
48
15.50
7.50
20
7
Note: 1 acre = 43,560 ft^

66
concrete block is heavy enough to withstand all the flow velocities in
this study. Therefore it was not required to anchor them to the flume
bottom. Scales of the model are then determined from the horizontal
dimension of a concrete block (15.5 in x 7.5 in) and the dimensions of
prototype buildings, as shown in Table 6. It is noted that in order to
scale the prototype buildings into the 8 foot wide flume, it was
necessary to use a distorted model.
Model Setup
Mangrove Stems and Roots
Based on the average parameters obtained from the prototype,
patterns of red and black mangroves were designed and shown in Figures
21 and 22. These two patterns were the best arrangements that could be
achieved in the modeling in order to insure that the stems and roots
were distributed evenly and yet still maintained their own natural
characteristics in dispositions. For example, the prop roots were
arranged in a hexagon pattern, which was found to be the most common
disposition found in natural. The staggered pattern used for the stems
of black mangroves and the root system of red mangroves was considered
to be the best regular pattern to simulate the fully random distribution
found in the prototype. The legends listed in Figures 21 and 22 were
the actual dimensions used in model setup.
The stems of red and black mangroves were simulated by dowels of
the specified diameters and heights. The substems and prop roots were
simulated by galvanized nails with the caps removed. The air roots of
black mangroves would be very hard to model on a one by one basis due to

67
o
o o
0+0
o o
o
O o
0 + 0
o o
o
o O
o + o
O r«
o
o o
o+o
o O
o
o
o o
0*0
o
o
O o
0+0
o o
o
o o
0+0
o O
o
o O
0 + 0
o o
o
0
o
o
O O
0+0
o o
o
o o
0+0
o o
o
o
o o
0 • 0
o
o
15’
h3ui
o + o
o o
O Main-Stern
• Sub-Stem
0 Prop Root
FIGURE 21: Model Setup for Red Mangroves

68
o Stem
• Air Root
FIGURE 22: Model Setup for Black Mangroves

69
their high density and small dimensions. Therefore, a manufactured nylon
door mat whose strings have the same height (0.6 inch) and the same
thickness (0.025 inch) as the design model dimensions of air roots was
used. Density of the strings is 44 per inch square area which is 91 l
of the average design density (48 per inch square area). The only
deficiency in using this mat is that the strings are blade-shaped, which
may cause a higher resistance to the flow than cylindrical air root.
However, considering the advantages of using the mat, this deficiency is
considered to be insignificant.
These dowels, nails and mats were fixed on three 8 feet by 4 feet
marine plywood sheets, which were coated with latex paint to prevent
swelling (Figures 23 and 24). The plywood sheets were secured to the
false bottom by a row of concrete blocks and by a 24 feet long L shaped
steel beam attached to two sides of the plywood sheets (Figure 25).
The row of concrete blocks stacked 15 inches high was placed in the main
flume, starting from the last flow straightener and extending a distance
of 80 feet. Thus only half of the flume width was used in these
experiments.
Canopy of Red Mangroves
During the second part of the experiments a canopy was constructed
in the red mangrove area. Strips of galvanized metal plates were used
to simulate 'leaf strips'. This assumes that the leaves are closely
connected to each other. Strips with three different widths, 0.2, 0.3
and 0.4 inch, represent three different sizes of leaves, as shown in
Table 5. Each stripe has a height of 9.6 inches, which covers 539, 360
and 308 leaves for section #1, #2 and #3, respectively. Stripe numbers

FIGURE 23: Stems and Roots of Red Mangroves
FIGURE 24: Stems and Roots of Black Mangroves

71
FIGURE 25: Overview of Setup for Mangroves
FIGURE 26: Setup of Model Equivalent of Canopy of Red Mangroves

72
for each section, which is 5 feet long, 34.2 feet wide and 8 feet high
in the prototype, can be calculated from the densities measured (cf.
Table 2). As a result, stripe numbers needed in the model for section
#1 to #3 are found to be 25, 20 and 10, respectively. These stripes
were also arranged in a staggered pattern, as shown in Figure 27.
Figure 26 shows the setup of the stripes in which the stripes are sus¬
pended from the top of the supporters and fixed to the plywood bottom.
Buildings
In this part of the experiments, the whole width of the flume was
used. As mentioned in the last section, the concrete blocks with
dimensions 7.5 inches x 7.5 inches x 15.5 inches were used to simulate
the buildings for the three different types of developed areas. Figure
28 shows 21 patterns to be tested in which no. 1 to 13 were designed to
simulate high-rise building areas, while no. 14 to 21 were for medium-
rise building and residential areas. As can be seen in these patterns,
both the aligned and staggered dispositions were included for each
density of the buildings. The design densities are started from low to
high and will at least cover the average densities obtained from the
prototype for the three developed areas (cf. Table 6). No extra work
was needed to anchor these concrete blocks except to move them into the
desired positions, since each concrete block weighs about 38.5 pounds
and two layers of blocks are steady enough to withstand all the flows
used in this study.

SECTION # I SECTION# 2 SECTION # 3
73
X 0.2" x 9.6"
° 0.3" x 9.6"
A 0.4" x 9.6"
FIGURE 27
Distribution of Leaf Stripes in the Model
OPEN AREA 4*—ROOT AND STEM AREA-*)

74
FLOW
â–¡ â–¡ â–¡ â–¡
(2)
â–¡â–¡â–¡â–¡â–¡â–¡
(3)
â–¡â–¡â–¡â–¡â–¡â–¡â–¡â–¡
(4)
ODDOOOOOa]
L
FIGURE 28: Building Patterns Designed for the Tests

75
uu
TT
nn
M
FLOW
V
(9)
(10)
U
U
TL
â–¡ u u
n n â–  -
(ID
(12)
(13)
8'
i l
â–¡ â–¡ â–¡
â–¡ â–¡ â–¡
8'
â–¡ â–¡ â–¡
â–¡ â–¡ â–¡
11
â–¡ â–¡ â–¡
â–¡ â–¡ â–¡
(14)
(15)
FIGURE 28: CONTINUED

76
FLOW
i—
,
â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ a â–¡
0
â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ G
â–¡ â–¡ â–¡ â–¡
(16)
(17)
â–¡ â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡â–¡ â–¡
â–¡ â–¡â–¡â–¡â–¡
-
â–¡ â–¡ â–¡ â–¡ D
0 Q â–¡ â–¡ â–¡
0 D â–¡ â–¡ â–¡
D â–¡ â–¡ â–¡ â–¡
(18)
(19)
â–¡â–¡â–¡â–¡â–¡â–¡
â–¡â–¡â–¡â–¡â–¡â–¡
â–¡â–¡â–¡â–¡â–¡â–¡
â–¡â–¡â–¡â–¡â–¡â–¡
â–¡â–¡â–¡â–¡â–¡â–¡
â–¡â–¡â–¡â–¡â–¡â–¡
ODOGDO
0 0 D â–¡ D â–¡
(20) (21)
FIGURE 28: CONTINUED

CHAPTER VI
EXPERIMENTAL TEST SERIES
Experimental Procedure
Calibration of Velocity Meter
After the construction of the apparatus, the first step in the
experimental procedure was to calibrate the velocity meter. Mounted
with its normal support on the carriage, the Nixon meter was pulled
through still water at constant velocity over a distance of 20 to 40
feet with the trolley. By operating the specific keys on the HP 9825A
calculator to execute program statements which read initial and final
values of propeller revolutions and time, the average frequency of the
current meter, the true velocity and the percent error of the calibra¬
tion curve were then computed and printed out. If the absolute error
was greater than 5 % the instrument was recalibrated.
The meter was checked in the range of 5 to 60 cm/sec and no less
than 20 points were used to determine a linear least square fit of
frequency versus velocity. Appendix C contains a complete program
listing for the HP 9825A.
Measurements of Mean Velocities
The velocity and depth obtained in this study were measured in the
center line of the test sections where the influence of sidewall was not
felt. The experimental run begins when the main pump is started. It
77

78
usually takes about twenty minutes for flow to reach steady state for
each set of discharge values. After the flow became stabilized,
velocities were taken at one section 30 feet downstream of the last
flow straighteners. Nine points on a vertical at the relative depths
of: z/d = 0.1504, 0.1881, 0.2352, 0.2492, 0.3679, 0.4601, 0.5754,
0.7197, 0.9000, were sampled to best describe the vertical velocity
profile (Christensen, 1978). The velocity at each depth was then
determined by the velocity program (Appendix B) from the calibration
formula and was printed out for immediate checking. Each velocity
obtained is on average velocity over a time span of 30 seconds which is
the maximum time interval that can be used with the HP 9825A. Plotting
the vertically distributed velocities on graph paper and integrating
over the water depth yields the discharge per unit width /udy. The
spatial mean velocity for each run then was obtained from the value of
unit-width area divided by the water depth. Even though this method is
time consuming for the large number of runs, it is still the best way
to determine the mean velocity for the mangrove part of the experiments
in which the test channel occupies one half of the main channel.
For the building part of the experiments, in which the entire main
channel was used, a discharge formula for the Thomson weir derived by
the hydraulic laboratory of Civil Engineering Department, University of
Florida, was applied to determine the mean velocities, i.e.,
2.514 , ,
Q = 2.840 H (82)
where Q is the discharge from the Thomson weir in cubic feet per second,
and H is the vertical distance in feet between the elevation of the

79
lowest part of the notch or the vertex and the elevation of the weir
pond. Eighteen runs with mean velocities from 17 cm/sec to 53 cm/sec
were tested by both methods to determine the accuracy of the weir
formula; the results are shown in Figure 29. It is apparent that the
mean velocities obtained using the Nixon meter in the center line of the
flume Un is slightly larger than that given by weir formula Uw, but is
within a limit of 5 %. This small error is considered to be insigni¬
ficant and may be compensated for by the advantages of using the weir
formula. For instance, the fluctuating water level above the weir vertex
due to the instability of the pump was often observed, therefore, the
mean velocity obtained from an average value of H over a longer period
of time should be more representative than that measured by the Nixon
meter over a 30 second period.
Measurements of Water Depths
For each run three water depths were measured by using the device
shown in Figure 19. Two water depths were taken at the two sections
which covered the roughness area and one was taken at the section where
velocities were measured out of the roughness area, located 30 feet
downstream of the last flow straighteners. Since the water head losses
between the two sections in the model tests were in the range from less
than 1 to a few centimeters, the water depths were measured to an
accuracy of one hundredth of a centimeter for a precise and reliable
result. To implement this fine measurement, all the siphon hoses used
in the tests were kept free of air bubbles and the well graduated
electronic point gages were used. Before each run the still water

80
FIGURE 29: Relation Between U and U
n w

81
depth was measured and its scale reading for water surface elevation on
the point gage was recorded. The same reading was performed for each
section after the flow became stable. From the difference of these two
readings for water surface elevations and the initial still water depth
the flowing water depth can be calculated.
In general, it takes about 5 to 20 minutes for a new water level in
the tube (cf. Figure 19) to reach its equilibrium state, which can be
observed by moving the vernier on the point gage to see whether any
change in the water level is detected. This water depth measuring
device worked very well through the entire experiment and provided
consistent and reliable data.
Experimental Runs
Mangrove Areas
The total model lengths of the red and black mangrove areas were 5
and 15 feet, respectively (Figure 25). For the red mangrove area the
water depths were measured at the two ends of the 5 feet long area. For
the black mangrove area, the first section was chosen 4 feet from the
front end, and the second section was located 3.5 feet from the rear end
of the black mangrove region so that the influences, including the
disturbance caused by the red mangroves in the front, and the depth drop
due to the end of the plywood sheets in the rear could be eliminated.
Therefore, a total length of 7.5 feet centered in the middle section of
the black mangrove area was used to measure the energy loss.
During the first part of experiments, 7 runs were conducted for the
air roots of the black mangrove area to determine its apparent roughness

82
height and friction factor. In the second part of the experiments, 38
runs were performed for the red mangrove areas (without canopy) and
black mangrove areas by adjusting the discharge value and changing the
still water depths so that the flow Reynolds number (R^ = UgR^/ v)
covered a range from 20,000 to 55,000 while the Froude numbers varied
from 0.14 to 0.44. An additional 32 runs were conducted for the red
mangroves with canopy at the later stage to determine the importance of
a canopy in reducing the flow energy.
Building Areas
At least 10 runs were conducted for each of the 21 patterns shown
in Figure 28. These runs for each pattern were controlled by adjusting
the flumes discharge valve and the tail gate so that they covered a
range of Reynolds number (R^) from 20,000 to 70,000, while the Froude
number varied from 0.1 to 0.5. Figure 30 shows 20 pictures of the
designed patterns in which pattern No. 9 is not included due to the
faulty picture. The results obtained for medium-rise building areas
can be converted using appropriate scaling factors to use in residential
areas since these two areas are presumed to have the same relative
distributions and have only dimensional differences.

83
FIGURE 30: Designed Building Patterns in the Tests

CHAPTER VII
PRESENTATION AND ANALYSIS OF DATA
The data obtained from each experiment include two water depths,
d.| and d^, and two depth averaged velocities, U-| and U^. Other
parameters, such as the energy loss AH, equivalent friction factor
f', apparent roughness k , Reynolds number R', Froude number F
(U //gd ) and the averaged drag coefficient Cn are calculated using
their specific definitions as given by equations (78), (79), (31) and
(48). Mean values and the corresponding standard deviation of f', k
G 3
and Cp were also determined for each set of tests. A summary of these
results is listed in Appendix C, in which II = (U, + U9)/2 and d= =
(d.j + d2)/2. Three relations between the friction factor, the drag
coefficient and Reynolds number, and the friction factor and the water
depth were found from the data for each area and plotted as shown.
Mangrove Areas
Figures 31 to 33 show the three relations for red mangroves with¬
out canopy, while Figure 34 to 36 show those for red mangroves with
canopy. The relations for black mangroves are shown in Figures 37
through 39. Table 7 lists the mean values and their corresponding
standard deviations for the equivalent friction factors, drag coeffi¬
cients and apparent roughnesses.
84


2.000
1.600
1.200
0.800
0.732
0.400-
0.000
J- 1 1 1
i ! I í ! !
i i i i i i
I I i i i i
¡ill!!
i i ! ! ! !
iiiiii
iiiiii
0
l r\ n j
I l l I
I l i I
till
l l l I
till
l l l i
O OOI O 1
0
U U i
r V/ U
0 0
WW Q ÍOO• .00
n i Mean
0
0^
o
o
o
o
0 !
0
0 0 0
O l
00
OO 1
1
1
1
1
1
1
1
1
1
I
1
1
1
1
1
1
1
1
1
■ —
i —H
1
1
1
1 ■——
20,000 25,000 30,000 35,000 40,000 45,000 50,000 5,000
R'
e
03
cr>
FIGURE 32: Relation Between and R/ for the Red Mangrove Areas (Without Canopy)

FIGURE 33: Relation Between f and Water Depth d for the Red Mangrove Areas
(Without Canopy) e

FIGURE 34: Relation Between and R^ for Red Mangrove Areas (with Canopy)

FIGURE 35: Relation Between CD and R^ for Red Mangrove Areas (with Canopy)

d (cm)
FIGURE 36:
Relation Between f and Water Depth d for Red Mangrove Areas
(with Canopy)

FIGURE 37: Relation Between f and R' for Black Mangrove Areas
e e

FIGURE 38: Relation Between and R^ for Black Mangrove Areas

d (cm)
FIGURE 39:' Relation Between f and Water Depth d for Black Mangrove Areas

94
TABLE 7: Statistical Values of Experimental Results for Mangrove
Areas
Statistical
r»
CD
ka(cm)
Mangroves^-
Values
1 e
Model
Prototype
Red
Mean
0.129
0.732
42.69
426.9
Standard
Deviation
0.013
0.077
6.14
61.4
Red +
Mean
0.134
0.758
44.10
441.0
Canopy
Standard
Deviation
0.012
0.078
6.48
64.8
Black
Mean
0.132
1.001
38.53
385.3
Stems
Standard
Deviation
0.020
0.090
9.72
97.2
Black
Mean
0.028
5.26
52.6
Air Roots
Standard
Deviation
0.003
0.65
6.5
It is clear that the experimental results show a good degree of
consistency in the relations between the equivalent friction factor,
drag coefficient and Reynolds number. This is especially true for the
drag coefficients obtained from the experimental data, which have only
about 10 percent standard deviation. This consistency not only proves
that Eq. (48) proposed for the evaluation of the equivalent friction
factor is acceptable, but also implies that the accuracy of the
measurements of the water depths and flow velocities was fairly high.
The increase of the equivalent friction factor and drag coefficient
due to the existence of canopy in red mangroves is less than 4 percent,
which is obtained by comparing their mean values as listed in Table 7.

95
The relations between the friction factor and water depth for the
mangrove areas are shown in Figures 33, 36 and 39, which provide an
important clue that the equivalent friction factor for protruding
obstructions like black mangroves is linearly proportional to the water
depth as proposed in equation (48). For red mangroves, however, the
linear relation due to the protruding stem is eliminated by the sub¬
merged prop roots which give an inverse relation between the equivalent
friction factor and water depth as described by equation (30). It
shall be noted that the drag coefficients obtained for red mangroves
represent the mean values of stems, substems and prop roots. For black
mangroves, the mean value of the drag coefficient only takes into
account the stems, while the air roots were considered to be the bottom
friction and their influences on the energy losses were excluded through
drag coefficient calculations.
Building Areas
The experimental results for 21 different distribution patterns
for building areas are shown in Figures 40 through 42. The statistical
values of these results are listed in Table 8, in which the data for
residential areas were converted from those for medium-rise building
areas. Since all these runs were conducted in the false-bottomed
section which is made of the smooth metal plate, the energy loss
between two cross sections was hardly measured. Comparing the small
energy loss to the experimental errors, the bottom friction, therefore,
was neglected through the entire tests for building areas. In general,
the relations between the equivalent friction factor, drag coefficient
and Reynolds number as shown in Figures 40 and 41 are similar to those

e v
(1)
UUU |
20,000 45,000 70,000
e v
(2) '
FIGURE 40: Relation Between f and R' for Building Areas
e e
(Number in the Parenthesis is the Pattern Number)

0.150'
0.00Q
f
e
0.07S +
00
CD O O
20,000
45,000
R‘
e
70,000
(3)
FIGURE 40:
CONTINUED

e
(8)
FIGURE 40
0.020
r
e
0.100-
0.000
20,000
O 00 0
o
45,000
R'
e
(7)
70,000
e
(9)
CONTINUED

e
(12)
FIGURE 40
0.060-
♦
f*
e
0.030'
0 0
O.OOGl
20,000
O
o
0
45,000
70,000
0.10Q
f
e
0.050'
0.000
0
0
20,000
0 0
o
0
0 o
o
45,000
R'
(13)
70,000
CONTINUED

e
(16)
FIGURE 40
0.050
r
e
0.025
0.000
0 0
0
0 0
0 0 0
0 0
0
20,000
45,000
r;
(15)
70,000
e
(17)
o
o
CONTINUED

0.050
0.025
0.000
20,000
0.120
f
e
0.060
70,000
0.000
20,000
7ÃœTÃœD0
FIGURE 40:
0.150'
1
1
1
f
e
0.0 / b
o
o
o
o
o •
o
o
o
o
o
L
0.000
â–  â– â–  -
20,000 45,000 70,000
R'
e
(19)
CONTINUED

FIGURE 41: Relation Between and R^ for Building Areas
(Number in the Parenthesis is the Pattern Number)

(4)
(5)
FIGURE 41: CONTINUED

1.110
0.000
Me on o
20,000
45,000
R¿
70,000
(6)
e
(8)
FIGURE 41
20,000
5.000
CD
2.500
1.600
0.000
Mean
TT
20,000
45,000
R'
e
(7)
70,000
u
0 0 00
U
0 0
o
-c*
45,000
R'
e
(9)
70,000
CONTINUED

e
(12)
FIGURE 41
(ID
o
cn
CONTINUED

20,000
45,000
R'
(14)
70,000
FIGURE 41
5.000
2.500
2.080
0.000
Mean
-tr
ooo
o oo
—o-o-o
20,000
45,000
R'
e
(15)
20,000
45,000
R'
e
(17)
70,000
70,000
CONTINUED

(18)
(20)
FIGURE 41
5.000
D
3.360
2.500
0.000
Mean
1
1
1
1
1
1
O OI 0 0
O" O" o
1—O" o—o
1
1
1
1
i
i
1
1
1
1
1
20,000
45,000
R'
(19)
5.000
3.930
CD
2.500Í-
O.OOQl
20,
n o
o o
.QQQQ
70,000
Mean
o
^1
000
45,000
R'
e
(21)
70,000
CONTINUED

0.012
0.024
d(cm)
(10)
f
e
0.012
0.000
0.0
15.0 30.0
d(cm)
(ID
FIGURE 42: Relation Between -T and Water depth d for Building Areas

TABLE 8: Statistical Values of Experimental Results for Building Areas
Pattern
No.
f7
e
CD
Buil ding
Area
Mean
Standard
Deviation
Mean
Standard
Deviation
val ue
% of
mean
val ue
% of
mean
1
0.0013
0.0003
23
0.788
0.163
21
High-Rise
2
0.0064
0.0006
38
1.965
0.254
13
II
3
0.0180
0.0030
17
3.406
0.181
5
II
4
0.0460
0.0090
20
7.014
1.121
16
II
5
0.1580
0.0260
16
17.813
1.298
7
II
6
0.0036
0.0006
17
1.108
0.123
11
II
7
0.0046
0.0011
24
0.907
0.230
25
II
8
0.0090
0.0010
11
1.346
0.119
9
II
9
0.0130
0.0010
8
1.597
0.189
12
II
10
0.0080
0.0014
18
2.369
0.275
12
II
11
0.0154
0.0029
19
2.965
0.227
8
II
12
0.0230
0.0040
17
3.662
0.221
6
II
13
0.0350
0.0050
14
4.315
0.389
9
II
14
0.0074
0.0011
15
1.232
0.170
14
Medium-Rise
15
0.0127
0.0030
24
2.082
0.173
8
II
16
0.0117
0.0020
17
1.088
0.096
9
II
17
0.0329
0.0067
20
3.040
0.114
4
II
18
0.0158
0.0018
11
1.181
0.097
8
II
19
0.0448
0.0072
16
3.355
0.098
3
II
20
0.0265
0.0038
14
1.586
0.089
6
II
21
0.0631
0.0112
18
3.926
0.120
3
II
14
0.0123
0.0018
15
1.232
0.170
14
Residential
15
0.0211
0.0050
24
2.082
0.173
8
II
16
0.0195
0.0034
17
1.088
0.096
9
II
17
0.0548
0.0112
20
3.040
0.114
4
II
18
0.0264
0.0030
11
1.181
0.097
8
II
19
0.0747
0.0120
16
3.355
0.098
3
II
20
0.0441
0.0063
14
1.586
0.089
6
II
21
0.1052
0.0186
18
3.926
0.120
3
II

no
in mangrove areas. The drag coefficient obtained from the experimental
data for each of these 21 patterns also show a good consistency for all
the flow conditions in which the standard deviations of 13 patterns are
less than 10 percent of the mean drag coefficients. For all the
patterns in building areas, the equivalent friction factors are found
to have similar linear relation to the water depths, and four examples,
taken from patterns 10 and 13, to express this relation are shown in
Figure 42.
Determination of Drag Coefficient
The consistent results of the drag coefficients obtained from the
experimental data imply that the mean value of the drag coefficients
can be represented for each building pattern as listed in Table 8.
Based on these mean drag coefficients and their corresponding building
densities, dispositions and area types, the drag coefficient may be
determined empirically through analyses of these relations.
Drag Coefficient - Building Density Relations
The data obtained from pattern 1 to 13 were analyzed and plotted
in Figure 43, which shows three possible relations between the drag
coefficients and densities for high-rise building areas. Figure 44
shows two relations between drag coefficients and densities for the
medium-rise building area and residential area, respectively. The
drag coefficients obtained for the high-rise building area were also
converted in order that they can be used in the medium-rise building
and residential areas. These interpolated data from the two row high-
rise building areas, however, agree very well with the data obtained
for the other two building areas (Figure 44). The drag coefficients

Ill
FIGURE 43: Relation Between CD and Density m for High-Rise
Building Areas

112
RESIDENTIAL AREA, m(no./ft2) (x 1.47 xIO5)
MEDIUM-RISE BUILDING AREA,m(no./ft2) (x 4.10 x I05)
FIGURE 44: Relation Between CD and Density m for Medium-Rise
Building and Residential Areas

113
as a function of densities were then derived by using a least square
curve fitting method. Seven such equations were derived, including
three for the high-rise building areas and two for medium-size building
and residential areas, respectively. It is apparent that the drag
coefficient for the staggered disposition is about two to three times
larger than that for the aligned disposition under the same building
density for all the three building areas. The drag coefficient is
found to vary linearly with the building density for all the cases
except for the one row high-rise building case, which shows an expo¬
nential relation between the drag coefficient and building density.
Using the equations shown in Figure 43 and 44, the drag coefficient
can easily be determined for developed areas if the building density
is known.
Drag Coefficient- Disposition Parameter Relations
Though the drag coefficient were shown to be explicitly related to
the building densities and dispositions in the previous section, it is
desired to represent the drag coefficient in a simple equation which
takes both the dependent parameters, density and disposition, into con¬
sideration. As shown in Figure 45, the lateral spacing between two
roughness elements (edge to edge) is S^. and the longitudinal spacing
between two successive roughness elements (edge to edge) is S^. The
corner-to-corner distance between the roughness elements in adjacent
transverse raws is S^, which is related to and by
S - D
2 2 u 2
5^ = 5^+ (—2 ) ln staggered arrangement (83)
in aligned arrangement
(84)

114
[Is'n. â–¡
% sd
ti
â–¡ â–¡ â–¡
t
FLOW
\JS'T\ â–¡
Sg, Sd
^0 Ü □
HD I*-
â–¡ â–¡ T â–¡
FLOW
(a) Aligned Arrangement (b) Staggered Arrangement
FIGURE 45: Position Spacings. Definition Sketch
in which D is the projected with of the roughness element in the
direction of the flow. A result of attempting to relate the drag
coefficient to the three disposition spaces, S^, S¿ and Sd is listed
in Table 9. By plotting these relations between the drag coefficients
and non-dimensional disposition parameters, St/D, S /D, Sd/D and
Sj/S^, it is found that only Sd/D and show a good correlation,
while the others either do not yield any relation (S^/D and C^) or
show a poor correction (S^/D, Sd/S^ and C^). The relations between
the drag coefficient and the disposition parameter, Sd/D, are shown in
Figures 46. Comparing Figures 46 and 44, it is clear that the drag
coefficient is closely related to the density or to the disposition
parameter, Sd/D, for the aligned and staggered patterns, respectively.
However, by combining the two relations for the aligned and staggered
patterns in Figure 46, it is found that the drag coefficient can be
solely expressed as a function of the disposition parameter, Sd/D, for
both the aligned and staggered cases, ie.,

TABLE 9: Relations Between Disposition Parameters and Drag Coefficients
Pattern
No.
Spacings (in)
Disposition
Parameters
CD
st
sd
St/D
VD
Sd/D
Vs 1
27.0
3.60
0.788
2
13.2
1.76
1.965
3
7.3
0.97
3.406
4
4.0
0.53
7.014
5
1.9
0.25
17.813
6
27.0
3.44
27.22
3.60
0.46
3.63
7.91
1.108
7
18.4
3.44
18.72
2.45
0.46
2.50
5.44
0.907
8
13.2
3.44
13.64
1.76
0.46
1.82
3.97
1.346
9
9.8
3.44
10.39
1.30
0.46
1.38
3.02
1.597
10
33.9
3.44
13.64
4.52
0.46
1.82
3.97
2.369
11
22.1
3.44
8.07
2.95
0.46
1.08
2.35
2.965
12
15.5
3.44
5.28
2.07
0.46
0.70
1.53
3.662
13
11.3
3.44
3.93
1.51
0.46
0.52
1.14
4.315
14
18.4
12.38
22.18
2.45
1.65
2.96
1.79
1.232
15
22.1
12.38
14.37
2,95
1.65
1.92
1.16
2.082
16
13.2
6.80
14.85
1.76
0.91
1.98
2.18
1.088
17
15.5
6.80
7.89
2.07
0.91
1.05
1.16
3.040
18
9.8
6.80
11.93
1.30
0.91
1.59
1.75
1.181
19
11.3
6.80
7.06
1.51
0.91
0.94
1.04
3.355
20
7.3
6.80
9.98
0.97
0.91
1.33
1.47
1.586
21
8.4
6.80
6.85
1.12
0.91
0.91
1.01
3.926

116
FIGURE 46: Relation Between C~ and S ,/D in Aligned and
Staggered Patternsu

117
S -2*0
CD = 1.0 + 1.8 (-§-) (85)
which is shown in Figure 47. This is attributed to the nature of the
diagonal spacing, S^, whose magnitude not only provides a measurement
of density but also reveals a difference of disposition for the evenly
distributed roughness elements. Therefore, the higher the building
density, the smaller the disposition parameter, and therefore a larger
drag coefficient results. For buildings with the same dimensions and
density, the disposition parameters, S^/D, of staggered patterns is
smaller than that of aligned patterns, which also results in a larger
drag coefficient.
The same result was also substantiated by Shen (1973) who measured
the mean drag coefficient of two cylinder patterns (aligned and
staggered) in open channel flow. Flow resistance through these
roughness elements varies row by row. In general, the drag on the
second row element is much smaller than that on the first row. It is
due to the result of decreased wake dynamic pressure and the increased
turbulence level of flow approaching the second row element. Therefore,
since the elements of the second and following rows for aligned pattern
are located at the low wake pressure areas, the mean drag is smaller
than that for staggered pattern, in which the elements are not or only
partially located at the wake areas.

118
FIGURE 47: Relation Between CD and Sd/D

CHAPTER VIII
DISCUSSIONS AND CONCLUSIONS
In the preceding chapters, both the wind stress and the bed shear
stress used in storm surge equations for the case of hurricane-induced
flooding in coastal areas were discussed. Special emphases were placed
on the friction characteristics of developed coastal areas and mangrove
areas. In the air section, the main reduction of wind shear stress due
to wind drag forces on the parts of the obstructions, which include
buildings and vegetation, above the water surface in flooding has been
developed in equations (19) and (20). In the water section, the equi¬
valent friction factor, which incorporates both the bottom friction
coefficient and the drag coefficient due to the submerged parts of obs¬
truction was presented in equation (49) and verified. The drag coef¬
ficient in equations (20) and (49) for mangroves and buildings were
determined empirically according to the typical distribution found in
coastal areas. The results obtained in this study will assist coastal
engineers and coastal planners in providing a better and more accurate
method for predicting hurricane-induced flooding. Discussions given
below are primarily based on engineering applications, these determina¬
tions of the equivalent friction factors for ocean bottom, grassy areas
and forested areas are also described.
119

120
Mangrove Areas
The proposed equation for the equivalent friction factor can be
expressed in the following form for mangrove areas
fe
0.32
[ln(I%Md + l )]2
n
(1-e) + £ m•cndDi
1=1 1 u 1
(86)
Typical values of k, e, m, CQ and D obtained in the previous chapters are
summarized in Table 10. The value of k for red mangrove open areas is
converted from the value of Manning's n, 0.025, which corresponds to a
clean, straight stream on plain in open channel flow, by using the
Strickler equation.
TABLE 10: Typical Values for Mangrove Areas
Mangrove
k
(m)
£
ml
(nr2)
(nr2)
m_
(nf2)
CD
(cm)
D2
(cm)
D3
(cm)
Red
0.07
0.013
0.3
1.0
6.1
0.73
15
5
3
Black
0.53
0.005
0.9
1.00
8
For black mangrove areas, the value of k is obtained from the
experimental results for air roots. The subscripts 1, 2 and 3 of m and D
represent the main-stems, sub-stems and prop roots, respectively. It has
been shown that the canopy on mangrove fringes causes only a 4 percent
increase in the resistance to water flow, and thus, may be neglected. It
shall be noted that red mangroves, in general, only extend, about 15 meters
inland from the shoreline and it is not possible to distinguish them from

121
black mangroves on aerial photographs and conventionally used city maps.
Examining the case of fine grid system for numerical models whose grid
elements are usually chosen to be 1,610 meters square, the average
friction factor for each grid element on the mangrove fringes should be
represented bt the sum of 1 % of the friction factor obtained from red
mangroves plus 99 % of that obtained from black mangroves. For
engineering applications, therefore, it is not necessary to consider the
difference between red and black mangroves. Therefore, the equivalent
friction factor derived for black mangroves can be used for all the
mangrove areas, i.e.,
f' = — + o.07d (37)
e [1n(21d + 1)]
in which the depth in meters.
Developed Areas
In developed areas, seven drag coefficient equations for most of
the building arrangements observed in Broward and Dade counties, Florida,
were determined and listed in Figures 43 and 44. To apply these equa¬
tions, however, the building density and the type of area must be given.
The most realistic way to determine these is to estimate them using
aerial photographs of the study areas. County Land Use Plan Map published
by each county are also recommended. If both the aerial photographs and
the County Land Use Plan Map are not available, or only a rough estimation
is needed, the average densities obtained for Dade and Broward counties
for the different developed areas can be utilized.

122
The following three drag coefficient equations are suggested for
friction factor cal cualtions in developed areas:
r . „ ocr _2.7m x 105 . _ r_T no.
Cp = 0.355 e ; m L=J —f^r
no.
or
CD = 0.355 e2,9m x 104 > m £ ^ ~¡ÍF
(88)
for the one row high-rise building areas;
Cp = 1.05 + 4.8m x 104 ; m [=]
or Cp = 1.05 + 5.2m x 103 ; m [=] (89)
for the staggered medium-rise building areas; and
Cp = 1.04 + 1.7m x 104 ; m [=]
or Cp = 1.04 + 1.8m x 103 ; m [=] -^- (90)
for the staggered residential areas. The particular equation is recom¬
mended because one row high-rise buildings are the most predominate type
of buildings found in highly developed coastal fringes such as those in
Broward and Dade counties, Florida. For medium-rise buildings and resi¬
dential areas, most buildings can be assumed to be staggered, since
aligned buildings are scarce and the probability that floods will flow
normal to the aligned building areas is very small in the field.
An alternative way of determining the drag coefficient is to use
equation (85) in which the diagonal spacings between buildings should be
measured from aerial photographs. It is convenient in this case to

123
apply equation (85) to areas whose dispoition parameters, S^/D are nearly
same so that the drag coefficient can be determined without considering
the building types and dispositions.
The equivalent sand roughness, k, for developed areas cannot be
determined by a single value, since the space between the buildings
cover many different roughnesses. These roughnesses, in general,
consist of pavements, grasses, light brush and trees, whose values of
Manning's n are 0.013, 0.025 and 0.040, respectively. Assuming these
roughnesses are evenly distributed, then an average value of n equals to
0.026 is assigned for these developed areas which gives k = 0.09 m using
the Manning Strickler equation. As a result, the equivalent friction
factor for developed areas may be represented as
fâ–  = ^32 (1 - e) + m CR d D (91)
e [In (122d + 1)] ü
with d in meters. The mean values of e, m and D can be determined
from information in Table 6 and the equations for are shown in
Equations (88) to (90).
Ocean Bottom
To determine the friction factor and the equivalent sand roughness
for the ocean bottom, the characteristic morphologic features of the
ocean bottom in the study area should be determined first. The ocean
bottom is mainly composed of different sizes of sands, on which vegeta¬
tion and sand ripples vary with location. Sand ripples are generated
and modified by wind-generated waves. Their profiles are controlled by
the nature of the near-bottom wave motion and by the size of the bed

124
material. Numerous investigations have been conducted to determine the
friction factor and the equivalent sand roughness by using experimental
results (cf. Vitale, 1979) and field measured data (cf. Nielsen, 1983).
However, the applicability of these results obtained for normal wave con¬
ditions to storm waves, during which strong current exists, is greatly in
doubt since not only the characteristics of storm waves are not able to
predicted, but the mechanism of ocean bed formation under storm conditions
is not fully understood at the present time. This uncertainty, plus the
varying distributed vegetation on the ocean bottom makes the direct esti¬
mation of the friction factor by using the aforementioned results obtained
for normal wave conditions questionable.
Therefore, an indirect calibration method is sometimes used in this
case. As suggested by Corps of Engineers (1977), typical bottom condi¬
tions result in a value of f in the range 0.004 to 0.01. For a first
estimate, a value of f = 0.005 may be assumed. This coefficient is used
in calibrating the storm surge numerical model. It not only accounts for
energy dissipation at the bed, but may be used to adjust for inexact
modeling and deficiencies caused by ignoring some of the hydrodynamic
processes involved. However, it shall be noted that the calibration
method will also reflect inconsistencies in the numerical method in the
f-values. Therefore, f-values found this way may not be physically
correct. In other words, they can only be used with the specific algori¬
thm used in the calibration.
Nevertheless, the hydraulic measurement in the field is still the
most reliable way to determine the bed friction characteristics, but since
field measurements of velocity distributions and water depths are nearly
impossible during a storm, a field measurement under normal condition

125
which is similar to the storm tide is greatly needed. One possible
instance is the flood and ebb flows through a tidal entrance in which the
flow is in the fully rough range, and the effect of the temporal accelera¬
tion of the flow is generally of a lesser magnitude than the effect due to
bed friction; therefore a correspondence between entrance flow and storm
flood is expected. Three tidal entrances located on the Gulf coast of
Florida were selected for hydraulic measurements to determine the bed
friction characteristics (Mehta, 1978). Table 11 lists some of the
results obtained from field data, in which the bed friction characteris¬
tics derived are based on near-bed velocity profiles in John's Pass and
Blind Pass, and water surface slope in the channel at O'Brine's Lagoon
Entrance.
The values of Manning's n obtained for these three entrances are in
the range from 0.020 (k = 2.1 cm) to 0.026 (k = 9.5 cm), which corres¬
ponds to the following bed morphologic features in open channel flows:
clean, straight stream on plain (n = 0.025-), gravel uniform excavated
channel (n = 0.022), and uniform dredged earth channel with short grass
(n = 0.022). This correspondence not only implies that the values of n
in the Manning's n table can be applied to tidal entrance, but that they
can also be used for the ocean bottom during storm events. Therefore,
for a rough ocean bottom, where the vegetation is significant or other
roughnesses such as rocks or reefs exist, the n value is assumed to equal
to 0.035 (k = 0.55m), which represents an irregular, rough stream, a
clean, winding stream on plain with some weeds and stones, a dredged
channel with light brush on banks, or a flood plain with scattered brush
and heavy weeds in open channel flows.

TABLE 11: Bed Friction Characteristics of Three Entrances
Entrance Tide Mean Depth Friction Roughness, Manning's n Bed Features
at the Throat Factor,f k (cm)
(m)
John's Pass
flood
4.9
0.007
9.5
0.026
Shells (predominate)
and Fine Sands
Blind Pass
ebb
1.6
0.005
2.1
0.020
Shells and Find Sand
O'Brien's
Lagoon
flood/
ebb
0.44
0.013
2.6
0.021
Sand, Shells and
Ripples

127
As a result, the k values corresponding to the recommended n values
are 0.03 m (n = 0.022) and 0.55 m (n = 0.035) for smooth and rough ocean
bottoms, respectively. The friction factors for ocean bottoms can then
be estimated using the equations
f1 = for smooth ocean bottom (92)
[1n(365d + l)]2
0 32
and f1 = ; for rough ocean bottom (93)
[ln(20d + l)]2
in which the depth d must be in meters.
Forested Areas
Forested areas may consist of many different species, like oaks,
magnolia, cedar, palm, pine and cypress, in which each species has
different representative feature. However, for a general view, some
typical values of the parameters for evaluating the equivalent friction
factor are suggested by Christensen and Walton (1980)
_2
k = 0.5 m ; m = 0.1 m ; D = 0.6 m
e = 0.028 ; Cp <* 0.8
It is worthwhile to examine the drag coefficient recommended for forested
areas. In Figure 47, it is seen that the drag coefficient has a tendency
to become constant and equal to 1.0 when the disposition parameter SyD
is larger then 4. This relation derived initially for the rectangular
roughness elements may also be used for circular roughness elements and

128
can be proved by comparing the measured drag coefficient (1.0) for black
mangroves as seen in Table 10. For black mangroves, the disposition
parameters syD is about 17, and the corresponding drag coefficient is
1.0 according to Figure 46, which agrees well with the measured results.
Applying the same reasoning to forested areas, whose disposition parameter
Sd/D equals to 12, the drag coefficient can be expected to be approxi¬
mately 1.0. Therefore, the drag coefficient 0.8 proposed for forested
areas by Christensen and Walton is reasonable. Adopting these values, the
equivalent friction factor can be expressed in the form
f' = — 2- + 0.048d (94)
e [1n(22d +1)]
for the forested areas, where d is in meters.
Grassy Areas
The vast expanse of saw grass marshes are the dominate plant
community occurring in the world's coastal zones where water stands all
or part of a year. Direct observations of the roughness characteristics
of saw grass in a completely submerged condition do not seem to be as
readily available as are measurements of the roughness, as expressed by
Manning's n, of the more common but smaller grasses used in lawns and as
ground covers for protection against erosion. Therefore, the results of
studies of the flow of water over various grass covers by Palmer (1946)
may provide some information to evaluate the roughness characteristics of
saw grass. Palmer found that a completely submerged surface of Bermuda
grass has a Manning's n of about 0.04, which corresponds to a k-value of

129
about 1.25m. If it is assumed that there is geometric similitude between
Bermuda grass and about three times taller saw grass, Palmer's experiment
with Bermuda grass may be considered as a model test of the prototype saw
grass. As a results, a k-value of about 4 m for saw grass is expected
based on the 1:3 length scale between the Bermuda grass and the saw grass.
Christensen's (1980) indirect observations of the equivalent roughness of
grass covered beds of estuaries on Florida's west coast have indicated a
k-value ranging from 3 m to 5 m, which is in good agreement with the
value derived from Palmer's experiments with Bermuda grass. It should be
noted that the apparent roughness measured Palmer may not depend only on
the size, shape and distribution of the vegetation, but also on the addi¬
tional energy losses induced by the rhythmic motion or vibration of the
usually flexible vegetation elements. This motion may be attributed to
the Karman vortex street trailing each individual grass element. Intro¬
ducing a k-value of 4 m in the friction factor equation, as suggested by
Christensen and Walton (1980), yields
f = 0.32
[1n(2.7d + 1)]Z
in which the depth d is in meters.
(95)
Cone!usions
The research has developed the capability of describing the friction
factor in coastal areas for improved representation in numerical modeling
of storm tides. Special emphasis was placed on the friction charac¬
teristics of mangrove areas and developed coastal areas. The proposed

130
analytical results of friction factors for these two areas were verified
by carrying out physical experiments in a hydraulic flume. In addition,
the reduction of wind shear stress due to the protruding obstructions
above the water surface is derived based on recent data of wind stress
coefficient - wind speed relations.
The friction characteristics for ocean bottoms, grassy areas and
forested areas were also presented by adopting the results from existing
field and laboratory measurements, which are considered as the most
reliable data available at present. It is felt that this research has
provided the most realistic data base for evaluations of wind stress and
bed stress in numerical models of storm surges.
Further verification of the wind stress reduction due to the
obstructions and the canopy is recommended. Effects due to the inter¬
action between the wind and the sea surface, which causes deviation of
the velocity profile from the ordinary logarithmic distribution, are not
well quantified. The use of the quadratic bottom friction, predicted in
steady open channel flow, can probably be improved upon.
Under unsteady conditions, the drag coefficient CQ should be
corrected to include the influence of added mass or virtial mass. It
is of particular importance in the cases of rapidly accelerating or
decelerating floodings, therefore, further study of the virtual mass
effect is encouraged.
Field data, especially during hurricane conditions, is practically
non-existent. Therefore, it is most imperative that work continue into
the measurement of field data, which would enhance the reliability of
the physical model results and give a more complete understanding of
the subject.

APPENDIX A
FIELD RECORDED DATA FOR MANGROVES

132
AREA # |
FIGURE A1: Field Data Record for Red Mangroves (Area #1)

133
AREA # 2
FIGURE A2: Field Data Record for Red Mangroves (Area #2)

134
AREA # 3
FIGURE A3: Field Data Record for Red Mangroves (Area #3)

135
AREA # 4
FIGURE A4: Field Data Record for Red Mangroves (Area #4)

136
AREA # 6
FIGURE A5: Field Data Record for Red Mangroves (Area #6)

137
AREA # 8
FIGURE A6: Field Data Record for Black Mangroves (Area #8)

138
AREA # 9
FIGURE A7: Field Data Record for Black Mangroves (Area #9)

139
AREA # 10
FIGURE A8: Field Data Record for Black Mangroves (Area #10)

140.
AREA #
12
O
i
1
TH i
1 I o
1 °
1 o ,
1
1
1 O
1~ “L
O I
| ° I O o
1
i I °
o 1
I â– 
i i
L _
-I-
1
I 0 I
1
I
o
1
1 _
J
1
o ! ° i
. lo
I
i q
i i
d
1
• i
! I
O
6
(FT.)
12
FIGURE A9: Field Data Record for Black Mangroves (Area #11)

141
TABLE Al: Parameters of Sampling Red Mangroves
AreaK
Parameters
No.
(12ft)¿
Diameter
(in)
Height
(ft)
%
Occupied
main-stem
4
7.5
8.0
1.11
1
sub-stem
9
2.0
1.5
0.14
canopy
8.0
root
48
1.0
1.5
0.18
main-stem
3
3.7
8.0
0.16
2
sub-stem
10
2.0
1.5
0.15
canopy
8.0
root
63
1.0
1.5
0.24
main-stem
1
6.5
12.0
0.16
3
sub-stem
13
2.0
1.5
0.20
canopy
6.0
root
108
1.0
1.5
0.41
main-stem
5
2.7
8.0
0.14
4
sub-stem
16
2.0
1.5
0.24
canopy
10.0
root
95
1.0
1.5
0.36
main-stem
6
5.0
8.0
0.63
5
sub-stem
17
r\D
•
o
1.5
0.26
canopy
10.0
root
97
1.0
1.5
0.37
main-stem
3
10.7
14.0
2.81
6
sub-stem
10
2.0
1.5
0.15
canopy
6.0
root
74
1.0
1.5
0.23

TABLE A2: Parameters of Sampling Black Mangroves

143
TABLE A3: Characteristics of Canopy of Red Mangroves
Section
Area
No. of Leaves
Leaf
Sizes (%)
#
#
(1'x 5'x 6')
2"xl"
3"xl.5"
4"xl.75"
4.5"x2"
1
262
2
392
3
416
1
4
138
90
10
5
296
6
288
2
154
29
51
20
3
63
35
37
28

APPENDIX B
COMPUTER PROGRAM LISTINGS

145
Novonic-Nixon Velocity Meter Calibration
0
1
2
3
4
5
•6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
"Velocity Calibration Program for Nixon Meter # 696":
fmt 0, c2, f6.2, c4
fmt 1, cl6
ent X
dim A$[16], T[2], N[2]
wtb 2,21,1,17
wtc 2,0
stp
"Start": red 2.1, A$; prt A$
val (A$[l,4])+N[l]; val (A$[10])->T[l]
stp
"End": red 2.1, A$; prt A$
val (A$[l,4])+N[2]; val (A$[10])^T[2]
(T[2]-T[l])/6->T
(N[2]-N[l])/T+N
X*30.48/T->-V
wrt 16, "T=", T, "sec"; wrt 16,"N=", N, "Hz"
wrt 16, "V=", V, "cm/s"
wtb 2,1,17
spc
if N<30; 1.026393N-16.654312->A
if N>=30; 0.659483N-5.648766->A
if A<6; 0->A
(A-V)*100/V+E
wrt 16, "E=", E, "%"
spc
stp
end
Definition of Variables:
X = distance in feet over which the calibration occurs
A$ = string variable from pink box containing the raw data

146
T = time in seconds to travel distance X
N = frequency in Hz
V = true velocity in cm/s
A = calculated velocity from previous calibration curve
E = percent error

147
Linear Regression for Velocity Meter Calibration
0: "Part I: Linear regression Program":
1: fxd 6
2: ent P
3: dim NIP], V[P], UfPJ, E[P]
4: 0->M; 0-+r0; 0-*rl; 0->-r2; 0-»-r3; 0->-r4
5: M+l+M
6: ent N[M],V[M]
7: rO+V[M]+rO; rl+N[M]*V[M]-*-rl; r2+N[M]+r2; r3+N [M]+2-*r3; r4+N[M]+2+r4
8: if M 9: (rl-r2*rO/P)/(r3-r2+2/P)+A; (rO-A*R2)/P+B
10: prt "B+A*N=U"; spc; prt "A=", A, "B=", B
11: spc
12: "":
13: "Part II: Percent Error Program":
14: fmt c2, f2.0, c4, f6.2, cl
15: 0->M
16: M+l+M
17: B+A*N [M ]-MJ [M]
18: (V[M]-U[Mj)/V[M]*100-^E[M]
19: wrt 16, "E[", M, "]=", EfM],
20: if M 21: end
Definition of Variables:
P = number of points
N = frequency in Hz
V = true velocity in cm/s
U = calculated velocity from curve determined in this program
E = percent error
M = loop counter
r = summation parameter
The calibration curve is of the following form:
B + A * N = U

148
Velocities at a Section
0: "Measure velocities at a Section by using Nixon Meter # 696"
1: dim A$[16], V[5,5], N[2], T[2]
2: fmt 2, c2, f6.2, c3
3: fmt 1, cl6
4: fmt c2, fl.O, cl, fl.O, c2, f6.2
5: ent T
6: prt "T=", T
7: T/2+T
8: wtb 2,21
9: 0+P
10: P+l-vP
11: 0+S
12: wtb 2,1
13: S+l+S
14: for 1=1 to 2
15; wtb 2,17
16: red 2.1, A$
17: val (A$[l,4]M[lj; val ( A$ [10] )-^T[ I ]
18: if I<2; wait T*1000
19: next I
20: 6(N[2]-N[1])/(T[2]-T[1])-N
21: P-*-R
22: if N<30; 1.026393N-16.654312-^V[R,S]
23: if N>=30; 0.659483N-5.648766+V[R,S]
24: if [R,S]<6; 0+V[R,S]
25: wrt 16.2, "N=", N, "Hz"
26: wrt 16, "V[", R, S, "]=", V[R,S], "cm/s"
27: spc
28: dsp "change the prob vertically"; stp; wtb 2,1
29: if S<5; gto 13
30: dsp "change the prob laterally"; stp
31: if P<5; gto 10

149
32:
spc
33:
stp
34:
end
Definition of Variables:
V = measured velocity in cm/s
T = averaging time
T. = first and last raw times (1/6 sec)
N = frequency in Hz
N.j = first and last counts of frequency
P = column loop counter
S = row loop counter
R,I = dummy counters

APPENDIX C
TABLES OF EXPERIMENTAL DATA

151
TABLE Cl: Experimental Data for Red Mangroves (without canopy)
Run
U
d
AH
f
k.
R'
F
Cn
a
a
e
a
e
r
D
No.
(cm/s)
(cm)
(cm)
(cm)
1
30.60
14.84
0.683
0.109
29.85
27188
0.25
0.698
2
32.87
15.65
0.818
0.117
33.24
30421
0.27
0.738
3
34.09
16.88
0.958
0.135
39.53
33430
0.27
0.824
4
37.53
18.39
0.820
0.102
33.23
39237
0.28
0.598
5
37.37
18.85
0.869
0.111
36.44
39779
0.27
0.643
6
38.74
19.84
1.105
0.125
41.86
42804
0.28
0.709
7
39.01
20.65
1.036
0.130
44.30
44363
0.27
0.720
8
22.40
18.21
0.359
0.125
39.18
23251
0.17
0.734
9
25.19
18.38
0.492
0.136
42.24
26327
0.19
0.796
10
30.49
18.87
0.722
0.139
43.75
32477
0.22
0.803
11
32.85
19.43
0.811
0.137
44.30
35752
0.24
0.783
12
34.18
19.91
0.940
0.149
48.16
37858
0.24
0.844
13
37.83
19.97
0.927
0.120
40.73
41998
0.27
0.680
14
42.45
20.79
1.140
0.121
42.10
48521
0.30
0.670
15
23.74
18.11
0.387
0.119
37.53
24542
0.18
0.702
16
26.69
18.36
0.586
0.144
44.15
27872
0.20
0.844
17
31.82
18.71
0.746
0.131
41.50
33686
0.24
0.760
18
35.57
19.24
0.933
0.133
43.07
38440
0.26
0.766
19
36.92
19.65
0.966
0.130
42.92
40510
0.27
0.741
20
39.04
20.13
1.098
0.135
44.84
43582
0.28
0.758
21
39.62
20.59
1.236
0.150
49.37
44960
0.28
0.832
22
23.78
23.13
0.332
0.121
45.42
29296
0.16
0.634
23
26.99
23.34
0.387
0.110
42.24
33465
0.18
0.576
24
31.01
23.54
0.527
0.115
43.89
38674
0.20
0.596
25
32.82
23.77
0.725
0.142
52.38
41206
0.22
0.732
26
36.13
24.09
0.926
0.151
55.39
45764
0.24
0.774
27
25.43
23.21
0,418
0.134
49.31
31393
0.17
0.701
28
27.29
23.45
0.406
0.114
43.48
33947
0.18
0.592
29
29.47
23.41
0.644
0.154
55.34
36619
0.19
0.805
30
32.88
23.72
0.753
0.146
53.66
41227
0.22
0.758
31
35.06
23.96
0.748
0.129
48.93
44258
0.23
0.664
32
32.52
14.71
0.880
0.123
32.94
28694
0.27
0.793
33
33.45
15.41
0.991
0.136
36.93
30597
0.27
0.859
34
36.07
16.71
1.123
0.141
40.38
35095
0.28
0.860
35
38.00
17.78
0.954
0.113
35.39
38737
0.29
0.672
36
39.45
18.65
1.034
0.T17
37.93
41660
0.29
0.684
37
40.57
19.35
1.196
0.132
42.86
44013
0.29
0.756
38
40.82
20.10
1.145
0.128
43.10
45541
0.29
0.722

152
TABLE C2: Experimental Data for Red Mangroves (with canopy)
Run
U
d
AH
r
k.
R'
F
Cr,
No.
d
a
e
a
e
r
D
(cm/s)
(cm)
(cm)
(cm)
1
20.59
18.21
0.361
0.148
44.85
21373
0.15
0.872
2
21.54
18.34
0.351
0.132
41.28
22467
0.16
0.776
3
26.46
18.82
0.587
0.149
46.22
28140
0.19
0.866
4
29.53
19.48
0.668
0.140
45.12
32200
0.21
0.799
5
30.59
19.79 •
0.714
0.141
45.87
33734
0.22
0.799
6
31.50
20.65
0.808
0.155
50.83
35819
0.22
0.862
7
33.86
20.98
0.869
0.146
49.16
38947
0.24
0.805
8
35.07
21.82
0.886
0.143
49.66
41483
0.24
0.771
9
36.96
22.20
0.975
0.143
50.39
44239
0.25
0.767
10
35.29
16.73
0.868
0.114
34.04
34372
0.28
0.695
11
36.88
18.32
0.989
0.127
39.90
38432
0.28
0.745
12
40.18
20.05
1.028
0.119
40.38
44735
0.29
0.669
13
42.11
21.27
1.134
0.124
43.77
4890&
0.29
0.681
14
44.62
22.10
1.190
0.119
43.47
55234
0.30
0.642
15
27.40
14.53
0.691
0.135
35.13
23947
0.23
0.874
16
27.36
15.28
0.788
0.160
41.59
24870
0.22
1.018
17
27.72
16.78
0.667
0.142
40.84
27063
0.22
0.867
18
29.85
18.10
0.703
0.137
41.93
30838
0.22
0.806
19
33.36
19.19
0.715
0.116
38.38
35980
0.24
0.668
20
35.93
20.33
0.860
0.126
42.71
40401
0.25
0.703
21
36.02
21.11
0.825
0.123
43.15
41615
0.25
0.676
22
29.86
15.81
0.688
0.121
34.28
27860
0.24
0.755
23
27.11
14.65
0.607
0.122
32.56
23841
0.23
0.786
24
26.12
13.79
0.551
0.114
29.27
21900
0.22
0.751
25
21.13
23.19
0.274
0.127
47.34
26078
0.14
0.666
26
24.64
23.34
0.389
0.133
49,37
30546
0.16
0.696
27
25.03
23.57
0.409
0.136
50.60
31246
0.16
0.709
28
28.19
23.77
0.502
0.133
49.90
35388
0.18
0.688
29
31.45
24.17
0.671
0.144
53.80
39934
0.20
0.741
30
31.54
24.57
0.700
0.151
56.37
40504
0.20
0.770
31
35.25
24.82
0.816
0.142
54.18
45579
0.23
0.720
32
20.60 •
23.05
0.236
0.114
43.16
25324
0.14
0.601

153
TABLE C3: Experimental Data for Black Mangroves
Run
U
d
AH
f'
k.
R'
F
Cn
3
a
e
a
e
r
D
No.
(cm/s)
(cm)
(cm)
(cm)
1
39.47
11.62
2.020
0.105
23.73
28800
0.37
1.063
2
42.65
12.17
2.034
0.094
22.28
32315
0.39
0.871
3
44.88
13.00
2.669
0.117
28.50
35839
0.40
1.103
4
50.85
13.81
2.985
0.106
27.58
42581
0.44
0.921
5
47.88
14.91
2.951
0.126
33.91
42589.
0.40
1.065
6
50.84
15.35
3.239
0.125
34.48
46242
0.41
1.028
7
50.88
16.04
3.124
0.125
35.64
47876
0.41
0.980
8
25.29
16.15
0.959
0.159
43.10
23971
0.20
1.288
9
29.06
15.95
1.097
0.136
38.02
27292
0.23
1.079
10
35.99
16.03
1.706
0.138
38.62
33912
0.29
1.097
11
40.44
15.86
2.076
0.132
36.88
37786
0.32
1.047
12
41.91
16.33
2.413
0.146
40.73
40041
0.33
1.153
13
47.27
16.10
2.515
0.118
34.04
44664
0.38
0.896
14
53.30
16.78
3.276
0.124
36.53
51897
0.42
0.923
15
26.65
16.15
0.955
0.142
39.72
25270
0.21
1.125
16
30.30
16.20
1.263
0.146
40.55
28790
0.24
1.157
17
36.95
16.15
1.590
0.123
35.38
35018
0.29
0.938
18
42.20
16.30
2.248
0.134
38.10
40268
0.33
1.038
19
45.07
16.21
2.520
0.131
37.23
42805
0.36
1.016
20
48.41
16.38
2.818
0.127
36.72
46319
0.38
0.975
21
50.01
16.50
3.094
0.131
37.86
48087
0.39
1.011
22
25.66
21.44
0.862
0.171
55.95
29979
0.18
1.059
23
29.57
21.32
1.140
0.169
55.40
34401
0.20
1.055
24
34.22
21.35
1.293
0.144
49.14
39855
0.24
0.861
25
36.75
21.26
1.628
0.156
52.11
42661
0.25
0.959
26
40.90
21.32
1.932
0.150
50.62
47572
0.28
0.909
27
27.56
21.41
0.915
0.157
52.63
32172
0.19
0.958
28
29.85
21.46
1.101
0.161
53.72
34883
0.21
0.988
29
32.52
21.24
1.243
0.152
51.13
37731
0.23
0.930
30
36.85
21.20
1.626
0.155
51.68
42697
0.26
0.951
31
39.86
21.13
1.911
0.155
51.60
46056
0.28
0.957
32
41.04
11.75
1.895
0.092
21.23
30228
0.38
0.874
33
42.91
12.14
2.240
0.102
23.94
32450
0.39
0.978
34
46.20
13.21
2.572
0.107
26.92 "
37364
0.41
0.973
35
48.57
14.10
2.803
0.111
29.26
41354
0.41
0.959
36
50.56
14.75
2.914
0.111
30.14
44592
0.42
0.909
37
52.11
15.30
3.243
0.119
32.95
47266
0.43
0.966
38
52.55
15.85
3.256
0.121
34.31
48986
0.42
0.950

154
TABLE C4: Experimental Data for Building Areas
Run
u
d
AH
a
a
f'
K
F
Cn
No.
(cm/s)
(cm)
(cm)
e
(cm)
e
r
D
Pattern no.
1
1
39.20
22.71
0.084
0.0013
0.00
57737
0.26
0.578
2
41.69
20.06
0.146
0.0019
0.00
55242
0.30
0.912
3
57.59
13.60
0.346
0.0016
0.00
54295
0.50
1.178
4
49.42
13.35
0.171
0.0011
0.00
45757
0.43
0.794
5
44.74
14.60
0.129
0.0011
0.00
44881
0.37
0.723
6
49.06
14.97
0.135
0.0010
0.00
50316
0.41
0.630
7
49.33
16.65
0.127
0.0010
0.00
55580
0.39
0.580
8
46.25
19.95
0.160
0.0016
0.00
60047
0.33
0.813
9
68.10
13.78
0.343
0.0012
0.00
64849
0.59
0.837
10
59.64
15.59
0.238
0.0012
0.00
63431
0.48
0.749
11
56.54
14.53
0.231
0.0012
0.00
56459
0.47
0.815
12
50.88
14.16
0.194
0.0012
0.00
49653
0.43
0.848
Pattern no.
2
1
43.85
19.25
0.602
0.0067
0.22
56082
0.32
1.704
2
42.43
21.26
0.557
0.0073
0.31
59096
0.29
1.662
3
45.50
17.89
0.785
0.0077
0.32
54618
0.34
2.084
4
46.97
15.94
0.714
0.0059
0.12
50944
0.38
1.805
5
44.98
15.13
0.769
0.0066
0.17
46553
0.37
2.132
6
42.12
14.75
0.624
0.0060
0.11
4 2 643
0.35
1.979
7
45.90
13.55
0.866
0.0065
0.14
43070
0.40
2.331
8
48.55
13.61
0.906
0.0061
0.11
45717
0.42
2.180
9
54.55
13.89
1.140
0.0062
0.12
52315
0.47
2.168
10
44.20
17.10
0.566
0.0056
0.10
50987
0.34
1.603
Pattern no.
3
1
35.44
13.58
1.105
0.0139
1.37
33305
0.31
3.327
2
39.70
13.70
1.404
0.0142
1.45
37612
0.34
3.365
3
44.58
14.02
1.959
0.0159
1.95
43111
0.38
3.714
4
40.83
15.10
1.518
0.0158
2.04
42198
0.34
3.404
5
43.91
15.55
1.834
0.0169
2.44
46566
0.36
3.545
6
44.88
16.74
1.789
0.0169
2.59
50800
0.35
3.282
7
43.52
18.58
1.801
0.0198
3.99
53975
0.32
3.467
8
41.79
20.77
1.509
0.0198
4.41
57043
0.29
3.104
9
40.65
22.91
1.519
0.0229
6.38
60295
0.27
3.252
10
43.33
18.58
1.856
0.0206
4.32
53732
0.32
3.605

155
TABLE C4: CONTINUED
Run
U
d
AH
f
k*
R'
F
C p.
a
a
e
a
e
r
D
No.
(cm/s)
(cm)
(cm)
(cm)
Pattern no.
4
1
55.29
17.44
9.082
0.0527
18.24
64170
0.42
8.107
2
39.11
20.65
3.041
0.0451
17.42
53102
0.27
5.356
3
39.68
16.05
4.554
0.0517
16.68
43180
0.32
8.039
4
31.35
14.74
2.218
0.0382
10.24
31687
0.26
6.349
5
26.25
13.70
1.509
0.0348
8.34
24870
0.23
6.205
6
39.33
14.13
3.464
0.0360
9.01
38241
0.33
6.318
7
36.61
14.99
2.935
0.0374
10.05
37544
0.30
6.142
8
38.68
15.61
4.285
0.0497
15.48
41085
0.31
7.936
9
42.35
16.66
5.697
0.0579
19.78
47568
0.33
8.776
10
31.54
21.32
2.565
0.0603
25.79
44006
0.22
6.912
Pattern no.
5
1
35.54
21.41
11.400
0.1951
72.61
49283
0.25
19.174
2
28.42
18.63
7.139
0.1769
60.67
35158
0.21
19.254
3
28.70
18.46
7.158
0.1724
59.13
35234
0.21
18.943
4
22.76
22.27
4.100
0.1910
74.69
32937
0.15
16.857
5
23.96
17.56
4.563
0.1548
52.59
28234
0.18
17.490
6
25.61
16.29
5.447
0.1498
48.01
28233
0.20
18.413
7
27.50
15.50
6.082
0.1368
42.80
28966
0.22
17.922
8
40.19
21.60
13.910
0.1795
68.83
55850
0.28
18.163
9
31.30
18.43
8.531
0.1691
58.15
38283
0.23
18.951
10
30.19
16.84
7.492
0.1483
48.93
34173
0.24
18.106
11
26.44
16.14
5.534
0.1411
45.51
28900
0.21
17.577
12
23.12
15.47
3.820
0.1248
39.83
24375
0.19
15.973
13
24.85
14.53
4.439
0.1176
35.83
24752
0.21
16.170
14
18.33
14.13
2.205
0.1068
32.23
17848
0.16
14.820
15
21.57
19.56
3.630
0.1686
61.43
27937
0.16
16.941
16
27.01
19.59
6.445
0.1868
65.83
34951
0.19
19.127
17
27.84
17.75
6.698
0.1662
55.74
33046
0.21
18.947
Pattern no.
6
*
1
40.38
14.46
0.328
0.0034
0.01
40160
0.34
1.136
2
48.21
15.09
0.439
0.0033
0.01
49801
0.40
1.059
3
45.36
19.30
0.356
0.0037
0.02
58138
0.33
0.943
4
29.09
13.20
0.196
0.0036
0.01
26661
0.26
1.320
5
36.63
13.64
0.225
0.0027
0.00
34582
0.32
0.951
6
54.94
14.54
0.651
0.0036
0.01
54913
0.46
1.215
7
52.76
15.15
0.569
0.0036
0.01
54692
0.43
1.147
8
50.69
17.57
0.536
0.0041
0.03
59863
0.39
1.150
9
49.11
17.19
0.428
0.0035
0.01
56918
0.38
0.982
10
48.11
20.71
0.505
0.0050
0.08
65499
0.34
1.177

156
TABLE C4: CONTINUED
Run
U
d
AH
f
ka
R'
F
Cn
a
a
e
a
e
r
D
No.
(cm/s)
(cm)
(cm)
(cm)
-
Pattern no.
7
1
42.54
21.86
0.421
0.0056
0.12
60663
0.29
0.830
2
38.89
20.47
0.397
0.0060
0.15
52444
0.27
0.946
3
45.16
18.53
0.398-
0.0040
0.03
55418
0.34
0.714
4
47.51
16.35
0.422
0.0035
0.01
52706
0.38
0.692
5
46.05
15.35
0.472
0.0039
0.02
48307
0.38
0.831
6
43.72
14.80
0.643
0.0057
0.10
44388
0.36
1.261
7
38.23
14.34
0.493
0.0056
0.09
37738
0.32
1.268
8
32.31
13.96
0.305
0.0047
0.04
31146
0.28
1.100
9
25.88
13.77
0.114
0.0027
0.00
24636
0.22
0.642
10
46.16
17.47
0.455
0.0042
0.03
54262
0.35
0.786
11
45.10
19.63
0.402
0.0043
0.04
58662
0.33
0.715
Pattern no.
8
1
50.93
13.79
1.334
0.0082
0.32
48530
0.44
1.456
2
45.63
13.64
1.081
0.0082
0.32
43055
0.39
1.472
3
38.67
13-.44
0.789
0.0083
0.32
36018
0.34
1.499
4
36.18
14.28
0.608
0.0077
0.27
35580
0.31
1.311
5
40.44
14.60
0.788
0.0081
0.33
40569
0.34
1.358
6
45.10
15.09
1.027
0.0088
0.42
46597
0.37
1.416
7
46.36
16.60
0.899
0.0077
0.31
52660
0.37
1.137
8
43.98
19.72
0.963
0.0109
1.02
57433
0.32
1.352
9
42.71
21.79
0.832
0.0109
1.11
60726
0.29
1.220
10
44.49
18.47
0.900
0.0094
0.63
54896
0.33
1.245
Pattern no.
9
1
52.89
14.28
2.201
0.0129
1.19
51977
0.45
1.775
2
48.78
14.02
1.894
0.0129
1.17
47173
0.42
1.800
3
41.30
13.60,
1.256
0.0116
0.87
38860
0.36
1.671
4
31.55
13.34
0.684
0.0107
0.68
29180
0.28
1.562
5
62.24
14.72
3.050
0.0131
1.27
62785
0.52
1.769
6
58.12
15.50
2.652
0.0138
1.52
61424
0.47
1.755
7
38.74
22.39
0.969
0.0158
2.88
56366
0.26
1.376
8
42,56
17.61
1.092
0.0120
1.19
50375
0.32
1.328
9
42.29
15.41
1.285
0.0126
1.20
44700
0.35
1.594
10
40.72
20.00
1.025
0.0137
1.87
53819
0.29
1.339

157
TABLE C4: CONTINUED
Run
U
d
AH
f'
k
R'
F
r
a
a
e
a
e
r
D
No.
(cm/s)
(cm)
(cm)
(cm)
Pattern no.
10
•
1
47.84
17.34
1.021
0.0088
0.48
55859
0.37
2.463
2
46.89
18.67
0.898
0.0086
0.47
58393
0.35
2.233
3
47.44
20.83
0.961
0.0098
0.79
64916
0.33
2.300
4
45.76
22.89
0.907
0.0108
1.12
67832
0.31
2.299
5
50.91
15.35
0.968
0.0066
0.17
53399
0.42
2.093
6
48.82
14.77
0.927
0.0066
0.17
49487
0.41
2.188
7
44.86
14.50
0.798
0.0066
0.17
44727
0.38
2.235
8
39.10
14.20
0.672
0.0072
0.22
38259
0.33
2.484
9
32.63
13.91
0.562
0.0085
0.36
31343
0.28
2.991
10
25.95
13.73
0.342
0.0081
0.31
24633
0.22
2.879
11
44.54
19.65
0.826
0.0091
0.60
57969
0.32
2.262
12
48.56
15.58
0.895
0.0068
0.19
51597
0.39
2.214
13
40.23
14.52
0.647
0.0067
0.17
40157
0.34
2.252
Pattern no.
11
1
25.04
13.83
0.439
0.0113
0.80
23991
0.21
2.640
2
30.15
14.04
0.626
0.0112
0.81
29207
0.26
2.595
3
36.18
14.38
1.107
0.0140
1.48
35797
0.30
3.178
4
40.24
14.71
1.402
0.0147
1.68
40615
0.34
3.249
5
44.12
15.01
1.640
0.0145
1.67
45353
0.36
3.152
6
46.69
15.50
1.824
0.0148
1.80
49386
0.38
3.118
7
46.77
16.83
1.718
0.0150
1.99
53185
0.36
2.899
8
44.52
18.99
1.545
0.0165
2.77
56275
0.33
2.834
9
42.61
21.11
1.615
0.0207
4.87
58986
0.30
3.186
10
41.93
22.21
1.397
0.0193
4.42
60580
0.28
2.826
11
44.62
18.87
1.607
0.0170
2.94
56066
0.33
2.938
Pattern no.
12
1
39.48
20.50
2.306
0.0335
11.18
53271
0.28
3.992
2
42.02
16.92
2.368
0.0256
6.04
48007
0.33
3.711
3
40.01
15.47
2.182
0.0241
4.98
42219
0.32
3.814
4
35.21
14.80
1.619
0.0222
4.12
35739
0.29
3.672
5
31.50
14.44
1.302
0.0219
3.90
31282
0.26
3.698
6
33.41
13.63
1.430
0.0203
3.18
31497
0.29
3.632
7
38.25
13.81
1.989
0.0217
3.69
36489
0.33
3.848
8
44.93
14.13
2.597
0.0208
3.47
43721
0.38
3.630
9
50.57
14.56
3.102
0.0201
3.30
50520
0.42
3.411
10
55.46
15.13
3.528
0.0195
3.24
57311
‘0.46
3.212

153
TABLE C4: CONTINUED
Run
No.
U
a
(cm/s)
d
a
(cm)
AH
(cm)
f
e
(cm)
R'
e
F
r
CD
Pattern no.
13 (Medium-Rise
Buildings)
1
38.55
20.99
2.264
0.0352
12.33
53079
0.27
3.277
2
39.37
19.01
3.355
0.0456
16.45
49751
0.29
4.719
3
38.88
15.84
3.087
0.0366
10.25
41871
0.31
4.555
4
34.85
15.05
2.408
0.0346
9.00
35608
0.28
4.518
5
27.12
14.38
1.450
0.0327
7.91
26818
0.23'
4.448
6
28.12
13.70
1.496
0.0301
6.61
26641
0.24
4.290
7
35.49
13.98
2.488
0.0317
7.34
34209
0.30
4.466
8
43.72
14.63
3.510
0.0304
7.13
43863
0.37
4.129
9
49.19
15.27
4.568
0.0320
8.05
51206
0.40
4.218
10
44.58
16.60
3.801
0.0355
10.20
50027
0.35
4.239
11
40.58
17.96
3.454
0.0420
13.96
48791
0.31
4.606
Pattern no.
14 (Medi
um-Rise
Buildings)
1
43.92
23.07
0.722
0.0073
0.34
65532
0.29
0.880
2
42.91
21.88
0.886
0.0090
0.64
61223
0.29
1.141
3
43.89
19.68
1.044
0.0093
0.63
57202
0.32
1.305
4
46.23
17.63
1.043
0.0076
0.30
54754
0.35
1.192
5
48.30
15.67
1.162
0.0070
0.21
51499
0.39
1.233
6
45.43
15.14
1.153
0.0076
0.27
47060
0.37
1.389
7
42.26
14.81
0.939
0.0070
0.20
42932
0.35
1.312
8
36.74
14.30
0.741
0.0071
0.20
36174
0.31
1.373
9
30.46
13.89
0.391
0.0053
0.07
29216
0.26
1.059
10
23.89
13.63
0.325
0.0070
0.19
22529
0.21
1.434
Pattern no.
15 (Medium-Rise
Buildings)
1
23.08
13.73
0.355
0.0083
0.33
21916
0.20
1.677
2
28.21
13.79
0.631
0.0099
0.56
26886
0.24
1.991
3
34.90
14.10
1.058
0.0110
0.78
33929
0.30
2.176
4
39.24
14.31
1.362
0.0114
0.86
38654
0.33
2.214
5
43.12
14.62
1.707
0.0120
1.01
43282
0.36
2.291
6
47.77
15.32
2.018
0.0121
1.07
49981
0.39
2.196
7
45.82
17.90
1.742
0.0130
1.49
54995
0.35
2.023
8
43.96
19.94
1.676
0.0149
2.29
57935
0.31
2.084
9
42.60
22.05
1.677
0.0174
3.50
61195
0.29
2.187
10
43.27
23.44
1.709
0.0181
4.02
65424
0.29
2.141
11
46.85
16.65
1.713
0.0115
1.01
52769
0.37
1.920

159
TABLE C4: CONTINUED
Run
U
d
AH
f
ka
R'
F
cn
d
a
e
e
r
D
No.
(cm/s)
(cm)
(cm)
(cm)
Pattern no.
16 (Medium-Rise
Buildings)
1
44.26
22.91
1.442
0.0143
2.32
65651
0.30
0.975
2
43.21
21.00
1.519
0.0147
2.29
59538
0.30
1.092
3
44.21
19.65
1.635
0.0142
2.01
57541
0.32
1.133
4 .
45.92
17.60
1.606
0.0118
1.13
54329
0.35
1.047
5
47.63
15.76
1.485
0.0092
0.51
51111
0.38
0.912
6
47.11
15.01
1.911
0.0115
0.93
48419
0.39
1.205
7
44.40
14.77
1.632
0.0109
0.80
44995
0.37
1.161
8
40.65
14.56
1.397
0.0110
0.81
40665
0.34
1.188
9
34.78
14.14
0.977
0.0103
0.65
33905
0.30
1.139
10
28.07
13.85
0.575
0.0091
0.45
26867
0.24
1.030
Pattern no.
17 (Medium-Rise
Buildings)
1
27.15
14.19
1.506
0.0261
5.35
26541
0.23
2.880
2
22.96
14.01
1.058
0.0254
5.02
22190
0.20
2.831
3
31.63
14.44
2.156
0.0278
6.07
31399
0.27
3.028
4
35.56
14.76
2.757
0.0286
6.49
35978
0.30
3.056
5
40.91
15.32
3.732
0.0300
7.25
42749
0.33
3.111
6
45.13
16.35
4.408
0.0307
7.96
49931
0.36
2.995
7
43.48
18.18
4.246
0.0352
10.87
52825
0.33
3.069
8
42.25
19.54
4.225
0.0396
13.79
54640
0.31
3.205
9
41.13
21.11
3.976
0.0421
16.14
56867
0.29
3.148
10
40.52
22.16
3.799
0.0433
17.48
58393
0.27
3.077
Pattern no.
18 (Medi
um-Rise
Buildings)
1
42.12
21.55
1.823
0.0189
4.14
59330
0.29
1.099
2
42.89
20.63
1.936
0.0187
3.87
58188
0.30
1.132
3
44.10
19.01
1.765
0.0150
2.22
55778
0.32
0.988
4
45.13
17.79
2.155
0.0165
2.60
53860
0.34
1.162
5
45.93
16.40
2.302
0.0158
2.20
51051
0.36
1.210
6
45.29
15.57
2.277
0.0153
1.96
48047
0.37
1.239
7
43.11
15.24
2.058
0.0150
1.83
44910
0.35
1.238
8
39.54
14.83
1.740
0.0148
1.72
40210
0.33
1.248
9
33.99
14.33
1.377
0.0154
1.83
33523
0.29
1.341
10
27.74
14.02
0.784
0.0129
1.17
26842
0.24
1.149

160
TABLE C4: CONTINUED
Run
No.
U
a
(cm/s)
d
a
(cm)
AH
(cm)
ka
(cm)
%
F
r
CD
Pattern no.
19 (Medium-Rise
Bui1 dings)
1
26.40
14.45
2.054
0.0381
10.04
26226
0.22
3.313
2
29.03
14.60
2.494
0.0386
10.30
29107
0.24
3.322
3
33.01
14.92
3.267
0.0396
10.90
33724
0.27
3.356
4
36.66
15.37
4.135
0.0414
11.91
38407
0.30
3.432
5
40.33
16.00
5.123
0.0435
13.24
43748
0.32
3.493
6
43.36
16.42
5.493
0.0410
12.45
48085
0.34
3.229
7
43.40
17.47
5.499
0.0435
14.27
50844
0.33
3.202
8
41.79
18.90
5.435
0.0499
18.40
52448
0.31
3.381
9
40.98
20.25
5.190
0.0530
21.06
54616
0.29
3.329
10
40.29
21.80
5.322
0.0599
26.05
57193
0.28
3.492
Pattern no.
20 (Medium-Rise
Buildings)
1
40.13
22.86
2.626
0.0315
11.15
59405
0.27
1.440
2
40.88
21.69
2.811
0.0311
10.41
57880
0.28
1.497
3
42.13
20.16
3.049
0.0297
9.09
56035
0.30
1.544
4
42.79
18.68
3.230
0.0285
7.93
53248
0.32
1.603
5
44.63
17.12
3.754
0.0281
7.15
51445
0.34
1.731
6
44.88
16.02
3.670
0.0255
5.71
48810
0.36
1.686
7
43.01
15.56
3.259
0.0241
5.02
45606
0.35
1.636
8
39.76
15.15
2.752
0.0234
4.63
41172
0.33
1.623
9
35.17
14.69
2.093
0.0222
4.09
35459
0.29
1.584
10
29.28
14.31
1.380
0.0207
3.49
28847
0.25
1.511
Pattern no.
21 (Medi
um-Rise
Buildings)
1
28.50
14.71
3.397
0.0545
16.49
28735
0.24
3.909
2
22.48
14.28
2.169
0.0550
16.26
22098
0.19
4.028
3
31.16
14.89
4.154
0.0559
17.19
31751
0.26
3.990
4
36.33
15.36
5.538
0.0553
17.44
37986
0.30
3.894
5
41.25
15.94
6.977
0.0546
17.70
44467
0.33
3.781
6
43.57
16.86
7.629
0.0559
19.12
49301
0.34
3.677
7
42.26
18.24
7.768
0.0654
24.62
51244
0.32
3.943
8
40.84
19,71
7.519
0.0735
29.95
53036
0.29
4.050
9
38.99
21.12
6.718
0.0776
33.66
53780
0.27
3.937
10
38.21
22.07
6.677
0.0836
37.73
54746
0.26
4.047

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BIOGRAPHICAL SKETCH
Shang-Yih Wang was born on April 13, 1952, in Taipei, Taiwan.
He graduated from Provincial Wu Lin High School in 1970. He then
enrolled at Taiwan National College of Marine and Oceanic Technology.
He received the degree of Bachelor of Science with a major in
oceanography in June of 1974.
After two years military service as a logistic officer in the
Army, he came to the United States in September, 1977, to continue
his studies in the Coastal and Oceanographic Engineering Department at
the University of Florida. He was awarded the degree of Master of
Science and a graduate school fellowship in August, 1979. Following
this, he joined the Hydraulic Laboratory, Department of Civil
Engineering in pursuit of the doctoral degree. He had worked as a
graduate research assistant during the six years of graduate study.
165

I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philosophy.
B. A. Christensen, Chairman
Professor of Civil Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philosophy.
T. V. Chiu, Cochairman
Professor of Coastal and Oceano¬
graphic Engineering
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for
the degree of Doctor of Philosophy.
D. P. Spangler,
Associate Professor of Geology
This dissertation was submitted to the Graduate Faculty of the
College of Engineering and to the Graduate School, and was accepted
as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
December 1983
/b-S-U~f G. -
Dean, College of Engineering
Dean for Graduate Studies and
Research

UNIVERSITY OF FLORIDA
3 1262 08556 9860




PAGE 1

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