Friction in hurricane-induced flooding


Material Information

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


Subjects / Keywords:
Hurricanes   ( lcsh )
Floods   ( lcsh )
Flood forecasting   ( lcsh )
bibliography   ( marcgt )
non-fiction   ( marcgt )


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

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
oclc - 11479424
System ID:

Full Text








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 extended 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.









Storm Surge Prediction

Objectives of Present Work


Mangrove Areas
General View of Mangroves in Florida
Sampling of Mangroves

Developed Areas
High-Rise Building Areas
Medium-Rise Building Areas
Residential Areas


Hydrodynamic Equations for Storm Surges

Wind Shear Stress
Wind Velocity Profile in Vertical
Proposed Wind Shear Stress on Obstructed Areas

Bed Shear Stress
Post Approach, Friction Factor for Surges in
Unobstructed Areas
Proposed Approach, Friction Factor for Surge
in Obstructed Areas








Distorted Model for Buildings 45

Undistorted Model for Mangrove Stems and Roots 49

Distorted Model for Canopy 50


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


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


Mangrove Areas 84

Building Areas 95

Determination of Drag Coefficient 110
Drag Coefficient-Building Density Relation 110
Drag Coefficient-Disposition Parameter Relation 113



Mangrove Areas 120

Developed Areas 121

Ocean Bottom 123

Forested Areas 127

Grassy Areas 128

Conclusions 129








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

Al. Parameters of Sampling Red Mangroves 141

A2. Parameters of Sampling Black Mangroves 142

A3. Characteristics of Canopy of Red Mangroves 143

C1. 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

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






Model Setup for Red Mangroves

Model Setup for Black Mangroves

Stems and Roots of Red Mangroves

Stems and Roots of Black Mangroves

25. Overview of Setup for Mangroves

26. Setup of Model Equivalent of Canopy of Red Mangroves

27. Distribution of Leaf Stripes in the Model

28. Building Patterns Designed for the Tests

29. Relation Between U, and U

30. Designed Building Patterns in the Tests

31. Relation Between f and R' for Red Mangrove Areas
(Without Canopy) e e

32. Relation Between CD and R' for Red Mangrove Areas
(Without Canopy) e

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

34. Relation Between f'
(with Canopy) e

35. Relation Between CD
(with Canopy)

36. Relation Between f'
Areas (with Canopye)

37. Relation Between f'
38. Relation Between CD
39. Relation Between f
Mangrove Areas e

and R' for Red Mangrove Areas

and Re for Red Mangrove Areas

and Water Depth d for Red Mangrove

and R' for Black Mangrove Areas
and Re for Black Mangrove Areas
and Water Depth d for Black
and Water Depth d for Black

40. Relation Between f' and R' for Building Areas
e e








and R' for Building Areas
and Water Depth d for Building

and Density m for High-Rise

and Density m for Medium-Rise
itial Areas

Definition Sketch

and Sd/D in Aligned and Staggei

41. Relation Between CD

42. Relation Between f'

43. Relation Between CD
Building Areas

44. Relation Between C.
Building and Residpe

45. Position Spacings.

46. Relation Between CD

47. Relation Between CD

Al. Field Recorded Data

A2. Field Recorded Data

A3. Field Recorded Data

A4. Field Recorded Data

A5. Field Recorded Data

A6. Field Recorded Data

A7. Field Recorded Data

A8. Field Recorded Data

A9. Field Recorded Data


Red Mangroves (Area #1)

Red Mangroves (Area #2)

Red Mangroves (Area #3)

Red Mangroves (Area #4)

Red Mangroves (Area #6)

Black Mangroves (Area #8)

Black Mangroves (Area #9)

Black Mangroves (Area #10)

Black Mangroves (Area #11)





























A area

A ,A leaf area for prototype and model, respectively

B horizontal width

CD drag coefficient

Cf skin friction coefficient

D average diameter of obstruction

D ,D vertical depth for prototype and model, respectively

d water depth

dl,d2 water depth at section 1 and section 2, respectively

da average water depth of d1 and d2

d diameter of pipe

FD drag force

Fe elastic force

Ff skin friction

F gravitational force

Fi inertial force

Fr Froude number =/U / gda
Fs surface tension force

Fv viscous force

f Darcy-Weisbach friction factor based on diameter of the pipe

f' friction factor based on hydraulic radius

fe equivalent friction factor

fq total friction factor

g gravitational acceleration

H vertical depth above the vertex of Thomson weir

h protruding height of obstructions above water surface

i indicial functional parameter

K wind stress coefficient

Kd drag force scale

Kf skin force scale

K gravity force scale

K. inertial force scale
Ks shear force scale

k equivalent sand roughness

ka apparent roughness

L horizontal length

ap ,m leaf length for prototype and model, respectively

m density = no. of obstruction elements/area

m,p indicating the subscripted parameters for model and prototype,

N total number of obstruction elements

Nd vertical length scale

N horizontal length scale

Nf force scale

Nt time scale

n Manning's n

P wetted perimeter of flow cross-section

p pressure

PS pressure on the water surface
Q discharge from the Thomson weir

qx, qy discharge per unit width
R hydraulic radius = A/P

R1,R2 hydraulic radius at section 1 and section 2, respectively

Ra average hydraulic radius of R1 and R2

Rd reduction factor for wind stress

Re Reynolds number based on depth = Uada/v

Re Reynolds number based on hydraulic radius = UaR a/

R* wall Reynolds number = ufk/v

Re,x Reynolds number = Ux/v

r radius of the pipe

S slope of energy grade line = AH/L

Sd corner to corner distance between the roughness elements in
adjacent transverse raws
S longitudinal spacing between two successive roughness elements

St 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

U1, U2 spatial mean flow velocity at section 1 and 2, respectively
Ua average spatial mean velocity of U1 and U2

Ux, Uy spatial mean flow velocity in x and y directions, respectively

U. free stream velocity
u,v,w instantaneous components of the water velocity in the x, y, z
coordinate directions, respectively


U', V'






w wM

x, y, Z





TI 1T2


K. :




Tbx' 'by


time-mean velocity in the direction of flow

turbulent velocity fluctuations in the x and z directions,

friction velocity based on bottom friction

friction velocity based on total friction

time-mean wind velocity at the elevation z above water surfa

critical wind velocity

time-mean wind velocity at the elevation 10 meters above wat

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 nld and n2d from bottom to the total
depth d, respectively


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 = T

bottom shear stress in the x and y directions, respectively









hydrodynamic drag

viscous stress

wind shear stress on open area

wind shear stress on obstructed area

Reynolds stress

obstruction correction factor

the earth's angular velocity

wind friction velocity = /a
s a
Laplacian operator

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



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,

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



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


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).

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


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).

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)


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


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

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.


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

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 frequency and the
amount of tidal flushing and freshwater runoff from the upland. The

coastal fringe forest including red and black mangroves, which are the

FIGURE 1: Prop Roots of Red Mangroves

Am wx.nin -

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

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
of leaves in a unit volume (1 ft.). Dimensions of leaves were also

FIGURE 3: Red Mangrove Area (Sampling Area #4)


FIGURE 4: Black Mangrove Area (Sampling Area #7)



01 000 I

O o o 0






0 c





o -- .
o I
0 I


0 I

0 0

0 0
0 0
o *




0 .


) 3 6 9 I



FIGURE 5: Field Data Record for Red Mangroves

*0 0 0 0
o o o
1 *o o ,o
O 1

o 0

o o

S oa o
0 0

0 0




AREA # 7

o I





oI o 'o
0 I

i I


0 Stem

FIGURE 6: Field Data Record for Black Mangroves

j -





9~--- -- -~



TABLE 1: Average Parameters of Sampling Mangroves

Red Black
Average Parameters
Mangroves Mangroves

no. 4 12
Main- (12 ft.)
tem diameter 6.0 in. 3.1 In.

height 10.0 ft. 11.2 ft.

% occupied 0.84 % 0.49%

2 13
(12 ft.)
Stem diameter 2.0 in.

height 18.0 in.

% occupied 0.19 %

Canopy height 8.0 ft. 9.4 ft

Sno 81 10,800
(12 ft.)

diameter 1.0 in. 0.25 in
height 18.0 in. 6.0 in.

% occupied 0.25 % 2.39%

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.

FIGURE 7: Measurement of Density of Canopy

L5' _L5' _L5' L Stem Prop Root
#1 '#2 1 #37

FIGURE 8: Section View of Survey Area (Red Mangroves)

TABLE 2: Average Characteristics of Canopy

Sections #1 #2 #3
Parameters # 1- 2 #

Density no. of leaves) 10 5 2
Density( 3 ) 1 5 2
Leaf Size 2" x 1" 3" x 1.5" 4" x 1.75"

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

larger than 10,000 ft An estimation of the dimensions and densities

of these high-rise buildings from Broward and Dade Counties are listed 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

Sea Side


- width

Sea Side
.". ...-.... : .. ..-


j---width ---



.-. -,-...


Top View of Building Shapes on Coastal Areas

--r. -



TABLE 3: Average Parameters of High-Rise Buildings

Range density % of County
No. length width no. land occupied
(ft.) (ft.) 1000'x500' 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


TABLE 4: Average Parameters of Medium-Rise Buildings

Range density % of County
No. length width no. land occupied
(ft.) (ft.) (1000'x500' 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

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 ft2. 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.


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 2w(sine)q = d s gd as 1 (1)
2w (sine)q p (sx bx (1)b

_ay + 2m(sine)qx d as gd as (+ ) (2)
at x a+y p sy by

ax y as = (3)
ax ay at

where t is time, w 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 qy 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) (4)

q (x,y,t) = u(x,y,t) dz (5)

q (x,y,t) = v(x,y,t) dz (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, T and bed shear stress,

T, for coastal areas in tropical storm induced flooding are presented

in the following sections.

Wind Shear Stress

In general, the wind stress (r ) on a water surface may be expressed

in terms of the mean wind speed (W10) at anemometer level (10 meters above

water surface), the air density (p ) and a wind-stress coefficient (K),


T = Pa K W2 (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 Ts is given as

Ts = a [K1W0 + K2(W10 )2] (8)

where K1 and K2 are the constants and W 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 = p0.0000026 + (1.0 7.0)2 x 0.0000030] W0 (9)

where W10 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 W10) x 10"3 (10)

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 W10) 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) = In () (12)

where W(z) is the wind velocity at a height z above mean sea level, is

the von Karman's turbulence constant, zo is the dynamic roughness of
the logarithmic velocity profile, and w, = (r /pa )/2 = friction velocity.




g 2 0 *< K=(0.8+0.065Wo)x103

0 20 40 60


FIGURE 10: Wind Stress Coefficient over Sea Surface

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

(z) = 2.5 In (- + 1) (13)
W* 0

By substituting equation (7) into equation (12), an equation for

determining the dynamic roughness, z is obtained

(z) = /K (2.5) In (- + 1) (14)
"10 zo

Applying the boundary condition

W(10) = W10 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 = 10 (15)
(2.5 Jv
e 1

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

I //////bstruction
Wind b o


FIGURE 11: Plan View for Wind Stress over an Obstructred Area

__ .. .;.. .: .. .... .. ...-' .:.. 7 .7 -..-.

FIGURE 12: Elevation View for Wind Stress over an Obstructed.Area

FD CPD h W2(z) dz (16)

where CD is the drag coefficient. Substituting equation (14) into

equation (16) gives

(2.5)2CDp DK rh 2
F a-W1( + 1)]2 dz
S2 10


FD = 3.13 C'paD W2 K f(h + z )1n( -+ 1)[1n( -+ 1) 2] + 2h } (17)
o o

Recall the equation for wind stress on an open water surface, i.e.,

Ts = K p WO .(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 (FD) 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

so = Rd s = Rd K a W0 0 < R <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.

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 Xxt, i.e., (Gee and Jenson, 1974)

T x
Rd -- 2 (19)
[(1 mSBD2 + mFD] x s

in which m = density = N/ 2, 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 p W 2
Rd a 10
(1 maD2)Kp 0W + 3.13 (mCD aDW20) K{(h+Z )ln(-L+ 1)(In( h +1)-2]+2hI
o0 o0

Rd (20)
(1-m8D2) + 3.13 (mDCD) {(h+z ) In(h-+1l) [In(h +1)-2] + 2h}
0 o

where K = (0.8 + 0.65 W10) x 10-3. W10 is in meters per second.

Based on the result presented in equations (18) and (20), the

reduction factor Rd 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 CD in equation (20) for

vegetations and buildings will be determined and discussed in Chapter


Bed Shear Stress

A space averaged bed shear stress, Tb, usually can be expressed as

To, i.e.,
Tb = T = yRS = yR (21)

in which To =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

U2 L
AH = f .(22)
2 0 d

where f = friction factor based on depth; U = spatial mean flow
velocity; do = diameter of the pipe. Since do = 4R, the above equation

may be written for an arbitrary cross section as

AH = f' U (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:

f'pU|Uu f'pu~uU
SfplUIUx T = xL y=JIU (24)
bx 2 by 2

or in terms of volume transport

f'plqq = qq (25)
Tbx 2d2 by 2d2 (

where f = 4f'; U = /U x2 + U 2; /q q = q2; Ux and Uy = spatial
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

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

Re = Ud/v = 104 to 106 has been widely used in flow fields and will be

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

(ufk/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, uf = friction velocity = T/p.

In range III (rough turbulent flow) the thickness of the viscous

sublayer 6 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

S= 8.48 + 2.5 In Z = 2.5 In 29.7z (26)
uf K k

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

U 8 29.73z
u= 8.48 + 2.5 In ( z + 0.0338) = 2.5 In ( 29 + 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

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)

U 29.73
-= 2.5 ln[973 (0.368d) + 1]
uf k
U d
--= 2.5 In [10.94 -+ 1] (28)
uf k
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 = V/F7p = /gRS, leading to the result

U 2 1
Uf f


Solving equation (29) for f' and introducing equation (28) gives the

following expression for the friction factor

f, =0.32
[n(10.94d + )]2


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 R
S1.171 + log R (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)
= / (32)
n k6

given k in meters, or in the English units

1.486 8.25 ,/-
S1866 (33)
kn 6

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

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 1i1 be the velocity at nl depth, that is,

at a distance nld from the bottom of a wide rough channel, where d is the

depth of flow. By equation (26), the velocity may be expressed as

ul 29.7nld
f 2.5 in k (34)

Similarly, let i2 be the velocity at n2 depth; then

u2 29.7n2d
2.5 In k (35)

Subtracting equation (34) by equation (35) and solving for uf,

u2 U1
uf = 2 (36)
2.5 In -

Introducing equation (36) into equation (35) and solving for k,
k = (37).

u2- U1

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, T in the

direction of flow may be written as equation (21), i.e.,

To = 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 theshearstress

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.

&Ii A






cn -

* r-

C 0

-0 u--







0 r_
*r- *


s- *

" ..,

' 0

du U2
yRS = p (1-e) pu'v' (1-) + mCDdDy g- (38)

(To ) (Tt) (Td)
Viscous Turbulent Hydrodynamic
Stress Stress Drag

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, CD = 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 = -pd'v (1-e) + mC dD (39)

(To) (Tt) (Td)

The turbulent stress, T, 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

-pu'v' = = To 2 (40)
1 E

where d /2
1 /2
S= ( ) 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 Td 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),

To. 2 1/2 T 1/2
f ( ) = ( -- ) = uft (42)

where uft is the friction velocity based on the total bed shear stress
To, i.e., uft = V7T.
Similarly, introducing f and ft 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
-u ( ,) for the obstruction-free area (43)

U 2 /2
and = ( ) for the total obstruction area (44)

Eliminating U from the two equations above, relation between f' and fV
is given as

f' = f 2


Substituting equations (40), (42) and (44) into equation (39), the
equilibrium equation becomes

S= j 2 0 ^U2
PR 2t (l-e) + mCDdDy 2

or HU2
YR = f'U2 (1-e) + mCDdDv (46)
'T 2 D 2g
The head loss, AH, then may be written as
2 2
AH = f U L (l-E) + mCdD L (47)
2g R D 2g R
U2 L
and AH = [f'(l-E) + mCDdD] (48)

which is the form of the Darcy-Weisbach formula shown in equation (23).

An equivalent friction factor, fe, which includes the effects of bottom

friction and form drag based on equation (48) is introduced

f; = f' (1-e) + mCDdD (49)

where, according to equation (30),

f= 0.32
L n ok
[In (~ d + 1)]2
Therefore, the equivalent friction factor, fe, 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.,

1 R
1.171 + eog R
4f-k a


The formula proposed in Chapter III for the friction factor

(equation (49)) has left some unsolved questions: What is the drag

coefficient (CD) 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,

velocity, is similar to the corresponding quantities in the prototype,

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

(F (F ) (F ) (Fe) (Fs)
==TF T = TF (50)
m m m m

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,


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

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:
Length Scale (horizontal): N = ~-= -L (51)
m m

Depth Scale (vertical): Nd = (52)

Time Scale: N = -p- (53)

Force Scale: N = -F (54)
m (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

K g L BpD (D /L) p g 2
K p() (M) N N (55)
9 P9m LmBmDm (Dm/Lm) pm Nd

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

Fi = pu'' (area) (56)

and the inertial force scale can be written as

L D r--
p( -))3
P T'')''( p) p N N
K. = LP T () Nt2N (57)
pm ()( mm)(LmBm) m Nt

In order to have dynamic similarity between model and prototype,

Kgravity should be equal to Kinertial, i.e., equation (55) should be the
same as equation (57). This condition is expressed by

g N
( )Nd =

or Nt (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)

Fd = Td A = mCDdDp 2- A (59)

The drag force scale in the flow direction can then be written as

d (L ) (CD) pDpBp P(m )2LpBp

K CD pp) (60)
(CD) 2
or Kd = ) NdN (60)
(CD m d t

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 = NI/(Nd)/2, into
equation (60) gives

(CD) = (CD) (61)
p m

Shear forces generally may be expressed by the Darcy-Weisbach form

U2 U2
Fs = fy U A = fp A (62)

where f is the friction factor: and can be substituted by the equivalent
friction factor, fe, or bottom friction factor, f', later for the present
use. The shear force scale is given as

(F) fpp P 2LpBp
K P T- pp
s (F) fm( )2L mB
m m) () ( m

f p (t) (63)
or. K= (E) ('.2P) N 1 2 (63)
m Pm t

Dynamic similarity requires that Ks = K Substituting the time

scale, equation (58) into the required equality, i.e.,

f p 2 p (64)
(m) ()N4 = ()( )N Nd (64)

gives the expression for the scale ratio of the friction factor

f N
S= d (65)
fm NI

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, Nd/N in the Froude distorted model. The distortion

ratio usually is defined as

D = (66)
r N

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, = Nd, equation (58) reduces to

Nt = (N )1/2 (67)

which is the time scale for an undistorted Froude model. Both the

dimensionless coefficients CD 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
(Rexcrit ) ( ) (68)
e,x crit v crit

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
(--) = 520 (69)
v crit

where 6 = displacement thickness and

6 = 1.7208 (-) (Schlichting, 1979) (70)

Combining the last two equations give (Re,x)crit = 9.1 x 104.
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
( crit = 3.5 x 105 to 5 x 105 (Schlichting, 1979) (71)
v crit

Similar experiments done by Schubauer and Skramstad in 1947 also yielded
results which indicated that the critical value of Re,x is in a range
from 9.5 x 105 to 3 x 106 depending on the relative intensity of the
free-stream turbulence, (1/U )(- ur/3j/2 (Hinze, 1975). Therefore, the
minimum value of (R e,xcrit is chosen as 3.5 x 105 for the present study.
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 (Re,x)crit of about
2.98 x 105, which is still lower than but near the minimum value
(Re,x)crit. This shows that a turbulent boundary layer has very little
chance to be formed over such a short length, and that a laminar
boundary layer should prevail over the entire leaf area.

Skin friction can be expressed in terms of a dimensionless skin
friction coefficient, Cf, times the stagnation pressure, pU2/2, and area
of the plate, A, as follows

F = Cfp A (72)

in which, for a laminar boundary layer,

Cf 1.328 (73)

where R, = Ut/v denotes the Reynolds number formed by the product of the
plate length and the free-stream velocity (Schlichting, 1979). The skin
friction scale in the flow direction may be written as

L / 2 L 2
(Ff) ( ) p (T) AP
K f= = PP P
(Ff) L /M2 L 2
m V m (!) Am
mm m

or Kf= ( () (Nt) (NJ i ( ~ A) (74)

where k and A are the leaf length and leaf area, respectively. Following
the undistorted Froude law, Nt = (N Y2 p ad 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

/4 -/2 A
Kf = (N ) () () (75)
S .m m

For dynamic similarity, Kf has to be set equal to K i.e.,

/4 Y-/2 A 3
(Ni) () (2) = (Ne) (76)
m m

giving the length scale

A2 -9/2
P (Lk) (N,) (77)
Lm Am z

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.


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 m3/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.




0614 r =

S--4 --

Oo a,
-I' LL

0 o .
~ 0 I0- 0
8s 4

Q 0

E E 0 _

4 a,, 5 -

* cx o,
0.03~~ ~~ C: iiue*r


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
U1 U2
d + d + +AH
1 2g 2 2g

or AH = (d1 d2) + ( U U2 ) (78)

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 = f L (79)
e 2g Ra

R + R
in which Ua = (U1 + U2) and Ra 2 the equivalent friction
a 2 t en a f t
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.


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.lmm 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 00 to 500C (320 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.


0 I 2cm

FIGURE 16: Novonic Nixon Velocity Meter


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 for 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-like

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,

FIGURE 17: Input Box of Data Acquisition System

HP 9825A Programmable Calculator


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.


Flow S

S Electric
D Point Gage -Flume Bed

Valve -Tube -S iphc

FIGURE 19: Setup of Water Depth Measuring Device

FIGURE 20: Point Gage and Tube

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 Nd = 10) is selected

through the entire study for both distorted and undistorted models.


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.,

S A2 9/2
m m

in which N, = Nd = 10, a 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 a and a width w.

Equation (77) is then reduced to

( p ( ) O 10 (80)
m m

By choosing 1:10 for the length scale for this study, the width scale of

the leaf becomes

-p= 10 7/ 56


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

This part includes the three previously discussed kinds of

buildings: high-rise, medium-rise and residential buildings. Table 6
shows a summary 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

Section No. (1) (2) (3)

prototype length () 2.00 3.00 4.00
(in) width (w) 1.00 1.50 1.75

Model length (k) 0.20 0.30 0.40
(in) width (w) 0.018 0.027 0.031

p Om 10 10 10
Wp/m 56 56 56






r- r-.

C0 O 0
co cD CM

-C 0 0 O
4- L LO LO
O *- ,. r-
C)/C C
C *- -
O)-.-E -C O 0
E 4-) ) L) U)
o ,C L LA

> 00
m r-

CO *r-

1.0 t o
ci C%1 e

en to o
0 w, LO O

> CE U -0 CJ 0 0


0 I : "
o r0 o

0 M, 0 tO

o ..' a o o 0
eas 0 om n
&. 4- C 4- )-
10 E
E *r-_____
X C .
0@a- 4) Ln Cn( C4J
SL 2- 01 C 0 10


0 -0

m -
a) **- *1
C0 C 4 *-

i- *- *- 3-

>9 *- 4-' .C *y r


U U 0r

301 20







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


- 15"

0 Main-Stem


0 Prop Root

FIGURE 21: Model Setup for Red Mangroves

0.3" 0

0 0 0

0 0

n= 432 o o 3"

15 -1

o Stem

SAir Root

FIGURE 22: Model Setup for Black Mangroves


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 %

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


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












FIG R ,2: v 'iI of. S

FIGURE 25: Overview of Setup for

Setup of Model Equivalent of Canopy of Red Mangroves



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.


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.

% )



x x

(n x x

x 0.2" x 9.6"

0 0.3" x 9.6"

a 0.4" x 9.6"

FIGURE 27: Distribution of Leaf Stripes in the Model


2.87' 0
10 0 DU

/00 0 00 0



0 0 0 0




FIGURE 28: Building Patterns Designed for the Tests


U (9)



---- 8' --

0 0

D 0


O0 O














I00I 00



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

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 fudy. 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.,

Q = 2.840 H2 (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

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 U 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



Relation Between U and U
n w


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

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 (Re = UaRd 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 (Re) 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.

(1) (2)

(5) C6)

(10) (11)

(7) (8)

(12) (13)

(15) (16)




FIGURE 30: Designed Building Patterns in the Tests





(3) (4)


The data obtained from each experiment include two water depths,

dl and d2, and two depth averaged velocities, U1 and U2. Other

parameters, such as the energy loss AH, equivalent friction factor

fe, apparent roughness ka, Reynolds number R Froude number Fr

(U a//a) and the averaged drag coefficient CD are calculated using

their specific definitions as given by equations (78), (79), (31) and

(48). Mean values and the corresponding standard deviation of f ka

and CD were also determined for each set of tests. A summary of these

results is listed in Appendix C, in which Ua = (U1 + U2)/2 and da =

(dl + 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.