Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00011146/00001
## Material Information- Title:
- Friction in hurricane-induced flooding
- Creator:
- Wang, Shang-Yih, 1952-
- Publication Date:
- 1983
- Language:
- English
- Physical Description:
- xvi, 165 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Buildings ( jstor )
Canopy ( jstor ) Dehydration ( jstor ) Drag coefficient ( jstor ) Floods ( jstor ) Friction factor ( jstor ) Shear stress ( jstor ) Surface water ( jstor ) Velocity ( jstor ) Water depth ( jstor ) Flood forecasting ( lcsh ) Floods ( lcsh ) Hurricanes ( lcsh ) City of Gainesville ( local ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1983.
- Bibliography:
- Includes bibliographical references (leaves 161-164).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Shang-Yih Wang.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 11479424 ( OCLC )
ocm11479424 00458303 ( ALEPH )
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FRICTION IN HURRICANE-INDUCED FLOODING By SHANG-YIH WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983 FRICTION IN HURRICANE-INDUCED FLOODING By SHANG-YIH WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983 ACKNOWLEDGEMENTS The author wishes to express his sincerest gratitude to the chairman of his supervisory committee, Dr. B. A. Christensen, for all his expert guidance, the tremendous benefit of his professional competence and vast practical experience throughout this study. He also wishes to thank Dr. T. Y. Chiu for his advice, understanding, encouragement and support during the author's six years of graduate study at the University of Florida. Without their untiring patience and help this dissertation would not have been possible. Thanks are also due to Dr. D. P. Spangler, Dr. B. A. Benedict, Dr. T. G. Curtis and Dr. H. Rubin for serving on the author's super visory committee and for their consulting and assistance. Appreciation is extented to Drs. A. J. Mehta, D. L. Harris, F. Morris and P. Nielsen for their suggestions and providing reference, which contributed greatly to this study. The author is indebted to Mr. E. Dobson for his technical assis tance, Ms. L. Pieter for her drafting and Ms. D. Butler for her typing. Special thanks are due to Mr. E. Hayter for his help in the preparation of this dissertation. Finally, the author wishes to thank his wife, Fu-Mei, whose partici pation in every phase of this study has made these years more joyful. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES vi LIST OF FIGURES vii LIST OF SYMBOLS x ABSTRACT xv CHAPTER I. INTRODUCTION 1 Storm Surge Prediction 4 Objectives of Present Work 6 II. FIELD EXPLORATION OF PHYSICAL ENVIRONMENT 7 Mangrove Areas 7 General View of Mangroves in Florida 7 Sampling of Mangroves 8 Developed Areas 15 High-Rise Building Areas 15 Medium-Rise Building Areas 17 Residential Areas 21 III. THEORETICAL BACKGROUND AND DEVELOPMENT 22 Hydrodynamic Equations for Storm Surges 22 Wind Shear Stress 23 Wind Velocity Profile in Vertical 25 Proposed Wind Shear Stress on Obstructed Areas 27 Bed Shear Stress 31 Post Approach, Friction Factor for Surges in Unobstructed Areas 33 Proposed Approach, Friction Factor for Surge in Obstructed Areas 38 iii Page IV.EXPERIMENTAL VERIFICATION OF FRICTION FACTORMODEL LAWS 43 Distorted Model for Buildings 45 Undistorted Model for Mangrove Stems and Roots 49 Distorted Model for Canopy 50 V.MODEL DESIGN 54 Recirculating Flume 54 Instrumentation 57 Velocity Meter 57 Data Acquisition System (DAS) 59 Depth-Measuring Device 59 Selection of Model Scales 63 Mangroves 63 Buildings 64 Model Setup 66 Mangrove Stems and Roots 66 Canopy of Red Mangroves 69 Buildings 72 VI.EXPERIMENTAL TEST SERIES 77 Experimental Procedure 77 Calibration of Velocity Meter 77 Measurements of Mean Velocities 77 Measurements of Water Depths 79 Experimental Runs 81 Mangrove Areas 81 Building Areas 82 VII.PRESENTATION AND ANALYSIS OF DATA 84 Mangrove Areas 84 Building Areas 95 Determination of Drag Coefficient 110 Drag Coefficient-Building Density Relation 110 Drag Coefficient-Disposition Parameter Relation 113 i v Page VIII. DISCUSSIONS AND CONCLUSIONS 119 Mangrove Areas 120 Developed Areas 121 Ocean Bottom 123 Forested Areas 127 Grassy Areas 128 Conclusions 129 APPENDICES A. FIELD RECORDED DATA FOR MANGROVES 132 B. COMPUTER PROGRAM LISTINGS 145 C. TABLES OF EXPERIMENTAL DATA 151 BIBLIOGRAPHY 161 BIOGRAPHICAL SKETCH 165 v LIST OF TABLES Table Page 1. Average Parameters of Sampling Mangroves 14 2. Average Characteristics of Canopy 16 3. Average Parameters of High-Rise Buildings 19 4. Average Parameters of Medium-Rise Buildings 20 5. Scale Selection for Canopy 64 6. Average Parameters of Prototype and Model for Building Areas 65 7. Statistical Values of Experimental Results for Mangrove Areas 94 8. Statistical Values of Experimental Results for Building Areas 109 9. Relations Between Disposition Parameters and Drag Coefficients 115 10. Typical Values for Mangrove Areas 120 11. Bed Friction Characteristics of Three Entrances 126 A1. Parameters of Sampling Red Mangroves 141 A2. Parameters of Sampling Black Mangroves 142 A3. Characteristics of Canopy of Red Mangroves 143 Cl. Experimental Data for Red Mangroves (Without Canopy) 151 C2. Experimental Data for Red Mangroves (with Canopy) 152 C3. Experimental Data for Black Mangroves 153 C4. Experimental Data for Building Areas 154 LIST OF FIGURES Figure Page 1. Prop Roots of Red Mangroves 9 2. Air Roots of Black Mangroves in 1 Foot Square Areas 9 3. Red Mangrove Area (Sampling Area #4) 11 4. Black Mangrove Area (Sampling Area #7) 11 5. Field Data Record for Red Mangroves 12 6. Field Data Record for Black Mangroves 13 7. Measurement of Density of Canopy 16 8. Section View of Survey Area (Red Mangroves) 16 9. Top View of Building Shapes on Coastal Areas 18 10. Wind Stress Coefficient over Sea Surface 26 11. Plan View for Wind Stress over an Obstructed Area 28 12. Elevation View for Wind Stress over an Obstructed Area 28 13. Distribution of Horizontal Apparent Shear Stress and of its Drag, Inertial and Viscous Components 39 14. Plan of Flume 55 15. Cross Section of Flume 55 16. Novonic-Nixon Velocity Meter 58 17. Input Box of Data Acquisition System 60 18. HP 9825A Programmable Calculator 60 19. Setup of Water Depth Measuring Device 62 20. Point Gage and Tube 62 vii Figure Page 21. Model Setup for Red Mangroves 67 22. Model Setup for Black Mangroves 68 23. Stems and Roots of Red Mangroves 70 24. Stems and Roots of Black Mangroves 70 25. Overview of Setup for Mangroves 71 26. Setup of Model Equivalent of Canopy of Red Mangroves 71 27. Distribution of Leaf Stripes in the Model 73 28. Building Patterns Designed for the Tests 74 29. Relation Between U and U 80 n w 30. Designed Building Patterns in the Tests 83 31. Relation Between f' and R' for Red Mangrove Areas (Without Canopy) 85 32. Relation Between Cn and R' for Red Mangrove Areas (Without Canopy) 86 33. Relation Between f' and Water Depth d for Red Mangrove Areas e (Without Canopy) 87 34. Relation Between f and R for Red Mangrove Areas (with Canopy) e e 88 35. Relation Between Cn and R1 for Red Mangrove Areas (with Canopy) 89 36. Relation Between f' and Water Depth d for Red Mangrove Areas (with Canopye) 90 37. Relation Between f and R' for Black Mangrove Areas 91 e e 38. Relation Between Cp and R^ for Black Mangrove Areas 92 39. Relation Between f1 and Water Depth d for Black Mangrove Areas e 93 40. Relation Between f and R' for Building Areas 94 e e Figure Page 41. Relation Between Cn and R' for Building Areas 102 D e 3 42. Relation Between f1 and Water Depth d for Building Areas e 108 43. Relation Between Cn and Density m for High-Rise Building Areas 111 44. Relation Between Cn and Density m for Medium-Rise Building and Residential Areas 112 45. Position Spacings. Definition Sketch 114 46. Relation Between Cn and S,/D in Aligned and Staggered Patterns 116 47. Relation Between and S^/D 118 Al. Field Recorded Data for Red Mangroves (Area #1) 132 A2. Field Recorded Data for Red Mangroves (Area #2) 133 A3. Field Recorded Data for Red Mangroves (Area #3) 134 A4. Field Recorded Data for Red Mangroves (Area #4) 135 A5. Field Recorded Data for Red Mangroves (Area #6) 136 A6. Field Recorded Data for Black Mangroves (Area #8) 137 A7. Field Recorded Data for Black Mangroves (Area #9) 138 A8. Field Recorded Data for Black Mangroves (Area #10) 139 A9. Field Recorded Data for Black Mangroves (Area #11) 140 i x LIST OF SYMBOLS area leaf area for prototype and model, respectively horizontal width drag coefficient skin friction coefficient average diameter of obstruction vertical depth for prototype and model, respectively water depth water depth at section 1 and section 2, respectively average water depth of d-j and d^ diameter of pipe drag force elastic force skin friction gravitational force inertial force Froude number =/ U / gd d d surface tension force viscous force Darcy-Weisbach friction factor based on diameter of the pipe friction factor based on hydraulic radius equivalent friction factor total friction factor gravitational acceleration vertical depth above the vertex of Thomson weir protruding height of obstructions above water surface indicial functional parameter wind stress coefficient drag force scale skin force scale gravity force scale inertial force scale shear force scale equivalent sand roughness apparent roughness horizontal length leaf length for prototype and model, respectively density = no. of obstruction elements/area indicating the subscripted parameters for model and prototype, respectively total number of obstruction elements vertical length scale horizontal length scale force scale time scale Manning's n wetted perimeter of flow cross-section pressure Ps pressure on the water surface Q discharge from the Thomson weir q q discharge per unit width x y R hydraulic radius = A/P R-|,R2 hydraulic radius at section 1 and section 2, respectively Rfl average hydraulic radius of R-j and R^ R^ reduction factor for wind stress Rq Reynolds number based on depth = U d /v R' Reynolds number based on hydraulic radius = UR /v 6 ad R* wall Reynolds number = u^.k/v R v Reynolds number = U^x/v C )A r radius of the pipe S slope of energy grade line = aH/L S corner to corner distance between the roughness elements in adjacent transverse raws SÂ£ longitudinal spacing between two successive roughness elements S^. laternal spacing between two roughness elements s free surface displacement from mean sea level S. Dev. standard deviation t time variable U spatial mean flow velocity U-j, U2 spatial mean flow velocity at section 1 and 2, respectively Ua average spatial mean velocity of U-| and U2 U U spatial mean flow velocity in x and y directions, respectively x y Uoo free stream velocity u,v,w instantaneous components of the water velocity in the x, y, z coordinate directions, respectively u u1 V' uf,t W(z) w m x, y, z z 0 0 Y 6 AH ^1*^2 e K u V e P pa 7b Tbx Tby time-mean velocity in the direction of flow turbulent velocity fluctuations in the x and z directions, respectively friction velocity based on bottom friction friction velocity based on total friction time-mean wind velocity at the elevation z above water surface critical wind velocity time-mean wind velocity at the elevation 10 meters above water surface leaf width for prototype and model, respectively Cartesian coordinate directions dynamic roughness shape factor specific weight of water = pg displacement thickness energy loss per unit weight of fluid fractions of distances nid and n2d from bottom to the total depth d, respectively latitude Von Karman's constant molecular viscosity of water kinematic viscosity of water percentage of the measured area occupied by obstructions water density air density spatial mean bottom shear stress = tq bottom shear stress in the x and y directions, respectively xi i i Tj hydrodynamic drag viscous stress wind shear stress on open area t wind shear stress on obstructed area so Reynolds stress 10 the earth's angular velocity to* wind friction velocity = /t./p S d V2 Laplacian operator xiv Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FRICTION IN HURRICANE-INDUCED FLOODING By Shang-Yih Wang December 1983 Chairman: B. A. Christensen Cochairman: T. Y. Chiu Major Department: Civil Engineering With the increasing development of coastal areas, it is necessary to have a sound method for predicting hurricane-induced flooding in these areas, especially for studies such as the coastal construction set-back line, flood insurance rate-making and county land use planning. The purpose of this study is to develop the capability of describing the friction factor in coastal areas for improved representation in numerical models of storm surges. Five types of areas are considered: A, ocean bottom with bed- forms and some vegetation; B, mangrove fringe and areas; C, grassy areas; D, forested areas; and E, developed areas. The friction factors, which incorporate both the bottom friction coefficient and drag coefficient due to the submerged parts of obstructions were verified by conducting laboratory experiments for mangrove and developed areas, xv using the typical distribution found in each of these coastal areas. Analysis of the experimental data revealed that the drag coef ficient for each case is invariant with the Reynolds number in the most possible flooding flow ranges, but that it is related strongly to the density and distribution of the roughness elements. Formulae expressing these relations were derived for the evaluation of the friction factor for different coastal areas. In addition, it is found that the drag coefficient for a staggered disposition is about two to three times larger than that for an aligned disposition under the same density for all building areas. A relationship between the drag coefficient and the disposition parameter of the evenly distributed roughness elements was developed. The principal reduction of the wind stress due to wind drag forces on the parts of the obstructions, including-buildings and vegetation, above the water surface during overland flooding was determined. Accounting for this reduction of the wind stress provides a realistic view of wind generation forces in coastal areas. Finally, the formulae of the friction factor for the ocean bottom, forested areas and grassy areas are presented by adopting results from previous investigations and discussed with the results of the current study. xvi CHAPTER I INTRODUCTION The rapid growth of population and industry in very low coastal areas in recent years has resulted in increased concern and attention to the potential hazard to these areas from tropical storms and hur ricanes. A severe tropical storm is called a hurricane when the maximum sustained wind speeds reach 75 mph or 65 knots (U.S. Army, Corps of Engineers, 1977). During a hurricane, the wind-driven storm waves are superimposed on the storm surge, which is the rise above normal water level due to the action of storm, and sometimes the low coastal areas are flooded. The worst natural disaster in the history of the United States came as the result of a hurricane which struck Galveston, Texas, in 1900. The storm, which hit the Texas coast on September 8, with winds of 125 mph caused a storm surge 15 feet in height above the usual two-foot tidal range. The fifteen-foot surge, accompanied by wave action, demolished the city and caused more than 5,000 deaths (Bascom, 1980). Weather warnings were ineffectual. The people of Galveston, unprepared for a storm of such intensity, were helpless in the face of the hurricane. But the hurricane is no longer the unheralded killer it once was. The years of progress in weather forecasting and wave research have now made it possible to predict such surges. Hurricane Donna, for example, which crossed the Florida Keys and then moved northeastward across the state of Florida from near Fort Myers to Daytona Beach on September 9-10, 1960, is thought to have been the most destructive 1 2 storm ever experienced in Florida. Fortunately, she was detected in advance. Thus, even though this hurricane caused an estimated $300 million in damage, only 13 fatalities occurred. In 1961, Hurricane Carla struck along the Gulf coast of Florida. The area was evacuated before its arrival, and there was no loss of life. From the facts mentioned above, it is clear that the great value of the modern storm warning service is in its reducing the loss of life. However, it is also apparent that the potential damage to property and structures has increased dramatically with the rapid development of coastal areas, if no concession is made to the storms. Therefore, the future development plans of these coastal counties must take into consideration this threat to life and property. As a result, the 1971 session of the Florida State Legislature passed a law (Chapter 16,053, Florida Statutes), requiring the Department of Natural Resources to establish a coastal construction set-back line (SBL) along Florida's sandy beaches fronting the Atlantic Ocean and the Gulf of Mexico. Based upon comprehensive engineering studies and topographical surveys, such a line, where deemed necessary, is intended to protect upland properties and control beach erosion. Basically, construction and excavation seaward of the SBL is prohibited, though a provision for variances is included in the law (Chiu, 1981). In 1973 the Congress of the United States enacted the Flood Disaster Protection Act (Public Law No. 93-234, 87 Statutes 983) which greatly expanded the available limits of federal flood insurance coverage. The act also imposed new requirements on property owners and communities desiring to participate in the National Flood Insurance Program (NFIP) (Chiu et al., 1979). 3 The Flood Disaster Protection Act of 1973 requires the Department of Housing and Urban Development (HUD) to notify those communities that have been designated as flood hazard areas. Such areas are defined as having a one percent annual chance of flooding at any location within the areas. Such a community must either make prompt application for participation in the flood insurance program or must satisfy the Secretary of HUD that the area is no longer flood prone. Participation in the program is mandatory (as of July 1,1975) or the community would be denied both federally related financing and most mortgage money. Individuals and businesses located in identified areas of special flood hazard are required to purchase flood insurance as a prerequisite for receiving any type of federally insured or regulated financial assistance for acquisition or construction purposes. Effective July 1, 1975, such assistance to individuals and businesses was predicated on the adoption of effective land use and land management controls by the community. Federally subsidized insurance for flood hazard is authorized only within communities where future development is controlled through adequate flood plain management. Management may include a comprehensive program of corrective and preventative measures for reducing flood damage, such as land use controls, emergency preparedness plans and flood control works. Participating communities may be suspended from the program for failure to adopt or to enforce land use regulation (National Flood Insurers Association, 1974). 4 Storm Surge Prediction To implement either the coastal construction setback line or the flood insurance program the flood elevation has to be determined on the basis of different time intervals. Accumulation of data over many years in areas of the Old World, such as regions near the North Sea, has led to relatively accurate empirical techniques of storm surge prediction for these locations. However, these empirical methods are not applicable to other locations. In general, not enough storm surge observations are available in the New World to make accurate prediction of the 100 year storm surge. Therefore, the general practice has been to use hypothetical design storms, and to estimate the storm-induced surge by numerical models, since it is difficult to represent some of the storm-surge generating processes (such as the direct wind effects and Coriolis effects) in physical laboratory models. With the use of digital com puters, numerical models have been able to analytically describe storm surges to much greater detail than was ever possible with the other methods. As a result, many numerical models for the prediction of surges have been proposed to investigate practical cases including irregular coastlines, irregular bathymetry, islands and arbitrary wind stress patterns (Mungall and Mattews, 1970; Reid and Bodine, 1968; Platzman, 1958; Platzman and Rao, 1963). Moreover, Pearce (1972) and Reid and Bodine (1968) developed their models to evaluate the inland extent of flooding by using a moveable boundary. For different emphases on off shore and nearshore areas, varied grid systems are used for most finite- difference models. An orthogonal curvilinear coordinate system with telescoping computing cells has also been introduced by Wanstrath (1978) 5 to solve the flooding problems of Louisiana. Regardless of the purposes and differences in approaches of all these numerical storm surge models, the Navier-Stokes and continuity equations which incor porate terms accounting for wind stress, bottom friction, inertia, Coriolis effect, pressure distribution, and other physical parameters are solved numerically in space and time to determine localized surge hydrographs. In order to obtain a more realistic and accurate predic tion of storm surge, these physical parameters should be carefully determined and incorporated into the numerical models. Modern achievements in meterology and oceanography have led to an increased understanding of hurricane to the extent that a model hurricane can be characterized to a satisfactory degree by certain parameters. A list of these variables includes central pressure deficit, radius to maximum winds, speed of hurricane system translation, hurricane direction and landfall location (or some other descriptor of hurricane track). Surprisingly little work has been done in measuring another important term-bottom friction. As suggested by Pearce (1972), future work on actual hurricane surges and currents is especially needed for improved representations of bottom friction that would be achieved with a better understanding of the dissipation mechanism (i.e. friction) during a hurricane. Objectives of Present Work Based on the necessities for a more accurate prediction of hurricane-induced flooding in coastal areas (especially for studies like the construction set-back line, flood insurance rate determination 6 and county land use plan), this study will develop a method of describing the friction factor in coastal areas for improved repre sentation of numerical storm-surge models. Special emphasis will be placed on the friction characteristics of mangroves and buildings which are the two most important causes of frictional resistance in vegetated and developed land areas, respectively. The effect of these two roughnesses in reducing overall wind stress on the water is also introduced. The friction factors for other roughnesses such as the ocean bottom, the forested and the grassy areas are determined by the results from previous investigations and present study and are discussed in the last chapter. CHAPTER II FIELD EXPLORATION OF PHYSICAL ENVIRONMENT As stated in Chapter I, mangroves and buildings are the two major flow retarding objects and are therefore being investigated in this study. However, information on the density, dimensions and typical distributions of these two forms of roughness in the coastal areas is very scarce. Thus, field trips were taken to a mangrove area and developed areas in southern Florida in order to collect the most representative data for use in the model tests. Mangrove Areas General View of Mangroves in Florida Mangrove is a kind of salt-resistant plant that usually grows densely on sub-tropic shorelines around the world. This special feature may be an inherent gift from nature in that the mangroves enable exposed shorelines to resist severe attacks of hurricanes. Basically, there are three species of mangroves, the red mangrove (Rhizophora mangle), the black mangrove (Aricennia nitida) and the white mangrove (Laguncularia racemosa). Each of these three species occupies a distinct zone within the forest, depending on the degree of salinity and length of inundation that each species can tolerate. Red mangroves usually are found at the outer or seaward zone. They are distinctive in appearance, with arching prop roots that project 7 8 from the trunk or branches down into the water (Figure 1). The root and trunk systems of red mangroves, which spreads in shallow offshore areas and onshore areas serve as a soil producer and stabilizer as well as a storm buffer. In their role as buffers against storm winds and tides, they prevent devastation of the coastline (Lugo et al., 1974). The middle zone, at slightly higher elevations, is dominated by the black mangroves in association with salt marsh plants. This zone is usually submerged at high tide, but is otherwise exposed. The roots produce pneumatophores (fingerlike extensions above the soil surface),as shown in Figure 2. Black mangroves may also be found in pure stands in shallow basins where sea water remains standing between tides. The heat from the sun evaporates some of the water, leaving slightly concentrated salt water behind. The black mangroves are also important for shoreline stabilization as they present a secondary defense behind the red mangroves. The white mangrove, which can be found in the most landward zone that is affected only by the highest spring and storm tides will not be discussed here since it is not as important in defending against storm flooding and its usual appearance may be categorized into buttonwood or other common types of vegetation. Sampling of Mangroves Five mangrove forest types Fringe, Riverine, Basin, Overwash and Dwarf Forests-have been found by Snedaker and Pool (1973) in southern Florida, with distinctive differences in structure. The pattern is strongly related to the action of water, both the freguency and the amount of tidal flushing and freshwater runoff from the upland. The coastal fringe forest including red and black mangroves, which are the 9 FIGURE 1: Prop Roots of Red Mangroves FIGURE 2: Air Roots of Black Mangroves in 1 Foot Square Area 10 most important species, was investigated in San Carlos Bay on the southwest coast of Florida. Eleven sampling areas which included six red mangrove fringe areas and five black mangrove areas were selected at random. Each sampling area was framed by survey poles to form a 12 by 12 foot square area in which the locations and dimensions of mangrove trunks, roots, and canopies were recorded. Figures 1 through 4 show some of the features of both types of mangroves in the surveyed area. Figures 5 and 6 show two examples of data recorded from red and black mangrove sampling areas, respectively. Data for the other nine sampling areas are shown in Appendix A. From these data it is clear that the density and dimensions of the trunk and root systems of mangroves are quite random; therefore, averaged characteristics are chosen to describe these samples as shown in Table 1. Red mangroves in the surveyed areas extend from the low tidal water line to about 50 feet inland which is the same distance Veri et al., (1975) recommended for fringe mangroves in order to form a protective buffer zone. Thus, this value of 50 feet can be considered to be a standard distribution distance for red mangroves and is used in the present study. Although the average height of the lower edge of canopy was found to be about 8 feet above the ground for red mangroves, the canopy along the water edge was found to generally have a distribution from the water surface to a few feet high. This feature may be important in resisting storm surges. Therefore, a detailed measurement of canopy distribution was done at a later time in Sarasota, Florida. Figure 7 depicts the measuring of the density of leaves by counting the number 3 of leaves in a unit volume (1 ft.). Dimensions of leaves were also 11 FIGURE 3: Red Mangrove Area (Sampling Area #4) FIGURE 4: Black Mangrove Area (Sampling Area #7) 12 AREA #5 FIGURE 5: Field Data Record for Red Mangroves 13 AREA #7 O Stem FIGURE 6: Field Data Record for Black Mangroves 14 TABLE 1: Average Parameters of Sampling Mangroves Average Parameters Red Mangroves Black Mangroves Main- Stem no. (12 ft.)2 4 12 diameter 6.0 in. 3.1 In. height 10.0 ft. 11.2 ft. % occupied 0.84 % 0.49% Sub- Stem no. (12 ft.)2 13 diameter 2.0 in. height 18.0 in. % occupied 0.19 % Canopy height 8.0 ft. 9.4 ft Root no. (12 ft.)2 81 10,800 diameter 1.0 in. 0.25 in height 18.0 in. 6.0 in. % occupied 0.25 % 2.39% 15 measured and recorded. Totally six sampling areas along the coastal fringe were randomly selected. As shown in Figure 8, the survey area contains three sections in which each section covers a distance of five feet. The data collected are shown in Appendix A. Table 2 lists the average densities and dimensions of leaves obtained from these six sampling areas. Developed Areas In developed areas, buildings constitute the principal roughness elements which would significantly affect the apparent bottom shear stress as well as the wind shear stress during a storm induced flood. Buildings are not, in general, arranged in a uniform manner but are strongly dependent on the environment where they are located. A common feature found in the coastal counties of Florida, especially in Broward and Dade counties, is that high-rise buildings are predominant along the beaches while residential houses are predominant a few miles inland from the coastline. Three areas, a high-rise building area, medium- rise building area and residential area, are defined to represent a developed area in this study. High-Rise Building Area Aerial photographs of Broward and Dade counties, Florida, made by the State Topographic Office, Florida Department of Transportation in 1980 were used to analyze the dimensions and densities of buildings in the coastal areas. Dade county is divided into 113 ranges while Broward county is divided into 128 ranges in the aerial photographs. 16 FIGURE 7: Measurement of Density of Canopy TABLE 2: Average Characteristics of Canopy ^Sections Parameters #1 n #3 ... no. of leaves\ Density! = ) ftJ 10 5 2 Leaf Size 2" x 1" 3" x 1.5" 4" x 1.75" 17 Each range has a length of 1,000 feet approximately parallel to the shoreline and is marked by monuments both in the field and on the aerial photographs. Typical shapes and orientation of buildings found in these two counties are shown in Figure 9. Category (a) is the most common type found (more than 50 percent) which may be dictated by the high cost of land per foot along seashore, and is chosen to present all the buildings in the study. High-rise buildings are defined as buildings having a surface area 2 larger than 10,000 ft An estimation of the dimensions and densities of these high-rise buildings from Broward and Dade Counties are 1isted in Table 3. High rises and hotels/motels are predominant in the area. Medium-Rise Building Area Madium-rise buildings cover all buildings which do not belong to either the high-rise or residential types. They can include two and more story semidetached houses, row houses, garden apartments and other buildings which are lower than ten stories. The surface area occupied 2 2 by medium-rise buildings is defined from 2,400 ft to 10,000 ft in this study. An investigation of buildings in this category was also made from aerial photographs of Broward and Dade Counties. Table 4 shows the average densities and dimensions of buildings from existing field data. The values obtained from these two counties at least present some general views of buildings in highly developed areas despite their irregularities in distributions found in the field. To apply these 18 Sea Side Sea Side (b) FIGURE 9: Top View of Building Shapes on Coastal Areas 19 TABLE 3: Average Parameters of High-Rise Buildings Range No. Buildings County length (ft.) width (ft.) density ( ^ ) 1000'x500' % of land occupied by buildings 42 280 80 10 45 Broward 43 290 60 4 14 Broward 45 290 110 4 26 Broward 50 140 140 10 39 Broward 54 300 100 3 18 Broward 72 200 70 8 22 Broward 82 200 50 19 38 Broward 83 200 180 5 36 Broward 84 200 230 5 46 Broward 118 140 90 8 20 Broward 119 150 70 12 25 Broward 121 230 90 6 25 Broward 8 240 200 3 29 Dade 11 300 50 8 24 Dade 12 320 70 7 31 Dade 14 240 50 9 22 Dade 15 230 50 8 18 Dade 17 230 60 9 25 Dade 18 250 50 10 25 Dade 19 200 125 5 25 Dade 36 220 65 4 11 Dade 42 190 65 7 17 Dade 43 270 200 6 65 Dade 44 200 200 5 40 Dade 48 120 200 6 29 Dade 52 220 210 4 37 Dade 56 190 160 9 55 Dade Mean 224 112 7 30 S.Dev. 53 62 3 13 20 TABLE 4: Average Parameters of Medium-Rise Buildings Range No. Buildings County length (ft.) width (ft.) density l no> ) U000'x500' 1 % of land occupied by buildings 26 65 65 23 19 Broward 36 120 40 30 29 Broward 37 100 35 33 23 Broward 46 150 40 27 32 Broward 49 80 60 31 30 Broward 51 100 40 22 18 Broward 52 120 40 22 21 Broward 62 80 30 33 16 Broward 64 100 40 20 16 Broward 66 70 40 43 24 Broward 67 80 70 14 16 Broward 101 60 50 35 21 Broward 109 70 70 18 18 Broward 110 70 40 24 13 Broward 111 70 70 25 25 Broward 116 100 70 17 24 Broward 1 120 75 14 25 Dade 2 90 60 13 14 Dade 3 80 50 15 12 Dade 4 75 75 11 12 Dade 5 100 65 15 20 Dade 33 160 49 24 38 Dade 35 125 35 32 28 Dade 68 130 40 21 22 Dade 69 130 40 24 25 Dade 70 125 40 28 28 Dade Mean 99 51 24 22 S.Dev. 28 15 8 7 21 data in the prediction of storm surge, it is recommended that the County Land Use Plan Map published by each county be used so that the most realistic results can be expected. Residential Area Residential houses are usually located behind the commercial areas ? and have a surface area less than 4,000 ft A typical density value of detached, one story houses is given as six units per acre (43,560 sq. ft.) (DeChiara and Callender, 1980). Density ranges in residential areas can also be found in the Land Use Map of each county which categorizes these single family houses in the density range of 0-8 units per acre (Reynolds, Smith and Hills, 1972). The significant difference between residential, medium-rise building areas and high-rise building areas is that the former two areas usually have a matrix type distribution while the latter one has only one or two rows distributed in the coastal fringe area. The importance of this variation in the building distribu tion will be shown later in the discussion of modeling studies. Dimensions of the typical residential house are chosen as 30 feet by 62 feet, 1,860 sq. ft., which are convenient for the model tests and also realistic for most single family houses. CHAPTER III THEORETICAL BACKGROUND AND DEVELOPMENT Hydrodynamic Equations for Storm Surges The equations governing incompressible fluid flows are the Navier-Stokes equations of motion and the equation of continuity. In the case of storm surges these equations may be written: 3qx 3 t 2o)(sin0)qy d_ 3Ps p 3 x gd H + p (Tsx -Tbx> (1) 2o)(sine)qx d_ 3Ps p 3y gd |S- + 1 a 9y p (2) 3qx + 3^y_ + 3S 3 x 3 y 3t 0 (3) where t is time, u is the angular velocity of the earth, e is the lati tude, p is the pressure, g is the acceleration of gravity, p is the water density, s is the free surface displacement from mean sea level, the subscripts s and b indicate that the subscripted quantities are to be evaluated at the surface and bed, respectively, d is the total depth, qx and q^ represent the time mean transport component, i.e. discharge per unit width, in x and y directions, respectively, i.e. d(x,y,t) = h(x,y) + s(x,y,t) qx(x,y,t) fs(x,y,t) u(x,y,t) dz -h(x,y) (4) (5) 22 23 qy(x,y,t) rs(x,y,t) v(x,y,t) dz -h(x,y) (6) in which h = water depth referenced to mean sea level, u and v are the instantaneous components of water velocity in the x and y coordinate directions, respectively. Expressions for the wind shear stress, and bed shear stress, x^, for coastal areas in tropical storm induced flooding are presented in the following sections. Wind Shear Stress In general, the wind stress (t ) on a water surface may be expressed in terms of the mean wind speed (W-^) at anemometer level (10 meters above water surface), the air density (p ) and a wind-stress coefficient (K), a as T s K W 10 (7) The problem of evaluating the wind stress is therefore reduced to estimating the wind-stress coefficient, K, at different wind speeds, if the reference wind speed and air density are known. Numerous studies have found the quadratic wind speeds relation to be appropriate for a wide range of wind speeds (Wilson, 1960). A wind-stress relation more physically satisfying the quadratic law correlation was developed by Keulegan (1951) and Van Dorn (1953) in the low winds range (<15 ms"^). The Keulegan-Van Dorn relation for x$ is given as T s Pa [K^o K2(W10 w/] (8) 24 where K-| and are the constants and Wc is critical wind speed. Although there are uncertainties in applying the Keulegan-Van Dorn relation to hurricane winds, it has been applied widely in hurricane- induced surge cases. To eliminate this deficit, the wind-stress relation has to be extended to higher wind ranges. Whitaker, Reid and Vastano (1975) investigated the wind-stress coefficient at hurricane wind speeds using a numerical simulation of dynamical water changes in Lake Okeechobee, Florida. Results of their numerical experiments showed that the Keulegan-Van Dorn wind-stress relation was superior to the more commonly used quadratic relation for wind speeds in the range of 20 to 40 meters per second. The relation they found for the wind stress ts is given by: t = p[0.0000026 + (1.0 ^.)2 x 0.0000030] W?n (9) s wio 1u where W^q and 7.0 are in meters per second and p is the water density. Unfortunately, though this result was verified by a simulation of the surge associated with a hurricane which occurred in October, 1950, it still has some deficiencies such as the limited range of applicability (Whitaker et al., 1975). Recent studies of the wind-stress coefficient over the sea surface have produced more complete and perhaps more accurate results with the refinement of measurements and analysis techniques. Garratt (1977) reviewed and averaged 17 selected sets of data and proposed an empirical expression for 'light' winds: K = (0.75 + 0.67 W1Q) x 10"3 (10) 25 Wu (1980) suggested a similar result for the wind-stress coef ficient from 33 averaged data sets under 'light* winds K = (0.8 + 0.065 W1Q) x 10"3 (11) Furthermore, Wu (1982) compiled and averaged all available data for 'strong' winds. The data were obtained from independent investigations either cited or reported in the following sources: Wu (1969), Kondo (1975), Garratt (1977), Smith (1980), Wu (1980), and Large and Pond (1981). All the data sets selected were obtained under nearly neutral conditions of atmospheric surface layer. Additional factors which affect the wind-stress coefficient, such as rainfall and sea spray, are neglected due to their minor importance compared to the major factor of wind speeds. As a result, the empirical formula proposed, given by equation (11), for 'light' winds appears to be applicable even in 'strong' winds. Light and strong winds are defined as those less than and greater than 15 meters per second, respectively. The averaged data obtained from those sources and the formula proposed, equation (11), are shown in Figure 10. Wind Velocity Profile in Vertical A vertical profile of wind velocities, usually expressed by the following logarithmic law, is regarded by meteorologists as a superior representation of strong winds in the lower atmosphere (Tennekes, 1973): W(z) = 1, In (f-) (12) 0 where W(z) is the wind velocity at a height z above mean sea level, is the von Karman's turbulence constant, z is the dynamic roughness of 1 /2 the logarithmic velocity profile, and w* = (t /pa) = friction velocity. FIGURE 10: Wind Stress Coefficient over Sea Surface 27 The wind velocity profile given by equation (12) is well defined except that it fails next to the bed where z approaches zero, and the wind velocity W(z) approaches minus infinity. This discrepancy can be corrected by using a modified mixing length approach as proposed by Christensen (1971) for the flow of water over a rough bed, resulting in modified logarithmic law for wind velocity profile is given in the form = 2.5 In (^- + 1) (13) * zo By substituting equation (7) into equation (12), an equation for determining the dynamic roughness, zq, is obtained 4ft = /K (2.5) In (j- + 1) (14) w10 o Applying the boundary condition W(10) = W1Q at z = 10m The dynamic roughness, zq, in all the wind speeds is found to be a function of wind-stress coefficient, or on the wind velocity, i.e., z 0 10 e'2.5 , (15) Proposed Wind Shear Stress on Obstructed Areas Consider an obstruction which has an effective width, D, and protruding height, h, as shown in Figures 11 and 12. The wind drag force on such an obstruction can be expressed as 28 I T D -L PLAN VIEW FIGURE 11: Plan View for Wind Stress over an Obstructred Area ELEVATION VIEW Protruding Obstruction Wind => FIGURE 12: Elevation View for Wind Stress over an Obstructed Area 29 F D CPpaD 2 h 0 W2(z) dz 06) where CQ is the drag coefficient. Substituting equation (14) into equation (16) gives (2.5) F0 " CDpa DK 10 [In { + 1 )]2 dz o or Fd = 3.13 Cdp3D W20 K {(h + zQ)1n(^-+ l)[ln(^- + 1) 2] + 2h } (17) o o Recall the equation for wind stress on an open water surface, i.e., K a *4 .(7) This wind stress acts on the water surface and causes a rise of the elevation of water surface which is called wind setup. The wind energy is being transformed from the wind field to the water flow by the wind shear. When the same wind field moves from the open water area to the obstructed area, the wind setup will be reduced. This is due to the extra form drag (FQ) acting on the protruding obstruction that can be contributed to the wind setup per unit area. As a result, this reduced wind stress causing a wind setup on an obstructed area may be expressed as: t = R t = R, K p W._ so d s d pa 10 , 0 < Rj < 1 (18) in which Rd is a reduction factor which represents the ratio of the wind stress on the obstructed area to that on an open area under the same wind condition. 30 Since the reduction of wind stress on an obstructed area is due to the extra form drag, or wind energy loss caused by the obstruction, the reduction factor may also be defined as the ratio of the total loss of wind energy per unit length on an open area to that on an obstructed area with dimensions 1*1, i.e., (Gee and Jenson, 1974) 2 T. X l Rd = K 2 (19) [(1 mgD)xs + mFp] x C 2 in which m = density = N/Â£ N = total number of obstructions, 8 = a shape factor defined as the horizontal cross-sectional area of average obstruction element at surface level divided by D. Substituting equations (7) and (17) into equation (19) gives K W 10 (1 m8D2)KpaW^0 + 3.13 (mCDPaDW2Q) K{(h+zo)ln(^-+l)[ln(^-+l)-2]+2h} o o or Rd = 1 (l-m6D2) + 3.13 (mDCD) {(h+ZQ) ln(^-+l) [ln(^-+ 1 )-2] + 2h} o o (20) _ o where K = (0.8 + 0.65 W^q) x 10 W^q is in meters per second. Based on the result presented in equations (18) and (20), the reduction factor R^ can be determined and incorporated into the storm surge model to produce more realistic results for the wind stress on water in flooding areas. The drag coefficient in equation (20) for vegetations and buildings will be determined and discussed in Chapter VII and VIII. 31 Bed Shear Stress A space averaged bed shear stress, 7b, usually can be expressed as V i-e Tb = To = yRS = yR T- (21) in which tq =average bed shear stress along the wetted perimeter; y = pg = unit weight of water; R = A/P = hydraulic radius = depth in sheet flow; A = cross-section area; P = wetted perimeter of flow cross-section; S = slope of the energy grade line; AH = energy loss per unit weight of fluid over a bed length of L. Primarily developed for flow in pipes, the energy loss term, aH, is defined by the Darcy- Weisbach formula as iH f 55 T (22) 3 0 where f = friction factor based on depth; U = spatial mean flow velocity; dQ = diameter of the pipe. Since dQ = 4R, the above equation / may be written for an arbitrary cross section as (23) where f = friction factor based on hydraulic radius. Incorporating equation (21) into equation (23) in the x and y directions, respectively, gives the following quadratic forms for the bed shear stresses: fp|U|U bx f'p|U|U by 2 2 (24) 32 or in terms of volume transport _ f,phlqx flp|qlqy Tbx 2d2 Tby 2d2 (25) where f = 4f'; U = /U 2 + U 2; q = /q 2 + q 2; U and U = spatial x y x y x y mean flow velocity in x and y directions, respectively. It is assumed that these steady state relationships for the two shear stress components are valid for storm surge propagation, which is generally considered to be quasi-steady, i.e., the velocity variation with time or the temporal acceleration is very small. The quadratic Darcy-Weisbach form of bed shear stress is the best formula available to account for the effect of bottom friction. The friction factor in the Darcy-Weisbach formula, f, has been studied by many investigators in both pipe flows and open channel flows. From the abundant experimental data, numerous empirical formulae have been established to express the relationship between the friction factor and the dependent parameters, such as bed roughness, Reynolds number, Froude number and Strouhal number. For example, the well-known Stanton diagram (1914), Moody diagram (1944) and many others (to be discussed in the next section) have enabled determination of the friction factor in varied flow conditions. In general, a surge could be expected to travel over five different terrains (Christensen and Walton, 1980): A. Ocean (river) bottom with flow induced bed form and completely submerged vegetation, B. Mangrove fringes and areas, C. Forested areas and cypress swamps, D. Grassy areas, and 33 E. Developed areas. Each of these five categories has unique roughness characteristics. However, in evaluating the friction factor in hurricane-induced flooding, these five terrains can be divided into two major categories, unobstructed and obstructed areas, based on their distinct functions to retard flow. Post Approach, Friction Factor for Surges in Unobstructed Areas Unobstructed areas include the ocean (river) bottom and grassy areas, the latter of which are assumed to be completely submerged in water during floods. Friction factors in these kinds of areas can be determined from the results of previous research which will be discussed below and used as basis for the present work. Overland flooding in this study is considered to be turbulent and in the hydraulically rough range, i.e., the wall Reynolds number is in excess of about 70. The effect of wall roughness on turbulent flow in pipes has been studied during the last century by many investigators. An important result obtained by Nikuradse (1933) in steady flow using six different values of the relative roughness k/r with Reynolds numbers ranging from 4 6 R = Ud /v = 10 to 10 has been widely used in flow fields and will be e o applied in this study (k is the equivalent sand roughness, r is the radius of pipe; U is the average velocity, v is the kinematic viscosity). Nikuradse divided flow conditions into three ranges, smooth flow range I (u^k/v < 4), transition flow range II (5 <_ ufk/v Â£ 68), and rough turbulent flow range III (ufk/v > 68) in which u^k/v = R* = wall Reynolds number, u^ = friction velocity = /x/p. 34 In range III (rough turbulent flow) the thickness of the viscous sublayer <5 is negligible compared to the equivalent sand roughness, k, and the friction factor is independent of the Reynolds number. The distribution of the time-mean velocity obtained using Prandtls' mixing length approach in combination with Nikuradse's experimental results is given by the general expression jj- = 8.48 + 2.5 In | = 2.5 In (26) in which u = time-mean velocity in the direction of flow at a distance z from the theoretical bed. Theoretical bed is defined as the plane located such that the volume of grains above the plane equals to the volume of pores below the plane but above the center of grains. The classic velocity profile given by equation (26) is well defined at moderate to large distances from the bed and for roughnesses much smaller than the depth. However, it falls next to the bed whereas z approaches zero, the time-mean velocity U approaches minus infinity. This is especially true in flows where the roughness is not significantly smaller than the depth. Because of the above-mentioned discrepancies, Christensen (1971) introduced a new law for the velocity profile by using a modified mixing length approach over a rough bed in the rough range = 8.48 + 2.5 In uf |+ 0.0338) = 2.5 In ( ?973z + 1) (27) The form of this equation is the same as that of classic equation (26) except for the +1 term in the argument of the logarithmic function which 35 makes the time-mean velocity u equal to zero at the theoretical bed. As the distance from the bed increases to more than a few times k, very little difference exists between these two velocity profiles. For practical purposes, the time-mean velocity profile is trans formed to a depth averaged velocity profile using the fact that the mean velocity (depth averaged), U, occurs theoretically at a distance z = 0.368d from the bed also for the modified logarithmic vertical velocity profile, where d is the water depth, and d/k is larger than 1. It shall be noted that the k value used here is the equivalent roughness height for bottom friction only. Therefore, at z = 0.368d equation (27) yields = 2.5 ln[%^ (0.368d) + 1] U K or = 2.5 In [10.94 Â£ + 1] U K (28) where U = time and depth averaged velocity. The friction factor may in general be related to the velocity profile by introducing the Darcy-Weisbach formula into the definition of the friction velocity, i.e., uf = /tq/p = /gRS, leading to the result (29) Solving equation (29) for f and introducing equation (28) gives the following expression for the friction factor 0.32 f (30) 36 This depth dependent friction factor is proposed for areas where the surge moves over bottoms at moderate depths. Another equation for f obtained from Nikuradse's experimental result (1933) for rough turbulent flow in circular pipes, is given by = 1.171 + log Â£ (31) 4/Tr K It seems quite clear from equation (30) and equation (31) that determination of friction factor in unobstructed areas is just a matter of finding the value of the equivalent sand roughness k. This k value can be related to Manning's n by using a Strickler-type formula (Henderson, 1966, Christensen, 1978) in metric units 1 8.25 /q (32) given k in meters, or in the English units 1.486 8.25 /q (33) with k in feet. Values for n may be determined from various sources such as textbooks by Henderson (1966) and Chow (1959), charts and graphs by the Soil Conservation Service (1954), and photographs of a number of typical channels by the U. S. Geological Survey (Barnes, 1967). Other specific studies, for instance, the experiments conducted by Palmer (1946), also provide valuable information on the flow of water through various grass and leguminous covers. Based on the theoretical velocity 37 distribution in rough channels, the value of Manning's n can also be determined by analytical methods such as that presented by Boyer (1954). It should be noted, however, that these values of n from previous sources may not be applied to every case under natural conditions. It should be also careful in selecting the values of n, since a small error on n will be amplified substantially on k by using the Strickler-type formula. Therefore, a method to determine the k-value from the vertical velocity distribution in turbulent flow over rough surface is recom mended (Christensen, 1978). Let Hi be the velocity at m depth, that is, at a distance md from the bottom of a wide rough channel, where d is the depth of flow. By equation (26), the velocity may be expressed as Ui 29.7nid = 2.5 In Â£ (34) Similarly, let u2 be the velocity at r\2 depth; then u2 29.7r)2d = 2.5 In u (35) Subtracting equation (34) by equation (35) and solving for uf, uf = U2 Ux *12 2.5 In fii (36) Introducing equation (36) into equation (35) and solving for k, 29.7md (37) n2 u2- ux 38 Proposed Friction Factors for Surges in Obstructed Areas Obstructions in these areas are defined as roughness elements with significant heights which either protrude through the water layer or consist of relatively rigid elements with heights that are sufficient to cause form drag that are much larger than surface friction on the same area. The two major forms of obstructed areas, mangroves and buildings, to be discussed in this study are often higher than the storm surge level so that the influence of hydrodynamic drag on the individual elements should be taken into consideration together with other factors of resistances in overland floods. The theoretical analysis presented here is based on the assumption of steady or quasi-steady flow in the rough flow range. Consider a design flow that passes over an obstructed area whose bottom is horizontal. The total averaged shear stress, x in the direction of flow may be written as equation (21), i.e., - dc d aH t0 yRS = yR For the steady state case, assume that the viscous stresses, turbulent stresses and hydrodynamic drag acting on the obstructions contribute independently to the flow system without any interaction among them. Following Prandtl's assumption (Schlichting, 1979), that the shear stress in the x-direction is constant and equal to the wall shear stress at all distance z from the wall, an equilibrium equation is given in the following and shown in Figure 13. t FIGURE 13: Distribution of Horizontal Apparent Shear Stress and of its Drag, Inertial and Viscous Components, (Christensen, 1976). 40 du YRS = ^ (l-e) - pu'v' (l-e) + mCDdDY S| (to} Viscous Stress (Tt) Turbulent Stress (Td> Hydrodynamic Drag (38) where e = fraction of total area occupied by obstructions, u', v' are the turbulent velocity fluctuations in the x and z directions, respectively, m = number of obstructions per unit bed area, CQ = average drag coefficient, D = average diameter of the obstructions in the pro jected plane normal to the flow. For fully developed turbulence, the viscous term is negligible compared to the turbulent term and may be omitted. Consequently pgRS = -pu1v' (1-e) + mCDdD | (39) The turbulent stress, x^, which expresses the rate of flow of x momentum in the z direction, was first derived by 0. Reynolds from the equation of motion in fluid dynamics, and is termed the Reynolds stress or the inertia stress. The Reynolds stress on the right hand side of equation (39) must be 1 e (40) where 1 41 = ( Id T 1 e V2 ) , 0 < $ < 1 (41) reflects the reduction of the Reynolds stress pu1v1 due to the presence of the obstructions. If the drag is equal to zero, implying that no obstruction exists (e = 0), the obstruction correction factor will of course become equal to 1. The corresponding friction velocity depends on the bottom friction only. It is defined from equation (40), \'2 ) uft * (42) where uft is the friction velocity based on the total bed shear stress V i>e uft = /T7p* Similarly, introducing f and fj. as the friction factors of bottom friction and total friction including bottom friction and drag acting on the obstructions, respectively. Therefore, these two friction factors can be related to the mean velocity as shown in equation (29), U 2 1^2 = (7.) for the obstruction-free area Uf T (43) and U uft for the total obstruction area (44) Eliminating U from the two equations above, relation between f and fj. is given as f fi *2 (45) 42 Substituting equations (40), (42) and (44) into equation (39), the equilibrium equation becomes yR A = Â¡L 2 (1-e) + mCDdDY ^ or AH = Â£. f ,.,2 YR y = f f'U" (1-0 + mCpdDy ^ The head loss, AH, then may be written as 4K= f' <-*> +raCDd0f?s- and 2 L AH = [f (1-e) + mCDdD] (46) (47) (48) which is the form of the Darcy-Weisbach formula shown in equation (23). An equivalent friction factor, f^, which includes the effects of bottom friction and form drag based on equation (48) is introduced f* = f (1-e) + mCpdD (49) where, according to equation (30), 0.32 [In total friction factor, f|, according to the above definitions, as can be proved easily by substituting equation (45) into equation (49). An apparent roughness height ka which represents the sum of bottom roughness and rigid drag element can then be expressed and calculated from equation (31), i.e., -7= 1.171 + eog if- 4/f7 a CHAPTER IV EXPERIMENTAL VERIFICATION OF FRICTION FACTORSMODEL LAWS The formula proposed in Chapter III for the friction factor (equation (49)) has left some unsolved questions: What is the drag coefficient (C^) for the kinds of obstructions being studied? What kind of relationship do drag coefficient and density of obstruction have? Does the friction factor depend on unspecified flow conditions like the Reynolds number? All these problems may not be satisfactorily answered without experimental verification. Therefore, laboratory measurements were carried out in this study to verify the analytical results and to build a data base for further development. The requirements of similarity between hydraulic scale-models and their prototypes are found by the application of several relationships generally known as the laws of hydraulic similitude, i.e., geometrical, kinematic and dynamic similitude. These laws, which are based on the principles of fluid mechanics, define the requirements necessary to ensure correspondence between model and prototype. Complete similarity between model and prototype requires that the system in question be geometrically, kinematically and dynamically similar. Geometric similarity implies that the ratio of all corres ponding lengths in the two systems must be the same, kinematic similarity exists if all kinematic quantities in the model, such.as velocity, is similar to the corresponding quantities in the prototype, 43 44 and dynamic similarity requires that two systems with geometrically similar boundaries have the same ratios of all forces acting on corresponding fluid element of mass. Following the basic dynamic law of Newton, which states that force is equal to rate of momentum, dynamic similarity is achieved when the ratio of inertial forces in the two systems equals that of the vector sum of the various active forces, which include gravitational forces, viscous forces, elastic forces and surface tension forces in fluid-motion phenomena. In other words, the ratios of each and every force must be the same, as given in the equation form V (F) (Fv) (F.) (Fs) P a P P- P. P (50) where subscripts p and m refer to prototype and model, relatively. Since it is almost always impossible to obtain exact dynamic similitude, it becomes necessary to examine the flow situation being modeled to determine which forces contribute little or nothing to the phenomenon. These forces can then be safely neglected with the goal of reducing the flow to an interplay of two major forces from which the pertinent similitude criterion may be analytically developed (Rouse, 1950). For a model of hurricane-induced flooding of coastal areas, elastic forces and surface tension forces are sufficiently small and can be neglected. The condition for dynamic similitude reduces to equating the ratio of inertial forces to the ratio of either gravity 45 forces or viscous forces. Viscous forces are only considered in the model of canopies whose surface friction effects are investigated. In models for measuring form drags in turbulent flow with high Reynolds numbers (inertia force/viscous force), viscous forces are small compared to the major forces due to the turbulent fluctuations and this can be neglected in this instance. Since the vertical dimension scale (involving flow depth) cannot follow the horizontal dimension scale in building models as the flow depth would be much too small for measurements to be made, or the viscous force would become important and cannot be neglected for the small flow depth if the same fluid is used for the prototype and the model. Therefore, a model with a different vertical dimension scale than horizontal dimension scale is used to keep the Reynolds numbers in the turbulent flow range. For simplicity, such distorted models will be introduced first since undistorted models with the same length scale in both the vertical and horizontal dimensions can be regarded as a special case of the former. Distorted Model for Buildings The fundamental model scale ratio may be written as: L B Length Scale (horizontal): (51) Lm Bm D Depth Scale (vertical): Nd = D^ (52) T Time Scale: (53) 46 Force Scale: (54) where L = horizontal length, B = horizontal width, D = vertical depth, T = time and F = force. Following the development of Christensen and Snyder (1975), the force scale for the gravity component in the nearly horizontal direction of the principal flow may be written as unit sine weight volume of slope i ^ i r~> K = g L B D (D /L p g 2 P P P P P = (_Â£) (_Â£.) N N Pmgm CCtTU V V Vd P 9 P^P m m m m' nr m Jm (55) where p is the fluid density, g is the gravitational acceleration and D/L is the bed slope ar the slope of the energy grade line. In a unidirectional flow the inertial force can be expressed as a horizontal, or nearly horizontal area multiplied by the Reynolds shear stress, which is proportional to the fluid density and the time mean value of the product of a vertical velocity fluctuation and the corresponding velocity fluctuation in the direction of the time mean flow. Consequently, the inertial force is F.. = puv1 (area) (56) and the inertial force scale can be written as 47 area D_ r Pp pn m m Pn N! N . m N, (57) In order to have dynamic similarity between model and prototype, Kgravity should be equal to Kn-nertial i-e,} equation (55) should be the same as equation (57). This condition is expressed by or (58) where gp is assumed equal to g^. The ratio between a gravitational force and an inertial force is commonly known as the Froude number, and the resulting time scale (equation (58)) is the similarity criterion of the Froude law for distorted models. The scale ratios of the drag coefficient and friction factor in distorted Froude models have to be determined before experimental data can be interpreted correctly. The drag force proposed for the present study is given by equation (39) ^d = Td* A = m^gd^P jr- A (59) The drag force scale in the flow direction can then be written as 48 1 L B JLÂ£ 1 L B m m L 2 )(Cn)ABmpJr) L B DmmmmT mm m or K, * d ,r \ vp "d l (Cpj m m (60) To satisfy the ratio of the force scale in the Froude model law, Kd must be equal to K^, and by substituting the time scale, Nt = NÂ£/(Nd^2, into equation (60) gives (CD> (61) Shear forces generally may be expressed by the Darcy-Weisbach form F s (62) where f is the friction factor and can be substituted by the equivalent friction factor, T, or bottom friction factor, f', later for the present use. The shear force scale is given as or Ks \ m pm t (63) 49 Dynamic similarity requires that = K^. Substituting the time scale, equation (58) into the required equality, i.e., Â£><>Â¡ (i)2. m pm (64) gives the expression for the scale ratio of the friction factor f N. _B. = A fm (65) Comparing equation (61) with equation (65) it is noted that the drag coefficient is the same in the model as in the prototype, however, the friction factor of the prototype should be modified by an inverse distortion ratio, N^/N^, in the Froude distorted model. The distortion ratio usually is defined as D = r Nd (66) Undistorted Model for Mangrove Stems and Roots Due to the fact that the dimensions of mangrove stems and roots are one to two orders of magnitude less than the water depth, an undistorted model can be used in this part of the study. All the methodology applied in the previous section for a distorted model is also applicable for this analysis. In the case of an undistorted model where = N^, equation (58) reduces to Nt 50 which is the time scale for an undistorted Froude model. Both the dimensionless coefficients and f are the same in the prototype and model, and the distortion ratio becomes unified in this instance. Distorted Model for Canopy Before the model law for canopies is derived, it is necessary to determine what kind of boundary layer forms over the surface of a mangrove leaf. A prototype red mangrove canopy was tested in the hydraulic laboratory flume and it was quite apparent that all the leaves bent in the direction of flow even at a flow velocity less than 10 cm/sec. Such high flexibility makes the leaves more resistant to a storm attack. As a result, leaves offer only skin friction and no form drag to resist the flow. The surface of a leaf is assumed to be smooth in this study so that theoretical and empirical results on the behaviour of a boundary layer on a smooth flat plate can be applied. In general, the point of instability on a flat plate at zero incidence to the flow is determined by the critical Reynolds number U x (R e.x^crit 'crit (68) in which U is the free stream velocity and x is the distance from the leading edge of the plate measured along the plate. An analytical stability criterion developed by R. Jordinson, based on W. Tollmien's theory, is given by U 6 / (69) 51 where = displacement thickness and (Schlichting, 1979) (70) Combining the last two equations give (R ) = 9.1 x 104. c jA Ci I U In reality, the position of the point of transition from laminar to turbulent flow will depend on the intensity of the turbulence in the external flow field. This has been investigated experimentally by J. M. Burgers, B. G. Van der Hagge Zijnen and M. Hansen in 1924. These measurements led to the result that the critical Reynolds number was contained in the range U x ()crit = 3-5 x 105 t0 5 x 105 (Schlichting, 1979) (71) Similar experiments done by Schubauer and Skramstad in 1947 also yielded results which indicated that the critical value of R is in a range G jX from 9.5 x 105 to 3 x 106 depending on the relative intensity of the free-stream turbulence, (l/U^Ku'u'/sj/2 (Hinze, 1975). Therefore, the minimum value of (R ) .. is chosen as 3.5 x 105 for the present study. cjX crit In the prototype, the maximum value of x is the largest leaf length and was found to be 5 inches; the highest flow velocity is assumed to be 10 ft/sec, which results in a maximum value of (R ) .. of about c)A Li I l 2.98 x 105, which is still lower than but near the minimum value (R ) ... This shows that a turbulent boundary layer has very little g ,x cn l chance to be formed over such a short length, and that a laminar boundary layer should prevail over the entire leaf area. 52 Skin friction can be expressed in terms of a dimensionless skin friction coefficient, C^., times the stagnation pressure, pU2/2, and area of the plate, A, as follows Ff = CfP I A (72) in which, for a laminar boundary layer, C f 1.328 (73) where R = lk/v denotes the Reynolds number formed by the product of the Xj * plate length and the free-stream velocity (Schlichting, 1979). The skin friction scale in the flow direction may be written as K f LnV1/2 (-P P) TnV JELÂ£ f III III \ m m m L 2 (/) A T^~ (j51) A P m (74) where z and A are the leaf length and leaf area, respectively. Following the undistorted Froude law, = (N^2 p and v are the same in proto type as in the model since the same water properties are assumed in the two systems. Equation (74) is then reduced to 3A -V2 A V (75) 53 For dynamic similarity, Kp has to be set equal to K i.e. % Â£ -V2 A 3 = <"*> m m (76) giving the length scale Â£_ An2 f(Ni) m m (77) It is obvious that the dimensions of a leaf need to be distorted according to the scale ratio shown in equation (77), which is the result of inclusion of viscous effects on a leaf surface in a Froude law controlled flow model. CHAPTER V MODEL DESIGN Recirculating Flume The present model tests were conducted in the hydraulic laboratory flume of the Civil Engineering Department at the University of Florida. Figures 14 and 15 show the primary elements of the flume geometry. The main channel is 120 feet (36.58 meters) long, 8 feet (2.44 meters) wide, and 2.7 feet (0.81 meter) deep. A false-bottomed section 20 feet (6.1 meters) in length and 13.4 inches (34 cm) deep is located at the longitudinal center of the flume. Centered in the false-bottomed area, observation windows cover a length of 12 feet (3.66 meters) and are 2 feet high (starting at the bed level). The 74 kW (100 HP) flume pump has a maximum discharge of 40 cfs (1.1 m /sec). Between the pump and the overflow weir are two sets of 8 inch long, 2 inch diameter poly vinyl chloride pipes arranged in a honeycomb fashion. Two more sets of these pipes, which act as flow straighteners, are located just beyond the outlet weir. By adjusting two gate valves at the main delivery pipe and return pipe, the flow rate and depth over the Thomson V-notch weir can be regulated. A Poncelet rectangular weir is also available for high discharges. A motor-driven sluice gate at the downstream end on the main channel serves to regulate the water depth in the main flume and to moderately regulate the discharge. 54 FIGURE 14: Plan of Flume f Power Source for Trolley Rail for Movable Trolley Return f Channel 38 (l.llm) iv > \ Main Channel i*r- 8' (2.44m) T 28"(0.8lm) . (1.22m) FIGURE 15: Cross Section of Flume 56 A movable trolley which spans the entire width of the flume and which has a maximum towing speed of 2 feet per second provides the work-deck for calibrating velocity meters as well as collecting data. To determine the drag coefficient and equivalent friction factor for a given roughness in the rough turbulent flow, the energy loss, aH, has to be measured (cf. equation (47)). According to the principle of conservation of energy, the total energy head at the upstream section 1 should be equal to the total energy head at the downstream section 2 plus the two sections, i.e., 2 2 u, u dl + 2g = d2 + 2? + aH or AH = (d1 is horizontal and a value of unity is assumed for the energy coefficient (Henderson, 1966). Therefore, the energy loss aH due to the friction in turbulent flow can be measured by knowing the water depths and mean velocities at the two sections. Relating the measured results of energy loss to the Darcy-Weisbach equation AH ,, Hi fe 2g Ra (79) 1 ^1 ^2 in which Ua = 2 (U-j + U2) and = 2- the equivalent friction factor f' can be determined for the designed roughness elements. To determine the water depths and velocities, some instruments are employed for this study and described in the following section. 57 Instrumentation Velocity Meter A Novonic-Nixon type velocity meter was employed for all velocity measurements. The probe consists of a measuring head supported by a thin shaft 18 inches long with an electrical lead connection. The head consists of a five blade, impeller mounted on a stainless steel spindle, terminating in conical pivots (Figure 16). These pivots run in jewels mounted in a sheathed frame. The impeller is 1 cm in diameter, machined from solid PVC and balanced. An insulated gold wire within the shaft support terminates 0.1mm from each rotor tip. As the rotor is rotated by the motion of a conductive fluid, the small clearance between the blades and the shaft slightly varies the impedance between the shaft and the gold wire. This impedance variation modulates a 15KHz carrier signal, which in turn is used to detect rotor rotations. The range of this velocity meter is from 2.5 to 150 cms-1 (0.08 to 4.92 fps) with an advertised accuracy of + 1 % of true velocity. Its operating temperature is from 0 to 50C (32 to 122F) with an operating medium of water or other fluids having similar conductive properties. The shaft of the current meter was clamped to the rack of a point gage. The point gage bracket was then bolted to the trolley carriage so that the instrument could be easily removed from its bracket with no deviation in the vertical setting. Also all the accuracy and ease of a point gage and vernier is accrued. 58 O I 2 cm > I I FIGURE 16: Novonic Nixon Velocity Meter 59 Data Acquisition System (DAS) The data acquisition system is composed of two pieces of equipment: an input box and an HP 9825A desk-top programmable calculator (Figure 17 and 18). The input box, which is specially designed for coupling with the HP 9825A, has connectors for fifteen thermistors, ten Cushing electromagnetic current meters, two Ott velocity meters and two Novonic- Nixon velocity meters. It contains the electronic circuitry which takes the raw transmission from the measuring devices and converts it into usable signals tor the programmable calculator. An electronic timer which registers six counts per second is also contained in the input box. The HP 9825A interfaces with the input box to provide program control and data storage capabilities. The calculator has a 32-character LED display, 16-character thermal strip printer, and a typewriter-!ike keyboard with upper and lower-case alphnumerics. A tape cartridge with the capacity of 250,000 bytes is used with this calculator to store and access the programs. Based on the manual of the HP 9825A and the instructions provided by Morris (1979), programs designed to calibrate the Novonic-Nixon meter, measure the velocities and perform linear regression are listed in Appendix B. Through the DAS a substantial amount of time usually used in experiments and data reduction was saved and the accuracy of results was greatly enhanced. Depth-Measuring Device Determination of the flowing water depth by measuring the difference of water surface elevations is the most important part, except for the measurement of the flow velocity, of the laboratory experiments. However, 60 FIGURE 17: Input Box of Data Acquisition System FIGURE 18: HP 9825A Programmable Calculator 61 the measurement of water surface elevations is not easy due to the rough water surface of turbulent flow. In addition, an accuracy of one millimeter or better is needed for the depth measurement, since the difference of water surface elevations at two sections is less than one centimeter in many test cases. Therefore, a stable and sensitive depth measuring device is required for the present study. Figure 19 shows the schematic diagram of the device designed, in which the hoses connected to the tube, which have a diameter of 0.2 inch, were extended to the desired cross-sections in the flume. A 10 inch long, 0.15 inch diameter glass tube was attached to the end of each hose and positioned perpendicular to the water surface. In high velocity flows some weights were added to the 0.15 inch diameter tube in order to maintain its vertical position. The diameter of the tube is 2 inches which is large enough to allow the point gage to be able to contact the plane water surface without the influence of surface tension on the side wall of the tube. The point gage was attached to the top of the tube, and the still water level is indicated when a white ball on the gage appears, which indicates that the sharp tip of the gage is touching the water surface (Figure 20). A manual hand operated vacuum pump was used to help initiate a siphon between the water in the flume and in the tube at the beginning of each test and to pump air bubbles out of the hoses periodically during the test. 62 FIGURE 19: Setup of Water Depth Measuring Device FIGURE 20: Point Gage and Tube 63 Selection of Model Scales In the last chapter the scale-model relationships based upon the Froude law were derived. The scale to which the model should be constructed depends on the following factors: the size of the flume (length, width and depth), the discharge capacity of pump, the accuracy of instrumentation and the dimensions of the prototype. According to these factors, the vertical length scale of 1:10 (or = 10) is selected through the entire study for both distorted and undistorted models. Mangroves The dimensions of all stems and roots, including height and diameter of the prototype, are reduced to 1/10 for the model based on the undistorted Froude law. However, for the canopy some distortion of scale between prototype and model is required according to equation (77) in the Chapter IV, i.e., it. A* m <\> in which N = Nd = 10, i is the length of a leaf and A is the surface area of the leaf. For simplicity, the elliptic shape of mangrove leaves are approximated by a rectangular area with a length n and a width w. Equation (77) is then reduced to (80) By choosing 1:10 for the length scale for this study, the width scale of 64 the leaf becomes w 7/ = 10/4= 56 (81) m Therefore, the dimensions of the leaves used in the model can be estimated from the derived relations and are shown in Table 5, which are based upon the prototype data listed in Table 2. TABLE 5: Scale Selection for canopy Section No. (1) (2) (3) prototype length (Â£) 2.00 3.00 4.00 (in) width (w) 1.00 1.50 1.75 Model length (Â£) 0.20 0.30 0.40 (in) width (w) 0.018 0.027 0.031 p' m 10 10 10 w /w p m 56 56 56 Buildings This part includes the three previously discussed kinds of buildings: high-rise, medium-rise and residential buildings. Table 6 shows a sumnary of the average parameters for these three categories. In searching for material to be used in constructing the buildings in the model, it was found that the ratio of length and width of a standard block was very close to that of the prototype. Another advantage in using concrete blocks is that they are easy to set up, since each TABLE 6: Average Parameters of Prototype and Model for Building Areas PROTOTYPE MODEL (Nd = 10) Type of Average Dimension (ft) Approximate Dimension (ft) Density % Dimension (in) Density Buildings 1ength width 1ength width no. no. Deve- 1 oped 1ength width no. no. l000'x500' acre 8'x8' 8'x2,871 High-Rise 224 112 225 109 7.19 0.63 36 174 15.50 7.50 28 10 Medium-Rise 99 51 103 ' 50 23.62 2.06 24 80 15.50 7.50 20 7 Residential 62 30 68.87 6.00 26 48 15.50 7.50 20 7 Note: 1 acre = 43,560 ft^ 66 concrete block is heavy enough to withstand all the flow velocities in this study. Therefore it was not required to anchor them to the flume bottom. Scales of the model are then determined from the horizontal dimension of a concrete block (15.5 in x 7.5 in) and the dimensions of prototype buildings, as shown in Table 6. It is noted that in order to scale the prototype buildings into the 8 foot wide flume, it was necessary to use a distorted model. Model Setup Mangrove Stems and Roots Based on the average parameters obtained from the prototype, patterns of red and black mangroves were designed and shown in Figures 21 and 22. These two patterns were the best arrangements that could be achieved in the modeling in order to insure that the stems and roots were distributed evenly and yet still maintained their own natural characteristics in dispositions. For example, the prop roots were arranged in a hexagon pattern, which was found to be the most common disposition found in natural. The staggered pattern used for the stems of black mangroves and the root system of red mangroves was considered to be the best regular pattern to simulate the fully random distribution found in the prototype. The legends listed in Figures 21 and 22 were the actual dimensions used in model setup. The stems of red and black mangroves were simulated by dowels of the specified diameters and heights. The substems and prop roots were simulated by galvanized nails with the caps removed. The air roots of black mangroves would be very hard to model on a one by one basis due to 67 o o o 0+0 o o o O o 0 + 0 o o o o O o + o o o o o o+o o O o o o o 0*0 o o O o 0+0 o o o o o 0+0 o O o o O 0+0 o o o 0 o o O O 0+0 o o o o o 0+0 o o o o o o 0 0 o o 15 h3ui ^r?ir o + o o o O Main-Stern Sub-Stem 0 Prop Root FIGURE 21: Model Setup for Red Mangroves 68 o Stem Air Root FIGURE 22: Model Setup for Black Mangroves 69 their high density and small dimensions. Therefore, a manufactured nylon door mat whose strings have the same height (0.6 inch) and the same thickness (0.025 inch) as the design model dimensions of air roots was used. Density of the strings is 44 per inch square area which is 91 l of the average design density (48 per inch square area). The only deficiency in using this mat is that the strings are blade-shaped, which may cause a higher resistance to the flow than cylindrical air root. However, considering the advantages of using the mat, this deficiency is considered to be insignificant. These dowels, nails and mats were fixed on three 8 feet by 4 feet marine plywood sheets, which were coated with latex paint to prevent swelling (Figures 23 and 24). The plywood sheets were secured to the false bottom by a row of concrete blocks and by a 24 feet long L shaped steel beam attached to two sides of the plywood sheets (Figure 25). The row of concrete blocks stacked 15 inches high was placed in the main flume, starting from the last flow straightener and extending a distance of 80 feet. Thus only half of the flume width was used in these experiments. Canopy of Red Mangroves During the second part of the experiments a canopy was constructed in the red mangrove area. Strips of galvanized metal plates were used to simulate 'leaf strips'. This assumes that the leaves are closely connected to each other. Strips with three different widths, 0.2, 0.3 and 0.4 inch, represent three different sizes of leaves, as shown in Table 5. Each stripe has a height of 9.6 inches, which covers 539, 360 and 308 leaves for section #1, #2 and #3, respectively. Stripe numbers FIGURE 23: Stems and Roots of Red Mangroves FIGURE 24: Stems and Roots of Black Mangroves 71 FIGURE 25: Overview of Setup for Mangroves FIGURE 26: Setup of Model Equivalent of Canopy of Red Mangroves 72 for each section, which is 5 feet long, 34.2 feet wide and 8 feet high in the prototype, can be calculated from the densities measured (cf. Table 2). As a result, stripe numbers needed in the model for section #1 to #3 are found to be 25, 20 and 10, respectively. These stripes were also arranged in a staggered pattern, as shown in Figure 27. Figure 26 shows the setup of the stripes in which the stripes are sus pended from the top of the supporters and fixed to the plywood bottom. Buildings In this part of the experiments, the whole width of the flume was used. As mentioned in the last section, the concrete blocks with dimensions 7.5 inches x 7.5 inches x 15.5 inches were used to simulate the buildings for the three different types of developed areas. Figure 28 shows 21 patterns to be tested in which no. 1 to 13 were designed to simulate high-rise building areas, while no. 14 to 21 were for medium- rise building and residential areas. As can be seen in these patterns, both the aligned and staggered dispositions were included for each density of the buildings. The design densities are started from low to high and will at least cover the average densities obtained from the prototype for the three developed areas (cf. Table 6). No extra work was needed to anchor these concrete blocks except to move them into the desired positions, since each concrete block weighs about 38.5 pounds and two layers of blocks are steady enough to withstand all the flows used in this study. SECTION # I SECTION# 2 SECTION # 3 73 X 0.2" x 9.6" 0.3" x 9.6" A 0.4" x 9.6" FIGURE 27 Distribution of Leaf Stripes in the Model OPEN AREA 4*ROOT AND STEM AREA-*) 74 FLOW (2) (3) (4) FIGURE 28: Building Patterns Designed for the Tests 75 uu IT nn M FLOW V (9) (10) U U _n u u n n - (ID (12) (13) 8' i l 8' 11 (14) (15) FIGURE 28: CONTINUED 76 FLOW i , 0 00 0 (16) (17) - 0 0 0 D 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0 (18) (19) D 0 0 0 DOOODO D 0 0 D 0 OOOODO (20) (21) FIGURE 28: CONTINUED CHAPTER VI EXPERIMENTAL TEST SERIES Experimental Procedure Calibration of Velocity Meter After the construction of the apparatus, the first step in the experimental procedure was to calibrate the velocity meter. Mounted with its normal support on the carriage, the Nixon meter was pulled through still water at constant velocity over a distance of 20 to 40 feet with the trolley. By operating the specific keys on the HP 9825A calculator to execute program statements which read initial and final values of propeller revolutions and time, the average frequency of the current meter, the true velocity and the percent error of the calibra tion curve were then computed and printed out. If the absolute error was greater than 5 % the instrument was recalibrated. The meter was checked in the range of 5 to 60 cm/sec and no less than 20 points were used to determine a linear least square fit of frequency versus velocity. Appendix C contains a complete program listing for the HP 9825A. Measurements of Mean Velocities The velocity and depth obtained in this study were measured in the center line of the test sections where the influence of sidewall was not felt. The experimental run begins when the main pump is started. It 77 78 usually takes about twenty minutes for flow to reach steady state for each set of discharge values. After the flow became stabilized, velocities were taken at one section 30 feet downstream of the last flow straighteners. Nine points on a vertical at the relative depths of: z/d = 0.1504, 0.1881, 0.2352, 0.2492, 0.3679, 0.4601, 0.5754, 0.7197, 0.9000, were sampled to best describe the vertical velocity profile (Christensen, 1978). The velocity at each depth was then determined by the velocity program (Appendix B) from the calibration formula and was printed out for immediate checking. Each velocity obtained is on average velocity over a time span of 30 seconds which is the maximum time interval that can be used with the HP 9825A. Plotting the vertically distributed velocities on graph paper and integrating over the water depth yields the discharge per unit width /udy. The spatial mean velocity for each run then was obtained from the value of unit-width area divided by the water depth. Even though this method is time consuming for the large number of runs, it is still the best way to determine the mean velocity for the mangrove part of the experiments in which the test channel occupies one half of the main channel. For the building part of the experiments, in which the entire main channel was used, a discharge formula for the Thomson weir derived by the hydraulic laboratory of Civil Engineering Department, University of Florida, was applied to determine the mean velocities, i.e., 2.514 , Q = 2.840 H (82) where Q is the discharge from the Thomson weir in cubic feet per second, and H is the vertical distance in feet between the elevation of the 79 lowest part of the notch or the vertex and the elevation of the weir pond. Eighteen runs with mean velocities from 17 cm/sec to 53 cm/sec were tested by both methods to determine the accuracy of the weir formula; the results are shown in Figure 29. It is apparent that the mean velocities obtained using the Nixon meter in the center line of the flume Un is slightly larger than that given by weir formula Uw, but is within a limit of 5 %. This small error is considered to be insigni ficant and may be compensated for by the advantages of using the weir formula. For instance, the fluctuating water level above the weir vertex due to the instability of the pump was often observed, therefore, the mean velocity obtained from an average value of H over a longer period of time should be more representative than that measured by the Nixon meter over a 30 second period. Measurements of Water Depths For each run three water depths were measured by using the device shown in Figure 19. Two water depths were taken at the two sections which covered the roughness area and one was taken at the section where velocities were measured out of the roughness area, located 30 feet downstream of the last flow straighteners. Since the water head losses between the two sections in the model tests were in the range from less than 1 to a few centimeters, the water depths were measured to an accuracy of one hundredth of a centimeter for a precise and reliable result. To implement this fine measurement, all the siphon hoses used in the tests were kept free of air bubbles and the well graduated electronic point gages were used. Before each run the still water 80 FIGURE 29: Relation Between U and U n w 81 depth was measured and its scale reading for water surface elevation on the point gage was recorded. The same reading was performed for each section after the flow became stable. From the difference of these two readings for water surface elevations and the initial still water depth the flowing water depth can be calculated. In general, it takes about 5 to 20 minutes for a new water level in the tube (cf. Figure 19) to reach its equilibrium state, which can be observed by moving the vernier on the point gage to see whether any change in the water level is detected. This water depth measuring device worked very well through the entire experiment and provided consistent and reliable data. Experimental Runs Mangrove Areas The total model lengths of the red and black mangrove areas were 5 and 15 feet, respectively (Figure 25). For the red mangrove area the water depths were measured at the two ends of the 5 feet long area. For the black mangrove area, the first section was chosen 4 feet from the front end, and the second section was located 3.5 feet from the rear end of the black mangrove region so that the influences, including the disturbance caused by the red mangroves in the front, and the depth drop due to the end of the plywood sheets in the rear could be eliminated. Therefore, a total length of 7.5 feet centered in the middle section of the black mangrove area was used to measure the energy loss. During the first part of experiments, 7 runs were conducted for the air roots of the black mangrove area to determine its apparent roughness 82 height and friction factor. In the second part of the experiments, 38 runs were performed for the red mangrove areas (without canopy) and black mangrove areas by adjusting the discharge value and changing the still water depths so that the flow Reynolds number (R^ = UgR^/ v) covered a range from 20,000 to 55,000 while the Froude numbers varied from 0.14 to 0.44. An additional 32 runs were conducted for the red mangroves with canopy at the later stage to determine the importance of a canopy in reducing the flow energy. Building Areas At least 10 runs were conducted for each of the 21 patterns shown in Figure 28. These runs for each pattern were controlled by adjusting the flumes discharge valve and the tail gate so that they covered a range of Reynolds number (R^) from 20,000 to 70,000, while the Froude number varied from 0.1 to 0.5. Figure 30 shows 20 pictures of the designed patterns in which pattern No. 9 is not included due to the faulty picture. The results obtained for medium-rise building areas can be converted using appropriate scaling factors to use in residential areas since these two areas are presumed to have the same relative distributions and have only dimensional differences. 83 FIGURE 30: Designed Building Patterns in the Tests |

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FRICTION IN HURRICANE-INDUCED FLOODING By SHANG-YIH WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY â€¢ UNIVERSITY OF FLORIDA 1983 ACKNOWLEDGEMENTS The author wishes to express his sincerest gratitude to the chairman of his supervisory committee, Dr. B. A. Christensen, for all his expert guidance, the tremendous benefit of his professional competence and vast practical experience throughout this study. He also wishes to thank Dr. T. Y. Chiu for his advice, understanding, encouragement and support during the author's six years of graduate study at the University of Florida. Without their untiring patience and help this dissertation would not have been possible. Thanks are also due to Dr. D. P. Spangler, Dr. B. A. Benedict, Dr. T. G. Curtis and Dr. H. Rubin for serving on the author's superÂ¬ visory committee and for their consulting and assistance. Appreciation is extented to Drs. A. J. Mehta, D. L. Harris, F. Morris and P. Nielsen for their suggestions and providing reference, which contributed greatly to this study. The author is indebted to Mr. E. Dobson for his technical assisÂ¬ 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 particiÂ¬ pation in every phase of this study has made these years more joyful. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES vi LIST OF FIGURES vii LIST OF SYMBOLS x ABSTRACT xv CHAPTER I. INTRODUCTION 1 Storm Surge Prediction 4 Objectives of Present Work 6 II. FIELD EXPLORATION OF PHYSICAL ENVIRONMENT 7 Mangrove Areas 7 General View of Mangroves in Florida 7 Sampling of Mangroves 8 Developed Areas 15 High-Rise Building Areas 15 Medium-Rise Building Areas 17 Residential Areas 21 III. THEORETICAL BACKGROUND AND DEVELOPMENT 22 Hydrodynamic Equations for Storm Surges 22 Wind Shear Stress 23 Wind Velocity Profile in Vertical 25 Proposed Wind Shear Stress on Obstructed Areas 27 Bed Shear Stress 31 Post Approach, Friction Factor for Surges in Unobstructed Areas 33 Proposed Approach, Friction Factor for Surge in Obstructed Areas 38 iii Page IV.EXPERIMENTAL VERIFICATION OF FRICTION FACTORâ€”MODEL LAWS 43 Distorted Model for Buildings 45 Undistorted Model for Mangrove Stems and Roots 49 Distorted Model for Canopy 50 V.MODEL DESIGN 54 Recirculating Flume 54 Instrumentation 57 Velocity Meter 57 Data Acquisition System (DAS) 59 Depth-Measuring Device 59 Selection of Model Scales 63 Mangroves 63 Buildings 64 Model Setup 66 Mangrove Stems and Roots 66 Canopy of Red Mangroves 69 Buildings 72 VI.EXPERIMENTAL TEST SERIES 77 Experimental Procedure 77 Calibration of Velocity Meter 77 Measurements of Mean Velocities 77 Measurements of Water Depths 79 Experimental Runs 81 Mangrove Areas 81 Building Areas 82 VII.PRESENTATION AND ANALYSIS OF DATA 84 Mangrove Areas 84 Building Areas 95 Determination of Drag Coefficient 110 Drag Coefficient-Building Density Relation 110 Drag Coefficient-Disposition Parameter Relation 113 i v Page VIII. DISCUSSIONS AND CONCLUSIONS 119 Mangrove Areas 120 Developed Areas 121 Ocean Bottom 123 Forested Areas 127 Grassy Areas 128 Conclusions 129 APPENDICES A. FIELD RECORDED DATA FOR MANGROVES 132 B. COMPUTER PROGRAM LISTINGS 145 C. TABLES OF EXPERIMENTAL DATA 151 BIBLIOGRAPHY 161 BIOGRAPHICAL SKETCH 165 v LIST OF TABLES Table Page 1. Average Parameters of Sampling Mangroves 14 2. Average Characteristics of Canopy 16 3. Average Parameters of High-Rise Buildings 19 4. Average Parameters of Medium-Rise Buildings 20 5. Scale Selection for Canopy 64 6. Average Parameters of Prototype and Model for Building Areas 65 7. Statistical Values of Experimental Results for Mangrove Areas 94 8. Statistical Values of Experimental Results for Building Areas 109 9. Relations Between Disposition Parameters and Drag Coefficients 115 10. Typical Values for Mangrove Areas 120 11. Bed Friction Characteristics of Three Entrances 126 A1. Parameters of Sampling Red Mangroves 141 A2. Parameters of Sampling Black Mangroves 142 A3. Characteristics of Canopy of Red Mangroves 143 Cl. Experimental Data for Red Mangroves (Without Canopy) 151 C2. Experimental Data for Red Mangroves (with Canopy) 152 C3. Experimental Data for Black Mangroves 153 C4. Experimental Data for Building Areas 154 LIST OF FIGURES Figure Page 1. Prop Roots of Red Mangroves 9 2. Air Roots of Black Mangroves in 1 Foot Square Areas 9 3. Red Mangrove Area (Sampling Area #4) 11 4. Black Mangrove Area (Sampling Area #7) 11 5. Field Data Record for Red Mangroves 12 6. Field Data Record for Black Mangroves 13 7. Measurement of Density of Canopy 16 8. Section View of Survey Area (Red Mangroves) 16 9. Top View of Building Shapes on Coastal Areas 18 10. Wind Stress Coefficient over Sea Surface 26 11. Plan View for Wind Stress over an Obstructed Area 28 12. Elevation View for Wind Stress over an Obstructed Area 28 13. Distribution of Horizontal Apparent Shear Stress and of its Drag, Inertial and Viscous Components 39 14. Plan of Flume 55 15. Cross Section of Flume 55 16. Novonic-Nixon Velocity Meter 58 17. Input Box of Data Acquisition System 60 18. HP 9825A Programmable Calculator 60 19. Setup of Water Depth Measuring Device 62 20. Point Gage and Tube 62 vii Figure Page 21. Model Setup for Red Mangroves 67 22. Model Setup for Black Mangroves 68 23. Stems and Roots of Red Mangroves 70 24. Stems and Roots of Black Mangroves 70 25. Overview of Setup for Mangroves 7T 26. Setup of Model Equivalent of Canopy of Red Mangroves 71 27. Distribution of Leaf Stripes in the Model 73 28. Building Patterns Designed for the Tests 74 29. Relation Between U and U, 80 n w 30. Designed Building Patterns in the Tests 83 31. Relation Between f' and R' for Red Mangrove Areas (Without Canopy) 85 32. Relation Between Cn and R' for Red Mangrove Areas (Without Canopy) 86 33. Relation Between f' and Water Depth d for Red Mangrove Areas e (Without Canopy) 87 34. Relation Between f and Râ€˜ for Red Mangrove Areas (with Canopy) e e 88 35. Relation Between Cn and R1 for Red Mangrove Areas (with Canopy) 89 36. Relation Between f' and Water Depth d for Red Mangrove Areas (with Canopye) 90 37. Relation Between f and R' for Black Mangrove Areas 91 e e 38. Relation Between CQ and R^ for Black Mangrove Areas 92 39. Relation Between f1 and Water Depth d for Black Mangrove Areas e 93 40. Relation Between f and R' for Building Areas 94 e e Figure Page 41. Relation Between Cn and R' for Building Areas 102 D e 3 42. Relation Between f1 and Water Depth d for Building Areas e 108 43. Relation Between Cn and Density m for High-Rise Building Areas 111 44. Relation Between Cn and Density m for Medium-Rise Building and Residential Areas 112 45. Position Spacings. Definition Sketch 114 46. Relation Between Cn and S,/D in Aligned and Staggered Patterns 116 47. Relation Between and S^/D 118 Al. Field Recorded Data for Red Mangroves (Area #1) 132 A2. Field Recorded Data for Red Mangroves (Area #2) 133 A3. Field Recorded Data for Red Mangroves (Area #3) 134 A4. Field Recorded Data for Red Mangroves (Area #4) 135 A5. Field Recorded Data for Red Mangroves (Area #6) 136 A6. Field Recorded Data for Black Mangroves (Area #8) 137 A7. Field Recorded Data for Black Mangroves (Area #9) 138 A8. Field Recorded Data for Black Mangroves (Area #10) 139 A9. Field Recorded Data for Black Mangroves (Area #11) 140 i x LIST OF SYMBOLS area leaf area for prototype and model, respectively horizontal width drag coefficient skin friction coefficient average diameter of obstruction vertical depth for prototype and model, respectively water depth water depth at section 1 and section 2, respectively average water depth of d-j and d2 diameter of pipe drag force elastic force skin friction gravitational force inertial force Froude number =/ U / gd d d surface tension force viscous force Darcy-Weisbach friction factor based on diameter of the pipe friction factor based on hydraulic radius equivalent friction factor total friction factor gravitational acceleration vertical depth above the vertex of Thomson weir protruding height of obstructions above water surface indicial functional parameter wind stress coefficient drag force scale skin force scale gravity force scale inertial force scale shear force scale equivalent sand roughness apparent roughness horizontal length leaf length for prototype and model, respectively density = no. of obstruction elements/area indicating the subscripted parameters for model and prototype, respectively total number of obstruction elements vertical length scale horizontal length scale force scale time scale Manning's n wetted perimeter of flow cross-section pressure Ps pressure on the water surface Q discharge from the Thomson weir q , q discharge per unit width x y R hydraulic radius = A/P R-|,R2 hydraulic radius at section 1 and section 2, respectively Rfl average hydraulic radius of R-j and R^ R^ reduction factor for wind stress Rq Reynolds number based on depth = U d /v R' Reynolds number based on hydraulic radius = UR /v 6 ad R* wall Reynolds number = u^.k/v R v Reynolds number = Uâ€žx/v C )A r radius of the pipe S slope of energy grade line = aH/L S , corner to corner distance between the roughness elements in adjacent transverse raws SÂ£ longitudinal spacing between two successive roughness elements S^. laternal spacing between two roughness elements s free surface displacement from mean sea level S. Dev. standard deviation t time variable U spatial mean flow velocity U-j, U2 spatial mean flow velocity at section 1 and 2, respectively Ua average spatial mean velocity of U-| and U2 U , U spatial mean flow velocity in x and y directions, respectively x y Uoo free stream velocity u,v,w instantaneous components of the water velocity in the x, y, z coordinate directions, respectively u u1 , V' uf,t W(z) w m x, y, z z 0 8 Y 6 AH Til >fl2 e K u V e P pa 7b Tbxâ€™ Tby time-mean velocity in the direction of flow turbulent velocity fluctuations in the x and z directions, respectively friction velocity based on bottom friction friction velocity based on total friction time-mean wind velocity at the elevation z above water surface critical wind velocity time-mean wind velocity at the elevation 10 meters above water surface leaf width for prototype and model, respectively Cartesian coordinate directions dynamic roughness shape factor specific weight of water = pg displacement thickness energy loss per unit weight of fluid fractions of distances nid and n2d from bottom to the total depth d, respectively latitude Von Karman's constant molecular viscosity of water kinematic viscosity of water percentage of the measured area occupied by obstructions water density air density spatial mean bottom shear stress = tq bottom shear stress in the x and y directions, respectively xi i i Tj hydrodynamic drag viscous stress wind shear stress on open area t wind shear stress on obstructed area so Tj. Reynolds stress 10 the earth's angular velocity to* wind friction velocity = /t./p S 3 V2 Laplacian operator xiv Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FRICTION IN HURRICANE-INDUCED FLOODING By Shang-Yih Wang December 1983 Chairman: B. A. Christensen Cochairman: T. Y. Chiu Major Department: Civil Engineering With the increasing development of coastal areas, it is necessary to have a sound method for predicting hurricane-induced flooding in these areas, especially for studies such as the coastal construction set-back line, flood insurance rate-making and county land use planning. The purpose of this study is to develop the capability of describing the friction factor in coastal areas for improved representation in numerical models of storm surges. Five types of areas are considered: A, ocean bottom with bed- forms and some vegetation; B, mangrove fringe and areas; C, grassy areas; D, forested areas; and E, developed areas. The friction factors, which incorporate both the bottom friction coefficient and drag coefficient due to the submerged parts of obstructions were verified by conducting laboratory experiments for mangrove and developed areas, xv using the typical distribution found in each of these coastal areas. Analysis of the experimental data revealed that the drag coefÂ¬ ficient for each case is invariant with the Reynolds number in the most possible flooding flow ranges, but that it is related strongly to the density and distribution of the roughness elements. Formulae expressing these relations were derived for the evaluation of the friction factor for different coastal areas. In addition, it is found that the drag coefficient for a staggered disposition is about two to three times larger than that for an aligned disposition under the same density for all building areas. A relationship between the drag coefficient and the disposition parameter of the evenly distributed roughness elements was developed. The principal reduction of the wind stress due to wind drag forces on the parts of the obstructions, including-buildings and vegetation, above the water surface during overland flooding was determined. Accounting for this reduction of the wind stress provides a realistic view of wind generation forces in coastal areas. Finally, the formulae of the friction factor for the ocean bottom, forested areas and grassy areas are presented by adopting results from previous investigations and discussed with the results of the current study. xvi CHAPTER I INTRODUCTION The rapid growth of population and industry in very low coastal areas in recent years has resulted in increased concern and attention to the potential hazard to these areas from tropical storms and hurÂ¬ ricanes. A severe tropical storm is called a hurricane when the maximum sustained wind speeds reach 75 mph or 65 knots (U.S. Army, Corps of Engineers, 1977). During a hurricane, the wind-driven storm waves are superimposed on the storm surge, which is the rise above normal water level due to the action of storm, and sometimes the low coastal areas are flooded. The worst natural disaster in the history of the United States came as the result of a hurricane which struck Galveston, Texas, in 1900. The storm, which hit the Texas coast on September 8, with winds of 125 mph caused a storm surge 15 feet in height above the usual two-foot tidal range. The fifteen-foot surge, accompanied by wave action, demolished the city and caused more than 5,000 deaths (Bascom, 1980). Weather warnings were ineffectual. The people of Galveston, unprepared for a storm of such intensity, were helpless in the face of the hurricane. But the hurricane is no longer the unheralded killer it once was. The years of progress in weather forecasting and wave research have now made it possible to predict such surges. Hurricane Donna, for example, which crossed the Florida Keys and then moved northeastward across the state of Florida from near Fort Myers to Daytona Beach on September 9-10, 1960, is thought to have been the most destructive 1 2 storm ever experienced in Florida. Fortunately, she was detected in advance. Thus, even though this hurricane caused an estimated $300 million in damage, only 13 fatalities occurred. In 1961, Hurricane Carla struck along the Gulf coast of Florida. The area was evacuated before its arrival, and there was no loss of life. From the facts mentioned above, it is clear that the great value of the modern storm warning service is in its reducing the loss of life. However, it is also apparent that the potential damage to property and structures has increased dramatically with the rapid development of coastal areas, if no concession is made to the storms. Therefore, the future development plans of these coastal counties must take into consideration this threat to life and property. As a result, the 1971 session of the Florida State Legislature passed a law (Chapter 16,053, Florida Statutes), requiring the Department of Natural Resources to establish a coastal construction set-back line (SBL) along Florida's sandy beaches fronting the Atlantic Ocean and the Gulf of Mexico. Based upon comprehensive engineering studies and topographical surveys, such a line, where deemed necessary, is intended to protect upland properties and control beach erosion. Basically, construction and excavation seaward of the SBL is prohibited, though a provision for variances is included in the law (Chiu, 1981). In 1973 the Congress of the United States enacted the Flood Disaster Protection Act (Public Law No. 93-234, 87 Statutes 983) which greatly expanded the available limits of federal flood insurance coverage. The act also imposed new requirements on property owners and communities desiring to participate in the National Flood Insurance Program (NFIP) (Chiu et al., 1979). 3 The Flood Disaster Protection Act of 1973 requires the Department of Housing and Urban Development (HUD) to notify those communities that have been designated as flood hazard areas. Such areas are defined as having a one percent annual chance of flooding at any location within the areas. Such a community must either make prompt application for participation in the flood insurance program or must satisfy the Secretary of HUD that the area is no longer flood prone. Participation in the program is mandatory (as of July 1,1975) or the community would be denied both federally related financing and most mortgage money. Individuals and businesses located in identified areas of special flood hazard are required to purchase flood insurance as a prerequisite for receiving any type of federally insured or regulated financial assistance for acquisition or construction purposes. Effective July 1, 1975, such assistance to individuals and businesses was predicated on the adoption of effective land use and land management controls by the community. Federally subsidized insurance for flood hazard is authorized only within communities where future development is controlled through adequate flood plain management. Management may include a comprehensive program of corrective and preventative measures for reducing flood damage, such as land use controls, emergency preparedness plans and flood control works. Participating communities may be suspended from the program for failure to adopt or to enforce land use regulation (National Flood Insurers Association, 1974). 4 Storm Surge Prediction To implement either the coastal construction setback line or the flood insurance program the flood elevation has to be determined on the basis of different time intervals. Accumulation of data over many years in areas of the Old World, such as regions near the North Sea, has led to relatively accurate empirical techniques of storm surge prediction for these locations. However, these empirical methods are not applicable to other locations. In general, not enough storm surge observations are available in the New World to make accurate prediction of the 100 year storm surge. Therefore, the general practice has been to use hypothetical design storms, and to estimate the storm-induced surge by numerical models, since it is difficult to represent some of the storm-surgeÂ¬ generating processes (such as the direct wind effects and Coriolis effects) in physical laboratory models. With the use of digital comÂ¬ 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 offÂ¬ shore and nearshore areas, varied grid systems are used for most finite- difference models. An orthogonal curvilinear coordinate system with telescoping computing cells has also been introduced by Wanstrath (1978) 5 to solve the flooding problems of Louisiana. Regardless of the purposes and differences in approaches of all these numerical storm surge models, the Navier-Stokes and continuity equations which incorÂ¬ 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 predicÂ¬ tion of storm surge, these physical parameters should be carefully determined and incorporated into the numerical models. Modern achievements in meterology and oceanography have led to an increased understanding of hurricane to the extent that a model hurricane can be characterized to a satisfactory degree by certain parameters. A list of these variables includes central pressure deficit, radius to maximum winds, speed of hurricane system translation, hurricane direction and landfall location (or some other descriptor of hurricane track). Surprisingly little work has been done in measuring another important term-bottom friction. As suggested by Pearce (1972), future work on actual hurricane surges and currents is especially needed for improved representations of bottom friction that would be achieved with a better understanding of the dissipation mechanism (i.e. friction) during a hurricane. Objectives of Present Work Based on the necessities for a more accurate prediction of hurricane-induced flooding in coastal areas (especially for studies like the construction set-back line, flood insurance rate determination 6 and county land use plan), this study will develop a method of describing the friction factor in coastal areas for improved repreÂ¬ sentation of numerical storm-surge models. Special emphasis will be placed on the friction characteristics of mangroves and buildings which are the two most important causes of frictional resistance in vegetated and developed land areas, respectively. The effect of these two roughnesses in reducing overall wind stress on the water is also introduced. The friction factors for other roughnesses such as the ocean bottom, the forested and the grassy areas are determined by the results from previous investigations and present study and are discussed in the last chapter. CHAPTER II FIELD EXPLORATION OF PHYSICAL ENVIRONMENT As stated in Chapter I, mangroves and buildings are the two major flow retarding objects and are therefore being investigated in this study. However, information on the density, dimensions and typical distributions of these two forms of roughness in the coastal areas is very scarce. Thus, field trips were taken to a mangrove area and developed areas in southern Florida in order to collect the most representative data for use in the model tests. Mangrove Areas General View of Mangroves in Florida Mangrove is a kind of salt-resistant plant that usually grows densely on sub-tropic shorelines around the world. This special feature may be an inherent gift from nature in that the mangroves enable exposed shorelines to resist severe attacks of hurricanes. Basically, there are three species of mangroves, the red mangrove (Rhizophora mangle), the black mangrove (Aricennia nitida) and the white mangrove (Laguncularia racemosa). Each of these three species occupies a distinct zone within the forest, depending on the degree of salinity and length of inundation that each species can tolerate. Red mangroves usually are found at the outer or seaward zone. They are distinctive in appearance, with arching prop roots that project 7 8 from the trunk or branches down into the water (Figure 1). The root and trunk systems of red mangroves, which spreads in shallow offshore areas and onshore areas serve as a soil producer and stabilizer as well as a storm buffer. In their role as buffers against storm winds and tides, they prevent devastation of the coastline (Lugo et al., 1974). The middle zone, at slightly higher elevations, is dominated by the black mangroves in association with salt marsh plants. This zone is usually submerged at high tide, but is otherwise exposed. The roots produce pneumatophores (fingerlike extensions above the soil surface),as shown in Figure 2. Black mangroves may also be found in pure stands in shallow basins where sea water remains standing between tides. The heat from the sun evaporates some of the water, leaving slightly concentrated salt water behind. The black mangroves are also important for shoreline stabilization as they present a secondary defense behind the red mangroves. The white mangrove, which can be found in the most landward zone that is affected only by the highest spring and storm tides will not be discussed here since it is not as important in defending against storm flooding and its usual appearance may be categorized into buttonwood or other common types of vegetation. Sampling of Mangroves Five mangrove forest typesâ€”â– Fringe, Riverine, Basin, Overwash and Dwarf Forests-have been found by Snedaker and Pool (1973) in southern Florida, with distinctive differences in structure. The pattern is strongly related to the action of water, both the freguency and the amount of tidal flushing and freshwater runoff from the upland. The coastal fringe forest including red and black mangroves, which are the 9 FIGURE 1: Prop Roots of Red Mangroves FIGURE 2: Air Roots of Black Mangroves in 1 Foot Square Area 10 most important species, was investigated in San Carlos Bay on the southwest coast of Florida. Eleven sampling areas which included six red mangrove fringe areas and five black mangrove areas were selected at random. Each sampling area was framed by survey poles to form a 12 by 12 foot square area in which the locations and dimensions of mangrove trunks, roots, and canopies were recorded. Figures 1 through 4 show some of the features of both types of mangroves in the surveyed area. Figures 5 and 6 show two examples of data recorded from red and black mangrove sampling areas, respectively. Data for the other nine sampling areas are shown in Appendix A. From these data it is clear that the density and dimensions of the trunk and root systems of mangroves are quite random; therefore, averaged characteristics are chosen to describe these samples as shown in Table 1. Red mangroves in the surveyed areas extend from the low tidal water line to about 50 feet inland which is the same distance Veri et al., (1975) recommended for fringe mangroves in order to form a protective buffer zone. Thus, this value of 50 feet can be considered to be a standard distribution distance for red mangroves and is used in the present study. Although the average height of the lower edge of canopy was found to be about 8 feet above the ground for red mangroves, the canopy along the water edge was found to generally have a distribution from the water surface to a few feet high. This feature may be important in resisting storm surges. Therefore, a detailed measurement of canopy distribution was done at a later time in Sarasota, Florida. Figure 7 depicts the measuring of the density of leaves by counting the number 3 of leaves in a unit volume (1 ft.). Dimensions of leaves were also 11 FIGURE 3: Red Mangrove Area (Sampling Area #4) FIGURE 4: Black Mangrove Area (Sampling Area #7) 12 AREA #5 FIGURE 5: Field Data Record for Red Mangroves 13 AREA #7 O Stem FIGURE 6: Field Data Record for Black Mangroves 14 TABLE 1: Average Parameters of Sampling Mangroves Average Parameters Red Mangroves Black Mangroves Main- Stem no. (12 ft.)2 4 12 diameter 6.0 in. 3.1 In. height 10.0 ft. 11.2 ft. % occupied 0.84 % 0.49% Sub- Stem no. (12 ft.)2 13 diameter 2.0 in. height 18.0 in. % occupied 0.19 % Canopy height 8.0 ft. 9.4 ft Root no. (12 ft.)2 81 10,800 diameter 1.0 in. 0.25 in height 18.0 in. 6.0 in. % occupied 0.25 % 2.39% 15 measured and recorded. Totally six sampling areas along the coastal fringe were randomly selected. As shown in Figure 8, the survey area contains three sections in which each section covers a distance of five feet. The data collected are shown in Appendix A. Table 2 lists the average densities and dimensions of leaves obtained from these six sampling areas. Developed Areas In developed areas, buildings constitute the principal roughness elements which would significantly affect the apparent bottom shear stress as well as the wind shear stress during a storm induced flood. Buildings are not, in general, arranged in a uniform manner but are strongly dependent on the environment where they are located. A common feature found in the coastal counties of Florida, especially in Broward and Dade counties, is that high-rise buildings are predominant along the beaches while residential houses are predominant a few miles inland from the coastline. Three areas, a high-rise building area, medium- rise building area and residential area, are defined to represent a developed area in this study. High-Rise Building Area Aerial photographs of Broward and Dade counties, Florida, made by the State Topographic Office, Florida Department of Transportation in 1980 were used to analyze the dimensions and densities of buildings in the coastal areas. Dade county is divided into 113 ranges while Broward county is divided into 128 ranges in the aerial photographs. 16 FIGURE 7: Measurement of Density of Canopy TABLE 2: Average Characteristics of Canopy â€”^Sections Parameters #1 n #3 â€ž .... , no. of leaves\ Density! = ) ftJ 10 5 2 Leaf Size 2" x 1â€ 3" x 1.5" 4" x 1.75" 17 Each range has a length of 1,000 feet approximately parallel to the shoreline and is marked by monuments both in the field and on the aerial photographs. Typical shapes and orientation of buildings found in these two counties are shown in Figure 9. Category (a) is the most common type found (more than 50 percent) which may be dictated by the high cost of land per foot along seashore, and is chosen to present all the buildings in the study. High-rise buildings are defined as buildings having a surface area 2 larger than 10,000 ft . An estimation of the dimensions and densities of these high-rise buildings from Broward and Dade Counties are 1isted in Table 3. High rises and hotels/motels are predominant in the area. Medium-Rise Building Area Madium-rise buildings cover all buildings which do not belong to either the high-rise or residential types. They can include two and more story semidetached houses, row houses, garden apartments and other buildings which are lower than ten stories. The surface area occupied 2 2 by medium-rise buildings is defined from 2,400 ft to 10,000 ft in this study. An investigation of buildings in this category was also made from aerial photographs of Broward and Dade Counties. Table 4 shows the average densities and dimensions of buildings from existing field data. The values obtained from these two counties at least present some general views of buildings in highly developed areas despite their irregularities in distributions found in the field. To apply these 18 Sea Side Sea Side (b) FIGURE 9: Top View of Building Shapes on Coastal Areas 19 TABLE 3: Average Parameters of High-Rise Buildings Range No. Buildings County length (ft.) width (ft.) density ( ^ ) 1000'x500' % of land occupied by buildings 42 280 80 10 45 Broward 43 290 60 4 14 Broward 45 290 110 4 26 Broward 50 140 140 10 39 Broward 54 300 100 3 18 Broward 72 200 70 8 22 Broward 82 200 50 19 38 Broward 83 200 180 5 36 Broward 84 200 230 5 46 Broward 118 140 90 8 20 Broward 119 150 70 12 25 Broward 121 230 90 6 25 Broward 8 240 200 3 29 Dade 11 300 50 8 24 Dade 12 320 70 7 31 Dade 14 240 50 9 22 Dade 15 230 50 8 18 Dade 17 230 60 9 25 Dade 18 250 50 10 25 Dade 19 200 125 5 25 Dade 36 220 65 4 11 Dade 42 190 65 7 17 Dade 43 270 200 6 65 Dade 44 200 200 5 40 Dade 48 120 200 6 29 Dade 52 220 210 4 37 Dade 56 190 160 9 55 Dade Mean 224 112 7 30 S.Dev. 53 62 3 13 20 TABLE 4: Average Parameters of Medium-Rise Buildings Range No. Buildings County length (ft.) width (ft.) density ( no- ) U000'x500' 1 % of land occupied by buildings 26 65 65 23 19 Broward 36 120 40 30 29 Broward 37 100 35 33 23 Broward 46 150 40 27 32 Broward 49 80 60 31 30 Broward 51 100 40 22 18 Broward 52 120 40 22 21 Broward 62 80 30 33 16 Broward 64 100 40 20 16 Broward 66 70 40 43 24 Broward 67 80 70 14 16 Broward 101 60 50 35 21 Broward 109 70 70 18 18 Broward 110 70 40 24 13 Broward 111 70 70 25 25 Broward 116 100 70 17 24 Broward 1 120 75 14 25 Dade 2 90 60 13 14 Dade 3 80 50 15 12 Dade 4 75 75 11 12 Dade 5 100 65 15 20 Dade 33 160 49 24 38 Dade 35 125 35 32 28 Dade 68 130 40 21 22 Dade 69 130 40 24 25 Dade 70 125 40 28 28 Dade Mean 99 51 24 22 S.Dev. 28 15 8 7 21 data in the prediction of storm surge, it is recommended that the County Land Use Plan Map published by each county be used so that the most realistic results can be expected. Residential Area Residential houses are usually located behind the commercial areas 2 and have a surface area less than 4,000 ft . A typical density value of detached, one story houses is given as six units per acre (43,560 sq. ft.) (DeChiara and Callender, 1980). Density ranges in residential areas can also be found in the Land Use Map of each county which categorizes these single family houses in the density range of 0-8 units per acre (Reynolds, Smith and Hills, 1972). The significant difference between residential, medium-rise building areas and high-rise building areas is that the former two areas usually have a matrix type distribution while the latter one has only one or two rows distributed in the coastal fringe area. The importance of this variation in the building distribuÂ¬ tion will be shown later in the discussion of modeling studies. Dimensions of the typical residential house are chosen as 30 feet by 62 feet, 1,860 sq. ft., which are convenient for the model tests and also realistic for most single family houses. CHAPTER III THEORETICAL BACKGROUND AND DEVELOPMENT Hydrodynamic Equations for Storm Surges The equations governing incompressible fluid flows are the Navier-Stokes equations of motion and the equation of continuity. In the case of storm surges these equations may be written: 3qx 3 t 2o)(sin0)qy d_ 3Ps p 3 x 9â€œ H + 7 (Tsx â€¢Tbx> (1) 2Ã¼)(sine)qx d_ 3Ps p 3y gd |S- + 1 a 3y p (2) 3qx + 3^y_ + 3S 3 x 3 y 3t 0 (3) where t is time, to is the angular velocity of the earth, e is the latiÂ¬ tude, p is the pressure, g is the acceleration of gravity, p is the water density, s is the free surface displacement from mean sea level, the subscripts s and b indicate that the subscripted quantities are to be evaluated at the surface and bed, respectively, d is the total depth, qx and q^ represent the time mean transport component, i.e. discharge per unit width, in x and y directions, respectively, i.e. d(x,y,t) = h(x,y) + s(x,y,t) qx(x,y,t) fs(x,y,t) u(x,y,t) dz -h(x,y) (4) (5) 22 23 qy(x,y,t) rs(x,y,t) v(x,y,t) dz -h(x,y) (6) in which h = water depth referenced to mean sea level, u and v are the instantaneous components of water velocity in the x and y coordinate directions, respectively. Expressions for the wind shear stress, and bed shear stress, x^, for coastal areas in tropical storm induced flooding are presented in the following sections. Wind Shear Stress In general, the wind stress (t ) on a water surface may be expressed in terms of the mean wind speed (W-^) at anemometer level (10 meters above water surface), the air density (p ) and a wind-stress coefficient (K), a as T s K W 10 (7) The problem of evaluating the wind stress is therefore reduced to estimating the wind-stress coefficient, K, at different wind speeds, if the reference wind speed and air density are known. Numerous studies have found the quadratic wind speeds relation to be appropriate for a wide range of wind speeds (Wilson, 1960). A wind-stress relation more physically satisfying the quadratic law correlation was developed by Keulegan (1951) and Van Dorn (1953) in the low winds range (<15 ms"^). The Keulegan-Van Dorn relation for x$ is given as T s Pa [K^o â™¦ K2(W10 - w/] (8) 24 where K-| and are the constants and Wc is critical wind speed. Although there are uncertainties in applying the Keulegan-Van Dorn relation to hurricane winds, it has been applied widely in hurricane- induced surge cases. To eliminate this deficit, the wind-stress relation has to be extended to higher wind ranges. Whitaker, Reid and Vastano (1975) investigated the wind-stress coefficient at hurricane wind speeds using a numerical simulation of dynamical water changes in Lake Okeechobee, Florida. Results of their numerical experiments showed that the Keulegan-Van Dorn wind-stress relation was superior to the more commonly used quadratic relation for wind speeds in the range of 20 to 40 meters per second. The relation they found for the wind stress ts is given by: T = p[0.0000026 + (1.0 - ^.)2 x 0.0000030] W2 (9) s wio 1u where W^q and 7.0 are in meters per second and p is the water density. Unfortunately, though this result was verified by a simulation of the surge associated with a hurricane which occurred in October, 1950, it still has some deficiencies such as the limited range of applicability (Whitaker et al., 1975). Recent studies of the wind-stress coefficient over the sea surface have produced more complete and perhaps more accurate results with the refinement of measurements and analysis techniques. Garratt (1977) reviewed and averaged 17 selected sets of data and proposed an empirical expression for 'light' winds: K = (0.75 + 0.67 W1Q) x 10"3 (10) 25 Wu (1980) suggested a similar result for the wind-stress coefÂ¬ ficient from 33 averaged data sets under 'light* winds K = (0.8 + 0.065 W1Q) x 10"3 (11) Furthermore, Wu (1982) compiled and averaged all available data for 'strong' winds. The data were obtained from independent investigations either cited or reported in the following sources: Wu (1969), Kondo (1975), Garratt (1977), Smith (1980), Wu (1980), and Large and Pond (1981). All the data sets selected were obtained under nearly neutral conditions of atmospheric surface layer. Additional factors which affect the wind-stress coefficient, such as rainfall and sea spray, are neglected due to their minor importance compared to the major factor of wind speeds. As a result, the empirical formula proposed, given by equation (11), for 'light' winds appears to be applicable even in 'strong' winds. Light and strong winds are defined as those less than and greater than 15 meters per second, respectively. The averaged data obtained from those sources and the formula proposed, equation (11), are shown in Figure 10. Wind Velocity Profile in Vertical A vertical profile of wind velocities, usually expressed by the following logarithmic law, is regarded by meteorologists as a superior representation of strong winds in the lower atmosphere (Tennekes, 1973): W(z) = 1Â», In (Â£-) (12) 0 where W(z) is the wind velocity at a height z above mean sea level, is the von Karman's turbulence constant, z is the dynamic roughness of 1 /2 the logarithmic velocity profile, and w* = (Ts/Pa) = friction velocity. FIGURE 10: Wind Stress Coefficient over Sea Surface 27 The wind velocity profile given by equation (12) is well defined except that it fails next to the bed where z approaches zero, and the wind velocity W(z) approaches minus infinity. This discrepancy can be corrected by using a modified mixing length approach as proposed by Christensen (1971) for the flow of water over a rough bed, resulting in modified logarithmic law for wind velocity profile is given in the form = 2.5 In (^- + 1) (13) â€œ* zo By substituting equation (7) into equation (12), an equation for determining the dynamic roughness, zq, is obtained 4ft = /K (2.5) In (y- + 1) (14) w10 o Applying the boundary condition W(10) = W1Q at z = 10m The dynamic roughness, z , in all the wind speeds is found to be a function of wind-stress coefficient, or on the wind velocity, i.e., z 0 10 e'2.5 . , (15) Proposed Wind Shear Stress on Obstructed Areas Consider an obstruction which has an effective width, D, and protruding height, h, as shown in Figures 11 and 12. The wind drag force on such an obstruction can be expressed as 28 I T D Jl PLAN VIEW FIGURE 11: Plan View for Wind Stress over an Obstructred Area ELEVATION VIEW Protruding Obstruction Wind => FIGURE 12: Elevation View for Wind Stress over an Obstructed Area 29 F D CPpaD 2 â€¢h â– 0 W2(z) dz 06) where CQ is the drag coefficient. Substituting equation (14) into equation (16) gives (2.5)â€˜ F0 " CDpa DK 10 [In + 1 )]2 dz o or Fd = 3.13 CDPaD W20 K {(h + zQ)1n(^-+ l)[ln(^- + 1) - 2] + 2h } (17) o o Recall the equation for wind stress on an open water surface, i.e., â– K â€œa *4 .(7) This wind stress acts on the water surface and causes a rise of the elevation of water surface which is called wind setup. The wind energy is being transformed from the wind field to the water flow by the wind shear. When the same wind field moves from the open water area to the obstructed area, the wind setup will be reduced. This is due to the extra form drag (FQ) acting on the protruding obstruction that can be contributed to the wind setup per unit area. As a result, this reduced wind stress causing a wind setup on an obstructed area may be expressed as: t = R , t = R, K p W._ so d s d pa 10 , 0 < Rj < 1 (18) in which Rd is a reduction factor which represents the ratio of the wind stress on the obstructed area to that on an open area under the same wind condition. 30 Since the reduction of wind stress on an obstructed area is due to the extra form drag, or wind energy loss caused by the obstruction, the reduction factor may also be defined as the ratio of the total loss of wind energy per unit length on an open area to that on an obstructed area with dimensions 1*1, i.e., (Gee and Jenson, 1974) 2 T. * l Rd = K 2 (19) [(1 - mgDÂ¿)xs + mFp] x C 2 in which m = density = N/Â£ , N = total number of obstructions, 8 = a shape factor defined as the horizontal cross-sectional area of average obstruction element at surface level divided by D. Substituting equations (7) and (17) into equation (19) gives K W 10 (1 - m8D2)KpaW^0 + 3.13 (mCDPaDW2Q) K{(h+zo)ln(^-+l)[ln(^-+l)-2]+2h} o o or Rd = 1 (l-m6D2) + 3.13 (mDCD) {(h+ZQ) ln(^-+l) [ln(^-+l)-2] + 2h} o o (20) _ o where K = (0.8 + 0.65 W^q) x 10 . W^q is in meters per second. Based on the result presented in equations (18) and (20), the reduction factor R^ can be determined and incorporated into the storm surge model to produce more realistic results for the wind stress on water in flooding areas. The drag coefficient in equation (20) for vegetations and buildings will be determined and discussed in Chapter VII and VIII. 31 Bed Shear Stress A space averaged bed shear stress, 7b, usually can be expressed as V i-eâ€ Tb = To = yRS = yR T- (21) in which tq =average bed shear stress along the wetted perimeter; y = pg = unit weight of water; R = A/P = hydraulic radius = depth in sheet flow; A = cross-section area; P = wetted perimeter of flow cross-section; S = slope of the energy grade line; AH = energy loss per unit weight of fluid over a bed length of L. Primarily developed for flow in pipes, the energy loss term, aH, is defined by the Darcy- Weisbach formula as iH â– f ' | â€¢ T (22) 3 0 where f = friction factor based on depth; U = spatial mean flow velocity; dQ = diameter of the pipe. Since dQ = 4R, the above equation / may be written for an arbitrary cross section as = <23> where f = friction factor based on hydraulic radius. Incorporating equation (21) into equation (23) in the x and y directions, respectively, gives the following quadratic forms for the bed shear stresses: fp|U|U bx f'p|U|U by 2 2 (24) 32 or in terms of volume transport f'p|q|qx _ f'p|q|qy Tbx = -^~ ; Tby= 2d2 (25) where f = 4f'; U = /U 2 + U 2; q = /q 2 + q 2; U and U = spatial x y x y x y mean flow velocity in x and y directions, respectively. It is assumed that these steady state relationships for the two shear stress components are valid for storm surge propagation, which is generally considered to be quasi-steady, i.e., the velocity variation with time or the temporal acceleration is very small. The quadratic Darcy-Weisbach form of bed shear stress is the best formula available to account for the effect of bottom friction. The friction factor in the Darcy-Weisbach formula, f, has been studied by many investigators in both pipe flows and open channel flows. From the abundant experimental data, numerous empirical formulae have been established to express the relationship between the friction factor and the dependent parameters, such as bed roughness, Reynolds number, Froude number and Strouhal number. For example, the well-known Stanton diagram (1914), Moody diagram (1944) and many others (to be discussed in the next section) have enabled determination of the friction factor in varied flow conditions. In general, a surge could be expected to travel over five different terrains (Christensen and Walton, 1980): A. Ocean (river) bottom with flow induced bed form and completely submerged vegetation, B. Mangrove fringes and areas, C. Forested areas and cypress swamps, D. Grassy areas, and 33 E. Developed areas. Each of these five categories has unique roughness characteristics. However, in evaluating the friction factor in hurricane-induced flooding, these five terrains can be divided into two major categories, unobstructed and obstructed areas, based on their distinct functions to retard flow. Post Approach, Friction Factor for Surges in Unobstructed Areas Unobstructed areas include the ocean (river) bottom and grassy areas, the latter of which are assumed to be completely submerged in water during floods. Friction factors in these kinds of areas can be determined from the results of previous research which will be discussed below and used as basis for the present work. Overland flooding in this study is considered to be turbulent and in the hydraulically rough range, i.e., the wall Reynolds number is in excess of about 70. The effect of wall roughness on turbulent flow in pipes has been studied during the last century by many investigators. An important result obtained by Nikuradse (1933) in steady flow using six different values of the relative roughness k/r with Reynolds numbers ranging from 4 6 R = Ud /v = 10 to 10 has been widely used in flow fields and will be e o applied in this study (k is the equivalent sand roughness, r is the radius of pipe; U is the average velocity, v is the kinematic viscosity). Nikuradse divided flow conditions into three ranges, smooth flow range I (u^k/v < 4), transition flow range II (5 <_ ufk/v Â£ 68), and rough turbulent flow range III (ufk/v > 68) in which ufk/v = R* = wall Reynolds number, u^ = friction velocity = /x/p. 34 In range III (rough turbulent flow) the thickness of the viscous sublayer <5 is negligible compared to the equivalent sand roughness, k, and the friction factor is independent of the Reynolds number. The distribution of the time-mean velocity obtained using Prandtls' mixing length approach in combination with Nikuradse's experimental results is given by the general expression jj- = 8.48 + 2.5 In | = 2.5 In ^iZl (26) in which u = time-mean velocity in the direction of flow at a distance z from the theoretical bed. Theoretical bed is defined as the plane located such that the volume of grains above the plane equals to the volume of pores below the plane but above the center of grains. The classic velocity profile given by equation (26) is well defined at moderate to large distances from the bed and for roughnesses much smaller than the depth. However, it falls next to the bed whereas z approaches zero, the time-mean velocity U approaches minus infinity. This is especially true in flows where the roughness is not significantly smaller than the depth. Because of the above-mentioned discrepancies, Christensen (1971) introduced a new law for the velocity profile by using a modified mixing length approach over a rough bed in the rough range = 8.48 + 2.5 In uf Â£ + 0.0338) = 2.5 In ( ?9Â¿73z + 1) (27) The form of this equation is the same as that of classic equation (26) except for the +1 term in the argument of the logarithmic function which 35 makes the time-mean velocity u equal to zero at the theoretical bed. As the distance from the bed increases to more than a few times k, very little difference exists between these two velocity profiles. For practical purposes, the time-mean velocity profile is transÂ¬ formed to a depth averaged velocity profile using the fact that the mean velocity (depth averaged), U, occurs theoretically at a distance z = 0.368d from the bed also for the modified logarithmic vertical velocity profile, where d is the water depth, and d/k is larger than 1. It shall be noted that the k value used here is the equivalent roughness height for bottom friction only. Therefore, at z = 0.368d equation (27) yields â€” = 2.5 ln[%^ (0.368d) + 1] U K or â€” = 2.5 In [10.94 Â£ + 1] U K (28) where U = time and depth averaged velocity. The friction factor may in general be related to the velocity profile by introducing the Darcy-Weisbach formula into the definition of the friction velocity, i.e., uf = /tq/p = /gRS, leading to the result (29) Solving equation (29) for f and introducing equation (28) gives the following expression for the friction factor 0.32 f (30) 36 This depth dependent friction factor is proposed for areas where the surge moves over bottoms at moderate depths. Another equation for f obtained from Nikuradse's experimental result (1933) for rough turbulent flow in circular pipes, is given by â€” = 1.171 + log Â£ (31) 4/Tr K It seems quite clear from equation (30) and equation (31) that determination of friction factor in unobstructed areas is just a matter of finding the value of the equivalent sand roughness k. This k value can be related to Manning's n by using a Strickler-type formula (Henderson, 1966, Christensen, 1978) in metric units 1 _ 8.25 /q (32) given k in meters, or in the English units 1.486 8.25/q (33) with k in feet. Values for n may be determined from various sources such as textbooks by Henderson (1966) and Chow (1959), charts and graphs by the Soil Conservation Service (1954), and photographs of a number of typical channels by the U. S. Geological Survey (Barnes, 1967). Other specific studies, for instance, the experiments conducted by Palmer (1946), also provide valuable information on the flow of water through various grass and leguminous covers. Based on the theoretical velocity 37 distribution in rough channels, the value of Manning's n can also be determined by analytical methods such as that presented by Boyer (1954). It should be noted, however, that these values of n from previous sources may not be applied to every case under natural conditions. It should be also careful in selecting the values of n, since a small error on n will be amplified substantially on k by using the Strickler-type formula. Therefore, a method to determine the k-value from the vertical velocity distribution in turbulent flow over rough surface is recomÂ¬ mended (Christensen, 1978). Let Hi be the velocity at m depth, that is, at a distance md from the bottom of a wide rough channel, where d is the depth of flow. By equation (26), the velocity may be expressed as Ui 29.7nid â€” = 2.5 In Â£ (34) Similarly, let u2 be the velocity at r\2 depth; then u2 29.7r)2d â€” = 2.5 In â€”u (35) Subtracting equation (34) by equation (35) and solving for uf, uf = U2 - Ux *12 2.5 In â€” fii (36) Introducing equation (36) into equation (35) and solving for k, 29.7md (37) n2 u2- ux 38 Proposed Friction Factors for Surges in Obstructed Areas Obstructions in these areas are defined as roughness elements with significant heights which either protrude through the water layer or consist of relatively rigid elements with heights that are sufficient to cause form drag that are much larger than surface friction on the same area. The two major forms of obstructed areas, mangroves and buildings, to be discussed in this study are often higher than the storm surge level so that the influence of hydrodynamic drag on the individual elements should be taken into consideration together with other factors of resistances in overland floods. The theoretical analysis presented here is based on the assumption of steady or quasi-steady flow in the rough flow range. Consider a design flow that passes over an obstructed area whose bottom is horizontal. The total averaged shear stress, x , in the direction of flow may be written as equation (21), i.e., - dc d aH t0 - yRS = yR â€” For the steady state case, assume that the viscous stresses, turbulent stresses and hydrodynamic drag acting on the obstructions contribute independently to the flow system without any interaction among them. Following Prandtl's assumption (Schlichting, 1979), that the shear stress in the x-direction is constant and equal to the wall shear stress at all distance z from the wall, an equilibrium equation is given in the following and shown in Figure 13. t FIGURE 13: Distribution of Horizontal Apparent Shear Stress and of its Drag, Inertial and Viscous Components, (Christensen, 1976). 40 du YRS = ^ (l-e) - pu'v' (l-e) + mCDdDY S| (to} <*Â£> Viscous Stress (Tt) Turbulent Stress (Td> Hydrodynamic Drag (38) where e = fraction of total area occupied by obstructions, u', v' are the turbulent velocity fluctuations in the x and z directions, respectively, m = number of obstructions per unit bed area, CQ = average drag coefficient, D = average diameter of the obstructions in the proÂ¬ jected plane normal to the flow. For fully developed turbulence, the viscous term is negligible compared to the turbulent term and may be omitted. Consequently pgRS = -pu1v1 (1-e) + mCDdD | (39) The turbulent stress, x^, which expresses the rate of flow of x momentum in the z direction, was first derived by 0. Reynolds from the equation of motion in fluid dynamics, and is termed the Reynolds stress or the inertia stress. The Reynolds stress on the right hand side of equation (39) must be 1 e (40) where 1 41 â™¦ = ( Id T 1 - e V2 ) , 0 < $ < 1 (41) reflects the reduction of the Reynolds stress - pu'vâ€˜ due to the presence of the obstructions. If the drag is equal to zero, implying that no obstruction exists (e = 0), the obstruction correction factor will of course become equal to 1. The corresponding friction velocity depends on the bottom friction only. It is defined from equation (40), \'2 ) * uft * (42) where uft is the friction velocity based on the total bed shear stress V i>eâ€ uft = /T7P' Similarly, introducing f and fj. as the friction factors of bottom friction and total friction including bottom friction and drag acting on the obstructions, respectively. Therefore, these two friction factors can be related to the mean velocity as shown in equation (29), U 2 1^2 â€” = (7.) for the obstruction-free area Uf T (43) and U uft for the total obstruction area (44) Eliminating U from the two equations above, relation between f and fj. is given as f - â€¢ *2 (45) 42 Substituting equations (40), (42) and (44) into equation (39), the equilibrium equation becomes yR AÃ¼ = Â¡L 2 (1-e) + mCpdDy ^ or AH = Â£. f ,.,2 YR y = f f'U" (1-0 + mCpdDy ^ The head loss, AH, then may be written as 4H= f i Ã ('-*> +raCDdDi?S- and Ãœ2 L AH = [f (1-0 + mCDdD] (46) (47) (48) which is the form of the Darcy-Weisbach formula shown in equation (23). An equivalent friction factor, f^, which includes the effects of bottom friction and form drag based on equation (48) is introduced f' = f (1-e) + mCpdD (49) where, according to equation (30), 0.32 [In + l)]2 Therefore, the equivalent friction factor, f^, must be equal to the total friction factor, f|, according to the above definitions, as can be proved easily by substituting equation (45) into equation (49). An apparent roughness height ka which represents the sum of bottom roughness and rigid drag element can then be expressed and calculated from equation (31), i.e., -7= - 1.171 + eog if- 4/f7 a CHAPTER IV EXPERIMENTAL VERIFICATION OF FRICTION FACTORSâ€”MODEL LAWS The formula proposed in Chapter III for the friction factor (equation (49)) has left some unsolved questions: What is the drag coefficient (C^) for the kinds of obstructions being studied? What kind of relationship do drag coefficient and density of obstruction have? Does the friction factor depend on unspecified flow conditions like the Reynolds number? All these problems may not be satisfactorily answered without experimental verification. Therefore, laboratory measurements were carried out in this study to verify the analytical results and to build a data base for further development. The requirements of similarity between hydraulic scale-models and their prototypes are found by the application of several relationships generally known as the laws of hydraulic similitude, i.e., geometrical, kinematic and dynamic similitude. These laws, which are based on the principles of fluid mechanics, define the requirements necessary to ensure correspondence between model and prototype. Complete similarity between model and prototype requires that the system in question be geometrically, kinematically and dynamically similar. Geometric similarity implies that the ratio of all corresÂ¬ ponding lengths in the two systems must be the same, kinematic similarity exists if all kinematic quantities in the model, such.as velocity, is similar to the corresponding quantities in the prototype, 43 44 and dynamic similarity requires that two systems with geometrically similar boundaries have the same ratios of all forces acting on corresponding fluid element of mass. Following the basic dynamic law of Newton, which states that force is equal to rate of momentum, dynamic similarity is achieved when the ratio of inertial forces in the two systems equals that of the vector sum of the various active forces, which include gravitational forces, viscous forces, elastic forces and surface tension forces in fluid-motion phenomena. In other words, the ratios of each and every force must be the same, as given in the equation form Â«V (F) (Fv) (F.) (Fs) P _ a P . P- P. P (50) where subscripts p and m refer to prototype and model, relatively. Since it is almost always impossible to obtain exact dynamic similitude, it becomes necessary to examine the flow situation being modeled to determine which forces contribute little or nothing to the phenomenon. These forces can then be safely neglected with the goal of reducing the flow to an interplay of two major forces from which the pertinent similitude criterion may be analytically developed (Rouse, 1950). For a model of hurricane-induced flooding of coastal areas, elastic forces and surface tension forces are sufficiently small and can be neglected. The condition for dynamic similitude reduces to equating the ratio of inertial forces to the ratio of either gravity 45 forces or viscous forces. Viscous forces are only considered in the model of canopies whose surface friction effects are investigated. In models for measuring form drags in turbulent flow with high Reynolds numbers (inertia force/viscous force), viscous forces are small compared to the major forces due to the turbulent fluctuations and this can be neglected in this instance. Since the vertical dimension scale (involving flow depth) cannot follow the horizontal dimension scale in building models as the flow depth would be much too small for measurements to be made, or the viscous force would become important and cannot be neglected for the small flow depth if the same fluid is used for the prototype and the model. Therefore, a model with a different vertical dimension scale than horizontal dimension scale is used to keep the Reynolds numbers in the turbulent flow range. For simplicity, such distorted models will be introduced first since undistorted models with the same length scale in both the vertical and horizontal dimensions can be regarded as a special case of the former. Distorted Model for Buildings The fundamental model scale ratio may be written as: L B Length Scale (horizontal): (51) Lm Bm D Depth Scale (vertical): Nd = D^ (52) T Time Scale: (53) 46 Force Scale: (54) where L = horizontal length, B = horizontal width, D = vertical depth, T = time and F = force. Following the development of Christensen and Snyder (1975), the force scale for the gravity component in the nearly horizontal direction of the principal flow may be written as unit sine weight volume of slope I ( I â€˜ > K = g L B D (D /L p g 2 P P P P P = (_Â£) (IE.) N N Pmgm CCtTÃœU V V Vd p q P^P m m m m' nr m Jm (55) where p is the fluid density, g is the gravitational acceleration and D/L is the bed slope ar the slope of the energy grade line. In a unidirectional flow the inertial force can be expressed as a horizontal, or nearly horizontal area multiplied by the Reynolds shear stress, which is proportional to the fluid density and the time mean value of the product of a vertical velocity fluctuation and the corresponding velocity fluctuation in the direction of the time mean flow. Consequently, the inertial force is F.. = puâ€˜v1 â€¢ (area) , (56) and the inertial force scale can be written as 47 area D_ r Pp m m Pn N! m N, (57) In order to have dynamic similarity between model and prototype, Kgravity should be equal to Kn-nertial * i-6*Â» equation (55) should be the same as equation (57). This condition is expressed by or (58) where gp is assumed equal to g^. The ratio between a gravitational force and an inertial force is commonly known as the Froude number, and the resulting time scale (equation (58)) is the similarity criterion of the Froude law for distorted models. The scale ratios of the drag coefficient and friction factor in distorted Froude models have to be determined before experimental data can be interpreted correctly. The drag force proposed for the present study is given by equation (39) ^d = Td* A = m^gd^P jr- * A (59) The drag force scale in the flow direction can then be written as 48 1 L B JLÂ£ 1 L B m m â€¢ L 2 )(Cn)mDmBmpJr) L B DmmmmT mm m or K, * Â«A d ,r \ vp ' "d l (Cpj m m (60) To satisfy the ratio of the force scale in the Froude model law, Kd must be equal to K^, and by substituting the time scale, Nt = NÂ£/(Nd^2, into equation (60) gives (CD> - (61) Shear forces generally may be expressed by the Darcy-Weisbach form F s (62) where f is the friction factor and can be substituted by the equivalent friction factor, T, or bottom friction factor, f', later for the present use. The shear force scale is given as or Ks â– <#<Â£> < m pm t (63) 49 Dynamic similarity requires that = K^. Substituting the time scale, equation (58) into the required equality, i.e., (i)2. (j(^)NA2 p m pm (64) gives the expression for the scale ratio of the friction factor f N. _B. = -A fm (65) Comparing equation (61) with equation (65) it is noted that the drag coefficient is the same in the model as in the prototype, however, the friction factor of the prototype should be modified by an inverse distortion ratio, N^/N^, in the Froude distorted model. The distortion ratio usually is defined as D = r Nd (66) Undistorted Model for Mangrove Stems and Roots Due to the fact that the dimensions of mangrove stems and roots are one to two orders of magnitude less than the water depth, an undistorted model can be used in this part of the study. All the methodology applied in the previous section for a distorted model is also applicable for this analysis. In the case of an undistorted model where N = N^, equation (58) reduces to Nt â– 50 which is the time scale for an undistorted Froude model. Both the dimensionless coefficients and f are the same in the prototype and model, and the distortion ratio becomes unified in this instance. Distorted Model for Canopy Before the model law for canopies is derived, it is necessary to determine what kind of boundary layer forms over the surface of a mangrove leaf. A prototype red mangrove canopy was tested in the hydraulic laboratory flume and it was quite apparent that all the leaves bent in the direction of flow even at a flow velocity less than 10 cm/sec. Such high flexibility makes the leaves more resistant to a storm attack. As a result, leaves offer only skin friction and no form drag to resist the flow. The surface of a leaf is assumed to be smooth in this study so that theoretical and empirical results on the behaviour of a boundary layer on a smooth flat plate can be applied. In general, the point of instability on a flat plate at zero incidence to the flow is determined by the critical Reynolds number U x (R e.x^crit 'crit (68) in which U is the free stream velocity and x is the distance from the leading edge of the plate measured along the plate. An analytical stability criterion developed by R. Jordinson, based on W. Tollmien's theory, is given by U 6 , co (69) 51 where Ã³ = displacement thickness and 6 = 1.7208 ( co (Schlichting, 1979) (70) Combining the last two equations give (R ) .. = 9.1 x 104. c )A Li I L In reality, the position of the point of transition from laminar to turbulent flow will depend on the intensity of the turbulence in the external flow field. This has been investigated experimentally by J. M. Burgers, B. G. Van der Hagge Zijnen and M. Hansen in 1924. These measurements led to the result that the critical Reynolds number was contained in the range U x (â€”)crit = 3-5 x 105 t0 5 x 105 (Schlichting, 1979) (71) Similar experiments done by Schubauer and Skramstad in 1947 also yielded results which indicated that the critical value of Râ€ž is in a range G jX from 9.5 x 105 to 3 x 106 depending on the relative intensity of the free-stream turbulence, (l/U^Jiu'u'/sj^2 (Hinze, 1975). Therefore, the minimum value of (R ) .. is chosen as 3.5 x 105 for the present study. C J X Cllt In the prototype, the maximum value of x is the largest leaf length and was found to be 5 inches; the highest flow velocity is assumed to be 10 ft/sec, which results in a maximum value of (R ) ,. of about c )a Li I L 2.98 x 105, which is still lower than but near the minimum value (R ) ... This shows that a turbulent boundary layer has very little g ,x cn L chance to be formed over such a short length, and that a laminar boundary layer should prevail over the entire leaf area. 52 Skin friction can be expressed in terms of a dimensionless skin friction coefficient, C^., times the stagnation pressure, pU2/2, and area of the plate, A, as follows Ff = CfP I A (72) in which, for a laminar boundary layer, C f 1.328 (73) where R = lk/v denotes the Reynolds number formed by the product of the Xj * plate length and the free-stream velocity (Schlichting, 1979). The skin friction scale in the flow direction may be written as K f LnV1/2 (-P P) TnV -P-P - f III III \ m m m L 2 (/) A T^~ (j51) A P m (74) where z and A are the leaf length and leaf area, respectively. Following the undistorted Froude law, = (N^2 , p and v are the same in protoÂ¬ type as in the model since the same water properties are assumed in the two systems. Equation (74) is then reduced to 3A -V2 A V (75) 53 For dynamic similarity, Kp has to be set equal to K , i.e. % i -V2 An 3 m m (76) giving the length scale An An2 f(Ni) m m (77) It is obvious that the dimensions of a leaf need to be distorted according to the scale ratio shown in equation (77), which is the result of inclusion of viscous effects on a leaf surface in a Froude law controlled flow model. CHAPTER V MODEL DESIGN Recirculating Flume The present model tests were conducted in the hydraulic laboratory flume of the Civil Engineering Department at the University of Florida. Figures 14 and 15 show the primary elements of the flume geometry. The main channel is 120 feet (36.58 meters) long, 8 feet (2.44 meters) wide, and 2.7 feet (0.81 meter) deep. A false-bottomed section 20 feet (6.1 meters) in length and 13.4 inches (34 cm) deep is located at the longitudinal center of the flume. Centered in the false-bottomed area, observation windows cover a length of 12 feet (3.66 meters) and are 2 feet high (starting at the bed level). The 74 kW (100 HP) flume pump has a maximum discharge of 40 cfs (1.1 m /sec). Between the pump and the overflow weir are two sets of 8 inch long, 2 inch diameter polyÂ¬ vinyl chloride pipes arranged in a honeycomb fashion. Two more sets of these pipes, which act as flow straighteners, are located just beyond the outlet weir. By adjusting two gate valves at the main delivery pipe and return pipe, the flow rate and depth over the Thomson V-notch weir can be regulated. A Poncelet rectangular weir is also available for high discharges. A motor-driven sluice gate at the downstream end on the main channel serves to regulate the water depth in the main flume and to moderately regulate the discharge. 54 FIGURE 14: Plan of Flume f Power Source for Trolley Rail for Movable Trolley l Return f Channel 38 (l.llm) iv > \ Main Channel Iâ€”*â€”r- 8' (2.44m) T 2â€˜8"(0.8lm) . (1.22m) FIGURE 15: Cross Section of Flume 56 A movable trolley which spans the entire width of the flume and which has a maximum towing speed of 2 feet per second provides the work-deck for calibrating velocity meters as well as collecting data. To determine the drag coefficient and equivalent friction factor for a given roughness in the rough turbulent flow, the energy loss, aH, has to be measured (cf. equation (47)). According to the principle of conservation of energy, the total energy head at the upstream section 1 should be equal to the total energy head at the downstream section 2 plus the two sections, i.e., 2 2 u, u dl + 2g = d2 + 2? + aH or AH = (d1 - is horizontal and a value of unity is assumed for the energy coefficient (Henderson, 1966). Therefore, the energy loss aH due to the friction in turbulent flow can be measured by knowing the water depths and mean velocities at the two sections. Relating the measured results of energy loss to the Darcy-Weisbach equation AH ,, Hi fe 2g Ra (79) 1 ^1 ^2 in which = 2 (U-j + U2) and = 2â€”- , the equivalent friction factor f' can be determined for the designed roughness elements. To determine the water depths and velocities, some instruments are employed for this study and described in the following section. 57 Instrumentation Velocity Meter A Novonic-Nixon type velocity meter was employed for all velocity measurements. The probe consists of a measuring head supported by a thin shaft 18 inches long with an electrical lead connection. The head consists of a five blade, impeller mounted on a stainless steel spindle, terminating in conical pivots (Figure 16). These pivots run in jewels mounted in a sheathed frame. The impeller is 1 cm in diameter, machined from solid PVC and balanced. An insulated gold wire within the shaft support terminates 0.1mm from each rotor tip. As the rotor is rotated by the motion of a conductive fluid, the small clearance between the blades and the shaft slightly varies the impedance between the shaft and the gold wire. This impedance variation modulates a 15KHz carrier signal, which in turn is used to detect rotor rotations. The range of this velocity meter is from 2.5 to 150 cms-1 (0.08 to 4.92 fps) with an advertised accuracy of + 1 % of true velocity. Its operating temperature is from 0Â° to 50Â°C (32Â° to 122Â°F) with an operating medium of water or other fluids having similar conductive properties. The shaft of the current meter was clamped to the rack of a point gage. The point gage bracket was then bolted to the trolley carriage so that the instrument could be easily removed from its bracket with no deviation in the vertical setting. Also all the accuracy and ease of a point gage and vernier is accrued. 58 O I 2 cm > I I FIGURE 16: Novonic - Nixon Velocity Meter 59 Data Acquisition System (DAS) The data acquisition system is composed of two pieces of equipment: an input box and an HP 9825A desk-top programmable calculator (Figure 17 and 18). The input box, which is specially designed for coupling with the HP 9825A, has connectors for fifteen thermistors, ten Cushing electromagnetic current meters, two Ott velocity meters and two Novonic- Nixon velocity meters. It contains the electronic circuitry which takes the raw transmission from the measuring devices and converts it into usable signals tor the programmable calculator. An electronic timer which registers six counts per second is also contained in the input box. The HP 9825A interfaces with the input box to provide program control and data storage capabilities. The calculator has a 32-character LED display, 16-character thermal strip printer, and a typewriter-!ike keyboard with upper and lower-case alphnumerics. A tape cartridge with the capacity of 250,000 bytes is used with this calculator to store and access the programs. Based on the manual of the HP 9825A and the instructions provided by Morris (1979), programs designed to calibrate the Novonic-Nixon meter, measure the velocities and perform linear regression are listed in Appendix B. Through the DAS a substantial amount of time usually used in experiments and data reduction was saved and the accuracy of results was greatly enhanced. Depth-Measuring Device Determination of the flowing water depth by measuring the difference of water surface elevations is the most important part, except for the measurement of the flow velocity, of the laboratory experiments. However, 60 FIGURE 17: Input Box of Data Acquisition System FIGURE 18: HP 9825A Programmable Calculator 61 the measurement of water surface elevations is not easy due to the rough water surface of turbulent flow. In addition, an accuracy of one millimeter or better is needed for the depth measurement, since the difference of water surface elevations at two sections is less than one centimeter in many test cases. Therefore, a stable and sensitive depthÂ¬ measuring device is required for the present study. Figure 19 shows the schematic diagram of the device designed, in which the hoses connected to the tube, which have a diameter of 0.2 inch, were extended to the desired cross-sections in the flume. A 10 inch long, 0.15 inch diameter glass tube was attached to the end of each hose and positioned perpendicular to the water surface. In high velocity flows some weights were added to the 0.15 inch diameter tube in order to maintain its vertical position. The diameter of the tube is 2 inches which is large enough to allow the point gage to be able to contact the plane water surface without the influence of surface tension on the side wall of the tube. The point gage was attached to the top of the tube, and the still water level is indicated when a white ball on the gage appears, which indicates that the sharp tip of the gage is touching the water surface (Figure 20). A manual hand operated vacuum pump was used to help initiate a siphon between the water in the flume and in the tube at the beginning of each test and to pump air bubbles out of the hoses periodically during the test. 62 FIGURE 19: Setup of Water Depth Measuring Device FIGURE 20: Point Gage and Tube 63 Selection of Model Scales In the last chapter the scale-model relationships based upon the Froude law were derived. The scale to which the model should be constructed depends on the following factors: the size of the flume (length, width and depth), the discharge capacity of pump, the accuracy of instrumentation and the dimensions of the prototype. According to these factors, the vertical length scale of 1:10 (or = 10) is selected through the entire study for both distorted and undistorted models. Mangroves The dimensions of all stems and roots, including height and diameter of the prototype, are reduced to 1/10 for the model based on the undistorted Froude law. However, for the canopy some distortion of scale between prototype and model is required according to equation (77) in the Chapter IV, i.e., Â£ An m m <\> in which N = Nd = 10, Â£ 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 Â£ and a width w. Equation (77) is then reduced to (80) By choosing 1:10 for the length scale for this study, the width scale of 64 the leaf becomes w 7/ = 10/4= 56 , (81) m Therefore, the dimensions of the leaves used in the model can be estimated from the derived relations and are shown in Table 5, which are based upon the prototype data listed in Table 2. TABLE 5: Scale Selection for canopy Section No. (1) (2) (3) prototype length (Â£) 2.00 3.00 4.00 (in) width (w) 1.00 1.50 1.75 Model length (Â£) 0.20 0.30 0.40 (in) width (w) 0.018 0.027 0.031 p' m 10 10 10 w /w p m 56 56 56 Buildings This part includes the three previously discussed kinds of buildings: high-rise, medium-rise and residential buildings. Table 6 shows a sumnary of the average parameters for these three categories. In searching for material to be used in constructing the buildings in the model, it was found that the ratio of length and width of a standard block was very close to that of the prototype. Another advantage in using concrete blocks is that they are easy to set up, since each TABLE 6: Average Parameters of Prototype and Model for Building Areas PROTOTYPE MODEL (Nd = 10) Type of Average Dimension (ft) Approximate Dimension (ft) Density % Dimension (in) Density Buildings 1ength width 1ength width no. no. Deve- 1 oped 1ength width no. no. l000'x500' acre 8'x8' 8'x2,871 High-Rise 224 112 225 109 7.19 0.63 36 174 15.50 7.50 28 10 Medium-Rise 99 51 103 ' 50 23.62 2.06 24 80 15.50 7.50 20 7 Residential 62 30 68.87 6.00 26 48 15.50 7.50 20 7 Note: 1 acre = 43,560 ft^ 66 concrete block is heavy enough to withstand all the flow velocities in this study. Therefore it was not required to anchor them to the flume bottom. Scales of the model are then determined from the horizontal dimension of a concrete block (15.5 in x 7.5 in) and the dimensions of prototype buildings, as shown in Table 6. It is noted that in order to scale the prototype buildings into the 8 foot wide flume, it was necessary to use a distorted model. Model Setup Mangrove Stems and Roots Based on the average parameters obtained from the prototype, patterns of red and black mangroves were designed and shown in Figures 21 and 22. These two patterns were the best arrangements that could be achieved in the modeling in order to insure that the stems and roots were distributed evenly and yet still maintained their own natural characteristics in dispositions. For example, the prop roots were arranged in a hexagon pattern, which was found to be the most common disposition found in natural. The staggered pattern used for the stems of black mangroves and the root system of red mangroves was considered to be the best regular pattern to simulate the fully random distribution found in the prototype. The legends listed in Figures 21 and 22 were the actual dimensions used in model setup. The stems of red and black mangroves were simulated by dowels of the specified diameters and heights. The substems and prop roots were simulated by galvanized nails with the caps removed. The air roots of black mangroves would be very hard to model on a one by one basis due to 67 o o o 0+0 o o o O o 0 + 0 o o o o O o + o O rÂ« o o o o+o o O o o o o 0*0 o o O o 0+0 o o o o o 0+0 o O o o O 0 + 0 o o o 0 o o O O 0+0 o o o o o 0+0 o o o o o o 0 â€¢ 0 o o 15â€™ h3ui o + o o o O Main-Stern â€¢ Sub-Stem 0 Prop Root FIGURE 21: Model Setup for Red Mangroves 68 o Stem â€¢ Air Root FIGURE 22: Model Setup for Black Mangroves 69 their high density and small dimensions. Therefore, a manufactured nylon door mat whose strings have the same height (0.6 inch) and the same thickness (0.025 inch) as the design model dimensions of air roots was used. Density of the strings is 44 per inch square area which is 91 l of the average design density (48 per inch square area). The only deficiency in using this mat is that the strings are blade-shaped, which may cause a higher resistance to the flow than cylindrical air root. However, considering the advantages of using the mat, this deficiency is considered to be insignificant. These dowels, nails and mats were fixed on three 8 feet by 4 feet marine plywood sheets, which were coated with latex paint to prevent swelling (Figures 23 and 24). The plywood sheets were secured to the false bottom by a row of concrete blocks and by a 24 feet long L shaped steel beam attached to two sides of the plywood sheets (Figure 25). The row of concrete blocks stacked 15 inches high was placed in the main flume, starting from the last flow straightener and extending a distance of 80 feet. Thus only half of the flume width was used in these experiments. Canopy of Red Mangroves During the second part of the experiments a canopy was constructed in the red mangrove area. Strips of galvanized metal plates were used to simulate 'leaf strips'. This assumes that the leaves are closely connected to each other. Strips with three different widths, 0.2, 0.3 and 0.4 inch, represent three different sizes of leaves, as shown in Table 5. Each stripe has a height of 9.6 inches, which covers 539, 360 and 308 leaves for section #1, #2 and #3, respectively. Stripe numbers FIGURE 23: Stems and Roots of Red Mangroves FIGURE 24: Stems and Roots of Black Mangroves 71 FIGURE 25: Overview of Setup for Mangroves FIGURE 26: Setup of Model Equivalent of Canopy of Red Mangroves 72 for each section, which is 5 feet long, 34.2 feet wide and 8 feet high in the prototype, can be calculated from the densities measured (cf. Table 2). As a result, stripe numbers needed in the model for section #1 to #3 are found to be 25, 20 and 10, respectively. These stripes were also arranged in a staggered pattern, as shown in Figure 27. Figure 26 shows the setup of the stripes in which the stripes are susÂ¬ pended from the top of the supporters and fixed to the plywood bottom. Buildings In this part of the experiments, the whole width of the flume was used. As mentioned in the last section, the concrete blocks with dimensions 7.5 inches x 7.5 inches x 15.5 inches were used to simulate the buildings for the three different types of developed areas. Figure 28 shows 21 patterns to be tested in which no. 1 to 13 were designed to simulate high-rise building areas, while no. 14 to 21 were for medium- rise building and residential areas. As can be seen in these patterns, both the aligned and staggered dispositions were included for each density of the buildings. The design densities are started from low to high and will at least cover the average densities obtained from the prototype for the three developed areas (cf. Table 6). No extra work was needed to anchor these concrete blocks except to move them into the desired positions, since each concrete block weighs about 38.5 pounds and two layers of blocks are steady enough to withstand all the flows used in this study. SECTION # I SECTION# 2 SECTION # 3 73 X 0.2" x 9.6" Â° 0.3" x 9.6" A 0.4" x 9.6" FIGURE 27 Distribution of Leaf Stripes in the Model OPEN AREA 4*â€”ROOT AND STEM AREA-*) 74 FLOW â–¡ â–¡ â–¡ â–¡ (2) â–¡â–¡â–¡â–¡â–¡â–¡ (3) â–¡â–¡â–¡â–¡â–¡â–¡â–¡â–¡ (4) ODDOOOOOa] L FIGURE 28: Building Patterns Designed for the Tests 75 uu TT nn M FLOW V (9) (10) U U TL â–¡ u u n n â– - (ID (12) (13) 8' i l â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ 8' â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ 11 â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ (14) (15) FIGURE 28: CONTINUED 76 FLOW iâ€” , â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ a â–¡ 0 â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ G â–¡ â–¡ â–¡ â–¡ (16) (17) â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡ â–¡â–¡ â–¡ â–¡ â–¡â–¡â–¡â–¡ - â–¡ â–¡ â–¡ â–¡ D 0 Q â–¡ â–¡ â–¡ 0 D â–¡ â–¡ â–¡ D â–¡ â–¡ â–¡ â–¡ (18) (19) â–¡â–¡â–¡â–¡â–¡â–¡ â–¡â–¡â–¡â–¡â–¡â–¡ â–¡â–¡â–¡â–¡â–¡â–¡ â–¡â–¡â–¡â–¡â–¡â–¡ â–¡â–¡â–¡â–¡â–¡â–¡ â–¡â–¡â–¡â–¡â–¡â–¡ ODOGDO 0 0 D â–¡ D â–¡ (20) (21) FIGURE 28: CONTINUED CHAPTER VI EXPERIMENTAL TEST SERIES Experimental Procedure Calibration of Velocity Meter After the construction of the apparatus, the first step in the experimental procedure was to calibrate the velocity meter. Mounted with its normal support on the carriage, the Nixon meter was pulled through still water at constant velocity over a distance of 20 to 40 feet with the trolley. By operating the specific keys on the HP 9825A calculator to execute program statements which read initial and final values of propeller revolutions and time, the average frequency of the current meter, the true velocity and the percent error of the calibraÂ¬ tion curve were then computed and printed out. If the absolute error was greater than 5 % the instrument was recalibrated. The meter was checked in the range of 5 to 60 cm/sec and no less than 20 points were used to determine a linear least square fit of frequency versus velocity. Appendix C contains a complete program listing for the HP 9825A. Measurements of Mean Velocities The velocity and depth obtained in this study were measured in the center line of the test sections where the influence of sidewall was not felt. The experimental run begins when the main pump is started. It 77 78 usually takes about twenty minutes for flow to reach steady state for each set of discharge values. After the flow became stabilized, velocities were taken at one section 30 feet downstream of the last flow straighteners. Nine points on a vertical at the relative depths of: z/d = 0.1504, 0.1881, 0.2352, 0.2492, 0.3679, 0.4601, 0.5754, 0.7197, 0.9000, were sampled to best describe the vertical velocity profile (Christensen, 1978). The velocity at each depth was then determined by the velocity program (Appendix B) from the calibration formula and was printed out for immediate checking. Each velocity obtained is on average velocity over a time span of 30 seconds which is the maximum time interval that can be used with the HP 9825A. Plotting the vertically distributed velocities on graph paper and integrating over the water depth yields the discharge per unit width /udy. The spatial mean velocity for each run then was obtained from the value of unit-width area divided by the water depth. Even though this method is time consuming for the large number of runs, it is still the best way to determine the mean velocity for the mangrove part of the experiments in which the test channel occupies one half of the main channel. For the building part of the experiments, in which the entire main channel was used, a discharge formula for the Thomson weir derived by the hydraulic laboratory of Civil Engineering Department, University of Florida, was applied to determine the mean velocities, i.e., 2.514 , , Q = 2.840 H (82) where Q is the discharge from the Thomson weir in cubic feet per second, and H is the vertical distance in feet between the elevation of the 79 lowest part of the notch or the vertex and the elevation of the weir pond. Eighteen runs with mean velocities from 17 cm/sec to 53 cm/sec were tested by both methods to determine the accuracy of the weir formula; the results are shown in Figure 29. It is apparent that the mean velocities obtained using the Nixon meter in the center line of the flume Un is slightly larger than that given by weir formula Uw, but is within a limit of 5 %. This small error is considered to be insigniÂ¬ ficant and may be compensated for by the advantages of using the weir formula. For instance, the fluctuating water level above the weir vertex due to the instability of the pump was often observed, therefore, the mean velocity obtained from an average value of H over a longer period of time should be more representative than that measured by the Nixon meter over a 30 second period. Measurements of Water Depths For each run three water depths were measured by using the device shown in Figure 19. Two water depths were taken at the two sections which covered the roughness area and one was taken at the section where velocities were measured out of the roughness area, located 30 feet downstream of the last flow straighteners. Since the water head losses between the two sections in the model tests were in the range from less than 1 to a few centimeters, the water depths were measured to an accuracy of one hundredth of a centimeter for a precise and reliable result. To implement this fine measurement, all the siphon hoses used in the tests were kept free of air bubbles and the well graduated electronic point gages were used. Before each run the still water 80 FIGURE 29: Relation Between U and U n w 81 depth was measured and its scale reading for water surface elevation on the point gage was recorded. The same reading was performed for each section after the flow became stable. From the difference of these two readings for water surface elevations and the initial still water depth the flowing water depth can be calculated. In general, it takes about 5 to 20 minutes for a new water level in the tube (cf. Figure 19) to reach its equilibrium state, which can be observed by moving the vernier on the point gage to see whether any change in the water level is detected. This water depth measuring device worked very well through the entire experiment and provided consistent and reliable data. Experimental Runs Mangrove Areas The total model lengths of the red and black mangrove areas were 5 and 15 feet, respectively (Figure 25). For the red mangrove area the water depths were measured at the two ends of the 5 feet long area. For the black mangrove area, the first section was chosen 4 feet from the front end, and the second section was located 3.5 feet from the rear end of the black mangrove region so that the influences, including the disturbance caused by the red mangroves in the front, and the depth drop due to the end of the plywood sheets in the rear could be eliminated. Therefore, a total length of 7.5 feet centered in the middle section of the black mangrove area was used to measure the energy loss. During the first part of experiments, 7 runs were conducted for the air roots of the black mangrove area to determine its apparent roughness 82 height and friction factor. In the second part of the experiments, 38 runs were performed for the red mangrove areas (without canopy) and black mangrove areas by adjusting the discharge value and changing the still water depths so that the flow Reynolds number (R^ = UgR^/ v) covered a range from 20,000 to 55,000 while the Froude numbers varied from 0.14 to 0.44. An additional 32 runs were conducted for the red mangroves with canopy at the later stage to determine the importance of a canopy in reducing the flow energy. Building Areas At least 10 runs were conducted for each of the 21 patterns shown in Figure 28. These runs for each pattern were controlled by adjusting the flumes discharge valve and the tail gate so that they covered a range of Reynolds number (R^) from 20,000 to 70,000, while the Froude number varied from 0.1 to 0.5. Figure 30 shows 20 pictures of the designed patterns in which pattern No. 9 is not included due to the faulty picture. The results obtained for medium-rise building areas can be converted using appropriate scaling factors to use in residential areas since these two areas are presumed to have the same relative distributions and have only dimensional differences. 83 FIGURE 30: Designed Building Patterns in the Tests CHAPTER VII PRESENTATION AND ANALYSIS OF DATA The data obtained from each experiment include two water depths, d.| and d^, and two depth averaged velocities, U-| and U^. Other parameters, such as the energy loss AH, equivalent friction factor f', apparent roughness k , Reynolds number R', Froude number F (U //gd ) and the averaged drag coefficient Cn are calculated using their specific definitions as given by equations (78), (79), (31) and (48). Mean values and the corresponding standard deviation of f', k G 3 and Cp were also determined for each set of tests. A summary of these results is listed in Appendix C, in which II = (U, + U9)/2 and d= = (d.j + d2)/2. Three relations between the friction factor, the drag coefficient and Reynolds number, and the friction factor and the water depth were found from the data for each area and plotted as shown. Mangrove Areas Figures 31 to 33 show the three relations for red mangroves withÂ¬ 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 coeffiÂ¬ cients and apparent roughnesses. 84 2.000 1.600 1.200 0.800 0.732 0.400- 0.000 J- 1 1 1 i ! I Ã ! ! i i i i i i I I i i i i Â¡ill!! i i ! ! ! ! iiiiii iiiiii 0 l r\ n j I l l I I l i I till l l l I till l l l i O OOI O 1 0 U U i r V/ U 0 0 WW Q ÃOOâ€¢ .00 n i Mean 0 0^ o o o o 0 ! 0 0 0 0 O l 00 OO 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 â– â€” i â€”H 1 1 1 1 â– â€”â€” 20,000 25,000 30,000 35,000 40,000 45,000 50,000 5,000 R' e 03 cr> FIGURE 32: Relation Between and R/ for the Red Mangrove Areas (Without Canopy) FIGURE 33: Relation Between f and Water Depth d for the Red Mangrove Areas (Without Canopy) e FIGURE 34: Relation Between and R^ for Red Mangrove Areas (with Canopy) FIGURE 35: Relation Between CD and R^ for Red Mangrove Areas (with Canopy) d (cm) FIGURE 36: Relation Between f and Water Depth d for Red Mangrove Areas (with Canopy) FIGURE 37: Relation Between f and R' for Black Mangrove Areas e e FIGURE 38: Relation Between and R^ for Black Mangrove Areas d (cm) FIGURE 39:' Relation Between f and Water Depth d for Black Mangrove Areas 94 TABLE 7: Statistical Values of Experimental Results for Mangrove Areas Statistical rÂ» CD ka(cm) Mangroves^- Values 1 e Model Prototype Red Mean 0.129 0.732 42.69 426.9 Standard Deviation 0.013 0.077 6.14 61.4 Red + Mean 0.134 0.758 44.10 441.0 Canopy Standard Deviation 0.012 0.078 6.48 64.8 Black Mean 0.132 1.001 38.53 385.3 Stems Standard Deviation 0.020 0.090 9.72 97.2 Black Mean 0.028 5.26 52.6 Air Roots Standard Deviation 0.003 0.65 6.5 It is clear that the experimental results show a good degree of consistency in the relations between the equivalent friction factor, drag coefficient and Reynolds number. This is especially true for the drag coefficients obtained from the experimental data, which have only about 10 percent standard deviation. This consistency not only proves that Eq. (48) proposed for the evaluation of the equivalent friction factor is acceptable, but also implies that the accuracy of the measurements of the water depths and flow velocities was fairly high. The increase of the equivalent friction factor and drag coefficient due to the existence of canopy in red mangroves is less than 4 percent, which is obtained by comparing their mean values as listed in Table 7. 95 The relations between the friction factor and water depth for the mangrove areas are shown in Figures 33, 36 and 39, which provide an important clue that the equivalent friction factor for protruding obstructions like black mangroves is linearly proportional to the water depth as proposed in equation (48). For red mangroves, however, the linear relation due to the protruding stem is eliminated by the subÂ¬ merged prop roots which give an inverse relation between the equivalent friction factor and water depth as described by equation (30). It shall be noted that the drag coefficients obtained for red mangroves represent the mean values of stems, substems and prop roots. For black mangroves, the mean value of the drag coefficient only takes into account the stems, while the air roots were considered to be the bottom friction and their influences on the energy losses were excluded through drag coefficient calculations. Building Areas The experimental results for 21 different distribution patterns for building areas are shown in Figures 40 through 42. The statistical values of these results are listed in Table 8, in which the data for residential areas were converted from those for medium-rise building areas. Since all these runs were conducted in the false-bottomed section which is made of the smooth metal plate, the energy loss between two cross sections was hardly measured. Comparing the small energy loss to the experimental errors, the bottom friction, therefore, was neglected through the entire tests for building areas. In general, the relations between the equivalent friction factor, drag coefficient and Reynolds number as shown in Figures 40 and 41 are similar to those e v (1) UUU | 20,000 45,000 70,000 e v (2) ' FIGURE 40: Relation Between f and R' for Building Areas e e (Number in the Parenthesis is the Pattern Number) 0.150' 0.00Q f e 0.07S + 00 CD O O 20,000 45,000 Râ€˜ e 70,000 (3) FIGURE 40: CONTINUED e (8) FIGURE 40 0.020 r e 0.100- 0.000 20,000 O 00 0 o 45,000 R' e (7) 70,000 e (9) CONTINUED e (12) FIGURE 40 0.060- â™¦ f* e 0.030' 0 0 O.OOGl 20,000 O 0 45,000 70,000 0.10Q f e 0.050' 0.000 0 0 20,000 0 0 o 0 0 o o 45,000 R' (13) 70,000 CONTINUED e (16) FIGURE 40 0.050 r e 0.025 0.000 0 0 0 0 0 0 0 0 0 0 0 20,000 45,000 r; (15) 70,000 e (17) o o CONTINUED 0.050 0.025 0.000 20,000 0.120 f e 0.060 70,000 0.000 20,000 7ÃœTÃœD0 FIGURE 40: 0.150' 1 1 1 f e 0.0 / b o o o o o â€¢ o o o o o L 0.000 â– â– â– - 20,000 45,000 70,000 R' e (19) CONTINUED FIGURE 41: Relation Between and R^ for Building Areas (Number in the Parenthesis is the Pattern Number) (4) (5) FIGURE 41: CONTINUED 1.110 0.000 Me on o 20,000 45,000 RÂ¿ 70,000 (6) e (8) FIGURE 41 20,000 5.000 CD 2.500 1.600 0.000 Mean TT 20,000 45,000 R' e (7) 70,000 u 0 0 00 U 0 0 o -c* 45,000 R' e (9) 70,000 CONTINUED e (12) FIGURE 41 (ID o cn CONTINUED 20,000 45,000 R' (14) 70,000 FIGURE 41 5.000 2.500 2.080 0.000 Mean -tr ooo o oo â€”o-o-o 20,000 45,000 R' e (15) 20,000 45,000 R' e (17) 70,000 70,000 CONTINUED (18) (20) FIGURE 41 5.000 D 3.360 2.500 0.000 Mean 1 1 1 1 1 1 O OI 0 0 O" O" o 1â€”O" oâ€”o 1 1 1 1 i i 1 1 1 1 1 20,000 45,000 R' (19) 5.000 3.930 CD 2.500Ã- O.OOQl 20, n o o o .QQQQ 70,000 Mean o ^1 000 45,000 R' e (21) 70,000 CONTINUED 0.012 0.024 d(cm) (10) f e 0.012 0.000 0.0 15.0 30.0 d(cm) (ID FIGURE 42: Relation Between -T and Water depth d for Building Areas TABLE 8: Statistical Values of Experimental Results for Building Areas Pattern No. f7 e CD Buil ding Area Mean Standard Deviation Mean Standard Deviation val ue % of mean val ue % of mean 1 0.0013 0.0003 23 0.788 0.163 21 High-Rise 2 0.0064 0.0006 38 1.965 0.254 13 II 3 0.0180 0.0030 17 3.406 0.181 5 II 4 0.0460 0.0090 20 7.014 1.121 16 II 5 0.1580 0.0260 16 17.813 1.298 7 II 6 0.0036 0.0006 17 1.108 0.123 11 II 7 0.0046 0.0011 24 0.907 0.230 25 II 8 0.0090 0.0010 11 1.346 0.119 9 II 9 0.0130 0.0010 8 1.597 0.189 12 II 10 0.0080 0.0014 18 2.369 0.275 12 II 11 0.0154 0.0029 19 2.965 0.227 8 II 12 0.0230 0.0040 17 3.662 0.221 6 II 13 0.0350 0.0050 14 4.315 0.389 9 II 14 0.0074 0.0011 15 1.232 0.170 14 Medium-Rise 15 0.0127 0.0030 24 2.082 0.173 8 II 16 0.0117 0.0020 17 1.088 0.096 9 II 17 0.0329 0.0067 20 3.040 0.114 4 II 18 0.0158 0.0018 11 1.181 0.097 8 II 19 0.0448 0.0072 16 3.355 0.098 3 II 20 0.0265 0.0038 14 1.586 0.089 6 II 21 0.0631 0.0112 18 3.926 0.120 3 II 14 0.0123 0.0018 15 1.232 0.170 14 Residential 15 0.0211 0.0050 24 2.082 0.173 8 II 16 0.0195 0.0034 17 1.088 0.096 9 II 17 0.0548 0.0112 20 3.040 0.114 4 II 18 0.0264 0.0030 11 1.181 0.097 8 II 19 0.0747 0.0120 16 3.355 0.098 3 II 20 0.0441 0.0063 14 1.586 0.089 6 II 21 0.1052 0.0186 18 3.926 0.120 3 II no in mangrove areas. The drag coefficient obtained from the experimental data for each of these 21 patterns also show a good consistency for all the flow conditions in which the standard deviations of 13 patterns are less than 10 percent of the mean drag coefficients. For all the patterns in building areas, the equivalent friction factors are found to have similar linear relation to the water depths, and four examples, taken from patterns 10 and 13, to express this relation are shown in Figure 42. Determination of Drag Coefficient The consistent results of the drag coefficients obtained from the experimental data imply that the mean value of the drag coefficients can be represented for each building pattern as listed in Table 8. Based on these mean drag coefficients and their corresponding building densities, dispositions and area types, the drag coefficient may be determined empirically through analyses of these relations. Drag Coefficient - Building Density Relations The data obtained from pattern 1 to 13 were analyzed and plotted in Figure 43, which shows three possible relations between the drag coefficients and densities for high-rise building areas. Figure 44 shows two relations between drag coefficients and densities for the medium-rise building area and residential area, respectively. The drag coefficients obtained for the high-rise building area were also converted in order that they can be used in the medium-rise building and residential areas. These interpolated data from the two row high- rise building areas, however, agree very well with the data obtained for the other two building areas (Figure 44). The drag coefficients Ill FIGURE 43: Relation Between CD and Density m for High-Rise Building Areas 112 RESIDENTIAL AREA, m(no./ft2) (x 1.47 xIO5) MEDIUM-RISE BUILDING AREA,m(no./ft2) (x 4.10 x I05) FIGURE 44: Relation Between CD and Density m for Medium-Rise Building and Residential Areas 113 as a function of densities were then derived by using a least square curve fitting method. Seven such equations were derived, including three for the high-rise building areas and two for medium-size building and residential areas, respectively. It is apparent that the drag coefficient for the staggered disposition is about two to three times larger than that for the aligned disposition under the same building density for all the three building areas. The drag coefficient is found to vary linearly with the building density for all the cases except for the one row high-rise building case, which shows an expoÂ¬ nential relation between the drag coefficient and building density. Using the equations shown in Figure 43 and 44, the drag coefficient can easily be determined for developed areas if the building density is known. Drag Coefficient- Disposition Parameter Relations Though the drag coefficient were shown to be explicitly related to the building densities and dispositions in the previous section, it is desired to represent the drag coefficient in a simple equation which takes both the dependent parameters, density and disposition, into conÂ¬ sideration. As shown in Figure 45, the lateral spacing between two roughness elements (edge to edge) is S^. and the longitudinal spacing between two successive roughness elements (edge to edge) is S^. The corner-to-corner distance between the roughness elements in adjacent transverse raws is S^, which is related to and by S - D 2 2 u 2 5^ = 5^+ (â€”2 ) ln staggered arrangement (83) in aligned arrangement (84) 114 [Is'n. â–¡ % sd ti â–¡ â–¡ â–¡ t FLOW \JS'T\ â–¡ Sg, Sd ^0 Ãœ â–¡ HD I*- â–¡ â–¡ T â–¡ FLOW (a) Aligned Arrangement (b) Staggered Arrangement FIGURE 45: Position Spacings. Definition Sketch in which D is the projected with of the roughness element in the direction of the flow. A result of attempting to relate the drag coefficient to the three disposition spaces, S^, SÂ¿ and Sd is listed in Table 9. By plotting these relations between the drag coefficients and non-dimensional disposition parameters, St/D, S /D, Sd/D and Sj/S^, it is found that only Sd/D and show a good correlation, while the others either do not yield any relation (S^/D and C^) or show a poor correction (S^/D, Sd/S^ and C^). The relations between the drag coefficient and the disposition parameter, Sd/D, are shown in Figures 46. Comparing Figures 46 and 44, it is clear that the drag coefficient is closely related to the density or to the disposition parameter, Sd/D, for the aligned and staggered patterns, respectively. However, by combining the two relations for the aligned and staggered patterns in Figure 46, it is found that the drag coefficient can be solely expressed as a function of the disposition parameter, Sd/D, for both the aligned and staggered cases, ie., TABLE 9: Relations Between Disposition Parameters and Drag Coefficients Pattern No. Spacings (in) Disposition Parameters CD st sd St/D VD Sd/D Vs 27.0 3.60 0.788 2 13.2 1.76 1.965 3 7.3 0.97 3.406 4 4.0 0.53 7.014 5 1.9 0.25 17.813 6 27.0 3.44 27.22 3.60 0.46 3.63 7.91 1.108 7 18.4 3.44 18.72 2.45 0.46 2.50 5.44 0.907 8 13.2 3.44 13.64 1.76 0.46 1.82 3.97 1.346 9 9.8 3.44 10.39 1.30 0.46 1.38 3.02 1.597 10 33.9 3.44 13.64 4.52 0.46 1.82 3.97 2.369 11 22.1 3.44 8.07 2.95 0.46 1.08 2.35 2.965 12 15.5 3.44 5.28 2.07 0.46 0.70 1.53 3.662 13 11.3 3.44 3.93 1.51 0.46 0.52 1.14 4.315 14 18.4 12.38 22.18 2.45 1.65 2.96 1.79 1.232 15 22.1 12.38 14.37 2,95 1.65 1.92 1.16 2.082 16 13.2 6.80 14.85 1.76 0.91 1.98 2.18 1.088 17 15.5 6.80 7.89 2.07 0.91 1.05 1.16 3.040 18 9.8 6.80 11.93 1.30 0.91 1.59 1.75 1.181 19 11.3 6.80 7.06 1.51 0.91 0.94 1.04 3.355 20 7.3 6.80 9.98 0.97 0.91 1.33 1.47 1.586 21 8.4 6.80 6.85 1.12 0.91 0.91 1.01 3.926 116 FIGURE 46: Relation Between C~ and S ,/D in Aligned and Staggered Patternsu 117 S -2*0 CD = 1.0 + 1.8 (-Â§-) (85) which is shown in Figure 47. This is attributed to the nature of the diagonal spacing, S^, whose magnitude not only provides a measurement of density but also reveals a difference of disposition for the evenly distributed roughness elements. Therefore, the higher the building density, the smaller the disposition parameter, and therefore a larger drag coefficient results. For buildings with the same dimensions and density, the disposition parameters, S^/D, of staggered patterns is smaller than that of aligned patterns, which also results in a larger drag coefficient. The same result was also substantiated by Shen (1973) who measured the mean drag coefficient of two cylinder patterns (aligned and staggered) in open channel flow. Flow resistance through these roughness elements varies row by row. In general, the drag on the second row element is much smaller than that on the first row. It is due to the result of decreased wake dynamic pressure and the increased turbulence level of flow approaching the second row element. Therefore, since the elements of the second and following rows for aligned pattern are located at the low wake pressure areas, the mean drag is smaller than that for staggered pattern, in which the elements are not or only partially located at the wake areas. 118 FIGURE 47: Relation Between CD and Sd/D CHAPTER VIII DISCUSSIONS AND CONCLUSIONS In the preceding chapters, both the wind stress and the bed shear stress used in storm surge equations for the case of hurricane-induced flooding in coastal areas were discussed. Special emphases were placed on the friction characteristics of developed coastal areas and mangrove areas. In the air section, the main reduction of wind shear stress due to wind drag forces on the parts of the obstructions, which include buildings and vegetation, above the water surface in flooding has been developed in equations (19) and (20). In the water section, the equiÂ¬ valent friction factor, which incorporates both the bottom friction coefficient and the drag coefficient due to the submerged parts of obsÂ¬ truction was presented in equation (49) and verified. The drag coefÂ¬ ficient in equations (20) and (49) for mangroves and buildings were determined empirically according to the typical distribution found in coastal areas. The results obtained in this study will assist coastal engineers and coastal planners in providing a better and more accurate method for predicting hurricane-induced flooding. Discussions given below are primarily based on engineering applications, these determinaÂ¬ tions of the equivalent friction factors for ocean bottom, grassy areas and forested areas are also described. 119 120 Mangrove Areas The proposed equation for the equivalent friction factor can be expressed in the following form for mangrove areas fe 0.32 [ln(I%Md + l )]2 n (1-e) + Â£ mâ€¢cndDi 1=1 1 u 1 (86) Typical values of k, e, m, CQ and D obtained in the previous chapters are summarized in Table 10. The value of k for red mangrove open areas is converted from the value of Manning's n, 0.025, which corresponds to a clean, straight stream on plain in open channel flow, by using the Strickler equation. TABLE 10: Typical Values for Mangrove Areas Mangrove k (m) Â£ ml (nr2) (nr2) m_ (nf2) CD (cm) D2 (cm) D3 (cm) Red 0.07 0.013 0.3 1.0 6.1 0.73 15 5 3 Black 0.53 0.005 0.9 1.00 8 For black mangrove areas, the value of k is obtained from the experimental results for air roots. The subscripts 1, 2 and 3 of m and D represent the main-stems, sub-stems and prop roots, respectively. It has been shown that the canopy on mangrove fringes causes only a 4 percent increase in the resistance to water flow, and thus, may be neglected. It shall be noted that red mangroves, in general, only extend, about 15 meters inland from the shoreline and it is not possible to distinguish them from 121 black mangroves on aerial photographs and conventionally used city maps. Examining the case of fine grid system for numerical models whose grid elements are usually chosen to be 1,610 meters square, the average friction factor for each grid element on the mangrove fringes should be represented bt the sum of 1 % of the friction factor obtained from red mangroves plus 99 % of that obtained from black mangroves. For engineering applications, therefore, it is not necessary to consider the difference between red and black mangroves. Therefore, the equivalent friction factor derived for black mangroves can be used for all the mangrove areas, i.e., f' = â€” + o.07d (37) e [1n(21d + 1)] in which the depth in meters. Developed Areas In developed areas, seven drag coefficient equations for most of the building arrangements observed in Broward and Dade counties, Florida, were determined and listed in Figures 43 and 44. To apply these equaÂ¬ tions, however, the building density and the type of area must be given. The most realistic way to determine these is to estimate them using aerial photographs of the study areas. County Land Use Plan Map published by each county are also recommended. If both the aerial photographs and the County Land Use Plan Map are not available, or only a rough estimation is needed, the average densities obtained for Dade and Broward counties for the different developed areas can be utilized. 122 The following three drag coefficient equations are suggested for friction factor cal cualtions in developed areas: r . â€ž ocr _2.7m x 105 . _ r_T no. Cp = 0.355 e ; m L=J â€”f^r no. or CD = 0.355 e2,9m x 104 > m Â£ ^ ~Â¡ÃF (88) for the one row high-rise building areas; Cp = 1.05 + 4.8m x 104 ; m [=] or Cp = 1.05 + 5.2m x 103 ; m [=] (89) for the staggered medium-rise building areas; and Cp = 1.04 + 1.7m x 104 ; m [=] or Cp = 1.04 + 1.8m x 103 ; m [=] -^- (90) for the staggered residential areas. The particular equation is recomÂ¬ mended because one row high-rise buildings are the most predominate type of buildings found in highly developed coastal fringes such as those in Broward and Dade counties, Florida. For medium-rise buildings and resiÂ¬ dential areas, most buildings can be assumed to be staggered, since aligned buildings are scarce and the probability that floods will flow normal to the aligned building areas is very small in the field. An alternative way of determining the drag coefficient is to use equation (85) in which the diagonal spacings between buildings should be measured from aerial photographs. It is convenient in this case to 123 apply equation (85) to areas whose dispoition parameters, S^/D are nearly same so that the drag coefficient can be determined without considering the building types and dispositions. The equivalent sand roughness, k, for developed areas cannot be determined by a single value, since the space between the buildings cover many different roughnesses. These roughnesses, in general, consist of pavements, grasses, light brush and trees, whose values of Manning's n are 0.013, 0.025 and 0.040, respectively. Assuming these roughnesses are evenly distributed, then an average value of n equals to 0.026 is assigned for these developed areas which gives k = 0.09 m using the Manning Strickler equation. As a result, the equivalent friction factor for developed areas may be represented as fâ– = ^32 (1 - e) + m CR d D (91) e [In (122d + 1)] Ã¼ with d in meters. The mean values of e, m and D can be determined from information in Table 6 and the equations for are shown in Equations (88) to (90). Ocean Bottom To determine the friction factor and the equivalent sand roughness for the ocean bottom, the characteristic morphologic features of the ocean bottom in the study area should be determined first. The ocean bottom is mainly composed of different sizes of sands, on which vegetaÂ¬ tion and sand ripples vary with location. Sand ripples are generated and modified by wind-generated waves. Their profiles are controlled by the nature of the near-bottom wave motion and by the size of the bed 124 material. Numerous investigations have been conducted to determine the friction factor and the equivalent sand roughness by using experimental results (cf. Vitale, 1979) and field measured data (cf. Nielsen, 1983). However, the applicability of these results obtained for normal wave conÂ¬ ditions to storm waves, during which strong current exists, is greatly in doubt since not only the characteristics of storm waves are not able to predicted, but the mechanism of ocean bed formation under storm conditions is not fully understood at the present time. This uncertainty, plus the varying distributed vegetation on the ocean bottom makes the direct estiÂ¬ mation of the friction factor by using the aforementioned results obtained for normal wave conditions questionable. Therefore, an indirect calibration method is sometimes used in this case. As suggested by Corps of Engineers (1977), typical bottom condiÂ¬ tions result in a value of f in the range 0.004 to 0.01. For a first estimate, a value of f = 0.005 may be assumed. This coefficient is used in calibrating the storm surge numerical model. It not only accounts for energy dissipation at the bed, but may be used to adjust for inexact modeling and deficiencies caused by ignoring some of the hydrodynamic processes involved. However, it shall be noted that the calibration method will also reflect inconsistencies in the numerical method in the f-values. Therefore, f-values found this way may not be physically correct. In other words, they can only be used with the specific algoriÂ¬ thm used in the calibration. Nevertheless, the hydraulic measurement in the field is still the most reliable way to determine the bed friction characteristics, but since field measurements of velocity distributions and water depths are nearly impossible during a storm, a field measurement under normal condition 125 which is similar to the storm tide is greatly needed. One possible instance is the flood and ebb flows through a tidal entrance in which the flow is in the fully rough range, and the effect of the temporal acceleraÂ¬ tion of the flow is generally of a lesser magnitude than the effect due to bed friction; therefore a correspondence between entrance flow and storm flood is expected. Three tidal entrances located on the Gulf coast of Florida were selected for hydraulic measurements to determine the bed friction characteristics (Mehta, 1978). Table 11 lists some of the results obtained from field data, in which the bed friction characterisÂ¬ tics derived are based on near-bed velocity profiles in John's Pass and Blind Pass, and water surface slope in the channel at O'Brine's Lagoon Entrance. The values of Manning's n obtained for these three entrances are in the range from 0.020 (k = 2.1 cm) to 0.026 (k = 9.5 cm), which corresÂ¬ ponds to the following bed morphologic features in open channel flows: clean, straight stream on plain (n = 0.025-), gravel uniform excavated channel (n = 0.022), and uniform dredged earth channel with short grass (n = 0.022). This correspondence not only implies that the values of n in the Manning's n table can be applied to tidal entrance, but that they can also be used for the ocean bottom during storm events. Therefore, for a rough ocean bottom, where the vegetation is significant or other roughnesses such as rocks or reefs exist, the n value is assumed to equal to 0.035 (k = 0.55m), which represents an irregular, rough stream, a clean, winding stream on plain with some weeds and stones, a dredged channel with light brush on banks, or a flood plain with scattered brush and heavy weeds in open channel flows. TABLE 11: Bed Friction Characteristics of Three Entrances Entrance Tide Mean Depth Friction Roughness, Manning's n Bed Features at the Throat Factor,f k (cm) (m) John's Pass flood 4.9 0.007 9.5 0.026 Shells (predominate) and Fine Sands Blind Pass ebb 1.6 0.005 2.1 0.020 Shells and Find Sand O'Brien's Lagoon flood/ ebb 0.44 0.013 2.6 0.021 Sand, Shells and Ripples 127 As a result, the k values corresponding to the recommended n values are 0.03 m (n = 0.022) and 0.55 m (n = 0.035) for smooth and rough ocean bottoms, respectively. The friction factors for ocean bottoms can then be estimated using the equations f1 = for smooth ocean bottom (92) [1n(365d + l)]2 0 32 and f1 = ; for rough ocean bottom (93) [ln(20d + l)]2 in which the depth d must be in meters. Forested Areas Forested areas may consist of many different species, like oaks, magnolia, cedar, palm, pine and cypress, in which each species has different representative feature. However, for a general view, some typical values of the parameters for evaluating the equivalent friction factor are suggested by Christensen and Walton (1980) _2 k = 0.5 m ; m = 0.1 m ; D = 0.6 m e = 0.028 ; Cp <* 0.8 It is worthwhile to examine the drag coefficient recommended for forested areas. In Figure 47, it is seen that the drag coefficient has a tendency to become constant and equal to 1.0 when the disposition parameter SyD is larger then 4. This relation derived initially for the rectangular roughness elements may also be used for circular roughness elements and 128 can be proved by comparing the measured drag coefficient (1.0) for black mangroves as seen in Table 10. For black mangroves, the disposition parameters syD is about 17, and the corresponding drag coefficient is 1.0 according to Figure 46, which agrees well with the measured results. Applying the same reasoning to forested areas, whose disposition parameter Sd/D equals to 12, the drag coefficient can be expected to be approxiÂ¬ mately 1.0. Therefore, the drag coefficient 0.8 proposed for forested areas by Christensen and Walton is reasonable. Adopting these values, the equivalent friction factor can be expressed in the form f' = â€” 2- + 0.048d (94) e [1n(22d +1)] for the forested areas, where d is in meters. Grassy Areas The vast expanse of saw grass marshes are the dominate plant community occurring in the world's coastal zones where water stands all or part of a year. Direct observations of the roughness characteristics of saw grass in a completely submerged condition do not seem to be as readily available as are measurements of the roughness, as expressed by Manning's n, of the more common but smaller grasses used in lawns and as ground covers for protection against erosion. Therefore, the results of studies of the flow of water over various grass covers by Palmer (1946) may provide some information to evaluate the roughness characteristics of saw grass. Palmer found that a completely submerged surface of Bermuda grass has a Manning's n of about 0.04, which corresponds to a k-value of 129 about 1.25m. If it is assumed that there is geometric similitude between Bermuda grass and about three times taller saw grass, Palmer's experiment with Bermuda grass may be considered as a model test of the prototype saw grass. As a results, a k-value of about 4 m for saw grass is expected based on the 1:3 length scale between the Bermuda grass and the saw grass. Christensen's (1980) indirect observations of the equivalent roughness of grass covered beds of estuaries on Florida's west coast have indicated a k-value ranging from 3 m to 5 m, which is in good agreement with the value derived from Palmer's experiments with Bermuda grass. It should be noted that the apparent roughness measured Palmer may not depend only on the size, shape and distribution of the vegetation, but also on the addiÂ¬ tional energy losses induced by the rhythmic motion or vibration of the usually flexible vegetation elements. This motion may be attributed to the Karman vortex street trailing each individual grass element. IntroÂ¬ ducing a k-value of 4 m in the friction factor equation, as suggested by Christensen and Walton (1980), yields f = 0.32 [1n(2.7d + 1)]Z in which the depth d is in meters. (95) Cone!usions The research has developed the capability of describing the friction factor in coastal areas for improved representation in numerical modeling of storm tides. Special emphasis was placed on the friction characÂ¬ teristics of mangrove areas and developed coastal areas. The proposed 130 analytical results of friction factors for these two areas were verified by carrying out physical experiments in a hydraulic flume. In addition, the reduction of wind shear stress due to the protruding obstructions above the water surface is derived based on recent data of wind stress coefficient - wind speed relations. The friction characteristics for ocean bottoms, grassy areas and forested areas were also presented by adopting the results from existing field and laboratory measurements, which are considered as the most reliable data available at present. It is felt that this research has provided the most realistic data base for evaluations of wind stress and bed stress in numerical models of storm surges. Further verification of the wind stress reduction due to the obstructions and the canopy is recommended. Effects due to the interÂ¬ action between the wind and the sea surface, which causes deviation of the velocity profile from the ordinary logarithmic distribution, are not well quantified. The use of the quadratic bottom friction, predicted in steady open channel flow, can probably be improved upon. Under unsteady conditions, the drag coefficient CQ should be corrected to include the influence of added mass or virtial mass. It is of particular importance in the cases of rapidly accelerating or decelerating floodings, therefore, further study of the virtual mass effect is encouraged. Field data, especially during hurricane conditions, is practically non-existent. Therefore, it is most imperative that work continue into the measurement of field data, which would enhance the reliability of the physical model results and give a more complete understanding of the subject. APPENDIX A FIELD RECORDED DATA FOR MANGROVES 132 AREA # | FIGURE A1: Field Data Record for Red Mangroves (Area #1) 133 AREA # 2 FIGURE A2: Field Data Record for Red Mangroves (Area #2) 134 AREA # 3 FIGURE A3: Field Data Record for Red Mangroves (Area #3) 135 AREA # 4 FIGURE A4: Field Data Record for Red Mangroves (Area #4) 136 AREA # 6 FIGURE A5: Field Data Record for Red Mangroves (Area #6) 137 AREA # 8 FIGURE A6: Field Data Record for Black Mangroves (Area #8) 138 AREA # 9 FIGURE A7: Field Data Record for Black Mangroves (Area #9) 139 AREA # 10 FIGURE A8: Field Data Record for Black Mangroves (Area #10) 140. AREA # 12 O i 1 TH i 1 I o 1 Â° 1 o , 1 1 1 O 1~ â€œL O I | Â° I O o 1 i I Â° o 1 I â– i i L _ -I- 1 I 0 I 1 I o 1 1 _ J 1 o ! Â° i . lo I i q i i d 1 â€¢ i ! I O 6 (FT.) 12 FIGURE A9: Field Data Record for Black Mangroves (Area #11) 141 TABLE Al: Parameters of Sampling Red Mangroves AreaK Parameters No. (12ft)Â¿ Diameter (in) Height (ft) % Occupied main-stem 4 7.5 8.0 1.11 1 sub-stem 9 2.0 1.5 0.14 canopy 8.0 root 48 1.0 1.5 0.18 main-stem 3 3.7 8.0 0.16 2 sub-stem 10 2.0 1.5 0.15 canopy 8.0 root 63 1.0 1.5 0.24 main-stem 1 6.5 12.0 0.16 3 sub-stem 13 2.0 1.5 0.20 canopy 6.0 root 108 1.0 1.5 0.41 main-stem 5 2.7 8.0 0.14 4 sub-stem 16 2.0 1.5 0.24 canopy 10.0 root 95 1.0 1.5 0.36 main-stem 6 5.0 8.0 0.63 5 sub-stem 17 r\D â€¢ o 1.5 0.26 canopy 10.0 root 97 1.0 1.5 0.37 main-stem 3 10.7 14.0 2.81 6 sub-stem 10 2.0 1.5 0.15 canopy 6.0 root 74 1.0 1.5 0.23 TABLE A2: Parameters of Sampling Black Mangroves 143 TABLE A3: Characteristics of Canopy of Red Mangroves Section Area No. of Leaves Leaf Sizes (%) # # (1'x 5'x 6') 2"xl" 3"xl.5" 4"xl.75" 4.5"x2" 1 262 2 392 3 416 1 4 138 90 10 5 296 6 288 2 154 29 51 20 3 63 35 37 28 APPENDIX B COMPUTER PROGRAM LISTINGS 145 Novonic-Nixon Velocity Meter Calibration 0 1 2 3 4 5 â€¢6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 "Velocity Calibration Program for Nixon Meter # 696": fmt 0, c2, f6.2, c4 fmt 1, cl6 ent X dim A$[16], T[2], N[2] wtb 2,21,1,17 wtc 2,0 stp "Start": red 2.1, A$; prt A$ val (A$[l,4])+N[l]; val (A$[10])->T[l] stp "End": red 2.1, A$; prt A$ val (A$[l,4])+N[2]; val (A$[10])^T[2] (T[2]-T[l])/6->T (N[2]-N[l])/T+N X*30.48/T->-V wrt 16, "T=", T, "sec"; wrt 16,"N=", N, "Hz" wrt 16, "V=", V, "cm/s" wtb 2,1,17 spc if N<30; 1.026393N-16.654312->A if N>=30; 0.659483N-5.648766->A if A<6; 0->A (A-V)*100/V+E wrt 16, "E=", E, "%" spc stp end Definition of Variables: X = distance in feet over which the calibration occurs A$ = string variable from pink box containing the raw data 146 T = time in seconds to travel distance X N = frequency in Hz V = true velocity in cm/s A = calculated velocity from previous calibration curve E = percent error 147 Linear Regression for Velocity Meter Calibration 0: "Part I: Linear regression Program": 1: fxd 6 2: ent P 3: dim NIP], V[P], UfPJ, E[P] 4: 0->M; 0-+r0; 0-*rl; 0->-r2; 0-Â»-r3; 0->-r4 5: M+l+M 6: ent N[M],V[M] 7: rO+V[M]+rO; rl+N[M]*V[M]-*-rl; r2+N[M]+r2; r3+N [M]+2-*r3; r4+N[M]+2+r4 8: if M
9: (rl-r2*rO/P)/(r3-r2+2/P)+A; (rO-A*R2)/P+B
21: end |