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Hydraulics and stability of mulitple inlet-bay systems: St. Andrew Bay, Florida

University of Florida Institutional Repository

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HYDRAULICS AND STABILITY OF MULITPLE INLET-BAY SYSTEMS: ST. ANDREW BAY, FLORIDA By MAMTA JAIN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2002

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Copyright 2002 by Mamta Jain

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ACKNOWLEDGMENTS The author would like to express her deepest and heartiest thanks to her advisor and chairman of the supervisory committee, Dr. Ashish Mehta, for his assistance, encouragement, moral support, guidance and patience throughout this study. Special thanks go to committee member Dr. Robert Dean for his help and advice in solving the hydraulic model equations. Gratitude and thanks are also extended to the other members of the committee, Dr. Robert Thieke and Dr. Andrew Kennedy, for their guidance and assistance. Thanks go to Dr. J. van de Kreeke for his help in solving the linearized lumped parameter model for the stability of inlets. Assistance provided by Michael Dombrowski of Coastal Tech, for whom the hydrographic surveys were carried out is sincerely acknowledged. Thanks go to Sidney Schofield and Vic Adams, for carrying out the fieldwork. The author wishes to acknowledge the assistance of Kim Hunt, Becky Hudson, and the entire Coastal and Oceanographic Engineering Program faculty and staff for their encouragement and emotional support. The author would like to thank her husband, Parag Singal, for his love, encouragement and support, and her parents and family for providing her with mind, body and soul. Last, but not least, the author would like to thank the eternal and undying Almighty who provides the basis for everything and makes everything possible. iii

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix LIST OF SYMBOLS.........................................................................................................xi ABSTRACT.....................................................................................................................xiv CHAPTER 1 INTRODUCTION...........................................................................................................1 1.1 Problem Definition....................................................................................................1 1.2 Objective and Tasks..................................................................................................4 1.3 Thesis Outline...........................................................................................................4 2 HYDRAULICS OF A MULTIPLE INLET BAY SYSTEM..........................................5 2.1 Governing Equations of an Inlet-Bay System..........................................................5 2.1.1 System Definition...........................................................................................5 2.1.2 Energy Balance...............................................................................................6 2.1.3 Continuity Equation........................................................................................7 2.2 The Linearized Method.............................................................................................9 2.3 Multiple Inlet-Bay System......................................................................................11 2.3.1 Two Inlets and Two Bays with One Inlet Connected to Ocean....................11 2.3.2 Three Inlets and Two Bays with Two Inlets Connected to Ocean................16 2.3.3 Three Inlets and Three Bays with One Inlet connected to Ocean.................19 2.3.4 Four Inlets and Three Bays with Two Inlets Connected to Ocean...............24 3 STABILITY OF MULTIPLE INLET-BAY SYSTEMS...............................................29 3.1 Stability Problem Definition...................................................................................29 3.2 Stability Criteria......................................................................................................29 3.2.1 Stability Analysis for One-Inlet Bay System................................................30 3.2.2 Stability of Two Inlets in a Bay....................................................................32 3.3 Stability Analysis with the Linearized Model........................................................34 3.3.1 Linearized lumped parameter model for N Inlets in a Bay...........................35 iv

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3.4 Application to St. Andrew Bay System..................................................................40 4 APPLICATION TO ST. ANDREW BAY COMPLEX AND ENTRANCES..............42 4.1 Description of Study Area......................................................................................42 4.2 Summary of Field Data...........................................................................................44 4.2.1 Bathymetry....................................................................................................46 4.2.2 Tides..............................................................................................................48 4.2.3 Current and Discharge...................................................................................51 4.3 Tidal Prism..............................................................................................................52 5 RESULTS AND DISCUSSION....................................................................................54 5.1 Introduction.............................................................................................................54 5.2 Hydraulics of St. Andrew Bay................................................................................54 5.2.1 Solution of Equations....................................................................................55 5.2.1.1 One-inlet one-bay system....................................................................55 5.2.1.2 Three inlets and three bays with one inlet connected to ocean............56 5.2.1.3 Three inlets and three bays with two inlets connected to ocean..........57 5.2.2 Input Parameters............................................................................................59 5.2.3 Model Results and Comparison with Data...................................................60 5.3 Stability Analysis....................................................................................................62 5.3.1 Input Parameters............................................................................................62 5.3.2 Results and Discussion..................................................................................63 6 CONCLUSIONS............................................................................................................73 6.1 Summary.................................................................................................................73 6.2 Conclusions.............................................................................................................74 6.3 Recommendations for Further Work......................................................................74 APPENDIX A ALGORITHMS FOR MULTIPLE INLET-BAY HYDRAULICS..............................76 A.1 Introduction............................................................................................................76 A.2 Program-1...............................................................................................................76 A.3 Program-2...............................................................................................................77 B INLET HYDRAULICS RELATED DERIVATIONS.................................................80 B.1 Linearization of Damping Term.............................................................................80 B.2 Shear Stress Dependence on Area..........................................................................81 B.3 General Equation for hydraulic radius...................................................................82 B.3.1 Rectangular...................................................................................................83 B.3.2 Triangular.....................................................................................................83 B.4 Hydraulic Radius for Triangular Cross-Section.....................................................83 v

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C CALCULATION OF BAY TIDE AND LINEAR DISCHARGE COEFFICIENTS...85 D CALCULATIONS FOR STABILITY ANALYSIS.....................................................89 D.1 Introduction............................................................................................................89 D.2 Calculations............................................................................................................89 D.2.1 Equilibrium velocity.....................................................................................89 D.2.2 Constant for Triangular schematization.......................................................89 D.3 Relationship between Flow Curves and Stability of Two Inlets............................90 D.4 Matlab Programs....................................................................................................91 D.4.1 Program-1.....................................................................................................91 D.4.2 Program-2.....................................................................................................93 LIST OF REFERENCES...................................................................................................96 BIOGRAPHICAL SKETCH.............................................................................................98 vi

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LIST OF TABLES Table page 1.1 Cross-sectional areas of Johns Pass and Blind Pass in Boca Ciega Bay.......................3 1.2 Cross-sectional areas of St. Andrew Bay Entrance and East Pass.................................3 1.3 Cross-sectional areas of Pass Cavallo and Matagorda Inlet..........................................3 4.1 Locations of St. Andrew Bay channel cross-sections..................................................45 4.2 Locations of East Pass channel cross-sections.............................................................45 4.3 Cross-section area, mean depths and width.................................................................46 4.4 Tidal ranges in September 2001, December 2001 and March 2002............................51 4.5 Phase lags between the stations and the ocean tide.....................................................51 4.6 Characteristic peak velocity and discharge values.......................................................52 4.7 Flood and ebb tidal prisms...........................................................................................53 5.1 List of input and output parameters for one-inlet one-bay model...............................55 5.2 List of input and output parameters for the three inlets and three bays model............56 5.3 List of Input and Output Parameters for the four inlets and three bays model............58 5.4 Input parameters for the hydraulic model....................................................................59 5.5 Model results and measurements.................................................................................60 5.6 Input parameters for stability analysis.........................................................................63 5.7 Effect of change in bay area and length of East Pass..................................................65 5.8 Stability observations for St. Andrew Bay Entrance and East Pass............................72 C.1 Weighted-average bay tide ranges and phase differences...........................................85 C.2 Calculation of ( o B1 ) max ( B1 B2 ) max and ( B1 B3 ) max ........................................87 vii

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D.1 Calculation of equilibrium velocity............................................................................89 D.2 Calculation of a i ..........................................................................................................89 viii

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LIST OF FIGURES Figure page 2.1 One bay and one inlet system........................................................................................5 2.2 Two bays and two inlets with one inlet connected to ocean........................................12 2.3 Two bays and three inlets, two inlets are connected to ocean.....................................16 2.4 Three bays and three inlets with one inlet connecting to the ocean.............................21 2.5 Three bays and four inlets, two inlets connect to ocean..............................................25 3.1 Closure curves..............................................................................................................31 3.2 Escoffier diagram.........................................................................................................31 3.3 Closure surfaces...........................................................................................................33 3.4 Equilibrium flow curve for Inlet 2...............................................................................33 3.5 Possible configurations of equilibrium flow curves for a two-inlet bay system..........34 3.6 Equilibrium flow curves for two inlets in a bay..........................................................41 4.1 Map showing the three bays and two inlets and bathymetry of the study area...........43 4.2 Aerial view of St. Andrew Bay Entrance in 1993. Jetties are ~430 m apart...............43 4.3 East Pass channel before its opening in December 2001...........................................44 4.4 St. Andrew Bay Entrance bathymetry and current measurement cross-sections.........46 4.5 Cross-section A in St. Andrew Bay Entrance..............................................................47 4.6 Cross-section F in East Pass measured by ADCP.......................................................47 4.7 Measured tide in Grand Lagoon on September18-19, 2001........................................49 4.8 NOS predicted tide at St. Andrew Bay Entrance on September18-19, 2001..............49 4.9 NOS predicted tide in St. Andrew Bay Entrance on December 18-19, 2001..............50 ix

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4.10 Tide at all selected NOS stations in March 2002.......................................................50 5.1 Equilibrium flow curves for rectangular cross-sections, Run No. 1............................66 5.2 Equilibrium flow curves for rectangular cross-sections, Run No. 2............................66 5.3 Equilibrium flow curves for rectangular cross-sections, Run No. 3............................67 5.4 Equilibrium flow curves for rectangular cross-sections, Run No. 4............................67 5.5 Equilibrium flow curves for rectangular cross-sections, Run No. 5............................68 5.6 Equilibrium flow curves for rectangular cross-sections, Run No. 6............................68 5.7 Equilibrium flow curves for triangular cross-sections, Run No. 7..............................69 5.8 Equilibrium flow curves for triangular cross-sections, Run No. 8..............................69 5.9 Equilibrium flow curves for triangular cross-sections, Run No. 9..............................70 5.10 Equilibrium flow curves for triangular cross-sections, Run No. 10..........................70 5.11 Equilibrium flow curves for triangular cross-sections, Run No. 11..........................71 5.12 Equilibrium flow curves for triangular cross-sections, Run No. 12..........................71 B.1 Trapezoidal Cross-section...........................................................................................83 B.2 Triangular cross-section..............................................................................................84 C.1 Head difference between ocean (Gulf) and bay 1.......................................................88 C.2 Head difference between bay1 and bay 2....................................................................88 D.1 General configuration of equilibrium flow curve.......................................................90 D.2 General configuration of equilibrium flow curve.......................................................90 x

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LIST OF SYMBOLS Symbols A B A B1 A B2 A B3 bay water surface areas at MSL A c A c1 A c2 A c3 A c4 flow cross-sectional areas of inlets a o ocean (Gulf) tide amplitude a B a B1 a B2 a B3 bay tide amplitudes Ba a, dimensionless bay tide amplitudes 1B 2Ba 3Ba a i constant that relates hydraulic radius with area of triangular cross-section a, b, c, A, B constants defined to solve system of equations B i dimensionless resistance factor C, C 1 C 2 C 3 C 4 coefficients in linear relations of inlet hydraulics C D C DL1 C DL2 C DL3 C DL4 linear discharge coefficients C K prism correction coefficient of Keulegan f Darcy-Weisbach friction factor F friction coefficient g acceleration due to gravity h k kinetic head i subscript specifying the inlet under consideration K Keulegan coefficient of filling or repletion k bottom roughness xi

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k en entrance loss coefficient k ex exit loss coefficient L c L 1 L 2 L 3 L 4 channel lengths m sum of entrance and exit losses. P tidal prism Q, Q 1 Q 2 Q 3 Q 4 discharges through inlets Q m peak discharge R, R 1 R 2 R 3 R 4 hydraulic radii R t bay tide range R o ocean (Gulf) tide range r 1 r 2 r 3 polar representation of the bay tides T tidal period t time u velocity u B bay current velocity u c u c1 u c2 u c3 u c4 velocities through inlets u eqi equilibrium velocity of inlet u max1 u max2 u max3 u max4 maximum velocities through inlets u o ocean (Gulf) current velocity X distance between UF and NOS tide stations o B velocity coefficients B1, B2, B3 high water (HW) or low water (LW) lags v1 v2 v3 v4 inlet velocity lags xii

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specific time when sea is at MSL dimensionless time water elevation o ocean (Gulf) tide elevation with respect to MSL B B1 B2 B3 bay tide elevations with respect to MSL B 1B 2B 3B dimensionless bay tide elevations maximum bottom shear stress eq equilibrium shear stress xiii

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science HYDRAULICS AND STABILITY OF MULTIPLE-INLET BAY SYSTEMS: ST. ANDREW BAY, FLORIDA By Mamta Jain December 2002 Chairman: Ashish J. Mehta Major Department: Civil and Coastal Engineering Tidal inlets on sandy coasts are subject to the continuous changes in their geometry and as a result influence shorelines in the vicinity. Since engineering modifications carried out at one inlet can affect the long-term stability of others in the vicinity of the modified inlet, it is important to understand the stability of all inlets connecting a bay to the ocean. Inlet stability is related to the equilibrium between the inlet cross-sectional area and the hydraulic environment. St. Andrew Bay on the Gulf of Mexico coast of Floridas panhandle is part of a three-bay and two(ocean) inlet complex. One of these inlets is St. Andrew Bay Entrance and the other is East Pass, both of which are connected to St. Andrew Bay on one side and the Gulf on the other. Historically, East Pass was the natural connection between the bay and the Gulf. In 1934, St. Andrew Bay Entrance was constructed 11 km west of East Pass to provide a direct access between the Gulf and Panama City. Due to the long-term effect of this opening of St. Andrew Bay Entrance, East Pass closed xiv

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naturally in 1998. A new East Pass was dredged open in December 2001, and the objective of the present study was to examine the hydraulics and stability of this system of two sandy ocean inlets connected to interconnected bays. To study the system as a whole, a linearized hydraulic model was developed for a three-bay and four-inlet (two ocean and the other two connecting the bays) system and applied to the St. Andrew Bay system. To investigate the stability of the ocean inlets, the hydraulic stability criterion was extended to the two-ocean inlets and one (composite) bay system using the linearized lumped parameter model. The following conclusions are drawn from this analysis. The linearized hydraulics model is shown to give good results--the amplitudes of velocities and bay tides are within %. The percent error for St. Andrew Bay is almost zero, and for the other bays it is within %. The stability model gives the qualitative results. The bay area has a significant effect on the stability of the two inlets. At a bay area of 74 km 2 (the actual area of the composite bay), both inlets are shown to be unstable. Increasing the area by 22% to 90 km 2 stabilizes St. Andrew Bay Entrance, and by 42% to 105 km 2 stabilizes East Pass as well. Keeping the bay area at 105 km 2 and increasing the length of East Pass from 500 m to 2000 m destabilizes this inlet because as the length increases the dissipation in the channel increases as well. xv

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CHAPTER 1 INTRODUCTION 1.1 Problem Definition Tidal inlets are the relative short and narrow connections between bays or lagoons and the ocean or sea. Inlets on sandy coasts are subject to the continuous changes in their geometry. Predicting the adjustment of the inlet morphology after a storm event in particular, i.e., whether the inlet will close or will remain open, requires knowledge of the hydraulic and sedimentary processes in the vicinity of the inlet. These processes are governed by complex interactions of the tidal currents, waves, and sediment. In spite of recent advances in the description of flow field near the inlet and our understanding of sediment transport by waves and currents (Aubrey and Weishar 1988), it is still not possible to accurately predict the morphologic adjustment of the inlet to hydrodynamic forcing. Inlet stability is dependent upon the cumulative result of the actions of two opposing factors, namely, a) the near-shore wave climate and associated littoral drift, and b) the flow regime through the inlet. Depending on the wave climate and the range of the tide, one of these two factors may dominate and cause either erosion or accumulation of the sand in the inlet. However, on a long-term basis, a stable inlet can be maintained only if the flow through the inlet has enough scouring capacity to encounter the obstruction against the flow due to sand accumulation, and to maintain the channel in the state of non-silting, non-scouring equilibrium. If such is not the case and waves dominate, then the accumulated sand will begin to constrict the inlet throat, thereby reducing the tidal 1

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2 prism. The resulting unstable inlet may migrate or orient itself at an angle with the shoreline depending on the predominant direction of the littoral drift; the channel may elongate, thereby increasing the frictional resistance to the flow, and finally, a stage may be reached when perhaps a single storm could close the inlet in a matter of hours. Stability criteria based on inlet hydraulics and sediment transport for single inlets have been proposed by, among others, OBrien (1931), Escoffier (1940), OBrien and Dean (1972), Bruun (1978) and Escoffier and Walton (1979). All criteria assume that sufficient sand is available to change the inlet channel geometry in response to the prevailing hydrodynamic conditions. These investigators found various stability parameters to describe the stability of the inlet. It should be noted, however, that while it is relatively easy to deal with the stability of single inlets, the problem becomes complex when, as is commonly the case, more than one inlet connect the ocean to a single bay or more than one interconnected bays. Some examples of such systems are as follows. Three cases of the history of two inlets in a bay are worthy of citation. One case is that of Boca Ciega Bay on the Gulf coast of Florida, where the co-dependency of two inlets, Blind Pass and Johns Pass, appears to be reflected in the history of their cross-sectional areas. While Blind Pass has historically been narrowing due to shoaling, Johns Pass has been increasing in size, as shown in Table 1.1. As a result, Blind Pass now requires regular dredging for its maintenance while severe bed erosion has occurred at Johns Pass (Mehta, 1975; Becker and Ross, 2001). Another example is that of St. Andrew Bay Entrance and the East Pass. As mentioned previously, East Pass used to be a large inlet and was the only natural connection between the Gulf of Mexico and the St. Andrew Bay. In 1934, St. Andrew

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3 Bay entrance was constructed 11 km west of East Pass through the barrier island by the federal government to provide a direct access between the Gulf and Panama City. Table 1.2 gives the cross-sectional area of each inlet over time. Table 1.1 Cross-sectional areas of Johns Pass and Blind Pass in Boca Ciega Bay Area (m 2 ) Hydraulic Radius (m) Year Johns Pass Blind Pass Johns Pass Blind Pass 1873 474 538 3.7 3.5 1883 432 496 3.8 3.0 1926 531 209 3.9 1.9 1941 636 225 4.1 1.4 1952 849 157 4.6 2.7 1974 883 411 4.9 1.6 1998 950 230 5.3 d 0.9 d d Estimated by assuming no change in channel width since 1974. Table 1.2 Cross-sectional areas of St. Andrew Bay Entrance and East Pass Area (m 2 ) Year St. Andrew Bay Entrance East Pass 1934 1,835 3,400 1946 3,530 2,146 1983 3,943 1,392 1988 Closed 2001 5,210 Reopened The third example is that of Pass Cavallo and Matagorda Inlet connecting Matagorda Bay, Texas, to the Gulf. Stability analysis carried out by van de Kreeke (1985) on this system showed that Pass Cavallo is an unstable inlet, which is decreasing in cross-section, whereas Matagorda Inlet is increasing in size. The areas of cross-sections of the two inlets are listed in Table 1.3. Table 1.3 Cross-sectional areas of Pass Cavallo and Matagorda Inlet Area (m 2 ) Year Pass Cavallo Matagorda Inlet 1959 8,000 Closed 1970 7,500 3,600 The above sets of complex problems are dealt with in this study in a simplified manner, with the following objective and associated tasks.

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4 1.2 Objective and Tasks The main objective of this study is to examine the hydraulics and thence the stability of a system of two sandy ocean inlets connected to interconnected bays. The sequence of tasks carried out to achieve this goal is as follows: 1 Deriving the basic hydraulic equations using the linearized approach for a complex four inlets and three bays system. 2 Solving these equations, applying them to the St. Andrew Bay system, and comparing the results with those obtained from the hydrographic surveys. 3 Developing stability criteria using the basic Escoffier (1940) model for one inlet and one bay and then extending this model to the two inlets and a bay. 4 Carrying out stability analysis for N inlets and a bay using the linearized lumped parameter model of van de Kreeke (1990), and then applying it to the St. Andrew Bay system. 1.3 Thesis Outline Chapter 2 describes the hydraulics of the multiple inlet-bay system. It progresses from the basic theory to the development of linearized models for simple and complex systems. Chapter 3 describes the stability of the system, including an approximate method to examine multiple inlets in a bay. Chapter 4 includes details of hydrographic surveys and summarizes the data. Chapter 5 discusses the input and output parameters required for the calculation. It also presents the results. All calculations are given in the appendices. Conclusions are made in Chapter 6, followed by a bibliography and a biographical sketch of the author.

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CHAPTER 2 HYDRAULICS OF A MULTIPLE INLET BAY SYSTEM 2.1 Governing Equations of an Inlet-Bay System 2.1.1 System Definition The governing equations for a simple inlet-bay system may be derived by considering the inlet connecting the ocean and the bay as shown in Figure 2.1. Bay Ocean o L A B Figure 2.1 One bay and one inlet system These equations are derived subjected to the following assumptions. 1 The inlet and bay banks are vertical. 2 The range of tide is small as compare to the depth of water everywhere. 3 The bay surface remains horizontal at all times, i.e., the tide is in phase across the bay. That means the longest dimension of the bay be small compared to the travel time of tide through the bay. 4 The mean water level in the bay equals that in the ocean. 5 The acceleration of mass of water in the channel is negligible. 6 There is no fresh water inflow into the bay. 5

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6 7 There is no flow stratification due to salinity. 8 Ocean tides are represented by a periodical function. 2.1.2 Energy Balance Applying the energy balance between ocean and bay one gets 2222oBooBBuuhgg (2.1) where o = Ocean tide elevation with respect to mean sea level, B = Bay tide elevation with respect to mean sea level, u o = Ocean current velocity, u B = Bay current velocity, o and B = Coefficients greater than one which depend on the spatial distribution of u o and u B respectively, h = Total head loss between the ocean and the bay, and g = acceleration due to gravity. It is also assumed that ocean and bay are relatively deep; thus u o and u B are small enough to be neglected. Then Eq. (2.1) becomes oh B (2.2) There are generally two types of head losses. One includes concentrated or minor losses due to convergence and divergence of streamlines in the channel. The second type is gradual loss due to bottom friction in the channel. The entrance and exit losses may be written in terms of the velocity head 22cug in the channel, with the entrance loss coefficient k en and the exit loss coefficient k ex i.e.,

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7 Entrance loss 22cenukg (2.3) Exit loss 22cexukg (2.4) where u c is the velocity through the inlet. Gradual energy losses per unit length depend on the channel roughness and are given in form of Darcy-Weisbach friction factor Gradual loss 242cufL R g (2.5) where f = Darcy-Weisbach friction coefficient, R = hydraulic radius of channel, and L = Length of channel. Substitution of Eqs. (2.3), (2.4) and (2.5) into (2.2) gives 224coBenexu f LkkgR (2.6) or 2||.4coBenexgufLkkR oBsign (2.7) The sign( 0 B ) term must be included since the current reverses in direction every half tidal cycle. 2.1.3 Continuity Equation The equation of continuity, which relates the inlet flow discharge to the rate of rise and fall of bay water level, is given as

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8 ccBBdQuAAdt (2.8) where Q = flow rate through the inlet, A c = Inlet flow cross-sectional area, and A B = bay surface area. Therefore Eq. (2.8) becomes BccAduAdt B (2.9) Eliminating u c between Eq. (2.7) and (2.9) leads to 2||.4cBoBoBBenexAdgsignfLdtAkkR (2.10) Next, we introduce the dimensionless quantities oooa ; BBoa ; 2ttT (2.11) where a o = ocean tide amplitude (one-half the ocean tidal range), T = tidal period and = tidal (angular) frequency. Substitution into Eq. (2.10) gives ||.BoBoBdKsignd (2.12) where 224cooBenexAgaTK f LaAkk R (2.13)

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9 in which K is referred to as the coefficient of filling or repletion (Keulegan, 1967). Keulegan solved the first order differential equation, Eq. (2.12), for B in terms of the repletion coefficient K and dimensionless time using numerical integration. 2.2 The Linearized Method A linear method was suggested by Dean (1983) for solving Eq. (2.12). For this approach it was assumed that the velocity u c in Eq. (2.7), is proportional to the head difference ( o B ) rather than the square root of the head difference, according to 2cDLooguCa B (2.14) where C DL = linear discharge coefficient. This coefficient is defined as max14oDLoBenexaCfLkkR (2.15) where ( o B ) max is the maximum head difference across the inlet. Now, combining Eqs. (2.14), (2.9) and (2.11), Eq. (2.12) can be written in terms of the linear relationship as BoBdCd (2.16) where 2cDLBoAgCCAa (2.17) Under assumption (8) the ocean tide is assumed to be periodic. Because of the linear assumption the bay tide is also periodic, it can be written as coso (2.18) 11cos()BBBa 1 (2.19)

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10 where BBoaaa a B = one-half the bay tide range (i.e., bay tide amplitude) and B = lag between high water (HW) or low water (LW) in the ocean and the corresponding HW or LW in the bay. Eq. (2.18) and Eq. (2.19) are next substituted into Eq. (2.16) and the following complex number technique is used to solve for a B and B : 1 Define the following constants: aC Reioe 2 Let the following variables be represented in the polar form: 1()11ReBiBBaer 1 3 Therefore 1Bdird 4 So the equations are reduced to (2.20) 11(1)air 111rai ; 121Re()1ra ; 12Im()1ara where Re(r 1 ) = is the real part of the solution, and Im(r 1 ) = is the imaginary part of the solution. The magnitude of r 1 represents and the phase lag 1Ba B1 is represented by the angle of r 1 : 1211Baa (2.21) 11tanBa (2.22)

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11 The velocity u c1 through inlet 1 is therefore given by 1max11coscuu v (2.23) where u max1 is the maximum velocity through inlet 1, v1 is the phase lag between the velocity in inlet 1 and HW or LW in the ocean. Substituting for o and B1 from Eq. (2.18) and Eq. (2.19) in Eq. (2.14) and combining Eqs. (2.23) and (2.14) we get the required expression for u max1. It should be noted that velocity is out of phase with respect to displacement by /2. Therefore, v1 = B1 -/2. 2.3 Multiple Inlet-Bay System. 2.3.1 Two Inlets and Two Bays with One Inlet Connected to Ocean In the case of two bays with one inlet connecting to the ocean and the second connecting the bays as shown in Figure 2.2, the eight assumptions mentioned in section 2.1.1 and the linear relationship both hold. In a manner similar to that employed for a single inlet-bay case, the velocity relationship and the equation of continuity for two-bay system may be written with reference to the notation of Figure 2.2. Thus the following relationships are obtained: 112cDLooguCa 1B (2.24) 111112BccBBddQuAAAdtdt 2B (2.25) 22112cDLBBguCa 2B (2.26) 22222BccBdQuAAdt (2.27)

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12 L 2 A B2 B2 A B1 B1 L 1 Ocean Bay2 Bay1 o Figure 2.2 Two bays and two inlets with one inlet connected to ocean. where u c1 u c2 = velocities through the inlets 1 and 2, Q 1 Q 2 = discharges through inlets 1 and 2, A c1 A c2 = inlet flow cross-sectional areas, and A B1 A B2 = bay water surface areas. 111max114oDLoBenexaCfLkkR (2.28) 12122max214BDLBBenexaCfLkkR (2.29) where L 1 L 2 = inlet lengths, and R 1 R 2 = hydraulic radii of the channels. Eliminating u c1 between Eq. (2.24) and Eq. (2.25) gives 121111BBBoBBdAdCdtAdt 2 (2.30) where

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13 11112cDLBoAgCCAa (2.31) Combining Eq. (2.26) and Eq. (2.27) yields 21221BBBdCdt (2.32) where 222212cDLBBAgCCAa (2.33) The dimensionless ocean tide is given by Eq. (2.18), and the dimensionless tides in bays 1 and 2 now become 11cos()BBBa 1 (2.34) 22cos()BBBa 2 (2.35) where 11BBoaaa a B1 = one-half the tide range (i.e., amplitude) in bay 1. 22BBoaaa a B2 = one-half the tide range (i.e., amplitude) in bay 2. B1 = lag between high water (HW) or low water (LW) in the ocean and corresponding HW or LW in the bay 1. B2 = lag between high water (HW) or low water (LW) in the ocean and corresponding HW or LW in the bay 2. Eq. (2.30) and Eq. (2.32) can be expressed in the dimensionless form as

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14 12111BBBoBBdAdCdAd 2 (2.36) 2122BBBdCd (2.37) The above equations are solved by the matrix method assuming the variables to be complex numbers. The solution is obtained as follows: 1 Define the following constants 1aC 2bC 21BBAAA Reioe 2 Let 1()11ReBiBBaer 1 2()22ReBiBBaer 2 11Bdird 22Bdird 3 So the equations are reduced to (2.38) 11(1)airaAir 221)1 (2.39) 10(rbir 4 In the matrix form they become (2.40) 121101raiaAirbi 5 The solution is 1ibirX (2.41) 21r X (2.42) where (1)( ) X abiaAab ; (1)( ) X abiaAab

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15 22(1)( ) X XabaAab 1(1Re()bbaaAabrXX ) ; 11Im()bababaArXX 2(1Re()abr ) X X ; 2()Im()abaArXX The amplitudes (and) of bays 1 and 2 are the magnitudes of the complex numbers r 1Ba 2Ba 1 and r 2 and the corresponding phase lags are the angles of the complex numbers: 121Re()Im()Bar 21r (2.43) 1111Im()tanRe()Brr (2.44) 222Re()Im()Bar 22r (2.45) 1222Im()tanRe()Brr v (2.46) The velocities u c1 and u c2 through inlets 1 and 2, respectively, are therefore given by 1max11coscuu v (2.47) 2max22coscuu (2.48) where u max1 and u max2 are the maximum velocities through inlets 1 and 2, respectively, v1 and v2 are the phase lags between the velocity in inlet 1 and HW or LW in the ocean, and in inlet 2 and HW or LW in the ocean. Substituting for o and B1 from Eqs. (2.18) and (2.34) in Eq. (2.24) and combining Eqs. (2.47) and (2.24) we get the required expression for u max1 Similarly we

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16 can obtain the expression for u max2 It should be noted that velocity is out of phase with respect to displacement by /2. Therefore, v1 = B1 -/2 and. v2 = B2 -/2. 2.3.2 Three Inlets and Two Bays with Two Inlets Connected to Ocean The inlet bay system is defined in Figure 2.3. In this system two bays are connected to each other with inlets 2 and inlet 3 and 1 connects bay 1 to the ocean. L 2 A B2 B2 A B1 B1 L 1 L 3 o Figure 2.3 Two bays and three inlets, two inlets are connected to ocean. The velocity in inlets 1 and 2 is given by Eq. (2.24) and Eq. (2.26) respectively. The velocity in inlet 3 is given by Eq. (2.49): 332cDLooguCa 1B (2.49) where u c3 = velocity through the inlet 3 and 313max314oDLoBenexaCfLkkR (2.50) where L 3 = inlet 3 length, and R 3 = hydraulic radius of inlet 3 channel. The governing equations of continuity are

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17 113113312BccccBBddQQuAuAAAdtdt 2B (2.51) 22222BccBdQuAAdt (2.52) where Q 1 Q 2 Q 3 = discharges through inlets 1, 2 and 3, A c1 A c2 A c3 = flow cross-sectional areas at inlets 1, 2 and 3, and A B1 A B2 = bay water surface areas. Substituting for the velocity expressions in the above equations we obtain 12211311BBBoBBdAdCCdtAdt (2.53) 21221BBBdCdt (2.54) where C 1 and C 2 are expressed by Eqs. (2.31) and (2.33), and 33312cDLBoAgCCAa (2.55) Stating Eqs. (2.53) and (2.54) in the dimensionless form we obtain 1221131BBBoBBdAdCCdAd (2.56) 2122BBBdCd (2.57) where o 1B and 2B are defined in Eqs. (2.18), (2.34) and (2.35), respectively. The solution of the system of Eqs. (2.56) and (2.57) is given below.

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18 1 Define the following constants 13aCC 2bC 21BBAAA Re()ioe 2 Let 1()11ReBiBer 2()22ReBiBer ; 11Bdird 22Bdird 3 So the equations are reduced to (2.58) 11(1)airaAir 221)1 (2.59) 10(rbir 4 Solve these equations by the matrix method. (2.60) 121101raiaAirbi 5 Solving the above equations yields 1ibirX (2.61) 21r X (2.62) (1)( ) X abiaAab ; (1)( ) X abiaAab 22(1)( ) X XabaAab 1(1Re()bbaaAabrXX ) ; 11Im()bababaArXX 2(1Re()abr ) X X ; 2()Im()abaArXX The amplitudes (and) of bays 1 and 2 are the magnitudes of the complex numbers r 1Ba 2Ba 1 and r 2 and the phase lags are the corresponding angles:

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19 121Re()Im()Bar 21r (2.63) 1111Im()tanRe()Brr (2.64) 222Re()Im()Bar 22r (2.65) 1222Im()tanRe()Brr v (2.66) The velocities u c1 and u c2 through inlets 1 and 2, respectively, are given by Eqs. (2.47) and (2.48), and u c3 through inlet 3 is obtained from 3max33coscuu (2.67) where u max3 is the maximum velocity through inlet 3 and v3 is the phase lag between velocity in inlet 3 and HW or LW in the ocean. Substituting for o and B1 from Eqs. (2.18) and (2.34) into Eq. (2.49) and combining Eqs. (2.49) and (2.67) we get the required expression for u max3 Then the phase lag v3 = B1 -/2. 2.3.3 Three Inlets and Three Bays with One Inlet connected to Ocean. This inlet bay system as defined in Figure.2.4 has three interconnected bays with inlets 2 and 4, while inlet 1 connects bay 1 to the ocean. The velocities in inlets 1 and 2 are given by Eqs. (2.24) and (2.26), respectively. The velocity in inlet 4 is given by Eq. (2.68): 44112cDLBBguCa 3B (2.68) where u c4 = velocity through the inlet 4 and

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20 14134max414BDLBBenexaCfLkkR with L 4 = inlet length and R 4 = hydraulic radius of inlet 4 channel. The governing continuity equations are 312111123BBBccBBBdddQuAAAAdtdtdt (2.69) 22222BccBdQuAAdt (2.70) 34443BccBdQuAAdt (2.71) Q 1 Q 2 Q 4 = discharges through inlets 1, 2 and 4 A c1 A c2 A c4 = flow cross-sectional areas at inlets 1, 2 and 4. A B1 A B2 A B3 = bay water surface areas. Substituting the velocity expressions in the above equations we obtain 3312211111BBBBBoBBBAddAdCdtAdtAdt (2.72) 21221BBBdCdt (2.73) 31341BBBdCdt (2.74) where C 1 and C 2 are as expressed by Eqs. (2.31) and (2.33), and 444312cDLBBAgCCAa (2.75)

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21 L 2 A B2 B2 A B1 B1 L 1 L 4 A B3 B3 o Figure 2.4 Three bays and three inlets with one inlet connecting to the ocean. Stating the above equations in the dimensionless form the desired solution is obtained by solving the following three equations: 331221111BBBBBoBBBAddAdCdAdAd (2.76) 2122BBBdCd (2.77) 3134BBBdCd (2.78) where o 1B and 2B are defined by Eqs. (2.18), (2.34) and (2.35), respectively, and 3B is 33cos()BBBa 3 (2.79) As before the above equations are solved by using complex numbers as follows:

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22 1 Define the following constants 1aC 2bC 4cC 21BBAAA 31BBA B A Re()ioe 2 Let 1()11ReBiBer 2()22ReBiBer 3()33ReBiBer 11Bdird 22Bdird 33Bdird 3 So the equations are reduced to (2.80) 121(1)airaAiraBir 3331)r01 (2.81) 120(1)rbir 0r (2.82) 1200(rrci 4 Solving the equations by matrix method: (2.83) 12311011010raiaAiaBirbicir yields 1cibirX (2.84) 2icirX (2.85) 3ibirX (2.86) (1)( ) X acabbcaBbaAciaAabcabcaB (1)( ) X acabbcaBbaAciaAabcabcaB 22(1)( ) X XacabbcaBbaAcaAabcabcaB

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23 222111Re()baBcccaAbabABcrXX 1 22222211Im()aABbcbcBbcAcBbbcBrXX 2221Re()abaBbaBccabcrXX 222()Im()abaAaBaBbcacbcaAcrXX 2 2231Re()acaAcabAabcbrXX 2223()Im()acaAaBabcAabcbaBbrXX The amplitudes (, and ) of bay1, bays 2 and 3 are the magnitudes of the complex numbers r 1Ba 2Ba 3Ba 1 r 2 and r 3 and the corresponding phase lags are the angles of the complex numbers: 121Re()Im()Bar 21r (2.87) 1111Im()tanRe()Brr (2.88) 222Re()Im()Bar 22r (2.89) 1222Im()tanRe()Brr (2.90) 323Re()Im()Bar 23r (2.91) 1333Im()tanRe()Brr (2.92)

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24 The velocities u c1 and u c2 through inlets 1 and 2, respectively, are given by Eqs. (2.47) and (2.48), and u c4 through inlet 4 is given by 4max44coscuu v (2.93) where u max4 is the maximum velocity through inlet 4 and v4 is the corresponding phase lags between this velocity and HW or LW in the ocean. Substituting for B1 and B3 from Eqs. (2.34) and (2.79)into Eq. (2.68) and combining Eqs. (2.93) and (2.68) we get the desired expression for u max4 Phase lag v4 = B3 -/2. 2.3.4 Four Inlets and Three Bays with Two Inlets Connected to Ocean. This system as defined in Figure 2.5 has three interconnected bays with inlets 2 and 4, while and inlets 1 and 3 connect bay 1 to the ocean. The velocities in inlets 1, 2, 3 and 4 are given by Eqs. (2.24), (2.26), (2.49) and (2.68), respectively. The governing continuity equations are written as follows. 312131133123BBBccccBBBdddQQuAuAAAAdtdtdt (2.94) 22222BccBdQuAAdt (2.95) 34443BccBdQuAAdt (2.96) Next, substituting the velocity expressions in the above equations yields 33122113111BBBBBoBBBAddAdCCdtAdtAdt (2.97) 21221BBBdCdt (2.98)

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25 L 2 A B2 B2 A B1 B1 L 1 L 3 L 4 A B3 B3 o Figure 2.5 Three bays and four inlets, two inlets connect to ocean. 31341BBBdCdt (2.99) where C 1 C 2 C 3 and C 4 are as expressed by Eqs. (2.31), (2.33), (2.55) and (2.75), respectively. Now we may state the above equations in the dimensionless form as 3312211311BBBBBoBBBAddAdCCdAdAd (2.100) 2122BBBdCd (2.101) 3134BBBdCd (2.102) where o 1B 2B and 3B are defined by Eqs. (2.18), (2.34), (2.35) and (2.79), respectively. These equations are solved as follows:

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26 1 Define the following constants 13aCC 2bC 4cC 21BBAAA 31BBA B A Re()ioe 2 Let 1()11ReBiBer 2()22ReBiBer 3()33ReBiBer 11Bdird 22Bdird 33Bdird 3 So the equations are reduced to 121(1)airaAiraBir 330r31)r01 (2.103) 120(1)rbir (2.104) 1200(rrci (2.105) 4 Solve these equations by matrix method: 12311011010raiaAiaBirbicir (2.106) 5 Thus we obtain 1cibirX (2.107) 2icirX (2.108) 3ibirX (2.109) (1)( ) X acabbcaBbaAciaAabcabcaB (1)( ) X acabbcaBbaAciaAabcabcaB

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27 22(1)( ) X XacabbcaBbaAcaAabcabcaB 222111Re()baBcccaAbabABcrXX 1 22222211Im()aABbcbcBbcAcBbbcBrXX 2221Re()abaBbaBccabcrXX 222()Im()abaAaBaBbcacbcaAcrXX 2 2231Re()acaAcabAabcbrXX 2223()Im()acaAaBabcAabcbaBbrXX The amplitudes (, and ) of bays1, 2 and 3 are the magnitudes of the complex numbers r 1Ba 2Ba 3Ba 1 r 2 and r 3 and the corresponding phase lags are the angles of the complex numbers: 121Re()Im()Bar 21r (2.110) 1111Im()tanRe()Brr (2.111) 222Re()Im()Bar 22r (2.112) 1222Im()tanRe()Brr (2.113) 323Re()Im()Bar 23r (2.114)

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28 1333Im()tanRe()Brr (2.115) Then the velocities u c1 u c2 u c3 and u c4 are given by Eqs. (2.47), (2.48), (2.67) and (2.93), respectively.

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CHAPTER 3 STABILITY OF MULTIPLE INLET-BAY SYSTEMS 3.1 Stability Problem Definition An inlet is considered stable when after a small change the cross-sectional area returns to its equilibrium value. Each inlet is subject to two opposing forces, the waves on one hand, which tend to push sand into the inlet, and the tidal current on the other hand, which tries to carry sand out of the channel back to the sea or the bay. The size of the inlet and its stability are determined by the relative strengths of these two opposing forces. 3.2 Stability Criteria Inlet stability as considered here basically deals with the equilibrium between the inlet cross-section area and the hydraulic environment. The pertinent parameters are the actual tide-maximum bottom shear stress and the equilibrium shear stress eq The equilibrium shear stress is defined as the bottom stress induced by the tidal current required to flush-out sediment carried into the inlet. When equals eq the inlet is considered to be in equilibrium. When is larger than eq the inlet is in the scouring mode, and when is smaller eq the inlet is in the shoaling mode. The value of equilibrium shear stress depends on the waves and associated littoral drift and sediment. Considering inlets at equilibrium on various coasts, Bruun (1978) found the value of equilibrium stress in fairly narrow range: 3.55.5eqPaPa 29

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30 The value of actual shear stress is obtained from maxmax|Fuu | (3.1) where F is the friction coefficient, a function of bottom roughness, k, u max is the maximum tidal velocity in the inlet, a function of area and length of the inlet, as discussed in Chapter 2 and is the fluid density. Therefore, can be written as a function of following form (,,,) f ALkm where m is the sum of entrance and exit losses. The plotted function is called a closure curve, as shown in Figure 3.1. It is clear from the calculation shown in the Appendix B that A is a strong function of A and a weak function of L, m, k. The strong dependence of on A explains why inlets adjust to changes in the hydraulic environment primarily via a change in the cross-sectional area. 3.2.1 Stability Analysis for One-Inlet Bay System Making use of the Escoffier (1940) diagram, Figure 3.2, one can study the response of the inlet to change in area. In the Figure, A I and A II both represent equilibrium flow areas, with A I representing unstable equilibrium and A II representing stable equilibrium. If the inlet cross-sectional area A were reduced but remained larger than A I the actual shear stress would be larger than the equilibrium shear stress and A would return to the value A II If the cross-sectional area were reduced below A I the shear stress would become lower than its equilibrium value and the inlet would close. If A became larger than A II the actual shear stress would become larger than equilibrium value and A would return to A II Note that the equilibrium condition only exists if the line =eq intersects the closure curve =()A

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31 Figure 3.1 Closure curves (source: van de Kreeke, 1985) A (L ,k, m)2 (L, k, m)1 Figure 3.2 Escoffier diagram (source: van de Kreeke, 1985) AI Unstable equilibrium AII Stable equilibrium Equilibrium interval A

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32 The equilibrium interval for the stable cross-section, A II ranges from A I to infinity. 3.2.2 Stability of Two Inlets in a Bay Similar to a single inlet, it can be shown that shear stresses 1 and 2 for two inlets in a bay strongly depend on A 1 and A 2 and are weak functions of (L 1 k 1 m 1 L 2 k 2 m 2 ). The functions and 112,AA 212,AA are referred to as a closure surfaces. The shape of is qualitatively illustrated in Figure 3.3. For a constant A 21,AA 2 1 the curve is similar to the closure curve shown in Figure 3.1. The value of 11A 2 decreases with increasing A 1 With the help of a closure surface in Figure 3.3, the loci of (A 1 A 2 ) for which 2eq 21eq 21eq are plotted in Figure 3.4. The locus of 2eq is referred to as the equilibrium flow curve for Inlet 2. Using the same reasoning as for a single inlet and assuming that the cross-sectional area of Inlet 1 is constant, it follows that if A 2 = A I Inlet 2 will shoal and close; if A 2 = A II Inlet 2 will scour until the cross-sectional area attains a value A s and if A 2 = A III Inlet 2 will shoal until the cross-sectional area attains the value A s The locus of (A 1 A 2 ) for which Inlet 2 has a stable equilibrium flow area is the enhanced (by a thicker line) part of the equilibrium flow curve for Inlet 2. Similarly, the locus of (A 1 A 2 ) for which Inlet 1 has a stable equilibrium flow area is the enhanced part of the equilibrium flow curve for Inlet 1. The condition for the existence of stable equilibrium flow areas for both Inlet 1 and Inlet 2 is that the enhanced parts of the equilibrium flow curves intersect. The common equilibrium interval of the two is

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33 2 2eq A 2 A 1 Figure 3.3 Closure surfaces (source: van de Kreeke, 1985) A 1 eq 1 eq 1 eq A I A II A III A 2 A s Figure 3.4 Equilibrium flow curve for Inlet 2 (source: van de Kreeke, 1985)

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34 A 1 A 2 2 1 A 2 (a) (b) 1 2 A 1 A 2 A 2 ( c ) (d) 1 2 A 1 1 2 A 1 Figure 3.5 Possible configurations of equilibrium flow curves for a two-inlet bay system. Stable equilibrium flow area is represented by and unstable equilibrium is represented by The hatched area in (a) represents the domain of the stable equilibrium flow area (source: van de Kreeke, 1990) represented by the hatched rectangle in Figure 3.5 (a). The general shapes of the equilibrium flow curves and their relative positions in the (A 1 A 2 ) plane are presented in Figure 3.5. The detailed explanations to the Figure 3.5 are given in Appendix D. 3.3 Stability Analysis with the Linearized Model Due to the complex nature of sediment transport by waves and currents it is difficult to carry out an accurate analysis of the stability of single or multiple inlet

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35 systems. We will therefore attempt to carry out an approximate analysis based on the van de Kreeke (1990) linearized lumped parameter model. The justification for use of simple model is that for purpose of this study the stability analysis serves to illustrate a concept rather than to provide exact numerical results. Accurate numerical values can only be obtained by using a full-fledged two-dimension tidal model to describe the hydrodynamics of the bay. 3.3.1 Linearized lumped parameter model for N Inlets in a Bay The basic assumptions of the Linearized lumped parameter model are as follows: 1 The linearized model assumes that the ocean tide and the velocity are simple harmonic functions. 2 The water level in the bay fluctuates uniformly and the bay surface area remains constant. 3 Hydrostatic pressure, and shear stress distribution along the wetted perimeter of the inlet cross-section is uniform. 4 For a given bay area and inlet characteristics, the tidal amplitude and/or tidal frequency must be sufficiently large for equilibrium to exist. Similarly, larger the littoral drift due to waves, larger the equilibrium shear stress required to balance it and therefore the equilibrium velocity, the larger the required bay surface area, tidal amplitude and the tidal frequency or, in other words, Eq. (3.17) and Eq. (3.19) must be satisfied for the existence of equilibrium areas. 5 There is no fresh water discharge in the bays. 6 In a shallow bay the effect of dissipation of tidal energy cannot be ignored, especially if the bay is large. Inlet flow dynamics of the flow in the inlets are governed by the longitudinal pressure gradient and the bottom shear stress, van de Kreeke (1967), 10p x h (3.2) in which p is the pressure, is the water density, h is the depth and is the bottom shear stress. This stress is related to the depth mean velocity u

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36 |Fuu | (3.3) where F= f/8, is the friction coefficient. Integration of Eq. (3.2) (with respect to the longitudinal x-coordinate) between the ocean and the bay yields (van de Kreeke 1988). 2||2iiioBiiiigRuumRFL (3.4) In Eq. (3.4), u i refers to the cross-sectional mean velocity of the i th inlet, g is the acceleration due to gravity, m i is the sum of exit and entrance losses, R i is the hydraulic radius of the inlet, L i is the length of the inlet, o is the ocean tide, and B is the bay tide. The velocity u i is positive when going from ocean to bay. Assuming the bay surface area to fluctuate uniformly, flow continuity can be expressed as 1NBiiBiduAAdt (3.5) in which A i is the cross-sectional area, A B is the bay surface area and t is time. Considering u i to be a simple harmonic function of t, Eq. (3.4) is linearized as shown in Appendix B to yield max28(32iiioBiiiigRuumRFL ) (3.6) in which u maxi is the amplitude of the current velocity in the i th inlet. It follows from Eq. (3.5) and Eq. (3.6) that for a simple harmonic ocean tide (in complex notation) ()jtootea (3.7) and assuming A i and A B to be constant, we obtain (maxvjtiiuue ) (3.8)

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37 where the phase angle v is considered to be the same for all inlets. Differentiating Eq. (3.6) with respect to t, eliminating d B /dt between Eq. (3.5) and Eq. (3.6), and making use of the expressions for u i and o yields an equation for u i and v 2maxmax11823vNjiiBiiBoiuAABujAajeg (3.9) in which the dimensionless resistance factor B i is defined as 12iiiiiiRBmRFL (3.10) where B i is the function of A i Now, equating the real and imaginary parts of Eq. (3.9) and eliminating the phase angle v yields the equation for u maxi 2222224maxmax11823NBiiBoiiiABuAauAg (3.11) For equilibrium flow ie qi Using linearized version in Eq. (3.6) and Eq. (3.3), the equilibrium velocity can be written as 8/3eqieqiiuF (3.12) where the approximate value of eqi can be taken from Mehta and Christensen (1983). For equilibrium flow areas u maxmaxiu eqi substituting this value Eq. (3.11) becomes: 2222224maxmax11823NBieqiBoeqiiiABuAauAg (3.13) When the maximum tidal velocity in all the inlets equals the corresponding equilibrium value, i.e., for i =1,2..N, the difference between the bay and the ocean tides becomes constant. So from Eq. (3.4) it follows that maxmax.iuu eqi

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38 22221max12max2maxmax......eqeqieqiNeqNBuBuBuBu (3.14) Eq. (3.13) and Eq. (3.14) constitute a set of N simultaneous equations with N unknowns [A 1 A 2 ,A N ]. In general, more than one set of equilibrium flow areas [A 1 A 2 ,A N ] will satisfy these equations. Since the dimensionless resistance factor B i is a function of A i Therefore, whether for a given ocean tide (a ,o ) and bay surface area (A B ), Eq. (3.13)and Eq. (3.14) yield sets of solutions [A 1 A 2 .A N ] that are real and positive depends on the particular form of R i =f(A i ). The function R i =f(A i ) plays an important role in the hydrodynamic efficiency of an inlet. For a given head difference, exit and entrance loss coefficients, friction factor and inlet length, the maximum tidal velocity increases with the increasing value of R, see Eq. (3.4). Therefore, larger the value of R, for a given value of A, larger the discharge. For a rectangular channel, iiiARW and for triangular channel ii i R aA (See Appendix B). Analytical solutions to equation Eq. (3.13) and Eq. (3.14) can be found by restricting attention to the friction-dominated flow in the inlets, i.e. m=0 From Eq. (3.10) with m = 0, we obtain 2iiiiFLB R (3.15) For rectangular inlets, substituting iiiARW in Eq. (3.15) and then in Eq. (3.13)and Eq. (3.14) we get

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39 2222222423311112331111814...322...iiiieqiBoBoBeqNNNeqNeqNNNeqNAFLWuAaAaAFLWuFLWugFLWuFLWu (3.16) When any A i (from Eq. (3.16)) is known, the cross-sectional areas of the other inlets follow from Eq. (3.14), with B i given by Eq. (3.15), provided that 231111812...3BoeqNNNeqNAaFLWuFLWug 3 (3.17) This is a quadratic equation in A i 2 for which we have two sets of real and positive roots and two sets of complex roots. For the triangular cross-section, ii i R aA substituting this in Eq. (3.13) and Eq. (3.14) we get, 222311112224....803iiNNieqeqNiBoiiiiiNiiBeqiFLaFLauuAAaFLaFLaFLAuag A (3.18) in which sets of A i are given by Eq. (3.18) (as we have two real and positive solution for A i ). When any A i is known, the cross-sectional areas of the other (N-1) inlets follow from Eq. (3.14) with B i given by Eq. (3.15). One root of Eq. (3.18) is always negative. The other two are real and positive roots provided that 2223551111338.....23NNBoeqNeqNFLFLAauuagag (3.19)

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40 The above stability concept, when applied to a multiple-bay inlet system, becomes complicated because the loci of the set of the values [A 1 A 2 .A N ] for which the tidal maximum of the bottom shear stress equals the equilibrium stress, are rather complicated surfaces and make it difficult to determine whether inlets are in a scouring mode or shoaling mode. With some simplifying assumptions, the stability analysis for a multiple-inlet system can be reduced to that for a two-inlet system. This is considered next in the context of the St. Andrew Bay system. 3.4 Application to St. Andrew Bay System In the above model if N=2, the model can be applied to the two inlet system. The equilibrium flow curves for Inlet 1 and Inlet 2 are calculated from Eq. (3.11) with u=u eq The equilibrium flow areas are given by the solution of Eq. (3.16) for rectangular inlet and Eq. (3.18) for triangular cross-section. Figure 3.6 illustrates the equilibrium flow curve. A line can be drawn passing from the intersection of two equilibrium flow areas. Above the line B 1 >B 2 and therefore u 1
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41 The St. Andrew Bay system is similar to the case of two inlets in a bay. In reality there are three interconnected bays, but only one is connected with the Gulf. So there is no forcing due to ocean tide from the other two bays. Thus, all the bays collectively behave as if there is only one bay connected by two inlets. So the linear model for N inlets can be applied to the St. Andrew system, where N = 2. The development of equilibrium curves for this case is discussed in Chapter 5. A 1 Inlet 1 Inlet 2 Zone-2 Zone-4 Zone-1 Zone-3 A 2 Figure 3.6 Equilibrium flow curves for two inlets in a bay (source: van de Kreeke, 1990)

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CHAPTER 4 APPLICATION TO ST. ANDREW BAY COMPLEX AND ENTRANCES 4.1 Description of Study Area St. Andrew Bay is located in Bay County on the Gulf of Mexico coast of Floridas panhandle. It is part of a three-bay and two-inlet complex. One of these inlets is St. Andrew Bay Entrance and the other is East Pass, which are connected to St. Andrew Bay on one side and the Gulf on the other. The other two bays are West Bay and the East Bay, which connect to St. Andrew Bay, as shown in the Figure 4.1 Note that West Bay as shown also includes a portion called North Bay. Prior to 1934, East Pass was the natural connection between St. Andrew Bay and the Gulf. In 1934, St. Andrew Bay Entrance (Figure. 4.2) was constructed 11 km west of East Pass through the barrier island by the federal government to provide a direct access between the Gulf and Panama City. The entrance has since been maintained by the U.S Army Corps of Engineers (USACE), Mobile District. The St. Andrew Bay State Recreational Area is located on both sides of this entrance, which has two jetties 430 m apart to prevent the closure of the inlet. The interior shoreline of the entrance has continually eroded since its opening. An environmentally sensitive fresh water lake located in the St. Andrew Bay State Recreational Area is vulnerable to the shoreline erosion and USACE has placed dredged soil to mitigate shoreline erosion. East Pass finally closed in the 1998, due to the long-term effect of the opening of St. Andrew Entrance. In December 2001, a new East Pass was opened (Figure 4.3), and the effect of this new inlet is presently being monitored over the entire system. 42

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43 7 3 6 2 5 4 Dupont Bridge Hathaway Bridge St. Andrew Ba y Gulf of Mexico East Pass St. Andrew Channel East Bay North Bay West Bay N 0 5 km 0 1 m 1 3 m 3 6 m 6 9 m > 9 m Figure 4.1 Map showing the three bays and two inlets and bathymetry of the study area. Dots show location of tide stations. N Figure 4.2 Aerial view of St. Andrew Bay Entrance in 1993. Jetties are ~430 m apart.

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44 E & F Figure 4.3 East Pass channel before its opening in December 2001. Plan view (pre-construction) design geometry and then anticipated current measurement transects are shown. The dots show the new cross-section (source: Jain et al., 2002) 4.2 Summary of Field Data Three hydrographic surveys were done by the University of Floridas Department of Civil and Coastal Engineering in the years 2001 and 2002. Figure 4.4 shows the bathymetry of St. Andrew Bay Entrance and the different cross-sections measured during the surveys. Cross-sections A-1, A-2 and B-1, B-2 were measured in September 2001, A-1, A-2, B-1, B-2, C-1, C-2, in December 2001, and D-1, D-2 in March 2002. Flow discharges, vertical velocity profiles and tide were also recorded. The tide gage (in the September 2001 survey only) was located in waters (Grand Lagoon) close to the entrance channel. The discharge and velocity data was measured with a vessel-mounted Acoustic Doppler Current Profiler, or ADCP (Workhorse 1200 kHz, RD Instruments, San Diego, CA), and the tide with an ultrasonic recorder (Model #220, Infinities USA, Daytona Beach, FL). The coordinates of the cross-section end-points are given in Table 4.1.

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45 Table 4.1 Locations of St. Andrew Bay channel cross-sections Section Side Latitude Longitude Northing Easting Date A A-1 30 07.70 -85 43.36 412452.62 1613441.90 09/18/01 A A-2 30.07.44 -85 43.28 410875.80 1613857.60 09/18/01 B B-1 30 07.35 -85 43.91 410315.83 1610524.00 09/18/01 B B-2 30 07.17 -85 43.71 409240.00 1611584.60 09/18/01 A A-1 30 07.18 -85 43.72 409256.63 1611563.75 12/18/01 A A-2 30 07.40 -85 43.91 410626.10 1610534.09 12/18/01 B B-1 30 07.43 -85 43.30 410766.60 1613757.91 12/18/01 B B-2 30 07.68 -85 43.44 412309.71 1613034.11 12/18/01 C C-1 30 07.06 -85 43.90 408542.02 1610606.43 12/18/01 C C-2 30 07.27 -85 44.01 409822.96 1610030.59 12/18/01 D D-1 30 07.42 -85 43 .32 410714.20 1613635.15 03/28/02 D D-2 30 07.65 -85 43. 58 412134.85 1612294.58 03/28/02 Measurements were also taken at the new East Pass after its reopening in December 2001. The locations of the East Pass cross-section coordinate end points are given in Table 4.2. Flow cross-section and vertical velocity profiles were measured along cross-section E in December 2001 and F in March 2002. Table 4.2 Locations of East Pass channel cross-sections Section Side Latitude Longitude Northing Easting Date E E-1 30 03.78 -85 37.07 388325.56 1646376.03 12/19/01 E E-2 30 03.79 -85 37.12 388371.27 1646103.36 12/19/01 F F-1 30 03.78 -85 37 07 388325.55 1646376.03 03/27/02 F F-2 30 03 79 -85 37 12 388371.26 1646103.35 03/27/02

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46 D-1 D-2 Figure 4.4 St. Andrew Bay Entrance bathymetry and current measurement cross-sections. Depths are in feet below MLLW (source: Jain et al. 2002) 4.2.1 Bathymetry The bathymetry of the study area is shown in Figure 4.1. During the hydrographic surveys the bottom depth was measured by the ADCP at all cross-sections shown in Figure 4.4. These have been compared with a bathymetric survey of 2000. Figures. 4.5 and 4.6 are example of measurements along cross-sections A and F, respectively. The trends in the two sets of depths are qualitatively (although not entirely) comparable. Areas, mean depths and widths are summarized in Table 4.3. Table 4.3 Cross-section area, mean depths and width Section Cross-section Area (m 2 ) Width (m) Mean Depth (m) A 6250 493 11.0 B 6600 457 10.6 A 5210 525 10.0 B 5640 544 11.0 C 5220 425 11.5 D 5970 528 11.9 E 255 109 3.0 F 300 85 2.5

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47 Batymetry side-A-18-16-14-12-10-8-6-4-200100200300400500600Distance (m)Depth(m) ADCP Bathymetry chartSide-A-1SideA-2 Figure 4.5 Cross-section A in St. Andrew Bay Entrance measured and compared with 2000 bathymetry. Distance is measured from point A-1. The datum is mean tide level (source: Jain and Mehta, 2001) Bottom Contour-4-3.5-3-2.5-2-1.5-1-0.5003.51118.22431.53746.65972.67584.5Distance (m) from F-1Depth(m) ADCPF-1F-2 Figure 4.6 Cross-section F in East Pass measured by ADCP. Distance is measured from point F-1. The datum is mean tide level (source: Jain and Mehta, 2002)

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48 4.2.2 Tides As noted, tide was measured in September 2001 in Grand Lagoon close to the entrance channel, at Lat: 30 07.9667, Long: -85 43.6667. Tide variation in the channel was compared with the predicted National Ocean Service (NOS) tide at St. Andrew Bay channel with reference station at Pensacola after applying the correction factors for the range and the lag. The measured tide is shown in Figure 4.7 and the corresponding NOS tide in Figure 4.8. Both show general similarities, although the measured one should be deemed more accurate. The data indicate a weak semi-diurnal signature with a range variation of 0.11 to 0.18 m. In the month of December and March no tides were measured, only the NOS tides were reported using the tide at Pensacola; see Figure 4.9 and Figure 4.10. For East Pass the same tide was assumed as for St. Andrew Bay Entrance. Five other NOS stations are also located in the study area as shown in Figure 4.1. The ranges of tides for September 2001, December 2001 and March 2002 at these stations are given in the Table 4.4. These tides were found by applying correction factors for the range and for the lag (see Appendix C). The Gulf tidal range, 2a o was obtained by applying an amplitude correction factor to the tide measured at the Grand Lagoon gauge (see calculations in Appendix C). Semi-diurnal tides were reported in September 2001 with the tidal period of 12.42 h. The tides in December 2001 were of mixed nature with a period of approximately 18 h. In contrast, diurnal tides were reported in March 2002 with the period of 25.82 h. The approximate tide level in each bay was then found by weighted-averaging the tide over the number of stations in that bay. The phase lag between the tides of all the stations were calculated by plotting all the tides in Figure 4.10, and the results are summarized in Table 4.5.

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49 Tide at St Andrew Bay Entrance00.10.20.30.40.50.6 09:00:00 10:45:00 12:30:00 14:15:00 16:00:00 17:45:00 19:30:00 21:15:00 23:00:00 00:45:00 02:30:00 04:15:00 06:00:00 07:45:00 09:30:00 11:15:00 13:00:00 14:45:00 16:30:00 18:15:00 20:00:00Water Level (m) Tide at St Andrew Bay Entrance 09/18/01 Time (hrs) 09/19/01 Figure 4.7 Measured tide in Grand Lagoon on September18-19, 2001. The datum is MLLW (source: Jain and Mehta, 2001) NOS Tides 00.050.10.150.20.250.30.350.40.450:472:055:089:3812:5916:0018:4321:140:063:219:2017:550:38Time (hr:min)Water level (m) Tides 09/19/01 09/18/01 Figure 4.8 NOS predicted tide at St. Andrew Bay Entrance on September18-19, 2001; reference station is Pensacola. The datum is MLLW.

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50 NOS Tides-0.2-0.100.10.20.30.40.51:063:446:218:5812:2015:3318:5522:291:063:446:218:5812:2015:3318:5522:291:06Time (hr:min)Water Level (m) Tides 12/18/0112/19/01 Figure 4.9 NOS predicted tide in St. Andrew Bay Entrance on December 18-19, 2001; reference station is Pensacola. The datum is MLLW. Tides in all the Stations00.050.10.150.20.250.30.350.40.4512:001:002:003:004:005:006:007:008:009:0010:0011:0012:0013:0014:0015:0016:0017:0018:0019:0020:0021:0022:0023:000:00Time (hrs)Water Level (m) Gulf and Channel Entrance Laird Bayou Panama City Parker Lynn Haven West Bay Creek Figure 4.10 Tide at all selected NOS stations in March 2002.

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51 Table 4.4 Tidal ranges in September 2001, December 2001 and March 2002. S No Station Name September Range (m) December Range (m) March Range (m) 1 Gulf of Mexico (Ocean tide) 0.216 0.572 0.425 2 Laird Bayou, East Bay 0.236 0.624 0.465 3 Parker, East Bay 0.236 0.624 0.465 4 Lynn Haven North Bay 0.236 0.624 0.465 5 Panama City, St. Andrew Bay 0.203 0.535 0.397 6 Channel Entrance, St. Andrew Bay 0.197 0.520 0.386 7 West Bay Creek 0.236 0.624 0.465 Table 4.5 Phase lags between the stations and the ocean tide. S No Stations Time Lag 1 Gulf of Mexico (Ocean tide) 0 h 2 Laird Bayou, East Bay + 2 h 3 Parker, East Bay +2 h 4 Lynn Haven North Bay +2 h 5 Panama City, St. Andrew Bay +1 h 6 Channel Entrance, St. Andrew Bay +1 min 7 West Bay Creek + 3h 4.2.3 Current and Discharge Currents and discharges were measured with the ADCP at all the six cross-sections in St. Andrew Bay Entrance (Figure 4.4) and at two cross-sections in East Pass (Figure.4.3). The detailed velocity and discharge curves are shown in Jain and Mehta (2001), Jain et al. (2002) and Jain and Mehta (2002). The measurements are summarized in the Table 4.6. From Table 4.6 it is observed that the average peak velocity in St. Andrew Bay channel was approximately 0.63 m/s (at or close to the throat section) and at East Pass it was approximately 0.50 m/s. The peak discharge value at St. Andrew was 4200 m 3 /s and at East Pass it was 139 m 3 /s.

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52 Table 4.6 Characteristic peak velocity and discharge values Quantity Velocity (m/s) Discharge (m 3 /s) Cross-section Peak Flood Peak Ebb Peak Flood Peak Ebb A 0.63 -0.62 4200 3620 B 0.45 -0.34 2980 2250 A 0.68 -0.69 3620 3920 B 0.69 -0.66 4061 3876 C 0.67 -0.77 3480 3750 D 0.42 -0.49 2509 2777 E 0.51 -0.49 139 165 F 0.43 -0.38 114 101 4.3 Tidal Prism Tidal prism is the volume of water that enters the bay during flood flow. Tidal prism for St. Andrew Bay system was calculated using the approximate formula m K QTPC (4.1) where Q m is the peak discharge (Table 4.6), T is the tidal period (12.42 hrs for September 2001, 18 hrs for December 2001 and 25.82 hrs for March 2002) and the coefficient C K = 0.86 (Keulegan, 1967). This tidal prism was compared with the OBrien (1969) relationship of Eq. (4.2), where A c is the throat area, P the tidal prism on the spring range for sandy inlets in equilibrium, and a and b are the constants: A c = a P b (4.2) For inlets with two jetties, a = 7.49x10 -4 and b = 0.86 (Jarrett, 1976). And for inlets without jetty (East Pass), a = 3.83x10 -5 and b = 1.03. The values of the tidal prism are summarized in Table 4.7. Spring ranges are reported in Table 4.4. It should be noted that the prism values from the OBrien relationship are mere estimates.

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53 Table 4.7 Flood and ebb tidal prisms Quantity Prism (m 3 ) from peak discharge Prism (m 3 ) from OBrien Cross-sections Flood Ebb Peak Flood Peak Ebb A 7.0x10 7 6.0x10 7 11.4x10 7 10.3x10 7 A 8.6x10 7 9.4x10 7 09.0x10 7 10.4x10 7 D 8.6x10 7 9.4x10 7 10.0x10 7 09.7x10 7 E 3.3x10 6 3.9x10 6 03.8x10 6 04.6x10 6 F 3.9x10 6 3.5x10 6 03.6x10 6 03.6x10 6

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CHAPTER 5 RESULTS AND DISCUSSION 5.1 Introduction There are two aspects of this chapter, one dealing with the hydraulics of the St. Andrew Bay system and the other with its stability. The linearized approach developed in Chapter 2 is used to examine the hydraulics of St. Andrew Bay under different conditions. The model is run as one-inlet/one-bay system for both September 2001 and March 2002. It is also run as a three-inlets/three-bays system in September 2001 when East Pass was closed, and as a three-bays/four-inlets system when East Pass was open in March 2002. Hydraulic parameters related to tides and currents thus obtained are then compared with values from the hydrographic surveys done in September 2001 and March 2002. In contrast to hydraulics, the (linearized lumped paramter model) inlet stability model developed in Chapter 3 is applied only to St. Andrew Bay. A qualitative approach is developed to discuss the results and graphs have been plotted to show stability variation. 5.2 Hydraulics of St. Andrew Bay The solution of equations for the linear model, derived in Chapter 2, forms the basis of calculation of the hydraulic parameters characterizing the system. One begins with the basic model of one-inlet (St. Andrew Bay Entrance) and one-bay (St. Andrew Bay) system, when East Pass was closed. As noted the model is then extended to the complete system of three bays (St. Andrew Bay, East Bay and North + West Bays) and 54

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55 three inlets when East Pass was closed in September 2001, and finally as three bays and four inlets when East Pass was open in March 2002. 5.2.1 Solution of Equations The solutions of the relevant hydraulic equations are given in Chapter 2. A Matlab program (see Appendix A) was developed to solve the one-inlet bay system as well as the multiple-inlet bay system. The input and output parameters for each system are listed in the tabular form. 5.2.1.1 One-inlet one-bay system The one-inlet one-bay system is based on solving Eq. (5.1): BoBdCd (5.1) The required input and output parameters for this case are given in Table 5.1. Table 5.1 List of input and output parameters for one-inlet one-bay model. Input Parameters a o Ocean tide amplitude (Gulf of Mexico) T Time period of tide a B1 Bay 1 tide amplitude (St. Andrew Bay) A B1 Bay 1 surface area L 1 Length of inlet 1 (St. Andrew Bay Entrance) R 1 Hydraulic radius of inlet 1 A c1 Inlet 1 cross-section area k Entrance and exit losses f Friction factor ( o B1 ) max Maximum ocean-bay tide difference Output Parameters B1 Bay 1 tide a B1 Bay 1 tide amplitude B1 Phase difference between bay 1 and ocean tides u max1 Maximum velocity through Inlet 1 v1 Phase difference between velocity in Inlet1 and ocean tide ( o B1 ) max Maximum ocean-bay tide difference

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56 5.2.1.2 Three inlets and three bays with one inlet connected to ocean This system is based on solving Eq. (5.2), Eq. (5.3) and Eq. (5.4): 331221111BBBBBoBBBAddAdCdAdAd (5.2) 2122BBBdCd (5.3) 3134BBBdCd (5.4) The required input and output parameters for this case are given in Table 5.2 Table 5.2 List of input and output parameters for the three inlets and three bays model. Input Parameters a o Ocean Tide amplitude (Gulf of Mexico) T Time period of the tide a B1 Bay 1 tide amplitude (St. Andrew Bay) a B2 Bay 2 tide amplitude (East Bay) a B3 Bay 3 tide amplitude (West Bay) A B1 Bay 1 surface area A B2 Bay 2 surface area A B3 Bay 3 surface area L 1 Length of inlet 1 (St. Andrew Bay Entrance) R 1 Hydraulic radius of inlet 1 A c1 Inlet 1 cross-section area L 2 Length of inlet 2 (connecting East Bay and St. Andrew Bay) R 2 Hydraulic radius of inlet 2 A c2 Inlet 2 cross-section area L 4 Length of inlet 4 (connecting West Bay and St. Andrew Bay) R 4 Hydraulic radius of inlet 4 A c4 Inlet 4 cross-section area k Entrance and exit losses f Friction factor ( o B1 ) max Maximum ocean-bay tide difference ( B1 B2 ) max Maximum Bay 1 and Bay 2 tide difference ( B1 B3 ) max Maximum Bay 1 and Bay 3 tide difference

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57 Table 5.2 (continued) Output Parameters B1 Bay 1 tide a B1 Bay 1 tide amplitude B1 Phase lag between bay 1 and ocean tide B2 Bay 2 tide a B2 Bay 2 tide amplitude B2 Phase lag between bay 2 and ocean tide B3 Bay 3 tide a B3 Bay 3 tide amplitude B3 Phase lag between bay 3 and ocean tide u max1 Maximum velocity through Inlet 1 v1 Phase difference between velocity of Inlet 1 and the ocean tide u max2 Maximum velocity through Inlet 2 v2 Phase difference between velocity of Inlet 2 and the ocean tide u max4 Maximum velocity through inlet 4 v4 Phase difference between velocity of Inlet 4 and the ocean tide ( o B1 ) max Maximum ocean-bay tide difference ( B1 B2 ) max Maximum Bay 1 and Bay 2 tide difference ( B1 B3 ) max Maximum Bay 1 and Bay 3 tide difference 5.2.1.3 Three inlets and three bays with two inlets connected to ocean This system is based on solving Eq. (5.5), Eq. (5.6) and Eq. (5.7): 3312211311BBBBBoBBBAddAdCCdAdAd (5.5) 2122BBBdCd (5.6) 3134BBBdCd (5.7) The required input and output parameters for this case are given in Table 5.3.

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58 Table 5.3 List of Input and Output Parameters for the four inlets and three bays model. Input Parameters a o Ocean Tide Amplitude (Gulf of Mexico) T Time period of the tide a B1 Bay 1 tide amplitude (St. Andrew Bay) a B2 Bay 2 tide amplitude (East Bay) a B3 Bay 3 tide amplitude (West Bay) A B1 Bay 1 surface area A B2 Bay 2 surface area A B3 Bay 3 surface area L 1 Length of inlet 1 (St. Andrew Bay Entrance) R 1 Radius of inlet 1 A c1 Inlet 1 cross-section area L 2 Length of inlet 2 (connecting East Bay and St. Andrew Bay) R 2 Radius of inlet 2 A c2 Inlet 2 cross-section area L 3 Length of inlet 3 (East Pass) R 3 Radius of inlet 3 A c3 Inlet 3 cross-section area L 4 Length of inlet 4 (connecting West Bay and St. Andrew Bay) R 4 Radius of inlet 4 A c4 Inlet 4 cross-section area k Entrance and exit losses f Friction factor ( o B1 ) max Maximum ocean-bay tide difference ( B1 B2 ) max Maximum Bay 1 and Bay 2 tide difference ( B1 B3 ) max Maximum Bay 1 and Bay 3 tide difference Output Parameters B1 Bay 1 tide a B1 Bay 1 tide amplitude B1 Phase lag between bay 1 and ocean tide B2 Bay 2 tide a B2 Bay 2 tide amplitude B2 Phase lag between bay 2 and ocean tide B3 Bay 3 tide a B3 Bay3 tide amplitude B3 Phase lag between bay 3 and ocean tide u max1 Maximum velocity through Inlet 1 v1 Phase difference between velocity of Inlet 1 and the ocean tide u max2 Maximum velocity through Inlet 2

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59 Table 5.3 Continued) Output Parameters v2 Phase difference between velocity of Inlet 2 and the ocean tide u max3 Maximum velocity through Inlet 3 v3 Phase difference between velocity of Inlet 3 and the ocean tide u max4 Maximum velocity through Inlet 4 v4 Phase difference between velocity of Inlet 4 and the ocean tide ( o B1 ) max Maximum ocean-bay tide difference ( B1 B2 ) max Maximum Bay 1 and Bay 2 tide difference ( B1 B3 ) max Maximum Bay 1 and Bay 3 tide difference 5.2.2 Input Parameters Table 5.4 provides the input values for all the three cases of the model as described in Section 5.2. 1 The amplitude in each bay is found by applying a weighting factor proportional to the tide station contribution to the total bay area. 2 Initial values are assumed for ( o B1 ) max ( B1 B2 ) max ( B1 B3 ) max for the initial calculation. The September 2001 tide showed a semidiurnal signal, with a period of 12.42 h. The tide in March 2002 showed diurnal signature with a period of 25.82 h. The model was run three times for three different cases as described in Section 5.2. Details regarding all input parameters are found in Jain and Mehta (2002), and are also summarized in Chapter 4. Table 5.4 gives values of all input parameters required for the model. Table 5.4 Input parameters for the hydraulic model. Values Input Parameters Sept 2001 March 2002 Remarks a o 0.109 m 0.212 m Calculated from UF tide gauge data, calculations shown in Appendix C. T 12.42 h 25.82 h NOS Tides Tables. a B1 0.103 m 0.201 m a B2 0.115 m 0.226 m a B3 0.118 m 0.233 m Calculated in proportion to the contributing tide at station. A B1 74 km 2 A B2 54 km 2 A B3 155 km 2 From the USGS topographic maps.

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60 Table 5.4 (continued) Values Input Parameters Sept 2001 March 2002 Remarks L 1 1340 m R 1 10 m A c1 6300 m 2 Measured in survey L 2 1000 m R 2 9 m A c2 1.9x10 4 m 2 L 4 1000 m R 4 12 m A c4 9.7x103 m2 From the USGS topographic maps. A c2, A c3 A c4 are zero for one inlet bay case L 3 400 m R 3 3 m A c3 255 m2 Measured in survey. A c3 is zero for three-bays and three-inlets case. k 1.05 f 0.025 ( o B1 ) max 0.037 0.036 ( B1 B2 ) max 0.060 0.063 ( B1 B3 ) max 0.099 0.998 Assumed initial values. Calculations are shown in appendix C 5.2.3 Model Results and Comparison with Data Model results are given in Table 5.5. Table 5.5 Model results and measurements. One Inlet One Bay System, September 2001 Output parameters Model Measurement %error a B1 0.10 m 0.10 m 0% B1 0.36 rad 0.34 rad 6% u c1 max 0.65 m/s 0.63 m/s 3% v1 -1.20 rad -1.22 rad 2% ( o B1 ) max 0.038 0.036 6% Three Bay Three Inlets System, September 2001 a B1 0.10 m 0.10 m 0% B1 0.34 rad 0.34 rad 0% a B2 0.10 m 0.11 m 9% B2 0.37 rad 0.91 rad 59% a B3 0.10 m 0.12 m 17% B3 0.54 rad 1.26 rad 57%

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61 Table 5.5 (continued) Three Bay Three Inlets System, September 2001 Output parameters Model Measurement %error u c1 max 0.62 m/s 0.63 m/s 2% v1 -1.11 rad -1.20 rad 7% u c2 max 0.04 m/s Not measured v2 -1.21 rad Not measured u c4 max 0.20 m/s Not measured v4 -1.04 rad Not measured ( o B1 ) max 0.037 0.037 0% ( B1 B2 ) max 0.003 0.060 95% ( B1 B3 ) max 0.020 0.098 80% One Inlet One Bay System, March 2002 a B1 0.20 m 0.20 m 0% B1 0.17 rad 0.17 rad u c1 max 0.63 m/s 0.65 m/s 3% v1 -1.40 rad -1.40 rad 0% ( o B1 ) max 0.036 0.036 0% Three Bay Four Inlets System, March 2002 a B1 0.21 m 0.20 m 5% B1 0.16 rad 0.16 rad 0% a B2 0.21 m 0.22 m 5% B2 0.18 rad 0.44 rad 59% a B3 0.21 m 0.23 m 9% B3 0.26 rad 0.60 rad 57% u c1 max 0.60 m/s 0.65 m/s 8% v1 -1.35 rad -1.40 rad 4% u c2 max 0.04 m/s Not measured v2 -1.40 rad Not measured u c3 max 0.60 m/s 0.55 m/s 9% v3 -1.35 rad -1.40 rad 4% u c4 max 0.22 m/s Not measured v4 -1.31 rad Not measured ( o B1 ) max 0.035 0.035 0% ( B1 B2 ) max 0.003 0.063 95% ( B1 B3 ) max 0.012 0.010 20% 0% It is evident from Table 5.5 that the linear model gives good results. The percent error decreases if the system is modeled as a three-bay system, which is actually the case. Velocity and tide amplitudes are within reasonably small error limits. The phase differences between ocean (Gulf) and bay tides from data are very approximate as they

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62 are calculated based on weighted-average tides at selected stations. Moreover, there are very few stations to yield a good value of tide for a bay. Note that the input values for ( o B1 ) max ( B1 B2 ) max ( B1 B3 ) max is also approximate. Sample calculation for ( o B1 ) max ( B1 B2 ) max ( B1 B3 ) max is given in Appendix C. 5.3 Stability Analysis The stability analysis developed in Chapter 3 is now applied to St. Andrew Bay system. This analysis is done for a two-inlet bay system using van de Kreekes (1990) linearized lumped parameter model. The two inlets, to which the model is applied, are St. Andrew Bay Entrance and the new East Pass opened in December 2002. Calculations related to stability are given in Appendix D. A Matlab program (Appendix D) has also been developed for doing the analysis and generating equilibrium flow curves for the two inlets. There are two programs, one for rectangular channel cross-section and another for triangular channel cross-sections. 5.3.1 Input Parameters Input parameters required for the Matlab program (Appendix D) are listed in Table 5.6. Since the objective was to study the effect of bay area on the stability because the results are sensitive to it, it is held constant for a particular set of calculation, but is varied for generating different sets of equilibrium flow curves. Similarly the length of East Pass, believed to have an uncertain value due to the complex bay shoreline and bathymetry in that region is also varied to study its effect on the system.

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63 Table 5.6 Input parameters for stability analysis. Input Parameters for December 2001 a o 0.26 m Amplitude of ocean tide T 18.0 hrs Time period of tide A B 74-105 km2 Area of bay (St. Andrew Bay) (varied from 74 to 105 km2) Inlet 1 u eq1 0.40 m/s Equilibrium velocity for Inlet 1 (see Appendix D) W 1 525 m Width of Inlet 1 L 1 1340 m Length of Inlet 1 a 1 0.138 Constant for triangular cross-section for Inlet 1 (see Appendices C and D) F 1 0.004 Friction coefficient for Inlet 1 Inlet 2 u eq2 0.45 m/s Equilibrium velocity for Inlet 2 (see Appendix D) W 2 300 m Width of Inlet 2 L 2 500-2000 m Length of Inlet 2 (East Pass) (varied from 500 m to 2000 m) a 2 0.187 Constant for triangular cross-section for Inlet 2 (see Appendices C and D) F 2 0.004 Friction coefficient for Inlet 2 5.3.2 Results and Discussion As noted, it is found that two inlets can never be unconditionally stable simultaneously in one bay. The bay area has a large effect on the stability of the inlets. Table 5.7 summarizes this effect. It is clear that with a small increase in bay area the inlets become stable. This is also demonstrated with the help of equilibrium flow curve in the Figure 5.1, Figure 5.2 and Figure 5.3 for rectangular cross-section and Figure 5.7 and Figure 5.9 for triangular cross-section. The cross-sectional area pair during December 2001 (Table 4.3) [5210, 255] is shown by the dot. Figure 5.1 and Figure 5.7 have small bay areas, and the dot lies outside the equilibrium flow curve indicating that both inlets are unstable. As the bay area increases St. Andrew becomes stable (Figure 5.2 and Figure 5.7), and a further increase in bay area also stabilizes East Pass (Figure 5.3 and Figure 5.9). However, in reality we cannot increase the bay area beyond a reasonable limit, because then the basic assumption of bay tide fluctuating evenly in the bay does not hold.

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64 Moreover, in a shallow bay the effect of dissipation of tidal energy cannot be ignored, especially if the bay is large. Also as per Figure 3.5 two inlets are not stable simultaneously. An increase in the length of East Pass has a destabilizing effect on that inlet as shown in the Table 5.7. Note also that for a rectangular cross-section (Figure 5.3) with the length of East Pass of 500m, this inlet is stable, whereas with a length of 2000 m (Figure 5.6) the inlet is instable. This is because as the length increases the dissipation increases. Friction dominated losses, (F = 0.004, R = 3m 2/FLR ) for East Pass with 500 m length is 1.33, where as that for 2000 m length it is 5.33. The same cases occur in Figure 5.9 and Figure 5.12. The other effects on the stability model are the approximation in the cross-section of the inlet. It is clear that triangular cross-section is a better approximation than rectangular section, because with the same parameters for rectangular cross-section in Figure 5.6, East Pass is predicted to be unstable whereas in Figure 5.12 for triangular cross-section, East Pass is stable even though barely, which is not believed to be the case for this newly opened inlet. Table 5.8 gives the qualitative indication of the stability. The various zones mentioned in the Table 5.8 are described in Section 3.4 and Figure 3.6. It is clear from these results that St. Andrew is a stable inlet (for a realistic bay area) as opposed to East Pass. This is also evident from the Figure 3.5, which shows that two inlets cannot be stable simultaneously, because we for unconditional stability, need four real points of intersection of equilibrium flow curve and none of the solutions (neither rectangular cross-section nor triangular cross-section) gives four real solution.

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65 The model does not yield an analytic solution for a more realistic parabolic cross-section. Another weakness is due to the assumptions made in Chapter 3 including a bay area in which the tide is spatially always in-phase, and simple a harmonic function for tide. These assumptions are not always satisfied. Table 5.7 Effect of change in bay area and length of East Pass. Rectangular cross-section Run No. Bay area (km2) East Pass Length (m) Result 1 74 500 Both inlets unstable (Figure 5.1) 2 90 500 St. Andrew becomes stable (Figure 5.2) 3 105 500 St. Andrew stable, East Pass barely stable (Figure 5.3)* 4 74 2000 Both inlets unstable (Figure 5.4) 5 90 2000 St. Andrew barely stable (Figure 5.5) 6 105 2000 St. Andrew stable, East Pass unstable (Figure 5.6) Triangular cross-section 7 74 500 Both inlets unstable Figure (5.7) 8 90 500 St. Andrew becomes stable (Figure 5.8) 9 105 500 Both inlets stable (Figure 5.9)* 10 74 2000 Both inlets unstable (Figure 5.10) 11 90 2000 St. Andrew stable (Figure 5.11) 12 105 2000 St. Andrew stable, East Pass just stable (Figure 5.12)* Two inlets cannot be simultaneously stable, because according to Figure 3.5, for unconditional stability we need four real points of intersection of equilibrium flow curve, which is not possible in either rectangular cross-section solution nor triangular cross-section solution.

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66 Inlet Stability -1 (rectangular section)050010001500200025003000350040004500010002000300040005000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.1 Equilibrium flow curves for rectangular cross-sections, Run No. 1. Inlet Stability 2 (rectangular section)01000200030004000500060000100020003000400050006000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.2 Equilibrium flow curves for rectangular cross-sections, Run No. 2.

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67 Inlet Stability 3 (rectangular section)010002000300040005000600001000200030004000500060007000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.3 Equilibrium flow curves for rectangular cross-sections, Run No. 3. Inlet Stability -4(rectangular section)050010001500200025003000350040004500010002000300040005000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.4 Equilibrium flow curves for rectangular cross-sections, Run No. 4.

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68 Inlet Stability 5(rectangular section)01000200030004000500060000100020003000400050006000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.5 Equilibrium flow curves for rectangular cross-sections, Run No. 5. Inlet Stability 6 (rectangular section)010002000300040005000600001000200030004000500060007000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.6 Equilibrium flow curves for rectangular cross-sections, Run No. 6.

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69 Inlet Stability -7 (triangular section)0500100015002000250030003500400045005000010002000300040005000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.7 Equilibrium flow curves for triangular cross-sections, Run No. 7. Inlet Stability 8 (triangular section)01000200030004000500060000100020003000400050006000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.8 Equilibrium flow curves for triangular cross-sections, Run No. 8.

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70 Inlet Stability 9 (triangular section)0100020003000400050006000700001000200030004000500060007000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.9 Equilibrium flow curves for triangular cross-sections, Run No. 9. Inlet Stability 10 (triangular section)0500100015002000250030003500400045005000010002000300040005000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.10 Equilibrium flow curves for triangular cross-sections, Run No. 10

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71 Inlet Stability 11 (triangular section)01000200030004000500060000100020003000400050006000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.11 Equilibrium flow curves for triangular cross-sections, Run No. 11. Inlet Stability 12 (triangular section)0100020003000400050006000700001000200030004000500060007000A1, St Andrew (m2)A2, East Pass (m2) East Pass St Andrew Figure 5.12 Equilibrium flow curves for triangular cross-sections, Run No. 12.

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72 Table 5.8 Stability observations for St. Andrew Bay Entrance and East Pass. Figure Placement of cross-sectional area pair [A 1 A 2 ], (black dot) Observations Figure 5.1 Zone-1 Both inlets are unstable Figure 5.2 Zone-2 St. Andrew Bay Entrance is stable Figure 5.3 Zone-4 Only one is stable i.e. St. Andrewa Figure 5.4 Zone-1 Both inlets are unstable Figure 5.5 Zone-2 St. Andrew Bay Entrance is stable Figure 5.6 Zone-2 St. Andrew Bay Entrance is stable Figure 5.7 Zone-1 Both inlets are unstable Figure 5.8 Zone-2 St. Andrew Bay Entrance is stable Figure 5.9 Zone-4 Only one is stable i.e. St. Andrewa Figure 5.10 Zone-1 Both inlets are unstable Figure 5.11 Zone-2 St. Andrew Bay Entrance is stable Figure 5.12 Zone-4 Only one is stable i.e. St. Andrewa a As per Figure 3.6, it is clear that even in Zone-4 only one inlet is stable, this is further clarified from Figure 3.5, which shows that only one inlet can be stable at one time.

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CHAPTER 6 CONCLUSIONS 6.1 Summary St. Andrew Bay, which is a composite of three interconnected bays (St. Andrew Bay proper, West Bay + North Bay and East Bay) is located in Bay County on the Gulf of Mexico coast of Floridas panhandle. It is part of a three-bay and two-inlet complex. One of these inlets is St. Andrew Bay Entrance and the other is East Pass, which are both connected to St. Andrew Bay on one side and the Gulf on the other. Prior to 1934, East Pass was the natural connection between St. Andrew Bay and the Gulf. In 1934, St. Andrew Bay Entrance (Figure 4.2) was constructed 11 km west of East Pass through the barrier island to provide a direct access between the Gulf and Panama City. The interior shoreline of the entrance has continually eroded since its opening. East Pass was closed in 1998, which is believed to be due to the opening of the St. Andrew Bay Entrance. In December 2001, a new East Pass was opened (Figure 4.3), and the effect of this new inlet is presently being monitored over the entire system. Accordingly, the objective of the present work was to examine the hydraulics of the newly formed two-(ocean) inlet/three-bay system and its hydraulic stability, especially as it relates to East Pass. The first aspect of the tasks performed to meet this objective was the development of equations for the linearized hydraulic model for the system of three bays and four inlets (two ocean and two between bays), and solving and applying them to the St. Andrew Bay system. The second aspect was the development of the ocean inlet stability criteria using the Escoffier (1940) model for one inlet and one bay and extending this 73

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74 model to the two ocean inlets and a bay. Stability analysis for the St. Andrew Bay system was then carried out using the linearized lumped parameter model of van de Kreeke (1990). 6.2 Conclusions The following are the main conclusions of this study: 1 If the system is modeled as a three-bay system as compare to a one-bay system, the error in the phase difference, B1, decreases from 6% to 0% and in the velocity amplitude from 3% to 2%. Moreover the error in maximum head difference, ( o B1 ) max also decreases from 6% to 0%. 2 The amplitudes of velocities and bay tides are within %, which is a reasonably small error band. The percent error for St. Andrew Bay is almost 0%, and for the other bays it is within %. 3 The bay area has a significant effect on the stability of the two inlets. At a bay area of 74 km2 both inlets are unstable. Increasing it by 22% to 90 km2 stabilizes St. Andrew Bay Entrance, and by 42% to 105 km2 stabilizes East Pass as well. 4 Two inlets can never be simultaneously unconditionally stable. 5 Keeping the bay area at 105 km2 and increasing the length of East Pass from 500 m to 2000 m destabilizes this inlet because as the length increases the dissipation in the channel increases as well. 6 A triangular channel cross-section is a better approximation than a rectangular one, because given the same values of all other hydraulic parameters, St. Andrew Bay Entrance with a rectangular cross-section is found to be barely stable, whereas with a triangular cross-section it is found to be stable, as is the case. 6.3 Recommendations for Further Work Accurate numerical values required for the stability analysis of a complex inlet-bay system can only be obtained by using a two(or three)-dimensional tidal model to describe the hydrodynamics of the bay. Freshwater discharges from the rivers into the bay should be incorporated through numerical modeling.

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75 Including a more realistic assumption for the channel cross-section can improve the stability analysis.

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APPENDIX A ALGORITHMS FOR MULTIPLE INLET-BAY HYDRAULICS A.1 Introduction The linearized approach described in Chapter 2 has been used to evaluate the hydraulic parameters of the multiple inlet bay system. The differential equations, developed by this approach (Chapter 2), Eq. (2.100), Eq. (2.101) and Eqs (2.102), are solved in Matlab Program-1 (given below). These are the general equations for four inlets and three bays system. These equations can be used to solve from one bay system to the complex three bays system. Note that for solving Program-1, the Matlab version should have a symbolic toolbox. The present program is solved in Matlab release 6.1. The solution from Program-1 is used as input to Program-2 (given below). The required input parameters and output for Program-2 are listed in Table 5.3 of Chapter 5. A.2 Program-1 %UNIVERSITY OF FLORIDA %CIVIL AND COASTAL ENGINEERING DEPARTMENT %PROGRAM FOR SOLVING THE EQS 2.100, 2.101, 2.102 % ALL CONSTANTS DEFINED IN CHAPTER 2 clear all syms a b c A B t1=sym('theta1') t2=sym('theta2') t3=sym('theta3') r1=sym('a1*exp(-i*t1)') r2=sym('a2*exp(-i*t2)') r3=sym('a3*exp(-i*t3)') C=[a*i+1 a*A*i a*B*i;-1 b*i+1 0;-1 0 c*i+1] D=[1;0;0] %END 76

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77 A.3 Program-2 %UNIVERSITY OF FLORIDA %CIVIL AND COASTAL ENGINEERING DEPARTMENT %PROGRAM FOR CALCULATION OF MULTIPLE INLET-BAY HYDRUALICS %FOR ONE -INLET BAY CASE, FOR Ac2, Ac3, Ac4 EQUAL TO ZERO %INLET 1 AND INLET 3 CONNECTS BAY1 TO THE OCEAN clear all g=9.81; ao=0.212;%ocean tide amplitude theta=0;%ocean tide phase etao=ao*cos(theta);%ocean tide T=25.82;%time period q=2*pi/(T*3600)%sigma k=1.05;% entrance and exit loss f=0.025;%friction factor aB1=0.201;%approximate amplitude of bays aB2=0.226; aB3=0.2325; %m1=max(eta0-etab1),m2=max(etab1-etab2),m3=max(etab1-etab3) m1=0.023; m2=0.0527; m3=0.123; %Inlet 1 L1=1340;%Length of inlet R1=10;%hydraulic radius Ac1=6300;%CROSS-SECTION AREA of the inlet F1=k+(f*L1)/(4*R1);%friction factor F includes ken kex fL/4R %Inlet 2 L2=1000;%Length of inlet R2=9;%hydraulic radius Ac2=1.9*10^4;%CROSS-SECTION AREA of the inlet, it is zero for one inlet bay case F2=k+(f*L2)/(4*R2);%friction factor F includes ken kex fL/4R %Inlet 3 L3=400;%Length of inlet R3=3;%hydraulic radius Ac3=255;%CROSS-SECTION AREA of the inlet F3=k+(f*L3)/(4*R3);%friction factor F includes ken kex fL/4R %Inlet 4 L4=1000;%Length of inlet R4=12;%hydraulic radius Ac4=9.7*10^3;%CROSS-SECTION AREA of the inlet F4=k+(f*L4)/(4*R4);%friction factor F includes ken kex fL/4R

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78 %bay1 area AB1=74*10^6; %bay2 area AB2=54*10^6; %bay3 area AB3=155*10^6; %calculations CDL1=sqrt(ao/(m1*F1)) CDL2=sqrt(aB1/(m2*F2)) CDL3=sqrt(ao/(m1*F3)) CDL4=sqrt(aB1/(m3*F4)) C1=CDL1*Ac1/AB1*sqrt(2*g/ao) C2=CDL2*Ac2/AB2*sqrt(2*g/aB1) C3=CDL3*Ac3/AB1*sqrt(2*g/ao) C4=CDL4*Ac4/AB3*sqrt(2*g/aB1) %ALL THE CONSTANTS ARE DEFINED IN THE THESIS a=q/(C1+C3) if Ac2==0 b=0 else b=q/C2 end if Ac4==0 c=0 else c=q/C4 end A=AB2/AB1 B=AB3/AB1 r1=(c-i)*(b-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b-1+c*b-i*a*B+a*b-i*a+a*B*b+a*A*c)%SOLUTIONS ARE OBTAINED FROM ANOTHER r2=-i*(c-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b-1+c*b-i*a*B+a*b-i*a+a*B*b+a*A*c)%MATLAB PROGRAM WHICH HAS SYMBOLLIC TOOLBOX. r3=-i*(b-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b-1+c*b-i*a*B+a*b-i*a+a*B*b+a*A*c) aB1=abs(r1)*ao eB1=-angle(r1) aB2=abs(r2)*ao eB2=-angle(r2) aB3=abs(r3)*ao eB3=-angle(r3) etaB1=aB1*cos(theta-eB1) etaB2=aB2*cos(theta-eB2)

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79 etaB3=aB3*cos(theta-eB3) CDL11=sqrt(ao/(max(etao-etaB1)*F1)) CDL22=sqrt(aB1/(max(etaB1-etaB2)*F2)) CDL33=sqrt(ao/(max(etao-etaB1)*F3)) CDL44=sqrt(aB1/(max(etaB1-etaB3)*F4)) C11=CDL1*Ac1/AB1*sqrt(2*g/ao) C22=CDL2*Ac2/AB2*sqrt(2*g/aB1) C33=CDL3*Ac3/AB1*sqrt(2*g/ao) C44=CDL4*Ac4/AB3*sqrt(2*g/aB1) %velocity in the inlet uc1=sqrt(2*g/ao)*CDL1*(etao-ao*r1) uc1max=abs(uc1) ev1=-angle(uc1) uc2=sqrt(2*g/aB1)*CDL2*(ao*r1-ao*r2) uc2max=abs(uc2) ev2=-angle(uc2) uc3=sqrt(2*g/ao)*CDL3*(etao-ao*r1) uc3max=abs(uc1) ev3=-angle(uc1) uc4=sqrt(2*g/aB1)*CDL4*(ao*r1-ao*r3) uc4max=abs(uc4) ev4=-angle(uc4) %END

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APPENDIX B INLET HYDRAULICS RELATED DERIVATIONS B.1 Linearization of Damping Term The linearization of the damping term in Eq. (3.6) is done as given in Bruun (1978). The bay tide response is represented by sin()BBBa (B.1) where 2ttT dimensionless time. a B = one-half the tide range (i.e., amplitude) in the bay, and B = lag between high water (HW) or low water (LW) in the ocean and corresponding HW or LW in the bay. Also, sin()ooa (B.2) from the continuity equation we further have BcBdAuAdt (B.3) where A c is the area of cross-section of the inlet and A B is the surface area of the bay. The time of HW or LW in the bay, i.e., when 0Bddt coincides with time of slack water, i.e., u, so that 0 B is also the lag of slack water after HW or LW in the ocean. Thus it can be written as 22coscosBBBBddadtdt B (B.4) 80

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81 or in terms of Fourier series Eq. (B.4) can be written as 22218sin2cos4nBBBnnddadtdtnn Bn (B.5) where n takes only odd integral values. For linearization purposes n=1, so that Eq. (B.5) becomes 228cos3BBBddandtdt B (B.6) The amplitude of the tidal velocity is given by maxBcaAuA B (B.7) Therefore, it can be written as max83uuuu (B.8) where u is the amplitude of the u. max B.2 Shear Stress Dependence on Area For each inlet discharge is defined as a time varying function: ()BiBiiAdQtAdt (B.9) 2()2iiioiiiigRQtAmRFL Bi (B.10) The expression for maximum tidal velocity can be obtained by the solution of the above equations with the simplifying assumptions mentioned in Chapter 2. max2()sinBoiiAaKuCKATK (B.11) where K is the coefficient of repletion,

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82 222iiioBiiiiAgRaTKaAmRFL oK (B.12) and iK (B.13) is summation is over all the inlets. The functionCK ()sin is a monotonically increasing function with C=0 for K=0 and C=1 for K=, is a specific time when sea is at MSL, as defined by Kuelegan (1951) It is seen below that the bottom shear stress, varies strongly with the cross-sectional area. This can be shown with the help of approximate analytical solution carried out by Keulegan (1951). Substituting the value of u from Eq. (B.11) in Eq. (3.1), and taking CK ()sin1 ; and F = 0.003: 221oBaAFTA; (B.14) It is clear from the above equation that has a strong dependence on A. B.3 General Equation for hydraulic radius. Consider the general trapezoidal cross-section: Area, 0011122 B ABBhBh B Wetted perimeter, 2222000012()144BBhPBBBhBBBB Hydraulic radius, 0220011214BhABRPBBh B BB

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83 B h B 0 Figure B.1 Trapezoidal Cross-section Now consider two cases: 1) Rectangular cross-section, i.e., B 0 = B, and 2) Triangular cross-section, i.e., B 0 = 0. B.3.1 Rectangular B=B 0 Therefore hydraulic radius for a rectangle is 12AhRhP B B.3.2 Triangular For triangular section, B 0 =0 21214hARPh B B.4 Hydraulic Radius for Triangular Cross-Section For a triangular cross-section the hydraulic radius is related as a square root of the area, as shown below:

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84 Figure B.2 is a triangular cross-section where is the angle with the horizontal on both the sides: Area 12tan2Ahh Wetted perimeter 2coshP Hydraulic radius 12sincosA R AaAP (B.15) h Figure B.2 Triangular cross-section.

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APPENDIX C CALCULATION OF BAY TIDE AND LINEAR DISCHARGE COEFFICIENTS This appendix contains sample calculations of input data in Table 5.4 for the Matlab Program (Appendix A) in Chapter 5. Estimation of bay tide amplitude (a B1, a B2, a B3 ) input for Table 5.4 was made by taking the weighted-averages of the NOS tide amplitudes at reported stations in the bay. Let us take the case of St. Andrew Bay (Table C.1). This bay has three stations where tide is reported. The weighting factor for the range at a given station was estimated by selecting the approximate area of influence of tide (range) surrounding that station. Given the tidal period of 12.42 h, the phase difference between Gulf and the bay could be converted in to degree or radians. Table C.1 Weighted-average bay tide ranges and phase differences St. Andrew Bay Station Weighting factor Sept. tide range (m) Weighted-average (m) Phase difference (h) Weighted-average (h) Channel Entrance 0.48 0.197 0.0945 0.0017 0.008 Panama City 0.37 0.203 0.0751 1.0000 0.370 Parker 0.15 0.236 0.0354 2.0000 0.300 Total 0.2050 0.678 h=19.65o East Bay Laird Bayou 0.40 0.236 0.0940 2.0000 0.800 Parker 0.40 0.236 0.0940 2.0000 0.800 Panama city 0.20 0.203 0.0406 1.0000 0.200 0.2300 1.8 h=52.17o West Bay West Bay Creek 0.50 0.236 0.118 3.0000 1.5 hrs Lynn Haven 0.50 0.236 0.118 2.0000 1.0 hrs 0.236 2.5 h=72.46o The Gulf tide range had to be estimated, as there was no open coast tidal station near to the study site. The procedure was as follows: 85

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86 1 The tidal range at the gauge installed by UF in September 2001 in the bay was R = 0.18m. 2 The kinetic head in the channel was calculated by using the peak velocity value, 0.63 m/s: 220.630.0222*9.81kuhg m. 3 Thus the Gulf tide range was obtained as 0.020.18okRRh m. So the amplitude correction factor 0.201.110.18oRR 4 The distance between the open coast (Gulf-ward tips of the entrance jetties) and the UF gauge was X = 673 m and the depth of the deepest channel (thalweg) between the two locations h =12 m. 5 Therefore the time difference between calculated ocean tide and the bay is calculated as 67362.03s1.03min9.81*12Xgh The maximum driving heads, ( o B1 ) max ( B1 B2 ) max ( B1 B3 ) max are calculated as suggested by OBrien and Clark (1973). 1 The driving head between ocean and bay 1 ( o B1 ) max at any time t during the tidal cycle is 11coscos()oBoBBatat 1 (C.1) slack water occurs at 1Bt Maximum head occurs at 12Bt 1maxsinoBoBa 1 (C.2) 2 The driving head between bay1 and bay 2 ( B1 B2 ) max at any time t during the tidal cycle is 121122coscos()BBBBBBatat (C.3) slack water occurs at 21BBt

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87 Maximum head occurs at 212BBt 1212112212maxcoscos22BBBBBBBBBBaa (C.4) 121212maxsin2sinBBBBBBBaa 1 (C.5) For maximum head difference we have to take absolute value of the head difference. Examples of the use of the above equations are shown in Figures C.1 and C.2. Figure C.1 gives the ocean and bay 1 tide head difference, while Figure C.2 gives the head difference between bay 1 and bay 2. Table C.2 provides the values used to calculate the linear discharge coefficients given by Eqs (2.28), (2.29) and (2.50). Table C.2 Calculation of ( o B 1 ) max ( B 1 B 2 ) max ( B 1 B 3 ) max Sept. 2001 tide Ampl (m) Phase diff. with Gulf (deg) Tide functions Max. tide difference (m) Ocean 0.1090a 00.00 0.109cos()o Bay 1 0.1025 19.65 010.1025cos(19.65)B 1max0.0366oB Bay 2 0.1150 52.17 020.115cos(52.17)B 1max0.0601oB Bay 3 0.1180 72.46 030.118cos(72.46)B 1max0.0989oB a Amplitude correction factor of 1.11 applied.

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88 Definition sketch of Ocean and Bay TIdes-0.15-0.1-0.0500.050.10.1500.310.630.941.261.571.882.22.512.833.143.463.774.084.44.715.035.345.655.976.28Angle (radians)Water Level (m) ocean bay1 ( o B1)max HW-Bay1, Slack water -inlet HW Ocean Figure C.1 Head difference between ocean (Gulf) and bay 1. Definition sketch of tides in two bays-0.15-0.1-0.0500.050.10.1500.310.630.941.261.571.882.22.512.833.143.463.774.084.44.715.035.345.655.976.28Angle (radians)Water Level (m) bay1 bay2 HW Bay1 HW-Bay2Slack water -inlet (etaB1 -etaB2)max MSL Figure C.2 Head difference between bay1 and bay 2.

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APPENDIX D CALCULATIONS FOR STABILITY ANALYSIS D.1 Introduction van de Kreekes (1990) linearized lumped model is used for development of the equilibrium flow curves for the two inlets, namely St. Andrew Entrance and East Pass. This appendix includes the calculation of some of the input parameters listed in Table 5.6 D.2 Calculations D.2.1 Equilibrium velocity The equilibrium velocity is calculated using Eq. (D.1), which is same as Eq. (3.12). The equilibrium shear stress, eq is taken from Mehta and Christensen (1983). 8/3eqieqiiuF (D.1) Table D.1 Calculation of equilibrium velocity Friction Coefficient, F=f/8 Range of eq (Pa) Density (kg/m3) Range of u eq (m/s) 0.003 0.48-0.72 1024 0.43-0.52 0.004 0.48-0.72 1024 0.37-0.45 D.2.2 Constant for Triangular schematization. The constant a i [Appendix B, Eq (B.15)] is calculated for St. Andrew Bay Entrance and East Pass. Table D.2 Calculation of a i Inlet Cross-sectional Area (m2) Hydraulic Radius, R (m) iii R aA St. Andrew Bay Entrance 5,250 10 0.138 East Pass 255 3 0.187 89

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90 D.3 Relationship between Flow Curves and Stability of Two Inlets In general, equilibrium areas of the two inlets are defined by the sets of the areas where the two inlet equilibrium flow curves intersect. The hatched rectangle (common equilibrium zone) is formed by the intersection of two equilibrium intervals. Each interval is defined from the first equilibrium area to Consider the following two cases: 1 For both inlets to be in equilibrium, it is necessary that the enhanced parts of both the curves intersect. The hatched portion is the common equilibrium region where both the inlets are stable. Suppose after a storm, point a (Figure. D.1) marked by a square (in hatched region) represent the areas of the two inlets. If we follow the arrows, which show how the areas of the inlets try to adjust after the storm, it is clear that both the areas lie in the enhanced parts, so both the inlets will be stable. Figure D.1 General configuration of equilibrium flow curve. a A 1 A 2 2 1 A 2 1 a 2 A 1 Figure D.2 General configuration of equilibrium flow curve.

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91 2 In this set of equilibrium flow curves, although the enhanced parts of the equilibrium curves intersect, the inlets need not necessarily come back to the equilibrium region. Lets draw a rectangle around the intersection of enhanced part of the equilibrium flow curve. Let us take a point a (Figure. D.2), which represents the areas of the inlets just after a storm event. In this case following the arrows, it is clear that Inlet 2 will reach stable equilibrium but Inlet 1 will not. So this set of equilibrium flow curves does not always represent a stable equilibrium. Similarly we can argue with the other Figures (Figure 3.5 (c) and Figure 3.5 (d) in Chapter 3) D.4 Matlab Programs For solving the linearized model for stability, two Matlab programs were developed, Program for solving the inlets with rectangular cross-section and Program for triangular cross-section. D.4.1 Program-1 %UNIVERSITY OF FLORIDA %DEPARTMENT OF CIVIL AND COASTAL ENGINEERING %INLET STABILITY CALCULATION ASSUMING THE CROSS-SECTION TO BE RECTANGULAR clear all Ab=90*10^6;%Bay surface area, St. Andrew eta_o=0.26;%Ocean Tide amplitude ueq1=0.40;%Equilibriuim velocity in Inlet 1 as calculated in Appendix D. ueq2=0.45;%Equilibriuim velocity in Inlet 2 as calculated in Appendix D. T=18;%Tidal Period sigma=(2*pi)/(T*3600); g=9.81;%acceleration due to gravity % Inlet -1 W1=525;% Width of the Inlet-1 L1=1340;% Length of the Inlet-1 F1=4*10^(-3);%Friction coefficient %a1=0.044; % Inlet-2 B=[]; W2=300; %Width of Inlet-2 L2=2000;%Length of the Inlet-2 F2=0.004;%Friction Coefficient of Inlet-2 %a2=0.197;

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92 A=[]; P1=Ab*sigma*(eta_o)^2; P2=(2*8*(F1*L1*W1*(ueq1^3)+F2*L2*W2*(ueq2^3)))/(3*pi*g); if P1>P2 sprintf('the solution to the inlet system is possible') else sprintf('the solution to the inlet system is not possible') end a=(F1*L1*W1*(ueq1^3)+F2*L2*W2*(ueq2^3))^2; b=-(Ab*sigma*(eta_o))^2; c=((Ab*sigma*8)/(3*pi*g))^2; p=[a b c ]; roots(p) A1eq=sqrt(((F1*L1*W1*(ueq1^2))^2)*roots(p)) A2eq=sqrt(((F2*L2*W2*(ueq2^2))^2)*roots(p)) %B1=(A1.^-1).*2*F1*L1*W1; newm1=[]; %Solve for inlet 1 for i=1:80 A2=i*100; % for equilibirium case B1*ueq1^2=B2*ueq2^2 a1=ueq1^2; b1=2*A2*ueq1*ueq2; c1=(ueq1^2)*(A2^2)-(Ab*sigma*eta_o)^2; d1=0; e1=((8*Ab*sigma*F1*L1*W1*ueq1^2)/(3*pi*g))^2; p1=[a1 b1 c1 d1 e1]; Af(1:4)=roots(p1); B=[A2 A2 A2 A2;Af(1:4)]; newa=B(3:6); newm=reshape(newa,2,2); newm1=cat(2,newm,newm1); fid = fopen('r1.txt','w'); fprintf(fid,'%6.2f %12.8f\n',newm1); fclose(fid); end newm12=[] %Solve for inlet 2 for i=1:80 A1=i*100;

PAGE 108

93 % for equilibirium case B1*ueq1^2=B2*ueq2^2 a2=ueq2^2; b2=2*A1*ueq1*ueq2; c2=(ueq2^2)*A1^2-(Ab*sigma*eta_o)^2; d2=0; e2=((8*Ab*sigma*F2*L2*W2*ueq2^2)/(3*pi*g))^2; p2=[a2 b2 c2 d2 e2]; As(1:4)=roots(p2); B2=[A1 A1 A1 A1;As(1:4)]; newa2=B2(3:6); newm2=reshape(newa2,2,2); newm12=cat(2,newm2,newm12); fid = fopen('r2.txt','w'); fprintf(fid,'%6.2f %12.8f\n',newm12); fclose(fid); end %Line A2L=0:100:2000; A1L=(F1*L1*W1*A2L)./(F2*L2*W2); eq1=[A2L;A1L] fid = fopen('r3.txt','w'); fprintf(fid,'%6.2f %12.8f\n',eq1); fclose(fid); %End D.4.2 Program-2 %UNIVERSITY OF FLORIDA %DEPARTMENT OF CIVIL AND COASTAL ENGINEERING %INLET STABILITY CALCULATION ASSUMING THE CROSS-SECTION TO BE TRIANGULAR clear all %Input Parameters Ab=90*10^6;%Bay surface area, St. Andrew eta_o=0.26;%Ocean Tide amplitude ueq1=0.4;%Equilibriuim velocity in Inlet 1 as calculated in Appendix D. ueq2=0.45;%Equilibriuim velocity in Inlet 2 as calculated in Appendix D T=18;%Tidal Period sigma=2*pi/(T*3600); g=9.81;%acceleration due to gravity

PAGE 109

94 %Inlet 1 L1=1340;%Length of Inlet R1=10;% Hydraulic Radius F1=4*10^-5%k+(f*L1)/(4*R1);%friction factor F includes ken kex fL/4R alpha1=0.138;%factor for triangular cross-section calculated as alpha=R/sqrt(Area) B=[]; %Inlet 2 L2=2000;%Length of Inlet R2=3;% Hydraulic Radius F2=0.004;%friction factor F includes ken kex fL/4R alpha2=0.187;%factor for triangular cross-section calculated as alpha=R/sqrt(Area) A=[]; %calculations P1=Ab*sigma*(eta_o)^3; P2=((3*sqrt(3))/2)*((8/(3*pi))^2)*(((F1*L1*(ueq1^5))/(alpha1*g))^2+((F2*L2*(ueq2^5))/(alpha2*g))^2); if P1>P2 sprintf('the solution to the inlet system is possible') else sprintf('the solution to the inlet system is not possible') end %Solve for the 2 roots its a third degree polynomial so lets define the coefficients a=(ueq1*(((F1*L1*alpha1)/(F1*L1*alpha1))^2)+ueq2*(((F2*L2*alpha1)/(F1*L1*alpha2))^2))^2; b=0; c=-(Ab*sigma*eta_o)^2; d=((8/(3*pi))*(Ab*sigma)*((F1*L1)/(alpha1*g))*(ueq1^2))^2; p=[a b c d]; roots(p) A(1:3)=roots(p); A1=A(2:3); % for equilibirium case B1*ueq1^2=B2*ueq2^2 A2=((((alpha1*F2*L2)/(alpha2*F1*L1))*((ueq2/ueq1)^2))^2)*A1; %B1=(A1.^-1).*2*F1*L1*W1; newm1=[]; %Equilibrium curve for Inlet 1 for i=1:100

PAGE 110

95 A2=i*100; a1=ueq1^2; b1=2*A2*ueq1*ueq2; c1=(ueq1^2)*(A2^2)-(Ab*sigma*eta_o)^2; d1=((8*Ab*sigma*F1*L1*ueq1^2)/(alpha1*3*pi*g))^2; p1=[a1 b1 c1 d1]; roots(p1); Af(1:3)=roots(p1); B=[A2 A2 A2;Af(1:3)]; newa=B(3:6); newm=reshape(newa,2,2); newm1=cat(2,newm,newm1); fid = fopen('t1.txt','w'); fprintf(fid,'%6.2f %12.8f\n',newm1); fclose(fid); end newm12=[]; %Equilibrium curve for Inlet 2 for i=1:100 A1=i*100; a2=ueq2^2; b2=2*A1*ueq1*ueq2; c2=(ueq2^2)*(A1^2)-(Ab*sigma*eta_o)^2; d2=((8*Ab*sigma*F2*L2*ueq2^2)/(alpha2*3*pi*g))^2; p2=[a2 b2 c2 d2]; roots(p2); As(1:3)=roots(p2); B=[A1 A1 A1;As(1:3)]; newa2=B(3:6); newm2=reshape(newa2,2,2); newm12=cat(2,newm2,newm12); fid = fopen('t2.txt','w'); fprintf(fid,'%6.2f %12.8f\n',newm12); fclose(fid); end %Line A2L=0:100:7000; A1L=((F1*L1*alpha2*(A2L.^0.5))./(F2*L2*alpha1)).^2; eq1=[A2L;A1L] fid = fopen('t3.txt','w'); fprintf(fid,'%6.2f %12.8f\n',eq1); fclose(fid); %end.

PAGE 111

LIST OF REFERENCES Aubrey, D. G., and Weishar, L. (eds), 1988. Hydrodynamics and Sediment Dynamics of Tidal Inlets Lecture Notes on Coastal and Estuarine Studies Vol. 29, SpringerVerlag, New York. Becker, M. L., and Ross, M. A., 2001. Interaction of tidal inlets in a multi-inlet bay system: A case study along the central Gulf Coast of Florida. Journal of Coastal Research 17(4), pp. 836-849 Brunn, P., 1978. Stability of Tidal Inlets Elsevier, Amsterdam. Dean, R. G., 1983. Hydraulics of inlets, Report UFL/COEL-83/004 Coastal and Oceanographic Engineering Department, University of Florida, Gainesville. Escoffier, F. F. 1940. The stability of tidal inlets. Shore and Beach 8(4), 114-115. Escoffier, F. F., and Walton, T. L., 1979. Inlet stability solutions for tributary inflow. Journal of the Waterways, Port, Coastal and Oceanographic Division, ASCE, 105(4), 341-355. Jain, M., and Mehta, A. J., 2001. Hydrographic measurements at St. Andrews Bay Entrance, Bay County, Florida. Report UFL-COEL-2001/014 Coastal and Oceanographic Engineering Program, Department of Civil and Coastal Engineering, University of Florida, Gainesville. Jain, M., and Mehta, A. J., 2002. Hydrographic measurements at St. Andrews Bay Entrance and East Pass, Bay County, Florida, Part II. Report UFL-COEL2002/010 Coastal and Oceanographic Engineering Program, Department of Civil and Coastal Engineering, University of Florida, Gainesville. Jain, M., Paramygin, V. A., and Mehta, A. J., 2002. Hydrographic measurements at St. Andrews Bay Entrance and East Pass, Bay County, Florida, Part I. Report UFL/COEL-2002/001 Coastal and Oceanographic Engineering Program, Department of Civil and Coastal Engineering, University of Florida, Gainesville. Jarrett, J. T., 1976 Prism-inlet area relationships. G.I.T.I Report No. 3 U.S. Army Engineering Coastal Engineering Re search Center, Ft. Belvoir, VA. 96

PAGE 112

97 Keulegan, G. H., 1951, Third Progress Report on Tidal flow in entrances. Water Level fluctuations of basins in communication with seas. Report No. 1146, National Bureau of Standards, U. S Department of Commerce, Washington, D. C. pp 32. Keulegan, G. H., 1967, Tidal flows in entrances: Water level fluctuations of basins in communication with the seas. Committee on Tidal Hydraulics Technical Bulletin No. 14, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Mehta, A. J., 1975. A long term stability criterion for inlets on sandy coasts. Report UFL/COEL-1975/0018, Coastal and Oceanographic Engineering Laboratory, University of Florida, Gainesville. Mehta, A. J. and Christensen, B. A., 1983, Initiation of sand transport over coarse beds in tidal entrances. Coastal Engineering, 7, 61-75. OBrien, M. P., 1931. Estuary tidal prisms related to entrance area, Civil Engineering, 1(8), 738-739. OBrien, M. P., 1969. Equilibrium flow areas of inlets on sandy coasts. Journal of Waterways and Harbors Division, ASCE, 95(1), 43-52. OBrien, M. P. and Clark, R. P., 1973. Hydraulic constants of tidal entrances I: Data from NOS Tide Tables, Current Tables and navigational charts. Technical Report No. 21, Coastal and Oceanographic Engineering Laboratory, University of Florida, Gainesville. OBrien, M. P. and Dean, R. G., 1972. Hydraulic and sedimentary stability of coastal inlets. Proceedings of the 13th International Conference on Coastal Engineering, ASCE, Vol. 2, New York, 761-780. van de Kreeke, J., 1967. Water level fluctuations and flow in tidal inlets. Journal of Waterways and Harbors Division, ASCE, 93(4), 97-106. van de Kreeke, J., 1985. Stability of tidal inlets Pass Cavallo, Texas. Estuarine, Coastal and Shelf Science, 21, 33-43. van de Kreeke, J., 1988. Hydrodynamics of tidal inlets. In: Hydrodynamics and Sediment Dynamics of Tidal Inlets, Aubrey, D.G. and Weishar, L., eds, Springer-Verlag, New York, 1-23. van de Kreeke, J., 1990. Can multiple tidal inlets be stable? Estuarine, Coastal and Shelf Science, 30, pp 261-273.

PAGE 113

BIOGRAPHICAL SKETCH Ma m t a Jain was born in 1976, as the only daughter of Kanta and Mohan Jain in Kathmandu, Nepal. She did her bachelors in civil engineering from Delhi College of Engineering in 1998. After that she worked fo r three years as Ocean Engineer in the oil sector consultancy, Engineers India Ltd. Her main area of specialization is design of oil terminals. The craving for more knowledge made her take a break from the job and she applied to graduate school. In fall 2001 she was admitted to the Graduate School of the University of Florida. She married Parag Singal in December 2001, who encouraged and supported her to continue her academic work in the Coastal and Oceanographic Engineering Program of the Department of Civil and Coastal Engineering. 98


Permanent Link: http://ufdc.ufl.edu/UFE0000545/00001

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Title: Hydraulics and stability of mulitple inlet-bay systems: St. Andrew Bay, Florida
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Publication Date: 2002
Copyright Date: 2002

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Material Information

Title: Hydraulics and stability of mulitple inlet-bay systems: St. Andrew Bay, Florida
Physical Description: Mixed Material
Creator: Jain, Mamta ( Author, Primary )
Publication Date: 2002
Copyright Date: 2002

Record Information

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Holding Location: University of Florida
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HYDRAULICS AND STABILITY OF MULITPLE INLET-BAY SYSTEMS:
ST. ANDREW BAY, FLORIDA


















By

MAMTA JAIN


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2002




























Copyright 2002

by

Mamta Jain















ACKNOWLEDGMENTS

The author would like to express her deepest and heartiest thanks to her advisor

and chairman of the supervisory committee, Dr. Ashish Mehta, for his assistance,

encouragement, moral support, guidance and patience throughout this study. Special

thanks go to committee member Dr. Robert Dean for his help and advice in solving the

hydraulic model equations. Gratitude and thanks are also extended to the other members

of the committee, Dr. Robert Thieke and Dr. Andrew Kennedy, for their guidance and

assistance. Thanks go to Dr. J. van de Kreeke for his help in solving the linearized

lumped parameter model for the stability of inlets.

Assistance provided by Michael Dombrowski of Coastal Tech, for whom the

hydrographic surveys were carried out, is sincerely acknowledged. Thanks go to

Sidney Schofield and Vic Adams, for carrying out the fieldwork.

The author wishes to acknowledge the assistance of Kim Hunt, Becky Hudson,

and the entire Coastal and Oceanographic Engineering Program faculty and staff for their

encouragement and emotional support.

The author would like to thank her husband, Parag Singal, for his love,

encouragement and support, and her parents and family for providing her with mind,

body and soul.

Last, but not least, the author would like to thank the eternal and undying

Almighty who provides the basis for everything and makes everything possible.
















TABLE OF CONTENTS
page

A C K N O W L E D G M E N T S ......... .................................................................................... iii

L IST O F T A B L E S ......... .. .......................... ......... ... .............. .. vii

LIST OF FIGURES ......... ........................................... ............ ix

LIST OF SYMBOLS ......................................................... xi

ABSTRACT ........ .............. ............. ...... ...................... xiv

CHAPTER

1 IN TR OD U CTION .................. ............................ ............. .............. .

1.1 Problem D definition ................................................................... ... .. 1
1.2 O bjectiv e and T ask s ....................................................... ....... .... ...... ........ .. 4
1.3 T h esis O utlin e ..................................... .................. ..... ................ 4

2 HYDRAULICS OF A MULTIPLE INLET BAY SYSTEM............... ................5

2.1 Governing Equations of an Inlet-Bay System .......................................................... 5
2.1.1 System D definition ..................... ................. ...................... .............. 5
2.1.2 Energy Balance ........................................... ........ 6
2.1.3 C ontinuity E quation ............................................... ............... .............. 7
2.2 The Linearized M ethod ................................................................ .............. 9
2.3 M multiple Inlet-Bay System ......... ............ ... .......... ................. 11
2.3.1 Two Inlets and Two Bays with One Inlet Connected to Ocean.................... 11
2.3.2 Three Inlets and Two Bays with Two Inlets Connected to Ocean.............. 16
2.3.3 Three Inlets and Three Bays with One Inlet connected to Ocean ............... 19
2.3.4 Four Inlets and Three Bays with Two Inlets Connected to Ocean. .............. 24

3 STABILITY OF MULTIPLE INLET-BAY SYSTEMS.................... ..................29

3.1 Stability Problem Definition ............ .... ......... ........................ 29
3.2 Stability Criteria ..................................... ...... ... ................... 29
3.2.1 Stability Analysis for One-Inlet Bay System .......................................... 30
3.2.2 Stability of Two Inlets in a Bay .............. ...................................... ......... 32
3.3 Stability Analysis with the Linearized Model ............................. ............ 34
3.3.1 Linearized lumped parameter model for N Inlets in a Bay ........................... 35









3.4 Application to St. Andrew Bay System ....................................................... 40

4 APPLICATION TO ST. ANDREW BAY COMPLEX AND ENTRANCES.............42

4 .1 D description of Study A rea ...................................................................................... 42
4 .2 Sum m ary of F ield D ata .................................................................................. 44
4.2.1 B athym etry ............................................................................. 46
4 .2 .2 T id e s ............................................................................................ 4 8
4.2.3 Current and D ischarge............................................. .......................... 51
4.3 T idal P rism ............................ ............... ..... 52

5 RESULTS AND DISCUSSION........................................................ ............. 54

5 .1 Intro du action .................................................... .... ......... ...... 54
5.2 Hydraulics of St. Andrew Bay ................................................... ................ 54
5.2 .1 Solution of E qu ation s.......................................................... .... ................ 55
5.2.1.1 O ne-inlet one-bay system ........................ ........... ............... .... 55
5.2.1.2 Three inlets and three bays with one inlet connected to ocean............56
5.2.1.3 Three inlets and three bays with two inlets connected to ocean..........57
5.2.2 Input Param eters ................... .. .......................... .... ...... .. .. .. ...... .. 59
5.2.3 Model Results and Comparison with Data ............................................ 60
5.3 Stability A naly sis .. ................ ............. ................ ... ............... ....... .................. 62
5.3.1 Input Param eters ............................................. ........... ........ .. 62
5.3.2 R results and D iscu ssion ........................................................ .... .. .............. 63

6 C O N C L U SIO N S........ .......................................................................... .................73

6 .1 S u m m a ry ...................................................................................................... 7 3
6.2 Conclusions................... ............... ............................ 74
6.3 Recommendations for Further Work ................................................................... 74

APPENDIX

A ALGORITHMS FOR MULTIPLE INLET-BAY HYDRAULICS.............................76

A 1 Introduction .......................................... 76
A.2 Program- ................................................. ... ..... ..................... 76
A .3 P program -2............................. .............. ...... 77

B INLET HYDRAULICS RELATED DERIVATIONS ...............................................80

B .1 L inearization of D am ping Term ................................................... .................... 80
B .2 Shear Stress D ependence on A rea................................................ ... ................. 81
B.3 General Equation for hydraulic radius. ...................................................... 82
B .3.1 R ectangular ............................ .............. 83
B .3.2 Triangular ............................................................................. ................ 83
B.4 Hydraulic Radius for Triangular Cross-Section...................... ................ 83



v









C CALCULATION OF BAY TIDE AND LINEAR DISCHARGE COEFFICIENTS...85

D CALCULATIONS FOR STABILITY ANALYSIS .......................... .....................89

D 1 Intro du action ...................................................... ............... 89
D .2 Calculations ............................... .............. 89
D .2.1 E quilibrium velocity ................................................ .............. ... 89
D.2.2 Constant for Triangular schematization .................................. ................. 89
D.3 Relationship between Flow Curves and Stability of Two Inlets........................ 90
D .4 M atlab Program s .......................................................... .. .......... 91
D .4. 1 Program- .................................... .......................... .... ........ 91
D .4 .2 P program -2 ............................................................................. 93

LIST OF REFEREN CES ............................................................................. 96

B IO G R A PH IC A L SK E TCH ..................................................................... ..................98
















LIST OF TABLES


Table page

1.1 Cross-sectional areas of Johns Pass and Blind Pass in Boca Ciega Bay ..................

1.2 Cross-sectional areas of St. Andrew Bay Entrance and East Pass.............................3

1.3 Cross-sectional areas of Pass Cavallo and Matagorda Inlet .............. ..................3

4.1 Locations of St. Andrew Bay channel cross-sections ...........................................45

4.2 Locations of East Pass channel cross-sections...................... ...................45

4.3 Cross-section area, mean depths and width........................ .............................. 46

4.4 Tidal ranges in September 2001, December 2001 and March 2002 ..........................51

4.5 Phase lags between the stations and the ocean tide. .................................................51

4.6 Characteristic peak velocity and discharge values...........................................52

4.7 Flood and ebb tidal prism s................................................ ............................... 53

5.1 List of input and output parameters for one-inlet one-bay model. ...........................55

5.2 List of input and output parameters for the three inlets and three bays model............56

5.3 List of Input and Output Parameters for the four inlets and three bays model............58

5.4 Input parameters for the hydraulic model.................................. ..............59

5.5 M odel results and m easurem ents. ........................................ ........................... 60

5.6 Input param eters for stability analysis ....................................................................... 63

5.7 Effect of change in bay area and length of East Pass. ..............................................65

5.8 Stability observations for St. Andrew Bay Entrance and East Pass. .........................72

C. 1 Weighted-average bay tide ranges and phase differences..................... ................85

C.2 Calculation of (ro 7B1)max, (7B1 -7B2).max and (7 -17B3)max ...................... ............... 87









D 1 Calculation of equilibrium velocity ........................... ....... ............................... 89

D.2 Calculation of a, ............... ................... ............................ 89





















































viii
















LIST OF FIGURES


Figure p

2.1 O ne bay and one inlet system ............................................................................ 5

2.2 Two bays and two inlets with one inlet connected to ocean................... .......... 12

2.3 Two bays and three inlets, two inlets are connected to ocean. ...................................16

2.4 Three bays and three inlets with one inlet connecting to the ocean...........................21

2.5 Three bays and four inlets, two inlets connect to ocean. ...........................................25

3.1 Closure curves ....................... ......... ....................................................... ....31

3.2 Escoffier diagram ............... ................. ................... .............3.. 31

3.3 Closure surfaces. ................................................................33

3.4 Equilibrium flow curve for Inlet 2. ........................................ .......................... 33

3.5 Possible configurations of equilibrium flow curves for a two-inlet bay system..........34

3.6 Equilibrium flow curves for two inlets in a bay. ................................. ............... 41

4.1 Map showing the three bays and two inlets and bathymetry of the study area ..........43

4.2 Aerial view of St. Andrew Bay Entrance in 1993. Jetties are -430 m apart. ..............43

4.3 East Pass channel before it's opening in December 2001 ..................................44

4.4 St. Andrew Bay Entrance bathymetry and current measurement cross-sections.........46

4.5 Cross-section A in St. Andrew Bay Entrance.................. .......... ............... 47

4.6 Cross-section F in East Pass measured by ADCP ....................................... .......... 47

4.7 Measured tide in Grand Lagoon on Septemberl8-19, 2001....................................49

4.8 NOS predicted tide at St. Andrew Bay Entrance on Septemberl8-19, 2001. .............49

4.9 NOS predicted tide in St. Andrew Bay Entrance on December 18-19, 2001 ..............50









4.10 Tide at all selected NOS stations in March 2002..................... ............. ............... 50

5.1 Equilibrium flow curves for rectangular cross-sections, Run No. 1...........................66

5.2 Equilibrium flow curves for rectangular cross-sections, Run No. 2...........................66

5.3 Equilibrium flow curves for rectangular cross-sections, Run No. 3...........................67

5.4 Equilibrium flow curves for rectangular cross-sections, Run No. 4 ........................67

5.5 Equilibrium flow curves for rectangular cross-sections, Run No. 5.........................68

5.6 Equilibrium flow curves for rectangular cross-sections, Run No. 6 ...........................68

5.7 Equilibrium flow curves for triangular cross-sections, Run No. 7. ...........................69

5.8 Equilibrium flow curves for triangular cross-sections, Run No. 8. ...........................69

5.9 Equilibrium flow curves for triangular cross-sections, Run No. 9. ...........................70

5.10 Equilibrium flow curves for triangular cross-sections, Run No. 10 .........................70

5.11 Equilibrium flow curves for triangular cross-sections, Run No. 11 ........................71

5.12 Equilibrium flow curves for triangular cross-sections, Run No. 12.......................71

B 1 Trapezoidal Cross-section .......................................................... ............... 83

B .2 T riangular cross-section. ..................................................................... .................. 84

C.1 Head difference between ocean (Gulf) and bay ................................................. 88

C.2 Head difference between bay and bay 2....................................... ............... 88

D. 1 General configuration of equilibrium flow curve. ................... ............................. 90

D.2 General configuration of equilibrium flow curve. ................... ............................. 90
















LIST OF SYMBOLS


Symbols

AB, AB1, AB2, AB3

Ac Ac, Ac2, Ac3, Ac4

ao

aB, aB1, aB2, aB3

^B ^aB, aB2 B3


a,



a, b, c, A, B

B,

C, C1, C2, C3, C4

CD, CDL1, CDL2, CDL3, CDL4

CK

f

F

g

hk

i

K

k


bay water surface areas at MSL

flow cross-sectional areas of inlets

ocean (Gulf) tide amplitude

bay tide amplitudes

dimensionless bay tide amplitudes


constant that relates hydraulic radius with area of

triangular cross-section

constants defined to solve system of equations

dimensionless resistance factor

coefficients in linear relations of inlet hydraulics

linear discharge coefficients

prism correction coefficient of Keulegan

Darcy-Weisbach friction factor

friction coefficient

acceleration due to gravity

kinetic head

subscript specifying the inlet under consideration

Keulegan coefficient of filling or repletion

bottom roughness










ken

kex

Lc, L1, L2, L3, L4

m

P

Q, 01, Q2, 03, 04

Qm

R, R1, R2, R3, R4

Rt

Ro

ri, r2, r3

T

t

u

UB

Uc, Ucl, Uc2, Uc3, Uc4

Ueqi

Umaxl, Umax2, 1Umax3, 1max4

Uo

X

ao, aB

sB1, EB2, 8B3

5vl, 5v2, 5v3, 5v4


entrance loss coefficient

exit loss coefficient

channel lengths

sum of entrance and exit losses.

tidal prism

discharges through inlets

peak discharge

hydraulic radii

bay tide range

ocean (Gulf) tide range

polar representation of the bay tides

tidal period

time

velocity

bay current velocity

velocities through inlets

equilibrium velocity of inlet

maximum velocities through inlets

ocean (Gulf) current velocity

distance between UF and NOS tide stations

velocity coefficients

high water (HW) or low water (LW) lags

inlet velocity lags










Y

0

77











Seq


specific time when sea is at MSL

dimensionless time

water elevation

ocean (Gulf) tide elevation with respect to MSL

bay tide elevations with respect to MSL

dimensionless bay tide elevations

maximum bottom shear stress

equilibrium shear stress















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

HYDRAULICS AND STABILITY OF MULTIPLE-INLET BAY SYSTEMS:
ST. ANDREW BAY, FLORIDA

By

Mamta Jain

December 2002


Chairman: Ashish J. Mehta
Major Department: Civil and Coastal Engineering

Tidal inlets on sandy coasts are subject to the continuous changes in their

geometry and as a result influence shorelines in the vicinity. Since engineering

modifications carried out at one inlet can affect the long-term stability of others in the

vicinity of the modified inlet, it is important to understand the stability of all inlets

connecting a bay to the ocean. Inlet stability is related to the equilibrium between the

inlet cross-sectional area and the hydraulic environment.

St. Andrew Bay on the Gulf of Mexico coast of Florida's panhandle is part of a

three-bay and two- ("ocean") inlet complex. One of these inlets is St. Andrew Bay

Entrance and the other is East Pass, both of which are connected to St. Andrew Bay on

one side and the Gulf on the other. Historically, East Pass was the natural connection

between the bay and the Gulf. In 1934, St. Andrew Bay Entrance was constructed 11 km

west of East Pass to provide a direct access between the Gulf and Panama City. Due to

the long-term effect of this opening of St. Andrew Bay Entrance, East Pass closed









naturally in 1998. A new East Pass was dredged open in December 2001, and the

objective of the present study was to examine the hydraulics and stability of this system

of two sandy ocean inlets connected to interconnected bays.

To study the system as a whole, a linearized hydraulic model was developed for a

three-bay and four-inlet (two ocean and the other two connecting the bays) system and

applied to the St. Andrew Bay system. To investigate the stability of the ocean inlets, the

hydraulic stability criterion was extended to the two-ocean inlets and one (composite) bay

system using the linearized lumped parameter model. The following conclusions are

drawn from this analysis.

The linearized hydraulics model is shown to give good results--the amplitudes of

velocities and bay tides are within 5%. The percent error for St. Andrew Bay is almost

zero, and for the other bays it is within 20%.

The stability model gives the qualitative results. The bay area has a significant

effect on the stability of the two inlets. At a bay area of 74 km2 (the actual area of the

composite bay), both inlets are shown to be unstable. Increasing the area by 22% to 90

km2 stabilizes St. Andrew Bay Entrance, and by 42% to 105 km2 stabilizes East Pass as

well. Keeping the bay area at 105 km2 and increasing the length of East Pass from 500 m

to 2000 m destabilizes this inlet because as the length increases the dissipation in the

channel increases as well.














CHAPTER 1
INTRODUCTION

1.1 Problem Definition

Tidal inlets are the relative short and narrow connections between bays or lagoons

and the ocean or sea. Inlets on sandy coasts are subject to the continuous changes in their

geometry. Predicting the adjustment of the inlet morphology after a storm event in

particular, i.e., whether the inlet will close or will remain open, requires knowledge of the

hydraulic and sedimentary processes in the vicinity of the inlet. These processes are

governed by complex interactions of the tidal currents, waves, and sediment. In spite of

recent advances in the description of flow field near the inlet and our understanding of

sediment transport by waves and currents (Aubrey and Weishar 1988), it is still not

possible to accurately predict the morphologic adjustment of the inlet to hydrodynamic

forcing.

Inlet stability is dependent upon the cumulative result of the actions of two

opposing factors, namely, a) the near-shore wave climate and associated littoral drift, and

b) the flow regime through the inlet. Depending on the wave climate and the range of the

tide, one of these two factors may dominate and cause either erosion or accumulation of

the sand in the inlet. However, on a long-term basis, a stable inlet can be maintained only

if the flow through the inlet has enough scouring capacity to encounter the obstruction

against the flow due to sand accumulation, and to maintain the channel in the state of

non-silting, non-scouring equilibrium. If such is not the case and waves dominate, then

the accumulated sand will begin to constrict the inlet throat, thereby reducing the tidal









prism. The resulting unstable inlet may migrate or orient itself at an angle with the

shoreline depending on the predominant direction of the littoral drift; the channel may

elongate, thereby increasing the frictional resistance to the flow, and finally, a stage may

be reached when perhaps a single storm could close the inlet in a matter of hours.

Stability criteria based on inlet hydraulics and sediment transport for single inlets

have been proposed by, among others, O'Brien (1931), Escoffier (1940), O'Brien and

Dean (1972), Bruun (1978) and Escoffier and Walton (1979). All criteria assume that

sufficient sand is available to change the inlet channel geometry in response to the

prevailing hydrodynamic conditions. These investigators found various stability

parameters to describe the stability of the inlet. It should be noted, however, that while it

is relatively easy to deal with the stability of single inlets, the problem becomes complex

when, as is commonly the case, more than one inlet connect the ocean to a single bay or

more than one interconnected bays. Some examples of such systems are as follows.

Three cases of the history of two inlets in a bay are worthy of citation. One case is

that of Boca Ciega Bay on the Gulf coast of Florida, where the co-dependency of two

inlets, Blind Pass and Johns Pass, appears to be reflected in the history of their cross-

sectional areas. While Blind Pass has historically been narrowing due to shoaling, John's

Pass has been increasing in size, as shown in Table 1.1. As a result, Blind Pass now

requires regular dredging for its maintenance while severe bed erosion has occurred at

John's Pass (Mehta, 1975; Becker and Ross, 2001).

Another example is that of St. Andrew Bay Entrance and the East Pass. As

mentioned previously, East Pass used to be a large inlet and was the only natural

connection between the Gulf of Mexico and the St. Andrew Bay. In 1934, St. Andrew









Bay entrance was constructed 11 km west of East Pass through the barrier island by the

federal government to provide a direct access between the Gulf and Panama City. Table

1.2 gives the cross-sectional area of each inlet over time.

Table 1.1 Cross-sectional areas of Johns Pass and Blind Pass in Boca Ciega Bay
Year Area (m2) Hydraulic Radius (m)
John's Pass Blind Pass John's Pass Blind Pass
1873 474 538 3.7 3.5
1883 432 496 3.8 3.0
1926 531 209 3.9 1.9
1941 636 225 4.1 1.4
1952 849 157 4.6 2.7
1974 883 411 4.9 1.6
1998 950 230 5.3 0.9d
d Estimated by assuming no change in channel width since 1974.

Table 1.2 Cross-sectional areas of St. Andrew Bay Entrance and East Pass
Year Area (m2)
St. Andrew Bay Entrance East Pass
1934 1,835 3,400
1946 3,530 2,146
1983 3,943 1,392
1988 Closed
2001 5,210 Reopened

The third example is that of Pass Cavallo and Matagorda Inlet connecting

Matagorda Bay, Texas, to the Gulf. Stability analysis carried out by van de Kreeke

(1985) on this system showed that Pass Cavallo is an unstable inlet, which is decreasing

in cross-section, whereas Matagorda Inlet is increasing in size. The areas of cross-

sections of the two inlets are listed in Table 1.3.

Table 1.3 Cross-sectional areas of Pass Cavallo and Matagorda Inlet
Year Area (m2)
Pass Cavallo Matagorda Inlet
1959 8,000 Closed
1970 7,500 3,600

The above sets of complex problems are dealt with in this study in a simplified

manner, with the following objective and associated tasks.









1.2 Objective and Tasks

The main objective of this study is to examine the hydraulics and thence the

stability of a system of two sandy ocean inlets connected to interconnected bays. The

sequence of tasks carried out to achieve this goal is as follows:

1 Deriving the basic hydraulic equations using the linearized approach for a
complex four inlets and three bays system.

2 Solving these equations, applying them to the St. Andrew Bay system, and
comparing the results with those obtained from the hydrographic surveys.

3 Developing stability criteria using the basic Escoffier (1940) model for one inlet
and one bay and then extending this model to the two inlets and a bay.

4 Carrying out stability analysis for N inlets and a bay using the linearized lumped
parameter model of van de Kreeke (1990), and then applying it to the St. Andrew
Bay system.

1.3 Thesis Outline

Chapter 2 describes the hydraulics of the multiple inlet-bay system. It progresses

from the basic theory to the development of linearized models for simple and complex

systems. Chapter 3 describes the stability of the system, including an approximate

method to examine multiple inlets in a bay. Chapter 4 includes details of hydrographic

surveys and summarizes the data. Chapter 5 discusses the input and output parameters

required for the calculation. It also presents the results. All calculations are given in the

appendices. Conclusions are made in Chapter 6, followed by a bibliography and a

biographical sketch of the author.














CHAPTER 2
HYDRAULICS OF A MULTIPLE INLET BAY SYSTEM

2.1 Governing Equations of an Inlet-Bay System

2.1.1 System Definition

The governing equations for a simple inlet-bay system may be derived by

considering the inlet connecting the ocean and the bay as shown in Figure 2.1.




Bay

AB








11o Ocean
Figure 2.1 One bay and one inlet system

These equations are derived subjected to the following assumptions.

1 The inlet and bay banks are vertical.

2 The range of tide is small as compare to the depth of water everywhere.

3 The bay surface remains horizontal at all times, i.e., the tide is "in phase" across
the bay. That means the longest dimension of the bay be small compared to the
travel time of tide through the bay.

4 The mean water level in the bay equals that in the ocean.

5 The acceleration of mass of water in the channel is negligible.

6 There is no fresh water inflow into the bay.









7 There is no flow stratification due to salinity.

8 Ocean tides are represented by a periodical function.

2.1.2 Energy Balance

Applying the energy balance between ocean and bay one gets

2 2
7o +ao ,= r7 +BaU +B Ah (2.1)
2g 2g

where

ro = Ocean tide elevation with respect to mean sea level,

7B = Bay tide elevation with respect to mean sea level,

Uo = Ocean current velocity,

UB = Bay current velocity,

a, and as = Coefficients greater than one which depend on the spatial distribution

of Uo and UB, respectively,

EAh = Total head loss between the ocean and the bay, and

g = acceleration due to gravity.

It is also assumed that ocean and bay are relatively deep; thus Uo and uB are small enough

to be neglected. Then Eq. (2.1) becomes

Ah = 7o -rB (2.2)

There are generally two types of head losses. One includes concentrated or

"minor losses" due to convergence and divergence of streamlines in the channel. The

second type is gradual loss due to bottom friction in the channel. The entrance and exit

U2
losses may be written in terms of the velocity head in the channel, with the entrance
2g


loss coefficient ken and the exit loss coefficient kex, i.e.,









/2

2g

2
Exit loss = ke (2.4)
2g

where uc is the velocity through the inlet. Gradual energy losses per unit length depend

on the channel roughness and are given in form of Darcy-Weisbach friction factor


Gradual loss = (2.5)
4R 2g

where

f= Darcy-Weisbach friction coefficient,

R = hydraulic radius of channel, and

L = Length of channel.

Substitution of Eqs. (2.3), (2.4) and (2.5) into (2.2) gives


7o U k + x + (2.6)
2g 4R

or


uc= 2g o l .sign(o B) (2.7)
ke +k +
4R

The sign( ro-r/B) term must be included since the current reverses in direction every half

tidal cycle.

2.1.3 Continuity Equation

The equation of continuity, which relates the inlet flow discharge to the rate of

rise and fall of bay water level, is given as











dt
=ucAc = d(4,B) (2.8)


where

Q = flow rate through the inlet,

Ac = Inlet flow cross-sectional area, and

AB = bay surface area.

Therefore Eq. (2.8) becomes


u = Bd (2.9)
A dt

Eliminating uc between Eq. (2.7) and (2.9) leads to

dB= Ac 2g 1lo- .sign(o- ) (2.10)
dt AR k + kex +f
S 4R

Next, we introduce the dimensionless quantities


io = o; 'L r O 2zt = t (2.11)
a a T
0 0

where ao = ocean tide amplitude (one-half the ocean tidal range), T= tidal period and

a = tidal (angular) frequency. Substitution into Eq. (2.10) gives


d =i K 4|o iOn.sign(i -r0 ) (2.12)
dO

where

T Ac 2 ga
K= TA 2ga (2.13)
2zao AB ke +k
4R









in which K is referred to as the "coefficient of filling or repletion" (Keulegan, 1967).

Keulegan solved the first order differential equation, Eq. (2.12), for b, in terms of the

repletion coefficient K and dimensionless time using numerical integration.

2.2 The Linearized Method

A linear method was suggested by Dean (1983) for solving Eq. (2.12). For this

approach it was assumed that the velocity u, in Eq. (2.7), is proportional to the head

difference ()r7 -re) rather than the square root of the head difference, according to


u = CDL o, J cB ) (2.14)


where CDL = "linear discharge coefficient." This coefficient is defined as


CDL = 1 a (2.15)
ke +k +f (7o -B )max
4R

where (77 -r7B),ax is the maximum head difference across the inlet. Now, combining Eqs.

(2.14), (2.9) and (2.11), Eq. (2.12) can be written in terms of the linear relationship as

r drB 7
G-o 7- I (2.16)
C dO

where


C = CD 2 (2.17)
C DL A ao


Under assumption (8) the ocean tide is assumed to be periodic. Because of the

linear assumption the bay tide is also periodic, it can be written as

,o = cos (2.18)


(2.19)


r), = aS cos(0 E1)









a
where a= aB = one-half the bay tide range (i.e., bay tide amplitude) and Ea = lag
ao

between high water (HW) or low water (LW) in the ocean and the corresponding HW or

LW in the bay.

Eq. (2.18) and Eq. (2.19) are next substituted into Eq. (2.16) and the following

complex number technique is used to solve for aB and SB:

1 Define the following constants:


C=a, o =Re el)
C

2 Let the following variables be represented in the polar form:

n, =Re(a e' "B -r

3 Therefore

diB
-= ir
dO

4 So the equations are reduced to

1= (1 + ai)r (2.20)

1 1 -a
r = Re(r,) = Im(r, --
1+ai 1 +a 1+a2

where

Re(ri) = is the real part of the solution, and

Im(ri) = is the imaginary part of the solution.

The magnitude of ri represents aB1 and the phase lag aB1 is represented by the angle of ri:

1
aB = -- (2.21)


(2.22)


EB = tan a









The velocity uc through inlet 1 is therefore given by

Uci = Umaxi Cos(0 ,) (2.23)

where Umax is the maximum velocity through inlet 1, evi is the phase lag between the

velocity in inlet 1 and HW or LW in the ocean.

Substituting for ro and r7B1 from Eq. (2.18) and Eq. (2.19) in Eq. (2.14) and

combining Eqs. (2.23) and (2.14) we get the required expression for Umaxl. It should be

noted that velocity is out of phase with respect to displacement by r/2. Therefore, vi =

SB 1-z/2.

2.3 Multiple Inlet-Bay System.

2.3.1 Two Inlets and Two Bays with One Inlet Connected to Ocean

In the case of two bays with one inlet connecting to the ocean and the second

connecting the bays as shown in Figure 2.2, the eight assumptions mentioned in section

2.1.1 and the linear relationship both hold. In a manner similar to that employed for a

single inlet-bay case, the velocity relationship and the equation of continuity for two-bay

system may be written with reference to the notation of Figure 2.2.

Thus the following relationships are obtained:


U =c CDL1 (ro rB1) (2.24)


dul duB2
Q, = uclAc = ABI + AB2 (2.25)


UC2 = gCDL B-(B2) (2.26)


Q2 = Uz2Ac2 = A2 d(2.27)
dt





















/0o Ocean
Figure 2.2 Two bays and two inlets with one inlet connected to ocean.

where

uc1, uc2 = velocities through the inlets 1 and 2,

Qi, Q2 = discharges through inlets 1 and 2,

Ac~, Ac2 = inlet flow cross-sectional areas, and

AB1, AB2 = bay water surface areas.


CL1 a (2.28)
1 ( o B1)max
+k + ( -L
e, x 4R,

1 a
CDL2 = 1 a (2.29)
k +k + A2 (--7 2)max
en e4R


where

L1, L2 = inlet lengths, and

R1, R2 = hydraulic radii of the channels.

Eliminating uc between Eq. (2.24) and Eq. (2.25) gives


7 17 I dA, AB2 dBt (2.30)
C, dt ABI dt


where










A /2g
CI = CDLI c, g (2.31)
A4, Va

Combining Eq. (2.26) and Eq. (2.27) yields


S- 7B2 = [dB (2.32)


where


AC2 2g (2.33)
C2 = CDL2 g


The dimensionless ocean tide is given by Eq. (2.18), and the dimensionless tides

in bays 1 and 2 now become

~7s = a1 l cos(O- Es,) (2.34)

7B2 = aB2 cos(O- )B2 (2.35)

where

a =
aBl
a

aBl = one-half the tide range (i.e., amplitude) in bay 1.

SaB2
aB2 B2
ao
0a

aB2 = one-half the tide range (i.e., amplitude) in bay 2.

B1 = lag between high water (HW) or low water (LW) in the ocean and

corresponding HW or LW in the bay 1.

EB2 = lag between high water (HW) or low water (LW) in the ocean and

corresponding HW or LW in the bay 2.

Eq. (2.30) and Eq. (2.32) can be expressed in the dimensionless form as










a- F= g i d, A, dB2 (2.36)
o0 Rl C, dO A4, dO (


C -7 F drB2 (2.37)
71 R2 dO

The above equations are solved by the matrix method assuming the variables to be

complex numbers. The solution is obtained as follows:

1 Define the following constants

a" a" AB2
=a, =b =A, o =Re e')
C1 C2 AB

2 Let

dB1 i, dRB2
71, = Re(aBel B1))=r, ,B =Re(aBe'(O 'B2 ) =r, =i, dO
Ri ) '7RdO6 dO

3 So the equations are reduced to

1 = (ai + 1)r, + aAir2 (2.38)

0 = -r, +(bi + )r, (2.39)

4 In the matrix form they become

S (r ai +1b aAi
I= rj l i (2.40)
0 r2 -1 bi+ 1

5 The solution is

-i(b-i)
r, = (2.41)
X

-1
r2 = (2.42)
r X

where


X = (ab-1)-i(aA+a+b); X =(ab-1)+i(aA+a+b)









XX = (ab -1)2 +(aA + a + b)2

b((b+a+aA)-(ab-1) -b(ab-1)-(a+b+aA)
Re(r=) =; Im(r)=
XX XX

-(ab 1) (a + b + aA)
Re(r, )= ; Im(r, )=
XX XX

The amplitudes (aB and ., ) of bays 1 and 2 are the magnitudes of the complex

numbers rl and r2 and the corresponding phase lags are the angles of the complex

numbers:

a1 = Re(t)2 +Im()2 (2.43)


S= -tan' (2.44)
Re(r1i)

S=Re(r2 + Im(r)2 (2.45)

1Im(r, )(
EB2 -tan1 Im(r2) (2.46)
Re(r2) )

The velocities uc and uc2 through inlets 1 and 2, respectively, are therefore given by

Uc1 = Umaxi Cos(0 ,) (2.47)

U2 = Umax2 COS(0 E,2) (2.48)

where Umaxi and Umax2 are the maximum velocities through inlets 1 and 2, respectively, evi

and 82 are the phase lags between the velocity in inlet 1 and HW or LW in the ocean, and

in inlet 2 and HW or LW in the ocean.

Substituting for ro and 7B1 from Eqs. (2.18) and (2.34) in Eq. (2.24) and

combining Eqs. (2.47) and (2.24) we get the required expression for Umaxl. Similarly we









can obtain the expression for Umax2. It should be noted that velocity is out of phase with

respect to displacement by z-/2. Therefore, v1 = EB1--/l2 and. v2 = SB2-/2.

2.3.2 Three Inlets and Two Bays with Two Inlets Connected to Ocean

The inlet bay system is defined in Figure 2.3. In this system two bays are

connected to each other with inlets 2 and inlet 3 and 1 connects bay 1 to the ocean.




AB1 I AB2
I7B 1 77B2






rio
L2

L,1 L3



Figure 2.3 Two bays and three inlets, two inlets are connected to ocean.

The velocity in inlets 1 and 2 is given by Eq. (2.24) and Eq. (2.26) respectively.

The velocity in inlet 3 is given by Eq. (2.49):


Uc3 = 2 g L3 o -1) (2.49)


where Uc3 = velocity through the inlet 3 and


DL3 a (2.50)
k + k + 3 (7, -7 1)max
c e4R3

where

L3 = inlet 3 length, and

R3 = hydraulic radius of inlet 3 channel.

The governing equations of continuity are









Q1 +3 = uclAc +uc3A,3 = Ad AB2 dB2 (2.51)
dt dt


Q2 = 2A2 = AB2 dB2 (2.52)
dt

where

Qi, Q2, Q3 = discharges through inlets 1, 2 and 3,

Ac1, Ac2, Ac3 = flow cross-sectional areas at inlets 1, 2 and 3, and

AB1, AB2 = bay water surface areas.

Substituting for the velocity expressions in the above equations we obtain

1 do7, AB2 d17B2 (2.53)
ro- = r1 dr +4 + (2.53)
CI +C3, dt AB, dt


I 7B2 B2 (2.54)


where C1 and C2 are expressed by Eqs. (2.31) and (2.33), and


C3 = DL3 (2.55)
ABI VFo

Stating Eqs. (2.53) and (2.54) in the dimensionless form we obtain


7o = g : dj +i AB2 d B2 (2.56)
S C,+C3 dO A1 dO


l -2= [dB22 (2.57)
C2 dO

where oj, sB1 and RB2 are defined in Eqs. (2.18), (2.34) and (2.35), respectively. The

solution of the system of Eqs. (2.56) and (2.57) is given below.









1 Define the following constants
C C AB
=a, -=b, =A, o =Re(e")
C, +C3 C2 A B

2 Let


= Re(e( e' B= r,, 2= Re e'"B2)= r2; d =i dB- i2

3 So the equations are reduced to

1= (ai + 1) + aAir2 (2.58)

0 = -r +(bi + )r, (2.59)

4 Solve these equations by the matrix method.

1 (r ai +1 aAi
o rbi + ) (2.60)
0 r2 -1 bi+l

5 Solving the above equations yields

-i(b-i)
i, = (2.61)
X

-1
r = (2.62)
X

X = (ab -1)- i(aA+a+b); X = (ab-1)+i(aA+a+b)

XX = (ab- 1)2 +(aA + a + b)2

b( (b+a+aA)-(ab-1) -b(ab-1)-(a+b+aA)
Re(r) =; Im(r)=
XX X

S-(ab 1) ) -(a+ b + aA)
Re(r,) = m(r,) =

The amplitudes (aB1 and ) of bays 1 and 2 are the magnitudes of the complex

numbers rl and r2 and the phase lags are the corresponding angles:









B =Re(t +Im() (2.63)


'B = -tan- 111m() (2.64)
lRe(rj)


a^B = Re(r2) + Im(r)2 (2.65)


B2 tan 1 (I(r2) (2.66)
Re(r2))

The velocities uc and Uc2 through inlets 1 and 2, respectively, are given by Eqs.

(2.47) and (2.48), and u,3 through inlet 3 is obtained from

"c3 = 2max3 COS ( v,3) (2.67)

where Umax3 is the maximum velocity through inlet 3 and 8v3 is the phase lag between

velocity in inlet 3 and HW or LW in the ocean. Substituting for ro and r7 from Eqs.

(2.18) and (2.34) into Eq. (2.49) and combining Eqs. (2.49) and (2.67) we get the

required expression for Umax3. Then the phase lag Ev3 = 8B1-i2.

2.3.3 Three Inlets and Three Bays with One Inlet connected to Ocean.

This inlet bay system as defined in Figure.2.4 has three interconnected bays with

inlets 2 and 4, while inlet 1 connects bay 1 to the ocean. The velocities in inlets 1 and 2

are given by Eqs. (2.24) and (2.26), respectively. The velocity in inlet 4 is given by Eq.

(2.68):


4 = 1CDL4 (B1 B3) (2.68)


where Uc4 = velocity through the inlet 4 and










S1 a/I
n +k L4 (7B1 7B3)m,,
4R4

with L4 = inlet length and R4 = hydraulic radius of inlet 4 channel.

The governing continuity equations are

dr]B drB3
Q1 = u1Ac, = A, + AB2 + A, (2.69)
at at dt


Q2 = UcAc2 = A2 d (2.70)
dt


Q4 = uc4Ac4 = A3 dB3 (2.71)
dt

Q1, Q2, Q4 = discharges through inlets 1, 2 and 4

Ac1, Ac2, Ac4 = flow cross-sectional areas at inlets 1, 2 and 4.

AB1, AB2, AB3 = bay water surface areas.

Substituting the velocity expressions in the above equations we obtain

1 =I dB, + AB2 dB2 + AB3 d;B3J (2.72)
o m C, dt A4, dt A,1 dt


1B O = 2'- drh (2.73)
C2 L dt J


B -1 dr=B3 ] (2.74)
=C4 dt

where C1 and C2 are as expressed by Eqs. (2.31) and (2.33), and


S DL4 FA 2 (2.75)
C4 = CDL4 g
4B3 B ,i































rio
Figure 2.4 Three bays and three inlets with one inlet connecting to the ocean.

Stating the above equations in the dimensionless form the desired solution is obtained by

solving the following three equations:

S 7- dB AB2 dBR2 + AB3 dB3 (2.76)
SC, dO A, dO AB, dO


o-i- [2 B2 (2.77)
B2 dO


B- lB3 dB3 (2.78)
C4 dO

where uoj, 'B1 and 7B2 are defined by Eqs. (2.18), (2.34) and (2.35), respectively, and

7B3 is

=B3 = B3 cos( CB3) (2.79)

As before the above equations are solved by using complex numbers as follows:









1 Define the following constants
C C C AR2 AB
a AA, AB B, Re(e"0)
-=a, -= b, -=c, B2 =A, =B, o =Re(e' )
C, C2 C4 A,, A I

2 Let

B, = Re(el rB1))= r, B2= =Re(e'( B2))= r, 7q,3 = R(el( B)--)= r3

d B1 dB 2 di 3
dO = dO dO 3

3 So the equations are reduced to

1 = (ai +l)r, + aAir2 +aBir, (2.80)

0= -r, +(bi + 1)r +Or3 (2.81)

0 = -r, + Or + (ci +1)r, (2.82)

4 Solving the equations by matrix method:

1' (r ai +1 aAi aBi
= r -1 bi+1 0 (2.83)
S r3 -1 0 ci +1

yields

(c -i)(b -i)
r, = (2.84)
X

-i(c-i
r = (2.85)


r3 = (2.86)
X

X = (ac + ab + bc + aBb + aAc -1)+i(-aA + abc -a-b-c- aB)

X = (ac + ab + bc + aBb + aAc -1)- i(-aA + abc -a-b-c- aB)

XX = (ac + ab + bc + aBb + aAc -1)2 + (-aA + abc -a-b-c- aB)2









b2( + aBc+c2)+c2 (1aAb)+a(bA+Bc)+1
Re(r,) =
XX

-a( +A+B+ + +c2 + bc +Bbc + Ac2 +Bb bcB)
Im(r,) =
xA

-Re ab -aBb + aBc +1+ c2 abc2
Re(r,) = -


m(r2 -(a+b+aA+aB+aBbc + ac2 +bc2 +aAc2)
Im(r, ) =
xA

Re -ac aAc + abA ab2c + b +1
Re(r,) =
XX


imr -(a+c+aA+aB+abcA+ab2 +cb2 +aBb2)
Im(r?,) =
xA

The amplitudes (aB, a2 and aB) of bayl, bays 2 and 3 are the magnitudes of the

complex numbers rl, r2 and r3, and the corresponding phase lags are the angles of the

complex numbers:

aB1 =Re(r )+Im()2 (2.87)


B = tan Im(rj) (2.88)
lRe(r))


aB2 = FRe(r2) + Im(r2) (2.89)

Im (Jmr)
2 -tan1 I (2.90)
Re(r2)


aB3 = Re(r) + Im(r3) (2.91)


B3 =-tan- 11(r3) (2.92)
Re(r3)









The velocities uc and uc2 through inlets 1 and 2, respectively, are given by Eqs. (2.47)

and (2.48), and uc4 through inlet 4 is given by

Uc4 = cmax4 COS(O E,4) (2.93)

where Umax4 is the maximum velocity through inlet 4 and vE4 is the corresponding phase

lags between this velocity and HW or LW in the ocean. Substituting for B1 and q7B3 from

Eqs. (2.34) and (2.79)into Eq. (2.68) and combining Eqs. (2.93) and (2.68) we get the

desired expression for Umax4. Phase lag Ev4 = SB3-ZI2.

2.3.4 Four Inlets and Three Bays with Two Inlets Connected to Ocean.

This system as defined in Figure 2.5 has three interconnected bays with inlets 2

and 4, while and inlets 1 and 3 connect bay 1 to the ocean. The velocities in inlets 1, 2, 3

and 4 are given by Eqs. (2.24), (2.26), (2.49) and (2.68), respectively.

The governing continuity equations are written as follows.

1 + 3 = ulAc + uA3 = AB dB + A B2 + AB dB3 (2.94)
dt dt dt


Q2 = cAc2 = A2 d (2.95)
dt


Q4 = Uc4c4 = AB3 (2.96)
dt

Next, substituting the velocity expressions in the above equations yields


o B1 = d + B1 c AR d7 + AR dt (2.97)
C,+C, dt AB, dt AB1 dt


BI 1 = d B2 (2.98)
C B2 L dt ]






























r77o
Figure 2.5 Three bays and four inlets, two inlets connect to ocean.


7B 7B3 = LdBJ (2.99)


where C1, C2, C3 and C4 are as expressed by Eqs. (2.31), (2.33), (2.55) and (2.75),

respectively.

Now we may state the above equations in the dimensionless form as

dFiB AB2 diB2 A3 di B3
-+ (2.100)
S C, +C, dO A, dO A ,, dO


B -B2 7 drB2 (2.101)
C2L dO]

B OB d=LB3 (2.102)
C4 dO

where o, 71I, jB2 and 7B3 are defined by Eqs. (2.18), (2.34), (2.35) and (2.79),

respectively. These equations are solved as follows:









1 Define the following constants

a O O AB2 A B
=a, = b, =c, = A, 3 B, = Re(e")
C, +C3 C2 4 IA, AI

2 Let

B, = Re(e )=B1 ,, B2 = Re(e e(B2o)- 2) =, B =Re(el(e B )=r- 3

di .= diB2 d B3
dO dO 2' d 3

3 So the equations are reduced to

1 = (ai + )r, + aAir2 +aBir3 (2.103)

0 = -r, + (bi + 1)r2 + Or3 (2.104)

0 =-r, + Or2 +(ci +)r3 (2.105)

4 Solve these equations by matrix method:

1' r, r ai +1 aAi aBi
= r2 bi+1 0 (2.106)
S r3 -1 0 ci +1

5 Thus we obtain

(c -i)(b- i)
r = (2.107)
X

-i(c-i
r2 = (2.108)
X


r3 = (2.109)
X

X = (ac + ab + bc + aBb + aAc -1)+i(-aA + abc -a-b-c- aB)

X = (ac + ab + bc + aBb + aAc -1)- i(-aA + abc -a-b-c- aB)









XX = (ac + ab + bc + aBb + aAc -1)2 +(-aA + abc -a-b-c- aB)2

b2 (1+aBc+c2)+ c2 (1+aAb)+a(bA +Bc)+l
Re(ri) =
XX

-a(l + A + B+b2 + 2 + c2 + Bb2 +B Ac2 + Bb bcB)
Im()-) =
xA

-R ) ab aBb + aBc +1+ c2 -abc2
Re(r,) = -


m(r2 -(a + b + aA + aB + aBbc + ac2 +bc2 +aAc2)
Im(r ) =
XX
xA

Re -ac aAc + abA ab2c + b +1
Re(r ) =
XX

-(a+c+aA+aB+abcA+ab2 +cb2 +aBb2)
Im(r,) =
xA

The amplitudes (aB, a2 and aB) ofbaysl, 2 and 3 are the magnitudes of the complex

numbers rl, r2 and r3, and the corresponding phase lags are the angles of the complex

numbers:

aB1 = Re( +Im()2 (2.110)


B = tan -'Im(Q) (2.111)
lRe(i))


aB = Re(7) + Im(() (2.112)


B2 =-tan (11Im(r2 (2.113)
Re(r2 )

caB3 = Re(r) + Im(r)2 (2.114)






28


EB3 -tan -' 1 I (2.115)
Re(r3)

Then the velocities ucl, uc2, Uc3 and uc4 are given by Eqs. (2.47), (2.48), (2.67) and

(2.93), respectively.














CHAPTER 3
STABILITY OF MULTIPLE INLET-BAY SYSTEMS

3.1 Stability Problem Definition

An inlet is considered stable when after a small change the cross-sectional area

returns to its equilibrium value. Each inlet is subject to two opposing forces, the waves on

one hand, which tend to push sand into the inlet, and the tidal current on the other hand,

which tries to carry sand out of the channel back to the sea or the bay. The size of the

inlet and its stability are determined by the relative strengths of these two opposing

forces.

3.2 Stability Criteria

Inlet stability as considered here basically deals with the equilibrium between the

inlet cross-section area and the hydraulic environment. The pertinent parameters are the

actual tide-maximum bottom shear stress i and the equilibrium shear stress i eq. The

equilibrium shear stress is defined as the bottom stress induced by the tidal current

required to flush-out sediment carried into the inlet. When i equals i eq the inlet is

considered to be in equilibrium. When i is larger than i eq the inlet is in the scouring

mode, and when i is smaller i eq the inlet is in the shoaling mode. The value of

equilibrium shear stress depends on the waves and associated littoral drift and sediment.

Considering inlets at equilibrium on various coasts, Bruun (1978) found the value of

equilibrium stress in fairly narrow range:

3.5Pa < iq < 5.5Pa









The value of actual shear stress is obtained from

ZT= PFUmax Umax 1 (3.1)

where F is the friction coefficient, a function of bottom roughness, k, Um,, is the

maximum tidal velocity in the inlet, a function of area and length of the inlet, as

discussed in Chapter 2 and p is the fluid density. Therefore, i can be written as a

function of following form

i = f(A, L, k, m)

where m is the sum of entrance and exit losses. The plotted function i(A) is called a

closure curve, as shown in Figure 3.1. It is clear from the calculation shown in the

Appendix B that i is a strong function of A and a weak function of L, m, k. The strong

dependence of i on A explains why inlets adjust to changes in the hydraulic environment

primarily via a change in the cross-sectional area.

3.2.1 Stability Analysis for One-Inlet Bay System

Making use of the Escoffier (1940) diagram, Figure 3.2, one can study the

response of the inlet to change in area. In the Figure, Ai and An both represent equilibrium

flow areas, with Ai representing unstable equilibrium and An representing stable

equilibrium. If the inlet cross-sectional area A were reduced but remained larger than AI,

the actual shear stress would be larger than the equilibrium shear stress and A would

return to the value An. If the cross-sectional area were reduced below Ai, the shear stress

would become lower than its equilibrium value and the inlet would close. If A became

larger than AH, the actual shear stress would become larger than equilibrium value and A

would return to A,,. Note that the equilibrium condition only exists if the line z= req


intersects the closure curve i= i(A).













7->





A
Figure 3.1 Closure curves (source: van de Kreeke, 1985)





A*


AI A,
Unstable Stable A
equilibrium equilibrium
Figure 3.2 Escoffier diagram (source: van de Kreeke, 1985)









The equilibrium interval for the stable cross-section, AII, ranges from Ai to infinity.

3.2.2 Stability of Two Inlets in a Bay

Similar to a single inlet, it can be shown that shear stresses r, and for two

inlets in a bay strongly depend on A1 and A2 and are weak functions of (Li, kl, mi, L2, k2,

m2). The functions (A1, A,) and (A, A,) are referred to as a closure surfaces. The

shape of ,2 (A, A,) is qualitatively illustrated in Figure 3.3. For a constant A1, the curve

i, (A1) is similar to the closure curve shown in Figure 3.1. The value of i2 decreases

with increasing Ai.

With the help of a closure surface in Figure 3.3, the loci of (A1, Az) for which

'2 = q,,, I2 = eq, + 1, i2 = ~,q -1 are plotted in Figure 3.4. The locus of i; = eq is

referred to as the equilibrium flow curve for Inlet 2. Using the same reasoning as for a

single inlet and assuming that the cross-sectional area of Inlet 1 is constant, it follows that

if A2 = A, Inlet 2 will shoal and close; if A2 = AH, Inlet 2 will scour until the cross-

sectional area attains a value As, and if A2 = AHz, Inlet 2 will shoal until the cross-sectional

area attains the value As.

The locus of (A1, A2) for which Inlet 2 has a stable equilibrium flow area is the

enhanced (by a thicker line) part of the equilibrium flow curve for Inlet 2. Similarly, the

locus of (A, A2) for which Inlet 1 has a stable equilibrium flow area is the enhanced part

of the equilibrium flow curve for Inlet 1. The condition for the existence of stable

equilibrium flow areas for both Inlet 1 and Inlet 2 is that the enhanced parts of the

equilibrium flow curves intersect. The common equilibrium interval of the two is
















2 = e


Figure 3.3 Closure surfaces (source: van de Kreeke, 1985)



A, A,, A, A,,, A2



i e / I

: eq l i
o\ I I

S\ I
\ / Teq


A 3\ //
A\ /



Figure 3.4 Equilibrium flow curve for Inlet 2 (source: van de Kreeke, 1985)









A2 A2

2 2










(a) (b)


A2 A2

2 2










(c) (d)

Figure 3.5 Possible configurations of equilibrium flow curves for a two-inlet bay system.
Stable equilibrium flow area is represented by and unstable equilibrium is
represented by o. The hatched area in (a) represents the domain of the stable
equilibrium flow area (source: van de Kreeke, 1990)

represented by the hatched rectangle in Figure 3.5 (a). The general shapes of the

equilibrium flow curves and their relative positions in the (Ai, A2) plane are presented in

Figure 3.5. The detailed explanations to the Figure 3.5 are given in Appendix D.

3.3 Stability Analysis with the Linearized Model

Due to the complex nature of sediment transport by waves and currents it is

difficult to carry out an accurate analysis of the stability of single or multiple inlet









systems. We will therefore attempt to carry out an approximate analysis based on the van

de Kreeke (1990) linearized lumped parameter model.

The justification for use of simple model is that for purpose of this study the

stability analysis serves to illustrate a concept rather than to provide exact numerical

results. Accurate numerical values can only be obtained by using a full-fledged two-

dimension tidal model to describe the hydrodynamics of the bay.

3.3.1 Linearized lumped parameter model for N Inlets in a Bay

The basic assumptions of the Linearized lumped parameter model are as follows:

1 The linearized model assumes that the ocean tide and the velocity are simple
harmonic functions.

2 The water level in the bay fluctuates uniformly and the bay surface area remains
constant.

3 Hydrostatic pressure, and shear stress distribution along the wetted perimeter of
the inlet cross-section is uniform.

4 For a given bay area and inlet characteristics, the tidal amplitude and/or tidal
frequency must be sufficiently large for equilibrium to exist. Similarly, larger the
littoral drift due to waves, larger the equilibrium shear stress required to balance it
and therefore the equilibrium velocity, the larger the required bay surface area,
tidal amplitude and the tidal frequency or, in other words, Eq. (3.17) and Eq.
(3.19) must be satisfied for the existence of equilibrium areas.

5 There is no fresh water discharge in the bays.

6 In a shallow bay the effect of dissipation of tidal energy cannot be ignored,
especially if the bay is large.

Inlet flow dynamics of the flow in the inlets are governed by the longitudinal

pressure gradient and the bottom shear stress, van de Kreeke (1967),

0 (3.2)
p ix ph

in which is the pressure, p is the water density, h is the depth and r is the bottom shear

stress. This stress is related to the depth mean velocity u









r = pFu u (3.3)

where F=f/8, is the friction coefficient. Integration of Eq. (3.2) (with respect to the

longitudinal x-coordinate) between the ocean and the bay yields (van de Kreeke 1988).


u I U -= 2gR (o) (3.4)
mjRm + 2F1L

In Eq. (3.4), u, refers to the cross-sectional mean velocity of the ith inlet, g is the

acceleration due to gravity, m, is the sum of exit and entrance losses, R, is the hydraulic

radius of the inlet, L, is the length of the inlet, ro is the ocean tide, and 7B is the bay tide.

The velocity u, is positive when going from ocean to bay.

Assuming the bay surface area to fluctuate uniformly, flow continuity can be

expressed as


u, A,= d4B (3.5)
=1 dt

in which A, is the cross-sectional area, AB is the bay surface area and t is time.

Considering u, to be a simple harmonic function of t, Eq. (3.4) is linearized as

shown in Appendix B to yield

8 2gR,
-8 mau = 2 (o ) (3.6)
37r mjR + 2FL

in which uax, is the amplitude of the current velocity in the ith inlet. It follows from Eq.

(3.5) and Eq. (3.6) that for a simple harmonic ocean tide (in complex notation)

ro (t) = acoe (3.7)

and assuming A, and AB to be constant, we obtain

Uz = umaxej(-t') (3.8)









where the phase angle ,v is considered to be the same for all inlets. Differentiating Eq.

(3.6) with respect to t, eliminating drB/dt between Eq. (3.5) and Eq. (3.6), and making

use of the expressions for u, and royields an equation for ui and E,

N 18
u A +--ABB u2 j = ABao jue e (3.9)
1 2g 37;

in which the dimensionless resistance factor B, is defined as


B = 21- L (3.10)
= mR ++2FL_

where B, is the function of A,. Now, equating the real and imaginary parts of Eq. (3.9) and

eliminating the phase angle c, yields the equation for Umaxi


S [ABC]2 B, max, =[ABao] f umaxA. (3.11)


For equilibrium flow = eq,. Using linearized version in Eq. (3.6) and Eq. (3.3), the

equilibrium velocity can be written as


eq eq' (3.12)
S 8/3 7pF7

where the approximate value of ;eq, can be taken from Mehta and Christensen (1983).

For equilibrium flow areas umax, = umaxeqi, substituting this value Eq. (3.11) becomes:


J[AB- BZJU maxq =eq[ABao]2 L ItmaxeqA (3.13)


When the maximum tidal velocity in all the inlets equals the corresponding equilibrium

value, i.e., umax, = 1max eq for i 1,2........... N, the difference between the bay and the

ocean tides becomes constant. So from Eq. (3.4) it follows that









Bu2 maxeql B2 2maxeq2... ~U2 max eq BNU2maxeqN (3.14)

Eq. (3.13) and Eq. (3.14) constitute a set of N simultaneous equations with Nunknowns

[A1, A2...,AN]. In general, more than one set of equilibrium flow areas [A1, A2...,AN] will

satisfy these equations. Since the dimensionless resistance factor B, is a function of A,.

Therefore, whether for a given ocean tide (ao, c) and bay surface area (AB), Eq. (3.13)

and Eq. (3.14) yield sets of solutions [A1, A2....AN] that are real and positive depends on

the particular form of R, f(A,).

The function R,=f(A,) plays an important role in the hydrodynamic efficiency of an inlet.

For a given head difference, exit and entrance loss coefficients, friction factor and inlet

length, the maximum tidal velocity increases with the increasing value of R, see Eq. (3.4).

Therefore, larger the value of R, for a given value of A, larger the discharge. For a


rectangular channel, R, = and for triangular channel R1 = a (See Appendix B).


Analytical solutions to equation Eq. (3.13) and Eq. (3.14) can be found by

restricting attention to the friction-dominated flow in the inlets, i.e. m=0

From Eq. (3.10) with m = 0, we obtain

B, 21 (3.15)
R

A
For rectangular inlets, substituting R, = in Eq. (3.15) and then in Eq. (3.13)
W,


and Eq. (3.14) we get








A2
(F, LU2eq, )2
L8ue \21 8
(ABaoJ)2 j(ABOa)4 -~ 2 2 (ABg)2 (FLIu3eq1 .FLW 3eqN2 (3.16)

2 (F1 W Ie NUN+...+F W 3e)2


When any A, (from Eq. (3.16)) is known, the cross-sectional areas of the other inlets

follow from Eq. (3.14), with B, given by Eq. (3.15), provided that

ABa2o >2 j ( u3eql +...+FL' 3eq ) (3.17)

This is a quadratic equation n in A2 for which we have two sets of real and positive roots

and two sets of complex roots.

For the triangular cross-section, R = a ,,, substituting this in Eq. (3.13) and

Eq. (3.14) we get,


ueql r1Ll J" "+ eqN A3 (ABaoC)2 A +
(3.18)
8 (A )2 [ 2 u4 =0
3a g eq,

in which sets of A, are given by Eq. (3.18) (as we have two real and positive solution for

A,). When any A, is known, the cross-sectional areas of the other (N-l) inlets follow from

Eq. (3.14) with B, given by Eq. (3.15). One root of Eq. (3.18) is always negative. The

other two are real and positive roots provided that

Au.a3o 3"8 2 L5 >F LlL2 U FL312)
A > 3u1q5 ...... U N (3.19)
2 37r eqa ag g j









The above stability concept, when applied to a multiple-bay inlet system,

becomes complicated because the loci of the set of the values [A1, A2....AN] for which the

tidal maximum of the bottom shear stress equals the equilibrium stress, are rather

complicated surfaces and make it difficult to determine whether inlets are in a scouring

mode or shoaling mode. With some simplifying assumptions, the stability analysis for a

multiple-inlet system can be reduced to that for a two-inlet system. This is considered

next in the context of the St. Andrew Bay system.

3.4 Application to St. Andrew Bay System

In the above model if N=2, the model can be applied to the two inlet system. The

equilibrium flow curves for Inlet 1 and Inlet 2 are calculated from Eq. (3.11) with u= eq.

The equilibrium flow areas are given by the solution of Eq. (3.16) for rectangular inlet

and Eq. (3.18) for triangular cross-section. Figure 3.6 illustrates the equilibrium flow

curve. A line can be drawn passing from the intersection of two equilibrium flow areas.

Above the line Bi>B2 and therefore u
1 When the point defined by the actual cross-sectional areas [Ai, A2] is located in
the vertically hatched zone or anywhere outside the curves, (Zone-1), both inlets
close.

2 When the point is located in the crosshatched zone, (Zone-2), Inlet 1 will remain
open and Inlet 2 will close.

3 When the point is located in the diagonally hatched zone, (Zone-3), Inlet 1 will
close and Inlet 2 will remain open.

4 Finally, when the point is located in the blank zone, (Zone-4), one inlet will close
and the other will remain open. However, in this case which one closes depends
on the relative rates of scouring and or/shoaling.









The St. Andrew Bay system is similar to the case of two inlets in a bay. In reality

there are three interconnected bays, but only one is connected with the Gulf. So there is

no forcing due to ocean tide from the other two bays. Thus, all the bays collectively

behave as if there is only one bay connected by two inlets. So the linear model for N

inlets can be applied to the St. Andrew system, where N = 2. The development of

equilibrium curves for this case is discussed in Chapter 5.


















Zone-4
IFnlet

Zone-1.- A,

Figure 3.6 Equilibrium flow curves for two inlets in a bay (source: van de Kreeke, 1990)














CHAPTER 4
APPLICATION TO ST. ANDREW BAY COMPLEX AND ENTRANCES

4.1 Description of Study Area

St. Andrew Bay is located in Bay County on the Gulf of Mexico coast of Florida's

panhandle. It is part of a three-bay and two-inlet complex. One of these inlets is St.

Andrew Bay Entrance and the other is East Pass, which are connected to St. Andrew Bay

on one side and the Gulf on the other. The other two bays are West Bay and the East Bay,

which connect to St. Andrew Bay, as shown in the Figure 4.1 Note that West Bay as

shown also includes a portion called North Bay. Prior to 1934, East Pass was the natural

connection between St. Andrew Bay and the Gulf. In 1934, St. Andrew Bay Entrance

(Figure. 4.2) was constructed 11 km west of East Pass through the barrier island by the

federal government to provide a direct access between the Gulf and Panama City. The

entrance has since been maintained by the U.S Army Corps of Engineers (USACE),

Mobile District. The St. Andrew Bay State Recreational Area is located on both sides of

this entrance, which has two jetties 430 m apart to prevent the closure of the inlet.

The interior shoreline of the entrance has continually eroded since it's opening.

An environmentally sensitive fresh water lake located in the St. Andrew Bay State

Recreational Area is vulnerable to the shoreline erosion and USACE has placed dredged

soil to mitigate shoreline erosion.

East Pass finally closed in the 1998, due to the long-term effect of the opening of

St. Andrew Entrance. In December 2001, a new East Pass was opened (Figure 4.3), and

the effect of this new inlet is presently being monitored over the entire system.






43



0-1m

Nor i 1-3m
3-6m
6-9m

Hitha imai Br&Il >9m

S0 5 km

Dupont Bridge

St. Andrew
Channel 2


East Pass


Figure 4.1 Map showing the three bays and two inlets and bathymetry of the study area.
Dots show location of tide stations.


Figure 4.2 Aerial view of St. Andrew Bay Entrance in 1993. Jetties are -430 m apart.




















act location
O throat)




Figure 4.3 East Pass channel before it's opening in December 2001. Plan view (pre-
construction) design geometry and then anticipated current measurement
transects are shown. The dots show the new cross-section (source: Jain et al.,
2002)

4.2 Summary of Field Data

Three hydrographic surveys were done by the University of Florida's Department

of Civil and Coastal Engineering in the years 2001 and 2002. Figure 4.4 shows the

bathymetry of St. Andrew Bay Entrance and the different cross-sections measured during

the surveys. Cross-sections A-i, A-2 and B-l, B-2 were measured in September 2001,

A'-1, A'-2, B'-1, B'-2, C'-1, C'-2, in December 2001, and D-l, D-2 in March 2002. Flow

discharges, vertical velocity profiles and tide were also recorded. The tide gage (in the

September 2001 survey only) was located in waters (Grand Lagoon) close to the entrance

channel. The discharge and velocity data was measured with a vessel-mounted Acoustic

Doppler Current Profiler, or ADCP (Workhorse 1200 kHz, RD Instruments, San Diego,

CA), and the tide with an ultrasonic recorder (Model #220, Infinities USA, Daytona

Beach, FL). The coordinates of the cross-section end-points are given in Table 4.1.









Table 4.1 Locations of St. Andrew Bay channel cross-sections
Section Side Latitude Longitude Northing Easting Date
A A-i 30 07.70 -8543.36 412452.62 1613441.90 09/18/01
A A-2 30.07.44 -85 43.28 410875.80 1613857.60 09/18/01
B B-l 30 07.35 -8543.91 410315.83 1610524.00 09/18/01
B B-2 30 07.17 -8543.71 409240.00 1611584.60 09/18/01
A' A'-1 3007.18 -8543.72 409256.63 1611563.75 12/18/01
A' A'-2 3007.40 -8543.91 410626.10 1610534.09 12/18/01
B' B'-1 3007.43 -8543.30 410766.60 1613757.91 12/18/01
B' B'-2 3007.68 -8543.44 412309.71 1613034.11 12/18/01
C' C'-1 3007.06 -8543.90 408542.02 1610606.43 12/18/01
C' C'-2 3007.27 -8544.01 409822.96 1610030.59 12/18/01
D D-1 3007.42 -8543.32 410714.20 1613635.15 03/28/02
D D-2 3007.65 -8543.58 412134.85 1612294.58 03/28/02

Measurements were also taken at the new East Pass after it's reopening in

December 2001. The locations of the East Pass cross-section coordinate end points are

given in Table 4.2. Flow cross-section and vertical velocity profiles were measured along

cross-section E in December 2001 and F in March 2002.

Table 4.2 Locations of East Pass channel cross-sections
Section Side Latitude Longitude Northing Easting Date

E E- 30 03.78 -85 37.07 388325.56 1646376.03 12/19/01
E E-2 3003.79 -85 37.12 388371.27 1646103.36 12/19/01
F F-1 3003.78 -85 3707 388325.55 1646376.03 03/27/02
F F-2 3003 79 -85 37 12 388371.26 1646103.35 03/27/02











D2


C'D,
111 O





409000.00 \ :



1610000M00 1611000.00 1612000~00 1613000,00 1614000.00 1615000.00


Figure 4.4 St. Andrew Bay Entrance bathymetry and current measurement cross-sections.
Depths are in feet below MLLW (source: Jain et al. 2002)

4.2.1 Bathymetry

The bathymetry of the study area is shown in Figure 4.1. During the hydrographic

surveys the bottom depth was measured by the ADCP at all cross-sections shown in

Figure 4.4. These have been compared with a bathymetric survey of 2000. Figures. 4.5

and 4.6 are example of measurements along cross-sections A and F, respectively. The

trends in the two sets of depths are qualitatively (although not entirely) comparable.

Areas, mean depths and widths are summarized in Table 4.3.

Table 4.3 Cross-section area, mean depths and width
Section Cross-section Area (m2) Width (m) Mean Depth (m)
A 6250 493 11.0
B 6600 457 10.6
A' 5210 525 10.0
B' 5640 544 11.0
C' 5220 425 11.5
D 5970 528 11.9
E 255 109 3.0
F 300 85 2.5








47



Batymetry side-A
Side-A-1 Side- A-2
0
0 100 200 300 400 500 600
-2
-4

-6

-8
S-10

-12

-14

-16
-18
--ADCP ---Bathymetry chart Distance (m)



Figure 4.5 Cross-section A in St. Andrew Bay Entrance measured and compared with
2000 bathymetry. Distance is measured from point A-1. The datum is mean
tide level (source: Jain and Mehta, 2001)



Bottom Contour

F-1 F-2
0 -
-0.5- 0 3.5 11 18.2 24 31.5 37 46.6 59 72.6 75 84.5
-1
E -1.5 -
-2
$ -2.5
-3
-3.5
-4
Distance (m) from F-1

---ADCP



Figure 4.6 Cross-section F in East Pass measured by ADCP. Distance is measured from
point F-1. The datum is mean tide level (source: Jain and Mehta, 2002)









4.2.2 Tides

As noted, tide was measured in September 2001 in Grand Lagoon close to the

entrance channel, at Lat: 30 07.9667, Long: -85 43.6667. Tide variation in the channel

was compared with the predicted National Ocean Service (NOS) tide at St. Andrew Bay

channel with reference station at Pensacola after applying the correction factors for the

range and the lag. The measured tide is shown in Figure 4.7 and the corresponding NOS

tide in Figure 4.8. Both show general similarities, although the measured one should be

deemed more accurate. The data indicate a weak semi-diurnal signature with a range

variation of 0.11 to 0.18 m. In the month of December and March no tides were

measured, only the NOS tides were reported using the tide at Pensacola; see Figure 4.9

and Figure 4.10.

For East Pass the same tide was assumed as for St. Andrew Bay Entrance. Five

other NOS stations are also located in the study area as shown in Figure 4.1. The ranges

of tides for September 2001, December 2001 and March 2002 at these stations are given

in the Table 4.4. These tides were found by applying correction factors for the range and

for the lag (see Appendix C). The Gulf tidal range, 2ao, was obtained by applying an

amplitude correction factor to the tide measured at the Grand Lagoon gauge (see

calculations in Appendix C). Semi-diurnal tides were reported in September 2001 with

the tidal period of 12.42 h. The tides in December 2001 were of mixed nature with a

period of approximately 18 h. In contrast, diurnal tides were reported in March 2002 with

the period of 25.82 h. The approximate tide level in each bay was then found by

weighted-averaging the tide over the number of stations in that bay. The phase lag

between the tides of all the stations were calculated by plotting all the tides in Figure

4.10, and the results are summarized in Table 4.5.







49



Tide at St Andrew Bay Entrance
-Tide at St Andrew Bay Entrance
0.6
0.5 -
S0.4
g 0.3
0.2
0.1
0
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
oi o t io o oi o i o oi o to i o oi o oi o i
O O 0 0 0 0" 0 0 0 0i 0 C 0 0 0 0 0
O O O O O O O 0 0 0
Time (hrs)
09/18/01 Time09/19/01
--------- -----"-



Figure 4.7 Measured tide in Grand Lagoon on Septemberl8-19, 2001. The datum is
MLLW (source: Jain and Mehta, 2001)




NOS Tides Tides

0.45
0.4
E 0.35
0.3
> 0.25
0.2
0.15
0.1
0.05 -

0- I I

Time (hr:min)
09/18/01 09/19/01

Y

Figure 4.8 NOS predicted tide at St. Andrew Bay Entrance on Septemberl8-19, 2001;
reference station is Pensacola. The datum is MLLW.








50




NOS Tides


0.5

0.4

E 0.3

> 0.2
a)
-j
S0.1

U 0

-0.1

-0.2




- Tides


S( n 0 ct In tO o "0 0 00 N cO Ion C o




12/18/01 12/19/01
y y-


Figure 4.9 NOS predicted tide in St. Andrew Bay Entrance on December 18-19, 2001;
reference station is Pensacola. The datum is MLLW.






Tides in all the Stations


0 45 -- Gulf and Channel Entrance --- Laird Bayou
-- Parker Lynn Haven


-*- Panama City
---West Bay Creek


035

S03

' 025
,-
. 02

015

01

005


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
M T L0 (D D D O D D D D D D 0) C O O 2 N o O

Time (hrs)


Figure 4.10 Tide at all selected NOS stations in March 2002.









Table 4.4 Tidal ranges in September 2001, December 2001 and March 2002.
S Station Name September December March
No Range (m) Range (m) Range (m)
1 Gulf of Mexico ("Ocean" tide) 0.216 0.572 0.425
2 Laird Bayou, East Bay 0.236 0.624 0.465
3 Parker, East Bay 0.236 0.624 0.465
4 Lynn Haven North Bay 0.236 0.624 0.465
5 Panama City, St. Andrew Bay 0.203 0.535 0.397
6 Channel Entrance, St. Andrew Bay 0.197 0.520 0.386
7 West Bay Creek 0.236 0.624 0.465

Table 4.5 Phase lags between the stations and the ocean tide.
S No Stations Time Lag
1 Gulf of Mexico ("Ocean" tide) 0 h
2 Laird Bayou, East Bay + 2h
3 Parker, East Bay +2 h
4 Lynn Haven North Bay +2 h
5 Panama City, St. Andrew Bay +1 h
6 Channel Entrance, St. Andrew Bay +1 min
7 West Bay Creek + 3h

4.2.3 Current and Discharge

Currents and discharges were measured with the ADCP at all the six cross-

sections in St. Andrew Bay Entrance (Figure 4.4) and at two cross-sections in East Pass

(Figure.4.3). The detailed velocity and discharge curves are shown in Jain and Mehta

(2001), Jain et al. (2002) and Jain and Mehta (2002). The measurements are summarized

in the Table 4.6.

From Table 4.6 it is observed that the average peak velocity in St. Andrew Bay

channel was approximately 0.63 m/s (at or close to the throat section) and at East Pass it

was approximately 0.50 m/s. The peak discharge value at St. Andrew was 4200 m3/s and

at East Pass it was 139 m3/s.









Table 4.6 Characteristic peak velocity and discharge values
Quantity Velocity (m/s) Discharge (m3/s)
Cross-section Peak Flood Peak Ebb Peak Flood Peak Ebb

A 0.63 -0.62 4200 3620
B 0.45 -0.34 2980 2250
A' 0.68 -0.69 3620 3920
B' 0.69 -0.66 4061 3876
C' 0.67 -0.77 3480 3750
D 0.42 -0.49 2509 2777
E 0.51 -0.49 139 165
F 0.43 -0.38 114 101

4.3 Tidal Prism

Tidal prism is the volume of water that enters the bay during flood flow. Tidal

prism for St. Andrew Bay system was calculated using the approximate formula

P QmT (4.1)
KC,

where Qm is the peak discharge (Table 4.6), Tis the tidal period (12.42 hrs for September

2001, 18 hrs for December 2001 and 25.82 hrs for March 2002) and the coefficient CK =

0.86 (Keulegan, 1967). This tidal prism was compared with the O'Brien (1969)

relationship of Eq. (4.2), where Ac is the throat area, P the tidal prism on the spring range

for sandy inlets in equilibrium, and a and b are the constants:

Ac = aPb (4.2)

For inlets with two jetties, a = 7.49x10-4 and b = 0.86 (Jarrett, 1976). And for

inlets without jetty (East Pass), a = 3.83x105 and b = 1.03. The values of the tidal prism

are summarized in Table 4.7. Spring ranges are reported in Table 4.4.

It should be noted that the prism values from the O'Brien relationship are mere

estimates.









Table 4.7 Flood and ebb tidal prisms
Quantity Prism (m3) from peak Prism (m3) from O'Brien
discharge
Cross-sections Flood Ebb Peak Flood Peak Ebb

A 7.0x107 6.0x107 11.4x107 10.3x107
A' 8.6x107 9.4x107 09.0x107 10.4x107
D 8.6x107 9.4x107 10.0x107 09.7x107
E 3.3x106 3.9x106 03.8x106 04.6x106
F 3.9x106 3.5x106 03.6x106 03.6x106














CHAPTER 5
RESULTS AND DISCUSSION

5.1 Introduction

There are two aspects of this chapter, one dealing with the hydraulics of the St.

Andrew Bay system and the other with its stability. The linearized approach developed in

Chapter 2 is used to examine the hydraulics of St. Andrew Bay under different

conditions. The model is run as one-inlet/one-bay system for both September 2001 and

March 2002. It is also run as a three-inlets/three-bays system in September 2001 when

East Pass was closed, and as a three-bays/four-inlets system when East Pass was open in

March 2002. Hydraulic parameters related to tides and currents thus obtained are then

compared with values from the hydrographic surveys done in September 2001 and March

2002.

In contrast to hydraulics, the linearizedd lumped paramter model) inlet stability

model developed in Chapter 3 is applied only to St. Andrew Bay. A qualitative approach

is developed to discuss the results and graphs have been plotted to show stability

variation.

5.2 Hydraulics of St. Andrew Bay

The solution of equations for the linear model, derived in Chapter 2, forms the

basis of calculation of the hydraulic parameters characterizing the system. One begins

with the basic model of one-inlet (St. Andrew Bay Entrance) and one-bay (St. Andrew

Bay) system, when East Pass was closed. As noted the model is then extended to the

complete system of three bays (St. Andrew Bay, East Bay and North + West Bays) and









three inlets when East Pass was closed in September 2001, and finally as three bays and

four inlets when East Pass was open in March 2002.

5.2.1 Solution of Equations

The solutions of the relevant hydraulic equations are given in Chapter 2. A Matlab

program (see Appendix A) was developed to solve the one-inlet bay system as well as the

multiple-inlet bay system. The input and output parameters for each system are listed in

the tabular form.

5.2.1.1 One-inlet one-bay system

The one-inlet one-bay system is based on solving Eq. (5.1):


o -n = (5.1)
C dO

The required input and output parameters for this case are given in Table 5.1.

Table 5.1 List of input and output parameters for one-inlet one-bay model.
Input Parameters
ao Ocean tide amplitude (Gulf of Mexico)
T Time period of tide
aB1 Bay 1 tide amplitude (St. Andrew Bay)
AB1 Bay 1 surface area
L1 Length of inlet 1 (St. Andrew Bay Entrance)
R1 Hydraulic radius of inlet 1
Ac, Inlet 1 cross-section area
k Entrance and exit losses
,f Friction factor
(rio r7B)max Maximum ocean-bay tide difference
Output Parameters
/7B1 Bay 1 tide
aB1 Bay 1 tide amplitude
EB1 Phase difference between bay 1 and ocean tides
Umaxl Maximum velocity through Inlet 1
Evi Phase difference between velocity in Inletl and ocean tide
(rio rIB)max Maximum ocean-bay tide difference









5.2.1.2 Three inlets and three bays with one inlet connected to ocean

This system is based on solving Eq. (5.2), Eq. (5.3) and Eq. (5.4):

a d B, AB2 dB2 AB3 dB3
C, dO A,, dO A,, dO


(5.2)



(5.3)


ad
m Bi 7R2 C2 dO


S- 3=7 [ d3 (5.4)
7C4L dOJ

The required input and output parameters for this case are given in Table 5.2

Table 5.2 List of input and output parameters for the three inlets and three bays model.


Input Parameters
ao Ocean Tide amplitude (Gulf of Mexico)
T Time period of the tide
aB1 Bay 1 tide amplitude (St. Andrew Bay)
aB2 Bay 2 tide amplitude (East Bay)
aB3 Bay 3 tide amplitude (West Bay)
AB1 Bay 1 surface area
AB2 Bay 2 surface area
AB3 Bay 3 surface area
Li Length of inlet 1 (St. Andrew Bay Entrance)
R1 Hydraulic radius of inlet 1
Ac, Inlet 1 cross-section area
L2 Length of inlet 2 (connecting East Bay and St. Andrew Bay)
R2 Hydraulic radius of inlet 2
Ac2 Inlet 2 cross-section area
L4 Length of inlet 4 (connecting West Bay and St. Andrew Bay)
R4 Hydraulic radius of inlet 4
Ac4 Inlet 4 cross-section area
k Entrance and exit losses
,f Friction factor
(ro -r) max Maximum ocean-bay tide difference
(QB1 -7IB2) maxMaximum Bay 1 and Bay 2 tide difference
(rQB -7/B3) max Maximum Bay 1 and Bay 3 tide difference


)









Table 5.2 (continued)
Output Parameters
Br/1 Bay 1 tide
aBl Bay 1 tide amplitude
EB1 Phase lag between bay 1 and ocean tide
r7B2 Bay 2 tide
aB2 Bay 2 tide amplitude
SB2 Phase lag between bay 2 and ocean tide
r/B3 Bay 3 tide
aB3 Bay 3 tide amplitude
EB3 Phase lag between bay 3 and ocean tide
Umaxl Maximum velocity through Inlet 1
Evi Phase difference between velocity of Inlet 1 and the ocean tide
Umax2 Maximum velocity through Inlet 2
Ev2 Phase difference between velocity of Inlet 2 and the ocean tide
Umax4 Maximum velocity through inlet 4
Ev4 Phase difference between velocity of Inlet 4 and the ocean tide
(ro -r/B1) max Maximum ocean-bay tide difference
(/B1 -r1B2) max Maximum Bay 1 and Bay 2 tide difference
(rB1 -7B3) max Maximum Bay 1 and Bay 3 tide difference


5.2.1.3 Three inlets and three bays with two inlets connected to ocean

This system is based on solving Eq. (5.5), Eq. (5.6) and Eq. (5.7):


SBm + d B d + AB2 d B2 AB3 dB3
C, +C3 dO ABI dO ABI dO


u- drB2
7B1 B2 C dO



m B3 d= _B3
7B1U3C4L dO


(5.5)



(5.6)


(5.7)


The required input and output parameters for this case are given in Table 5.3.









Table 5.3 List of Input and Output Parameters for the four inlets and three bays model.
Input Parameters
ao Ocean Tide Amplitude (Gulf of Mexico)
T Time period of the tide
aBl Bay 1 tide amplitude (St. Andrew Bay)
aB2 Bay 2 tide amplitude (East Bay)
aB3 Bay 3 tide amplitude (West Bay)
AB1 Bay 1 surface area
AB2 Bay 2 surface area
AB3 Bay 3 surface area
Li Length of inlet 1 (St. Andrew Bay Entrance)
R1 Radius of inlet 1
Ac1 Inlet 1 cross-section area
L2 Length of inlet 2 (connecting East Bay and St. Andrew Bay)
R2 Radius of inlet 2
Ac2 Inlet 2 cross-section area
L3 Length of inlet 3 (East Pass)
R3 Radius of inlet 3
Ac3 Inlet 3 cross-section area
L4 Length of inlet 4 (connecting West Bay and St. Andrew Bay)
R4 Radius of inlet 4
Ac4 Inlet 4 cross-section area
k Entrance and exit losses
f Friction factor
(r/o -rB) max Maximum ocean-bay tide difference
(r7 -i7B2) max Maximum Bay 1 and Bay 2 tide difference
(/1 -17B3) max Maximum Bay 1 and Bay 3 tide difference
Output Parameters
r/i Bay 1 tide
aBl Bay 1 tide amplitude
EB1 Phase lag between bay 1 and ocean tide
/7B2 Bay 2 tide
aB2 Bay 2 tide amplitude
SB2 Phase lag between bay 2 and ocean tide
r7B3 Bay 3 tide
aB3 Bay3 tide amplitude
EB3 Phase lag between bay 3 and ocean tide
Umaxl Maximum velocity through Inlet 1
Evi Phase difference between velocity of Inlet 1 and the ocean tide
Umax2 Maximum velocity through Inlet 2









Table 5.3 Continued)
Output Parameters
8v2 Phase difference between velocity of Inlet 2 and the ocean tide
Umax3 Maximum velocity through Inlet 3
8v3 Phase difference between velocity of Inlet 3 and the ocean tide
lUmax4 Maximum velocity through Inlet 4
8v4 Phase difference between velocity of Inlet 4 and the ocean tide
(7o -r7B1) max Maximum ocean-bay tide difference
(urB1 -7B2) max Maximum Bay 1 and Bay 2 tide difference
(TB1 -17B3) max Maximum Bay 1 and Bay 3 tide difference

5.2.2 Input Parameters

Table 5.4 provides the input values for all the three cases of the model as

described in Section 5.2.

1 The amplitude in each bay is found by applying a weighting factor proportional to
the tide station contribution to the total bay area.

2 Initial values are assumed for (ro -ri1) max, (iB1 -7B2) max (1B1 -17B3) maxfor the
initial calculation. The September 2001 tide showed a semidiurnal signal, with a
period of 12.42 h. The tide in March 2002 showed diurnal signature with a period
of 25.82 h. The model was run three times for three different cases as described in
Section 5.2. Details regarding all input parameters are found in Jain and Mehta
(2002), and are also summarized in Chapter 4. Table 5.4 gives values of all input
parameters required for the model.

Table 5.4 Input parameters for the hydraulic model.
Input Values Remarks
Parameters Sept 2001 March 2002
ao 0.109 m 0.212 m Calculated from UF tide gauge data,
calculations shown in Appendix C.
T 12.42 h 25.82 h NOS Tides Tables.
aBl 0.103 m 0.201 m
aB 0. m 0.2 m Calculated in proportion to the
aB2 0.115 m 0.226 m
aB 0.11 m 02 m contributing tide at station.
aB3 0.118 m 0.233 m
AB1 74 km2
AB2 54 km2 From the USGS topographic maps.
AB3 155 km2









Table 5.4 (continued)
Input Values Remarks
Parameters Sept 2001 March 2002
Li 1340 m
RI 10 m Measured in survey
Ac1 6300 m2
L2 1000 m
R2 9 m
Ac2 1.9x104 m From the USGS topographic maps. Ac2,
L4 1000 m Ac3, Ac4 are zero for one inlet bay case
R4 12 m
Ac4 9.7x103 m2
L3 400 m
43 m Measured in survey. Ac3 is zero for three-
R3 3m
A3 255 M2 bays and three-inlets case.
Ac3 255 m
k 1.05
f 0.025
(ro -4Bl) max 0.037 0.036
Assumed initial values. Calculations are
(l -r 2) max 0.060 0.063
B1 17B2) max 0.060 0.063 shown in appendix C
(rB1 -rB3) max 0.099 0.998

5.2.3 Model Results and Comparison with Data

Model results are given in Table 5.5.

Table 5.5 Model results and measurements.
One Inlet One Bay System, September 2001
Output parameters Model Measurement %error
aB1 0.10 m 0.10 m 0%
EB1 0.36 rad 0.34 rad 6%
Ucl max 0.65 m/s 0.63 m/s 3%
Evl -1.20 rad -1.22 rad 2%
(7o -r1) max 0.038 0.036 6%
Three Bay Three Inlets System, September 2001
aB1 0.10 m 0.10 m 0%
EB1 0.34 rad 0.34 rad 0%
aB2 0.10 m 0.11 m 9%
EB2 0.37 rad 0.91 rad 59%
aB3 0.10 m 0.12 m 17%
CB3 0.54 rad 1.26 rad 57%









Table 5.5 (continued)
Three Bay Three Inlets System, September 2001
Output parameters Model Measurement %error
Ucl max 0.62 m/s 0.63 m/s 2%
v1 -1.11 rad -1.20 rad 7%
Uc2 max 0.04 m/s Not measured
Sv2 -1.21 rad Not measured
Uc4 max 0.20 m/s Not measured -
8 4 -1.04 rad Not measured -
(17 -IB) max 0.037 0.037 0%
(7 -17B2) max 0.003 0.060 95%
(7B1 -B3) max 0.020 0.098 80%
One Inlet One Bay System, March 2002
aB1 0.20 m 0.20 m 0%
EB1 0.17 rad 0.17 rad 0%
Ucl max 0.63 m/s 0.65 m/s 3%
Ev1 -1.40 rad -1.40 rad 0%
(17 -B1) max 0.036 0.036 0%
Three Bay Four Inlets System, March 2002
aB1 0.21 m 0.20 m 5%
EB1 0.16 rad 0.16 rad 0%
aB2 0.21 m 0.22 m 5%
CB2 0.18 rad 0.44 rad 59%
aB3 0.21 m 0.23 m 9%
EB3 0.26 rad 0.60 rad 57%
Ucl max 0.60 m/s 0.65 m/s 8%
v1l -1.35 rad -1.40 rad 4%
Uc2 max 0.04 m/s Not measured
Ev2 -1.40 rad Not measured
Uc3 max 0.60 m/s 0.55 m/s 9%
Ev3 -1.35 rad -1.40 rad 4%
Uc4 max 0.22 m/s Not measured -
Ev4 -1.31 rad Not measured
(7o -B1) max 0.035 0.035 0%
( -MB2) max 0.003 0.063 95%
(1 -B3) max 0.012 0.010 20%

It is evident from Table 5.5 that the linear model gives good results. The percent

error decreases if the system is modeled as a three-bay system, which is actually the case.

Velocity and tide amplitudes are within reasonably small error limits. The phase

differences between ocean (Gulf) and bay tides from data are very approximate as they









are calculated based on weighted-average tides at selected stations. Moreover, there are

very few stations to yield a good value of tide for a bay. Note that the input values for (r7

-rB1) max, (rB1 -rB2) max (i1 -rB3) maxiS also approximate. Sample calculation for (7ro -r/B)

max, (iB1 -772) max (7B1 -1B3) max is given in Appendix C.

5.3 Stability Analysis

The stability analysis developed in Chapter 3 is now applied to St. Andrew Bay

system. This analysis is done for a two-inlet bay system using van de Kreeke's (1990)

linearized lumped parameter model. The two inlets, to which the model is applied, are St.

Andrew Bay Entrance and the new East Pass opened in December 2002. Calculations

related to stability are given in Appendix D. A Matlab program (Appendix D) has also

been developed for doing the analysis and generating equilibrium flow curves for the two

inlets. There are two programs, one for rectangular channel cross-section and another for

triangular channel cross-sections.

5.3.1 Input Parameters

Input parameters required for the Matlab program (Appendix D) are listed in

Table 5.6. Since the objective was to study the effect of bay area on the stability because

the results are sensitive to it, it is held constant for a particular set of calculation, but is

varied for generating different sets of equilibrium flow curves. Similarly the length of

East Pass, believed to have an uncertain value due to the complex bay shoreline and

bathymetry in that region is also varied to study its effect on the system.









Table 5.6 Input parameters for stability analysis.
Input Parameters for December 2001
ao 0.26 m Amplitude of ocean tide
T 18.0 hrs Time period of tide
AB 74-105 km2 Area of bay (St. Andrew Bay) (varied from 74 to 105 km2)
Inlet 1
ueql 0.40 m/s Equilibrium velocity for Inlet 1 (see Appendix D)
Wi 525 m Width of Inlet 1
L1 1340 m Length of Inlet 1
al 0.138 Constant for triangular cross-section for Inlet 1 (see Appendices
C and D)
F, 0.004 Friction coefficient for Inlet 1
Inlet 2
ueq2 0.45 m/s Equilibrium velocity for Inlet 2 (see Appendix D)
W2 300 m Width of Inlet 2
L2 500-2000 m Length of Inlet 2 (East Pass) (varied from 500 m to 2000 m)
a2 0.187 Constant for triangular cross-section for Inlet 2 (see Appendices
C and D)
F2 0.004 Friction coefficient for Inlet 2

5.3.2 Results and Discussion

As noted, it is found that two inlets can never be unconditionally stable

simultaneously in one bay. The bay area has a large effect on the stability of the inlets.

Table 5.7 summarizes this effect. It is clear that with a small increase in bay area the

inlets become stable. This is also demonstrated with the help of equilibrium flow curve in

the Figure 5.1, Figure 5.2 and Figure 5.3 for rectangular cross-section and Figure 5.7 and

Figure 5.9 for triangular cross-section. The cross-sectional area pair during December

2001 (Table 4.3) [5210, 255] is shown by the dot. Figure 5.1 and Figure 5.7 have small

bay areas, and the dot lies outside the equilibrium flow curve indicating that both inlets

are unstable. As the bay area increases St. Andrew becomes stable (Figure 5.2 and Figure

5.7), and a further increase in bay area also stabilizes East Pass (Figure 5.3 and Figure

5.9). However, in reality we cannot increase the bay area beyond a reasonable limit,

because then the basic assumption of bay tide fluctuating evenly in the bay does not hold.









Moreover, in a shallow bay the effect of dissipation of tidal energy cannot be ignored,

especially if the bay is large. Also as per Figure 3.5 two inlets are not stable

simultaneously.

An increase in the length of East Pass has a destabilizing effect on that inlet as

shown in the Table 5.7. Note also that for a rectangular cross-section (Figure 5.3) with

the length of East Pass of 500m, this inlet is stable, whereas with a length of 2000 m

(Figure 5.6) the inlet is instable. This is because as the length increases the dissipation

increases. Friction dominated losses, (F 0.004, R 3m (2FL /R)) for East Pass with

500 m length is 1.33, where as that for 2000 m length it is 5.33. The same cases occur in

Figure 5.9 and Figure 5.12.

The other effects on the stability model are the approximation in the cross-section

of the inlet. It is clear that triangular cross-section is a better approximation than

rectangular section, because with the same parameters for rectangular cross-section in

Figure 5.6, East Pass is predicted to be unstable whereas in Figure 5.12 for triangular

cross-section, East Pass is stable even though barely, which is not believed to be the case

for this newly opened inlet.

Table 5.8 gives the qualitative indication of the stability. The various zones

mentioned in the Table 5.8 are described in Section 3.4 and Figure 3.6. It is clear from

these results that St. Andrew is a stable inlet (for a realistic bay area) as opposed to East

Pass. This is also evident from the Figure 3.5, which shows that two inlets cannot be

stable simultaneously, because we for unconditional stability, need four real points of

intersection of equilibrium flow curve and none of the solutions (neither rectangular

cross-section nor triangular cross-section) gives four real solution.









The model does not yield an analytic solution for a more realistic parabolic cross-

section. Another weakness is due to the assumptions made in Chapter 3 including a bay

area in which the tide is spatially always in-phase, and simple a harmonic function for

tide. These assumptions are not always satisfied.

Table 5.7 Effect of change in bay area and length of East Pass.
Rectangular cross-section
Run Bay East Pass Result
No. area (km2) Length
(m)
1 74 500 Both inlets unstable (Figure 5.1)
2 90 500 St. Andrew becomes stable (Figure 5.2)
3 105 500 St. Andrew stable, East Pass barely stable (Figure
5.3)*
4 74 2000 Both inlets unstable (Figure 5.4)
5 90 2000 St. Andrew barely stable (Figure 5.5)
6 105 2000 St. Andrew stable, East Pass unstable (Figure 5.6)
Triangular cross-section
7 74 500 Both inlets unstable Figure (5.7)
8 90 500 St. Andrew becomes stable (Figure 5.8)
9 105 500 Both inlets stable (Figure 5.9)
10 74 2000 Both inlets unstable (Figure 5.10)
11 90 2000 St. Andrew stable (Figure 5.11)
12 105 2000 St. Andrew stable, East Pass just stable (Figure 5.12)*
Two inlets cannot be simultaneously stable, because according to Figure 3.5, for
unconditional stability we need four real points of intersection of equilibrium flow curve,
which is not possible in either rectangular cross-section solution nor triangular cross-
section solution.







66





Inlet Stability -1 (rectangular section)


4500
4000
S3500
3000
2500
S2000
w 1500
1000
500
0


2000


3000


A1, St Andrew (m2)


- East Pass St Andrew


Figure 5.1 Equilibrium flow curves for rectangular cross-sections, Run No. 1.








Inlet Stability 2 (rectangular section)


6000


5000 -
E
S4000

a 3000

w 2000

1000


0 1000


2000 3000 4000 5000
A1, St Andrew (m2)


6000


- East Pass St Andrew


Figure 5.2 Equilibrium flow curves for rectangular cross-sections, Run No. 2.


5000


-::2-2-._-_








67




Inlet Stability 3 (rectangular section)


6000


S5000 -
E
4000

a 3000-

w 2000 -

S1000-

0


a- --


0 1000 2000 3000 4000 5000 6000 7000
A1, St Andrew (m2)


East Pass St Andrew



Figure 5.3 Equilibrium flow curves for rectangular cross-sections, Run No. 3.









Inlet Stability -4(rectangular section)


4500
4000
S3500
E
3000
S 2500
- 2000
L 1500
- 1000
500
0-
0


2000


3000


4000


5000
5000


A1, St Andrew (m2)


- East Pass St Andrew


Figure 5.4 Equilibrium flow curves for rectangular cross-sections, Run No. 4.


I- ------. - ----


=-.


-













Inlet Stability 5(rectangular section)


6000

_5000 --

4000

3000 -

j 2000 -
l O I 3 _
1000 -
S----- i -------------------------------
0 1000 2000 3000 4000 5000 6000
A1, St Andrew (m2)


East Pass St Andrew


Figure 5.5 Equilibrium flow curves for rectangular cross-sections, Run No. 5.








Inlet Stability 6 (rectangular section)


6000

S5000 ---

4000 -

S3000 --- -

L 2000 --

S1000 -
---- : -----
01.-----------------------------------------
0 1000 2000 3000 4000 5000 6000 7000
A1, St Andrew (m2)


East Pass St Andrew


Figure 5.6 Equilibrium flow curves for rectangular cross-sections, Run No. 6.







69



Inlet Stability -7 (triangular section)


5000
4500 -_
R 4000 --
S3500 -
S3000 --
S2500-
2000 -
J 1500 :--
S1000 --
500-- _

0 1000 2000 3000 4000 5000
A1, St Andrew (m2)


East Pass St Andrew


Figure 5.7 Equilibrium flow curves for triangular cross-sections, Run No. 7.







Inlet Stability 8 (triangular section)


6000

5000 --

S4000 -
4000 .-- "-- -.---_
n 3000

Iu 2000 .

S1000 --

0
0 1000 2000 3000 4000 5000 6000
A1, St Andrew (m2)


East Pass St Andrew


Figure 5.8 Equilibrium flow curves for triangular cross-sections, Run No. 8.








70




Inlet Stability 9 (triangular section)


7000
6000
E 5000 ""--- --
i 4000 --
S3000 "---
2000 "--
1000----
01 ------------W--.^~- -
0 1000 2000 3000 4000 5000 6000 7000
A1, St Andrew (m2)


East Pass St Andrew


Figure 5.9 Equilibrium flow curves for triangular cross-sections, Run No. 9.








Inlet Stability 10 (triangular section)


5000
4500 --
ir 4000
3500 -
3000 --.-
2500
2000
u 1500
1000
500 ---

0 1000 2000 3000 4000 5000
A1, St Andrew (m2)


East Pass St Andrew


Figure 5.10 Equilibrium flow curves for triangular cross-sections, Run No. 10













Inlet Stability 11 (triangular section)


6000

S5000 --_
E
4000 "----

a 3000 -

u 2000

S1000 --
0 ---------------"----.........---- --
0 1000 2000 3000 4000 5000 6000
A1, St Andrew (m2)


East Pass St Andrew


Figure 5.11 Equilibrium flow curves for triangular cross-sections, Run No. 11.







Inlet Stability 12 (triangular section)


7000
6000 -
5000 "---"--.
t 4000
S3000 "
L 2000 ""----
1000---


0 1000 2000 3000 4000 5000 6000 7000
A1, St Andrew (m2)


East Pass St Andrew

Figure 5.12 Equilibrium flow curves for triangular cross-sections, Run No. 12.









Table 5.8 Stability observations for St. Andrew Bay Entrance and East Pass.
Figure Placement of cross-sectional Observations
area pair [A1, A2], (black dot)
Figure 5.1 Zone-1 Both inlets are unstable
Figure 5.2 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.3 Zone-4 Only one is stable i.e. St. Andrewa
Figure 5.4 Zone-1 Both inlets are unstable
Figure 5.5 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.6 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.7 Zone-1 Both inlets are unstable
Figure 5.8 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.9 Zone-4 Only one is stable i.e. St. Andrewa
Figure 5.10 Zone-1 Both inlets are unstable
Figure 5.11 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.12 Zone-4 Only one is stable i.e. St. Andrewa
a As per Figure 3.6, it is clear that even in Zone-4 only one inlet is stable, this is further
clarified from Figure 3.5, which shows that only one inlet can be stable at one time.














CHAPTER 6
CONCLUSIONS

6.1 Summary

St. Andrew Bay, which is a composite of three interconnected bays (St. Andrew

Bay proper, West Bay + North Bay and East Bay) is located in Bay County on the Gulf

of Mexico coast of Florida's panhandle. It is part of a three-bay and two-inlet complex.

One of these inlets is St. Andrew Bay Entrance and the other is East Pass, which are both

connected to St. Andrew Bay on one side and the Gulf on the other. Prior to 1934, East

Pass was the natural connection between St. Andrew Bay and the Gulf. In 1934, St.

Andrew Bay Entrance (Figure 4.2) was constructed 11 km west of East Pass through the

barrier island to provide a direct access between the Gulf and Panama City. The interior

shoreline of the entrance has continually eroded since it's opening. East Pass was closed

in 1998, which is believed to be due to the opening of the St. Andrew Bay Entrance.

In December 2001, a new East Pass was opened (Figure 4.3), and the effect of this

new inlet is presently being monitored over the entire system. Accordingly, the objective

of the present work was to examine the hydraulics of the newly formed two-("ocean")

inlet/three-bay system and its hydraulic stability, especially as it relates to East Pass.

The first aspect of the tasks performed to meet this objective was the development

of equations for the linearized hydraulic model for the system of three bays and four

inlets (two ocean and two between bays), and solving and applying them to the St.

Andrew Bay system. The second aspect was the development of the ocean inlet stability

criteria using the Escoffier (1940) model for one inlet and one bay and extending this









model to the two ocean inlets and a bay. Stability analysis for the St. Andrew Bay system

was then carried out using the linearized lumped parameter model of van de Kreeke

(1990).

6.2 Conclusions

The following are the main conclusions of this study:

1 If the system is modeled as a three-bay system as compare to a one-bay system,
the error in the phase difference, SB1, decreases from 6% to 0% and in the velocity
amplitude from 3% to 2%. Moreover the error in maximum head difference, (/o -
tB1i) m, also decreases from 6% to 0%.

2 The amplitudes of velocities and bay tides are within 5%, which is a reasonably
small error band. The percent error for St. Andrew Bay is almost 0%, and for the
other bays it is within 20%.

3 The bay area has a significant effect on the stability of the two inlets. At a bay
area of 74 km2 both inlets are unstable. Increasing it by 22% to 90 km2 stabilizes
St. Andrew Bay Entrance, and by 42% to 105 km2 stabilizes East Pass as well.

4 Two inlets can never be simultaneously unconditionally stable.

5 Keeping the bay area at 105 km2 and increasing the length of East Pass from 500
m to 2000 m destabilizes this inlet because as the length increases the dissipation
in the channel increases as well.

6 A triangular channel cross-section is a better approximation than a rectangular
one, because given the same values of all other hydraulic parameters, St. Andrew
Bay Entrance with a rectangular cross-section is found to be barely stable,
whereas with a triangular cross-section it is found to be stable, as is the case.

6.3 Recommendations for Further Work

Accurate numerical values required for the stability analysis of a complex inlet-

bay system can only be obtained by using a two- (or three)-dimensional tidal model to

describe the hydrodynamics of the bay.

Freshwater discharges from the rivers into the bay should be incorporated through


numerical modeling.






75


Including a more realistic assumption for the channel cross-section can improve

the stability analysis.














APPENDIX A
ALGORITHMS FOR MULTIPLE INLET-BAY HYDRAULICS

A.1 Introduction

The linearized approach described in Chapter 2 has been used to evaluate the

hydraulic parameters of the multiple inlet bay system. The differential equations,

developed by this approach (Chapter 2), Eq. (2.100), Eq. (2.101) and Eqs (2.102), are

solved in Matlab Program-1 (given below). These are the general equations for four inlets

and three bays system. These equations can be used to solve from one bay system to the

complex three bays system. Note that for solving Program-i, the Matlab version should

have a symbolic toolbox. The present program is solved in Matlab release 6.1. The

solution from Program-1 is used as input to Program-2 (given below). The required input

parameters and output for Program-2 are listed in Table 5.3 of Chapter 5.

A.2 Program-1

%UNIVERSITY OF FLORIDA
%CIVIL AND COASTAL ENGINEERING DEPARTMENT
%PROGRAM FOR SOLVING THE EQS 2.100, 2.101, 2.102
% ALL CONSTANTS DEFINED IN CHAPTER 2

clear all
syms a b c AB
tl=sym('thetal')
t2=sym('theta2')
t3=sym('theta3')
rl=sym('al*exp(-i*tl)')
r2=sym('a2*exp(-i*t2)')
r3 =sym('a3 *exp(-i*t3)')
C=[a*i+l a*A*i a*B*i;-l b*i+l 0;-1 0 c*i+l]
D=[1;0;0]
%END









A.3 Program-2

%UNIVERSITY OF FLORIDA
%CIVIL AND COASTAL ENGINEERING DEPARTMENT
%PROGRAM FOR CALCULATION OF MULTIPLE INLET-BAY HYDRUALICS
%FOR ONE -INLET BAY CASE, FOR Ac2, Ac3, Ac4 EQUAL TO ZERO
%INLET 1 AND INLET 3 CONNECTS BAY1 TO THE OCEAN

clear all
g=9.81;
ao=0.212;%ocean tide amplitude
theta=0;%ocean tide phase
etao=ao*cos(theta);%ocean tide
T=25.82;%time period
q=2*pi/(T*3600)%sigma
k=1.05;% entrance and exit loss
f=0.025;%friction factor
aB 1=0.201 ;%approximate amplitude of bays
aB2=0.226;
aB3=0.2325;
%ml= 1 max(eta0-etab 1),m2=max(etab 1-etab2),m3=max(etab 1-etab3)
ml=0.023;
m2=0.0527;
m3=0.123;
%Inlet 1
L1=1340;%Length of inlet
R1=10;%hydraulic radius
Acl=6300;%CROSS-SECTION AREA of the inlet
Fl=k+(f*L1)/(4*R1);%friction factor F includes ken kex fL/4R

%Inlet 2
L2=1000;%Length of inlet
R2=9;%hydraulic radius
Ac2=1.9* 104;%CROSS-SECTION AREA of the inlet, it is zero for one inlet bay case
F2=k+(f*L2)/(4*R2);%friction factor F includes ken kex fL/4R

%Inlet 3
L3=400;%Length of inlet
R3=3;%hydraulic radius
Ac3=255;%CROSS-SECTION AREA of the inlet
F3=k+(f*L3)/(4*R3);%friction factor F includes ken kex fL/4R

%Inlet 4
L4=1000;%Length of inlet
R4=12;%hydraulic radius
Ac4=9.7* 10A3;%CROSS-SECTION AREA of the inlet
F4=k+(f*L4)/(4*R4);%friction factor F includes ken kex fL/4R









%bayl area
AB1=74*10A6;
%bay2 area
AB2=54*10A6;
%bay3 area
AB3=155*10A6;

%calculations
CDL =sqrt(ao/(m 1*F 1))
CDL2=sqrt(aB 1/(m2*F2))
CDL3 =sqrt(ao/(ml *F3))
CDL4=sqrt(aB 1/(m3 *F4))

C1=CDL1*Acl/AB1*sqrt(2*g/ao)
C2=CDL2*Ac2/AB2*sqrt(2*g/aB 1)
C3 =CDL3 *Ac3/AB 1 *sqrt(2*g/ao)
C4=CDL4*Ac4/AB3 *sqrt(2*g/aB 1)

%ALL THE CONSTANTS ARE DEFINED IN THE THESIS
a=q/(C1+C3)
if Ac2==0
b=0
else b=q/C2
end

if Ac4==0
c=0
else
c=q/C4
end

A=AB2/AB1
B=AB3/AB1
rl=(c-i)*(b-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b- +c*b-i*a*B+a*b-
i*a+a*B*b+a*A*c)%SOLUTIONS ARE OBTAINED FROM ANOTHER
r2=-i*(c-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b-l+c*b-i*a*B+a*b-
i*a+a*B*b+a*A*c)%MATLAB PROGRAM WHICH HAS SYMBOLLIC TOOLBOX.
r3=-i*(b-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b-l+c*b-i*a*B+a*b-i*a+a*B*b+a*A*c)
aB l=abs(rl)*ao
eB l=-angle(rl)
aB2=abs(r2)*ao
eB2=-angle(r2)
aB3=abs(r3)*ao
eB3=-angle(r3)
etaB l=aB l*cos(theta-eB 1)
etaB2=aB2*cos(theta-eB2)









etaB3=aB3*cos(theta-eB3)
CDL1 1=sqrt(ao/(max(etao-etaB 1)*F 1))
CDL22=sqrt(aB 1/(max(etaB 1-etaB2)*F2))
CDL33=sqrt(ao/(max(etao-etaB 1)*F3))
CDL44=sqrt(aB 1/(max(etaB 1 -etaB3)*F4))

C11=CDL1*Acl/AB1*sqrt(2*g/ao)
C22=CDL2*Ac2/AB2*sqrt(2*g/aB 1)
C33=CDL3*Ac3/AB l*sqrt(2*g/ao)
C44=CDL4*Ac4/AB3 *sqrt(2*g/aB 1)

%velocity in the inlet
ucl=sqrt(2*g/ao)*CDL1*(etao-ao*rl)
uclmax=abs(ucl)
evl=-angle(ucl)
uc2=sqrt(2*g/aB1)*CDL2*(ao*rl-ao*r2)
uc2max=abs(uc2)
ev2=-angle(uc2)
uc3=sqrt(2*g/ao)*CDL3*(etao-ao*rl)
uc3max=abs(ucl)
ev3=-angle(ucl)
uc4=sqrt(2*g/aB1)*CDL4*(ao*rl-ao*r3)
uc4max=abs(uc4)
ev4=-angle(uc4)
%END














APPENDIX B
INLET HYDRAULICS RELATED DERIVATIONS

B.1 Linearization of Damping Term

The linearization of the damping term in Eq. (3.6) is done as given in Bruun

(1978). The bay tide response is represented by

rq =asin( 0- B) (B.1)

where

0 = rt = at, dimensionless time.
T

aB = one-half the tide range (i.e., amplitude) in the bay, and

B = lag between high water (HW) or low water (LW) in the ocean and corresponding

HW or LW in the bay. Also,

ro = ao sin(0) (B.2)

from the continuity equation we further have


Au = A dqB (B.3)
dt

where Ac is the area of cross-section of the inlet and AB is the surface area of the bay.


The time of HW or LW in the bay, i.e., when d = 0, coincides with time of
dt

slack water, i.e., u = 0, so that SB is also the lag of slack water after HW or LW in the

ocean. Thus it can be written as

d7Bd7 = cr2 B2 Cs(0-EB) cos(0-E) (B.4)
dt t dt









or in terms of Fourier series Eq. (B.4) can be written as


8 sin n -
dq^ d^ 2 2" 2\
dt, dt =raB 2 cosn(8-eB) (B.5)
dt dt nnir4 --nI

where n takes only odd integral values. For linearization purposes n=l, so that Eq. (B.5)

becomes

dB dB caB 8 cosn(8- ,) (B.6)
dt dt 3rt

The amplitude of the tidal velocity is given by


max = (B.7)
A

Therefore, it can be written as
8
uu = max (B.8)
3)r

where umax is the amplitude of the u.

B.2 Shear Stress Dependence on Area

For each inlet discharge is defined as a time varying function:


Q (t) = A d (B.9)
Sdt


Q,(t) = ( +tA, (B.10)
m R, +2FL

The expression for maximum tidal velocity can be obtained by the solution of the above

equations with the simplifying assumptions mentioned in Chapter 2.


um = C(K)siny 2ABaOK (B.11)
AT K

where K is the coefficient of repletion,









T A4 2gR a
K T A= 2g a (B.12)
K 2;ra Ao m,R, +2FL,

and

K = K (B.13)

is summation is over all the inlets. The function C(K) sin 7 is a monotonically increasing

function with C=0 for K=0 and C 1 for K=co, y is a specific time when sea is at MSL, as

defined by Kuelegan (1951)

It is seen below that the bottom shear stress, r, varies strongly with the cross-

sectional area. This can be shown with the help of approximate analytical solution carried

out by Keulegan (1951). Substituting the value of u from Eq. (B.11) in Eq. (3.1), and

taking C(K) sin ; 1 and F = 0.003:


A; pF 2 2 (B.14)


It is clear from the above equation that i has a strong dependence on A.

B.3 General Equation for hydraulic radius.

Consider the general trapezoidal cross-section:


Area, A=I (B+Bo)h=I Bh 1+BO
2 2 B


Wetted perimeter, P = Bo + 2 (B B2 + h2 = B ++



I h 1+ Bo
A 2 BB
Hydraulic radius, R =-
P B BO h2
B FLB) [,B






83



K B


hA







Bo


Figure B.1 Trapezoidal Cross-section

Now consider two cases: 1) Rectangular cross-section, i.e., Bo = B, and 2)

Triangular cross-section, i.e., Bo = 0.

B.3.1 Rectangular

B=Bo, Therefore hydraulic radius for a rectangle is

R A h

B
R-



B.3.2 Triangular

For triangular section, Bo= 0

1h
A 2
RT
P 1+4 h


B.4 Hydraulic Radius for Triangular Cross-Section

For a triangular cross-section the hydraulic radius is related as a square root of the

area, as shown below:






84


Figure B.2 is a triangular cross-section where /is the angle with the horizontal on both

the sides:

1
Area A = h2h tan /
2


Wetted perimeter P



Hydraulic radius R-


2h
cos f


1 -A = aA
/2 sin/ fcos/


Figure B.2 Triangular cross-section.


(B.15)














APPENDIX C
CALCULATION OF BAY TIDE AND LINEAR DISCHARGE COEFFICIENTS

This appendix contains sample calculations of input data in Table 5.4 for the

Matlab Program -2 (Appendix A) in Chapter 5. Estimation of bay tide amplitude (aB1, aB2,

aB3), input for Table 5.4 was made by taking the weighted-averages of the NOS tide

amplitudes at reported stations in the bay. Let us take the case of St. Andrew Bay (Table

C. 1). This bay has three stations where tide is reported. The weighting factor for the

range at a given station was estimated by selecting the approximate area of influence of

tide (range) surrounding that station. Given the tidal period of 12.42 h, the phase

difference between Gulf and the bay could be converted in to degree or radians.

Table C.1 Weighted-average bay tide ranges and phase differences
St. Andrew Bay
Station Weighti Sept. tide Weighted Phase Weighted-
ng factor range -average difference average
(m) (m) (h) (h)
Channel Entrance 0.48 0.197 0.0945 0.0017 0.008
Panama City 0.37 0.203 0.0751 1.0000 0.370
Parker 0.15 0.236 0.0354 2.0000 0.300
Total 0.2050 0.678 h=19.650
East Bay
Laird Bayou 0.40 0.236 0.0940 2.0000 0.800
Parker 0.40 0.236 0.0940 2.0000 0.800
Panama city 0.20 0.203 0.0406 1.0000 0.200
0.2300 1.8 h=52.170
West Bay
WestBay Creek 0.50 0.236 0.118 3.0000 1.5 hrs
Lynn Haven 0.50 0.236 0.118 2.0000 1.0 hrs
0.236 2.5 h=72.460

The Gulf tide range had to be estimated, as there was no open coast tidal station

near to the study site. The procedure was as follows: