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HYDRAULICS AND STABILITY OF MULITPLE INLETBAY 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 InletBay 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 InletBay 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 INLETBAY SYSTEMS.................... ..................29 3.1 Stability Problem Definition ............ .... ......... ........................ 29 3.2 Stability Criteria ..................................... ...... ... ................... 29 3.2.1 Stability Analysis for OneInlet 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 neinlet onebay 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 INLETBAY 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 CrossSection...................... ................ 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 Crosssectional areas of Johns Pass and Blind Pass in Boca Ciega Bay .................. 1.2 Crosssectional areas of St. Andrew Bay Entrance and East Pass.............................3 1.3 Crosssectional areas of Pass Cavallo and Matagorda Inlet .............. ..................3 4.1 Locations of St. Andrew Bay channel crosssections ...........................................45 4.2 Locations of East Pass channel crosssections...................... ...................45 4.3 Crosssection 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 oneinlet onebay 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 Weightedaverage 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 twoinlet 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 crosssections.........46 4.5 Crosssection A in St. Andrew Bay Entrance.................. .......... ............... 47 4.6 Crosssection F in East Pass measured by ADCP ....................................... .......... 47 4.7 Measured tide in Grand Lagoon on Septemberl819, 2001....................................49 4.8 NOS predicted tide at St. Andrew Bay Entrance on Septemberl819, 2001. .............49 4.9 NOS predicted tide in St. Andrew Bay Entrance on December 1819, 2001 ..............50 4.10 Tide at all selected NOS stations in March 2002..................... ............. ............... 50 5.1 Equilibrium flow curves for rectangular crosssections, Run No. 1...........................66 5.2 Equilibrium flow curves for rectangular crosssections, Run No. 2...........................66 5.3 Equilibrium flow curves for rectangular crosssections, Run No. 3...........................67 5.4 Equilibrium flow curves for rectangular crosssections, Run No. 4 ........................67 5.5 Equilibrium flow curves for rectangular crosssections, Run No. 5.........................68 5.6 Equilibrium flow curves for rectangular crosssections, Run No. 6 ...........................68 5.7 Equilibrium flow curves for triangular crosssections, Run No. 7. ...........................69 5.8 Equilibrium flow curves for triangular crosssections, Run No. 8. ...........................69 5.9 Equilibrium flow curves for triangular crosssections, Run No. 9. ...........................70 5.10 Equilibrium flow curves for triangular crosssections, Run No. 10 .........................70 5.11 Equilibrium flow curves for triangular crosssections, Run No. 11 ........................71 5.12 Equilibrium flow curves for triangular crosssections, Run No. 12.......................71 B 1 Trapezoidal Crosssection .......................................................... ............... 83 B .2 T riangular crosssection. ..................................................................... .................. 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 crosssectional areas of inlets ocean (Gulf) tide amplitude bay tide amplitudes dimensionless bay tide amplitudes constant that relates hydraulic radius with area of triangular crosssection 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 DarcyWeisbach 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 MULTIPLEINLET 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 longterm 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 crosssectional area and the hydraulic environment. St. Andrew Bay on the Gulf of Mexico coast of Florida's panhandle is part of a threebay 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 longterm 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 threebay and fourinlet (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 twoocean 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 resultsthe 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 nearshore 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 longterm 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 nonsilting, nonscouring 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 codependency 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 crosssectional area of each inlet over time. Table 1.1 Crosssectional 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 Crosssectional 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 crosssection, 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 Crosssectional 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 inletbay 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 InletBay System 2.1.1 System Definition The governing equations for a simple inletbay 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 DarcyWeisbach friction factor Gradual loss = (2.5) 4R 2g where f= DarcyWeisbach 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( ror/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 crosssectional 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 (onehalf the ocean tidal range), T= tidal period and a = tidal (angular) frequency. Substitution into Eq. (2.10) gives d =i K 4o 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 Go 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 = onehalf 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 1z/2. 2.3 Multiple InletBay 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 inletbay case, the velocity relationship and the equation of continuity for twobay 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 crosssectional 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 = onehalf the tide range (i.e., amplitude) in bay 1. SaB2 aB2 B2 ao 0a aB2 = onehalf 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(bi) r, = (2.41) X 1 r2 = (2.42) r X where X = (ab1)i(aA+a+b); X =(ab1)+i(aA+a+b) XX = (ab 1)2 +(aA + a + b)2 b((b+a+aA)(ab1) b(ab1)(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 crosssectional 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(bi) i, = (2.61) X 1 r = (2.62) X X = (ab 1) i(aA+a+b); X = (ab1)+i(aA+a+b) XX = (ab 1)2 +(aA + a + b)2 b( (b+a+aA)(ab1) b(ab1)(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 = 8B1i2. 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 crosssectional 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 oi [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(ci r = (2.85) r3 = (2.86) X X = (ac + ab + bc + aBb + aAc 1)+i(aA + abc abc aB) X = (ac + ab + bc + aBb + aAc 1) i(aA + abc abc aB) XX = (ac + ab + bc + aBb + aAc 1)2 + (aA + abc abc 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 = SB3ZI2. 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(ci r2 = (2.108) X r3 = (2.109) X X = (ac + ab + bc + aBb + aAc 1)+i(aA + abc abc aB) X = (ac + ab + bc + aBb + aAc 1) i(aA + abc abc aB) XX = (ac + ab + bc + aBb + aAc 1)2 +(aA + abc abc 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 INLETBAY SYSTEMS 3.1 Stability Problem Definition An inlet is considered stable when after a small change the crosssectional 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 crosssection area and the hydraulic environment. The pertinent parameters are the actual tidemaximum 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 flushout 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 crosssectional area. 3.2.1 Stability Analysis for OneInlet 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 crosssectional 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 crosssectional 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 crosssection, 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 crosssectional 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 crosssectional 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 twoinlet 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 fullfledged 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 crosssection 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 xcoordinate) 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 crosssectional 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 crosssectional 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 frictiondominated 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 crosssectional 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 crosssection, 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 crosssectional areas of the other (Nl) 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 multiplebay 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 multipleinlet system can be reduced to that for a twoinlet 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 crosssection. 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 crosssectional areas [Ai, A2] is located in the vertically hatched zone or anywhere outside the curves, (Zone1), both inlets close. 2 When the point is located in the crosshatched zone, (Zone2), Inlet 1 will remain open and Inlet 2 will close. 3 When the point is located in the diagonally hatched zone, (Zone3), Inlet 1 will close and Inlet 2 will remain open. 4 Finally, when the point is located in the blank zone, (Zone4), 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. Zone4 IFnlet Zone1. 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 threebay and twoinlet 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 longterm 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 01m Nor i 13m 36m 69m 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 crosssection (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 crosssections measured during the surveys. Crosssections Ai, A2 and Bl, B2 were measured in September 2001, A'1, A'2, B'1, B'2, C'1, C'2, in December 2001, and Dl, D2 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 vesselmounted 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 crosssection endpoints are given in Table 4.1. Table 4.1 Locations of St. Andrew Bay channel crosssections Section Side Latitude Longitude Northing Easting Date A Ai 30 07.70 8543.36 412452.62 1613441.90 09/18/01 A A2 30.07.44 85 43.28 410875.80 1613857.60 09/18/01 B Bl 30 07.35 8543.91 410315.83 1610524.00 09/18/01 B B2 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 D1 3007.42 8543.32 410714.20 1613635.15 03/28/02 D D2 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 crosssection coordinate end points are given in Table 4.2. Flow crosssection and vertical velocity profiles were measured along crosssection E in December 2001 and F in March 2002. Table 4.2 Locations of East Pass channel crosssections Section Side Latitude Longitude Northing Easting Date E E 30 03.78 85 37.07 388325.56 1646376.03 12/19/01 E E2 3003.79 85 37.12 388371.27 1646103.36 12/19/01 F F1 3003.78 85 3707 388325.55 1646376.03 03/27/02 F F2 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 crosssections. 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 crosssections 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 crosssections 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 Crosssection area, mean depths and width Section Crosssection 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 sideA SideA1 Side A2 0 0 100 200 300 400 500 600 2 4 6 8 S10 12 14 16 18 ADCP Bathymetry chart Distance (m) Figure 4.5 Crosssection A in St. Andrew Bay Entrance measured and compared with 2000 bathymetry. Distance is measured from point A1. The datum is mean tide level (source: Jain and Mehta, 2001) Bottom Contour F1 F2 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 F1 ADCP Figure 4.6 Crosssection F in East Pass measured by ADCP. Distance is measured from point F1. 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 semidiurnal 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). Semidiurnal 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 weightedaveraging 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 Septemberl819, 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 Septemberl819, 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 1819, 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 crosssections 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) Crosssection 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 = 7.49x104 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 Crosssections 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 oneinlet/onebay system for both September 2001 and March 2002. It is also run as a threeinlets/threebays system in September 2001 when East Pass was closed, and as a threebays/fourinlets 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 oneinlet (St. Andrew Bay Entrance) and onebay (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 oneinlet bay system as well as the multipleinlet bay system. The input and output parameters for each system are listed in the tabular form. 5.2.1.1 Oneinlet onebay system The oneinlet onebay 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 oneinlet onebay 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 crosssection area k Entrance and exit losses ,f Friction factor (rio r7B)max Maximum oceanbay 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 oceanbay 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 crosssection area L2 Length of inlet 2 (connecting East Bay and St. Andrew Bay) R2 Hydraulic radius of inlet 2 Ac2 Inlet 2 crosssection area L4 Length of inlet 4 (connecting West Bay and St. Andrew Bay) R4 Hydraulic radius of inlet 4 Ac4 Inlet 4 crosssection area k Entrance and exit losses ,f Friction factor (ro r) max Maximum oceanbay 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 oceanbay 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 crosssection area L2 Length of inlet 2 (connecting East Bay and St. Andrew Bay) R2 Radius of inlet 2 Ac2 Inlet 2 crosssection area L3 Length of inlet 3 (East Pass) R3 Radius of inlet 3 Ac3 Inlet 3 crosssection area L4 Length of inlet 4 (connecting West Bay and St. Andrew Bay) R4 Radius of inlet 4 Ac4 Inlet 4 crosssection area k Entrance and exit losses f Friction factor (r/o rB) max Maximum oceanbay 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 oceanbay 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 threeinlets 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 threebay 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 weightedaverage 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 twoinlet 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 crosssection and another for triangular channel crosssections. 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 74105 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 crosssection 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 5002000 m Length of Inlet 2 (East Pass) (varied from 500 m to 2000 m) a2 0.187 Constant for triangular crosssection 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 crosssection and Figure 5.7 and Figure 5.9 for triangular crosssection. The crosssectional 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 crosssection (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 crosssection of the inlet. It is clear that triangular crosssection is a better approximation than rectangular section, because with the same parameters for rectangular crosssection in Figure 5.6, East Pass is predicted to be unstable whereas in Figure 5.12 for triangular crosssection, 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 crosssection nor triangular crosssection) 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 inphase, 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 crosssection 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 crosssection 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 crosssection 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 crosssections, 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 crosssections, Run No. 2. 5000 ::22.__ 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 crosssections, 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 crosssections, 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 crosssections, 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 crosssections, 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 crosssections, 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 crosssections, 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 crosssections, 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 crosssections, 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 crosssections, 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 crosssections, Run No. 12. Table 5.8 Stability observations for St. Andrew Bay Entrance and East Pass. Figure Placement of crosssectional Observations area pair [A1, A2], (black dot) Figure 5.1 Zone1 Both inlets are unstable Figure 5.2 Zone2 St. Andrew Bay Entrance is stable Figure 5.3 Zone4 Only one is stable i.e. St. Andrewa Figure 5.4 Zone1 Both inlets are unstable Figure 5.5 Zone2 St. Andrew Bay Entrance is stable Figure 5.6 Zone2 St. Andrew Bay Entrance is stable Figure 5.7 Zone1 Both inlets are unstable Figure 5.8 Zone2 St. Andrew Bay Entrance is stable Figure 5.9 Zone4 Only one is stable i.e. St. Andrewa Figure 5.10 Zone1 Both inlets are unstable Figure 5.11 Zone2 St. Andrew Bay Entrance is stable Figure 5.12 Zone4 Only one is stable i.e. St. Andrewa a As per Figure 3.6, it is clear that even in Zone4 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 threebay and twoinlet 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/threebay 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 threebay system as compare to a onebay 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 crosssection 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 crosssection is found to be barely stable, whereas with a triangular crosssection 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 crosssection can improve the stability analysis. APPENDIX A ALGORITHMS FOR MULTIPLE INLETBAY 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 Program1 (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 Programi, the Matlab version should have a symbolic toolbox. The present program is solved in Matlab release 6.1. The solution from Program1 is used as input to Program2 (given below). The required input parameters and output for Program2 are listed in Table 5.3 of Chapter 5. A.2 Program1 %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 Program2 %UNIVERSITY OF FLORIDA %CIVIL AND COASTAL ENGINEERING DEPARTMENT %PROGRAM FOR CALCULATION OF MULTIPLE INLETBAY 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(eta0etab 1),m2=max(etab 1etab2),m3=max(etab 1etab3) ml=0.023; m2=0.0527; m3=0.123; %Inlet 1 L1=1340;%Length of inlet R1=10;%hydraulic radius Acl=6300;%CROSSSECTION 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;%CROSSSECTION 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;%CROSSSECTION 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;%CROSSSECTION 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=(ci)*(bi)/(i*a*A+i*a*c*b+a*ci*ci*b +c*bi*a*B+a*b i*a+a*B*b+a*A*c)%SOLUTIONS ARE OBTAINED FROM ANOTHER r2=i*(ci)/(i*a*A+i*a*c*b+a*ci*ci*bl+c*bi*a*B+a*b i*a+a*B*b+a*A*c)%MATLAB PROGRAM WHICH HAS SYMBOLLIC TOOLBOX. r3=i*(bi)/(i*a*A+i*a*c*b+a*ci*ci*bl+c*bi*a*B+a*bi*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(thetaeB 1) etaB2=aB2*cos(thetaeB2) etaB3=aB3*cos(thetaeB3) CDL1 1=sqrt(ao/(max(etaoetaB 1)*F 1)) CDL22=sqrt(aB 1/(max(etaB 1etaB2)*F2)) CDL33=sqrt(ao/(max(etaoetaB 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*(etaoao*rl) uclmax=abs(ucl) evl=angle(ucl) uc2=sqrt(2*g/aB1)*CDL2*(ao*rlao*r2) uc2max=abs(uc2) ev2=angle(uc2) uc3=sqrt(2*g/ao)*CDL3*(etaoao*rl) uc3max=abs(ucl) ev3=angle(ucl) uc4=sqrt(2*g/aB1)*CDL4*(ao*rlao*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 = onehalf 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 crosssection 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(0EB) cos(0E) (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(8eB) (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 crosssection: 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 Crosssection Now consider two cases: 1) Rectangular crosssection, i.e., Bo = B, and 2) Triangular crosssection, 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 CrossSection For a triangular crosssection the hydraulic radius is related as a square root of the area, as shown below: 84 Figure B.2 is a triangular crosssection 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 crosssection. (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 weightedaverages 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 Weightedaverage 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: 