Hydraulics and stability of multiple inlet-bay systems

UFL/COEL-2002/014

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

by

Mamta Jain

Thesis

2002

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

By

MAMTA JAIN

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

UNIVERSITY OF FLORIDA

2002

Copyright 2002

by

Mamta Jain

ACKNOWLEDGMENTS

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

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

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

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

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

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

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

lumped parameter model for the stability of inlets.

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

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

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

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

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

encouragement and emotional support.

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

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

body and soul.

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

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

TABLE OF CONTENTS
page

ACKNOWLEDGMENT S ..___. ... ... .............. iii
LICKNST OF TABLEDGM EN... ..... ......................................................... ............. ii
L IS T O F T A B L E S ............................................................................................................ v ii

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

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

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

CHAPTER

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

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

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

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

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

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

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

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

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

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

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

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

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

APPENDIX

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

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

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

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

v

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

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

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

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

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

LIST OF TABLES

Table p

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

viii

LIST OF FIGURES

Figurege

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

LIST OF SYMBOLS

Symbols

AB, AB1, AB2, AB3

Ac Ac, Ac2, Ac3, Ac4

ao

aB, aB1, aB2, aB3

^B ^aB, aB2 B3

a,

a, b, c, A, B

B,

C, C1, C2, C3, C4

CD, CDL1, CDL2, CDL3, CDL4

CK

f

F

g

hk

i

K

k

bay water surface areas at MSL

flow cross-sectional areas of inlets

ocean (Gulf) tide amplitude

bay tide amplitudes

dimensionless bay tide amplitudes

constant that relates hydraulic radius with area of

triangular cross-section

constants defined to solve system of equations

dimensionless resistance factor

coefficients in linear relations of inlet hydraulics

linear discharge coefficients

prism correction coefficient of Keulegan

Darcy-Weisbach friction factor

friction coefficient

acceleration due to gravity

kinetic head

subscript specifying the inlet under consideration

Keulegan coefficient of filling or repletion

bottom roughness

ken

kex

Lc, L1, L2, L3, L4

m

P

Q, 01, Q2, 03, 04

Qm

R, R1, R2, R3, R4

Rt

Ro

ri, r2, r3

T

t

u

UB

Uc, Ucl, Uc2, Uc3, Uc4

Ueqi

Umaxl, Umax2, 1Umax3, 1max4

Uo

X

ao, aB

sB1, EB2, 8B3

5vl, 5v2, 5v3, 5v4

entrance loss coefficient

exit loss coefficient

channel lengths

sum of entrance and exit losses.

tidal prism

discharges through inlets

peak discharge

hydraulic radii

bay tide range

ocean (Gulf) tide range

polar representation of the bay tides

tidal period

time

velocity

bay current velocity

velocities through inlets

equilibrium velocity of inlet

maximum velocities through inlets

ocean (Gulf) current velocity

distance between UF and NOS tide stations

velocity coefficients

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

inlet velocity lags

y

0

77o

77B, 77B1, 77B2, 77B3

7B I7B1' 77B2' 7B3

eq

specific time when sea is at MSL

dimensionless time

water elevation

ocean (Gulf) tide elevation with respect to MSL

bay tide elevations with respect to MSL

dimensionless bay tide elevations

maximum bottom shear stress

equilibrium shear stress

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

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

By

Mamta Jain

December 2002

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

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

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

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

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

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

inlet cross-sectional area and the hydraulic environment.

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

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

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

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

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

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

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

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

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

of two sandy ocean inlets connected to interconnected bays.

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

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

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

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

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

drawn from this analysis.

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

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

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

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

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

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

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

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

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

channel increases as well.

CHAPTER 1
INTRODUCTION

1.1 Problem Definition

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

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

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

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

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

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

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

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

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

forcing.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

prevailing hydrodynamic conditions. These investigators found various stability

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Table 1.1 Cross-sectional areas of Johns Pass and Blind Pass in Boca Ciega Bay
Year Area (m) 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 5.3d 0.9d
d Estimated by assuming no change in channel width since 1974.

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

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

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

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

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

sections of the two inlets are listed in Table 1.3.

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

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

manner, with the following objective and associated tasks.

1.2 Objective and Tasks

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

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

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

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

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

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

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

1.3 Thesis Outline

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

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

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

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

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

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

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

biographical sketch of the author.

CHAPTER 2
HYDRAULICS OF A MULTIPLE INLET BAY SYSTEM

2.1 Governing Equations of an Inlet-Bay System

2.1.1 System Definition

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

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

SBay

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
77 +a, = r7 + a, +-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 = o7 -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

2
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
Entrance loss = ke (2.3)
2g

2
Exit loss = ke (2.4)
2g

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

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

J u2
Gradual loss = (2.5)
4R 2g

where

f= Darcy-Weisbach friction coefficient,

R = hydraulic radius of channel, and

L = Length of channel.

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

1o 7 -i k + ex + (2.6)
2g 4R

or

uc= -g 7o l .sign(o Bn) (2.7)
ke +k +
4R

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

tidal cycle.

2.1.3 Continuity Equation

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

rise and fall of bay water level, is given as

dt
=ucAc = dA(B) (2.8)

where

Q = flow rate through the inlet,

Ac = Inlet flow cross-sectional area, and

AB = bay surface area.

Therefore Eq. (2.8) becomes

AB drlB
u 4 d (2.9)
A dt

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

d,= Ac 2g _lo-B I.sign(o- ) (2.10)
dt A )
ken + kex +
4R

Next, we introduce the dimensionless quantities

-;e = 7o = ; 0n 2t = t (2.11)
a a T
0 0

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

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

di=- K | i.sign(iO-B) (2.12)
dO

where

K = A 2ga (2.13)
2;a ABkn +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

uI, -= gcDL (o ) (2.14)

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

CDL =a (2.15)
k, + k +_(_ o -'B 7)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
Bo c- = L (2.16)
C dO

where

C = D g (2.17)
C DL AR 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~7 = a S cos(0 E1)

10

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:

= a, = Re e)
C

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

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

3 Therefore

= =ir
d0B
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
aBI = (2.21)
a l+a2

(2.22)

EB = tan 1 a

The velocity uc through inlet 1 is therefore given by

Uci = Umaxi Cos(0 ,) (2.23)

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

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

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

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

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

SB 1-z/2.

2.3 Multiple Inlet-Bay System.

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

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

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

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

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

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

Thus the following relationships are obtained:

Uc CDL1 (ro rB) (2.24)

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

UC2 = gCDLrB-B2) (2.26)

Q2 = Uz2Ac2 = A2 d(2.27)
dt

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

where

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

Qi, Q2 = discharges through inlets 1 and 2,

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

AB1, AB2 = bay water surface areas.

CL1 a (2.28)
en1 o B1)max
k +k + ll
e, x 4R,

1 a
CDL2 = 1 a (2.29)
k +k + (2 7B1 7 B2max
e e4R2

where

L1, L2 = inlet lengths, and

R1, R2 = hydraulic radii of the channels.

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

17 I d,7t AB2 dBt 2 (2.30)
C, dt AL- dt

where

CI = CDL1 A,, (2.31)
Al Va

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

BI- 17B2 = L d7B2 (2.32)

where

C2 DL2 (2.33)
AB2 V aBI

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

in bays 1 and 2 now become

7sB = aBl cos(O- EsB) (2.34)

JB2 = aB2 cos(O- B2) (2.35)

where

a =
aBl
a

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

aB2
aB2 B2
ao
0a

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

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

corresponding HW or LW in the bay 1.

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

corresponding HW or LW in the bay 2.

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

a diR AR dR
C = drB A d (2.36)
o0 'T C, dO A4, dO

7Bl B =- d (2.37)
C2 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, = b, A2 =A, o =Re(e'")
C1 C2 AB

2 Let

71, = Re(aBle ( l))=r, ,2 =Re(aBe'<( 'B2))= r, =, d d=2 i
Ri ) 'dO dO

3 So the equations are reduced to

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

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

4 In the matrix form they become

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

5 The solution is

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

-1
r2 = (2.42)
X

where

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

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

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

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

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:

aB1 =Re(t)2 +Im()2 (2.43)

-'B tan -' t (2.44)
Re(rji)

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

1Im(r, )(
EB2 -tan I'(r2) (2.46)
Re(r2))

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

UcI = Umaxi Cos(0- cE) (2.47)

Uc2 = Umax2 COS(0 E,2) (2.48)

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

and 8v2 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 r7B1 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 zr/2. Therefore, v1 = EB1-r/l2 and. v2 = SB2-r2.

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 A B2

rio
L2

L1 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.I L3 (o 71) (2.49)

where Uc3 = velocity through the inlet 3 and

DL3 a (2.50)
ke +k +L3 (7,o B1)max
c e4R3

where

L3 = inlet 3 length, and

R3 = hydraulic radius of inlet 3 channel.

The governing equations of continuity are

Q1 +Q3 = ucAc +c3A,3 = AB d +AB2 d2 (2.51)
dt dt

Q2 = 2A2 = AB2 dB2 (2.52)
dt

where

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

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

AB1, AB2 = bay water surface areas.

Substituting for the velocity expressions in the above equations we obtain

1 d, AB2 dB2 (2.53)
7o r1 dCI + A L2 (2.53)
SCI +C3 dt AR dt

7B- = IL dtB2 (2.54)

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

C3 =DL3 A (2.55)
AR Vo

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

o g [ dB1R A B2 dB2 (2.56)
SC, +C3 dO A 4,1 dO

B B2 = 7 2 (2.57)
C2 dO

where oj, '1s 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 AB2
=a, -=b, =A, o =Re(e")
C, +C3 C2 A B

2 Let

=R el /d l r dv 2 r2
B, =Re(e'" ,B1) B, = Re(e'("'-B2)) r2; dI I7B d iX

3 So the equations are reduced to

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

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

4 Solve these equations by the matrix method.

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

5 Solving the above equations yields

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

-1
r- = (2.62)
X

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(2.68):

4 = 1CDL4 (B1- B3 ) (2.68)

where Uc4 = velocity through the inlet 4 and

S1 aBi
CDL4 I
ken +k L4 (7B1 -7B3 )max
4R4

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

The governing continuity equations are

dr]B drB3
Q1 = u1A,1 = AB, +AB2 dB+A B (2.69)
at at dt

Q2 = UcAc2 = A2 d (2.70)
dt

Q4 = uc4Ac4 = AB3 (2.71)
dt

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

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

AB1, AB2, AB3 = bay water surface areas.

Substituting the velocity expressions in the above equations we obtain

7o 1 C dI A,2 dB AB3 dyB (2.72)
SmI C, dt A,, dt A,, dt

BI =72 dU7B2 (2.73)
1 dt

1B 17B d7B3J (2.74)
S-B3 =C dt

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

C4 =CDL4A (2.75)
A^vtni

r/o
Figure 2.4 Three bays and three inlets with one inlet connecting to the ocean.

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

solving the following three equations:

S 7 [ d=Bl + AB2 dB2 + AB3 dB3 (2.76)
S C, dO A, dO A4, dO

B1 -B2 7 (2.77)
C2 dO

BI B3 B 3L (2.78)
C4 dO

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

B3 is

=B3 = 3 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 A
-=a b,-= c, B =A, A-B B, oRe(e"')
C, C2 C4 A,, A B

2 Let

B, = Re(el(rB1=)) B2 = =Re(e'( B2))= r, 7q,3 = R(el( B3ba)= r3

d9B 1 dB 2 di 3
dO dO dO

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)
0 r3 -1 0 ci +1

yields

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

-i(c-i
r = (2.85)

r3 = (2.86)
X

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

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

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

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

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

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

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

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

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

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)
gB2 -tan-1 1 (r2 (2.90)
Re(r2)

aB = Re(r3) + Im(r)2 (2.91)

B3 =-tan 1 1(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 = max4 COS ( ,4) (2.93)

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

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

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

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

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

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

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

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

The governing continuity equations are written as follows.

1 + 3 = clAc + 3A3 = AB, d + A B2 +4 ABB3 (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

7o B = I [1 d + AB2,, dyB2, AB3 dyB3 (2.97)
C0 +C3 dt A,, dt AB1 dt

B1- B2 = 1 L dB2 (2.98)
IC L dt I

L L3

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

B = 1B L 7BJ (2.99)
3C4 dt

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

S= c[ d- + AB2 dB2 A3 dB3 (2.100)
- -+ (2.100)
C, +C, dO AI, dO AB, dO

-l B2 7-d (2.101)
C2 dO

-Bl -4 dLB3 (2.102)
C4 dO

where o, 71I, rB2 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 OC A ABB
=a, = b, = c, =A, B3 = B, = Re(e)
C, +C3 A2 C4 A,, A, I

2 Let

B, = Re(e' B1 )= ,, B2 = Re(e'l B2= 2 rB3 = Re(e(e B--)=r 3

d 1 = r diB2 d i r3
dO dO dO 3

3 So the equations are reduced to

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

0 = -r, + (bi + )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)
r1 = (2.107)
X

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

r3 = (2.109)
X

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

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

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

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

-a( +A+ B+b2 +c2 + 2 +Bbc+ Ac2 +Bb2 bcB)
Im()i) =

-Re ab -aBb + aBc +1+ c2 abc2
Re(r,) = 7
AX

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

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

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

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- 1m(r2 (2.113)
R e(r) )

aB3 = 4Re(r) + Im(r)2 (2.114)

28

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

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

(2.93), respectively.

CHAPTER 3
STABILITY OF MULTIPLE INLET-BAY SYSTEMS

3.1 Stability Problem Definition

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

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

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

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

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

forces.

3.2 Stability Criteria

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

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

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

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

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

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

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

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

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

equilibrium stress in fairly narrow range:

3.5Pa < iq < 5.5Pa

The value of actual shear stress is obtained from

ZT= PFUmax Umax 1 (3.1)

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

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

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

function of following form

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

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

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

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

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

primarily via a change in the cross-sectional area.

3.2.1 Stability Analysis for One-Inlet Bay System

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

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

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

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

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

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

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

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

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

intersects the closure curve i= i(A).

7->

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

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

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

3.2.2 Stability of Two Inlets in a Bay

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

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

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

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

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

with increasing Ai.

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

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

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

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

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

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

area attains the value As.

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

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

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

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

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

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

,T2 = e

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

A1 A,, A, A,,, A2

It I I
+1 I
\ I i

F\ / i v
\,,, /

\ /

A, /

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

A2 A2

2 2

(a) (b)

A2 A2

2 2

(c) (d)

Figure 3.5 Possible configurations of equilibrium flow curves for a two-inlet bay system.
Stable equilibrium flow area is represented by 0 and unstable equilibrium is
represented by 0 The hatched area in (a) represents the domain of the stable
equilibrium flow area (source: van de Kreeke, 1990)

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

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

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

3.3 Stability Analysis with the Linearized Model

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

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

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

de Kreeke (1990) linearized lumped parameter model.

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

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

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

dimension tidal model to describe the hydrodynamics of the bay.

3.3.1 Linearized lumped parameter model for N Inlets in a Bay

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

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

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

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

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

5 There is no fresh water discharge in the bays.

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

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

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

0 =- (3.2)
p ix ph

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

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

r = pFu u (3.3)

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

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

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

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

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

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

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

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

expressed as

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

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

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

shown in Appendix B to yield

8 2gR,
Umaxu,( = (7 ) (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 +- -ABBu2m ju = ABao jue e (3.9)
1 2g 37;

in which the dimensionless resistance factor B, is defined as

B = 21- (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 E, yields the equation for Umaxi

L [ABC]2 BUax = [ABa] 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 8/337rpF

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

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

2[AB] BZ-U maxq =eq[ABao]2 maxeqA (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 R a = a (See Appendix B).

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

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

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

B,21L (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
(^ ^e,)2
( \ 22 eq )2

(ABao2) ABoO)4 -~ 2 I 2 (ABg)2 (FLIu3eq1 ...F NLNWNu3e 2 (3.16)
2 (FIWil + Fl+...+F W 3e)2
Wu 3q1 +...+ FLWu

When any A, (from Eq. (3.16)) is known, the cross-sectional areas of the other inlets
follow from Eq. (3.14), with B, given by Eq. (3.15), provided that

ABa2o > 2i) 1(Li 3eql +...+FL 3eq ) (3.17)

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

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

Eq. (3.14) we get,

Lr a 2 NLN 3 (ABaF)2 L+(3
u eql L eqN Az -(Agao[) Az +
(3.18)
-~ (A ))2 2 4 0
3h a g eq

in which sets of A, are given by Eq. (3.18) (as we have two real and positive solution for
A,). When any A, is known, the cross-sectional areas of the other (N-l) inlets follow from
Eq. (3.14) with B, given by Eq. (3.15). One root of Eq. (3.18) is always negative. The
other two are real and positive roots provided that
Aa3 3 2 5 FlL2 5 FNLN 21
A 3 > u15eq +....N (3.19)
2 37r eqa ag g j

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

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

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

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

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

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

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

3.4 Application to St. Andrew Bay System

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

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

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

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

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

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

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

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

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

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

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

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

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

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

equilibrium curves for this case is discussed in Chapter 5.

Zone-4
IFigure 3.6 Equilibrium flow curves for two inlet Kreeke, 1990)

Zone-1 A,

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

CHAPTER 4
APPLICATION TO ST. ANDREW BAY COMPLEX AND ENTRANCES

4.1 Description of Study Area

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

soil to mitigate shoreline erosion.

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

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

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

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.

lect ion
Will lo
ofthmet)

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

4.2 Summary of Field Data

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

D-2

11 \ -\ n r,

410000.00 cA

4
40900000

161000000 1611000.00 1612000.00 1613000,00 1614000.0 1615000.00

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

4.2.1 Bathymetry

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

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

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

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

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

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

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

47

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

-6

-8
S-10

-12

-14

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

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

Bottom Contour

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

--ADCP

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

4.2.2 Tides

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

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

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

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

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

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

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

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

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

and Figure 4.10.

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

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

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

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

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

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

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

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

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

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

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

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

4.10, and the results are summarized in Table 4.5.

49

Tide at St Andrew Bay Entrance
-Tide at St Andrew Bay Entrance
0.6
0.5 -
S0.4
g 0.3
0.2
0.1
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

O i- i- i- - 0 O O O O O O 0

Time (hrs)
09/18/01 Time09/19/01

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

NOS Tides Tides

0.45
0.4
E 0.35
0.3
> 0.25
0.2
0.15
0.1
0.05 -
0> I t o

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

Y Y

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

50

NOS Tides

0.5

0.4

E 0.3

> 0.2
a)
-j
S0.1

U 0

-0.1

-0.2

- Tides

0I )" to C O t0 0') 0 0 Cc) LO t o C .0
o d ok In sh cr In C L o C- D,, In CM o
Sc "\". inC Od L. c N O d \O O N

Time (hr:min)
12/18/01 12/19/01
y y-

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

Tides in all the Stations

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

-*- Panama City
---West Bay Creek

035

S03

' 025
,-
. 02

015

01

005

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

Time (hrs)

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

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

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

4.2.3 Current and Discharge

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

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

(Figure.4.3). The detailed velocity and discharge curves are shown in Jain and Mehta

(2001), Jain et al. (2002) and Jain and Mehta (2002). The measurements are summarized

in the Table 4.6.

From Table 4.6 it is observed that the average peak velocity in St. Andrew Bay

channel was approximately 0.63 m/s (at or close to the throat section) and at East Pass it

was approximately 0.50 m/s. The peak discharge value at St. Andrew was 4200 m3/s and

at East Pass it was 139 m3/s.

Table 4.6 Characteristic peak velocity and discharge values
Quantity Velocity (m/s) Discharge (m /s)
Cross-section Peak Flood Peak Ebb Peak Flood Peak Ebb

A 0.63 -0.62 4200 3620
B 0.45 -0.34 2980 2250
A' 0.68 -0.69 3620 3920
B' 0.69 -0.66 4061 3876
C' 0.67 -0.77 3480 3750
D 0.42 -0.49 2509 2777
E 0.51 -0.49 139 165
F 0.43 -0.38 114 101

4.3 Tidal Prism

Tidal prism is the volume of water that enters the bay during flood flow. Tidal

prism for St. Andrew Bay system was calculated using the approximate formula

P= QT (4.1)

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 = a Pb (4.2)

For inlets with two jetties, a = 7.49x10-4 and b = 0.86 (Jarrett, 1976). And for

inlets without jetty (East Pass), a = 3.83x105 and b = 1.03. The values of the tidal prism

are summarized in Table 4.7. Spring ranges are reported in Table 4.4.

It should be noted that the prism values from the O'Brien relationship are mere

estimates.

Table 4.7 Flood and ebb tidal prisms
Quantity Prism (m ) from peak Prism (m ) from O'Brien
discharge
Cross-sections Flood Ebb Peak Flood Peak Ebb

A 7.0x107 6.0x107 11.4x107 10.3x107
A' 8.6x107 9.4x107 09.0x107 10.4x107
D 8.6x107 9.4x107 10.0x107 09.7x107
E 3.3x106 3.9x106 03.8x10 04.6x106
F 3.9x106 3.5x106 03.6x106 03.6x106

CHAPTER 5
RESULTS AND DISCUSSION

5.1 Introduction

There are two aspects of this chapter, one dealing with the hydraulics of the St.

Andrew Bay system and the other with its stability. The linearized approach developed in

Chapter 2 is used to examine the hydraulics of St. Andrew Bay under different

conditions. The model is run as one-inlet/one-bay system for both September 2001 and

March 2002. It is also run as a three-inlets/three-bays system in September 2001 when

East Pass was closed, and as a three-bays/four-inlets system when East Pass was open in

March 2002. Hydraulic parameters related to tides and currents thus obtained are then

compared with values from the hydrographic surveys done in September 2001 and March

2002.

In contrast to hydraulics, the linearizedd lumped paramter model) inlet stability

model developed in Chapter 3 is applied only to St. Andrew Bay. A qualitative approach

is developed to discuss the results and graphs have been plotted to show stability

variation.

5.2 Hydraulics of St. Andrew Bay

The solution of equations for the linear model, derived in Chapter 2, forms the

basis of calculation of the hydraulic parameters characterizing the system. One begins

with the basic model of one-inlet (St. Andrew Bay Entrance) and one-bay (St. Andrew

Bay) system, when East Pass was closed. As noted the model is then extended to the

complete system of three bays (St. Andrew Bay, East Bay and North + West Bays) and

three inlets when East Pass was closed in September 2001, and finally as three bays and

four inlets when East Pass was open in March 2002.

5.2.1 Solution of Equations

The solutions of the relevant hydraulic equations are given in Chapter 2. A Matlab

program (see Appendix A) was developed to solve the one-inlet bay system as well as the

multiple-inlet bay system. The input and output parameters for each system are listed in

the tabular form.

5.2.1.1 One-inlet one-bay system

The one-inlet one-bay system is based on solving Eq. (5.1):

o = (5.1)
C dO

The required input and output parameters for this case are given in Table 5.1.

Table 5.1 List of input and output parameters for one-inlet one-bay model.
Input Parameters
ao Ocean tide amplitude (Gulf of Mexico)
T Time period of tide
aBl Bay 1 tide amplitude (St. Andrew Bay)
AB1 Bay 1 surface area
Li Length of inlet 1 (St. Andrew Bay Entrance)
R1 Hydraulic radius of inlet 1
Ac, Inlet 1 cross-section area
k Entrance and exit losses
,f Friction factor
(r/o riB)max Maximum ocean-bay tide difference
Output Parameters
17B1 Bay 1 tide
aB1 Bay 1 tide amplitude
EB1 Phase difference between bay 1 and ocean tides
Umaxi Maximum velocity through Inlet 1
Evi Phase difference between velocity in Inletl and ocean tide
(r/o r/B1)max Maximum ocean-bay tide difference

5.2.1.2 Three inlets and three bays with one inlet connected to ocean

This system is based on solving Eq. (5.2), Eq. (5.3) and Eq. (5.4):

a do l, A 2 dB2 A B3 dbB3
C, dO A,, dO AI, dO

(5.2)

(5.3)

2 dOja
n -n^=---

=B 3 dB3 (5.4)

The required input and output parameters for this case are given in Table 5.2

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

Input Parameters
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)
RI Hydraulic radius of inlet 1
Ac, Inlet 1 cross-section area
L2 Length of inlet 2 (connecting East Bay and St. Andrew Bay)
R2 Hydraulic radius of inlet 2
Ac2 Inlet 2 cross-section area
L4 Length of inlet 4 (connecting West Bay and St. Andrew Bay)
R4 Hydraulic radius of inlet 4
Ac4 Inlet 4 cross-section area
k Entrance and exit losses
,f Friction factor
(ro -B) max Maximum ocean-bay tide difference
(iB1 -iB2) max Maximum Bay 1 and Bay 2 tide difference
(/ -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
r/B2 Bay 2 tide
aB2 Bay 2 tide amplitude
SB2 Phase lag between bay 2 and ocean tide
r/B3 Bay 3 tide
aB3 Bay 3 tide amplitude
EB3 Phase lag between bay 3 and ocean tide
Umaxl Maximum velocity through Inlet 1
Evi Phase difference between velocity of Inlet 1 and the ocean tide
Umax2 Maximum velocity through Inlet 2
Ev2 Phase difference between velocity of Inlet 2 and the ocean tide
Umax4 Maximum velocity through inlet 4
Ev4 Phase difference between velocity of Inlet 4 and the ocean tide
(ro -r/B1) max Maximum ocean-bay tide difference
(1/i -r1B2) max Maximum Bay 1 and Bay 2 tide difference
(7i -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):

o =CB +c [ d + A2 d B2 AB3 dB3
C, +C3, dO ABI dO ABI dO

Bla- Bdr -
udO4

7B B3 = d
m 3C4 d0

(5.5)

(5.6)

(5.7)

The required input and output parameters for this case are given in Table 5.3.

Table 5.3 List of Input and Output Parameters for the four inlets and three bays model.
Input Parameters
ao Ocean Tide Amplitude (Gulf of Mexico)
T Time period of the tide
aBl Bay 1 tide amplitude (St. Andrew Bay)
aB2 Bay 2 tide amplitude (East Bay)
aB3 Bay 3 tide amplitude (West Bay)
AB1 Bay 1 surface area
AB2 Bay 2 surface area
AB3 Bay 3 surface area
Li Length of inlet 1 (St. Andrew Bay Entrance)
R1 Radius of inlet 1
Ac1 Inlet 1 cross-section area
L2 Length of inlet 2 (connecting East Bay and St. Andrew Bay)
R2 Radius of inlet 2
Ac2 Inlet 2 cross-section area
L3 Length of inlet 3 (East Pass)
R3 Radius of inlet 3
Ac3 Inlet 3 cross-section area
L4 Length of inlet 4 (connecting West Bay and St. Andrew Bay)
R4 Radius of inlet 4
Ac4 Inlet 4 cross-section area
k Entrance and exit losses
.f Friction factor
(r/o -rB) max Maximum ocean-bay tide difference
(ri 7B2) max Maximum Bay 1 and Bay 2 tide difference
(ri -17B3) max Maximum Bay 1 and Bay 3 tide difference
Output Parameters
7/1 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
8B3 Phase lag between bay 3 and ocean tide
Umaxl Maximum velocity through Inlet 1
vE, 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
Ulmax3 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 -r/B1) max Maximum ocean-bay tide difference
(/B1 -17B2) max Maximum Bay 1 and Bay 2 tide difference
(B1 -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 -/7B3) 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 )arameters 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 0.1 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 km
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
R1 10 m Measured in survey
Ac1 6300 m2
L2 1000 m
R2 9 m
Ac2 1.9x104 m2 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.
40 m Measured in survey. Ac3 is zero for three-
R3 3 m
A3 255 M2 bays and three-inlets case.
Ac3 255 m
k 1.05
f 0.025
(qo -r~1),,,, 0.037 0.036
mx Assumed initial values. Calculations are
( -2) max 0.060 0.063
1 max shown in appendix C
(B1 -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
aBl 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%
8vl -1.20 rad -1.22 rad 2%
(r7o -r1) max 0.038 0.036 6%
Three Bay Three Inlets System, September 2001
aBl 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%
v1l -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 -
(7o -rB1)max 0.037 0.037 0%
(71 -7B2) max 0.003 0.060 95%
(r7B1 -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%
v1i -1.40 rad -1.40 rad 0%
(7o -r 1) 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%
EB2 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%
vl -1.35 rad -1.40 rad 4%
Uc2 max 0.04 m/s Not measured
8v2 -1.40 rad Not measured
Uc3 max 0.60 m/s 0.55 m/s 9%
6g3 -1.35 rad -1.40 rad 4%
Uc4 max 0.22 m/s Not measured -
4v4 -1.31 rad Not measured -
(r7o -1) max 0.035 0.035 0%
(01 -7B2) max 0.003 0.063 95%
(1 -1B3) max 0.012 0.010 20%

It is evident from Table 5.5 that the linear model gives good results. The percent

error decreases if the system is modeled as a three-bay system, which is actually the case.

Velocity and tide amplitudes are within reasonably small error limits. The phase

differences between ocean (Gulf) and bay tides from data are very approximate as they

are calculated based on weighted-average tides at selected stations. Moreover, there are

very few stations to yield a good value of tide for a bay. Note that the input values for (r7

-rB1) max, (rB1 -rB2) max (i1 -rB3) maxiS also approximate. Sample calculation for (7ro -r/B)

max, (iB1 -772) max (7B1 -1B3) max is given in Appendix C.

5.3 Stability Analysis

The stability analysis developed in Chapter 3 is now applied to St. Andrew Bay

system. This analysis is done for a two-inlet bay system using van de Kreeke's (1990)

linearized lumped parameter model. The two inlets, to which the model is applied, are St.

Andrew Bay Entrance and the new East Pass opened in December 2002. Calculations

related to stability are given in Appendix D. A Matlab program (Appendix D) has also

been developed for doing the analysis and generating equilibrium flow curves for the two

inlets. There are two programs, one for rectangular channel cross-section and another for

triangular channel cross-sections.

5.3.1 Input Parameters

Input parameters required for the Matlab program (Appendix D) are listed in

Table 5.6. Since the objective was to study the effect of bay area on the stability because

the results are sensitive to it, it is held constant for a particular set of calculation, but is

varied for generating different sets of equilibrium flow curves. Similarly the length of

East Pass, believed to have an uncertain value due to the complex bay shoreline and

bathymetry in that region is also varied to study its effect on the system.

Table 5.6 Input parameters for stability analysis.
Input Parameters for December 2001
ao 0.26 m Amplitude of ocean tide
T 18.0 hrs Time period of tide
AB 74-105 km 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
Li 1340 m Length of Inlet 1
al 0.138 Constant for triangular cross-section for Inlet 1 (see Appendices
C and D)
F, 0.004 Friction coefficient for Inlet 1
Inlet 2
Ueq2 0.45 m/s Equilibrium velocity for Inlet 2 (see Appendix D)
W2 300 m Width of Inlet 2
L2 500-2000 m Length of Inlet 2 (East Pass) (varied from 500 m to 2000 m)
a2 0.187 Constant for triangular cross-section for Inlet 2 (see Appendices
C and D)
F2 0.004 Friction coefficient for Inlet 2

5.3.2 Results and Discussion

As noted, it is found that two inlets can never be unconditionally stable

simultaneously in one bay. The bay area has a large effect on the stability of the inlets.

Table 5.7 summarizes this effect. It is clear that with a small increase in bay area the

inlets become stable. This is also demonstrated with the help of equilibrium flow curve in

the Figure 5.1, Figure 5.2 and Figure 5.3 for rectangular cross-section and Figure 5.7 and

Figure 5.9 for triangular cross-section. The cross-sectional area pair during December

2001 (Table 4.3) [5210, 255] is shown by the dot. Figure 5.1 and Figure 5.7 have small

bay areas, and the dot lies outside the equilibrium flow curve indicating that both inlets

are unstable. As the bay area increases St. Andrew becomes stable (Figure 5.2 and Figure

5.7), and a further increase in bay area also stabilizes East Pass (Figure 5.3 and Figure

5.9). However, in reality we cannot increase the bay area beyond a reasonable limit,

because then the basic assumption of bay tide fluctuating evenly in the bay does not hold.

Moreover, in a shallow bay the effect of dissipation of tidal energy cannot be ignored,

especially if the bay is large. Also as per Figure 3.5 two inlets are not stable

simultaneously.

An increase in the length of East Pass has a destabilizing effect on that inlet as

shown in the Table 5.7. Note also that for a rectangular cross-section (Figure 5.3) with

the length of East Pass of 500m, this inlet is stable, whereas with a length of 2000 m

(Figure 5.6) the inlet is instable. This is because as the length increases the dissipation

increases. Friction dominated losses, (F 0.004, R 3m (2FL /R)) for East Pass with

500 m length is 1.33, where as that for 2000 m length it is 5.33. The same cases occur in

Figure 5.9 and Figure 5.12.

The other effects on the stability model are the approximation in the cross-section

of the inlet. It is clear that triangular cross-section is a better approximation than

rectangular section, because with the same parameters for rectangular cross-section in

Figure 5.6, East Pass is predicted to be unstable whereas in Figure 5.12 for triangular

cross-section, East Pass is stable even though barely, which is not believed to be the case

for this newly opened inlet.

Table 5.8 gives the qualitative indication of the stability. The various zones

mentioned in the Table 5.8 are described in Section 3.4 and Figure 3.6. It is clear from

these results that St. Andrew is a stable inlet (for a realistic bay area) as opposed to East

Pass. This is also evident from the Figure 3.5, which shows that two inlets cannot be

stable simultaneously, because we for unconditional stability, need four real points of

intersection of equilibrium flow curve and none of the solutions (neither rectangular

cross-section nor triangular cross-section) gives four real solution.

The model does not yield an analytic solution for a more realistic parabolic cross-

section. Another weakness is due to the assumptions made in Chapter 3 including a bay

area in which the tide is spatially always in-phase, and simple a harmonic function for

tide. These assumptions are not always satisfied.

Table 5.7 Effect of change in bay area and length of East Pass.
Rectangular cross-section
Run Bay East Pass Result
No. area (km2) Length
(m)
1 74 500 Both inlets unstable (Figure 5.1)
2 90 500 St. Andrew becomes stable (Figure 5.2)
3 105 500 St. Andrew stable, East Pass barely stable (Figure
5.3)*
4 74 2000 Both inlets unstable (Figure 5.4)
5 90 2000 St. Andrew barely stable (Figure 5.5)
6 105 2000 St. Andrew stable, East Pass unstable (Figure 5.6)
Triangular cross-section
7 74 500 Both inlets unstable Figure (5.7)
8 90 500 St. Andrew becomes stable (Figure 5.8)
9 105 500 Both inlets stable (Figure 5.9)
10 74 2000 Both inlets unstable (Figure 5.10)
11 90 2000 St. Andrew stable (Figure 5.11)
12 105 2000 St. Andrew stable, East Pass just stable (Figure 5.12)
Two inlets cannot be simultaneously stable, because according to Figure 3.5, for
unconditional stability we need four real points of intersection of equilibrium flow curve,
which is not possible in either rectangular cross-section solution nor triangular cross-
section solution.

66

Inlet Stability -1 (rectangular section)

4500
4000
3500
3000
2500
- 2000
L 1500
1000
500
0

2000

3000

A1, St Andrew (m2)

- East Pass St Andrew

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

Inlet Stability 2 (rectangular section)

6000

_5000 .

4000 n

S3000- o
" I
L 2000- *

1000 -

0 1000

2000

3000

A1, St Andrew (m2)

5000

6000

- East Pass St Andrew

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

5000

-," .

E
E *. -
-
-
- - - - ** * * * *

I
I -
I

I
I
I
I
I

I -
I -
I
I -
I -
I
I -
I -

-.

67

Inlet Stability 3 (rectangular section)

6000

5000 -

4000 a"*-.-"-
E
v 4000"

3000-

31000 "--

o ----..--......----
0 1000 2000 3000 4000 5000 6000 7000
A1, St Andrew (m2)

East Pass St Andrew

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

Inlet Stability -4(rectangular section)

4500
4000
S3500
E
3000
S 2500
- 2000
L 1500
1000-
500
0
0

2000

3000

4000

5000

5000

A1, St Andrew (m2)

- East Pass St Andrew

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

'" .-
X
C . .. .

E -
I -
E -

c- -- --------------- ********

-

Inlet Stability 5(rectangular section)

6000

5000 -
.. *_-_
E 4000 '-.

1000
S-------.. -- ------------------------
0 1000 2000 3000 4000 5000 6000
A1, St Andrew (m2)

East Pass St Andrew

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

Inlet Stability 6 (rectangular section)

6000

S5000- -

4000 g .

3000 "------

4 2000 "*-"--.

1000
6000 | "'----

0 --.................................

0 1000 2000 3000 4000 5000 6000 7000
A1, St Andrew (m2)

East Pass St Andrew

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

69

Inlet Stability -7 (triangular section)

5000
4500 -_
R 4000 --
S3500 -
3000 "-.-.
2500
S2000 -
L 1500
S1000--_
500

0 1000 2000 3000 4000 5000
A1, St Andrew (m2)

East Pass St Andrew

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

Inlet Stability 8 (triangular section)

6000

5000 --

g4000 -_

S3000 -. -
-Th-
w" 2000

1000 -

0 1000 2000 3000 4000 5000 6000
A1, St Andrew (m2)

East Pass St Andrew

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

70

Inlet Stability 9 (triangular section)

7000
6000
E 5000
S4000
3000
L 2000 ---

1000
0
0 1000 2000 3000 4000
A1, St Andrew (m2)

5000 6000 7000

East Pass St Andrew

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

Inlet Stability 10 (triangular section)

1000

2000 3000
A1, St Andrew (m2)

4000

5000
5000

- East Pass St Andrew

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

5000
4500
i 4000
S3500
D 3000
a 2500
S2000
W 1500
S1000
500
0

'---

Inlet Stability 11 (triangular section)

3000
A1, St Andrew (m2)

East Pass St Andrew

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

Inlet Stability 12 (triangular section)

7000
6000
S5000
S4000
S3000
- 2000
1000

0 1000 2000 3000 4000
A1, St Andrew (m2)

5000 6000 7000

- East Pass St Andrew

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

6000

5000
CN
E
4000

. 3000

w 2000

S1000

0

2000

4000

5000

6000

Table 5.8 Stability observations for St. Andrew Bay Entrance and East Pass.
Figure Placement of cross-sectional Observations
area pair [A1, A2], (black dot)
Figure 5.1 Zone-1 Both inlets are unstable
Figure 5.2 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.3 Zone-4 Only one is stable i.e. St. Andrewa
Figure 5.4 Zone-1 Both inlets are unstable
Figure 5.5 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.6 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.7 Zone-1 Both inlets are unstable
Figure 5.8 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.9 Zone-4 Only one is stable i.e. St. Andrewa
Figure 5.10 Zone-1 Both inlets are unstable
Figure 5.11 Zone-2 St. Andrew Bay Entrance is stable
Figure 5.12 Zone-4 Only one is stable i.e. St. Andrewa
a As per Figure 3.6, it is clear that even in Zone-4 only one inlet is stable, this is further
clarified from Figure 3.5, which shows that only one inlet can be stable at one time.

CHAPTER 6
CONCLUSIONS

6.1 Summary

St. Andrew Bay, which is a composite of three interconnected bays (St. Andrew

Bay proper, West Bay + North Bay and East Bay) is located in Bay County on the Gulf

of Mexico coast of Florida's panhandle. It is part of a three-bay and two-inlet complex.

One of these inlets is St. Andrew Bay Entrance and the other is East Pass, which are both

connected to St. Andrew Bay on one side and the Gulf on the other. Prior to 1934, East

Pass was the natural connection between St. Andrew Bay and the Gulf. In 1934, St.

Andrew Bay Entrance (Figure 4.2) was constructed 11 km west of East Pass through the

barrier island to provide a direct access between the Gulf and Panama City. The interior

shoreline of the entrance has continually eroded since it's opening. East Pass was closed

in 1998, which is believed to be due to the opening of the St. Andrew Bay Entrance.

In December 2001, a new East Pass was opened (Figure 4.3), and the effect of this

new inlet is presently being monitored over the entire system. Accordingly, the objective

of the present work was to examine the hydraulics of the newly formed two-("ocean")

inlet/three-bay system and its hydraulic stability, especially as it relates to East Pass.

The first aspect of the tasks performed to meet this objective was the development

of equations for the linearized hydraulic model for the system of three bays and four

inlets (two ocean and two between bays), and solving and applying them to the St.

Andrew Bay system. The second aspect was the development of the ocean inlet stability

criteria using the Escoffier (1940) model for one inlet and one bay and extending this

model to the two ocean inlets and a bay. Stability analysis for the St. Andrew Bay system

was then carried out using the linearized lumped parameter model of van de Kreeke

(1990).

6.2 Conclusions

The following are the main conclusions of this study:

1 If the system is modeled as a three-bay system as compare to a one-bay system,
the error in the phase difference, SB1, decreases from 6% to 0% and in the velocity
amplitude from 3% to 2%. Moreover the error in maximum head difference, (/o -
tB1i) m, also decreases from 6% to 0%.

2 The amplitudes of velocities and bay tides are within 5%, which is a reasonably
small error band. The percent error for St. Andrew Bay is almost 0%, and for the
other bays it is within 20%.

3 The bay area has a significant effect on the stability of the two inlets. At a bay
area of 74 km2 both inlets are unstable. Increasing it by 22% to 90 km2 stabilizes
St. Andrew Bay Entrance, and by 42% to 105 km2 stabilizes East Pass as well.

4 Two inlets can never be simultaneously unconditionally stable.

5 Keeping the bay area at 105 km2 and increasing the length of East Pass from 500
m to 2000 m destabilizes this inlet because as the length increases the dissipation
in the channel increases as well.

6 A triangular channel cross-section is a better approximation than a rectangular
one, because given the same values of all other hydraulic parameters, St. Andrew
Bay Entrance with a rectangular cross-section is found to be barely stable,
whereas with a triangular cross-section it is found to be stable, as is the case.

6.3 Recommendations for Further Work

Accurate numerical values required for the stability analysis of a complex inlet-

bay system can only be obtained by using a two- (or three)-dimensional tidal model to

describe the hydrodynamics of the bay.

Freshwater discharges from the rivers into the bay should be incorporated through

numerical modeling.

75

Including a more realistic assumption for the channel cross-section can improve

the stability analysis.

APPENDIX A
ALGORITHMS FOR MULTIPLE INLET-BAY HYDRAULICS

A.1 Introduction

The linearized approach described in Chapter 2 has been used to evaluate the

hydraulic parameters of the multiple inlet bay system. The differential equations,

developed by this approach (Chapter 2), Eq. (2.100), Eq. (2.101) and Eqs (2.102), are

solved in Matlab Program-1 (given below). These are the general equations for four inlets

and three bays system. These equations can be used to solve from one bay system to the

complex three bays system. Note that for solving Program-i, the Matlab version should

have a symbolic toolbox. The present program is solved in Matlab release 6.1. The

solution from Program-1 is used as input to Program-2 (given below). The required input

parameters and output for Program-2 are listed in Table 5.3 of Chapter 5.

A.2 Program-1

%UNIVERSITY OF FLORIDA
%CIVIL AND COASTAL ENGINEERING DEPARTMENT
%PROGRAM FOR SOLVING THE EQS 2.100, 2.101, 2.102
% ALL CONSTANTS DEFINED IN CHAPTER 2

clear all
syms a b c AB
tl=sym('thetal')
t2=sym('theta2')
t3=sym('theta3')
rl=sym('al*exp(-i*tl)')
r2=sym('a2*exp(-i*t2)')
r3 =sym('a3 *exp(-i*t3)')
C=[a*i+l a*A*i a*B*i;-l b*i+l 0;-1 0 c*i+l]
D=[1;0;0]
%END

A.3 Program-2

%UNIVERSITY OF FLORIDA
%CIVIL AND COASTAL ENGINEERING DEPARTMENT
%PROGRAM FOR CALCULATION OF MULTIPLE INLET-BAY HYDRUALICS
%FOR ONE -INLET BAY CASE, FOR Ac2, Ac3, Ac4 EQUAL TO ZERO
%INLET 1 AND INLET 3 CONNECTS BAY1 TO THE OCEAN

clear all
g=9.81;
ao=0.212;%ocean tide amplitude
theta=0;%ocean tide phase
etao=ao*cos(theta);%ocean tide
T=25.82;%time period
q=2*pi/(T*3600)%sigma
k=1.05;% entrance and exit loss
f=0.025;%friction factor
aB 1=0.201 ;%approximate amplitude of bays
aB2=0.226;
aB3=0.2325;
%ml= 1 max(eta0-etab 1),m2=max(etab 1-etab2),m3=max(etab 1-etab3)
ml=0.023;
m2=0.0527;
m3=0.123;
%Inlet 1
L1=1340;%Length of inlet
R1=10;%hydraulic radius
Acl=6300;%CROSS-SECTION AREA of the inlet
Fl=k+(f*L1)/(4*R1);%friction factor F includes ken kex fL/4R

%Inlet 2
L2=1000;%Length of inlet
R2=9;%hydraulic radius
Ac2=1.9* 104;%CROSS-SECTION AREA of the inlet, it is zero for one inlet bay case
F2=k+(f*L2)/(4*R2);%friction factor F includes ken kex fL/4R

%Inlet 3
L3=400;%Length of inlet
R3=3;%hydraulic radius
Ac3=255;%CROSS-SECTION AREA of the inlet
F3=k+(f*L3)/(4*R3);%friction factor F includes ken kex fL/4R

%Inlet 4
L4=1000;%Length of inlet
R4=12;%hydraulic radius
Ac4=9.7* 10A3;%CROSS-SECTION AREA of the inlet
F4=k+(f*L4)/(4*R4);%friction factor F includes ken kex fL/4R

%bayl area
AB1=74*10A6;
%bay2 area
AB2=54*10A6;
%bay3 area
AB3=155*10A6;

%calculations
CDL =sqrt(ao/(m 1*F 1))
CDL2=sqrt(aB 1/(m2*F2))
CDL3 =sqrt(ao/(ml *F3))
CDL4=sqrt(aB 1/(m3 *F4))

C1=CDL1*Acl/AB1*sqrt(2*g/ao)
C2=CDL2*Ac2/AB2*sqrt(2*g/aB 1)
C3 =CDL3 *Ac3/AB 1 *sqrt(2*g/ao)
C4=CDL4*Ac4/AB3 *sqrt(2*g/aB 1)

%ALL THE CONSTANTS ARE DEFINED IN THE THESIS
a=q/(C1+C3)
if Ac2==0
b=0
else b=q/C2
end

if Ac4==0
c=0
else
c=q/C4
end

A=AB2/AB1
B=AB3/AB1
rl=(c-i)*(b-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b- +c*b-i*a*B+a*b-
i*a+a*B*b+a*A*c)%SOLUTIONS ARE OBTAINED FROM ANOTHER
r2=-i*(c-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b-l+c*b-i*a*B+a*b-
i*a+a*B*b+a*A*c)%MATLAB PROGRAM WHICH HAS SYMBOLLIC TOOLBOX.
r3=-i*(b-i)/(-i*a*A+i*a*c*b+a*c-i*c-i*b-l+c*b-i*a*B+a*b-i*a+a*B*b+a*A*c)
aB l=abs(rl)*ao
eB l=-angle(rl)
aB2=abs(r2)*ao
eB2=-angle(r2)
aB3=abs(r3)*ao
eB3=-angle(r3)
etaB l=aB l*cos(theta-eB 1)
etaB2=aB2*cos(theta-eB2)

etaB3=aB3*cos(theta-eB3)
CDL1 1=sqrt(ao/(max(etao-etaB 1)*F 1))
CDL22=sqrt(aB 1/(max(etaB 1-etaB2)*F2))
CDL33=sqrt(ao/(max(etao-etaB 1)*F3))
CDL44=sqrt(aB 1/(max(etaB 1 -etaB3)*F4))

C11=CDL1*Acl/AB1*sqrt(2*g/ao)
C22=CDL2*Ac2/AB2*sqrt(2*g/aB 1)
C33=CDL3*Ac3/AB l*sqrt(2*g/ao)
C44=CDL4*Ac4/AB3 *sqrt(2*g/aB 1)

%velocity in the inlet
ucl=sqrt(2*g/ao)*CDL1*(etao-ao*rl)
uclmax=abs(ucl)
evl=-angle(ucl)
uc2=sqrt(2*g/aB1)*CDL2*(ao*rl-ao*r2)
uc2max=abs(uc2)
ev2=-angle(uc2)
uc3=sqrt(2*g/ao)*CDL3*(etao-ao*rl)
uc3max=abs(ucl)
ev3=-angle(ucl)
uc4=sqrt(2*g/aB1)*CDL4*(ao*rl-ao*r3)
uc4max=abs(uc4)
ev4=-angle(uc4)
%END

APPENDIX B
INLET HYDRAULICS RELATED DERIVATIONS

B.1 Linearization of Damping Term

The linearization of the damping term in Eq. (3.6) is done as given in Bruun

(1978). The bay tide response is represented by

rq =asin( 0- B) (B.1)

where

0 = rt = at, dimensionless time.
T

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

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

HW or LW in the bay. Also,

ro = ao sin(0) (B.2)

from the continuity equation we further have

Au = A dqB (B.3)
dt

where Ac is the area of cross-section of the inlet and AB is the surface area of the bay.

The time of HW or LW in the bay, i.e., when dB = 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

d7Bd r7 C7 2 aB2 a cs(0-EB) cos(0- E) (B.4)
dt dtc

or in terms of Fourier series Eq. (B.4) can be written as

8 sini -
dq^ d^ 2 2" 2\
dt dt =caB 2 2)cosn(O-gB) (B.5)
dt dt nnr 4 nI

where n takes only odd integral values. For linearization purposes n=l, so that Eq. (B.5)

becomes

dB dB =ca -8cosn(8- ,) (B.6)
dt dt 3"

The amplitude of the tidal velocity is given by

mx a (B.7)
A

Therefore, it can be written as
8
SI = -umaxu (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) = +A, 2gR- 17, (B.10)
m R, +2FL, a

The expression for maximum tidal velocity can be obtained by the solution of the above

equations with the simplifying assumptions mentioned in Chapter 2.

umx = C(K)siny 2ABaOK (B.11)
AT K

where K is the coefficient of repletion,

T A4 2gR a
K T A = 2g=R (B.12)
K 2aa A, 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 Koo, 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:

-2a,A 1
A; pF 2QAB1 2 (B.14)

It is clear from the above equation that i has a strong dependence on A.

B.3 General Equation for hydraulic radius.

Consider the general trapezoidal cross-section:

1 1 B>
Area, A= -(B +B)h= -Bh 1+-B
2 2 B)

Wetted perimeter, P = B + 2 (B B)2 + h2 =B L+ + \ 2
4 B B

I h A+ B1
Hydraulic radius, R = = 2
P Bo B1 h\
B FBI B

83

Bo

Figure B.1 Trapezoidal Cross-section

Now consider two cases: 1) Rectangular cross-section, i.e., Bo = B, and 2)

Triangular cross-section, i.e., Bo = 0.

B.3.1 Rectangular

B=Bo, Therefore hydraulic radius for a rectangle is

A h
R-

B

B.3.2 Triangular

For triangular section, Bo= 0

1h
A 2

P 1+4 Ih2

B.4 Hydraulic Radius for Triangular Cross-Section

For a triangular cross-section the hydraulic radius is related as a square root of the

area, as shown below:

84

Figure B.2 is a triangular cross-section where /is the angle with the horizontal on both

the sides:

1
Area A = h2h tan /
2

Wetted perimeter P

Hydraulic radius R-

2h
cos f

1 V-Ja= v
2 sin/ cos

Figure B.2 Triangular cross-section.

(B.15)

MILLISECOND	CLASS.METHOD	MESSAGE
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