
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00029732/00001
Material Information
 Title:
 Tideinduced mass transport in shallow lagoons.
 Added title page title:
 Mass transport in shallow lagoons
 Creator:
 Van de Kreeke, Jacobus
 Publication Date:
 1971
 Language:
 English
 Physical Description:
 x, 112 leaves. : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Average linear density ( jstor )
Bodies of water ( jstor ) Boundary conditions ( jstor ) Inlets ( jstor ) Lagoons ( jstor ) Lakes ( jstor ) Ocean tides ( jstor ) Oceans ( jstor ) Sea transportation ( jstor ) Velocity ( jstor ) Fluid dynamics ( lcsh ) Lagoons ( lcsh ) Tides ( lcsh ) Transport theory ( lcsh ) City of Lake Worth ( local )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 109111.
 General Note:
 Manuscript copy.
 General Note:
 Vita.
 Statement of Responsibility:
 By Jacobus Van de Kreeke.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright J. Van de Kreeke. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 021864215 ( ALEPH )
18249307 ( OCLC )

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Full Text 
TIDEINDUCED MASS TRANSPORT IN SHALLOW LAGOONS
By
JACOBUS VAN DE KREEKE
A Dissertation Presented to the Graduate Council of
the University of Florida in Partial
Fulfillment of the Requirements of the Degree of
Doctor of Philosophy
UNIVERSITY OF FLORIDA 1971
ACKNOWLEDGEMENT
First, I would like to thank my wife, Hetty, for her understanding and patience during the time of deprived family life accompanying the preparation of this dissertation.
A dissertation to a large extent is the result of the interplay between the candidate and his academic environment; in this case, the faculty, staff and student body of the Department of Coastal and Oceanographic Engineering. Among those, a special word of thanks is due to Dr. R. G. Dean, Chairman and Professor of the Department, who supervised the dissertation; the many hours of discussion with Dr. Dean have greatly contributed to a better understanding of the subject. The helpful criticism and suggestions by Dr. B. A. Christensen, Professor of Civil Engineering, concerning the experiments is greatly appreciated.
The study was in part supported by the Office of the Water Resources Center and in part by the Florida Department of Natural Resources. All computations were carried out using the University of Florida IBM 360 computer.
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT ............... .............................. ii
LIST OF FIGURES ...... ....................................... iv
LIST OF TABLES................................................ vi
LIST OF SYMBOLS............................................... vii
ABSTRACT........................................................ ix
CHAPTERS:
1. INTRODUCTION.......................................... 1
2. THE TIDAL EQUATIONS FOR THE LAGOON AND THE INLETS..... 4 3. NUMERICAL SOLUTION OF THE TIDAL EQUATIONS............. 13
4. DETERMINATION OF THE FRICTION FACTOR F ................ 24
5. MASS TRANSPORT IN LAGOONS DUE TO TIDAL ACTION......... 30
a. Analytic Solution; Physical Concepts.............. 30
b. Influence of Inlets on the Mass Transport........ 47
6. LABORATORY EXPERIMENTS ................................. 52
a. Equipment and Procedure.......................... 52
b. Results........ ................................. 59
c. Computation of Mass Transport; Comparison with
Measured Data .................................... 76
d. Summary.................................. ....... 86
7. APPLICATION.......................................... 90
8. SUMMARY AND CONCLUSIONS............................... 96
APPENDIX ......... ............................................ 98
A. THE MATHEMATICAL DESCRIPTION OF LONG PERIOD GRAVITY
WAVES; EULERIAN DESCRIPTION ........................... 99
REFERENCES...................... .............................. 109
BIOGRAPHICAL SKETCH........................................... 112
iii
LIST OF FIGURES
Figure No. Page
1 Reference Frame................................ 5
2 Inlet Schematization........................... 8
3 Location of Variables in the Numerical Grid.... 14 4 Grid Scheme for Different Inlet Configurations. 18
5 Curves Represented by Equations (3.3) and
(3.4) ...................... ...... ............. 21
6 Tide in a Sea Level Canal....................... 27
7 Mean Water Level in a Lagoon in the Presence
of a Damped Progressive Wave................... 38
8 Effect of Higher Order Terms on q ............. 41
9 Effect of "Linearization" of the Friction Term
on q .......................................... 42
10 Effect of Convective Acceleration and Nonlinear Part of the Surface Gradient on the Net
Discharge q ................................... 44
11 Lagoon Connected to the Ocean by Inlets ........ 47
12 Net Discharge in Lagoon Versus Width of
Inlet II....................................... 50
13 ExperimentalSet Up ............................ 53
14 Measured Tide Curves for Experiment 5.......... 55 15 Measured Float Positions for Experiment 4...... 61 16 Measured Float Positions for Experiment 5a..... 62 17 Measured Float Positions for Experiment 13..... 63 18 Measured Float Positions for Experiment 13a.... 64
19 Determination of Net Discharge from Measured
Float Path ..................................... 68
20 Measured Float Positions for Experiment 22..... 69
iv
LIST OF FIGURES (Continued)
Figure No. Page
21 Measured Float Positions for Experiment 24 ....... 70 22 Weir; Flow from Lagoon to Tidal Basin............. 77
23 Grid Scheme for Numerical Computations of
Laboratory Experiments........................... 80
24 Computed and Measured Net Discharges.............. 82
25 Measured and Computed Float Positions for
Experiment 22..................................... 84
26 Measured and Computed Float Positions for
Experiment 25..................................... 85
27 Variation of Net Discharge with Depth............ 87
28 Lake Worth, Florida; Location of Tide Recorders.. 91 29 Schematization of Lake Worth ..................... 93
30 Measured and Computed Water Levels in
Lake Worth........................................... 95
V
LIST OF TABLES
Table No. Page
I Summary of Experiments; Test Series 1 ........... 58
II Summary of Experiments; Test Series 2........... 59
III Reproducibility of Experiments; Particle
Excursion Between Successive Slack Tides
for Experiments 13 and 13a........... ......... 65
IV Measured Drift Velocities and Net Discharges.... 72
V Computed Net Discharges; Evaluation of
Empirical Method of Determining Net
Discharges...................................... 83
VI Variation of Net Discharge in Experiment
20 for Different Combinations of Weir
Coefficients ............. ............................... 86
vi
LIST OF SYMBOLS
a tidal amplitude c wave celerity in case of no friction d total depth f DarcyWeisbach resistance coefficient g gravitational acceleration h mean depth k wave number k wave number in case of no friction
0
m coefficient which accounts for entrance losses and nonuniform
velocity distribution
q discharge per unit width q* net discharge per unit width s particle trajectory; in Appendix A used as strength of a source
per unit volume t time
u velocity in x direction u, net drift velocity v velocity in y direction w velocity in z direction x particle position at t = to
x horizontal Cartesian coordinate y horizontal Cartesian coordinate z vertical Cartesian coordinate, positive upward A crosssectional area Ab lagoon area
vii
B width of the water body C contraction coefficient
c
D total depth F resistance coefficient used when considering quadratic friction F resistance coefficient used when considering linear friction L length of the water body M lateral flow, rainfall P wetted perimeter Q total discharge R hydraulic radius S slope of the water surface X free surface stress/(mass density of water) in the x direction Y free surface stress/(mass density of water) in the y direction a phase angle C relative error In water surface elevation 'n mean water surface elevation
factor related to the resistance; also used as weir coefficient
in Chapter 6
"linear" weir coefficient p density of water a angular frequency of the tide T shear stress w angular frequency of the earth rotation
latitude, positive for Northern Hemisphere
0 Coriolis factor
vii1
Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TIDEINDUCED MASS TRANSPORT IN
SHALLOW LAGOONS
By
Jacobus van de Kreeke
December, 1971
Chairman: Bent A. Christensen
CoChairman: Robert G. Dean
Major Department: Civil Engineering
Analytical, numerical and experimental approaches are employed to describe the hydromechanics of lagoons connected to the ocean by two or more inlets. Because special attention is given to the tideinduced mass transport, all second order terms in the hydrodynamic equations are retained. The study is restricted to lagoons with a onedimensional flow pattern and water of uniform density. In designing a numerical solution to the equations, the inlet equations are regarded as implicit boundary conditions to the equations describing the flow in the lagoon proper. The advantages of this approach are
(1) the size of the computational grid in the lagoon can be chosen independently of the relative small dimensions of the inlets, (2) the flow at branching inlets (an inlet connecting a lagoon to the ocean such that branching of the inlet flow can occur) still can be described by a onedimensional tidal model.
To gain physical insight into the phenomenon of tideinduced
ix
mass transport, an analytic solution to a simplified set of hydrodynamic equations is presented. This solution demonstrates that the magnitude and direction of the mass transport depend on the difference in mean levels at the two inlets imposed by the ocean and the hydraulic characteristics of the inlets and the lagoon. In particular the mass
2
transport (1) increases with the ratio  (a = amplitude, h = depth),
(2) is sensitive to changes in the friction factor F which must be established by calibration. It may be inferred from (1) and (2), respectively, that a significant tideinduced mass transport should only be expected in shallow lagoons and that accurate calibration procedures are necessary to arrive at a reliable magnitude of the mass transport. In presenting the analytic solution, special attention is given to the role of the inlets; it is demonstrated that by properly choosing the dimensions of the inlet, a considerable increase in net transport of water can be obtained.
The predictive capability of the numerical model with regard
to mass transport is confirmed by favorable comparison between measured and computed mass transport for a series of experiments carried out in the laboratory. In the experiments, the tideinduced mass transport was determined in a lagoon of uniform width and depth; at one end the lagoon was connected to a tidal basin and at the other end was connected to the same basin via a submerged weir.
The numerical model is used to compute the mass transport in
one of Florida's shallow lagoons, Lake Worth. The order of magnitude of the computed mass transport is such that it is of considerable importance when regarding renewal of lagoon waters and transport of pollutants.
x
1. INTRODUCTION
Estuaries traditionally have been centers of population concentration. Unfortunately for reasons of economy and convenience, the domestic and commercial wastes accompanying many of these urban areas are discharged into coastal waters thereby degrading the water quality. An improved understanding of the mechanisms responsible for the transport of constituents in estuaries is necessary in order to assess the capability of these systems to assimilate waste loads, to renew their waters by flushing due to fresh water inflow or by exchange with oceanic waters and the possible improvement due to engineering structures, such as inlets. In general, two modes of transport exist, convection and mixing. In tidal waters, often the convective transport is solely attributed to fresh water inflow or winddriven circulation, while it is generally regarded that the only effect of the tidal motion is to generate the turbulence associated with the mixing process. It will be shown, based on theoretical considerations, that for shallow lagoons (a lagoon is defined here as a body of water connected to the ocean by two or more narrow constricted inlets),the convective transport or mass transport due to the tidal motion only can be an important factor in the renewal of water in the lagoon. Because the magnitude of the mass transport is mainly governed by the location and the dimensions of the different inlets, a considerable increase in mass transport can be obtained by properly designing the inlets. This concept of using tidal inlets for environmental controls has been discussed in a wider context by Lockwood and Carothers [11].
I
2
In general, a tidal wave, when entering a lagoon, will be
modified by the inlets and the main water body. The special case for which the tidal propagation in the main water body may be neglected will not be discussed here. The reader is referred to Keulegan [9], Van de Kreeke [25], Shemdin [201, and Mota Oliveira [15]. In describing the flow field, different sets of equations are used for the flow in the lagoon proper and the flow in the inlets. The reason for this is that certain assumptions made in deriving the equations for the lagoon do not hold for the flow in the inlets. In the equations for both the lagoon and the inlets the nonlinear terms must be retained because in a mathematical sense the convective transport results from these nonlinear terms. The inlet equations used here are algebraic equations while the flow in the lagoon is described by the wellknown long wave equations. In the solution, the inlet equations are regarded as implicit boundary conditions for the long wave equations. This implies that it is not necessary to extend the numerical grid used for the main water body to the inlets which, because of their relatively small horizontal dimensions, would lead to small grid sizes and consequently a considerable increase in computational effort. The study is restricted to lagoons having a definite onedimensional flow pattern. The onedimensional numerical model used is based on a space and time staggered explicit finite difference scheme similar to the one described by Reid and Bodine [18].
As stated before, the nonlinear terms in the equations, which include the friction term, are important when regarding convective transport. An accurate value of the friction factor, therefore, is one
3
of the first requirements to prediction of reliable values for the convective transport. In view of this a special discussion is devoted to the determination of the friction factor.
To evaluate the predictive capability of the computational model, measured and computed mass transport in a lagoon are compared for a series of laboratory experiments. The experiments were carried out in the Coastal Engineering Laboratory of the University of Florida.
Finally, the results of the study are applied to an actual lagoon, Lake Worth, located in southeast Florida.
2. THE TIDAL EQUATIONS FOR THE LAGOON AND THE INLETS
In the Eulerian frame, the velocities and water surface
elevations are related to time and the various locations in geometric space by two types of equations, one expressing the conservation of mass, the other expressing the conservation of momentum. The equations are derived in Appendix A. The discharge per unit width and the water levels rather than the velocity and the water levels are chosen as the dependent variables because this leads to a simpler form of the conservation of mass equation. Because the considerations on mass transport will be restricted to those lagoons having a definite onedimensional flow pattern, only the onedimensional form of the equations and the assumptions made in deriving these equations will be summarized here.
It may be seen from Appendix A that the equation of conservation of mass takes the form
S+ x = M (2.1) at ax
in which x is the horizontal Cartesian coordinate, see Figure 1.
n = water surface elevation q = discharge per unit width
M = net inflow per unit surface area due to rainfall, lateral
inflow, etc.
The variables q and n are mean values over a time interval, which is large compared to the turbulence time scale and small compared to the period of the tide.
4
5
In deriving the above equation, the fluid was assumed to be incompressible.
2(x,t) z
~ 4 1 STILL WATER LEVEL w x
 xh(x)
Figure 1. REFERENCE FRAME
The equation of conservation of momentum may be written as
+ +1 q + g(h + )  =  (2.2)
St h 8x Sx 2 (h + n)
in which
h = mean depth
F = resistance coefficient.
The main assumptions made in deriving Equation (2.2) are
 incompressible fluid
 uniform density
 vertical accelerations are negligible
 tidegenerating forces in the lagoon are negligible
 no free surface stresses.
6
The assumptions made in deriving the long wave equations, in general, hold very well for large bodies of water but are less justified when dealing with transitions and regions in which the flow is restricted, e.g. inlets. Unfortunately, the complete equations describing the flow in inlets are still very complex and difficult to solve and therefore, recourse is taken to a semiempirical representation.
The equation commonly used to describe flow in inlets is
2 2
g F Q L mR Q g R(ni  no) = 2
P R 2P R
(2.3)
+ sign for Q > 0  sign for Q < 0.
See for example Keulegan [9]. In Equation (2.3)
P = wetted perimeter of inlet crosssection measured at
mean ocean level
R = hydraulic radius for the inlet crosssection measured at
mean ocean level
Q = total discharge L = length of inlet
S= ocean tide
ni = lagoon tide
m = coefficient which accounts for entrance losses and the nonuniform velocity distribution.
In Equation (2.3) second order effects have been neglected. A revised version of the inlet equation, including all second order terms, will be derived in the next paragraphs.
7
It is assumed that the flow in the inlet region is one
dimensional and storage is neglected. The equation of conservation of mass then becomes
S= 0 (2.4) ax
in which Q = total discharge.
To describe the dynamics of the flow, the inlet is divided
into three regions as indicated in Figure 2. During flood, the flow in Region I is governed by the convective acceleration and the pressure forces; bottom friction and local acceleration are considered small. This leads, when neglecting the velocity head in the ocean, to the Bernoulli equation (the Bernoulli equation is the same as the energy equation when neglecting energy losses)
2
mlQ
n = + n 2gA1
in which
Al = P1R1 = crosssectional area at location 1 (see Figure 2). For most inlets the bank slopes are steep and thus
R R + B 11
1
P P
in which
Al = crosssectional area at location 1 measured at Mean
Ocean Level
MEAN OCEAN 'i
LEVEL'
STATION I STATION 2 OCEAN B Bi LAGOON REON GION REGION  REGION m PLAN VIEW
FIGURE 2. INLET SCHEMATIZATION
9
B = width of inlet measured at Mean Ocean Level.
Substituting the expressions for Al, P1 and R1 in the Bernoulli equation yields
mlQ
nm =1Q2 + n (2.5)
2  B 2
2gP (R + n)
P
It is assumed that during ebb all the kinetic energy present at location
1 is lost. Neglecting bottom friction it then follows that for ebb
no = n1 (2.6)
The flow conditions in Region III are the reverse of those in Region I. Bernoulli's equation holds during ebb while energy dissipation takes place during flood. The only difference with Region III is that, in general, the velocity head at location i can no longer be neglected as was the case for the corresponding location 0 in Region I. For ebb flows, the equation for Region III now reads
m2Q miQ n2 + = n. + (2.7)
2g P (R +_ )  i 2 7F 2 2g Pi(Ri + n.) P.
Again neglecting bottom friction in Region III, the equation describing the flow during flood becomes n2 = ni (2.8)
10
In Region II, the actual restricted part of the inlet, only pressure forces and bottom friction are taken into account and thus
A  P  (2.9) ax pg
in which A = A + Bn = crosssectional area
P P = wetted perimeter
T = bottom shear stress
o
p = density of water.
The bottom shear stress is related to the discharge and crosssectional area by the empirical relation
= p (2.10) o = p 2
Furthermore A = R P.
Substituting these expressions for P, To and A in Equation (2.9) and rearranging slightly yields
g(R + B n) Fqn q (2.11) ax  2 P 2 B 2 P (R + n)
P
This equation holds for both ebb and flood. Integrating Equation (2.11) with respect to x between Stations 1 and 2, the resulting expression can be written, to 0(n2) as
B + 2 3 FQQL(2.12)
g(R + 2 2 )3 (n n) = + (2.12)
P P
11
This particular form of Equation (2.12) was chosen to be comparable
with Equation (2.3). Equations (2.5), (2.8) and (2.12) are then
combined. It is assumed that the velocity head in Equation (2.5) is
of 0(n). When neglecting terms of 0(n3) and higher, the following
equation describing the dynamics of the flow in the inlet during flood,
Q < 0, is obtained.
B 0o + Bi FQ2L g( +  2 )(n.  no) =
P 2 B o 2 P (R + )
2
2
mlQ R
_2 _ ml 2 m 1Q 2
2P [R +  (n )22 p 2gP R
2 2
m1Q B B 3 B 1
S72 (2 + n + 22 (2.13)
2P R P P P 2gP R
Similarly for ebb, Q > 0, it follows by combining Equations (2.6),
(2.7) and (2.12)
g(R+B o + ni = FQ2L
2 )( i  n+) =
22
2 m.Q2
+ m 2 B o i
2 2
mQ RQ B 2 B 2 2 2 +i 2
2 2[+ (n + 22 22 ] 2P(R + n) P 2gP R 2gP R P.
ii i
12
m2Q2 miQ2 B B 3 B m2Q2
22 2 2 o i 2 2g2 2PR 2PR P P P 2gPR ii
miQ
2 ) (2.14)
22
2gPiRi
1i
Equations (2.13) and (2.14) are the same as Equation (2.3) when neglecting the second order terms and also the velocity head in the lagoon and assuming mI = m2 = m. Finally, it is noted that in deriving Equations (2.13) and (2.14) the actual restricted part of the inlet, Region II, is assumed to be a prismatic channel. Many inlet channels do not fulfill this condition, because the width and depth vary from point to point. In that case, the irregular channel may be replaced by a prismatic one having the same discharge for a given head difference. For a detailed outline of this procedure, see Keulegan [91, page 19.
3. NUMERICAL SOLUTION OF THE TIDAL EQUATIONS
The numerical procedure presented here is based upon an
explicit finite difference scheme; the unknowns depend only on values previously computed at a lower time level. This, in general, leads to a more stringent stability requirement (see for instance Vreugdenhil [30]) as compared to implicit methods but has the advantage that the different steps in the computation are easier to trace.
The numerical scheme is space and time staggered. Water levels are computed at n.At and discharges at (n + 1/2)At. The water levels apply at the center of the grid blocks and the discharges are computed at the gridlines; see Figure 3. The mean depth h and the lateral inflow or rainfall are given at the time level and location of n. The basic recurrence equations for the onedimensional tidal equations are
q'(i) 1 [ + (D(i) + D(il))(r(il)  n(i))] (3.1)
G(il1) 2Ax
(i) = n(i) + At(q'(i)  q'(i+l)) + M(i)At (3.2) Ax
in which
D(i) = n(i) + h(i)
G(il) = + 4F At jq(i) + 2 At (g(i+l)  q(il)) (D(i) + D(il))2 Ax (h(i) + h(i1))
Primed symbols denote values of the variables at time step at lazer. Equations (3.1) and (3.2) are based respectively upon the differential Equations (2.2) and (2.1). The differential quotients in these
13
h(i) h(i+l) q(i) 4 (i) q (i+) + ,( i I) M(i) M (i+ 1)
FIGURE 3. LOCATION OF VARIABLES IN THE NUMERICAL GRID
15
equations are replaced by different quotients using central differences. The difference quotients for Equation (2.1) are centered about (n + 1/2)Lt and the location of n. The difference quotients for Equation (2.2) are centered about time level nAt and are centered in space about the location of q. Starting from the initial conditions, all the q's are computed for the next time level by means of Equation (3.1), then the n's are computed using Equation (3.2). It is noted that because of the convective acceleration, the recurrence formula (3.1) includes values as far apart as 2 space steps; see expression for G(il). This leads to difficulties when the boundary conditions at open boundaries are given as water levels. In that case, the convective acceleration is taken off center for the grids adjacent to those boundaries. For a more detailed outline of the numerical scheme, the reader is referred to Reid and Bodine [18] and Verma and Dean [28].
Because of the nonlinearities in the differential equations, the existing mathematical theory is inadequate to determine the exact criteria for stability and convergence of the numerical scheme. However, some insight might be obtained by regarding a simplified set of equations. The equations describing the tidal flow, Equations (2.1) and (2.2),may be replaced by two wave equations in respectively n and q, when neglecting the nonlinear terms and the source term, and assuming a horizontal bottom. It then can be shown, starting from the recurrence formulae for the simplified set of equations that,for this case, the difference scheme is equivalent to solving the wave equation on the mesh Ax, At. The stability and convergence criterion then is (see Platzman [161])
16
Ax
(For the twodimensional equations the condition is A> 2gh.) At
When regarding the solution of the tidal equations as a sum of Fourier components, stability as used here implies that there should be a limit to the extent to which any component of the solution is amplified in the numerical procedure. Platzman [16] showed that even before the stability criterion is reached, the amplitude of especially the highest modes might be greatly magnified and might obscure the true solution. To study the eventual magnification of higher harmonies, a few computations were carried out for a 17milelong and 7ft.deep channel, open at one end and closed at the other. In addition, these computations provided some insight as to whether the terms neglected when discussing the stability criteria in the previous paragraph give rise to instabilities. The tide at the open end of the channel was sinusoidal with an amplitude of 1.5 ft. and a period of 12.5 hours. The computations were carried out for values of the friction factor F = 0.002 and F = 0.004 and for the time steps At = 100 sec., At = 200 sec. and At = 300 sec. The space step Ax = 5000 ft. was the same for all computations. The stability criterion for the case Ax
discussed here is At = A = 333 sec. Each computation extended over
4 tidal cycles. The results were judged visually; no overflow occurred, and amplification of higher modes was not apparent. The source term and the free surface forces were not included in the computations.
17
In the computational procedure,the inlet equations, Equations (2.13) and (2.14) may be regarded as an implicit boundary condition for the flow in the lagoon. The way in which this boundary condition is incorporated in the numerical scheme depends on the inlet configuration. The following two cases are considered:
 An inlet connecting a lagoon to the ocean such that no
branching of the inlet flow occurs; see Figure 4a.
 An inlet connecting a lagoon to the ocean such that
branching of the inlet flow can occur; see Figure 4b.
Consider the "nonbranching inlet;" see Figure 4a. The total discharge Q rather than the discharge per unit width, q, is used as a dependent variable. An auxiliary water level n. is introduced which is computed at the same time level as the water levels in the lagoon. Starting from the initial conditions, all the discharges in the lagoon except Q(i + 1) can be computed using the procedure described before. The value of Q(i + 1) is then computed as follows. Q(i + 1) is related to the known ocean level n0 and the auxiliary level n. by means of the
O 1
inlet equations. The difference form of the inlet equations used here is based on the following simplified form of Equations (2.13) and (2.14).
g(R B o + i + FQ2L + m2QR ( + )(n,  no) = +
2 i o 2 B 0 + 2 2 B o +i P (+ + 2 ) 2P (R +
2 2
P P
 sign for ni < no + sign for ni > n .
OCEAN
o(i+l) (71 77)) 77(+2
+  +  71+
+77(i) 0 (1) Q (i.) 0 (i.2) LAGOON
0 (i) LAGOON
BI
+"7') Ic
LAGOON
ONEDIMENSIONAL FLOW IN LAGOON
C
A ONE DIMENSIONAL FLOW IN LAGOON LAGOON WITH VARYING WIDTH
"NONBRANCHING" INLET o OCEAN
o
  +  (it2) 'T)(i*3) Qi) Oi4ï¿½ + +.
(i2) Q i) m LAGOON LAGOON '
ONEDIMENSIONAL FLOW IN LAGOON INLET
B
S BRANCHING" INLET
FIGURE 4. GRID SCHEME FOR DIFFERENT INLET CONFIGURATIONS
19
It is assumed that mi = m2 = m. After some algebraic manipulation, the following result is obtained.
Q = DD /W Ini  no
+ sign for ni > n  sign for ni < n
in which
Q'(i+l) + Q(i+l) (3.3) Q= 22
2 n + .
DD = (R +  3/2 2FL + mR P 2
It is noted that for many practical cases the modifications made in Equations (2.13) and (2.14) are justified. They were introduced here to simplify the algebra; however, the computational procedure applies equally well when using the complete Equations (2.13) and (2.14).
A second equation relating Q(i + 1) and ni is found by applying the dynamic equation, Equation (2.2),between the discharge stations Q(i + 1) and Q(i). Note that when computing the flow in the lagoon, the dynamic equations are applied between two water level stations. The difference form of the dynamic equation applied between Q(i + 1) and Q(i) yields
Q = AA(ni  n ) + BB + AA lo (3.4)
in which
20
SQ'(i + 1) + q(i + 1)
AA= g D(i) B1 At GAx
BB = {Q(i+l) + Q(i)  Q'(i) * G + g Bl D(i) At Ax
[n(i) + n(i1)]} /2G + Q(i+l)
2
G = 1 + FAtIQ(i+l) + q(i) + 2At[Q(i+l)  q(i)] 2D(i) 2 B Axh(i)Bl In determining the difference form, the terms in both the inlet and dynamic equation are centered about n * At.
The general shapes of the curves Q = f(ni  no) represented
by the Equations (3.3) and (3.4) are indicated in Figure 5. Equation (3.4) represents a straight line. The slope of this line, AA, is for most practical cases negative. Therefore, the sign of (BB + AAn ), which is a known quantity, determines the sign of (ni  no) which in turn determines the sign to be used in Equation (3.3). Eliminating (i  no) between Equations (3.3) and (3.4) yields AA
1  J1  4 A (AAn + BB)
2 AA
DD
2
for (BB + AAno) 0 AA
1  + 4 AA (AAno + BB) and Q = 0 (3.6)
2
DD
TF <
BB 5. CURVES REPRESENTED BY EQUATIONS (3.AA3) AND (34)
FIGURE 5. CURVES REPRESENTED BY EQUATIONS (3.3) AND (3.4)
22
for (BB + AAn ) < 0.
Note that only the first order terms in (ni  no) are eliminated because second order terms are still present in the factor DD. Equations (3.5) and (3.6) therefore may be regarded as being quasilinear, which suggests finding a solution by means of a perturbation method. First the value of (ni  ro) in DD is taken equal to the value at the previous time step. The value of Q can then be found from Equation (3.5) or (3.6) depending on the sign of (BB + AAno). Knowing Q, the value of (ni  no) is determined from either Equation (3.3) or (3.4). This value of (ni  no) is substituted in DD. The procedure then is repeated until the difference between the computed and previously computed value of Q is within certain limits.
The numerical scheme for the "branching inlet" (see Figure 4b) involves four unknowns Q'(i+l), Q'(i+2), Q1 and n as compared to only two, Q(i+l) and ni, for the "nonbranching inlet." The four unknowns are related by the following four equations
 the inlet equation which takes the form of Equation (3.3) Q! + Qi
with Q = 2
 the dynamic equation applied between the locations of Q(i)
and Q(i+l); this equation takes the form of Equation (3.4).
 the dynamic equation applied between the locations of
Q(i+2) and Q(i+3); this equation takes the form of Equation
(3.4) with Q(i) replaced by Q(i+3), Q(i+l) replaced by
Q(i+2), n(i) replaced by n(i+2), n(il) replaced by n(i+3),
h(i) replaced by h(i+2), Ax replaced by Ax, and Bl
replaced by B2.
23
 the continuity condition which, when assuming, m.t the water
level in the hatched area is the same everywhere, taies the
form
Q(i+l) = Qi + Q(i+2)
Elimination of Q(i+l) and Q(i+2) between the two dynamic equations and the continuity equation yields a relation between Qi and ni similar to Equation (3.4). This equation together with the inlet equation then can be solved following the procedure described before.
Finally, it is noted that in onedimensional flow con.putations, it is often necessary to divide the lagoon into parts with differenL widths; see Figure 4c. The flow at the boundary of two such parts may be computed following a procedure similar to the one applied to the "nonbranching inlet," replacing the inlet equation by a second dynamic equation between the location of Q(i+l) and Q(i+2).
The foregoing procedure of incorporating the inlet equations in the numerical scheme, including the case with twodimensional flow in the lagoon, can also be found in Chiu, Van de Kreeke and Dean [2 .
4. DETERMINATION OF THE FRICTION FACTOR F
The factor F in the friction term of the conservation of
momentum equation is difficult to estimate in advance; therefore, an appropriate value is usually based on measurements. Two methods for determining F are compared. One method is based on matching computed and measured water levels, the other is based on matching computed and measured discharges. Special attention is given to the influence of errors in the measured water levels and discharges on the value of F.
To calibrate a tidal model, i.e., to determine the value of F, the area covered by the model is divided into sections each with approximately uniform geometry for which the values of F may be assumed constant. It is assumed that at both ends of each section the water levels are known from measurements. Given these water levels and assuming a value of F, the water levels and discharges within each section can be computed. The computations are carried out for different values of F until a good match exists between computed and measured water levels or discharges. The F value obtained in this manner reflects, in addition to friction, the effects of schematization, neglected terms in the equations and measurement errors. These effects may cause large variations in the value of F during a tidal cycle especially during periods of slack tide when the inertia forces are predominant. Therefore, in determining F these periods should be disregarded. Furthermore, a realistic value of F, that is a value mainly determined by friction and more or less constant during a tidal cycle can only be obtained in cases for which the flow is frictiondominated during the larger part of the tidal cycle (see Dronkers [ 5 1, page 425). It should be noted that in
24
25
many actual cases not all of the field data necessary for this ideal procedure are available. In that case recourse should be taken to trial and error methods.
As stated before, measurement errors result in errors in F.
In this paragraph, some general considerations concerning these errors are put forward. In the next paragraph an example will be presented showing the order of magnitude of the errors in F resulting from errors in the "calibration curves" (a calibration curve is defined as measured water levels or discharges within a section and used to determine F by matching with computed water levels or discharges). The water surface in each section may, as a first approximation, be regarded as a straight line because in practice the length of the sections are small compared to the length of the tidal wave. The end points of the straight line are determined by the water levels at the boundaries. Therefore a change in F will hardly affect the computed water level in the section. From this it may be inferred that, when determining F by matching computed and measured water levels, an error in the latter or in the boundary conditions will lead to a large error in F. When calibrating on discharges the effect of measuring errors can be expressed as follows. During periods of frictiondominated flow a first approximation of the equation of conservation of momentum is
gh r
ax h 2
and thus
26
e(F)= ( an ) + e(qjqj)
in which c = relative error.
The error in F thus is of the same order as the error in the boundary
an
conditions ( x ) and the calibration curve (q).
The following example serves to illustrate the order of magnitude of errors in F due to errors in the calibration curve. Consider a section with a constant depth h = 7 ft. and a length L = 50,000 ft. The boundary condition at x = 0 is n1 = 1.3 sin (2rt/45000) and at x = L is n2 =.1.3 sin (27(t2700)/45000); see Figure 6a. The water levels and discharges in the middle of the section are computed for F = 0.002, F = 0.0025 and F = 0.003. The results are plotted in Figures 6b and 6c. The differences between the water levels for the different values of F were too small to be reproduced in the figure. In addition, for the same location and for F = 0.002, the values of the different terms in the equation of conservation of momentum are plotted; see Figure 6d. The latter results provide some insight in the relative magnitude of the forces and accelerations represented by these terms and their phase relationship with water levels and discharges. It may be seen from Figure 6d that the flow is frictiondominated during the larger part of the tidal cycle, which as stated before is a condition to arrive at a realistic value of F. When using a measured water level in the section as calibration curve, the effect of an error in this curve on the F value can be demonstrated as follows. Let the curve corresponding to F = 0.002 in Figure 6b represent the correct water level (calibration curve) in the section. Let the curve
27
X=oF T. '..
X0 X 50,000 FT.
(a) TIDE AT X= 0 FT. AND X 50,000 FT.
F= 0,002
F = 0.0025
w 6 12
14 TIME IN HOURS
(b) TIDE AT X 25,000 FT. 10
TIME IN HOURS
10 C) DISCHARGE AT X 25,000 FT.
u Fq{q O.0
V)F F0.002
2.10 aQ.t ) X
.0 NI . "
o. O 6 /" .2 12
L , TIME IN HOURS ,
(d) MAGNITUDE OF TERMS IN CONSERVATION OF MOMENTUM EQUATION AT X= 25,000 FT.
FIGURE 6. TIDE IN A SEA LEVEL CANAL
FIGURE 6. TIDE IN A SEA LEVEL CANAL
28
corresponding to F = 0.003 in the same figure be the measured calibration curve. This curve deviates from the correct curve at the most by
0.03 ft.; the relative error (maximum deviation divided by tidal amplitude times hundred) is 2%. An error of 2% in the water level (calibration curve) would thus lead to a ( 0.003  0.002 . 100%= )
0.002
50% error in F and as may be seen from Figure 6c an error of 20% in q. The influence of errors in the calibration curve on the F value when using discharges can be demonstrated in a similar way. Let the curve in Figure 6c corresponding with F = 0.002 be the correct discharge curve. Consider the curve corresponding with F = 0.0025 in the same figure to be the measured discharge curve. The relative error in the measured discharge is approximately 10%. The resulting error in F is 25% and the error in the water level appears to be 1.2%.
To summarize for the example presented here, when calibrating
on the water level a maximum error of 2% in the water level leads to an error of 50% in F and a maximum error in q of 20%. When calibrating on a discharge curve an error of 10% in the calibration curve leads to an error of 25% in F and a maximum error in the water level of 1.2%.
The relative errors in the measured water levels and discharges of respectively 2% and 10% used in the previous example are realistic values for conventional types of equipment and measuring procedures. Also the example is a typical case for flow in lagoons of the type considered here. Therefore it may be safely concluded that, when
This value cannot be read from Figure 6b; it was taken directly from the computer output.
29
calibrating on water levels and assuming that the boundary conditions for a section are given as water levels, small inaccuracies in the measured calibration curve lead to large errors in F and q. When calibrating on discharges, the errors resulting from inaccuracies in the calibration curves are far less.
5. MASS TRANSPORT IN LAGOONS DUE TO TIDAL ACTION
In this chapter an analytic solution is presented for the mass transport due to tidal waves in lagoons. The analytic approach is restricted to lagoons and inlets of relatively simple geometry, which prohibits its application to the often irregular shaped tidal waters in nature. In such cases, recourse has to be taken to numerical techniques. Therefore the analytic model cannot replace the numerical procedure; it merely serves to build faith in the numerical results and to obtain a better understanding of the physics underlying the phenomenon of mass transport. In the final part of the chapter the effect of inlets on the net transport is discussed.
a. Analytic Solution; Physical Concepts
Consider a straight lagoon of finite length Z in which the ocean tide can freely enter at both ends. The depth and width of the lagoon are uniform. The conservation of mass equation neglecting the source term is
an +  = 0 (5.1) at 8x
The conservation of momentum equation, when neglecting the nonlinear terms except the friction term, reduces to
aq + gh D= Fq (5.2) at x (h + n)
The term on the righthand side of Equation (5.2) was
derived (see Appendix A) by relating the bottom shear stress and the mean velocity by means of a DarcyWeisbach type friction law.
30
31
S=f1 l (5.3)
4R 2g
in which S = slope of the water level
f = DarcyWeisbach friction factor
u = mean velocity over depth
R = hydraulic radius. Furthermore
T
=  gRS (5.4)
in which To = bottom shear stress. From Equations (5.3) and (5.4), it then follows
To f
P f ulu (5.5)
The friction term in Equation (5.2) will now be rederived by replacing Equation (5.5) by its linearized version. Assume that as a first approximation the velocity u in the lagoon is simple harmonic in t
u = u cos(at + a) (5.6) in which a = a phase angle. Substituting this expression for u in Equation (5.5) and expanding in a Fourier series yields
o f 8
   uu + third order term + ... p 8 37r
The second, fourth, etc. harmonics are zero. Thus neglecting third order terms and higher yields
32
8
T ( u)q o f 3r
S(5.7)
p 8 h + n.7) When approximating
1 1 n
h + n h h2
and introducing f 8 ^ 8
F 8 u = F u (5.8) Equation (5.7) may be written as
To Fzq F qn
 + (5.9)
p h 2 With this expression for the bottom friction, the conservation of momentum equation becomes
q + gh 9= Fq + (5.10)
at ax h 2
Equations (5.1) and (5.10) are used to describe the flow in the lagoon. Equation (5.1) is a linear equation while Equation (5.10) may be regarded as quasilinear because the nonlinear term is small compared to the first term on the righthand side. The ocean tides at both ends of the lagoon serve as the boundary conditions. To avoid lengthy computations, the amplitude and phase of the ocean tides are taken such that when neglecting the nonlinear term in Equation (5.10) the resulting solution represents a damped progressive wave. The water levels and discharges for such a wave as a function of x and t are
33
n = a epxcos(at  kx) (5.11)
k
q = ac excos(at  kx + a) (5.12)
2 2
+ k
in which i, k and a are defined by
2 2 2
o
F 2iaok
h 2 2
k
a = tan ( ) with k = o/c and c = /g. Thus when
no = a cos at (5.13) is the boundary condition for x = 0, it then follows from Equation (5.11) that the boundary condition at x = L should be
nL = a ewLcos(ot  kL) (5.14) It is furthermore assumed that the mean level at both ends of the lagoon is the same and equal to the mean ocean level. A trial solution for Equations (5.1) and (5.10) when including the nonlinear term in the last equation and considering the boundary conditions (5.13) and (5.14) is
n = n,(x) + a eyxcos(at  kx) + ... (5.15)
34
k
q = q,(x) + a co 2O e~xcos(at  kx + a) + ... (5.16)
2 2
2 + k
Substituting this trial solution in Equation (5.1) and averaging over the tidal period yields aq,
= 0 (5.17)
(Note: This must be true for any periodic forcing function.) Substituting in Equation (5.10) and neglecting higher order terms yields after time averaging
2
an, F F a c k k gh   q + h 2 e (5.18) x h + h 2h 2 2 x +k
For x = 0 and x = L, n* = 0 because of the assumption that at both ends of the lagoon the mean level equals the mean ocean level. With these boundary values of n* the solutions for n, and q* from Equations (5.17) and (5.18) are
2
ac kk
o 1 o 1 2(5.19)
q (c0 k1k  (e ) (5.20)
* 2 2h 2 2 2 L(5.19)
2
F ac kk
n0 1 = 2[   1)  (e  1 )] (5.20)
gh p +k
Equation (5.19) is the expression for the net discharge in a straight lagoon when imposing the boundary conditions (5.13) and (5.14) and assuming the same mean level at both ends of the lagoon. Equation (5.20) represents the mean water level in the lagoon.
35
The following serves as an explanation of the physics
underlying the expression for the net transport and mean water level in a lagoon derived in the preceding paragraph. Considered is a damped progressive wave in an infinitely long channel. The velocity u, rather than the discharge per unit width q, is introduced as a dependent variable. Assuming a horizontal bottom Equations (5.1) and (5.10), which represent the conservation of mass and conservation of momentum, can be written as respectively
D'_ + h au = un (5.21) at ax ax
and
h u + gh ax "  F u (5.22)
The expressions for the damped progressive wave and the accompanying velocity field, satisfying the linear part of Equations (5.21) and (5.22), are
n = a e cos(ot  kx) ac k
S= ek Xcos(at  kx + a) h 2 2
h1 + k
See, for example, Ippen [8 ], page 507.
The net transport accompanying such a wave, q, , may be found
w
from
T n
q, =  u dz dt
0 h
36
Substituting the expression for u in this equation and integrating yields
2
ac kk
* = 2h 0 e x (5.23)
* 2h 2 2 w . +k
This expression shows the net transport to decrease with increasing x. Continuity requires the net discharge to be the same for every crosssection and equal to the net discharge at x = 0. Conveniently taking the net discharge at x = 0 equal to zero, the net discharge in the channel, q, is written as
q, = q, + q, = 0 (5.24)
w s
in which q* results from the slope in the mean water level
s
q*= Fh x (5.25)
* F /h ax
Substituting the expression for q, and q, , respectively Equations
w s
(5.23) and (5.25), in Equation (5.24) yields
2
ac k k an
o o k 2x gh = 0 (5.26)
2h 2 2 F /h ax
Integration of Equation (5.26) and arbitrarily setting n~ = 0 at x = 0 yields
2
F ac kk F 1 1 1 o o (  e2x) (5.27) S h gh 2p 2h 2 +2
37
Equation (5.27) represents the mean water level in an infinite long channel in the presence of a free damped progressive wave and zero net transport. See also Figure 7a.
When regarding a channel or lagoon of finite length L, at both ends connected to the ocean, the mean level at x = 0 and x = L is the same and equal to the mean level in the ocean. The resulting net flow and mean water level in the channel of finite length can be derived from the mean flow and mean water level for the infinite long channel, as follows. A slope
n,(L)
S )
m L
is added to the mean water surface in the infinite channel, which renders the mean level at x = L zero. The expression for the mean water surface becomes
n*(L) F ac kk
*,(L F 1 1 o o 2px
n x + (1  e (5.28)
* L h gh 2p 2h 2 2 (  (5.28) y +k
See Figures (7b) and (7c). Substituting the expression for n*(L) from Equation (5.27) in Equation (5.28) yields Equation (5.20). The discharge q* accompanying the slope Sm is
mm
q*gh n (L)
q =   (5.29) m F /h L
and the total net transport q* is
q =0  F/h L (5.30) q* ,0 F k/h L
38
at X co 77: I1 2o ko h gh 2)1 2h U2+ k2
*. Eq (5.27) MEAN OCEAN LEVEL
0 L
a. MEAN WATER LEVEL IN AN INFINITELY LONG CHANNEL
77(L)
MEAN OCEAN LEVEL
/11//////11/// ///////711 /11//117//1//////////////11117//
b. SLOPE , Sm
o~t Eq (5.20) MEAN OCEAN LEVEL
O x L
/11///II///////////7 //////////////////////1////////
c. MEAN WATER LEVEL IN CHANNEL OF FINITE LENGTH, L
FIGURE 7, MEAN WATER LEVEL IN A LAGOON IN
THE PRESENCE OF A DAMPED
PROGRESSIVE WAVE
39
Substituting the expression for n,(L) as given by Equation (5.27) yields Equation (5.19).
It may be inferred from the previous considerations that the net transport in the finite channel may be regarded as the result of forcing the mean level at the boundaries to be different from the mean level at these boundaries in the case of a free wave. Note also that in case the friction approaches zero the expression for the net transport, Equation (5.19), approaches
2
ac
q* = 2h (5.31)
which is the expression for the net transport accompanying a free undamped wave with amplitude a, traveling in shallow water.
When substituting the trial solutions Equations (5.15)
and (5.16) in Equation (5.10), all higher order terms were neglected. To evaluate the effect of these terms on the net transport the q, as given by Equation (5.19) is compared with the true q*, by taking into account the third and higher order terms. The true q* is found by numerically integrating Equations (5.1) and (5.10) with the boundary conditions (5.11) and (5.12) following the procedure described in Chapter 3 and then averaging q over a tidal period. The computations were carried out for a lagoon with a length L = 70,000 ft. and a depth h = 7 ft., a tidal period T = 45,000 sec., a time step At = 100 sec., and a space step Ax = 5,000 ft. Two values for the amplitude a and three values for the linear friction factor F are considered. The results of both the analytic and numerical (true) solution are presented
40
in Figure 8. From this figure, it may be seen that neglecting the third and higher order terms introduces only a very slight error.
The question might arise whether the "linearization" of the friction term as outlined on pages 31 and 32 is a valid procedure when discussing effects caused by second order terms in the equations. To shed some light on this, the net discharges when using Equations (5.1) and (5.10) and the net discharges when using Equations (5.1) and (5.2), which include the quadratic friction, are compared. The same lagoon is considered as in the previous paragraph. The boundary conditions again are presented by Equations (5.11) and (5.12). The "linear" friction factor F and the "nonlinear" friction factor F are related by Equation (5.8). The value of u in this equation is taken equal to the average of the u values at both ends and the u in the middle of the lagoon. The equations are integrated numerically using a time step At = 100 sec. and a space step Ax = 5000 ft. To find q, q is averaged over a tidal period. Two different values for the amplitude a and three values for the friction factor F are considered. The results, which are plotted in Figure 9, compare well especially considering the way in which u was determined. The agreement is better for larger than for smaller values of F which probably stems from the fact that for larger values of the net flow and thus smaller values of F, the approximation of u as given by Equation (5.6) becomes less justified. It is noted that methods do exist to arrive at a better estimate of the representative average value u. However, incorporating these methods would lead to a considerable increase in computational effort which is not warranted by the purpose of the computations. The
41
a : 0.5 ft.
0.15  O ANALYTIC (EQUATION ( 5.19) ) O NUMERICAL (EQUATIONS ( 5.1) a (5.10))
0r
0.10
00
w O0
z
0.05 "LINEAR" FRICTION FACTOR F
0.001 0.002 0.003
1.5
o 0:1.5 ft.
O ANALYTIC (EQUATION (5.19))
z
0 NUMERICAL (EQUATIONS (5.1) cr a (5.10))
1.0 _ w
z 0
0.5 "LINEAR" FRICTION FACTOR F
0.001 0002 0.003
FIGURE 8. EFFECT OF HIGHER ORDER TERMS ON q.
42
0.3
.a = 0.5 ft.
0 EQUATIONS (51) a (52) Z 0 EQUATIONS (5*1) 8 (5.10)
0.2  0
w
0
cn
0 O
0.1
z 0
"NONLINEAR" FRICTION FACTOR F
0.0
0.001 0.004 0.008
a=1.5 ft. IS O0 EQUATION (51) 8 (52)
0 EQUATIONS (51) 8 (510)
z
2 
I
0
 O
I
z
"NONLINEAR" FRICTION FACTOR F
0.001 0.002 0.003
FIGURE 9. EFFECT OF "LINEARIZATION" OF THE FRICTION
TERM ON q .
43
reader is referred to Dronkers [ 6 ], Ippen [ 8] and Van de Kreeke
[26].
In deriving Equation (5.19) the nonlinear terms associated with the convective acceleration and surface gradient have been neglected. These terms can easily be taken into account. In doing so, the only change in the differential Equation (5.18) will be in the expression preceding e . In the initial derivation these nonlinear terms were omitted to simplify the algebra as much as possible. They should, however, be taken into account when requiring quantitative accurate results. This may be seen from Figure 10 in which the net transport with and without the convective acceleration and the nonlinear part of the surface gradient is presented for a lagoon at both ends freely connected to the ocean. The results plotted in Figure 10 were obtained numerically.
Also in driving the expression for q,, the boundary
conditions were chosen such that to a first order of approximation, the water motion in the lagoon was a single damped progressive wave. However, expressions for q* can be derived in a similar way for arbitrary boundary conditions. For example, consider a lagoon connected to the ocean by inlets at both ends. The implicit boundary conditions, formed by the inlet Equations (2.13) and (2.14), and the field Equation (5.2) first are made quasilinear. This is done by linearizing the factors Q2 and qjql in respectively the inlet and dynamic equation. A first order solution is obtained by omitting the second order terms (the nonlinear terms) from both the field equations and the boundary condition. The trial solution given by Equations (5.15)
L = 70,0001.
0.18
0.16  DEPTH 7 ft.
S71 = 0.5 Sin t
0.14  12 = 0.5 Sin(O't TT
u 25
V TIDAL PERIOD T = 45,000 Sec.
0.12 u2
0.10 FqIqI ,h (h +7 )
at ax
0.08 a4+ I aq2 a FqIqI 0t 6 +h x + gh+ (h +.7)2
0.06  0
I
0.04
002  " NONLINEAR " FRICTION FACTOR F
0.00 I I II I II
0.000 0002 0.003 0.004
FIGURE 10. EFFECT OF CONVECTIVE ACCELERATION AND NONLINEAR PART
OF THE SURFACE GRADIENT ON THE NET DISCHARGE q
45
and (5.16) is then substituted in the complete equations and boundary conditions including the nonlinear terms. The resulting expressions are time averaged which yields two linear differential equations in * and q*, corresponding to Equations (5.17) and (5.18), and two linear boundary conditions. It is thus possible to solve for n, and q,. Preliminary computations showed the resulting expressions for n, and q* to be of such a nature (long algebraic forms with implicit expressions for amplitude and phase angles) that they defy the purpose of presenting the analytic solution, that is, gaining physical insight.
When considering a straight lagoon of uniform depth and
width at both ends freely connected to the ocean, one obvious conclusion can be arrived at without any algebra; when the tides at both ends of the lagoon are the same, the net transport is zero because of the symmetry of the problem. Therefore, in order to have mass transport in a straight lagoon of uniform depth and width, the ocean tide at both ends of the lagoon must be different. This is an important consideration when discussing the effect of inlets on the mass transport; see Chapter 5b.
So far the vertical profile of the net drift has not been discussed; an important factor when designing an experimental apparatus for measuring the mass transport. Neglecting viscosity and thus neglecting bed friction, the net drift is the same for all levels. For example, Stokes' [22] second order wave theory gives for shallow water waves in an inviscid fluid a mass transport velocity
46
2
ac
* 2
2h
However, when accounting for viscosity the net drift is no longer uniformly distributed as shown theoretically by LonguetHiggins [12] for the case of null net transport.
Experimental results on drift profiles in a progressive shallow water wave and null net transport, U. S. Beach Erosion Board [23],Bagnold [1], Russell and Osorio [19] are in qualitative agreement with LonguetHiggins socalled conduction solution even though the conditions imposed on the solution (wave amplitude small compared to the thickness of the boundary layer) are not satisfied in the experiments. Generally speaking, the theory and experimental results show a forward drift, that is in the direction of wave propagation, near the bottom and near the surface with a return flow in the middle of the water column. It is noted here than the k h value, being one of the characteristic parameters describing the "shallowness" of the waves, was not less than 0.3 for any of the experiments presented in references (23], [1] and [19]. LonguetHiggins' [12] conduction solution for the net drift in a pure standing wave shows a net drift near the bottom towards the location of the anti node while the net drift at the surface is from the anti node towards the nodes. The total picture is a circulation of mass between the vertical planes through nodes and anti nodes. To the writer's knowledge, no experimental results exist to verify this net drift pattern.
47
b. Influence of Inlets on the Mass Transport
Consider a lagoon of uniform depth and width connected to the ocean by inlets at each end; see Figure 11.
L7
FIGURE 11. LAGOON CONNECTED TO THE OCEAN BY INLETS
It is assumed that the depth of the lagoon is so large that the propagation of the tide in the lagoon itself can be neglected. The continuity equation then may be written as
QI + QII = Q = Ab  (5.32) in which
Ab = lagoon area
Q = discharge into lagoon
ni = lagoon level.
Equation (2.3) is used to describe the dynamics of the flow in the inlet. From this equation, it follows
Q= A 2gR ï¿½ [o
2FL + mR
or in Keulegan's [9] notation
48
Q = K 2 /io ji (5.33)
in which
T A 2gRH
2 H b 2FL + mR
K = coefficient of repletion 2H = tidal range in the ocean
A = crosssectional area of the inlet measured at mean
ocean level.
The solution to this problem, for a simple harmonic ocean tide, is given by Keulegan [9]. For an ocean tide of the form 2nt
0 = H sinT
in which H = tidal amplitude in the ocean, the corresponding bay level ni is
i = A1 sin( T + E ) + A sin 3( T + E3) + ... (5.34)
The amplitudes Al and A3... and the phase angles E1, E3... are a function of the coefficient of repletion K = K1 + K2; K1 and K2 are the coefficients of repletion, respectively, for the Inlets I and II. It may be seen from Equation (5.34) that both amplitude and phase of the ocean tide are modified while in addition, higher harmonics are generated.
The foregoing solution pertains to relative deep lagoons. No analytic solution is known in case of a shallow lagoon. However, it is likely that each inlet again will modify the ocean tide but not to the
49
same extent (except when the inlets have the same dimensions), as was the case for the deep lagoon, because the propagation of the tide in the lagoon can no longer be neglected. In general, therefore, the amplitude and phase of the lagoon tide at one inlet will be different from the amplitude and phase of the lagoon tide at the other inlet, even though the ocean tide is the same for both inlets. A difference in tide at both ends of the lagoon is a requirement for net transport because in case of equal tides, the net transport is zero as a result of the symmetry of the problem. See also Chapter 5a.
It may be inferred from the foregoing general considerations that inlets may play an important role in the flushing of the lagoon. By properly choosing the dimensions of the inlets, the net transport of water in the lagoon can be considerably increased. This is illustrated for the bay system presented in Figure 11. The ocean tide is assumed simple harmonic and the same at both inlets; the amplitude is 1.30 ft. and the tidal period is 45,000 sec. For the dimensions of the lagoon and the inlets, see Figure 12. In the same figure, the results of the computations are presented; the net discharge per tidal cycle is plotted versus the width of Inlet II. A definite maximum in the net discharge occurs when the width of Inlet II is in the order of 300  400 ft. The computations were carried out for both the complete inlet Equations (2.13) and (2.14), neglecting terms resulting from the velocity head in the bay and the ocean, and for the simplified version of the inlet equation, Equation (2.31 again neglecting the velocity heads. The results for the two inlet equations differ only slightly. Equation (2.2) in which the convective term was neglected, was used to
400 1
0
320
.* 4 0
I
0
2O
160
2 .O INLET EQUATION (2.3)
" INLET EQUATION (2.13) 8 (2.14) 80
so
WIDTH OF INLET n IN FTE
0 100 200 300 400 500 600\ 700 800 FOR LAGOON SEE FIGURE II DIMENSIONS OF LAGOON DIMENSIONS OF INLET I DIMENSIONS OF INLET I"
LENGTH = 80,000 ft , DEPTH = 7 ft. LENGTH = 2500 ft , DEPTH = 20ft, LENGTH = 2500 ft, DEPTH = lOft.
WIDTH = 2500 ft , F= 0.0025 WIDTH = 800 ft, F = 0.0025 WIDTH = VARIABLE, F = 0.0025
FIGURE 12. NET DISCHARGE IN LAGOON VERSUS WIDTH OF INLET I1
51
describe the dynamics of the flow in the lagoon. The equations were solved by applying the numerical method discussed in Chapter 3. The net discharge was found by integrating the computed discharge over a tidal period. The time step in the computations was At = 100 sec. and the space step was Ax = 5000 ft.
6. LABORATORY EXPERIMENTS
A series of laboratory experiments was designed to measure the net drift in a lagoon of uniform width and depth connected at both ends to a tidal basin by openings of different dimensions. The purpose of the experiments was to evaluate the capability of the computational model described in the previous chapters to predict mass transport in tidal lagoons. The experiments were carried out in the Coastal Engineering Laboratory of the University of Florida.
a. Equipment and Procedure
Test SetUp and Measuring Devices  A straight canal of uniform depth and width, simulating a lagoon, was constructed in a tidal basin; see Figure 13. The canal was 488 cm long and 30 cm wide. As outlined in Chapter 5, a necessary condition for mass transport is that the geometry of a lagoon be asymmetric. In nature, the asymmetry in many cases results from the different dimensions of the inlets connecting the lagoon to the ocean. The asymmetry in the experimental lagoon was introduced by providing one end of the lagoon with a submerged sharp crested weir. (A weir was preferred over an inlet because it led to a more regular flow pattern.) The weir was made of 1.5 mm steel plate. The edge of the weirwas 3.35 cm above the bottom of the canal. The average water depths used in the tests were 5 cm and 6.5 cm.
The dimensions of the tidal basin were 600 x 600 cm, and the average water depth in the basin was 30.5 cm. Two weir boxes, one at each side of the basin provided a constant discharge of 22 liters/sec. A filter of honeycomb placed across the basin and wave damping material
52
610
136 252
FILTER MADE OF GATE
STEEL SCRAP
HONEYCOMB
WEIR BOX WEIR BOX Ln
EL. 0.0
/,BARRIER ISLAND
e LAGOON EL. 25 1: /,
SUBMERGED WEIR 61 488
LEGEND
0 WATER LEVEL RECORDERS
O  POINT GAUGE DIMENSIONS IN CMI. FIGURE 13. EXPERIMENTAL SETUP
54
attached to the walls served to absorb small disturbances caused by the inflow from the weir boxes. The tide in the basin (later referred to herein as the ocean tide) was generated by letting the discharge gate move up and down at approximately constant speed (the downward speed of the gate was about 4% larger than the upward speed) between two fixed points. The distance between the two fixed points and the speed of the gate could be varied to generate tides of different amplitude and period.
An example of the ocean tide is presented in Figure 14. As
demonstrated in this figure the ocean tide can very well be approximated by two half sines with slightly different periods (a result of the discharge gate moving faster down than up). In all of the tests the periods of the two half sines differed by less than 15%. The corresponding tide at the lagoon side of the weir is also presented in Figure 14. The weir tide lags the ocean tide by 12 sec while the tidal ranges are approximately the same.
Water surface elevations were measured by vibrating point level recorders. For the location of these recorders, see Figure 13. A quick indication of the water level was provided by a conventional point gage located in the lagoon. The same gage also served to determine the maximum and minimum water depth in each experiment. The mean depth was taken equal to the average of the maximum and minimum water depth. Net discharges were determined from measured float paths using a method that will be described in Section b of this chapter. Floats of different lengths were used. The floats which were made either of wood or styrofoam were weighted so that the top of the cone just
j 2
w Ln
O 100
TIME (SEC) LAGOON (STATION I) APPROXIMATION OF OCEAN TIDE USED IN COMPUTATIONS.
FOR LOCATION OF STATIONS SEE FIGURE 13.
FIGURE 14. MEASURED TIDE CURVES FOR EXPERIMENT 5
56
contacted the water surface. All floats were cylindrical and had a diameter of approximately 0.5 cm, the top was given a conical shape to minimize effects of surface tension. For positioning of the floats a righthanded coordinate system was introduced, the x axis coinciding with the longitudinal axis of the lagoon and the zero located at the open end of the lagoon.
In addition to floats, dye (Rhodamine B) was used to determine the horizontal motion of the water in the lagoon. It was noticed that solutions with a high dye concentration, visible to the eye, when released in the lagoon water had a tendency to stick together and to settle (like a turbidity current) thus indicating only the motion of the bottom layers. An attempt was made to avoid this problem of settling by using dye solutions of a low concentration. After injection of the dye, water samples were taken from the lagoon by means of a rack provided with seven 1 cm glass tubes, spaced 10 cm apart. The glass tubes were placed vertically in the lagoon and sealed off at the top, by means of corks. The rack was then lifted out and the samples collected in test tubes. The samples thus obtained cover the entire water column. The dye concentrations in the test tubes were determined using a Fluorometer. It was hoped that when plotting concentration versus location of the samples a maximum could be indicated, the location of the maximum being the coordinate of the water mass initially located at the location of dye release. Unfortunately, many of the measured distributions did not have a clearly defined maximum and
57
it appeared that test results were not reproducible. It is believed that the dye still tended to stick together and settle rather than dispersethe reason being the low turbulence intensity in the lagoon. (The Reynolds number Re = 9 in the experiments was on the order of
V
1000.) It might well be that in experiments with a higher turbulence level, the previously described technique would be successful. In view of the negative results, the test procedures and test results for the low concentration dye techniques will not be discussed further.
Test Procedure  First the discharge gate was left in a
position resulting in the required average depth and the lagoon checked for circulation induced by the inflow from the weir boxes. This circulation could be compensated for by varying the length of the adjustable walls; see Figure 13. The discharge gate then was started and run for at least two hours to dampen possible disturbances caused by the sudden start of the gate.
Actual testing began with measuring water surface fluctuations at Stations I and III (see Figure 13). Initially also the surface fluctuations at Station II were measured. They appeared not to differ noticeably from those at Station III, and therefore have not been measured in the later experiments. Floats were released one at a time either in the middle of the lagoon or 5 cm from the side walls. Positions of the floats were marked at each slack tide for a period of at least five tidal cycles. In case the lateral displacement of the floats was more than 5 cm, the tests were disregarded and started again until a good run was obtained. The water motion near the bottom was determined by releasing dye in sufficient concentrations to be visible by eye.
58
Summary of Tests  Two series of experiments were carried out. A preliminary series, Test Series 1, to study the variation of the drift velocities in a cross section, the possible side and end effects and the reproducibility of the experiments, and Test Series 2 in which an attempt was made to determine the average net discharge. A summary of the experiments in each test series is presented in respectively Table I and Table II. Test Series 1 consists of 14 experiments numbered 1 through 14. The experiments 1, 2, 6, 7, 9, 10, 12 and 14 only served to develop the experimental procedure and therefore are omitted in Table I. Test Series 2 consists of 11 experiments numbered 15 through 25. Water depths in the experiments range from 5 to 6.5 cm, wave periods range from 70 to 100 sec. and amplitudes range between 0.75 to 1.15 cm. No attempt was made to systematically study the effect of different parameters on the net discharge because of the difficulty of varying water depth, amplitude and period independently.
TABLE I
SUMMARY OF EXPERIMENTS; TEST SERIES 1
Tidal Tidal
Experiment Amplitude (cm) Period (sec) Ave. Depth (cm)
3 0.77 71 5.10 4 0.92 72 5.00 5 1.05 100 4.90 5a 1.05 104 4.90 8 1.00 69 5.00
11 1.10 70 6.45 13 1.15 71 5.30 13a 1.15 71 5.30
59
TABLE II
SUMMARY OF EXPERIMENTS; TEST SERIES 2
Tidal Tidal
Experiment Amplitude (cm) Period (sec) Ave. Depth (cm)
15 1.13 70 5.00 16 0.87 70 5.05 17 1.13 70 6.60 18 1.03 70 6.60 19 0.91 69 6.55 20 0.81 70 6.45 21 0.95 70 6.70 22 1.09 71 6.60 23 0.96 70 5.00 24 0.94 70 4.80 25 1.05 70 4.95
b. Results
Test Series 1  Introductory tests with confetti and dye showed the flow in the middle part of the lagoon to be regular but indicated a somewhat irregular flow pattern at both ends, especially during periods of reversal of the tide. Therefore tracking of the floats was restricted to the region between x = 50 cm and x = 450 cm. Maximum velocities in the experiments varied between 3 cm/sec and
5 cm/sec, depending on water depth, amplitude and period. Reynolds numbers (Re = 1 ) were larger than 800 (the limit for fully turbulent flow) 60%  80% of the time (the exact percentage depending on depth, period and amplitude).
Actual experiments were carried out with three different
float lengths. 3 cm, 2 cm and 1 cm. Floats were placed in the middle
60
of the lagoon, y = 0 cm, and 5 cm from the side wall, y = +10 cm. In addition, in Experiments 13 and 13a floats were also placed at y = 10 cm. Typical examples of measured float positions are presented in Figures 15, 16, 17 and 18. In these figures the positions of the floats, at the time they first reversed their path, are reduced to the same coordinate, which is taken equal to the average of the positions of the floats at the first reversal point (the positions usually did not differ more than 20 cm). The time a float arrives at the first reversal point is set equal to zero. Because of the difficulty of determining the exact time a float reverses its path, the subsequent times of reversal are set equal to the average of the reversal times of all the floats used in an experiment (reversal times for different floats at corresponding slack tides differed at the most by 10 sec). Positions of the floats are plotted at these average reversal times.
Figures 17 and 18 show the results of two identical
Experiments 13 and 13a. From these figures it may be seen that the positions of corresponding floats differ somewhat, but the range covering the positions of the floats at corresponding slack tides is approximately the same for both experiments. For a better comparison, the particle excursion between successive slack tides of the two longer floats in the middle of the canal are tabulated for both experiments in Table III.
Most floats when placed in the water moved parallel to the
side walls; in unusual cases, a float was traced again because it had displaced laterally more than 5 cm and when this happened it nearly always occurred for those floats initially placed 5 cm from the wall.
LEGEND
ZERO OF COORDINATE SYSTEM IS AT OPEN END OF LAGOON
LENGTH FLOATS PLACED AT
OF
FLOAT y O0 CM. y =+1O CM. y 10 CM.
I cm. V v
2 cm A A
350
3 cm. 0 o
L AVERAGE OF THE POSITIONS OF THE FLOATS AT A THE FIRST REVERSAL POINT O V O * j 300 v A
O
j 250
0
o
200
150 '  I a
0 50 100 150 200
+ TIME (SEC.)
FIGURE 15. MEASURED FLOAT POSITIONS FOR EXPERIMENT 4
FOR LEGEND SEE FIGURE 15 400
0
350
VA
VA
 300 o VA
o
A
o o z 250 o VA
200
150 I I I I
0 50 100 150 200
 TIME (SEC.) FIGURE 16. MEASURED FLOAT POSITIONS FOR EXPERIMENT 50
FOR LEGEND SEE FIGURE 15 400
350
<
0 300
IL
, o v' A z * o 0 i 250  v OAA O 0
o v ved
200  A
150 I I I I I
O 50 100 150 200
~ TIME (SEC.)
FIGURE 17. MEASURED FLOAT POSITIONS FOR EXPERIMENT 13
FOR LEGEND SEE FIGURE 15
400
350
< 300
o v
J v I_ o, e" V oA
0 ï¿½
S250 VVA
 0 Sv
VO
vA A 200
150
0 50 100 150 200
TIME (SEC.)
FIGURE 18. MEASURED FLOAT POSITIONS FOR EXPERIMENT 13a
65
TABLE III
REPRODUCIBILITY OF EXPERIMENTS;
PARTICLE EXCURSION BETWEEN SUCCESSIVE SLACK TIDES
FOR EXPERIMENTS 13 AND 13a
Experiment 13 Experiment 13a
3 cm 2 cm 3 cm 2 cm Slack Tide Float Float Float Float
1  2 +72 +69 +74 +69 2  3 41 48 41 58 3  4 +62 +60 +62 +69 4  5 35 43 38 50 5  6 +66 +61 +63 +67
Floats were placed in the middle of the lagoon. Tabulated values are excursions in cm.
66
From this it may be inferred that the flow pattern in the experimental lagoon is one dimensional in a sense that no significant meandering of the flow occurs.
It may be seen from the examples presented in Figures 15, 16, 17 and 18 that the net drift differs for each float length which is not surprising in view of the discussion of the vertical drift profile on page 45. For all experiments the following was noted: the direction of the net drift in each experiment was the same for all floats, however, the magnitude of the net drift varied with the length of the float. This and the fact that injected dye clouds always moved in the same direction as the floats suggests that the direction of the net drift is the same for each point in the vertical. The experiments do not define the shape of the drift profile over depth.
Results of Experiments 13 and 13a in which floats were tracked at both sides of the lagoon show that the net drift is not uniformly distributed over the width of the lagoon; the net drift at the north side is considerably larger than in the middle and at the south side. This probably is a result of the flow in and out the lagoon not being symmetric with respect to the longitudinal axis of the lagoon.
Determination of Net Discharge from Float Paths  Before presenting the results of Test Series 2, the method used to determine the net discharge q* from a float path s will be discussed. Mainly because of the different nature of the two parameters, q, is an Eulerian parameter and s is a Lagrangian parameter, the mathematical relation between these two quantities gets rather complicated and unmanageable. Therefore for the determination of q* from s recourse is taken to the
67
following empirical method. Assume that the path of a float is representative of the average horizontal particle motion. As an example the (fictitious) measured positions of such a float at successive slack tides are indicated in Figure 19. The successive float positions are connected by straight lines. The midpoints of these straight lines are determined and a straight line is drawn through these points. The slope of this line represents a velocity u,, which may be interpreted as an average net drift velocity. The net discharge q, then is found by multiplying this average velocity and the average depth. Numerical experiments to be discussed in Section c show this method to yield good results.
Test Series 2  The range of hydraulic parameters used in Test Series 2 (see Table II) is approximately the same as those used in Test Series 1 and thus also the velocities and Reynolds numbers are on the same order as earlier mentioned.
In contrast to the Test Series 1 experiments, the Test Series 2 program was carried out with only one float length. The length of the float was chosen as large as possible to arrive at an average over depth drift velocity. This and the requirement that the float not contact the bottom at all times resulted in float lengths of 3.5 and 4.5 cm, respectively, for depths of 5 and 6.5 cm. Because of the irregular flow pattern at both ends of the lagoon (see discussion of result of Test Series 1) the tracking of the floats was limited also to the lagoon section between x = 50 cm and x = 450 cm. Typical results of float tests for Test Series 2 are presented in Figures 20 and 21. In both figures straight lines representing the "average net drift velocity" u,
SLOPE OF THIS LINE REPRESENTS U
350 '
300 P,
0 0 MEASURED FLOAT POSITIONS
z
 * MIDPOINTS " 200
0
CL
150 50 100 150 200
 TIME (SEC.)
FIGURE 19. DETERMINATION OF NET DISCHARGE FRONT MEASURED FLOAT PATH
FOR LEGEND SEE FIGURE 15
250
I
0
1 200
LL FLOAT AT Y=lOcm. LL r U = 0.286 cm./sec.
O
0 FLOAT AT Y= Ocm U = 0.365 cm./sec. S150
FLOAT AT Y t IOcm.
U&= 0.426 cm./sec.
100 I I
0 50 100 150 200
  TIME (SE C.)
FIGURE 20. MEASURED FLOAT POSITIONS FOR EXPERIMENT 22
FOR LEGEND SEE FIGURE 15 400
FLOAT AT Y=41Ocm. = 0.69 cmlo. ./sec. 350o
O FLOAT AT Y =l10cm. 0 u U*= 0.71 cm./sec. o 300 g I "FLOAT AT Y Ocm.
SU,(= 0.59 cm./sec.
0
O
250
20 50 100 150 200
TIME (SEC.)
FIGURE 21. MEASURED FLOAT POSITIONS FOR EXPERIMENT 24
71
are drawn using the previously described method. A complete summary of the "average net drift velocities" and net discharges for the different experiments is presented in Table IV.
The net discharge q,, presented in the last column of Table IV, is found by multiplying the average of the three net drift velocities by the average depth h. Velocities and discharges are also listed for Experiments 13 and 13a, which were the only experiments in Test Series 1 in which floats were used in the middle and both sides of the lagoon.
The results listed in Table IV show that the net drift velocity strongly depends on the lateral position of the float; in general, for the experiments with 5 cm depth the net drift is largest for the floats placed at y = 10 cm, for the experiments with 6.5 cm depth the net drift is largest for the floats placed at y = +10 cm. As mentioned earlier the lateral variation in net drift should probably be attributed to the asymmetry of the experimental setup; see page 66.
In each of the Test Series 2 experiments, in addition to the tracking of floats, the water motion was also studied by observing the displacement of a dye cloud released in the middle of the experimental lagoon. The dye cloud, initially about 30 cm long and extending over the full width of the channel, stretched longitudinally and at the same time gradually moved in the same direction as the floats. In all cases, the net displacement of the dye patch was observed to be in the same direction as that of the floats.
One of the more striking results of the experiments is (see Table IV) that the net drift in the experiments with a water depth of
TABLE IV
MEASURED DRIFT VELOCITIES AND NET DISCHARGES
Tidal Tidal Aver. u (cm/sec) u*(cm/sec) u*(cm/sec) 2
Exper. Ampl(cm) Period(sec) Depth(cm) y = +10 cm y = 0 cm y = 10 cm q*(cm/sec)
15 1.13 70 5.00 1.04 1.30 1.60 6.40 16 0.87 70 5.05 0.13 0.09 0.37 1.00 17 1.13 70 6.60 0.47 0.47 0.27 2.68 18 1.03 70 6.60 0.42 0.43 0.42 2.79 19 0.91 69 6.55 0.33 0.39 0.27 2.16 20 0.81 70 6.45 0.25 0.31 0.19 1.61 21 0.95 70 6.70 0.29 0.40 0.27 2.18 22 1.09 71 6.60 0.36 0.42 0.28 2.34 23 0.96 70 5.00 0.43 0.27 0.39 1.81 24 0.94 70 4.80 0.69 0.59 0.71 3.18 25 1.05 70 4.95 0.95 1.02 0.95 4.80 13 1.15 71 5.30 0.29 0.37 0.58 2.18 13a 1.15 71 5.30 0.43 0.33 0.55 2.32
73
about 5 cm (Experiments 15, 16, 23, 24, 25, 13 and 13a) is toward the weir while the direction of the net drift is reversed for the experiments with a water depth of approximately 6.5 cm (Experiments 17, 18, 19, 20, 21 and 22). An explanation for this might be the following. Consider the hydrodynamic Equations (2.1) and (2.2) neglecting friction and storage. (This simplification is justified because in the experiments the storage is zero and friction is of relatively little importance.) Equations (2.1) and (2.2) then read:
0 + L = 0 (6.1) at ax
2(6.2)
'q + 1 'q + g(h + n) 0 (6.2)
at h ax ax
The boundary condition at x = 0, the open end of the experimental lagoon is
n = a cos at (6.3) in which
2,
a = angular frequency of the tide
T = tidal period.
A linearized version of the weir equation is used for the boundary condition at the weir side of the lagoon.
q = Ii2ghw(ni  no) + 2gn o(ni  no) (6.4)
in which
74
ni = water level at the lagoon side of the weir
Il = "linear" weir coefficient
h = average water depth over the weir.
w
The last term on the righthand side of Equation (6.4) represents the influence of varying depth over the weir on the discharge.
A trial solution for the system of Equations (6.1) through (6.4) is
n = n,(x) + nk(x,t) + . . . (6.5) q = q,(x) + qt(x,t) + . . . (6.6) in which n and qz are the solutions of the system of equations when neglecting the nonlinear terms in the Equations (6.1) and (6.2) and the last term on the righthand side of Equation (6.4). In view of Equation (6.3), nQ and qz are periodic functions with period T. Substituting the trial solution, Equation (6.5), in Equations (6.1) and (6.2) and time averaging yields
q, = const. (6.7)
an* 2 3q z n
gh + q  + gn = 0 (6.8) ax h 18x a x
Equation (6.8) with the boundary condition, n~ = 0 at x = 0, yields a solution n~(x). Substituting Equation (6.5) in Equation (6.4) yields after time averaging
2
q = I2ghw n + 1Q2g n no  1 2g no (6.3)
in which n, and n are evaluated at the lagoon side of the weir. The
75
first term on the righthand side of Equation (6.9) results from the nonlinear terms in the field Equations (6.1) and (6.2); the last two terms result from the term in the weir equation representing the influence of the time varying depth over the weir. It is known that for a channel closed at one end n, at the closed end is positive; see LonguetHiggins [13]. It seems reasonable to assume that n, is also positive in case of a partially closed channel. Furthermore, because of the phase shift between no and n at the lagoon side of the weir see Figure 14) it follows that in Equation (6.9) no n < no. The first term on the righthand side of Equation (6.9) thus is positive while the sum of the last two terms is negative. In the experiments, the net drift results from two opposing effects.
 the net drift resulting from the nonlinear terms in the field
equations
 the net drift resulting from the variation in depth over the
weir during a tidal cycle.
The resultant direction of the net drift thus depends on which of these effects is predominant.
It seems tempting to translate the experimental data to a
prototype scale. In this context, however, it should be mentioned that translating to realistic prototype dimensions is not possible because the relative magnitude of the terms in the dynamic equation is entirely different for model and prototype. In the experiments the friction term is very small compared to the total acceleration while in a natural lagoon these terms are of the same order of magnitude; see e.g. Figure 6.
76
c. Computation of Mass Transport; Comparison with Measured
Data
Equations (2.1) and (2.2) are used to describe the conservation of mass and momentum in the experimental lagoon. In Equation (2.1) the term M representing the storage is set equal to zero. The flow over the weir is described by an equation of the form
qw=  n (hw + ni) V2g(n  i) (6.10) for flow from the ocean to the lagoon, and
qw= + pp(h + n o) /2g(ni  no) (6.11) for flow from the lagoon to the ocean. In these equations
qw = discharge per unit width over the weir
91n and p = weir coefficients; the subscript n refers to flow from the ocean to the lagoon; the subscript p refers to flow
from lagoon to ocean
h = average depth over weir
no = water level in ocean
ni = water level at Station III; see Figure 23.
The weir equations involve empirical constants 9n and p . A n p
literature search [27],[29],[14], [10] for values of these socalled weir coefficients did not show any data for the combination of parameters (downstream water level, upstream energy level and weir height) encountered in the experiments. In addition, all tests described in the literature deal with steadystate conditions while also the turbulence level is much higher than found in the experiments described here. Therefore,
77
to arrive at an order of magnitude value for the coefficients .n and Pp the following theoretical approach is taken. Bernoulli's equation is applied upstream of the weir between Stations I and II; see Figure 22. This yields for positive flow (flow from the lagoon to the ocean).
I II
di LAGOON {
TIDAL BASIN{
Figure 22. WEIR; FLOW FROM LAGOON TO TIDAL BASIN
2 2
ni + q n2 w + qw (6.12) 2gd 2gd C
in which
d. and d represent the total depths at respectively Stations
1 W
I and II; see Figure 22.
nw = the time varying water level at the weir measured with
respect to mean level of the weir
C = coefficient to account for the stream lines over the weir
not being horizontal.
At the down stream side of the weir, the momentum equation is applied between Stations II and III which yields
78
2 Cd = 1 c w 1] (6.13)
nw no g Cdd d
c wo o
in which
d = total depth at Station III; see Figure 22.
Eliminating n between (6.12) and (6.13) and writing the result in the
form of Equation (6.11) yields
q 1 d /2g(n n ) (6.14)
d 2d Cd
1 w w cw
[ + W ( 1 W )]
2 2 Cd h C d. co o
c 1
and thus
S1 (6.15) P 2
d 2d C d
1 w w c w [  d +   (  1)]
2 2 Cd d C d. co o c 2
Similarly, for flow from the ocean to the lagoons it is found
n = 1 (6.16) d 2d C d
1 w w c w
2 2 C d d C d ci 1 )
c o
Values of 9p and pn are computed from, respectively, Equations (6.15)
and (6.16) for two typical sets of values for d., do and d w. The
results are listed below.
79
d. = 5 cm d = 1.65 cm d = 30 cm
I w o
C = 0.7 p 0.75
c p
pn = 0.87
C = 0.8 p= 0.87
C p
pn = 1.02
Cc = 0.9 p = 1.00 pn = 1.18
d. = 6.5 cm d = 3.15 cm d = 31.5 cm
1 W o
C = 0.7 p = 0.81 pn = 0.95
C = 0.8 p = 0.95 p = 1.12
n
C = 0.9 = 1.13
c p
n = 1.34
The foregoing "theoretical" values of the weir coefficients indicate that
 the order of magnitude of the weir coefficients is one
 the coefficient pn is always larger than p .
Within this framework the final combination of values for p and p n will be chosen such that measured and computed net discharges for the experiments are in good agreement.
The equations describing the flow in the lagoon and the flow over the weir are solved using the numerical technique described in Chapter 3. The grid scheme used in the computations is indicated in Figure 23. The value of the friction factor F = 0.005 (this value
472.5 17.5 ,
2 3 4 5 6 7 8 9 10 11
r0 + +  +  +  +  +  +  +  +  +
2 3 4 5 6 7 8 9 10 11 12
WEIRC
 DISCHARGE STATIONS
+ WATER LEVEL STATIONS
DIMENSIONS IN CM.
FIGURE 23. GRID SCHEME FOR NUMERICAL COMPUTATIONS OF LABORATORY EXPERIMENTS
81
was obtained from experiments carried out in a hydraulic flume with a bottom of the same roughness as the experimental lagoon). The space and time steps are: Ax = 45 cm and At = 0.1 sec. Recorded tides in the basin (Station I; see Figure 13) are approximated by two half sines. In the computational procedure, first the discharges q and the water levels n in the lagoon are determined, starting from the given tide in the basin. The net discharge q, then follows from
T
q = q dt (6.17)
The integral is evaluated numerically.
Trial and error procedures indicate that for the combination of weir coefficients, lp = 0.8 and pn = 1.2,computed and measured net discharges are in good agreement as shown in Figure 24. In Figure 24, the net discharges computed with this combination of weir coefficients and measured net discharges (see Table IV) are presented for all experiments of Test Series 2 and for Experiments 13 and 13a of Test Series 1. For each of these experiments, also the particle path s(xo,t) is computed from
t q(s(xo),t)
s(x ,t) J h+(s(xo,t) dt (6.18) o h + n(s(x,,t)
The integral is evaluated numerically. The computed particle paths are used to verify the empirical method of determining net discharges from particle trajectories; see page 66. Net discharges are determined from the computed particle paths using the empirical method and are
6
*1
6
0
24 I
2 2
(7
01 0 42

6NUMERALS REFER 10 EXPERIMENT
02
716
MEASURED NET DISCHARGE q, (CM./SEC.) 39
FIGURE 24. COMPUTED AND MEASURED NET DISCHARGES
4
83
compared with the net discharges obtained by integration of Equation (6.17). For results, see Table V. The good agreement proves the empirical method to be a reliable tool in deriving net discharges from particle trajectories.
TABLE V
COMPUTED NET DISCHARGES; EVALUATION OF EMPIRICAL
METHOD OF DETERMINING NET DISCHARGES
Tidal Tidal Aver. q*(cm 2/sec) q*(cm 2/sec) Exper. Ampl(cm) Period(sec) Depth(cm) Eq. (6.18) Empirical
15 1.13 70 5.00 7.10 7.40 16 0.87 70 5.05 0.27 0.30 17 1.13 70 6.60 2.60 2.70 18 1.03 70 6.60 2.16 2.26 19 0.91 69 6.55 1.87 1.90 20 0.81 70 6.45 1.83 1.87 21 0.95 70 6.70 2.26 2.28 22 1.09 71 6.60 2.14 2.24 23 0.96 70 5.00 1.92 2.00 24 0.94 70 4.80 4.54 4.70 25 1.05 70 4.95 5.10 5.35 13 1.15 71 5.30 1.60 1.65 13a 1.15 71 5.30 1.60 1.65
Examples of computed particle paths for Experiments 22 and 25 are presented in Figures 25 and 26. In the same figures, the measured float positions at slack tide are plotted also.
Finally, it is noted that a comparison between computed and
recorded data can only be as good as the quality of the individual data.
FOR LEGEND SEE FIGURE 15
 COMPUTED
I
0
O
_
200
a. O
O
V
z
0
o"
100   '0 50 100 150
  TIME (SEC.)
FIGURE 25. MEASURED AND COMPUTED FLOAT POSITIONS FOR EXPERIMENT 22
FOR LEGEND SEE FIGURE 15
 COMPUTED
400
0
350
o 300 L
o
o
I.
E 250
o
0
2 0 0  .. . . . J  I I I 1 _.0 50 100 150 200
 TIME (SEC.) FIGURE 26. MEASURED AND COMPUTED FLOAT POSITIONS FOR EXPERIMENT 25
86
In this respect, it should be kept in mind that the measured net discharges are empirically derived from float trajectories which are assumed to represent the average horizontal particle motion. The validity of the latter assumption cannot be judged with any certainty until more detailed experiments on the distribution of the net drift over the vertical have been carried out. With regard to the computational model, it should be mentioned that inaccuracies in measured depth, especially for the smaller depths, lead to relatively large errors in the net discharge as demonstrated in Figure 27. Also, the trial and error procedure necessary to determine the weir coefficients is a poor substitute for a welldefined description of the flow over the weir, especially when variations in those weir coefficients lead to rather large changes in the net discharge as shown for Experiment 20 in Table VI. It is admitted that this to a certain extent reduces the value of a comparison between computed and recorded data.
TABLE VI
VARIATION OF NET DISCHARGE IN EXPERIMENT 20
FOR DIFFERENT COhBINATIONS OF WEIR COEFFICIENTS
Pp  10.750.95 P.751.05 0.81.1 0.81.2 i0.91.15 0.91.2 11.3
q,(cm2/sec) 0.65 1.37 1.23 .83 0.67 0.96 0.68
d. Summary
The displacement of water masses in an idealized lagoon constructed in the laboratory has been studied (1) by tracking floats
87
I0
4
AMPLITUDE= 0.8 CM.
2
3
4.8 52 5.6 6.0 6.4 DEPTH (CM.) FIGURE 27. VARIATION OF NET DISCHARGE WITH DEPTH
88
of different lengths and tracing of injected dye clouds and (2) by conducting calculations to represent this phenomenon. The lagoon which had a uniform width and depth was at one end freely connected to a tidal basin and on the other end was connected to the same basin via a submerged sharp crested weir. The experimental results showed that for a given set of hydraulic conditions (1) the magnitude of the net drift of a float depends on its length, (2) the direction of the net drift is the same for each float length and (3) the dye clouds move in the same direction as the floats. These observations suggest that the direction of the net drift is the same for each point over the entire depth.
The direction of the net drift is not the same for all experiments. An explanation for this is that the net drift is the result of two opposing effects: (1) the net drift due to the nonlinear terms in the field equations and (2) the net drift resulting from the nonlinear terms in the weir equation.
Net discharges are derived from measured float paths using an
empirical method described on page 66. To account for lateral variation in the drift, net discharges are based on the average path of three floats placed in the middle and at the two sides of the lagoon.
The computational model describing the flow in the lagoon
involves two weir coefficients. The value of these weir coefficients is determined by matching computed and measured net discharges. This admittedly reduces the value of the computation model; however, the good agreement between measured and computed net discharges and thus
89
between measured and computed mass transport for all experiments for the same combination of weir coefficients indicates the validity of the theory on which the computations are based.
7. APPLICATION
To demonstrate that the convective transport in lagoons
resulting from tides only is not merely a matter of academic interest but of practical significance as well, the net discharge is calculated for Lake Worth, Florida, which is one of a number of typical lagoons of the U. S. east coast and Gulf of Mexico. The computations were carried out using the model described in Chapters 2, 3 and 4. Because the purpose of the computations is to demonstrate the effect of the tide only, wind stresses and fresh water inflow are not included in this treatment.
Lake Worth, located on the lower east coast of Florida, is a narrow tidal lagoon; see Figure 28. The lake is generally parallel to the coastline and is about 20 miles long and 5 to 8 ft. deep. The actual flowconveying portion is approximately 2000 ft. wide; the shallow areas at the east side of the lake serve primarily as storage. Near the north end, Lake Worth is connected to the ocean by North Lake Worth Inlet, which has a width of 800 ft. between the two entrance jetties and is about 35 ft. deep. At the south end, Lake Worth is connected to the ocean by a much smaller inlet, South Lake Worth Inlet, with dimensions of 130 ft. wide and 10 ft. deep. Lake Worth is part of the Intracoastal Waterway for its entire length.
The tides in the Lake Worth area have a predominantly semidiurnal character. According to the tide tables [24], the mean tidal range in the Atlantic Ocean at Palm Beach is 2.8 ft. and the spring tidal range is 3.3 ft. Going inland the corresponding tidal ranges at the
90

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TIDEINDUCED MASS TRANSPORT IN SHALLOW LAGOONS By JACOBUS VAN DE KREEKE A Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements of the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 1971
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ACKNOWLEDGEMENT First, I would like to thank my wife, Hetty, for her understanding and patience during the time of deprived family life accompanying the preparation of this dissertation. A dissertation to a large extent is the result of the interplay between the candidate and his academic environment; in this case, the faculty, staff and student body of the Department of Coastal and Oceanographic Engineering. Among those, a special word of thanks is due to Dr. R. G. Dean, Chairman and Professor of the Department, who supervised the dissertation; the many hours of discussion with Dr. Dean have greatly contributed to a better understanding of the subject. The helpful criticism and suggestions by Dr. B. A. Christensen, Professor of Civil Engineering, concerning the experiments is greatly appreciated, The study was in part supported by the Office of the Water Resources Center and in part by the Florida Department of Natural Resources. All computations were carried out using the University of Florida IBM 360 computer. 11
PAGE 3
TABLE OF CONTENTS Pa 3 e ACKNOWLEDGEMENT ii LIST OF FIGURES iv LIST OF TABLES vi LIST OF SYMBOLS vii ABSTRACT ix CHAPTERS : 1. INTRODUCTION 1 2. THE TIDAL EQUATIONS FOR THE LAGOON AND THE INLETS 4 3 . NUMERICAL SOLUTION OF THE TIDAL EQUATIONS 13 A. DETERMINATION OF THE FRICTION FACTOR F 24 5. MASS TRANSPORT IN LAGOONS DUE TO TIDAL ACTION 30 a. Analytic Solution; Physical Concepts 30 b. Influence of Inlets on the Mass Transport 47 6 . LABORATORY EXPERIMENTS 52 a . Equipment and Procedure 52 b. Results 59 c. Computation of Mass Transport; Comparison with Measured Data 76 d . Summary 86 7. APPLICATION 90 8 . SUMMARY AND CONCLUSIONS 96 APPENDIX 98 A. THE MATHEMATICAL DESCRIPTION OF LONG PERIOD GRAVITY WAVES ; EULERIAN DESCRIPTION 99 REFERENCES 109 BIOGRAPHICAL SKETCH 112 in
PAGE 4
LIST OF FIGURES Figure No. Page 1 Reference Frame 5 2 Inlet Schematization 8 3 Location of Variables in the Numerical Grid.... 14 4 Grid Scheme for Different Inlet Configurations. 18 5 Curves Represented by Equations (3.3) and (3.4) 21 6 Tide in a Sea Level Canal 27 7 Mean Water Level in a Lagoon in the Presence of a Damped Progressive Wave 38 8 Effect of Higher Order Terms on q^ 41 9 Effect of "Linearization" of the Friction Term on q 42 10 Effect of Convective Acceleration and Nonlinear Part of the Surface Gradient on the Net Discharge q^ 44 11 Lagoon Connected to the Ocean by Inlets 47 12 Net Discharge in Lagoon Versus Width of Inlet II 50 13 ExperimentalSet Up 53 14 Measured Tide Curves for Experiment 5 55 15 Measured Float Positions for Experiment 4 61 16 Measured Float Positions for Experiment 5a 62 17 Measured Float Positions for Experiment 13 63 18 Measured Float Positions for Experiment 13a.... 64 19 Determination of Net Discharge from Measured Float Path 68 20 Measured Float Positions for Experiment 22 69 IV
PAGE 5
LIST OF FIGURES (Continued) Figure No. Page 21 Measured Float Positions for Experiment 24 70 22 Weir; Flow from Lagoon to Tidal Basin 77 23 Grid Scheme for Numerical Computations of Laboratory Experiments 80 24 Computed and Measured Net Discharges 82 25 Measured and Computed Float Positions for Experiment 22 84 26 Measured and Computed Float Positions for Experiment 25 . 85 27 Variation of Net Discharge with Depth 87 28 Lake Worth, Florida; Location of Tide Recorders.. 91 29 Schematization of Lake Worth 93 30 Measured and Computed Water Levels in Lake Worth 95
PAGE 6
LIST OF TABLES Table No. Pa Â§ e I Summary of Experiments; Test Series 1 II Summary of Experiments; Test Series 2 Ill Reproducibility of Experiments; Particle Excursion Between Successive Slack Tides for Experiments 13 and 13a IV V Computed Net Discharges; Evaluation of Empirical Method of Determining Net Discharges VI Variation of Net Discharge in Experiment 20 for Different Combinations of Weir Coefficients 58 59 65 Measured Drift Velocities and Net Discharges.... ' 2 83 86 vi
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LIST OF SYMBOLS a tidal amplitude c wave celerity in case of no friction o d total depth f DarcyWeisbach resistance coefficient g gravitational acceleration h mean depth k wave number k wave number in case of no friction o m coefficient which accounts for entrance losses and nonuniform velocity distribution q discharge per unit width q A net discharge per unit width s particle trajectory; in Appendix A used as strength of a source per unit volume t time u velocity in x direction u^ net drift velocity v velocity in y direction w velocity in z direction x particle position at t = t o f f Q x horizontal Cartesian coordinate y horizontal cartesian coordinate z vertical Cartesian coordinate, positive upward A crosssectional area A, lagoon area vn
PAGE 8
B width of the water body C contraction coefficient c D total depth F resistance coefficient used when considering quadratic friction F resistance coefficient used when considering linear friction L length of the water body M lateral flow, rainfall P wetted perimeter Q total discharge R hydraulic radius S slope of the water surface X free surface stress/ (mass density of water) in the x direction Y free surface stress/ (mass density of water) in the y direction a phase angle e relative error n water surface elevation h . mean water surface elevation u factor related to the resistance; also used as weir coefficient in Chapter 6 u "linear" weir coefficient i p density of water a angular frequency of the tide x shear stress a) angular frequency of the earth rotation <> latitude, positive for Northern Hemisphere 0, Coriolis factor vm
PAGE 9
Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment or the Requirements for the Degree of Doctor of Philosophy TIDEINDUCED MASS TRANSPORT IN SHALLOW LAGOONS By Jacobus van de Kreeke December, 1971 Chairman: Bent A. Christensen CoChairman: Robert G. Dean Major Department: Civil Engineering Analytical, numerical and experimental approaches are employed to describe the hydromechanics of lagoons connected to the ocean by two or more inlets. Because special attention is given to the tideinduced mass transport, all second order terms in the hydrodynamic equations are retained. The study is restricted to lagoons with a onedimensional flow pattern and water of uniform density. In designing a numerical solution to the equations, the inlet equations are regarded as implicit boundary conditions to the equations describing the flow in the lagoon proper. The advantages of this approach are (1) the size of the computational grid in the lagoon can be chosen independently of the relative small dimensions of the inlets, (2) the flow at branching inlets (an inlet connecting a lagoon to the ocean such that branching of the inlet flow can occur) still can be described by a onedimensional tidal model. To gain physical insight into the phenomenon of tideinduced ix
PAGE 10
mass transport, an analytic solution to a simplified set of hydrodynamic equations is presented. This solution demonstrates that the magnitude and direction of the mass transport depend on the difference in mean levels at the two inlets imposed by the ocean and the hydraulic characteristics of the inlets and the lagoon. In particular the mass 2 transport (1) increases with the ratio Â— (a = amplitude, h = depth), (2) is sensitive to changes in the friction factor F which must be established by calibration. It may be inferred from (1) and (2) , respectively, that a significant tideinduced mass transport should only be expected in shallow lagoons and that accurate calibration procedures are necessary to arrive at a reliable magnitude of the mass transport. In presenting the analytic solution, special attention is given to the role of the inlets; it is demonstrated that by properly choosing the dimensions of the inlet, a considerable increase in net transport of water can be obtained. The predictive capability of the numerical model with regard to mass transport is confirmed by favorable comparison between measured and computed mass transport for a series of experiments carried out in the laboratory. In the experiments, the tideinduced mass transport was determined in a lagoon of uniform width and depth; at one end the lagoon was connected to a tidal basin and at the other end was connected to the same basin via a submerged weir. The numerical model is used to compute the mass transport in one of Florida's shallow lagoons, Lake Worth. The order of magnitude of the computed mass transport is such that it is of considerable importance when regarding renewal of lagoon waters and transport of pollutants.
PAGE 11
1. INTRODUCTION Estuaries traditionally have been centers of population concentration. Unfortunately for reasons of economy and convenience, the domestic and commercial wastes accompanying many of these urban areas are discharged into coastal waters thereby degrading the water quality. An improved understanding of the mechanisms responsible for the transport of constituents in estuaries is necessary in order to assess the capability of these systems to assimilate waste loads, to renew their waters by flushing due to fresh water inflow or by exchange with oceanic waters and the possible improvement due to engineering structures, such as inlets. In general, two modes of transport exist, convection and mixing. In tidal waters, often the convective transport is solely attributed to fresh water inflow or winddriven circulation, while it is generally regarded that the only effect of the tidal motion is to generate the turbulence associated with the mixing process. It will be shown, based on theoretical considerations, that for shallow lagoons ^ lagoon is defined here as a body of water connected to the ocean by two or more narrow constricted inlets), the convective transport or mass transport due to the tidal motion only can be an important factor in the renewal of water in the lagoon. Because the magnitude of the mass transport is mainly governed by the location and the dimensions of the different inlets, a considerable increase in mass transport can be obtained by properly designing the inlets. This concept of using tidal inlets for environmental controls has been discussed in a wider context by Lockwood and Carothers [11 ] .
PAGE 12
In general, a tidal wave, when entering a lagoon, will be modified by the inlets and the main water body. The special case for which the tidal propagation in the main water body may be neglected will not be discussed here. The reader is referred to Keulegan [9], Van de Kreeke [25], Shemdin [20], and Mota Oliveira [15]. In describing the flow field, different sets of equations are used for the flow in the lagoon proper and the flow in the inlets. The reason for this is that certain assumptions made in deriving the equations for the lagoon do not hold for the flow in the inlets. In the equations for both the lagoon and the inlets the nonlinear terms must be retained because in a mathematical sense the convective transport results from these nonlinear terms. The inlet equations used here are algebraic equations while the flow in the lagoon is described by the wellknown long wave equations. In the solution, the inlet equations are regarded as implicit boundary conditions for the long wave equations. This implies that it is not necessary to extend the numerical grid used for the main water body to the inlets which, because of their relatively small horizontal dimensions, would lead to small grid sizes and consequently a considerable increase in computational effort. The study is restricted to lagoons having a definite onedimensional flow pattern. The onedimensional numerical model used is based on a space and time staggered explicit finite difference scheme similar to the one described by Reid and Bodine [18]. As stated before, the nonlinear terms in the equations, which include the friction term, are important when regarding convective transport. An accurate value of the friction factor, therefore, is one
PAGE 13
of the first requirements to prediction of reliable values for the convective transport. In view of this a special discussion is devoted to the determination of the friction factor. To evaluate the predictive capability of the computational model, measured and computed mass transport in a lagoon are compared for a series of laboratory experiments. The experiments were carried out in the Coastal Engineering Laboratory of the University of Florida. Finally, the results of the study are applied to an actual lagoon, Lake Worth, located in southeast Florida.
PAGE 14
2. THE TIDAL EQUATIONS FOR THE LAGOON AND THE INLETS In the Eulerian frame, the velocities and water surface elevations are related to time and the various locations in geometric space by two types of equations, one expressing the conservation of mass, the other expressing the conservation of momentum. The equations are derived in Appendix A. The discharge per unit width and the water levels rather than the velocity and the water levels are chosen as the dependent variables because this leads to a simpler form of the conservation of mass equation. Because the considerations on mass transport will be restricted to those lagoons having a definite onedimensional flow pattern, only the onedimensional form of the equations and the assumptions made in deriving these equations will be summarized here. It may be seen from Appendix A that the equation of conservation of mass takes the form a + a = M (2.D 3t 8x in which x is the horizontal Cartesian coordinate, see Figure 1. n = water surface elevation q = discharge per unit width M = net inflow per unit surface area due to rainfall, lateral inflow, etc. The variables q and n are mean values over a time interval, which is large compared to the turbulence time scale and small compared to the period of the tide.
PAGE 15
In deriving the above equation, the fluid was assumed to be incompressible . / TjU.t) h(x) STiLL WATER LEVEL Figure 1. REFERENCE FRAME The equation of conservation of momentum may be written as tz + r tt + g(h 3t h 9x n) oil 3x .EslaJ (h + n )' in which h = mean depth F = resistance coefficient. The main assumptions made in deriving Equation (2.2) are incompressible fluid uniform density vertical accelerations are negligible tidegenerating forces in the lagoon are negligible no free surface stresses. (2.2)
PAGE 16
The assumptions made in deriving the long wave equations, in general, hold very well for large bodies of water but are less justified when dealing with transitions and regions in which the flow is restricted, e.g. inlets. Unfortunately, the complete equations describing the flow in inlets are still very complex and difficult to solve and therefore, recourse is taken to a semiempirical representation. The equation commonly used to describe flow in inlets is Â™/ \ + / F Q 2 L . m R Q 2 N g R(n n ) = + ( z^r + Â— T=Z ) 10 P R 2P R (2.3) + sign for Q >_ sign for Q <_ 0. See for example Keulegan [9]. In Equation (2.3) P = wetted perimeter of inlet crosssection measured at mean ocean level R = hydraulic radius for the inlet crosssection measured at mean ocean level Q = total discharge L = length of inlet n = ocean tide o n . = lagoon tide m = coefficient which accounts for entrance losses and the nonuniform velocity distribution. In Equation (2.3) second order effects have been neglected. A revised version of the inlet equation, including all second order terms, will be derived in the next paragraphs.
PAGE 17
It is assumed that the flow in the inlet region is one dimensional and storage is neglected. The equation of conservation of mass then becomes go in which Q = total discharge. To describe the dynamics of the flow, the inlet is divided into three regions as indicated in Figure 2. During flood, the flow in Region I is governed by the convective acceleration and the pressure forces; bottom friction and local acceleration are considered small. This leads, when neglecting the velocity head in the ocean, to the Bernoulli equation (the Bernoulli equation is the same as the energy equation when neglecting energy losses) 2 n = r + n, O A 2 1 in which A = P..R = crosssectional area at location 1 (see Figure 2). For most inlets the bank slopes are steep and thus R, R + Â— H, P l P K P 1 1 in which A = crosssectional area at location 1 measured at Mean Ocean Level
PAGE 19
B = width of inlet measured at Mean Ocean Level. Substituting the expressions for A , P.. and R in the Bernoulli equation yields 2 m Q n n = + n. (2.5) 2gP (R + Â± nJ P It is assumed that during ebb all the kinetic energy present at location 1 is lost. Neglecting bottom friction it then follows that for ebb n = n, (2.6) o 1 The flow conditions in Region III are the reverse of those in Region I. Bernoulli's equation holds during ebb while energy dissipation takes place during flood. The only difference with Region III is that, in general, the velocity head at location i can no longer be neglected as was the case for the corresponding location in Region I. For ebb flows, the equation for Region III now reads 2 2 m Q m Q n + i _ n _ + 1 _ (2.7) 28 P < R + =V 2gP 2 (R. + ^n.) 2 X 1 2. _ J. i Again neglecting bottom friction in Region III, the equation describing the flow during flood becomes n 2 = r\. (2.8)
PAGE 20
10 In Region II, the actual restricted part of the inlet, only pressure forces and bottom friction are taken into account and thus A iEL = p 1Â° (2.9) A 3x pg in which A = A + B n = crosssectional area P P = wetted perimeter t = bottom shear stress o p = density of water. The bottom shear stress is related to the discharge and crosssectional area by the empirical relation _ = D FQ l Ql (2.10) A 2 Furthermore A = R P. Substituting these expressions for P, x o and A in Equation (2.9) and rearranging slightly yields p 3x _2 B ,2 r P (R + n) P This equation holds for both ebb and flood. Integrating Equation (2.11) with respect to x between Stations 1 and 2, the resulting A , 2. expression can be written, to 0(.n ; as iff + Ii^Â£,3 (n2 , 1 >. + a!Â§^ C2.u>
PAGE 21
11 This particular form of Equation (2.12) was chosen to be comparable with Equation (2.3). Equations (2.5), (2.8) and (2.12) are then combined. It is assumed that the velocity head in Equation (2.5) is 3 of 0(n). When neglecting terms of 0(n ) and higher, the following equation describing the dynamics of the flow in the inlet during flood, Q < 0, is obtained. ,p + B n o + \ . . FQ 2 L g(R + = ^ )^i ~ O M * p _2 Â« n + n. P (RH^^^) 2 P ^2 m x Q R 2 2P [r + ^ (n Â±55)] z P Â° 2gP R 2 2 2 + 12=2 (" 2 = ^n + = ^ + f = Â—=5=9" > ^ 2 ' 13) 2PR PÂ°P 1Z P 2gP R Similarly for ebb, Q > 0, it follows by combining Equations (2.6), (2.7) and (2.12) tv * B n o + n i s , , FQ 2 L g(R + ) (n . n ) = * 2 /Vl i V P Â„ Â— n + n . o p2 (I + l_o_x )2 P 22mÂ„Q R m.Q R 2 l 2 2 2P Z [R + Â£ ( n , +~ Z?" Â— Z9TZ9 J 2Pt(R, 4^ n,) Z P X 2gP RT 2gP R P. 1 1 1
PAGE 22
12 ra 2 Q2 m i Q2 w B Â„ B 3 B m 2 q2 + ( Â— sÂ— o ^r o ) ( n + 2 Â— n . 2P _2 R 2 2P 2 R 2 ' P Â° P X 2 P 2gP 2 R 2 m.Q Â— ) Â—2Â—2 ; 2gP R Â° i i (2.14) Equations (2.13) and (2.14) are the same as Equation (2.3) when neglecting the second order terms and also the velocity head in the lagoon and assuming m = mÂ„ = m. Finally, it is noted that in deriving Equations (2.13) and (2.14) the actual restricted part of the inlet, Region II, is assumed to be a prismatic channel. Many inlet channels do not fulfill this condition, because the width and depth vary from point to point. In that case, the irregular channel may be replaced by a prismatic one having the same discharge for a given head difference. For a detailed outline of this procedure, see Keulegan [9], page 19.
PAGE 23
3. NUMERICAL SOLUTION OF THE TIDAL EQUATIONS The numerical procedure presented here is based upon an explicit finite difference scheme; the unknowns depend only on values previously computed at a lower time level. This, in general, leads to a more stringent stability requirement (see for instance Vreugdenhil [30]) as compared to implicit methods but has the advantage that the different steps in the computation are easier to trace. The numerical scheme is space and time staggered. Water levels are computed at n.At and discharges at (n + l/2)At. The water levels apply at the center of the grid blocks and the discharges are computed at the gridlinesj see Figure 3. The mean depth h and the lateral inflow or rainfall are given at the time level and location of n . The basic recurrence equations for the onedimensional tidal equations are q ' (i) = c(ii) [q(i) + l^ (D(i) + D ( i 1 ))( r '( i D " l(i))] (3.1) n'(i) = n(Â±) + Â£ (q'(i) q*(i+D) + M(i)At (3.2) in which D(i) = n(i) + h(i) ( }
PAGE 24
14 en < o en ui CO UJ _1 ca < < > o < O o PO LlI CC C9
PAGE 25
15 equations are replaced by different quotients using central differences. The difference quotients for Equation (2.1) are centered about (n + l/2)Lt and the location of n. The difference quotients for Equation (2.2) are centered about time level nAt and are centered in space about the location of q. Starting from the initial conditions, all the q's are computed for the next time level by means of Equation (3.1), then the n's are computed using Equation (3.2). It is noted that because of the convective acceleration, the recurrence formula (3.1) includes values as far apart as 2 space steps; see expression for G(il). This leads to difficulties when the boundary conditions at open boundaries are given as water levels. In that case, the convective acceleration is taken off center for the grids adjacent to those boundaries. For a more detailed outline of the numerical scheme, the reader is referred to Reid and Bodine [18] and Verma and Dean [28]. Because of the nonlinearities in the differential equations, the existing mathematical theory is inadequate to determine the exact criteria for stability and convergence of the numerical scheme. However, some insight might be obtained by regarding a simplified set of equations. The equations describing the tidal flow, Equations (2.1) and (2.2), may be replaced by two wave equations in respectively ri and q, when neglecting the nonlinear terms and the source term, and assuming a horizontal bottom. It then can be shown, starting from the recurrence formulae for the simplified set of equations that, for this case, the difference scheme is equivalent to solving the wave equation on the mesh Ax, At. The stability and convergence criterion then is (see Platzman [16])
PAGE 26
16 Ax /TT~ > * gh At 6 Ax i Â— r~ (For the twodimensional equations the condition is Â— > /2gh. ) When regarding the solution of the tidal equations as a sum of Fourier components, stability as used here implies that there should be a limit to the extent to which any component of the solution is amplified in the numerical procedure. Platzman [16] showed that even before the stability criterion is reached, the amplitude of especially the highest modes might be greatly magnified and might obscure the true solution. To study the eventual magnification of higher harmonies, a few computations were carried out for a 17milelong and 7ft.deep channel, open at one end and closed at the other. In addition, these computations provided some insight as to whether the terms neglected when discussing the stability criteria in the previous paragraph give rise to instabilities. The tide at the open end of the channel was sinusoidal with an amplitude of 1.5 ft. and a period of 12.5 hours. The computations were carried out for values of the friction factor F = 0.002 and F = 0.004 and for the time steps At = 100 sec, At = 200 sec. and At = 300 sec. The space step Ax = 5000 ft. was the same for all computations. The stability criterion for the case Ax discussed here is At = = 333 sec. Each computation extended over Â•gh A tidal cycles. The results were judged visually; no overflow occurred, and amplification of higher modes was not apparent. The source term and the free surface forces were not included in the computations .
PAGE 27
17 In the computational procedure, the inlet equations, Equations (2.13) and (2.14) may be regarded as an implicit boundary condition for the flow in the lagoon. The way in which this boundary condition is incorporated in the numerical scheme depends on the inlet configuration. The following two cases are considered: An inlet connecting a lagoon to the ocean such that no branching of the inlet flow occurs; see Figure 4a. An inlet connecting a lagoon to the ocean such that branching of the inlet flow can occur; see Figure 4b. Consider the "nonbranching inlet;" see Figure 4a. The total discharge Q rather than the discharge per unit width, q, is used as a dependent variable. An auxiliary water level n . is introduced which is computed at the same time level as the water levels in the lagoon. Starting from the initial conditions, all the discharges in the lagoon except Q(i + 1) can be computed using the procedure described before. The value of Q(i + 1) is then computed as follows. Q(i + 1) is related to the known ocean level r\ and the auxiliary level n . by means of the inlet equations. The difference form of the inlet equations used here is based on the following simplified form of Equations (2.13) and (2.14). rÂ» 4. B n o + n i w , + FQ 2 L + mQ 2 R g(R + s ><"! V "" P X Â° 1 ,B n o + n i,2 2,B n o P (R + == Â— ) 2P (R + = sign for n . <_ n + sign for n . Â£_ H Â•
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18 z o o CO < (S> o o UJ hu. oc UJ UJ u_ _J u_ o z Q en = = en s o 1 X o z o u_ til < z CE UJ o CD 2 UJ X o m CO Q cr C9 UJ o
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19 It is assumed that m = m = m. After some algebraic manipulation, the following result is obtained. Q = Â± DD / n i n c + sign for n . > n i Â— o sign for n . < n i Â— o in which m Q'(i+1) + Q(i+1) (3i3) Q n Â° ^ ) 3/2 It is noted that for many practical cases the modifications made in Equations (2.13) and (2.14) are justified. They were introduced here to simplify the algebra; however, the computational procedure applies equally well when using the complete Equations (2.13) and (2.14). A second equation relating Q(i + 1) and n. is found by applying the dynamic equation, Equation (2. 2), between the discharge stations Q(i + 1) and Q(i). Note that when computing the flow in the lagoon, the dynamic equations are applied between two water level stations. The difference form of the dynamic equation applied between Q(i + 1) and Q(i) yields Q = AA(n. n ) + BB + AA r\ (3.4) 10 o in which
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20 Q'(i + 1) + Q(l + 1) 2 AA = B D ^> B1 At GAx BB = {Q(l+1) + Q(i) Q' (i) Â• G + Â§ B1 ^ i} At [n(i) + n(il)]}/2G +5Ii+il G 1 + FAtlQ(i+l) + Q(j) 2At[Q(i+l) Q(i)] 2D(i) 2 Bl Axh(i)B1 In determining the difference form, the terms in both the inlet and dynamic equation are centered about n Â• At. The general shapes of the curves Q = f(n. n ) represented by the Equations (3.3) and (3.4) are indicated in Figure 5. Equation (3.4) represents a straight line. The slope of this line, AA, is for most practical cases negative. Therefore, the sign of (BB + AAn ), which is a known quantity, determines the sign of (n . r, ) which in turn determines the sign to be used in Equation (3.3). Eliminating (n . n ) between Equations (3.3) and (3.4) yields 1 /l 4 ^~ (AAn + BB) Â— t: <"> 2 DD for (BB + AAn ) > o Â— + 4 ^~ (AAn Q + BB) * Â« Â• Â— : rfr Â— Â«Â•Â« DD 2
PAGE 31
21 to < ro TO CO o r< O UJ >OQ Q UJ CO Ul tx o. ui or z> o lT> Ul C5
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22 for (BB + AAn ) <_ 0. Note that only the first order terms in (n . n ) are eliminated because second order terms are still present in the factor DD. Equations (3.5) and (3.6) therefore may be regarded as being quasilinear, which suggests finding a solution by means of a perturbation method. First the value of (n . n ) in DD is taken equal to the x o value at the previous time step. The value of Q can then be found from Equation (3.5) or (3.6) depending on the sign of (BB + AAn ). Knowing Q, the value of (n. n ) is determined from either Equation (3.3) or (3.4). This value of (n . n ) is substituted in DD. The procedure then is repeated until the difference between the computed and previously computed value of Q is within certain limits. The numerical scheme for the "branching inlet" (see Figure 4b) involves four unknowns Q'(i+1), Q'(i+2), Q! and r\. as compared to only two, Q(i+1) and n.Â» for the "nonbranching inlet." The four unknowns are related by the following four equations the inlet equation which takes the form of Equation (3.3) Q! + Q. X X with Q = ^ the dynamic equation applied between the locations of Q(i) and Q(i+1); this equation takes the form of Equation (3.4). the dynamic equation applied between the locations of Q(i+2) and Q(i+3); this equation takes the form of Equation (3.4) with Q(i) replaced by Q(i+3), Q(i+1) replaced by Q(i+2), n(i) replaced by n(i+2), n(il) replaced by r,(i+3), h(i) replaced by h(i+2), Ax replaced by Ax, and Bl replaced by B2.
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23 the continuity condition which, when assumin , th; t C le water level in the hatched area is the same everywhere, takes the form Q(i+1) = Q. + Q(i+2) 1 Elimination of Q(i+1) and Q(i+2) between the two dynamic equations and the continuity equation yields a relation between Q. ana n. similar to Equation (3.4). This equation together with the inlet equation then can be solved following the procedure described before. Finally, it is noted that in onedimensional flow computations, it is often necessary to divide the lagoon into parts with different widths; see Figure 4c. The flow at the boundary of two such parts may be computed following a procedure similar to the one applied to the "nonbranching inlet," replacing the inlet equation by a second dynamic equation between the location of Q(i+1) and Q(i+2). The foregoing procedure of incorporating the inlet equations in the numerical scheme, including the case with twodimensional flow in the lagoon, can also be found in Chiu, Van de Kreeke and Dean [2 ]Â•
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4. DETERMINATION OF THE FRICTION FACTOR F The factor F in the friction term of the conservation of momentum equation is difficult to estimate in advance; therefore, an appropriate value is usually based on measurements. Two methods for determining F are compared. One method is based on matching computed and measured water levels, the other is based on matching computed and measured discharges. Special attention is given to the influence of errors in the measured water levels and discharges on the value of F. To calibrate a tidal model, i.e., to determine the value of F, the area covered by the model is divided into sections each with approximately uniform geometry for which the values of F may be assumed constant. It is assumed that at both ends of each section the water levels are known from measurements. Given these water levels and assuming a value of F, the water levels and discharges within each section can be computed. The computations are carried out for different values of F until a good match exists between computed and measured water levels or discharges. The F value obtained in this manner reflects, in addition to friction, the effects of schematization, neglected terms in tne equations and measurement errors. These effects may cause large variations in the value of F during a tidal cycle especially during periods of slack tide when the inertia forces are predominant. Therefore, in determining F these periods should be disregarded. Furthermore, a realistic value of F, that is a value mainly determined by friction and more or less constant during a tidal cycle can only be obtained in cases for which the flow is frictiondominated during the larger part of the tidal cycle (see Dronkers [ 5 ] , page 425) . It should be noted that in 24
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25 many actual cases not all of the field data necessary for this ideal procedure are available. In that case recourse should be taken to trial and error methods . As stated before, measurement errors result in errors in F. In this paragraph, some general considerations concerning these errors are put forward. In the next paragraph an example will be presented showing the order of magnitude of the errors in F resulting from errors in the "calibration curves" (a calibration curve is defined as measured water levels or discharges within a section and used to determine F by matching with computed water levels or discharges) . The water surface in each section may, as a first approximation, be regarded as a straight line because in practice the length of the sections are small compared to the length of the tidal wave. The end points of the straight line are determined by the water levels at the boundaries. Therefore a change in F will hardly affect the computed water level in the section. From this it may be inferred that, when determining F by matching computed and measured water levels, an error in the latter or in the boundary conditions will lead to a large error in F. When calibrating on discharges the effect of measuring errors can be expressed as follows. During periods of frictiondominated flow a first approximation of the equation of conservation of momentum is gn 9x u 2 h and thus
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26 e(P) e( J) + e(qq) in which e = relative error. The error in F thus is of the same order as the error in the boundary conditions ( Â— ) and the calibration curve (q) . The following example serves to illustrate the order of magnitude of errors in F due to errors in the calibration curve. Consider a section with a constant depth h = 7 ft. and a length L = 50,000 ft. The boundary condition at x = is ru = 1.3 sin (2irt/45000) and at x = L is n 2 = 1.3 sin (2TT(t2700) /45000); see Figure 6a. The water levels and discharges in the middle of the section are computed for F = 0.002, F = 0.0025 and F = 0.003. The results are plotted in Figures 6b and 6c. The differences between the water levels for the different values of F were too small to be reproduced in the figure. In addition, for the same location and for F = 0.002, the values of the different terms in the equation of conservation of momentum are plotted; see Figure 6d. The latter results provide some insight in the relative magnitude of the forces and accelerations represented by these terms and their phase relationship with water levels and discharges, It may be seen from Figure 6d that the flow is frictiondominated during the larger part of the tidal cycle, which as stated before is a condition to arrive at a realistic value of F. When using a measured water level in the section as calibration curve, the effect of an error in this curve on the F value can be demonstrated as follows. Let the curve corresponding to F = 0.002 in Figure 6b represent the correct water level (calibration curve) in the section. Let the curve
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27 X= OFT TIME IN HOURS (o) TIDE AT X= FT. AND X = 50,000 FT. TIME IN HOURS 10 i~~ (b) TIDE ATX= 25,000 FT. FÂ«O,O025 5 " o 5 (e) DISCHARGE AT X = 25,000 FT. .Fqlq 2.10 1.10 OT 1. 10 2.10 3 . (d) MAGNITUDE OF TERMS IN CONSERVATION OF MOMENTUM EQUATION AT X= 25,000 FT. FIGURE 6. TIDE IN A SEA LEVEL CANAL
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28 corresponding to F = 0.003 in the same figure be the measured calibration curve. This curve deviates from the correct curve at the most by 0.03 ft.; the relative error (maximum deviation divided by tidal amplitude times hundred) is 2%. An error of 2% in the water level , , ., . v , Â„ , , i f 0. 003 0.002 ._.Â„ s (calibration curve) would thus lead to a ( n nn? " luu/ Â° = ' 50% error in F and as may be seen from Figure 6c an error of 20% in q. The influence of errors in the calibration curve on the F value when using discharges can be demonstrated in a similar way. Let the curve in Figure 6c corresponding with F = 0.002 be the correct discharge curve. Consider the curve corresponding with F = 0.0025 in the same figure to be the measured discharge curve. The relative error in the measured discharge is approximately 10%. The resulting error in F is 25% and the error in the water level appears to be 1.2%. To summarize for the example presented here, when calibrating on the water level a maximum error of 2% in the water level leads to an error of 50% in F and a maximum error in q of 20%. When calibrating on a discharge curve an error of 10% in the calibration curve leads to an error of 25% in F and a maximum error in the water level of 1.2%. The relative errors in the measured water levels and discharges of respectively 2% and 10% used in the previous example are realistic values for conventional types of equipment and measuring procedures. Also the example is a typical case for flow in lagoons of the type considered here. Therefore it may be safely concluded that, when This value cannot be read from Figure 6b; it was taken directly from the computer output.
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29 calibrating on water levels and assuming that the boundary conditions for a section are given as water levels, small inaccuracies in the measured calibration curve lead to large errors in F and q. When calibrating on discharges, the errors resulting from inaccuracies in the calibration curves are far less.
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5. MASS TRANSPORT IN LAGOONS DUE TO TIDAL ACTION In this chapter an analytic solution is presented for the mass transport due to tidal waves in lagoons. The analytic approach is restricted to lagoons and inlets of relatively simple geometry, which prohibits its application to the often irregular shaped tidal waters in nature. In such cases, recourse has to be taken to numerical techniques. Therefore the analytic model cannot replace the numerical procedure; it merely serves to build faith in the numerical results and to obtain a better understanding of the physics underlying the phenomenon of mass transport. In the final part of the chapter the effect of inlets on the net transport is discussed. a. Analytic Solution; Physical Concepts Consider a straight lagoon of finite length % in which the ocean tide can freely enter at both ends. The depth and width of the lagoon are uniform. The conservation of mass equation neglecting the source term is 3t dX The conservation of momentum equation, when neglecting the nonlinear terms except the friction term, reduces to (h + n) The term on the righthand side of Equation (5.2) was derived (see Appendix A) by relating the bottom shear stress and the mean velocity by means of a DarcyWeisbach type friction law. 30
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31 Bffeap (5.3) 4R 2g in which S = slope of the water level f = DarcyWeisbach friction factor u = mean velocity over depth R = hydraulic radius. Furthermore Â— = gRS (5.4) P in which x = bottom shear stress. From Equations (5.3) and (5.4), it then follows T Â— = Â— u u (5.5) p 8 ' ' The friction term in Equation (5.2) will now be rederived by replacing Equation (5.5) by its linearized version. Assume that as a first approximation the velocity u in the lagoon is simple harmonic in t u = u cos(at + a) (5.6) in which a = a phase angle. Substituting this expression for u in Equation (5.5) and expanding in a Fourier series yields T o f 8 Â— = Â— tÂ— uu + third order term + . . . p 8 3tt The second, fourth, etc. harmonics are zero. Thus neglecting third order terms and higher yields
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32 la. f ( 37 u)q p 8 h + n (5.7) When approximating and introducing h + n h u 2 h F n = 4Itu = fu (5.8) 'Â£ 8 3tt 3tt Equation (5.7) may be written as T o F A q F Â£ qn ^ = ^+ ^~ (5 ' 9) P h h 2 With this expression for the bottom friction, the conservation of momentum equation becomes it + gh ^ Â— r + ~r (5 10) h Equations (5.1) and (5.10) are used to describe the flow in the lagoon. Equation (5.1) is a linear equation while Equation (5.10) may be regarded as quasilinear because the nonlinear term is small compared to the first term on the righthand side. The ocean tides at both ends of the lagoon serve as the boundary conditions. To avoid lengthy computations, the amplitude and phase of the ocean tides are taken such that when neglecting the nonlinear term in Equation (5.10) the resulting solution represents a damped progressive wave. The water levels and discharges for such a wave as a function of x and t are
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33 ri = a e cos(ot kx) (5.11) k q = a c ' Â— e cos (at kx + a) (5.12) Vm 2 + k in which p, k and a are defined by v 2 v 2 2 k = k y o Ik 2yak h = ,2 2 1 / U \ a = tan ( f) with k = a/c and c = vgh. ooo Thus when n = a cos at (5.13) o is the boundary condition for x = 0, it then follows from Equation (5.11) that the boundary condition at x = L should be n T = a e" yL cos(ot kL) (5.14) It is furthermore assumed that the mean level at both ends of the lagoon is the same and equal to the mean ocean level. A trial solution for Equations (5.1) and (5.10) when including the nonlinear term in the last equation and considering the boundary conditions (5.13) and (5.14) is 1 = ^(x) + a e cos(at kx) + ... (5.15)
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34 k q = q.(x) + a c , Â— e cos(at kx + a) + ... (5.16) Substituting this trial solution in Equation (5.1) and averaging over the tidal period yields 8 q* 7Â— = (5.17) dX (Note: This must be true for any periodic forcing function.) Substituting in Equation (5.10) and neglecting higher order terms yields after time averaging 3n . F F. a 2 c k k Â„ , * I I o o 2yx ,,. .. QN gh ^ h~ q * + h" "Sir "T~ 72 e <5 ' 18) y + k For x = and x = L, n A = because of the assumption that at both ends of the lagoon the mean level equals the mean ocean level. With these boundary values of r\ A the solutions for n A and q^ from Equations (5.17) and (5.18) are o 1 o 1 /, 2uL. ,c , Q v q * = ^T2yTÂ— 2L (1 " e } (5ll9) y + k FÂ„ a c k k Â„ I o o 1 r x , 2yL . Â» , /yx lN1 /on x n * = ~2 "ITT ^TTT 2y" [ I (e 1) " ( Â£ Â«] ^ 20 > gh y + k Equation (5.19) is the expression for the net discharge in a straight lagoon when imposing the boundary conditions (5.13) and (5.14) and assuming the same mean level at both ends of the lagoon. Equation (5.20) represents the mean water level in the lagoon.
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35 The following serves as an explanation of the physics underlying the expression for the net transport and mean water level in a lagoon derived in the preceding paragraph. Considered is a damped progressive wave in an infinitely long channel . The velocity u, rather than the discharge per unit width q, is introduced as a dependent variable. Assuming a horizontal bottom Equations (5. 1) and (5.10), which represent the conservation of mass and conservation of momentum, can be written as respectively and 111+ h Â£i = _ lm (5.21) St 3x 3x hÂ£+shfa = ff V Â«Â•Â«> The expressions for the damped progressive wave and the accompanying velocity field, satisfying the linear part of Equations (5.21) and (5.22), are n = a e cos (at kx) ac k o o ux , , N u = Â— rÂ— Â— e cos (at kx + a) h \/y 2 + k' See, for example, Ippen [ 8 ], page 507. The net transport accompanying such a wave, q^ , may be found w from T n 1 w h u dz dt
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36 Substituting the expression for u in this equation and integrating yields a c k k Â„ o o 2yx / q oos Â•Â»* =2h2~T e (5 ' 23) w u + k This expression shows the net transport to decrease with increasing x. Continuity requires the net discharge to be the same for every crosssection and equal to the net discharge at x = 0. Conveniently taking the net discharge at x = equal to zero, the net discharge in the channel, q^, is written as q* = q* + q* =0 (5.24) w s in which q^ results from the slope in the mean water level s Â„ =i& * (5.25) q * F Â„/h Sx s a Substituting the expression for q^ and q^ , respectively Equations w s (5.23) and (5.25), in Equation (5.24) yields 2 a c k k , 3n,,, Â° 2 e" 2 ^ _ 3h = o (5.26) 2h 2^,2 F./h 3x u + k I Integration of Equation (5.26) and arbitrarily setting n* = at x = yields 2 F , , a c k k r. I 1 1 O Q /, ~ 2 ^ x \ ,'S 97^ ^* = h"ih 2M"1hT7 72 (1 " 6 > ^ ,27) Â° u + k
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37 Equation (5.27) represents the mean water level in an infinite long channel in the presence of a free damped progressive wave and zero net transport. See also Figure 7a. When regarding a channel or lagoon of finite length L , at both ends connected to the ocean, the mean level at x = and x = L is the same and equal to the mean level in the ocean. The resulting net flow and mean water level in the channel of finite length can be derived from the mean flow and mean water level for the infinite long channel, as follows. A slope n*a> s = m is added to the mean water surface in the infinite channel, which renders the mean level at x = L zero. The expression for the mean water surface becomes 2 n * (L) < ** 1 1 a C o k o k n 2yx. rs Â„. u + k See Figures (7b) and (7c). Substituting the expression for n^(L) from Equation (5.27) in Equation (5.28) yields Equation (5.20). The discharge q A accompanying the slope S is m gh n * (L) ** = "VhÂ— (5 29) m I and the total net transport q^ is gh n * (L) **Â°VhL(5 ' 30)
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38 at X Â— co * " h gk 2)1 21, ^2 + k 2 MEAN OCEAN LEVEL 2. //// / / / / / inn n i /nui/ n /// / ii i / i /////// u 1 1 // mi/i/n ii ii // a. MEAN WATER LEVEL IN AN INFINITELY LONG CHANNEL MEAN OCEAN LEVEL VJM 777777777777777 1 1 1 / / / / / 1 1 / 1 1 1 / 1 1 / / / 1 1/ 1 / 1 1 / 1 / / / / / / / / / / / / / / b. SLOPE , S m ^ MEAN OCEAN LEVEL V jC Eq (5.20) in/mi > in i mm 1 1 mi ///i in/// / ii 1 1/// //// / ii n in////// C. MEAN WATER LEVEL IN CHANNEL OF FINITE LENGTH, L FIGURE 7. MEAN WATER LEVEL IN A LAGOON IN THE PRESENCE OF A DAMPED PROGRESSIVE WAVE
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39 Substituting the expression for n, v (L) as given by Equation (5.27) yields Equation (5.19). It may be inferred from the previous considerations that the net transport in the finite channel may be regarded as the result of forcing the mean level at the boundaries to be different from the mean level at these boundaries in the case of a free wave. Note also that in case the friction approaches zero the expression for the net transport, Equation (5.19), approaches 2 q* = 2ir (5 31) which is the expression for the net transport accompanying a free undamped wave with amplitude a, traveling in shallow water. When substituting the trial solutions Equations (5.15) and (5.16) in Equation (5.10), all higher order terms were neglected. To evaluate the effect of these terms on the net transport the q^ as given by Equation (5.19) is compared with the true q^, by taking into account the third and higher order terms. The true q A is found by numerically integrating Equations (5.1) and (5.10) with the boundary conditions (5.11) and (5.12) following the procedure described in Chapter 3 and then averaging q over a tidal period. The computations were carried out for a lagoon with a length L = 70,000 ft. and a depth h = 7 ft., a tidal period T = 45,000 sec, a time step At = 100 sec, and a space step Ax = 5,000 ft. Two values for the amplitude a and three values for the linear friction factor F are considered. The results of both the analytic and numerical (true) solution are presented
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40 in Figure 8. From this figure, it may be seen that neglecting the third and higher order terms introduces only a very slight error. The question might arise whether the "linearization" of the friction term as outlined on pages 31 and 32 is a valid procedure when discussing effects caused by second order terms in the equations. To shed some light on this, the net discharges when using Equations (5.1) and (5.10) and the net discharges when using Equations (5.1) and (5.2), which include the quadratic friction, are compared. The same lagoon is considered as in the previous paragraph. The boundary conditions again are presented by Equations (5.11) and (5.12). The "linear" friction factor F and the "nonlinear" friction factor F are related by Equation (5.8). The value of u in this equation is taken equal to the average of the u values at both ends and the u in the middle of the lagoon. The equations are integrated numerically using a time step At = 100 sec. and a space step Ax = 5000 ft. To find q^, q is averaged over a tidal period. Two different values for the amplitude a and three values for the friction factor F are considered. The results, which are plotted in Figure 9, compare well especially considering the way in which u was determined. The agreement is better for larger than for smaller values of F which probably stems from the fact that for larger values of the net flow and thus smaller values of F, the approximation of u as given by Equation (5.6) becomes less justified. It is noted that methods do exist to arrive at a better estimate of the representative average value u. However, incorporating these methods would lead to a considerable increase in computational effort which is not warranted by the purpose of the computations. The
PAGE 51
41 0.15 0.10 0.05 * cr UJ < I o IQa LINEAR FRICTION FACTOR 0.001 a 0.5 ft. O ANALYTIC (EQUATION (519)) O NUMERICAL (EQUATIONS (5.1) a (510) ) Oq 0.002 CO 0.003 * _ ui o cr < x o 0.5 "linear" FRICTION FACTOR Fj 0.001 Q = 1.5 ft. O ANALYTIC (EQUATION (5l9)) O NUMERICAL (EQUATIONS (5.1) a (5.10)) I 0002 0003 FIGURE 8. EFFECT OF HIGHER ORDER TERMS ON
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42 0.3 0.2 0.0 Â£\N = 0.5 ff. O EQUATIONS (51 ) a ( 52 ) Â© EQUATIONS (51) a (510) a < O o CO o o NONLINEAR FRICTION FACTOR F o 0.001 0.004 0.008 o 0) 0=1.5 ft. O EQUATION (51) a (52) Â«^ O EQUATIONS (5) a (5io) 2 2 * o 1 n 1 NET DISCHARGE o "nonlinear" 1 O O FRICTION FACTOR F 1 3 i 0.001 0.002 0.003 FIGURE 9. EFFECT OF "LINEARIZATION" OF THE FRICTION TERM ON q #
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43 reader is referred to Dronkers [ 6 ] , Ippen [ 8 ] and Van de Kreeke [26]In deriving Equation (5.19) the nonlinear terms associated with the convective acceleration and surface gradient have been neglected. These terms can easily be taken into account. In doing so, the only change in the differential Equation (5.18) will be in the expression preceding e . In the initial derivation these nonlinear terms were omitted to simplify the algebra as much as possible. They should, however, be taken into account when requiring quantitative accurate results. This may be seen from Figure 10 in which the net transport with and without the convective acceleration and the nonlinear part of the surface gradient is presented for a lagoon at both ends freely connected to the ocean. The results plotted in Figure 10 were obtained numerically. Also in dsriving the expression for q,, the boundary conditions were chosen such that to a first order of approximation, the water motion in the lagoon was a single damped progressive wave. However, expressions for q A can be derived in a similar way for arbitrary boundary conditions. For example, consider a lagoon connected to the ocean by inlets at both ends. The implicit boundary conditions, formed by the inlet Equations (2.13) and (2.14), and the field Equation (5.2) first are made quasilinear. This is done by 2 i i linearizing the factors Q and qq in respectively the inlet and dynamic equation. A first order solution is obtained by omitting the second order terms (the nonlinear terms) from both the field equations and the boundary condition. The trial solution given by Equations (5.15)
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44 (VJCM OJ cr Pcr + C Â— 1 pII + Pjx p+ o o o o" t> b h II o> Â—  CM c e 10 o PX CD + CM . P N in o 5 TO cr x I 6 O uj .cr lc ii ii + + I j . tCL 111 a P^ < PP Cr (0 %F O G * P G cr o ho < Ll. z o ho cr Ul cr c 51 iu u > 2 LU o o o < lice u. r> O CO (M o Q 1LU O o x UJ H u_ u. u_ Lu O 6 o d o o o o o o o o o b UJ u.
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45 and (5.16) is then substituted in the complete equations and boundary conditions including the nonlinear terms. The resulting expressions are time averaged which yields two linear differential equations in n* and q } , { , corresponding to Equations (5.17) and (5.18), and two linear boundary conditions. It is thus possible to solve for r\. and q. . Preliminary computations showed the resulting expressions for n^ and q^ to be of such a nature (long algebraic forms with implicit expressions for amplitude and phase angles) that they defy the purpose of presenting the analytic solution, that is, gaining physical insight. when considering a straight lagoon of uniform depth and width at both ends freely connected to the ocean, one obvious conclusion can be arrived at without any algebra; when the tides at both ends of the lagoon are the same, the net transport is zero because of the symmetry of the problem. Therefore, in order to have mass transport in a straight lagoon of uniform depth and width, the ocean tide at both ends of the lagoon must be different. This is an important consideration when discussing the effect of inlets on the mass transport; see Chapter 5b. So far the vertical profile of the net drift has not been discussed; an important factor when designing an experimental apparatus for measuring the mass transport. Neglecting viscosity and thus neglecting bed friction, the net drift is the same for all levels. For example, Stokes' [22] second order wave theory gives for shallow water waves in an inviscid fluid a mass transport velocity
PAGE 56
46 2 a c o 2h 2 However, when accounting for viscosity the net drift is no longer uniformly distributed as shown theoretically by LonguetHiggins [12] for the case of null net transport. Experimental results on drift profiles in a progressive shallow water wave and null net transport, U. S. Beach Erosion Board [23],Bagnold [1], Russell and Osorio [19] are in qualitative agreement with LonguetHiggins socalled conduction solution even though the conditions imposed on the solution (wave amplitude small compared to the thickness of the boundary layer) are not satisfied in the experiments. Generally speaking, the theory and experimental results show a forward drift, that is in the direction of wave propagation, near the bottom and near the surface with a return flow in the middle of the water column. It is noted here than the k h value, being one of the characteristic parameters describing the "shallowness" of the waves, was not less than 0.3 for any of the experiments presented in references [23], [1] and [19]. LonguetHiggins' [12] conduction solution for the net drift in a pure standing wave shows a net drift near the bottom towards the location of the anti node while the net drift at the surface is from the anti node towards the nodes. The total picture is a circulation of mass between the vertical planes through nodes and anti nodes. To the writer's knowledge, no experimental results exist to verify this net drift pattern.
PAGE 57
47 b. Influence of Inlets on the Mass Transport Consider a lagoon of uniform depth and width connected to the ocean by inlets at each end; see Figure 11. E FIGURE 11. LAGOON CONNECTED TO THE OCEAN BY INLETS It is assumed that the depth of the lagoon is so large that the propagation of the tide in the lagoon itself can be neglected. The continuity equation then may be written as 3n. in which *! + Q n " Q " *b IT A, = lagoon area Q = discharge into lagoon n. = lagoon level. Equation (2.3) is used to describe the dynamics of the flow in the inlet. From this equation, it follows 1 (5.32) Q = Â± A iÂ£R_ 2FL + mR n n. o l 1 or in Keulegan's [9] notation
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48 Â±K^^, T ^V"o 'iin which r = JL_ A_ / 2gRH K 9^H A 1/ 2FL + mR 2ttH A^ K = coefficient of repletion 2H = tidal range in the ocean A = crosssectional area of the inlet measured at mean ocean level. The solution to this problem, for a simple harmonic ocean tide, is given by Keulegan [9], For an ocean tide of the form Â« Â• 2rrt n = H sin =o l in which H = tidal amplitude in the ocean, the corresponding bay level n . is i n Â± = A x sin( ^ + e Â± ) + A 3 sin 3( ^p + Â£3) + ... (5.34) The amplitudes A and A ... and the phase angles e., Â£Â«... are a function of the coefficient of repletion K = K + KÂ„; K and K_ are the coefficients of repletion, respectively, for the Inlets I and II. It may be seen from Equation (5.34) that both amplitude and phase of the ocean tide are modified while in addition, higher harmonics are generated . The foregoing solution pertains to relative deep lagoons. No analytic solution is known in case of a shallow lagoon. However, it is likely that each inlet again will modify the ocean tide but not co the
PAGE 59
49 same extent (except when the inlets have the same dimensions) , as was the case for the deep lagoon, because the propagation of the tide in the lagoon can no longer be neglected. In general, therefore, the amplitude and phase of the lagoon tide at one inlet will be different from the amplitude and phase of the lagoon tide at the other inlet, even though the ocean tide is the same for both inlets. A difference in tide at both ends of the lagoon is a requirement for net transport because in case of equal tides, the net transport is zero as a result of the symmetry of the problem. See also Chapter 5a. It may be inferred from the foregoing general considerations that inlets may play an important role in the flushing of the lagoon. By properly choosing the dimensions of the inlets, the net transport of water in the lagoon can be considerably increased. This is illustrated for the bay system presented in Figure 11. The ocean tide is assumed simple harmonic and the same at both inlets; the amplitude is 1.30 ft. and the tidal period is 45,000 sec. For the dimensions of the lagoon and the inlets, see Figure 12. In the same figure, the results of the computations are presented; the net discharge per tidal cycle is plotted versus the width of Inlet II. A definite maximum in the net discharge occurs when the width of Inlet II is in the order of 300 400 ft. The computations were carried out for both the complete inlet Equations (2.13) and (2.14), neglecting terms resulting from the velocity head in the bay and the ocean, and for the simplified version of the inlet equation, Equation (2.3), again neglecting the velocity heads. The results for the two inlet equations differ only slightly. Equation (2.2) in which the convective term was neglected, was used to
PAGE 60
50 o o o o Â• *o*Â— oÂ• *oo * o o s o o o o ro Si3 H\ 30UVH3SIQ 13N O o o M O 10 o CO O o u. O O o *>n o in c\j o :Â• o O o < I K0. C\J o o LJ o o (S> z ir o o u. < IS < n o o Â«i CO o o o o CJÂ» CD o in !Â» <\j ro U) < X _ Ixl tX UJ o o _l $ CM a:
PAGE 61
51 describe the dynamics of the flow in the lagoon. The equations were solved by applying the numerical method discussed in Chapter 3. The net discharge was found by integrating the computed discharge over a tidal period. The time step in the computations was At = 100 sec. and the space step was Ax 5000 ft.
PAGE 62
6. LABORATORY EXPERIMENTS A series of laboratory experiments was designed to measure the net drift in a lagoon of uniform width and depth connected at both ends to a tidal basin by openings of different dimensions. The purpose of the experiments was to evaluate the capability of the computational model described in the previous chapters to predict mass transport in tidal lagoons. The experiments were carried out in the Coastal Engineering Laboratory of the University of Florida. a. Equipment and Procedure Test SetUp and Measuring Devices A straight canal of uniform depth and width, simulating a lagoon, was constructed in a tidal basin; see Figure 13. The canal was 488 cm long and 30 cm wide. As outlined in Chapter 5, a necessary condition for mass transport is that the geometry of a lagoon be asymmetric. In nature, the asymmetry in many cases results from the different dimensions of the inlets connecting the lagoon to the ocean. The asymmetry in the experimental lagoon was introduced by providing one end of the lagoon with a submerged sharp crested weir. (A weir was preferred over an inlet because it led to a more regular flow pattern.) The weir was made of 1.5 mm steel plate. The edge of the weir was 3.35 cm above the bottom of the canal. The average water depths used in the tests were 5 cm and 6.5 cm. The dimensions of the tidal basin were 600 x 600 cm, and the average water depth in the basin was 30.5 cm. Two weir boxes, one at each side of the basin provided a constant discharge of 22 liters/sec. A filter of honeycomb placed across the basin and wave damping material 52
PAGE 63
53 CO o CO 2 CL i CO cc UI Q. X UJ UJ cc ZD
PAGE 64
54 attached to the walls served to absorb small disturbances caused by the inflow from the weir boxes. The tide in the basin (later referred to herein as the ocean tide) was generated by letting the discharge gate move up and down at approximately constant speed (the downward speed of the gate was about 4% larger than the upward speed) between two fixed points. The distance between the two fixed points and the speed of the gate could be varied to generate tides of different amplitude and period. An example of the ocean tide is presented in Figure 14. As demonstrated in this figure the ocean tide can very well be approximated by two half sines with slightly different periods (a result of the discharge gate moving faster down than up). In all of the tests the periods of the two half sines differed by less than 15%. The corresponding tide at the lagoon side of the weir is also presented in Figure 14. The weir tide lags the ocean tide by 12 sec while the tidal ranges are approximately the same. Water surface elevations were measured by vibrating point level recorders. For the location of these recorders, see Figure 13. A quick indication of the water level was provided by a conventional point gage located in the lagoon. The same gage also served to determine the maximum and minimum water depth in each experiment. The mean depth was taken equal to the average of the maximum and minimum water depth. Net discharges were determined from measured float paths using a method that will be described in Section b of this chapter. Floats of different lengths were used. The floats which were made either of wood or styrofoam were weighted so that the top of the cone just
PAGE 65
55 z o z < o z o c/> z o o o < uj 2 Â°S u_ 5 o o u z 2 z Â« o S uj J "> o = < hI i z o < => CC UJ O w U. U5 UO UJ cr uj G_ X UJ O Q UJ CC CO < UJ UJ ir CD (WO) 13A31 U31VM
PAGE 66
56 contacted the water surface. All floats were cylindrical and had a diameter of approximately 0.5 cm, the top was given a conical shape to minimize effects of surface tension. For positioning of the floats a righthanded coordinate system was introduced, the x axis coinciding with the longitudinal axis of the lagoon and the zero located at the open end of the lagoon. In addition to floats, dye (Rhodamine B) was used to determine the horizontal motion of the water in the lagoon. It was noticed that solutions with a high dye concentration, visible to the eye, when released in the lagoon water had a tendency to stick together and to settle (like a turbidity current) thus indicating only the motion of the bottom layers. An attempt was made to avoid this problem of settling by using dye solutions of a low concentration. After injection of the dye, water samples were taken from the lagoon by means of a rack provided with seven 1 cm glass tubes, spaced 10 cm apart. The glass tubes were placed vertically in the lagoon and sealed off at the top, by means of corks. The rack was then lifted out and the samples collected in test tubes. The samples thus obtained cover the entire water column. The dye concentrations in the test tubes were determined using a Fluorometer. It was hoped that when plotting concentration versus location of the samples a maximum could be indicated, the location of the maximum being the coordinate of the water mass initially located at the location of dye release. Unfortunately, many of the measured distributions did not have a clearly defined maximum and
PAGE 67
57 it appeared that test results were not reproducible. It is believed that the dye still tended to stick together and settle rather than disperse Â— the reason being the low turbulence intensity in the lagoon. (The Reynolds number Re = * in the experiments was on the order of 1 v 1000.) It might well be that in experiments with a higher turbulence level, the previously described technique would be successful. In view of the negative results, the test procedures and test results for the low concentration dye techniques will not be discussed further. Test Procedure First the discharge gate was left in a position resulting in the required average depth and the lagoon checked for circulation induced by the inflow from the weir boxes. This circulation could be compensated for by varying the length of the adjustable walls; see Figure 13. The discharge gate then was started and run for at least two hours to dampen possible disturbances caused by the sudden start of the gate. Actual testing began with measuring water surface fluctuations at Stations I and III (see Figure 13). Initially also the surface fluctuations at Station II were measured. They appeared not to differ noticeably from those at Station III, and therefore have not been measured in the later experiments. Floats were released one at a time either in the middle of the lagoon or 5 cm from the side walls. Positions of the floats were marked at each slack tide for a period of at least five tidal cycles. In case the lateral displacement of the floats was more than 5 cm, the tests were disregarded and started again until a good run was obtained. The water motion near the bottom was determined by releasing dye in sufficient concentrations to be visible by eye.
PAGE 68
58 Summary of Tests Two series of experiments were carried out. A preliminary series, Test Series 1, to study the variation of the drift velocities in a cross section, the possible side and end effects and the reproducibility of the experiments, and Test Series 2 in which an attempt was made to determine the average net discharge. A summary of the experiments in each test series is presented in respectively Table I and Table II. Test Series 1 consists of 14 experiments numbered 1 through 14. The experiments 1, 2, 6, 7, 9, 10, 12 and 14 only served to develop the experimental procedure and therefore are omitted in Table I. Test Series 2 consists of 11 experiments numbered 15 through 25. Water depths in the experiments range from 5 to 6.5 cm, wave periods range from 70 to 100 sec. and amplitudes range between 0.75 to 1.15 cm. No attempt was made to systematically study the effect of different parameters on the net discharge because of the difficulty of varying water depth, amplitude and period independently. TABLE I SUMMARY OF EXPERIMENTS; TEST SERIES 1 Tidal Tidal Experiment Amplitude (cm) Period (sec) Ave. Depth (cm) 3 0.77 71 5.10 4 0.92 72 5.00 5 1.05 100 4.90 5a 1.05 104 4.90 8 1.00 69 5.00 11 1.10 70 6.45 13 1.15 71 5.30 13a 1.15 71 5.30
PAGE 69
59 TABLE II SUMMARY OF EXPERIMENTS; TEST SERIES 2 Tidal Tidal Experiment Amplitude (cm) Period (sec) Ave. Depth (cm) 15 1.13 70 5.00 16 0.87 70 5.05 17 1.13 70 6.60 18 1.03 70 6.60 19 0.91 69 6.55 20 0.81 70 6.45 21 0.95 70 6.70 22 1.09 71 6.60 23 0.96 70 5.00 24 0.94 70 4.80 25 1.05 70 4.95 b. Results Test Series 1 Introductory tests with confetti and dye showed the flow in the middle part of the lagoon to be regular but indicated a somewhat irregular flow pattern at both ends, especially during periods of reversal of the tide. Therefore tracking of the floats was restricted to the region between x = 50 cm and x = 450 cm. Maximum velocities in the experiments varied between 3 cm/sec and 5 cm/sec, depending on water depth, amplitude and period. Reynolds numbers (Re = *) were larger than 800 (the limit for fully turbulent flow) 60% 80% of the time (the exact percentage depending on depth, period and amplitude). Actual experiments were carried out with three different float lengths, 3 cm, 2 cm and 1 cm. Floats were placed in the middle
PAGE 70
60 of the lagoon, y = cm, and 5 cm from the side wall, y = +10 cm. In addition, in Experiments 13 and 13a floats were also placed at y = 10 cm. Typical examples of measured float positions are presented in Figures 15, 16, 17 and 18. In these figures the positions of the floats, at the time they first reversed their path, are reduced to the same coordinate, which is taken equal to the average of the positions of the floats at the first reversal point (the positions usually did not differ more than 20 cm). The time a float arrives at the first reversal point is set equal to zero. Because of the difficulty of determining the exact time a float reverses its path, the subsequent times of reversal are set equal to the average of the reversal times of all the floats used in an experiment (reversal times for different floats at corresponding slack tides differed at the most by 10 sec) . Positions of the floats are plotted at these average reversal times. Figures 17 and 18 show the results of two identical Experiments 13 and 13a. From these figures it may be seen that the positions of corresponding floats differ somewhat, but the range covering the positions of the floats at corresponding slack tides is approximately the same for both experiments. For a better comparison, the particle excursion between successive slack tides of the two longer floats in the middle of the canal are tabulated for both experiments in Table III. Most floats when placed in the water moved parallel to the side walls; in unusual cases, a float was traced again because it had displaced laterally more than 5 cm and when this happened it nearly always occurred for those floats initially placed 5 cm from the wall.
PAGE 71
61 z o o CD < I < a a. O o o r< Q 2 o o 1 II V ^ >. O < a. in r< o s o o + ii 4 Â• IL. E o o ii > < I (19 Z III _l 3 3 6 E E o o " Â— oa 10 dwÂ« o o C\J 40 Â«! 4 < o z o <9 K < IT UJ 1:1 > X 4 1* cxjo* o IT) o UJ CD o " o IT) UJ or uj a. x or o ljco o CO o Q. < o Q UJ or CO < UJ in u or o in rO BO O O m o o
PAGE 72
62 >^ te UJ O UJ co Ul c o O Â£>< o o CM O m o UJ CO o ><Â• a2 A o o o o o o o IT) o o C\J o IT) o m UJ or UJ o. x UJ or o CO o CO o o. < o Q UJ en n CO < UJ CD UJ or r> CÂ£ u_ CWO) 1V01J JO NOIllSOd
PAGE 73
63 * Â«< o o ro ID CO Q UJ o V >> o m UJ LU 0X UJ en o CO o o UJ to UJ 2 CO O CL < o _1 u. << Q Ul CC =5 CO < Ul o m v > UJ on ID oo o o m ro O O ro O m O O Cwo) ivo~u jo Nomsod o m
PAGE 74
64 V*< o o CM tr r> o UQ LU O UJ tr o ** ,4 > o m ** o o V^ o u w UJ o hz. Ld IE CC LJ Ql X Ul en o u_ co o en O < o ^ Â• o^< o lO LU tr r> CO < LJ O o o in ro o O o in ro CM ao o CM O in CO <^* P* > (WO) IVOId JO NOIllSOd
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65 TABLE III REPRODUCIBILITY OF EXPERIMENTS; PARTICLE EXCURSION BETWEEN SUCCESSIVE SLACK TIDES FOR EXPERIMENTS 13 AND 13a Experiment 13 Experiment 13a 3 cm 2 cm 3 cm 2 cm Slack Tide Float Float Float Float 12 +72 +69 +74 +69 23 41 48 41 58 34 +62 +60 +62 +69 45 35 43 38 50 56 +66 +61 +63 +67 Floats were placed in the middle of the lagoon. Tabulated values are excursions in cm.
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66 From this it may be inferred that the flow pattern in the experimental lagoon is one dimensional in a sense that no significant meandering of the flow occurs. It may be seen from the examples presented in Figures 15, 16, 17 and 18 that the net drift differs for each float length which is not surprising in view of the discussion of the vertical drift profile on page 45. For all experiments the following was noted: the direction of the net drift in each experiment was the same for all floats, however, the magnitude of the net drift varied with the length of the float. This and the fact that injected dye clouds always moved in the same direction as the floats suggests that the direction of the net drift is the same for each point in the vertical. The experiments do not define the shape of the drift profile over depth. Results of Experiments 13 and 13a in which floats were tracked at both sides of the lagoon show that the net drift is not uniformly distributed over the width of the lagoon; the net drift at the north side is considerably larger than in the middle and at the south side. This probably is a result of the flow in and out the lagoon not being symmetric with respect to the longitudinal axis of the lagoon. Determination of Net Discharge from Float Paths Before presenting the results of Test Series 2, the method used to determine the net discharge q^ from a float path s will be discussed. Mainly because of the different nature of the two parameters, q A is an Eulerian parameter and s is a Lagrangian parameter, the mathematical relation between these two quantities gets rather complicated and unmanageable. Therefore for the determination of q^ from s recourse is taken to the
PAGE 77
67 following empirical method. Assume that the path of a float is representative of the average horizontal particle motion. As an example the (fictitious) measured positions of such a float at successive slack tides are indicated in Figure 19. The successive float positions are connected by straight lines. The midpoints of these straight lines are determined and a straight line is drawn through these points. The slope of this line represents a velocity u^, which may be interpreted as an average net drift velocity. The net discharge q Vc then is found by multiplying this average velocity and the average depth. Numerical experiments to be discussed in Section c show this method to yield good results. Test Series 2 The range of hydraulic parameters used in Test Series 2 (see Table II) is approximately the same as those used in Test Series 1 and thus also the velocities and Reynolds numbers are on the same order as earlier mentioned. In contrast to the Test Series 1 experiments, the Test Series 2 program was carried out with only one float length. The length of the float was chosen as large as possible to arrive at an average over depth drift velocity. This and the requirement that the float not contact the bottom at all times resulted in float lengths of 3.5 and 4.5 cm, respectively, for depths of 5 and 6.5 cm. Because of the irregular flow pattern at both ends of the lagoon (see discussion of result of Test Series 1) the tracking of the floats was limited also to the lagoon section between x = 50 cm and x = 450 cm. Typical results of float tests for Test Series 2 aie presented in Figures 20 and 21. In both figures straight lines representing the "average net drift velocity" u.
PAGE 78
68 x < 0. O Q CC 3 < o CC Ld <Â£) CC < X o CO o o t< CC LU LU Q Ld x> o CWO) 1V0"IJ dO NOIllSOd
PAGE 79
69 ce UJ a. x UJ o CO o W g o Â£ ^2 Â— i= W 1 Â° 1 Qh< o _J Li_ Q UJ cc 3 CO o < m LU o CM UJ o U_ ('WD) IVOld JO NOIllSOd
PAGE 80
70 CO LU cr LU Q. X LU CC o U_ CO o CO o < o Q LU r> CO < LU CM LU Cl/\I0) 1V0~1J JO NOIllSOd
PAGE 81
71 are drawn using the previously described method. A complete summary of the "average net drift velocities" and net discharges for the different experiments is presented in Table IV. The net discharge q,, presented in the last column of Table IV, is found by multiplying the average of the three net drift velocities by the average depth h. Velocities and discharges are also listed for Experiments 13 and 13a, which were the only experiments in Test Series 1 in which floats were used in the middle and both sides of the lagoon. The results listed in Table IV show that the net drift velocity strongly depends on the lateral position of the float; in general, for the experiments with 5 cm depth the net drift is largest for the floats placed at y = 10 cm, for the experiments with 6.5 cm depth the net drift is largest for the floats placed at y = +10 cm. As mentioned earlier the lateral variation in net drift should probably be attributed to the asymmetry of the experimental setup; see page 66. In each of the Test Series 2 experiments, in addition to the tracking of floats, the water motion was also studied by observing the displacement of a dye cloud released in the middle of the experimental lagoon. The dye cloud, initially about 30 cm long and extending over the full width of the channel, stretched longitudinally and at the same time gradually moved in the same direction as the floats. In all cases, the net displacement of the dye patch was observed to be in the same direction as that of the floats. One of the more striking results of the experiments is (see Table IV) that the net drift in the experiments with a water depth of
PAGE 82
72 w J PQ H M Pi Q CO Â•"N O CD n o O 00 a\ vO H CO 1 Â•Â— ** B o u Q) CO o dCO r>Â« CM O0 lO CTi vO CO 0\ lO as CO ^^ H O iH > yÂ— N e Â• a Si >Â•_Â• O LO o o LO LO o o o o LO o o CU J3 O o vO vO lO 4J CO as O C7\ as o rH iH Â•H iH H ft iH O il H O O o rH o o iH rH rH M c a LO VO r~ CO a\ O rH CM CO
PAGE 83
73 about 5 cm (Experiments 15, 16, 23, 24, 25, 13 and 13a) is toward the weir while the direction of the net drift is reversed for the experiments with a water depth of approximately 6.5 cm (Experiments 17, 18, 19, 20, 21 and 22). An explanation for this might be the following. Consider the hydrodynamic Equations (2.1) and (2.2) neglecting friction and storage. (This simplification is justified because in the experiments the storage is zero and friction is of relatively little importance.) Equations (2.1) and (2.2) then read: t + Â£Â° < 6 X)  + ia! +g(h + n) a.O (6.2) The boundary condition at x = 0, the open end of the experimental lagoon is n = a cos at (6.3) o in which 2ir a = Â— = angular frequency of the tide T = tidal period. A linearized version of the weir equation is used for the boundary condition at the weir side of the lagoon. q = U^2gh w (n. n o > + ^2gn o (n. n Q ) (6.4) in which
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74 n . = water level at the lagoon side of the weir y = "linear" weir coefficient h = average water depth over the weir. The last term on the righthand side of Equation (6.4) represents the influence of varying depth over the weir on the discharge. A trial solution for the system of Equations (6.1) through (6.4) is n = n*(x) + n Â£ (x,t) + . . . (6.5) q = q*(x) + q Â£ (x,t) + . . . (6.6) in which n. and q are the solutions of the system of equations when neglecting the nonlinear terms in the Equations (6.1) and (6.2) and the last term on the righthand side of Equation (6.4). In view of Equation (6.3), t\ and q are periodic functions with period T. Substituting the trial solution, Equation (6.5), in Equations (6.1) and (6.2) and time averaging yields q. = const. (6.7) 3n * 2 dq SL 3 \ * asr + Kai* + *>Â£ ar Â° C6.a> Equation (6.8) with the boundary condition, n* = at x = 0, yields a solution n*(x). Substituting Equation (6.5) in Equation (6.4) yields after time averaging q * = V 8h w n * + U J1 2S n o\ " M iL 2g n o (6 ' ;) in which n A and r\ are evaluated at the lagoon side of the weir. The
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75 first term on the righthand side of Equation (6.9) results from the nonlinear terms in the field Equations (6.1) and (6.2); the last two terms result from the term in the weir equation representing the influence of the time varying depth over the weir. It is known that for a channel closed at one end n, at the closed end is positive; see LonguetHiggins [13]. It seems reasonable to assume that n^ is also positive in case of a partially closed channel. Furthermore, because of the phase shift between n and r\ at the lagoon side of the weir (see Figure 14) it follows that in Equation (6.9) n nÂ„ < r\ . The first o x o term on the righthand side of Equation (6.9) thus is positive while the sum of the last two terms is negative. In the experiments, the net drift results from two opposing effects. the net drift resulting from the nonlinear terms in the field equations the net drift resulting from the variation in depth over the weir during a tidal cycle. The resultant direction of the net drift thus depends on which of these effects is predominant. It seems tempting to translate the experimental data to a prototype scale. In this context, however, it should be mentioned that translating to realistic prototype dimensions is not possible because the relative magnitude of the terms in the dynamic equation is entirely different for model and prototype. In the experiments the friction term is very small compared to the total acceleration while in a natural lagoon these terms are of the same order of magnitude; see e.g. Figure 6.
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76 c. Computation of Mass Transport; Comparison with Measured Data Equations (2.1) and (2.2) are used to describe the conservation of mass and momentum in the experimental lagoon. In Equation (2.1) the term M representing the storage is set equal to zero. The flow over the weir is described by an equation of the form q w = " p n (h w + V /2 Â§ (n o " n i> (6 ' 10) for flow from the ocean to the lagoon, and % + y h w + v /2 s ( \ v (6 ix) for flow from the lagoon to the ocean. In these equations q = discharge per unit width over the weir u and u = weir coefficients; the subscript n refers to flow from n p the ocean to the lagoon; the subscript p refers to flow from lagoon to ocean h = average depth over weir n = water level in ocean o n . = water level at Station III; see Figure 23. The weir equations involve empirical constants u and u . A literature search [27], [29], [14], [10] for values of these socalled weir coefficients did not show any data for the combination of parameters (downstream water level, upstream energy level and weir height) encountered in the experiments. In addition, all tests described in the literature deal with steadystate conditions while also the turbulence level is much higher than found in the experiments described here. Therefore,
PAGE 87
77 to arrive at an order of magnitude value for the coefficients \i and n u the following theoretical approach is taken. Bernoulli's equation is applied upstream of the weir between Stations I and II; see Figure 22. This yields for positive flow (flow from the lagoon to the ocean) I n EI LAGOON A i< w TiDAL BASIN y > />//////> ///r//////Â»///////;//>s// '/ y////s/>i/sj/r>'r/// J Figure 22. WEIR; FLOW FROM LAGOON TO TIDAL BASIN n i + . .2 n w + . , 2 2gd i 2gd w C c (6.12) in which d. and d represent the total depths at respectively Stations I and II; see Figure 22. n = the time varying water level at the weir measured with respect to mean level of the weir C = coefficient to account for the stream lines over the weir not being horizontal. At the down stream side of the weir, the momentum equation is applied between Stations II and III which yields
PAGE 88
78 2 C d n n 2tt^[Â§*1] (6.13) w o g C d d d c w o o in which d = total depth at Station III; see Figure 22. o Eliminating n between (6.12) and (6.13) and writing the result in the w form of Equation (6.11) yields q = X d /2g(n. n ) (6.14) 1 w b i o d 2d C d w j. w l g w iM d . co o i and thus (6.15) Similarly, for flow from the ocean to the lagoons it is found p = x (6.16) M n i 1 2d C d c i i Values of y and y are computed from, respectively, Equations (6.15) P n and (6.16) for two typical sets of values for d. , d and d . The results are listed below.
PAGE 89
79 d. = 5 cm i d w = 1.65 C = 0.7 c U P = 0.75 M n = 0.87 C = 0.8 c P P = 0.87 M n = 1.02 C = 0.9 c P P = 1.00 p n = 1.18 d. = 6.5 cm d = 3.15 cm d = 31.5 cm x w o C = 0.7 c P P Â— 0.81 y n = 0.95 C = 0.8 c M P = 0.95 y n = 1.12 C = 0.9 c P P = 1.13 M n 1.34 The foregoing "theoretical" values of the weir coefficients indicate that the order of magnitude of the weir coefficients is one the coefficient u is always larger than u . n p Within this framework the final combination of values for u and u will be chosen such that measured and computed net discharges n for the experiments are in good agreement. The equations describing the flow in the lagoon and the flow over the weir are solved using the numerical technique described in Chapter 3. The grid scheme used in the computations is indicated in Figure 23. The value of the friction factor F = 0.005 (this value
PAGE 90
80 pÂ° CO in sT> JÂ£VI CVJ in = + 2 + a> + en a> + r+ CO + w + lO t + ro + m CVJ + CO o CO z o UJ o UJ > to UJ 2 a: _l < X o cc UJ io CO < CO Q 5 z Ul CC UJ Q_ X UJ >cc o f< cc o CQ < o CO O 3 a. o o _j < o cc UJ o u. UJ Ul X a CO Q CC CD ro c\j Ul cc o Â£
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81 was obtained from experiments carried out in a hydraulic flume with a bottom of the same roughness as the experimental lagoon) . The space and time steps are: Ax = 45 cm and At = 0.1 sec. Recorded tides in the basin (Station I; see Figure 13) are approximated by two half sines. In the computational procedure, first the discharges q and the water levels n in the lagoon are determined, starting from the given tide in the basin. The net discharge q, then follows from T q dt (6.17) _ 1 1* x The integral is evaluated numerically. Trial and error procedures indicate that for the combination of weir coefficients, u = 0.8 and u = 1.2, computed and measured net p n ' discharges are in good agreement as shown in Figure 24. In Figure 24, the net discharges computed with this combination of weir coefficients and measured net discharges (see Table IV) are presented for all experiments of Test Series 2 and for Experiments 13 and 13a of Test Series 1. For each of these experiments, also the particle path s (x ,t) is computed from s(x ,t) = o L q(s(xj,t) dt (6.18) h + n(s(x Q ,t) o The integral is evaluated numerically. The computed particle pazhs are used to verify the empirical method of determining net discharges from particle trajectories; see page 66. Net discharges are determined from the computed particle paths using the empirical method and are
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82 Â— rZ UJ CO 2 UJ a. O UJ a. cz X < UJ O X o " o rCO o: Q UJ u. UJ tx UJ CO _J o 2 Â«I Ixl IX CO III 3 ' 00 r> Q 2 UJ Z * cr CO < Â— ^v UJ 1 1 CM X 01 Â• \^ o X 2 X. o VrO*X C\J X 10 V Â• ID t CM UJ C3 or < X o CO o hUJ z a UJ or n CO < UJ 2 FIGURE 24. COMPUTED AND M oo
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83 compared with the net discharges obtained by integration of Equation (6.17). For results, see Table V. The good agreement proves the empirical method to be a reliable tool in deriving net discharges from particle trajectories. TABLE V COMPUTED NET DISCHARGES; EVALUATION OF EMPIRICAL METHOD OF DETERMINING NET DISCHARGES 2 2 q >v (cm /sec) q,, c (cm /sec) Exper. Ampl(cm) Period(sec) Depth(cm) Eq. (6.18) Empirical 15 1.13 70 5.00 7.10 7.40 16 0.87 70 5.05 0.27 0.30 17 1.13 70 6.60 2.60 2.70 18 1.03 70 6.60 2.16 2.26 19 0.91 69 6.55 1.87 1.90 20 0.81 70 6.45 1.83 1.87 21 0.95 70 6.70 2.26 2.28 22 1.09 71 6.60 2.14 2.24 23 0.96 70 5.00 1.92 2.00 24 0.94 70 4.80 4.54 4.70 25 1.05 70 4.95 5.10 5.35 13 1.15 71 5.30 1.60 1.65 13a 1.15 71 5.30 1.60 1.65 Examples of computed particle paths for Experiments 22 and 25 are presented in Figures 25 and 26. In the same figures, the measured float positions at slack tide are plotted also. Finally, it is noted that a comparison between computed end recorded data can only be as good as the quality of the individual data.
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84 UJ a o a: o o en z o tco o Q. < o O o CM C\J Q "_' bl UJ OL rj zz rr u ) hi <~ CL UJ X 2 UJ to CM UJ a: r> CD CWO) IVCHJ JO NOUISOd
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85 CWO) IVOld dO NOIllSOd
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86 In this respect, it should be kept in mind that the measured net discharges are empirically derived from float trajectories which are assumed to represent the average horizontal particle motion. The validity of the latter assumption cannot be judged with any certainty until more detailed experiments on the distribution of the net drift over the vertical have been carried out. With regard to the computational model, it should be mentioned that inaccuracies in measured depth, especially for the smaller depths, lead to relatively large errors in the net discharge as demonstrated in Figure 27. Also, the trial and error procedure necessary to determine the weir coefficients is a poor substitute for a welldefined description of the flow over the weir, especially when variations in those weir coefficients lead to rather large changes in the net discharge as shown for Experiment 20 in Table VI. It is admitted that this to a certain extent reduces the value of a comparison between computed and recorded data. TABLE VI VARIATION OF NET DISCHARGE IN EXPERIMENT 20 FOR DIFFERENT COMBINATIONS OF WEIR COEFFICIENTS y v _J2 2_ 0.750.95 p. 751. 05 0.81.1 10.81.2 i0. 91.15 10.91.2 111.3 q.(cm /sec) 0.65 1.37 1.23 .83 0.67 0.96 10.68 d. Summary The displacement of water masses in an idealized lagoon constructed in the laboratory has been studied (1) by tracking floats
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87 LlJ CO s o 2 < x o CO Q AMPLITUDE: 0.8 CM. 3 4.8 5.2 5.6 DEPTH (CM.) 6.0 6.4 FIGURE 27. VARIATION OF NET DISCHARGE WITH DEPTH
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88 of different lengths and tracing of injected dye clouds and (2) by conducting calculations to represent this phenomenon. The lagoon which had a uniform width and depth was at one end freely connected to a tidal basin and on the other end was connected to the same basin via a submerged sharp crested weir. The experimental results showed that for a given set of hydraulic conditions (1) the magnitude of the net drift of a float depends on its length, (2) the direction of the net drift is the same for each float length and (3) the dye clouds move in the same direction as the floats. These observations suggest that the direction of the net drift is the same for each point over the entire depth. The direction of the net drift is not the same for all experiments. An explanation for this is that the net drift is the result of two opposing effects: (1) the net drift due to the nonlinear terms in the field equations and (2) the net drift resulting from the nonlinear terms in the weir equation. Net discharges are derived from measured float paths using an empirical method described on page 66. To account for lateral variation in the drift, net discharges are based on the average path of three floats placed in the middle and at the two sides of the lagoon. The computational model describing the flow in the lagoon involves two weir coefficients. The value of these weir coefficients is determined by matching computed and measured net discharges. This admittedly reduces the value of the computation model; however, the good agreement between measured and computed net discharges and thus
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89 between measured and computed mass transport for all experiments for the same combination of weir coefficients indicates the validity of the theory on which the computations are based.
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7. APPLICATION To demonstrate that the convective transport in lagoons resulting from tides only is not merely a matter of academic interest but of practical significance as well, the net discharge is calculated for Lake Worth, Florida, which is one of a number of typical lagoons of the U. S. east coast and Gulf of Mexico. The computations were carried out using the model described in Chapters 2, 3 and 4. Because the purpose of the computations is to demonstrate the effect of the tide only, wind stresses and fresh water inflow are not included in this treatment. Lake Worth, located on the lower east coast of Florida, is a narrow tidal lagoon; see Figure 28. The lake is generally parallel to the coastline and is about 20 miles long and 5 to 8 ft. deep. The actual flowconveying portion is approximately 2000 ft. wide; the shallow areas at the east side of the lake serve primarily as storage. Near the north end, Lake Worth is connected to the ocean by North Lake Worth Inlet, which has a width of 800 ft. between the two entrance jetties and is about 35 ft. deep. At the south end, Lake Worth is connected to the ocean by a much smaller inlet, South Lake Worth Inlet, with dimensions of 130 ft. wide and 10 ft. deep. Lake Worth is part of the Intracoastal Waterway for its entire length. The tides in the Lake Worth area have a predominantly semidiurnal character. According to the tide tables [24] , the mean tidal range in the Atlantic Ocean at Palm Beach is 2.8 ft. and the spring tidal range is 3.3 ft. Going inland the corresponding tidal ranges at the 90
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91 . 5. Â«&J 2 < UJ o o o \z. < _J < Â— < "IVNVO H3W39Â», WlVd 1S3M ,, N: .'3 nvNvo 2 < U o o o < _l t<
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92 Port of Palm Beach decrease to respectively, 2.6 ft. and 3.1 ft. Velocities in Lake Worth are on the order of 12 ft. /sec. Velocities can be as high as 24 ft. /sec. in North Lake Worth Inlet and 78 ft. /sec. in South Lake Worth Inlet. For the purpose of computation, the actual geometry of Lake Worth is schematizied as presented in Figure 29. The lagoon is divided into three different sections for which the friction factor F is assumed constant. Each of the sections has a conveying channel and a storage area. Lake Worth is assumed to be closed at both the northern end (North Palm Beach) and the southern end (Boynton Beach) . In formulating the mathematical description, the flow is assumed to be one dimensional, that is the current direction is always along the longitudinal axis of the lagoon. Equations (2.1) and (2.2) are employed to describe the time varying flow in the main water body of the lagoon system. The flow in the inlets is described by the simplified version of the inlet equations presented on page 17. The equations are solved using the numerical scheme described in Chapter 3. The space step used in the computations is Ax = 5000 ft., the time step is At = 180 sec. Simultaneous tide recordings at the tide stations, LAKE WORTH, NORTH PALM BEACH, PALM BEACH and BOYNTON BEACH (for locations; s^e Figure 28), serve to calibrate the numerical model. (The tide stations are operated by National Ocean Survey. The recorders are Fisher and Porter punched tape level recorders, Model 1550. Water levels are measured in a stilling well by means of a cablefloat assembly and recorded on punched paper tape every 6 min.) No discharge measurements
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93 P c*,. J 2 . 300' ;; 2 o < CE
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94 are available. It will be clear that these data are not sufficient for the ideal calibration procedure described in Chapter 4 and a trial and error procedure must be employed, flasea on experience and sound judgment, friction factors for the different sections and inlets are chosen and adjusted until a good match between measured and computed tides is obtained. This trial and error procedure is carried out for six different 24hour periods. The tidal range for these periods varies between 2.00 ft. and 3.90 ft., covering both neap and spring tides. Wind speeds for all periods are less than 10 knots while fresh water inflow is negligible. The values of the friction factors leading to a good fit are: North Lake Worth Inlet, F = 0.01; South Lake Worth Inlet, F = 0.08; each of the lagoon sections F = 0.004. The standard deviation of the differences between computed and measured tides at 30min. time intervals is on the order of 0.1 ft. or less for each of the six different tides; see [ 4 ] . The good agreement is also demonstrated in Figure 30 in which the measured and computed tides are plotted for one of the 24hour periods. The relative high F values for the inlets can be attributed to the flow contraction in the inlets. With the values of the friction factors described in the previous paragraph, the discharges and water levels in Lake Worth are computed for a sinusoidal ocean tide with an amplitude of 1.4 ft. and a period of 45000 sec. (12.5 hours). The corresponding net discharge is found by time averaging the discharge in a cross section and is found to be equal to 250 cfs to the south. The total volume of water below mean low water between North Lake Worth Inlet and South Lake Worth Inlet is 9 3 estimated to be on the order of 0.86 10 ft . It would thus take , 0.86 * 10 9 ' N ,Â„ J UJ C 250 * 24 * 3600 = ' days to renew this volume.
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95 Â•2.0 + 1.0 SLO 1.0 2.0 J i I i I , I i__L 1612 1812 2012 2212 0012 0212 0412 0S12 0812 1012 1212 AUG. 22, 1970 Â£>TIME .+ 2.0 + 1.0 SLD 1.0 2.0 Â— > Â— r ' Â— r i Â— i Â— i Â— r ' ; Â« 1 ' i ' 1 1 1 Â• '/ \ 7 / ^>Â«Â« r ' Â• NORTH i 1 PALM i l_ BEACH i 1 i 1 i 1 1 1,1,1, . 1 + 2.0 + 1.0 SLO 1.0 2.0 I Â— Â— 1 Â— T " 1 1 ' i Â» i ' I ' i " i Â• i ; Â• ^V ^r / Â™ N_^/ . PALM BEACH 1 1 i 1 i i.i.i.i, I.I.I.I + 2 + 1 Q SLO 1 .0 < Â— i Â— Â• Â— i Â— Â« Â— i Â— Â— i Â— Â« Â— i Â— Â« Â— i Â— Â« Â— i Â— < Â— i Â— " Â— i Â— ' Â— r 2.0 B0YNT0N BEACH ' i I . L J i I i I i I s I i L MEASURED COMPUTED FIGURE 30. MEASURED AND COMPUTED WATER LEVELS IN LAKE WORTH
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8. SUMMARY AND CONCLUSIONS The net displacement of water masses in a lagoon connected to the ocean by narrow, constricted inlets is studied by (1) developing a mathematical description of the phenomenon and (2) carrying out laboratory experiments. Equations describing the flow in the lagoon and in the inlets are derived. Because the mass transport results from the higher order (nonlinear) terms in the equations, only third order terms and higher are neglected. A numerical model is developed to solve for the equations and the boundary conditions. The flow in the lagoon is assumed to be one dimensional, that is the current direction is always along the longitudinal axis of the lagoon. In the model, the algebraic inlet equations are regarded as implicit boundary conditions for the partial differential equations describing the flow in the lagoon. The model is based on an explicit finite difference scheme. The instantaneous discharges and, therefore, the mass transport are very sensitive to changes in the friction factor F. It is shown that, with the conventional measuring equipment and field procedures, the required accuracy in F can only be obtained by calibrating the numerical model on discharges. That is, determining F by matching computed and measured discharges, rather than matching computed and measured water levels. An analytic expression is derived for the mass transport due to tidal motion in lagoon systems with a relatively simple geometry. The net transport computed with the analytic model agrees well with the numerically computed transport. The analytic solution is used tc 96
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97 explain the physical concepts of the net transport. It is shown that to satisfy the nonlinear terms in the field equations (the equations describing the flow in the lagoon) , the mean surface level in the lagoon must vary with x. The condition imposed by the ocean that at both ends of the lagoon the mean level equals the mean ocean level then introduces a slope in the mean surface level which in turn is responsible for the mass transport. Because the location and dimensions of the inlets play an important role in the net transport, the magnitude of the convective transport can be increased considerably by properly designing these inlets. Net discharges were measured in an idealized lagoon of uniform width and depth constructed in the laboratory. One end of the lagoon was freely connected to a tidal basin while the other end was connected to the same basin via a submerged weir. Measured and numerically computed net discharges for the different experiments are in good agreement (see Figure 24) which proves the long wave and inlet equations including the second order nonlinear terms to predict well the mass transport. The results of the study are applied to Lake Worth, Florida. For an average tide (amplitude 1.4 ft., period 12.5 hours), a net discharge southward on the order of 250 cfs was found. Using this figure, it would take approximately 40 days to renew the volume of water below mean low water between North Lake Worth Inlet and South Lake Worth Inlet showing the tideinduced mass transport to be of practical significance.
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APPENDIX
PAGE 109
APPENDIX A THE MATHEMATICAL DESCRIPTION OF LONG PERIOD GRAVITY WAVES; EULERIAN DESCRIPTION The Eulerian method describes the velocities and water surface elevations as a function of time and space. The Cartesian coordinate system shown in Figure 1 will be used. ^(x.y.t) h(x,y) <' Â•' > > /^V STILL WATER LEVEL Figure 1. Reference Frame Assuming incompressible flow, the equation of conservation of mass may be written as 9u _3y_ Â£W __ . 8x 8y 8z (1) See PrandtlTietjens [17],page 100. Instead of the velocities, the discharges per unit width q and q defined as y n q x = u dz (2) h 99
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100 n q y = j v dz (2) h are introduced. Integration of Equation (1) with respect to z between the limits h and n and substituting the foregoing expressions for q_ and q yields y x _, y / n Sn . , 9h , N on 37" + jf "k*Â™^ ^ u(x,y,h,t) Â— v(x,y,n,t) jj v(x,y,h,t) ~ + w(x,y,n,t) w(x,y,h,t) = (3) Â°y The kinematic free surface boundary condition (KFSBC), expressing the fact that a water particle cannot leave the surface, may be written as 3n Â°n dr\ //N TT+uÂ— + v Â— = w (4) 3t dX dy where u, v and w are evaluated at the free surface. The expression for the kinematic bottom boundary condition (KBBC) is u im + v Â£iÂ±). = w (5 ) 3x dy where u, v and w are evaluated at the bottom. Substituting the expressions for w at the free surface and w at the bottom as given, respectively, by Equations (4) and (5) in Equation (3) yields Q + T 2L + T X= Â° (6: 3t 3x 3y
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101 In this equation, the variables n , q and q refer to the x y instantaneous water level fluctuations and discharges. These instantaneous quantities are composed basically of a mean flow component and small turbulent fluctuations and as such can be represented as q = q + q' x x x q = q + q* (7) y y y n = n + n 1 where the bar refers to the short term mean motion and the prime refers to the fluctuations about this mean. In describing the tidal flow only the mean component is of interest. The equation of conservation of mass with the dependent variables pertaining to the basic motion is obtained by substituting the expressions given by Equations (7) in Equation (6) and then time averaging the result. This yields _8q 3q U + tt^ + it*(8) 9t 8x dy Time averaging here means averaging over a period large compared to the time scale of the turbulence and short compared to the tidal period, For the derivation of the conservation of mass equation, see also Dronkers [5 ], page 179 and Henderson [7 ]Â» page 4. The starting point for the description of the dynamics of the flow are the NavierStokes equations for an incompressible fluid as presented in Daily and Harleman [3 ], page 113.
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102 2 2 2 3u 3u , 3u . 3u 13p,y,3u,3u,3u. /r, \ Â— +u Â— +V Â— + w Â— = r^ + ( 7 + ~ + ~ ) (5a; 3t 3x 3y 3z p 3x p v Â„ 2 . 2 Â„ 2 y 3x 3y oz 2 2 2 3v , 3v , 3v 3v 1 3p y . 3 v 3 v I v . , Q , s + u __ + v Â— + WTÂ— T^ + T ( Â— o + Â— o + Â— o ) ( &b > 3t 3x 3y 3z p 3y p Â„ 2 2 Â„ 2 ' J 3x 3y 3z 2 2 2 3w , 3w , 3w , 3w 1 3p y . 3 w , 3 w 3 w . N Â— + u Â— +VÂ— + w Â— = TT _ S + T ( Â— 5" + Â— o + Â— o ) ( 5c ) 3t 3x 3y 3z p 3z p ^2 8y 2 ^2 in which y = dynamic viscosity p = pressure = arithmetic mean of the three normal stresses. The foregoing equations pertain to an inertial frame of reference. When using a coordinate system fixed to the earth, the fluid particles are affected by the Coriolis acceleration, which results from the earth rotation. These accelerations are in x direction fiv in y direction fiu in which fi = 2co sin
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103 of normal forces, forces on the vertical planes or lateral forces and forces on the horizontal planes. In tidal flow, the normal viscous forces are small compared to the pressure gradients. When dealing with bodies of water in which the horizontal dimensions are large compared to the vertical dimensions, the lateral viscous forces are generally small and will be neglected here. Upon introducing the Coriolis acceleration and neglecting the normal and lateral viscous forces, Equations (9a) and (9b) become, respectively ou , 3u , 3u , ou n 1 3_p_ 1 zx nn ^ Â— + u Â— + v Â— + w Hv = T^ + 5 Â— (10a) 3t 3x 3y 3z p 3x p 3z 3v 3v 3v ov 1 op 1 zy /,Â„,% Â— + u Â— + v Â— + w Â— + Qu = r*+ t^UOa) 3t 3x 3y 3z p ay p 3z in which , 3u , 3w . x = u ( Â— + Â— ) zx oz ox , 3v , 3w x zy 3z 3y By virtue of the definition of long waves, the vertical acceleration Â— is small compared to the acceleration g and thus Equation (9c) may be written as = I 2 g (10c) p 3z The conservation of mass equation, Equation (1), times u is added to Equation (10a) . This yields
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104 2 3 t 9u , 3u 3uv . 3uw _ 1 3p , 1 zx ,.,., N __ + _ Â— + _ Â— + _ q w = Â—c ^ . (11a; 8t 3x 3y 3z p ax p 3z Integration of Equation (10c) with respect to z and requiring that p = at the surface, yields the usual hydrostatic relationship p = pgz + pgn This expression for p is substituted in Equation (11a) and the resulting equation then is integrated with respect to z between the limits h and n. After application of Leibnitz rule, the result is n n _ u dz 3 u dz 3  uv dz h h h / \ I / \ I I 3 r i 2 1 + : H r + (uw) (uw) u] ru 3t 3x 3y v 'n N W h 'n9t 'r, 3x X ox n 2 1 3(h) , vi 9n , vi 3(h) _ f , f. , Â» 3n + u v, ~^ Â— ~ ( uv ) a + ( uv ) i, \ " fi v dz = g(n r n) tÂ— 'h 3x ' n 3y 'h 3y J jx + JÂ™\ _ Isil (12a) p 'n p ' h The terms in Equation (12a) marked with "o" constitute the KFSBC given by Equation (A). The terms marked with "x" constitute the KBBC as expressed by Equation (5). Therefore, Equation (12a) may be written as 3 u dz 3 J J n n 2 f u dz 3 uv dz Â± + _^ + \ n v dz = g (h + n) t 1 3t 3x 9y J B ox h + t  x I (ua) zx zx , 'n 'h
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105 The variables q and q defined by Equation (2) are now x y introduced. In doing so, the following approximations are used for the terms resulting from the convective acceleration. 2 a ~ 1 ! u dz ( u dz) = ; Â— n h h n n n 1 f a f , Vy uv dz Â— \ u dz v dz = h h h Based on the foregoing, the equation of conservation of momentum in the x direction becomes 2 3q 3q 3q q x T x , ,, N 9n 1 tc , 1 x y _, zxi zx ,. Â» + s 9t & 3x h 3x h 3y ^y p ' r, p ' h Introducing the expressions for q_, q and n given by Equation (7) and time averaging yields _2 3q x 1 3q x 1 d _%^L oft u "^ ^ ^^ i ^x" 1 x y . zx I zx . ' i c \ Â— rÂ— *+ ^15a) h 3y p ' n p ' h Note that as a result of the application of Leibnitz rule and the kinematic boundary condition, the term, puw (one component of the Reynolds stress) has disappeared. Usually, the second, third and fourth terms on the righthand side are small compared to the first term on that side and will be neglected; therefore
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106 _2 3q , 3q . 3q q X T x . > 3n 1 x , 1 n x^y zx +g(h + n) "TT + 7tÂ— + 7Â— r^ S2qÂ„ = Â— 3t &N " 3x h 3x h 3y y p The corresponding equation for the y direction is ZX  h p l n (16a) 3q . 3q . 3q q x x 7^ + s( h + n) IL + r4 + rf JSL +"i a + ^ (i6b) 3t Â° 3yhÂ„2hdx n x Pi P 3 3y h 'n The terms on the righthand side of Equations (16a) and (16b) result from the external forces. The first term times p represents the bottom stress, the last term times +p represents the free surface stress. Assuming quasisteady flow, the bottom shear stress can be related to the characteristics of the main flow, by one of the wellknown resistance laws. Here use is made of the DarcyWeisbach formula which yields (17a) X zx p f \\ h" 8 _ f SI h" 8 Ll 2 V q x + % (h + n) 2 X zy P 12 2 A x + q y (h + n) 2 (17b) f = DarcyWeisbach resistance coefficient. For a more detailed derivation of Equations (17a) and (17b), see Dronkers [5 ], page 184, The second term on the righthand side of Equations (16a) and (16b) results from the stresses exerted by the wind on the water. For expressions relating these stresses and the windspeed and direction, see Dronkers [5 ], page 184. In view of the foregoing, the dynamic
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107 Equations (16a) and (16b) take the form 2 _ _ _ . /2 2 , 3q x 3n 1 3q x 1 3q x q v qJV^ + O t^+ g(h + n) l^ + rÂ— i + r~^ JL "q = F x Â— ^ +x 3t 3x h 3x h 3y y ,, , N z 7 (h + n) (18a) _2 _ .2 _2 8q v 3n 1 3q v 1 3q x q v VV q x + q v ! J+ g(h + n) r + ^r I 4T J + "q = F Â— ^ + Y 3t 3y h 3y h 3x x ,, N z 7 (h + r,; (18b) in which F = f/8 T x= zx  p n T P 'n The twodimensional equations of conservation of mass and momentum as derived here do not account for rainfall or evaporation. The effect of these phenomena can be introduced into the twodimensional equations but only in an approximate way; proper formulation of the problem requires a threedimensional description. In accounting for the effects of rainfall and evaporation, it is assumed that the supplementary discharge may be thought to originate from sources evenly distributed throughout the flow field. In that case, the equation of conservation of mass reads Hi + i + w = (19) 3x 3y dz in which s = strength of source per unit volume.
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108 Integrating with respect to z between the limits h and n and introducing the variables q and q yields after time averaging 3q 3q _i + i + a M (20) 3x 3y 3t in which M = net inflow per unit surface area due to rainfall, lateral inflow, etc. Determination of the effect of rainfall and evaporation on the dynamics of the flow requires a semiempirical approach. In case the dynamic effects are important, measurements are needed to determine the empirical constants. The reader is referred to Dronkers [5 ], page 194 and Stoker [21], page 452. A similar problem as described in the previous paragraph is encountered when dealing with storage areas in tidal models, frequently found in onedimensional schematizations of tidal waters. Assuming that the discharge from the storage region may be thought to originate from evenly distributed sources in the flow field, the equation of conservation of mass is the same as Equation (20). M now is defined as the discharge per unit width from the storage region, divided by the width of the conveying channel. For the effect of inflow from the storage area on the dynamics of the flow in the conveying channel, the reader again is referred to Dronkers [ 5 ], page 194 and Stoker [21], page 452.
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REFERENCES 1. Bagnold, R. A., "Sand Movement by Waves: Some Small Scale Experiments with Sand of Very Low Density," Inst, of Civ. Eng. , Vol. 4 , February, 1947. 2. Chiu, T. Y. , Van de Kreeke, J. and Dean, R. G. , "Residence Times of Waters Behind Barrier Islands," Department of Coastal and Oceanographic Engineering, University of Florida, Technical Report No. 7 , October, 1970. 3. Daily, J. W. and Harleman, D. R. F. , Fluid Dynamics , AddisonWesley Publishing Company, 1965. 4. Department of Coastal and Oceanographic Engineering, University of Florida, "The Use of Tidal Computations in Establishing Tidal Datums; Results of a Pilot Study," July, 1971. 5. Dronkers, J. J., "T idal Computations in Rivers and Coastal Waters ," NorthHolland Publishing Company, 1964. 6. Dronkers, J. J., Discussion of "WaterLevel Fluctuations and Flow in Tidal Inlets," by J. Van de Kreeke, Journal of the Waterways and Harbors Division, ASCE, Vol. 94, No. WW3 , August, 1968. 7. Henderson, F. M. , Open Channel Flow , The Macmillan Company, 1967. 8. Ippen, A. T. , Estuary and Coastline Hydrodynamics , McGrawHill Book Company, New York, 1966. 9. Keulegan, G. H., "Third Progress Report on Tidal Flow in Entrances, Water Level Fluctuations of Basins in Communication with Seas," Report No. 1146 , National Bureau of Standards , U. S. Department of Commerce , September, 1951. 10. King, H. W. and Brater, E. F. , Handbook of Hydraulics , McGrawHill, Inc., 1963. 11. Lockwood, M. G. and Carothers, H. P., "Preservation of Estuaries by Tidal Inlets," Journal of the Waterways and Harbors Division , ASCE, Vol. 93, No. WW4 , November, 1967. 12. LonguetHiggins , M. S., "Mass Transport in Water Waves," Phil. Trans. Roy. Soc. London , February, 1953. 13. LonguetHiggins, M. S. and Stewart, R. W. , "Radiation Stresses in Water Waves; a Physical Discussion with Applications," Deep Sea Research, Vol. II , pp. 529562, 1964. 14. Mavis, F. T. , "Submerged Thin Plate Weirs," Engineering News Record , July 7, 1949. 109
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110 15. Mota Oliveira, T. B., "Natural Flushing Ability in Tidal Inlets," Proceedings of Xllth Conference on Coastal Engineering , Washington, D.C. , 1970. 16. Platzman, G. W. , "The Computational Stability of Meteorological Prediction Equations," Report , Department of Meteorology , University of Chicago , 1958 17. Prandtl, L. and Tietjens, 0. G. , " Fundamentals of Hydroand Aeromechanics , " Dover Publications, Inc., 1934. 18. Reid, R. 0. and Bodine, B. R. , "Numerical Model for Storm Surges in Galveston Bay," Journal of the Waterways and Harbors Division, ASCE, Vol. 94, No. WW1 , February, 1969. 19. Russell, R. C. H. and Osorio, J. D. C. , "An Experimental Investigation of Drift Profiles in a Closed Channel," Proceedings, Vlth Conference on Coastal Engineering , Florida, 1958. 20. Shemdin, 0. H. and Forney, R. M. , "Tidal Motion in Bays," Proceedings of Xllth Conference on Coastal Engineering , Washington, D.C, 1970. 21. Stoker, J. J., Water Waves , Interscience Publishers, Inc., 1966. 22. Stokes, G. G. , "On the Theory of Oscillatory Waves," Trans. Cambridge Phil. Soc , Vol. VIII , March, 1847. 23. U. S. Beach Erosion Board, "A Study of Progressive Oscillatory Waves in Water," Tech. Report No. 1 , May, 1941. 24. U. S. Department of Commerce, Coast and Geodetic Survey, Tide Tables. 25. Van de Kreeke, J., "Water Level Fluctuations and Flow in Tidal Inlets," Journal of the Waterways and Harbors Division, ASCE, Vol. 93, No. WW4, November, 1967. 26. Van de Kreeke, J., Closure of "WaterLevel Fluctuations and Flow in Tidal Inlets," Journal of the Waterways and Harbors Division, ASCE, Vol. 95, No. WW1 , February, 1969. 27. Vennard, J. K. and Weston, R. F. , "Submergence Effect on SharpCrested Weirs," Engineering News Record , June 3, 1943. 28. Verma, A. P. and Dean, R. G., "Numerical Modeling of Hydromechanics of Bay Systems," Proceedings, Civil Engineering in the Ocean II , Miami, 1969. 29. Villemonte, J. R. , "Submerged Weir Discharge Studies," Engineering News Record , December 25, 1947.
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Ill 30. Vreugdenhil, C. B. , "The Influence of the Frictionterm on the Stability of DifferenceMethods for Hydraulic Problems," De Ingenieur, ,jrg. 78, nr 20 , May, 1966. (Dutch text)
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BIOGRAPHICAL SKETCH Jacobus van de Kreeke was born in Holland where he received his primary and secondary education. He attended the Technical University in Delft and received the Ingenieurs degree from that institution in 1959. From 1959 to 1961, he served in the Royal Dutch Navy. Most of this time he spent in New Guinea where he designed and supervised construction of marine structures. When discharged as a Lieutenant, j.g. he joined the Hydraulics Division of Deltaworks, the Hague, Holland. While employed with the Delta service he was involved in coastal engineering research and economic analysis mainly pertaining to the design of the breakwaters of Europoort Harbor (Rotterdam) . As part of an exchange program, he worked for six months at the Laboratoire National d'Hydraulique, Chatou, France, carrying out research on wave forces on submerged oil reservoirs. In 1965, he came to the United States where he was employed with the Department of Coastal and Oceanographic Engineering at the University of Florida. Besides his teaching duties he was involved in breakwater design, sediment transport, beach erosion, cidal computations, and laboratory and field experiments . In 1967, he joined NESCO and later SEA, where he was involved in studies on sediment pollution. In 1969, he returned to the University of Florida where he was employed as a Research Associate, while pursuing his doctoral degree He is the author of several publications in the field of coastal engineering. He is a member of the American Society of Civil Engineers, and a member of the Dutch Institution of Graduate Engineers. 112
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bent A. Chris tensen, Chairman Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. fM;6L Robert G. Dean, CoChairman Professor of Coastal and Oceanographic Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert G. Blake Associate Professor of Mathematics This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy, December, 1971 Dean, College of Engineering Dean, Graduate School
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UNIVERSITY OF FLORIDA 3 1262 08553 7875

