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FLOW SEPARATION' AN.) RELATED
PHENOMENA AT TIDAL INLETS
By
EMIN OZSOY
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
ACKNOWLEDGMENTS
The author would like to thank Dr. Umit A. Unluata,
who supervised the research. His guidance, encouragement,
and enthusiasm will not be forgotten, in spite of the fact
that he could not be present at the time of submission of
this dissertation. Suggestions and enthusiastic support
received from Dr. Ulrich H. Kurzweg, Dr. Joseph L. Hammack,
and Dean Morton Smutz are deeply appreciated.
Mrs. Nancy V. McDavid worked many arduous hours to
type a highly professional final draft of the manuscript.
The talent of Mrs. Lillean Pieter,who did the drafting,
and Mr. Ron Franklin's photographical work were also of
great value. Special thanks are given to Mrs. Evelyn Hill
who helped the author in every respect. Support received
from Dr. Ashish J. Mehta and the efforts made by Mr. Howard
Groger made possible the experiments reported in Part I.
During his graduate studies the author was supported
through a NATO Scholarship awarded by the Scientific and
Technical Research Council of Turkey. Part II of this
research was supported by the National Science Foundation
through a research initiation grant to the Coastal and
Oceanographic Engineering Laboratory, University of Florida.
The use of COEL facilities and the funds made available
for the publication of this report are greatly appreciated.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS. .
LIST OF FIGURES. .
ABSTRACT .
. xii
PART I
FLOW AND MASS TRANSPORT IN THE
VICINITY OF TIDAL INLETS
CHAPTER 1 INTRODUCTION. .
1.1 Background .
1.2 Objective and Scope of the Present
Study. .
CHAPTER 2 NEARSHORE HYDRODYNAMICS IN THE VICINITY OF
TIDAL INLETS .. ..
2.1 Shallow Water Equations .
2.2 Turbulent Jets in Shallow Water. .
2.2.1 Simplified Equations for Jets
2.2.2 General Solution for Bottom
Frictional Jets Over Variable
Topography .
2.2.3 Jets on a Frictionless Bottom
of Constant Depth .
2.2.4 Bottom Frictional Jets with
Constant Depth .
2.2.5 Bottom Frictional Jets on a
Linearly Varying Topography
2.2.6 Jets on Arbitrary Bottom
Topography. .
2.3 Curved Jets in the Presence of Cross-
Currents .
2.3.1 Equations for Deflected Jets.
2.3.2 Flows in the Presence of Cross-
Currents .
vii
2
S 2
. 8
TABLE OF CONTENTS (continued)
Page
CHAPTER 3 MASS TRANSPORT IN THE VICINITY OF TIDAL
INLETS ..
3.1 Shallow Water Diffusion Equation
3.2 Diffusion in Turbulent Jets .
3.2.1 Simplified Equations for Jets
3.2.2 General Solution. .
3.2.3 Transport of Pollutants .
3.2.4 Transport of Suspended Sedi-
ments .
3.2.5 Effects of Ambient Concentra-
tion .
3.2.6 Effects of Cross-currents
3.3 Tidal Exchange and Mixing in Bays. .
CHAPTER 4 THE STARTING JET .
CHAPTER 5 CONCLUSIONS .
APPENDIX A MEASUREMENT OF JET WIDTH FROM AERIAL
PHOTOGRAPHS .
REFERENCES . .
PART II
DISSIPATION AND WAVE SCATTERING
BY NARROW OPENINGS
CHAPTER 1
CHAPTER 2
CHAPTER 3
INTRODUCTION .
1.1 Background .
1.2 Objective and Scope of the Present
Study .
THEORETICAL MODEL FOR WAVE TRANSMISSION
APPROXIMATE SOLUTIONS FOR SCATTERING AND
DISSIPATION .
3.1 Approximate Analysis for Monochromatic
Wave Incidence .
3.2 Approximate Analysis for Multiple
Harmonic Wave Incidence. .
83
84
89
89
91
102
108
141
143
148
172
185
193
198
198
206
207
218
218
234
TABLE OF CONTENTS (continued)
CHAPTER 4
CHAPTER 5
EXPERIMENTS .
4.1 Experimental Set-Up and Procedures
4.2 Ranges of the Parameters a and B
4.3 Shallow Water Nonlinearity of the
Incident and Scattered Waves ..
4.4 Transmission, Reflection and Dissipa-
tion Characteristics of Narrow
Openings .
4.4.1 Sharp Edged (Type I) Openings
4.4.2 Long (Type II) Openings .
4.5 Determination of the Drag Coefficient
from Experiments .
4.6 Parametric Dependence of Wave Scatter-
ing Observed in Other Experimental
Studies .......
CONCLUSIONS .
APPENDIX A EXPERIMENTAL PARAMETERS .
APPENDIX B
INCIDENT, TRANSMITTED, AND REFLECTED WAVE
AMPLITUDES FOR EXPERIMENTS 1-10 .
REFERENCES .
. 328
Page
242
242
250
251
256
256
274
276
281
285
290
295
325
VITA .
LIST OF FIGURES
Figure Page
PART I
2.1 Definition sketch. .... .. 13
2.2 Definition sketch for shallow water jet flows. 25
2.3 Variation of centeriine velocity for jets on a
frictionless, constant bottom. .. 38
2.4 Variation of half-width for jets on a friction-
less, constant bottom. ... 39
2.5 Volume flux and energy flux variations for
jets on a frictionless, constant bottom. 41
2.6 Bottom frictional jet over constant bottom;
jet half-width, centerline velocity. 44
2.7 Ebbing flow at Redfish Pass, Florida .. 46
2.8 Jet characteristics over a linearly varying
bottom; jet half-width, centerline velocity. 52
2.9 Laboratory layout of the bay-inlet-ocean
model. ... 56
2.10 Centerline velocity for a jet on linearly
varying bottom topography. ... 57
2.11 An example of arbitrary depth variations;
depth variations, jet half-width, centerline
velocity . 59
2.12 Jet development at Jupiter Inlet, Florida. 61
2.13 Bathymetric map of Jupiter Inlet and vicinity. 62
2.14 Three dimensional plot of the bottom topog-
raphy near Jupiter Inlet 63
vii
LIST OF FIGURES (continued)
Figure Page
2.15 Comparison with the jet characteristics at
Jupiter Inlet; depth variations, jet half-
width, centerline velocity ... .64
2.16 Polar coordinates. ... 66
2.17 Definition sketch for jets in cross-currents 70
2.18 Cross-sectional pattern of flow in the y,-z
plane. ...... .. 73
2.19 Jet characteristics for jets in cross-currents;
jet half-width, centerline trajectory. 79
2.20 Curved jet at Bakers Haulover, Florida 81
2.21 Strongly curved jet at South Lake Worth Inlet,
Florida. . ... 82
3.1 Definition sketch for diffusion in shallow
water jets . 92
3.2 Jet centerline concentration for constant
depth. . 107
3.3 Jet centerline concentration for linearly
varying depth. 107
3.4 Lateral variation of the bottom deposition rate
K(E,S) in the ZOEF, in the ZOFE. .. 112
3.5 Diffusion and settling characteristics; center-
line concentration, centerline deposition rate,
integrated deposition rate ... 118
3.6 Diffusion and settling characteristics; center-
line concentration, centerline deposition rate,
integrated deposition rate ... 120
3.7 Diffusion and settling characteristics; center-
line concentration, centerline deposition rate,
integrated deposition rate ... 121
3.8 Diffusion and settling characteristics; center-
line concentration, centerline deposition rate,
integrated deposition rate ... 123
viii
LIST OF FIGURES (continued)
Figure Page
3.9 Contours of bottom deposition rate K(E,C)
for 1 = 0.05, v = 0,y = 0.1, = 1.0 125
3.10 Contours of bottom deposition rate K(E,c)
for V = 0.05, v = O,y = 0.1, p = 1.4 126
3.11 Sand depositions near four inlets in the vi-
cinity of Big Marco Pass, Marco Island,
Florida. . .. 128
3.12 Sand depositions and jet interaction at
Little Hickory Pass and Big Hickory Pass,
Florida. . ... 129
3.13 Sand depositions at East Pass, Destin,
Florida. . 130
3.14 Terminology for tidal inlet sedimentation
patterns .. ... 133
3.15 River mouth sedimentation patterns ... .. 135
3.16 Land area of distributaries at the Mississippi
River delta. ... 138
3.17 Thermal infrared imagery of Mississippi River
effluents . 139
3.18 Schematic diagram showing sand pockets and
distributaries of the Mississippi River delta. 140
3.19 Contours of bottom deposition rate K(E,C) for
= 0.05, v = 0, y = 0.1, p = 1.0 and
CA = (E/2)exp(-2/2). .. 142
3.20 Centerline concentration for jets in cross-
currents, y = 0.02, v = 0. ... 145
3.21 Contours of bottom deposition rate K(,S) for
a jet in a cross-current p = 0.02, v = 0,
6 = 0.1, p = 1.0, UA = 0.4 ... 146
3.22 Curved shoals at Captiva Pass, Florida 147
3.23 Schematic representations of a bay-inlet-ocean
system, jet flow during ebb tide, sink flow
during flood tide. ... 149
LIST OF FIGURES (continued)
Figure Page
3.24 Jets issuing from three bays on the Texas and
Louisiana Coast. . 151
3.25 Definition sketch for ocean mixing .. .159
3.26 The ocean mixing coefficient, T, as functions
of Tf and p. ... 162
3.27 Schematic representations of water surface
displacement in bay, velocity at inlet, evolu-
tion during flood tide, evolution during ebb
tide . 164
3.28 Total amount of rhodamine dye in Card Sound
during continuous injection test, T = 0.3. 171
4.1 Unsteady development of a starting jet 173
4.2 Starting flow at the beginning of ebb tide at
Cape Hatteras, North Carolina. .. 175
4.3 Starting jet at Fort Pierce Inlet, Florida 176
4.4 Starting jet at Ponce de Leon Inlet, Florida 177
4.5 Jet development at Little Pass, Clearwater,
Florida. . 178
4.6 Frames from motion picture showing jet
development. .. 180
4.7 Distance to jet front QF versu' time T 183
A.1 Guide for the measurement of jet width from
photographs. ... 191
PART II
1.1 Tsunami induced vortex shedding at the harbor
entrance of SA-ME District, Hachinohe Port,
Japan. . 203
2.1 Definition sketch, sharp edged (type I)
opening with L/d << 1, long (type II) opening
with R/d >> 1. ... 209
LIST OF FIGURES (continued)
Figure Page
2.2 Definition sketch illustrating the concept of
effective mass; plan view, side view ..... 214
2.3 Inertial reactance for sharp edged (type I)
openings, long (type II) openings. ... 216
3.1 Equivalent block diagram for equation (3.1),
block diagram for first approximation quasi-
linear system. ... 219
3.2 T1 as a function of a and B. ... 228
3.3 R1 as a function of a and B. ... 229
3.4 e as a function of a and a. .. 230
3.5 T1 as a function of p2, p3, and B. ... 238
3.6 T2 as a function of P2 and B ... 239
3.7 T3 as a function of p3 and B ... 240
4.1 Laboratory layout of the wave tank .. 244
4.2 Narrow opening placed in the wave tank .. 246
4.3 Array of capacitance type wave gauges. ... 248
4.4 Strip chart recorder and controls for the
wave-maker . 249
4.5 Data acquisition system and magnetic tape
recorders. . ... 249
4.6 Visualization of the wave-induced flow through
a narrow opening ... 258
4.7 T as a function of 8 ... 262
4.8 R as a function of 8 .... .263
4.9 e as a function of 8 .. ... .264
4.10 Incident, transmitted and reflected wave spec-
tra for experiment 1 ... .266
LIST OF FIGURES (continued)
Figure Page
4.11 T as a function of a for different ranges of
the parameter . 268
4.12 T1, R1, and E as functions of a1l/h. .. 271
4.13 T1, R1, and e as functions of all/h. ... 272
4.14 TI, R1, and e as functions of all/h. ... 273
4.15 T and R1 as functions of Kb for a long (type
I ) opening. .. 275
4.16 The drag coefficient f as a function of d/b. 280
4.17 T, and R1 versus B for experiments due to
Hayashi et al 282
xii
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
FLOW SEPARATION AND RELATED
PHENOMENA AT TIDAL INLETS
By
Emin Ozsoy
August, 1977
Chairman: Dr. Ulrich H. Kurzweg
Major Department: Engineering Sciences
Various phenomena associated with flow separation
at tidal inlets and other narrow entrances are analyzed.
The solutions are then used in discussing relevant appli-
cations.
In Part I, flow and mass transport are analyzed for
turbulent jets issuing from tidal inlets during the ebbing
phase. Frictional resistance and topographical variations
of the bottom and ambient cross-currents are taken into
account. In studying mass transport, settling of materials
to the bottom and ambient concentrations are also allowed.
The flow is analogous to that of a classical two-
dimensional turbulent jet when bottom friction and topo-
graphical variations are neglected; and the solutions com-
pare favorably with available theories. When bottom friction
is present, the jet expands rapidly and gets retarded.
Effects due to an offshore-sloping bottom counteract the
rapid expansion due to friction. For arbitrary depth
xii i
variations, the jet may go through a series of expansions
and contractions with distance. Comparing these solutions
with experimental and field evidence gives satisfactory
results. In the case of pollutants, bottom friction re-
duces dilution within the jet due to a decrease in convec-
tive transport. An offshore sloping bottom opposes this
effect. The transport of suspended sediments depends on
the fall velocity of sediments and the velocity at the
inlet, in addition to bottom friction and depth variations.
Finer sediments are transported to larger distances off-
shore as compared to coarser sediments. The largest depo-
sition occurs at the location of two marginal shoals
flanking the jet centerline. When the velocity at the
inlet exceeds a critical value, deep scour holes are formed
near the inlet mouth. These results and their implications
on the geomorphology near tidal inlets and river mouths are
discussed qualitatively.
Exchange and mixing in a bay-inlet-ocean system is
studied, based on the results of the previous analysis.
Materials within the bay are jetted to great distances off-
shore during the ebbing phase and this provides an efficient
flushing mechanism.
Finally, the unsteady development of a tidal jet is
described qualitatively and a simple method for calculating
the speed of the jet front is proposed. Preliminary experi-
ments are also provided.
xiv
In Part II, an experimental approach is used in
determining the wave scattering by narrow openings such
as tidal inlets, tsunami breakwaters, and harbor entrances.
Flow separation and generation of jets and vortices in
the near field of the opening determines the subsequent
loss of energy and the scattering of waves in the far field.
Approximate methods are used in analyzing the non-
linear behavior of a narrow opening in the presence of flow
separation. Experimental results are interpreted by making
use of these approximate solutions.
Experimental procedures are complicated because of
the shallow water nonlinearity, which causes the incident
waves to be almost periodic in space and to have harmonic
distortion. Therefore, the measured waveforms are Fourier
analyzed and the spatial amplitude variations of the inci-
dent and reflected harmonics are obtained.
The transmission of waves is reduced by two main
factors; one being due to the inertial reactance resulting
from the disturbance of the flow by the opening, and the
other being due to the drag resistance caused by flow
separation. When inertial reactance is dominant, the re-
sponse is selective of frequency; whereas, when separation
resistance is the dominating factor, wave transmission
becomes dependent on amplitude only. The dissipation of
energy is maximal in the second case.
It is also shown that the scattering of the funda-
mental component is not affected significantly by the
presence of higher harmonics in the incident wave. How-
ever, the same result is not true for the scattering of
higher harmonics.
xvi
PART I
FLOW AND MASS TRANSPORT IN THE
VICINITY OF TIDAL INLETS
CHAPTER 1
INTRODUCTION
1.1 Background
Tidal inlets play important roles in a broad range
of coastal activities such as commerce, nagivation, small
craft operations, and achievement of adequate water quality
standards. Since they act as interfaces between estuarine
or lagoonal waters and the coastal ocean, they deserve,
and to some extent have received, utmost attention. It
is on these grounds that the prediction and understanding
of tidal currents in the vicinity of inlets gains particu-
lar significance.
Tidal currents at inlets often transport pollutants
or suspended sediments. Mass transport of these materials
determines the ultimate distribution of pollutants and
the bathymetric changes near tidal inlets.
Tide-induced currents in coastal waters become es-
pecially significant near tidal inlets and estuary mouths.
Patterns of flow change with time during a tidal period.
More specifically, the flow on the ocean side separates from
boundaries during ebb, as opposed to the non-separated flow
3
during flood. Therefore a turbulent jet is often formed
during ebb-flows. Depending on the strength of inlet
currents, the jet may extend to distances of considerable
magnitude, hence, contributing significantly to the small-
scale mixing and circulation on the continental shelf.
During flood, the currents in the semi-infinite ocean
region may be modeled as a sink flow, since water is drawn
towards the inlet.
There has been one serious attempt by French (1960)
in modeling the ocean flow patterns during both stages of
the tide, but the model is inadequate in that the effects
of bottom friction and variable bottom topography are
neglected. In reality, these factors may become important
especially during ebb-tide, when a jet flow is formed.
In the presence of bottom friction, the jet loses
its momentum against the resistance of the bottom and
therefore expands more rapidly in width as compared to
a frictionless jet. Bottom topographical variations
seaward of tidal inlets also contribute to the jet develop-
ment. For example, if the bottom slopes downward in the
offshore direction, the expansion rate of the jet is
reduced due to the increased vertical extent of the
flow area.
Bottom frictional effects have been hinted at in a
number of other studies. Taylor and Dean (1974) and
Borichansky and Mikhailov (1966) have included bottom
friction in their analyses, but have been led to results
that are of limited value since they have utilized various
inaccurate assumptions. The latter have also accounted
for depth variations. Experimental studies by Savage
and Sobey (1975) and Gadgil's (1971) analysis of the effect
of Ekman friction on laminar jets also indicate the fast
jet growth.
Tidal flows near inlets may have a complex nature
due to a number of other factors. For example, waves
incident on the coast and local winds often set up circu-
lation patterns in addition to tidal currents. These cur-
rents in the ambient waters often change the behavior of
tidal jets. Furthermore, secondary circulations within the
jet may also occur.
Density stratification is also a commonly observed
phenomenon at tidal inlets, especially when fresh water
outflows are present in the adjacent bay areas. In the
event of stratification, the jet exhibits buoyant spread-
ing and thus a salt water wedge may be generated at the
inlet. An example of a tidal inlet where this phenomenon
occurs is reported by Wright and Sonu (1974).
On a speculative basis, the earth's rotation may
be expected to influence tidal jets, since they often reach
to large distances offshore. However, Savage and Sobey
(1975) have shown both analytically and experimentally
that shallow water jets (extending through the full depth
of the fluid) can support pressure forces arising due to
rotation and therefore are not deflected. This result
appears to be a confirmation of the Taylor-Proudman theorem
(see Savage and Sobey, 1975). On the contrary, for deep
water jets, i.e., for jets which do not cover the whole
depth, they have shown that the jet encounters deflection
due to Coriolis forces. Therefore, for jets issuing from
tidal inlets, deflection may occur when the flow separates
from the bottom either due to a steep bottom slope or due
to buoyancy of effluents.
A tidal inlet serves as an interface between the
interior waters and the ocean and thus plays an important
role in the circulation and flushing characteristics of
the interior waters. To date much effort has been directed
toward the analysis of tide-induced circulation in bays
and estuaries (see Wang and Connor, 1975). However, while
the exchange and flushing characteristics of inlets and
entrance regions have been generally recognized as having
a significant effect on the quality of interior waters,
there has been very little effort made to understand
related phenomena. An approximate analysis of inlet ex-
change characteristics has been done by Taylor and Dean
(1974).
With regard to sedimentation, a basic knowledge of
the hydrodynamic phenomena in the vicinity of an inlet,
a river, or an estuary mouth is crucial to the understanding
of the evolution, persistence, and maintenance of the
geomorphology of the neighboring regions, including inner
and outer shoals, swash channels, navigation channels and
the configuration of the adjacent shorelines. Indeed, a
significant amount of sand in a littoral system can be
trapped by an inlet. Briefly, the flow issuing from an
inlet in the ebb-cycle is in the form of a turbulent jet,
laterally entraining the adjacent waters and, therefore,
carrying littoral material into the inlet zone. During
flood cycle the flow radially converges toward the inlet
carrying sand with it. The sand in the adjacent littoral
system is thus transported toward the inlet in all phases
of the tidal cycle and is either jetted off-shore or
carried into the interior waters to form extensive shoals.
It has been found in some areas of Florida that the ma-
terial extracted in this manner from a littoral system has
caused 40 feet retrogressions of the shoreline each year
for a period of 50 years (Dean and Walton, 1974).
Deposition on the seaward side of an inlet often
creates bar structures that encircle the inlet mouth.
Sedimentation patterns mainly depend on tidal currents,
but other effects due to waves, stratification, etc. may
also have an influence. For instance, Dean and Walton
(1974) have discussed the effect of wave energy on limit-
ing shoal volumes. They have shown that in areas of low
wave energy, the bar structure consists of two massive
shoals flanking a central channel; whereas, on high
energy coasts, the shoals take a crescent shape due to
the shoreward transport of sediments. This balance be-
tween tidal currents in the offshore direction and wave
induced onshore currents has also been discussed by Oertel
(1974); who also incorporated the effects of longshore
currents.
The above cited features of the flow and mass transport
are by no means reserved to tidal inlets and in fact ex-
tend to the case of a river that terminates at a coast.
Fresh water discharging from a river into coastal waters
also forms a turbulent jet. However, a buoyant spreading is
associated with the flow in this case due to the lower
density effluents. Essential features of river effluents
have been thoroughly outlined and documented by Wright and
Coleman (1974) and Garvine (1977). The buoyant spreading
of river plumes has been analyzed by Bondar (1970) (see
Wright and Coleman, 1974) who neglected lateral entrainment
and lateral shear stresses. An analysis including buoyancy,
lateral shear stresses, and earth's rotation has been done
earlier by Takano (1954) (see Officer, 1976) but his
analysis applies only in the far field of the river mouth;
since,the nonlinear convective terms in the equations of
motion have been neglected.
River mouth sediments also exhibit deposition pat-
terns that are similar to those at tidal inlets, even
though buoyant spreading of the jet influences the deposi-
tion of sediments to some degree. Sedimentary forms
gain particular significance at river deltas since they
contribute to delta building. Computer simulations of
the deposition of suspended sediments at river mouths have
been made by Bonham-Carter and Sutherland (1967); however,
their model is limited since they have assumed the effluent
to be initially separated from the bottom. Experimental
studies by Butakov (1971) and Mikhailov (1972) have shown
that bars similar to the marginal shoals of a tidal inlet
are formed in the vicinity of a river mouth. Wright
and Coleman (1974) also report similar features for
Mississippi River mouths.
1.2 Objective and Scope of the Present Study
While inlet hydrodynamics and the mass transport
of pollutants and sediments has been explained on a quali-
tative basis in the available literature, the quantitative
aspects of such phenomena are virtually nonexistent or
simply inadequate.
The present work is an attempt in analyzing the
phenomena associated with flow separation at tidal inlets.
The approach herein is based on the theory of turbulent
jets; however, the effects of bottom friction and topo-
graphical changes have been included. Diffusion of pollu-
tants and the diffusion-settling of suspended sediments are
also analyzed. Several additional factors such as cross-
currents and distributions of pollutants or sediments in
the ambient waters are accounted for. The results of the
analysis are then used in studying the exchange mechanisms
of a bay-inlet-ocean system.
Some of the conclusions drawn from the analysis
also have a bearing on river mouth processes.
Hydrodynamics of the flow is studied in Chapter 2.
Mass transport of pollutants and suspended sediments is
analyzed in Chapter 3. Chapter 4 addresses itself to the
unsteady development of the flow during the early stages of
the ebb tide. Conclusions reached by the present study
are outlined in Chapter 5.
CHAPTER 2
NEARSHORE HYDRODYNAMICS IN THE VICINITY
OF TIDAL INLETS
In this chapter,the hydrodynamics of the flow issu-
ing from an inlet is studied in the ebbing phase of the
tide. During this phase, the flow is similar to a two-
dimensional turbulent jet. Tidal flows in the neighbor-
hood of inlets are ultimately determined by the ocean tides
and hence have an unsteady character. However, it will be
shown through an order of magnitude analysis that the
unsteadiness is of little importance in the near-field
of the inlet. Therefore, a steady jet is assumed.
The analysis also applies to the near-field flow
caused by a river terminating at a coast, although river
outflows may have a slightly different structure. Fresh
water discharging from a river mouth gradually rises above
the sea water due to its buoyancy. Nevertheless, river
effluents also form jet flows which may in some cases
"touch" the bottom in the near field of the mouth and
therefore the present theory may still be applicable.
The analysis in this chapter covers the hydro-
dynamic aspects of the jet flows in shallow water. The
depth averaged equations of motion are derived and later
solved analytically in some cases. The effects of
bottom friction, variable depth and cross-currents are
included in the theory.
2.1 Shallow Water Equations
Consider the long period oscillatory motions in
a fluid medium of limited vertical extent. In such cases,
vertical motions are usually much less significant than
the horizontal motions and thus the flow can be con-
sidered to be vertically uniform. This assumption implies
a horizontal, two-dimensional flow and a hydrostatic
pressure distribution. In this section, the shallow
water equations are derived, based on this approximation.
The continuity equation states the conservation
of mass within the flow domain and for an incompressible
flow (Schlichting, 1968, p. 44) it is
+ + 0 (2.1)
3x ay az
where u, v, and w are components of the velocity vector
in the cartesian coordinate system (x, y, z).
12
The conservation of momentum is stated by the
Navier-Stokes equations of motion (Schlichting, 1968,
p. 61). Rearranging the convective terms by making use
of equation (2.1), the Navier-Stokes equations can alterna-
tively be written as
u + +u2 + vu + w+1 X I + + ] (2.2a)
at ax @y z p ax p x 8y [z
+v .uv v2 8wv 1 [ + + yz] (2.2b)
-t ax ay 2t p ay p x ay 2z
+w + auw + avw + w2 -g- 1p + 1 zx z + ] (2.2c)
at x y az p a-z p ax +y az
where t denotes the time, p the pressure, p the density,
g the gravitational acceleration pointed in the negative
z-direction. Here TaB is the deviatoric stress tensor
(Schlichting, 1968, p. 55) with symmetrical properties
where the subscript B denotes the direction of a stress
component and a the direction of the normal to its plane.
Depth and time averaging of equations (2.1) and (2.2)
yields the corresponding shallow water equations.
13
Vertical averaging. Consider the fluid domain
sketched in figure 2.1, where the free surface and the
bottom topography are described by the equations
Z=n(x, y, t) and z = h(x, y) respectively. The velocity
field is described by the components u, v, and w in the
coordinate system (x, y, z) shown in the figure.
x T(x,y,t)
[ {x,y)
h(x, y) H(x, y,t)
7= +h
Figure 2.1 Definition sketch
Since the volume flux through the bottom and
surface boundaries has to be zero, the kinematic free
surface and bottom boundary conditions
w(z=n) = + u(z=n) + v(z=n) (2.3)
at ax ay
w(z=-h) = -u(z=-h) h -v(z=-h) A (2.4)
ax ay
have to be satisfied in addition to the dynamic equations
(2.1) and (2.2).
In the forthcoming derivations, the shorthand
X will be used to denote the vertical average
in
X = Xdz (2.5)
X = -h
of a quantity X, such that HX is the integrated quantity.
Similarly, the deviations of X from the vertically aver-
aged quantity X will be denoted by
X = X X. (2.6)
Vertically integrating equation (2.1) and making
use of Leibniz's rule together with equations (2.3) and
(2.4) yields
+ (Hu) + (Hv) = 0. (2.7)
at ax u y
Before integrating the momentum equations (2.2)
some a priori assumptions have to be made in regard to
the shallow water approximations. Assuming that the
water depth is small compared to the horizontal length
scale (say, the tidal wavelength) but large compared to
the vertical amplitude of the motion, it follows that
<< g (2.8)
15
and therefore the vertical particle accelerations are
negligibly small. Also neglecting the vertical components
of the viscous stresses, the z-momentum equation (2.2c)
reduces to
0 = -g (2.9)
p az
When integrated, the above equation yields the hydrostatic
pressure distribution
p = pg (n-z) + Po (2.10)
where po is the atmospheric pressure of the water surface.
Integrating equations (2.2a,b) vertically by
making use of equations (2.10), (2.3), (2.4) and of
Leibniz's rule one arrives at
aHu a. H2 + aHuv ar 1 aH02 aH09 H 3Txx xy (2
+ + -gH + + (2+11a)
at ax ay ax p bx ax -y P ax +y
aHv a3Huv + Hv2 Ta 1 3Hv9 9aH2 H yx y
W+ .+ = ~ y yx x+y (2.11b)
at x y -gHy p -by ax ay p ax -y
where
Tx Txz (z=-h) (2.12a)
by yz ( h) (212
T T (z=-h) (2.12b)
Tby yz
are the bottom shear stresses. It has been assumed
here that no shear stresses or atmospheric pressure
variations are acting on the free surface.
Time averaging. Since turbulent flows are con-
sidered here, equations (2.7) and (2.11) have to be
averaged over a representative turbulence time scale T in
order to obtain equations describing time mean properties.
For this purpose, the time average of a quantity X will
be denoted by
T
= x t X dt (2.13)
0
and deviations from this time mean property by
X' = X X (2.14)
Some preliminary assumptions will be made before
carrying out the time averaging. In free turbulent flows
(Schlichting, 1968, p. 531), the molecular diffusion terms
are usually much smaller than their turbulent counterparts.
Based on this fact, the laminar shear stress terms for
momentum (last two terms in equations 2.11a,b) can be
omitted. It will also be assumed that the water surface
displacement is small compared to the depth, that is,
n/h << 1. Therefore the total depth H will be replaced
by the still water depth h.
17
After the time averaging, the shallow water
equations (2.7) and (2.11) take the following form:
S+ (h-) + (hV) = 0 (2.15a)
at ax ay
h U x- (hu2) + (huv) = -gh (h 2)_
at ax ay ax p bx ax ay
(hu,2) -- (h -'') (2.15b)
ax ay
-~
h + (hv) + (h2) = -gh 1 p by (h) (hz)
at ax ay ay p byax ay
(hu'V )- ) (hV ) (2.15c)
ax ay
It may be noted here that similar forms of the shallow
water equations (2.15) have also been derived elsewhere
(for example see Wang and Connor, 1975).
The bottom shear stresses in equations (2.15) are
usually expressed in terms of empirical formulae. Follow-
ing Dronkers (1964), the average bottom stresses in shallow
water are given by
1 f (2.16)
p bx u q
1 f
P by v q (2.17)
p Tby -
where
q = (u' + V2) (2.18)
represents the magnitude of the horizontal velocity and
f is the Darcy-Weisbach friction coefficient. The friction
factor f is in general a function of the Reynolds number
and the ratio of the bottom roughness to the hydraulic radius
except in fully rough regimes, where the dependence on
Reynolds number vanishes. The values of f for free
surface flows can be calculated through a procedure
analogous to that in Daily and Harleman (1966) used for
pipe flows.
2.2 Turbulent Jets in Shallow Water
2.2.1 Simplified Equations for Jets
For turbulent jet flows in shallow water (sketched
in Figure 2.2), the equations derived in the last section
can be simplified considerably. Jets, in being confined
to coherent flow regions, are similar to boundary layers
in which the lateral velocity and length scales are much
smaller than the longitudinal ones. In addition, equa-
tions (2.15) can be simplified further by considering
shallow water motions in the near field of the inlet,
19
following an order of magnitude analysis. For this pur-
pose the scales of the motion are selected as in Table 2.1.
Table 2.1
Variable Scale Variable Scale
t T1 u uo
x L v Su
y 6Lo 0 pu0
h ho v 61uo
n ao u' euO
a0 eu 0
v' EU0
The parameters T L h a u correspond re-
spectively to the time, length, surface displacement, and
velocity scales that characterize the flow. The coordinates
are selected such that the x-axis coincides with the jet
axis. The scale UL is a measure of the boundary layer
thickness which corresponds in this case to the width of
the jet. It is therefore obvious from equation (2.15a)
that the small number 6 has to be the ratio of the veloci-
ties v and u. In later sections it will become appar-
ent that the lateral velocity v = O(au ), where the
20
entrainment coefficient a is of order a = 0(10-2).
(Here, as well as in later usage, the notation x = 0(10")
will imply 5 x 10n-1 < x < 5 x 10n.) Therefore, the
order of magnitude for 6 is estimated as
6 = = 0(a) = 0(10 2). (2.19)
u
The number p characterizes the deviations of the flow
properties from their vertical means and e characterizes
the ratio of turbulent fluctuations to the time mean
properties. These are small numbers roughly of the orders
S = 0(10-1) (2.20)
S = 0(10-1). (2.21)
After scaling equations (2.15) based on the
parameters in Table 2.1, the following dimensionless
parameters appear:
a L gho fL0
K X = Too A = 2 r 2 8h (2.22)
Since the application here addresses itself to jets issu-
ing from tidal inlets, magnitudes of these parameters
can be estimated at the outset. For tidal periods,
T 0 105 sec., typically. Based on field observations (and
the discussions in Chapter 4) it may be argued that these
tidal jets may extend to offshore distances of around
L 0 10 Km. The tidal velocities usually observed at
inlets are on the order of u 0 Im/sec. Assuming also
h 10 m, a 1 m, f = 10-2 and taking g = 10 m/sec
it is obvious that 6, C, e, K, and X constitute a set of
small numbers < 0(10-1), while the numbers A and r may
be 0(1) or larger.
An additional small parameter arises due to the
boundary layer structure of turbulence since the apparent
shear stresses may be one order larger than the apparent
normal stresses, that is,
I2
O(u) (2.23a)
v,2
= 0( ) (2.24b)
U'V'
where p is a small number.
Finally, the relative magnitudes of the terms in
the governing equations (2.15) and the auxiliary expres-
sions (2.16) and (2.17) are shown below each term in the
following:
+ (hu) + (hv) : 0 (2.25a)
ut ax xy
KX 1 1
h u + (hu'i) + (huv) = -gh u q (h) 2 (h0^)
t ax ay 3x 8 ax ay
x 1 1 KA r 2 p2
(hu'2) (h V') (2.25b)
ax ay
E2 i2
h + T (huv)+ ( v (h (ay
6X 6 6 KA Fr 2 6p2
S(hu'v') (h '2) (2.25c)
ax ay
E2 2
Terms which are < 0(10- ) are in general negligible with
respect to the 0(1) or larger terms. However, note that
the ratio E2/6 of two small numbers is not necessarily
small. In fact, by virtue of equations (2.19) and (2.21)
S = 0(1) (2.26)
and therefore the corresponding terms cannot be
neglected.
Since each individual term in equation (2.25c)
is of small order, say 0(6), the pressure gradient term
as a reactional force has to be the same order:
S= 0(6) (2.27)
and by virtue of this result, the pressure gradient term
in equation (2.25b) is even smaller, with an order of
magnitude
KA = 0(62). (2.28)
Again, since fv/u = 0(6), the local magnitude of
the velocity is approximately equal to the longitudinal
velocity,
62
= 1 + 0(62) 1 + 0( ) (2.29)
lul
and therefore q can be replaced by uij in the governing
equations. Thus, neglecting all small terms in equations
(2.25), the simplified equations for shallow water
turbulent jets read as
(hu) +-- (hI ) = 0 (2.30a)
ax 3y
S(h2) + (hu) = -f j (hu'v'). (2.30b)
2.2.2 General Solution for Bottom Frictional
Jets Over Variable Topography
Consider the turbulent jet flow produced by water
ebbing from a tidal inlet into a semi-infinite ocean as
shown in figure 2.2. The water issuing from the inlet
entrains and is mixed with the ambient waters due to the
turbulent exchange of eddies. The flow is retarded by
the frictional resistance of the bottom and the depth
variations also affect the expansion of the jet. The
equations of motion (2.30) for this flow were derived in
the last section. Omitting the notations used before for
the vertical and time averaging these can be written as
S(hu2) + (huv) u2 1 F (2.31a)
ax y( 8 (p y xy
' (hu) + (hv) = 0 (2.31b)
ax 3y
(x,y)
.. -- -s
SXxs
ZOFE ZOEF
Z
F"igr2 D n th (xyt)
h(x,y)
Figure 2.2 Definition sketch for shallow water jet flows
where F = ph u'v' (using the old notation) is the
xy
depth integrated turbulent shear stress acting laterally
on the flow. Since the flow is in the ebbing phase,
ujul has been set equal to u2. These equations which
account for the effects of bottom friction and variable
depth can be reduced to the classical two-dimensional
turbulent jet equations (Schlichting, 1968, p. 682) for
a constant, frictionless bottom by simply taking f = 0 and
h = constant.
In obtaining solutions to equations (2.31),
the flow variables of interest are the velocity distribu-
tion u(x,y) and the characteristic width b(x) in which
the motion is confined (see figure 2.2). Solutions to
these equations are then possible by assuming the velocity
profiles to be self-similar. The similarity hypothesis
has a firm basis in the case of classical turbulent jets
and wakes as demonstrated by both theory and experiments
(for general information on the subject see, for example,
Abramovich, 1963, and Schlichting, 1968). The self-
similarity of velocity profiles will also be assumed in
the present case although it may be argued that this may
not hold in the case of a frictional bottom and variable
depth. Nevertheless, this assumption may be justified
intuitively if one reconsiders the shallow water approxi-
mations. On this basis it is expected that any possible
vertical gradients in velocity are not as strong to affect
the horizontal velocity profiles. Therefore it may be
assumed that the velocity profiles u(x,y) are self-similar.
This facilitates the integration of the equations of
motion (2.31) across the jet. The velocity u and the
shear force Fy vanish as y b, but there is a lateral
xy
velocity ve due to the entrainment of the surrounding fluid
into the jet. Therefore, allowing only one dimensional
variations in depth: h = h(x) and integrating equations
(2.31) across the jet yields
b b
d (h udy) u2dy (2.32a)
dx -b -b
d (h J u2dy) = 2h v (2.32b)
-b
Various different similarity functions have been
developed based on different sets of hypotheses. Tollmien
(1945) has used Prandtl's mixing length hypothesis whereas
Reichardt (1951) and Goertler (1942) have found a differ-
ent solution based on Prandtl's second hypothesis. Abramo-
vich (1963) has shown that a simpler similarity function
can be used to represent the velocity distribution and has
compared it with experimental data. The functional form
he used was adopted from Schlichting (1930) who initially
derived it for turbulent wakes (see Schlichting, 1968,
p. 691). Later, Stolzenbach and Harleman (1971) have
generalized the same functional form to the core region
of the jet and seem to get reasonable results from their
theory.
The profiles used here will be the ones adopted
by Stolzenbach and Harleman (1971). Taking b(x) and r(x)
to be the jet width and the core width respectively, and
xs to be the length of the core region as shown in
figure 2.2, the velocity profiles are
u = F(y) (2.33)
where the function F(y) is defined as follows: in the
Zone of Flow Establishment (ZOFE), x < xs:
0 ; -y > b
F(y) = (1- -5)2 ; r < |yl < b = J (2.34a)
( o o b r
1 ; 0 < lyl < r
while in the Zone of Established Flow (ZOEF), x > xs:
F(y) = 0 2 C = b (2.34b)
S ; 0 < ly < b. b
The entrainment velocity v can be related to the center-
line velocity uc by
e = auc (2.35)
where a is defined as the entrainment coefficient. The
above statement, which implies a proportionality between
the mean inflow velocity across the edge of the jet and
the local jet centerline velocity was initially introduced
by G. I. Taylor and was used later by Morton et al.
(1956). It has consequently been shown to be a useful
approximation in a variety of applications. The numerical
values of a will be obtained later by comparing the results
with experiments.
Now, using the self-similar velocity profiles
(equations 2.34) and the entrainment assumption (equation
2.35) in equations (2.32) yields the two ordinary differ-
ential equations
d Uf 22 T2
d
(h b u2 TI) uc T (2.36a)
d (h b uc I) = h uc (2.36b)
df xc1c
where
1
T = f (-u-) d
0 c
12 1
0
(2.37a)
(2.37b)
2
(U) d .
uC
The inlet is modeled by a rectangular slot of width 2bo
and depth ho as in figure 2.2. The initial velocity of the
jet at the inlet is u Using these as the appropriate
scales, equations (2.36) can be written in the normalized
form:
d (2 H B U2) = T2 B U2 (2.38a)
d E 2
(I1 H B U)
= a H U
where E= x/bo and
fb
0
8ho
h
b=
0
B(e) = -
u
U() uc
o
(2.38b)
(2.39a)
(2.39b)
(2.39c)
(2.39d)
The core width r(x) will also be normalized as
R() r
o
Noting by virtue of equations (2.33) and (2.34) that
= -R/B R R
o 1-R/B o- 0 o +
equations (2.37) can be written as
R R+
Il 1(1-
R R
12 + 12(1- )
where
I1 = I
0
12 = f
d
(1C1.5) dC
are numerical constants. Therefore it is seen from equa-
tions (2.41) that the quantities 1T and T2 may vary with
distance only within the ZOFE since, as E Es = xs/b ,
R 0. For E > s, the core width R = 0, and therefore
(2.39e)
(2.40)
(2.41a)
(2.41b)
= 0.450
(2.42a)
= 0.316
(2.42b)
Il and 12 take on the constant values I, and 12 by virtue
of equations (2.41). The integration of equations (2.38)
for the unknowns R and B in the ZOFE and U and B in the
ZOEF is quite straightforward using the initial conditions
H(O) = 1, R(O) = 1, B(O) = 1, U(O) = 1. (2.43)
Thus, equation (2.38a) can be integrated at once to
yield
12 H B U2 = exp[- p J H1T d '] J(() 2.44)
0
in which the initial conditions (2.43) have been invoked
and it has also been noted that 12B 1 as -* 0.
Equation (2.38b) has to be integrated separately
in the two zones of the jet. In the ZOFE, the centerline
velocity u = u by virtue of equation (2.33) and the
initial conditions (2.43). Therefore, by setting U = 1
in the ZOFE, equation (2.38b) is integrated to yield
I1 H B = a H(E') dE' + 1 = G() (2.45)
o
where it has been noted that I1B + 0 as E + 0. Therefore
the solutions for the two unknowns R and B in the ZOFE fol-
low from equations (2.44) and (2.45) by setting U = 1 in
the former equation and utilizing equations (2.41):
llJ( ) 12G(5)
R = G ) (2.46a)
B = -(2.46b)
(11-12) H(E)
To obtain solutions for the unknowns U and B in
the ZOEF, equation (2.44) is first utilized to put equa-
tion (2.38b) into the form
al HJ
d 2J J (2.47)
d-C Il T U
where it has been noted that I = I, and 12 = 12 by
virtue of equations (2.41) since R( > Es) = 0. Then,
equation (2.47) can readily be integrated from x = xs/b0
to E > E,, yielding
1 2 ( 2 I H(') J(S') d' + 2 (s) (2.48)
2 s
The constant of integration y (Ts) is determined by
requiring continuity of the centerline velocity U at = (s.
Thus by evaluating equation (2.48) at E = Es it is seen
that
1 1 J2(5s) 1
T (2 ( = J '), (2.49)
s 2 U2( s)
since U(Es) = U( < s) = 1. Substituting equation (2.49)
into (2.48) there follows
U = J(- ) (2.50a)
[J2( s) + S()]r2
and substituting this result back into equation (2.44) and
taking 12 = 12 gives
[J2(s) + S(0)]
B = s(2.50b)
12 H(E) J(S)
where
2ctI2 /f
S( ) I= 2 H((') J(E') d '. (2.51)
1 s
This completes the general similarity solutions to
equations (2.31), with equations (2.46) and (2.50)
constituting the respective solutions in the ZOFE and the
ZOEF. The point Es = xs/bo where the ZOFE terminates is
found from the roots of the equation R(Es) = 0 with R given
by equation (2.46a).
2.2.3 Jets on a Frictionless Bottom
of Constant Depth
It has been remarked earlier that the governing
equations (2.31) can be reduced to the classical jet equa-
tions in the case of a frictionless bottom with constant
depth. Similarly the solutions in equations (2.46) and
(2.50) can be reduced to those describing classical two-
dimensional jets by substituting h = const and f = 0. Thus
in the ZOFE, the simplified solutions are
al2
R = 1 2 (2.52a)
I 1-I2
2-1
B = 1 + 1a 2 (2.52b)
11 2
whereas in the ZOEF they take the following form:
2al -2
U = [1 + 2-2 (-s)] (2.53a)
2aI
1 2al?
B = [1 + (2-~ )] (2.53b)
12 I1
Thus, the dependence on E of the jet properties observed
from the above solutions are identical to those obtained
from classical turbulent jet solutions; i.e., the core
and the jet widths are shown to obey linear variations,
whereas the centerline velocity decays as (" (see for
example Daily and Harleman, 1966, p. 415).
The numerical value of the entrainment coefficient
c, which has been left unknown earlier, can now be obtained
by comparing the simplified solutions in equations (2.52)
and (2.53) with the available experimental data and/or
analytical solutions for classical jets. For example, if
one had empirical knowledge of the growth rate e of the
shear layer thickness B-R (in the ZOFE) and the growth
rate 2 of the jet width B (in the ZOEF); by virtue of
equations (2.52) and (2.53b), the relations
d(B-R) (2.54a)
in the ZOFE and
dB 2a
dB = 2 = 2 (2.54b)
in the ZOEF could be used for obtaining the numerical
values of a. Stolzenbach and Harleman (1971) have also
used this approach in obtaining their entrainment coeffi-
cient and have arrived at the same results as in equations
(2.54a) and (2.54b). For the growth rates they have
assumed l = E2 = 0.22. Abramovich (1963) has observed
different growth rates in the two regions of the jet and
reports them as el = 0.27 and E2 = 0.22. By using results
due to Abramovich, it follows from equations (2.54) that
a = a1 = 0.036 (2.55a)
in the ZOFE and
a = a2 = 0.050 (2.55b)
in the ZOEF.
Next, the simplified solutions in equations (2.52)
and (2.53) are compared with the available information on
two-dimensional turbulent jets. The predictions of the
centerline velocity and the jet width shown in figures
2.3 and 2.4 indicate a fair agreement with the previous
theories and experimental results. Thus, it can be said
that the present theory predicts jet properties accurately
when the bottom is frictionless and has constant depth.
Other possible comparisons include jet properties such
as volume and energy flux variations with offshore distance.
The normalized forms of the volume flux Q/Q0 and the energy
flux E/Eo as given by Albertson et al. (1950) are calculated
0.8- a\
U() 0.6
0.4-
0.2
0
Figure 2.3
200
Variation of centerline velocity for jets on a frictionless, constant
bottom (f = 0, h = constant)
eU
0 Dota by F6rthmonn (after Abramovich)
Abramovich
Present Solution
8() 10-
0 10 20 30 40 50
Figure 2.4 Variation of half-width for jets on a frictionless, constant bottom
(f = 0, h = constant)
from the present solutions as
b
u dy
Q o
Q u= BUIl
Qo u b
o o
b
f (u2 + v2) u dy
E o
0 u3b
0 0
S(1 + 2) B U3 T6.
Here 11 is the previously calculated function given by
equation (2.41a) in the ZOFE and 16 is a similar function
which can be expressed as
R R
6 = B+ (1 )
6 B 6 B
(2.57)
Here
16 =
(1- 1"s) de = 0.251
(2.58)
is a numerical constant corresponding also to the value of
the function T6 in the ZOEF. The fluxes of volume and
energy calculated from equations (2.56a) and (2.56b) are
compare: in figures 2.5(a) and 2.5(b) with the experimental
and theretical results due to Albertson et al. (1950).
(2.56a)
(2.56b)
41
1.0
& Experiments, Albertson, et. al.
--- Theory, Albertson, et. al.
5.- Present Solution
Q .01
Qo
1.0
2 5 10 20 50 100 300
(a)
1.0 --
0.5-
E
E 0 Experiments, Albertson,et.oal.
---- Theory, Albertson, et. al.
2 -Present Solution
0.2 -
0.1 I
2 5 10 20 50 100 300
S(b)
Figure 2.5 (a) Volume flux and (b) energy flux variations
for jets on a frictionless, constant bottom
(f = 0, h = constant)
The accuracy of the present solution is also demonstrated
here.
2.2.4 Bottom Frictional Jets
with Constant Depth
In this section bottom frictional effects are
examined for a constant depth. Therefore, taking H = 1
in equations (2.46) and (2.50), these solutions simplify
to more specific forms. Thus, in the ZOFE, E < (s:
I1 e 12 (1 + cE)
R = -
I1 12
-1J
(-12 )(1+ac ) (1-11) e
B =
I 2
1112
and in the ZOEF, 5 > (s:
U = e
S-2S + 2al2-
e + e ee
_1
B = [-2e5 + 2
12L I1
(2.59a)
(2.59b)
(2.60a)
(2.60b)
- e
-P)]
The above solutions reveal the exponential
dependence of the centerline velocity and the jet width on
distance. In fact, the jet width grows and the centerline
velocity decays exponentially for sufficiently large
distances away from the inlet as opposed to the linear
growth of the classical two-dimensional turbulent jet
in which the centerline velocity would decay as E -,
hence much slower. In the presence of bottom friction,
a linear behavior is found only near the inlet and in the
ZOFE since, .s E + 0, B 1+[(1-I2) + (1-1 )] /( 2)
Nevertheless, even in this situation the rate of growth
of the jet width is altered from its frictionless value
(1-I2)a/(Ii-12) because i = 0(a), typically.
The features of the bottom-frictional jet are
displayed in figure 2.6 where p = fb /8ho is kept as a
parameter. The parameter p combines a number of effects,
such that when the friction is large or the inlet width
is large compared to depth, the jet merely explodes as
it faces the bottom resistance and loses its momentum.
Since a value of i = 1/20 may be considered as typical, it
may be inferred from figure 2.6 that the bo,:tom friction
plays an important role in the jet dynamics provided that
the variations in depth are sufficiently small. One also
notes in figure 2.6 that the important features of the jet
8(() 20
.= 0.05
10
0
0 10 20 30 40 50
1.0
U(b)
0.6-
0.2 -=0.10
0 10 20 30 40 50
(b)
Figure 2.6 Bottom frictional jet over constant bottom
(a) jet half-width, (b) centerline velocity
characteristics can be inferred without considering the
ZOFE whose extent is suppressed by the frictional re-
sistance. This will be done in the later analysis.
A possible field evidence for the explosive growth
of the bottom-frictional jet may be seen in an aerial
photograph of Redfish Pass, Florida, shown in figure 2.7.
The ebb currents which are superposed onto the photograph
indicate a rapid widening of the jet. This is perhaps
further strengthened by the sedimentary features.
The fast growth rate of the bottom-frictional jet
has been demonstrated in a number of other studies. In
investigating the structure of a jet in a rotating system,
Gadgil (1971) has examined the effect of laminar Ekman
boundary layers which develop at the top and bottom solid
boundaries of a two-dimensional laminar jet. It has been
found that when the rotation effects are sufficiently
large, the jet spreads out much faster than the classical
jet in the same manner as in the present solution. It
has also been shown that the jet, having lost all of its
momentum against friction, begins to eject fluid into the
surrounding. In experiments carried out for jets in
rotating basins, Savage and Sobey (1975) have found that
jets in shallow waters exhibit faster spreading rates than
jets in deep waters. In a study concerning exchange
Figure 2.7 Ebbing flow at Redfish Pass, Florida
(University of Florida, Coastal Engineering
Archives)
characteristics of inlets, Taylor and Dean (1974) have
also found an exponentially growing jet which expands as
e p/2 whereas the growth in the present theory is as e"
hence, faster. This is because they neglect the lateral
entrainment and assume the velocity to be uniform across
the jet. Borichansky and Mikhailov (1966) have analyzed
flows near river mouths in a tideless sea. Their theory
includes both the bottom friction and the variations in
depth, however, in accounting for the lateral friction on
the jet they have used a friction factor analogous to the
bottom friction coefficient, and have also neglected the
lateral entrainment into the jet. It is interesting to
note that due to the bottom friction Borichansky and Mikhai-
lov also find an exponentially decaying mean velocity and
an exponential increase of the jet width, but, they are
led to erroneous results since they have assumed the
integral momentum of the jet to be conserved as opposed to
the present theory.
2.2.5 Bottom Frictional Jets on a
Linearly Varying Topography
The effects of depth variations are investigated
by taking a linearly varying depth:
h(x) = h + mx (2.61)
where m is the slope. In normalized form this is
H(E) = 1 + v~ (2,62)
where v = mb /h is the slope parameter. Using equation
(2.62), the general solutions in equations (2.46) and
(2.50) can be expressed as
I H v I2(1+ca+ 0" 2)
R =1 12) H (2.63a)
1 1
(1-I2)(1+at + 2) (1-11) H v
B = (I1 12) H (2.63b)
in the ZOFE (5 < ) and
H-lVy
U = 2 l (2.64a)
[H-2p/+ 2 (H -1 H )]
s Il(2v-') 2(Hp H2
S2a2 a2-P/ 2-U/V
H -2 + 2 (H H2s )
s I (2v--) s
B = -(2.64b)
12 H1Iv
in the ZOEF ( > Es), where Hs is the depth when E = Cs"
In the solutions (2.64), the effect of variations
in depth, the bottom friction and the lateral entrainment
onthe dynamics of the jet are expressed, respectively,
by v, p, and a. The roles of parameters p and a have al-
ready been considered in the last section by setting
v = 0. It is therefore appropriate to first inquire in
the effect of the variable bathymetry by neglecting fric-
tion and the lateral entrainment. Taking p = 0, a = 0 it
follows from equations (2.64) that
B = 1/12H, U = 1 (2.65)
that is, with increasing depth, the jet contracts and the
centerline velocity remains a constant. This agrees with
Arthur's (1962) results. The contraction of the jet with
increasing depth is due to mass conservation because the
velocity U does not vary with distance.
Next, taking a f 0 but keeping 0 = 0, that is,
allowing for lateral entrainment, the jet width in equa-
tion (2.64b) simplifies to
B = 1 1 2 (H2 H ) (2.66)
12H 1Is
It can be inferred from equation (2.66) that near S = s
(i.e., H = Hs) the effect due to the depth change will
dominate while, as E + -, B (aC/I1 + a/vll) so that
the jet growth is linear as in the case of the classical
jet; however, it is worth remarking that the rate of growth
is only half that given by equation (2.53). Consequently,
the jet will tend to contract first and then expand. Note
that this result will hold for v > 0. For v < 0 (decreas-
ing depth), the bottom will intersect the surface at a
distance E = 1/IJv and near this region the jet will grow
as B u 1/H. Thus, for negative slopes the jet expands both
due to decrease in depth and the lateral entrainment.
On the other hand, if the entrainment can be
neglected (a = 0) but not the bottom friction, then it
follows from equation (2.64b) that
B = Hs2/v / IH v (2.67)
It is now seen that for positive slopes (v > 0) the jet
will contract if v > p and expand if v < p. In the former
situation the depth effect, which contracts the jet, is
suppressing the jet growth resulting from the bottom fric-
tion while in the latter case the bottom friction effect
dominates. If p = v the two effects are in balance so
the width of the jet remains the same with distance off-
shore. Finally, for v < 0 the jet expands since the bottom
friction effect is enhanced by the decrease in depth.
It is clear from the preceding discussions of
the various limiting cases that, under general circum-
stances, the effect of increasing depth is to counteract
the effects of the bottom friction and the lateral entrain-
ment. For decreasing depth with distance offshore, the
three effects will be acting the same way to lead to the
expansion of the jet. These points are further demon-
strated in figure 2.8 keeping v as a parameter and fixing
the values of i and a. The values of a are used as given
by equations (2.55). It is seen that the effect of bottom
topographical variations on jet expansion are overwhelming
because, even for slopes as small as that corresponding
to v = 0.01, the exponentially growing extent of the
bottom-frictional jet is altered significantly. The
velocity, however, does not exhibit any significant vari-
ations with v.
1.0
0.8-
0.6-
V=0
0.4- V = 0.01
S= 0.05= A
0.2
L :=-0o03
O 10 20 30 40 50
(b)
Figure 2.8 Jet characteristics over a linearly varying
bottom, p = 0.05 (a) jet half-width
(b) centerline velocity (Dotted line is an
assymptote for v = -0.03)
It is worth noting that for p =vequations
(2.64) lead to
U = H -2+ (H Hs) (2.68a)
B H 2 + 2 (H Hs) (2.68b)
12 s I I
that is, the bottom-friction effect is in balance with
the effect of the depth increase so that the jet expands
linearly with distance as a result of the lateral entrain-
ment alone as in the case of the classical jet. In fact,
the jet width given by equation (2.68b) is identical to
that given by equation (2.53b) apart from a slight shift
in the coordinate E. The velocity variations, however,
differ from the classical jet because the decay here is
as H"3/2 hence, faster.
Finally, it can be seen from equation (2.64b) that
there is an apparent singularity in jet expansion for
2v = p. Upon carrying out the limit as p + 2v it can
be shown that
4 4 2al2 H
B H + 2c i n (2.69)
that is, the jet grows linearly near the inlet (H HS)
but the growth is logarithmic as (H 9n H) at far distances
away from the inlet.
In conclusion, it can be stated that depending on
the relative orders of magnitude of the parameters i and v
and a, the behavior of the jet is changed dramatically
from an exponential to a linear to logarithmic behavior.
Experiments. Although experimental data on jet
properties are highly desirable for comparing with the
present solutions, undertaking such detailed measurements
are not attempted within the context of the present work.
Therefore measurements of only a preliminary nature were
conducted to complement the an lytical results. The model
used in the experiments consisted of an inlet constructed
on a sloping bottom. The general layout of the model is
shown in figure 2.9. The inc ined bottom pre-existed in
the modal basin and had a slcje of m ~ 0.025. The inlet
half-width and depth were b = 10 cm and h = 5 cm, re-
spectively. Thus, assuming a friction factor of f = 0.02
yields p ~ 0.05 and v ~ 0.05 for the values of the friction
and the slope parameters. This assumption for the friction
coefficient is not unreasonable when considering a rough-
ness height of Ks = 10-2 cm (for concrete bottom) and
assuming that the flow is in the fully rough regime. The
value of f = 0.02 can then be calculated from Moody dia-
grams that are available, for example, in Daily and
Harleman (1966).
The measurements of the jet centerline velocity
were done as follows: A jet flow is formed on the ocean
side of the inlet model by discharging water on one side
of the basin and withdrawing the same amount from the other
end. A steady flow was assured by adjusting the discharge
rates and letting the water levels reach constant values
at the points Hi and H2 (see figure 2.9) where they were
measured by point gauges. Then the velocity was measured
along the centerline A-A' (figure 2.9). The velocities
were obtained with a Type 265 Kent Miniflometer kit that
had previously been used and documented by Jones (1975).
Measurements of the centerline velocity obtained
in five individual experiments demonstrate a fair agree-
ment with the analytical solution as shown in figure 2.10.
The importance of the bottom friction is also displayed
in figure 2.10 since the analytical solution for a fric-
tionless bottom (p = 0) differs significantly from the
observed variation of the centerline velocity.
OUTFLOW OUTFLOW OUTFLOW
Figure 2.9 Laboratory layout of the bay-inlet-ocean model
1.0.8 ----
0.8 o 8,^ o "- 1
U(C) 0.6 -
04 -0
Analytical Solutions
02 -- v = 0.05, p= 0.05 8
.Z -- 0.05, .= 0
0 10 20
Figure 2.10 Centerline velocity for a jet on linearly
varying bottom topography
2.2.6 Jets on Arbitrary
Bottom Topography
The general solutions obtained in section 2.2.2
are valid for arbitrary one-dimensional depth variations
H = H(). Therefore, only a minimal effort has to be
made, i.e., the solutions have to be evaluated by numeri-
cal quadrature, when the depth varies in an arbitrary manner.
Such example calculations are performed in this section.
An example: Now, consider an example where the
depth changes are modeled by
H = 0.826 {1 + 0.02 + 4.0 [ 1 tan-1 0.2 (&-30) + 0.5]} (2.70)
as shown in figure 2.11a. The variation in equation
(2.70) is such that the bottom slope changes smoothly from
a mild to a steeper one and then back to the same mild
slope; with increasing distance E. The solutions obtained
from equations (2.46) and (2.50) are shown in figures 2.11b
and 2.11c. The jet expands initially, stays of constant
width over the steep section and starts expanding when
the slope becomes mild again. Over the steep slope the
jet expansion due to the bottom friction is counter-
balanced by the contraction due to the increase in depth.
The velocity variation does not demonstrate significant
changes.
0 10 20 30 40 50 60 70
I0
S-------- (a)
(a)
20
B(C) 10
O
(
1.0
0.8
U(C)0.6
0.4
0.2
0.0
Figure 2.11
An example of arbitrary depth variations,
i = 0.05;(a) depth variations, (b) jet half-
width, (c) centerline velocity
Calculations for a prototype inlet: In this
section the present results are compared with what can be
obtained from the aerial photographs of an actual inlet.
A series of aerial photographs of Jupiter Inlet,
Florida, taken on a relatively calm day in June 1973 are
unique in showing the jet boundary clearly (figures
2.12ab). The bathymetric map of the inlet's vicinity is
shown in figure 2.13. A three-dimensional view of the bottom
topography is provided in figure 2.14. Using figures 2.13
or 2.14, it can be inferred from the aerial photographs
that the jet expands in passing over the shoals and con-
tracts afterwards as the depth increases (unfortunately,
the view of the section of the jet beyond the region where
its width is varying slowly was not photographed).
A sample calculation is made by finding an average
one-dimensional bottom variation shown on the top of
figure 2.15. In the calculation, the inlet half width
and the depth are taken to be bo = 50 m and h = 3 m,
respectively. The friction coefficient f was set equal
to 0.02 implying p = 0.04. The variation of the jet width
calculated from the theory is compared in figure 2.15 with
its actual values estimated from the aerial photographs
(see Appendix A). In view of the uncertainties, the
result seems to be satisfactory.
Figure 2.12(a)(b) Jet development at Jupiter Inlet,
Florida (1973 photographs, Ziya
Ceylanli)
.1?C
-a
:-Y-~.-.a~-C~
..:.
i-
kc~r. c
~L~i~-z~j
;:s~o-~2
:-~3
t--
Figure 2.13 Bathymetric map of Jupiter Inlet and vicinity
(depth in meters based on 1967 data)
----- -C-, -__ ___-. -
c ~--- _2m. 200m.
00m.
.< --- _____..- _
Figure 2.14 Three dimensional plot of the bottom topography near Jupiter Inlet
(1967) C
-- -
H(e)
B(C)
08
0.6
o ----
0
Figure 2.15
Comparison with the jet characteristics at
Jupiter Inlet; (a) depth variations,
(b) jet half-width, (c) centerline velocity
-----1975 Data
-.-1967 Data
Used in the Calculations
I I
2.3 Curved Jets in the Presence
of Cross-Currents
2.3.1 Equations for Deflected Jets
Jets in cross-currents will be deflected sideways
in reaction to the forces exerted on them. These cross-
currents may have originated in the receiving water body
due to a multiplicity of reasons such as wind stresses
acting on the water surface, waves breaking at an angle
with the shoreline or alongshore components of tidal cur-
rents. In the last section equations (2.30) were derived
for shallow water jets; however, these did not account for
the jet curvature due to cross-currents. Since gently
curving jets are most suitably described in locally cylin-
drical coordinates, a change of coordinates is performed
in this section.
In the last section, equations (2.25) were simpli-
fied by omitting terms pertaining to the unsteadiness,
pressure gradients, dispersion and normal components of
the turbulent diffusion. The lateral component of the
momentum equation was also dropped. However, in deflected
jets not all of these terms can be neglected. Pressure
gradient terms must be retained since the deflection of
the jet materializes in response to pressure forces
exerted by the cross-current. In addition, since the
Jet is curving, both components of the momentum equation
and the turbulent diffusion terms will be kept. However,
the unsteady terms and dispersion terms may be omitted at
the outset, since they contribute nothing. Furthermore,
since the dispersion terms are neglected, the overbars
denoting vertical averages may also be dropped. Thus,
equations (2.25) can be written as
V (ho) = 0 (2.71a)
(. V)h = ghVi | ('.V)h (2.71b)
where u = (u,v) and u' = u',v') are vectors, representing
respectively the mean and the fluctuating components of
velocity and V =i + j denotes the two-dimensional
ax dy
gradient operator.
Y
ee
rer
e r
Figure 2.16 Polar coordinates
Now consider the system of polar coordinates
shown in figure 2.16. Equation (2.71a) can be expressed
as
_htr hQr aha-
r+ + 1 (2.72)
ar r r 38
where r and u0 are the velocity components parallel to
the unit vectors er and e respectively (see figure 2.16)
and r and e the new coordinates.
For writing equation (2.71b) in polar coordinates,
first note that for any arbitrary vector a = (ar,a ) it
can be shown that
aha ha a a )ha
(I.v)ht = (a, + er e aI ,
r r r r 0 ae
2
aha, a, aha ha
+ (ar ar +- r r ) er (2.73)
Thus by virtue of equation (2.73) the components of the
momentum equation in polar coordinates are
68
ahU hU ahU B ahu' hu'u' u;' hu'
S + + fu q -u (2.74a)
r ar r r ae 8 r ar r r e
ah r hr 6 fhu' r 6
3hu U 3hiU hUh2 hu' u; hu'
"r + r -u + (2.74b)
r 3r r ae r Urq r ar r 36 r
The above equations may then be put into a more compact
form by making use of the continuity equations for the
mean and fluctuating velocity components. The continuity
equation for fluctuations may be obtained by subtracting
equation (2.15a) from (2.7). Thus, by neglecting the
unsteady terms and converting into polar coordinates,
ahu' hu' ahu
rr r + r H =0. (2.75)
Next, equations (2.72) and (2.75) are multiplied by bu and
u' respectively, time averaged, and then added to equation
(2.74a). The same is done for the radial component of
the momentum equation but this time equations (2.72) and
(2.75) are multiplied by Ur and ur respectively, time
averaged, and added to equation (2.74b). Thus,
69
ahWu 3h uD 2hU U ahu ;2
1 eOr Or _gh f~~ 1 h
+ + uf- 1q
r -9 3r r r 0 8 uq r g9
ahu'ur' 2hu;u'
Sr (2.76a)
ar r
DhU U 3hu2 hU2 hU2 hu'
1 rO r + r-gh fT 1 rU
r 8 ar r r ar 8 Ur r Be
Shu 2 h'2 hu,2
-+ r (2.76b)
ar r r
are the final forms of the momentum equations (2.74).
2.3.2 Flows in the Presence
of Cross-Currents
The equations derived in section 2.3.1 can now be
applied to jets bending in response to cross-currents.
The cross-currents are assumed to be constant in the
alongshore direction but variations in the offshore direc-
tion are allowed. The velocity profiles are modified
locally as shown in figure 2.17 due to the co-flowing
Y*
-u -(x)
;Vy(
Figure 2.17 Definition sketch for jets in cross-currents
p~h4-
E:: -,b~, *T
71
component of the cross-current velocity. The following
additional assumptions are made:
1. The jet bending is assumed to be gradual,
such that
u, cos 0
a << 1 (2.77)
Uc
uc
2. It is assumed that the velocity profiles and
other jet properties are symmetrical with respect to the
jet centerline.
3. The entrainment by the jet is assumed suf-
ficiently small with
v sin e
e<< 1 (2.78)
U
a
such that the entrained flow does not modify the cross-
current in a significant manner. These assumptions are
tantamount to the assertion that the interactions between
the jet and the cross-flow are weak. In other words, the
approach used herein is not applicable to strongly bending
or attached jets. For a gradually bending jet (see
Stolzenbach and Harleman, 1971) one can define a local
cylindrical coordinate system
dx, = r dO
dy. = dr
(2.79)
(2.80)
(see figure 2.17), where the coordinates x,, y, are
attached to the centerline of the jet. After setting
= uu ur u' = v' (2.81)
r 6 r
making use of equations (2.79) and (2.80) and neglecting
small terms, equations (2.72) and (2.77) can be written
as
ahO + ah ( d
S h .-L hV = 0
ax, vy, dx
(2.82a)
9h U 2 MO 2 9 do ) U2 hu'v' d
+2 + 2 (d+) h2 = d- hu'v +2 ( ) hu'v' (2.82b)
ax*, @y dxy 8 ay, dx,
hN +ah02 deO dO aU f ahu'v'
+ + ( h) h ( ) h2 -gh uv -
ax* ay Tx, d, ay ax
do de
hu'2 + ( ) hv'2 (2.82c)
73
These equations can now be integrated across the jet
cross-section assuming that the velocity profiles are
self-similar. Furthermore, a potential flow is assumed
external to the jet.
The entrainment velocity Ve is pr-
portional to the centerline velocity Uc as before:
Se uc
(2.83)
The pressure force which deflects the jet can be calculated
from
+b
-b
-gh
Dy*
+b +b
dy = -gh b = (-h )b
\-b = _-b
(2.84)
where p = pgn is the dynamic pressure. Taking a cross
section as shown in figure 2.18 and writing the Bernoulli
v -(** +Co)
Ua,sine -^Ve
---*--
v(y,---- -4
:-Z. sine +oe
7/ 7 7-,,
Figure 2.18 Cross-sectional pattern of flow in the y,-z plane
equation between the jet boundaries y* = +b and points
away from the jet at y. + + -one obtains:
+) + 2 +2) ( v2) (2.85a)
S=y+b 2 '3+ b y,4+
( +(2 u2) = v2) (2.85b)
= =-b y= r- b y*+- 0
Noting that
V(y* = +b) = -
V(y* = -b) = + Ve
u(y* = +b) = (y* = -b) = Ua cose (2.86)
v(y.++~) = a sine e
v(y.-_) = Ua sine + e
and substituting equations (2.85) into equation (2.84)
yields
f+b -gh dy = 2 Ve Ua sine (2.87)
-be a*
Thus, with the substitution of the pressure force
term in equation (2.87), the integrated equations (2.82)
read:
b
d- (h f U dy*) = a h
o
d (h f
dchf
_R* 0
do
dx*
b b
U2dy*) = f 2dy, + ah Uc Ua cose
0
b
h I L2dy*) = ah UcUa sine
o
(2.88a)
(2.88b)
(2.88c)
Now, making use of the notations for I1 and 12 in equa-
tions (2.37) and dropping the unnecessary time averaging
notation these can be written as
d (h b u, T ) = h u
dx, c 1 c
d f 2 2
-dxd (h b Uc T2) = u hu uc cose b uc
dr, c 2 ac8 c2
de
dx*
(2.89a)
(2.89b)
(2.89c)
(hbuc T2) = h ua u sine
S2u a c
or in the normalized form as
d H B U) = HU (2.90a)
d (12 H B U2) = H UA U cose pl2 B U2 (2.90b)
do
I2 BU = U sin (2.90c)
2 d, "A
where UA = Ua/Uo and x. = x,/b has been defined in addi-
tion to the normalized variables in equations (2.39). Two
extra relations arise by relating the cartesian coordinates
S= x/bo and X = y/bo to the curvilinear coordinate E,:
S= sine (2.90d)
dE*
dX = cose .(2.90e)
dS*
Equations (2.90) constitute five simultaneous ordinary
differential equations to be solved for the sets of five
unknown variables R, B, e, E, X in the ZOFE and U, B, 0,
E, x in the ZOEF. Solutions to equations (2.90) will
be obtained by numerical integration.
The functions T1 and 12 can be obtained by carrying
out the integration in equations (2.37), but this procedure
is slightly different than before since the self-similar
velocity profiles are altered by the co-flowing component
of the cross-current as seen in figure 2.16. Following
Abramovich (1963) and Stolzenbach and Harleman (1971),
the similarity profiles in a co-flowing stream can be
expressed as
u u, cose
u = F(y ) (2.91)
uc ua cos
c a
where the function F(y,) is defined by equations (2.34)
when y is replaced by y*. Thus, it follows from equation
(2.91) that
u
u u
u- a cose [1-F(yj)] + F(y*)
c c
U
= cose [I-F(C)] + F() (2.92)
where r = y,/b is the normalized coordinate. Using equation
(2.93) in equations (2.37) yields expressions for T1 and
12. In the ZOFE, making a change of variables as in equa-
tion (2.40) yields
U2
S= B+ [I+(1-II) cos 6] (1-R) (2.93a)
2 B + [12+2(1-12) coso + (1-21 +2) cs28](l-) (2.93b)
U2
where I1 and 12 are constants defined in equations (2.42).
It is worth noting that in the absence of cross-currents
(UA = 0), equations (2.93) reduce to equations (2.41).
The centerline velocity is constant in the ZOFE, therefore
it follows that U = 1 in equations (2.93). The correspond-
ing expressions in the ZOEF can be obtained carrying out
the integration in equations (2.37) using (2.92) and
(2.34b), or by simply setting R = 0 in equations (2.93).
Thus in the ZOEF
UA
T1 = + (1-1I) cos6 (2.94a)
U U2
T +2(11-12) A1 2
2 + 2(1-1^ U cos9 + (1-21+12) cos e (2.94b)
A set of example calculations obtained by numeri-
cally integrating equations (2.90) is displayed in figure
2.19. The equations were solved on IBM 370 using the IBM
Scientific Subroutine Package routine DHPCG. The
20-
B() 10 -
0
0
X -10
Figure 2.19
Jet characteristics for jets in cross-currents,
p = 0.02; (a) jet half-width, (b) centerline
trajectory
routine solves sets of first order differential equations
with given initial values using Hammings Modified Predictor-
Correc:or Method. The cross-current is assumed to be
constant with offshore distance. With increasing magnitude
of the cross-current, the jet bends more and the jet expan-
sion is greatly reduced. The suppression of the jet expan-
sion is due to the addition of momentum by the co-flowing
component of the ambient current. The jet centerline
velocity however does not seem to be influenced to a great
extent and therefore it is not shown.
Finally, photographical evidence for jet bending
is presented in figures 2.20 and 2.21. Wind driven cur-
rents are probably responsible for the gentle curvature
observed in the first picture. In the second photograph,
a shoreline attached jet is observed. The inclination
with respect to the shoreline of the breaking waves indi-
cates the presence of a strong longshore current which
bends over the jet and attaches it to the beach. The
curved jetty on the north side of the inlet may have also
helped the jet bending. The present theory obviously
does not apply to such recirculating flows.
Figure 2.20 Curved jet at Bakers Haulover, Florida (University of Florida,
Coastal Engineering Archives)
00
Figure 2.21 Strongly curved jet at South Lake Worth Inlet, Florida (University
of Florida, Coastal Engineering Archives)
CHAPTER 3
MASS TRANSPORT IN THE VICINITY
OF TIDAL INLETS
Effluents from tidal inlets and fresh water outflows
from river mouths usually carry materials in solution or sus-
pension. The diffusion of these materials and the mixing
with the ambient water determine their ultimate distribu-
tion. The material may consist of pollutants or suspended
sediments which may originate in the bay or lagoon areas or
may be entrained into the effluent from the adjacent littoral
system. Therefore a generalized formulation will be used
in this chapter in analyzing the turbulent jet diffusion
processes in shallow water based on the hydrodynamic aspects
studied in Chapter 2. The effects of bottom friction, vari-
able depth, cross-currents, as well as ambient concentra-
tions of material in the receiving water are included.
The theoretical analysis is applied to the diffusion of
pollutants and suspended sediments. In the case of sedi-
ments or other pollutants that may settle, the bottom
accumulation or depletion rates are also predicted. Re-
sults of the theory are then used in studying the mixing
and exchange mechanisms of a bay-inlet-ocean system.
3.1 Shallow Water Diffusion Equation
The conservation of mass of the materials in solu-
tion or suspension is expressed by the diffusion equation.
For a homogenous fluid carrying passive constituents which
experience gravitational settling, the diffusion equation
(Nihoul and Adam, 1975; Csanady, 1973, p. 7 and p. 43) is
ac + uc +uc + wc 2 c +2c +2c Ws (
+ -x + D (z +-+ ) + z (3.1)
Tt ax ay @z -5 { 3 W 3 z
where c is the concentration (in arbitrary units) of the
substance, D is the molecular diffusivity, and ws the
settling velocity. The last term in equation (3.1) ex-
presses the rate of change of the vertical flux due to
settling. When the material in suspension consists of
particles with a density slightly greater than that of
the fluid medium, they settle to the bottom with a terminal
velocity ws. One of the main assumptions used in deriving
equation (3.1) is that the concentration (in units of mass/
volume) of the material in solution is small with respect
to the density of the original fluid; hence, the density
of the mixture is nearly uniform throughout the fluid domain
and the small density variations do not contribute to the
momentum balance in equation (2.2).
|
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