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Title:
Notes a short course in coastal engineering, University of Florida, November 19-21, 1975
Series Title:
UFLCOEL
Portion of title:
Short course in coastal engineering, University of Florida, November 19-21, 1975
Creator:
Mehta, A. J ( Ashish Jayant ), 1944-
University of Florida -- Coastal and Oceanographic Engineering Laboratory
Place of Publication:
Gainesville Fla
Publisher:
Coastal and Oceanographic Engineering Laboratory, University of Florida
Publication Date:
Language:
English
Physical Description:
34, 24 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Inlets ( lcsh )
Hydraulic models ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references.
Funding:
This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
Statement of Responsibility:
by A.J. Mehta.

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University of Florida
Holding Location:
University of Florida
Rights Management:
All rights reserved, Board of Trustees of the University of Florida
Resource Identifier:
39032875 ( OCLC )

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Full Text
PART I
HYDRAULICS OF TIDAL INLETS SIMPLE ANALYTIC MODELS FOR THE ENGINEER1 By
A. J. Mehta 2
75/019
I. INTRODUCTION
Inlet-Bay Regime
Inlets are common coastal features around the world. Essentially an
inlet connects, a lagoon.' a bay or an estuary to the ocean (or sea), and the flow through the inlet channel is primarily induced by the tidal rise and fall of water level in the ocean. When speaking of the hydraulics of an inlet, one is interested. mainly in determining the flow through the inlet and the tidal variation in the bay, given the following:
(1 ) Inlet geometry
(2) Bay geometry
(3) Bottom sediment characteristics in the inlet
(4) Fresh water inflow into the bay (and out through the inlet)
(5) Ocean tide characteristics
A combination of all these factors can produce a rather complex situation. For example,a significant fresh water outflow through the inlet may result in two distinct layers of flow a saline bottom layer underneath a fresh water surface layer. Such a situation will give rise to a stronger ebb than flood at 1Notes A Short Course in Coastal Engineering, University of Florida, November 19-21, 1975.
2 Assistant Professor, Coastal and Oceanographic Engineering Laboratory, University of Florida.




the surface, and a stronger flood than ebb at the bottom. Waves entering through the entrance can complicate the events further by generating their own flow oscillations and associated sediment transport.
From an engineering point of view, it is convenient to approach the
problem of inlet hydraulics with reference to the simplest possible system, i.e., an inlet connecting the ocean to a well defined bay, as shown in Fig. 1. The inlet is assumed to have a certain bottom friction, but sedimentary aspects will not be considered in what follows. Salinity induced stratification of the flow is also ignored. Some of the major contributors to the development of the hydraulics of such a system are Brown (1925), O'Brien (1931), Keulegan (1951), Baines (1957), Keulegan (1967), van de Kreeke (1967), Mota Oliveira (1970), Huval and Wintergerst (1972), Dean (1971), O'Brien and Clark (1973), King (1974) and Escoffier (1975).
Returning to Fig. 1, it is noted that most inlets have a well defined
throat section, i.e., a minimum flow cross-sectional area. This is analogous to the Vena Contracta of such flow measuring devices as the venturi meter. O'Brien (1931) and others have shown that the throat section is a characteristic feature of an inlet, and that measurement of the current at the throat can yield information on the size of the throat and also on the relationship between the flow and the ocean and bay tides. Another important aspect of
an inlet is the fact that its cross-section resembles a wide channel, such that the hydraulic radius can be approximated by the depth. This fact is generally lost when one looks at the commonly distorted depiction of the crosssection. This is illustrated by the example shown in Fig. 2.
II. HYDRAULICS OF A SIMPLE INLET-BAY SYSTEM
Problem Formulation
The governing equations for a simple inlet-bay system will be derived

-2-




OCEAN

' TIDAL FLOW

Fig. 1. A Simple Inlet-Bay System.




0 100' 200' 300' 400' 500' 600'

CROSS- SECTIONAL

200' 300' 400'

500' 600'
l i

700' 800' 900' 1000'
l l i t

IEAN DEPTH. ROESL E -R
WET PERIMETER
HYDRAU LC RADIUS CROSS-SECTIONAL AREA MEAN DEPTH
WET PERIMETER
Fig. 2. Comparison between Distorted and Ldistorted
Depictions of Inlet cross-section.

-30'
-40'
-50'-

0 100'

0]

I I

700' 800' 900' 1000'




subject to the following assumptions:
1. The inlet and bay banks are vertical.
2. The range of tide is small compared to the depth of water everywhere.
3. The bay surface remains horizontal at all timesi.e., the tide
is "in phase" across the bay.
4. The meanwater level in the bay equals that in the ocean.
5. The acceleration of the mass of water in the channel is negligible.
6. No fresh water inflow into the bay.
7. No flow stratification due to salinity.
8. Ocean tide is represented by a sine curve.
The consequence of these assumptions is a deep bay connected to the
ocean via an inlet of a short lengthL as shown in Fig. 3. Assumption 3 requires that the longest dimension of the bay be small compared to the time of travel of the tide through the bay.
With reference to the notation of Fig. 3, the Bernoulli equation can be written between the ocean and the bay, S+ + + --+Ah (2-1)
where
no = Ocean tide elevation with respect to meanwater level,
nB = Bay tide elevation with respect to mean water level,
Vo = Ocean current velocity,
VB = Bay current velocity,
a 0 B= Coefficients (greater than unity) which depend on the special
distributions of V and V, respectively,
Ah : Total head loss between the ocean and the bay.
The ocean and bay are large bodies of water so that V0 and VB can be




BAY

2
n2h k- V
en F
h -fL V2 2 4R 2g
5h -k V
3 ex ig

INLET

Fig. 3. Energy Losses Across an Inlet of length L.

OCEAN

WATER MEAN W

......................
.........................................
........... ........................................
...........................................
.............................................
....................................................
................................
..................................... ......
. .. . . . . . . .. .. .. . . . . . . .
. ............ ... ...... ............. .. .....
- I ........... .......... .. ..... .. .........................
......... .. ..... .. ............ ... .......
............................................................................
..................... ........................
.............
........................
..........
.. .................. ......
.................................................................
.. .....................................................................
.........................................................................
.... ... .......... .......... .... .......... ............ .... ... .......
.....................................................................
............................................................
.............................................
. . . .. .. . . .. . . . . . .. . . .. .. .. . . . .. . . . . . . .




considered to be negligible. Therfore Eq. (24) becomes

Ah = no nB (2-2)
As shown in Fig. 3, Ah can be considered to have three contributions, i.e.,
Ah = Ah1 + Ah2 + Ah3 (2-3)
Here,
V2
Ah = ken 2 (2-4)
is the head loss at the flow entrance due to the convergence of the flow streamlines into the inlet, to generate a velocity V in the inlet and ken is the-entrance loss coefficient.
Ah fL V2 (2-5)
2 R- 2g
is the gradual head loss due to bottom friction in the channel of length L. R is the hydraulic radius and f is the Darcy-Weisbach friction factor.
h=k V2 (2-6)
Ah3 kex 2g
is the head loss at the exit due to the expansion of the flow out of the channel. k ex is the exist loss coefficient.
Substitution of Eqs. (2-4), (2-5) and (2-6) into (2-3) and of (2-3) into (2-2) yields,
V2 (k +k fL n B (2-7)
2g en ex+ 4R) =n (27
or taking square roots,
V k2 fL In nBI sign (no nB (2-8)
(ken + kex + )




Sign (no riB) must be included since the current reverses in direction every one-half tidal cycle.
The rate of rise and fall of the bay tide is specified by the following
continuity equation:
dnB
Q = VA = AB dt (2-9)
where Q is the flow rate through the inlet, A the inlet flow cross-sectional area and AB the bay surface area. Eliminating V between Eqs. (2-8) and (2-9) leads to,
dnB A 2 /1n nB sign (n- n (2-10)
dt AB vk + k +fL
en ex 4R
we introduce the dimensionless quantities
no = no/ao nB = nB/ao 0 = 27T/t (2-11)
where a is the ocean tide amplitude (one-half the tidal range) and T is the tidal period. Substitution into Eq. (2-10) gives,
dr =KiIn -I 1 -I
do K In- nB sign (no nB) (2-12)
where
K T A 2-gao
K T A (2-13)
2a 0 AB k + k + fL en ex 4R
K is referred to as the "coefficient of filling or repletion" as defined by Keulegan (1951). Note that Eq. (2-12) is a first order differential equation for nB, the solution for which must be expected to be a function of K and e only, for a known sinusoidal variation of the ocean tide no (or no).

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A definition sketch for the tides and current is shown in Fig. 4. Solution
Selecting
no = a sin (e T) (2-14)
Eq. (2-12) can be solved for nB as a function of ao, e and K. Note that T represents the time lagas indicated in Fig. 4. Because of the non-linearity of Eq. (2-12), nB is not a sine curve, but has higher harmonics.
To an engineer, three aspects regarding the hydraulics of inlet-bay are of greatest significance.
1. Lag of Slack Water e, After HW and LW in the Ocean
Slack water is the time of zero current just prior to current reversal. According to the simple case depicted in Fig. 4, this occurs when the ocean and bay tide curves intersect, i.e., There is no head difference necessary for flow. The time lags (in radians) of slack after HW and after LW are observed to be the same, inthis idealized case.
Keulegan's solution for F in degrees as a function of the repletion coefficient K is presented in Fig. 5. Note that the time of slack water is also the time of maximum bay elevation according to this model, as seen in Fig. 4. As the lag s increases, the bay tide becomes smaller until E approaches 900 when there is no tidal fluctuation in the bay. This limiting situation occurs when K 0, which can occur when the bay is so large that A/AB 0, or when the friction term under the square root sign in the denomenator (Eq. 2-13) tends to be very large.
The other limiting situation is when K when e 00. This is the
case of a very wide inlet (A/AB large) or negligible friction. Fig. 4 shows that in this case the bay tide approaches the ocean tide.




SLACK

EBB,
2 wr

HW IN OCEAN

LAG OF SLACK WATER AFTER HW IN OCEAN
V fHW IN BAY,
SLACK WATER IN INLET

LW IN BAY SLACK WATER IN INLET

LW IN OCEAN

- --I LAG OF SLACK WATER
AFTER LW IN OCEAN

Fig. 4. Ocean
Inlet

Tide, Bay tide and Current Through the as Functions of Time at in Radians.

-10-

FLOOD




-cO
80-
700
Io
o -T- ----------0. 0.2 0.3 0.4 0.5 0.7 2 3 4 5
Repletion Coefficient, K

Fig. 5. Lag s in Degrees as a Function of Keulegan's Repletion Coefficient K.

1O




2. Maximum Bay Tide Range
As noted, this occurs at the time of slack water in the inlet channel. Keulegan's solution of the ratio aB/ao, where aB is the bay tide amplitude, as a function of K is presented in Fig. 6.
At this stage it is worthwhile to look at a result obtained by Dean (1971) using a linearized approach. Dean assumed an approximation of Eq. (2-8) which resulted in a linear relationship between the velocity V and the head difference qo nB' and the definition of a "linear discharge coefficient" CDL defined as,
_________0(2-15) CDL fL
C L k + k +-f (n Bmx(-5
en ex 4R
where (n0 nBmax is the maximum head difference across the inlet. Omitting calculations, it can be shown, using this approach, that,
a B (C1/a)
(2-16)
ao 0 I + (Ci/)2
where o = 2ir/T is the tidal frequency in radians and AC_2- (2-17)
C1 CDL /a- AB
0
Eq. (2-16) allows calculcation of a B' if an estimate of CDL is available, in addition to a0, A, AB and o. We will return to Eq. (2-16) later.
3. Maximum Current
The maximum flood and ebb currents, Vmax, are defined in Fig. 4. Vmax
is also referred to as the strength of current. Fig. 7 shows Keulegan's solution presented as a dimensionless maximum current V' as a function of K, where
max
V' T A V(2-18) max 2rra0 AB max

-12-




Repletion Coefficient, K
Fig. 6. Ratio bf Bay Tide Amplitude to the Ocean Tide Amplitude, aB/ao as a
Function of Keulegan's Repletion Coefficient K.

0.8
04 -40.21--- -004
0.2
Cl ~~ W3. ..

a5
r1.
id

v.

0.11

0.3 0.4 0.5

2 3 4 5

7 o10




-~ I I liii IIIIIiIIIWIIiWIIWIIIIWiWUIHIIHIflHI I III

.1I

0.2 0.3 0.4 0.5 0.7

2 3 4 5

Repletion Coefficient, K.
Fig. 7. Dimensionless Maximum Velocity, V'max, as a Function of Keulegans Repletion
mCoefficiaxent.
Coefficient.

I..4
'0. 0
0

0
8
6 .1
.2 - -

0

0 O
C




Note that Vmax occurs when the head difference q is a maximum. This
happens when the bay tide is at mean sea level, before HW in the ocean and again before LW in the ocean.
Impedance and Equivalent Channel
In theterm,
ken + kex + fL (2-19)
the value of kex is usually taken as unity, whereas ken depends on the shape of the entrance. Generally, ken ranges from 0.05 to 0.25 (Dean, 1971). From an engineering point of view, however, it is sufficiently accurate to assume the sum ken + kex to be equal to unity.
A problem with real inlets is that they do not have a constant flow
cross-section, and therefore the velocity V changes from point to point along the channel. In order to overcome this problem, O'Brien and Clark (1973) suggested defining a channel with a constant cross-sectional area equal to the throat area A at mean sea level, with the additional requirement that the
c
total head loss Ah2 be equal in the two channels. Omitting details, this leads to the definition of a length Lc of the equivalent channel as,(See Fig. 8(a), (b)) L X L dx (2-20)
Ikc = RCA c 0 RX2 (-0
0 RAX
x x
where
Ac = Throat cross-sectional area at mean sea level,
Ax = Channel cross-sectional area at mean sea level as a function of
distance x,
Rx = Channel hydraulic radius at mean sea level as a function of distance
X,
x = Distance measured along the deepest part of the channel,

-15-




DEEPEST CHANNEL (a)
R-- A2 xx
0

0 X XL

Fig. 8. Definition and Computation of Equivalent Length
of an Inlet.

-16-

N
N
N
N
N
N
N
N
N
N
N
N
N

(b)

,AX)Rx




xL = Length of the channel (maximum value of x), along the deepest part of the channel.
In view of the above, one may write,
fL c
F en ex 4c- (2-21)
fL
= 1 + c (2-22)
Rc
Where F may be referred to as the "impedance" of the inlet-bay system. We will now redefine the repletion coefficient as, T Ac f2gao
K=- A (2-23)
2Tra0 A B F
Also A in Eq. (2-18) should be replaced by Ac, and V by Vc in Eq. (2-8) and elsewhere. Vmax in Fig. 7 is now the maximum cross-sectional average current at the throat.
Note that in the absence of geometric data on the channel, i.e., Rx
Rx(X) and Ax = Ax(x), it is best to select the length of the inlet L between the entrance and exit, to compute F according to Eq. (2-22). Example 2-1
Consider the case of Rockaway Inlet on the south shore of Long Island
(Mehta and Hou, 1974). This inlet connects Jamaica Bay to the Atlantic Ocean. The procedure is as follows:
1. Planimetering Jamaica Bay3 on National Ocean Survey (NOS) chart 1215
gives AB = 6.20 x 108 ft2.
2. From NOS chart 542, the following data on the channel cross-sections are
obtained:
3To be consistent with the definition of the tidal prism as defined in the Section III, the bay should include the body of water bayward of the throat section, rather than the section at the bayward end of the inlet.

-17-




*Hydraulic radius assumed to be equal to

mean depth.

3. The throat section is observed to be at section 4.

Ac = 7.18 x 104 ft2
Rc = 26 ft
XL = 24600 ft
Graphical integration using the above data according to Eq. (2-20)4
gives Lc = 20240 ft.
4. Select f = 0.022 (this will be discussed in Section IV). This gives
F = 5.2 according to Eq. (2-22).
5. Since the tide gage at an inlet records a tide which is affected by
the flow through the inlet, the effective "ocean" tide range 2a0 must
be obtained by interpolation between two outer coast tide gage station,
one to the left and the other to the right of the inlet. NOS Tide Tables
give the following data:

4Compute I/R A 2, and plot it on y axis, for the corresponding x on the x-axis. Join the p~ifts by straight lines and determine the area under the curve thus obtained by a planimeter. Multiply this area by RcAcZ (See Fig. 8 (c)).

-18-

x Rx A
(ft)x x2
Section (ft) (ft) (ft2)
1 0 17 1.85 x 105
2 6000 16 8.88 x 104
3 10000 14 9.11 x 1O4
4 15600 26 7.18 x 104
5 24600 26 1.25 x 105

Station Mean Tide Range
Coney Island 4.70
Long Beach, Outer Coast 4.50




Linear intropolation between these two stations at the location
of Rockaway Inlet gives 2ao = 4.65 ft.
6. For a semi-diurnal range of tide T = 12.4 hours, Eq. (2-23) gives the
repletion coefficient
K = 1.9
7. From Fig. 5,
= 80
T
= 80 x -To = 0.27 hour = x360
From Fig. 6,
aB/ao = 0.99 2aB = 0.99 x 4.65 = 4.60 ft.
From Fig. 7,
Vax = 0.93 2rra A
Vax= 0.93 x T _B = 2.62 ft/sec
max T A
c
III. THE TIDAL PRISM Hydraulically Determined Prism
The tidal prism is the volume of water that enters the bay during flood, and leaves during ebb. The prism is significant because it is a measure of the rate at which waters in the bay are renewed by oceanic flushing. By definition the prism PH is,
T/2
PH f Qdt (3-1)
0If we assume that the flow rate Q is expressed as a sinefunction with an amplitude
If we assume that the flow rate Q is expressed as a sinefunction with an amplitude

-19-




Qmax' i.e.,

Q = Qmax sin at (3-2)
then substituting Eq. (3-2) in (3-1) gives, Qmax T
PH ( -3
Keulegan (1967) noted that in order to account for the non-linearities in the actual flow rate Q, Eq. (3-3) must be divided by a coefficient CK. It was found that CK varies with the repletion coefficient K and ranges from
0.81 to 1. From an engineering point of view, an average value of C K = 0.86 will suffice. Thus, noting that Qmax = Vmax Ac' Eq. (3-3) becomes, VmaxT Ac (3-4)
PH r CK
where PH may be referred to as the hydraulically computed tidal prism. Volumentrically Determined Prism
Another definition of the prism is
Pv = 2aBAB (3-5)
This is the volumetrically determined prism, and for a single inlet-bay system, PH and Pv are clearly identical. Example 3-1
Consider the inlet connecting O'Brien's Lagoon to the Gulf of Mexico, at Treasure Island, Florida (Sedwick and Mehta, 1974; Sedwick, 1974). Given:
A B = 9.1 x 105 ft2
2a B = 2.0 ft

-20-




Parameter John's Pass Blind Pass
P m/Ac (ft) 3.79 x 104 4.98 x 104
Ac (ft2) variable variable
Pa (ft-lb/ft/hr) 1.04 x 105 1.04 x 105
T (hr) 12 12
Rc = Wc/Ac (ft) variable variable
C (ft1/2/sec) 32 37
y (lb/ft3) 64 64
The prism Pm is obtained by multiplying Pm/Ac by the Ac at the given year. M values are computed through Eqs. (2-15) and (2-11).

-21-

Year John's Pass Year Blind Pass
Pm M Pm M
(ft3) (ft3)
1873 1.93 x 108 0.19 1873 2.88 x 108 0.094
1883 1.76 x 108 0.22 1883 2.67 x 108 0.089
1926 2.17 x 108 0.18 1926 1.12 x 108 0.14
1941 2.60 x 108 0.16 1936 1.21 x 108 0.094
1952 3.46 x 108 0.14 1952 8.42 x 107 0.25
8 7
1974 3 .60 x 10 0.14 1974 2.19 x 10 0.58




M versus P Mfrom the above table are plotted in Fig. 9. The trajectory of the data for John's Pass shows that in the 100 year period covered, this inlet has improved its stability, moving from a state of intermediate to good stability during 1880's. Conversely, Blind Pass, which was a stable inlet up until the early 1920's became rapidly unstable from then on, in direct response to dredge fill operations near its bayward end, as indicated by the overall reduction in the bay area, in Fig. 8.'

-22-




REGION OF POO 1STABILITYL
E.. . ... ... ..
................. .....
.. ........... .. .. ... I.
_- .....
REGIO. .O... OO S
REGION OF GOOD STABILITY

AVERAGE TIDAL PRISM PM(ft3)
Fig. 9. Historical Stability Progression of John's
Pass and Blind Pass (after Mehta and Adams, 1975).

-23-

0.01

I07,




REFERENCES

Bruun, P., Tidal Inlets and Littoral Drift, Universitetsforlaget,
Trondheim, Norway, 1966.
Dean, R. G., "Hydraulics of Tidal Inlets," Unpublished Notes, Coastal and
Oceanographic Engineering Laboratory, University of Florida, 1971a.
Dean, R. G., "Coastal Engineering Study of Proposed Navarre Pass," Coastal
and Oceanographic Engineering Laboratory Report No. UFL/COEL-73/006,
University of Florida, Gainesville, Florida, 1971b.
Escoffier, F. F., "The Stability of Tidal Inlets," Shore and Beach, Journal
of the American Shore and Beach Preservation Association, Vol.8,
No. 4, 1940, pp. 114-115.
Johnson, J. W., "Characteristics and Behavior of Pacific Coast Tidal Inlets,"
Journal of the Waterways, Harbors and Coastal Engineering Division,
ASCE, Vol. 99, No. WW3, Proc. Paper 9927, August, 1973, pp. 325-339.
Keulegan, G. H., "Tidal Flow in Entrances: Water Level Fluctuations of
Basins in Communication with the Seas," Committee on Tidal Hydraulics
Technical Bulletin No. 14, U.S. Army Engineers Waterways Experiment
Station, Vicksburg, Mississippi, July, 1967.
Mehta, A. J., "A Long-term Stability Criterion for Inlets on Sandy Coasts,"
Coastal and Oceanographic Engineering Laboratory Report UFL/COEL-75/015,
University of Florida, August, 1975.
Mehta, A. J., and Brooks, H. K., "Mosquito Lagoon Barrier Beach Study,"
Shore and Beach, Journal of the American Shore and Beach Preservation
Association, Vol. 41, No. 2, October, 1973, pp. 26-34.
Mehta, A. J., and Adams, Wm. D., "Glossary of Inlets Report John's Pass
and Blind Pass," Sea Grant Publication (in Press), University of
Florida, August, 1975.
Mehta, A. J., and Adams, Wm. D., "Glossary of Inlets Report Sebastian
Inlet," Sea Grant Publication (in Press), University of Florida,
December, 1975.
O'Brien, M. P., "Estuary Tidal Prisms Related to Entrance Area," Civil
Engineering, ASCE, Vol. 1, No. 8, 1931, pp. 738-739.
O'Brien, M. P., "Notes on Tidal Inlets on Sandy Shores," Hydraulic Engineering
Laboratory Report No. HEL-24-5, University of California, Berkeley,
California, May, 1971.
O'Brien, M. P., and Dean, R. G., "Hydraulic and Sedimentary Stability of
Coastal Inlets," Proceedings of the 13th Coastal Engineering Conference,
ASCE, Vol. II, Vancouver, Canada, July, 1972, pp. 761-780.

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