HYDRAULICS OF TIDAL INLETS
SIMPLE ANALYTIC MODELS FOR THE ENGINEER1
A. J. Mehta2
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 in-
let, 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
(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
Notes A Short Course in Coastal Engineering, University of Florida, November
2Assistant Professor, Coastal and Oceanographic Engineering Laboratory, Univ-
ersity 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 develop-
ment 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 be-
tween 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 cross-
section. This is illustrated by the example shown in Fig. 2.
II. HYDRAULICS OF A SIMPLE INLET-BAY SYSTEM
The governing equations for a simple inlet-bay system will be derived
Fig. 1. A Simple Inlet-Bay System.
0 100' 200' 300' 400' 500' 600'
200' 300' 400'
500' 600' 700'
800' 900' 1000'
MEAN DEPTH *,
TRUESCALE L k,
HYDRAULIC RADIUS R- .MEAN DEPTH
Fig. 2. Comparison between Distorted and Uldistorted
Depictions of Inlet cross-section.
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 re-
quires 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,
o + o B + B + Ah (2-1)
S0 2g B B 2g
no = Ocean tide elevation with respect to meanwater level,
nB = Bay tide elevation with respect to mean water level,
V = Ocean current velocity,
VB = Bay current velocity,
a Ba= Coefficients (greater than unity) which depend on the special
distributions of V and VB, 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
n2h k- V
th fL V2
nh ~k y2
Fig. 3. Energy Losses Across an Inlet of length L.
......................;';" "'' '''
............ ... ...... ............. .. .....
.................... ............ ....... ... .. ...... .... ...
. .. .. . .. .. .. . .. .. .. .. . .. . .. . .. ..
................ ... ................... ..... ..........
............ .. .......... ... ........ .. ...
. . . . . . . . . . . . . . ..........
. . . . . . . . . . . . . .. .. .....................
. . .. .. . .. . . . . .. .. .. .. . .
considered to be negligible. Therfore Eq. (2-4) becomes
Ah = no nB (2-2)
As shown in Fig. 3, Ah can be considered to have three contributions,
Ah = Ah1 + Ah2 + Ah3 (2-3)
Ah = k V (2-4)
1 en 2g
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.
h fL V2 (2-5)
2 4R 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.
VAh = k (2-6)
3 ex 2g
is the head loss at the exit due to the expansion of the flow out of the
channel. kex 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,
V + + fL (2-7)
2g en ex 4R) =0 o
or taking square roots,
V k= 29 fL n nBi sign (no nB) (2-8)
(k en+ kex + )
en ex 4R
Sign (n nB) 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
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 g / no nB sign (n nB) (2-10)
t k + k +-
en ex 4R
we introduce the dimensionless quantities
no = n/ao nB = nB/ao 0 = 2T/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,
d K /no n = sign (n ng) (2-12)
T A 2gao
K A (2-13)
2ra. An i------
a B 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 a
only, for a known sinusoidal variation of the ocean tide no (or n ).
A definition sketch for the tides and current is shown in Fig. 4.
no = a sin (e r) (2-14)
Eq. (2-12) can be solved for nB as a function of ao, e and K. Note that
T represents the time lag as 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 ob-
served to be the same, in this idealized case.
Keulegan's solution for e in degrees as a function of the repletion coef-
ficient 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 e increases, the bay tide becomes smaller until e approaches 90
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 0. 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.
HW IN OCEAN
LAG OF SLACK WATER
AFTER HW IN OCEAN
' ^HW IN BAY.
SLACK WATER IN INLET
LW IN BAY
SLACK WATER IN INLET
LW IN OCEAN
I- e-I LAG OF SLACK WATER
AFTER LW IN OCEAN
Fig. 4. Ocean
Tide, Bay tide and Current Through the
as Functions of Time at in Radians.
H I 1
f i l l
50- ----- ---- ---
tJ I 5II 1 :- :.
0. .. .
. _ _ _ -. _ _... .... t i i i i i__
O. 0.2 0.3 0.4 0.5 0.7 2 3 4 5
Repletion Coefficient, K
Fig. 5. Lag e in Degrees as a Function of Keulegan's Repletion Coefficient K.
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/a 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 no nB, and the definition of a "linear discharge coef-
ficient" CDL defined as,
D k + k + f-R (n nB)max
en ex 4R
where (no nB)max is the maximum head difference across the inlet. Omitting
calculations, it can be shown, using this approach, that,
ao /I + (Ci/I)2
where o = 2ir/T is the tidal frequency in radians and
AC 2 -g (2-17)
1 CDL /a- AB
Eq. (2-16) allows calculation of a Bif an estimate of CDL is available, in
addition to a 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
V T A v (2-18)
max 2ira AB max
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.
0.2 -- ----
_ _ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~ L-^ - ^ ^ _ ^ .
0.3 0.4 0.5
2 3 4 5
0.2 0.3 0.4 0.5 0.7
2 3 4 5
Repletion Coefficient, KI
Fig. 7. Dimensionless Maximum Velocity, V' max as a Function of Keulegans Repletion
^S'-l^IffiffFT^IiiWtgS+S FI' Jill
......i - --
8 : :: : :II :HIM:I.I
0 ^ ^ ------- _ .. .. .
6 = : : : : ::::::::::::-::^ .:.. = = : : : :::::
=^ = : : ::::::: :::::-:;.'::^^= = : : -::::::- ::::-: :::::::
LET I I
- -- -- -
Note that Vmax occurs when the head difference n n is a maximum. This
max 0 B
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
kn + kx + (2-19)
en ex 4R
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) sug-
gested 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
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))
Lc = RA2 L _dx (2-20)
Ac = Throat cross-sectional area at mean sea level,
Ax = Channel cross-sectional area at mean sea-level as a function of
Rx = Channel hydraulic'radius at mean sea level as a function of distance
x = Distance measured along the deepest part of the channel,
x = Distance measured along the deepest part of the channel,
0 X .
Fig. 8. Definition and Computation of Equivalent Length
of an Inlet.
-' I 1--".. .1-1- 1
xL = Length of the channel (maximum value of x), along the deepest part of the
In view of the above, one may write,
F = k + k + (2-21)
en ex 4R
= 1 + c (2-22)
Where F may be referred to as the "impedance" of the inlet-bay system. We
will now redefine the repletion coefficient as,
T Ac |2gao
K=- A (2-23)
2na0 AB 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).
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
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.
*Hydraulic radius assumed to be equal to
3. The throat section is observed to be at section 4.
A = 7.18 x 104 ft2
Rc = 26 ft
XL = 24600 ft
Graphical integration using the above data according to Eq. (2-20)
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 2ao 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 /R A 2, and plot it on y axis, for the corresponding x on the x-axis.
Join the poiits by straight lines and determine the rea under the curve thus
obtained by a planimeter. Multiply this area by RA c (See Fig. 8 (c)).
x Rx A
Section (ft) (ft) (ft2)
1 0 17 1.85 x 105
2 6000 16 8.88 x 104
3 10000 14 9.11 x 104
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
K = 1.9
7. From Fig. 5,
=8 x- = 0.27 hour
From Fig. 6,
aB/a = 0.99
2aB = 0.99 x 4.65 = 4.60 ft.
From Fig. 7,
Vax = 0.93 x T B = 2.62 ft/sec
max T A
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,
PH Qdt (3-1)
If we assume that the flow rate Q is expressed as a sinefunction with an amplitude
Q = Qmax sin at (3-2)
then substituting Eq. (3-2) in (3-1) gives,
P -ma (3-3)
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 CK = 0.86
will suffice. Thus, noting that Qmax = Vmax Ac Eq. (3-3) becomes,
VmaxT A (3-4)
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.
Consider the inlet connecting O'Brien's Lagoon to the Gulf of Mexico,
at Treasure Island, Florida (Sedwick and Mehta, 1974; Sedwick, 1974).
AB = 9.1 x 105 ft2
2aB = 2.0 ft
Therefore, according to Eq. (3-5),
P = 1.82 x 106 ft3
Vmax = 1.90 ft/sec
A = 42 ft2
T = 18 hours (average of diurnal and semi-
Therefore, according to Eq. (3-4),
PH = 1.91 x 106 ft3
which compares reasonably with P v
When there are more than one connections between the bay and the ocean,
the bay is filled by all the inlets, and therefore Pv becomes the sum of
the tidal prisms of each inlet. Thus for N inlets,
SPHi = P = 2aAB (3-6)
c P Ti V maxiA (3-7)
Hi iC maxi ci
i=l PHi CK i=l
The range of tide in the bay may be different from point to point. For
engineering estimation, it is generally sufficient to assume a range which is the
average of these values.
John's Pass and Blind Pass connect the Gulf of Mexico to north Boca
Ciega Bay. Current measurements have yielded the following values for the
average hydraulically determined tidal prisms (Mehta and Adams, 1975).
PHJhns Pas 3.60 x 108 ft3
HBlind Pass = 2.20 x 107 ft3
cP =P+P= 3.82 x 108 ft3
SHi = HJohn's Pass + HBlind Pass 382 x 10
For north Boca Ciega Bay,
2aB = 1.7 ft. (average of tide at several points
around the bay)
AB = 2.21 x 108 ft3
P = 3.76 x 108 ft3
EPH. and P are thus observed to be in reasonable agreement.
O'Brien (1931) spent a summer studying Pacific Coast inlets and made
the observation that the ratio of the tidal prism Ps based on the spring
range of tide divided by the throat section Ac is either constant or a slowly
varying function of the tidal prism, depending on whether the inlets have no
jetties or two jetties, respectively. This observation pertains to inlets which
are in stable sedimentary equilibrium, and has found worldwide applicability5
Recent studies have also shown that a similar rule may be used with reasonable
accuracy if Ps is replaced by the prism Pm based on the mean range of tide.
The ratio Ps/Ac has been found to depend on the number of jetties (zero,
one or two) and on whether the inlet is on the Atlantic, Gulf or Pacific
It is not applicable to newly cut or improved inlets.
Coast. This adds up to nine relationships, and nine more for P /A Further-
more, in some cases, different investigators have proposed slightly different
relationships for the same category (e.g., two jettied Pacific Coast inlets).
Ignoring these details, the following relationships give first order accuracy
for engineers(P in cubic feet, A in square feet):
Pm = 5.3 x 104 A, ft (unimproved) (3-8)
Pm = 1.3 x 104 A 1 ft (one and two jetties) (3-9)
Eq. (3-9) was proposed by Johnson (1973). A convenient approximation
for converting PS to Pm or vice versa is,
P a 1/2
pm (aom) (3-10)
where aom is the mean and aos the spring ocean tide amplitudes. Eq. (3-10)
is obtained by combining Eqs. (2-16) and (2-17) with (3-5), (Mehta, et al., 1975).
Considering Eq. (3-8), upon substitution into Eq. (3-4) yields,
max- 5.3 x 104 ft
with T = 12.4 hours and Ck = 0.86
Vmax = 3.2 ft/sec
Eq. (3-8) is therefore essentially a statement that under semi-diurnal
tides, maximum cross-sectional average currents at the throats of unimproved
inlets are of the order of 3 feet per second. Indeed it has been found that
in general maximum currents do infact range from 2.5 to 3.5 feet per second.
IV. DEVIATIONS FROM THE SIMPLE SYSTEM
A Semi-Empirical Approach
Although the model described in Section II is based on sound hydrodynamic
principles, the assumptions inherent in its derivation limit its applicability
to relatively small inlet-bay systems. O'Brien and Clark (1973) therefore
used the available data on ocean tides in NOS Tide Tables, and inlet currents
in NOS Tidal Current Tables, to examine the behavior of a number of real inlets.
They considered the bay tide to be approximated by a sine curve (no higher
harmonics) which results in the following two simple relationships,
V = 2a (4-1)
aB cos (4-2)
Because of the asymmetry of real tides and currents, unlike Fig. 4, the
strengths Vmax of flood and of ebb are not in general equal in magnitude, nor
are the lags e. Furthermore, because the ocean tide range 2ao changes some-
what from day to day, Vmax and s also vary correspondingly, and F, as computed
from Eq. (4-1) is found to vary as well. O'Brien and Clark however noted
that two week averages, e and F, of the lag e and impedance F, respectively,
for a given inlet are nearly constant. We may thus write Eq. (4-1) as
max F o
Fma gI (4-3)
Vmax = CD 1 (4-4)
where CD = g sin /F may be referred to as a characteristic velocity coef-
ficient of the inlet. Once CD is evaluated for a given inlet, it may be used
to approximately estimate Vmax under a given range of ocean tide 2a .
Before proceeding, a point should be noted with reference to the strength
of current reported in the Current Tables. This generally is not a cross-
sectional average, but is close to what may be measured near the surface
in the center of the channel at the throat. If this value is Vmaxs, then
it has been found to be reasonable to obtain Vmax (cross-sectional average)
V = 0.85 V (4-5)
where the 0.85 factor is an empirically determined constant.
Consider Indian River Inlet, connecting the Atlantic to Indian River Bay,
Delaware. NOS Tide Table and Current Table data may be combined to yield
the following biweekly average values (O'Brien and Clark, 1973),
e = 65.80
F = 15.3
This gives CD = 1.39, so that Eq. (4-4) for this inlet is (g = 32.2 ft/sec2)
Vmax= 1.39 2 (4-6)
Eq. (4-6) can now be used to approximately determine Vmax under any given range
2ao (in feet) at Indian River Inlet. It is interesting to note that Keulegan
(1967) obtained the same value of CD using his method and his data. For larger
inlet-bay systems, however, Keulegan's results deviate from predictions based
on measurements. Note that Eq. (4-2) gives aB/ao = 0.41. Tide Tables yield
aB/ao = 0.21.. The agreement is not very close.
If one is interested in differences in magnitude between the strengths
of flood and ebb, Eq. (4-3) may be evaluated for the flood and ebb phases separately.
Consider Chesapeake Bay Entrance. Tide Tables and Current Tables data,
and application of Eq. (4-1) yields for the flood phase subscriptt F) and
ebb phase subscriptt E)
EF = 110.20
E = 113.80
FF = 51.6
VmaxF = 0.76 (4-7)
maxE = 1.19 o
Note that the lags exceed 90. Eq. (4-2) can not therefore be used to
predict the range of tide in the bay.
The approach described in this section is useful for determining CD in
Eq. (4-4), provided ocean tide and inlet current records, atleast two weeks
long, are available. Tide Tables are generally sufficient for tides, because
predictions at a large number of outer coast stations are available. However,
current predictions are not available at all the inlets. It is moreover
expensive and often cumbersome to install a current meter at an inlet. Times
of slack water can however be recorded by an onshore observer without difficulty.
This yields e values and the average, s. To calculate CD we also need F.
This can be estimated from Eq. (2-22) if the friction-factor f is available in
addition to Lc and R .
f values in inlets have been generally found to be larger than those for
rivers, and therefore, friction factors or Manning's n for rivers can not be
used for inlets. To estimate f in inlet, the procedure described in this section
was used to calculate the average impedance F and lag e for a number of inlets
(e.g., Rockaway Inlet by Mehta and Hou (1974)). Eq. (2-22) then gave f, which
was converted to the Chezy coefficient C according to,
C = (4-8)
An observation that larger inlets have higher Chezy coefficients led
to the empirical plot of Fig. 9, where C has been plotted against Ls/R The
straight line relationship expressed according to this plot is similar to the
relationship between C and Ac proposed by Bruun (1966), but yields comparatively
lower C values. Ls is defined in Fig. 10 as the distance along the inlet channel
between the outer bar and the point where the channel ends or is divided by the
inner shoal. Ls can be estimated from charts or surveys with relative ease.
Clearly, because of the nature of actual inlet channels, a degree of engineering
judgement is often required in such an estimation. Fig. 9 is then entered with
the ratio Ls/Rc and C determined on the vertical axis. Then f = 8g/C2
Variable Bay Area
Often the bay has gently sloping rather than vertical banks, in which
case the surface area at low water may be significantly smaller than at high
AB = ABMWL (1 + 'nB) (4-9)
where ABMWL is the bay surface area at mean water level in the bay and is a
coefficient which depends on the bay geometry. The continuity expression is,
Q dt (ABnB) (4-10)
Substitution of Eq. (4-9) into (4-10) gives,
Q = ABMWL (1 + 2TnB) dt (4-11)
This analogous to Eq. (2-9). Following a procedure similar to Section II,
I i I I
1~T~sVr] 17 I ~ ~~1=4zI
-i--I ~j-~--FffiL h'1- I 4-~ -~--,---,- + -,--T TTh V LI~ZffiL liLA Eli
liii I I
I I i I I I I i I I I a I I
100 200 300 400 500 600 700 800 900 1000 1100 12001300 1400 1500 1600 1700 1800 1900
Fig. 9. Empirical Relationship between the Chezy Coefficient C (ft1/2/sec) and the Ratio Ls/R .
~-~i:Fi~f+--H--+i-t ~-~ ttt--itittf-T--Tf fsy-- -~RI----r-~i=F;L-f
:= 1: zi 1 "
;:..;.. DEEPEST CHANNEL ,//
:...:t. i .-.,Ai/ .L
., Rc ,. *2 \.. ,
Fig. 10. Definition of the Length Ls to be used in Fig. 9.
K sign (q- (4-12)
d- (1 + 2aoBn) n sign n) (4-12)
Here, in the repletion coefficient, K, AB = ABMWL. The hydraulics of the in-
let are observed to depend on the parameter 8, in addition to K. In general,
Eq. (4-12) must be solved on a computer. Huval and Wintergest (1972), Mota
Oliveira (1970), King (1974) and others have treated this problem.
Effect of Inertia
Under certain conditions, as for example when the inlet channel is very
long, the acceleration ;of the mass of water in the channel may have a measurable
effect on the hydraulics. One such effect is to make the bay tide range larger
than the ocean range. This case of aB/ao greater than unity is beyond the
scope of the described methods. For example, in Example 2-1, aB/a = 0.99 is
predicted for Rockaway Inlet. Tide Tables however yield aB/ao = 1.12.
The inclusion of the acceleration term aV/at in the momentum equation leads to,
V = In nB a sign (no nB -g a) (4-13)
At slack water, V = 0, and
(no "B)slack _= aVI (4-14)
Where aV/at at slack is also its maximum value6. Usually, this head difference
King (1974) has defined a number
a F AB
1 2L AV
He notes that when K1 < 100, the inertia term assumes significance. King has
solved Eq. (4-13) along with the continuity equation on an analog computer,
and has presented a family of curves analogous to that in Fig. 6. Recently,
6Unlike the Keulegan model, in this case the ocean and bay will become
horizontal (no head) after slack water in the inlet.
Escoffier (1975) has solved the same equations analytically.
A Numerical Approximation of the Basic Equations
There are times when one.is interested in obtaining the bay tide and
the flow discharge as functions of time, rather than merely the maximum
values from the plots of Figs. 6 and 7. This becomes tedious when the
ocean tide is not sinusoidal as for example when the "ocean" is the Gulf of
Mexico. In such a situation, Eqs. (2-8) and (2=9) should be solved on a
n = n (t) (4-16)
which is the known ocean tide input.
n c o nB sign (n n ) At (4-17)
TB AB F Fno TB' B0- B
which is Eq. (2-10) in difference form. no and nB are at time level n.
The time level shift is carried out through,
n+1 n n
n = n + AnB (4-18)
It is sufficient to select, as initial condition, at n = 0,
B = 0 (4-19)
since the solution, eventually after about two tidal cycles, becomes in-
dependent of the effects of this initial condition. Nan de Kreeke (1967)
has solved a similar system of equations with an additional contribution
of a time varying fresh water outflow through the inlet.
Eqs. (4-17) and (4-18) must be solved successively, with the input Eq.
(4-16), on a computer, for a selected time increment At. At = 5 minutes may
be selected as a reasonable first choice. Solution of the above equations
essentially yields nn+ at the end of each time increment. Eq. (2-9) may be
approximated according to,
n+l B n+1 n 0
V At n) (4-20)
c A At B B
Qn+l = n+1A (4-21)
At each time level, Eqs. (4-20) and (4-21) give the throat velocity Vn+ and
the flow rate Q
the flow rate Q +l
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Blind Pass," Sea Grant Publication (in Press), University of Florida,
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D.C., 1970, pp. 1827-1845.
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