HYDRAULICS OF TIDAL INLETS SIMPLE ANALYTIC MODELS FOR THE ENGINEER1 By
A. J. Mehta 2
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
The governing equations for a simple inlet-bay system will be derived
' TIDAL FLOW
Fig. 1. A Simple Inlet-Bay System.
0 100' 200' 300' 400' 500' 600'
200' 300' 400'
700' 800' 900' 1000'
l l i t
IEAN DEPTH. ROESL E -R
HYDRAU LC RADIUS CROSS-SECTIONAL AREA MEAN DEPTH
Fig. 2. Comparison between Distorted and Ldistorted
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 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+ B + --+ Ah (2-1)
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
n2h k- V
h -fL V2 2 4R 2g
5h -k V
3 ex ig
Fig. 3. Energy Losses Across an Inlet of length L.
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)
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
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)
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).
A definition sketch for the tides and current is shown in Fig. 4. Solution
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.
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
Tide, Bay tide and Current Through the as Functions of Time at in Radians.
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.
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)
ao 0 I + (Ci/)2
where o = 2ir/T is the tidal frequency in radians and AC_2- (2-17)
C1 CDL /a- AB
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
V' T A V(2-18) max 2rra0 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.
04 -40.21--- -004
Cl ~~ W3. ..
0.3 0.4 0.5
2 3 4 5
-~ I I liii IIIIIiIIIWIIiWIIWIIIIWiWUIHIIHIflHI I III
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
.2 - -
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
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
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
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,
DEEPEST CHANNEL (a)
R-- A2 xx
0 X XL
Fig. 8. Definition and Computation of Equivalent Length
of an Inlet.
xL = Length of the channel (maximum value of x), along the deepest part of the channel.
In view of the above, one may write,
F en ex 4c- (2-21)
= 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 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
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.
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)).
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 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
K = 1.9
7. From Fig. 5,
= 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
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 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
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
Therefore, according to Eq. (3-5),
Pv = 1.82 x 106 ft3 Given:
Vmax = 1.90 ft/sec Ac = 42 ft2
T = 18 hours (average of diurnal and semidiurnal periods) Therefore, according to Eq. (3-4),
PH = 1.91 x 106 ft3 which compares reasonably with Pv'
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, i=N
i= PHi = P = 2aB AB (3-6)
EP T 1 V mxA (3-7)
Hi Cma i c
i=l Hi 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).
P HJohns Pass = 3.60 x 108 ft3 PHBlind Pass = 2.20 x 107 ft3 i=2
Z P Hi = PHJohn's Pass + PHBlind Pass = 3.82 x 10 ft i=l
For north Boca Ciega Bay,
2aB = 1.7 ft. (average of tide at several points around the bay)
AB = 2.21 x 10 8 ft 3
P v = 3.76 x 108 ft3
EPHi and Pv 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 A c 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 applicability 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 P s/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 Pm/Ac. Furthermore, 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 (Pm in cubic feet, A in square feet):
Pm = 5.3 x 104 Ac ft (unimproved) (3-8)
Pm = 1.3 x 104 AI. 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, PM aOM 1/2
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,
VmaxT = 5.3 x l04 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, Vma = 2aogsinE: (4-1)
aB CO 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 c. Furthermore, because the ocean tide range 2a 0 changes somewhat from day to day, V max and 6 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, 6 and T, of the lag c and impedance F, respectively, for a given inlet are nearly constant. We may thus write Eq. (4-1) as Vmax sin c (4-3)
V max = CD *P (4-4)
where CD i / may be referred to as a characteristic velocity coefficient of the inlet. Once C D is evaluated for a given inlet, it may be used to approximately estimate V max under a given range of ocean tide 2a 0.
Before proceeding, a point should be noted with reference to the strength of current reported in the Current Tables. This generally is not a crosssectional 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) from
V = 0.85 Vmaxs (4-5)
where the 0.85 factor is an empirically determined constant. Example 4-1
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),
This gives CD = 1.39, so that Eq. (4-4) for this inlet is (g = 32.2 ft/sec2)
Vmax = 1.39 2ao (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 a B/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 (subscript F) and ebb phase (subscript E)
F = 110.20
E = 113.80
FF = 51.6
VmaxF = 0.76 42ao (4-7)
VmaxE = 1.19 ao
Note that the lags exceed 900. 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 c 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 Rcf 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 P and lag : 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,
An observation that larger inlets have higher Chezy coefficients led
to the empirical plot of Fig. 9, where C has been plotted against Ls/Rc* 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. L 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/C 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 water.
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= d (ABnB) (4-10)
Substitution of Eq. (4-9) into (4-10) gives,
Q = ABMWL (1 + 2rnB) dt (4-11)
This analogous to Eq. (2-9). Following a procedure similar to Section II,
FF44 y E
~4---+-4---4----~--~--+ A I LI
-ii i l
I I I 1 I I I I I I I I I I
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 LS/ Rc
Fig. 9. Empirical Relationship between the Chezy Coefficient C (ftl/2/sec) and the Ratio LS/R .
- 01 KI
I i i -i
% 16.1 i 1 1 i 1
. ....., -LS< .". .
., ., ~ ~ ~ ~ ~~~.... .............' . :. ..- !:
-;,. DEEPEST CHANNEl
- \\ ;
THROAT,.' Ai, RL4*2.
Fig. 10. Definition of the Length L to be used in Fig. 9..
K sig (- (4 12
d- (1 + 2aoSn ) o n19sg (o-n (4-12)
Here, in the repletion coefficient, K, AB = ABMWL. The hydraulics of the inlet are observed to depend on the parameter 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/ao = 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 = 2 jn nB t sign (n nB g at) (4-13)
At slack water, V 0 0, and
(no nB)slack = L c tVslack (4-14)
Where aV/at at slack is also its maximum value6. Usually, this head difference
King (1974) has defined a number
a0F A B
K= c A (4-15)
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 computer.
n= (t) (4-16)
which is the known ocean tide input.
A c 'n n n n
c n nj sign (no n _iB) At (4-17)
T1 ABJF n o T' (i- 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+l n n
n = nB + AnB (4-18)
It is sufficient to select, as initial condition, at n = 0,
0 0 (4-19)
since the solution, eventually after about two tidal cycles, becomes independent 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 inlt.
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+1 at the end of each time increment. Eq. (2-9) may be approximated according to,
n+l AB n+1
V AA (n+l n) (4-20)
c A At B 8
Qn+l = Vn+1A (4-21)
At each time level, Eqs. (4-20) and (4-21) give the throat velocity Vn+l and
cthe flow rate Qn+ the flow rate Q ~
Baines, W. D., "Tidal Currents in Constricted Inlets," Proceedings of the
6th Conference on Coastal Engineering, ASCE, Gainesville, Florida, 1958,
Brown, E. I., "Inlets on Sandy Coasts," Proc. ASCE, Vol. 54, 1928, pp. 505-553.
Bruun, P., "Tidal Inlets and Littoral Drift," Printed in Trondheim, Norway, 1966.
Dean, R. G., "Hydraulics of Inlets," Coastal and Oceanographic Engineering
Laboratory, UFL/COEL-71/O19, (Unpublished), 1971.
Escoffier, F. F., "Effect of Inertia on Tidal Flow Through Inlets," Unpublished
Huval, C. J., and Wintergerst, G. L., "Coastal Inlet Mechanics and Design,
Preliminary Notes on Numerical Modeling," Unpublished Notes, Waterways
Experiment Station, Vicksburg, Mississippi, 1972.
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., "Third Progress Report on Tidal Flow in Entrances: Water
Level Fluctuations of Basins in Communication with Seas," National
Bureau of Standards Report No. 1146, Washington, D.C., September, 1951.
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.
King, D. B., "Dynamics of Inlets and Bays," Coastal and Oceanographic Engineering
Laboratory Technical Report No. 22, University of Florida, Gainesville,
Florida, March, 1974.
Mehta, A. J., and Hou, H. S., "Hydraulic Constants of Tidal Entrances II:
Stability of Long Island Inlets," Coastal and Oceanographic Engineering Laboratory Technical Report No. 23, University of Florida, Gainesville,
Florida, November, 1974.
Mehta, A. J., Byrne, R. J., and DeAlteris, J.,"Hydraulic Constants of Tidal
Entrances III: Bed Friction Measurements at John's Pass and Blind Pass,"
Coastal and Oceanographic Engineering Laboratory Technical Report No. 26,
University of Florida, Gainesville, Florida, March, 1975.
Mehta, A. J., and Adams, Wm. D., "Glossary of Inlet Report John's Pass and
Blind Pass," Sea Grant Publication (in Press), University of Florida,
Mota Oliveira, I. B., "Natural Flushing Ability in Tidal Inlets," Proceedings
of 12th Coastal Engineering Conference, ASCE, Vol. 3, Ch. 111, Washington,
D.C., 1970, pp. 1827-1845.
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., and Clark, R. R., "Hydraulic Constants of Tidal Entrances I:
Data from NOS Tide Tables, Current Tables and Navigation Charts," Coastal and Oceanographic Engineering Laboratory Technical Report No. 21, University of Florida, Gainesville, November, 1973.
Sedwick, E. A., and Mehta, A. J., "Data from Hydrographic Study at MOB Inlet,"
Coastal and Oceanographic Engineering Laboratory Report UFL/COEL-66/005,
University of Florida, July, 1974.
Sedwick, E. A., "Hydraulic Constants and Stability Criterion for MOB Inlet,"
Department of Engineering Sciences Master of Science Thesis, University
of Florida, Gainesville, Florida, 1974.
van de Kreeke, J., "Water Level Fluctuations and Flows in Tidal Inlets," Journal
of the Waterways, Harbors and Coastal Engineering Division, ASCE, No. WW4,
November, 1967, pp. 97-106.