Depth-Averaged 2-D Model of Tidal Flow in Estuaries

Depth-Averaged 2-D Model of Tidal Flow in Estuaries

Weiming Wu, Honghai Qi and Sam S.Y. Wang

National Center for Computational Hydroscience and Engineering
The University of Mississippi, MS 38677

Abstract

A depth-averaged 2-D numerical model for unsteady tidal flow in estuaries is
established using the finite volume method on non-staggered, curvilinear grid. The
2-D shallow water equations are solved by the SIMPLEC algorithm with the Rhie and
Chow’s momentum interpolation technique. The convection terms are discretized by
one of the hybrid upwind/central difference scheme, exponential difference scheme,
QUICK scheme and HLPA scheme. The algebraic equations are solved using the
strongly implicit procedure (SIP). The model is capable of handling the drying and
wetting problem due to the variation of water surface elevation. The model has been
tested in Tokyo Bay and San Francisco Bay. The tests show that the present model is
very stable and efficient. The simulated water elevation and flow velocity are in good
agreement with the measured data.

Introduction

Estuarine regions are often densely populated, so navigation, flood protection, water
supply and environment quality control are highly demanded. However, many very
complex phenomena, such as river flow, tidal flow, wave-induced currents, wind-
driven flow, salinity transport and cohesive sediment transport, are involved there.
These make modeling of the estuarine processes extremely difficult. Several 2-D and
3-D numerical models for tidal flow have been establishedin last decades (Casulli,
1990; Blumberg and Mellor, 1987; Kodama et al., 1991; and others). Most of these
models are based on the finite difference method or finite element method. The aim
of this study is to establish a depth-averaged 2-D numerical model for unsteady flow
in estuaries based on the finite volume method on a non-staggered, curvilinear grid.
Several new numerical techniques widely used in Computational Fluid Dynamics,
such as Rhie and Chow’s (1983) momentum interpolation method and the SIMPLEC
algorithm on non-staggered grid, are adopted to enhance the computation efficiency
of the model.

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Governing Equations

The depth-integrated continuity and momentum equations of shallow water flow in

open channels are:

h + (hU ) + (hV ) = 0 (1)
tx y

(hU ) + (hUU ) + (hVU )= gh zs + 1 (hTxx ) + 1 (hTxy )
tx y xx y
(2)
+ 1 ( sx )bx + fchV

(hV ) + (hUV ) + (hVV )= gh zs + 1 (hTyx ) + 1 (hTyy )
tx y yx y
(3)
( )+ 1 sy by fchU

where t is the time; x and y are the horizontal Cartesian coordinates; h is the flow

depth; U and V are the depth-averaged flow velocities in x- and y-directions; zs is the
water surface elevation; g is the gravitational acceleration; is the density of flow;

Txx, Txy, Tyx and Tyy are the depth-averaged turbulent stresses; sx and sy are the shear

stresses on water surface due to wind driving, determined by sx = ac faU w U 2 + Vw2
w

and sy = ac faVw U 2 + Vw2 , in which a is the density of air, cfa is the friction
w

coefficient of wind, Uw and Vw are the components of wind velocity in x- and y-

directions; bx and by are the bed shear stresses determined by bx = c fU U 2 + V 2

and by = c fV U 2 + V 2 , in which c f = gn2 h1/ 3 and n is the Manning’s roughness
coefficient; fc is the Coriolis coefficient.

Five turbulence models, including the depth-averaged parabolic eddy viscosity
model, the mixing length model, the standard k- turbulence model, the non-
equilibrium k- turbulence model and the RNG k- turbulence model, are adopted to
close Eqs. (1)-(3). Considering the simulation of tidal flow needs relatively coarse
mesh, only the former two zero-equation turbulence models are used in this study.
Therefore the turbulent shear stresses are determined as

Txx = 2 ( + t) U; Txy = Tyx = ( + t) U+ V ; Tyy = 2 ( + t) V (4)
x y x y

where is the kinematic viscosity of water; t is the eddy viscosity due to

turbulence. In the depth-averaged parabolic model, t is calculated by t = tU h , in

which U is the bed shear velocity, and t is an coefficient between 0.3 ~ 1.0 . In the

( )mixing length model, ( 2
t is calculated by t= 0U h)2 + lh2 S
, in which

2

[ ]S = 2( U 1/ 2
x)2 + 2( V y)2 + ( U y+ V )x 2 and lh = min(cmh, y) , with
,

being the von Karman’s constant and y being the distance to the nearest rigid wall.

0 and cm are empirical coefficients, with 0 being set as 0.0667 and cm being

calibrated in the range of 0.4-1.2.

Boundary Conditions

Near rigid wall boundaries, such as banks and islands, the wall-function apprroach is
employed. The resultant wall shear stress r to the flow velocity VP at the
w is related

center P of the control volume close to the walrl by the following relation:
r VP
w = (5)
,E
( )where is a coefficient, determined by = u ln Ey + , with yP+ = u yP
P

being the coefficient of 8.432, yP being the distance from P to the wall, and u being

the shear velocity on the wall.

In the numerical simulation of the flow in open channels with sloped banks, sand bars
and islands, the water edges change with time, with part of nodes being possibly wet
or dry. Even for steady flow, the water edges are not known until the computation is
finished. In present model, a threshold flow depth (a small value such as 0.02m in
natural rivers) is used to judge drying and wetting. If the flow depth in a node is
larger than the threshold value, this node is considered to be wet, and if the flow
depth is lower than the threshold value, this node is dry. Because the fully implicit
solver is used, all the wet and dry nodes are needed to participate in the solution. The
dry nodes are given zero velocity. On the water edges, the wall-function approach is
applied.

At the outlet boundary, the water surface elevation is needed in the case of subcritical
flow conditions. It can be specified as either time series of water elevations or open
boundary in present model.

Numerical Methods

The governing equations are discretized using the finite volume method in a
curvilinear non-orthogonal grid with a non-staggered variable arrangement. In a
curvilinear coordinate system, Eqs. (1)-(3) can be written in the common tensor
notation form:

(J h ) + J h uˆm mn = J hS (6)
t jj
m
n

where stands for 1, U and V, respectively, depending on the equation considered;

= + t is the diffusivity of the quantity ; S is the source term in the

equation of ; J is the Jacobian of the transformation between the Cartesian

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coordinate system xi (x1=x and x2=y) and the computational curvilinear coordinate

system m (m=1,2); uˆm = imUi ; m = m xi .
i

Eq. (6) is integrated over a control volume. The convection terms in Eq. (6) are
discretized by one of the hybrid upwind/central difference scheme (Spalding, 1972),
exponential difference scheme (Spalding, 1972), QUICK scheme (Leonard, 1979)
and HLPA scheme (Zhu, 1992). The HLPA scheme is used in this study, because it
has better accuracy than the hybrid scheme and exponential scheme, and better
numerical stability than the QUICK scheme. The diffusion terms are discretized by
the central difference scheme. The time derivative term is discretized by a three-level
implicit scheme with second accuracy in time. The discretized equations are solved
by the strongly implicit procedure (SIP) of Stone (1968).

The flow calculation adopts the SIMPLEC algorithm in conjunction with Rhie and
Chow’s (1983) momentum interpolation method to acquire the coupling of velocity
and pressure (water level). The details of this method can be found in Wu (2003).
This method is very stable and efficient.

Model Tests

Tidal Flow in Tokyo Bay. Tokyo bay is located off the southeast coast of Honshu
Island, Japan, and connected to west Pacific Ocean. It is about 48 km long and 37 km
wide, as shown in Figure 1(a). From August 25 to October 25, 1983, the data of tidal
levels, wind velocity and etc. in the bay were collected, which are used to test the
established hydrodynamic model. The computational mesh consists of 151×50
quadrilateral cells in longitudinal and transverse directions, shown in Figure 1(b). The
wind field is interpolated at each computational node from the averaged values
measured in the neighboring wind stations in this period. The maximum wind speed
is about 10 m/s. The wind drag coefficient is 0.00015. The Coriolis coefficient is
0.000084. The bay receives water from the Edogawa River, the Arakawa River, the
Tamagawa River and the Tsurumigawa River, but the inflow discharges from these
rivers are not considered in the simulation. The time series of tidal level generated by
using four major astronomical constituents M2, S2, K1 and O1 with an identification
method (Kodama et al., 1991) are imposed at the entrance of the bay. The
computational time step is 30 minutes, which is almost the maximum value allowed
to represent the temporal variation of a semi-diurnal tide.

Figure 2 shows the close-up view of the simulated flow fields in the region around
Futtsu at flood tide and ebb tide. Because of the effect of two protrusions in two sides
of the bay, the flow separates behind each protrusion. The simulated flow patterns are
apparently reasonable.

Figure 3 shows the comparison of the observed and simulated tidal levels at eight
gauge stations. The observed tidal levels are determined by a Fourier series of four
major astronomical constituents M2, S2, K1 and O1 (Kodama et al., 1991):

4

4 2! (t t0 ) km (7)

zs (t ) = am sin Tm

m =1

where zs (t) is the tidal level at time t; t0 is the initial time, equal to 8.8 hr; am , Tm and km

are the amplitude, period and phase delay of each constituent. The periods of M2, S2, K1 and

O1 are 12.42 hr, 12.0 hr, 23.92 hr, and 25.82 hr, respectively. The amplitudes and phase

delays of the four constituents are listed in Table 1.

Table 1. Tidal constituents of tidal levels at eight tide gauge stations

Gauge Amplitudes am (m) Phase delays km (rad)
Stations
M2 K1 S2 O1 M2 K1 S2 O1
Funabashi 0.4842 0.2645 0.2270 0.1935 2.6957 3.1367 3.1142 2.7981
Samugawa 0.5220 0.2540 0.2680 0.1900 2.6721 3.1206 3.0508 2.7471
0.5110 0.2640 0.2480 0.1960 2.8990 3.0543 2.9112 2.9147
Haneda 0.5600 0.2700 0.2900 0.2300 2.7227 3.0631 3.1206 2.9845
Anesaki 0.4725 0.2490 0.2293 0.1960 2.6803 3.1187 3.0938 2.7997
Yokohama 0.4409 0.2442 0.2148 0.1902 2.6300 3.0997 3.1301 2.7887
Kimitsu 0.4500 0.2500 0.2100 0.1800 2.6005 3.1067 3.1416 2.7576
Futtsu 0.4124 0.2405 0.1986 0.1875 2.6663 3.1161 3.1168 2.7997
Yokosuka

The comparison is carried out for a period of 6 days. The agreement between the
measured and simulated tidal levels is generally good. The amplitudes of tidal level
are slightly over-predicted at the first three days in several stations, and well predicted
in the late three days. No significant phase difference is found.

(a) (b)
Figure 1. (a) Tide Gauge Stations and (b) Computational Mesh in Tokyo Bay

5

Futtsu h 1 m/s
50 Futtsu
40
Yokosuka 30
(a) 20
10
0

Yokosuka

(b)

Figure 2. Close-Up View of Simulated Flow Fields around Futtsu Station in
Tokyo Bay: (a) Flood Tide; (b) Ebb Tide.

1.5 1.5

Yokosuka1 Futtsu1

0.5 0.5
0 0

-0.5 -0.5
-1
Simulated-1 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Tidal Level (m) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Tidal Level (m)

+ Measured

-1.5 0 48 96 144 -1.5 0 48 96 144

Time (hour) Time (hour)

1.5 1.5

Yokohama1 Kimitsu1

0.5 0.5
0 0

-0.5 -0.5
-1 -1
Tidal Level (m) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Tidal Level (m) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

-1.5 0 48 96 144 -1.5 0 48 96 144

Time (hour) Time (hour)

1.5 1.5

Haneda1 Anesaki1

0.5 0.5
0 0

-0.5 -0.5
-1 -1
Tidal Level (m) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Tidal Level (m) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

-1.5 0 48 96 144 -1.5 0 48 96 144

Time (hour) Time (hour)

1.5 Funabashi 1.5 Samugawa

Tidal Level (m) 1 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Tidal Level (m) 1 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0.5 0.5

0 0
-0.5 -0.5

-1 -1

-1.5 0 48 96 144 -1.5 0 48 96 144

Time (hour) Time (hour)

Figure 3. Observed vs. Simulated Tidal Levels in Tokyo Bay

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Tidal Flow in San Francisco Bay. San Francisco Bay is home to three major ocean
shipping ports, and the largest estuary on the west coast of U.S.A. It includes four
bays: Suisun Bay, San Pablo Bay, Central Bay and South Bay, as shown in Figure 4.
The tidal flow in the about 150km-long full bay is simulated. The bathymetry data
covering the full bay area was downloaded from USGS’s San Francisco Bay
Bathymetry Web Site (http://sfbay.wr.usgs.gov/access/Bathy/grids.html). This data
set was composed of the topographies measured from 1951 to 1993, most of which
were colleted after 1980. Because the domain is very irregular, a quadrilateral mesh is
used, which consists of 810×60 cells in longitudinal and transverse directions. The
tidal boundary is set at the Golden Gate Bridge, where the tidal level has been
continuously recorded. The inflows from the Sacramento River, etc. are considered.
The Coriolis Coefficient is 0.00089. The effect of wind driving is not included in the
simulation. The simulation period is 120 hours long, from April 25 to April 30, 2003.
The simulation starts from a static condition (zero flow velocity), but in order to get
reasonable initial tidal flow field, one-day period is pre-simulated before the actual
simulation is accounted. The computational time step is 30 minutes.

Figure 4. Simulation Domain and Measurement Stations in San Francisco Bay.
Figure 5 shows the comparison of the measured and simulated water surface
elevations in four observation stations: Port Chicago, Richmond, Alameda and
Redwood. The amplitude and phase of the tidal level are well predicted. There is no
significant phase difference between the measurement and simulation. Figure 6 shows
the comparison of the measured and simulated flow velocities in two observation
stations: Richmond and Oakland. The measured velocities are those at upper, middle

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and bottom layers, while the simulated velocities are the depth-averaged values. The
general trend of the temporal variation of the velocity is reasonably well obtained.

Tide Level (m) 01..5501 Port Chicago SMimeausluarteedd+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Tide Level (m) 2 Richmond

Tide Level (m) 0 24 48 72 96 120 Tide Level (m) 10..5510 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0 24 48 72 96 120
Time (hour)
Time (hour)
2 Alameda
2.5 Redwood
01..0155 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0 24 48 72 96 120 01..10255 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
0 24 48 72 96 120
Time (hour)
Time (hour)

Figure 5. Measured vs. Simulated Tide Levels in San Francisco Bay.

1 Oakland Measured, at Upper (-1.52 m)
Measured, at Middle (- 3.35 m)
0.5 Measured, at Bottom (- 4.26 m)
Simulated

Velocity (m/s) 0

-0.5

-1 0 24 48 72 96 120
Measured, at Upper (-2.44 m)
1 Richmond Time (hour) Measured, at Middle (-4.27 m)
0.5 Measured, at Bottom ( -13.41 m)
24 Simulated
0
Velocity (m/s)-0.5

-1 48 72 96 120
0
Time (hour)

Figure 6. Measured vs. Simulated Velocities in San Francisco Bay.

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Conclusion

A depth-averaged 2-D model for tidal flow is established. The governing equations of
flow, salinity and sediment transport are discretized using finite volume method on
non-staggered, curvilinear grid. The time derivative terms are discretized by a full
implicit scheme, the convective terms are by the HLPA scheme, and the diffusion
terms are by the central difference scheme. The 2-D shallow water equations are
solved by the SIMPLEC algorithm with the Rhie and Chow’s momentum
interpolation method, which acquires the coupling of velocity and pressure and
eliminates the potential numerical oscillation often existed on the non-staggered grid.
The discretized algebraic equations are solved by using SIP method with fast
convergence.

The present model has been tested in Tokyo Bay and San Francisco Bay. The
simulated water elevation, flow velocity, salinity and sediment concentration are in
good agreement with the measured data. The tests show that the model is very stable
and efficient. Large time steps, such as 30 minutes, have been successfully used in the
simulation of semi-diurnal tidal flow. This model can thus be used for the long-term
simulation of estuarine processes.

Acknowledgement

This work is a result of research sponsored by the US State Department Agency for
International Development under Agreement No. EE-G-00-02-00015-00 and by the
USDA Agriculture Research Service under Specific Research Agreement No.58-
6408-2-0062 (monitored by the USDA-ARS National Sedimentation Laboratory) and
the University of Mississippi.

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