(1)
Li
Xi
Graduate student, Dept. of Civil Engineering, National University of Singapore,
10 Kent Ridge Crescent, Singapore-119260.
Address (after 12 Dec. 2000): Building 3 Unit 302,
Guangzhou Road 252, Nanjing 210029, China.
E-mail:
engp7844@nus.edu.sg
Abstract: In this paper, a new expression about sediment-loaded flow is given to predict the topography change under wave climate. The sediment transport and wave induced hydrodynamics are solved separately, which is more convenient to predict long-term beach evolution than solving a coupled equation. A total sediment transport rate is created based on filed survey and experiments. In applying the sediment transport equation to wave model, the topography change is considered to give a new bathymetry as input for the wave equations, which in return will give a time-dependent hydrodynamic environment. The wave transformation including wave breaking is simulated by solving Boussinesq-type wave equations, together with some verification and comparison of computational results with measurements in a physical model (BERKHOFF et al. 1982). The longshore currents due to eddy viscosity is simulated by the model under uni-directional waves, which have been applied in erosion control of reclaimed land by headland breakwaters. The shoreline is found in equilibrium by headland breakwaters and the equilibrium process is simulated with comparison to the engineering practice in East Coast of Singapore.
Keywords: topography change, sediment-loaded flow, Boussinesq equation, wave transformation, shoreline, viscosity coefficient
The geomorphology of beach is controlled by natural and unnatural effects. In coastal reclaimed land, the topography change is based on suspended sediment transport and near-bed sediment transport. No matter the topography change is in equilibrium or not, the exchange between the suspended sediment and bed-load sediment happens, and will be influenced by wave conditions and many other factors. Theoretically the determination of sediment concentration and flow characteristics is essential to give the solution of the sediment-flow system, and the prediction of sediment transport rate is important to the control of the erosion of the reclaimed land. Numerous equations have been suggested for predicting sediment transport rate, a good review is given by Nian-Sheng Cheng and Yee-Meng Chiew (1999). For wave-sediment system, when wave dominates current, the sediment movement and long-term topography change will mainly be influenced by the wave induced hydrodynamic environment, where the improved Boussinesq equations are applied.
In the case of reclaimed land, East Coast of Singapore, bathymetry changes are measured before the carryout of constructing a system of headland breakwaters and in the consecutive years after construction of the breakwaters. The model is calibrated and verified to provide the long-term prediction of evolution of bays between the breakwaters, which can be applied in the erosion control and to predict the equilibrium of the reclaimed land.
Differential
equation of the sediment-loaded flow can be expressed as following:
(1)
where
S is sediment concentration; Q
is flux per width in the length scale l
direction; h is water depth;
w is settling velocity of the sediment; S* is the sediment
carrying capacity; and a is the sediment
settling probability.
The
annual average depositional rate can be calculated by the following:
(2)
where n is the full period wave number per year. The
sediment settling probability is related to the grain diameter and water
velocity. The average value is about 0.67 when d50 is less than 0.03
mm and the water velocity is about 0.4 m/s (Luo, 1987). When the settling
sediment are very fine with diameter varying between 0.015 mm and 0.03 mm in the
field measurement, the settling velocity is considered as a constant when we
calculate the sediment transport rate. Since the water depth in reclamation area
is small, ωT can be replaced by k0H.
The constant, k0 is less
than 1, calibrated by the field measurement. Then the equation (3) can be
expressed as
(3)
Much more important is to calculate the exact value of
sediment transport rate, it can be expressed as the following:
(4)
where k and m
are determined by the local field data. The regression analysis can be used to
establish the sediment transport rate formula associated with the local
conditions. A semi-empirical formula is obtained by the regression analysis on
the basis of the local field data (Wang Yi Gang,1996, Li Xi, 1997)：
(5)
where U is
depth integrated water velocity; g is the local gravitational acceleration; A=290.0;
and m=1.28. The transport rate can be calculated by using formula (3)
and (4), if velocities are provided by the numerical model. Let k0
= 1, then following equation can be obtained from the equation (8) and (10)
(6)
The U
is depth-averaged velocity of a combined effect of wave induced mean current.
U1, U2 are the mean water velocity in a wave cycle before and
after the breakwaters system construction.
Continuity equation
(7a)
X-direction momentum equation
(7b)
Y-direction momentum equation
(7c)
In
which, x, y and z
form a rectangular coordinate system, with the plane coordinates of x,
y on the still water level, z
measured vertical upwards,
,
,
, D is the varying water depth or still
water depth, h
is the free surface elevation, u and v is the depth average velocity.
Where
dispersion terms y1
y2
in equations
8-(a,b) equals to
(8a)
(8b)
Fbx, Fby are the bottom friction term, they can be written as
(9a)
(9b)
where friction coefficient fb is introduced for the wave-current coexistent system by assuming a constant depending on breaking condition, for no-breaking situation a value as 0.015 is used and for breaking condition a value as 0.03 is used for calibration.
Considering
the diffusion term, MDx, MDy
, they can be written as
(10a)
(10b)
Though for sediment-loaded flow we generally use the depth-averaged diffusion coefficients for horizontal viscosity and a constant can and has been used instead. In this paper, laminar viscosity coefficient vL combined with the eddy viscosity coefficient vE due to wave breaking are used. They are coefficients calibrated from experiments and numerical model application.
The
finite difference method is used to solve the partial different equations as
controlling equations (P.A.Madsen, 1991). Defining the variables h,
P and Q
on a space-staggered rectangular grid system so called as ARAKAWA C grid, then
“side-feeding” and ADI algorithm is used to solve the problem.
Calculation of reflection coefficient enable the model to calculate the effect
of wave energy dissipation, it also includes porosity calculation to simulation
non-Darcy flow through porous media. With respect to the partial reflection,
transmission, absorption or dissipation of wave propagation energy, the flow
resistance inside the porous structure is described by the non-linear term in
the model.
For the verification of the validation of the combined model for refraction-diffraction of waves, simulation calculation is done for the topographic data of a hydraulic scale model (BERKHOFF et al., 1982). Computational parameters are H=0.05m, T=1.0s, and Dx=0.125 m, Dy=0.25 m. computation area is 25 m X 20 m (see Fig.1). The differences between the computational results of the present model and measurements in the hydraulic scale model (BERKHOFF et al., 1982) are shown in Fig.2(a)~(h). From Fig.2(a) ~ (e), it can be seen that the computation results are in agreement with measurements for sections 1~5. From Fig.2(f)~(h), it can be seen that the model is satisfactory in agreement with the laboratory measurement with proper accuracy.
Fig.1 Bottom configuration of hydraulic model (Berkhoff, 1982)
|
section (a) |
section (b) |
|
section (c) |
section (d) |
|
section (e) |
section (f) |
|
section (g) |
section (h) |
Fig.2 (a~h) Comparison between measured and computed wave heights following the
combined refraction-diffraction model in this paper
In
the past one hundred years, ten percent of industrial and commercial land of
Singapore has been reclaimed from sea. The shore protection of reclaimed land is
always of its concerns. A major erosion control method has been carried out is
the headlands control method. The beaches in East Coast of 8.3 km length have
been protected by constructions a system of breakwaters in headland, some data
validate the numerical model is from the measurements of Port Authority of
Singapore, and House Development Board. The growth of this headland-bay beach
were reported and concluded that current has no significant influences on
littoral drift at this area. From the wave observation, the predominant wave is
from southeast, based
on the wave extrapolation method, the wave height for 3 months monsoon return to
be 0.495m with a 30 degree to the shoreline.
The procedure to apply the numerical model is
(1) Give out the hydrodynamic environment before construction the breakwaters by solving equations (8a,b,c) and obtain mean depth velocity field U1;
(2) Calculate the mean flow field U2 after construction the breakwater;
(3) Predict the topography change by equation (7) and calibrate the model with field measurement;
(4) The changed topography is applied in new water depth for solving the Boussinesq equations, which give the new flow field as U2 at end of first year;
(5) Compare the flow field in step 4 as U2and the one in step 2 as U1 to give the second year topography change, then repeat the procession to predict the third year topography.
The
growing process is of minor change in these bays in 1990s' from Figure (3b),
which starting from the Figure (3a) as the initial condition. And it is shown
the effect of construction of detached breakwater to protect the stability of
beach.
Fig.3 (a,b) Flow field of the initial condition after and before construction of the
breakwaters in 03-07-1992
Fig.4 The predicted topography in the end of 1992 after construction of the brakwaters system
Fig.5 (a,b) The measured and computed topography comparison in 1998
Based on the analysis, it is concluded that the sediment and flow system can be solved numerically by introduction a new expression on sediment carrying capacity and also worthy to note the relationship among wave velocity, sediment carrying capacity, and diffusion coefficient. The solution proved valuable to engineering practice, and more verification and calibration is expected in the future.
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