DEVELOPMENT AND APPLICATION OF A 2D HORIZONTAL BOUNDARY

Abstract: In order to compute tidal currents in a lake with a narrow tidal channel or boundary-complicated shoreline, a 2D horizontal boundary-fitted model is developed in combination with Alternating-Direction-Implicit method. The model¡¯s utility is shown by the application to Lake Notoro, Hokkaido, Japan. Changes on tidal currents in Lake Notoro are reproduced based on the process of the construction of breakwaters. According to the velocities of computation results, the scour mechanism of channel bed can well be understood and further field surveys or countermeasures can be directed. As a result, this model can be applied to simulate the tidal currents in a boundary-complicated gulf with a narrow channel, and can provide useful suggestions on scour and sedimentation problems.

1 INTRODUCTION

The stability of a tidal inlet must be quantitatively evaluated in the detailed design procedures of channel structures. Numerical computation provides a quantitative understanding than a macroscopic method (O¡¯Brein,1931), though it needs more detailed field data.

For a numerical model to solve the motion of estuarine and near-shore waters, there are some problems in properly resolving the narrow channel or the complex shape of the shoreline contour. The most widely applied approach is to employ a finite-difference grid, which approximates the boundary of the flow domain by a series of line segments parallel to the Cartesian coordinate axes (Leendertse and Gritton, 1971). The method is an Alternating-Direction-Implicit solution procedure for the non-linear shallow water equations in two dimensions. A drawback of such an approach is that the solution may poorly represent the flow conditions close to the coastline and the alignment of a narrow channel. Although nested models are used to reproduce the flow in an estuary, the boundary-fitted model proposed in this study is more direct to solve the complex morphology. This is because that the model allows the use of a finite-difference scheme, and accurately follows the boundaries.

Johnson and Thompson (1978) first presented the non-orthogonal boundary-fitted versions of vertically integrated equations of motion and pollutant transport. In the present study, by combining the scheme of Leendertse¡¯s and non-orthogonal coordinate, an improved boundary-fitted model is developed in order to solve the problem when computing the narrow channel of an inlet and irregular shoreline. And as an example, the model is applied to reproduce tidal currents and research the effect of construction of breakwaters in Lake Notoro, Hokkaido, Japan.

2 NUMERICAL METHOD

The two-dimensional vertically-averaged equations of continuity and motion with curvilinear coordinate system can be written, respectively, as:

Where U and V are the depth-averaged velocity components in the x- and y- directions; is the surface elevation above a fixed horizontal datum; g is the gravitational acceleration; f is the Coriolis parameter; H is the total water depth;

and

are the wind stresses in the o- and y- directions, respectively.

and

are the curvilinear coordinates. And J is the Jacobian of the transformation given by

. As the boundary-fitted methodology is to transform the governing equations from the physical domain into a curvilinear domain, the space interval of grid near boundaries can be properly adjusted and not affect the stability of computation. The equations are solved with staggered-mesh in space by Alternating-Direction-Implicit method. The main computational scheme is presented as follows.

One time step is divided into two steps. If k is defined as a general time step, from k to k+1/2 velocities

and water levels

along axis

are calculated using implicit method by Thomas Algorithm. Velocities

along axis

are computed using explicit method. Similarly, from k+1/2 to k+1 the velocities

and water levels

along axis

are calculated using implicit method by Thomas Algorithm. And velocities

along axis

are computed using explicit method.

3 TIDAL CURRENTS IN LAKE NOTORO

3.1 Introduction on lake notoro

As shown in Fig. 1, Lake Notoro with surface area 58.4 km²is located on the northern-east coast of Hokkaido, and faced to the Okhotsk Sea. The breakwater that surrounds the tidal inlet was constructed from June 1971, and was completed in Oct. 1986. Its construction process is shown in Fig. 2. The Outer Channel is 370m wide, 690m long. And the Inner Channel is 200m wide, 200m long. In the Inner Channel, the breakwater is constructed with double sheet piles. The depth of sheet piles is among ¨C10.7~ ¨C13m.

It has been reported that the sheet pile of the east breakwater (shown in Fig. 2) was destroyed and the upper structures fell into the channel on July 15-16, 1997. In order to understand the mechanism of the erosion process, tidal currents are investigated in this research using the developed model above.

3.2 Relation between erosion and flow pattern

Fig. 3 shows the erosion situation in the Inner Channel from 1974 to 1978. The channel bed close to the west breakwater was obviously scoured, and the east shoreline retreated remarkably. That happened before the east breakwater was constructed in 1979. There were no significant changes in the Inner Channel from 1979 to 1983.

Fig. 4 shows the erosion situation from 1983 to 1997. During the period from 1983 to 1986, which is in the construction process of offshore breakwater, erosion of channel bottom close to the east breakwater had begun. That suggests the construction of offshore breakwater changed the flow situation in the Inner Channel, especially during flood tides. The final shape of the cross-section in the Inner Channel is like the character ¡®W¡¯. That is, the scour close to the breakwater is serious, as the maximum depth of bottom has exceeded the depth of sheet piles. Hence, it is necessary to apply the developed model to understand the erosion process at the bottom close to the breakwaters.

Firstly, using the bathymetry in 1974, maximum velocities during a flood tide are shown in Fig. 5. It shows large velocities close to the center of the Inner Channel, where is close to the west breakwater after the east breakwater was constructed. And as the east shore of the Inner Channel was not fixed with any structure, sands near the shore scoured away because of fast flow. The flow patterns agree well with the scouring as shown in Fig. 3.

And then, in order to reproduce the flow situation due to the offshore breakwater construction, bathymetry in 1985 is applied. As shown in Fig. 6, the maximum velocities in the Inner Channel during a flood tide become greater than those in 1974. And large velocities close to the two breakwaters correspond to the erosion situation near the two breakwaters. As the bottom close to the west breakwater had been scoured before, shallow bottom close to the east breakwater could be scoured considerably. Thus, the simulation results properly reflected the construction effect of the offshore breakwater on the streamline changes in the Inner Channel.

4 CONCLUSIONS

(1) A 2D horizontal boundary¨Cfitted model is developed to compute tidal currents in a lake with a narrow inlet channel and complicated shoreline.

(2) The model is capable to reproduce the ¡®W¡¯ type scour in the Inner Channel of Lake Notoro, Japan.

[1] Johnson, B.H. and J.F Thompson (1978). A Discussion of Boundary-fitted Coordinate Systems and their Applicability to the Numerical Modelling of Hydraulic Problems, Paper H78-9, U.S. Army Engineer Waterways Experiment Station Hydraulics Laboratory, Vicksburg, USA.

[2] Leendertse, J.J. and E.C. Gritton (1971). A Water-Quality Simulation Model for well-fixed Estuaries and Coastal Seas: Vol 2: Computation Procedures, The Rand Corporation, R-708-NYC.

[3] O¡¯Brein, M.P. (1931): Estuary Tidal Prism Related to Entrance Areas, Civil Eng., vol. 1, No. 8, 738-739.

Fig. 3 Scouring in Notoro Inlet from 1974 to 1978 Fig. 4 Scouring in Notoro Inlet from 1983 to 1997

(5)

Fig. 5 Velocity Distribution during A Flood Tide in 1974

(6)

Fig. 6 Velocity Distribution during A Flood Tide in 1985