(1)
Zhang
Zheng 1
,
Li Mengguo and Li Bei
Tianjin Research Institute of Water Transport Engineering, Tanggu,Tianjin, 300456, China
1 Tel: +86 22 25707168-382, Fax: +86 22 25795125,
E-mail:
zhang1_zheng1@sina.com
Abstract: Through continuous research of many years, a very applicable 2-D tidal flow mathematical model based on explicit difference numerical method of irregular triangular grids in coastal and estuarine waters has been gradually improved and perfected, and a practical product visualization software for tidal flow field has been developed. Combining the mathematical model with the visualization software constitutes a visualization mathematical model, which not only makes mathematical model visualized and directly perceived, but also accelerates the debugging of the mathematical model and makes the comparison work of various project schemes much convenient, thus highly raising the efficiency of the mathematical model in solving real engineering problems.
Keywords: coast and estuary, visualization, mathematical model, tidal flow
Tidal flow is one of major hydrodynamic factors in coastal and estuarine waters. In coastal and environmental works, such as construction of ports and harbors, dredging of navigation channels, reclamation of land, protection of beaches, discharge of sewerage into the sea, environmental protection and aqueous farming, it is necessary to have a comprehensive understanding of the tidal flow field. At present, mathematical model of tidal flow is one of the effective means to understand the tidal flow field, which has been widely used in computation of tidal flow field in various coastal and estuarine engineering problems.
The features in the coastal and estuarine areas are characterized by shoaling waters, complex submarine topography and shorelines, islands and man-made structures of various types and dimensions, such as quays, wharves, and breakwaters. These features often make the tidal field in this area even more complicated. Consequently, it is quite necessary to build a practical and effective mathematical model. The authors, based on their concrete practices in the past ten-odd years, believe that a mathematical model of 2-D tidal flow, which is established with the numerical method of explicit difference on the basis of irregular triangular grids, is very practical and useful in understanding of coastal and estuarine waters. The model is advantageous in many aspects. In the pattern of grids, the model can very well simulate the submarine topography and complicated boundary feature and the grids can be established freely. In discrete method, as explicit difference is used, the discrete equations in the model are simple to understand and can be easily programmed with a short procedure and debugged. In versatility, the model can be not only applicable to the calculation of irregular triangular grids but also applicable to the calculation of rectangular grids or square grids. Both rectangular and square grids can be considered as triangular grids composed of several right triangles, which are special patterns of irregular triangular grids.
One of the reasons why the mathematical model of tidal flow has been widely used is that it works fast. Production managers, designers and decision makers often demand that mathematical models should produce conclusive results as fast as possible so that the results can be timely consulted. Sometimes they ask for the results in one month and some even ask for preliminary conclusions in a week's time. Therefore it is quite necessary to increase the working speed of a mathematical model. A mathematical model is generally operated in three parts, pre-processing(division of grids and preparation of information and data); debugging of program and calculation, and post-processing(interpretation and analysis of the calculated data). Although the development of computer technique may increase the speed of the debugging of the program and calculation, the speed of the overall mathematical model can not be increased. Furthermore, the larger the domain of calculation is, the smaller the grids are, thus partially offsetting to some extent the increased processing speed of computer. In order to increase the running speed of the mathematical model, the authors have compiled programs and software for the three parts of the mathematical model.
From the first time when mathematical models were developed, they have been noted to have the shortcomings that they are not visible, in comparison with the physical models. The results produced by a mathematical model are insipid data and figures of great amount and static tidal flow field pictures based on these data and figures as well as some dull curves. To overcome these defects in mathematical models, the authors have developed a software to make the flow field visual. The visualization of a flow field is to reproduce on the screen the actual tidal movement on the basis of the calculated results of a mathematical model. The software is an executable program, separating the source program. The software is based on the operating platform of Windows 9X and the visualized flow field is then completed with interactive windows. The software for visualized flow field may be incorporated in the program of mathematical models and may also make visible the computed final results, thus facilitating the selection of various engineering proposals, so that the operational speed of the mathematical model may be increased and the practicability of the mathematical model is enhanced.
The paper presents a visualization mathematical model which combines a mathematical model with a flow field visualization software, which can not only make (the computed results of) the mathematical model visible but also increase the operating speed of the mathematical model. Compared with the traditional mathematical models, the visualization mathematical model is more practicable and runs much faster. The following is an introduction to the visualization mathematical model and its applications.
Generally, the basic equations for 2-D tidal flow may be written as follows.
Continuity equation,
(1)
Motion equations,
(2)
(3)
where, x
and y are coordinates in the
rectangular coordinate system coinciding with the still sea level ( a certain
datum level ); u and
v are tidal flow components in the directions of x
and y respectively; h is water depth ( distance between the datum level and the sea bed
);
is tide level ( distance between the datum level and the free water surface
); H is total water depth, H
= h +
; f is Coriolis coefficient; g
is gravitational acceleration; C is
Chezy coefficient,
, n is Manning roughness
coefficient; t is time;
is coefficient of horizontal
eddy viscosity.
The definite conditions of the basic equations include boundary conditions and initial conditions. For boundary conditions, measured value or analytical value of the flow velocity or tide level are taken for open boundaries while normal component of the flow velocity will be naught for fixed boundaries. For initial conditions, naught is taken for flow velocity while the initial value of the tide is taken for tide level.
2.2.1 Discretion of partial derivatives of first and second order in triangular grids
Any physical quantity,
, which represents any one of
and
, in any triangular element on a plane can be written into the following
approximate expression of first order.
(4)
in which
and
are three coefficients. If the coordinates of the
three vertexes,
,
and
arranged anti-clockwise, in a triangle are known to be (
),(
) and
(
) respectively and the values of the physical
quantity,
,
and
are also known, the following equations are obtained.
，
，
(5)
The three coefficients can then be derived from equations (5) as below
,
，
(6)
The area of a
triangular element,
is given below in (7)
(7)
From Equation (4), the partial derivatives of first order for physical quantity, F, may be obtained as
,
If point M is the common vertex of m triangles, the partial derivative of first order for this point can be the weighted average of the partial derivatives of first order for m triangles at the same point, that is,
，
(8)
After the partial derivatives of first order are obtained, the partial derivatives of second order may be obtained in imitation of the procedure for solution of the partial derivatives of first order.
2.2.2 Difference equations
Forward difference is used for time derivatives, while the pattern of the above discretion is used for spatial derivatives, thus the above 2-D basic equations (1), (2) and (3) may be discreted separately and re-arranged into the following equations.
(9)
(10)
(11)
where, K
is the number of time layer, and
is the time step.
Thus far, a mathematical model of 2-D tidal flow by means of the numerical method of explicit difference of irregular triangular grids is given and it may be seen that the numerical method is very simple. With difference equations and equations for spatial partial derivatives, the program can be easily made. The difference method of irregular triangular grids (may be regarded as) is the generalization of the finite difference method of traditional rectangular grids on the nodes of triangular grids[1] .
The visualization software can be run directly on Windows 9X and the main menu window of the software (version 1.0) is as shown in Fig 1. The software has the following functions and characteristics.
Fig. 1 Main menu of the Visualization software for flow field ( version 1.0)
(1) The visualization software for flow field can complete the creation of visualized tidal flow fields with the data files obtained from mathematical models just by means of simple operations of interactive windows and it will not be necessary to touch the source program which is used to make the visualized flow fields, thus making it a common software. Personnel who deal with general mathematical models can easily master the use of the software if they follow the operational manual of the software.
(2) The visualized flow field generated by the software is the Lagrange field in which the motion of particles is indicated by arrows, thus actually realizing the reproduction of real tidal movement on the screen and completely overcoming the defects of non-visualization of mathematical models.
(3) The visualized flow fields exist in a certain form of data file and can be displayed at any time. While this data file can be inserted into Powerpoint, any static frame of it may be intercepted.
(4) The visualization software for flow fields can be used to directly process the data and results of the mathematical models of square grids, triangular grids and rectangular grids and can generate visualized flow fields accordingly.
(5) The visualization software for flow field can not only be used to make the flow fields of the whole computation domain visualized, but can also be used to make the fields of any sector in the computation domain visualized by independent amplification.
(6) Information such as titles, tidal curves, dates, name of location, compass, scales of pictures and velocity, etc. may be inserted in the screen displaying the visualized flow field. The location of the inserted information may be shifted from one location to another and the size of some inserted information may also be changed. The topographical outlines of the sea bottom may also be inserted.
(7) The displaying frame of the visualized tidal field and the size of the arrows of tidal flow can be freely readjusted as per scales and the density of the arrows can also be freely readjusted.
(8) A quantity value of characteristic flow velocity, which shall not be a naught, may be assumed. When the actual flow-velocity is greater than the given characteristic flow velocity, the arrows indicating the flow-velocity may change their color to show the difference.
(9) All the information in the main menu of the software may be saved and stored in the form of a data file, which may be also read directly in the already saved data file from the main menu.
The visualization software for flow fields has almost covered all the aspects for making the visualization of flow fields.
Incorporating the mathematical model with the visualization software, as described above, will create a visualization mathematical model. Like conventional mathematical model, the working procedure of the visualization mathematical model also include pre-processing, debugging of program and computation, and post-processing.
The difference between visualization mathematical model and conventional one is that the visualization mathematical model incorporates the visualization software for flow fields in the debugging and post-processing of a mathematical model. During the debugging and verification in the conventional mathematical model, static flow field pictures are used to verify the plane flow situation, which is not very convenient when the computation domain is large and there are many nodes of grids. But, in a visualization mathematical model, dynamic flow fields are used to verify the plane flow situation, which significantly overcomes the defects in static flow field pictures, and the open boundary conditions may be amended at any time in accordance with the observed change of the flow fields, thus greatly increasing the speed of the debugging of the model verification and the working efficiency. During the post-processing, the proposed project schemes can be evaluated in accordance with their dynamic flow fields, thus significantly improving the efficiency of the mathematical model in solving practical engineering problems.
The visualization mathematical model of 2-D tidal flow has been satisfactorily used in the numerical simulation of tidal flow in Shanghai Yangshan Harbour area [2].
The proposed Shanghai Yangshan Harbour area is located between Da Yangshan and Xiao Yangshan in Qiqu Archipelago off Shanghai, which is about 86 km away from the city center of Shanghai. To optimize the layout of Yangshan Harbor and the proposed scheme of navigation channels, it is necessary to carry out the studies on the tidal flow in the harbor area by means of mathematical models. The problems faced by the mathematical models are wide calculation domain, which is 56 km by 34 km, a lot of islands (more than 40) in the calculation domain, complicated topographical conditions with wavy terrain running ups and downs, swift and disorderly flow, open boundaries on all the sides, many comparable alternatives of the layout (more than 10 alternatives to be compared), and short duration for carrying out the mathematical model studies and providing the results of the studies. Fortunately, by using the visualization mathematical model presented herein, the above problems are successfully solved, manifesting the model in the following three major aspects. First, the irregular triangular grids have not only generalized, in a better way, the complicated topography, numerous islands and complex configuration of various layout schemes, but also the features of the model allowing the triangular grids to be freely arranged have made the total number of nodes in the grids in the calculation domain to be not too many, thus cutting down the calculation time. Secondly, the numerical method of explicit difference based on irregular triangular grids can be easily used to solve the problems of swift and disorderly flow and open boundaries on all the sides. Thirdly, visualization of the flow fields has helped and thus expedited the debugging and verification of the program, thus achieving rapid comparison among and selection from various scheme proposals. Figure 2 shows a static picture of the visualized flow field for one of the scheme proposals.
Fig. 2 A static picture of the visualized flow field taken from the numerical tide model of Shanghai Yanshan Harbor
The success in the numerical simulation of tidal flow in Shanghai Yangshan Harbour area has proven the high-efficiency and practicality of the visualization mathematical model presented in the paper.
The paper presents a visualization mathematical model of 2-D tidal flow based on explicit difference method of irregular triangular grids, which is a combination of the conventional mathematical model and the visualization software for flow fields. The mathematical model based on explicit difference of irregular triangular grids is advantageous in many aspects. The model can very well simulate the topography and complicated boundary feature, the grids can be established freely, the numerical method is simple to understand and can be easily programmed with a short procedure and easily debugged, and the model is applicable to the calculation of various patterns of grids, such as triangular grids, rectangular grids and square grids. The successful development of the software for visual flow field has brought a visualization mathematical model of tidal flow to reality. The visualization mathematical model is one great step forward from a conventional mathematical model. It has not only made mathematical models visualized, overcoming the defects of invisibility of the conventional mathematical models, but also improved the running speed of mathematical models. The practical use of the visualization mathematical model presented in this paper has proven its high-efficiency and practicality. It is worth popularizing and applying the visualization mathematical model to numerical simulation studies of tidal flow in coastal and estuarine waters.
References
[1] Li Mengguo and Cao Zude, 1999 . A review on tidal current numerical modeling in coastal and estuarine waters. Acta Oceanologica Sinica , 21(1), 111-125. (in Chinese).
[2] Li Bei and Zhang Zheng, 1999. Research on Numerical Simulation of 2-D Tidal Flow for Phase I of Yangshan Harbour Project of Shanghai International Shipping Center. Technical Report, Tianjin Research Institute of Water Transport Engineering, Ministry of Communications. (in Chinese).