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Bai Li1
Department of Mechanics, Dalian university of technology, Dalian 116023, China
Tel:0411-4708391-8003(O) E-mail: baili@student.dlut.edu.cn
Zhou Zhu2, Qiu Xiuyun2 and Hou Jie2
School of Water conservancy and Civil Engineering,
Xinjiang Agricultural University, Urimqi 830052, China
Tel(Fax):0991-4541793(O) E-mail: xjsgs@sina.com
Abstract: A high-performance finite volume method (FVM) is adopted to solve hydrodynamics characteristic of Bosten Large Lake District in this paper. Equations of Unsteady 2D shallow water circulation flow expressing by conservational variables is educed. Unstructured mesh which can be arbitrary triangle, quadrilateral, or combinations of them is implemented to fit irregular boundaries. The cell-center formula of variables is defined in principle of FVM. The formulation of Wind force in FVM discretions is discussed in specific. Based on the principle of characteristic of variables in governing equations, the discreted equation can be solved effectively and satisfactorily by introducing a High-performance Algorithm—Osher scheme. It can be considered as a high efficiency and practical implement for the solution of shallow circulating flow problems especially concerning complex geometries. The solutions show that the characteristic of current field is obviously different under different wind direction. We can revealed that the input fresh water flows toward outlet quickly before it blends with lake water adequately when wind blows from Northwestern, which is the main reason why the water quality distributing has misproportion property.
Keywords: Characteristic FVM, unstructured mesh, shallow water equations, water quality wind force
Environment protection is one most important and basically question in carrying on the Western Develop Strategy. Bosten Lake is the largest freshwater lake in Xinjiang and plays an important role in local economical development. Bosten Lake has multi-function such as regulating poundage, providing aquatic organism resources and improving local climate. But during the past twenties years, salinity in Bosten Lake increases continually. The water quality is worsening quickly with receiving large quantities of water polluted by papermaking and tan industry. In order to recover its freshwater quality and contain natural condition, the study on hydrodynamics characteristic should be carried on because it is hydrodynamics process in lake that plays a dominant role in varieties material and energy transfer in lake.
Bosten Lake lies in the bottom of Yanqi basin at southern of Tianshan Mountain where often blows Northwest wind. The main source comes from the Kaiduhe river and flow into the Kongque river. It composed of the large Lake District (LLD) and the small Lake District (SLD). The former is mainly discussed in this paper. Both inflow and outflow local on Westside of LLD. The LLD area is about 1002.4km2 with the water surface elevation 1048.75m. The mean water depth is 8.2m.
Numerical method is one of the most important approaches for environmental fluid flow problem. The general shallow water circulation model has been implemented successfully. But the studies of complex problems such as irregular boundaries, free surface are still difficult to solve effectively. It is necessary to find an expectable method to resolve this problem because most of natural lakes have irregular boundaries and complex topographies.
The cell-center formula of variables is defined in principle of FVM in this paper. That is mesh node is dividing from the element, the obtained discrete equation from which means mass and momentum keeping conservation in the element. The mesh can be arbitrary triangle, quadrilateral, or combinations of them. It is possible to simulate irregular boundaries smoothly. The explicit form of the disctete equation both in space and time scheme are implemented also. According to the theory of air similar to water, the normal flux formula—Osher formula has be transported, it can simplify the calculation and improve the accuracy.
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The conservational form of equations should be adopted with vector member. The unsolved variables are replaced by conservational variable named U (h, hu, hv)T. The vector form is adopted:
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and replace b by hb’ the equation was transferred as follows:
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in which,
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resource item are composed by:
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in which, S0 is the slope of bottom, Sfx is bottom fiction resistance.
It can be expressed by Manning empirical formula:
; in which, n means roughness coefficient;
τs wind
force on surface.
,
, the atmosphere density
，CD is wind dragging coefficient in empirical formula.
, w10 means the wind velocity
of ten meters above water surface. f means Coriolis coefficient, and
the rotating angular velocity of
the earth ,
.
F (U) and G (U) in equations have rotational invariance property. The solutions of equations have index property which can simply described that a consistent equation should be contented when a dimensional wave always spreads along the typical line. That is why it called characteristic method (see Ref. [1] in detail).
The Large Lake District is divided 365 element combined by triangles and quadrilaterals (Fig. 1). Define the control volume on each element and the unsolved variables are node-node arrangements (Fig. 2). The node corresponds to the middle of an element and the middle states between two conjunction nodes correspond to the interface states of public side between two elements. To solve an arbitrary control element is related to the three or four elements surrounding it. If the number of all the elements and nodes is known, the topological relations between the solving element and its neighbor can be predetermined. Then the numerical fluxes of all the sides of solving elements can be determined by Osher scheme (see Ref. [1] in detail).
For the element i, the integral form of equation for the inner region and the boundary can be written as:
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where A represents the area of the region Ω, dl denotes the arc length of the boundary ¶Ω, and n is a unit outward vector normal to the boundary ¶Ω. The vector U is assumed to be constant over an element. Applied Gauss-Green formula and transferred Equation. (10) by transformation matrix T(f) (see Fig.3), the basic equation of the finite volume method can be obtained:
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T=
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in which, suppose
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At last, for the element i, the explicit form both on space and time of discretion equations can be written as:
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where A means the area of each element, j represents the public side between conjunction element, m the sides of each element. Tj-1 (f) is inverse transformation matrix.
How to get normal numerical flux that named FLR is critical for realizing solution. The state variables of the interface middle point between conjunction elements need to obtain. A high performance formlua named Osher scheme is introduced (see Ref. [1] in detail). The formula of normal numerical flux according to the principle of Osher scheme is:
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Where L is lift or inside corresponding to R is right or outside of the public side for element i. lk represents eigenvalue while rk represents eigenvector (k=1,2,3) of characteristic equation in Osher scheme.
In generally, the boundaries of the computational domain have solid boundaries and open boundaries. For land boundaries slip conditions are supposed. But the open boundary conditions need to be discussed specifically. The local value of the Froude number or whether the flow is subcritical or supercritical is the basis of determining the number of boundary conditions. For supercritical flow, three conditions at the inflow boundary and none at the outflow boundary must be specified. For subcritical flow, two external conditions are specified at inflow boundary and one is required at the outflow boundary (see Ref. [2] in detail).
The outside flow state of boundaries can be defined according to the given discharge duration or elevation duration and given flow state at inside of boundaries, then defined out-normal numerical flux as same as inter element. the depth can be obtained by the following iterative formula:
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The simulation is implement during the non-freeze-up period, the wind force play an important role in lake current circulating activity. Fig.4 shows a main clockwise circulation over computation domain at northwest wind at the average perennial speed of 3m/s. Current velocity is centimeter magnitude basically. In Fig.5 shows a reverse circulation at northwest wind with same speed. The difference isn’t obvious weather bottom fiction resistance is considered or not considered.
Different from the hydrodynamics characteristics of nonfreezeup period, lake has been covered by icecap in freeze up period, at the same time decreased input discharge caused to the lower self-purification capability of lake. So the main motion form include shear discreteness, molecule diffusion and turbulent diffusion in winter. All this results in water quality worsening and being very higher nutritious.
It showed that CFVM is a high efficiency and practical implement for the solution of shallow circulating flow problems concerning complex geometries and irregular boundaries. The wind force plays an important role in lake current circulating activity. The results indicated that the input fresh water flows toward outlet quickly before it blends with lake water adequately, it doesn’t benefit decreasing salinity but cause water quality in a inhomogeneous. The water quality including salinity demonstrates the misproportion characteristic can be explained clearly.
[1] W. Y. Tan, S. Y. Hu, 1991: a General High-performance scheme of 2D shallow water flow—finite volume Osher scheme, Advances in Water Science, 2(3), 154-161.
[2] W. Y. Tan, S. Y. Hu, 1994: The realization of one order FVM with Osher scheme of computation of shallow water, Advances in Water Science, 5(4) (in Chinese), 262-269.
[3] K. Anastasion, C. T. Chan, 1997: Solution of the 2D shallow water equation using the FVM on unstructured triangular meshes, International Journal for Numerical Method in Fluids, 24,1225-1245.
[4] J. S. Wang, H. G. Ni, 2000: A High resolution Finite volume Method for solving shallow water equations, Journal of Hydrodynamics, Ser, B, 1,35-41.
Fig. 1 The division mesh of Large Lake District
Fig. 2 The rotation of coordinate
Fig. 3 The rotation of coordinate by transformation matrix T
Fig. 4 The velocity field of Bosten Lake under northwest wind at the speed of 3m/s
