(1a)
Zhao Mingdeng，Li
Yitian and Cao Zhifang
Wuhan University, Wuhan, China, Tel: 86-27-67803461
Abstract:
This paper presents new weighting functions in
grid generation and new discretizing scheme of momentum equations in numerical
Simulation of river flow. By using the new weighting functions, the curvilinear
grid can be concentrated as desired near assigned points or lines in physical
plane. By using the new discretizing scheme, the problems of movable boundary
and dry riverbed can be solved. As an application, the influence on flood
control by Hankou flood-plain
regulation is studied. The new method is applicable to the numerical simulation
of river flow with irregular region and moveable boundary.
Keywords:
gird generation, moveable
boundary, flood-plain regulation
In natural rivers, the boundary and section shapes are often irregular, which is a problem for numerical simulation. To solve this problem, a coordinate transformation method is often used to transform the irregular region in physical plane (X, Y) to the regular region in computational plane (x,h). In this paper, we present some new formulas. By using these new formulas, we can generate grid automatically, and concentrate the grid as desired near assigned points or lines in physical plane. On the flood-plain, the boundary moves with the flood level. To solve this movable boundary problem, we rearrange the resistance item and present a new discretizing scheme, which can avoid the inaccuracy from the method of “false depth ”.
As an example, we apply the mathematical model and grid generation to the Yangtze River flow in Hankou flood-plain regulation engineering and research the influence on the flood control. Hankou flood-plain is about 200m wedth, 12km long from Hanjiang River Mouth to Zhujiahe River Mouth. It is valuable to exploit the flood-plain. To prettify the appearance of Wuhan City, improve the living environment of residents and build Wuhan City as an open, multi-function, internationalizing city, it is necessary to exploit the flood-plain. On the other hand, to exploit the flood-plain will reduce the cross-sectional area and influence on flood control of Wuhan City. With reference to the measured data in Wuhan Station, the maximum flood is listed in Table 1, four times of which are taken place in resent years. So it is essential to study on the influence on the flood control.
Table 1 Flood lever in Wuhan Station
|
Flood lever (m) |
29.73 |
29.43 |
28.89 |
28.66 |
28.28 |
28.11 |
27.79 |
|
Year |
1954 |
1998 |
1999 |
1996 |
1931 |
1983 |
1995 |
The governing equations of coordinate transformation are Poisson equations:
(1a)
(1b)
Where P(x,h)
and
Q(x,h) are weighting functions which cause the coordinate
line to be concentrated as desired. P(x,h)
and
Q(x,h) can be expressed as[1,2]:
(2a)
(2b)
Where P and Q are depended on x ,h. The coordinate lines can be concentrated as desired near point (xl ,h l) or line (x=x k , h=h h). In fact, what we desire is that the coordinate lines can be concentrated near fixed point (Xn ,Yn) or line Y=fm (X) (x ,h¹constant) in physical plane. It is difficult to concentrate the coordinate lines as desired by using the formulas (2), new formulas are presented as follows.
Because it is easier to transform rectangular grid in computational plane (x,h) into curvilinear grid in physical plane (X,Y), the equations (1) should be transformed into the following equations:
(3a)
(3b)
Where
To concentrate the coordinate lines as desired in fixed point (Xn ,Yn) and line (Y=kmX+bm) in physical plane, we propose two new formulas as follows:
(4a)
(4b)
Where Xn and Yn are coordinate values in physical plane, Y=kmX–bm is the equation of line in physical plane(Fig.1).
As an example, we transform a rectangular region (00≤X, Y≤1) in physical plane to a computational region (0≤x, h≤1), and we desire that the coordinate lines will be concentrated near point (Xn = 0.25, Yn =0.25) and line Y +1.8X – 1.93=0. By using the new method proposed above, a satisfactory curvilinear grid is generated (Fig.2). In the computational plane (x,h), the grid is uniform (21´21), Dx=Dh=1/20. In the physical plane (X,Y), the grid is uniform on the boundary line, DX=DY =1/20, but in the inside of the region, the grid is nonuniform, the grid near point (0.25,0.25) and line Y +1.8X–1.93=0 is concentrated as desired, DX,DY <1/20.
Fig.
1 Fixed point and line in physical plane
Fig. 2 Grid in physical plane
For large ratio of width to depth in river flow, the governing equations are 2-D depth-averaged shallow water equations.
Continuity equation:
(5)
Momentum equations:
(6)
Momentum equations in X-direction
(7)
Momentum equations in Y-direction
(8)
Where Ux and Uy are depth-averaged velocities along the two coordinate axis, H and z are water depth and water level.
Using the boundary-fitted grid, the governing equations must be transformed to the transformed plane (x,h)[3].
Continuity equation can be written as:
(9)
Momentum equations can be transformed as:
(10)
Where
Momentum equations (10) are connective and diffusion equations, which can be solved by the finite analytic method. In an element, the finite analytic solution can be written as
(11)
Where Cn ,Cp are finite analytic coefficients[4],
the continuity equation can be solved
by the finite difference method.
Riverbank boundary changes with water lever. When riverbed is over the water lever, the depth should be zero, but when the depth is equal to zero in computation, the mathematical overflow will take place. To solve this problem, many scholars assume a false depth (e.g. H=5~10cm), but the false depth will bring inaccuracy and unsteadiness in computation. We give a new discretizing scheme with high accuracy and good stability, in which the false depth is very small (H=10-20cm). The new discretizing scheme of momentum equation is:
(12)
This new discretizing scheme can be used in
moveable boundary and dry riverbed.
The computational domain is 50km long from Zhuankou to Yangluo, and 1000m~2000m wide from south bank to north bank. The lines of Hankou flood-plain regulation are shown as in fig.3. For project A, B and C, the distance from the regulation line to flood wall is 120m,150m and 190m.We desire to transform the irregular region to regular region in computational plane (x,h), and to concentrate the coordinate grid near the regulation line. It is difficult to concentrate the coordinate line as desired by using the formulas (2). By using the new formulas (4) proposed above, the desired boundary-fitted curvilinear coordinate grid (317´61) is generated (Fig.4).
For
model verification, the measured data in 1998~1999 were selected as verification
data. Fig.5 shows the comparison of water surface curve between measured and
computational results. Fig.6 shows the comparison of the velocity distribution
between measured and computational results. It shows that the computational
results are in good agreement with that of the measured. The maximum relative
errors are smaller than 1% for velocity, Which shows that the present model is
suitable for the simulation of flow in natural rivers.
Fig. 3 The computational domain
Fig. 4 Grid in physical plane
Fig. 5 Comparison of water surface curve
Fig. 6 Comparison of velocity
distribution Fig.
7 Celocity distribution for different
project
With reference to the measured data, the maximum flood takes place in 1954(Qmax=75900 m3/s,zmax=29.73m), so the measured data in 1954 were selected as Computational data. Under the influence of the flood-plain regulation engineering, the raising of water level in Wuhanguan section is about 2cm, the raising of water level is increase with the distance from the regulation line to flood wall. Fig.7 shows the velocity distribution. Under the influence of the flood-plain regulation engineering, the computational discharge is 75900m3/s. The computational result shows that the influence of the flood-plain regulation engineering on flood control is very small.
(1) By using the new weighting functions in grid generation, the curvilinear grid can be concentrated as desired near assigned points or lines in physical plane.
(2) By using the new discretizing scheme of momentum equations in numerical simulation of river flow, the problems of movable boundary and dry riverbed can be solved.
(3) The new method is applied to the Hankou
Section of Yangtze River; the computational results are in good agreement with
the measured results. The new method is applicable to the numerical simulation
of river flow with irregular region and moveable boundary.
Acknowledgments
The research is supported by the “Chenguang Jihua” Foundation of
Wuhan City (985003080) and the
National Natural Science Foundation of China (No. 59890200 ) .
References
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[2] Wang Jia-he, Wang Chao, 1994: Numerical Simulation of 2-D Tidal Flow and Water Qualit under the Curvilinear Coordinates, Journal of Hydrodynamics, Ser.B Vol.6, No.3, 78-84.
[3] Zheng Bong-min, Zhao Ming-deng, 1991: Grid Generation in Space and Calculation for 3-d Turbulent Flow, Journal of Hydrodynamics, Ser.B Vol.3, No.11,66-71.
[4] C.J.Chen, Hnaseri-Neshat & K.S.Ho, 1981:Finite Analytic Numerical Solution of Heat Transfer in 2-D Cavety Flow, Numerical Heat Transfer,Vol.4,179-197.