(1)
Jiang
Enhui, Zhao Lianjun
Institute of Hydraulic Research, Yellow River Conservancy Commission,
Zhengzhou 450000, China
Huang Yuandong, Zhang Hongwu
Qinghua University, Beijing 100084, China
Abstract: Based on the fitting method of boundary coordinate, the solution by dispersion for flow continuity equation and flow motion equation is worked out with ADI method on alternative grids. The solution by dispersion for sediment continuity equation is worked out with the method on control volume. The basic frame of model is set up by the newest research results of sediment continuity equation, bed deformation equation, capacity of sediment-laden flow and bed roughness etc introduced. This model is verified by 1982’s flood. It is shown from the results that the process of peak propagation, the process of water level variation and the process of sediment concentration calculated by the model tally with measured results well. The velocity field, flooded depth and range, and sediment concentration field etc are identical with the fact.
Keywords: sediment mathematic model, two dimensions of plane, velocity field, flood stage
Yellow River is a sediment-laden river known at home and abroad. On one hand, it forms a wide plain on the Yellow River, Huaihe River and Haihe River, on the other hand, the special secondary suspended river is formed step by step in the process of deposition development flowing a plain region with Yellow River, Huaihe River and Haihe River, it has brought heavy disaster for people on both banks. Up to now, the bed in the lower reaches is still rising continuously and threat of flood hazard is still existed.
With the subject development of calculated fluid mechanics, two-phase flowing with liquid and solid, mechanics of sediment movement and evolution of river bed etc., theoretical research level of two dimensional sediment mathematic model of plane and its utilized range and depth in engineering have been raised as well. At present, more and more scholars throw themselves into the development of two-dimensional sediment mathematic model of plane.In respect of studying on two-dimensional sediment mathematic model of plane in the lower Yellow River, the models established by the scholars of Wei Zhilin, Cheng Xiaotao, Yang Guolu, Li Dongfeng and Zhang Shiqi successively are used in many projects of scientific-research production in the lower Yellow River. But there is still certain error in many main factors of calculated processes of flood propagation and stage variation, and deformation of bed erosion-deposition with natural measured results. Based on summing up research results of predecessors, according to the characteristics of incoming water with high sediment concentration and serious deformation of bed erosion-deposition in the lower Yellow River, two-dimensional sediment mathematic model of plane has been developed and verified systematically by 1982’s flood.
The basic equations of flow and sediment movement are used in model’s calculation:
Flow continuity equation:
(1)
Flow momentum equation (ignoring wind stress and Coriolis force):
(2)
(3)
Sediment continuity equation:
(4)
Bed deformation equation:
(5)
Where z—water level; h—water depth; u and v—average velocity of vertical in direction of x, y; g—acceleration of gravity; C—Chezy’s coefficient; vt—coefficient of turbulent viscosity. S—sediment concentration (kg/m3); V—velocity of flow; ω—sedimentation velocity;α*—recovery coefficient of average concentration; K1—additional coefficient; f1—unsaturated coefficient; S*—sediment carrying-capacity of flow; Zb—bed elevation; γ0—dry unit weight of sediment deposits.
The recovery coefficient of equilibrium
concentrationα*
can be calculated by the formula below[1]:
(6)
In which
(7)
Where κ—Karman constant of muddy water; cn—parameter of eddy (cn=0.375κ); u*—friction velocity.
The research results of reference literature [2], additional coefficient K1 and unsaturated coefficient f1 are separately calculated by the formula below:
(8)
(9)
The selection of sediment-carrying capacity formula of flow is one of key factors of calculating on channel mathematic model successfully in the lower Yellow River. The calculating results introduced sediment -carrying capacity formula of flow are smaller, in order to make deposition amount of bed not be overloaded, the sediment recovery saturated coefficient multiplied above becomes much smaller. Thereby the process of alternative development on channel erosion-deposition is not affected. Thus sediment-carrying capacity is calculated by the formula of sediment-laden river [2] in order to ensure this model more suitable to Yellow River, which is such a special river, especially suitable to hyper-concentrated flood.
(10)
In forecasting and calculating flood in the lower Yellow River, the water stage along the river is one of the most important contents. Therefore the formula of roughness calculation not only can describe the effect of hydraulic sediment factor variation on friction characteristics, but also affect various additional roughness in natural channel. After changing the conditions of flow and sediment greatly, the calculated results are agreement with measured one, i.e[3].
(11)
Where δ*—friction thickness, which is equivalent coarseness on river flood-plain, according to the vegetation of plain, it is looked out by the booklet of hydraulics calculation. But in a main channel, the formula below can be calculated:
(12)
In order to overcome
the difficulties caused by boundary complex and scale between the length and
width in calculation domain, using the boundary coordinate fitting method
pointed out by American scholar J. F. Thompson transforms irregular area on the
plane of physics into regular one on the calculation plane, at the same time,
the basic equation on physical plane is transformed into corresponding one on
plane of orthogonal curvilinear coordinate
,
the
boundary condition of flow is adduced on calculating grids exactly, thereby the
equation of water-sediment movement is derived under the system of orthogonal
curvilinear coordinate. In order to ensure the calculation stability and reduce
fluctuation, flow continuity equation and flow momentum equation according to
and
directions are disperseed with
operator-splitting method on alternate grids, which are divided into two groups
of equations with ADI method. The derived process is completed by four steps:
time t is calculated by n→n+1/2
in x
direction: (1) zn+1/2 and
are derived by implicit difference format;
by reveal format of convection adverse the wind; time t is calculated by n→n+1/2
in
direction:
(3) zn+1/2 and
are derived by implicit difference
format; (4)
by reveal format of convection adverse the wind. In this way, the flow field
computation of whole interval from n to n+1 of time t is completed along the
direction of
and
through four steps around double-half
intervals. With the time’s increase, the alternation is rolling forward. Under
the condition of suitable fixed solution, the computation of two-dimensional
unsteady flow field of plane can be completed.
The sediment continuity equation is derived by control volume method; the control volume is arranged under the system of orthogonal curvilinear coordinate. The sediment continuity equation is integrated in control volume and subsumed into flow continuity equation; the dispersed mode meeting with flow continuity equation is simultaneously obtained.
In
order to save internal storage and computational time, the grids produced by
this verified computation only include flooded landform boundary below high
floodplain in 1982’s flood. The initial landform uses measured big cross
section and many small cross sections interpolated before 1982’s flood,
grid’s nodes are 16×100 (16—number along river width, 100—number along
flow direction).
After
the measured discharge, sediment concentration process at Huayuankou station and
relationship between water stage and discharge at Jiahetan station in 1982 are
given, the simulated computation work on flood’s water-sediment evolution on
two-dimensional model of plane and variation of bed’s erosion-deposition can
be carried out. The computational results of discharge process at Jiahetan
station are drawn in Fig. 1. It is found in contrast with measured values that
the peak pattern and flood propagation time etc are close to measured results.
It is seen from computational condition of velocity field and flood inundation
depth etc that the velocity and depth in the whole computational domain increase
to different extents with the enlargement of inlet discharge. The computational
velocity field results in flood peak period are shown in Fig.2, in order to
contrast with measured data, the main flow line of river regime in flood season
in natural channel is drawn in the figure. It can be seen that the computational
main flow line tallies well with that in natural channel and the velocity field
also tallies with the fact qualitatively.
The
comparison of computational values of stage with measured values at both
Huayuankou and Xinzhai stations is shown in Fig.3 and Fig.4 respectively. From
this, it can be seen that in the whole process of flood evolution, the
computational results of stage are very close to the measured one and the
computational results of stage at each station can meet the demands of accuracy
as well.
The
comparative results of computational values in sediment concentration process at
Jiahetan station are shown in Fig.5. It is seen as a whole that the both tally
well so that ensure the computational accuracy of bed deformation. In view of
the distribution of sediment concentration field at different moments and it can
be found in contrast with computational results of velocity and inundation depth
that the regional sediment concentration with common velocity and high depth is
higher, which is in basic agreement on the law of natural measured data.
The study on two-dimensional sediment mathematical model of plane in the lower Yellow River is a difficult problem in the world with much concern. It is of significance in academic and production. Based on summing up research results of predecessors, the special study is carried out. The conclusions are obtained as follows:
(1) The flow continuity equation, flow movement equation and sediment continuity equation under the system of rectangular coordinate are transformed into corresponding equations under the system of orthogonal curvilinear coordinate with boundary fitting coordinate method. The computational difficulties caused by boundary complication and length-width in computational domain have been overcome, the computational figure is simplified. In introducing the newest research results in sediment continuity equation, bed deformation equation, sediment-carrying capacity with hyper-concentrated flow and bed roughness etc, the solution by dispersion for flow continuity equation and flow motion equation is derived by ADI method on alternative grids. The solution by dispersion for sediment continuity equation is derived by the method on control volume, which supply essential condition for mathematical model's calculation.
(2) This model has been verified by 1982's flood. The results show that the peak propagation process, stage variation process and sediment concentration process calculated by this model are close to measured results. For the former two-dimensional model in the Yellow River, this model is an important breakthrough. In addition, the velocity field, flood inundation depth, inundation extent and sediment concentration field etc are also agreement on the fact. It is further shown that it is feasible to simulate water-sediment evolution and bed deformation in the lower Yellow River with this computational method.
References
[1] ZHANG Hongwu & LU Xin. Bend Hydraulics. Press of Water Resources and Electric Power, 1993.
[2] ZHANG Hongwu & JIANG Enhui et al. The Similarity Law of Hyper-concentrated flood Model in Yellow River. Henan Scientific and Technical Press, 1994.
[3] ZHAO Lianjun & ZHANG Hongwu. Study on Flow Friction Characteristics in the Lower Yellow River. Journal of Yellow River, 1997.
Fig.1 Verified results of flooddischarge process at Jiahetan station in 1982
Main-flow line on August 2~3, 1982
Fig. 2 Flow field at 20:00 on August 2, 1982
(Huayuankou: 15300 m3/s; Jiahetan: 11600 m3/s)
Fig.3 Verified results of flood stage at Huayuankou gate station
Fig.4 Verified results of flood stage at Xinzhai station measured
Fig.5 Verified results of sediment concentration in 1982’s flood