(1)
Xiangsan Wang
Dept.
of River Engineering, Wuhan University, China
Abstract: The water current of river network belongs to the unsteady flow. One-dimentional unsteady flow can be described by the famous Saint-Venant System of Equations. The demountable finite element method of unsteady flow of river network is based on the numerical value calculation of the open channel flow, discretizes the flow area into several small units, changes the Saint-Venant System of Equations into finite element equations by using Galerkin Method, and combines the analysis and the constrained disposition of the spreading of wave at the mouth of river arms, makes an embedded programme in BASIC language adopting the two-step condensable finite method of demonstration. By this means, the water stage and discharge hydrograph of the mouth of river arm in the network can be obtained by the electronic computer.
China is a large country with wide range of territory and rich water resourses. Lots of rivers spread in the mainland crosswise, which can be called "river network". The water currents of each river in the river network differs from one another. It is quite necessary to use the numerical method to simulate the unsteady flow of river network in order to make full use of the inland river resources to work out the programming of the agro-fields, water conservancy, as well as to formulate the plan of flood diversion and flood alternation.
Because the river network is consisted of open channel flows, it should be based on the numerical value calculation of unsteady flow in the open channel. Unsteady flow in open channel flow refers to the one in which the elements of the spatial spot movement (such as flow velocity , water pressure, discharge etc) changes with the time. Onedimensional unsteady flow of open channel can be described by the famous Saint-Venant system of equations:
(1)
(2)
x——spatial variable
t——time variable
u——average current velocity
Q——cross section discharge
H——water stage
c——chezy coefficient
R——hydrualic radius
g——acceleration of gravity
W——breadth of water current
Since Saint-Venant put first proposed the famous system of equations in 1871, more than one hundred years has passed achievement which should the most worth praising is the numerical value solution, even though some developments in terms of basic theories have been made.
For instance, the finite difference method commonly used in the recent years is a very useful method to solve the complex unsteady flow problem.
Finite difference method is a linear equation formed on the basis of discretization of the approximate difference pattern and some additional conditions. In the straight river without the mouth of the river arm, the coefficient matrix of the linear equations has the characteristics of symmetry and sparse. In the river network, the movement of the flow in the mouth of river arm are complicated. Through the analysing of the law of movement of the flow of the mouth of river arm , we know, althought this law should satisfy with the Stokes Assumption:
ΣQINPUT=ΣQOUTPUT (3)
Hi=Hj=Hk=…… (4)
The coefficient matrix of the simultaneous equations combining the difference equations of each straight river and equation(3) & (4) at the mouth of the river arm loses the characteristics of symmetry and sparse and it also makes the calculation difficult. Even through the electronic computer system has big capacity, its lay-in quantity will increase greatly. Therefor, it will be in convenient in the real word. On the other hand, finite difference method is a useful way to solve the problems of regular solution regions, but for the problems of the irregular solution regions, it is difficulty of even impossible to get the solution.
Finite element method is an approaching method developed on the basis of the finite difference method. It considers the solution region(any form) as the assemble of the division regions (definite units), solves each unit equation (in the way called method oder weighted residual), then conbines the units and finally turns the question into an algebraic problem again.
The steps to use definite element method may be summarized as follows:
(1)To discretize the solution region into several units
(2)To choose the inserting value function
(3)To decide the unit characteristics and the method to work out the unit equation
(4)To combine the units and to get the solution of the algebraic equation.
According to the foregoing steps, we can make alternations to equations(1), (2) of Saint-Venant System of Equations.
If
B=
then
(5)
(6)
According to the Galerkin's approaching method, supposing that S is the definite element division space, is the fundamental system of functions. The variable of H, Q, u, F, B in S can be unfolded.i.e.
α——the node number of variable
then (5), (6) become definite element equations:
(7)
(8)
in which,
using the two-step demountable method to get the integral of the numerical value of (7),(8).
1st step:
(9)
(10)
2nd step:
(11)
(12)
in which,
n——time step number
——choice
condensation matrix
——quality
condensation matrix
The characteristics of water current at the river mouths will be analysed in the following, which is a simple network formed by three straight rivers.
as show in Fig.1
Upon dealing with the river network problem shown by Fig. 1 in difference method. The stokes continuous equation of the mouth of resulting in the increase of the rank number of the equations. Moreover, the coefficient matrix of the system of equations has lost its characteristics of symmetry and sparse. The solution of the system of the linear equation is getting more difficulty. If the average flow velocity value U and the river level value H in the nth step are given, the U and H in the (n+1)+h step, can be calculated directly by the use of demountable definite element method.
Fig. 1 A simple network formed by three straight rivers
In the open channel flow, the wave length is larger than that of the water depth, that is, it belongs to long wave. In the river mouth, though water currents interfere with one another, the curvature of the water particles obits is very small and the vertical acceleration can be negnetness, in other words, the current velocity along the vertical line is uniform distributed. Additionally, in the river mouth, the wave effect can not spread very far. Therefor, it is assumed that the water level and discharge of every neighbouring nodes keep moving continuously in the straight river reach, these nodes are called the set of mutually affected nodes in the river mouth, the area enclosed by the set of nodes are called affected area in the mouth of river arm. In these affected area, the items of local force of inertia and motive force of inertia in the dynamic equation can be ignored. So the equation can be simplified as:
(13)
If the mouth of the river arm links to the first unit of the straight river reach, the formula of the force of resistance can be shown as follows:
(14)
in which:
K——number of mouths of river arms
E1——unit length whose number is one
——the
water depth of the first unit in the mouth of the river arm
As for the first unit of the straight river reach which links to the mouth of the river arm, according to Chezy Formula:
(15)
in which:
V——the average velocity of the water flow in the 1st unit
n——rate of coarseness
C——chezy
coefficient for 1st unit C=
J——gradient of the 1st unit
R——hydraulic radius of the 1st unit
Let equ. (14)& (15) into(13), and set:
Thus:
If it is in the case of the last unit of the straight river section, a similar formula can also be obtained, and generally expressed as:
K——number of the mouth of river arm
i——number of unit in the affected area of the mouth of river arm
n——the total number of units
according to Stokes Equation:
or
(16)
Equation (16) is therefore the constraint equation in the mouth of the river arm. This is an algebraic equation depending on the water level in the mouth of river arm. Solving this equation can get the correction value of water level in the mouth. In addition, the correction value of discharge of various river section in the mouth can be traced from formula (12). Obviously the correction value of the constrained water level and discharge satisfy the Stokes Equation.
The fundament equations for shallow long waves may be described by demountable discretization model. If the equations are solved by discretization finite element method, the stability condition must be met. From a lot of actual examples the stability condition may be:
(17)
in which
is element length, it varies with
the salinity gradient. H is depth,
which is a function of location. Hence eq.(17)is indeterminable. In order to
meet the stability condition and accuracy requirements, the time-step
should be given from
. Usually, the element length 1 should be smaller and the time-step
t shorter for larger depth h.
From above analysis the maximum and minimum element length of divded solution space can be given by
(18)
For the solution stability the value C should be smaller, in the paper C is adopted as 2.3.
Example 1:
There is a small river named Gutian
Brook in Fujian Province, of which there is a straight river reach, 55
Kilometres long, 82 meter wide, the river course was cutted into two at the spot
K (see Fig.2), one reach is 30
Kilimeters long, the other is 25 km long. Then, the river reach are divided into
55 units, each unit is one kilometer long. Supposing on the boundary line of the
upper reach, the water level equation Z
and Q=f relational curves are given,
the lower reach is a drop regarded as non-reflectional boundary line, the first
water level Z & the first velocity
is known: i.e. Z=4.6 metres, V=1.9
m/s, Manin friction drag coefficient K=1/n=40
/s (n is the rate of coarseness), water level adopts the "pass" of the
boundary condition, t=10 s, Through
the calculating of the demountable element method. We can link the two river
reach at spot k using the constrained
equations and output the water level process, and it coincides the measured
water level process. See Fig 3.
Fig.2 River course cut into two at the Fig. 3 Calculated and measured water level
spot K hydroFig in K of the straight river
Example 2:
Here is the Fig of Gutian Brook. See Fig.4.
Fig. 4 Fig of Gutian Brook
L1=29 km, L1——length of the river
L2=21 km, L2——length of the river
B1=78 m, B——breadth of water surface
B2=71 m, B——breadth of water surface
B3=82 m, B——breadth of water surface
L3=6 km, L3——length of the river
The process of the river stage in the
upper reach is known, and Mannin friction drag coefficient K=1/n=40
/s, t=13 seconds. Every unit is one km long, L1,L2
and L3 consists of 29, 21
and 6 units respectively. K represent
the mouth of the river arm. The process of river stage at the mouth can be
calculated by electronic computer, in which the demountable definite element
method and the constraint equation at spot K
are used. After compared the calculated process with the actual measured process
of water stage, it is found that the largest derivation between is only 3 cm.
See Fig.5.
Fig. 5 Comparing between the calculated process and the actual measured process of water stage
(1) In calculating unsteady flow of network, any solution region problem can be solved by using the two-step demountable choice condensation definite element method. It gets rid of the difficulty in approaching the complicated solution regions by the commonly used difference pattern (method).
(2) The actual experiment shows that the calculating method mentioned above can actually imitate the spreading process of the long wave in the network. This method outstanding advantages nowdays because the electronic computers are commonly employed.
(3) When this method is used to calculate the value of river network, the calculation value of each part of the rivers is considered separately, there is not any connection between each numbered information. The calculation of each single rivers reach is programmed into subroutine. Employing the constrained programme, some correction arrangements of the discharge and river stage in the mouth of the river network, the programmd need not change greatly. It's very convenient to make the water conservancy arrangement, to make out the numerical value imitation and to decide the scheme of flood diversion and alternation.
Bibliography
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[2] J.J. Jansen etal. "Principles of River Engineering, 1979.
[3] Agmon, S., "Lecture on Elliptie Boundary Value Problems", D. Von Nosirand Co. Inc.(1965).
[4] I. Babuska, J. Osborn, "Analysis of finite element methods for second order boundary value problems using mash dependent norms" Numer. Math. 34, 41-62(1980).