in motion equation can be neglected usually, it can be regarded as
two-dimensional gradient current(i.e. the horizontal and vertical gradient of
salinity). The salinity along vertical coordinates can be divided in two
layers(see Fig.1).
Damei Li
Dept.
of River Engineering, Wuhan University, Wuhan 430072, China
Abstract: In this paper the finite element method is used for solving the two-dimensional gradient current, and its solution is obtained two-steep explicit method. It is verified by the example field data offered by Xiamen E.P.A, Fujian, P.R.C. with good agreement. The authors advance that the proposed method is suitable for the analysis and computation of long waves of epeiric sea at the seaward firth.
Keywords: gradient current, finite element method, shallow long waves
The different salinity between river and sea causes the water to appear layering in the seaward firth. The difference breads variance of flow condition. The type of flow can be treated as a gradient current. The difference in density used to be taken for a constant. The results were unstable, when the density difference was substituted as constant into the motion equation with different factors. Thereby the calculating accuracy is affected. In addition, the density difference has a wide selection of range. It leads to a tremendous calculation. In order to the selected factor tallys with actual situation, we have to recalculate over and over again. It is inconvenient to actual engineering problems. The paper regards the density difference as a function of space and time in the seaward firth. The finite element method can be used here.
Different salinities lead to density difference
in the mixing and interchanging region of salt and fresh water in seaward firth.
From the analysis of current type, the region can be regarded as a spreading and
interchanging region of shallow long waves. When the nonlinear item u
in motion equation can be neglected usually, it can be regarded as
two-dimensional gradient current(i.e. the horizontal and vertical gradient of
salinity). The salinity along vertical coordinates can be divided in two
layers(see Fig.1).
The fundament equations for shallow long waves:
Upper layer
Motion equation:
(1)
continuity equation:
(2)
Fig. 1 A sketch for two layers shallow long waves
diffusion
equation:
(3)
lower layer
motion equation:
(4)
continuity
equation:
(5)
diffusion
equation:
(6)
in which
velocity;
h depth; d
thickness of upperlayer;
salinity of upper layer;
salinity of lower layer;
eddy viscosity; Kij
diffusion coefficient;
r maximum difference of density; Su salinity rate of upper layer;
salinity rate of lower layer;
wave height; g
acceleration of gravity;
;
frictional resistance along sea
level (it rises from wind);
frictional resistance from lower
layer to upper layer;
frictional resistance from upper
loyer to lower layer;
frictional resistance along
seafloor;
velocity of upper layer;
velocity of lower layer;
friction coefficient between two
layers;
friction coefficient on seafloor
,
The finite element method is a numerical approached method. It is developed from finite differences. The finite differences method regards solution space as an assembly of nets. The differential equations are replaced by difference equations for functional values of knots. Thereupon the differential equations become a set of algebroidal equations. The difference method has the following defects:
(1) it is not suited for complex boundary conditions;
(2) it is not suited for irregular nets;
(3) the velocity and salinity fields on a knot can representative only the mean velues in this element, where is the knot.
The finite element method overcomes the above defects. It regards solution space as a lot of connecting subspaces(i. e. finite elements). We can solute each element equation of them and combine the results, so that the problem change to solving algebroidal equations. This method divided solution space into finite elements and integrate step-by-step for time.
It should be emphasized, that the choice of the form and size of an element is based on the density gradient and the requirement of stability and accuracy of solution. It is a principle of the discretization of solution space, e.g. the larger the density gradient, the smaller size element. The optimum discretization should be based on a large number of calculation and comparison.
Moreover, the discretization affects directly the result accuracy. The key to the settlement of finite element method lies in the choice of an interpolating(form ) function usually a linear function.
In this paper the solution space is divided into
triangular elements. In a triangular element the velocity and density fields
have been described by linear interpolating functions. The equations for field
variable of every element are solved on the basis of Calerkin weighted remainder
law. Analysis the interpolating functions for every element should be satisfied
with the continuity equations for current velocity and density overall. In order
to determine the value of velocity and density at N knots, the N conditional
equations must be developed from above supposition, so as to develop N sets of
equations every time-step
t (i.e.t, t+
t).
Form analysis of current type at the seaword
firth the velocity v, the depth
increment
and the density S
are unknown field variable. The interpolating function (from function) is linear
equation for triangular element (see Fig.2).
Fig. 2 Definition of current at a knot
After considering weighting of unknown variable
we can integrate in solution space. The two-step demonstration is used on
integration for time. Supposing
and
are a fundamental system of
functions in finite element subspace, the variable
can be unfolded separately.
then their finite element discretizational equations are as follows:
first step
(7)
(8)
(9)
(10)
(11)
(12)
second step
(13)
(14)
(15)
(10)
(16)
(17)
(18)
in
which, the concentration mass matrix (i.e. condensation mass matrix)
can be yielded from follows:
It should be noted, that the concentration mass
matrix
is normal and sparse. Only when two
knots are part of an element, their matrixes are relative. The dot produce
·
can be obtained easy and systematically. Here, as long as you calculate the
weight of each triangle for all dot product, need not count one by one, i.e.
integrating on each triangle the element mass matrixes can be given. Every
element mass matrix
consists of these element matrixes.
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:
(19)
in which
is element length, it varies with
the salinity gradient.
is depth, which is a function of
location. Hence eq.(19)is indeterminable. In order to meet the stability
condition and accuracy requirements, the time-step
should be given from
. Usually, the element length
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
(20)
For the solution stability the value C should be smaller, in the paper C is adopted as 2.3.
This example data is from the river Jiulongjiang, Xiamen Fujian. Finite element method is applied. The mean depth h=5 ~30 m at the seaward firth. After analysis the water area belongs to current region of shallow long waves. The mixing type of salt and fresh water is weak and slow. Both horizontal and vertical density gradient should not be neglected. The element nets of solution area are as shown in Fig.3.
Fig. 3 Element nets of solution area
In Fig.3 at knot 10 and neighboring knots d=0.5
m;
=0.002;
=0.02; g=9.8m/s2;
Kij=0,
r=0.03 g/cm3;
=10 m2/2;
t=5 s
upper layer:S u=0.9;
, lower layer:S l=1.0;
The calculating result and actual measurement data are shown in Fig.4,5,6. It shows a fairly good agreement.
Fig. 4 Vertical velocity distribution at knot 83
Fig. 5 Vertical velocity distribution at knot 10
Fig. 6 Variasion of upper thickness(calculated) at knots 10 and 83
(1) Seaward firth can be regarded usually as a shallow long waves space. The mixing of salt and fresh water is weak and slow. Here, there are horizontal and vertical density gradient. Hence it may be considered as gradient current.
(2) The motion, continuity and diffusion equation can be dispersed by finite element method and integrated numerical for time. The process is effective, accurate and applicable with computer.
(3) Finite element method has superiority in gradient current at seaward firth space with complex boundary conditions. After analyzing actual example the result is satisfactory to accuracy requirement.
(4) This example has a stable motion programme.
The compute time is 0.5 hour by the computer PC586. The method can be applied
feasible to actual problem.
Acknowledgements
The authors thank the environmental protection office of Xiamen, Fujian for offering valuable data.
References
[1] Kaitai Li, Finite element method and its application I, the publishing house of the Jiaotong University Xi'an, 1984.12.
[2] C. G. Koutitas, Fundamentals of computative hydraulics, translated in Chinese by Zhongtang Hao, 1984.
[3] J.J.Jansen, Principles of River Engineering, pp.119-128. New York, 1979.