thehung-korea8-2009final
TRANSCRIPT
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International Sym posium on W ater City W ater Forum
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ON THE FINITE ELEMENT TECHNIQUE
FOR THE SHALLOW WATER EQUATIONS
APPLICATION TO URBAN FLOODING
NGUYEN The Hung(1)
, TO Thuy Nga(2)
, LE Hung(3)
, LE Van Thao(4)
(1) Faculty of water resources engineering, Danang University of Technology, 54 Street Nguyen
Luong Bang, Lien Chieu Districts, Da Nang City, VietNam,
Phone: (84).905233440 ; e-mail: [email protected] ; [email protected](2)
Faculty of water resources engineering, Danang University of Technology, 54 Street Nguyen
Luong Bang, Lien Chieu Districts, Da Nang City, VietNam,
Phone: (84).905586568 ; e-mail: [email protected]
(3) Faculty of water resources engineering, Danang University of Technology, 54 Street Nguyen
Luong Bang, Lien Chieu Districts, Da Nang City, VietNam,
Phone: (84).905888950 ; e-mail: [email protected](4)
Faculty of water resources engineering, Danang University of Technology, 54 Street Nguyen
Luong Bang, Lien Chieu Districts, Da Nang City, VietNam,
Phone: (84).973496313 ; e-mail: [email protected]
ABSTRACTA non standard Galerkin method for the solution of the shallow water equations in
conservative form with using the characteristic-based split algorithm is presented. The main
advantage of the model are high accuracy and ability solving the time discontinuous shock
waves, the supercritical and subcritical regimes as well as the transitions between the regimes;
the domain is discritization by triangular elements, so it is capable of handling complex
geometry. The results are verified by comparison with measurement data; a case study on
urban flooding, downstream of Han river, is showed.
Keywords: Finite element method, shallow water equations, characteristic-based split
algorithm
1 INTRODUCTION
The two dimensional horizontal flow model or shallow water equations system are very
popular for treatment of flow of lakes, rivers, oceans.
Many difficults issues for solving the two dimensional horizontal flow model are non-self
adjoint operator in governing equations and how to treat the nonlinear advective terms
(Cullen and Morton, 1980; Zienkiewicz and Taylor, 2002).
The finite element method when applied to hydrodynamic problems gives an accuratesolution. This method is conservative and therefore avoids aliasing errors associated with
nonlinear terms.
Moreover, it has the advantage over the finite difference method of being flexible in the
treatment of irregular domains and to allow a variable resolution, thus permitting a focus on
regions of interest.
Navon (1982, 1983) introduced the Numerov Galerkin finite element method for the
shallow water equations with an Augmented Lagrangian constrained optimization method to
enforce integral invariants conservation. Similar work was done by Zienkiewicz and
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Heinrich (1979) with a finite element penalty method, and by Zienkiewicz and others (1984).
Navon (1987 ), N. T. Hung (2004) applied a two-stage finite element Galerkin method
combined with a high-accuracy compact approximation to the first derivative for solving the
shallow water equations.
In this paper, Author develops the characteristic- based split (CBS) algorithm for the
temporal discretization and a non- standard Galerkin finite element method for space solving
the full shallow water equation.
2. THE CHARACTERISTIC BASED SPLIT ALGORITHM FOR THE SHALLOW
EQUATION
2.1. Governing equationThe shallow water equations in conservative form are as follow ( 7 ):
Where Q, Fx, Fy, H : are vectors, and
+=
yx
xxx
h
h
pq
hgh
h
p
p
F 22
2
1 (3)
+
=
yy
xy
y
hgh
h
q
h
h
pq qF
22
2
1
(4)
++
++
=
3/72
0
222
0
3/72
0
222
0
0
hC
qpqng
y
zgh
hC
qppng
x
zghH
(5)
where:
p = u.h, u = Being the depth-averaged x-direction component of velocity
q = v.h, v = Being the depth-averaged y-direction component of velocity
h = water depth
g = Celeration due to gravity
xx, xy, yx, yy = Reynolds stresses where the first subscript indicates the axis
)1(0=+
+
+
Hy
F
x
F
t
Q yx
)2(
=qp
h
Q
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International Sym posium on W ater City W ater Forum
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direction normal to the face on which the stress acts
= Fluid densityz0= Channel invert elevation
n = Mannings roughness coefficient
C0 = Dimensional constant (C0= 1 for SI units and C0= 1.486 for English units)
The Reynolds stresses are determined using the Boussinesq approach relating stress to
the gradient in the mean currents:
y
vv
x
v
y
uv
x
uv
tyy
txyxy
txx
=
+
==
=
2
)(
2
(6)
wheret= is the eddy viscosity, which varies spatially and is solved empirically as a functionof local flow variables (Rodi 1980):
6/1
0
22
8.hC
qpgnCt += (7)
Where C = is a coefficient that varies between 0.1 1.0
2.2 The split - temporal discretization
Equation (1) can rewrite as follow:
+
= nyx H
y
F
x
F
t
Q (8)
With Hn+
being treated as a known quantity evaluated at t = tn + .t in a time
increment t; and [0,1].
Equation (8) can discritize in time using the characteristic - Galerkin process(Zienkiewicz and Taylor, 2002)n
yx
k
k
nn
y
n
x
nn HFy
Fxx
ut
HFy
Fx
tQQ
+
+
+
= ++ )))()((..2
()()(1 (9)
where k = 1,2 ; u1 = u ; u2= v
At this stage we have to introduce the split in which we substitute a suitable
approximation for H.
First, we remove the term H from Eq. (9), and the next step we introduce an auxiliary
variable Q*such that
n
yx
k
kyx
n
Fy
Fxx
utFy
Fx
tQ
QQQ
++=
=
))()((..2
)()(*
**
(10)
Equation (10) will be solved subsequently by an explicit time step applied to the
discritized form. The correction step is as follow
k
n
k
nn
x
Hu
tQQQQ
== + ..2
2*1
(11)
The above equation is fully self-adjoint in the variable Q which is the unknown; so a
Galerkin type procedure can be optimally used for spatial approximation.
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International Sym posium on W ater City W ater Forum
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Figure 4:Upstream boundary condition of scenario A!Discharge Q "t
Figure 5:Upstream boundary condition of scenario B - Discharge Q "t
Figure 6:Downstream boundary condition!Tidal water level H "t
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4 CONCLUSIONIn comparison with the Galerkin finite element method , it is observed that the
characteristic - non Galerkin method with using the characteristic-based split algorithm for the
solution of the shallow water equations in conservative form is more accuracy and ability to
solve the time discontinuous shock waves, the supercritical and subcritical regimes as well as
the transitions between the regimes because of self-adjoint operator of governing equations
and using special test function. This model is good applying in urban flooding.
REFERENCES
Cullen, M. J. P., and Morton, K., K., 1980, Analysis of evolutionary error in finite element
and other methods,Jour. Comput. Phys., v.34, p. 245-267.
Navon, I. M., 1982, A Numerov-Galerkin technique applied to a finite element shallow
water equations model with exact conservation of integral invariants, in Kaway, T., ed.
Finite element flow analysis, Univ. Tokyo Press, Tokyo, p. 75-86.
Navon, I. M., 1983, A Numerov-Galerkin technique applied to a finite element shallow water
equations model with enforced conservation of integral invariants and selective lumping,
Jour. Comput. Phys., v.52, N0.2, p. 313-339.
Navon, I. M., 1987, A two stage, high accuracy, finite element Fortran program for solvingshallow water equations, Computers & Geosciences, v.13, n
0. 3, p. 255-285.
NGUYEN The Hung, Mathematical model of the two dimensional vertical flow,Journal of
Vietnam National Science & Technology, N07+8, Hanoi 1990.
NGUYEN The Hung, Mathematical modeling of sediment transport two dimensional
horizontal, Proceedings of International Conference on Engineering Mechanics Today,
Vol. 1, p. 541-548, Hanoi 1995.
NGUYEN The Hung, 2004, Finite element method in flow problems, Construction Pub.
House, Hanoi.
NGUYEN The Hung, 2005, A two stage, high accuracy, finite element technique of the two
dimensional horizontal flow model, Modeling, Simulation and Optimization of Complex
Processes: Springer- Verlag Berlin Heidelberg, p. 255-233.
NGUYEN The Hung & LE Van Hoi (2001), Applying mathematical models forecast
inundation of Han river, Danang City (Project: Treatement of pollution environment causing
flooded and intensity of capacity to cope of flooding damage problems in Danang City).
Zienkiewicz, O. C., and Heinrich, J. C., 1979, A unified treatment of steady stage shallow
water equations and two dimensional Navier-Stocks equations a finite element penalty
function approach : Computer Math. Appl. Mech. and Eng., v. 17/18, p. 673-688.
Zienkiewicz, O. C., Vilotte, J. P., Nakazawa, S., and Toyoshima, S., 1984, Iterative methods
for constrained and mixed approximation : an inexpensive improvement of F.E.M.
performance : Inst. Numerical methods in Engineering Report C/R/489/84, Swansea,
United Kingdom, 20p.
Zienkiewicz, O. C., and Taylor R. L., The Finite element method, 5th
edition Butterworth
Heinemann 2000.