COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Research on Numerical Simulation of Bend-Flows
Wenli Wei 1, Z. Liu 1, W. Y. He 2
1
2
Institute of Water Conservancy and Hydraulic Engineering, Xi’an University of Technology, Xi’an, 710048 China
Science school, Xi’an University of Technology, Xi’an, 710048 China
E-mail: weiwenli@xaut.edu.cn
Abstract This paper is concerned with the research on numerical simulation of bend-flows. The purpose of this
paper is to modify the commonly-used two-dimensional horizontal shallow water model in a curvilinear coordinate
system considering the transverse momentum exchanges caused by the secondary flows in channel bends according
to the past research results, and the adopted curvilinear coordinate system has been generated by using the proposed
boundary-fitted coordinate technology, with which a curvilinear gird being orthogonal near the boundaries can be
obtained. By using the modified shallow water mathematical model, the behaviors of the flows through the sequent
bends have been computed; and the computational results coincide well with the measured data.
Keywords: boundary-fitted coordinate technology, curvilinear gird, transverse momentum exchange, secondary
flow
INTRODUCTION
The research and application of two-dimensional horizontal shallow water model has been a long history, and many
research results about it have been applied to actual engineering, such as hydraulic engineering, environmental
engineering, flood defense, and etc; and play an important guidance role in the organization, development and
utilizations of water resources [1-4]. Because the natural streams are serpentine, narrow and long, the commonly
used two-dimensional depth-averaged model can’t accurately predict bend flows with obvious 3D properties.
Therefore, some researchers have investigated the influences of transverse momentum exchange caused by the
secondary flows on the distributions of water depths and velocities across the section in channel bends, and obtained
some useful results. The purpose of this paper is to modify the two-dimensional horizontal depth-averaged shallow
water model in a curvilinear coordinate system considering the transverse momentum exchanges caused by the
secondary flows in channel bends according to past research results [1], and the adopted curvilinear coordinate
system has been generated by using the proposed boundary-fitted coordinate technology in the Ref. [4], with which a
curvilinear gird being orthogonal near the boundaries can be obtained. By using the modified shallow water
mathematical model, the behaviors of the flows through the curved channel with an angle of 90° in front of the
sluice gate, and of the flows in a sequent bends have been computed; and the computational results of the two
examples coincide well with the measured data.
MATHEMATICAL MODEL
1. Generation of orthogonal curvilinear grids Generating a grid in an arbitrary physical domain involves a
coordinate transformation from the physical plane (x, y) to the computational plane ( ξ ,η ) (see Fig. 1). This is done
here by solving a system of Poisson equation
αxξξ − 2βxξη + γxηη + J 2 ( Pxξ + Qxη ) = 0
(1a)
αyξξ − 2βyξη + γyηη + J 2 ( Pxξ + Qxη ) = 0
(1b)
2
2
2
2
where α = xη + yη , β = xξ xη + yξ yη , γ = xξ + yξ , J = xξ yη − xη yξ .
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In Eq. (1), how to determine the functions of P and Q is often difficult. Liu Zhe and Wei Wenli [3] have proposed a
new method to determine them more efficiently, with which a desired boundary-fitted curvilinear coordinate grid can
be automatically generated. The details of the method are referred to the Ref. [3]. The expressions of the functions P
and Q are given below:
P = ϕ (ξ ,η )
(x
Q = ψ (ξ ,η )
+ yξ2
)
(2a)
ξ
(x
+ yη2
)
(2b)
2
2
η
where ϕ = −
xξ xξξ + yξ yξξ
xξ2 + yξ2
+
xξ xηη + yξ yηη
xη2 + yη2
,ψ = −
xη xηη + yη yηη
xη2 + yη2
+
xη xξξ + yη yξξ
xξ2 + yξ2
.
Now considering the functions P and Q in Eq. (2), and solving Eq. (1), we can generate a boundary-fitted orthogonal
curvilinear system. Eq. (1) are rewritten in finite difference scheme and then solved by the ADI method. If the points
on the boundaries are unreasonably selected, the generated grids may be less orthogonal. Therefore, we make the
points slip along the boundaries under the condition, β = xξ xη + yξ yη = 0 . Further more, all subsequent
hydrodynamic computations are performed in the coordinates ( ξ ,η ).
2. Modification of governing equations in curvilinear coordinates The commonly used two dimensional
governing water equations can be written in curvilinear coordinates in the following form as
Continuity equation:
zt +
1
1
[ yη (hu)ξ − xη (hv)ξ ] + [− yξ (hu)η + xξ (hv)η ] = 0
J
J
(3a)
Momentum transportation equations:
g
1
1
u t + [ ( yη u − xη v)u ξ + (− yξ u + xξ v)uη ] + [ yη z ξ − yξ zη ] −
J
J
J
n 2u u 2 + v 2
υ ⎪⎧⎡ 1
⎤ ⎫⎪
⎤
⎡1
− Fv = 0
⎨⎢ (αu ξ − βuη )⎥ + ⎢ (− βu ξ + γuη )⎥ ⎬ + g
4
J ⎪⎩⎣ J
⎦ η ⎪⎭
⎦ξ ⎣ J
3
h
1
1
g
vt +[ ( yηu − xηv)vξ + (−yξu + xξ v)vη ] + [−xη zξ + xξ zη ] −
J
J
J
2
2
2
υ ⎪⎧⎡ 1
⎤ ⎡1
⎤ ⎪⎫ n v u + v
+ Fu = 0
⎨⎢ (αvξ − βvη )⎥ + ⎢ (− βvξ + γvη )⎥ ⎬ + g
4
J ⎪⎩⎣ J
⎦ξ ⎣ J
⎦η ⎪⎭
3
h
(3b)
(3c)
where Z is water elevation; h is water depth; u and v are the velocity components in the x and y directions,
respectively; F is Coriolis force; n is roughness coefficient of river bed; υ is the effective viscosity.
The relations between the velocity components in curvilinear coordinates and in the Cartesian coordinates are as
follows:
⎧⎪u * = uyη − vxη
⎨ *
⎪⎩v = − uy ξ + vx ξ
(4)
where u * and v * are velocity components in curvilinear coordinates ξ - and η -directions, respectively.
Some researchers have found that the commonly used two-dimensional depth-averaged model for Eqs.(3) can not
accurately predict the behaviors of bend flows with obvious 3D properties, because they do not consider the
influences of transverse momentum exchange caused by the secondary flows on the distributions of water depths and
velocities across the section in channel bends when being derived. The purpose of this paper is to modify the
two-dimensional horizontal depth-averaged shallow water model in a curvilinear coordinate system in channel bends
according to past research results by DE Vriend (1981), and a modified shallow water mathematical model
considering the transverse momentum exchanges caused by the secondary flows can be obtained as follows [4]:
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zt +
1
1
[ yη ( hu )ξ − xη ( hv )ξ ] + [ − yξ ( hu )η + xξ ( hv )η ] = 0
J
J
(5a)
1
1
g
u t + [ (y η u − x η v)u ξ + (− y ξ u + x ξ v)u η ] + [ yη z ξ − yξ zη ] −
J
J
J
M
υ ⎧⎪⎡ 1
n 2u u 2 + v 2
⎡1
⎤ ⎫⎪
⎤
(
)
(
)
α
β
β
γ
+
− Fv + u = 0
+
−
+
u
−
u
u
u
g
⎨⎢
ξ
η ⎥
ξ
η ⎥ ⎬
4
⎢
J ⎪⎩⎣ J
h
⎦ η ⎪⎭
⎦ξ ⎣ J
h3
g
1
1
vt + [ ( yη u − xη v)vξ + (− yξ u + xξ v)vη ] + [− xη zξ + xξ zη ] −
J
J
J
M
n2v u 2 + v 2
υ ⎪⎧⎡ 1
⎤
⎡1
⎤ ⎪⎫
(
)
(
)
v
v
v
v
g
α
−
β
+
−
β
+
γ
+
+ Fu + v = 0
⎨⎢
ξ
η ⎥
ξ
η ⎥ ⎬
4
⎢
J ⎪⎩⎣ J
h
⎦ξ ⎣ J
⎦η ⎪⎭
h3
(5b)
(5c)
where： M u , M v are the correction terms caused by the secondary flows, can be written in curvilinear coordinates as
[(
)]
1 ∂
1 ∂
⎧
* *
* *
* *
⎪⎪ M u = J ∂ξ (−ξ y u u ϕ ) + J ∂η η y u u − 2ξ y u v ϕ
⎨
⎪ M v = 1 ∂ (ξ x u * u*ϕ ) + 1 ∂ − η x u * u* + 2ξ x u* v * ϕ
⎪⎩
J ∂ξ
J ∂η
[(
(6)
)]
with
ϕ=
γ h2
K TS
α R
(7)
where R is the η -lines curvature radius; K TS is the coefficient of transverse momentum exchanges caused by
the secondary flows, and can be written as [1]
2
K TS
3
⎛ g⎞
⎛ g⎞
g
⎟ + 37.5⎜
⎟ ;
=5
− 15.6⎜
⎜ kc ⎟
⎜ kc ⎟
kc
⎝
⎠
⎝
⎠
and K is the von Karman’s constant (about 0.4); C the Chezy coefficient written as C = H
γ
=
α
xξ2 + yξ2
xη2 + yη2
1
6
/n.
is the ratio of width to length of curvilinear girds.
In general, ξ x , ξ y are much smaller than η x ,η y , so Eqs.(5) for M u , M v can be simplified as
(
)
1 ∂
⎧
η y u * u *ϕ
M
=
u
⎪⎪
J ∂η
⎨
⎪ M v = 1 ∂ − η x u * u *ϕ
⎪⎩
J ∂η
(
(8)
)
3. Numerical discretization The modified shallow water governing equations (5) can split into two sets of equations
according to Yanenko operator splitting method.
the equations in ξ - direction：
[
]
1
1
Zt + yη (hu)ξ − xη (hv)ξ = 0
2
J
(9a)
g
1
1
1 ⎡Γ
n 2u u 2 + v 2
⎤
u t + ( yη u − xη v )u ξ + yη Z ξ − ⎢ (αu ξ − β uη )⎥ + g
− Fv = 0
4
2
J
J
J ⎣J
⎦ξ
3
h
⎯ 321 ⎯
(9b)
g
1
1
1 ⎡Γ
⎤
vt + ( yη u − xη v)vξ − xη Zξ − ⎢ (αvξ − βvη )⎥ = 0
2
J
J
J ⎣J
⎦ξ
(9c)
the equations in η -direction：
[
]
1
1
Z t + − y ξ (hu )η + xξ (hv )η = 0
J
2
(10a)
M
g
1
1
1 ⎡Γ
⎤
u t + (− y ξ u + xξ v )uη − y ξ Z η − ⎢ (− β u ξ + γuη )⎥ + u = 0
J
J
J ⎣J
2
h
⎦η
(10b)
M
1
1
g
1 ⎡Γ
n 2v u 2 + v 2
⎤
v t + (− y ξ u + xξ v )vη + xξ Z η − ⎢ (− β vξ + γvη )⎥ + g
+ Fu + v = 0
4
J
2
J
J ⎣J
h
⎦η
h3
(10c)
Each set of transformed governing equations (9, 10) are discretized on a staggered (ξ, η) grid and solved using
alternating direction implicit finite difference scheme. The velocity variables are fully staggered, and the water level
modes are located at the center of the continuity flow cell as illustrated in Fig. 1. When solving equations, a
two-order upwind scheme is used to discretize the convective term and a central difference scheme is used to
discretize the diffusion term in Eqs. (9, 10).
Figure 1:
Grid arrangement
Δξ and Δη are defined as the distances in the transformed domain between the velocity vector positions. Since the
range of the coordinates ξ and η in the computational plane is completely arbitrary, the mesh increments Δξ and Δη
are specified, for convenience, as unity.
The alternating direction algorithm splits each time step into two intervals. The procedure can be demonstrated with
respect to the transformed equations as follows. In Step 1,in the first half time step, the equations in ξ-direction are
solved; and in Step 2, the computed results of the first half-time step are prepared for the initial values of the second
step, and the equations inη - direction are solved. The values for second half-time step (un+1、vn+1 and zn+1) are those
for one time step (from t = nΔt to t = (n+1)Δt). The computation procedure will go until the computed results
converge.
4. Boundary Conditions The boundary conditions for flow are that the upstream and downstream water level is
specified and the flux through the solid boundaries is zero; and the initial conditions are that water level at any flow
element is the averaged value of upstream and downstream water level and velocity is set equal to zero.
5. Technique of moving boundary In the numerical solution of unsteady flow, the riverbed may be exposed to the
water surface. In order to deal with the changeable computational region, the technique of moving boundary (or the
method of condensation) is used in the computation. The basic idea of this method is to make the roughness of the
computational finite cell very large when its riverbed is exposed to the water surface [5]. This enables the velocity of
⎯ 322 ⎯
the fluid in the computational finite cell to be zero, just looking like the fluid being condensed to solid.
6. Stability condition The above-described numerical scheme is a time-marching method in which Δt must be
satisfied with Courant-Friedrichs-Levy Condition. For every point i ,j of the computational domain the Δt time step
is expresses by
ΔS min
Δt
≤α
2
gH max
(11)
with： H max h is the maximum water depth in computational region; ΔS min is the minimum length size of
curvilinear girds. In the computation of flows in large and wide domains, α ranges from 1 to3;and rivers, α is
smaller than 1 tin general.
COMPUTED EXAMPLE
This method has been used to predict the water flow fields in the sequent bend in Fig.2.The water flow fields have
been measured in the Ref. [6].
The computed region for the sequent bend is about 800 cm long and 100 cm wide. The curvilinear coordinate grid
for the computed region is shown in Fig. 3. The region concerned is divided into 13×83 elements. In the computation,
the time step Δt is about 0.5s; the Courant number is less than 5; the Coriolis parameter f is neglected, owing to the
experimental flow region being computed is small; the roughness n is about 0.012; the slop of the channel bed is
Jb=1‰; the discharge is 45.3L/s; the downstream water depth is h=9.4cm.
The important hydraulic parameters such as velocity and water level have been obtained. The comparisons of
computed water depths and velocities with measured data on different cross section are shown in Fig.4, and a fair
agreement demonstrates the validity of the method development.
Figure 2:
Figure 3:
A sequent curved rectangular channel
Curvilinear gird of computational region
⎯ 323 ⎯
velocity(m/s)
depth(cm)
11
9
7
5
0
1
2
3
4
5
0.6
0.4
0.2
0
0
6
1
2
3
4
5
6
N
N
cross section 2
velocity (m/s)
velocity(cm)
cross section 2
11
9
7
5
0
1
2
3
4
5
6
0.6
0.4
0.2
0
0
1
2
N
2
3
4
5
6
velocity (m/s)
velocity(cm)
11
9
7
5
1
5
6
N
0.6
0.4
0.2
0
0
1
2
3
4
5
6
N
N
cross section 4
cross section 4
◆measured values
4
cross section 3
cross section 3
0
3
——computed values
Figure 4: Comparisons of computed water depths and velocities with measured data
on measured point of different cross section
CONCLUSIONS
The boundary-orthogonal curvilinear grid and the technique of moving boundary, presented above are convenient
and effective in dealing with the complicated boundary of physical region and the computational finite cell exposed
to water surface; and also the boundary-orthogonal curvilinear grid can make the boundary conditions be accurately
used.
The modified shallow water mathematical model includes the influences of transverse momentum exchange caused
by the secondary flows on the distributions of water depths and velocities across the section in channel bends when
being derived. Therefore it can accurately predict the variation of physical parameters of bend flows with obvious
3D properties.
By using the modified shallow water mathematical model, the behaviors of the flows through the curved channel
with an angle of 90°in front of the sluice gate, and of the flows in a sequent bends have been computed; And the
computational results of the two examples coincide well with the measured data., and the fair agreement
demonstrates the validity of the method development. The study has provided a good basis for research of sediment
and water quality in rivers with complex boundaries.
REFERENCES
1.
De Vriend HJ. Velocity redistribution in curved rectangular channels. Fluid Mech., 1981; 107: 423-439.
2.
Xu Guiing. Experimental study on the behavior of flow through the curved channel in front of a sluice gate. J.
Wuhan Univ. of Hydr. and Elec. Eng., 1996; 28(3): 339-341 (in Chinese).
3.
Liu Zhe, Wei Wenli. Research on method of grid generation in boundary-fitted coordinate system. Journal of
Xi’An University of Technology, 2004; 20(4) (in Chinese).
4.
Liu Zhe. Research and Application on Two-dimensional Numerical Simulation of Flows with Complex
Boundaries. Master’s Degree Thesis, Xi’An University of Technology, China, 2004 (in Chinese).
5.
Wei Wenli. Computational Hydrodynamics Theory and Application. Shanxi Publishing Cooperation of
⎯ 324 ⎯
Technology, Xi’an, China, 2001 (in Chinese).
6.
Han Longxi. 3-D numerical simulation for water environment in the region of Three Gorge Project during
construction period. Advances in water science, 2002; 113(14): 427-432 (in Chinese).
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