( 1 )
J.G. Zhou, D.M. Causon, D.M. Ingram and C.G. Mingham
Centre
for Mathematical Modelling and Flow Analysis
The Manchester Metropolitan University
Manchester M1 5GD, U.K.
Tel: +44 161 2471548
Fax: +44 161 2471483
E-mail:
J.G.Zhou@mmu.ac.uk
Abstract:
In this paper, a numerical method for dam-break flows is presented. The method
is based on the 2D shallow water equations with bed shear stress source terms.
The equations are discretized and solved with the MUSCL-Hancock finite volume
upwind scheme, in which the fluxes at cell interfaces are determined by the HLL
approximate Riemann solver. The method is applied to predict a dam-break flow
into an open channel. This demonstrated that the model is able to predict water
surface with small amplitude waves as observed in real dam-break flows. The
results are compared with experimental data of the CADAM project, which shows a
good agreement. The method is conservative and robust.
Keywords:
dam break, godunov-type method, riemann solver, shallow water equations.
Dam-break, once occurring, can cause disasters to human beings, especially in the regions where there are high populations such as in Europe. In recent years, it becomes compulsory to set up emergency plans for large dams in the world, which include numerical and experimental investigation on dam-break flows or wave and their potential damages. In fact, dam-break analysis is playing more and more essential role in hydraulic and river engineerings in respect to reservoir safety. For example, in the Europe, there is a two-year CADAM (Concerted Action on Dame-Break Modelling) project (CADAM, 1998) for dam-break analysis with aims of both numerical and experimental studies, providing several benchmark test problems together with corresponding experimental data.
Dam-break flow is a two dimensional problem. Usually, the shallow water equations are chosen as the mathematical model. In the literature, there are several numerical methods for solution of the shallow water equations. Most of these solve the shallow water equations without any source terms with a Godunov-type method. Zhao et al. (1993) described a finite-volume method and solved the 2D dam-break occurring from a square reservoir. But no experimental data is available for comparison. Recently, the CADAM project carried out a few experimental investigations on dam-break flows, e.g. dam-break flow from a reservoir to an open channel with dry/wet bed downstream. A lot of experimental data are collected in the CADAM team. This provides reliable data for verification of a numerical model in simulating dam-break flows. Also different methods are developed for dam-break flows and the results are compared with the corresponding experimental data at the CADAM meeting, showing good agreement. However, it is reported that all of the numerical schemes always reproduce smoother water surface, instead of water surface with small amplitude waves observed in real dam-break flows.
Therefore, in this paper, we present a MUSCL-Hancock finite volume method for dam-break flows, which is second-order accurate and then is preferred in practical engineering, based on the 2D shallow water equations. The method is verified by solving a benchmark test problem of CADAM project and the numerical results are compared with the experimental data. The capability of the method in simulating water surface with small amplitude waves has been demonstrated.
The 2D shallow water equations may be written in a vector form as (Zhou et al., 2000)
( 1 )
where U is the vector of conserved variables, F is the flux vector function and S is the source term vector function, and
and
is a gradient operator. U,
F and S are
,
,
(
2 )
where
,
is the gravitational
acceleration, h is the water depth, u
and v are the x- and y-components of
flow velocity, respectively, V is the
velocity vector defined by
,
and
are the bed shear stress in x-
and y-directions, which can be expressed by depth-averaged velocities,
i.e.
,
(
3 )
where
is the bed friction coefficient,
which may be set to constant or be estimated from
in which
is Chezy constant.
The equation (1) is solved with the MUSCL-Hancock finite volume scheme (van Leer, 1985). The scheme is a second-order accurate, high resolution, upwind scheme of the Godunov type and consists of two steps: a predictor and a corrector step.
In the predictor step, a non-conservative approach is used to determine intermediate values over a half time step,
( 4 )
where A is the cell area,
is the cell side vector and
is the number of the Cartesian cell
sides. The flux
can be evaluated at each cell face m
based on the variable
at the cell face, which is decided
with a linear reconstruction method, i.e.
( 5 )
where
is the normal distance from the
cell centroid to the cell face m and the
is a gradient which is usually obtained with a slope limiter to avoid
non-physical oscillations in the reconstructed data.
In the corrector step, a fully conservative solution over the full time step is achieved by solving a series of local Riemann problems defined by the values from the predictor step,
( 6 )
where
and
are the variables on the left and
right hands of the cell interface m , providing a local Riemann problem. In this paper, the HLL
approximate Riemann solver proposed by Harten et al. (1983) is used and the flux
vector
is calculated as,
( 7 )
in which
and
are the wave speeds which can be
estimated by (Fraccarollo and Toro, 1995)
,
(
8 )
where
and
are estimated by
,
( 9 )
and
is the normalised side vector for
cell face m.
For shallow water flow involving a dry bed, the wave speed expression (8) is replaced by (Fraccarollo and Toro, 1995)
,
(Right
dry bed) (
10 )
,
(Left dry bed)
( 11 )
The time step is calculated at the start of each time step by
( 12 )
where
,
(
13 )
The benchmark test problem considered here is the TEST 1 from the CADAM project, for which experimental data is available for comparison. The flow domain is a rectangular reservoir connected to a L-shaped open channel which has a right bend as shown in Fig. 1.
Fig. 1 Flow domain of the TEST 1: reservoir and L-shaped channel.
Initially, there is a gate which separates the reservoir and the L-shaped channel. The water depth in the reservoir is 0.2 m and the depth in the channel is 0.01 m. All boundary conditions are solid reflective conditions but with the transmissive boundary condition at the outlet boundary. The Manning coefficient is 0.0095 and there is no bed slop in either reservoir or channel. The variations of water depths with time at the different locations shown in Fig. 1 are measured in the experiment study. This provides reliable data for comparison.
In the numerical simulation,
and
. Comparison of the water depths between the numerical results and experimental
data are shown in Figs. 2-7. Generally, the figures show good performance of the
current method. It is seen from these figures that the water surface with small
amplitude waves is successfully captured by the numerical method, indicating the
flow characteristics of highly unsteady dam-break flows, which is consistent
with the experimental data and observation. However, it is also noticeable that
some numerical results are overshot at Gauge 2 and Gauge 5, This may be due to
the treatment of boundary conditions because the effect of side wall friction is
not taken into account in the present computations.
Fig. 2 Comparison with experimental data for Gauge P1
Fig. 3 Comparison with experimental data for Gauge P2
Fig. 4 Comparison with experimental data for Gauge P3
Fig. 5 Comparison with experimental data for Gauge P4
Fig. 6 Comparison with experimental data for Gauge P5
Fig. 7 Comparison with experimental data for Gauge P6
A high resolution finite volume method for modelling dam-break flows is described. It is based on the 2D shallow water equations and MUSCL-Hancock scheme in which the HLL approximate Riemann solver is applied. The model is verified by comparing the predictions with experimental data. The agreement is reasonably good. The method successfully reproduce water surface with small amplitude waves occurring in real dam-break flows. This suggests that the present method be able to simulate dam-break flows.
References
[1] CADAM (1998) Proceedings of the CADAM meeting, HR Wallingford, U.K.
[2] Fraccarollo, L. and Toro, E. (1995) J. Hydr. Res. Vol 33: pp843.
[3] Harten, A; Lax, P. and van Leer, B. (1983) SIAM Review, Vol 25 (1): pp35.
[4] Mingham, C.G and Causon, D.M (1998) J. Hydr. Eng., Vol 124: pp605.
[5] van Leer, B (1985) SIAM J. Sci. Stat. Comput., Vol 5 (1): pp1.
[6] Zhao, D.H.; Shen, H.W.; Tabios, G.Q.; Lai, J.S. and Tan, W.Y. (1993) J. Hydr. Eng. Vol 120: pp863.
[7] Zhou, J.G.; Causon, D.M.; Mingham, C.G. and Ingram, D.M. (2000) J. Comp. Phys. (To appear)