Based on the Godunov-type cell-centered finite volume method, this paper presents a two-dimensional well-balanced shallow water model for simulating flows over arbitrary topography with wetting and drying. The central upwind scheme is used for the computation of mass and momentum fluxes on interface. The novel aspect of the present model is a robust and accurate nonnegative water depth reconstruction method which is implemented in the unstructured mesh to achieve second-order accuracy in space and to track the moving wet/dry fronts of the flow over irregular terrain. By defining the bed elevation and primary flow variables at the cell center in the nonstaggered grid system, all computational cells are either fully wet or dry to avoid the problem of being partially wetted. The developed model is capable of being well balanced and preserving the computed water depth to be nonnegative under a certain CFL restriction, which makes it robust and stable. The present model is validated against three benchmark tests and two laboratory dam-break cases. Finally, the good agreement between the numerical results by the established model and measured data of the Malpasset dam break event on a 1/400 scale physical model demonstrates the capability of the model for the real-life applications.
The two-dimensional (2D) shallow water equations (SWEs) have been widely used to mathematically describe free surface flows over complex topography, such as river and overland flows, dam-break floods, and estuarine and coastal circulation. Because the exact analytical solutions of SWEs are only available for some simple and specific cases, it is important to develop numerical methods of SWEs with good properties for general applications in hydraulic and coastal engineering. Various numerical methods have been developed to obtain the satisfactory solutions of SWEs, such as the finite difference [
In recent decades, Godunov-type cell-centered finite volume schemes have been applied to solve SWEs numerically due to its robustness [
Several mathematical and numerical treatments have been proposed to preserve the “lake at rest” steady states, which are known as well-balanced property [
To track the wetting and drying moving boundary accurately without numerical instability has attracted significant attention in solving of SWEs in recent years. This is because the velocity is calculated by dividing the discharge per unit width by water depth in finite-volume schemes; nonphysically high velocity might be produced near the wet/dry fronts, which yields negative water depths to crash the numerical simulation. Hence, a successful shallow water model should preserve the well-balanced property and the positivity of water depth at the same time. A variety of positivity preserving schemes for SWEs have been proposed and widely applied in numerical models with structure or unstructured meshes [
A two-dimensional model, which was developed by Bryson et al. [
Stationary state with dry boundaries (a) real situation and (b) reconstructed situation.
To conquer the difficulties above, this study presents an improved cell-centered finite volume model based on the Godunov-type central upwind scheme to simulate shallow water flow over arbitrary topography in this paper. The model is developed using the unstructured mesh to make it more applicable to the real-life engineering problems with complicated boundaries and bathymetries. The central upwind scheme is adopted to calculate the fluxes of mass and momentum because it is Riemann-problem-solver-free and provides a robust and accurate solution. The nonnegative water depth reconstruction proposed by Liang [
The paper is organized as follows. Section
Based on the hydrostatic pressure assumption, the depth-averaged 2D SWEs can be obtained by integrating 3D Navier-Stokes equations over water depth. Neglecting the kinetic and turbulent viscous terms, wind effects, and the Coriolis term, the conservative form of the depth-averaged 2D shallow water equations can be written as [
Sketch of water surface elevation, water depth, and bed elevation.
The computational domain is discretized using a number of triangular cells defined as control volumes, as shown in Figure
Definition of triangular mesh.
The limited central difference (LCD) scheme [
The limit function is given by [
In the framework of Godunov-type scheme, the determination of the interface fluxes requires the values at both sides of the edge. Thus, the left states of the variables at the midpoint
Similarly, the right states of the variables at the midpoint
In order to obtain the final states, a single value of bed elevation suggested by Audusse et al. [
Then, the reconstructed water depth at the midpoint
Hence, other flow variables at the midpoint
Local bed modification for dry-bed applications.
The Godunov-type central-upwind scheme is adopted in this study to calculate fluxes at interface since this scheme is Riemann-problem-solver-free with good robustness and accuracy (Kurganov and Levy [
The source terms include slope source terms and friction terms. The slope source terms need to be discretized reasonably to exactly balance the numerical fluxes to guarantee the well-balanced property of the scheme. A well-balanced slope source term discretization proposed by Bryson et al. [
To increase the stability of the numerical model, the friction terms are evaluated by a semi-implicit method [
It is crucial to choose an appropriate time step for an explicit scheme to maintain its stability. In this study, the CFL condition is used to estimate the time step on triangular grids (Bryson et al. [
In this section, the developed model is applied to a wide variety of problems including three benchmark tests, two experimental cases, and a field-scale application to test its performance. All the computational triangular grids are generated using an open grid generation package “Triangle,” developed by Shewchuk [
To verify the capability of the model to preserve the well-balanced property when the dry area is included in the computational domain, a stationary flow with wet/dry front interface over uneven bed is simulated. A frictionless rectangular computational domain
The initial condition is static state with a constant water level of 0.2 m. So the topography is partially submerged. The total simulation is 100 s, using a mesh of 15846 triangular grids (see Figure
The bed elevation, free surface, and flow velocity at
The classical subcritical flow over bump (Goutal and Maurel [
Definition of triangular grids with mesh size
Convergence curves in logarithm scale for the water level and the unit discharge in
Periodic flow in parabolic basins proposed by Thacker [
The test is performed in a square domain with a dimension of
Comparison between numerical and analytical solutions of contours of water depth at (a)
Comparison between numerical and analytical solutions of time history of velocity component in
A series of dam-break experiments were carried out by Bellos et al. [
Sketch of a dam-break flow experiment in a converging-diverging channel.
Comparisons between the observed and calculated water depths at gauges.
The present model is applied to reproduce the experimental dam-break flow over a triangular bottom sill, which is a part of IMPACT project [
Sketch of laboratory dam-break flow.
The simulation runs for 45 s on a mesh of 8914 triangular cells. A constant Manning’s coefficient
Time history of water depth at (a) gauge G1 (b) gauge G2, and (c) gauge G3.
Comparisons between numerical and experimental water surface profile in the centerline around the hump at (a)
The Malpasset dam was located on the Reyran River in southern France (see Figure
The elevation of topography with the locations of dam and experimental gauges P.
Because of the varying topography and availability of measurement data, this case is often simulated by many researchers [
Triangle cells near the dam.
Computed water depth at
Calculated maximum water depth and water front arrival time at the gauges.
A two-dimensional shallow water flow model has been developed based on cell-centered unstructured triangular grids to simulate the flow over arbitrary topography. The governing equation is discretized by the explicit finite volume method. The intercell fluxes are calculated using a Godunov-type central upwind scheme that is the Riemann-problem-solver-free method for hyperbolic conservation laws. In comparison with some other unstructured models based on central upwind scheme, the present model defines the bed elevation and primary flow variables at the cell center in the nonstaggered grid system to avoid the problem of partially wetted cells. The nonnegative reconstruction of the water depth method is implemented in the unstructured mesh to track the stationary or wet/dry fronts. The friction source term is discretized using a semi-implicit treatment to reduce the risk of numerical instability caused by small water depth near the wet/dry front. With the improvements above, the present model does not need to adjust the wet/dry fronts in the nonstaggered mesh and is able to ensure the static steady states in the presences of dry areas such as buildings, islands, and railroad foundation.
The developed model was tested using three benchmark cases with analytical solutions and two dam-break experiments with measured data. Good agreements were achieved between the numerical results by the established model and the data in the literature, which shows that the present model has the satisfying robustness and accuracy. The developed model was also applied to simulate a real-life dam-break case with complex topography, that is, Malpasset dam failure. The successful prediction demonstrates that the model is capable of simulating extreme flood over irregular topography with wet/dry fronts with well-balanced property and positivity preserving property. The established model can be a useful tool in flood management and engineering practice.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is part of the research project sponsored by the National Basic Research Program of China (973 Program) (no. 2013CB035900), Natural Science Foundation of China (51009120, 41376095), Zhejiang Province Ocean and Fisheries Bureau (2010210), and Zhejiang University (2012HY012B).