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The last two decades have seen great progress in mathematical modeling of fluvial processes and flooding in terms of either approximation of the physical processes or dealing with the numerical difficulties. Yet attention to simultaneously taking advancements of both aspects is rarely paid. Here a well-balanced and fully coupled noncapacity model is presented of dam-break flooding over erodible beds. The governing equations are based on the complete mass and momentum conservation laws, implying fully coupled interactions between the dam-break flow and sediment transport. A well-balanced Godunov-type finite volume method is used to solve the governing equations, facilitating satisfactory representation of the complex flow phenomena. The well-balanced property is attained by using the divergence form of matrix related to the static force for the bottom slope source term. Existing classical tests, including idealized dam-break flooding over irregular topography and experimental dam-break flooding with/without sediment transport, are numerically simulated, showing a satisfactory quantitative performance of this model.

Mathematical modeling of fluvial flows and flooding has been a routine work in support of flood risk management and river management. Nevertheless, efforts to improve the quality of mathematical modeling for fluvial flows and flooding have never stopped due to their complexities: hydraulic jump/drop, wet/drying, sediment transport, bed deformation and heterogeneous sediment sizes, and so forth. This work is motivated by the general recognition that physically meaningful results in agreement with observations depend on not only accurate numerical algorithms, but also physically well-grounded model formulations [

In terms of numerical algorithm, a mathematical model for dam-break flooding should be capable of capturing the transitions between subcritical and supercritical flow regimes such as hydraulic jumps/drops (i.e., shock) and the moving wet-dry fronts. One of the widely used methods dealing with these is the Riemann solver-based technique: Godunov-type finite volume method [

Indeed, the development of these well-balanced numerical techniques for Godunov-type finite volume method greatly improved confidence in mathematical representation of the complex phenomena of the fluvial flows and the dam-break flooding. However, most of those well-balanced models focus on clear water flow [

This paper presents a well-balanced and fully coupled noncapacity model for dam-break flooding over erodible beds. The governing equations are numerically solved using a second-order Godunov-type finite volume method: a predictor-corrector time stepping along with the HLLC approximate Riemann solver for flux estimation. The bed slope source term is rewritten in a divergence form of matrix related to the static force due to bottom slope, facilitating straightforward satisfaction of the

The governing equations of the model comprise the mass and momentum conservation equations for the water-sediment mixture flow and the mass conservation equations, respectively, for sediment and bed material. Two-dimensional governing equations are written in a matrix form as follows:

Based on (

Using the unstructured triangular mesh system, the governing equations are discretized by the finite volume method, and the interface numerical fluxes are estimated by the HLLC approximate Riemann solver. In order to achieve second-order accuracy in both space and time, the monotone upstream schemes for conservation laws (MUSCL) reconstruction are implemented before the numerical fluxes are estimated. In the MUSCL, the water surface level is used following the surface gradient method [

Essentially, the issue of the well-balanced property arises from two distinct terms of the momentum equation: the flux gradient/hydrostatic pressure term and the bed slope source term. Take the case of a body of static water as an example: the two terms in the momentum equation of the

The present paper makes use of the method of DFB (the divergence form for bed slope source term) to resolve this issue. The DFB method was proposed by Valiani and Begnudelli [

Make use of (

Figure

Sketch of the unstructured triangular mesh system.

Applying Green’s theorem to the two matrix terms, the second term in the LHS and the first term in the RHS, (

Estimation of the source vectors

For estimation of the intercell fluxes, that is, the second term in the LHS of (

The boundary conditions used in this model include two types, that is, closed and open boundaries. At a closed boundary, a slip condition is used, namely,

Cell-averaged bed evolution is computed from (

The time step is constrained by the Courant-Friedrichs-Levy condition

The friction slopes are determined using the Manning roughness

In this section, numerical case studies are presented to demonstrate the performance of the model.

This numerical case is used to test whether the present model can simulate steady flow over irregular topography, the main indication of satisfying the well-balanced property. The computational domain is a 25 m long one-dimensional channel. The bed elevation is defined as ^{2}/s is imposed at the upstream boundary (

Comparisons of the computed water level against the analytical solution.

The present model is now tested against a widely used idealized case of dam-break flow in a closed dry channel with irregular rough bed. Dam-break flow requires the model to be capable of capturing shock waves. The dry property of the channel requires the model to deal with wet/dry fronts satisfactorily. The irregular bed requires the model to be well-balanced. Dissipation of energy due to friction of the rough bed will finally lead to a stationary state of the dam-break flow, which combined with the irregular feature of the bed provides challenging testing for the well-balanced property.

The closed channel is 75 m long and 30 wide. A dam is arranged 16 m to the right of the left end (

Figure

Evolution of the dam-break flow: 3D view at different times as computed by the well-balanced model.

Velocity vectors of dam-break waves at different times as computed by the well-balanced model.

Evolution of the dam-break flow: 3D view at different times as computed by an unwell-balanced model.

Velocity vectors of dam-break waves at different times as computed by an unwell-balanced model.

Bellos et al. [

Sketch of dam-break flow in a converging-diverging channel.

Figure

Comparison of computed and measured water depth variation against time.

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At the Civil Engineering Laboratory of the University of Catholique de Louvain (UCL), Belgium, a series of experimental dam-break flows in an abruptly widening erodible channel are conducted [^{3}, and sediment porosity

Sketch of the UCL dam-break experiment in a widening channel (m).

Figures

Computed and measured stage hydrographs at six gauges.

Comparison of computed and measured bed profiles at CS1 and CS2.

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In this section the present model is further tested against an experimental partial dam-break flow in a straight erodible channel, which is also conducted at UCL, Belgium. This experiment differs from the one used in the previous section mainly in two points: ^{3}, and bed sediment porosity

Sketch of the UCL partial dam-break experiment.

Figure

Computed and measured stage hydrographs at distinct gauges.

Figure

Computed and measured bed profiles at three cross sections.

A well-balanced and fully coupled noncapacity 2D depth-averaged model for sediment-laden dam-break flooding over erodible beds is developed. The satisfaction of the C-property (well-balanced property) of the model is demonstrated through a case of steady flow over a bump and a case of dam-break flow over three humps. The numerical accuracy of the model in terms of reproducing key flow variables (i.e., water level) and bed deformation is demonstrated against three sets of experimental dam-break flows over movable beds.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Basic Research Program of Yangtze River Waterway Research Institute (2011-02-029), the Research Fund for Doctoral Program of Higher Education of China (20130101120152), Open Fund of the State Key Laboratory of Satellite Ocean Environment Dynamics (SOED1309), and the National Natural Science Foundation of China (11402231).