Mathematical Problems in Engineering

Volume 2015, Article ID 613853, 13 pages

http://dx.doi.org/10.1155/2015/613853

## A Well-Balanced and Fully Coupled Noncapacity Model for Dam-Break Flooding

^{1}Changjiang Waterway Planning Design and Research Institute, Wuhan 430011, China^{2}Ocean College, Zhejiang University, Hangzhou 310058, China^{3}State Key Laboratory of Satellite Ocean Environment Dynamics, The Second Institute of Oceanography, Hangzhou 310012, China^{4}Changjiang River Administration of Navigational Affairs, Wuhan 430011, China

Received 16 October 2014; Accepted 6 March 2015

Academic Editor: Jian Guo Zhou

Copyright © 2015 Zhiyuan Yue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1].

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 [2]. This, at the same time, gives rise to an important issue: the well-balanced property of a model, which refers to the ability of the model to reproduce a body of static water over irregular topographies [3, 4]. It is because the Riemann solver-based technique estimates the hydrostatic pressure term (i.e., flux gradients) and the bed slope source term differently. This subsequently may make the two otherwise equal terms in the case of a body of static water become unequal/unbalanced over irregular topography. Fortunately, the last several decades have seen numerous novel techniques resolving this issue and wide applications can be found. Those include the surface gradient method [4], the flux correction method [5], upwind discretization of bed slope [3], divergence form for bed slope source term [6], wave propagation algorithm/augmented Riemann solver [7], prebalanced shallow water equations [8], and hydrostatic reconstruction technique [9].

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 [3–8, 10, 11], in which the potential sediment transport and high erodibility of the flow on the bed are ignored. For these models that have taken into account sediment transport, a capacity description is often adopted, in which the sediment transport rate/concentration is directly estimated by empirical relations [9, 12, 13]. In fact, however, sediment transport is governed also by physical laws, that is, the mass conservation law. In the noncapacity model, sediment concentration at a specific control volume is computed as a result of the net flux across the control volume and the sediment exchange between the flow and the bed. The advantages of the noncapacity modelling have been recently well recognized [14].

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 (conservation)-property. Unstructured triangular grid system is used to represent the computational domain. The quantitative performance of the model was tested against classical idealized and experimental dam-break flooding benchmark tests.

#### 2. Numerical Model

##### 2.1. Governing Equations

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: where = vector of conserved variables; , = vectors of flux variables; , = the diffusive vectors; = vectors of bed slope source term; = vectors of friction source term; = vectors of source terms representing the feedback impacts of sediment transport on the flow; = time; = streamwise coordinate; = cross channel coordinate; = flow depth; , = depth-averaged velocities in the - and -directions; = depth-averaged volumetric sediment concentration; = gravitational acceleration; is the turbulent viscosity coefficient, = bed shear velocity, = bed elevation; , = friction slopes in - and -directions; , = sediment entrainment and deposition fluxes across the bottom boundary of the flow; = bed sediment porosity; , = densities of water and sediment; = density of water-sediment mixture; and = density of saturated bed.

Based on (1), (2a), (2b), (2c), (2d), (2e), (2f), (2g), (2h), and (3), the model is categorized as a coupled model. By “coupled,” the meaning is twofold. The first feature is the nonequilibrium description of sediment transport by the mass conservation of sediment carried by the flow (the fourth component of (1) and (2a), (2b), (2c), (2d), (2e), (2f), (2g), and (2h)), which determines the sediment concentration. This is in contrast with the commonly used equilibrium description in the existing well-balanced morphological models, which assume sediment concentration equal to the sediment transport capacity of the flow. The second feature refers to simplifications in the mass and momentum conservation of the flowing water-sediment mixture: the first, second, and third components of the source term were set equal to zero in previous well-balanced morphological models.

##### 2.2. Numerical Algorithm

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 [4]: the flow depths are calculated by applying the MUSCL to the water surface level and then subtracting the bed elevation. The well-balanced property is achieved by using the DFB form for the bed slope source term.

###### 2.2.1. Well-Balanced Property

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 -direction have the relation . While the left-hand-side (LHS) term (the flux gradient/hydrostatic pressure term) is computed by an approximate Riemann solver, the right-hand-side (RHS) term [the bed slope source term] is often estimated by centered discretization. The different estimation ways would easily give rise to inconsistency: the computed is not equal to in the case of a body of static water especially when the topography is irregular.

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 [6] following the recognition that “*the simplest way towards the numerical closure of physical balances is the divergence form of physical laws, which leads to conservation of physical properties such as mass, momentum, energy, and so on.*” By DFB, the bed slope source term is written as the divergence form of a proper matrix related to the static force (the DFB method). Following Valiani and Begnudelli [6], one haswhere is the free water level and is a constant value for the free water level. It has been proved that reformulating the bed slope source term in the form of (4), (5a), (5b), and (5c) facilitates straightforward satisfaction of the well-balanced property irrespective of the specific numerical algorithm. It is noted that the governing equations of the present model involve sediment transport. However, the extension of this DFB method from clear water flow to the present sediment-laden flow is straightforward and is justified, because in the simplest yet challenging case of a body of static water over an irregular topography, governing equations for sediment-laden flow become the same as those for clear water flow.

Make use of (4), (5a), (5b), and (5c) and let and ; the governing (1) can be rewritten as

###### 2.2.2. Godunov-Type Finite Volume Discretization

Figure 1 shows a sketch for the unstructured triangular mesh system. Bed elevation is defined at nodes of a cell. Flow variables are stored at the centre of each cell. Integrating (6) over the area of an arbitrary cell giveswhere = average free water level of the cell which is set equal to the flow depth plus the average bed elevation of the cell : , where = bed elevation of the th node of the cell , the subscript denotes the cell number, and the subscript denotes the edge number or node number of a cell. For a neighborhood description between cells, nodes of a cell, and neighboring cells of a cell, the model follows the integer mapping arrays used in Begnudelli and Sanders [5].