Keywords: computational fluid dynamics, finite-volume method, moving grid, river flow, shallow water equation.

There has been considerable interest in predicting flow in rivers from the viewpoint of environment or defense against disasters. There are a lot of papers dealing with steady and unsteady river-flow problems. As for the unsteady problem, a dam-break problem is popular and many papers deal with this problem. Almost all papers are treating this problem using the model such that the dam is instantaneously broken and the simulation is performed afterward. The dam, however, is not instantaneously collapsed. It takes some short time for dam to be broken. Another interesting unsteady flow problem is a flood flow. It is very interesting to simulate the flood flow with expanding its bank or a flow over the bank. Both unsteady flows accompany with movement of boundaries. In the simulations, it is necessary to model the process of extepanding the breach of the dam or moving banks. A body fitted grid method need to move and deform the grid system according to the movement of the boundary at dam collapse or bank breaking. The governing equation for river flow is shallow water equations. Although there are many papers which treat the moving boundary problem directly in mechanical and aeronautical field recently, it is difficult to find a paper which treats the unsteady problem with moving body fitted grid system in hydraulics field.

The objective of this paper is to present a moving finite-volume method for the shallow water equation in the moving and deforming body fitted coordinate system in order to predict the unsteady flow in rivers. Since the boundary is changing its shape according to the flow, the grid has to be dynamically moved according to the boundary movement. This means that the whole of the grid system is also moving and deforming, and thus grid coordinate is also non-stationary. It further means that a control volume of the finite volume method has to be dynamically changing its shape at every time step. In this paper, the finite-volume method is constructed on the moving and deforming grid system.

Since the present method is on the moving grid system, it is very important for unsteady flows that the numerical method need to satisfy the geometric conservation laws, which means that the method does not affect the uniform flow even if the grid is moving or deforming[1].

To assure geometric conservation laws, a finite-volume formulation in the complete space-time unified computational domain, of three dimension for two spatial dimension of shallow water equation, is adopted in the method. The resultant fully implicit method is solved iteratively at every time step by an inner iteration procedure with introducing pseudo-time approach in order to assure the physical and geometrical conservation laws. The base algorithm is originally proposed and developed for compressible flows[2,3] and the present method is the extension and development to the shallow water equations.

The basic feature of the present method is investigated using simple dam-break problems. Application to real river flows are also presented in the paper. As for the dam-break problem, the transient flow about the dam region is simulated according to the time change of the dam shape. It is demonstrated that the method is perfectly simulate the flow. The river flow simulation treats a kind of flood. The sudden increase of flow rate at upper reach of the river causes a kind of flood and large amount of flow with shock wave is traveling with expanding its bank. The flow field of this situation is simulated successfully.

1

H.Zhang, M.Reggio, J.Y.Trepanier and R.Camarero, Discrete Form of the GCL for Moving Meshes and Its Implementation in CFD Schemes, Computers and Fluids, Vol.22, No.1, pp.9-23, 1993. doi:10.1016/0045-7930(93)90003-R

2

K.Matsuno, K.Mihara, and N.Satofuka, A Moving-Mesh Finite-Volume Scheme for Compressible flows. Computational Fluid dynamics 2000, Springer, 2000, pp.705-710, Proceedings of the First International Conference on Computational Fluid dynamics, ICCFD, Kyoto, July, 2000.

3

K.Matsuno, An Adaptively-Moving-Grid Finite-Volume Scheme with Boundary-Grid Ellimination/Addition. Computational Fluid dynamics 2002, Springer, 2002, pp.447-452, Proceedings of the Second International Conference on Computational Fluid dynamics, ICCFD, Sydney, July, 2002.

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