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SUMMARYThere are two main di culties in numerical simulation calculations using FD/FV method for the ows in real rivers. Firstly, the boundaries are very complex and secondly, the generated grid is usually very non-uniform locally. Some numerical models in this ÿeld solve the ÿrst di culty by the use of physical curvilinear orthogonal co-ordinates. However, it is very di cult to generate an orthogonal grid for real rivers and the orthogonal restriction often forces the grid to be over concentrated where high resolution is not required. Recently, more and more models solve the ÿrst di culty by the use of generalized curvilinear co-ordinates ( ; Á). The governing equations are expressed in a covariant or contra-variant form in terms of generalized curvilinear co-ordinates ( ; Á). However, some studies in real rivers indicate that this kind of method has some undesirable mesh sensitivities. Sharp di erences in adjacent mesh size may easily lead to a calculation stability problem or even a false simulation result. Both approaches used presently have their own disadvantages in solving the two di culties that exist in real rivers. In this paper, the authors present a method for two-dimensional shallow water ow calculations to solve both of the main di culties, by formulating the governing equations in a physical form in terms of physical curvilinear non-orthogonal co-ordinates (s; n). Derivation of the governing equations is explained, and two numerical examples are employed to demonstrate that the presented method is applicable to nonorthogonal and signiÿcantly non-uniform grids.

Contour-integration-based residual distribution, CRD scheme is one of numerical computational methods to simulate flows with shock waves. Authors have applied the CRD scheme to the open channel flow, and shown validity as compared with the experiment-with-a-model result or the theoretical value. However, for applying to various flows, the boundary condition of dry-wet is required.In the paper, when the CRD scheme is applied to a steep gradient river, the boundary condition of dry-wet is shown paying attention to a discharge error. And the shock capturing method of two dimensional calculation for raising applicability the CRD scheme is shown.

Stream flows with steep gradient bed form complicating flow configurations, where super-critical and sub-critical flows co-exist. Computing numerically such flows are the key to successful river management. Contour-integral-based residual distribution, CRD, schemes are one of the numerical computation to simulate the flow with shock waves. In the last fiscal year, the Authors applied the CRD schemes to one dimension and two dimensions of the open channel flow, and showed validity as compared with the experiment-with-a-model result or the theoretical value. However, in the paper, application has not carried out the CRD schemes to a natural river. In this paper, the CRD schemes are applied for the Toyohira River of a steep slope municipal river, and the applicability in a natural river is shown.