Numerical modeling of debris flows using basic equations in generalized curvilinear coordinate system and its application to debris flows in Kinryu River Basin in Saga City, Japan

Abstract

Each year, structures installed in rivers and valleys (e.g., bank protections, bridges, and sabo dams) are damaged by debris flows; therefore, it is necessary to predict the flow characteristics of debris flows accurately to design river and valley structures that are robust against such debris flows. In recent years, several two-dimensional models have been developed to simulate debris flow. However, the basic equations in the numerical simulation models are in the Cartesian coordinate system, and the shape of each numerical analysis grid is square or rectangular. It is impossible to use numerical grids that match the horizontal shape of the river or valley structures, such as bank protections and sabo dams, when employing square or rectangular grids. This makes it difficult to calculate the fluid forces acting on river or valley structures. In this study, the basic equations of debris flow are provided in the generalized curvilinear coordinate system and a numerical simulation model of debris flow designed using an orthogonal curved grid along the horizontal shape of rivers and valleys is developed. Numerical simulations of a periodical meandering valley are conducted using the basic equations in the generalized curvilinear and Cartesian coordinate systems; the difference between the two results is discussed. Further, numerical simulations of the debris flow that occurred in the Kinryu River Basin in Saga City, Japan, in 2019 are performed using the basic equations in the generalized curvilinear and Cartesian coordinate systems, and the difference between the two simulation results is discussed. The numerical simulation results of debris flow in a periodical meandering channel suggest a slow propagation of debris flow when equations in the Cartesian coordinate system are used; this is attributed to the rough boundaries of debris flow when a square grid is used, which causes numerical energy dissipation that does not exist physically. The inundation areas and land surface deformation of debris flow in Kinryu River Basin were reproduced well by the developed numerical analysis model; the basic equations were written in the generalized curvilinear coordinate system. The development and decrescence of debris flows were reproduced using the developed model; the first debris flow in Kinryu River Basin was stopped in the upstream mild slope area, while the second flowed into the residential area. Reproducing the flow depth, velocity, and discharge could be improved by the model using basic equations in the generalized curvilinear coordinate system, as compared with those reproduced using basic equations in the Cartesian coordinate system. When the Cartesian coordinate system was used, the volume of the debris flow was evaluated to be small and the travel distance of the debris flow was evaluated to be short; thus, the debris flow does not arrive at the residential area and the risk of debris flows is underestimated.