Quasi-two-layer finite-volume scheme for modelling shallow water flows over an arbitrary bed in the presence of external force. I. Algorithm development and validation

Abstract

A finite-volume numerical method for studying shallow water flows over an arbitrary bed profile in the presence of external force is proposed. This method uses the quasi-two-layer model of hydrodynamic flows over a stepwise boundary with advanced consideration of the flow features near the step. A distinctive feature of the suggested model is a separation of the studied flow into two layers in calculating the flow quantities near each step and thus improving the approximation of depth-averaged solutions of the initial three-dimensional Euler equations.We are solving the shallow-water equations for one layer, introducing the fictitious lower layer only as an auxiliary structure in setting up the appropriate Riemann problems for the upper layer. Besides, the quasi-two-layer approach leads to the appearance of additional terms in one-layer finite-difference representation of the balance equations. These terms provide the mechanical work done by a nonhomogeneous bed interacting with flow. A notable advantage of the proposed method is the consideration of the properties of the waterfall process, namely the fluid flow on the step in which the fluid does not wet a part of the vertical wall of the step. The presence of dry zones in the vertical part of the step indicates a violation of the hydrostatic flow conditions. The quasi-two-layer approach determines the size of the dry zone of the vertical component of the step. Consequently, it gives an opportunity to figureout the amount of the flow kinetic energy dissipation on the complex boundary. Comprehensive analysis and comparison of different methods for modelling shallow water flows in presence of an external force is provided.