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Finite-volume numerical method for study shallow water flows over anarbitrary bed profile in the presence of external force is proposed. Thismethod uses the quasi-two-layer model of hydrodynamic flows over a stepwiseboundary with advanced consideration of the flow features near the step. Adistinctive feature of the suggested model is a separation of a studied flowinto two layers in calculating flow quantities near each step, and improving bythis means approximation of depth-averaged solutions of the initialthree-dimensional Euler equations. We are solving the shallow-water equationsfor one layer, introducing the fictitious lower layer only as an auxiliarystructure in setting up the appropriate Riemann problems for the upper layer.Besides quasi-two-layer approach leads to appearance of additional terms inone-layer finite-difference representation of balance equations. These termsprovide the mechanical work made by nonhomogeneous bed interacting with flow. Anotable advantage of the proposed method is the consideration of the propertiesof the process of the waterfall, namely the fluid flow on the step in which thefluid does not wet part of the vertical wall of the step. The presence of dryzones in the vertical part of the step indicates violation of the conditions ofhydrostatic flow. The quasi-two-layer approach determines the size of the dryzone of the vertical component of the step. Consequently it gives anopportunity to figure out the amount of flow kinetic energy dissipation oncomplex boundary. Numerical simulations are performed based on the proposedalgorithm of various physical phenomena, such as a breakdown of the rectangularfluid column over an inclined plane, large-scale motion of fluid in the gravityfield in the presence of Coriolis force over an mounted obstacle on underlyingsurface.

Quasi-two-layer finite-volume scheme for modeling shallow water flows with the presence of external forces