Quasi-two-layer finite-volume scheme for modeling shallow water flows   with the presence of external forces
  Finite-volume numerical method for study 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 a studied flow into two layers in calculating flow quantities near each step, and improving by this means 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 quasi-two-layer approach leads to appearance of additional terms in one-layer finite-difference representation of balance equations. These terms provide the mechanical work made by nonhomogeneous bed interacting with flow. A notable advantage of the proposed method is the consideration of the properties of the process of the waterfall, namely the fluid flow on the step in which the fluid does not wet part of the vertical wall of the step. The presence of dry zones in the vertical part of the step indicates violation of the conditions of hydrostatic flow. 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 figure out the amount of flow kinetic energy dissipation on complex boundary. Numerical simulations are performed based on the proposed algorithm of various physical phenomena, such as a breakdown of the rectangular fluid column over an inclined plane, large-scale motion of fluid in the gravity field in the presence of Coriolis force over an mounted obstacle on underlying surface. 
K. V. Karelsky, A. S. Petrosyan, A. G. Slavin
physics.flu-dyn cs.NA math-ph math.MP physics.ao-ph physics.comp-ph