  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.
