Numerical simulation of flows of a heavy nonviscous fluid with a free surface in the gravity field over a bed surface with an arbitrary profile

Abstract

Numerical simulation of flows of a heavy nonviscous fluid with a free surface in the gravity field over a bed surface with an arbitrary profile Abstract -A numerical method is proposed in this paper for studying hydrodynamic flows of heavy nonviscous fluid with a free surface over an arbitrary bed profile. This arbitrary bed profile is approximated by a piecewise-linear function splitting it into a finite number of domains with a step boundary. In order to implement this method, a quasi-two-layer model of fluid flow over a step boundary is used taking into account physical processes near the step. In this paper we implement numerical modelling of various physical phenomena, such as fluid flow over a sloping bed surface, shock wave creeping on a sloping bank, a fluid column fall over a bed surface given in various ways, fluid flow over a bed surface of a complex profile.

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

“…As a result, the problem of calculating of the shallow water flows in the presence of external force becomes identical to the problem of calculating the shallow water flows over a time-dependent nonhomogeneous bed. Papers [29,30,31,32,47,48] have offered the quasi-two-layer method for calculating shallow water flows over an nonhomogeneous bed. In present work we show how quasi-two-layer method can be applied in case of a time-dependent bed.…”

“…The method consists in reducing of the problem to successive solutions of classical shallow water equations on the flat plane using Godunov method with allowance for the vertical nonhomogeneity effect in calculating the fluxes through the boundaries of cells adjoining to stepwise boundaries. The vertical nonhomogeneity leads to the Riemann problem solution on a step based on the quasi-two-layer shallow water model developed in [29,30,31,32,47,48]. 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.…”

“…These terms provide the mechanical work made by nonhomogeneous bed interacting with flow. Quasi-two-layer model 29,30,31,32,47,48 is extended here for a generalized time-dependent bed which represents an external force. The term timedependent bed means that at each time moment bed can have different values.…”

“…Algorithm suggested in our work avoids this difficulty in a natural way. In present work we extend algorithms developed in [29,30,31,32,47,48] to the case of multiply-connected domains including partly flooded and dry regions. Section 2 presents the shallow water model over an arbitrary bed profile in the presence of external force.…”

Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid in the shallow water approximation

“…The Riemann problem solution found before forms a basis for development of finite-volume numerical methods to compute continuous and discontinuous solutions without capturing of discontinuities [22][23][24]. OA -magnetogravity shock wave; OB, OC -Alfvenic waves …”