A Staggered Method for Simulating Shallow Water Flows along Channels with Irregular Geometry and Friction

Abstract

We consider the shallow water equations along channels with non-uniform rectangular cross sections with source terms due to bottom topography, channel width, and friction factor. The system of equations consist of the mass and momentum conservation equations. We have two main goals in this paper. The first is to develop a numerical method for solving the model of shallow water equations involving those source terms. The second is to investigate effects of friction in water flows governed by the model. We limit our research to the flows of one-dimensional problems. The friction uses the Manning's formula. The mathematical model is solved numerically using a finite volume numerical method on staggered grids. We propose the use of this method, because the computation is cheap due to that no Riemann solver is needed in the flux calculation. Along with a detailed description of the scheme, in this paper, we show a strategy to include the discretization of the friction term in the staggered-grid finite volume method. Simulation results indicate that our strategy is successful in solving the problems. Furthermore, an obvious effect of friction is that it slows down water flows. We obtain that great friction values lead to slow motion of water, and at the same time, large water depth. Small friction values result in fast motion of water and small water depth.

A flux globalization based well-balanced path-conservative central-upwind scheme for the shallow water flows in channels

“…In the first test, which is a modification of the example from [40], we demonstrate the ability of the WB PCCU, WB CU, and PCCU schemes to preserve the "lake-at-rest" steady state.…”

“…Such systems are widely used to study the environmental water system, such as rivers and canals. The studied system (see, e.g., [2,34,40,42]) reads as…”

“…Several WB numerical methods for SW models in channels have been proposed. For example, schemes capable of preserving "lake-at-rest" steady states were introduced in [2,40,42] in the case of rectangular cross-section channels and in [16,20,21] in a more general case of arbitrary cross-section channels. In [34], a relaxation scheme that can exactly preserve moving-water equilibria for rectangular cross-section channels was proposed.…”

Numerical Simulation of Flood Routing through Channels with Area Variation using Staggered Grid Method

“…Meanwhile, the focus of other study is developing certain numerical methods to resolve complete Saint Venant equations without simpli cation that is ordinarily called the dynamic wave model. Numerical schemes that are ordinarily applied to solve dynamic model were the characteristic scheme (Chaudhry, 2008;Chow et al, 1988;Cunge, 1980), the scheme of nite di erence (Amein and Fang, 1970;Chaudhry, 2008;Nazari-Sharabian et al, 2020), the scheme of nite element (Cunge, 1980;Qureshi et al, 2014), the scheme of nite volume (Lai and Khan, 2018;Mungkasi et al, 2018), and the staggered scheme (Mungkasi et al, 2018;Sulistyono et al, 2020;Sulistyono and Wiryanto, 2019).…”

Numerical Simulation of Flood Routing using the Simplified Saint Venant Equations in Rectangular Channels

“…Meanwhile, other researchers developed certain numerical methods for solving Saint Venant's equations without simplification. Among the numerical methods used include the characteristic method [7], [8], the finite difference method [9], [10], [11], the finite element method [12], [13], the finite volume method [14], [15], and the finite volume method on the grid staggered [16], [17].…”