A numerical method for junctions in networks of shallow-water channels

Abstract

There is growing interest in developing mathematical models and appropriate numerical methods for problems involving networks formed by, essentially, one-dimensional (1D) domains joined by junctions. Examples include hyperbolic equations in networks of gas tubes, water channels and vessel networks for blood and lymph in the human circulatory system. A key point in designing numerical methods for such applications is the treatment of junctions, i.e. points at which two or more 1D domains converge and where the flow exhibits multidimensional behaviour. This paper focuses on the design of methods for networks of water channels. Our methods adopt the finite volume approach to make full use of the two-dimensional shallow water equations on the true physical domain, locally at junctions, while solving the usual one-dimensional shallow water equations away from the junctions. In addition to mass conservation, our methods enforce conservation of momentum at junctions; the latter seems to be the missing element in methods currently available. Apart from simplicity and robustness, the salient feature of the proposed methods is their ability to successfully deal with transcritical and supercritical flows at junctions, a property not enjoyed by existing published methodologies. Systematic assessment of the proposed methods for a variety of flow configurations is carried out. The methods are directly applicable to other systems, provided the multidimensional versions of the 1D equations are available.

A Solution of the Junction Riemann Problem for 1D Hyperbolic Balance Laws in Networks including Supersonic Flow Conditions on Elastic Collapsible Tubes

“…Numerical modeling of one-dimensional (1D) flow in networks of spatial domains joined by junctions offers a satisfactory compromise between the quality of the numerical predictions and the computational cost. There are a variety of 1D flow applications in the literature such as industrial piping networks, traffic flow, water flows in open channels or blood flow in the human circulatory system [1]. Flow physics at the bifurcation is complex in symmetric or asymmetric bifurcations and also in arbitrary junctions.…”

“…In the 1D framework the junction is a singular point, where the numerical scheme cannot be directly applied and therefore internal boundary conditions must be prescribed [12], leading to the Junction Riemann Problem (JRP). A shortcoming of existing methods is their inability to deal with discontinuities or high subcritical, transcritical and supercritical flow through junctions, as in 1D channel networks, or to deal with transonic and supersonic flow conditions at junctions in physiological flows, such as in the venous system because of postural changes [1,13]. Existing methods are based on coupling approaches for energy or momentum conservation to the continuity equation and the characteristic equations [12,14,15] considering only subcritical or subsonic flow conditions.…”

Open Water Flow in a Wet/Dry Multiply-Connected Channel Network: A Robust Numerical Modeling Algorithm

“…Yoshioka et al (2015Yoshioka et al ( , 2016 use as the internal boundary conditions at the junction the continuity equation discretized on a dual mesh around it and the momentum flux at the junction calculated as a weighted linear combination of the momentum fluxes inflowing into it. Sanders et al (2001) and Bellamoli et al (2018) simulated open water flow in channel networks by nesting a 2D model at junctions. Contarino et al (2016) developed the implicit solver for the junction-generalized Riemann problem which was applied to construct a high-order ADER scheme for stiff hyperbolic balance laws in networks.…”

One-Dimensional/Two-Dimensional Coupling Approach with Quadrilateral Confluence Region for Modeling River Systems

“…This approach was considered in [14,15,32] using a hexagonal region to represent the river junction. In [3], this approach was implemented for T-shaped regions. One advantage of the 1-D/2-D coupling approach is that the shallow water assumption is valid everywhere.…”

“…A special ghost cell technique will be developed for coupling the reaches to the confluence region, which is one of the most important parts of a good 1-D/2-D coupling method; see, e.g. [3,9,27,31,34,40]. The proposed approach does not only allow one to easily model the geometry of the river junction, including relative widths of the reaches, the angle of entry of the tributary and the bottom topography, but also leads to very significant computational savings compared to solving the full 2-D problem.…”