A comparative study of 1D and 2D approaches for simulating flows at right angled dividing junctions

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Abstract

This paper widens insight on the comparison between 1D and 2D approaches when simulating flow division at a 90° open-channel junction. For the 1D simulation, existing models regarding this issue are, on the one hand, of empirical nature and depend on the flow regime, thus not practical in the unsteady case. On the other hand, theoretical dividing models are strongly nonlinear, thus do not guarantee compatibility if combined with the shallow water model. By explicitly inducting the mathematical model of the side weir into the source terms of the conservative form of the 1D shallow water equations, the flow bifurcation is represented by considering a crest-free lateral spillway. For the 2D simulation, the whole system (branches and junction) is considered as one system and discretized into triangular cells forming an unstructured mesh. The numerical approximation of the two approaches is performed by a second order Runge–Kutta Discontinuous Galerkin (RKDG) scheme and tested through a pre-defined flow problem to illustrate the effects of the two approaches. The results are validated by experimental data. Comparisons are carried out for a super-, trans- and subcritical bifurcation, respectively, showing the practicality of the 1D approach and the advantage of the 2D approach.

Introduction

The division of a channel flow into two (or more) flows is a common occurrence within many fluvial systems. These fluvial forms are commonly called bifurcations, or diffluences, and they are fundamental forms within braided and distributary systems such as alluvial fans and river deltas.

The past 50 years have mostly witnessed substantial efforts to understand flow processes at intersections between channels. This focus has reflected the importance of river channel junctions as key components of both dendritic drainage networks and braided river systems.

The detailed hydrodynamics of junction flows is complex [1] and there are a number of parameters that characterise the flow’s physics. These include the size, shape, slope, and angle [2] between the two combining/dividing [3], [4], [5], [6] channels. Hence, scientific interest has increased, mostly toward theoretical considerations [1], [3], [7], [8], [9], [10], experiments conducted in laboratories [2], [4], [10], [11], [12] as well as recent 3D numerical modelling [4], [10], [13].

Confluence and bifurcation units are key elements within many river networks, having a major impact upon the routing of flow. Although much progress has been made in understanding river confluences, little recognition (compared to the confluence issue) has been given to bifurcations and dividing systems [3], [5], [6], [9], [14]. Focusing on the 1D bifurcation problem [15], [16], [17], [18], compared to the extensive research performed on confluences, there has been much less research on what happens in water flow separations, moreover, with a 90° T-junction type [3], [5].

Having an inflow divided into two parts – side and downstream – the lateral contribution is established by the use of a mathematical model of the side weir [19]. This flow diversion device is different from a junction because it has a lateral crest, and is usually used to divert water from the main waterway into irrigation systems, drainage networks or water distribution projects. The side weir is by definition an overflow and metering diversion structure installed on one or both sides of a main channel with the purpose of allowing part of the liquid to spill over the side and into another channel situated above the weir crest.

Following the idea of Rajaratnam and Pattabiramiah [17], we aim to calculate the flow separation at an intersection by considering a side weir without a weir crest (zero crest length). By including the Hager [19] side weir model into the conservative form of the shallow water equations, a new model is obtained allowing the calculation of the water level and the discharge in the downstream branches while taking into account the effect of the lateral outflow discharge. The main advantage of this model is that it allows the lateral overflow without prior knowledge of the flow regime, because it is introduced within an unsteady hyperbolic conservative model and thus it can handle the presence of flow discontinuities, such as hydraulic jumps, and transcritical flows with shocks [20].

A flow over a side weir can be accurately modelled [21] by incorporating the lateral overflow parameters within the source terms of the conservative form of the unsteady shallow water equations. By this means, if one considers a zero weir crest height, the coupling of the intersection model and the canal model becomes natural and the lateral outflow effects act routinely within the Saint-Venant model [22].

The aim of this paper is to compare the 1D and 2D approaches when simulating flows at a right angled dividing junction and to show the advantage of the 2D analysis of the junction. The numerical approximation of the two approaches is performed by a second order Runge–Kutta Discontinuous Galerkin (RKDG) scheme [23]. The flow bifurcation cases are defined considering super-, trans- and subcritical flows at the separation point. The numerical results are evaluated through a set of experiments that are performed at the INSA (Lyon). We are interested in this comparison of the upstream discharge distribution in the downstream branches.

Section snippets

1D mathematical model

Flow over a side weir is one of the most complex flows to simulate in a 1D analysis. A complete analytical solution for the equations governing flows over a side weir [24], [25] is very complicated. In real-life applications, the hydraulic behaviour of lateral overflows often encounters a discontinuous evolution of the depth line through the occurrence of a hydraulic jump (super- to subcritical flow regime passage), in addition to several cases of transcritical flows (sub- to supercritical

2D mathematical model

Assuming that the flow is homogenous, incompressible, two-dimensional, viscous with a hydrostatic pressure distribution and with the absence of Coriolis and wind forces, the non-linear partial differential equation system used to describe the 2D, depth-averaged, free-surface flow is as follows:Ut+F=S,in which F = (E, G) andU=hhuhv,E=huhu2+gh2/2huv,G=hvhuvhv2+gh2/2andS=S0+Sf=0ghS0xghS0y+0-ghSfx-ghSfyis the source term.

In the above relations h represents the water depth, u, v are the

Evaluations and discussions

In this section, the two approaches 1D and 2D are used for handling open-channel flow bifurcations. For the 1D simulation, we consider a zero crest height, while for the 2D simulation, we simply use the 2D Saint-Venant equations. The results depicted from the two approaches are validated with the experimental results that are performed at the INSA (Lyon). All the channels are rectangular, 2.54 m long and 0.3 m wide. A main inflow division, to lateral and downstream outflows, is investigated; the

Conclusion

Over the past few years, understanding of river junctions has improved. Compared to the confluence issue, dividing flow models and the bifurcation problem have been given less recognition. Attention has mostly focused on 1D theoretical 90° dividing models, empirical relationships, experimental designs and very recently 3D numerical simulations. Typically, the flow at a bifurcation point may be predicted by empirical formulas that, unfortunately, depend on the flow regime and are hence not

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