Modelling Undular Bores: A Comparative Study

The sudden stopping of the turbines used in hydropower production leads to the propagation of waves in both upstream and downstream directions. This phenomenon, referred to as triggering or disjunction, results in an elevation in water level in the headrace channel. The complex patterns of free-surface undulations in this channel depend on the triggering event characteristics and the geometry of the channel. Those patterns result from the combination of (i) the water level adaptation to the abrupt change in the discharge regime and momentum distribution, (ii) the pressure distribution within the wave, causing secondary undulations and (iii) the channel configuration, including profile geometry, the presence of secondary channels or hydraulic structures like locks, etc. Classical 1D numerical approach fails to accurately reproduce these complex patterns. Shallow Water Equation, based on the assumption of hydrostatic pressure distribution, might prove inadequate to model secondary waves; non hydrostatic approaches such as Boussinesq equations can deal with this phenomenon (Violeau in Contribution to the theory of undular bores. A journey around the Korteweg–de Vries equation, 2022 [1]), but both are unable to describe breaking bores phenomena. Furthermore, 1D modelling is unable to accurately capture wave reflection phenomena, as well as geometrical effects. In this study, we investigate the modelling of different disjunction test cases in a headrace channel, using various numerical and scale models. This paper describes the methodology used to correctly reproduce undular bores, compared with field measurements. A 1D computation with CNR’s Crue10 code (Balayn et al. in J l’Hydraul 36(1):1–8, 2014 [2]) is carried out as a first step. Then 2D modelling is performed with the 2-D code Basilisk (Page Web Basilisk, 2023 [3]), using three numerical solvers/approaches: Saint-Venant’s Shallow Water Equations, Green–Naghdi, and Multilayers. The results were validated against laboratory and field experiments (Alliau et al. in New modelling paradigms for water issues? Paris, 2023 [4]). For a “wave Froude number” Fro < 1.09 and a near-uniform channel geometry, the study shows that the Saint-Venant solver can accurately capture the dynamics of the observed wave patterns. In that case, Saint-Venant, Green–Naghdi, and Multilayers solvers provide close results. Intercomparison of the 2D results with in-situ measurements (for two different triggers) revealed a trendy for this Saint-Venant approached to underestimate the maximum water levels, with an average bias of 6 cm.