Undular and broken surges in dam-break flows: a review of wave breaking strategies in a Boussinesq-type framework
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Undular and broken surges in dam‑break flows: a review of wave breaking strategies in a Boussinesq‑type framework Oscar Castro‑Orgaz1 · Hubert Chanson2 Received: 11 November 2019 / Accepted: 23 May 2020 © Springer Nature B.V. 2020
Abstract The water waves resulting from the collapse of a dam are important unsteady free surface flows in civil and environmental engineering. Considering the basic case of ideal dam break waves in a horizontal and rectangular channel the wave patterns observed experimentally depends on the initial depths downstream (hd) and upstream (ho) of the dam. For r = hd/ho above the transition domain 0.4–0.55, the surge travelling downstream is undular, a feature described by the dispersive Serre–Green–Naghdi (SGN) equations. In contrast, for r below this transition domain, the surge is broken and it is well described by the weak solution of the Saint–Venant equations, called Shallow Water Equations (SWE). Hybrid models combining SGN–SWE equations are thus used in practice, typically implementing wave breaking modules resorting to several criteria to define the onset of breaking, frequently involving case-dependent calibration of parameters. In this work, a new set of higher-order depth-averaged non-hydrostatic equations is presented. The equations consist in the SGN equations plus additional higher-order contributions originating from the variation with elevation of the velocity profile, modeled here with a Picard iteration of the potential flow equations. It is demonstrated that the higher-order terms confer wave breaking ability to the model without using any empirical parameter, such while, for r > 0.4– 0.55, the model results are essentially identical to the SGN equations but, for r 0.4–0.55 the dam break surge is undular (Fig. 1b), a feature linked to the existence of vertical accelerations and non-hydrostatic pressures [14, 15]. This feature is well-known to be out of the capabilities of the SWE, but Boussinesq-type models are able to replicate such wave motion [16, 17], the accuracy of the solution depending of the terms retained while making an approximate depth-averaging process of the Euler equations [18]. Most river flood waves resulting from the collapse of a dam are long, and, thus, we limit this study to the frequent case. The Serre–Green–Naghdi (SGN) equations are especially well-suited, given that this is an extended (non-hydrostatic) system of SWE for long waves (weakly dispersive) preserving full non-linearity [19]. Simulations of dam break waves using the SGN equations do predict undular or dispersive surges and rarefactions influenced by vertical accelerations. One would be inclined to discard the SWE for hd/ho > 0.4–0.55 and simply solve the SGN. However, for r = hd/ho
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