Undular and broken surges in dam-break flows: a review of wave breaking strategies in a Boussinesq-type framework

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, wave breaking is automatically accounted for, thereby producing broken waves as part of the solution. The transition from undular to broken surges predicted by the high-order equations is gradual and in good agreement with experimental observations. Using the solution of the new higher-order equations it was further developed a new wave breaking index based on the acceleration at the free surface to its use in hybrid SGN–SWE models.