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.

Vertically Averaged and Moment Equations for Dam-Break Wave Modeling: Shallow Water Hypotheses

“…The features of this surge depend on the ratio r of downstream (h d ) to upstream (h u ) flow depths. If r is equal to or greater than the transition domain 0.4−0.55 [4,5,48], non-hydrostatic effects are important, producing an advancing train of waves, essentially formed by a leading solitary-like wave followed by a packet of shorter waves (Figure 1a). This type of front surge is called undular surge and reproduced by the SGNE [4].…”

“…flow depths and velocities before wave release, the resulting dam-break flows can lead to different phenomena, e.g. the front surge may be undular or broken, in addition to a rarefaction wave propagating into the upstream direction [4]. The prediction of dam-break waves is, therefore, of paramount importance in the river environment [5] as well as canals and their hydraulic structures.…”

“…Castro-Orgaz et al [7] demonstrated that the single-layer case is a special example of the Boussinesq-type model assuming a linear non-hydrostatic pressure distribution. The Serre-Green-Naghdi Equations (SGNE) [4,[35][36][37] is a Boussinesq type model obtained by assuming that (i) the horizontal velocity is independent of the vertical coordinate, (ii) vertical velocity is linear, (iii) pressure distribution is non-hydrostatic, following a parabolic law.…”

“…However, these improvements on the shallow water hypotheses inherent of the SGNE are not independent, e.g., assuming (i), improvement (ii) follows from continuity considerations, and (iii) by resorting to the vertical momentum balance. The SGNE are widely used for nearshore flow modelling [38][39][40], and more recently in dam break wave modeling [4,37]. The so-called hybrid shock-capturing Boussinesq models are able to simulate the wave propagation from deep water to the coastline, including wave breaking, by switching off the dispersive terms when and where the ratio between the wave height and the water depth is equal to a certain threshold value, usually taken in practice equal to 0.8.…”

“…A consequence is that wave breaking shall be mimicked in SGNE by introducing different types of breaking-sub-models, like an eddy viscosity term, a roller flow model, or a hybrid treatment switching to the SWE at breaking nodes. It was recently revealed that the lack of the wave-breaking ability in the SGNE is due to the lack of modeling of differential advection of momentum, e.g., the horizontal velocity profile is not modeled in the vertically integrated equations [4].…”