This paper presents the numerical simulation of the undular bore using a Boussinesq-type model. The Boussinesq model is solved numerically using a hybrid finite-difference and finite volume scheme to achieve shock-capturing ability, which is suitable for modelling dam-break induced undular bore. New experiments are also carried out to investigate the evolution of dam-breaking induced undular bore in a wave flume, and its runup on a vertical wall. The Boussinesq model is utilized to reproduce the available experiments in literature and the present experiments as well. The ability and deficiency of the present shock-capturing Boussinesq model are discussed.
Undular bore waves commonly exist in nature. Fig.1 illustrates the undular bores observed in the Qiantang River in China. Other better known bores appear in the Severn River in England and the Dordogne River in France (Ali and Kalisch, 2010). Undular bores have also been reported in coastal areas. For example, the leading waves from large tsunamis are found to undergo undular bore process (e.g., Matsuyama et al. 2007), the notable occurrence is recorded during the 2004 Indian Ocean tsunami (e.g., Grue et al. 2008). The undular bores are characterized by a large leading wave followed by a series wave trains, which can cause catastrophic damage to the engineering structures and residency in the low-lying coastal regions during their evolution process. Thus the investigation and prediction of undular bores have been an important issue for a very long time and great efforts have been devoted to developing the mathematical and numerical tools for this purpose (Kim and Lynett, 2011; Mitsotakis et al., 2017).
Though commonly found in shallow water in the river and along coast, the generation and evolution of undular bores are quite dispersive and nonhydrostatic pressure plays a key role. The traditionally used shallow water equations (SWE) therefore fails to accurately predict the undular bores due to its inherent hydrostatic pressure assumption. RANS model (e.g., Du et al., 2017) and SPH model (e.g., Liu et al., 2015) are expected to accurately predict the undular bores, they however require relatively heavy computation cost and this limits their application in the real scenarios. An optional simulation approach is the Boussinesq type model. An attractive attribute of a conventional Boussinesq model is that it is able to reduce a complex three-dimensional (3D) problem into a two-dimensional (2D) problem, thereby greatly reducing the computational cost (Liu and Fang, 2016; Liu et al., 2018). In essence, most of existing Boussinesq models could be regarded as the extensions to the SWE model by including high order correction terms to account for both frequency dispersion and nonlinearity (Kim and Lynett, 2011).
