High-order shallow water expansions in free surface flows: Application to steady overflow processes

Abstract

Free surface flow modeling in the natural environment is frequently conducted within a depth-integrated framework using the non-linear shallow water equations or Saint-Venant equations. However, these equations underestimate the discharge in overflow processes over obstacles, such as during the flooding of coastal levees. Alternative models are constructed by expansions in terms of a shallowness parameter σ, with the 2nd-order results being the so-called Serre-Green-Naghdi theory, widely used in ocean research. Higher-order expansions are required in some instances to obtain more accurate results or to set the validity limit of the leading order term of these perturbation results. However, most of the high-order results available apply to water wave problems over horizontal beds, yet not to the overflow over obstacles of generally uneven bathymetry. High-order shallow water expansions for open channel flows over uneven beds are developed in this work by an iterative procedure to generate the corresponding asymptotic expansions up to the high-order O(σ6), resulting in high-order ODEs when the Bernoulli equation at the free surface streamline is invoked to include gravity effects. These new equations are applied to the overflow process, as occurs during the coastal flooding in a levee-protected area. A perturbation solution to this problem is presented up to O(σ6) for the main overflow variables, which are successfully and systematically compared with a new set of experiments conducted in a large scale obstacle model. The novel theoretical solutions presented are further demonstrated to be better than former theories available in the literature.