A Gas-Kinetic model for Shallow Water flows in presence of wet/dry fronts

Recently the single relaxation time Lattice Boltzmann Method (hereinafter LBM) has considerably spread in Computational Hydraulics. The LBM consists of a mesoscopic representation of the flow, whose description is made in terms of a finite number of probability distribution functions, each one giving the probability to find a fluid particle in a given position and with a given velocity. The main reason for the diffusion of the LBM in Computational Hydraulics is that the corresponding numerical algorithm is much simpler than the usual ones derived from “classical” hydraulic models (such as e.g. the Shallow Water equations). The simplicity of the standard LBM lies on the discretization of the velocity space: i.e. lies on the fact that even a rather low number of velocities (9 in the standard LBM-based model of the Shallow Water equations) permits a satisfactory description of subcritical Shallow Water (hereinafter SW) flows. The main drawback of the standard LBM-based model of the SW equations is that it cannot simulate transcritical and supercritical SW flows, which always occur in realistic situations. Such a serious drawback depends on the low number of lattice velocities usually adopted in standard LBM-based models of SW equations. If the number of lattice velocity is arbitrarily increased (i.e. an infinite number of lattice velocity is adopted), numerical simulations of transcritical and supercritical SW flows are possible. This is the basic idea of the Gas Kinetic Method (hereinafter GKM), recently extended to the simulation of SW flows. A reliable and versatile numerical tool aimed to the simulation of realistic SW flows must own the capability of handling wet/dry fronts. SW equations do not account for this phenomenon automatically. Under strict mathematical point of view, wet/dry fronts are moving boundaries and should be treated just like that. On the other hand, the approach employing a moving boundary can become extremely cumbersome and many possibilities have been proposed and adopted in the recent past as remedies within numerical SW models. The aim of this work is to assess the ability of the finite volume formulation of the GKM recently proposed in literature by Ghidaoui et al. (2001) and Liang et al. (2007) when simulating SW flows in presence of wet-dry fronts. The assessment is performed through a comparison with a considerable number of benchmark cases, both theoretical and experimental, both 1D and 2D, in order to consider some of the main critical aspects of the SW numerical modeling. Results are promising. Some of the preliminary ones, aimed to the assessment of the correctness of the proposed numerical implementation of the GKM-based model for SW flows, are shown in Fig. 1, which shows several 1D steady SW flows over smooth bottom profiles. In the left panel depth profiles are shown, in the right panel velocity and Froude number profiles are shown. “Sub” and “Sup” stands for Subcritical and Supercritical regime respectively. Continuous traces are analytical solution. The ability in satisfactorily reproducing the hydraulic jump is evident in the left panel. Finally a qualitative idea on the ability of the proposed GKM model in simulating SW flows in presence of wet-dry fronts is given in Fig. 2, where the well-known experimental CADAM test of Hiver (2000) is shown. In the left panel the blue line represents a propagating wet-dry front, the position of the latter being highlighted by the circle, while the red dots represent the Froude number of the flow. In the right panel the comparison between experimental and GKM numerical results is given at x=19.5 for a time interval of 40 seconds. The agreement is quite good.