Vertically averaged and moment equations for free surface flows: Pressure-Poisson solver, breaking waves, and morphodynamic modeling

Simulating unsteady free surface flows with full three-dimensional (3D) Navier–Stokes (N–S) equations is computationally expensive, so simplified models beyond shallow flow approximations are needed for large-scale river and ocean dynamics. In this study, we focus on a Vertically Averaged and Moment (VAM) equations approach for modeling unsteady free surface flows. The VAM equations are a quasi-3D modeling method derived from a variational approximation of the N–S equations, and therefore are not limited by the shallow flow approximation. The VAM models can approximate the N–S equations in a variety of free surface flow problems with good accuracy, but there is a lack of systematic consideration of these methods in the literature, which has slowed their adoption and development. Here, the VAM equations are presented, and several advances are introduced. For instance, a novel and efficient numerical method that easily enables the transformation of a Saint Venant solver into a VAM solver is presented. The nonhydrostatic pressure field is determined by a projection method inspired by Chorin's technique, notably without requiring iterative processes. Key physical aspects of the VAM model are then systematically highlighted, namely form drag over bedforms, frequency dispersion of water waves under nonlinear conditions as in Favre waves and collision of surges, turbulence modeling in the canonical case of uniform flow, breaking waves including dam-break flows and hydraulic jumps, and morphodynamic modeling, which encompasses geomorphic dam-break waves, erosion of dikes due to overtopping, and the generation and migration of antidunes in fluvial streams.