Vertically averaged and moment equations: New derivation, efficient numerical solution and comparison with other physical approximations for modeling non-hydrostatic free surface flows

• Published in: Journal of computational physics Vol. 504; p. 112882
• Main Authors: Escalante, C., Morales de Luna, T., Cantero-Chinchilla, F., Castro-Orgaz, O.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2024
• Subjects: Depth-integrated modeling; Dispersive waves; Finite difference; Finite volume; Non-hydrostatic models; Weighted average; Finite difference; Depth-integrated modeling; Weighted average; Dispersive waves; Finite volume; Non-hydrostatic models
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2024.112882

Efficient modeling of flow physics is a prerequisite for a reliable computation of free-surface environmental flows. Non-hydrostatic flows are often present in shallow water environments, making the task challenging. In this work, we use the method of weighted residuals for modeling non-hydrostatic free surface flows in a depth-averaged framework. In particular, we focus on the Vertically Averaged and Moment (VAM) equations model. First, a new derivation of the model is presented using expansions of the field variables in sigma-coordinates with Legendre polynomials basis. Second, an efficient two-step numerical scheme is proposed: the first step corresponds to solving the hyperbolic part with a second-order path-conservative PVM scheme. Then, in a second step, non-hydrostatic terms are corrected by solving a linear Poisson-like system using an iterative method, thereby resulting in an accurate and efficient algorithm. The computational effort is similar to the one required for the well-known Serre-Green-Naghdi (SGN) system, while the results are largely improved. Finally, the physical aspects of the model are compared to the SGN system and a multilayer model, demonstrating that VAM is comparable in physical accuracy to a two-layer model. •New reformulation of the Vertically Averaged and Moment (VAM) equations model.•Discretization with high order accurate finite.•volume finite-difference scheme.•The proposed approach is computationally very efficient and robust.•Successful comparison with analytical solutions and test problem with experimental reference data.