Modelling Non-Hydrostatic Free-Surface Flows: A Discontinuous Galerkin Scheme for the Vertically Averaged and Moment Equations

This thesis aims to contribute in the advancement of non-hydrostatic free-surface modelling through the development and implementation of a new and robust numerical scheme for the Vertically Averaged and Moment (VAM) equations. Despite the Shallow Water Equations are widely used in free-surface flow modelling, their underlying assumptions may be violated in the presence of non-hydrostatic pressure distributions and/or non-uniform vertical velocities. For this reason, the thesis aims to fill the gap between the classical Shallow Water formulation and Navier-Stokes equations, overcoming the limitations of the shallow water assumptions, which neglect the vertical details of the velocity and pressure distributions, by introducing a formulation capable of capturing essential three-dimensional flow dynamics within a two-dimensional computational grid and thus avoiding the high computational cost. To achieve this, among the main non-hydrostatic free-surface models adopted in the literature, the VAM model emerges as the most complete and promising compromise between physical accuracy and computational effort. In particular, after a careful evaluation of the limitations of the numerical schemes adopted in the literature for solving the VAM system of equations, a novel fully implicit numerical scheme has been developed combining a Discontinuous Galerkin (DG) discretization with a local Taylor-based reconstruction. Specifically, the homogenous part of the governing equations is discretized using the DG formulation to guarantee stability without the need of any empirical tuning parameters, while the non-conservative terms are handled through a local Taylor based spatial reconstruction. The full system of equations is advanced in time using a single implicit step, resulting in a linear system that does not require any Newton-type linearization or other computationally expensive nonlinear iterative solvers. The new numerical scheme is validated against non-hydrostatic experimental benchmarks, demonstrating stability and good agreement in both steady and unsteady flow conditions even for Courant–Friedrichs–Lewy (CFL) numbers up to 10. Beyond the experimental test cases available in the literature, the thesis also includes a novel laboratory experiment specifically designed to create flow conditions for which the shallow-water hypotheses are no longer valid. This experiment represents a suitable two-dimensional experimental benchmark for the validation of non-hydrostatic numerical models providing detailed measurements of the free-surface elevation, pressure distribution, and velocity field. In conclusion, the present work provides a step in the non-hydrostatic free-surface modelling, introducing a new robust and efficient Discontinuous Galerkin finite element scheme for solving the VAM equations. The results demonstrate the model’s capability in reproducing the main non-hydrostatic flow features observed in the reference experiments, including cases where non-hydrostatic pressure effects dominate, as well as those where longitudinal velocity redistribution and secondary currents play a key role in the flow dynamics or where a delicate balance between nonlinearity and frequency dispersion is required. Compared to the classical shallow-water approach, the proposed model achieves a higher level of accuracy while maintaining reasonable computational times compared to full 3D models, thus providing a solid framework for future applications on large-scale problems, such as levee breaches, flood wave propagation in river bends, and flows in meandering channels.