Shallow Water and Navier-Stokes SPH-like numerical modelling of rapidly varying free-surface flows

Abstract

In coastal engineering, Lagrangian meshless numerical methods have reached a good popularity and they have been applied with success to describe wave breaking, impact of wave on structures and other rapid phenomena. This is due to the fact that they have a number of advantages in comparison with classical Eulerian schemes: no explicit treatment of the free surface and no computational grid mean that sophisticated meshing is not needed for complex geometries and therefore a number of problems that were considered largely intractable using classical Eulerian numerical methods such as finite volume or finite elements can now be simulated. As a relatively new method in Computational Fluid Dynamics, this kind of methods may be considered immature and many fundamental aspects and key characteristics remain to be fully investigated. The solid boundary condition is such an example: imposing closed boundary conditions in meshless methods in general, and in Smoothed Particle Hydrodynamic (SPH) in particular, is still an open problem. In the first chapter of this thesis an approximate Virtual Boundary Particle Method (VBP) for solid boundary conditions in two-dimensional (2-D) SPH models is presented; this is a development of the original VBP method recently proposed by Ferrari et al. (A new 3-D parallel SPH scheme for free-surface flows, Computers \& Fluids, 38(6), 1203-1217, 2009). The aim is to maintain the zeroth moment of the kernel function as closely as possible to unity, (a property referred to as zero-consistency), for particles close to solid boundaries. The main advantage of the MVBP in comparison with other methods such as Mirrored Particles is that curved boundaries or boundaries with angles can be easily reproduced. Some authors applied the Smoothed Particle Hydrodynamics (SPH) method to integrate the Shallow Water Equations (SWE) obtaining promising results for simple test cases where no open boundaries are present and the analytical formulation of source terms are applied: with SPH the wet-dry fronts do not need any special treatment, the equations are solved just where the fluid is present and this can potentially speed up the calculations if there are large dry areas in the domain. A 2D Shallow Water code based on the SPH interpolation is developed in the chapters 2 – 4 of this work, with the aim of further improving the capability of these numerical schemes of simulating real flooding events. The SPH-SWEs code is developed following the variational formulation, thanks to this approach the numerical scheme is robust and both the total mass and the momentum are conserved. Some major improvement has been introduced in the SPH-SWEs model in order to make the simulation of real floodings feasible. The Modified Virtual Boundary Particles (MVBP) is used to describe the closed boundaries, the bottom and the friction source term is described by a set of bottom particle. This discretization is effective not just for simple test case but also in for real bathymetries. Moreover, a particle splitting procedure has been inserted: it has the purpose to avoid the lack of resolution due to the variable kernel size being inversely proportional to water depth. This splitting procedure conserves mass and momentum by varying the smoothing length, velocity and acceleration of each refined particle. This improves predictions but does not necessarily provide good shock capturing. This is improved by treating particle interactions as a Riemann problem with MUSCL reconstruction providing stability. The last limitation that inhibits the use of the SPH-SWEs for real flooding simulation is the absence of any method to impose open boundary conditions. These are introduced in chapter 4 by adopting a simplified version of the Characteristic boundary method. Both supercritical and subcritical inflow and outflow boundary conditions can be simulated. Thanks to all the improvements described above, the simulation of two real events by a SPH-SWEs is presented in chapter 4, for the first time. The first case is the Okushiri tsunami occurred in Japan in 1993, whereas the second one is a flooding flood inundation at Thamesmead (UK). In Chapter 5 the simulation of rapidly varying flows is analysed removing the hypothesis of Shallow Water flows: a meshless Lagrangian numerical model called Finite Pointset Method (FPM) for the integration of Navier-Stokes equations in presence of free-surface flow is presented. The Finite Pointset Method (FPM) is a Lagrangian meshless method for numerical integration of pure incompressible Navier-Stokes equations, applied to date just to internal flows. It belongs to SPH like family because each particle carries a vector of field quantities such as pressure, density, velocity etc. and information and physical quantities are approximated using particles in a circular neighbourhood. FPM holds also some remarkable advantages in comparison with classical SPH methods: it is based on a moving least squares approach, where particles are just interpolation points without any associated mass and this means that any order of accuracy can be reached regardless to the particle’s position. In FPM the fluid is described as purely incompressible and the Navier-Stokes equation are solved numerically by means of the projection method therefore no spurious oscillations in the pressure field are present. Moreover in FPM boundary conditions can be analytically enforced using boundary particles and fluid particles can be added and removed in order to preserve the stability of the solution. This fact represents another fundamental advantage in comparison with classical SPH. Originally the FPM has been confined to single or two phase flow, but in chapter 5 it has extended also to free-surface flows by introducing a novel algorithm for free surface detection. In addition to that, a novel formulation of the Projection Method, called Incremental Pressure Projection Method, has been applied in order to preserve the hydrostatic condition.