Simulating open-channel flows and advective diffusion phenomena through SPH model

Abstract

The present thesis treats Computational Fluid Dynamics based on particle methods. The fully Lagrangian approach Smoothed Particle Hydrodynamics (SPH) is developed for two-phase flows. The model is extended to research fields of environmental hydraulic and open-channel flows. SPH is a Lagrangian, meshless and particle model. It was born about 30 years ago to solve gas-dynamics problems in open space (Lucy, 1977 [1]; Gingold and Monaghan, 1977 [2]). For many years, the SPH method has been applied to problems in the astrophysical field, as documented in the review paper by Benz (1990) [3]. During the last decades, the SPH method has been increasingly modified and extended to provide approximations to the partial difference equations (PDEs) in a wide range of scientific and engineering applications particularly in the hydrodynamic field. Monaghan (1994) [4] was the first to apply the SPH scheme to fluid-dynamics problems. After that, the SPH approach has been successfully extended to multiphase flows (see e.g. Grenier et al., 2009 [5]) and fluid-structure interaction problems (see e.g. Colagrossi and Landrini, 2003 [6]). Following the SPH method, the motion of a continuum medium is described using an interpolation technique which allows to approximate functions and differential operators on an irregular distribution of points. In the standard SPH, where a weakly compressible fluid is considered, the discretized continuity and momentum equations are linked via a state equation. Firstly, an algorithm is developed to treat upstream/downstream boundary conditions for 2D open-channel flows in SPH context. For this purpose two suitable sets of particles (in/out-flow particles) are defined allowing the enforcement of different upstream and downstream flow conditions. In particular this permits to avoid generation of unphysical pressure shock waves due to a direct creation/deletion of fluid particles. As first test case, the proposed algorithm is validated for a viscous laminar flow in open channel considering Reynolds numbers of order O(102). The obtained results are compared with analytical ones in order to heuristically check the convergence of the numerical scheme. The simulations are performed for a time interval long enough to reach steady state conditions. The suitability of the in/out-flow algorithm has been highlighted comparing the velocity field with the analytical Poiseuille solution. The second test case deals with a hydraulic jump for which different upstream and downstream conditions are needed. Several types of jumps, obtained varying the flow Froude number, are investigated with particular reference to the location of the jump and the velocity field. Comparisons between the numerical results and the classical theory of the hydraulic jump are provided, showing good agreements. In the second part of the thesis, the SPH model is applied to evaluate the concentration field of pollutants in water. A Lagrangian formalism is formulated to solve the fickian diffusion equation considering pollutants with the same density as the water. Furthermore, a SPH form of the advective diffusion equation is also developed for pollutant-water, taking into account the effects of molecular diffusion and natural advection induced vii by differences between the fluid densities. These equations are coupled with the fluid mechanics equations. Attention is paid to the numerical aspects involved in the solution procedure and to the optimization of the model parameters. Environmental engineering problems concerning diffusion and natural advection phenomena occur in the presence of a pollutant in still water. Numerical tests referring to a strip and a bubble of contaminant in a water tank with different initial concentration laws have been carried out. The results obtained by the proposed SPH models are compared with other available SPH formulations, showing an overall better agreement with standard analytical solutions in terms of spatial evolution of the concentration values. Capabilities and limits of the proposed SPH models to simulate advective diffusion phenomena for a wide range of density ratios are discussed. As future perspectives, coupling the two aspects considered in this thesis, it will be developed a numerical code for the simulation of the concentration field along a water stream by an intake of pollutants. viii