Variable resolution smoothed particle hydrodynamics schemes for 2-d and 3-d viscous flows

Smoothed Particle Hydrodynamics (SPH) is a Lagrangian particle-based method for the numerical solution of the partial differential equations that govern the motion of fluids. The main aim of this thesis work is to better enable the applicability of SPH to problems involving multi-scale fluid dynamics. In the first part of the thesis, the capability of the SPH method to simulate three-dimensional isotropic turbulence is investigated with a detailed comparison of Lagrangian and Eulerian SPH formulations. The main reason for this first investigation is to provide an assessment of the error introduced by the particle disorder on the SPH discrete operators when being purely Lagrangian. When the free decay of isotropic turbulence in a triple periodic box is studied, the Eulerian SPH formulation achieves a very good agreement with other well validated reference solutions, whereas Lagrangian SPH yields an inaccurate prediction of turbulent energy spectra. When considering linearly forced isotropic turbulence, the use of a Godunov-type SPH scheme becomes essential for the achievement of a stable solution. The efficacy of the particle shifting technique applied to turbulent SPH flows is also studied in this part of the thesis and numerical findings indicate that corrective terms derived from the arbitrary Lagrangian-Eulerian theory are essential for a proper estimation of turbulence characteristics. Subsequently, numerical analyses of a decaying isotropic turbulent flow are carried out for the first time using SPH schemes based on high-order kernels. A dramatic increase in the accuracy of the results is observed when high-order SPH is employed, especially for the description of the vorticity dynamics. Motivated by the findings of the computational investigations above, the second part of this thesis focuses on the implementation and testing of a novel SPH variable-resolution algorithm. A domain-decomposition approach is adopted to partition the computational domain into regions having different particle resolutions. Each numerical sub-problem is then closed by appending buffer regions to every sub-domain, and populating these regions with particles whose physical quantities are obtained by means of interpolations over adjacent sub-domains. These interpolations are carried out using a second-order kernel correction procedure to ensure the proper consistency and accuracy of the interpolation process. The mass transfer among sub-domains is modeled by evaluating the Eulerian mass flux at the domain boundaries. Particles that belong to a specific zone are created/destroyed in the buffer regions and do not interact with fluid particles that belong to a different resolution zone. The algorithm is implemented in the DualSPhysics open-source code [62] and optimized thanks to DualSPHysics\u27 parallel framework. The algorithm is tested on a series of different fluid dynamics problems: a 2-D hydrostatic tank case, a flow past cylinder for different values of the Reynolds number, a flow past an oscillating cylinder in the cross-flow direction, and the propagation of regular waves across a rectangular tank. The present algorithm is able to simulate efficiently fluid dynamics problems characterized by a wide range of spatial scales, achieving a ratio between the coarsest and the finest resolution up to a factor equal to 256. The investigation is then extended to 3-D fluid dynamics problems, such as the flow past a sphere with a Reynolds Number equal to 300 and 500, for which a SPH solution using a uniform resolution is unfeasible due to the high computational cost, showing a good agreement between the results obtained with the variable-resolution algorithm herein presented and relevant numerical investigations in the literature. The work is then concluded with the simulation and validation of a 3-D dam-breaking flow impacting a cubic obstacle