This thesis investigates turbulent wall-bounded flows using the Smoothed Particle Hydrodynamics (SPH) method. The first part focuses on the SPH method itself in the context of the Navier-Stokes equations with a special emphasis on wall boundary conditions. After discussing classical wall boundary conditions a detailed introduction to unified semi-analytical wall boundary conditions is given where the key parameter is a renormalization factor that accounts for the truncated kernel support in wall-bounded flows. In the following chapter it is shown that these boundary conditions fulfill energy conservation only approximately. This leads to numerical noise which, interpreted as form of Brownian motion, is treated using an additional volume diffusion term in the continuity equation where it is shown to be equivalent to an approximate Riemann solver. Two extensions to the boundary conditions are presented dealing with variable driving forces and a generalization to Robin type and arbitrary-order interpolation. Two modifications for freesurface flows are then presented, one for the volume diffusion term and the other for the algorithm that imposes Robin boundary conditions. The variable driving force is validated using a Poiseuille flow and the results indicate an error which is five orders of magnitude smaller than with the previous formulation. Discretising the wave equation with Robin boundary conditions proves that these are correctly imposed and that increasing the order of the interpolation decreases the error. The two modifications for flows under the influence of external forces significantly reduce the error at the free-surface. Finally, a dam break over a wedge demonstrates the capabilities of all the proposed modifications. With the aim of simulating turbulent flows in channels, the thesis moves on to extending the unified semi-analytical wall-boundary conditions to three dimensions. The thesis first presents the consistent computation of the vertex particle mass. Then, the computation of the kernel renormalization factor is considered, which in 3-D consists of solving an integral over a two dimensional manifold where the smoothing kernel intersects the boundary. Using a domain decomposition algorithm special integration areas are obtained for which this integral can be solved for the 5 th -order Wendland kernel. This algorithm is successfully applied to several validation cases including a dam break with an obstacle which show a significant improvement compared to other approximative methods and boundary conditions. The second part of this thesis investigates turbulent flows, in particular turbulent channel flow. This test case is introduced in detail showing both the physical properties as well as established numerical methods such as direct numerical simulation (DNS) and large eddy simulation (LES). In the penultimate chapter several SPH simulations of the turbulent channel flow are shown. The first section deals with a quasi DNS of the minimal-flow unit, a channel flow with a minimal domain size to sustain turbulent flow structures. The Eulerian statistics are compared to literature and show good agreement except for some wall-normal quantities. Furthermore, preliminary Lagrangian statistics are shown and compared to results obtained from a mesh-based DNS. The final simulation shows a LES of a full-sized channel at Reynolds number Re τ = 1000. The Eulerian statistics are compared to literature and the discrepancies found are explained using simulations of the Taylor-Green vortex, indicating that the momentum is not transferred appropriately due to an unresolved velocity-pressure-gradient tensor.

This thesis presents a two-phase liquid-solid numerical model using Smoothed Particle Hydrodynamics (SPH). The scheme is developed for multi-phase flows in industrial tanks containing sediment used in the nuclear industry for decommissioning. These two-phase liquid-sediments flows feature a changing interfacial profile, large deformations and fragmentation of the interface with internal jets generating resuspension of the solid phase. SPH is a meshless Lagrangian discretization scheme whose major advantage is the absence of a mesh making the method ideal for interfacial and highly non-linear flows with fragmentation and resuspension. Emphasis has been given to the yield profile and rheological characteristics of the sediment solid phase using a yielding, shear and suspension layer which is needed to predict accurately the erosion phenomena. The numerical SPH scheme is based on the explicit treatment of both phases using Newtonian and non-Newtonian Bingham-type constitutive models. This is supplemented by a yield criterion to predict the onset of yielding of the sediment surface and a suspension model at low volumetric concentrations of sediment solid. The multi-phase model has been compared with experimental and 2-D reference numerical models for scour following a dry-bed dam break yielding satisfactory results and improvements over well-known SPH multi-phase models. A 3-D case using more than 4 million particles, that is to the author’s best knowledge one of the largest liquid-sediment SPH simulations, is presented for the first time. The numerical model is accelerated with the use of Graphic Processing Units (GPUs), with massively parallel capabilities. With the adoption of a multi-phase model the computational requirements increase due to extra arithmetic operations required to resolve both phases and the additional memory requirements for storing a second phase in the device memory. The open source weakly compressible SPH solver DualSPHysics was chosen as the platform for both CPU and GPU implementations. The implementation and optimisation of the multi-phase GPU code achieved a speed up of over 50 compared to a single thread serial code. Prior to this thesis, large resolution liquid-solid simulations were prohibitive and 3-D simulations with millions of particles were unfeasible unless variable particle resolution was employed. Finally, the thesis addresses the challenging problem of enforcing wall boundary conditions in SPH with a novel extension of an existing Modified Virtual Boundary Particle (MVBP) technique. In contrast to the MVBP method, the extended MVBP (eMVBP) boundary condition guarantees that arbitrarily complex domains can be readily discretized ensuring approximate zeroth and first order consistency for all particles whose smoothing kernel support overlaps the boundary. The 2-D eMVBP method has also been extended to 3-D using boundary surfaces discretized into sets of triangular planes to represent the solid wall. Boundary particles are then obtained by translating a full uniform stencil according to the fluid particle position and applying an efficient ray casting algorithm to select particles inside the fluid domain. No special treatment for corners and low computational cost make the method ideal for GPU parallelization. The models are validated for a number of 2-D and 3-D cases, where significantly improved behaviour is obtained in comparison with the conventional boundary techniques. Finally the capability of the numerical scheme to simulate a dam break simulation is also shown in 2-D and 3-D.

With the algorithm’s suitability for exploiting current petascale and next-generation exascale supercomputers, stable and structure-preserving properties are necessary to develop predictive computational tools. In this dissertation, summation-by-parts (SBP) operators and a new relaxation Runge–Kutta (RRK) scheme are used to construct mimetic and structure-preserving full discretization for non-reactive compressible computational fluid dynamics (CFD) and reaction-diffusion models. In the first chapter, we provide the necessary background and a literature survey that forms the basis of this dissertation. Next, we provide a short overview of entropy stability for general conservation laws. The second chapter covers the analysis of the Eulerian model for compressible and heat-conducting flows. We provide the necessary background of the new system of parabolic partial differential equation (PDE). Then, we present the entropy stability analysis of the model at the continuous level. Subsequently, using the SBP, we construct an entropy-stable discretization of any order for unstructured grids with tensor-product elements. The third chapter discusses the implementation of RRK methods. We start by reviewing the RRK scheme constructed to guarantee conservation or stability with respect to any inner-product norm. Then, we present the extension and generalization of RRK schemes to general convex functionals and their application to compressible fluid flow problems. The final chapter demonstrates the far-reaching capabilities of the SBP operators and RRK schemes presenting the development of a novel fully discrete Lyapunov stable discretization for reaction models with spatial diffusion. Finally, we conclude this dissertation with an overview of our achievements and future research directions.