Semi-Lagrangian simulation of particle laden flows using an SPH framework

Particle-laden flows occur in many natural and industrial systems and simulating them can be particularly challenging. The coupling of smoothed particle hydrodynamics (SPH) with the discrete element method (DEM) can effectively simulate particle-laden multiphase fluid dynamical systems, due to the shared Lagrangian nature of both methods. However, this approach has some inherent shortcomings, including a prohibitively small time step when dealing with small particles. An alternative approach is to use an Eulerian-Eulerian reference frame, usually using finite element or finite volume discretisations. In this approach, momentum and continuity equations are solved for each discrete phase as well as the continuous phase, and particles are modelled by means of concentration and velocity fields. This approach can suffer from strong numerical diffusion in the advection of the concentrations when absolute velocities of the phases are high, whilst their relative velocities are small. This numerical diffusion can obscure important aspects of the behaviour as it can smooth out details, especially in the particle concentration fields. In order to mitigate the shortcomings of these existing techniques, we present a new SPH-based semi-Lagrangian framework for solving the momentum and continuity equations for all phases in particle-laden flows. In this framework, the discrete and continuous phases move relative to a reference frame that moves at a momentum-averaged velocity. By focusing on velocities relative to the reference frame our method substantially reduces numerical diffusion compared to traditional Eulerian-Eulerian approaches, at a lower computational cost compared to Lagrangian-Lagrangian approaches. The simulation approach is validated by means of comparisons to both computational and experimental results for a number of relevant systems including particle-liquid separation in an inclined channel, particle sedimentation in a liquid, and gas-particle fluidised beds. This new method is shown to compare very favourably in terms of both the accuracy of the results and the computational cost required to achieve them.