SPH energy conservation for fluid-solid interactions

In the present work, the energy conservation properties of the Smoothed Particle Hydrodynamics (SPH) are investigated in the presence of fluid-solid interactions. Similarly to the fluid phase, the solid bodies are modeled through solid particles so that the whole solid-fluid domain can be described as a unique particle system. The pressure and velocity fields are, then, extended over the solid particles in different ways, taking into account consistency issues related to the SPH differential operators and using the projection of the Navier-Stokes equations on the solid boundary. It is shown that, when solid particles are considered, the energy equation of the particle system contains some extra terms that depend on the pressure-velocity field extensions. The presence of these extra terms does not affect the consistency of the SPH model, since they tend to vanish when the spatial resolution is increased. Three prototypical numerical test cases are considered in order to provide a quantitative perspective of the matter and confirm the theoretical developments of the paper. These test cases show that the nature of these terms is globally dissipative, being thus in accordance with the Second Law of Thermodynamics. In particular, the viscous extra term displays a remarkably slow rate of convergence, this being a relevant outcome for those SPH practitioners that deal with these types of flows.