Consistency issues on SPH energy conservation when simulating viscous flows around solid bodies

The energy conservation properties of SPH in the presence of fluid-solid interactions are investigated in this paper. 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 equation since they tend to vanish when the spatial resolution is increased. Two prototypical numerical test-cases are considered in order to give a quantitative description of this topic