LES-SPH model for weakly-compressible Navier-Stokes equations

The Smoothed Particle Hydrodynamics (SPH) method is revisited within a Large Eddy Simulation (LES) perspective following the recent work of [1]. To this aim, LES filtering procedure is recast in a Lagrangian framework by defining a filter centred at the particle position that moves with the filtered fluid velocity. The Lagrangian formulation of LES is then used to re-interpret the SPH approximation of differential operators as a specific model based on the decomposition of the LES filter into a spatial and time filter. The derived equations represent a general LES-SPH scheme and contain terms that in part come from LES filtering and in part derive from SPH kernels. The last ones lead to additional terms (with respect to LES filtering) that contain fluctuations in space, requiring adequate modelling. Further, since the adopted LES filter differs from the classical Favre averaging for the density field, fluctuation terms also appear in the continuity equation. In the paper, a closure model for all the terms is suggested and some simplifications with respect to the full LES-SPH model are proposed. The simplified LES model is formulated in a fashion similar to the diffusive SPH scheme of Molteni & Colagrossi [2] and the diffusive parameter is reinterpreted as a turbulent diffusive coefficient, namely ? ? . In analogy with the turbulent kinetic viscosity ? T , the diffusive coefficient is modelled through a Smagorinsky-like model and both ? T and ? ? are assumed to depend on the magnitude of the local strain rate tensor D. Some examples of the simplified model are reported for both 2D and 3D free-decaying homogeneous turbulence and comparisons with the full LES-SPH model are provided.