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, LESfiltering procedure is recast in a Lagrangian framework bydefining a filter centred at the particle position that moves withthe filtered fluid velocity. The Lagrangian formulation of LES isthen used to re-interpret the SPH approximation of differentialoperators as a specific model based on the decomposition of theLES filter into a spatial and time filter.The derived equations represent a general LES-SPH schemeand contain terms that in part come from LES filtering and inpart derive from SPH kernels. The last ones lead to additionalterms (with respect to LES filtering) that contain fluctuations inspace, requiring adequate modelling. Further, since the adoptedLES filter differs from the classical Favre averaging for thedensity field, fluctuation terms also appear in the continuityequation.In the paper, a closure model for all the terms is suggested andsome simplifications with respect to the full LES-SPH model areproposed. The simplified LES model is formulated in a fashionsimilar to the diffusive SPH scheme of Molteni & Colagrossi[2] and the diffusive parameter is reinterpreted as a turbulentdiffusive coefficient, namely ? ? . In analogy with the turbulentkinetic viscosity ? T , the diffusive coefficient is modelled througha Smagorinsky-like model and both ? T and ? ? are assumed todepend on the magnitude of the local strain rate tensor D.Some examples of the simplified model are reported forboth 2D and 3D free-decaying homogeneous turbulence andcomparisons with the full LES-SPH model are provided.