In SPH, in fact, the building block is the substitution of the differential operators of the Navier–Stokes equations by their smoothed counterpart (see, for example, the works of Colagrossi et al. Here the differential operators are approximated by a space filtering approach that somehow recalls the space filtering used in LES modelling, although based on a totally different starting point and with a different goal. In addition to these considerations, there is a further crucial aspect that makes the LES approach most suited for modelling the Navier–Stokes equations in the SPH scaffold, i.e., the structure of the SPH itself. This (and the increased computer power) is probably the reason that determined the success of LES approaches in the last years, possibly in conjunction with zonal approaches, like detached eddy simulation (see the review in Spalart 12), that bypass the strict requirements for boundary layer simulations. LES modelling is expected to be easier than RANS modelling because, when the discretization is fine enough, the small sub-grid eddies should be closer to an “isotropic and homogeneous” scenario and therefore much less dependent on the specific problem under investigation.

The approach that lies in the middle between DNS and RANS is Large Eddy Simulation (LES), where only sub-grid turbulent eddies are modelled by space filtering, whereas the largest eddies are directly simulated (see, for instance, the classical reviews by Lesieur and Métais, 8 Piomelli, 9 Meneveau and Katz, 10 and Piomelli and Balaras 11).

Smoothed particle hydrodynamics book