The simplest decomposition of a turbulent field

In this paper we will present some recent results concerning the simplest decomposition of a turbulent field. The large scale filtering operator is simply given by the two-point sum in space and the associated fluctuation is given by the two-point difference. In the paper we will present the general properties of this simple decomposition and their particular relations with the subgrid stresses and the generalized central moments associated to a generic filtering operator.