  Obtaining reliable numerical simulations of turbulent fluids is a challenging
problem in computational fluid mechanics. The Large Eddy Simulations (LES)
models are efficient tools to approximate turbulent fluids and an important
step in the validation of these models is the ability to reproduce relevant
properties of the flow. In this paper we consider a fully discrete
approximation of the Navier-Stokes-Voigt model by an implicit Euler algorithm
(with respect to the time variable) and a Fourier-Galerkin method (in the space
variables). We prove the convergence to weak solutions of the incompressible
Navier-Stokes equations satisfying the natural local entropy condition, hence
selecting the so-called physically relevant solutions
