  We consider a 3D Approximate Deconvolution Model (ADM) which belongs to the
class of Large Eddy Simulation (LES) models. We work with periodic boundary
conditions and the filter that is used to average the fluid equations is the
Helmholtz one. We prove existence and uniqueness of what we call a "regular
weak" solution $(\wit_N,q_N)$ to the model, for any fixed order $N\in\N$ of
deconvolution. Then, we prove that the sequence $\{(\wit_N,q_N)\}_{N \in \N}$
converges -in some sense- to a solution of the filtered Navier-Stokes
equations, as $N$ goes to infinity. This rigorously shows that the class of ADM
models we consider have the most meaningful approximation property for averages
of solutions of the Navier-Stokes equations.
