If the Navier-Stokes equations are averaged with a local, spacial convolution type filter, φ = gδ ∗ φ , the resulting system is not closed due to the filtered nonlinear term uu. An approximate deconvolution operator D is a bounded linear operator which satisfies

u = D(u) + O(δ α ),

where δ is the filter width and α ≥ 2. Using a deconvolution operator as an approximate filter inverse yields the closure

uu = D(u)D(u) + O(δ α ).

We derive optimal approximate deconvolution models for 3D turbulence. Specifically, we find the optimal parameters that minimize the time averaged consistency error of approximate deconvolution operators and models for time averaged, fully developed, homogeneous, isotropic turbulence.

We answer important questions of How to adapt deconvolution procedures to velocities from homogeneous, isotropic turbulent flows? and What is the increase in accuracy that results?