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 termuu¯¯¯¯. An approximate deconvolution operator DD is a bounded linear operator which satisfies

u=D(u¯¯)+O(δα), Turn MathJaxon

where δδ is the filter width and α⩾2α⩾2. Using a deconvolution operator as an approximate filter inverse, yields the closure

uu¯¯¯¯=D(u¯¯)D(u¯¯)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯+O(δα). Turn MathJaxon

The residual stress of this model (and related models) depends directly on the deconvolution error,u−D(u¯¯). This report derives deconvolution operators yielding an effective turbulence model, which minimize the deconvolution error for velocity fields with finite kinetic energy. We also give a convergence theory of deconvolution as δ→0δ→0, an ergodic theorem as the deconvolution order N→∞N→∞, and estimate the increase in accuracy obtained by parameter optimization. The report concludes with numerical illustrations.