Chebychev optimized approximate deconvolution models of turbulence

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 DD is a bounded linear operator which satisfiesu=D(u¯)+O(δα),where δδ is the filter width and α⩾2α⩾2. Using a deconvolution operator as an approximate filter inverse, yields the closureuu¯=D(u¯)D(u¯)¯+O(δα).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.