Chebychev optimized approximate deconvolution models of turbulence

Abstract

If the Navier–Stokes equations are averaged with a local, spacial convolution type filter, ϕ ¯ = g δ ∗ 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 ) . 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 → 0 , an ergodic theorem as the deconvolution order N → ∞ , and estimate the increase in accuracy obtained by parameter optimization. The report concludes with numerical illustrations.