Attenuating and enhancing properties of the approximate deconvolution method based on higher-order explicit and compact filters

Abstract

We analyse the accuracy of a deconvolution (inverse filtering) method in 1D and 2D periodic domains. The deconvolution is performed by applying the iterative van Cittert method using explicit and compact filters of the 2nd to 8th order. We consider cases in which an approximate inverse filter G_a⁻¹ formulated to deconvolve a an original function from a filtered one (f‾ = G∗f ) is constructed based on: (a) G the same as used to define f‾= G∗f ; (b) G different than the one used to define f‾ = G∗f. In case (a), the convergence rate of the deconvolution process is much better when compact filters are used. This is attributed to a flatter transfer function of this type of filter and thus a smaller deterioration of the input function f‾. Case (b) reflects a real situation in which the precise definition of a basic filter G used in f‾ = G∗f is unknown. We found that when G_a⁻¹ is formulated based on G of a higher order a than the one used to define f‾ the reconstructed function f^* = G_a⁻¹∗f‾ is suppressed compared to the original function f. On the other hand, the deconvolution process performed with the use of G_a⁻¹ defined based on G of a lower order than the order a of the basic filter significantly amplifies the reconstructed function f^*. As a result, the function f^* contains more energy than the function f, especially in the range of small and high-frequency scales. This effect is particularly strong when explicit filters of different orders are used. The impact of the filter type in the practical application of deconvolution is demonstrated based on large eddy simulations (LES) of a 2D decaying homogenous turbulent flow. LES combined with an approximate deconvolution method (ADM) for the computation of sub-filter terms shows better accuracy than in the case when these terms are modelled using the classical Smagorinsky model or when they are neglected (no-model approach). This analysis consists of comparisons of the evolution of total energy, energy spectra, and higher-order moments (variance, skewness, kurtosis) of the velocity components and vorticity. We found that more accurate results are obtained when the deconvolution is performed using the explicit filters even if the deconvolution process based on the compact filters was found to converge faster in 1D and 2D test cases. Most likely this is because in the performed LES the explicit filters correspond better to an unknown filter induced by discretisation.