Enabling numerical accuracy of Navier-Stokes-α through deconvolution and enhanced stability
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 2, p. 277-307

We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly improves accuracy in simulations. Standard finite element schemes for NS-α suffer from two major sources of error if their solutions are considered approximations to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error on the velocity error that arises from the (necessary) use of the rotational form nonlinearity. The proposed scheme “fixes” these two numerical issues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We prove the scheme is stable, optimally convergent, and the effect of the pressure error on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate the effectiveness of the method.

DOI : https://doi.org/10.1051/m2an/2010042
Classification:  65M12,  65M60,  76D05
Keywords: ns-alpha, grad-div stabilization, turbulence, approximate deconvolution
@article{M2AN_2011__45_2_277_0,
     author = {Manica, Carolina C. and Neda, Monika and Olshanskii, Maxim and Rebholz, Leo G.},
     title = {Enabling numerical accuracy of Navier-Stokes-$\alpha $ through deconvolution and enhanced stability},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {2},
     year = {2011},
     pages = {277-307},
     doi = {10.1051/m2an/2010042},
     zbl = {1267.76021},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_2_277_0}
}
Manica, Carolina C.; Neda, Monika; Olshanskii, Maxim; Rebholz, Leo G. Enabling numerical accuracy of Navier-Stokes-$\alpha $ through deconvolution and enhanced stability. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 2, pp. 277-307. doi : 10.1051/m2an/2010042. http://www.numdam.org/item/M2AN_2011__45_2_277_0/

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