Enabling numerical accuracy of Navier-Stokes-α through deconvolution and enhanced stability^*

¹ Departmento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Brazil. carolina.manica@ufrgs.br; http://chasqueweb.ufrgs.br/~carolina.manica
² Department of Mathematics, University of Nevada, Las Vegas, USA. Monika.Neda@unlv.edu; http://www.pitt.edu/~mon5
³ Department of Mechanics and Mathematics, Moscow State M. V. Lomonosov University, Moscow 119899, Russia. Maxim.Olshanskii@mtu-net.ru; http://www.mathcs.emory.edu/~molshan
⁴ Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA. rebholz@clemson.edu; http://www.math.clemson.edu/~rebholz

Received: 4 May 2009
Revised: 29 October 2009
Revised: 29 March 2010

Abstract

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.