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A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations

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Abstract

This paper presents a variational multiscale residual-based stabilized finite element method for the incompressible Navier–Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multiscale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska–Brezzi (inf–sup) condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressure-velocity elements comprising 4-and 10-node tetrahedral elements and 8- and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-driven cavity flow problem.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Arif Masud & Ramon Calderer

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  1. Arif Masud
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  2. Ramon Calderer
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Correspondence to Arif Masud.

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Masud, A., Calderer, R. A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations. Comput Mech 44, 145–160 (2009). https://doi.org/10.1007/s00466-008-0362-3

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