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Finite Element Approximation of the Steady Smagorinsky Model

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Mathematical and Numerical Foundations of Turbulence Models and Applications

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Abstract

This chapter is devoted to the numerical approximation of the Smagorinsky model, in steady regime. We consider this model as a regularization of Navier–Stokes equations that includes the modeling of eddy diffusion effects by means of a discrete viscosity. We introduce Lagrange finite element spaces adapted to approximate the slip condition. We prove stability and strong convergence for solutions with the natural minimal regularity. We moreover study the asymptotic energy balance and in particular prove that the subgrid energy associated to the eddy diffusion asymptotically vanishes. We analyze the approximation of laminar flow by the SM, by means of error estimates for smooth solutions. These show a lack of optimality due to the Smagorinsky modeling of eddy viscosity.

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Authors and Affiliations

  1. Department of Differential Equations and Numerical Analysis and Institute of Mathematics (IMUS), University of Seville, Seville, Spain

    Tomás Chacón Rebollo

  2. Institute of Mathematical Research of Rennes, IRMAR - UMR 6625, CNRS, University of Rennes 1, Rennes, France

    Roger Lewandowski

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  1. Tomás Chacón Rebollo
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  2. Roger Lewandowski
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Chacón Rebollo, T., Lewandowski, R. (2014). Finite Element Approximation of the Steady Smagorinsky Model. In: Mathematical and Numerical Foundations of Turbulence Models and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-0455-6_9

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