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A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

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Abstract

Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: they are based on the variation al formulation of the incompressible Navier–Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used math ematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

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Acknowledgments

The research of Tomás Chacón Rebollo and Samuele Rubino has been partially supported by the Spanish Government Project MTM2012-36124-C02-01. Samuele Rubino would also gratefully acknowledge the financial support received from VPPI-US (V Plan Propio de Investigación-Universidad de Sevilla) and FSMP (Fondation Sciences Mathématiques de Paris) during his postdoctoral research involved in this article.

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117, Berlin, Germany

    Naveed Ahmed & Volker John

  2. Department EDAN and IMUS, University of Seville, C/Tarfia s/n., 41012, Seville, Spain

    Tomás Chacón Rebollo & Samuele Rubino

  3. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195, Berlin, Germany

    Volker John

  4. Laboratoire Jacques-Louis Lions, Sorbonne Universités, UPMC Univ. Paris 06, 75005, Paris, France

    Samuele Rubino

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  1. Naveed Ahmed
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  2. Tomás Chacón Rebollo
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Correspondence to Samuele Rubino.

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Ahmed, N., Chacón Rebollo, T., John, V. et al. A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows. Arch Computat Methods Eng 24, 115–164 (2017). https://doi.org/10.1007/s11831-015-9161-0

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