(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ahmed, Naveed; Rebollo, Tomás Chacón; John, Volker; Rubino, Samuele
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 variational 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 mathematical
tools. Furthermore, the literature presenting numerical studies with the VMS
methods is surveyed and the obtained results are summarized.
