Variational Multiscale Methods for incompressible flows

The present paper reviews the Variational Multiscale Method (VMM) as a framework enabling the development of computational methods for the simulation of incompressible flows, particularly in the form of Large Eddy Simulations (LESs) of turbulent flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of all scales of the flow problem into two and three scale groups, respectively, is shown. A particular numerical method emanating from both the two- and the three-scale separation, respectively, is described, one of them in the form of a Finite Element Method (FEM) and the other one in the form of a Finite Volume Method (FVM). Respective numerical examples illustrate the suitability of both variants of the VMM for the numerical simulation of turbulent flows.