The variational multiscale method for laminar and turbulent flow

The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.