ENERGY-PRESERVING DISCRETIZATIONS OF THE NAVIER-STOKES EQUATIONS. CLASSICAL AND MODERN APPROACHES

The nonlinear character of the convective terms in the Navier-Stokes equations is at the root of many of the interesting features of fluid flow phenomena, the turbulent regime that invariably establishes at high Reynolds numbers being the most studied example. Nonlinear convective terms also pose the most critical issues when a numerical discretization of the Navier-Stokes equations is performed, especially at high Reynolds numbers. They are indeed responsible for a nonlinear instability arising from the amplification of aliasing errors that come from the evaluation of the products of two or more variables on a finite grid. The classical remedy to this frustrating difficulty has been the construction of difference schemes able to reproduce on a discrete level some of the fundamental symmetry properties of the Navier-Stokes equations. The invariant character of quadratic quantities such as kinetic energy in inviscid incompressible flows is a particular symmetry, whose enforcement typically guarantees a sufficient control of aliasing errors that allows the fulfillment of long-time integration. In this paper, a survey of the most successful approaches developed in this field is presented. The incompressible and compressible cases are treated separately and both the topics of spatial and temporal energy conservation are discussed. The effectiveness of some of the approaches illustrated is documented by numerical simulations of canonical flows.