Discrete energy-conservation properties in the numerical simulation of the Navier-Stokes equations

Nonlinear convective terms 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 difficulty has been the construction of difference schemes able to reproduce at a discrete level some of the fundamental symmetry properties of the Navier-Stokes equations. The invariant character of quadratic quantities such as global 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 both covered, and treated separately, and the topics of spatial and temporal energy conservation are discussed. The theory and the ideas are exposed with full details in classical simplified numerical settings, and the extensions to more complex situations are also reviewed. The effectiveness of the illustrated approaches is documented by numerical simulations of canonical flows and by industrial flow computations taken from the literature.