GENERALIZED KINETIC-ENERGY PRESERVING FORMULATIONS FOR COMPRESSIBLE FLOWS

Numerical discretization of Navier-Stokes equations are widely known to be susceptible to nonlinear numerical instabilities at high values of the Reynolds number, when central non-dissipative schemes are used. One of the possible remedies which have shown to be effective in alleviating this problem is the development of numerical methods that are able to replicate in the discrete sense the conservation of linear and quadratic invariants of the flow. The origin of the nonlinear instability is in the spatial discretization of the products derivatives arising in the convective terms. The problem is hence present both in incompressible and in (low-Mach) compressible flows. However, while in the incompressible case the adoption of staggered meshes and/or of the skew-symmetric form for the (quadratic) nonlinearity in the momentum equation are usually sufficient to ensure stable simulations, in the compressible low-Mach case the situation is different. The presence of an additional energy equation and of cubic nonlinearities highly complicates the analysis, because of the increased number of degrees of freedom in the choice of the discretization procedure and of the additional constraints required. In past years, various energy-preserving integration strategies have been proposed for compressible flows (for recent reviews cfr. [1]-[2]), in which different choices for the discretization of the energy equation and of pressure terms have been adopted. Each of them has different merits and peculiar characteristics, and a systematic assessment of the various approaches is missing. In this work, a review of the recent and most successful proposals in this field is presented, together with a theoretical analysis of the discrete equations. The conservation properties yielded by different choices for the energy equation (i.e. total and internal energy, entropy) are analyzed thoroughly. Numerical tests are performed to confirm the theoretical analysis, and they show that a careful choice of both the splitting and the energy formulation can provide robust and accurate results.