Asymptotically entropy conservative and kinetic-energy and pressure-equilibrium preserving schemes based on economical algebraic fluxes

In the context of Direct and Large Eddy Simulations of (low-Mach) turbulent compressible flows, an important topic is the design of robust and accurate numerical methods, which can efficiently handle high Reynolds number and/or under resolved simulations by keeping under control the aliasing errors coming from spatial discretization. To this aim modern numerical methods are usually required to satisfy some physics-compatible constraints, which typically amount to the discrete enforcement of the induced balance of suitably selected secondary quantities. Kinetic Energy Preserving (KEP) schemes are the most used among them, and in recent years various KEP formulations have been proposed. Their use for the spatial discretization of the mass and momentum equations is now considered a necessity for a reliable simulation. Entropy Conservative (EC) methods have also been explored. As they guarantee a correct discrete balance of entropy, they provide an important additional property for both the reliability and robustness of the overall procedure. Finally, Pressure Equilibrium Preserving (PEP) schemes guarantee the correct discrete evolution of density waves.

Existing KEP, PEP and EC schemes require the specification of nonlinear fluxes involving the evaluation of costly transcendental functions, which are more expensive than the classical algebraic fluxes associated with FD formulations. In this contribution, we propose a hierarchy of numerical fluxes for the compressible flow equations which are KEP, PEP and Asymptotically Entropy Conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The main feature is that the fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. Numerical tests on benchmark 1D problems as well as on turbulent flows confirm the theoretical predictions and the favorable properties in terms of robustness and low computational cost of the proposed schemes.