In this paper, the entropy preserving (EP) scheme (which is introduced recently by Jameson) has been considered deeply and compared with the other artificial viscosity and upwind schemes. The discretization of the governing equations in the EP scheme is performed in such a way that the entropy is conserved in all those points with no shock. The purpose of this study was to introduce a stable numerical method that enters a minimum artificial dissipation only in the vicinity of shocks. In this paper, an inviscid one-dimensional flow through a convergent-divergent nozzle and a viscous two-dimensional flow with axial symmetry are considered. It is shown that the EP scheme is more accurate if the number of mesh points is increased; and in contrast to other schemes, there is no limit in increasing the number of points.

In this article, the kinetic energy-preserving (KEP) scheme and, also, the way of applying this scheme to the k-ε turbulent model are introduced. This study aims to introduce a stable method in which a few artificial dissipation terms are added to the governing equations in a way that the intensity of the solution fluctuations is reduced and, therefore, the problem stability increases. In accord with the importance of the study of turbulent flows, the effects of the fluctuating velocity terms on the calculation of all fluxes in the governing equations are scrutinized as well. Also, the influence from applying the KEP scheme on the k-ε turbulent model is investigated. This article reveals that by using the KEP scheme and, afterwards, improving the discretization method of the velocity fluctuation terms in k-ε equations, the accuracy of the results obtained is enhanced without a need to add artificial dissipation terms (or by minimizing their values).