Stable, non-dissipative and physically-consistent kinetic energy and entropy preserving (Keep) schemes for compressible flows

It is well known that it is effective to satisfy the conservation of secondary quantities, such as kinetic energy and entropy, for achieving stable numerical simulations of turbulent flows. Although the secondary conservative quantities can analytically be derived from the primary conservative quantities, kinetic energy preservation (KEP) or entropy preservation (EP) is not always guaranteed in a discrete sense. Some previous studies showed that KEP or EP schemes could enhance the numerical robustness of turbulent flow simulations without introducing additional numerical viscosity. Recently, the authors proposed kinetic energy and entropy preserving (KEEP) schemes for a Cartesian coordinate system and showed that the KEEP schemes could enhance numerical stability further, compared to typical KEP schemes. For the past three years, we have been working for three research subjects related to the KEEP schemes: a 2nd-order KEEP scheme on uniform Cartesian grids, a KEEP scheme for non-conforming block boundaries on Cartesian grids, and high-order KEEP schemes on generalized curvilinear grids. This study comprehensively discusses the recent progress in the development of the KEEP schemes.