In this paper, the differencing schemes for the simulation of incompressible flows in the OpenFOAM software are developed and tested. In order to select and apply eddy-resolved approaches, the dissipativity and stability of most of the numerical schemes implemented in the OpenFOAM are analyzed by solving the problems of decay of homogeneous isotropic turbulence and scalar transport. It is found that some schemes correctly describe the evolution of vortex structures but are not stable, so they are improved to eliminate oscillations and maintain an acceptable level of dissipation. Additionally, a streamline upwind numerical scheme that uses an equal-order finite-volume cell size to introduce additional stabilization is implemented. These numerical schemes have been compiled into an OpenFOAM library and are available on GitHub.