Symmetry-preserving discretization of turbulent flow

We propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators, i.e., the convective operator is represented by a skew-symmetric coefficient matrix and the diffusive operator is approximated by a symmetric, positive-definite matrix. Mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner. We give it a concrete form by noting that a symmetry-preserving discretization of the Navier-Stokes equations is stable on any grid, and conserves the total mass, momentum and kinetic energy (for the latter the physical dissipation is to be turned off, of coarse). Being stable on any grid, the choice of the grid may be based on the required accuracy solely, and the main question becomes: how accurate is a symmetry-preserving discretization? Its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies. The comparison is carried out for a Reynolds number of 5600, which is based on the channel width and the mean bulk velocity (based on the channel half-width and wall shear velocity the Reynolds number becomes 180). The comparison shows that with a fourth-order, symmetry-preserving method a 64x64x32 grid suffices to perform an accurate numerical simulation