Preserving Symmetry in Convection-Diffusion Schemes

We propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the corresponding differential operators. That is, the convective operator is represented by a skew-symmetric difference operator and the diffusive operator is approximated by a 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 conservative, i.e. it conserves the (total) mass, momentum and kinetic energy (when the physical dissipation is turned off); a symmetry-preserving discretization of the Navier-Stokes equations is stable on any grid. Because the numerical scheme is stable on any grid, the choice of the grid spacing can be based on the required accuracy. We investigate the accuracy of a fourth-order, symmetry-preserving discretization for the turbulent flow in a channel. The Reynolds number (based on the channel width and the mean bulk velocity) is equal to 5,600. It is shown that with the fourth-order, symmetry-preserving method a 64 × 64 × 32 grid suffices to perform an accurate simulation