High Order Finite Difference Schemes on Non-uniform Meshes with Good Conservation Properties

Numerical simulation of turbulent flows (DNS or LES) requires numerical methods that are both stable and free of numerical dissipation. One way to achieve this is to enforce additional constraints, such as discrete conservation of mass, momentum, and kinetic energy. The objective of this work is to generalize the high order schemes of Morinishi et al. to non-uniform meshes while maintaining conservation properties of the schemes as much as possible. This generalization is achieved by preserving symmetries of the uniform mesh case. The proposed schemes do not simultaneously conserve mass, momentum, and kinetic energy. However, depending on the form of the convective term, conservation of either momentum or energy in addition to mass can be achieved. It is shown that the conservation properties of the generalized schemes are as good as those of the standard second order finite difference scheme on non-uniform meshes, while the accuracy of the new schemes is definitely superior. The predicted conservation properties are demonstrated numerically in inviscid flow simulations.