Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids

Freestream and vortex preservation properties of a weighted essentially nonoscillatory scheme (WENO) and a weighted compact nonlinear scheme (WCNS) on curvilinear grids are investigated. While the numerical technique used for the compact difference scheme can be applied to WCNS, applying it to WENO is difficult. This difference is caused by difference in the formulation of numerical fluxes. WENO computed in the generalized coordinate system does not work well for either freestream or vortex preservation, whereas WENO computed in the Cartesian coordinate system works well for both freestream and vortex preservation, but its resolution is lower than that of WCNS. In addition, WENO in the Cartesian coordinate system costs three times as much as WENO or WCNS in the generalized coordinate system. Therefore, WENO in the Cartesian coordinate system is not suitable for solving Euler equations on a curvilinear grid. On the other hand, WCNS computed in the generalized coordinate system works well for freestream and vortex preservation when used with the numerical technique proposed for the compact difference scheme. The results show that WCNS with this numerical technique can be used for an arbitrary grid system. In this paper, the excellent freestream and vortex preservation properties of WCNS when used with the numerical technique, compared with those of WENO, are shown for the first time.