A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

The weighted essentially non-oscillatory (WENO) schemes have been extensively employed for the simulation of complex flow fields due to their high order accuracy and good shock-capturing properties. However, the standard finite difference WENO scheme cannot hold freestream automatically in general curvilinear coordinates. Numerical errors from non-preserved freestream can hide small scales such as turbulent flow structures; aero-acoustic waves which can make the results inaccurate or even cause the simulation failure. To address this issue, a new numerical strategy to ensure freestream preservation properties of the WENO schemes on stationary curvilinear grids is proposed in this paper. The essential idea of this approach is to offset the geometrically induced errors by proper discretization of the metric invariants. It includes the following procedures: (1) the metric invariants are retained in the governing equations and the full forms of the transformed equations on the general curvilinear coordinates are solved; (2) the symmetrical, conservative form of the metrics instead of the original ones are used; (3) the WENO schemes which are applied for the inviscid fluxes of the governing equations are employed to compute the outer-level partial derivatives of the metric invariants. In other words, the outer-level derivative operators for the metric invariants are kept the same with those for the corresponding inviscid fluxes. It is verified theoretically in this paper that by using this approach, the WENO schemes hold the freestream preservation properties naturally and thus work well in the generalized coordinate systems. For some well-known WENO schemes, the derivative operators for the metric invariants are explicitly expressed and thus this approach can be straightforwardly employed. The effectiveness of this strategy is validated by several benchmark test cases.