High-Order Fluxes for Conservative Skew-Symmetric-like Schemes in Structured Meshes: Application to Compressible Flows

Abstract

Developing high-order non-dissipative schemes is an important research task for both steady and unsteady flow computations. We take as a starting point the “built-in” de-aliasing property of the discretized skew-symmetric form for the non-linear terms of the Navier–Stokes equations, recalled in Kravchenko and Moin [1]. Two families of high-order locally conservative schemes matching this discretized skew-symmetric form are considered and rewritten in terms of telescopic fluxes for both finite difference and finite volume approximations in the context of compressible flows. The Jameson's scheme [2] is shown to be the second-order member of larger families of “skew-symmetric-like” centered schemes. The fourth-order finite volume and finite difference and the sixth-order finite difference schemes which belong to this family are provided. The proposed schemes are extended to shock capturing schemes, either by modifying the Jameson's artificial viscosity or by hybriding the centered flux with Weno [3] fluxes. An adapted interpolation is proposed to extend the use of the proposed schemes to non-regular grids. Several tests are provided, showing that the conjectured order is properly recovered, even with irregular meshes and that the shock capturing properties allow us to improve the second-order results for standard test cases. The improvement due to fourth-order is then confirmed for the estimation of the growth of two- (TS waves) and three- (Crow instability) dimensional unstable modes for both confined and free-shear flows. The last application concerns the steady computation using the Spalart–Allmaras model of a separated boundary layer: it confirms that the use of a high-order scheme improves the results, even in this type of steady applications.