On the Use of Higher-Order Finite-Difference Schemes on Curvilinear and Deforming Meshes

This study enables the use of very high-order finite-difference schemes for the solution of conservation laws on stretched, curvilinear, and deforming meshes. To illustrate these procedures, we focus on up to 6th-order Pade-type spatial discretizations coupled with up to 10th-order low-pass filters. These are combined with explicit and implicit time integration methods to examine wave propagation and wall-bounded flows described by the Navier–Stokes equations. It is shown that without the incorporation of the filter, application of the high-order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability. Inclusion of the discriminating low-pass high-order filter restores the advantages of high-order approach even in the presence of large grid discontinuities. When three-dimensional curvilinear meshes are employed, the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated. To overcome this problem, a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes. For dynamically deforming grids, an effective numerical treatment is described to evaluate expressions containing the time-varying transformation metrics. With these techniques, metric cancellation is guaranteed regardless of the manner in which grid speeds are defined. The efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel.