Nearly All-Speed, Stabilized Time-Accurate Upwind Scheme on Unstructured Grid

A time-accurate, upwind, finite volume method for computing compressible flows on unstructured grids is presented. The method is second-order-accurate in space and time and yields high resolution in the presence of discontinuities. In the basic Euler and Navier - Stokes upwind scheme, many concepts of high-order upwind schemes are adopted: the surface flux integrals are carefully treated, a Cauchy - Kowalewski time-stepping scheme is used in the time-marching stage, and a multidimensional limiter is applied in the reconstruction stage. However, even with these up-to-date improvements, the basic upwind scheme is still plagued by the so-called pathological behaviors (for example, the carbuncle, the expansion shock, etc.), which are mostly triggered due to some undesirable local numerical instability. A simple multidimensional dissipation model is used to systematically suppress such behaviors and stabilize the scheme for flows at high Mach numbers, whereas for flows at very low Mach number (for example, M = 0:02), it is found that computation can be directly carried out without invoking preconditioning. The modified, stabilized scheme is referred to as the enhanced time-accurate upwind scheme (Loh, C. Y., and Jorgenson, P. C. E., "A Time Accurate Upwind Unstructured Finite Volume Method for Compressible Flow with Cure of Pathological Behaviors," AIAA Paper 2007-4463, 2007.) in this paper. The unstructured grid capability renders flexibility for use in complex geometry, and the present enhanced time-accurate upwind Euler and Navier - Stokes scheme is capable of handling a broad spectrum of flow regimes from high supersonic to subsonic at very low Mach number, appropriate for both computational fluid dynamics and computational aeroacoustics. Numerous examples are included to demonstrate the robustness of the scheme. [PUBLICATION ABSTRACT]