Flux-limited schemes for the compressible Navier-Stokes equations

Abstract

Several high-resolution schemes are formulated with the goal of improving the accuracy of solutions to the full compressible Navier-Stokes equations. Calculations of laminar boundary layers at subsonic, transonic, and supersonic speeds are carried out to validate the proposed schemes. It is concluded that these schemes, which were originally tailored for nonoscillatory shock capturing, yield accurate solutions for viscous flows. The results of this study suggest that the formulation of the limiting process is more important than the choice of a particular flux splitting technique in determining the accuracy of computed viscous flows. Symmetric limited positive and upstream limited positive schemes hold the promise of improving the accuracy of the results, especially on coarser grids. HE calculation of compressible flows at transonic, supersonic, and hypersonic Mach numbers requires the implementation of nonoscillatory discrete schemes which combine high accuracy with high resolution of shock waves and contact discontinuities. These schemes must also be formulated in such a way that they facilitate the treatment of complex geometric shapes. In the past decade numerous schemes have been developed to meet these requirements in conjunction with the solution of the Euler equations.l More recently, the application of such schemes to the Navier-Stokes equations has produced algorithms which have progressively gained acceptance as analysis tools in the aerospace industry. There remains, however, a need to understand and improve Navier-Stokes schemes beyond the current state of the art. The most compelling reason for this rests on the fact that shock capturing requires the construction of schemes which are numerically dissipative, a requirement which could affect the global accuracy of the solution of the physical viscous problem. In a recent paper2 Jameson has shown that a theory of nonoscillatory schemes can be developed for scalar conservation laws based upon the local extremum diminishing (LED) principle that maxima should not increase and minima should not decrease. Moreover, although it is equivalent to the total variation diminishing principle (TVD) for one-dimensional problems, the LED principle can be applied naturally to multidimensional problems on both structured and unstructured meshes. This recent development has shed new light on the principles underlying the construction of both high-resolution switched and flux-limited dissipation schemes. In particular, it allowed the new formulation of two families of flux-limited schemes denominated, symmetric limited positive (SLIP) and upstream limited positive (USLIP), respectively. The present work merges several dissipation schemes based on both the SLIP and USLIP construction with a well-developed cell-centered, finite-volume formulation for solving the two-dimensional Navier-Stokes equations.3 The aim is to analyze and validate these new discretizations for the solution of viscous flow problems. In Sec. II the design principles of nonoscillator y discrete approximations to a scalar convection equation are reviewed together