A stable high-order finite difference scheme for the compressible Navier-Stokes equations, far-field boundary conditions

  • Authors:

  • Magnus Svärd;Mark H. Carpenter;Jan Nordström

  • Affiliations:

  • Center for Turbulence Research, Building 500, Stanford University, Stanford, CA 94305-3035, USA;Computational Methods and Simulation Branch, NASA Langley Research Center, Hampton, VA 23681-2199, USA;Computational Physics Department, Division of Systems Technology, The Swedish Defence Research Agency, SE-164 90 Stockholm, Sweden and Department of Information Technology, Uppsala University, SE- ...

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

We construct a stable high-order finite difference scheme for the compressible Navier-Stokes equations, that satisfy an energy estimate. The equations are discretized with high-order accurate finite difference methods that satisfy a Summation-By-Parts rule. The boundary conditions are imposed with penalty terms known as the Simultaneous Approximation Term technique. The main result is a stability proof for the full three-dimensional Navier-Stokes equations, including the boundary conditions. We show the theoretical third-, fourth-, and fifth-order convergence rate, for a viscous shock, where the analytic solution is known. We demonstrate the stability and discuss the non-reflecting properties of the outflow conditions for a vortex in free space. Furthermore, we compute the three-dimensional vortex shedding behind a circular cylinder in an oblique free stream for Mach number 0.5 and Reynolds number 500.