Journal of Mathematical Fluid Mechanics

, Volume 10, Issue 1, pp 45–70 | Cite as

Second Order Adaptive Boundary Conditions for Exterior Flow Problems: Non-Symmetric Stationary Flows in Two Dimensions

  • Sebastian Bönisch
  • Vincent Heuveline
  • Peter WittwerEmail author


We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude.

Mathematics Subject Classification (2000).

76D05 76D25 76M10 41A60 35Q35 


Navier–Stokes equations artificial boundary conditions drag lift 


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Copyright information

© Birkhaueser 2006

Authors and Affiliations

  • Sebastian Bönisch
    • 1
  • Vincent Heuveline
    • 2
  • Peter Wittwer
    • 3
    Email author
  1. 1.Numerical Analysis group, IWRUniversity of HeidelbergHeidelbergGermany
  2. 2.Computing Center and Institute for Applied MathematicsUniversity of KarlsruheKarlsruheGermany
  3. 3.Département de Physique ThéoriqueUniversité de GenèveGenèveSwitzerland

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