Advertisement

Multi-domain Finite Element — Spectral Chebyshev Parallel Navier-Stokes Solver for Viscous Flow Problems

  • W. Borchers
  • S. Kräutle
  • R. Pasquetti
  • R. Peyret
  • R. Rautmann
Conference paper
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM) book series (NNFM, volume 82)

Summary

The paper is concerned with a hybrid finite element — spectral Chebyshev parallel solver of the incompressible Navier-Stokes equations. A domain decomposition, well adapted to the computation of wake or jet type flow, is assumed. Subdomains with complex geometry are handled with finite elements and the other ones with the highly accurate spectral method. The iterative resolution of the multi-domain problem is carried out with the “Conjugate Gradient Boundary Iteration” method. Here we focus on its optimal preconditioning when Gauss-Lobatto type grids are involved. The resulting algorithm is then applied to the flow past a cylinder and over a backward facing step at higher Reynolds numbers. For the latter case the numerical results display some transitional laminar-turbulent behaviour of the flow arising from unstable steady states. Numerical results are presented for both convective and absolute instability regions.

Keywords

High Reynolds Number Domain Decomposition Strouhal Number Spectral Element Method Projection Step 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Borchers, W.; Forestier, M.Y.; Kräutle, S.; Pasquetti, R.; Peyret, R.; Rautmann, R.; Ross, N.; Sabbah, C.: A Parallel Hybrid Highly Accurate Elliptic Solver for Viscous Flow Problems Numerical Flow Simulation I, NNFM66 , Hirschel Ed., Springer, 1998, pp. 3–24Google Scholar
  2. [2]
    Borchers, W.; Kräutle, S.; Pasquetti, R.; Rautmann, R.; Wielage, K.; Xu, C.J.: Towards a parallel hybrid highly accurate Navier-Stokes solver, Numerical Flow Simulation II, NNFM75 , Hirschel Ed., Springer, 2001, pp. 3–18Google Scholar
  3. [3]
    Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A.: Spectral methods in fluid dynamics, Springer-Verlag 1987Google Scholar
  4. [4]
    Botella, O.: On the solution of the Navier-Stokes equations using Chebyshev projection schemes with third order accuracy in time, Computers & Fluids, 26 (1997), pp. 107–116zbMATHCrossRefGoogle Scholar
  5. [5]
    Farhat, C; Roux, F.-X.: An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems, SIAM J. Sci. Stat. Comput., Vol. 13, No. 1 (1992), pp. 379–396zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Forestier, M.Y. et al.: Computations of 3D wakes in stratified fluids, Computational Fluid Dynamics Conference ECCOMAS2000, proc. in CD (2000)Google Scholar
  7. [7]
    Kräutle, S.: A Navier-Stokes solver based on CGBI and the method of characteristics, Doctoral thesis, Erlangen, 2001 http://www.am.uni-erlangen.de/am1/publications/dipl_phd_thesis/PhD_Kraeutle.ps.gz Google Scholar
  8. [8]
    Kräutle, S.; Wielage, K.: The CGBI method for viscous channel flows and its preconditioning, Nonlinear Analysis: Theory, Methods & Applications, 47 (6) (2001) pp. 4193–4203zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Kaiktsis, L.; Karniadakis, G.; Orszag, S.: Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step, J.Fluid Mech., 321 (1996), pp. 157–187zbMATHCrossRefGoogle Scholar
  10. [10]
    Maday, Y. et al.: An operator-integration-factor splitting method for timedependent problems: application to incompressible fluid flow, J. of Sci. Comp., 5(4) (1990), pp. 263–292zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Phillips, R.M.; Phillips, T.N.: Flow past a cylinder using a semi-Lagrangian spectral element method, Appl. Num. Math, 33 (2000), pp. 251–257zbMATHCrossRefGoogle Scholar
  12. [12]
    Prohl, A.: Projektions- und Quasi-Kompressibilitätsmethoden zur Lösung der inkompressiblen Navier-Stokes Gleichungen. Ph.D, thesis, 1995Google Scholar
  13. [13]
    Shen, J.: On error estimates of projection methods for the Navier-Stokes Equations: First order schemes, SIAM J.Numer. Anal. 29 (1992), pp. 57–77zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Shen, J.: On error estimates of projection methods for the Navier-Stokes Equations: Second-order schemes, Math. Comput. 65 215 (July 1996), pp. 1039–1065zbMATHCrossRefGoogle Scholar
  15. [15]
    Schaefer, M.; Turek, S.: Benchmark computations of laminar flow around a cylinder.Hirschel, E.H. (ed.), Flow simulation with high-performance computers II. DFG priority research programme results 1993–1995. Vieweg, Wiesbaden. Notes Numer. Fluid Mech. 52 (1996), pp. 547–566CrossRefGoogle Scholar
  16. [16]
    Xu, C.J.; Pasquetti, R.: On the efficiency of semi-implicit and semi-Lagrangian spectral methods for the calculation of incompressible flows, Inter. J. Numer. Meth. Fluids, 35 (2001), pp. 319–340zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • W. Borchers
    • 1
  • S. Kräutle
    • 1
  • R. Pasquetti
    • 2
  • R. Peyret
    • 2
  • R. Rautmann
    • 3
  1. 1.Institut für Angewandte MathematikFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany
  2. 2.Laboratoire J.A. Dieudonné, UMR CNRS 6621Université de Nice-Sophia AntipolisNiceFrance
  3. 3.Fachbereich Mathematik InformatikUniversität-Gesamthochschule PaderbornGermany

Personalised recommendations