Advertisement

Computational Mechanics

, Volume 63, Issue 5, pp 821–833 | Cite as

A cut finite element method for the solution of the full-potential equation with an embedded wake

  • M. DavariEmail author
  • R. Rossi
  • P. Dadvand
  • I. López
  • R. Wüchner
Original Paper
  • 66 Downloads

Abstract

Potential flow solvers represent an appealing alternative for the simulation of non-viscous subsonic flows. In order to deliver accurate results, such techniques require prescribing explicitly the so called Kutta condition, as well as adding a special treatment on the “wake” of the body. The wake is traditionally modelled by introducing a gap in the CFD mesh, which requires an often laborious meshing work. The novelty of the proposed work is to embed the wake within the CFD domain. The approach has obvious advantages in the context of aeroelastic optimization, where the position of the wake may change due to evolutionary steps of the geometry. This work presents a simple, yet effective, method for the imposition of the embedded wake boundary condition. The presented method preserves the possibility of employing iterative techniques in the solution of the linear problems which stem out of the discretization. Validation and verification of the solver are performed for a NACA 0012 airfoil.

Keywords

Full potential solver Extended finite element method Kutta condition 

Notes

Acknowledgements

This work was done within the ExaQUte’s project. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 800898. The authors are grateful for the collaboration offered by Stepan Rechtik and Fernaß Daoud of Airbus Defence and Space. They also acknowledge the financial support of CIMNE via the CERCA programme/Generalitat de Catalunya.

References

  1. 1.
    Eller D (2012) Fast, unstructured-mesh finite-element method for nonlinear subsonic flow. J Airc 49(5):1471–1479CrossRefGoogle Scholar
  2. 2.
    Holst T-L (2000) Transonic flow computations using nonlinear potential methods. Prog Aerosp Sci 36(1):1–61MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dang TQ, Chen LT (1989) Euler correction method for two- and three-dimensional transonic flows. AIAA J 27(10):1377–1386CrossRefGoogle Scholar
  4. 4.
    Lipman B (1958) Mathematical aspects of subsonic and transonic gas dynamics, volume 164 of surveys in applied mathematics, vol 3. Wiley, New YorkGoogle Scholar
  5. 5.
    Huicheng Y, Chunhui Z (2009) On global transonic shocks for the steady supersonic euler flows past sharp 2-d wedges. J Differ Equ 246(11):4466–4496MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen GQ, Chen J, Feldman M (2016) Transonic flows with shocks past curved wedges for the full euler equations. Discrete Contin Dyn Syst 36(8):4179–4211MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dowell EH, Bliss D (2013) New look at unsteady supersonic potential flow aerodynamics and piston theory. AIAA J 51(8):2278–2281CrossRefGoogle Scholar
  8. 8.
    Zhang Y (1999) Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary. SIAM J Math Anal 31(1):166–183MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bae M, Chen GQ, Feldman M (2011) Prandtl–Meyer reflection for supersonic flow past a solid ramp. Q Appl Math 71:583–600MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Elling V, Liu TP (2008) Supersonic flow onto a solid wedge. Commun Pure Appl Math Syst 61:1347–1448MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nishida B, Drela M (1995) Fully simultaneous coupling for three-dimensional viscous/inviscid flows. In: 13th AIAA applied aerodynamics conference, San Diego, CA, AIAA paper, June 1995, pp 1995–1806Google Scholar
  12. 12.
    Bochev P, Gunzburger M (2009) Least-squares finite element methods, volume 166 of applied mathematical sciences. Springer, New YorkGoogle Scholar
  13. 13.
    Agathos K, Chatzi E, Bordas S (2016) Stable 3d extended finite elements with higher order enrichment for accurate non planar fracture. Comput Methods Appl Mech Eng 306:19–46MathSciNetCrossRefGoogle Scholar
  14. 14.
    Carraro T, Wetterauer S (2016) On the implementation of the extended finite element method (xfem) for interface problems. Arch Numer Softw 4(2):1–23Google Scholar
  15. 15.
    Henning S, Thomas-Peter F (2011) The extended finite element method for two-phase and free-surface flows: a systematic study. J Comput Phys 230(9):3369–3390MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sawada T, Tezuka A (2011) Llm and x-fem based interface modeling of fluid-thin structure interactions on a non-interface-fitted mesh. Comput Mech 48(3):319–332MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fries T, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304MathSciNetzbMATHGoogle Scholar
  18. 18.
    Sven G, Arnold R (2007) An extended pressure finite element space for two-phase incompressible flows with surface tension. J Comput Phys 224(1):40–58MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chessa J, Belytschko T (2003) An extended finite element method for two-phase fluid. J Appl Mech 70(1):10–17MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chessa J, Smolinski P, Belytschko T (2002) The extended finite element method (xfem) for solidification problems. Int J Numer Methods Eng 53(8):1959–1977MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50(4):993–1013CrossRefzbMATHGoogle Scholar
  22. 22.
    Wells GN, Sluys LJ (2001) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50(12):2667–2682CrossRefzbMATHGoogle Scholar
  23. 23.
    Davari M, Rossi R, Dadvand P (2017) Three embedded techniques for finite element heat flow problem with embedded discontinuities. Comput Mech 59(6):1003–1030MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Soghrati S, Aragón A, Duarte C, Geubelle P (2012) An interface-enriched generalized fem for problems with discontinuous gradient fields. Int J Numer Methods Eng 89(8):991–1008MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Soghrati S, Duarte C, Geubelle P (2015) An adaptive interface-enriched generalized fem for the treatment of problems with curved interfaces. Int J Numer Methods Eng 102(6):1352–1370MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Soghrati S (2014) Hierarchical interface-enriched finite element method: an automated technique for mesh-independent simulations. J Comput Phys 275:41–52MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Aragón A, Simone A (2017) The discontinuity-enriched finite element method. Int J Numer Methods Eng 112(11):1589–1613MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193(33–35):3523–3540MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Dadvand P, Rossi R, Oñate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Archives Comput Methods Eng 17(3):253–297CrossRefzbMATHGoogle Scholar
  30. 30.
    Drela M (2014) Flight vehicle aerodynamics. MIT Press, CambridgeGoogle Scholar
  31. 31.
    Hess J, Smith AMO (1964) Calculation of non-lifting potential flow about arbitrary three-dimensional bodies, the journal of ship research. J Ship Res 8(2):22–44Google Scholar
  32. 32.
    Gennaretti M, Calcagno G, Zamboni A, Morino L (1998) A high order boundary element formulation for potential incompressible aerodynamics. Aeronaut J 102:211–219Google Scholar
  33. 33.
    Rodriguez D, Sturdza P, Suzuki Y, Martins-Rivas H, Peronto A (2012) A rapid, robust, and accurate coupled boundary-layer method for cart3d. In: 50th AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition.  https://doi.org/10.2514/6.2012-302

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • M. Davari
    • 1
    Email author
  • R. Rossi
    • 1
    • 2
  • P. Dadvand
    • 2
  • I. López
    • 3
  • R. Wüchner
    • 3
  1. 1.Polytechnic University of Catalonia (UPC)BarcelonaSpain
  2. 2.International Centre for Numerical Methods in EngineeringBarcelonaSpain
  3. 3.Technical University of MunichMunichGermany

Personalised recommendations