A Cartesian Cut-Cell Solver for Compressible Flows

  • Daniel Hartmann
  • Matthias Meinke
  • Wolfgang Schröder
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 115)


A Cartesian cut-cell solver is presented to simulate two- and three-dimensional viscous, compressible flows on arbitrarily refined graded meshes. The finite-volume method uses cut cells at the boundaries rendering the method strictly conservative and is flexible in terms of shape and size of embedded boundaries. A linear least-squares method is used to reconstruct the cell center gradients in irregular regions of the mesh such that the surface flux can be formulated. The accuracy of the method is demonstrated for the three-dimensional laminar flow past a sphere.


Compressible Flow Ghost Cell Immerse Boundary Method Ghost Point Embed Boundary 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berger, M., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    De Zeeuw, D., Powell, K.: An adaptively refined Cartesian mesh solver for the Euler equations. J. Comput. Phys. 104, 56–68 (1993)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ghias, R., Mittal, R., Dong, H.: A sharp interface immersed boundary method for compressible viscous flows. J. Comput. Phys. 225, 528–553 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Hartmann, D., Meinke, M., Schröder, W.: An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods. Comput. Fluids 37, 1103–1125 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Hunt, J., Wray, A., Moin, P.: Eddies, stream, and convergence zones in turbulent flows. Center for turbulence research report CTR-S88 (1988)Google Scholar
  6. 6.
    Johnson, T., Patel, V.: Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70 (1999)CrossRefGoogle Scholar
  7. 7.
    Kim, J., Kim, D., Choi, H.: An immersed-boundary finite-volume method for simulations of flow in complex geometries. J. Comput. Phys. 171, 132–150 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)CrossRefGoogle Scholar
  9. 9.
    Liou, M.S., Steffen Jr., C.J.: A new flux splitting scheme. J. Comput. Phys. 107, 23–39 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Marella, S., Krishnan, S., Liu, H., Udaykumar, H.: Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations. J. Comput. Phys. 210, 1–31 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Meinke, M., Schröder, W., Krause, E., Rister, T.: A comparison of second- and sixth-order methods for large-eddy simulation. Comput. Fluids 31, 695–718 (2002)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mittal, R., Dong, H., Bozkurttas, M., Najjar, F., Vargas, A., von Loebbecke, A.: A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227, 4825–4852 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Mittal, R., Iaccarino, G.: Immersed boundary methods. Ann. Rev. Fluid Mech. 37, 239–261 (2005)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Peskin, C.: Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Ye, T., Mittal, R., Udaykumar, H., Shyy, W.: An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comput. Phys. 156, 209–240 (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daniel Hartmann
    • 1
  • Matthias Meinke
    • 1
  • Wolfgang Schröder
    • 1
  1. 1.Institute of AerodynamicsRWTH Aachen UniversityAachenGermany

Personalised recommendations