Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 052, 20 pages      physics/0605106      https://doi.org/10.3842/SIGMA.2006.052

Consequences of Symmetries on the Analysis and Construction of Turbulence Models

Dina Razafindralandy and Aziz Hamdouni
LEPTAB, Avenue Michel Crépeau, 17042 La Rochelle Cedex 01, France

Received October 28, 2005, in final form May 02, 2006; Published online May 12, 2006

Abstract
Since they represent fundamental physical properties in turbulence (conservation laws, wall laws, Kolmogorov energy spectrum, ...), symmetries are used to analyse common turbulence models. A class of symmetry preserving turbulence models is proposed. This class is refined such that the models respect the second law of thermodynamics. Finally, an example of model belonging to the class is numerically tested.

Key words: turbulence; large-eddy simulation; Lie symmetries; Noether's theorem; thermodynamics.

pdf (305 kb)   ps (728 kb)   tex (576 kb)

References

  1. Atherton R.W., Homsy G.M., On the existence and formulation of variational principles for nonlinear differential equations, Stud. Appl. Math., 1975, V.54, 31-60.
  2. Berselli L.C., Grisanti C.R., On the consistency of the rational large eddy simulation model, Comput. Vis. Sci., 2004, V.6, N 2-3, 75-82.
  3. Bytev V.O., Group-theoretical properties of the Navier-Stokes equations, Numerical Methods of Continuum Mechanics, 1972, V.3, N 3, 13-17 (in Russian).
  4. Cannone M., Karch G., About the regularized Navier-Stokes equations, J. Math. Fluid Mech., 2005, V.7, 1-28, math.AP/0305097.
  5. Cantwell B.J., Similarity transformations for the two-dimensional, unsteady, stream-function equation, J. Fluid Mech., 1978, V.85, 257-271.
  6. Chen Q., Jiang Y., Béhein C., Su M., Particulate dispersion and transportation in buildings with large eddy simulation, Technical Report, Massachusetts Institute of Technology, 2001.
  7. Danilov Yu.A., Group properties of the Maxwell and Navier-Stokes equations, Preprint, Khurchatov Inst. Nucl. Energy, Acad. Sci. USSR, 1967 (in Russian).
  8. Fushchych W.I., Popowych R.O., Symmetry reduction and exact solutions of the Navier-Stokes equations, J. Nonlinear Math. Phys., 1994, V.1, 75-113, 156-188, math-ph/0207016.
  9. Ibragimov N.H., Ünal G., Equivalence transformations of Navier-Stokes equations, Istanbul Tek. Üniv. Bül., 1994, V.47, 203-207.
  10. Ibragimov N.H., Kolsrud T., Lagrangian approach to evolution equations: symmetries and conservation laws, Nonlinear Dynam., 2004, V.36, 29-40.
  11. Iliescu T., John V., Layton W., Convergence of finite element approximations of large eddy motion, Numer. Methods Partial Differential Equations, 2002, V.18, 689-710.
  12. Iliescu T., John V., Layton W.J., Matthies G., Tobiska L., A numerical study of a class of LES models, Int. J. Comput. Fluid Dyn., 2003, V.17, 75-85.
  13. Kim P., Olver P.J., Geometric integration via multi-space, Regul. Chaotic Dyn., 2004, V.9, N 3, 213-226.
  14. Lilly D., A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids, 1992, V.4, 633-635.
  15. Lindgren B., Österlund J., Johansson A., Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressure-gradient turbulent boundary layers, J. Fluid Mech., 2004, V.502, 127-152.
  16. Méais O., Lesieur M., Spectral large-eddy simulation of isotropic and stably stratified turbulence, J. Fluid Mech., 1992, V.256, 157-194.
  17. Nielsen P., Restivo A., Whitelaw J., The velocity characteristics of ventilated rooms, J. Fluids Engrg., 1978, V.100, 291-298.
  18. Oberlack M., Symmetries, invariance and scaling-laws in inhomogeneous turbulent shear flows, Flow, Turbulence and Combustion, 1999, V.62, 111-135.
  19. Oberlack M., A unified approach for symmetries in plane parallel turbulent shear flows, J. Fluid Mech., 2001, V.427, 299-328.
  20. Olver P., Geometric foundations of numerical algorithms and symmetry, Appl. Algebra Engrg. Comm. Comput., 2001, V.11, 417-436.
  21. Razafindralandy D., Contribution à l'étude mathématique et numérique de la simulation des grandes échelles, PHD Thesis, Université de La Rochelle, 2005.
  22. Sagaut P., Large eddy simulation for incompressible flows. An introduction, Scientific Computation, Springer, 2004.
  23. Saveliev V., Gorokhovski M., Group-theoretical model of developed turbulence and renormalization of the Navier-Stokes equation, Phys. Rev. E, 2005, V.72, 016302, 6 pages.
  24. Ünal G., Application of equivalence transformations to inertial subrange of turbulence, Lie Groups Appl., 1994, V.1, 232-240.
  25. Ünal G., Constitutive equation of turbulence and the Lie symmetries of Navier-Stokes equations, in Modern Group Analysis VII, Editors N.H. Ibragimov, K. Razi Naqvi and E. Straume, Trondheim, Mars Publishers, 1997, 317-323.
  26. Winckelmans G.S., Wray A., Vasilyev O.V., Jeanmart H., Explicit filtering large-eddy simulation using the tensor-diffusivity model supplemented by a dynamic Smagorinsky term, Phys. Fluids, 2001, V.13, 1385-1403.

Previous article   Next article   Contents of Volume 2 (2006)