Lie Symmetries of the Lundgren−Monin−Novikov Hierarchy

  • N. Staffolani
  • M. Waclawczyk
  • Martin Oberlack
  • R. Friedrich
  • Michael Wilczek
Part of the Springer Proceedings in Physics book series (SPPHY, volume 149)

Abstract

In this work we consider the statistical approach to turbulence represented by the Lundgren-Monin-Novikov (LMN) hierarchy of equations for the probability density functions (PDFs). After a review of the properties that the PDFs have to satisfy, we first show the basic Galilean invariance of the LMN equations; then we discuss the extended Galilean one and finally we present a transformation of the PDFs and examine the conditions which have to be satisfied so that this transformation represents a symmetry of the LMN hierarchy corresponding in the Multi-Point Correlation (MPC) approach to one of the so called statistical symmetries found using the Lie symmetry machinery in [6] for the infinite hierarchy of equations satisfied by the correlation functions from which some decay exponents of turbulent scaling law could be worked out.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2010)MATHCrossRefGoogle Scholar
  2. 2.
    Hydon, P.E.: Symmetry Methods for Differential Equations: A Beginner’s Guide. Cambridge University Press (2000)Google Scholar
  3. 3.
    Lundgren, T.S.: Distribution functions in the statistical theory of turbulence. Physics of Fluids 10(5), 969–975 (1967)CrossRefGoogle Scholar
  4. 4.
    Monin, A.S.: Equations of turbulent motion. Prikl. Mat. Mekh. 31(6), 1057–(1967)MATHGoogle Scholar
  5. 5.
    Novikov, E.A.: Kinetic equations for a vortex field. Soviet Physics-Doklady 12(11), 1006–1008 (1968)Google Scholar
  6. 6.
    Oberlack, M., Rosteck, A.: New statistical symmetries of the multi–point equations and its importance for turbulent scaling laws. Discrete and Continuous Dynamical Systems Series S 3(3), 451–471 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Pope, S.B.: Turbulent Flows. Cambridge University Press (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • N. Staffolani
    • 1
  • M. Waclawczyk
    • 1
  • Martin Oberlack
    • 2
    • 3
  • R. Friedrich
    • 4
  • Michael Wilczek
    • 4
  1. 1.Chair of Fluid Dynamics, Department of Mechanical EngineeringTU DarmstadtDarmstadtGermany
  2. 2.Chair of Fluid Dynamics, Department of Mechanical Engineering and Center of Smart InterfacesTU DarmstadtDarmstadtGermany
  3. 3.GS Computational EngineeringTU DarmstadtDarmstadtGermany
  4. 4.Institute for Theoretical PhysicsUniversity of MünsterMünsterGermany

Personalised recommendations