New developments and classical theories of turbulence

  • E. Levich
Part of the International Centre for Mechanical Sciences book series (CISM, volume 415)


In this paper we review classical and modern concepts pertinent for the theory of developed turbulent flows. We begin by introducing basic facts concerning the properties of the Navier-Stokes equations with the emphasis on invariant properties of the vorticity field. Then we discuss classical semi-empirical approaches to developed turbulence which for a long time constitute a basis for engineering solutions of turbulent flows problems. We do it for two cases, homogeneous isotropic turbulence and flat channel turbulent flow. Next we discuss the insufficiency of classical semi-empirical approaches. We show that intermittency is an intrinsic feature of all turbulent flows and hence it should be accounted for in any reasonable theoretical approach to turbulence. We argue that intermittency in physical space is in one to one correspondence with certain phase coherence of turbulence in an appropriate dual space, e.g., Fourier space for the case of homogeneous isotropic turbulence. At the same time, the phase coherence has its origin in invariant topological properties of vortex lines in inviscid flows, modified by the presence of little molecular viscosity. This viewpoint is expounded again using the examples of homogenous isotropic turbulence and channel flow turbulence. Finally we briefly discuss the significance of phase coherence and intermittency in turbulence for the fundamental engineering challenge of turbulence control.


Vortex Line Vortex Tube Phase Coherence Vorticity Field Inertial Range 
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  1. Arnol’d, V. I. (1974). The asymptotic Hopf invariant and its applications. In Proc. of Summer School in Differential Equations, Erevan Armenian SSR Academy of Sciences, 229–256, (English Translation in Selecta Mathematica Sovietica, 5, 326–345 ).Google Scholar
  2. Batchelor, G. K., and Townsend, A. A. In Proc. Royal Society, A 199, 1057, 238–255.Google Scholar
  3. Berger, M. A., and Field, G. B. Journal of Fluid Mechanics, 147, 133.Google Scholar
  4. Handler, R. A., Levich, E., and Sirovich, L. Physics of Fluids, 5, 686.Google Scholar
  5. Kolmogorov A. A. Dokl. Akad. Nauk SSSR,30, 4 299–303.Google Scholar
  6. Landau, L. D., and Lifshitz, E. Fluid Mechanics,Pergamon Press.Google Scholar
  7. Levich, E. Anomalous helicity fluctuations and coherence in turbulence: Theory and Experiment. In Proc. of IUTAM Symposium on Topological Fluid Mechanics, Moffatt, H. K., and Tsinober, A., eds., Cambridge Press. Levich, E. Physics Report, 151, 120.Google Scholar
  8. Levich, E., and Malomed, B. Model of boundary layer turbulence. In Progress in Turbulence Research,28–56, 11.Google Scholar
  9. Branover and Y. Unger, eds., 162 Progress in Astronautics and Aeronautics,A.R., Seebas, Editor-in-Chief, Published by American Institute of Aeronautics and Astronautics.Google Scholar
  10. Levich, E., and Shtilman, L. Helicity fluctuations and coherence in developed turbulence. In Proc. Symposium on Scaling, Fractals and Non-linear Variability in Geophysics 1,D. Schertzer, D., and Lovejoy, S., eds., Kluwer.Google Scholar
  11. Levich, E., and Tsinober, A. Physics Review Letters, 93, 293.Google Scholar
  12. Levich, E., Tur, A. V., and Shtilman, L. Physica A, 176, 241–296.Google Scholar
  13. Meneveau, C., and Sreenivasan, K. R. Physics Review Letters, 59, 1424.Google Scholar
  14. Migdal, A. A. Turbulence as Statistics of Vortex Cells, PUPT, 1409.Google Scholar
  15. Moffatt, H. K. Journal of Fluid Mechanics, 35, 117–129.Google Scholar
  16. Moffatt, H. K. Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press. Moffatt, H. K. Journal of Fluid Mechanics, 159, 359.Google Scholar
  17. Moffatt, H. K. Fixed points of turbulent dynamical systems and suppression of nonlinearity. Whither Turbulence? Turbulence at the Crossroads, Lecture Notes in Applied Physics, Lumley, J. L., ed., SpringerVerlarg, Berlin, 357, 250–257.Google Scholar
  18. Moffatt, H. K., and Tsinober, A. Topological fluid mechanics. In Proceedings of the IUTAM Symposium,Cambridge, Cambridge University Press.Google Scholar
  19. Monin, A. S., and Yaglom, A. M. Statistical Fluid Mechanics,2 MIT Press.Google Scholar
  20. Moreau, J. J. C.R. Acad. Sci. Paris, 252, 2810–2818.Google Scholar
  21. Rajee, M., and Karllsson, S. Technical Report for ORLEV Scientific Computing Ltd.Google Scholar
  22. Sanada, T. Physics Review Letters, 70, 3035.Google Scholar
  23. Shtilman, L., and Polifke, W. Physics of Fluids, 2, 12.Google Scholar
  24. Sirovich, L. Private communication.Google Scholar
  25. Sirovich, L., Bali, K. S., and Handler, R. A. Theoretical and Computational Fluid Dynamics, 2, 308.Google Scholar
  26. Sirovich, L., Bali, K. S., and Handler, R. A. Physics of Fluids, 2, 2217.Google Scholar
  27. Sirovich, L., Levich, E., and Bronicki, L. Y. Method and Apparatus for Controlling Turbulence in a Wall-Bounded Fluid Flow, US Patent Number 5. 263. 793.Google Scholar
  28. Sirovich, L., Levich, E., and Bronicki, L. Y. Method and Apparatus for Controlling Turbulence in a Wall-Bounded Fluid Flow, US Patent Number 5. 362. 179.Google Scholar
  29. Sirovich, L., and Zhou, X. Physics Review Letters, 72, 340.Google Scholar
  30. irovich, L., and Zhou, X. Physics of Fluids, 6, 1579.Google Scholar
  31. Tur, A. V., and Yanovsky, V. V. Journal of Fluid Mechanics, 248, 67–106.Google Scholar
  32. Woltjer, L. In Proc. National Acadademy of Sciencs,USA, 44, 4890491.Google Scholar
  33. Zeldovich, Y. B., Molchanov, S. A., Ruzmaikin, A. A., and Sokolov, D. D. Soy. Jetp, 62, 1188.Google Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • E. Levich
    • 1
  1. 1.Division of ORMAT Industries Ltd.Benjamin Levich Center for Turbulent ResearchYavneIsrael

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