Skip to main content
SpringerLink
Log in

Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

Some rigorous results connected with the conventional statistical theory of turbulence in both the two- and three-dimensional cases are discussed. Such results are based on the concept of stationary statistical solution, related to the notion of ensemble average for turbulence in statistical equilibrium, and concern, in particular, the mean kinetic energy and enstrophy fluxes and their corresponding cascades. Some of the results are developed here in the case of nonsmooth boundaries and a less regular forcing term and for arbitrary stationary statistical solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Abergel: Attractor for a Navier-Stokes ow in an unbounded domain. Attractors, Inertial Manifolds and Their Approximation (Marseille-Luminy, 1987). RAIRO Modél. Math. Anal. Numér. 23 (1989), 359-370.

    Google Scholar 

  2. A. V. Babin: The attractor of a Navier-Stokes system in an unbounded channel-like domain. J. Dynam. Differential Equations 4 (1992), 555-584.

    Google Scholar 

  3. G. I. Barenblatt, A. J. Chorin: New perspectives in turbulence: scaling laws, asymptotics, and intermittency. SIAM Rev. 40 (1998), 265-291.

    Google Scholar 

  4. G. K. Batchelor: The Theory of Homogeneous Turbulence. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1953.

    Google Scholar 

  5. G. K. Batchelor: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. II 12 (1969), 233-239.

    Google Scholar 

  6. H. Bercovici, P. Constantin, C. Foias and O. P. Manley: Exponential decay of the power spectrum of turbulence. J. Statist. Phys. 80 (1995), 579-602.

    Google Scholar 

  7. A. J. Chorin: Vorticity and Turbulence. Applied Mathematical Sciences 103. Springer-Verlag, New York, 1994.

    Google Scholar 

  8. P. Constantin: Geometric statistics in turbulence. SIAM Rev. 36 (1994), 73-98.

    Google Scholar 

  9. P. Constantin, C. Foias: Navier-Stokes Equation. University of Chicago Press, Chicago, 1989.

    Google Scholar 

  10. P. Constantin, C. Foias and O. Manley: Effects of the forcing function spectrum on the energy spectrum in 2-D turbulence. Phys. Fluids 6 (1994), 427-429.

    Google Scholar 

  11. C. Foias: Statistical study of Navier-Stokes equations I. Rend. Sem. Mat. Univ. Padova 48 (1972), 219-348..

    Google Scholar 

  12. C. Foias: Statistical study of Navier-Stokes equations II. Rend. Sem. Mat. Univ. Padova 49 (1973), 9-123.

    Google Scholar 

  13. C. Foias, M. S. Jolly, O. P. Manley and R. Rosa: Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence. J. Statist. Phys. 108 (2002), 591-646.

    Google Scholar 

  14. C. Foias, O. P. Manley, R. Rosa and R. Temam: Navier-Stokes Equations and Turbulence. Encyclopedia of Mathematics and Its Applications, Vol. 83. Cambridge University Press, Cambridge, 2001.

    Google Scholar 

  15. C. Foias, O. P. Manley, R. Rosa and R. Temam: Cascade of energy in turbulent flows. C. R. Acad. Sci. Paris, Série I Math. 332 (2001), 509-514.

    Google Scholar 

  16. C. Foias, O. P. Manley, R. Rosa and R. Temam: Estimates for the energy cascade in three-dimensional turbulent flows. C. R. Acad. Sci. Paris, Série I Math. 333 (2001), 499-504.

    Google Scholar 

  17. C. Foias, G. Prodi: Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2. Rend. Sem. Mat. Univ. Padova 39 (1967), 1-34.

    Google Scholar 

  18. S. Friedlander, L. Topper: Turbulence. Classic Papers on Statistical Theory. Interscience Publisher, New York, 1961.

    Google Scholar 

  19. U. Frisch: Turbulence. The Legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  20. J. O. Hinze: Turbulence. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill, New York, 1975.

    Google Scholar 

  21. P. Holmes, J. L. Lumley and G. Berkooz: Turbulence, Coherent Structures, Dynamical Systems, and Symmetry. Cambridge University Press, Cambridge, 1996.

    Google Scholar 

  22. E. Hopf: Statistical hydromechanics and functional calculus. J. Rat. Mech. Analysis 1 (1952), 87-123.

    Google Scholar 

  23. A. A. Ilyin: Attractors for Navier-Stokes equations in domains with finite measure. Nonlinear Anal. 27 (1996), 605-616.

    Google Scholar 

  24. J. Jiménez, A. A. Wray, P. G. Saffman and R. S. Rogallo: The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255 (1993), 65-90.

    Google Scholar 

  25. T. von Karman, L. Howarth: On the statistical theory of isotropic turbulence. Proc. Roy. Soc. London A164. 1938, pp. 192-215.

    Google Scholar 

  26. A. N. Kolmogorov: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. (Dokl.) Acad. Sci. URSS 30 (1941), 301-305.

    Google Scholar 

  27. A. N. Kolmogorov: On degeneration of isotropic turbulence in an incompressible viscous liquid. C. R. (Dokl.) Acad. Sci. URSS 31 (1941), 538-540.

    Google Scholar 

  28. A. N. Kolmogorov: Dissipation of energy in locally isotropic turbulence. C. R. (Doklady) Acad. Sci. URSS 32 (1941), 16-18.

    Google Scholar 

  29. A. N. Kolmogorov: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13 (1962), 82-85.

    Google Scholar 

  30. R. H. Kraichnan: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (1967), 1417-1423.

    Google Scholar 

  31. R. H. Kraichnan: Some modern developments in the statistical theory of turbulence. Statistical Mechanics: New Concepts, New Problems, New Applications (S. A. Rice, K.F. Freed and J. C. Light, eds.). 1972, pp. 201-227.

  32. O. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow. Revised English edition. Gordon and Breach Science Publishers, New York-London-Paris, 1963.

    Google Scholar 

  33. O. Ladyzhenskaya: First boundary value problem for the Navier-Stokes equations in domains with non smooth boundaries. C. R. Acad. Sci. Paris, Sér. I Math. 314 (1992), 253-258.

    Google Scholar 

  34. L. Landau, E. Lifchitz: Mécanique des Fluides, Physique Théorique, Tome 6. Editions Mir, Moscow, 1971.

    Google Scholar 

  35. C. E. Leith: Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11 (1968), 671-673.

    Google Scholar 

  36. M. Lesieur: Turbulence in Fluids. Third edition. Fluid Mechanics and its Applications, 40. Kluwer Academic Publishers Group, Dordrecht, 1997.

    Google Scholar 

  37. J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris, 1969.

    Google Scholar 

  38. I. Moise, R. Rosa and X. Wang: Attractor for noncompact semigroups via energy equations. Nonlinearity 11 (1998), 1369-1393.

    Google Scholar 

  39. A. S. Monin, A. M. Yaglom: Statistical Fluid Mechanics: Mechanics of Turbulence 2. MIT Press, Cambridge, 1975.

    Google Scholar 

  40. E. A. Novikov, R. V. Stewart: The intermittency of turbulence and the spectrum of energy dissipation. Izv. Akad. Nauk SSSR, Ser. Geoffiz 3 (1964), 408-413.

    Google Scholar 

  41. A. M. Obukhoff: On the energy distribution in the spectrm of turbulent flow. C. R. (Dokl.) Acad. Sci. USSR 32 (1941), 19-21.

    Google Scholar 

  42. A. M. Obukhoff: Some specific features of atmospheric turbulence. J. Fluid Mech. 13 (1962), 77-81.

    Google Scholar 

  43. L. F. Richardson: Weather Prediction by Numerical Process. Cambridge University Press, Cambridge, 1922.

    Google Scholar 

  44. R. Rosa: The global attractor for the 2D Navier-Stokes flow on some unbounded domains. Nonlinear Anal. 32 (1998), 71-85.

    Google Scholar 

  45. Z. S. She, E. Jackson and S. A. Orszag: Structure and dynamics of homogeneous turbulence: models and simulations. Proc. Roy. Soc. London Ser. A 434, 101-124.

  46. G. I. Taylor: Diffusion by continuous movements. Proc. London Math. Soc. Ser. 2 20 (1921), 196-211.

    Google Scholar 

  47. G. I. Taylor: Statistical theory of turbulence. Proc. Roy. Soc. London Ser. A 151 (1935), 421-478.

    Google Scholar 

  48. G. I. Taylor: The spectrum of turbulence. Proc. Roy. Soc. London Ser. A 164 (1938), 476-490.

    Google Scholar 

  49. R. Temam: Navier-Stokes Equations. Theory and numerical analysis. Studies in Mathematics and its Applications, 2. 3rd edition. North-Holland Publishing Co., Amsterdam-New York, 1984. Reedition in 2001 in the AMS Chelsea series, AMS, Providence.

    Google Scholar 

  50. H. Tennekes, J. L. Lumley: A First Course in Turbulence. MIT Press, Cambridge, Mass., 1972.

    Google Scholar 

  51. C.V. Tran, T. G. Shepherd: Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence. Physica D. To appear.

  52. M. I. Vishik, A. V. Fursikov: Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations. Sibirsk. Mat. Zh. 19 (1978), 1005-1031.

    Google Scholar 

  53. M. I. Vishik, A. V. Fursikov: Mathematical Problems of Statistical Hydrodynamics. Kluwer, Dordrecht, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matematica Aplicada, Instituto de Matematica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Ilha do Fundao, RJ 21945-970, Rio de Janeiro, Brazil

    Ricardo M. S. Rosa

Authors
  1. Ricardo M. S. Rosa
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rosa, R.M.S. Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence. Applications of Mathematics 47, 485–516 (2002). https://doi.org/10.1023/A:1023297721804

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023297721804

Search

Navigation