Journal of Statistical Physics

, Volume 65, Issue 1–2, pp 33–52

Stationary turbulent closure via the Hopf functional equation

  • Hubert H. Shen
  • Alan A. Wray


Exact closed-form solutions are exhibited for the Hopf equation for stationary incompressible 3D Navier-Stokes flow, for the cases of homogeneous forced flow (including a solution with depleted nonlinearity) and inhomogeneous flow with arbitrary boundary conditions. This provides an exact method for computing two- and higher-point moments, given the mean flow.

Key words

Navier-Stokes turbulence closure generating functional Hopf equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Hopf, Statistical hydromechanics and functional calculus,J. Rat. Mech. Anal. 1:87 (1952).Google Scholar
  2. 2.
    C. Foias, O. Manley, and R. Temam, Self-similar invariant families of turbulent flow,Phys. Fluids 30:2008 (1987).Google Scholar
  3. 3.
    A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics, Vol. 2 (MIT Press, 1965), §28.1 and references therein.Google Scholar
  4. 4.
    M. M. Stanisic,The Mathematical Theory of Turbulence (Springer-Verlag, 1985), Chapter 12.Google Scholar
  5. 5.
    G. Rosen, Functional calculus theory for incompressible fluid turbulence,J. Math. Phys. 12:812 (1971).Google Scholar
  6. 6.
    T. Alankus, An exact representation of the space-time characteristic functional of turbulent Navier-Stokes flows with prescribed random initial states and driving forces,J. Stat. Phys. 54:859 (1989).Google Scholar
  7. 7.
    H. H. Shen, Boson Hamiltonians and stochasticity for the vorticity equation,Can. J. Phys., in press (1990).Google Scholar
  8. 8.
    R. M. Lewis and R. H. Kraichnan, A space-time functional formalism for turbulence,Commun. Pure Appl. Math. 15:397 (1962).Google Scholar
  9. 9.
    M. J. Vishik and A. V. Fursikov,Mathematical Problems of Statistical Hydromechanics (Kluwer, 1988), §9.6.Google Scholar
  10. 10.
    P. Constantin and C. Foias,Navier-Stokes Equation (University of Chicago Press, 1988), p. 84.Google Scholar
  11. 11.
    H. K. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrary complex topology,J. Fluid Mech. 159:359 (1985).Google Scholar
  12. 12.
    A. Tsinober, On one property of Lamb vector in isotropic turbulent flow,Phys. Fluids A 2:484 (1990), and references therein.Google Scholar
  13. 13.
    A. A. Townsend,Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, 1976), §3.10.Google Scholar
  14. 14.
    E. Levich, Certain problems in the theory of developed hydrodynamical turbulence,Phys. Rep. 151:149 (1987).Google Scholar
  15. 15.
    M. Sargent, M. Scully, and W. Lamb,Laser Physics (Addison-Wesley, 1974), p. 323.Google Scholar
  16. 16.
    H. H. Shen, Nonconservative dynamics of vortical structures: A statistical-mechanical perspective,Physica D, in press (1991).Google Scholar
  17. 17.
    A. K. M. F. Hussain, Coherent structures and turbulence,J. Fluid Mech. 173:303 (1986).Google Scholar
  18. 18.
    R. H. Kraichnan, Remarks on turbulence theory,Adv. Math. 16:305 (1975).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Hubert H. Shen
    • 1
  • Alan A. Wray
    • 2
  1. 1.Center for Turbulence ResearchNASA-Ames/Stanford UniversityUSA
  2. 2.NASA-Ames Research CenterUSA

Personalised recommendations