Journal of Statistical Physics

, Volume 65, Issue 1, pp 33–52

Stationary turbulent closure via the Hopf functional equation

  • Hubert H. Shen
  • Alan A. Wray

DOI: 10.1007/BF01329849

Cite this article as:
Shen, H.H. & Wray, A.A. J Stat Phys (1991) 65: 33. doi:10.1007/BF01329849


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 turbulenceclosuregenerating functionalHopf equation

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