• Batchelor, G. K., 1970: The Theory of Homogeneous Turbulence. Cambridge University Press, 197 pp.

  • Blackadar, A. K., 1962: The vertical distribution of wind and turbulent exchange in a neutral atmosphere. J. Geophys. Res., 67 , 30953102.

    • Search Google Scholar
    • Export Citation
  • Canuto, V. M., 2009: Turbulence in astrophysical and geophysical flows. Interdisciplinary Aspects of Turbulence, W. Hillebrandt and F. Kukpa, Eds., Lectures Notes in Physics Series, Vol. 756, Springer-Verlag, 107–160.

    • Search Google Scholar
    • Export Citation
  • Canuto, V. M., , and I. Goldman, 1985: Analytical model for large-scale turbulence. Phys. Rev. Lett., 54 , 430433.

  • Canuto, V. M., , and M. S. Dubovikov, 1996: A dynamical model for turbulence. II. Shear-driven flows. Phys. Fluids, 8 , 587598.

  • Chasnov, J. R., 1995: The decay of axisymmetric homogeneous turbulence. Phys. Fluids, 7 , 600605.

  • Davydov, B. I., 1961: On statistical dynamics of an incompressible turbulent fluid. Sov. Phys. Dokl., 6 , 1012.

  • Lumley, J. L., 1967: Similarity and the turbulence energy spectrum. Phys. Fluids, 10 , 855858.

  • Mellor, G. L., , and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys., 20 , 851875.

    • Search Google Scholar
    • Export Citation
  • Moeng, C-H., , and P. P. Sullivan, 1994: A comparison of shear- and buoyancy-driven planetary boundary layer flows. J. Atmos. Sci., 51 , 9991022.

    • Search Google Scholar
    • Export Citation
  • Pope, S. B., 2000: Turbulent Flows. Cambridge University Press, 771 pp.

  • Reynolds, W. C., 1987: Fundamentals of turbulence for turbulence modeling and simulation. Advisory Group for Aerospace Research, 66 pp.

  • Rubinstein, R., , and Y. Zhou, 2001: Schiestel’s derivation of the epsilon equation and two equation modeling of rotating turbulence. NASA/CR-2001-211060, ICASE Rep. 2001-24, 6 pp.

    • Search Google Scholar
    • Export Citation
  • Saffman, P. G., 1967: The large-scale structure of homogenous turbulence. J. Fluid Mech., 27 , 581593.

  • Schiestel, R., 1987: Multiple-time-scale modeling of turbulent flows in one-point closures. Phys. Fluids, 30 , 722731.

  • Umlauf, L., , and H. Burchard, 2005: Second-order turbulence closure models for geophysical boundary layers. A review of recent work. Cont. Shelf Res., 25 , 795827.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 20 20 1
PDF Downloads 9 9 1

An Attempt to Derive the ε Equation from a Two-Point Closure

View More View Less
  • 1 NASA Goddard Institute for Space Studies, and Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York
  • | 2 NASA Goddard Institute for Space Studies, and Center for Climate Systems Research, Columbia University, New York, New York
  • | 3 NASA Goddard Institute for Space Studies, and Department of Physical, Environmental and Computer Sciences, Medgar Evers College of CUNY and Graduate School of CUNY, New York, New York
© Get Permissions
Restricted access

Abstract

The goal of this paper is to derive the equation for the turbulence dissipation rate ε for a shear-driven flow. In 1961, Davydov used a one-point closure model to derive the ε equation from first principles but the final result contained undetermined terms and thus lacked predictive power. Both in 1987 (Schiestel) and in 2001 (Rubinstein and Zhou), attempts were made to derive the ε equation from first principles using a two-point closure, but their methods relied on a phenomenological assumption. The standard practice has thus been to employ a heuristic form of the ε equation that contains three empirical ingredients: two constants, c1,ε and c2,ε, and a diffusion term Dε. In this work, a two-point closure is employed, yielding the following results: 1) the empirical constants get replaced by c1, c2, which are now functions of K and ε; 2) c1 and c2 are not independent because a general relation between the two that are valid for any K and ε are derived; 3) c1, c2 become constant with values close to the empirical values c1,ε, c2,ε (i.e., homogenous flows); and 4) the empirical form of the diffusion term Dε is no longer needed because it gets substituted by the K–ε dependence of c1, c2, which plays the role of the diffusion, together with the diffusion of the turbulent kinetic energy DK, which now enters the new ε equation (i.e., inhomogeneous flows). Thus, the three empirical ingredients c1,ε, c2,ε, Dε are replaced by a single function c1(K, ε) or c2(K, ε), plus a DK term. Three tests of the new equation for ε are presented: one concerning channel flow and two concerning the shear-driven planetary boundary layer (PBL).

Corresponding author address: Ye Cheng, NASA GISS, 2880 Broadway, New York, NY 10025. Email: ycheng@giss.nasa.gov

Abstract

The goal of this paper is to derive the equation for the turbulence dissipation rate ε for a shear-driven flow. In 1961, Davydov used a one-point closure model to derive the ε equation from first principles but the final result contained undetermined terms and thus lacked predictive power. Both in 1987 (Schiestel) and in 2001 (Rubinstein and Zhou), attempts were made to derive the ε equation from first principles using a two-point closure, but their methods relied on a phenomenological assumption. The standard practice has thus been to employ a heuristic form of the ε equation that contains three empirical ingredients: two constants, c1,ε and c2,ε, and a diffusion term Dε. In this work, a two-point closure is employed, yielding the following results: 1) the empirical constants get replaced by c1, c2, which are now functions of K and ε; 2) c1 and c2 are not independent because a general relation between the two that are valid for any K and ε are derived; 3) c1, c2 become constant with values close to the empirical values c1,ε, c2,ε (i.e., homogenous flows); and 4) the empirical form of the diffusion term Dε is no longer needed because it gets substituted by the K–ε dependence of c1, c2, which plays the role of the diffusion, together with the diffusion of the turbulent kinetic energy DK, which now enters the new ε equation (i.e., inhomogeneous flows). Thus, the three empirical ingredients c1,ε, c2,ε, Dε are replaced by a single function c1(K, ε) or c2(K, ε), plus a DK term. Three tests of the new equation for ε are presented: one concerning channel flow and two concerning the shear-driven planetary boundary layer (PBL).

Corresponding author address: Ye Cheng, NASA GISS, 2880 Broadway, New York, NY 10025. Email: ycheng@giss.nasa.gov

Save