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Pressure Effects on Triple Correlations in Turbulent Convective Flows

  • J.-C. André
  • P. Lacarrère
  • K. Traoré

Abstract

The parameterization of pressure effects on triple correlations is shown to be an important part of the closure problem.

Based on arguments relevant to homogeneous and isotropic turbulence, it is argued that pressure effects are not only of a relaxative nature. In the case of turbulent thermal convection, a parameterization scheme including non-linear and rapid effects is proposed and used within the framework of a third-order model. Comparison with experimental results by Ferreira [10] indicates that the rapid part of pressure effects represents an important contribution to the budget of triple correlations and that it prevents the development of spurious stable stratification in the interior of the convective layer.

Keywords

Rayleigh Number Pressure Effect Planetary Boundary Layer Dissipative Scale Eddy Flux 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

C

Dimensionless constants

ē

Eddy kinetic energy (per unit mass)

h

Depth of the turbulent layer

i

Square root of (-1 )

k, p, q, r, s

Wave-vectors in Fourier space

k, p, q, r, s

Wave-numbers in Fourier space

t

Time

T

Temperature

T*

Convective temperature scale

u′=

\((u{{\prime }_{1}},u{{\prime }_{2}},u{{\prime }_{3}}) = (u\prime ,v\prime ,w\prime )\)= Fluctuating velocity

\(\hat{u}\)

\(({{\hat{u}}_{\alpha }},\alpha = 1 to 3)\) Fourier transform of the fluctuating velocity field

w*

Convective velocity scale

x =

(x 1,x 2, x 3) = (x, y, z) Position

z

Height

α

Coefficient of thermal expansion

β

Buoyancy parameter

δij

Kronecker delta

ε

Eddy kinetic energy dissipation rate

υ

Kinematic viscosity

ρ0

Constant density

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References

  1. 1.
    Deardorff, J.W.: The counter-gradient heat flux in the lower atmosphere and in the laboratory. J. Atmos. Sci. 23, 503–506 (1966)ADSCrossRefGoogle Scholar
  2. 2.
    Willis, G.E., Deardorff, J.W.: A laboratory model of the unstable planetary boundary layer. J. Atmos. Sci. 31, 1297–1307 (1974)ADSCrossRefGoogle Scholar
  3. 3.
    Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 3, 269–289 (1974)ADSMATHCrossRefGoogle Scholar
  4. 4.
    André, J.C., De Moor, G., Lacarrère, P., du Vachat, R.: Turbulence approximation for inhomogen-eous flows — Part II. The numerical simulation of a penetrative convection experiment. J. Atmos. Sci. 33, 482–491 (1976)ADSCrossRefGoogle Scholar
  5. 5.
    Lumley, J.L., Zeman, O., Siess, J.: The influence of buoyancy on turbulent transport. J. Fluid Mech. 84, 581–597 (1978)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Zeman, O., Lumley, J.L.: Modeling buoyancy driven mixed layers. J. Atmos. Sci. 33, 1974–1988 (1976)ADSCrossRefGoogle Scholar
  7. 7.
    Sun, W.Y., Ogura, Y.: Modelling the evolution of the convective planetary boundary layer. J. Atmos. Sci. 37, 1558–1572 (1980)ADSCrossRefGoogle Scholar
  8. 8.
    André, J.C., De Moor, G., Lacarrère, P., du Vachat, R.: Turbulence approximation for inhomogeneous flows — Part I. The clipping approximation. J. Atmos. Sci. 33, 476–481 (1976)Google Scholar
  9. 9.
    Warn-Varnas, A.C., Piacsek, S.A.: An investigation of the importance of third-order correlations and choice of length scale in mixed layer modelling. Geophysical and Astrophysical Fluid Dynamics, 13, 225–243 (1979)ADSMATHCrossRefGoogle Scholar
  10. 10.
    Ferreira, R.T.S.: “Unsteady turbulent thermal convection”; Ph. D. Thesis, University of Illinois, Ur-bana Champaign (1978)Google Scholar
  11. 11.
    André, J.C.: Irreversible interaction between cumulants in homogeneous, isotropic, two-dimensional turbulence theory. Phys. Fluids 17, 15–21 (1974)ADSMATHCrossRefGoogle Scholar
  12. 12.
    Orszag, S.A., Kruskal, M.D.: Formulation of the theory of turbulence. Phys. Fluids 11, 43–60 (1968)ADSMATHCrossRefGoogle Scholar
  13. 13.
    Orszag, S.A.: Analytical theories of turbulence. J. Fluid Mech. 45, 363–386 (1970)ADSCrossRefGoogle Scholar
  14. 14.
    Latour, J., Spiegel, E.A., Toomre, J., Zahn, J.P.: Stellar convection theory I. The anelastic modal equations. Astrophys. J. 207, 233–243 (1976)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    André, J.C., De Moor, G., Lacarrère, P., Therry, G., du Vachat, R.: Modeling the 24-hour evolution of the mean and turbulent structures of the planetary boundary layer. J. Atmos. Sci. 35, 1861–1883 (1978)ADSCrossRefGoogle Scholar
  16. 16.
    André, J.C., De Moor, G., Lacarrère, P., Therry, G., du Vachat, R.: “The Clipping Approximation and Inhomogeneous Turbulence Simulations”, in: Turbulent Shear Flows I, ed. by F. Durst, B.E. Launder, F.W. Schmidt, J.H. Whitelaw (Springer, Berlin, Heidelberg, New York 1979) pp. 307–318CrossRefGoogle Scholar
  17. 17.
    Launder, B.E.: Private communication (1976)Google Scholar
  18. 18.
    Launder, B.E.: On the effects of a gravitational field on the turbulent transport of heat and momentum. J. Fluid Mech. 67, 569–581 (1975)ADSCrossRefGoogle Scholar
  19. 19.
    Deardorff, J.W.: Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211–1213 (1970)ADSCrossRefGoogle Scholar
  20. 20.
    Hanjalic, K., Launder, B.E,: Fully developed asymmetric flow in a plane channel. J. Fluid Mech. 51, 301–335 (1972)ADSCrossRefGoogle Scholar
  21. 21.
    Wyngaard, J.C., Coté, O.R.: The evolution of a convective planetary boundary layer — A higher-order-closure model study. Boundary-Layer Meteorol. 7, 289–304 (1974)ADSCrossRefGoogle Scholar
  22. 22.
    Brown, R.A.: Longitudinal instabilities and secondary flows in the planetary boundary layer: a review. Rev. Geophys. Space Phys. 18, 683–697 (1980)ADSCrossRefGoogle Scholar
  23. 23.
    Deardorff, J.W.: A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41, 453–480 (1970)ADSMATHCrossRefGoogle Scholar
  24. 24.
    Kwak, D., Reynolds, W.C., Ferziger, J.H.: “Three-Dimensional Time-Dependent Computation of Turbulent Flow”, TF-5, Department of Mechanical Engineering, Stanford University CA (1975)Google Scholar
  25. 25.
    Rotta, J.C.: Statistische Theorie nichthomogener Turbulenz. Z. Phys. 129, 547–572 (1951)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Reynolds, W.C.: Computation of turbulent flows. Annu. Rev. Fluid Mech. 8, 183–208 (1976)ADSCrossRefGoogle Scholar
  27. 27.
    Deardorff, J.W.: Private communication (1980)Google Scholar
  28. 28.
    Cambon, C, Jeandel, D.: “Approach of Non-isotropic Homogeneous Turbulence’ submitted’ to Mean Velocity Gradients”, Proc. 3rd Symp. on Turbulent Shear Flows, University of California, Davis, Sept. 1981, pp. 17.7-17.11Google Scholar
  29. 29.
    Lumley, J.L.: Computational modeling of turbulent flows. Adv. Appl. Mech. 18, 123–176 (1978)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Domingos, J.J.D.: “Pressure Strain: Exact Results and Models”, Proc 3rd Symp. on Turbulent Shear Flows, University of California, Davis, Sept. 1981, pp. 19.26-19.30Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • J.-C. André
    • 1
    • 2
  • P. Lacarrère
    • 1
  • K. Traoré
    • 1
  1. 1.Direction de la MétéorologieEERM/GMDBoulogneFrance
  2. 2.Department of Atmospheric SciencesOregon State UniversityCorvallisUSA

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