A Simplified Spectral Closure for Inhomogeneous Turbulence: Application to the Boundary Layer

  • Jean-Pierre Bertoglio
  • Denis Jeandel
Conference paper

Abstract

A turbulence model is proposed for the prediction of inhomogeneous flows. A spectral closure is used to predict the energy transfer to the dissipative scales. The inhomogeneous transport terms are evaluated within the framework of one-point closures. Rough approximations are made concerning the anisotropy of turbulence. In the case of a flat plate boundary layer, the model is found to compare favorably with the experimental data. Possible improvements are discussed.

Keywords

Combustion Anisotropy Convection Covariance Advection 

Nomenclature

< >)

Statistical average

aKPQ)

\( = \frac{1}{2}({b_{KPQ}} + {b_{KQP}}) \)

b)

Constant

bKPQ)

Coefficient depending on the geometry of a triad in wave space

D(K, X, t))

Transport term in the energy spectrum equation

E(K, X, t))

Energy spectrum

g(K, Kmax))

Modeled spectral shape for the transport term D

K)

Wave-vector

K)

Wave-number, = |K|

Kmin)

Infrared cutoff for the spectral computation

Kmax)

Wave-number corresponding to the maximum of the energy spectrum

k)

Turbulent kinetic energy

K)

Constant

L)

Integral length scale

<q2>)

Twice the turbulent kinetic energy

r)

Constant

T(K, X, t))

Energy transfer

Ui(X, t))

Velocity field

u*)

Shear stress velocity

δ)

Boundary layer thickness

σ1, σ2)

Constants

Фij(K, X , t))

Spectral tensor, Fourier transform of the double velocity correlation at x and x′ with \(X = \frac{{x + x\prime }}{2}\)

φij(k, X, t))

Integral of Ф ij over a spherical shell of radius K

φij,l)

Fourier transform of the triple correlation

λ)

Constant

ω)

Dimensionless stream function, \( = \frac{{\psi - {\psi _1}}}{{{\psi _E} - {\psi _1}}} \)

πi)

Fourier transform of the pressure velocity correlation

ψ)

Stream function

µ)

Damping factor

v)

Kinematic viscosity

θKPQ)

E.D.Q.N.M. characteristic time

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References

  1. Bertoglio, J.-P. (1982): A model of three-dimensional transfer in non-isotropic homogeneous turbulence, in Turbulent Shear Flows 3, ed. by L. J. S. Bradbury, F. Durst, B. E. Launder, F. W. Schmidt, J. H. Whitelaw (Springer, Berlin, Heidelberg) p. 253Google Scholar
  2. Cambon, C., Jeandel, D., Mathieu, J. (1981 a): Spectral modeling of homogeneous non-isotropic Turbulence. J. Fluid Mech. 104, pp. 247–262ADSMATHCrossRefGoogle Scholar
  3. Cambon, C., Bertoglio, J.-P., Jeandel, D., (1981b); Spectral closure of homogeneous turbulence, AFOSR-HTTM-Stanford Conf. on Complex Turb. Flows, Sept. 1981Google Scholar
  4. Dannevik, W., (1984): Two-point closure study of covariance budgets for turbulent Rayleigh-Benard convection, Ph. D. Thesis, St. Louis Univ., MOGoogle Scholar
  5. Goldstein, M. E., Durbin, P. A. (1980): The effect of finite turbulence spatial scale on the amplification of turbulence by contracting stream. J. Fluid Mech. 98, (3), 473–508ADSMATHCrossRefGoogle Scholar
  6. Hamadiche, M. (1981): Modélisation spectrale des mécanismes linéaires dans les écoulements turbulents non homogènes. Thèse Univ. Cl. Bernard, Lyon I, Déc. 1981Google Scholar
  7. Herring, J. R. (1985): Some contributions of two-point closure to turbulence, in Frontiers in Fluid Mechanics (Springer, Berlin, Heidelberg) p. 68Google Scholar
  8. Hunt, J. C. R. (1978): A review of the theory of rapidly distorded turbulent flows and its applications. Fluid Dyn. Trans. 9, 121–152Google Scholar
  9. Jeandel, D. (1976): Une approche phénoménologique des écoulements turbulents inhomogènes. Thèse Univ. Cl. Bernard, Lyon I, Juillet 1976Google Scholar
  10. Jeandel, D., Brison, J. F., Mathieu, J. (1978): Modeling methods in physical and spectral space. Phys. Fluids 21, 169MathSciNetADSMATHCrossRefGoogle Scholar
  11. Klebanoff, P. S. (1955): Characteristics of turbulence in a boundary layer with zero pressure gradient. NACA Report 1247Google Scholar
  12. Kraichnan, R. H. (1972): Test-field model for inhomogeneous turbulence. J. Fluid Mech. 56, 287–304MathSciNetADSMATHCrossRefGoogle Scholar
  13. Lumley, J. L. (1983): Turbulence modeling, J. Appl. Mech. 50, 1097ADSCrossRefGoogle Scholar
  14. Maxey, M. R. (1982): Distortion of turbulence in flows with parallel streamlines. J. Fluid Mech. 124, 261–282ADSMATHCrossRefGoogle Scholar
  15. Ménoret, L. (1982): Contribution à l’étude spectrale des écoulements turbulents faiblement inhomogènes et anisotropes. Thèse Univ. Cl. Bernard, Lyon I, Juin 1982Google Scholar
  16. Ng, K. H., Spalding, D. B. (1972): Turbulence model for boundary layers near walls. Phys Fluids 15, 20ADSMATHCrossRefGoogle Scholar
  17. Ng, K. H., Spalding, D. B. (1976): Predictions of two-dimensional boundary layers on smooth walls with a two-equation model of turbulence. Int. J. Heat Mass Transf. 19, 1161–1172ADSCrossRefGoogle Scholar
  18. Orszag, S. A. (1970): Analytical theories of turbulence. J. Fluid Mech. 41, 363–386ADSMATHCrossRefGoogle Scholar
  19. Patankar, S.V., Spalding, D. B. (1967): Heat and Mass Transfer in Boundary Layers (Morgan-Grampian, London)Google Scholar
  20. Rodi, W., Spalding, D. B. (1970): Wärme Stoffübertrag. 3, 85ADSCrossRefGoogle Scholar
  21. Weinstock, J. (1981): Theory of pressure-strain-rate correlation for Reynolds-stress turbulence closures, Part 1. Off-diagonal element. J. Fluid. Mech. 105, 369–396ADSMATHCrossRefGoogle Scholar
  22. Weinstock, J. (1982): Theory of the pressure-strain-rate. Part 2. Diagonal elements. J. Fluid. Mech. 116, 1–29ADSMATHCrossRefGoogle Scholar
  23. Wu, C. T., Ferziger, J. H., Chapman, D. R. (1985): Simulations and modeling of homogeneous, compressed turbulence, Fifth Symposium on Turbulent Shear FlowsGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Jean-Pierre Bertoglio
    • 1
  • Denis Jeandel
    • 1
  1. 1.Laboratoire de Mécanique des FluidesEcully CedexFrance

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