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Interaction between different scales of turbulence over short times

Published online by Cambridge University Press:  26 April 2006

S. Kida  and
J. C. R. Hunt
S. Kida
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan
J. C. R. Hunt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

The distortion by large-scale random motions of small-scale turbulence is investigated by examining a model problem. The changes in energy spectra, velocity and vorticity moments, and anisotropy of small-scale turbulence are calculated over timescales short compared with the timescale of small-scale turbulence by applying rapid distortion theory with a random distortion matrix for different initial conditions: irrotational or rotational, and isotropic or anisotropic large-scale turbulence with or without mean strain, and isotropic or anisotropic small-scale turbulence.

We have obtained the following results: (1) Irrotational random strains broaden the small-scale energy spectrum and transfer energy to higher wavenumbers. (2) The rotational part of the large-scale strain is important for reducing anisotropy of turbulence rather than transferring energy to higher wavenumbers. (3) Anisotropy in small-scale turbulence is reduced by large-scale isotropic turbulence. The reduction of anisotropy of the velocity field depends on the initial value of the velocity anisotropy tensor of the small-scale velocity field ui defined by $\overline{u_iu_j}/\overline{u_lu_l}-\frac{1}{3}\delta_{ij}$, and also on the anisotropy of the distribution of the energy spectrum in wavenumber space. The reduction in anisotropy of the vorticity field ωi depends only on the vorticity anisotropy tensor. (4) The pressure-strain correlation is calculated for the change in Reynolds stress of the anisotropic small-scale turbulence. The correlation is proportional to time and depends on the difference between the velocity and wavenumber anisotropy tensors. These results (which are exact for small time) differ significantly from current turbulence models. (5) The effect of large-scale anisotropic turbulence on isotropic small-scale turbulence is calculated in general. Results are given for the case of axisymmetric large scales and are compared with the observed behaviour of small-scale turbulence near interfaces. (6) When a mean irrotational straining motion is applied to turbulence with distinct large-scale and small-scale components in their velocity field, the large-scale irrotational motions combine with the mean straining to increase further the anisotropy of the vorticity of the small scales, but the large-scale rotational motions reduce the small-scale anisotropy. For isotropic straining motion, the latter is weaker than the former. After the mean distortion ceases, both kinds of large-scale straining tend to reduce the anisotropy. This also has implications for modelling the rate of reduction of anisotropy.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 201 , April 1989 , pp. 411 - 445
Copyright
© 1989 Cambridge University Press

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References

Bertoglio, J. P. 1986 Etude d'une turbulence anisotrope: modelisation de sous maille et approach statistique. These d'Etat, Ecole Central de Lyon.
Biringen, S. & Reynolds, W. C. 1981 Large eddy simulation of the shear-free turbulent boundary layer. J. Fluid Mech. 103, 53–63.Google Scholar
Cambon, C., Jeandel, D. & Mathieu, J. 1981 Spectral modelling of homogeneous non-isotropic turbulence. J. Fluid Mech. 104, 247–262.Google Scholar
Carruthers, D. J. & Hunt, J. C. R. 1988 Turbulence waves and entrainment near density inversion layers. Proc. Symp. on Stratified Flow, Cal. Tech.
Durbin, P. A. 1981 Distorted turbulence in axisymmetric flow. Q. J. Mech. Appl. Maths 34, 489–500.Google Scholar
Edwards, S. F. 1964 The statistical dynamics of homogeneous turbulence. J. Fluid Mech. 18, 239–273.Google Scholar
Finnigan, J. T. & Einaudi, F. 1981 The interaction between an internal gravity wave and the planetary boundary layer. II. The effect of the wave on the turbulence structure. Q. J. R. Met. Soc. 107, 807–832.Google Scholar
Hunt, J. C. R. 1973 A theory of turbulent flow round two-dimensional bluff bodies. J. Fluid Mech. 61, 625–706.Google Scholar
Hunt, J. C. R. 1984 Turbulent structure in thermal convection and shear-free boundary layers. J. Fluid Mech. 138, 161–184.Google Scholar
Hunt, J. C. R., Kaimal, J. C. & Gaynor, J. E. 1988 Eddy structure in the convective boundary layer: new measurements and new concepts. Q. J. R. Met. Soc. 114, 827–858.Google Scholar
Hunt, J. C. R., Wray, A. A. & Buell, J. C. 1988 Big whirls carry little whirls. Proc. Stanford NASA Ames Center for Turb. Res. Summer Program 1987 (submitted for publication).Google Scholar
Hussain, A. K. M. F. 1983 Coherent structures-reality and myth. Phys. Fluids 26, 2816–2850.Google Scholar
Itsweire, E. C. & Van Atta, C. W. 1984 An experimental study of the response of nearly isotropic turbulence to a spectrally local disturbance. J. Fluid Mech. 145, 423–445.Google Scholar
Jeandel, D., Brison, J. F. & Mathieu, J. 1978 Modeling methods in physical and spectral space. Phys. Fluids 21, 169–182.Google Scholar
Kellogg, R. M. & Corrsin, S. 1980 Evolution of a spectrally local disturbance in grid generated nearly isotropic turbulence. J. Fluid Mech. 96, 641–669.Google Scholar
Kerr, R. M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31–48.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497–543.Google Scholar
Launder, B. E., Reece, G. J. & Rodi, W. 1975 Progress in the development of a Reynolds stress turbulence closure. J. Fluid Mech. 68, 537–566.Google Scholar
Lee, M. J. & Reynolds, W. C. 1985 Numerical experiments on the structure of homogeneous turbulence. Rep. TF-24. Thermosciences Div., Mech. Eng., Stanford University.Google Scholar
Lesieur, M. 1987 Turbulence in Fluids. Martinus Nijhoff.
Lumley, J. L. 1978 Computational modeling of turbulent flows. Adv. Appl. Mech. 18, 123–176.Google Scholar
Moffatt, H. K. 1967 The interaction of turbulence with strong wind shear. Proc. URSI-IUGG Intl Coll. on Atmospheric Turbulence and Radio Wave Propagation. Moscow: Nauka.
Moffatt, H. K. 1981 Some developments in the theory of turbulence. J. Fluid Mech. 106, 27–47.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Theory of Turbulence, Vol. II. MIT Press.
Pouquet, A., Frish, U. T. & Chollet, J. P. 1983 Turbulence with spectral gap. Phys. Fluids 26, 877–880.Google Scholar
Stewart, R. W. 1951 Triple velocity correlations in isotropic turbulence. Proc. Camb. Phil. Soc. 47, 146–157.Google Scholar
Tan-Atichat, J., Nagib, H. M. & Loehrke, R. J. 1982 Interaction of free-stream turbulence with screens and grids: a balance between turbulence scales. J. Fluid Mech. 114, 501–528.Google Scholar
Thomas, N. H. & Hancock, P. E. 1977 Grid turbulence near a moving wall. J. Fluid Mech. 82, 481–496.Google Scholar
Townsend, A. A. 1976 Structure of Turbulent Shear Flow. Cambridge University Press.
Townsend, A. A. 1980 The response of sheared turbulence to additional distortion. J. Fluid Mech. 98, 171–191.Google Scholar
Weinstock, J. & Burk, S. 1985 Theoretical pressure-strain term, experimental comparison, and resistance to large anisotropy. J. Fluid Mech. 154, 429–443.Google Scholar
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