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Homogeneous buoyancy-generated turbulence

Published online by Cambridge University Press:  26 April 2006

G. K. Batchelor ,
V. M. Canuto  and
J. R. Chasnov
G. K. Batchelor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
V. M. Canuto
Affiliation:
NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA
J. R. Chasnov
Affiliation:
NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA
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Abstract

We consider the statistically homogeneous motion that is generated by buoyancy forces after the creation of homogeneous random fluctuations in the density of infinite fluid at an initial instant. The mean density is uniform, and density fluctuations are smoothed by molecular diffusion. This turbulent flow system has interesting properties, and shows how self-generated motion contributes to the rate of mixing of an ‘active’ scalar contaminant.

If nonlinear terms in the governing equations are negligible, there is an exact solution which shows that the history of the motion depends crucially on the form of the buoyancy spectrum near zero wavenumber magnitude (κ). According to this solution the Reynolds number of the motion increases indefinitely, so the linear equations do not remain valid. There are indications of similar behaviour when the nonlinear terms are retained. The value of the three-dimensional buoyancy spectrum function at κ = 0 is shown to be independent of time, and this points to the existence of a similarity state of turbulence with decreasing mean-square velocity but increasing Reynolds number at large times.

We have made a numerical simulation of the flow field and have obtained the mean-square velocity and density fluctuations and the associated spectra as functions of time for various initial conditions. An estimate of the time required for the mean-square density fluctuation to fall to a specified small value is found. The expected similarity state at large times is confirmed by the numerical simulation, and there are indications of a second similarity state which develops asymptotically when the buoyancy spectrum is zero at κ = 0. The analytical and numerical results together give a comprehensive description of the birth, life and lingering death of buoyancy-generated turbulence.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 235 , February 1992 , pp. 349 - 378
Copyright
© 1992 Cambridge University Press

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References

Batchelor, G. K. 1963 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. & Proudman, I. 1956 The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. 248, 369–405.Google Scholar
Chasnov, J. R. 1991 Simulation of the Kolmogorov inertial subrange using an improved subgrid model. Phys. Fluids A3, 188–200.Google Scholar
Chollet, J. P. 1985 Two-point closure used for a sub-grid scale model in large eddy simulations. In Turbulent Shear Flows IV (ed. L. J. S. Bradbury et al.), pp. 62–72.
Chollet, J. P. & Lesieur, M. 1981 Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmos. Sci. 38, 2747–2757.Google Scholar
Gibson, C. H. 1986 Internal waves, fossil turbulence, and composite ocean microstructure spectra. J. Fluid Mech. 168, 89–117.Google Scholar
Kraichnan, R. H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521–1536.Google Scholar
Lesieur, M. & Rogallo, R. 1989 Large-eddy simulation of passive scalar diffusion in isotropic turbulence. Phys. Fluids A 1, 718–722.Google Scholar
MeAtas, O. & Chollet, J. P. 1989 Turbulence submitted to a stable density stratification: large-eddy simulations and statistical theory. In Turbulent Shear Flows VI (ed. J.-C. AndreA et al.), pp. 398–416. Springer.
Rogallo, R. S. 1981 Numerical experiments in homogeneous turbulence. NASA TM 81315.Google Scholar
Saffman, P. G. 1967 The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27, 581–593.Google Scholar