Journal of Fluid Mechanics (1992), 235:349-378. Cambridge University Press.

Copyright © 1992 Cambridge University Press

## Homogeneous buoyancy-generated turbulence

**G. K.** **BATCHELOR**^{1}, **V. M. ** **CANUTO**^{2} AND **J. R.** **CHASNOV**^{2}

^{1}Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK

^{2}NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA

(Received 25 April 1991)

**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 k. 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 k = 0 is shown to be indpendent 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 k = 0. The analytical and numerical results together give a comprehensive description of the birth, life and lingering death of buoyancy-generated turbulence.

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