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The unidirectional emptying box

Published online by Cambridge University Press:  01 September 2010

C. J. COFFEY  and
G. R. HUNT
C. J. COFFEY
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
G. R. HUNT*
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

A theoretical description of the turbulent mixing within and the draining of a dense fluid layer from a box connected to a uniform density, quiescent environment through openings in the top and the base of the box is presented in this paper. This is an extension of the draining model developed by Linden et al. (Annu. Rev. Fluid Mech. vol. 31, 1990, pp. 201–238) and includes terms that describe localized mixing within the emptying box at the density interface. Mixing is induced by a turbulent flow of replacement fluid into the box and as a consequence we predict, and observe in complementary experiments, the development of a three-layer stratification. Based on the data collated from previous researchers, three distinct formulations for entrainment fluxes across density interfaces are used to account for this localized mixing. The model was then solved numerically for the three mixing formulations. Analytical solutions were developed for one formulation directly and for a second on assuming that localized mixing is relatively weak though still significant in redistributing buoyancy on the timescale of the draining process. Comparisons between our theoretical predictions and the experimental data, which we have collected on the developing layer depths and their densities show good agreement. The differences in predictions between the three mixing formulations suggest that the normalized flux turbulently entrained across a density interface tends to a constant value for large values of a Froude number FrT, based on conditions of the inflow through the top of the box, and scales as the cube of FrT for small values of FrT. The upper limit on the rate of entrainment into the mixed layer results in a minimum time (tD) to remove the original dense layer. Using our analytical solutions, we bound this time and show that 0.2tE ≲ tD ≲ tE, i.e. the original dense layer may be depleted up to five times more rapidly than when there is no internal mixing and the box empties in a time tE.

JFM classification

Geophysical and Geological Flows: Stratified flows Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 660 , 10 October 2010 , pp. 456 - 474
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Baines, W. D. 1975 Entrainment by a plume or jet at a density interface. J. Fluid Mech. 68 (2), 309–320.CrossRefGoogle Scholar
Baines, W. D. & Turner, J. S. 1969 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 37, 51–80.CrossRefGoogle Scholar
Cardoso, S. S. S. & Woods, A. W. 1993 Mixing by a turbulent plume in a confined stratified region. J. Fluid Mech. 250, 277–305.CrossRefGoogle Scholar
Cenedese, C. & Dalziel, S. B. 1998 Concentration and depth field determined by the light transmitted through a dyed solution. In Proceedings of the Eighth International Symposium on Flow Visualization, Sorrento, Italy.Google Scholar
Coffey, C. J. & Hunt, G. R. 2007 Ventilation effectiveness measures based on heat removal. Part 1. Definitions. Build. Environ. 42 (6), 2241–2248.CrossRefGoogle Scholar
Etheridge, D. W. & Sandberg, M. 1984 A simple parametric study of ventilation. Build. Environ. 19 (3), 163–173.CrossRefGoogle Scholar
Fernando, J. & Smith, D. C. 2001 Vortex structures in geophysical convection. Eur. J. Mech. B – Fluids 20, 437–470.CrossRefGoogle Scholar
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic Press, ISBN 0-12-258150-4.Google Scholar
Hunt, G. R. & Coffey, C. J. 2010 Emptying boxes – classifying transient natural ventilation flows. J. Fluid Mech. 646, 137–168.CrossRefGoogle Scholar
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319–328.CrossRefGoogle Scholar
Kumagai, M. 1984 Turbulent buoyant convection from a source in a confined two-layered region. J. Fluid Mech. 147, 105–131.CrossRefGoogle Scholar
Lin, Y. J. P. & Linden, P. F. 2005 A model for an under floor air distribution system. Energy Build. 37 (4), 399–409.CrossRefGoogle Scholar
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60, 467–480.CrossRefGoogle Scholar
Linden, P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31, 201–238.CrossRefGoogle Scholar
Linden, P. F., Lane-Serff, G. F. & Smeed, D. A. 1990 Emptying filling boxes, the fluid mechanics of natural ventilation. J. Fluid Mech. 212, 309–335.CrossRefGoogle Scholar
Park, O. H., Seo, S. J. & Lee, S. H. 2001 Laboratory simulation of vertical plume dispersion within a convective boundary layer. Bound.-Layer Meteorol. 99, 159–169.CrossRefGoogle Scholar
Presley, J. D. & Telford, J. W. 1988 Turbulent entrainment at an inversion. J. Pure Appl. Geophys. 127, 117–141.CrossRefGoogle Scholar
Thorpe, S. A. 2005 The Turbulent Ocean. Cambridge University Press.CrossRefGoogle Scholar
Ward-Smith, A. J. 1980 Internal Fluid Flow – The Fluid Dynamics of Flow in Pipes and Ducts. Oxford University Press, ISBN 0–19–856325–6.Google Scholar
Willis, G. E. & Deardorff, J. W. 1981 A laboratory study of dispersion from a source in the middle of the convectively mixed layer. Atmos. Environ. 15, 109–117.CrossRefGoogle Scholar
Willis, G. E. & Deardorff, J. W. 1983 On plume rise within a convective boundary layer. Atmos. Environ. 17, 2435–2477.CrossRefGoogle Scholar
Willis, G. E. & Deardorff, J. W. 1987 Buoyant plume dispersion and inversion entrapment in and above a laboratory mixed layer. Atmos. Environ. 21, 1725–1735.CrossRefGoogle Scholar