Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T04:41:45.405Z Has data issue: false hasContentIssue false
  • English
  • Français

On the mixing of a confined stratified fluid by a turbulent buoyant plume

Published online by Cambridge University Press:  06 March 2009

RICHARD W. MOTT  and
ANDREW W. WOODS
RICHARD W. MOTT*
Affiliation:
BP Institute, Madingley Rise, Cambridge CB3 0EZ, UK
ANDREW W. WOODS
Affiliation:
BP Institute, Madingley Rise, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: richard.mott@bpi.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the mixing of a stratified fluid of finite volume by a turbulent buoyant plume. We develop a model to describe the mixing and apply this to both the cases of a two-layer stratification and a continuous stratification. With a two-layer stratification, the plume intrudes at the interface where it supplies an intermediate layer of fluid. This new layer gradually deepens, primarily mixing the original near-source layer of fluid through entrainment. Eventually, this intermediate layer becomes sufficiently buoyant that the plume penetrates into the more distal layer, leaving a partially mixed region between the original layers of fluid. Analysis of new experiments shows that the growth of the intermediate layer depends primarily on the ratio λ of (i) the filling box time, during which the plume entrains a volume of fluid equal to that in the near-source layer, and (ii) the time for the buoyancy of the near-source layer to increase to that of the more distal layer. For small values of λ, the near-source layer becomes approximately well mixed, and the penetration time of the plume scales with the buoyancy evolution time of the near-source layer. In the limit λ ~ O(1), however, the plume penetrates through into the distal layer long before the near-source layer becomes well mixed; instead, at the time of penetration, the plume leaves an intermediate partially mixed zone between the two original layers. We develop a new phenomenological model to account for the mixing in this intermediate layer based on the effective turbulent diffusion associated with the kinetic energy in the plume and compare this with the model for penetrative entrainment proposed by Kumagai (J. Fluid Mech., vol. 147, 1984, p. 105). In comparison with the experimental data, the models provide a reasonably accurate prediction of the plume penetration time, while the diffusive mixing model provides a somewhat more accurate description of the evolution of the density profile for a range 0 < λ < 1. The diffusive mixing model also leads to predictions which are consistent with some new experimental data for the case in which a plume mixes a continuously stratified layer. In particular, the model is able to predict the initial transient mixing of the region between the source and the height at which the plume intrudes laterally in the ambient fluid, thereby providing an advance on the late-time mixing model of Cardoso and Woods (J. Fluid Mech., vol. 250, 1993, p. 277). We consider the implications of these results on the turbulent penetrative entrainment associated with buoyant plumes.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 623 , 25 March 2009 , pp. 149 - 165
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ansong, J. K., Kyba, P. J. & Sutherland, B. R. 2008 Turblent fountains in a closed chamber. J. Fluid Mech. 595, 115–139.CrossRefGoogle Scholar
Baines, W. D., Corriveau, A. F. & Reedman, T. J. 1993 Turblent fountains in a closed chamber. J. Fluid Mech. 255, 621–646.CrossRefGoogle Scholar
Baines, W. D. & Turner, J. S. 1969 Turblent 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 plume in a confined stratified region. J. Fluid Mech. 250, 277–305.Google Scholar
Caulfield, C. P. & Woods, A. W. 1995, Plumes with non-monotonic mixing behaviour. Geophys. Astrophys. Fluid Dyn. 79 (1–4), 173–199.CrossRefGoogle Scholar
Conroy, D. T., Llewellyn Smith, S. G. & Caulfield, C. P. 2005, Evolution of a chemically reacting plume in a ventilated room. J. Fluid Mech. 537 221–253.CrossRefGoogle Scholar
Cooper, C. P., & Linden, P. F. 1996, Natural ventilation of an enclosure containing two buoyancy sources. J. Fluid Mech. 311, 153–176.Google Scholar
Fitzgerald, S. D. & Woods, A. W. 2007 Transient natural ventilation of a room with a distributed heat source. J. Fluid Mech. 591, 21–42.CrossRefGoogle Scholar
Hunt, G. R. & Kaye, N. E. 2001 Virtual origin correction for lazy turbulent plumes. J. Fluid Mech. 435, 377–396.CrossRefGoogle Scholar
Kumagai, M. 1984 Turbulent convection in a two-layered region. J. Fluid Mech. 147, 105–131.Google Scholar
Linden, P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31, 201–238.CrossRefGoogle Scholar
Linden, P. F & Cooper, P. 1996, Multiple sources of buoyancy in a naturally ventilated enclosure. J. Fluid Mech. 311, 177–192CrossRefGoogle Scholar
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convectoin from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Paluskiiewicz, T., Garwood, R. W. & Denbo, D. W. 1988 Deep convective plumes in the ocean. Oceanography 7, 37–44.CrossRefGoogle Scholar
Papanicolaou, P. N. & List, E. J. 1994 Investigations of round vertical turbulent buoyant jets. J. Fluid Mech. 195, 341–391.Google Scholar
Phillips, J. C. & Woods, A. W. 2001 Bubble plumes generated during recharge of a basaltic magma chamber. Earth Planet Sci. Lett. 186, 297–309.Google Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids,. pp. 165–206. Cambridge University Press.Google Scholar
Woods, A. W. & Phillips, J. C. 1999 Turbulent bubble plumes and CO2 driven lake eruptions. J. Volcanol. Geothermal Res. 92, 259–270.CrossRefGoogle Scholar
Woods, A. W., Caulfield, C. P. & Phillips, J. C. 2003 Blocked natural ventilation: the effect of a source mass flux. J. Fluid Mech. 495, 119–133.Google Scholar