Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-17T23:14:50.377Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixing by a turbulent plume in a confined stratified region

Published online by Cambridge University Press:  26 April 2006

Silvana S. S. Cardoso  and
Andrew W. Woods
Silvana S. S. Cardoso
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Andrew W. Woods
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

An experimental and theoretical study of the mixing produced by a plume rising in a confined stratified environment is presented. As a result of the pre-existing stable stratification, the plume penetrates only part way into the region; at an intermediate level it intrudes laterally forming a horizontal layer. As time evolves, this layer of mixed fluid is observed to increase in thickness. The bottom front advects downward in a way analogous to the first front in the filling box of Baines & Turner (1969), while the lateral spreading of the plume occurs at an ever-increasing level and an ascending top front results. We develop a model of this stratified filling box; the model predicts the rate at which the two fronts advance into the environment.

It is found that stratification in the environment, when smooth, has no significant influence on the dynamics of the descending front. We show that the rate of rise of the ascending front is determined by the turbulent mixing occurring at the spreading level. Entrainment of environmental fluid from above into the overshooting plume is significant; as a result, a density interface develops at this level. Asymptotically, the system reaches a state in which a bottom convecting layer, with an almost homogeneous density, deepens in a stratified background. The model proposed for this large-time behaviour is based on the simple energetic formulation that a constant fraction of the kinetic energy supplied by the plume, for mixing across the interface, is converted into potential energy of the convective layer. Our experimental results suggest an efficiency of approximately 50 % for this conversion.

We discuss our results in the light of previous studies on turbulent penetrative convection and conclude that the theory developed should be valid for an intermediate range of values of the Richardson number characterizing the dynamic conditions at the interface. The model is applied quantitatively to the process of cooling of a room wherein stratification is relevant. The geological problem of replenishment of a magma chamber by a light input of magma is also analysed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 250 , May 1993 , pp. 277 - 305
Copyright
© 1993 Cambridge University Press

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

Baines, W. D. 1975 Entrainment by a plume or jet at a density interface. J. Fluid Mech. 60, 467–480.Google Scholar
Baines, W. D. & Turner, J. S. 1969 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 37, 51–80.Google Scholar
Baines, W. D., Turner, J. S. & Campbell, I. H. 1990 Turbulent fountains in an open chamber. J. Fluid Mech. 212, 557–592.Google Scholar
Ball, F. K. 1960 Control of inversion height by surface heating. Q. J. R. Met. Soc. 86, 483–494.Google Scholar
Barnett, S. J. 1991 The dynamics of buoyant releases in confined spaces. PhD thesis, University of Cambridge.
Deardorff, J. W. & Willis, G. E. 1985 Further results from a laboratory model of the convective planetary boundary layer. Boundary-Layer Met. 32, 205–236.Google Scholar
Germeles, A. E. 1975 Forced plumes and mixing of liquids in tanks. J. Fluid Mech. 71, 601–623.Google Scholar
Holman, J. P. 1983 Heat Transfer. McGraw Hill. pp. 265–304.
Huppert, H. E., Sparks, R. S. J., Whitehead, J. A. & Hallworth, M. A. 1986 Replenishment of magma chambers by light inputs. J. Geophys. Res. 91, 6113–6122.Google Scholar
Kumagai, M. 1984 Turbulent buoyant convection from a source in a confined two-layered region. J. Fluid Mech. 147, 105–131.Google 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.Google Scholar
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 3–23.Google Scholar
Manins, R. C. 1979 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 91, 765–781.Google Scholar
Manins, R. C. & Turner, J. S. 1978 The relation between the flux ratio and the energy ratio in convectively mixed layers. Q. J. R. Met. Soc. 104, 39–44.Google Scholar
Morton, B., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Oster, G. 1965 Density gradients. Sci. A. 213, 70–76.Google Scholar
Papanicolaou, P. N. & List, E. J. 1988 Investigation of round vertical turbulent buoyant jets. J. Fluid Mech. 195, 341–391.Google Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids, pp. 165–206. Cambridge University Press.
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431–471.Google Scholar
Worster, M. G. & Huppert, H. E. 1983 Time-dependent density profiles in a filling box. J. Fluid Mech. 132, 457–466.Google Scholar
Zilitinkevich, S. S. 1991 Turbulent Penetrative Convection. Avebury Technical.