Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T21:36:43.678Z Has data issue: false hasContentIssue false
  • English
  • Français

Universal solutions for Boussinesq and non-Boussinesq plumes

Published online by Cambridge University Press:  11 February 2010

T. S. VAN DEN BREMER  and
G. R. HUNT
T. S. VAN DEN BREMER
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, Imperial College Road, London SW7 2AZ, UK
G. R. HUNT*
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, Imperial College Road, London SW7 2AZ, UK
*
†Email address for correspondence: gary.hunt@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Closed-form solutions describing the behaviour of buoyant axisymmetric turbulent rising plumes and fountains, emitted vertically from area sources in unconfined quiescent environments of uniform density, are proposed in a form that is universally applicable to Boussinesq and non-Boussinesq plumes. This paper, thereby, generalizes the results obtained separately for steady Boussinesq and non-Boussinesq plumes, including asymptotic virtual source corrections. The flux balance parameter Γ = Γ(z), a local Richardson number, is instrumental in describing the behaviour of steady plumes and the initial rise behaviour of fountains with height z. Non-dimensional graphs (cf. the ‘scale diagrams’ of Morton & Middleton, J. Fluid Mech., vol. 58, 1973, pp. 165–176) are plotted, showing certain characteristic heights for different source conditions, characterized by one single source flux balance parameter, giving a unique representation of the behaviour of Boussinesq fountains and both Boussinesq and non-Boussinesq plumes. Finally, a length scale has been identified that characterizes the height over which non-Boussinesq effects are important for lazy plumes rising from area sources.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 644 , 10 February 2010 , pp. 165 - 192
Copyright
Copyright © Cambridge University Press 2010

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

Baines, W. D. 1983 A technique for the direct measurement of volume flux of a plume. J. Fluid Mech. 132, 247–256.CrossRefGoogle Scholar
Bloomfield, L. J. & Kerr, R. C. 2000 A theoretical model of a turbulent fountain. J. Fluid Mech. 424, 197–216.CrossRefGoogle Scholar
Carazzo, G., Kaminski, E. & Tait, S. 2006 The route to self-similarity in turbulent jets and plumes. J. Fluid Mech. 547, 137–148.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
Carlotti, P. & Hunt, G. R. 2005 Analytical solutions for turbulent non-Boussinesq plumes. J. Fluid Mech. 538, 343–359.CrossRefGoogle Scholar
Caulfield, C. P. 1991 Stratification and buoyancy in geophysical flows. PhD Thesis, University of Cambridge, Cambridge, UK.Google Scholar
Caulfield, C. P. & Woods, A. W. 1998 Turbulent gravitational convection from a point source in a non-uniformly stratified environment. J. Fluid Mech. 360, 229–248.CrossRefGoogle Scholar
Cetegen, B. M., Zukoski, E. E. & Kubota, T. 1984 Entrainment in the near field and far field of fire plumes. Combust. Sci. Technol. 39, 305–331.CrossRefGoogle Scholar
Conroy, D. T. & Llewellyn Smith, S. G. 2008 Endothermic and exothermic chemically reacting plumes. J. Fluid Mech. 612, 291–310.CrossRefGoogle Scholar
Fannelop, T. K. & Webber, D. M. 2003 On buoyant plumes rising from area sources in a calm environment. J. Fluid Mech. 497, 319–334.CrossRefGoogle Scholar
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. (Eds.) 1979 Mixing in Inland and Coastal Waters. Academic.Google Scholar
Heskestad, G. 1998 Dynamics of the fire plume. Phil. Trans. R. Soc. Lond. A 356, 2815–2833.CrossRefGoogle Scholar
Hunt, G. R. & Kaye, N. G. 2001 Virtual origin correction for lazy turbulent plumes. J. Fluid Mech. 435, 377–396.CrossRefGoogle Scholar
Hunt, G. R. & Kaye, N. B. 2005 Lazy plumes. J. Fluid Mech. 533, 329–338.CrossRefGoogle Scholar
Kaminski, E., Tait, S. & Carazzo, G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361–376.CrossRefGoogle Scholar
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319–328.CrossRefGoogle Scholar
Kaye, N. B. 2008 Turbulent plumes in stratified environments: a review of recent work. Atmos.-Ocean 46, 433–441.CrossRefGoogle Scholar
Lin, W. & Armfield, S. W. 2000 Very weak fountains in a homogeneous fluid. Numer. Heat Transfer A 38, 377–396.Google Scholar
Linden, P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31, 201–238.CrossRefGoogle Scholar
List, E. J. & Imberger, J. 1973 Turbulent entrainment in buoyant jets and plumes. J. Hydraul. Div. ASCE 99, 1461–1474.CrossRefGoogle Scholar
Morton, B. R. 1959 Forced plumes. J. Fluid Mech. 5, 151–163.CrossRefGoogle Scholar
Morton, B. R. 1965 Modelling fire plumes. In Tenth Symposium (International) on Combustion, pp. 973–982.Google Scholar
Morton, B. R. & Middleton, J. 1973 Scale diagrams for forced plumes. J. Fluid Mech. 58, 165–176.CrossRefGoogle Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Papanicolaou, P. N., Papakonstantis, I. G. & Christodoulou, G. C. 2008 On the entrainment coefficient in negatively buoyant jets. J. Fluid Mech. 614, 447–470.CrossRefGoogle Scholar
Priestley, C. H. B. & Ball, F. K. 1955 Continuous convection from an isolated source of heat. Quart. J. R. Meteorol. Soc. 81, 144–157.CrossRefGoogle Scholar
Ricou, F. P. & Spalding, D. B. 1961 Measurements of entrainment by axisymmetrical turbulent jets. J. Fluid Mech. 11, 21–32.CrossRefGoogle Scholar
Rooney, G. G. 1997 Buoyant flows from fires in enclosures. PhD Thesis, University of Cambridge, Cambridge, UK.Google Scholar
Rooney, G. G. & Linden, P. F. 1996 Similarity considerations for non-Boussinesq plumes in an unstratified environment. J. Fluid Mech. 318, 237–250.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P. & Dalziel, S. B. 2006 a Boussinesq plumes and jets with decreasing source strengths in stratified environments. J. Fluid Mech. 563, 463–472.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P. & Dalziel, S. B. 2008 Temporal variation of non-ideal plumes with sudden reductions in buoyancy flux. J. Fluid Mech. 600, 181–199.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P., Dalziel, S. B. & Hunt, J. C. R. 2006 b Time-dependent plumes and jets with decreasing source strengths. J. Fluid Mech. 563, 443–461.CrossRefGoogle Scholar
Scase, M. M., Caulfield, C. P., Linden, P. F. & Dalziel, S. B. 2007 Local implications for self-similar turbulent plume models. J. Fluid Mech. 575, 257–265.CrossRefGoogle Scholar
Shabbir, A. & George, W. K. 1994 Experiments on a round turbulent buoyant plume. J. Fluid Mech. 275, 1–32.CrossRefGoogle Scholar
Thring, M. W. & Newby, M. P. 1953 Combustion length of enclosed turbulent jet flames. In Fourth Symposium (International) on Combustion, pp. 789–796.Google Scholar
Turner, J. S. 1966 Jets and plumes with negative or reversing buoyancy. J. Fluid Mech. 26, 779–792.CrossRefGoogle Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids, 2nd edn. Cambridge University Press.Google Scholar
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431–471.CrossRefGoogle Scholar
Wang, H. & Law, A. W-K. 2002 Second-order integral model for a round turbulent buoyant jet. J. Fluid Mech. 459, 397–428.CrossRefGoogle Scholar
Woods, A. W. 1997 A note on non-Boussinesq plumes in an incompressible stratified environment. J. Fluid Mech. 345, 347–356.CrossRefGoogle Scholar