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The turbulent Prandtl number in a pure plume is 3/5

  • John Craske (a1), Pietro Salizzoni (a2) and Maarten van Reeuwijk (a1)
Abstract

We derive a new expression for the entrainment coefficient in a turbulent plume using an equation for the squared mean buoyancy. Consistency of the resulting expression with previous relations for the entrainment coefficient implies that the turbulent Prandtl number in a pure plume is equal to 3/5 when the mean profiles of velocity and buoyancy have a Gaussian form of equal width. Entrainment can be understood in terms of the volume flux, the production of turbulence kinetic energy or the production of scalar variance for either active or passive variables. The equivalence of these points of view indicates how the entrainment coefficient and the turbulent Prandtl and Schmidt numbers depend on the Richardson number of the flow, the ambient stratification and the relative widths of the velocity and scalar profiles. The general framework is valid for self-similar plumes, which are characterised by a power-law scaling. For jets and pure plumes it is shown that the derived relations are in reasonably good agreement with results from direct numerical simulations and experiments.

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Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Email address for correspondence: john.craske07@imperial.ac.uk
References
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Antonia R. A. & Mi J. 1993 Temperature dissipation in a turbulent round jet. J. Fluid Mech. 250, 531551.
Batchelor G. K. 1954 Heat convection and buoyancy effects in fluids. Q. J. R. Meteorol. Soc. 80 (345), 339358.
Carazzo G., Kaminski E. & Tait S. 2006 The route to self-similarity in turbulent jets and plumes. J. Fluid Mech. 547, 137148.
Caulfield C. P. & Woods A. W. 1998 Turbulent gravitational convection from a point source in a non-uniformly stratified environment. J. Fluid Mech. 360, 229248.
Chang K.-A. & Cowen E. A. 2002 Turbulent Prandtl number in neutrally buoyant turbulent round jet. J. Engng Mech. 128 (10), 10821087.
Chen C. J. & Rodi W. 1980 Vertical Turbulent Buoyant Jets: A Review of Experimental Data. Pergamon.
Chua L. P. & Antonia R. A. 1990 Turbulent Prandtl number in a circular jet. Intl J. Heat Mass Transfer 33 (2), 331339.
Craske J. & van Reeuwijk M. 2015 Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets. J. Fluid Mech. 763, 500537.
Ezzamel A., Salizzoni P. & Hunt G. R. 2015 Dynamical variability of axisymmetric buoyant plumes. J. Fluid Mech. 765, 576611.
George W. K., Alpert R. L. & Tamanini F. 1977 Turbulence measurements in an axisymmetric buoyant plume. Intl J. Heat Mass Transfer 20 (11), 11451154.
Hunt G. R. & Kaye N. B. 2005 Lazy plumes. J. Fluid Mech. 533, 329338.
Kaminski E., Tait S. & Carazzo G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361376.
Linden P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31 (1), 201238.
Morton B. R. & Middleton J. 1973 Scale diagrams for forced plumes. J. Fluid Mech. 58 (01), 165176.
Morton B. R., Taylor G. I. & Turner J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234 (1196), 123.
Nakagome H. & Hirita M. 1977 The structure of turbulent diffusion in an axisymmetrical thermal plume. In 1976 ICHMT Seminar on Turbulent Buoyant Convection, pp. 361372.
Papanicolaou P. N. & List E. J. 1988 Investigations of round vertical turbulent buoyant jets. J. Fluid Mech. 195, 341391.
Pietri L., Amielh M. & Anselmet F. 2000 Simultaneous measurements of temperature and velocity fluctuations in a slightly heated jet combining a cold wire and laser doppler anemometry. Intl J. Heat Fluid Flow 21 (1), 2236.
Plourde F., Pham M. V., Kim S. D. & Balachandar S. 2008 Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction. J. Fluid Mech. 604, 99123.
Priestley C. H. B. & Ball F. K. 1955 Continuous convection from an isolated source of heat. Q. J. R. Meteorol. Soc. 81 (348), 144157.
van Reeuwijk M. & Craske J. 2015 Energy-consistent entrainment relations for jets and plumes. J. Fluid Mech. 782, 333355.
van Reeuwijk M., Salizzoni P., Hunt G. R. & Craske J. 2016 Turbulent transport and entrainment in jets and plumes: a DNS study. Phys. Rev. Fluids 1, 074301.
Rouse H., Yih C. S. & Humphreys H. W. 1952 Gravitational convection from a boundary source. Tellus 4 (3), 201210.
Shabbir A. & George W. K. 1994 Experiments on a round turbulent buoyant plume. J. Fluid Mech. 275, 132.
Taylor G. I.1945 Dynamics of a mass of hot gas rising in air. Tech. Rep. US Atomic Energy Commission. Los Alamos National Laboratory Research Library. Report 236.
Wang H. & Law W. K. 2002 Second-order integral model for a round turbulent buoyant jet. J. Fluid Mech. 459, 397428.
Woods A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42 (1), 391412.
Zel’dovich Y. B. 1937 The asymptotic laws of freely-ascending convective flows. Zh. Eksp. Teor. Fiz. 7, 14631465 (in Russian). English translation in Selected Works of Yakov Borisovich Zeldovich, vol. 1, 1992 (J. P. Ostriker ed.), pp. 82–85. Princeton University Press.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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