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The entrainment and energetics of turbulent plumes in a confined space

  • John Craske (a1) and Megan S. Davies Wykes (a2)

Abstract

We analyse the entrainment and energetics of equal and opposite axisymmetric turbulent air plumes in a vertically confined space at a Rayleigh number of $1.24\times 10^{7}$ using theory and direct numerical simulation. On domains of sufficiently large aspect ratio, the steady state consists of turbulent plumes penetrating an interface between two layers of approximately uniform buoyancy. As described by Baines & Turner (J. Fluid Mech., vol. 37(1), 1969, pp. 51–80), upon penetrating the interface the flow in each plume becomes forced and behaves like a constant-momentum jet, due to a reduction in its mean buoyancy relative to the local environment. To observe the behaviour of the plumes we partition the domain into sub-domains corresponding to each plume. Domains of relatively small aspect ratio produce a single primary mean-flow circulation between the sub-domains that is maintained by entrainment into the plumes. At larger aspect ratios the mean flow between the sub-domains bifurcates, indicating the existence of a secondary circulation within each layer associated with entrainment into the jets. The largest aspect ratios studied here exhibit an additional, tertiary, circulation in the vicinity of the interface. Consistency between independent calculations of an effective entrainment coefficient allows us to identify aspect ratios for which the flow can be modelled using plume theory, under the assumption of a two-layer stratification. To study the flow’s energetics we use a local definition of available potential energy (APE). For plumes with Gaussian velocity and buoyancy profiles, the theory we develop suggests that the kinetic energy dissipation is split equally between the jets and the plumes and, collectively, accounts for almost half of the input of APE at the boundaries. In contrast, 1/4 of the APE dissipation and background potential energy (BPE) production occurs in the jets, with the remaining 3/4 occurring in the plumes. These bulk theoretical predictions agree with observations of BPE production from simulations to within 1 % and form the basis of a similarity solution that models the vertical dependence of APE dissipation and BPE production. Unlike results concerning the dissipation of buoyancy variance and the strength of the circulations described above, the model for the flow’s energetics does not involve an entrainment coefficient.

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Corresponding author

Email address for correspondence: john.craske07@imperial.ac.uk

References

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Andrews, D. G. 1981 A note on potential energy density in a stratified compressible fluid. J. Fluid Mech. 107, 227236.
Baines, W. D. & Turner, J. S. 1969 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 37 (1), 5180.
Batchelor, G. K. 1954 Heat convection and buoyancy effects in fluids. Q. J. R. Meteorol. Soc. 80 (345), 339358.
Bonnebaigt, R., Caulfield, C. P. & Linden, P. F. 2018 Detrainment of plumes from vertically distributed sources. Environ. Fluid Mech. 18 (1), 325.
Burridge, H. C., Parker, D. A., Kruger, E. S., Partridge, J. L. & Linden, P. F. 2017 Conditional sampling of a high Péclet number turbulent plume and the implications for entrainment. J. Fluid Mech. 823, 2656.
Camassa, R., Lin, Z., McLaughlin, R. M., Mertens, K., Tzou, C., Walsh, J. & White, B. 2016 Optimal mixing of buoyant jets and plumes in stratified fluids: theory and experiments. J. Fluid Mech. 790, 71103.
Carazzo, G., Kaminski, E. & Tait, S. 2006 The route to self-similarity in turbulent jets and plumes. J. Fluid Mech. 547, 137148.
Craske, J. & van Reeuwijk, M. 2015a Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets. J. Fluid Mech. 763, 500537.
Craske, J. & van Reeuwijk, M. 2015b Energy dispersion in turbulent jets. Part 2. A robust model for unsteady jets. J. Fluid Mech. 763, 538566.
Craske, J. & van Reeuwijk, M. 2016 Generalised unsteady plume theory. J. Fluid Mech. 792, 10131052.
Craske, J., Salizzoni, P. & van Reeuwijk, M. 2017 The turbulent Prandtl number in a pure plume is 3/5. J. Fluid Mech. 822, 774790.
Davies Wykes, M. S., Hogg, C., Partridge, J. & Hughes, G. O. 2019 Energetics of mixing for the filling box and the emptying-filling box. Environ. Fluid Mech. 19 (4), 819831.
Fanneløp, T. & Webber, D. M. 2003 On buoyant plumes rising from area sources in a calm environment. J. Fluid Mech. 497, 319334.
Gayen, B., Hughes, G. O. & Griffiths, R. W. 2013 Completing the mechanical energy pathways in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 111 (12), 124301.
Gladstone, C. & Woods, A. W. 2014 Detrainment from a turbulent plume produced by a vertical line source of buoyancy in a confined, ventilated space. J. Fluid Mech. 742, 3549.
Gregg, M., D’Asaro, E., Riley, J. & Kunze, E. 2018 Mixing efficiency in the ocean. Annu. Rev. Mar. Sci. 10 (1), 443473; pMID: 28934598.
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.
Holliday, D. & Mcintyre, M. E. 1981 On potential energy density in an incompressible, stratified fluid. J. Fluid Mech. 107, 221225.
Hubner, J.2006 Buoyant plumes in a turbulent environment. PhD thesis, University of Cambridge.
Hughes, G. O., Gayen, B. & Griffiths, R. W. 2013 Available potential energy in Rayleigh–Bérnard convection. J. Fluid Mech. 729, R3.
Hunt, G. R. & Kaye, N. B. 2005 Lazy plumes. J. Fluid Mech. 533, 329338.
Hunt, G. R. & Kaye, N. G. 2001 Virtual origin correction for lazy turbulent plumes. J. Fluid Mech. 435, 377396.
Hunt, J. C. R., Eames, I. & Westerweel, J. 2006 Mechanics of inhomogeneous turbulence and interfacial layers. J. Fluid Mech. 554, 499519.
Kaminski, E., Tait, S. & Carazzo, G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361376.
Khorsandi, B., Gaskin, S. & Mydlarski, L. 2013 Effect of background turbulence on an axisymmetric turbulent jet. J. Fluid Mech. 736, 250286.
Lai, A. C. H., Law, A. W.-K. & Adams, E. E. 2019 A second-order integral model for buoyant jets with background homogeneous and isotropic turbulence. J. Fluid Mech. 871, 271304.
Linden, P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31 (1), 201238.
Linden, P. F., Lane-Serff, G. F. & Smeed, D. A. 1990 Emptying filling boxes: the fluid mechanics of natural ventilation. J. Fluid Mech. 212, 309335.
Lorenz, E. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7 (2), 157167.
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.
Novak, L. & Tailleux, R. 2018 On the local view of atmospheric available potential energy. J. Atmos. Sci. 75 (6), 18911907.
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. & Holzner, M. 2014 The turbulence boundary of a temporal jet. J. Fluid Mech. 739, 254275.
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.
Roullet, G. & Klein, P. 2009 Available potential energy diagnosis in a direct numerical simulation of rotating stratified turbulence. J. Fluid Mech. 624, 4555.
Scotti, A., Beardsley, R. & Butman, B. 2006 On the interpretation of energy and energy fluxes of nonlinear internal waves: an example from massachusetts bay. J. Fluid Mech. 561, 103112.
Scotti, A. & White, B. 2014 Diagnosing mixing in stratified turbulent flows with a locally defined available potential energy. J. Fluid Mech. 740, 114135.
da Silva, C. B., Hunt, J. C. R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 46 (1), 567590.
Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy. J. Fluid Mech. 638, 339382.
Tailleux, R. 2018 Local available energetics of multicomponent compressible stratified fluids. J. Fluid Mech. 842, R1.
Taylor, G. I. 1958 Flow induced by jets. J. Aerosp. Sci. 25, 464465.
Winters, K. B., Lombard, P. N., Riley, J. J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.
Worster, M. G. & Huppert, H. E. 1983 Time-dependent density profiles in a filling box. J. Fluid Mech. 132, 457466.
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The entrainment and energetics of turbulent plumes in a confined space

  • John Craske (a1) and Megan S. Davies Wykes (a2)

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