Skip to main content
SpringerLink
Log in

Determination of discharge coefficient in non-Boussinesq cases and application in emptying–filling boxes

  • Original Article
  • Published:
Environmental Fluid Mechanics Aims and scope Submit manuscript

Abstract

The effect of high buoyancy forces due to the density contrast between the fluids at either side of the vent on the discharge coefficient Cd was considered in emptying–filling boxes. Salt-water experiments were conducted at small scale in a large freshwater tank with saline to generate buoyancy force. In non-Boussinesq cases, a larger discharge parameter Γd at the vent may make plume-like flow contract further with a smaller value of the discharge coefficient. Simple draining flows with variant Cd at the upper vent is modeled when considering the emptying of an enclosure initially filled with a large amount of light fluid. The time in the non-dimensional form required for draining light fluid fully out of the space by the analysis with a variable value is twice as much as the time predicted with a constant value of Cd = 0.6. A theoretical model of displacement flow with a virtual source correction at the initial position of an internal source was developed to reveal a time-dependent movement of the layer interface, and comparisons were made with the experimental results. The oscillatory amplitude of the interface overshooting during the transient period was found to depend on a geometrical parameter Λ and a dimensionless parameter Θ that characterizes the source strength relative to the space height.

Graphical abstract

Highlights

  1. 1.

    Salt-water experiments with and without a buoyant source were performed in a lab.

  2. 2.

    The discharge coefficient in non-Boussinesq cases is fitted with experimental results.

  3. 3.

    For emptying flow the dropping process of the interface includes two distinctive stages.

  4. 4.

    The amplitude of the oscillatory interface is determined by parameters of Θ and Λ.

  5. 5.

    The dimensionless plume radius indicates dependence on the discharge parameter Γ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Availability of data and material

Not applicable.

Code availability

Not applicable.

References

  1. Higton TD, Burridge HC, Hughes GO (2021) Natural ventilation flows established by a localised heat source in a room with a doorway and a high-level vent. Build Environ 203:108093. https://doi.org/10.1016/j.buildenv.2021.108093

    Article  Google Scholar 

  2. Morton BR, Taylor G, Turner JS (1956) Turbulent gravitational convection from maintained and instantaneous sources. P Roy Soc Lond 234(1196):1–23. https://doi.org/10.1098/rspa.1956.0011

    Article  Google Scholar 

  3. Baines WD, Turner JS (1969) Turbulent buoyant convection from a source in a confined region. J Fluid Mech 37(1):51–80. https://doi.org/10.1017/s0022112069000413

    Article  Google Scholar 

  4. Linden PF, Lane-Serff GF, Smeed DA (1990) Emptying filling boxes: the fluid mechanics of natural ventilation. J Fluid Mech 212:309–335. https://doi.org/10.1017/s0022112090001987

    Article  Google Scholar 

  5. Partridge JL, Linden PF (2017) Steady flows in a naturally-ventilated enclosure containing both a distributed and a localised source of buoyancy. Build Environ 125:308–318. https://doi.org/10.1016/j.buildenv.2017.08.023

    Article  Google Scholar 

  6. Yu Z, Hunt GR (2021) On the stratification and induced flow in an emptying-filling box driven by a plane vertically distributed source of buoyancy. J Fluid Mech. https://doi.org/10.1017/jfm.2020.1034

    Article  Google Scholar 

  7. Parker DA, Burridge HC, Partridge JL, Hacker JN, Linden PF (2021) Vertically distributed wall sources of buoyancy. Part 2. Unventilated and ventilated confined spaces. J Fluid Mech. https://doi.org/10.1017/jfm.2020.809

    Article  Google Scholar 

  8. Hunt GR, Coffey CJ (2010) Emptying boxes - classifying transient natural ventilation flows. J Fluid Mech 646:137–168. https://doi.org/10.1017/S0022112009993028

    Article  Google Scholar 

  9. Shrinivas AB, Hunt GR (2014) Transient ventilation dynamics induced by heat sources of unequal strength. J Fluid Mech 738:34–64. https://doi.org/10.1017/jfm.2013.579

    Article  Google Scholar 

  10. Hunt GR, Linden PF (2001) Steady-state flows in an enclosure ventilated by buoyancy forces assisted by wind. J Fluid Mech 426:355. https://doi.org/10.1017/S0022112000002470

    Article  Google Scholar 

  11. Craske J, Hughes GO (2019) On the robustness of emptying filling boxes to sudden changes in the wind. J Fluid Mech 868:R3. https://doi.org/10.1017/jfm.2019.199

    Article  Google Scholar 

  12. Mehaddi R, Boulet P, Koutaiba M, Vauquelin O, Candelier F (2021) Emptying-filling boxes with non-Boussinesq plumes and fountains. Phys Rev Fluid 6(8):083801. https://doi.org/10.1103/PhysRevFluids.6.083801

    Article  Google Scholar 

  13. Madival DG (2021) Filling of smoke due to fire in a room with roof ventilation. Int J Therm Sci 160:106650. https://doi.org/10.1016/j.ijthermalsci.2020.106650

    Article  Google Scholar 

  14. Woods AW (1997) A note on non-Boussinesq plumes in an incompressible stratified environment. J Fluid Mech 345:347–356. https://doi.org/10.1017/S0022112097006332

    Article  Google Scholar 

  15. Rooney GG, Linden PF (1996) Similarity considerations for non-Boussinesq plumes in an unstratified environment. J Fluid Mech 318:237–250. https://doi.org/10.1017/S0022112096007100

    Article  Google Scholar 

  16. Morton BR (1965) Modeling fire plumes. Symp (Int) Combust 10(1):973–982. https://doi.org/10.1016/S0082-0784(65)80240-5

    Article  Google Scholar 

  17. Ricou FP, Spalding DB (1961) Measurements of entrainment by axisymmetrical turbulent jets. J Fluid Mech 11(1):21–32. https://doi.org/10.1017/s0022112061000834

    Article  Google Scholar 

  18. FannelØp TK, Webber DM (2003) On buoyant plumes rising from area sources in a calm environment. J Fluid Mech 497:319–334. https://doi.org/10.1017/S0022112003006669

    Article  Google Scholar 

  19. Carlotti P, Hunt GR (2005) Analytical solutions for turbulent non-Boussinesq plumes. J Fluid Mech 538(1):343–359. https://doi.org/10.1017/S0022112005005379

    Article  Google Scholar 

  20. Hunt GR, Kaye NG (2001) Virtual origin correction for lazy turbulent plumes. J Fluid Mech 435:377–396. https://doi.org/10.1017/s0022112001003871

    Article  Google Scholar 

  21. Van Den Bremer TS, Hunt GR (2010) Universal solutions for Boussinesq and non-Boussinesq plumes. J Fluid Mech 644:165–192. https://doi.org/10.1017/s0022112009992199

    Article  Google Scholar 

  22. Heiselberg PK, Svidt K, Nielsen PV (2000) Windows: engineering IE, measurements of air flow capacity. Aalborg University, Aalborg, pp 749–754

    Google Scholar 

  23. Chu CR, Chiu YH, Wang YW (2010) An experimental study of wind-driven cross ventilation in partitioned buildings. Energy Build 42(5):667–673. https://doi.org/10.1016/j.enbuild.2009.11.004

    Article  Google Scholar 

  24. Karava P, Stathopoulos T, Athienitis AK (2004) Wind driven flow through openings: a review of discharge coefficients. Int J Vent 3(3):255–266. https://doi.org/10.1080/14733315.2004.11683920

    Article  Google Scholar 

  25. Liao QH, Guan YL, Wang QN (2014) Research of window’s discharge coefficient in the natural ventilation room. Appl Mech Mater 525:420–426. https://doi.org/10.4028/www.scientific.net/AMM.525.420

    Article  Google Scholar 

  26. Hunt GR, Holford JM (2000) The discharge coefficient: experimental measurement of a dependence on density contrast. In: Proc 21st Intl AIVC Conf, 26–29 September 2000, Hague, Netherlands

  27. Holford JM, Hunt GR (2001) The dependence of the discharge coefficient on density contrast: experimental measurements. In: 14th Australasian fluid mechanics conference, 10–14 December 2001, Adeliade University, Adeliade, Australia

  28. Hunt GR, Linden PF, Mundt E, Malmstrom TG (1998) Time-dependent displacement ventilation caused by variations in internal heat gains: application to a lecture theatre. In: Proc ROOMVENT '98, the 6th Intl Conf on Air Distribution in Rooms, 20 February 1998, Stockholm, Sweden

  29. Kaye NB, Hunt GR (2004) Time-dependent flows in an emptying filling box. J Fluid Mech 520:135–156. https://doi.org/10.1017/s0022112004001156

    Article  Google Scholar 

  30. Vauquelin O, Koutaiba EM, Blanchard E, Fromy P (2017) The discharge plume parameter Γd and its implications for an emptying-filling box. J Fluid Mech 817:171–182. https://doi.org/10.1017/jfm.2017.130

    Article  Google Scholar 

  31. Turner JS (1973) Buoyancy effects in fluids. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511608827

    Book  Google Scholar 

  32. Morton BR (1959) Forced plumes. Int J Air Pollut 1(3):184–197. https://doi.org/10.1017/S002211205900012X

    Article  Google Scholar 

  33. Morton BR, Middleton J (1973) Scale diagrams for forced plumes. J Fluid Mech 58(1):165–176. https://doi.org/10.1017/S002211207300220X

    Article  Google Scholar 

  34. Hunt GR, Kaye NB (2005) Lazy plumes. J Fluid Mech 533:329–338. https://doi.org/10.1017/S002211200500457X

    Article  Google Scholar 

  35. Dalziel SB (1993) Rayleigh-Taylor instability: experiments with image analysis. Dyn Atmos Oceans 20(1–2):127–153. https://doi.org/10.1016/0377-0265(93)90051-8

    Article  Google Scholar 

  36. Liu Y, Li X, Huang L, Liu Z (2021) Determination of the entrainment coefficient of a pure plume using the salt-bath technique. Environ Fluid Mech 21(6):1303–1332. https://doi.org/10.1007/s10652-021-09824-3

    Article  Google Scholar 

  37. Vauquelin O (2015) Oscillatory behaviour in an emptying-filling box. J Fluid Mech 781:712–726. https://doi.org/10.1017/jfm.2015.518

    Article  Google Scholar 

  38. Etheridge DW, Sandberg M (1996) Building ventilation: theory and measurement. Wiley

    Google Scholar 

  39. Barnett SJ (1992) The dynamics of buoyant releases in confined spaces. PhD thesis. University of Cambridge, UK https://doi.org/10.17863/CAM.16149

  40. Kaye NB, Hunt GR (2007) Overturning in a filling box. J Fluid Mech 576:297–323. https://doi.org/10.1017/S0022112006004435

    Article  Google Scholar 

  41. Manins PC (1979) Turbulent buoyant convection from a source in a confined region. J Fluid Mech 91(4):765–781. https://doi.org/10.1017/S0022112079000434

    Article  Google Scholar 

  42. Wong ABD, Griffiths RW, Hughes GO (2001) Shear layers driven by turbulent plumes. J Fluid Mech 434:209–241. https://doi.org/10.1017/S002211200100355X

    Article  Google Scholar 

  43. Michaux G, Vauquelin O (2008) Solutions for turbulent buoyant plumes rising from circular sources. Phys Fluids 20(6):29. https://doi.org/10.1063/1.2926758

    Article  Google Scholar 

Download references

Funding

This research was supported by the National Natural Science Funds of China (No. 52068031) and Natural Science Foundation of Jiangxi Province of China (No. 20202BABL204062).

Author information

Authors and Affiliations

  1. School of Civil and Surveying & Mapping Engineering, Jiangxi University of Science and Technology, Ganzhou, China

    Yang Liu, Zhongwei Huang & Siyi Hou

Authors
  1. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhongwei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siyi Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Yang Liu. The first draft of the manuscript was written by Yang Liu and the figures and tables are prepared by Zhongwei Huang and Siyi Hou. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yang Liu.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Y., Huang, Z. & Hou, S. Determination of discharge coefficient in non-Boussinesq cases and application in emptying–filling boxes. Environ Fluid Mech (2023). https://doi.org/10.1007/s10652-023-09938-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10652-023-09938-w

Keywords

Search

Navigation