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Turbulent buoyant convection from a maintained source of buoyancy in a narrow vertical tank

  • Daan D. J. A. van Sommeren (a1) (a2), C. P. Caulfield (a1) (a2) and Andrew W. Woods (a1)

Abstract

We describe new experiments to examine the buoyancy-induced mixing which results from the injection of a small constant volume flux of fluid of density at the top of a long narrow vertical tank with square cross-section which is filled with fluid of density . The injected fluid vigorously mixes with the less dense fluid which initially occupies the tank, such that a dense mixed region of turbulent fluid propagates downwards during the initial mixing phase of the experiment. For an ideal source of constant buoyancy flux , we show that the height of the mixed region grows as and that the horizontally averaged reduced gravity at the top of tank increases as , where is the width of the tank. Once the mixed region reaches the bottom of the tank, the turbulent mixing continues in an intermediate mixing phase, and we demonstrate that the reduced gravity at each height increases approximately linearly with time. This suggests that the buoyancy flux is uniformly distributed over the full height of the tank. The overall density gradient between the top and bottom of the mixed region is hence time-independent for both the mixing phases before and after the mixed region has reached the bottom of the tank. Our results are consistent with previous models developed for the mixing of an unstable density gradient in a confined geometry, based on Prandtl’s mixing length theory, which suggest that the turbulent diffusion coefficient and the magnitude of the local turbulent flux are given by the nonlinear relations and , respectively. The constant relates the width of the tank to the characteristic mixing length of the turbulent eddies. Since the mixed region is characterized by a time-independent overall density gradient, we also tested the predictions based on a linear model in which the turbulent diffusion coefficient is approximated by a constant . We solve the corresponding nonlinear and linear turbulent diffusion equations for both mixing phases, and show a good agreement with experimental profiles measured by a dye attenuation technique, in particular for the solutions based on the nonlinear model.

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Corresponding author

Email address for correspondence: cpc12@cam.ac.uk

References

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1. Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
2. Baines, W. D. & Turner, J. S. 1969 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 37, 5180.
3. Baird, M. H. I., Aravamudan, K., Rao, N. V. R., Chadam, J. & Peirce, A. P. 1992 Unsteady axial mixing by natural convection in a vertical column. AIChE J. 38, 18251834.
4. Barnett, S. 1991 The dynamics of buoyant releases in confined spaces. PhD thesis, DAMTP University of Cambridge.
5. Caulfield, C. P. & Woods, A. W. 2002 The mixing in a room by a localized finite-mass-flux source of buoyancy. J. Fluid Mech. 471, 3350.
6. Cenedese, C. & Dalziel, S. B. 1998 Concentration and depth fields determined by the light transmitted through a dyed solution. In Proceedings of the 8th International Symposium on Flow Visualization (ed. Carlomango, G. M. & Grant, I. ), ISBN 0953399109, Paper 061.
7. Dalziel, S. B., Patterson, M. D., Caulfield, C. P. & Coomaraswamy, I. A. 2008 Mixing efficiency in high-aspect-ratio Rayleigh–Taylor experiments. Phys. Fluids 20, 065106.
8. Debacq, M., Fanguet, V., Hulin, J. P., Salin, D. & Perrin, B. 2001 Self-similar concentration profiles in buoyant mixing of miscible fluids in a vertical tube. Phys. Fluids 13, 30973100.
9. Dimonte, G., Youngs, D. L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M. J., Ramaprabhu, P., Calder, A. C., Fryxell, B., Biello, J., Dursi, L., MacNeice, P., Olson, K., Ricker, P., Rosner, R., Timmes, F., Tufo, H., Young, Y. N. & Zingale, M. 2004 A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: The Alpha-Group Collaboration. Phys. Fluids 16, 16681693.
10. Holmes, T. L., Karr, A. E. & Baird, M. H. I. 1991 Effect of unfavourable continuous phase density gradient on axial mixing. AIChE J. 37, 360366.
11. Karmis, M. 2001 Mine Health and Safety Management. SME.
12. Lide, D. R.  (Ed.) 2001 CRC Handbook of Chemistry and Physics, 82nd edn. CRC.
13. Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. R. Soc. Lond. Proc. Ser. A 234, 123.
14. Ryan, M. P. 1994 Magmatic Systems (International Geophysics Series). Academic.
15. Thakore, S. B. & Bhatt, B. I. 2007 Introduction to Process Engineering and Design. Tata McGraw-Hill.
16. Weiss, S. & Ahlers, G. 2011 Turbulent Rayleigh–Bernard convection in a cylindrical container with aspect ratio and Prandtl number . J. Fluid Mech. 676, 540.
17. Woods, A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42, 391412.
18. Zukoski, E. E. 1995 Review of flows driven by natural convection in adiabatic shafts. Tech. Rep. NIST-GCR-95-679. U.S. Department of Commerce, National Institute of Standard and Technology, Building and Fire Research Laboratory.
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Turbulent buoyant convection from a maintained source of buoyancy in a narrow vertical tank

  • Daan D. J. A. van Sommeren (a1) (a2), C. P. Caulfield (a1) (a2) and Andrew W. Woods (a1)

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