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Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe

Published online by Cambridge University Press:  12 February 2009

MURALI R. CHOLEMARI
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016, India
JAYWANT H. ARAKERI*
Affiliation:
Department of Mechanical Engineering, Indian Institute of ScienceBangalore 560012, Karnataka, India
*
Email address for correspondence: jaywant@mecheng.iisc.ernet.in

Abstract

We report an experimental study of a new type of turbulent flow that is driven purely by buoyancy. The flow is due to an unstable density difference, created using brine and water, across the ends of a long (length/diameter=9) vertical pipe. The Schmidt number Sc is 670, and the Rayleigh number (Ra) based on the density gradient and diameter is about 108. Under these conditions the convection is turbulent, and the time-averaged velocity at any point is ‘zero’. The Reynolds number based on the Taylor microscale, Reλ, is about 65. The pipe is long enough for there to be an axially homogeneous region, with a linear density gradient, about 6–7 diameters long in the midlength of the pipe. In the absence of a mean flow and, therefore, mean shear, turbulence is sustained just by buoyancy. The flow can be thus considered to be an axially homogeneous turbulent natural convection driven by a constant (unstable) density gradient. We characterize the flow using flow visualization and particle image velocimetry (PIV). Measurements show that the mean velocities and the Reynolds shear stresses are zero across the cross-section; the root mean squared (r.m.s.) of the vertical velocity is larger than those of the lateral velocities (by about one and half times at the pipe axis). We identify some features of the turbulent flow using velocity correlation maps and the probability density functions of velocities and velocity differences. The flow away from the wall, affected mainly by buoyancy, consists of vertically moving fluid masses continually colliding and interacting, while the flow near the wall appears similar to that in wall-bound shear-free turbulence. The turbulence is anisotropic, with the anisotropy increasing to large values as the wall is approached. A mixing length model with the diameter of the pipe as the length scale predicts well the scalings for velocity fluctuations and the flux. This model implies that the Nusselt number would scale as Ra1/2Sc1/2, and the Reynolds number would scale as Ra1/2Sc−1/2. The velocity and the flux measurements appear to be consistent with the Ra1/2 scaling, although it must be pointed out that the Rayleigh number range was less than 10. The Schmidt number was not varied to check the Sc scaling. The fluxes and the Reynolds numbers obtained in the present configuration are much higher compared to what would be obtained in Rayleigh–Bénard (R–B) convection for similar density differences.

Type
Papers
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Amati, G., Koal, K., Massaioli, F., Sreenivasan, K. R. & Verzicco, R. 2005 Turbulent thermal convection at high Rayleigh numbers for a Boussinesq fluid of constant Prandtl number. Phys. Fluids 17 (121701), 14.Google Scholar
Arakeri, J. H., Avila, F. E., Dada, J. M. & Tovar, R. O. 2000 Convection in a long vertical tube due to unstable stratification-a new type of turbulent flow? Curr. Sci. 79 (6), 859866.Google Scholar
Arakeri, J. H. & Cholemari, M. R. 2002 Fully developed buoyancy driven turbulence in a tube. In Proc. Ninth Asian Cong. Fluid Mech. (ed. Shirani, E. & Pishevar, A. R.). Isfahan University of Technology, Iran.Google Scholar
Aronson, D. Johansson, A. V. & Löfdahl, L. 1997 Shear free turbulence near a wall. J. Fluid Mech. 338, 363385.Google Scholar
Batchelor, G. K., Canuto, V. M. & Chasnov, J. R. 1991 Homogeneous buoyancy-generated turbulence. J. Fluid Mech. 235, 349378.Google Scholar
Calzavarini, E., Lohse, D., Toschi, F. & Tripiccione, R. 2005 Rayleigh and prandtl number scaling in the bulk of rayleigh-bnard turbulence. Phys. Fluids 17 (055107), 17.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover Publications.Google Scholar
Cholemari, M. R. 2004 Buoyancy driven turbulence in a vertical pipe. PhD thesis. Department of mechanical engineering, Indian Institute of Science, Bangalore, Karnataka, India.Google Scholar
Cholemari, M. R. 2007 Modelling and correction of peak-locking in digital PIV. Exp. Fluids 42 (6), 913922.Google Scholar
Cholemari, M. R. & Arakeri, J. H. 2005 Experiments and a model of turbulent exchange flow in a vertical pipe. Int. J. Heat Mass Transfer 48, 44674473.Google Scholar
Cholemari, M. R. & Arakeri, J. H. 2006 A model relating eulerian spatial and temporal velocity correlations. J. Fluid Mech. 551, 1929.Google Scholar
Christensen, K. T. 2004 On the influence of peaklocking errors on turbulence statistics compared from PIV ensembles. Exp. Fluids 36 (3), 484497.Google Scholar
Constantin, P. & Doering, C. R. 1999 Infinite prandtl number convection. J. Stat. Phys. 94 (1–2), 159172.Google Scholar
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, 3097.Google Scholar
Debacq, M., Hulin, J.-P. & Salin, D. 2003 Buoyant mixing of miscible fluids of varying viscosities in vertical tubes. Phys. Fluids 15 (12), 38463855.Google Scholar
Doering, C. R., Otto, F. & Reznikoff, M. G. 2006 Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Benard convection. J. Fluid Mech. 560, 229241.Google Scholar
Epstein, M. 1988 Buoyancy driven exchange flow through small openings in horizontal partitions. J. Heat Transfer 110, 885893.Google Scholar
Fitzjarrald, D. E. 1976 An experimental study of turbulent convection in air. J. Fluid Mech. 73 (pt. 4), 693719.Google Scholar
Gardener, G. C. 1977 Motion of miscible and immiscible fluids in closed horizontal and vertical ducts. Int. J. Multiphase Flow 3, 305318.Google Scholar
Gibert, M., Pabiou, H., Chilla, F. & Castaing, B. 2006 High-Rayleigh-number convection in a vertical channel. Phys. Rev. Lett. 96, 084501-1084501-4.Google Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.Google Scholar
Hunt, J. C. R. & Graham, J. M. R. 1978 Free stream turbulence near plane boundaries. J. Fluid Mech. 84, 209235.Google Scholar
Hyun, B. S., Balachander, R., Yu, K. & Patel, V. C. 2003 Assessment of PIV to measure mean velocity and turbulence in open channel flow. Exp. Fluids 35, 262267.Google Scholar
Keane, R. D. & Adrian, R. J. 1992 Theory of cross-correlation analysis of PIV images. Appl. Sci. Res. 49, 191215.Google Scholar
Kim, J. Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.Google Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary prandtl number. Phys. Fluids 5 (11), 13741389.Google Scholar
Lecordier, B., Demare, D., Vervisch, L. M. J., Réveillon, J. & Trinité, M. 2001 Estimation of the accuracy of PIV treatments for turbulent flow studies by direct numerical simulation of multi-phase flow. Meas. Sci. Technol. 12 (13821391).Google Scholar
Lohse, D. & Toschi, F. 2003 Ultimate state of thermal convection. Phys. Rev. Lett. 90 (3), 034502.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics: Mechanics of Turbulence (vol. 1). The MIT press.Google Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2000 Turbulent convection at high Rayleigh numbers. Nature 404, 837841.Google Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2001 The wind in confined thermal convection. J. Fluid Mech. 449, 169178.Google Scholar
Niemela, J. J. & Sreenivasan, K. R. 2003 Confined turbulent convection. J. Fluid Mech. 481, 355384.Google Scholar
Niemela, J. J. & Sreenivasan, K. R. 2006 Turbulent convection at high Rayleigh numbers and aspect ratio 4. J. Fluid Mech. 557, 411422.Google Scholar
Nikolaenko, A., Brown, E., Funfschilling, D. & Ahlers, G. 2005 Heat transport by turbulent Rayleigh-Bénard convection in cylindrical cells with aspect ratio one and less. J. Fluid Mech. 523, 251260.Google Scholar
Noullez, A., Wallace, G., Lempert, W., Miles, R. B. & Frisch, U. 1997 Transverse velocity increments in turbulent flow using the relief technique. J. Fluid Mech. 339, 287307.Google Scholar
Perot, B. & Moin, P. 1995 Shear free turbulent boundary layers. Part 1. Physical insights to near wall turbulence. J. Fluid Mech. 295, 199227.Google Scholar
Pope, Stephen B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Prasad, A. K. 2000 Particle image velocimetry. Curr. Sci. 79, 5160.Google Scholar
Raffel, M., Willert, C. E. & Kompenhans, J. 1998 Particle Image Velocimetry. Springer.Google Scholar
Roche, P.-E., Castaing, B., Chabaud, B. & Hébral, B. 2001 Observation of the 1/2 power law in Rayleigh–Bénard convection. Phys. Rev. E 63, 045303.Google Scholar
Saarenrinne, P. & Piirto, M. 2000 Turbulent kinetic energy dissipation rate estimation from PIV velocity vector fields. Exp. Fluids (suppl.) 29 (7), S300S307.Google Scholar
Saarenrinne, P., Piirto, M. & Eloranta, H. 2001 Experiences of turbulence measurement with PIV. Meas. Sci. Technol. 12, 19041910.Google Scholar
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137168.Google Scholar
Stringano, G. & Verzicco, R. 2006 Mean flow structure in thermal convection in a cylindrical cell of aspect ratio one half. J. Fluid Mech. 548, 116.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. The MIT press.Google Scholar
Theerthan, S. A. & Arakeri, J. H. 1998 A model for near-wall dynamics in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 373, 221254.Google Scholar
Theerthan, S. A. & Arakeri, J. H. 2000 Planform structure and heat transfer in turbulent free convection over horizontal surfaces. Phys. Fluids 12 (4), 884894.Google Scholar
Thomas, N. H. & Hancock, P. E. 1977 Grid turbulence near a moving wall. J. Fluid Mech. 82, 481496.Google Scholar
Uzkan, T. & Reynolds, W. C. 1967 A shear-free turbulent boundary layer. J. Fluid Mech. 28, 803821.Google Scholar
Verzicco, R. & Sreenivasan, K. R. 2008 Confined thermal convection in a ‘cigar box’ cylindrical cell. Private communication.Google Scholar
Westerweel, J. 1997 Fundamentals of digital particle image velocimetry. Meas. Sci. Technol. 8, 13791392.Google Scholar
Westerweel, J., Dabiri, D. & Gharib, M. 1997 the effect of a descrete window offset on the accuracy of cross-correlation analysis of digital PIV recordings. Exp. Fluids 23, 2028.Google Scholar