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Experimental and numerical investigation of turbulent convection in a rotating cylinder

Published online by Cambridge University Press:  23 December 2009

R. P. J. KUNNEN ,
B. J. GEURTS  and
H. J. H. CLERCX
R. P. J. KUNNEN*
Affiliation:
Fluid Dynamics Laboratory, Department of Physics, International Collaboration for Turbulence Research (ICTR) & J. M. Burgers Centre for Fluid Dynamics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
B. J. GEURTS
Affiliation:
Fluid Dynamics Laboratory, Department of Physics, International Collaboration for Turbulence Research (ICTR) & J. M. Burgers Centre for Fluid Dynamics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands Department of Applied Mathematics, International Collaboration for Turbulence Research (ICTR) & J. M. Burgers Centre for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
H. J. H. CLERCX
Affiliation:
Fluid Dynamics Laboratory, Department of Physics, International Collaboration for Turbulence Research (ICTR) & J. M. Burgers Centre for Fluid Dynamics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands Department of Applied Mathematics, International Collaboration for Turbulence Research (ICTR) & J. M. Burgers Centre for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
*
†Present address: Institute of Aerodynamics, RWTH Aachen University, Wüllnerstraße 5a, 52062 Aachen, Germany. Email address for correspondence: [email protected]
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Abstract

The effects of an axial rotation on the turbulent convective flow because of an adverse temperature gradient in a water-filled upright cylindrical vessel are investigated. Both direct numerical simulations and experiments applying stereoscopic particle image velocimetry are performed. The focus is on the gathering of turbulence statistics that describe the effects of rotation on turbulent Rayleigh–Bénard convection. Rotation is an important addition, which is relevant in many geophysical and astrophysical flow phenomena.

A constant Rayleigh number (dimensionless strength of the destabilizing temperature gradient) Ra = 109 and Prandtl number (describing the diffusive fluid properties) σ = 6.4 are applied. The rotation rate, given by the convective Rossby number Ro (ratio of buoyancy and Coriolis force), takes values in the range 0.045 ≤ Ro ≤ ∞, i.e. between rotation-dominated flow and zero rotation. Generally, rotation attenuates the intensity of the turbulence and promotes the formation of slender vertical tube-like vortices rather than the global circulation cell observed without rotation. Above Ro ≈ 3 there is hardly any effect of the rotation on the flow. The root-mean-square (r.m.s.) values of vertical velocity and vertical vorticity show an increase when Ro is lowered below Ro ≈ 3, which may be an indication of the activation of the Ekman pumping mechanism in the boundary layers at the bottom and top plates. The r.m.s. fluctuations of horizontal and vertical velocity, in both experiment and simulation, decrease with decreasing Ro and show an approximate power-law behaviour of the shape Ro0.2 in the range 0.1 ≲ Ro ≲ 2. In the same Ro range the temperature r.m.s. fluctuations show an opposite trend, with an approximate negative power-law exponent Ro−0.32. In this Rossby number range the r.m.s. vorticity has hardly any dependence on Ro, apart from an increase close to the plates for Ro approaching 0.1. Below Ro ≈ 0.1 there is strong damping of turbulence by rotation, as the r.m.s. velocities and vorticities as well as the turbulent heat transfer are strongly diminished. The active Ekman boundary layers near the bottom and top plates cause a bias towards cyclonic vorticity in the flow, as is shown with probability density functions of vorticity. Rotation induces a correlation between vertical vorticity and vertical velocity close to the top and bottom plates: near the top plate downward velocity is correlated with positive/cyclonic vorticity and vice versa (close to the bottom plate upward velocity is correlated with positive vorticity), pointing to the vortical plumes. In contrast with the well-mixed mean isothermal bulk of non-rotating convection, rotation causes a mean bulk temperature gradient. The viscous boundary layers scale as the theoretical Ekman and Stewartson layers with rotation, while the thermal boundary layer is unaffected by rotation. Rotation enhances differences in local anisotropy, quantified using the invariants of the anisotropy tensor: under rotation there is strong turbulence anisotropy in the centre, while near the plates a near-isotropic state is found.

JFM classification

Convection: Convection Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 642 , 10 January 2010 , pp. 445 - 476
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.CrossRefGoogle Scholar
Ashkenazi, S. & Steinberg, V. 1999 Spectra and statistics of velocity and temperature fluctuations in turbulent convection. Phys. Rev. Lett. 83, 4760–4763.CrossRefGoogle Scholar
Balachandar, S. & Sirovich, L. 1991 Probability distribution functions in turbulent convection. Phys. Fluids A 3, 919–927.CrossRefGoogle Scholar
van Bokhoven, L. J. A. 2007 Experiments on rapidly rotating turbulent flows. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands.Google Scholar
van Bokhoven, L. J. A., Clercx, H. J. H., van Heijst, G. J. F. & Trieling, R. R. 2009 Experiments on rapidly rotating turbulent flows. Phys. Fluids 21, 096601.CrossRefGoogle Scholar
Boubnov, B. M. & Golitsyn, G. S. 1986 Experimental study of convective structures in rotating fluids. J. Fluid Mech. 167, 503–531.CrossRefGoogle Scholar
Boubnov, B. M. & Golitsyn, G. S. 1990 Temperature and velocity field regimes of convective motions in a rotating plane fluid layer. J. Fluid Mech. 219, 215–239.CrossRefGoogle Scholar
Boubnov, B. M. & Golitsyn, G. S. 1995 Convection in Rotating Fluids. Kluwer Academic.CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2006 Effect of the Earth's Coriolis force on the large-scale circulation of turbulent Rayleigh–Bénard convection. Phys. Fluids 18, 125108.CrossRefGoogle Scholar
Brown, E. & Ahlers, G. 2007 Temperature gradients, and search for non-Boussinesq effects, in the interior of turbulent Rayleigh–Bénard convection. Europhys. Lett. 80, 14001.CrossRefGoogle Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Progr. Phys. 41, 1929–1967.CrossRefGoogle Scholar
Busse, F. H. 1994 Convection driven zonal flows and vortices in the major planets. Chaos 4, 123–134.CrossRefGoogle ScholarPubMed
Busse, F. H. & Carrigan, C. R. 1976 Laboratory simulation of thermal convection in rotating planets and stars. Science 191, 81–83.CrossRefGoogle ScholarPubMed
Camussi, R. & Verzicco, R. 2004 Temporal statistics in high Rayleigh number convective turbulence. Eur. J. Mech. B 23, 427–442.CrossRefGoogle Scholar
Castaing, B., Gagne, Y. & Hopfinger, E. J. 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46, 177–200.CrossRefGoogle Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh–Bénard convection. J. Fluid Mech. 204, 1–30.CrossRefGoogle Scholar
Chandrasekhar, S. 1953 The instability of a layer of fluid heated from below and subject to Coriolis forces. Proc. R. Soc. Lond. A 217, 306–327.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Choi, K.-S. & Lumley, J. L. 2001 The return to isotropy of homogeneous turbulence. J. Fluid Mech. 436, 59–84.CrossRefGoogle Scholar
Ecke, R. E. & Liu, Y. 1998 Traveling-wave and vortex states in rotating Rayleigh–Bénard convection. Intl J. Engng Sci. 36, 1471–1480.CrossRefGoogle Scholar
Ekman, V. W. 1905 On the influence of the Earth's rotation on ocean-currents. Arch. Math. Astron. Phys. 2, 1–52.Google Scholar
Fernando, H. J. S., Chen, R.-R. & Boyer, D. L. 1991 Effects of rotation on convective turbulence. J. Fluid Mech. 228, 513–547.Google Scholar
Gascard, J.-C., Watson, A. J., Messias, M.-J., Olsson, K. A., Johannessen, T. & Simonsen, K. 2002 Long-lived vortices as a mode of deep ventilation in the Greenland Sea. Nature (Lond.) 416, 525–527.CrossRefGoogle ScholarPubMed
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Glazier, J. A., Segawa, T., Naert, A. & Sano, M. 1999 Evidence against ‘ultrahard’ thermal turbulence at very high Rayleigh numbers. Nature (Lond.) 398, 307–310.CrossRefGoogle Scholar
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1993 Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers. J. Fluid Mech. 248, 583–604.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Hart, J. E. 2000 On the influence of centrifugal buoyancy on rotating convection. J. Fluid Mech. 403, 133–151.CrossRefGoogle Scholar
Hart, J. E., Kittelman, S. & Ohlsen, D. R. 2002 Mean flow precession and temperature probability density functions in turbulent rotating convection. Phys. Fluids 14, 955–962.CrossRefGoogle Scholar
Hart, J. E. & Ohlsen, D. R. 1999 On the thermal offset in turbulent rotating convection. Phys. Fluids 11, 2101–2107.CrossRefGoogle Scholar
Heslot, F., Castaing, B. & Libchaber, A. 1987 Transition to turbulence in helium gas. Phys. Rev. A 36, 5870–5873.CrossRefGoogle ScholarPubMed
Jones, C. A. 2000 Convection-driven geodynamo models. Phil. Trans. R. Soc. Lond. A 358, 873–897.CrossRefGoogle Scholar
Julien, K., Legg, S., McWilliams, J. & Werne, J. 1996 a Hard turbulence in rotating Rayleigh–Bénard convection. Phys. Rev. E 53, R5557–R5560.CrossRefGoogle ScholarPubMed
Julien, K., Legg, S., McWilliams, J. & Werne, J. 1996 b Rapidly rotating turbulent Rayleigh–Bénard convection. J. Fluid Mech. 322, 243–273.CrossRefGoogle Scholar
Julien, K., Legg, S., McWilliams, J. & Werne, J. 1999 Plumes in rotating convection. Part 1. Ensemble statistics and dynamical balances. J. Fluid Mech. 391, 151–187.CrossRefGoogle Scholar
Kerr, R. M. 1996 Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139–179.CrossRefGoogle Scholar
Kerr, R. M. & Herring, J. R. 2000 Prandtl number dependence of Nusselt number in direct numerical simulations. J. Fluid Mech. 419, 325–344.CrossRefGoogle Scholar
van de Konijnenberg, J. A., Andersson, H. I., Billdal, J. T. & van Heijst, G. J. F. 1994 Spin-up in a rectangular tank with low angular velocity. Phys. Fluids 6, 1168–1176.CrossRefGoogle Scholar
Krishnamurti, R. & Howard, L. N. 1981 Large-scale flow generation in turbulent convection. Proc. Natl. Acad. Sci. USA 78, 1981–1985.CrossRefGoogle ScholarPubMed
Kunnen, R. P. J. 2008 Turbulent rotating convection. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands.Google Scholar
Kunnen, R. P. J., Clercx, H. J. H. & Geurts, B. J. 2006 Heat flux intensification by vortical flow localization in rotating convection. Phys. Rev. E 74, 056306.CrossRefGoogle ScholarPubMed
Kunnen, R. P. J., Clercx, H. J. H. & Geurts, B. J. 2008 a Breakdown of large-scale circulation in turbulent rotating convection. Europhys. Lett. 84, 24001.CrossRefGoogle Scholar
Kunnen, R. P. J., Clercx, H. J. H. & Geurts, B. J. 2008 b Enhanced vertical inhomogeneity in turbulent rotating convection. Phys. Rev. Lett. 101, 174501.CrossRefGoogle ScholarPubMed
Kunnen, R. P. J., Clercx, H. J. H., Geurts, B. J., van Bokhoven, L. J. A., Akkermans, R. A. D. & Verzicco, R. 2008 c Numerical and experimental investigation of structure function scaling in turbulent Rayleigh–Bénard convection. Phys. Rev. E 77, 016302.CrossRefGoogle ScholarPubMed
Kunnen, R. P. J., Geurts, B. J. & Clercx, H. J. H. 2009 Turbulence statistics and energy budget in rotating Rayleigh–Bénard convection. Eur. J. Mech. B 28, 578–589.CrossRefGoogle Scholar
Küppers, G. & Lortz, D. 1969 Transition from laminar convection to thermal turbulence in a rotating fluid layer. J. Fluid Mech. 35, 609–620.CrossRefGoogle Scholar
Legg, S., Julien, K., McWilliams, J. & Werne, J. 2001 Vertical transport by convection plumes: modification by rotation. Phys. Chem. Earth B 26, 259–262.CrossRefGoogle Scholar
Lide, D. R. (Ed.) 2007–2008 CRC Handbook of Chemistry and Physics, 88th edn. CRC Press/Taylor and Francis.Google Scholar
Liu, Y. & Ecke, R. E. 1997 Heat transport scaling in turbulent Rayleigh–Bénard convection: effects of rotation and Prandtl number. Phys. Rev. Lett. 79, 2257–2260.CrossRefGoogle Scholar
Lucas, P. G. J., Pfotenhauer, J. M. & Donnelly, R. J. 1983 Stability and heat transfer of rotating cryogens. Part 1. Influence of rotation on the onset of convection in liquid 4He. J. Fluid Mech. 129, 251–264.CrossRefGoogle Scholar
Lumley, J. L. 1978 Computational modelling of turbulent flows. Adv. Appl. Mech. 18, 123–175.CrossRefGoogle Scholar
Lumley, J. L. & Newman, G. R. 1977 The return to isotropy of homogeneous turbulence. J. Fluid Mech. 82, 161–178.CrossRefGoogle Scholar
Marshall, J. & Schott, F. 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 1–64.CrossRefGoogle Scholar
Miesch, M. S. 2000 The coupling of solar convection and rotation. Solar Phys. 192, 59–89.CrossRefGoogle Scholar
Moeng, C.-H. & Rotunno, R. 1990 Vertical-velocity skewness in the buoyancy-driven boundary layer. J. Atmos. Sci. 47, 1149–1162.2.0.CO;2>CrossRefGoogle Scholar
Nakagawa, Y. & Frenzen, P. 1955 A theoretical and experimental study of cellular convection in rotating fluids. Tellus 7, 1–21.CrossRefGoogle Scholar
Oresta, P., Stringano, G. & Verzicco, R. 2007 Transitional regimes and rotation effects in Rayleigh–Bénard convection in a slender cylindrical cell. Eur. J. Mech. B 26, 1–14.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.CrossRefGoogle Scholar
Pfotenhauer, J. M., Lucas, P. G. J. & Donnelly, R. J. 1984 Stability and heat transfer of rotating cryogens. Part 2. Effects of rotation on heat-transfer properties of convection in liquid 4He. J. Fluid Mech. 145, 239–252.CrossRefGoogle Scholar
Pfotenhauer, J. M., Niemela, J. J. & Donnelly, R. J. 1987 Stability and heat transfer of rotating cryogens. Part 3. Effects of finite cylindrical geometry and rotation on the onset of convection. J. Fluid Mech. 175, 85–96.CrossRefGoogle Scholar
Prandtl, L. 1932 Meteorologische Anwendungen der Strömungslehre. Beitr. Phys. Atmos. 19, 188–202.Google Scholar
Prasad, A. K. 2000 Stereoscopic particle image velocimetry. Exp. Fluids 29, 103–116.CrossRefGoogle Scholar
Qiu, X.-L., Shang, X.-D., Tong, P. & Xia, K.-Q. 2004 Velocity oscillations in turbulent Rayleigh–Bénard convection. Phys. Fluids 16, 412–423.CrossRefGoogle Scholar
Qiu, X.-L. & Tong, P. 2001 Large-scale velocity structures in turbulent thermal convection. Phys. Rev. E 64, 036304.CrossRefGoogle ScholarPubMed
Raasch, S. & Etling, E. 1991 Numerical simulation of rotating turbulent thermal convection. Beitr. Phys. Atmos. 64, 185–199.Google Scholar
Raffel, M., Willert, C. & Kompenhans, J. 1998 Particle Image Velocimetry. Springer.CrossRefGoogle Scholar
Rogers, M. H. & Lance, G. N. 1960 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. J. Fluid Mech. 7, 617–631.CrossRefGoogle Scholar
Rossby, H. T. 1969 A study of Bénard convection with and without rotation. J. Fluid Mech. 36, 309–335.CrossRefGoogle Scholar
Sakai, S. 1997 The horizontal scale of rotating convection in the geostrophic regime. J. Fluid Mech. 333, 85–95.CrossRefGoogle Scholar
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42, 3650–3653.CrossRefGoogle ScholarPubMed
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137–168.CrossRefGoogle Scholar
Simonsen, A. J. & Krogstad, P.-Å. 2005 Turbulent stress invariant analysis: clarification of existing terminology. Phys. Fluids 17, 088103.CrossRefGoogle Scholar
Solomon, T. H. & Gollub, J. P. 1990 Sheared boundary layers in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 64, 2382–2385.CrossRefGoogle ScholarPubMed
Sprague, M., Julien, K., Knobloch, E. & Werne, J. 2006 Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech. 551, 141–174.CrossRefGoogle Scholar
Stevens, R. J. A. M., Zhong, J.-Q., Clercx, H. J. H., Ahlers, G. & Lohse, D. 2009 Transitions between turbulent states in rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 102, 044502.Google Scholar
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 17–26.CrossRefGoogle Scholar
Tilgner, A., Belmonte, A. & Libchaber, A. 1993 Temperature and velocity profiles of turbulent convection in water. Phys. Rev. E 47, R2253–R2256.CrossRefGoogle ScholarPubMed
Verdoold, J., van Reeuwijk, M., Tummers, M. J., Jonker, H. J. J. & Hanjalić, K. 2008 Spectral analysis of boundary layers in Rayleigh–Bénard convection. Phys. Rev. E 77, 016303.CrossRefGoogle ScholarPubMed
Verzicco, R. & Camussi, R. 1997 Transitional regimes of low-Prandtl thermal convection in a cylindrical cell. Phys. Fluids 9, 1287–1295.CrossRefGoogle Scholar
Verzicco, R. & Camussi, R. 1999 Prandtl number effects in convective turbulence. J. Fluid Mech. 383, 55–73.CrossRefGoogle Scholar
Verzicco, R. & Camussi, R. 2003 Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 19–49.CrossRefGoogle Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402–413.CrossRefGoogle Scholar
Vorobieff, P. & Ecke, R. E. 1998 a Transient states during spin-up of a Rayleigh–Bénard cell. Phys. Fluids 10, 2525–2538.CrossRefGoogle Scholar
Vorobieff, P. & Ecke, R. E. 1998 b Vortex structure in rotating Rayleigh–Bénard convection. Physica D 123, 153–160.CrossRefGoogle Scholar
Vorobieff, P. & Ecke, R. E. 2002 Turbulent rotating convection: an experimental study. J. Fluid Mech. 458, 191–218.CrossRefGoogle Scholar
Xi, H.-D., Zhou, Q. & Xia, K.-Q. 2006 Azimuthal motion of the mean wind in turbulent thermal convection. Phys. Rev. E 73, 056312.CrossRefGoogle ScholarPubMed
Zhong, F., Ecke, R. E. & Steinberg, V. 1993 Rotating Rayleigh–Bénard convection: asymmetric modes and vortex states. J. Fluid Mech. 249, 135–159.CrossRefGoogle Scholar
Zhong, J.-Q., Stevens, R. J. A. M., Clercx, H. J. H., Verzicco, R., Lohse, D. & Ahlers, G. 2009 Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 102, 044502.CrossRefGoogle ScholarPubMed
Zhou, Q., Sun, C. & Xia, K.-Q. 2007 Morphological evolution of thermal plumes in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 074501.CrossRefGoogle ScholarPubMed