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Rotating turbulent thermal convection at very large Rayleigh numbers. (English) Zbl 1461.76248

Summary: We report on turbulent thermal convection experiments in a rotating cylinder with a diameter \((D)\) to height \((H)\) aspect ratio of \(\Gamma =D/H=0.5\). Using nitrogen and pressurised sulphur hexafluoride we cover Rayleigh numbers \((Ra)\) from \(8\times 10^9\) to \(8\times 10^{14}\) at Prandtl numbers \(0.72\lesssim Pr\lesssim 0.94\). For these \(Ra\) we measure the global vertical heat flux (i.e. the Nusselt number \(-Nu\), as well as statistical quantities of local temperature measurements, as a function of the rotation rate, i.e. the inverse Rossby number \(-1/Ro\). In contrast to measurements in fluids with a higher \(Pr\) we do not find a heat transport enhancement, but only a decrease of \(Nu\) with increasing \(1/Ro\). When normalised with \(Nu(0)\) for the non-rotating system, data for all different Ra collapse and, for sufficiently large \(1/Ro\), follow a power law \(Nu/Nu_0\propto (1/Ro)^{-0.43}\). Furthermore, we find that both the heat transport and the temperature field qualitatively change at rotation rates \(1/Ro^*_1=0.8\) and \(1/Ro^*_2=4\). We interpret the first transition at \(1/Ro^*_1\) as change from a large-scale circulation roll to the recently discovered boundary zonal flow (BZF). The second transition at rotation rate \(1/Ro^*_2\) is not associated with a change of the flow morphology, but is rather the rotation rate for which the BZF is at its maximum. For faster rotation the vertical transport of warm and cold fluid near the sidewall within the BZF decreases and hence so does \(Nu\).

MSC:

76F35 Convective turbulence
76U05 General theory of rotating fluids
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[1] Ahlers, G.2000Effect of sidewall conductance on heat-transport measurements for turbulent Rayleigh-Bénard convection. Phys. Rev. E63, 015303.
[2] Ahlers, G., Bodenschatz, E., Funfschilling, D., Grossmann, S., He, X., Lohse, D., Stevens, R.J.A.M. & Verzicco, R.2012aLogarithmic temperature profiles in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett.109, 114501.
[3] Ahlers, G., Bodenschatz, E. & He, X.2014Logarithmic temperature profiles of turbulent Rayleigh-Bénard convection in the classical and ultimate state for a Prandtl number of 0.8. J.Fluid Mech.758, 436-467.
[4] Ahlers, G., Funfschilling, D. & Bodenschatz, E.2009aTransitions in heat transport by turbulent convection at Rayleigh numbers up to \(10^{15}\). New J. Phys.11, 123001. · Zbl 1183.76003
[5] Ahlers, G., Grossmann, S. & Lohse, D.2009bHeat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys.81, 503-538.
[6] Ahlers, G., He, X., Funfschilling, D. & Bodenschatz, E.2012bHeat transport by turbulent Rayleigh-Bénard convection for \(Pr\approx 0.8\) and \(3\times 10^{12}\lesssim Ra\lesssim 10^{15}\): aspect ratio \(\gamma =0.50\). New J. Phys.14 (10), 103012.
[7] Bajaj, K.M.S., Ahlers, G. & Pesch, W.2002Rayleigh-Benard convection-with rotation at small Prandtl numbers. Phys. Rev. E65, 056309.
[8] Bénard, H.1900Les tourbillons cellularies dans une nappe liquide. Rev. Gen. Sci. Pure Appl.11, 1261-1309.
[9] Boussinesq, J.1903Theorie analytique de la chaleur, vol. 2. Gauthier-Villars.
[10] Brown, E. & Ahlers, G.2007Large-scale circulation model of turbulent Rayleigh-Bénard convection. Phys. Rev. Lett.98, 134501. · Zbl 1183.76011
[11] Buell, J.C. & Catton, I.1983aThe effect of wall conduction on the stability of a fluid in a right circular cylinder heated from below. Trans. ASME: J. Heat Transfer105 (2), 255-260.
[12] Buell, J.C. & Catton, I.1983bEffects of rotation on the stability of a bounded cylindrical layer of fluid heated from below. Phys. Fluids26, 892-896. · Zbl 0511.76076
[13] Buffett, B.A.2000Earth’s core and the geodynamo. Science288 (5473), 2007-2012.
[14] Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.Z., Zaleski, S. & Zanetti, G.1989Scaling of hard thermal turbulence in Rayleigh-Bénard convection. J.Fluid Mech.204, 1-30.
[15] Chandrasekhar, S.1981Hydrodynamic and Hydromagnetic Stability. Dover.
[16] Chavanne, X., Chilla, F., Castaing, B., Hebral, B., Chabaud, B. & Chaussy, J.1997Observation of the ultimate regime in Rayleigh-Bénard convection. Phys. Rev. Lett.79, 3648-3651.
[17] Cheng, J.S., Aurnou, J.M., Julien, K. & Kunnen, R.P.J.2018A heuristic framework for next-generation models of geostrophic convective turbulence. Geophys. Astrophys. Fluid Dyn.112 (4), 277-300. · Zbl 1499.76130
[18] Ching, E.S.C.1991Probabilities for temperature differences in Rayleigh-Bénard convection. Phys. Rev. A44, 3622-3629.
[19] Daya, Z.A. & Ecke, R.E.2002Prandtl number dependence of interior temperature and velocity fluctuations in turbulent convection. Phys. Rev. E66, 045301.
[20] Ecke, R.E. & Niemela, J.J.2014Heat transport in the geostrophic regime of rotating Rayleigh-Bénard convection. Phys. Rev. Lett.113, 114301.
[21] Favier, B. & Knobloch, E.2020 Robust wall states in rapidly rotating Rayleigh-Bénard convection. J. Fluid Mech.895, R1. · Zbl 1460.76719
[22] Gilman, P.A.1977Nonlinear dynamics of Boussinesq convection in a deep rotating spherical shell-i. Geophys. Astrophys. Fluid Dyn.8 (1), 93-135. · Zbl 0354.76064
[23] Goldstein, H.F., Knobloch, E., Mercader, I. & Net, M.1993Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers. J.Fluid Mech.248, 583-604. · Zbl 0796.76037
[24] Goldstein, H.F., Knobloch, E., Mercader, I. & Net, M.1994Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers. J.Fluid Mech.262, 293-324. · Zbl 0800.76142
[25] Grossmann, S. & Lohse, D.2000Scaling in thermal convection: a unifying view. J.Fluid Mech.407, 27-56. · Zbl 0972.76045
[26] Grossmann, S. & Lohse, D.2001Thermal convection for large Prandtl number. Phys. Rev. Lett.86, 3316-3319.
[27] Grossmann, S. & Lohse, D.2004Fluctuations in turbulent Rayleigh-Bénard convection: the role of plumes. Phys. Fluids16, 4462-4472. · Zbl 1187.76190
[28] He, X., Funfschilling, D., Bodenschatz, E. & Ahlers, G.2012Heat transport by turbulent Rayleigh-Bénard convection for \(Pr \approx 0.8\) and \(4\times 10^{11} < Ra < 2\times 10^{14}\): ultimate-state transition for aspect ratio \(\gamma =1.00\). New J. Phys.14, 063030.
[29] He, X., Van Gils, D.P.M., Bodenschatz, E. & Ahlers, G.2014Logarithmic spatial variations and universal \({f}^{-1}\) power spectra of temperature fluctuations in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett.112, 174501.
[30] Herrmann, J. & Busse, F.H.1993Asymptotic theory of wall-attached convection in a rotating fluid layer. J.Fluid Mech.255, 183-194. · Zbl 0785.76027
[31] Holton, J.R.2004An Introduction to Dynamic Meteorology. Elsevier Academic Press.
[32] Holzmann, H. & Vollmer, S.2008A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU. AStA Adv. Stat. Anal.92 (1), 57-69. · Zbl 1171.62013
[33] Horn, S. & Aurnou, J.M.2018Regimes of coriolis-centrifugal convection. Phys. Rev. Lett.120, 204502.
[34] Horn, S. & Aurnou, J.M.2019Rotating convection with centrifugal buoyancy: numerical predictions for laboratory experiments. Phys. Rev. Fluids4, 073501.
[35] Horn, S. & Shishkina, O.2015Toroidal and poloidal energy in rotating Rayleigh-Bénard convection. J.Fluid Mech.762, 232-255.
[36] Julien, K., Knobloch, E., Rubio, A.M. & Vasil, G.M.2012aHeat transport in low-Rossby-number Rayleigh-Bénard convection. Phys. Rev. Lett.109, 254503.
[37] Julien, K., Rubio, A.M., Grooms, I. & Knobloch, E.2012bStatistical and physical balances in low Rossby number Rayleigh-Bénard convection. Geophys. Astrophys. Fluid Dyn.106 (4-5), 392-428. · Zbl 07649676
[38] Kadanoff, L.P.2001Turbulent heat flow: structures and scaling. Phys. Today54 (8), 34-39.
[39] King, E.M., Stellmach, S. & Aurnou, J.M.2012Heat transfer by rapidly rotating Rayleigh-Bńard convection. J.Fluid Mech.691, 568-582. · Zbl 1241.76269
[40] King, E., Stellmach, S., Noir, J., Hansen, U. & Aurnou, J.2009Boundary layer control of rotating convection systems. Nature457, 301-304.
[41] Kraichnan, R.H.1962Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids5, 1374-1389. · Zbl 0116.42803
[42] Kunnen, R.P.J., Stevens, R.J.A.M., Overkamp, J., Sun, C., Van Heijst, G.F. & Clercx, H.J.H.2011The role of Stewartson and Ekman layers in turbulent rotating Rayleigh-Bénard convection. J.Fluid Mech.688, 422-442. · Zbl 1241.76270
[43] Kuo, E.Y. & Cross, M.C.1993Traveling-wave wall states in rotating Rayleigh-Bénard convection. Phys. Rev. E47, R2245-R2248.
[44] Long, R.S., Mound, J.E., Davies, C.J. & Tobias, S.M.2020Scaling behaviour in spherical shell rotating convection with fixed-flux thermal boundary conditions. J.Fluid Mech.889, A7.
[45] Malkus, M.V.R.1954The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A225, 196-212. · Zbl 0058.20203
[46] Niemela, J.J., Skrbek, L., Sreenivasan, K.R. & Donnelly, R.2000Turbulent convection at very high Rayleigh numbers. Nature404, 837-840.
[47] Oberbeck, A.1879Über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem.243, 271-292. · JFM 11.0787.01
[48] Plumley, M., Julien, K., Marti, P. & Stellmach, S.2016The effects of ekman pumping on quasi-geostrophic Rayleigh-Bénard convection. J.Fluid Mech.803, 51-71. · Zbl 1462.76172
[49] Portegies, J.W., Kunnen, R.P.J., Van Heijst, G.J.F. & Molenaar, J.2008A model for vortical plumes in rotating convection. Phys. Fluids20, 066602. · Zbl 1182.76606
[50] Rayleigh, Lord1916On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag.32, 529. · JFM 46.1249.04
[51] Shishkina, O., Emran, M.S., Grossmann, S. & Lohse, D.2017Scaling relations in large-Prandtl-number natural thermal convection. Phys. Rev. Fluids2, 103502.
[52] Shishkina, O., Weiss, S. & Bodenschatz, E.2016Conductive heat flux in measurements of the Nusselt number in turbulent Rayleigh-Bénard convection. Phys. Rev. Fluids1, 062301.
[53] Sondak, D., Smith, L.M. & Waleffe, F.2015Optimal heat transport solutions for Rayleigh-Bénard convection. J.Fluid Mech.784, 565-595. · Zbl 1382.76237
[54] Spiegel, E.A. & Veronis, G.1960On the Boussinesq approximation for a compressible fluid. Astrophys. J.131, 442-447.
[55] Stellmach, S., Lischper, M., Julien, K., Vasil, G., Cheng, J.S., Ribeiro, A., King, E.M. & Aurnou, J.M.2014Approaching the asymptotic regime of rapidly rotating convection: boundary layers versus interior dynamics. Phys. Rev. Lett.113, 254501.
[56] Stevens, R.J.A.M., Clercx, H.J.H. & Lohse, D.2010Optimal Prandtl number for heat transfer enhancement in rotating turbulent Rayleigh-Bénard convection. New J. Phys.12, 075005.
[57] Stevens, R.J.A.M., Lohse, D. & Verzicco, R.2014Sidewall effects in Rayleigh-Bénard convection. J.Fluid Mech.741, 1-27.
[58] Stevens, R.J.A.M., Van Der Poel, E.P., Grossmann, S. & Lohse, D.2013The unifying theory of scaling in thermal convection: the updated prefactors. J.Fluid Mech.730, 295-308. · Zbl 1291.76301
[59] Stewartson, K.1957On almost rigid rotations. J.Fluid Mech.3 (1), 17-26. · Zbl 0080.39103
[60] Stewartson, K.1966On almost rigid rotations. Part 2. J.Fluid Mech.26 (1), 131-144. · Zbl 0139.44102
[61] Sun, C., Xi, H.D. & Xia, K.Q.2005Azimuthal symmetry, flow dynamics, and heat transport in turbulent thermal convection in a cylinder with an aspect ratio of 0.5. Phys. Rev. Lett.95, 074502.
[62] Taylor, G.I.1923Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A223 (605-615), 289-343. · JFM 49.0607.01
[63] Tobasco, I. & Doering, C.R.2017Optimal wall-to-wall transport by incompressible flows. Phys. Rev. Lett.118, 264502. · Zbl 1434.76036
[64] Verzicco, R. & Camussi, R.2003Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J.Fluid Mech.477, 19-49. · Zbl 1063.76572
[65] Wang, Y., He, X. & Tong, P.2019Turbulent temperature fluctuations in a closed Rayleigh-Bénard convection cell. J.Fluid Mech.874, 263-284. · Zbl 1419.76597
[66] Weiss, S. & Ahlers, G.2011aHeat transport by turbulent rotating Rayleigh-Bénard convection and its dependence on the aspect ratio. J.Fluid Mech.684, 407-426. · Zbl 1241.76054
[67] Weiss, S. & Ahlers, G.2011bThe large-scale flow structure in turbulent rotating Rayleigh-Bénard convection. J.Fluid Mech.688, 461-492. · Zbl 1241.76272
[68] Weiss, S. & Ahlers, G.2011cTurbulent Rayleigh-Bénard convection in a cylindrical container with aspect ratio \(\varGamma =0.50\) and Prandtl number \({P}r=4.38\). J.Fluid Mech.676, 5-40. · Zbl 1241.76053
[69] Weiss, S., Stevens, R.J.A.M., Zhong, J.-Q., Clercx, H.J.H., Lohse, D. & Ahlers, G.2010Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh-Bénard convection. Phys. Rev. Lett.105 (22), 224501.
[70] Weiss, S., Wei, P. & Ahlers, G.2016Heat-transport enhancement in rotating turbulent Rayleigh-Bénard convection. Phys. Rev. E93, 043102.
[71] Whitehead, J.P. & Wittenberg, R.W.2014A rigorous bound on the vertical transport of heat in Rayleigh-Bénard convection at infinite Prandtl number with mixed thermal boundary conditions. J.Math. Phys.55 (9), 093104. · Zbl 1366.76082
[72] De Wit, X.M., Guzmán, A.J.A., Madonia, M., Cheng, J.S., Clercx, H.J.H. & Kunnen, R.P.J.2020Turbulent rotating convection confined in a slender cylinder: the sidewall circulation. Phys. Rev. Fluids5, 023502.
[73] Zhang, X., Ecke, R.E. & Shishkina, O.2021 Boundary zonal flows in rapidly rotating thermal convection. J. Fluid Mech. (accepted for publication).
[74] Zhang, X., Van Gils, D.P.M., Horn, S., Wedi, M., Zwirner, L., Ahlers, G., Ecke, R.E., Weiss, S., Bodenschatz, E. & Shishkina, O.2020Boundary zonal flow in rotating turbulent Rayleigh-Bénard convection. Phys. Rev. Lett.124, 084505.
[75] Zhang, K. & Liao, X.2009The onset of convection in rotating circular cylinders with experimental boundary conditions. J.Fluid Mech.622, 63-73. · Zbl 1165.76331
[76] Zhong, J.-Q. & Ahlers, G.2010Heat transport and the large-scale circulation in rotating turbulent Rayleigh-Bénard convection. J.Fluid Mech.665, 300-333. · Zbl 1225.76033
[77] Zhong, F., Ecke, R. & Steinberg, V.1993Rotating Rayleigh-Bénard convection: asymmetric modes and vortex states. J.Fluid Mech.249, 135-159.
[78] Zhong, J.-Q., Stevens, R.J.A.M., Clercx, H.J.H., Verzicco, R., Lohse, D. & Ahlers, G.2009Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh-Bénard convection. Phys. Rev. Lett.102, 044502.
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