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$Nu\sim Ra^{1/2}$ scaling enabled by multiscale wall roughness in Rayleigh–Bénard turbulence

  • Xiaojue Zhu (a1) (a2), Richard J. A. M. Stevens (a1), Olga Shishkina (a3), Roberto Verzicco (a1) (a4) and Detlef Lohse (a1) (a3)...

Abstract

In turbulent Rayleigh–Bénard (RB) convection with regular, mono-scale, surface roughness, the scaling exponent $\unicode[STIX]{x1D6FD}$ in the relationship between the Nusselt number $Nu$ and the Rayleigh number $Ra$ , $Nu\sim Ra^{\unicode[STIX]{x1D6FD}}$ can be ${\approx}1/2$ locally, provided that $Ra$ is large enough to ensure that the thermal boundary layer thickness $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ is comparable to the roughness height. However, at even larger $Ra$ , $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ becomes thin enough to follow the irregular surface and $\unicode[STIX]{x1D6FD}$ saturates back to the value for smooth walls (Zhu et al., Phys. Rev. Lett., vol. 119, 2017, 154501). In this paper, we prevent this saturation by employing multiscale roughness. We perform direct numerical simulations of two-dimensional RB convection using an immersed boundary method to capture the rough plates. We find that, for rough boundaries that contain three distinct length scales, a scaling exponent of $\unicode[STIX]{x1D6FD}=0.49\pm 0.02$ can be sustained for at least three decades of $Ra$ . The physical reason is that the threshold $Ra$ at which the scaling exponent $\unicode[STIX]{x1D6FD}$ saturates back to the smooth wall value is pushed to larger $Ra$ , when the smaller roughness elements fully protrude through the thermal boundary layer. The multiscale roughness employed here may better resemble the irregular surfaces that are encountered in geophysical flows and in some industrial applications.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: xjzhu@g.harvard.edu

References

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Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503538.
van den Berg, T. H., Doering, C. R., Lohse, D. & Lathrop, D. 2003 Smooth and rough boundaries in turbulent Taylor–Couette flow. Phys. Rev. E 68, 036307.
Cadot, O., Couder, Y., Daerr, A., Douady, S. & Tsinober, A. 1997 Energy injection in closed turbulent flows: stirring through boundary layers versus inertial stirring. Phys. Rev.  E 56, 427433.
Calzavarini, E., Lohse, D., Toschi, F. & Tripiccione, R. 2005 Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence. Phys. Fluids 17, 055107.
Chavanne, X., Chilla, F., Castaing, B., Hebral, B., Chabaud, B. & Chaussy, J. 1997 Observation of the ultimate regime in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 36483651.
Chavanne, X., Chilla, F., Chabaud, B., Castaing, B. & Hebral, B. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids 13, 13001320.
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 58.
Ciliberto, S. & Laroche, C. 1999 Random roughness of boundary increases the turbulent convection scaling exponent. Phys. Rev. Lett. 82, 39984001.
Deluca, E. E., Werne, J., Rosner, R. & Cattaneo, F. 1990 Numerical simulations of soft and hard turbulence: preliminary results for two-dimensional convection. Phys. Rev. Lett. 64 (20), 2370.
Du, Y. B. & Tong, P. 2000 Turbulent thermal convection in a cell with ordered rough boundaries. J. Fluid Mech. 407, 5784.
Fadlun, E. A., Verzicco, R., Orlandi, P. & Mohd-Yusof, J. 2000 Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161, 3560.
Gibert, M., Pabiou, H., Chilla, F. & Castaing, B. 2006 High-Rayleigh-number convection in a vertical channel. Phys. Rev. Lett. 96, 084501.
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying view. J. Fluid. Mech. 407, 2756.
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 33163319.
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.
He, X., Funfschilling, D., Bodenschatz, E. & Ahlers, G. 2012a Heat transport by turbulent Rayleigh–Bénard convection for Pr = 0. 8 and 4 × 1011 < Ra < 2 × 1014 : ultimate-state transition for aspect ratio 𝛤 = 1. 00. New J. Phys. 14, 063030.
He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012b Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.
Johnston, H. & Doering, C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.
Kraichnan, R. H. 1962 Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 13741389.
Lepot, S., Aumaître, S. & Gallet, B. 2018 Radiative heating achieves the ultimate regime of thermal convection. Proc. Natl Acad. Sci. USA 115 (36), 89378941.
Lohse, D. & Toschi, F. 2003 The ultimate state of thermal convection. Phys. Rev. Lett. 90, 034502.
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.
MacDonald, M., Hutchins, N., Lohse, D. & Chung, D.2019 Heat transfer in fully-rough-wall-bounded turbulent flow in the ultimate regime. Phys. Rev. Fluids (submitted).
Malkus, M. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196212.
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239261.
Pawar, S. S. & Arakeri, J. H. 2018 Two regimes of flux scaling in axially homogeneous turbulent convection in vertical tube. Phys. Rev. Fluids 1 (4), 042401(R).
Peskin, C. S. 2002 The immersed boundary method. Acta Numer. 11, 479517.
van der Poel, E. P., Ostilla-Mónico, R., Donners, J. & Verzicco, R. 2015 A pencil distributed finite difference code for strongly turbulent wall–bounded flows. Comput. Fluids 116, 1016.
Priestley, C. H. B. 1954 Convection from a large horizontal surface. Aust. J. Phys. 7, 176201.
Qiu, X. L., Xia, K.-Q. & Tong, P. 2005 Experimental study of velocity boundary layer near a rough conducting surface in turbulent natural convection. J. Turbul. 6, 113.
Roche, P. E., Castaing, B., Chabaud, B. & Hebral, B. 2001 Observation of the 1/2 power law in Rayleigh–Bénard convection. Phys. Rev. E 63, 045303.
Rodriguez-Iturbe, I., Marani, M., Rigon, R. & Rinaldo, A. 1994 Self-organized river basin landscapes: fractal and multifractal characteristics. Water Resour. Res. 30 (12), 35313539.
Rusaouën, E., Liot, O., Castaing, B., Salort, J. & Chillà, F. 2018 Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries. J. Fluid Mech. 837, 443460.
Salort, J., Liot, O., Rusaouen, E., Seychelles, F., Tisserand, J.-C., Creyssels, M., Castaing, B. & Chillá, F. 2014 Thermal boundary layer near roughnesses in turbulent Rayleigh–Bénard convection: flow structure and multistability. Phys. Fluids 26, 015112.
Shen, Y., Tong, P. & Xia, K.-Q. 1996 Turbulent convection over rough surfaces. Phys. Rev. Lett. 76, 908911.
Shishkina, O. & Wagner, C. 2011 Modelling the influence of wall roughness on heat transfer in thermal convection. J. Fluid Mech. 686, 568582.
Spiegel, E. A. 1963 A generalization of the mixing-length theory of turbulent convection. Astrophys. J. 138, 216225.
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.
Tisserand, J. C., Creyssels, M., Gasteuil, Y., Pabiou, H., Gibert, M., Castaing, B. & Chilla, F. 2011 Comparison between rough and smooth plates within the same Rayleigh–Bénard cell. Phys. Fluids 23 (1), 015105.
Toppaladoddi, S., Succi, S. & Wettlaufer, J. S. 2017 Roughness as a route to the ultimate regime of thermal convection. Phys. Rev. Lett. 118, 074503.
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402413.
Villermaux, E. 1998 Transfer at rough sheared interfaces. Phys. Rev. Lett. 81 (22), 48594862.
Wagner, S. & Shishkina, O. 2015 Heat flux enhancement by regular surface roughness in turbulent thermal convection. J. Fluid Mech. 763, 109135.
Wei, P., Chan, T.-S., Ni, R., Zhao, X.-Z. & Xia, K.-Q. 2014 Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection. J. Fluid Mech. 740, 2846.
Xia, K.-Q. 2013 Current trends and future directions in turbulent thermal convection. Theor. Appl. Mech. Lett. 3 (5), 052001.
Xie, Y.-C. & Xia, K.-Q. 2017 Turbulent thermal convection over rough plates with varying roughness geometries. J. Fluid Mech. 825, 573599.
Yang, X. I. A. & Meneveau, C. 2017 Modelling turbulent boundary layer flow over fractal-like multiscale terrain using large-eddy simulations and analytical tools. Phil. Trans. R. Soc. Lond. A 375 (2091), 20160098.
Zhang, Y.-Z., Sun, C., Bao, Y. & Zhou, Q. 2018 How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 836, R2.
Zhu, X., Mathai, V., Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2018a Transition to the ultimate regime in two-dimensional Rayleigh–Bénard convection. Phys. Rev. Lett. 120 (14), 144502.
Zhu, X., Ostilla-Monico, R., Verzicco, R. & Lohse, D. 2016 Direct numerical simulation of Taylor–Couette flow with grooved walls: torque scaling and flow structure. J. Fluid Mech. 794, 746774.
Zhu, X., Phillips, E., Spandan, V., Donners, J., Ruetsch, G., Romero, J., Ostilla-Mónico, R., Yang, Y., Lohse, D., Verzicco, R., Massimiliano, F. & Stevens, R. J. A. M. 2018b AFiD-GPU: a versatile Navier–Stokes solver for wall-bounded turbulent flows on GPU clusters. Comput. Phys. Commun. 229, 199210.
Zhu, X., Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2017 Roughness-facilitated local 1/2 scaling does not imply the onset of the ultimate regime of thermal convection. Phys. Rev. Lett. 119 (15), 154501.
Zhu, X., Verschoof, R. A., Bakhuis, D., Huisman, S. G., Verzicco, R., Sun, C. & Lohse, D. 2018c Wall roughness induces asymptotic ultimate turbulence. Nat. Phys. 14 (4), 417423.10.1038/s41567-017-0026-3
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