## References

Ahlers, G., Grossmann, S. & Lohse, D.
2009
Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys.
81, 503–538.

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, 427–433.

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, 3648–3651.

Chavanne, X., Chilla, F., Chabaud, B., Castaing, B. & Hebral, B.
2001
Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids
13, 1300–1320.

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, 3998–4001.

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, 57–84.

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, 35–60.

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, 27–56.

Grossmann, S. & Lohse, D.
2001
Thermal convection for large Prandtl number. Phys. Rev. Lett.
86, 3316–3319.

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 × 10^{11} < *Ra* < 2 × 10^{14} : 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, 1374–1389.

Lepot, S., Aumaître, S. & Gallet, B.
2018
Radiative heating achieves the ultimate regime of thermal convection. Proc. Natl Acad. Sci. USA
115 (36), 8937–8941.

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, 335–364.

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, 196–212.

Mittal, R. & Iaccarino, G.
2005
Immersed boundary methods. Annu. Rev. Fluid Mech.
37, 239–261.

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, 479–517.

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, 10–16.

Priestley, C. H. B.
1954
Convection from a large horizontal surface. Aust. J. Phys.
7, 176–201.

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, 1–13.

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), 3531–3539.

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, 443–460.

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, 908–911.

Shishkina, O. & Wagner, C.
2011
Modelling the influence of wall roughness on heat transfer in thermal convection. J. Fluid Mech.
686, 568–582.

Spiegel, E. A.
1963
A generalization of the mixing-length theory of turbulent convection. Astrophys. J.
138, 216–225.

Stringano, G. & Verzicco, R.
2006
Mean flow structure in thermal convection in a cylindrical cell of aspect-ratio one half. J. Fluid Mech.
548, 1–16.

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, 402–413.

Villermaux, E.
1998
Transfer at rough sheared interfaces. Phys. Rev. Lett.
81 (22), 4859–4862.

Wagner, S. & Shishkina, O.
2015
Heat flux enhancement by regular surface roughness in turbulent thermal convection. J. Fluid Mech.
763, 109–135.

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, 28–46.

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, 573–599.

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, 746–774.

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, 199–210.

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), 417–423.10.1038/s41567-017-0026-3