Turbulence governs the transport of heat, mass and momentum on multiple scales. In real-world applications, wall-bounded turbulence typically involves surfaces that are rough; however, characterizing and understanding the effects of wall roughness on turbulence remains a challenge. Here, by combining extensive experiments and numerical simulations, we examine the paradigmatic Taylor–Couette system, which describes the closed flow between two independently rotating coaxial cylinders. We show how wall roughness greatly enhances the overall transport properties and the corresponding scaling exponents associated with wall-bounded turbulence. We reveal that if only one of the walls is rough, the bulk velocity is slaved to the rough side, due to the much stronger coupling to that wall by the detaching flow structures. If both walls are rough, the viscosity dependence is eliminated, giving rise to asymptotic ultimate turbulence—the upper limit of transport—the existence of which was predicted more than 50 years ago. In this limit, the scaling laws can be extrapolated to arbitrarily large Reynolds numbers.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


  1. 1.

    Nikuradse, J. Strömungsgesetze in rauhen Rohren. Forschung. Arb. Ing. Wes. 361, (1933).

  2. 2.

    Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. Logarithmic scaling of turbulence in smooth-and rough-wall pipe flow. J. Fluid. Mech. 728, 376–395 (2013).

  3. 3.

    Chan, L., MacDonald, M., Chung, D., Hutchins, N. & Ooi, A. A systematic investigation of roughness height and wavelength in turbulent pipe flow in the transitionally rough regime. J. Fluid. Mech. 771, 743–777 (2015).

  4. 4.

    Chung, D., Chan, L., MacDonald, M., Hutchins, N. & Ooi, A. A fast direct numerical simulation method for characterising hydraulic roughness. J. Fluid. Mech. 773, 418–431 (2015).

  5. 5.

    Squire, D. T. et al. Comparison of turbulent boundary layers over smooth and rough surfaces up to high Reynolds numbers. J. Fluid. Mech. 795, 210–240 (2016).

  6. 6.

    Jiménez, J. Turbulent flows of rough walls. Ann. Rev. Fluid Mech. 36, 173–196 (2004).

  7. 7.

    Flack, K. A. & Schultz, M. P. Roughness effects on wall-bounded turbulent flows. Phys. Fluids 26, 101305 (2014).

  8. 8.

    Pope, S. B. Turbulent Flow. (Cambridge Univ. Press, Cambridge, 2000).

  9. 9.

    Schlichting, H. & Gersten, K. Boundary Layer Theory 8th edn (Springer, Berlin, 2000).

  10. 10.

    Grossmann, S., Lohse, D. & Sun, C. High Reynolds number Taylor-Couette turbulence. Ann. Rev. Fluid Mech. 48, 53–80 (2016).

  11. 11.

    Eckhardt, B., Grossmann, S. & Lohse, D. Torque scaling in turbulent Taylor-Couette flow between independently rotating cylinders. J. Fluid. Mech. 581, 221–250 (2007).

  12. 12.

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

  13. 13.

    Lohse, D. & Xia, K.-Q. Small-scale properties of turbulent Rayleigh-Bénard convection. Ann. Rev. Fluid Mech. 42, 335–364 (2010).

  14. 14.

    Kraichnan, R. H. Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 1374–1389 (1962).

  15. 15.

    Chavanne, X. et al. Observation of the ultimate regime in Rayleigh-Bénard convection. Phys. Rev. Lett. 79, 3648–3651 (1997).

  16. 16.

    He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. Transition to the ultimate state of turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 108, 024502 (2012).

  17. 17.

    He, X., Funfschilling, D., Bodenschatz, E. & Ahlers, G. 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 (2012).

  18. 18.

    Huisman, S. G., van Gils, D. P. M., Grossmann, S., Sun, C. & Lohse, D. Ultimate turbulent Taylor-Couette flow. Phys. Rev. Lett. 108, 024501 (2012).

  19. 19.

    Ostilla-Mónico, R., van der Poel, E. P., Verzicco, R., Grossmann, S. & Lohse, D. Exploring the phase diagram of fully turbulent Taylor-Couette flow. J. Fluid. Mech. 761, 1–26 (2014).

  20. 20.

    Grossmann, S. & Lohse, D. Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108 (2011).

  21. 21.

    Ostilla-Mónico, R., Verzicco, R., Grossmann, S. & Lohse, D. The near-wall region of highly turbulent Taylor-Couette flow. J. Fluid. Mech. 788, 95–117 (2016).

  22. 22.

    Doering, C. & Constantin, P. Variational bounds on energy dissipation in incompressible flows: III. Convection. Phys. Rev. E 53, 5957–5981 (1996).

  23. 23.

    Nicodemus, R., Grossmann, S. & Holthaus, M. Variational bound on energy dissipation in turbulent shear flow. Phys. Rev. Lett. 79, 4170 (1997).

  24. 24.

    Plasting, S. C. & Kerswell, R. R. Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse’s problem and the Constantin-Doering-Hopf problem with one-dimensional background field. J. Fluid. Mech. 477, 363–379 (2003).

  25. 25.

    Toppaladoddi, S., Succi, S. & Wettlaufer, J. S. Roughness as a route to the ultimate regime of thermal convection. Phys. Rev. Lett. 118, 074503 (2017).

  26. 26.

    Xie, Y.-C. & Xia, K.-Q. Turbulent thermal convection over rough plates with varying roughness geometries. J. Fluid. Mech. 825, 573–599 (2017).

  27. 27.

    Zhu, X., Stevens, R. A. J. M., Verzicco, R. & Lohse, D. Roughness-facilitated local 1/2 scaling does not imply the onset of the ultimate regime of thermal convection. Phys. Rev. Lett. 119, 154501 (2017).

  28. 28.

    Lohse, D. & Toschi, F. The ultimate state of thermal convection. Phys. Rev. Lett. 90, 034502 (2003).

  29. 29.

    Gibert, M., Pabiou, H., Chilla, F. & Castaing, B. High-Rayleigh-number convection in a vertical channel. Phys. Rev. Lett. 96, 084501 (2006).

  30. 30.

    Cholemari, M. & Arakeri, J. Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe. J. Fluid. Mech. 621, 69–102 (2009).

  31. 31.

    von Kármán, T. Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1, 233–252 (1921).

  32. 32.

    Lathrop, D. P., Fineberg, J. & Swinney, H. S. Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68, 1515–1518 (1992).

  33. 33.

    Huisman, S. G. et al. Logarithmic boundary layers in strong Taylor-Couette turbulence. Phys. Rev. Lett. 110, 264501 (2013).

  34. 34.

    Ostilla-Mónico, R., van der Poel, E. P., Verzicco, R., Grossmann, S. & Lohse, D. Boundary layer dynamics at the transition between the classical and the ultimate regime of Taylor-Couette flow. Phys. Fluids 26, 015114 (2014).

  35. 35.

    Shen, Y., Tong, P. & Xia, K.-Q. Turbulent convection over rough surfaces. Phys. Rev. Lett. 76, 908–911 (1996).

  36. 36.

    Du, Y. B. & Tong, P. Turbulent thermal convection in a cell with ordered rough boundaries. J. Fluid. Mech. 407, 57–84 (2000).

  37. 37.

    Roche, P. E., Castaing, B., Chabaud, B. & Hebral, B. Observation of the 1/2 power law in Rayleigh-Bénard convection. Phys. Rev. E 63, 045303 (2001).

  38. 38.

    van den Berg, T. H., Doering, C., Lohse, D. & Lathrop, D. Smooth and rough boundaries in turbulent Taylor-Couette flow. Phys. Rev. E 68, 036307 (2003).

  39. 39.

    Tisserand, J. C. et al. Comparison between rough and smooth plates within the same Rayleigh-Bénard cell. Phys. Fluids 23, 015105 (2011).

  40. 40.

    Wei, P., Chan, T.-S., Ni, R., Zhao, X.-Z. & Xia, K.-Q. Heat transport properties of plates with smooth and rough surfaces in turbulent thermal convection. J. Fluid. Mech. 740, 28–46 (2014).

  41. 41.

    van Gils, D. P. M., Huisman, S. G., Bruggert, G. W., Sun, C. & Lohse, D. Torque scaling in turbulent Taylor-Couette flow with co- and counter-rotating cylinders. Phys. Rev. Lett. 106, 024502 (2011).

  42. 42.

    Brauckmann, H. J. & Eckhardt, B. Direct numerical simulations of local and global torque in Taylor-Couette flow up to Re = 30 000. J. Fluid. Mech. 718, 398–427 (2013).

  43. 43.

    Grossmann, S., Lohse, D. & Sun, C. Velocity profiles in strongly turbulent Taylor-Couette flow. Phys. Fluids 26, 025114 (2014).

  44. 44.

    Lewis, G. S. & Swinney, H. L. Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow. Phys. Rev. E 59, 5457–5467 (1999).

  45. 45.

    Zhu, X., Verzicco, R. & Lohse, D. Disentangling the origins of torque enhancement through wall roughness in Taylor-Couette turbulence. J. Fluid. Mech. 812, 279–293 (2017).

  46. 46.

    Moody, L. F. Friction factors for pipe flow. Trans. ASME 66, 671–684 (1944).

  47. 47.

    Taylor, G. I. Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. A 223, 289–343 (1923).

  48. 48.

    Brauckmann, H. J. & Eckhardt, B. Intermittent boundary layers and torque maxima in Taylor-Couette flow. Phys. Rev. E 87, 033004 (2013).

  49. 49.

    Huisman, S. G., van der Veen, R. C. A., Sun, C. & Lohse, D. Multiple states in highly turbulent Taylor-Couette flow. Nat. Commun. 5, 3820 (2014).

  50. 50.

    van Gils, D. P. M., Huisman, S. G., Grossmann, S., Sun, C. & Lohse, D. Optimal Taylor-Couette turbulence. J. Fluid. Mech. 706, 118–149 (2012).

  51. 51.

    Chouippe, A., Climent, E., Legendre, D. & Gabillet, C. Numerical simulation of bubble dispersion in turbulent Taylor-Couette flow. Phys. Fluids 26, 043304 (2014).

  52. 52.

    Martínez-Arias, B., Peixinho, J., Crumeyrolle, O. & Mutabazi, I. Effect of the number of vortices on the torque scaling in Taylor-Couette flow. J. Fluid. Mech. 748, 756–767 (2014).

  53. 53.

    Grossmann, S. & Lohse, D. Scaling in thermal convection: a unifying view. J. Fluid. Mech. 407, 27–56 (2000).

  54. 54.

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

  55. 55.

    Shang, X. D., Tong, P. & Xia, K.-Q. Scaling of the local convective heat flux in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett. 100, 244503 (2008).

  56. 56.

    Ni, R., Huang, S.-D. & Xia, K.-Q. Local energy dissipation rate balances local heat flux in the center of turbulent thermal convection. Phys. Rev. Lett. 107, 174503 (2011).

  57. 57.

    van Gils, D. P. M., Bruggert, G. W., Lathrop, D. P., Sun, C. & Lohse, D. The Twente turbulent Taylor-Couette (T3C) facility: strongly turbulent (multi-phase) flow between independently rotating cylinders. Rev. Sci. Instr. 82, 025105 (2011).

  58. 58.

    Verzicco, R. & Orlandi, P. A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402–413 (1996).

  59. 59.

    Fadlun, E. A., Verzicco, R., Orlandi, P. & Mohd-Yusof, J. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161, 35–60 (2000).

  60. 60.

    Yang, J. & Balaras, E. An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries. J. Comput. Phys. 215, 12–40 (2006).

  61. 61.

    van der Poel, E. P., Ostilla-Mónico, R., Donners, J. & Verzicco, R. A pencil distributed finite difference code for strongly turbulent wall-bounded flows.Comput. Fluids 116, 10–16 (2015).

  62. 62.

    Avila, M. Stability and angular-momentum transport of fluid flows between co-rotating cylinders. Phys. Rev. Lett. 108, 124501 (2012).

  63. 63.

    Ostilla-Mónico, R., Verzicco, R. & Lohse, D. Effects of the computational domain size on direct numerical simulations of Taylor-Couette turbulence with stationary outer cylinder. Phys. Fluids 27, 025110 (2015).

  64. 64.

    Landau, L. D. & Lifshitz, E. M. Fluid Mechanics (Pergamon, Oxford, 1987).

Download references


We gratefully acknowledge V. Mathai for insightful discussions. We thank G. W. Bruggert and M. Bos, as well as G. Mentink and R. Nauta, for their technical support and D.P.M. van Gils and R. Ezeta for various discussions and help with the experiments. The work is financially supported by NWO-I, NWO-TTW, the Netherlands Center for Multiscale Catalytic Energy Conversion (MCEC), and a VIDI grant (No. 13477), all sponsored by the Netherlands Organisation for Scientific Research (NWO). C.S. acknowledges the financial support from Natural Science Foundation of China under Grant No. 11672156. Part of the simulations were carried out on the Dutch national e-infrastructure with the support of SURF Cooperative. We also acknowledge PRACE for awarding us access to Marconi at CINECA, Italy under PRACE project number 2016143351 and DECI resource ARCHER UK National Supercomputing Service with the support from PRACE under project 13DECI0246.

Author information

Author notes

  1. These authors contributed equally: Xiaojue Zhu and Ruben A. Verschoof.


  1. Physics of Fluids Group and Max Planck Center Twente for Complex Fluid Dynamics, MESA+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, Enschede, The Netherlands

    • Xiaojue Zhu
    • , Ruben A. Verschoof
    • , Dennis Bakhuis
    • , Sander G. Huisman
    • , Roberto Verzicco
    • , Chao Sun
    •  & Detlef Lohse
  2. Dipartimento di Ingegneria Industriale, University of Rome Tor Vergata, Roma, Italy

    • Roberto Verzicco
  3. Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing, China

    • Chao Sun
    •  & Detlef Lohse
  4. Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany

    • Detlef Lohse


  1. Search for Xiaojue Zhu in:

  2. Search for Ruben A. Verschoof in:

  3. Search for Dennis Bakhuis in:

  4. Search for Sander G. Huisman in:

  5. Search for Roberto Verzicco in:

  6. Search for Chao Sun in:

  7. Search for Detlef Lohse in:


X.Z., S.G.H., R.A.V., R.V., C.S. and D.L. conceived the ideas. X.Z. performed the numerical simulations. R.A.V. and D.B. performed the measurements. X.Z. and R.A.V. analysed the data. X.Z., R.A.V. and D.L. wrote the paper. R.V., C.S. and D.L. supervised the project. All authors discussed the physics and proofread the paper.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Chao Sun or Detlef Lohse.

Supplementary information

  1. Supplementary Figures

    Supplementary Figures S1–S5

About this article

Publication history






Further reading