- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Transition to the ultimate regime of turbulent convection in stratified inclined duct flow

Rundong Zhou1, Adrien Lefauve2,3, Roberto Verzicco1,4,5, and Detlef Lohse1,6

PDF

Phys. Rev. Fluids 11, 044802 – Published 7 April, 2026

DOI: https://doi.org/10.1103/psz6-f48t

Abstract

The stratified inclined duct (SID) provides a canonical setup for sustained, buoyancy-driven exchange flow between two reservoirs of different density, and emerges as a paradigm in geophysical fluid dynamics. Yet, the flow dynamics remain unclear in the highly turbulent regime; laboratory experiments can access this regime but they lack resolution, while direct numerical simulations (DNSs) at realistically high Prandtl number, Pr=7 (for heat in water), have not achieved sufficiently high Reynolds numbers Re. We conduct three-dimensional DNSs up to Re=8000 and observe the transition to the so-called ultimate regime of turbulent convection as evidenced by the Nusselt number scaling Nu∼Ra1/2, indicating substantially enhanced transport. At the transition, the shear Reynolds number, a key parameter characterizing boundary layer (BL) dynamics, exceeds the threshold range of 420 for turbulent kinetic BLs with the emergence of logarithmic velocity profiles. The nature of the transition toward ultimate SID flow is non-normal-nonlinear, i.e., subcritical and hysteretic, as is typical for the transition to fully turbulent shear flows. Our work connects SID flow with the broader class of wall-bounded turbulent convection flows and gives insight into mixing properties in the vigorously turbulent regime encountered in oceanographic and industrial flows.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (66)

  1. I. R. Wood, A lock exchange flow, J. Fluid Mech. 42, 671 (1970).
  2. D. L. Wilkinson, Buoyancy driven exchange flow in a horizontal pipe, J. Eng. Mech. 112, 485 (1986).
  3. M. Deacon, Scientists and the Sea 1650-1900, a Study of Marine Science (Academic Press, London, 1971).
  4. H. B. Fischer, Mixing and dispersion in estuaries, Annu. Rev. Fluid Mech. 8, 107 (1976).
  5. P. MacCready, W. R. Geyer, and H. Burchard, Estuarine exchange flow is related to mixing through the salinity variance budget, J. Phys. Oceanogr. 48, 1375 (2018).
  6. P. F. Linden, The fluid mechanics of natural ventilation, Annu. Rev. Fluid Mech. 31, 201 (1999).
  7. O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Philos. Trans. R. Soc. 174, 935 (1883).
  8. G. I. Taylor, Effect of variation in density on the stability of superposed streams of fluid, Proc. R. Soc. A 132, 499 (1931).
  9. C. R. Meyer and P. F. Linden, Stratified shear flow: Experiments in an inclined duct, J. Fluid Mech. 753, 242 (2014).
  10. S. A. Thorpe, A method of producing a shear flow in a stratified fluid, J. Fluid Mech. 32, 693 (1968).
  11. A. Lefauve, J. L. Partridge, and P. F. Linden, Regime transitions and energetics of sustained stratified shear flows, J. Fluid Mech. 875, 657 (2019).
  12. A. Lefauve and P. F. Linden, Buoyancy-driven exchange flows in inclined ducts, J. Fluid Mech. 893, A2 (2020).
  13. A. Lefauve and P. F. Linden, Experimental properties of continuously forced, shear-driven, stratified turbulence. Part 1. Mean flows, self-organisation, turbulent fractions, J. Fluid Mech. 937, A34 (2022).
  14. A. Lefauve and P. F. Linden, Experimental properties of continuously forced, shear-driven, stratified turbulence. Part 2. Energetics, anisotropy, parameterisation, J. Fluid Mech. 937, A35 (2022).
  15. L. Zhu, A. Atoufi, A. Lefauve, J. R. Taylor, R. R. Kerswell, S. B. Dalziel, G. A. Lawrence, and P. F. Linden, Stratified inclined duct: Direct numerical simulations, J. Fluid Mech. 969, A20 (2023).
  16. A. Lefauve and M. M. P. Couchman, Data-driven classification of sheared stratified turbulence from experimental shadowgraphs, Phys. Rev. Fluids 9, 034603 (2024).
  17. A. Lefauve, Y. H. M. Cheung, X. Jiang, and M. M. P. Couchman, Routes to stratified turbulence and temporal intermittency revealed by a cluster-based network model of experimental data, Europhys. Lett. 149, 53001 (2025).
  18. A. Lefauve, Geophysical stratified turbulence and mixing in the laboratory, C. R. Phys. 25, 479 (2024).
  19. A. Blass, X. Zhu, R. Verzicco, D. Lohse, and R. J. A. M. Stevens, Flow organization and heat transfer in turbulent wall sheared thermal convection, J. Fluid Mech. 897, A22 (2020).
  20. Q. Zhou, J. R. Taylor, C. P. Caulfield, and P. F. Linden, Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate, J. Fluid Mech. 823, 198 (2017).
  21. Q. Zhou, J. R. Taylor, and C. P. Caulfield, Self-similar mixing in stratified plane Couette flow for varying Prandtl number, J. Fluid Mech. 820, 86 (2017).
  22. G. S. Yerragolam, C. J. Howland, R. J. A. M. Stevens, R. Verzicco, O. Shishkina, and D. Lohse, Scaling relations for heat and momentum transport in sheared Rayleigh–Bénard convection, J. Fluid Mech. 1000, A74 (2024).
  23. C. J. Howland, G. S. Yerragolam, R. Verzicco, and D. Lohse, Turbulent mixed convection in vertical and horizontal channels, J. Fluid Mech. 998, A48 (2024).
  24. G. Ahlers, S. Grossmann, and D. Lohse, Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection, Rev. Mod. Phys. 81, 503 (2009).
  25. D. Lohse and K.-Q. Xia, Small-scale properties of turbulent Rayleigh–Bénard convection, Annu. Rev. Fluid Mech. 42, 335 (2010).
  26. F. Chillà and J. Schumacher, New perspectives in turbulent Rayleigh–Bénard convection, Eur. Phys. J. E 35, 58 (2012).
  27. K.-Q. Xia, Current trends and future directions in turbulent thermal convection, Theor. Appl. Mech. Lett. 3, 052001 (2013).
  28. D. Lohse and O. Shishkina, Ultimate Rayleigh-Bénard turbulence, Rev. Mod. Phys. 96, 035001 (2024).
  29. M. G. Visakh and J. H. Arakeri, Convection in slender Rayleigh–Bénard cells is a combination of wall and tube components, J. Fluid Mech. 1016, A46 (2025).
  30. M. Gibert, H. Pabiou, F. Chillà, and B. Castaing, High-Rayleigh-number convection in a vertical channel, Phys. Rev. Lett. 96, 084501 (2006).
  31. L. E. Schmidt, E. Calzavarini, D. Lohse, F. Toschi, and R. Verzicco, Axially homogeneous Rayleigh–Bénard convection in a cylindrical cell, J. Fluid Mech. 691, 52 (2012).
  32. S. S. Pawar and J. H. Arakeri, Two regimes of flux scaling in axially homogeneous turbulent convection in vertical tube, Phys. Rev. Fluids 1, 042401(R) (2016).
  33. B. Castaing, E. Rusaouën, J. Salort, and F. Chillà, Turbulent heat transport regimes in a channel, Phys. Rev. Fluids 2, 062801(R) (2017).
  34. C. S. Ng, A. Ooi, D. Lohse, and D. Chung, Vertical natural convection: Application of the unifying theory of thermal convection, J. Fluid Mech. 764, 349 (2015).
  35. C. S. Ng, A. Ooi, D. Lohse, and D. Chung, Changes in the boundary-layer structure at the edge of the ultimate regime in vertical natural convection, J. Fluid Mech. 825, 550 (2017).
  36. C. S. Ng, A. Ooi, D. Lohse, and D. Chung, Bulk scaling in wall-bounded and homogeneous vertical natural convection, J. Fluid Mech. 841, 825 (2018).
  37. Q. Wang, H. Liu, R. Verzicco, O. Shishkina, and D. Lohse, Regime transitions in thermally driven high–Rayleigh number vertical convection, J. Fluid Mech. 917, A6 (2021).
  38. J. Ke, N. Williamson, S. W. Armfield, and A. Komiya, The turbulence development of a vertical natural convection boundary layer, J. Fluid Mech. 964, A24 (2023).
  39. B. Dubrulle and F. Hersant, Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection, Eur. Phys. J. B 26, 379 (2002).
  40. B. Eckhardt, S. Grossmann, and D. Lohse, Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders, J. Fluid Mech. 581, 221 (2007).
  41. S. Grossmann, D. Lohse, and C. Sun, High–Reynolds number Taylor–Couette turbulence, Annu. Rev. Fluid Mech. 48, 53 (2016).
  42. R. H. Kraichnan, Turbulent thermal convection at arbitrary Prandtl number, Phys. Fluids 5, 1374 (1962).
  43. R. Verzicco and P. Orlandi, A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates, J. Comput. Phys. 123, 402 (1996).
  44. E. P. van der Poel, R. Ostilla-Mónico, J. Donners, and R. Verzicco, A pencil distributed finite difference code for strongly turbulent wall-bounded flows, Comput. Fluids 116, 10 (2015).
  45. A. Ceci and S. Pirozzoli, Natural grid stretching for DNS of compressible wall-bounded flows, J. Comput. Phys.: X 17, 100128 (2023).
  46. R. Ostilla-Monico, Y. Yang, E. P. van der Poel, D. Lohse, and R. Verzicco, A multiple-resolution strategy for direct numerical simulation of scalar turbulence, J. Comput. Phys. 301, 308 (2015).
  47. A. Atoufi, L. Zhu, A. Lefauve, J. R. Taylor, R. R. Kerswell, S. B. Dalziel, G. A. Lawrence, and P. F. Linden, Stratified inclined duct: Two-layer hydraulics and instabilities, J. Fluid Mech. 977, A25 (2023).
  48. L. Zhu, A. Atoufi, A. Lefauve, R. R. Kerswell, and P. F. Linden, Long-wave instabilities of sloping stratified exchange flows, J. Fluid Mech. 983, A12 (2024).
  49. H. Stommel and H. G. Farmer, Control of salinity in an estuary by a transition, J. Mar. Res. 12, 13 (1953).
  50. L. Armi, The hydraulics of two flowing layers with different densities, J. Fluid Mech. 163, 27 (1986).
  51. S. B. Dalziel, Two-layer hydraulics: A functional approach, J. Fluid Mech. 223, 135 (1991).
  52. A. Choffrut, C. Nobili, and F. Otto, Upper bounds on Nusselt number at finite Prandtl number, J. Differ. Equations 260, 3860 (2016).
  53. M. C. Gregg, E. A. D'Asaro, J. J. Riley, and E. Kunze, Mixing efficiency in the ocean, Annu. Rev. Mar. Sci. 10, 443 (2018).
  54. C. P. Caulfield, Open questions in turbulent stratified mixing: Do we even know what we do not know? Phys. Rev. Fluids 5, 110518 (2020).
  55. F. Feraco, R. Marino, A. Pumir, L. Primavera, P. D. Mininni, A. Pouquet, and D. Rosenberg, Vertical drafts and mixing in stratified turbulence: Sharp transition with Froude number, Europhys. Lett. 123, 44002 (2018).
  56. T. R. Osborn, Estimates of the local rate of vertical diffusion from dissipation measurements, J. Phys. Oceanogr. 10, 83 (1980).
  57. R. O. Tovar, Estudios sobre transicion y turbulencia en flujos de conveccion natural, Ph.D. thesis, Centro de Investigacion en Energia, Universidad Nacional Autonoma de Mexico, Mexico, D.F., 2002 (unpublished).
  58. M. R. Cholemari and J. H. Arakeri, Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe, J. Fluid Mech. 621, 69 (2009).
  59. É. Rusaouën, Échanges turbulents en convection thermique, Ph.D. thesis, Ecole Normale Supérieure de Lyon (ENS Lyon).
  60. E. Rusaouen, X. Riedinger, J. C. Tisserand, F. Seychelles, J. Salort, B. Castaing, and F. Chillà, Laminar and intermittent flow in a tilted heat pipe, Eur. Phys. J. E 37, 4 (2014).
  61. X. Riedinger, J.-C. Tisserand, F. Seychelles, B. Castaing, and F. Chillà, Heat transport regimes in an inclined channel, Phys. Fluids 25, 015117 (2013).
  62. J.-C. Tisserand, M. Creyssels, M. Gibert, B. Castaing, and F. Chillà, Convection in a vertical channel, New J. Phys. 12, 075024 (2010).
  63. E. Calzavarini, C. R. Doering, J. D. Gibbon, D. Lohse, A. Tanabe, and F. Toschi, Exponentially growing solutions in homogeneous Rayleigh-Bénard convection, Phys. Rev. E 73, 035301(R) (2006).
  64. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., Course of Theoretical Physics Vol. 6 (Butterworth Heinemann, Exeter, 1987).
  65. A. J. Smits, B. J. McKeon, and I. Marusic, High–Reynolds number wall turbulence, Annu. Rev. Fluid Mech. 43, 353 (2011).
  66. S. J. Kline, W. C. Reynolds, F. A. Schraub, and P. W. Runstadler, The structure of turbulent boundary layers, J. Fluid Mech. 30, 741 (1967).

Outline

Information

Phys. Rev. Fluids 11, 044802– Published 7 April, 2026
  • Received 15 August 2025
  • Revised 5 December 2025
  • Accepted 2 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation