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Boundary layer structure in turbulent Rayleigh–Bénard convection in a slim box

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Abstract

Logarithmic boundary layers have been observed in different regions in turbulent Rayleigh–Bénard convection (RBC). However, how thermal plumes correlate with the logarithmic law of temperature and how the velocity profile changes with pressure gradient are not fully understood. Here, we perform three-dimensional (3D) simulations of turbulent Rayleigh–Bénard convection in a slim box without the front and back walls, with aspect ratio \(\mathrm {width}{:}\mathrm {depth}{:}\mathrm {height}=L{:}D{:}H=1{:}1/6{:}1\) (corresponding to the x, y, and z coordinates, respectively), in the Rayleigh number \({Ra} = [1\times 10^8, 1\times 10^{10}]\) for Prandtl number \({Pr}=0.7\). To investigate the structures of the viscous and thermal boundary layers, we examine the velocity profiles in the streamwise and vertical directions (i.e. U and W) along with the mean temperature profile throughout the plume-impacting, plume-ejecting, and wind-shearing regions. The velocity profile is successfully quantified by a two-layer function of a stress length,, \(\ell _u^+ = \ell _0^+(z^+)^{3/2} \left[ 1+\left( {z^+}/{z_\mathrm{sub}^+}\right) ^4\right] ^{1/4}\), as proposed by She et al. (J Fluid Mech, 2017), though it is neither  Prandtl–Blasius–Pohlhausen (PBP) type nor logarithmic. In contrast, the temperature profile in the plume-ejecting region is logarithmic for all simulated cases, attributed to the emission of thermal plumes. The coefficient of the temperature log law, A, can be described by the composition of the thermal stress length \(\ell ^*_{\theta 0}\) and the thicknesses of thermal boundary layer \(z^*_\mathrm{sub}\) and \(z^*_\mathrm{buf}\), i.e. \(A \simeq z^*_\mathrm{sub}/\left( \ell ^*_{\theta 0}{z^*_\mathrm{buf}}^{3/2}\right) \). The adverse pressure gradient responsible for turning the wind direction contributes to intensively emitting plumes and the logarithmic temperature profile at the plume-ejecting region. The Nusselt number scaling and the local heat flux in the slim box are consistent with previous results for confined cells. Therefore, the slim-box RBC is a preferred system for investigating in-box kinetic and thermal structures of turbulent convection with the large-scale circulation in a fixed plane.

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References

  1. Xia, K.Q.: Current trends and future directions in turbulent thermal convection. Theor. Appl. Mech. Lett. 3, 052001 (2013)

  2. Liu, C., Tang, S., Shen, L., et al.: Characteristics of turbulence transport for momentum and heat in particle-laden turbulent vertical channel flows. Acta Mech. Sinica 33, 833–845 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gao, Z.Y., Luo, J.H., Bao, Y.: Numerical study of heat-transfer in two- and quasi-two-dimensional Rayleigh–Bénard convection. Chin. Phys. B 27, 104702 (2018)

    Article  Google Scholar 

  4. Chen, J., Bao, Y., Yin, Z.X., et al.: Theoretical and numerical study of enhanced heat transfer in partitioned thermal convection. Int. J. Heat Mass Transf. 115, 556–569 (2017)

    Article  Google Scholar 

  5. Bao, Y., Chen, J., Liu, B.F., et al.: Enhanced heat transport in partitioned thermal convection. J. Fluid Mech. 784, R5 (2015)

  6. Shishkina, O., Horn, S., Wagner, S., et al.: Thermal boundary layer equation for turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 114, 114302 (2015)

    Article  Google Scholar 

  7. Wang, Y., He, X., Tong, P.: Boundary layer fluctuations and their effects on mean and variance temperature profiles in turbulent Rayleigh–Bénard convection. Phys. Rev. Fluids 1, 082301 (2016)

    Article  Google Scholar 

  8. Zhou, Q., Stevens, R.J.A.M., Sugiyama, K., et al.: Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 664, 297312 (2010)

    Article  MATH  Google Scholar 

  9. van der Poel, E.P., Stevens, R.J., Lohse, D.: Comparison between two-and three-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 736, 177–194 (2013)

    Article  MATH  Google Scholar 

  10. Wagner, S., Shishkina, O., Wagner, C.: Boundary layers and wind in cylindrical Rayleigh–Bénard cells. J. Fluid Mech. 697, 336–366 (2012)

    Article  MATH  Google Scholar 

  11. Shi, N., Emran, M.S., Schumacher, J.: Boundary layer structure in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 706, 5–33 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ahlers, G., Bodenschatz, E., He, X.: Logarithmic temperature profiles of turbulent convection in the classical and ultimate state for a Prandtl number of 0.8. J. Fluid Mech. 758, 436467 (2014)

    Article  MathSciNet  Google Scholar 

  13. Ahlers, G., Bodenschatz, E., Funfschilling, D., et al.: Logarithmic temperature profiles in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 109, 114501 (2012)

    Article  Google Scholar 

  14. Wei, P., Ahlers, G.: Logarithmic temperature profiles in the bulk of turbulent Rayleigh–Bénard convection for a Prandtl number of 12.3. J. Fluid Mech. 758, 809830 (2014)

    Article  Google Scholar 

  15. Shishkina, O., Wagner, C.: Local heat fluxes in turbulent Rayleigh–Bénard convection. Phys. Fluids 19, 085107 (2007)

    Article  MATH  Google Scholar 

  16. van der Poel, E.P., Ostilla-Mónico, R., Verzicco, R., et al.: Logarithmic mean temperature profiles and their connection to plume emissions in turbulent Rayleigh–Bénard convectionvan. Phys. Rev. Lett. 115, 154501 (2015)

    Article  Google Scholar 

  17. Huang, S.D., Wang, F., Xi, H.D., et al.: Comparitive experimental study of fixed temperature and fixed heat flux boundary condidtions in turbulent thermal convection. Phys. Rev. Lett. 115, 154502 (2015)

    Article  Google Scholar 

  18. Xi, H.D., Zhang, Y.B., Hao, J.T., et al.: High-order flow modes in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 805, 31–51 (2016)

    Article  MathSciNet  Google Scholar 

  19. Vasiliev, A., Sukhanovskii, A., Frick, P., et al.: High rayleigh number convection in a cubic cell with adiabatic sidewalls. Int. J. Heat Mass Transf. 102, 201–212 (2016)

    Article  Google Scholar 

  20. Podvin, B., Sergent, A.: Proper orthogonal decomposition investigation of turbulent Rayleigh–Bénard convection in a rectangular cavity. Phys. Fluids 24, 105106 (2012)

  21. Huang, S.D., Kaczorowski, M., Ni, R., et al.: Confinement-induced heat-transport enhancement in turbulent thermal convection. Phys. Rev. Lett. 111, 104501 (2013)

    Article  Google Scholar 

  22. Zhou, W.F., Chen, J.: Similarity model for corner roll in turbulent Rayleigh–Bénard convection. Phys. Fluids 30, 111705 (2018)

    Article  Google Scholar 

  23. van der Poel, E.P., Verzicco, R., Grossmann, S., et al.: Plume emission statistics in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 772, 5–15 (2015)

    Article  Google Scholar 

  24. Chong, K.L., Xia, K.Q.: Exploring the severely confined regime in Rayleigh–Bénard convection. J. Fluid Mech. 805, R4 (2016)

    Article  Google Scholar 

  25. She, Z.S., Chen, X., Hussain, F.: Quantifying wall turbulence via a symmetry approach: a lie group theory. J. Fluid Mech. 827, 322–356 (2017)

    Article  MathSciNet  Google Scholar 

  26. Grossmann, S., Lohse, D.: Logarithmic temperature profiles in the ultimate regime of thermal convection. Phys. Fluids 24, 125103 (2012)

  27. Verzicco, R., Camussi, R.: Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 19–49 (2003)

    Article  MATH  Google Scholar 

  28. Verzicco, R., Orlandi, P.: A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402–414 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kim, J., Moin, P.: Application of a fractional-step method to incompressible navier-stokes equations. J. Comput. Phys. 59, 308–323 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rai, M.M., Moin, P.: Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96, 15–53 (1991)

    Article  MATH  Google Scholar 

  31. Sun, X.H.: Application and accuracy of the parallel diagonal dominant algorithm. Parallel Comput. 21, 1241–1267 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shishkina, O., Stevens, R.J., Grossmann, S., et al.: Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J Phys. 12, 075022 (2010)

    Article  Google Scholar 

  33. Scheel, J.D., Emran, M.S., Schumacher, J.: Resolving the fine-scale structure in turbulent Rayleigh–Bénard convection. New J. Phys. 15, 113063 (2013)

    Article  Google Scholar 

  34. Shraiman, B.I., Siggia, E.D.: Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42, 3650 (1990)

    Article  Google Scholar 

  35. Sun, C., Cheung, Y.H., Xia, K.Q.: Experimental studies of the viscous boundary layer properties in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 605, 79–113 (2008)

    Article  MATH  Google Scholar 

  36. Wei, P., Xia, K.Q.: Viscous boundary layer properties in turbulent thermal convection in a cylindrical cell: the effect of cell tilting. J. Fluid Mech. 720, 140–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kaczorowski, M., Chong, K.L., Xia, K.Q.: Turbulent flow in the bulk of Rayleigh–Bénard convection: aspect-ratio dependence of the small-scale properties. J. Fluid Mech. 747, 73–102 (2014)

    Article  Google Scholar 

  38. Stevens, R.J.A.M., Lohse, D., Verzicco, R.: Prandtl and Rayleigh number dependence of heat transport in high rayleigh number thermal convection. J. Fluid Mech. 688, 31–43 (2011)

    Article  MATH  Google Scholar 

  39. du Puits, R., Resagk, C., Thess, A.: Mean velocity profile in confined turbulent convection. Phys. Rev. Lett. 99, 234504 (2007)

    Article  Google Scholar 

  40. Krishnamurti, R., Howard, L.N.: Large-scale flow generation in turbulent convection. In: Proceedings of the National Academy of Sciences, vol 78. National Academy of Sciences (1981)

  41. Qiu, X.L., Xia, K.Q.: Spatial structure of the viscous boundary layer in turbulent convection. Phys. Rev. E 58, 5816–5820 (1998)

    Article  Google Scholar 

  42. Niemela, J.J., Skrbek, L., Sreenivasan, K.R., et al.: The wind in confined thermal convection. J. Fluid Mech. 449, 169–178 (2001)

    Article  MATH  Google Scholar 

  43. Benzi, R., Verzicco, R.: Numerical simulations of flow reversal in Rayleigh–Bénard convection. Europhys. Lett. 81, 64008 (2008)

    Article  Google Scholar 

  44. Sugiyama, K., Calzavarini, E., Grossmann, S., et al.: Flow organization in two-dimensional non-Oberbeck–Boussinesq Rayleigh–Bénard convection in water. J. Fluid Mech. 637, 105–135 (2009)

    Article  MATH  Google Scholar 

  45. Zhou, Q., Sugiyama, K., Stevens, R.J.A.M., et al.: Horizontal structures of velocity and temperature boundary layers in two-dimensional numerical turbulent Rayleigh–Bénard convection. Phys. Fluids 23, 125104 (2011)

    Article  Google Scholar 

  46. van Reeuwijk, M., Jonker, H.J.J., Hanjalić, K.: Wind and boundary layers in Rayleigh–Bénard convection. I. Analysis and modeling. Phys. Rev. E 77, 036312 (2008)

    Article  Google Scholar 

  47. van Reeuwijk, M., Jonker, H.J.J., Hanjalić, K.: Wind and boundary layers in Rayleigh–Bénard convection. II. Boundary layer character and scaling. Phys. Rev. E 77, 036311 (2008)

    Article  Google Scholar 

  48. Xia, K.Q., Sun, C., Zhou, S.Q.: Particle image velocimetry measurement of the velocity field in turbulent thermal convection. Phys. Rev. E 68, 066303 (2003)

    Article  Google Scholar 

  49. Qiu, X.L., Tong, P.: Large-scale velocity structures in turbulent thermal convection. Phys. Rev. E 64, 036304 (2001)

    Article  Google Scholar 

  50. Shishkina, O., Horn, S., Emran, M.S., et al.: Mean temperature profiles in turbulent thermal convection. Phys. Rev. Fluids 2, 113502 (2017)

    Article  Google Scholar 

  51. He, X., Tong, P.: Space-time correlations in turbulent Rayleigh–Bénard convection. Acta Mech. Sinica 30, 457–467 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The Project was supported by the National Natural Science Foundation of China (Grants 11452002, 11521091, and 11372362) and MOST (China) 973 Project (Grant 2009CB724100).

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Authors and Affiliations

  1. State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871, China

    Hong-Yue Zou, Wen-Feng Zhou, Xi Chen, Jun Chen & Zhen-Su She

  2. Department of Mechanics, College of Engineering, Sun Yat-sen University, Guangzhou, 510275, China

    Yun Bao

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  1. Hong-Yue Zou
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Correspondence to Jun Chen.

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Zou, HY., Zhou, WF., Chen, X. et al. Boundary layer structure in turbulent Rayleigh–Bénard convection in a slim box. Acta Mech. Sin. 35, 713–728 (2019). https://doi.org/10.1007/s10409-019-00874-x

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