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High Grashof number turbulent natural convection on an infinite vertical wall

Published online by Cambridge University Press:  21 October 2021

Junhao Ke Open the ORCID record for Junhao Ke [Opens in a new window] ,
N. Williamson Open the ORCID record for N. Williamson [Opens in a new window] ,
S.W. Armfield Open the ORCID record for S.W. Armfield [Opens in a new window] ,
A. Komiya Open the ORCID record for A. Komiya [Opens in a new window]  and
S.E. Norris Open the ORCID record for S.E. Norris [Opens in a new window]
Junhao Ke*
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW2006, Australia
N. Williamson
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW2006, Australia
S.W. Armfield
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW2006, Australia
A. Komiya
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai980-8577, Japan
S.E. Norris
Affiliation:
Department of Mechanical Engineering, The University of Auckland, Auckland1010, New Zealand
*
†Email address for correspondence: [email protected]
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Abstract

The present study concerns a temporally evolving turbulent natural convection boundary layer (NCBL) adjacent to an isothermally heated vertical wall, with Prandtl number 0.71. Three-dimensional direct numerical simulations (DNS) are carried out to investigate the turbulent flow up to $\textit {Gr}_\delta =1.21\times 10^8$, where $\textit {Gr}_\delta$ is the Grashof number based on the boundary layer thickness $\delta$. In the near-wall region, there exists a constant heat flux layer, similar to previous studies for the spatially developing flows (e.g. George & Capp, Intl J. Heat Mass Transfer, vol. 22, 1979, pp. 813–826). Beyond a wall-normal distance $\delta _i$, the NCBL can be characterised as a plume-like region. We find that this region is well described by a self-similar integral model with profile coefficients (cf. van Reeuwijk & Craske, J. Fluid Mech., vol. 782, 2015, pp. 333–355) which are $\textit {Gr}_\delta$-independent after $\textit {Gr}_\delta =10^7$. In this Grashof number range both the outer plume-like region and the near-wall boundary layer are turbulent, indicating the beginning of the so-called ultimate turbulent regime (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56; Grossmann & Lohse, Phys. Fluids, vol. 23, 2011, 045108). Solutions to the self-similar integral model are analytically obtained by solving ordinary differential equations with profile coefficients empirically obtained from the DNS results. In the present study, we have found the wall heat transfer of the NCBL is directly related to the top-hat scales which characterise the plume-like region. The Nusselt number is found to follow $\textit {Nu}_\delta \propto \textit {Gr}_\delta ^{0.381}$, slightly higher than the empirical $1/3$-power-law correlation reported for spatially developing NCBLs at lower $\textit {Gr}_\delta$, but is shown to be consistent with the ultimate heat transfer regime with a logarithmic correction suggested by Grossmann & Lohse (Phys. Fluids, vol. 23, 2011, 045108). By modelling the near-wall buoyancy force, we show that the wall shear stress would scale with the bulk velocity only at asymptotically large Grashof numbers.

JFM classification

Convection: Buoyant boundary layers Turbulent Flows: Turbulent convection Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 929 , 25 December 2021 , A15
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abedin, M.Z., Tsuji, T. & Hattori, Y. 2009 Direct numerical simulation for a time-developing natural-convection boundary layer along a vertical flat plate. Intl J. Heat Mass Transfer 52 (19-20), 4525–4534.CrossRefGoogle Scholar
Abedin, M.Z., Tsuji, T. & Hattori, Y. 2010 Direct numerical simulation for a time-developing combined-convection boundary layer along a vertical flat plate. Intl J. Heat Mass Transfer 53 (9-10), 2113–2122.CrossRefGoogle Scholar
Bonnebaigt, R., Caulfield, C.P. & Linden, P.F. 2018 Detrainment of plumes from vertically distributed sources. Environ. Fluid Mech. 18 (1), 3–25.CrossRefGoogle ScholarPubMed
Cheesewright, R. 1968 Turbulent natural convection from a vertical plane surface. J. Heat Transfer 90 (1), 1–6.CrossRefGoogle Scholar
Cooper, P. & Hunt, G.R. 2010 The ventilated filling box containing a vertically distributed source of buoyancy. J. Fluid Mech. 646, 39–58.CrossRefGoogle Scholar
Craske, J. & van Reeuwijk, M. 2015 Energy dispersion in turbulent jets. Part 1. Direct simulation of steady and unsteady jets. J. Fluid Mech. 763, 500–537.CrossRefGoogle Scholar
Craske, J. & van Reeuwijk, M. 2016 Generalised unsteady plume theory. J. Fluid Mech. 792, 1013–1052.CrossRefGoogle Scholar
Del Alamo, J.C., Jiménez, J., Zandonade, P. & Moser, R.D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.CrossRefGoogle Scholar
Eckert, E.R. & Jackson, T.W. 1950 Analysis of turbulent free-convection boundary layer on flat plate. Tech. Rep.. National Aeronautics and Space Administration.Google Scholar
Gayen, B., Griffiths, R.W. & Kerr, R.C. 2016 Simulation of convection at a vertical ice face dissolving into saline water. J. Fluid Mech. 798, 284–298.CrossRefGoogle Scholar
George, W.K. & Capp, S.P. 1979 A theory for natural convection turbulent boundary layers next to heated vertical surfaces. Intl J. Heat Mass Transfer 22 (6), 813–826.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86 (15), 3316.CrossRefGoogle ScholarPubMed
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23 (4), 045108.CrossRefGoogle Scholar
He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012 Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108 (2), 024502.CrossRefGoogle ScholarPubMed
Hölling, M. & Herwig, H. 2005 Asymptotic analysis of the near-wall region of turbulent natural convection flows. J. Fluid Mech. 541, 383–397.CrossRefGoogle Scholar
Illingworth, C.R. 1950 Unsteady laminar flow of gas near an infinite flat plate. Math. Proc. Camb. Phil. Soc. 46, 603–613.CrossRefGoogle Scholar
Jonker, H., van Reeuwijk, M., Sullivan, P.P. & Patton, E.G. 2013 On the scaling of shear-driven entrainment: a DNS study. J. Fluid Mech. 732, 150–165.CrossRefGoogle Scholar
Kaye, N.B. & Cooper, P. 2018 Source and boundary condition effects on unconfined and confined vertically distributed turbulent plumes. J. Fluid Mech. 850, 1032–1065.CrossRefGoogle Scholar
Ke, J., Williamson, N., Armfield, S.W., McBain, G.D. & Norris, S.E. 2019 Stability of a temporally evolving natural convection boundary layer on an isothermal wall. J. Fluid Mech. 877, 1163–1185.CrossRefGoogle Scholar
Ke, J., Williamson, N., Armfield, S.W., Norris, S.E. & Komiya, A. 2020 Law of the wall for a temporally evolving vertical natural convection boundary layer. J. Fluid Mech. 902, A31.CrossRefGoogle Scholar
Kerr, R.C. & McConnochie, C.D. 2015 Dissolution of a vertical solid surface by turbulent compositional convection. J. Fluid Mech. 765, 211–228.CrossRefGoogle Scholar
Kozul, M., Chung, D. & Monty, J.P. 2016 Direct numerical simulation of the incompressible temporally developing turbulent boundary layer. J. Fluid Mech. 796, 437–472.CrossRefGoogle Scholar
Kraichnan, R.H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5 (11), 1374–1389.CrossRefGoogle Scholar
Krug, D., Chung, D., Philip, J. & Marusic, I. 2017 Global and local aspects of entrainment in temporal plumes. J. Fluid Mech. 812, 222–250.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels up to re $\tau =4200$. Phys. Fluids 26 (1), 011702.CrossRefGoogle Scholar
MacDonald, M., Chung, D., Hutchins, N., Chan, L., Ooi, A. & García-Mayoral, R. 2017 The minimal-span channel for rough-wall turbulent flows. J. Fluid Mech. 816, 5–42.CrossRefGoogle Scholar
McConnochie, C.D. & Kerr, R.C. 2016 The turbulent wall plume from a vertically distributed source of buoyancy. J. Fluid Mech. 787, 237–253.CrossRefGoogle Scholar
Morton, B.R., Taylor, G.I. & Turner, J.S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234 (1196), 1–23.Google Scholar
Nakao, K., Hattori, Y. & Suto, H. 2017 Numerical investigation of a spatially developing turbulent natural convection boundary layer along a vertical heated plate. Intl J. Heat Fluid Flow 63, 128–138.CrossRefGoogle Scholar
Ng, C.S., Ooi, A., Lohse, D. & Chung, D. 2017 Changes in the boundary-layer structure at the edge of the ultimate regime in vertical natural convection. J. Fluid Mech. 825, 550–572.CrossRefGoogle Scholar
Paillat, S. & Kaminski, E. 2014 Entrainment in plane turbulent pure plumes. J. Fluid Mech. 755, R2.CrossRefGoogle Scholar
Parker, D.A., Burridge, H.C., Partridge, J.L. & Linden, P.F. 2020 A comparison of entrainment in turbulent line plumes adjacent to and distant from a vertical wall. J. Fluid Mech. 882, A4.CrossRefGoogle Scholar
Ramaprian, B.R. & Chandrasekhara, M.S. 1989 Measurements in vertical plane turbulent plumes. Trans. ASME J. Fluids Engng 111, 69–77.CrossRefGoogle Scholar
van Reeuwijk, M. & Craske, J. 2015 Energy-consistent entrainment relations for jets and plumes. J. Fluid Mech. 782, 333–355.CrossRefGoogle Scholar
van Reeuwijk, M., Vassilicos, J.C. & Craske, J. 2021 Unified description of turbulent entrainment. J. Fluid Mech. 908, A12.CrossRefGoogle Scholar
Schetz, J.A. & Eichhorn, R. 1962 Unsteady natural convection in the vicinity of a doubly infinite vertical plate. J. Heat Transfer 84 (4), 334–338.CrossRefGoogle Scholar
Tsuji, T. & Nagano, Y. 1988 Characteristics of a turbulent natural convection boundary layer along a vertical flat plate. Intl J. Heat Mass Transfer 31 (8), 1723–1734.CrossRefGoogle Scholar
Tsuji, T., Nagano, Y. & Tagawa, M. 1992 Experiment on spatio-temporal turbulent structures of a natural convection boundary layer. J. Heat Transfer 114 (4), 901–908.CrossRefGoogle Scholar
Versteegh, T.A.M. & Nieuwstadt, F.T.M. 1999 A direct numerical simulation of natural convection between two infinite vertical differentially heated walls scaling laws and wall functions. Intl J. Heat Mass Transfer 42 (19), 3673–3693.CrossRefGoogle Scholar
Vliet, G.C. & Liu, C.K. 1969 An experimental study of turbulent natural convection boundary layers. J. Heat Transfer 91 (4), 517–531.CrossRefGoogle Scholar
Wells, A.J. & Worster, M.G. 2008 A geophysical-scale model of vertical natural convection boundary layers. J. Fluid Mech. 609, 111–137.CrossRefGoogle Scholar
Williamson, N., Armfield, S.W. & Kirkpatrick, M.P. 2012 Transition to oscillatory flow in a differentially heated cavity with a conducting partition. J. Fluid Mech. 693, 93–114.CrossRefGoogle Scholar