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Law of the wall for a temporally evolving vertical natural convection boundary layer

Published online by Cambridge University Press:  14 September 2020

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] ,
S. E. Norris Open the ORCID record for S. E. Norris [Opens in a new window]  and
A. Komiya Open the ORCID record for A. Komiya [Opens in a new window]
Junhao Ke*
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, New South Wales2006, Australia
N. Williamson
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, New South Wales2006, Australia
S. W. Armfield
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, New South Wales2006, Australia
S. E. Norris
Affiliation:
Department of Mechanical Engineering, The University of Auckland, Auckland1010, New Zealand
A. Komiya
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai 980-8577, Japan
*
†Email address for correspondence: [email protected]
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Abstract

The present study concerns a temporally developing parallel natural convection boundary layer with Prandtl number $\textit {Pr} = 0.71$ over an isothermally heated vertical plate. Three-dimensional direct numerical simulations (DNS) with different initial conditions were carried out to investigate the turbulent statistical profiles of mean velocity and temperature up to ${\textit {Gr}}_\delta =7.7\times 10^7$, where $Gr_\delta$ is the Grashof number based on the boundary layer thickness $\delta$. By virtue of DNS, we have identified a constant heat flux layer (George & Capp, Intl J. Heat Mass Transfer, vol. 22, issue 6, 1979, pp. 813–826; Hölling & Herwig, J. Fluid Mech., vol. 541, 2005, pp. 383–397) and a constant forcing layer in the near-wall region. In the close vicinity of the wall ($y^+<5$) a laminar-like sublayer has developed, and the temperature profile follows the linear relation, consistent with the studies of spatially developing flows (Tsuji & Nagano, Intl J. Heat Mass Transfer, vol. 31, issue 8, 1988, pp. 1723–1734); whereas such a linear relation cannot be observed for the velocity profile due to the extra buoyancy. Similar to earlier studies (Ng et al., J. Fluid Mech., vol. 825, 2017, pp. 550–572) we show that this buoyancy effect would asymptotically become zero if the ${\textit {Gr}}_\delta$ is sufficiently large. Further away from the wall ($y^+>50$), there is a log-law region for the mean temperature profile as reported by Tsuji & Nagano (1988). In this region, the turbulent length scale which characterises mixing scales linearly with the distance from the wall once ${\textit {Gr}}_\delta$ is sufficiently large. By taking the varying buoyancy into consideration with the robust mixing length model, a modified log-law for the mean velocity profile for $y^+>50$ is proposed. The effect of the initialization is shown to persist until relatively high ${\textit {Gr}}_\delta$ as a result of slow adjustment of the buoyancy (temperature) profile. Once these differences are accounted for, we find excellent agreement with our two DNS cases and with the spatially developing data of Tsuji & Nagano (1988). In the limit of higher ${\textit {Gr}}_\delta$ the velocity profile is expected to become asymptotic to momentum-dominated behaviour as buoyancy becomes increasingly weak in comparison with shear in the near-wall region.

JFM classification

Convection: Buoyant boundary layers Turbulent Flows: Turbulent convection Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 902 , 10 November 2020 , A31
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Copyright
© The Author(s), 2020. 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
Buschmann, M. H. & Gad-el Hak, M. 2005 New mixing-length approach for the mean velocity profile of turbulent boundary layers. Trans. ASME: J. Fluids Engng 127 (2), 393–396.Google Scholar
Chan, L., MacDonald, M., Chung, D., Hutchins, N. & Ooi, A. 2015 A systematic investigation of roughness height and wavelength in turbulent pipe flow in the transitionally rough regime. J. Fluid Mech. 771, 743–777.CrossRefGoogle Scholar
Cheesewright, R. 1968 Turbulent natural convection from a vertical plane surface. J. Heat Transf. 90 (1), 1–6.CrossRefGoogle Scholar
Fujii, T., Takeuchi, M., Fujii, M., Suzaki, K. & Uehara, H. 1970 Experiments on natural-convection heat transfer from the outer surface of a vertical cylinder to liquids. Intl J. Heat Mass Transfer 13 (5), 753–787.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
Granville, P. S. 1989 A modified van driest formula for the mixing length of turbulent boundary layers in pressure gradients. Trans. ASME: J. Fluids Engng 111, 94–97.Google 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), 3316–3319.CrossRefGoogle ScholarPubMed
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23 (4), 045108.CrossRefGoogle Scholar
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
Hutchins, N. & Marusic, I. 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. Lond. A 365 (1852), 647–664.CrossRefGoogle ScholarPubMed
Illingworth, C. R. 1950 Unsteady laminar flow of gas near an infinite flat plate. Math. Proc. Camb. Phil. Soc. 46, 603–613.CrossRefGoogle Scholar
Janssen, R. & Armfield, S. W. 1996 Stability properties of the vertical boundary layers in differentially heated cavities. Intl J. Heat Fluid Flow 17 (6), 547–556.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. & Kirkpatrick, M. P. 2018 Direct numerical simulation of a temporally developing natural convection boundary layer on a doubly infinite isothermal wall. In International Heat Transfer Conference Digital Library, IHTC-16. Begel House Inc.CrossRefGoogle Scholar
Kiš, P. & Herwig, H. 2012 The near wall physics and wall functions for turbulent natural convection. Intl J. Heat Mass Transfer 55 (9–10), 2625–2635.CrossRefGoogle Scholar
Klewicki, J. C. 2010 Reynolds number dependence, scaling, and dynamics of turbulent boundary layers. J. Fluids Engng 132 (9), 094001.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
Marusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 49–74.CrossRefGoogle Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle 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., Chung, D. & Ooi, A. 2013 Turbulent natural convection scaling in a vertical channel. Intl J. Heat Fluid Flow 44, 554–562.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
Nickels, T. B. 2004 Inner scaling for wall-bounded flows subject to large pressure gradients. J. Fluid Mech. 521, 217–239.CrossRefGoogle Scholar
Perry, A. E. & Chong, M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University.CrossRefGoogle Scholar
Prandtl, L. 1925 Bericht über untersuchungen zur ausgebildeten turbulenz. Z. Angew. Math. Mech. 5 (2), 136–139.CrossRefGoogle Scholar
Prandtl, L. 1932 Zur turbulenten strömung in rohren und längs platten. Ergebnisse der aerodynamischen versuchsanstalt zu göttingen 4, 18–29.Google Scholar
Sayadi, T., Hamman, C. W. & Moin, P. 2013 Direct numerical simulation of complete h-type and k-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480–509.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
Shiri, A. & George, W. K. 2008 Turbulent natural convection in a differentially heated vertical channel. In ASME 2008 Heat Transfer Summer Conference Collocated with the Fluids Engineering, Energy Sustainability, and 3rd Energy Nanotechnology Conferences, pp. 285–291. American Society of Mechanical Engineers Digital Collection.CrossRefGoogle Scholar
Townsend, A. A. 1951 The structure of the turbulent boundary layer. Math. Proc. Camb. Phil. Soc. 47, 375–395.CrossRefGoogle Scholar
Townsend, A. A. 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11 (1), 97–120.CrossRefGoogle Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University.Google 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
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
Von Kármán, T. 1930 Mechanische änlichkeit und turbulenz. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1930, 58–76.Google Scholar
Wei, T. 2019 Scaling of reynolds stresses in a differentially heated vertical channel. Phys. Rev. Fluids 4 (5), 051501.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
Yaglom, A. M. 1979 Similarity laws for constant-pressure and pressure-gradient turbulent wall flows. Annu. Rev. Fluid Mech. 11 (1), 505–540.CrossRefGoogle Scholar
Zhao, Y., Lei, C. & Patterson, J. C. 2013 Resonance of the thermal boundary layer adjacent to an isothermally heated vertical surface. J. Fluid Mech. 724, 305–336.CrossRefGoogle Scholar
Zhao, Y., Lei, C. & Patterson, J. C. 2017 The k-type and h-type transitions of natural convection boundary layers. J. Fluid Mech. 824, 352–387.CrossRefGoogle Scholar