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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

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