5 results
Non-Oberbeck–Boussinesq effects on the linear stability of a vertical natural convection boundary layer
- Junhao Ke, S.W. Armfield, N. Williamson
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- Journal:
- Journal of Fluid Mechanics / Volume 988 / 10 June 2024
- Published online by Cambridge University Press:
- 24 July 2024, A44
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The non-Oberbeck–Boussinesq effects on the stability of a vertical natural convection boundary layer are investigated using the linearised disturbance equations for air flows up to a temperature difference of $\Delta T=100\,{\rm K}$. Based on the linear stability results, the neutral curve is shown to be sensitive to the choice of reference temperature. When evaluated using the film temperature $T_f$, a lower film Grashof number is required to trigger the linear instability for larger $\Delta T$. The relative contributions of shear and buoyant production to the perturbation kinetic energy budget reveals that the marginally unstable modes are amplified based on different mechanisms: for lower wavenumbers at relatively small Grashof number, the instability is driven by buoyancy; whereas for higher wavenumbers and larger Grashof number, the flow becomes unstable due to a shear instability. The use of reference temperature is found to scale the shear- and buoyant-driven instabilities differently so that no single reference temperature definition would collapse the neutral curves. The linear stability result further demonstrates that at a given Grashof number a higher temperature difference would give a larger amplification rate of the perturbation, which then leads to an earlier onset of the nonlinearities when evaluated at $T_f$. Finally, by comparing the amplification rates obtained from direct numerical simulation and the linear stability results, the extent of the linear regime is determined for $\Delta T = 100\,{\rm K}$.
The turbulence development of a vertical natural convection boundary layer
- Junhao Ke, N. Williamson, S.W. Armfield, A. Komiya
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- Journal:
- Journal of Fluid Mechanics / Volume 964 / 10 June 2023
- Published online by Cambridge University Press:
- 30 May 2023, A24
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Results from direct numerical simulations of a vertical natural convection boundary layer (NCBL) with $Pr=0.71$ reveal that the turbulence development of such a thermally driven convective flow has two distinct stages: at relatively low Grashof number, the bulk flow is turbulent while the near-wall region is laminar-like or weakly turbulent; at sufficiently high Grashof number, the entire flow becomes turbulent in the sense of von Kármán (cf. Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56, for the ultimate turbulent regime). Investigations on the turbulence statistics show that the near-wall Reynolds shear stress is negligible in the weakly turbulent regime but will grow in magnitude as the flow transitions to the ultimate regime at higher Grashof number. Similar behaviour is also seen in the streamwise turbulence intensity, where it develops from a mono-peak profile into a dual-peak structure as the Grashof number increases. At higher Grashof number, the near-wall energetic site is shown to have an energy distribution similar to that of a canonical wall-bounded turbulence (e.g. Hutchins & Marusic, Phil. Trans. R. Soc. A, vol. 365, 2007, pp. 647–664), with a peak centred at fixed location and wavelength ($y^+=18$ and $\lambda ^+_x=1000$) in viscous coordinates. Investigation on the spanwise spectra also suggests that the turbulent near-wall streaks emerge only at sufficiently high Grashof number, with constant spacing $\lambda _z^+\approx 130$. The extent of the weakly turbulent regime is identified using the maximum velocity location $\delta_m$ and a laminar length scale $\delta_u$. The development of near-wall turbulence is also investigated by examining the turbulence kinetic energy budget. In the weakly turbulent regime, the near-wall turbulence is sustained predominantly by the pressure transport in addition to the shear production. At higher Grashof number, the flow becomes fully turbulent, and both turbulent transport and shear production become stronger, while the pressure transport is decreased. These results also reveal that the production–dissipation ratio $P/\varepsilon$ of the NCBL would follow a fundamentally different trend to the canonical wall-bounded flows, which further supports that the near-wall turbulence generation is affected by the bulk flow.
High Grashof number turbulent natural convection on an infinite vertical wall
- Junhao Ke, N. Williamson, S.W. Armfield, A. Komiya, S.E. Norris
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- Journal:
- Journal of Fluid Mechanics / Volume 929 / 25 December 2021
- Published online by Cambridge University Press:
- 21 October 2021, A15
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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.
Law of the wall for a temporally evolving vertical natural convection boundary layer
- Junhao Ke, N. Williamson, S. W. Armfield, S. E. Norris, A. Komiya
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- Journal:
- Journal of Fluid Mechanics / Volume 902 / 10 November 2020
- Published online by Cambridge University Press:
- 14 September 2020, A31
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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.
Stability of a temporally evolving natural convection boundary layer on an isothermal wall
- Junhao Ke, N. Williamson, S. W. Armfield, G. D. McBain, S. E. Norris
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- Journal:
- Journal of Fluid Mechanics / Volume 877 / 25 October 2019
- Published online by Cambridge University Press:
- 02 September 2019, pp. 1163-1185
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The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-steady assumption, whereby the unsteady base flow is frozen in time. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearized disturbance equations using a ‘frozen’ base flow as an initial-value problem at various $Gr_{\unicode[STIX]{x1D6FF}}$, where $Gr_{\unicode[STIX]{x1D6FF}}$ is the Grashof number based on the velocity integral boundary layer thickness $\unicode[STIX]{x1D6FF}$. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three-dimensional direct numerical simulations (DNS) of the temporally evolving flow. The amplification rate predicted by the linear theory compares well with the result of direct numerical simulation up to a transition point. The extent of the linear regime where the perturbations linearly interact with the base flow is thus identified. The value of the transition $Gr_{\unicode[STIX]{x1D6FF}}$, according to the three-dimensional DNS results, is dependent on the initial perturbation amplitude. Beyond the transition point, the DNS results diverge from the linear stability predictions as nonlinear mechanisms become important.