Law of the wall for a temporally evolving vertical natural convection boundary layer. (14th September 2020) - Ke, Junhao; Williamson, N.; Armfield, S. W.; Norris, S. E.; Komiya, A.
Abstract: Abstract : The present study concerns a temporally developing parallel natural convection boundary layer with Prandtl number $Pr = 0.71$ 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 ${Gr}_δ =7.7× 10^7$, where $Gr_δ$ is the Grashof number based on the boundary layer thickness $δ$ . 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 ${Gr}_δ$ 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 ${Gr}_δ$ 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 ${Gr}_δ$ 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 ${Gr}_δ$ 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.