Mean temperature profiles in turbulent thermal convection

Abstract

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al. [Phys. Rev. Lett. 114, 114302 (2015)], the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated, respectively, as a cubic and a linear function of the distance from the plate. In the present work, we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from 10(7) to 10(9)

Flow organisation in laterally unconfined Rayleigh–Bénard turbulence

“…When interpreting results for the BL thicknesses, it should be kept in mind that different definitions exist in the literature (du Puits et al 2007;Schmidt et al 2012;du Puits, Resagk & Thess 2013;Zhou & Xia 2013;Scheel & Schumacher 2014;Shishkina et al 2015Shishkina et al , 2017bChing et al 2019). We note that values may depend on the boundary layer definition that is employed.…”

Thermal boundary layer structure in low-Prandtl-number turbulent convection

“…The local thermal BL profiles have not been compared with the PBP profile in low- RBC, except for the horizontally-averaged profiles, which exhibit increasing deviation with decreasing (Scheel & Schumacher 2016; Shishkina et al. 2017; Ching et al. 2019).…”

Bulk temperature and heat transport in turbulent Rayleigh–Bénard convection of fluids with temperature-dependent properties

“…This limits the applicability of the extended Prandtl-Blasius BL theory to predict the centre temperature in NOB RBC. Note that the deviations between the predictions of the Prandtl-Blasius BL theory and real temperature profiles are larger for larger Ra and smaller Pr, see theoretical explanations in Shishkina et al (2015Shishkina et al ( , 2017b and Ching et al (2017), and results of direct numerical simulations in, e.g. Scheel, Kim & White (2012), Stevens et al (2012), , Shishkina, Wagner & Horn (2014).…”

“…From an analysis of the mean temperatures at the cross-overs from the BLs to the bulk in RBC for different Ra and Pr but fixed Γ , one can conclude that this temperature and, hence, also the value of β are influenced by Pr and depend negligibly weak on Ra, as soon as the convective flow is turbulent (Shishkina et al 2017b). The value of β 0 is slightly larger for smaller Pr, larger Ra and near the lateral walls, where stronger fluctuations are observed.…”