Values of B obtained in the DNS [8] (circles) and [26] (crosses). The data points for the same Pr, which ranges from 0.005 to 300, are connected by a solid line.

Values of B obtained in the DNS [8] (circles) and [26] (crosses). The data points for the same Pr, which ranges from 0.005 to 300, are connected by a solid line.

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In turbulent Rayleigh-Bénard convection, the boundary layers are nonsteady with fluctuations, the time-averaged large-scale circulating velocity vanishes far away from the top and bottom plates, and the motion arises from buoyancy. In this paper, we derive the full set of boundary layer equations for both the temperature and velocity fields from th...

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... the horizontal pressure gradient would play a direct role in (8) when B is at least of order 1. In Fig. 1 we show the values of B obtained in the DNS [8,24,26,27]. It can be seen that B decreases with Pr and B 1 at low Pr and high ...

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... Precisely, we define the conductive thermal boundary layer width δ T as the wall-normal location where the conductive heat flux −κ dT/dx is equal to the turbulent heat flux u T . In RBC at moderate Ra, there is a general consensus from existing literature (Ahlers et al. 2009;Ching et al. 2019) that, scaling-wise, the thickness of the boundary layers follows a laminar-like scaling according to Prandtl, Blasius and Pohlhausen, that is ...
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... Vishnu et al. 215 observed that the large-scale flow was generally aligned along one of the diagonals in a cubic container. Ching et al. 216 and Shishkina et al. 217 provided an asymptotic solution of the boundary layer profiles considering both viscous effect and fluctuations in the boundary layer of turbulent RBC. Wei and du Puits 218 further divided the RBC cell into four layers, i.e., linear viscous sub-layer, transition layer, power-law layer and well-mixed layer. ...
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... where U (x) and Θ(x) are the time-averaged velocity and temperature fields. In the literature, the turbulent viscosity is usually estimated by the flux-gradient method, according to which ν t = − u x u z /(∂U x /∂z) and the turbulent thermal diffusivity by κ t = − u z T /(∂Θ/∂z) [11,12,16,31,[48][49][50]. In turbulent convection, however, we observe that ν t and κ t computed using this method become undefined at various heights. ...
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... One expects that a theoretical understanding of heat flux in turbulent thermal convection can be achieved similarly using a boundary layer theory. We have recently developed a closed set of boundary layer equations for turbulent RB convection [14,15] that takes into account both buoyancy and fluctuations. In this work, we derive analytical results for the dependence of Nu on Re and Pr using this theory. ...
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... We have shown that the normalized mean temperature boundary layer profiles at high Prandtl number (Pr = 4.38 and Pr = 2547.9) [29] and the mean temperature and velocity boundary layer profiles at low Prandtl number (Pr = 0.1) [15] obtained in direct numerical simulation (DNS) can be well described by our theoretical results. Further support of our theoretical results has been reported in other numerical [30] and experimental studies [31]. ...
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Using a closed set of boundary layer equations [E. S. C. Ching et al., Phys. Rev. Research 1, 033037 (2019)] for turbulent Rayleigh-Bénard convection, we derive analytical results for the dependence of the heat flux, measured by the Nusselt number (Nu), on the Reynolds (Re) and Prandtl (Pr) numbers and two parameters that measure fluctuations in the regime where the horizontal pressure gradient is negligible. This regime is expected to be reached at sufficiently high Rayleigh numbers for a fluid of any given Prandtl number. In the high-Pr limit, Nu=F1(k1)Re1/2Pr1/3 and, in the low-Pr limit, Nu tends to π−1/2Re1/2Pr1/2, where F1(k1) has a weak dependence on the parameter k1 in the eddy viscosity that measures velocity fluctuations. These theoretical results further reveal a close resemblance of the scaling dependencies of heat flux in steady forced convection and turbulent Rayleigh-Bénard convection and this finding solves a puzzle in our present understanding of heat transfer in turbulent Rayleigh-Bénard convection.
... Wagner et al. (2012). 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. ...
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We investigate the large-scale circulation (LSC) of turbulent Rayleigh-Benard convection in a large box of aspect ratio Γ=32 for Rayleigh numbers up to Ra=10^9 and at a fixed Prandtl number Pr=1. A conditional averaging technique allows us to extract statistics of the LSC even though the number and the orientation of the structures vary throughout the domain. We find that various properties of the LSC obtained here, such as the wall-shear stress distribution, the boundary layer thicknesses and the wind Reynolds number, do not differ significantly from results in confined domains (Γ≈1). This is remarkable given that the size of the structures (as measured by the width of a single convection roll) more than doubles at the highest Ra as the confinement is removed. An extrapolation towards the critical shear Reynolds number of Re_s^{\textrm{crit}}≈420, at which the boundary layer (BL) typically becomes turbulent, predicts that the transition to the ultimate regime is expected at Ra_{\textrm{crit}} \approx \mathcal{O}(10^{15}) in unconfined geometries. This result is in line with the Göttingen experimental observations. Furthermore, we confirm that the local heat transport close to the wall is highest in the plume impacting region, where the thermal BL is thinnest, and lowest in the plume emitting region, where the thermal BL is thickest. This trend, however, weakens with increasing Ra.
... Nonetheless, persistent deviations even after using this dynamic rescaling have been reported in 3-D RBC for moderate-and high-Pr RBC (Scheel et al. 2012;Shi et al. 2012;Stevens et al. 2012). The local thermal BL profiles have not been compared with the PBP profile in low-Pr RBC, except for the horizontally-averaged profiles, which exhibit increasing deviation with decreasing Pr Shishkina et al. 2017;Ching et al. 2019). Therefore, we measure the temperature profiles at various horizontal positions in our low-Pr RBC and observe deviations from the PBP profile everywhere, with the degree of deviation depending on the measurement position. ...
... The reason for the deviation is that the RBC flow in a bounded domain does not satisfy the criteria for the PBP profile due to the presence of other effects, such as the emission of thermal plumes, buoyancy, pressure gradient, turbulent fluctuations, sidewalls, etc. Therefore, modified BL profiles in RBC have been suggested by incorporating these additional effects in the laminar BL equations (Shi et al. 2012;Shishkina et al. 2015;Ovsyannikov et al. 2016;Shishkina et al. 2017;Ching et al. 2019). Shishkina et al. (2017) and Ching et al. (2019) recently proposed a model of the horizontally-and temporally-averaged temperature profiles in the BL region by incorporating the effects of turbulent fluctuations in the laminar BL equations. ...
... Therefore, modified BL profiles in RBC have been suggested by incorporating these additional effects in the laminar BL equations (Shi et al. 2012;Shishkina et al. 2015;Ovsyannikov et al. 2016;Shishkina et al. 2017;Ching et al. 2019). Shishkina et al. (2017) and Ching et al. (2019) recently proposed a model of the horizontally-and temporally-averaged temperature profiles in the BL region by incorporating the effects of turbulent fluctuations in the laminar BL equations. They proposed that the temperature profile could be fitted with an equation of the form ...
Preprint
We study the structure of the thermal boundary layer (BL) in Rayleigh-B\'enard convection for Prandtl number ($Pr$) 0.021 by conducting direct numerical simulations in a two-dimensional square box for Rayleigh numbers ($Ra$) up to $10^9$. The large-scale circulation in the flow divides the horizontal plates into three distinct regions, and we observe that the local thermal BL thicknesses in the plume-ejection region are larger than those in the plume-impact and shear-dominated regions. Moreover, the local BL width decreases as $Ra^{-\beta(x)}$, with $\beta(x)$ depending on the position at the plate. We find that $\beta(x)$ are nearly the same in impact and shear regions and are smaller than those in the ejection region. Thus, the local BL width decreases faster in the ejection region than those in the shear and impact regions, and we estimate that the thermal BL structure would be uniform throughout the horizontal plate for $Ra \geq 8 \times 10^{12}$ in our low-$Pr$ convection. We compare the thermal BL profiles measured at various positions at the plate with the Prandtl-Blasius-Pohlhausen (PBP) profile and find deviations everywhere for all the Rayleigh numbers. However, the dynamically-rescaled profiles, as suggested by Zhou \& Xia ({\it Phys. Rev. Lett.}, vol. 104, 2010, 104301), agree well with the PBP profile in the shear and impact regions for all the Rayleigh numbers, whereas they still deviate in the ejection region. We also find that, despite the growing fluctuations with increasing $Ra$, thermal boundary layers in our low-$Pr$ convection are transitional and not yet fully turbulent.
... , as in RBC (Ching et al., 2019;Grossmann & Lohse, 2000Shishkina et al., 2015), and ...