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# 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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Many environmental flows arise due to natural convection at a vertical surface, from flows in buildings to dissolving ice faces at marine-terminating glaciers. We use three-dimensional direct numerical simulations of a vertical channel with differentially heated walls to investigate such convective, turbulent boundary layers. Through the implementation of a multiple-resolution technique, we are able to perform simulations at a wide range of Prandtl numbers ${Pr}$ . This allows us to distinguish the parameter dependences of the horizontal heat flux and the boundary layer widths in terms of the Rayleigh number $\mbox {{Ra}}$ and Prandtl number ${Pr}$ . For the considered parameter range $1\leq {Pr} \leq 100$ , $10^{6} \leq \mbox {{Ra}} \leq 10^{9}$ , we find the flow to be consistent with a ‘buoyancy-controlled’ regime where the heat flux is independent of the wall separation. For given ${Pr}$ , the heat flux is found to scale linearly with the friction velocity $V_\ast$ . Finally, we discuss the implications of our results for the parameterisation of heat and salt fluxes at vertical ice–ocean interfaces.
... where U (x) and (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 /(∂ /∂z) [12,13,17,32,[54][55][56]. In turbulent convection, however, both ν t and κ t computed using this method become undefined at some heights. ...
... 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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Natural convection arising over vertical and horizontal heated flat surfaces is one of the most ubiquitous flows at a range of spatiotemporal scales. Despite significant developments over more than a century contributing to our fundamental understanding of heat transfer in natural convection boundary layers, certain “hidden” characteristics of these flows have received far less attention. Here, we review scattered progress on less visited fundamental topics that have strong implications to heat and mass transfer control. These topics include the instability characteristics, laminar-to-turbulent transition, and spatial flow structures of vertical natural convection boundary layers and large-scale plumes, dome, and circulating flows over discretely and entirely heated horizontal surfaces. Based on the summarized advancements in fundamental research, we elaborate on the selection of perturbations and provide an outlook on the development of perturbation generators and methods of altering large-scale flow structures as a potential means for heat and mass transfer control where natural convection is dominant.
... As we have seen in this study, the lateral confinement of the convection cell can significantly influence all global response characteristics in the system and also the global structure of the convective flow. Therefore it is desired in the future to advance also the boundary-layer theory for Rayleigh-Bénard convection (see Shishkina et al. [165,166] and Ching et al. [167][168][169]) to the case of confined plates, in order to obtain accurate predictions of the profiles of the main flow characteristics in confined geometries. ...
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To study turbulent thermal convection, one often chooses a Rayleigh-Bénard flow configuration, where a fluid is confined between a heated bottom plate, a cooled top plate of the same shape, and insulated vertical sidewalls. When designing a Rayleigh-Bénard setup, for specified fluid properties under Oberbeck-Boussinesq conditions, the maximal size of the plates (diameter or area), and maximal temperature difference between the plates, Δmax, one ponders: Which shape of the plates and aspect ratio Γ of the container (ratio between its horizontal and vertical extensions) would be optimal? In this article, we aim to answer this question, where under the optimal container shape, we understand such a shape, which maximizes the range between the maximal accessible Rayleigh number and the critical Rayleigh number for the onset of convection in the considered setup, Rac,Γ. First we prove that Rac,Γ∝(1+cuΓ−2)(1+cθΓ−2), for some cu>0 and cθ>0. This holds for all containers with no-slip boundaries, which have a shape of a right cylinder, whose bounding plates are convex domains, not necessarily circular. Furthermore, we derive accurate estimates of Rac,Γ, under the assumption that in the expansions (in terms of the Laplace eigenfunctions) of the velocity and reduced temperature at the onset of convection, the contributions of the constant-sign eigenfunctions vanish, both in the vertical and at least in one horizontal direction. With that we derive Rac,Γ≈(2π)4(1+cuΓ−2)(1+cθΓ−2), where cu and cθ are determined by the container shape and boundary conditions for the velocity and temperature, respectively. In particular, for circular cylindrical containers with no-slip and insulated sidewalls, we have cu=j112/π2≈1.49 and cθ=(j̃11)2/π2≈0.34, where j11 and j̃11 are the first positive roots of the Bessel function J1 of the first kind or its derivative, respectively. For parallelepiped containers with the ratios Γx and Γy, Γy≤Γx≡Γ, of the side lengths of the rectangular plates to the cell height, for no-slip and insulated sidewalls we obtain Rac,Γ≈(2π)4(1+Γx−2)(1+Γx−2/4+Γy−2/4). Our approach is essentially different to the linear stability analysis, however, both methods lead to similar results. For Γ≲4.4, the derived Rac,Γ is larger than Jeffreys' result Rac,∞J≈1708 for an unbounded layer, which was obtained with linear stability analysis of the normal modes restricted to the consideration of a single perturbation wave in the horizontal direction. In the limit Γ→∞, the difference between Rac,Γ→∞=(2π)4 for laterally confined containers and Jeffreys' Rac,∞J for an unbounded layer is about 8.8%. We further show that in Rayleigh-Bénard experiments, the optimal rectangular plates are squares, while among all convex plane domains, circles seem to match the optimal shape of the plates. The optimal Γ is independent of Δmax and of the fluid properties. For the adiabatic sidewalls, the optimal Γ is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of Γ=1/2 in most Rayleigh-Bénard experiments is right and justified. For the given plate diameter D and maximal temperature difference Δmax, the maximal attainable Rayleigh number range is about 3.5 orders of magnitudes smaller than the order of the Rayleigh number based on D and Δmax. Deviations from the optimal Γ lead to a reduction of the attainable range, namely, as log10(Γ) for Γ→0 and as log10(Γ−3) for Γ→∞. Our theory shows that the relevant length scale in Rayleigh-Bénard convection in containers with no-slip boundaries is ℓ∼D/Γ2+cu=H/1+cu/Γ2. This means that in the limit Γ→∞, ℓ equals the cell height H, while for Γ→0, it is rather the plate diameter D.
... 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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Convection in the Sun occurs at Rayleigh numbers, $Ra$, as high as $10^{22}$, molecular Prandtl number, $Pr$, as low as $10^{-6}$, and occurs under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to $Pr$, up to $Pr$ as low as $10^{-4}$, of the turbulent Prandtl number, $Pr_t$, which is the ratio of turbulent viscosity to thermal diffusivity. The simulations are primarily two-dimensional but we draw upon some three-dimensional results as well. We focus on non-Oberbeck-Boussinesq (NOB) conditions of a certain type, but also study OB convection for comparison. The OB simulations are performed in a rectangular box of aspect ratio 2 by varying $Pr$ from $O(10)$ to $10^{-4}$ at fixed Grashof number $Gr \equiv Ra/Pr = 10^9$. The NOB simulations are done in the same box by letting only the thermal diffusivity depend on the temperature. Here, the Rayleigh number is fixed at the top boundary while the mean $Pr$ varies in the bulk from 0.07 to $5 \times 10^{-4}$. The three-dimensional simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh number of $10^5$, and $0.005 < Pr < 7$. The principal finding is that $Pr_t$ increases with decreasing $Pr$ in both OB and NOB convection: $Pr_t \sim Pr^{-0.3}$ for OB convection and $Pr_t \sim Pr^{-1}$ for the NOB case. The $Pr_t$-dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence.
... Based on the boundary-layer approximations and dimensional analysis, we find that the convection and diffusion terms in the vertical direction are much smaller than those in the horizontal direction, as shown in (4.2). As a result, only the pressure gradient and buoyancy terms remain in the viscous boundary-layer equation in the vertical direction, namely, ρ −1 ∂ z p = gα(T − T 0 ) (Ching et al. 2019). As shown in figure 3(a) and more quantitatively in figure 5(a), the pressure gradient term ρ −1 ∂ x p in (4.2) is negligibly small, so that the vertical pressure-buoyancy balance equation does not 918 A1-9 Figure 5. (a) Contributions of the mean convection term u∂ x u + w∂ z u, molecular momentum diffusion components −ν∂ 2 ...
... It can be seen from figure 5(a) that the boundary-layer approximations hold for the mean velocity equations in the near-wall region. We verify numerically that the horizontal pressure gradient ρ −1 ∂ x p is negligible at low Pr so that the viscous boundary-layer equation (4.2) is decoupled from the thermal boundary-layer equation (4.3) (Ching et al. 2019). In addition, we find numerically that the mean convection term, u∂ x u + w∂ z u, in (4.2) is negligibly small for this low-Pr RBC system. ...
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We report a direct numerical simulation (DNS) study of the mean velocity and temperature profiles in turbulent Rayleigh-Bénard convection (RBC) at low Prandtl numbers (Pr). The numerical study is conducted in a vertical thin disk with Pr varied in the range 0.17 ≤ Pr ≤ 4.4 and the Rayleigh number (Ra) varied in the range 5 × 10 8 ≤ Ra ≤ 1 × 10 10. By varying Pr from 4.4 to 0.17, we find a sharp change of flow patterns for the large-scale circulation (LSC) from a rigid-body rotation to a near-wall turbulent jet. We numerically examine the mean velocity equation in the bulk region and find that the mean horizontal velocity profile u(z) can be determined by a balance equation between the mean convection and turbulent diffusion with a constant turbulent viscosity ν t. This balance equation admits a self-similarity jet solution, which fits the DNS data well. In the boundary-layer region, we find that both the mean temperature profile T(z) and u(z) can be determined by a balance equation between the molecular diffusion and turbulent diffusion. Within the viscous boundary layer, both u(z) and T(z) can be solved analytically and the analytical results agree well with the DNS data. Our careful characterisation of the mean velocity † Email address for correspondence: penger@ust.hk
... 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. ...
... To consider the effects of turbulent fluctuations, one encounters the well-known closure problem, which is caused by more unknowns than equations, and turbulence closure models have to be introduced. By proposing a turbulence model which relates the eddy viscosity and diffusivity functions to 033501-2 the velocity stream function, we have developed a closed set of boundary layer equations [15]. This set of equations enables us to obtain both the mean velocity and temperature profiles, for fluid of all Prandtl number, in terms of two parameters that measure the size of the velocity and temperature fluctuations. ...
... 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 ...