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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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Atmospheric boundary layer (ABL) dynamics over glaciers mediate the response of glacier mass balance to large‐scale climate forcing. Despite this, very few ABL observations are available over mountain glaciers in complex terrain. An intensive field campaign was conducted in June 2015 at the Athabasca Glacier outlet of Columbia Icefield in the Canad...

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... Along the experimental, many numerical studies have been performed. All kinds of methods were used, such as direct numerical simulation [16,24,25,42,43], simulations of large eddies or averaged Navier-Stokes equations [44,45], with boundary layer treatment [46][47][48][49][50][51][52][53][54][55][56]. The Rayleigh-Bénard convection, as reported in [57], can be considered as "the forefather of the canonical examples that were used for the study of model forming and behavior in extended systems in space." ...

Naturally flows have been the scope of the scientific research for centuries, Rayleigh-B?nard convection being one of the leading. Many researchers have considered the flow patterns, boundary conditions, various cavities, nanofluids, theoretically, numerically, experimentally. The flow was investigated in atmosphere and in nano fluids, in air, water, molten metals, non - Newtonian fluids. Almost all research focuses on 2 or 3 - dimensional analysis of flow in laterally unlimited enclosures, as parallel plates or coaxial cylinders. In technical practice, only limited enclosures exist. This paper presents numerical and real experimental results for the test chamber with ratio 4?2?1 in x, y and z direction, respectfully. The measurements were taken at fifteen different positions on the faces of the tank. Probes used are PT100 elements. As the chamber is limited in all directions, the results have shown strong influence of the lateral walls. The results are compared with the those obtained by IR camera. Various fluids were tested, and results for motor oil will be presented.

... The most fundamental and ubiquitous coherent structure in thermal convection is the large-scale circulation (LSC) -approximately described as a quasi-two-dimensional overturning motion of the fluid bulk (Krishnamurti & Howard 1981;Villermaux 1995;Funfschilling & Ahlers 2004;Zhou et al. 2009), but also subject to a variety of complex dynamics, as we will discuss further in this work. The LSC is of central importance for convective turbulence theory; it shears the fluid where thermal plumes are ejected from boundary layers into the bulk, and numerous predictions for the transport properties of convective systems rely on this sweeping effect as a theoretical launching point (Grossmann & Lohse 2000;Shishkina et al. 2015;Ching et al. 2019). ...

We investigate the scaling properties of the primary flow modes and their sensitivity to aspect ratio in a liquid gallium (Prandtl number Pr = 0.02) convection system through combined laboratory experiments and numerical simulations. We survey cylindrical aspect ratios 1.4 ≤ Γ ≤ 3 and Rayleigh numbers 10 4 Ra 10 6. In this range the flow is dominated by a large-scale circulation (LSC) subject to low-frequency oscillations. In line with previous studies, we show robust scaling of the Reynolds number Re with Ra and we confirm that the LSC flow is dominated by a jump-rope vortex (JRV) mode whose signature frequency is present in velocity and temperature measurements. We further show that both Re and JRV frequency scaling trends are relatively insensitive to container geometry. The temperature and velocity spectra consistently show peaks at the JRV frequency, its harmonic and a secondary mode. The relative strength of these peaks changes and the presence of the secondary peak depend highly on aspect ratio, indicating that, despite having a minimal effect on typical velocities and frequencies, the aspect ratio has a significant effect on the underlying dynamics. Applying a bandpass filter at the secondary frequency to velocity measurements reveals that a clockwise twist in the upper half of the fluid layer coincides with a counterclockwise twist in the bottom half, indicating a torsional mode. For aspect ratio Γ = 3, the unified LSC structure breaks down into multiple rolls in both simulation and experiment.

... Our results are consistent with the fact that, for moderate Rayleigh number range, the viscous BL is still laminar albeit with fluctuations (Tilgner, Belmonte & Libchaber 1993;Verzicco & Camussi 2003;Xia et al. 2003;Funfschilling & Ahlers 2004;Stevens et al. 2012;Scheel & Schumacher 2014). In the recent BL model of Ching et al. (2019) for turbulent RBC, the scaling of the viscous BL thickness is also proportional to Re −0.5 . Outside the viscous BL at z/δ d u = 1.5, the magnitudes of u d and w become larger, which is due to the intense plume emissions outside the BL. ...

... Our result that the scaling exponent being 3.10 ± 0.05 indicates that, within the experimental uncertainty, the higher-order terms (larger than three) also make a non-negligible contribution. We also remark that the scaling exponent for the eddy viscosity is close to that for the eddy thermal diffusivity (Shishkina et al. 2015;Schumacher et al. 2016;Ching et al. 2019). ...

We report an experimental study of the viscous boundary layer (BL) properties of turbulent Rayleigh–Bénard convection in a cylindrical cell. The velocity profile with all three components was measured from the centre of the bottom plate by an integrated home-made particle image velocimetry system. The Rayleigh number $Ra$ varied in the range $1.82 \times 10^8 \le Ra \le 5.26 \times 10^9$ and the Prandtl number $Pr$ was fixed at $Pr = 4.34$ . The probability density function of the wall-shear stress indicates that using the velocity component in the mean large-scale circulation (LSC) plane alone may not be sufficient to characterise the viscous BL. Based on a dynamic wall-shear frame, we propose a method to reconstruct the measured full velocity profile which eliminates the effects of complex dynamics of the LSC. Various BL properties including the eddy viscosity are then obtained and analysed. It is found that, in the dynamic wall-shear frame, the eddy viscosity profiles along the centre line of the convection cell at different $Ra$ all collapse on a single master curve described by $\nu _t^d / \nu = 0.81 (z / \delta _u^d) ^{3.10 \pm 0.05}$ . The Rayleigh number dependencies of several BL quantities are also determined in the dynamic frame, including the BL thickness $\delta _u^d$ ( ${\sim } Ra^{-0.21}$ ), the Reynolds number $Re^d$ ( ${\sim }Ra^{-0.46}$ ) and the shear Reynolds number $Re_s^d$ ( ${\sim } Ra^{0.24}$ ). Within the experimental uncertainty, these scaling exponents are the same as those obtained in the static laboratory frame. Finally, with the measured full velocity profile, we obtain the energy dissipation rate at the centre of the bottom plate $\varepsilon _{w}$ , which is found to follow $\langle \varepsilon _{w} \rangle _t \sim Ra^{1.25}$ .

... A temperature BL equation that takes into account the influence of turbulent fluctuations has been derived along a semiinfinite horizontal heated plate. It has yielded mean temperature [28][29][30] and variance temperature profiles [31,32] . Recently, The temperature variance equations have been derived in the mixing zone and the log layer [33][34][35] . ...

We report the results of the direct numerical simulations of two-dimensional Rayleigh-Bénard convection(RBC) in order to study the influence of the periodic(PD) and confined(CF) samples on the heat transport Nu. The numerical study is conducted with the Rayleigh number(Ra) varied in the range 106≤Ra≤109 at a fixed Prandtl number Pr=4.3 and aspect ratio Γ=2 with the no-slip(NS) and free-slip(FS) plates. There exists a zonal flow for Ra≥3×106 with the free-slip plates in the periodic sample. In all the other cases, the flow is the closed large-scale circulation(closed LSC). The striking features are that the heat transport Nu is influenced and the temperature profiles do not be influenced when the flow pattern is zonal flow.

... 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 ...

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. ...

Convection in the Sun occurs at Rayleigh numbers, Ra, as high as 1022 and molecular Prandtl numbers, Pr, as low as 10−6, 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, to values as low as 10−4, of the turbulent Prandtl number, Prt, 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≡Ra/Pr=109. 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 5×10−4. The three-dimensional simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh number of 105, and 0.005≤Pr≤7. The principal finding is that Prt increases with decreasing Pr in both OB and NOB convection: Prt∼Pr−0.3 for OB convection and Prt∼Pr−1 for the NOB case. The Prt dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence.

... 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. ...

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. ...

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. ...

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. ...

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