Rayleigh–Bénard convection: Improved bounds on the Nusselt number

2011 ◽  
Vol 52 (8) ◽  
pp. 083702 ◽  
Author(s):  
Felix Otto ◽  
Christian Seis
Keyword(s):  
Nusselt Number ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

scholarly journals Bulk scaling in wall-bounded and homogeneous vertical natural convection

2018 ◽  
Vol 841 ◽  
pp. 825-850 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Nusselt Number ◽  
Natural Convection ◽  
Temperature Gradient ◽  
Power Law ◽  
Exact Relation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Previous numerical studies on homogeneous Rayleigh–Bénard convection, which is Rayleigh–Bénard convection (RBC) without walls, and therefore without boundary layers, have revealed a scaling regime that is consistent with theoretical predictions of bulk-dominated thermal convection. In this so-called asymptotic regime, previous studies have predicted that the Nusselt number ($\mathit{Nu}$) and the Reynolds number ($\mathit{Re}$) vary with the Rayleigh number ($\mathit{Ra}$) according to $\mathit{Nu}\sim \mathit{Ra}^{1/2}$ and $\mathit{Re}\sim \mathit{Ra}^{1/2}$ at small Prandtl numbers ($\mathit{Pr}$). In this study, we consider a flow that is similar to RBC but with the direction of temperature gradient perpendicular to gravity instead of parallel to it; we refer to this configuration as vertical natural convection (VC). Since the direction of the temperature gradient is different in VC, there is no exact relation for the average kinetic dissipation rate, which makes it necessary to explore alternative definitions for $\mathit{Nu}$, $\mathit{Re}$ and $\mathit{Ra}$ and to find physical arguments for closure, rather than making use of the exact relation between $\mathit{Nu}$ and the dissipation rates as in RBC. Once we remove the walls from VC to obtain the homogeneous set-up, we find that the aforementioned $1/2$-power-law scaling is present, similar to the case of homogeneous RBC. When focusing on the bulk, we find that the Nusselt and Reynolds numbers in the bulk of VC too exhibit the $1/2$-power-law scaling. These results suggest that the $1/2$-power-law scaling may even be found at lower Rayleigh numbers if the appropriate quantities in the turbulent bulk flow are employed for the definitions of $\mathit{Ra}$, $\mathit{Re}$ and $\mathit{Nu}$. From a stability perspective, at low- to moderate-$\mathit{Ra}$, we find that the time evolution of the Nusselt number for homogenous vertical natural convection is unsteady, which is consistent with the nature of the elevator modes reported in previous studies on homogeneous RBC.

Effect of Sidewall Conductance on Nusselt Number for Rayleigh-Bénard Convection: A Semi-Analytical and Experimental Correction

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
U. Madanan ◽  
R. J. Goldstein
Keyword(s):  
Thermal Conductivity ◽  
Nusselt Number ◽  
Analytical Model ◽  
Thermal Conductance ◽  
Bénard Convection ◽  
Benard Convection ◽  
Extended Surfaces ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Good Agreement

Abstract The effect of sidewall conductance on Nusselt number for the Rayleigh-Bénard convection is examined by performing nearly identical sets of experiments with sidewalls made of three different materials. These experimental results are utilized to extrapolate and estimate the Nusselt number for an ideal zero-thermal-conductivity sidewall case, which is the case when the sidewalls are perfectly insulating. A semi-analytical model is proposed, based on the concept of extended surfaces, to compute the discrepancy in Nusselt number caused by the presence of finite thermal conductance of the sidewalls. The predictions obtained using this model are found to be in good agreement with the corresponding experimentally determined values.

Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio Γ = 0.50 and Prandtl number Pr = 4.38

2011 ◽  
Vol 676 ◽  
pp. 5-40 ◽  
Author(s):  
STEPHAN WEISS ◽  
GUENTER AHLERS
Keyword(s):  
Nusselt Number ◽  
Aspect Ratio ◽  
Prandtl Number ◽  
State Transitions ◽  
Small Scale ◽  
Bénard Convection ◽  
Benard Convection ◽  
Run Time ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Measurements of the Nusselt number and properties of the large-scale circulation (LSC) are presented for turbulent Rayleigh–Bénard convection in water-filled cylindrical containers (Prandtl number Pr = 4.38) with aspect ratio Γ = 0.50. They cover the range 2 × 108 ≲ Ra ≲ 1 × 1011 of the Rayleigh number Ra. We confirm the occurrence of a double-roll state (DRS) of the LSC and focus on the statistics of the transitions between the DRS and a single-roll state (SRS). The fraction of the run time when the SRS existed varied continuously from about 0.12 near Ra = 2 × 108 to about 0.8 near Ra = 1011, while the fraction of the run time when the DRS could be detected changed from about 0.4 to about 0.06 over the same range of Ra. We determined separately the Nusselt number of the SRS and the DRS, and found the former to be larger than the latter by about 1.6% (0.9%) at Ra = 1010 (1011). We report a contribution to the dynamics of the SRS from a torsional oscillation similar to that observed for cylindrical samples with Γ = 1.00. Results for a number of statistical properties of the SRS are reported, and some are compared with the cases Γ = 0.50, Pr = 0.67 and Γ = 1.00, Pr = 4.38. We found that genuine cessations of the SRS were extremely rare and occurred only about 0.3 times per day, which is less frequent than for Γ = 1.00; however, the SRS was disrupted frequently by roll-state transitions and other less well-defined events. We show that the time derivative of the LSC plane orientation is a stochastic variable which, at constant LSC amplitude, is Gaussian distributed. Within the context of the LSC model of Brown & Ahlers (Phys. Fluids, vol. 20, 2008b, art. 075101), this demonstrates that the stochastic force due to the small-scale fluctuations that is driving the LSC dynamics has a Gaussian distribution.

Has the ultimate state of turbulent thermal convection been observed?

2015 ◽  
Vol 785 ◽  
pp. 270-282 ◽  
Author(s):  
L. Skrbek ◽  
P. Urban
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Ultimate State ◽  
Logarithmic Factor ◽  
Aspect Ratios ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

An important question in turbulent Rayleigh–Bénard convection is the scaling of the Nusselt number with the Rayleigh number in the so-called ultimate state, corresponding to asymptotically high Rayleigh numbers. A related but separate question is whether the measurements support the so-called Kraichnan law, according to which the Nusselt number varies as the square root of the Rayleigh number (modulo a logarithmic factor). Although there have been claims that the Kraichnan regime has been observed in laboratory experiments with low aspect ratios, the totality of existing experimental results presents a conflicting picture in the high-Rayleigh-number regime. We analyse the experimental data to show that the claims on the ultimate state leave open an important consideration relating to non-Oberbeck–Boussinesq effects. Thus, the nature of scaling in the ultimate state of Rayleigh–Bénard convection remains open.

Strongly turbulent Rayleigh–Bénard convection in mercury: comparison with results at moderate Prandtl number

1997 ◽  
Vol 335 ◽  
pp. 111-140 ◽  
Author(s):  
S. CIONI ◽  
S. CILIBERTO ◽  
J. SOMMERIA
Keyword(s):  
Nusselt Number ◽  
Prandtl Number ◽  
Scaling Laws ◽  
Convective Cell ◽  
Zone Model ◽  
Turbulent Regime ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

An experimental study of Rayleigh–Bénard convection in the strongly turbulent regime is presented. We report results obtained at low Prandtl number (in mercury, Pr = 0.025), covering a range of Rayleigh numbers 5 × 106 < Ra < 5 × 109, and compare them with results at Pr∼1. The convective chamber consists of a cylindrical cell of aspect ratio 1.Heat flux measurements indicate a regime with Nusselt number increasing as Ra0.26, close to the 2/7 power observed at Pr∼1, but with a smaller prefactor, which contradicts recent theoretical predictions. A transition to a new turbulent regime is suggested for Ra ≃ 2 × 109, with significant increase of the Nusselt number. The formation of a large convective cell in the bulk is revealed by its thermal signature on the bottom and top plates. One frequency of the temperature oscillation is related to the velocity of this convective cell. We then obtain the typical temperature and velocity in the bulk versus the Rayleigh number, and compare them with similar results known for Pr∼1.We review two recent theoretical models, namely the mixing zone model of Castaing et al. (1989), and a model of the turbulent boundary layer by Shraiman & Siggia (1990). We discuss how these models fail at low Prandtl number, and propose modifications for this case. Specific scaling laws for fluids at low Prandtl number are then obtained, providing an interpretation of our experimental results in mercury, as well as extrapolations for other liquid metals.

Bounds on Rayleigh–Bénard convection with an imposed heat flux

2002 ◽  
Vol 473 ◽  
pp. 191-199 ◽  
Author(s):  
JESSE OTERO ◽  
RALF W. WITTENBERG ◽  
RODNEY A. WORTHING ◽  
CHARLES R. DOERING
Keyword(s):  
Heat Flux ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Transport ◽  
Absolute Constant ◽  
Temperature Drop ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We formulate a bounding principle for the heat transport in Rayleigh–Bénard convection with fixed heat flux through the boundaries. The heat transport, as measured by a conventional Nusselt number, is inversely proportional to the temperature drop across the layer and is bounded above according to Nu [les ] cRˆ1/3, where c < 0.42 is an absolute constant and Rˆ = αγβh4/(νκ) is the ‘effective’ Rayleigh number, the non-dimensional forcing scale set by the imposed heat flux κβ. The relation among the parameter Rˆ, the Nusselt number, and the conventional Rayleigh number defined in terms of the temperature drop across the layer, is NuRa = Rˆ, yielding the bound Nu [les ] c3/2Ra1/2.

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