scholarly journals A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part III. Turbulent Rayleigh–Bénard convection

2017 ◽  
Vol 836 ◽  
pp. 560-598
Author(s):  
Gregory L. Eyink ◽  
Theodore D. Drivas
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Dissipation Rate ◽  
Mixing Time ◽  
Thermal Dissipation ◽  
Fluctuation Dissipation ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Near Wall

A Lagrangian fluctuation–dissipation relation has been derived in a previous work to describe the dissipation rate of advected scalars, both passive and active, in wall-bounded flows. We apply this relation here to develop a Lagrangian description of thermal dissipation in turbulent Rayleigh–Bénard convection in a right-cylindrical cell of arbitrary cross-section, with either imposed temperature difference or imposed heat flux at the top and bottom walls. We obtain an exact relation between the steady-state thermal dissipation rate and the time $\unicode[STIX]{x1D70F}_{mix}$ for passive tracer particles released at the top or bottom wall to mix to their final uniform value near those walls. We show that an ‘ultimate regime’ with the Nusselt number scaling predicted by Spiegel (Annu. Rev. Astron., vol. 9, 1971, p. 323) or, with a log correction, by Kraichnan (Phys. Fluids, vol. 5 (11), 1962, pp. 1374–1389) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time $\unicode[STIX]{x1D70F}_{free}$. Precisely, we show that $\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}=(RaPr)^{1/2}/Nu,$ with $Ra$ the Rayleigh number, $Pr$ the Prandtl number, and $Nu$ the Nusselt number. We suggest a new criterion for an ultimate regime in terms of transition to turbulence of a thermal ‘mixing zone’, which is much wider than the standard thermal boundary layer. Kraichnan–Spiegel scaling may, however, not hold if the intensity and volume of thermal plumes decrease sufficiently rapidly with increasing Rayleigh number. To help resolve this issue, we suggest a program to measure the near-wall mixing time $\unicode[STIX]{x1D70F}_{mix}$, which is precisely defined in the paper and which we argue is accessible both by laboratory experiment and by numerical simulation.

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.

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.

scholarly journals Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection

2010 ◽  
Vol 643 ◽  
pp. 495-507 ◽  
Author(s):  
RICHARD J. A. M. STEVENS ◽  
ROBERTO VERZICCO ◽  
DETLEF LOHSE
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Layer Structure ◽  
Three Dimensional ◽  
Thermal Dissipation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Boundary Layer Profile

Results from direct numerical simulation (DNS) for three-dimensional Rayleigh–Bénard convection in a cylindrical cell of aspect ratio 1/2 and Prandtl number Pr=0.7 are presented. They span five decades of Rayleigh number Ra from 2 × 106 to 2 × 1011. The results are in good agreement with the experimental data of Niemela et al. (Nature, vol. 404, 2000, p. 837). Previous DNS results from Amati et al. (Phys. Fluids, vol. 17, 2005, paper no. 121701) showed a heat transfer that was up to 30% higher than the experimental values. The simulations presented in this paper are performed with a much higher resolution to properly resolve the plume dynamics. We find that in under-resolved simulations the hot (cold) plumes travel further from the bottom (top) plate than in the better-resolved ones, because of insufficient thermal dissipation mainly close to the sidewall (where the grid cells are largest), and therefore the Nusselt number in under-resolved simulations is overestimated. Furthermore, we compare the best resolved thermal boundary layer profile with the Prandtl–Blasius profile. We find that the boundary layer profile is closer to the Prandtl–Blasius profile at the cylinder axis than close to the sidewall, because of rising plumes close to the sidewall.

Direct Numerical Simulation of a Low Prandtl Number Rayleigh–Bénard Convection in a Square Box

2019 ◽  
Vol 11 (6) ◽  
Author(s):  
Ojas Satbhai ◽  
Subhransu Roy ◽  
Sudipto Ghosh
Keyword(s):  
Nusselt Number ◽  
Steady State ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Dimensionless Temperature ◽  
Bénard Convection ◽  
Benard Convection ◽  
Wide Range ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Direct numerical simulations for low Prandtl number fluid (Pr = 0.0216) are used to study the steady-state Rayleigh–Bénard convection (RB) in a two-dimensional unit aspect ratio box. The steady-state RB convection is characterized by analyzing the time-averaged temperature-field, and flow field for a wide range of Rayleigh number (2.1 × 105 ⩽ Ra ⩽ 2.1 × 108). It is seen that the time-averaged and space-averaged Nusselt number (Nuh¯) at the hot-wall monotonically increases with the increase in Rayleigh number (Ra) and the results show a power law scaling Nuh¯∝Ra0.2593. The current Nusselt number results are compared with the results available in the literature. The complex flow is analyzed by studying the frequency power spectra of the steady-state signal of the vertical velocity at the midpoint of the box for different Ra and probability density function of dimensionless temperature at various locations along the midline of the box.

Sign in / Sign up

Export Citation Format

Share Document