Search for the "ultimate state" in turbulent Rayleigh-Bénard convection

Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Γ follows Ra_{c,Γ}∼Ra_{c,∞}(1+CΓ^{-2})^{2}, with C≲1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32)≤Γ≤32, the rescaling Ra→Ra_{ℓ}≡Ra[Γ^{2}/(C+Γ^{2})]^{3/2} collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Γ). Our findings predict the Γ dependence of the onset of the ultimate regime Ra_{u,Γ}∼[Γ^{2}/(C+Γ^{2})]^{-3/2} in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ=1, 1/2, and 1/3) for the transition in OB RB convection and explains why, in small-Γ cells, much larger Ra (namely, by a factor Γ^{-3}) must be achieved to observe the ultimate regime.

Supergravitational turbulent thermal convection

High-Rayleigh number convective turbulence is ubiquitous in many natural phenomena and in industries, such as atmospheric circulations, oceanic flows, flows in the fluid core of planets, and energy generations. In this work, we present a novel approach to boost the Rayleigh number in thermal convection by exploiting centrifugal acceleration and rapidly rotating a cylindrical annulus to reach an effective gravity of 60 times Earth's gravity. We show that in the regime where the Coriolis effect is strong, the scaling exponent of Nusselt number versus Rayleigh number exceeds one-third once the Rayleigh number is large enough. The convective rolls revolve in prograde direction, signifying the emergence of zonal flow. The present findings open a new avenue on the exploration of high-Rayleigh number turbulent thermal convection and will improve the understanding of the flow dynamics and heat transfer processes in geophysical and astrophysical flows and other strongly rotating systems.

Thermal radiation in Rayleigh-Bénard convection experiments

An important question in turbulent Rayleigh-Bénard convection (RBC) is the effectiveness of convective heat transport, which is conveniently described via the scaling of the Nusselt number (Nu) with the Rayleigh (Ra) and Prandtl (Pr) numbers. In RBC experiments, the heat supplied to the bottom plate is also partly transferred by thermal radiation. This heat transport channel, acting in parallel with the convective and conductive heat transport channels, is usually considered insignificant and thus neglected. Here we present a detailed analysis of conventional far-field as well as strongly enhanced near-field radiative heat transport occurring in various RBC experiments. A careful inclusion of the radiative transport appreciably changes the Nu=Nu(Ra) scaling inferred in turbulent RBC experiments near ambient temperature utilizing gaseous nitrogen and sulfur hexafluoride as working fluids. On the other hand, neither the conventional far-field radiation nor the strongly enhanced near-field radiative heat transport appreciably affects the heat transport law deduced in cryogenic helium RBC experiments.

Scaling of heat flux and energy spectrum for very large Prandtl number convection

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations θ(rms). We complement the analytical work with direct numerical simulations, and show that Nu ∼ Ra(γ) with γ ≈ (0.30-0.32), Pe ∼ Ra(η) with η ≈ (0.57-0.61), and θ(rms) ∼ const. The Nusselt number is observed to be an intricate function of Pe, θ(rms), and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum E(u)(k) ∼ k(-13/3), which is in a very good agreement with our numerical results. The entropy spectrum E(θ)(k), however, exhibits dual branches consisting of k(-2) and k(0) spectra; the k(-2) branch corresponds to the Fourier modes θ[over ̂](0,0,2n), which are approximately -1/(2 nπ). The scaling relations for Prandtl number beyond 10(2) match with those for infinite Prandtl number.

Effect of boundary layers asymmetry on heat transfer efficiency in turbulent Rayleigh-Bénard convection at very high Rayleigh numbers [corrected]

The heat transfer efficiency in turbulent Rayleigh-Bénard convection is investigated experimentally, in a cylindrical cell of height 0.3 m, diameter 0.3 m. We show that for Rayleigh numbers 10(12) < or approximately equal to Ra < or approximately equal to 10(15) the Nusselt number closely follows Nu is proportional to Ra(1/3 if the mean temperature of the working fluid-cryogenic helium gas-is measured by small sensors directly inside the cell at about half of its height. In contrast, if the mean temperature is determined in a conventional way, as an arithmetic mean of the bottom and top plate temperatures, the Nu(Ra) is proportional to Ra(γ) displays spurious crossover to higher γ that might be misinterpreted as a transition to the ultimate Kraichnan regime.

Scalings of field correlations and heat transport in turbulent convection

Using direct numerical simulations of Rayleigh-Bénard convection under free-slip boundary condition, we show that the normalized correlation function between the vertical velocity field and the temperature field, as well as the normalized viscous dissipation rate, scales as Ra-0.22 for moderately large Rayleigh number Ra. This scaling accounts for the Nusselt number Nu exponent of approximately 0.3, as observed in experiments. Numerical simulations also reveal that the aforementioned normalized correlation functions are constants for the convection simulation under periodic boundary conditions.

Efficiency of heat transfer in turbulent Rayleigh-Bénard convection

We present an experimental study of turbulent Rayleigh-Bénard convection (RBC) in a cylindrical cell of height 0.3 m, diameter 0.3 m. It is designed to minimize the influence of its structure on the convective flow of cryogenic (4)He gas of Prandtl number Pr≈1, with the aim of resolving existing contradictions in Nusselt (Nu) versus Rayleigh number (Ra) scaling. For 7.2×10(6)≤Ra≤10(11) our data agree with suitably corrected data from similar cryogenic experiments and are consistent with Nu∝Ra(2/7). On approaching Ra≈10(11) our data display a crossover to Nu∝Ra(1/3) that approximately holds up to Ra=4.6×10(13); there is no sign of a transition to the ultimate Kraichnan regime. Differences in Nu(Ra) scaling observed in similar RBC experiments for Ra≥10(11) cannot be explained due to the difference in Pr, but seem to depend also on experimental details.

Transition phenomena in unstably stratified turbulent flows

We study experimentally and theoretically the transition phenomena caused by external forcing from Rayleigh-Bénard convection with large-scale circulation (LSC) to the limiting regime of unstably stratified turbulent flow without LSC, where the temperature field behaves like a passive scalar. In the experiments we use the Rayleigh-Bénard apparatus with an additional source of turbulence produced by two oscillating grids located near the sidewalls of the chamber. When the frequency of the grid oscillations is larger than 2 Hz, the LSC in turbulent convection is destroyed, and the destruction of the LSC is accompanied by a strong change of the mean temperature distribution. However, in all regimes of the unstably stratified turbulent flow the ratio [(ℓ{x}∇{x}T)²+(ℓ{y}∇{y}T)² + (ℓ{z}∇{z}T)²]/<θ²> varies slightly (even in the range of parameters where the behavior of the temperature field is different from that of the passive scalar). Here ℓ{i} are the integral scales of turbulence along the x,y,z directions, and T and θ are the mean and fluctuating parts of the fluid temperature. At all frequencies of the grid oscillations we have detected long-term nonlinear oscillations of the mean temperature. The theoretical predictions based on the budget equations for turbulent kinetic energy, turbulent temperature fluctuations, and turbulent heat flux, are in agreement with the experimental results.

Structure of viscous boundary layers in turbulent Rayleigh-Bénard convection

Highly resolved profiles of the mean velocity in turbulent Rayleigh-Bénard convection in air are presented and discussed. The present work extends our recently performed experiments at constant aspect ratio [Phys. Rev. Lett. 99, 234504 (2007)] to variable aspect ratios. The experiments cover a range of Rayleigh numbers 10(9)<Ra<10(12) and aspect ratios 1.13<Gamma<11.3 whereas the Prandtl number is fixed at Pr=0.7. The major finding of the present work is that the profiles of the mean horizontal velocity and its fluctuations are virtually invariant against the variation in Ra or Gamma if the wall distance is scaled by the displacement thickness of the boundary layer. Furthermore we have studied typical length scales of the boundary layer and their scaling with Ra and Gamma. Regarding a potential transition of the heat transport toward the ultimate regime we found that the boundary layer Reynolds number remains below Redelta=250 which is significantly lower than the predicted limit.