Large-scale Circulation In Turbulent Rayleigh Research Articles

Stabilizing/destabilizing the large-scale circulation in turbulent Rayleigh–Bénard convection with sidewall temperature control

Abstract

Structure and Dynamics of Large-Scale Circulation in Turbulent Convection at High Prandtl Numbers

The structure and dynamics of the large-scale circulation in turbulent Rayleigh—Benard convection in a cubic cell are experimentally investigated for fluids with moderate and high Prandtl numbers (from 3.5 to 64). The decomposition of two-dimensional velocity fields into Fourier modes shows that with variation of the Prandtl number from 3.5 to 24 one large-scale mode predominates in the flow. With further increase in the Prandtl number the flow structure considerably changes and the energy is redistributed between several lower Fourier modes. The temporal dynamics of the predominant mode is also considerably changed.

Effects of tilt on the orientation dynamics of the large-scale circulation in turbulent Rayleigh–Bénard convection

We experimentally test the effects of tilting a turbulent Rayleigh–Bénard convection cell on the dynamics of the large-scale circulation (LSC) orientation θ0. The probability distribution of θ0 is measured and used to obtain a tilt-induced potential acting on θ0, which is used in a low-dimensional model of diffusion of θ0 in a potential. The form of the potential is sinusoidal in θ0 and linear in tilt angle for small tilt angles, which is explained by a simple geometric model of the vector direction of the mean buoyancy force acting on the LSC. However, the magnitude of the tilt-induced forcing is found to be two orders of magnitude larger than previously predicted. When this parameter is adjusted to match the values obtained from the probability distribution of θ0, the diffusive model can quantitatively predict the effects of tilt on θ0. In particular, tilt causes a change in potential barrier height between neighboring corners of a cubic cell, and changes in the barrier-crossing rate for θ0 to escape a corner are predicted with an accuracy of ±30%. As a cylindrical cell is tilted, the tilt-induced potential provides a restoring force that induces oscillations when it exceeds the strength of damping; this critical tilt angle is predicted within 20%, and the prediction is consistent with the measured oscillation frequencies. These observations show that a self-consistent low-dimensional model can be extended to include the dynamics of θ0 due to tilt. However, the underprediction of the effect of tilt on θ0 warrants revisiting the predicted magnitude.

Reduced flow reversals in turbulent convection in the absence of corner vortices

We report a comparative experimental study of the reversal of the large-scale circulation in turbulent Rayleigh–Benard convection in a quasi-two-dimensional corner-less cell where the corner vortices are absent and in a quasi-two-dimensional normal cell where the corner vortices are present. It is found that in the corner-less cell the reversal frequency exhibits a slow decrease followed by a fast decrease with increasing Rayleigh number , separated by a transitional ( ). The transition is similar to that in the normal cell, and is almost the same for both cells. Despite the similarities, the reversal frequency is greatly reduced in the corner-less cell. The reduction of the reversal frequency is more significant, in terms of both the amplitude and the scaling exponent, in the high- regime. In addition, we classified the reversals into main-vortex-led and corner-vortex-led, and found that both types exist in the normal cell while only the former exists in the corner-less cell. The frequency of main-vortex-led reversal in the normal cell is found to be in excellent agreement with the frequency of reversals in the corner-less cell. Our results reveal for the first time the quantitative role of the corner vortices in the occurrence of the reversals of the large-scale circulation.

Dynamics of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation

We present measurements of the azimuthal rotation velocity $\dot{{\it\theta}}(t)$ and thermal amplitude ${\it\delta}(t)$ of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation. Both $\dot{{\it\theta}}(t)$ and ${\it\delta}(t)$ exhibit clear oscillations at the modulation frequency ${\it\omega}$. Fluid acceleration driven by oscillating Coriolis force causes an increasing phase lag in $\dot{{\it\theta}}(t)$ when ${\it\omega}$ increases. The applied modulation produces oscillatory boundary layers and the resulting time-varying viscous drag modifies ${\it\delta}(t)$ periodically. Oscillation of $\dot{{\it\theta}}(t)$ with maximum amplitude occurs at a finite modulation frequency ${\it\omega}^{\ast }$. Such a resonance-like phenomenon is interpreted as a result of optimal coupling of ${\it\delta}(t)$ to the modulated rotation velocity. We show that an extended large-scale circulation model with a relaxation time for ${\it\delta}(t)$ in response to the modulated rotation provides predictions in close agreement with the experimental results.

Oscillations of the large-scale circulation in turbulent Rayleigh–Bénard convection: the sloshing mode and its relationship with the torsional mode

We report an experimental study of the large-scale circulation (LSC) in a turbulent Rayleigh–Bénard convection cell with aspect ratio unity. The temperature-extrema-extraction (TEE) method for obtaining the dynamic information of the LSC is presented. With this method, the azimuthal angular positions of the hot ascending and cold descending flows along the sidewall are identified from the measured instantaneous azimuthal temperature profile. The motion of the LSC is then decomposed into two different modes based on these two angles: the azimuthal mode and the translational or sloshing mode that is perpendicular to the vertical circulation plane of the LSC. Comparing to the previous sinusoidal-fitting (SF) method, it is found that both the TEE and the SF methods give the same information about the azimuthal motion of the LSC, but the TEE method in addition can provide information about the sloshing motion of the LSC. The sloshing motion is found to oscillate time-periodically around the cell's central vertical axis with an amplitude being nearly independent of the turbulent intensity and to have a π/2 phase difference with the torsional mode. It is further found that the azimuthal angular positions of the hot ascending and cold descending flows oscillate out of phase with each other by π, which leads to the observations of the torsional mode when these two flows are near the top and the bottom plates, respectively, and of the sloshing mode when they are both near the mid-height plane. A direct velocity measurement further confirms the existence of the bulk sloshing mode of the flow field.

Azimuthal asymmetries of the large-scale circulation in turbulent Rayleigh–Bénard convection

Previously we published a dynamical model [E. Brown and G. Ahlers, Phys. Fluids 20, 075101 (2008)] for the large-scale-circulation (LSC) dynamics of Rayleigh–Bénard convection in cylindrical containers. The model consists of a pair of stochastic ordinary differential equations, motivated by the Navier–Stokes equations, one each for the strength δ and the orientation θ0 of the LSC. Here we extend it to cases where the rotational invariance of the system is broken by one of several physically relevant perturbations. As an example of this symmetry breaking we present experimental measurements of the LSC dynamics for a container tilted relative to gravity. In that case the model predicts that the buoyancy of the thermal boundary layers encourages fluid to travel along the steepest slope, that it locks the LSC in this direction, and that it strengthens the flow, as seen in experiments. The increase in LSC strength is shown to be responsible for the observed suppression of cessations and azimuthal fluctuations. We predict and observe that for large enough tilt angles, the restoring force that aligns the flow with the slope is strong enough to cause oscillations of the LSC around this orientation. This planar oscillation mode is different from coherent torsional oscillations that have been observed previously. The model was applied also to containers with elliptical cross sections and predicts that the pressure due to the side wall forces the flow into a preferred orientation in the direction of the longest diameter. When the ellipticity is large enough, then oscillations around this orientation are predicted. The model shows that various azimuthal asymmetries will lock the LSC orientation. However, only those that act on the δ-equation (such as tilting relative to gravity) will enhance the LSC strength and suppress cessations and other azimuthal dynamics. Those that affect only the θ0 equation, such as an interaction with Earth’s Coriolis force, will align the flow but will not influence its strength and the frequency of cessations.

A model of diffusion in a potential well for the dynamics of the large-scale circulation in turbulent Rayleigh–Bénard convection

Experimental measurements of properties of the large-scale circulation (LSC) in turbulent convection of a fluid heated from below in a cylindrical container of aspect ratio 1 are presented and used to test a model of diffusion in a potential well for the LSC. The model consists of a pair of stochastic ordinary differential equations motivated by the Navier–Stokes equations. The two coupled equations are for the azimuthal orientation θ0 and for the azimuthal temperature amplitude δ at the horizontal midplane. The dynamics is due to the driving by Gaussian distributed white noise that is introduced to represent the action of the small-scale turbulent fluctuations on the large-scale flow. Measurements of the diffusivities that determine the noise intensities are reported. Two time scales predicted by the model are found to be within a factor of 2 or so of corresponding experimental measurements. A scaling relationship predicted by the model between δ and the Reynolds number is confirmed by measurements over a large experimental parameter range. The Gaussian peaks of probability distributions p(δ) and p(θ̇0) are accurately described by the model; however, the non-Gaussian tails of p(δ) are not. The frequency, angular change, and amplitude behavior during cessations are accurately described by the model when the tails of the probability distribution of δ are used as experimental input.

Torsional oscillations of the large-scale circulation in turbulent Rayleigh–Bénard convection

Measurements over the Rayleigh-number range 108 ≲ R ≲ 1011 and Prandtl-number range 4.4≲σ≲29 that determine the torsional nature and amplitude of the oscillatory mode of the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection are presented. For cylindrical samples of aspect ratio Γ=1 the mode consists of an azimuthal twist of the near-vertical LSC circulation plane, with the top and bottom halves of the plane oscillating out of phase by half a cycle. The data for Γ=1 and σ=4.4 showed that the oscillation amplitude varied irregularly in time, yielding a Gaussian probability distribution centred at zero for the displacement angle. This result can be described well by the equation of motion of a stochastically driven damped harmonic oscillator. It suggests that the existence of the oscillations is a consequence of the stochastic driving by the small-scale turbulent background fluctuations of the system, rather than a consequence of a Hopf bifurcation of the deterministic system. The power spectrum of the LSC orientation had a peak at finite frequency with a quality factor Q≃5, nearly independent of R. For samples with Γ≥2 we did not find this mode, but there remained a characteristic periodic signal that was detectable in the area density ρp of the plumes above the bottom-plate centre. Measurements of ρp revealed a strong dependence on the Rayleigh number R, and on the aspect ratio Γ that could be represented by ρp ~ Γ2.7±0.3. Movies are available with the online version of the paper.

Rotations and cessations of the large-scale circulation in turbulent Rayleigh–Bénard convection

We present a broad range of measurements of the angular orientation increased as the circulation strength of the LSC decreased. Tilting the sample relative to gravity significantly reduced the frequency of occurrence of both rotations and cessations.