Dynamics Of Large-scale Circulation Research Articles

Unravelling the large-scale circulation modes in turbulent Rayleigh-Bénard convection (a) Contribution to the Focus Issue Turbulent Thermal Convection edited by Mahendra Verma and Jörg Schumacher.

The large-scale circulation (LSC) is the most fundamental turbulent coherent flow structure in Rayleigh-Bénard convection. Further, LSCs provide the foundation upon which superstructures, the largest observable features in convective systems, are formed. In confined cylindrical geometries with diameter-to-height aspect ratios of , LSC dynamics are known to be governed by a quasi-two-dimensional, coupled horizontal sloshing and torsional (ST) oscillatory mode. In contrast, in cylinders, a three-dimensional jump rope vortex (JRV) motion dominates the LSC dynamics. Here, we use dynamic mode decomposition (DMD) on direct numerical simulation data of liquid metal to show that both types of modes co-exist in and cylinders but with opposite dynamical importance. Furthermore, with this analysis, we demonstrate that ST oscillations originate from a tilted elliptical mean flow superposed with a symmetric higher-order mode, which is connected to the four rolls in the plane perpendicular to the LSC in tanks.

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.

Dynamics of large-scale circulation and energy transfer mechanism in turbulent Rayleigh–Bénard convection in a cubic cell

We present the characteristics and dynamics of large-scale circulation (LSC) in turbulent Rayleigh–Bénard convection (RBC) inside a cubic cell. The simulations are carried out for a Rayleigh number range of 2 × 106 ≤ Ra ≤ 109 and using air (at Prandtl number Pr = 0.7) as the working fluid. Using the Fourier mode analysis, the strength, orientation, and associated dynamics of LSC are characterized. Following previous two-dimensional studies in RBC, we propose a mechanism of flow reversals based on the dynamics of corner vortices, which is less attempted in three-dimensional counterparts. We observe that the plane containing LSC is generally aligned along one of the diagonals of the box accompanied by a four-roll structure in the other. In addition to the primary roll, two secondary corner-roll structures are also observed in the LSC plane, which grow in size and destabilize the LSC, resulting in partial (ΔΦ1 ≈ π/2) and complete (ΔΦ1 ≈ π) reversals. In addition to previously reported rotation-led reorientations, we also observe cessation events that are rare in cubic cells. We observe that as the Rayleigh number is increased from Ra = 2 × 106 to 107, the number of reorientations reduces by one third. With an increase in Ra, the strength of LSC (SLSC) increases and the corner rolls reduce in size, which leads to the reduction in the occurrence of reorientations. At higher Rayleigh numbers (Ra > 108), the strength saturates around SLSC ≈ 0.75. To connect the dynamics between different coherent structures, we evaluate the turbulent kinetic energy (TKE) budget. Notably, our novel approach to study the variation of TKE along the azimuthal direction helps in identifying the dynamical coupling between the LSC and non-LSC planes. The analysis suggests that TKE is generally produced in localized regions in both the planes, while its dissipation mainly happens in the vicinity of the plane that contains LSC. The transport mechanism redistributes the energy between these planes and thus sustains the LSC and other coherent structures.

Dynamics and statistics of reorientations of large-scale circulation in turbulent rotating Rayleigh-Bénard convection

We present a direct numerical simulation to investigate the dynamics and statistics of reorientations of large-scale circulation (LSC) in turbulent rotating Rayleigh-Bénard convection for air (Pr = 0.7) contained in a cylindrical cell with unit aspect ratio. A wide range of rotation rates (0 ≤ Ro−1 ≤ 30) is considered for two different Rayleigh numbers Ra = 2 × 106 and 2 × 107. Using the Fourier mode analysis of time series data obtained from the different probes placed in the azimuthal direction of the container at the midplane, the orientation and associated dynamics of LSC are characterized. The amplitude of the first Fourier mode quantifies the strength of LSC, and its phase Φ1 gives the information on the azimuthal orientation of LSC. Based on the energy contained in the Fourier modes, different flow regimes are identified as the rotation rate is varied for a given Rayleigh number. The LSC structure is observed in the low rotation regime (Ro−1 ≲ 1), while the presence of other flow structures, namely, quadrupolar and sextupolar, is obtained at high rotation rates. In the LSC regime, a strong correlation between the orientation of LSC structure and the heat transfer and boundary layer dynamics is observed. At low rotation rates, the dissipation rates follow the log-normal behavior, while at higher rotation rates, a clear departure from log-normality is noted. Different types of reorientations, namely, rotation-led, cessation-led, partial, and complete reversal, are identified. The distribution of change in orientation of LSC follows a power law behavior as P(|ΔΦ1|) ∝|ΔΦ1|−m, with the exponent m ≈ 3.7. In addition, the statistics of time interval between successive reorientations follow a Poisson distribution. These observations are in good agreement with earlier experimental results.

On the nature of fluctuations in turbulent Rayleigh–Bénard convection at large Prandtl numbers

We report experimental results for the power spectra, variance, skewness and kurtosis of temperature fluctuations in turbulent Rayleigh–Bénard convection (RBC) of a fluid with Prandtl number $Pr=12.3$ in cylindrical samples with aspect ratios $\unicode[STIX]{x1D6E4}$ (diameter $D$ over height $L$) of 0.50 and 1.00. The measurements were primarily for the radial positions $\unicode[STIX]{x1D709}=1-r/(D/2)=1.00$ and $\unicode[STIX]{x1D709}=0.063$. In both cases, data were obtained at several vertical locations $z/L$. For all locations, there is a frequency range of about a decade over which the spectra can be described well by the power law $P(f)\sim f^{-\unicode[STIX]{x1D6FC}}$. For all $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D6E4}$, the $\unicode[STIX]{x1D6FC}$ value is less than one near the top and bottom plates and increases as $z/L$ or $1-z/L$ increase from 0.01 to 0.5. This differs from the finding for$Pr=0.8$ (He et al., Phys. Rev. Lett., vol. 112, 2014, 174501) and the expectation for the downstream velocity of turbulent wall-bounded shear flow (Rosenberg, J. Fluid Mech., vol. 731, 2013, pp. 46–63), where $\unicode[STIX]{x1D6FC}=1$ is found or expected in an inner layer ($0.01\lesssim z/L\lesssim 0.1$) near the wall but in the bulk. The variance is described better by a power law $\unicode[STIX]{x1D70E}^{2}\sim (z/L)^{-\unicode[STIX]{x1D701}}$ than by the logarithmic dependence found or expected for $Pr=0.8$ and for turbulent shear flow. For both $\unicode[STIX]{x1D6E4}$, we found that, independent of Rayleigh number, $\unicode[STIX]{x1D701}\simeq 2/3$ near the sidewall ($\unicode[STIX]{x1D709}=0.063$), where plumes primarily rise or fall and the large-scale circulation (LSC) dynamics is most influential. This result agrees with a model due to Priestley (Turbulent Transfer in the Lower Atmosphere, 1959, University of Chicago Press) for convection over a horizontal heated surface. However, we found $\unicode[STIX]{x1D701}\simeq 1$ along the sample centreline ($\unicode[STIX]{x1D709}=1.00$), where there are relatively few plumes moving vertically and the LSC dynamics is expected to be less important; that result is consistent with one of two possible interpretations by Adrian (Intl J. Heat Mass Transfer, vol. 39, 1996, pp. 2303–2310) of a model due to Libchaber et al. (J. Fluid Mech., vol. 204, 1989, pp. 1–30). We discuss the composite nature of fluctuations in turbulent RBC, with contributions from intrinsic background fluctuations, plumes, the stochastic dynamics of the LSC, and the sloshing and torsional mode of the LSC. None of the models advanced so far explicitly consider all of these contributions.

Large-scale circulation classification and its links to observed precipitation in the eastern and central Tibetan Plateau

The relationship between the large-scale circulation dynamics and regional precipitation regime in the Tibetan Plateau (TP) has so far not been well understood. In this study, we classify the circulation types using the self-organizing maps based on the daily field of 500 hPa geopotential height and link them to the precipitation climatology in the eastern and central TP. By virtue of an objective determining method, 18 circulation types are quantified. The results show that the large amount of precipitation in summer is closely related to the circulation types in which the enhanced and northward shifted subtropical high (SH) over the northwest Pacific and the obvious cyclconic circulation anomaly over the Bay of Bengal are helpful for the Indian summer monsoon and East Asian summer monsoon to take abundant low-latitude moisture to the eastern and southern TP. On the contrary, the dry winter in the central and eastern Tibet corresponds to the circulation types with divergence over the central and eastern TP and the water vapor transportations of East Asian winter monsoon and mid-latitude westerly are very weak. Some circulation types are associated with some well-known circulation patterns/monsoons influencing the TP (e.g. East Atlantic Pattern, El Nino Southern Oscillation, Indian Summer Monsoon and the mid-latitude westerly), and exhibit an overall good potential for explaining the variability of regional seasonal precipitation. Moreover, the climate shift signals in the late 1970s over the eastern Pacific/North Pacific Oceans could also be reflected by both the variability of some circulation types and their correspondingly composite precipitations. This study extends our understandings for the large-scale atmospheric dynamics and their linkages with regional precipitation and is beneficial for the climate change projection and related adaptation activities in the highest and largest plateau in the world.

Dynamics of large-scale circulation of turbulent thermal convection in a horizontal cylinder

AbstractA systematic study of the effects of cell geometry on the dynamics of large-scale flows in turbulent thermal convection is carried out in horizontal cylindrical cells of different lengths filled with water. Four different flow modes are identified with increasing aspect ratio$\Gamma $. For small aspect ratios ($\Gamma \leq 0.16$), the flow is highly confined in a thin disc-like cell with a quasi-two-dimensional (quasi-2D) large-scale circulation (LSC) in the circular plane of the cell. For larger aspect ratios ($\Gamma >0.16$), we observe periodic switching of the angular orientation$\theta $of the rotation plane of LSC between the two longest diagonals of the cell. The sides of the container along which the LSC oscillates changes at a critical aspect ratio$\Gamma _{c}\simeq 0.82$. The measured switching period is equal to the LSC turnover time for$\Gamma \leq \Gamma _c$, shows a sharp increase at$\Gamma _{c}$and decays exponentially to the LSC turnover time with increasing$\Gamma $. For$\Gamma \geq 1.3$, a periodic rocking of LSC along the long axis of the cylinder is also observed. The measured probability density function$P(\theta )$of the LSC orientation$\theta $peaks at the two diagonal positions, and its shape is described by a phenomenological model proposed by Brown & Ahlers (Phys. Fluids, vol. 20, 2008b, 075101; J. Fluid Mech., vol. 638, 2009, pp. 383–400). Using this model, we describe the dynamics of the LSC orientation$\theta $by stochastic motion in a double-well potential. The potential is predicted from a model in which the sidewall shape produces an orientation-dependent pressure on the LSC. This model also captures key features of the four flow modes. The experiment reveals an interesting array of rich dynamics of LSC in the horizontal cylinders, which are very different from those observed in the upright cylindrical convection cells. The success of the model for both upright and horizontal cylinders suggests that it can be applied to different geometries.

Rare Fluctuations and Large-Scale Circulation Cessations in Turbulent Convection

In turbulent Rayleigh-Bénard convection, a large-scale circulation (LSC) develops in a nearly vertical plane and is maintained by rising and falling plumes detaching from the unstable thermal boundary layers. Rare but large fluctuations in the LSC amplitude can lead to extinction of the LSC (a cessation event), followed by the reemergence of another LSC with a different (random) azimuthal orientation. We extend previous models of the LSC dynamics to include momentum and thermal diffusion in the azimuthal plane and calculate the tails of the probability distributions of both the amplitude and azimuthal angle. Our analytical results are in very good agreement with the experimental data.

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

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.

Experimental study of low-frequency oscillations and large-scale circulations in turbulent mixed convection

The formation and dynamics of large-scale circulations in forced and mixed convection has been studied at ambient and elevated fluid pressure by means of particle image velocimetry and temperature measurements. The study has been conducted in two rectangular containers of the same shape and aspect ratios of Γ xz = 1 and Γ yz = 5. For the measurements at high fluid pressure the dimensions of the cell have been scaled down by a factor of 5. Air with Pr = 0.7 has been used as fluid in both configurations. Forced convection has been investigated at Re = 1.01 × 10 4 and mixed convection has been studied at Ar = 3.3, Re = 1.01 × 10 4 and Ra = 2.4 × 10 8. In this configuration low-frequency oscillations in the heat transfer between the inlet and outlet have been found for mixed convection. Instantaneous velocity vector fields obtained from particle image velocimetry have been analysed using proper orthogonal decomposition and an algorithm to detect the core and the core centre position of large-scale circulations.

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.