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

Brown, Eric; Ahlers, Guenter

doi:10.1063/1.2919806

Cited by 73 publications

(206 citation statements)

References 45 publications

(132 reference statements)

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“…Broadly, these works involve either stochasticity (e.g., "stochastic resonance" [8,13]), or low-dimensional models with noise [14,15]. Mishra et al [10] studied the large-scale modes of RBC in a cylindrical geometry and showed that the dipolar mode decreases in amplitude and the quadrupolar mode increases during the cessation-led reversals.…”

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confidence: 99%

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Dynamics and symmetries of flow reversals in turbulent convection

2011

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Based on direct numerical simulations and symmetry arguments, we show that the large-scale Fourier modes are useful tools to describe the flow structures and dynamics of flow reversals in Rayleigh-Bénard convection (RBC). We observe that during the reversals, the amplitude of one of the large-scale modes vanishes, while another mode rises sharply, very similar to the "cessation-led" reversals observed earlier in experiments and numerical simulations. We find anomalous fluctuations in the Nusselt number during the reversals. Using the structures of the RBC equations in the Fourier space, we deduce two symmetry transformations that leave the equations invariant. These symmetry transformations help us in identifying the reversing and non-reversing Fourier modes.PACS numbers: 47.55. 47.27.De, Many experiments [1][2][3][4][5][6][7] and numerical simulations [5,[8][9][10] on turbulent convection reveal that the velocity field of the system reverses randomly in time (also see review articles [11]). This phenomenon, known as "flow reversal", remains ill understood. This process gains practical importance due to its similarities with the magnetic field reversals in geodynamo and solar dynamo [12]. In this letter, we study the dynamics and symmetries of flow reversals in turbulent convection using the large-scale Fourier modes of the velocity and temperature fields.The experiments and simulations performed to explore the nature of flow reversals are typically for an idealized convective system called Rayleigh-Bénard convection (RBC) in which a fluid confined between two plates is heated from below and cooled at the top. Detailed measurements show that the first Fourier mode vanishes abruptly during some reversals [3,4]. These reversals are referred to as "cessation-led". Recently Sugiyama et al. [5] performed RBC experiments on water in a quasi two-dimensional box, and observed flow reversals with the flow profile dominated by a diagonal large-scale roll and two smaller secondary rolls at the corners. They attribute the flow reversals to the growth of the two smaller corner rolls as a result of plume detachments from the boundary layers.Several theoretical studies performed to understand reversals in RBC provide important clues. Broadly, these works involve either stochasticity (e.g., "stochastic resonance" [8, 13]), or low-dimensional models with noise [14,15]. Mishra et al.[10] studied the large-scale modes of RBC in a cylindrical geometry and showed that the dipolar mode decreases in amplitude and the quadrupolar mode increases during the cessation-led reversals. Regarding dynamo, low-dimensional models constructed using the large-scale modes and symmetry arguments reproduced dynamo reversals successfully [7,16].The theoretical models described above only focus on the large-scale modes. Here too, they provide limited information about these modes due to small number of measuring probes. In this letter we compute large-scale and intermediate-scale Fourier modes accurately using the complete flow profile. This helps us ...

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“…Broadly, these works involve either stochasticity (e.g., "stochastic resonance" [8, 13]), or low-dimensional models with noise [14,15]. Mishra et al…”

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confidence: 99%

Dynamics and symmetries of flow reversals in turbulent convection

2011

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“…Recent studies also report that the LSC flow-reversals and its tilted orientation are strongly influenced by the corner-flows that form in a rectangular geometry [12]. The dynamics of a nearly-vertical, single LSC can be modeled by a set of nonlinear stochastic differential equations that describe the amplitude of azimuthal temperature variations, δ, and the azimuthal orientation angle, θ 0 [13,14]. This model is found to be in excellent agreement with the typical fluctuations of the LSC [13,14].…”

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confidence: 99%

“…The dynamics of a nearly-vertical, single LSC can be modeled by a set of nonlinear stochastic differential equations that describe the amplitude of azimuthal temperature variations, δ, and the azimuthal orientation angle, θ 0 [13,14]. This model is found to be in excellent agreement with the typical fluctuations of the LSC [13,14]. Yet, the cessations require an extension of the model, since boundary terms describing the thermal and viscous diffusion become dominant terms when the amplitudes are small, as they inevitably must be during a cessation.…”

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Effect of Weak Rotation on Large-Scale Circulation Cessations in Turbulent Convection

2012

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We investigate the effect of weak rotation on the large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection, using the theory for cessations in a low-dimensional stochastic model of the flow previously studied. We determine the cessation frequency of the LSC as a function of rotation, and calculate the statistics of the amplitude and azimuthal velocity fluctuations of the LSC as a function of the rotation rate for different Rayleigh numbers. Furthermore, we show that the tails of the reorientation probability distribution function remain unchanged for rotating systems, while the distribution of the LSC amplitude and correspondingly the cessation frequency are strongly affected by rotation. Our results are in close agreement with experimental observations. PACS numbers: 47.27.te, 05.65.+b, 47.27.eb The complex phenomenon of thermal turbulence, also known as turbulent Rayleigh-Bénard convection (RBC), develops in a heated fluid layer under gravity by a succession of instabilities in the thermal transport due to an interplay between different driving forces such as buoyancy, viscous drag, and diffusion. This balance between different transport mechanisms is quantified by the Rayleigh number Ra = α 0 g∆T L 3 /νκ determining the flow state. Here, α 0 is the isobaric thermal expansion coefficient, g is the gravity field, ∆T is the temperature gap between bottom and top plates, L is the height of the fluid container, κ is the thermal diffusivity and ν is the kinematic viscosity. When Ra is sufficiently large, the flow becomes turbulent and a large-scale circulation (LSC) is formed. The latter is maintained by the emission of plumes from the top and bottom surfaces which, due to buoyancy, move upwards (hot plumes) or downwards (cold plumes) [1][2][3]. This LSC is known to appear in various rotating natural systems, such as atmospheric [4] and oceanic flows [5], and the dynamo driving planetary magnetic fields [6].Such a circulating state does not persist indefinitely, however, and cessations followed by restarted flows at a new azimuthal angle occur sporadically [3,[7][8][9][10]. The complex dynamics of the LSC is influenced by the heat and momentum transport mechanisms, as well as by the geometry and aspect ratio Γ (diameter over height) of the fluid container. In experiments using a cylindrical geometry with Γ = 1, the LSC occurs in a nearly vertical plane, whereas for aspect ratio Γ = 0.5, two convection rolls may coexist [11]. Recent studies also report that the LSC flow-reversals and its tilted orientation are strongly influenced by the corner-flows that form in a rectangular geometry [12]. The dynamics of a nearly-vertical, single LSC can be modeled by a set of nonlinear stochastic differential equations that describe the amplitude of azimuthal temperature variations, δ, and the azimuthal orientation angle, θ 0 [13,14]. This model is found to be in excellent agreement with the typical fluctuations of the LSC [13,14]. Yet, the cessations require an extension of the model, since boundary terms describ...

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“…This ordering influence stops when the large-scale circulation decelerates strongly and becomes re-oriented. Such events have been studied statistically in experiments [27][28][29], numerically in two-dimensional [24,30,31] or three-dimensional [32] convection as well as in low-dimensional models [33].…”

Section: Plume Collision Due To Transition Of Large-scale Flowmentioning

confidence: 99%

Extreme dissipation event due to plume collision in a turbulent convection cell

Schumacher

Scheel

2016

Phys. Rev. E

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An extreme dissipation event in the bulk of a closed three-dimensional turbulent convection cell is found to be correlated with a strong reduction of the large-scale circulation flow in the system that happens at the same time as a plume emission event from the bottom plate. The reduction in the large-scale circulation opens the possibility for a nearly frontal collision of down-and upwelling plumes and the generation of a high-amplitude thermal dissipation layer in the bulk. This collision is locally connected to a subsequent highamplitude energy dissipation event in the form of a strong shear layer. Our analysis illustrates the impact of transitions in the large-scale structures on extreme events at the smallest scales of the turbulence, a direct link that is observed in a flow with boundary layers. We also show that detection of extreme dissipation events which determine the far-tail statistics of the dissipation fields in the bulk requires long-time integrations of the equations of motion over at least hundred convective time units.

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

Cited by 73 publications

References 45 publications

Dynamics and symmetries of flow reversals in turbulent convection

Dynamics and symmetries of flow reversals in turbulent convection

Effect of Weak Rotation on Large-Scale Circulation Cessations in Turbulent Convection

Extreme dissipation event due to plume collision in a turbulent convection cell

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