Turbulent Rayleigh–Bénard convection in gaseous and liquid He

Chavanne, Xavier; Chillà, Francesca; Chabaud, Benoît; Castaing, B.; Hébral, B.

doi:10.1063/1.1355683

Cited by 229 publications

(304 citation statements)

References 33 publications

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“…Other experimental results have indicated departure from the trend in Fig. 2 at Ra > 10 11 [28] or Ra > 10 12 -10 13 [29]. Our results are not significantly different from the latter of these.…”

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confidence: 38%

Completing the Mechanical Energy Pathways in Turbulent Rayleigh-Bénard Convection

2013

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A new, more complete view of the mechanical energy budget for Rayleigh-Bénard convection is developed and examined using three-dimensional numerical simulations at large Rayleigh numbers and Prandtl number of 1. The driving role of available potential energy is highlighted. The relative magnitudes of different energy conversions or pathways change significantly over the range of Rayleigh numbers Ra $ 10 7 -10 13 . At Ra < 10 7 small-scale turbulent motions are energized directly from available potential energy via turbulent buoyancy flux and kinetic energy is dissipated at comparable rates by both the largeand small-scale motions. In contrast, at Ra ! 10 10 most of the available potential energy goes into kinetic energy of the large-scale flow, which undergoes shear instabilities that sustain small-scale turbulence. The irreversible mixing is largely confined to the unstable boundary layer, its rate exactly equal to the generation of available potential energy by the boundary fluxes, and mixing efficiency is 50%. Introduction.-Flow in a horizontal layer of fluid heated uniformly from below and cooled from above-RayleighBénard convection (RBC)-is an idealized problem from which much can be learned about the nature of convective flow and heat transfer. Recent attention has focused on the relative importance of the boundary layer and interior behavior, including flow structures, at very large Rayleigh numbers [1][2][3][4][5][6]. The salient features of the flow can be usefully interpreted in terms of the energy budget, and previous work has considered the kinetic and thermal energy [2,3,7,8]. However, the existing framework is incomplete and provides limited insight into the exchange of mechanical energy among different forms or between fluid motions of different scales. In particular, the nature of the turbulent energy cascade in RBC is still not fully understood [3] and improved models are needed.Convection is caused by the generation of mechanical energy in the form of available potential energy (APE) by boundary buoyancy fluxes. APE is a subset of the total gravitational potential energy (GPE) and is the form essential for motions [9][10][11]. A complete mechanical energy framework has been recently developed for the case of horizontal convection forced by differential heating at one horizontal boundary [12,13] and recently extended to RBC [14]. In RBC there is a release of APE from thermal boundary layers adjacent to the top and bottom boundaries, driving flow at various scales, while complex feedbacks further distribute the potential and kinetic energy among the different scales or, through mixing, to the background potential energy. The role of APE in RBC was recognized in quantifying the overall rate of irreversible mixing [14]. Here we show that an account of GPE, its partition into available and background forms, and its links to kinetic energy at various scales are essential in a full understanding of the turbulent energy cascade and the relative roles of the boundary layer and the interior.

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confidence: 38%

Completing the Mechanical Energy Pathways in Turbulent Rayleigh-Bénard Convection

2013

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“…The obtained value of γ is slightly smaller than the Corrsin-Oboukhov value of γ = 5/3 for passive scalars in a turbulent flow at sufficiently large Reynolds numbers [6]. Possible reasons for this deviation have been discussed elsewhere [18,19].…”

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confidence: 75%

Kraichnan’s random sweeping hypothesis in homogeneous turbulent convection

Tong

2011

Phys. Rev. E

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We report an experimental study of the temperature space-time cross-correlation function, C T (r,τ ), in the central region of turbulent Rayleigh-Bénard convection. The measured C T (r,τ ) is found to have the scaling form C T (r E ,0) in turbulent Rayleigh-Bénard convection, where a fluid layer of thickness H is heated from below and cooled from the top. In the above, δT is the local temperature deviation from the mean and (σ T ) i is its standard deviation at position i. When the Rayleigh number [2] Ra 10 8 , the convective flow becomes turbulent. The flow in the closed convection cell is known to be inhomogeneous with a large-scale circulation (LSC) across the cell height [3]. In the rotation plane of LSC, the flow has a fly-wheel-like structure with a zero mean at the center and a linearly increasing velocity along the radial direction in the bulk region. After reaching its maximum value near the sidewall, the mean vertical velocity starts to drop quickly and becomes zero at the cell wall.As a result, the flow field near the sidewall is similar to that of a channel flow with a mean vertical velocity U 0 and a rms velocity σ v 0.6U 0 [3]. In this case, temperature is a passive scalar and follows the local flow [4,5]. Therefore the energy cascade picture can also be used to describe the spectrum of temperature variance [6], and C T (r,τ ) is expected to have the same functional form as the velocity counterpart C v (r,τ ). It was found [1] that the obtained C T (r,τ ) has the scaling formwith r E being of the elliptical form,where U is a characteristic convection velocity proportional to the local mean velocity U 0 and V is associated with a random sweeping velocity proportional to the local rms velocity σ v . Equation (3) incorporates both Taylor's frozen flow hypothesis [7,8] when V is small and Kraichnan's random sweeping hypothesis [9] for a homogenous and isotropic turbulent flow with a zero mean velocity. The experiment thus verified the theory by He and Zhang [10], in which Eq. (3) was derived for small values of r and τ and both U and V were calculated from the second derivatives of C v (r,τ ). The scaling theory by He and Zhang [10] has important practical implications for a large class of turbulent flows and thus it is essential to test the theory in different flow systems. The above experiment tested the theory in the sidewall region of a convective flow, where there still exists a dominant mean flow but the rms velocity is so large that Taylor's hypothesis [7] does not hold. In the experiment, r was varied only in the longitudinal direction along the mean flow. In this Brief Report, we present new measurements at the center of the convection cell, where the mean flow is zero and velocity fluctuations are approximately homogeneous [3]. Here we vary r in the lateral direction across a large-scale shear imposed by LSC. The experiment further confirms the theory by He and Zhang and demonstrates its applications to homogenous turbulent flows, where Kraichnan's hypothesis [9] holds approximately.The exper...

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“…11 They predict a transition to an "ultimate" regime similar to Kraichnan's for Ra 10 14 and argue that their log correction would yield an effective scaling 12 of about Nu ∼ Ra 0.38 in the range 10 12 < Ra < 10 15 . Some experimental results show transition to an ultimate regime [13][14][15][16] while others do not. 17,18 However, all the data are well-fitted by Nu ∼ 0.105 Ra 0.31 for Ra < 10 11 and that is the regime for which we report on unstable coherent states, in particular optimum transport solutions with Nu ∼ 0.115 Ra 0.31 .…”

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Heat transport by coherent Rayleigh-Bénard convection

2015

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Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number Ra ≈ 1708 has been calculated up to Ra ≈ 5.10 6 and its Nusselt number is N u ∼ 0.143 Ra 0.28 with a delicate spiral structure in the temperature field. Another solution that maximizes N u over the horizontal wavenumber has been calculated up to Ra = 10 9 and scales as N u ∼ 0.115 Ra 0.31 for 10 7 < Ra ≤ 10 9 , quite similar to 3D turbulent data that show N u ∼ 0.105 Ra 0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133 Ra 0.217 . That solution is unstable to larger scale perturbations and in particular to mean shear flows, yet it appears to be relevant as a backbone for turbulent solutions, possibly setting the scale, strength and spacing of elemental plumes.

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Turbulent Rayleigh–Bénard convection in gaseous and liquid He

Cited by 229 publications

References 33 publications

Completing the Mechanical Energy Pathways in Turbulent Rayleigh-Bénard Convection

Completing the Mechanical Energy Pathways in Turbulent Rayleigh-Bénard Convection

Kraichnan’s random sweeping hypothesis in homogeneous turbulent convection

Heat transport by coherent Rayleigh-Bénard convection

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