Observation of the Ultimate Regime in Rayleigh-Bénard Convection

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

doi:10.1103/physrevlett.79.3648

Cited by 264 publications

(327 citation statements)

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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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confidence: 99%

Heat transport by coherent Rayleigh-Bénard convection

2015

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“…In experiments, traditionally the Prandtl number was more or less kept fixed [2][3][4]. However, the recent experiments in the vicinity of the critical point of helium gas [5,6] and of SF 6 [7] or with various alcohols [8] allow to vary both Ra and P r and thus to explore a larger domain of the Ra−P r parameter space of Rayleigh-Benard (RB) convection, in particular that for P r ≫ 1. While the experiments of Steinberg's group [7] suggest a decreasing Nusselt number with increasing P r, namely Nu = 0.22Ra 0.3±0.03 P r −0.2±0.04 in 10 9 ≤ Ra ≤ 10 14 and 1 ≤ P r ≤ 93, the experiments of the Ahlers group suggest a saturation of Nu with increasing P r for fixed Ra, at least up to Ra = 10 10 [9].…”

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confidence: 99%

“…The relevant velocity at the edge between the thermal BL and the thermal bulk now is less than U, about Uλ θ /λ u . To describe the transition from λ u being smaller to being larger than λ θ we introduce the function f (x) = (1 + [5], diamonds those by Cioni et al [4], circles those by Niemela et al [6], the very thick lines those by Xu et al [8], and the triangles are those points for which Verzicco and Camussi did full numerical simulations [10]. The long-dashed line is the line λ u = λ θ .…”

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confidence: 99%

Thermal Convection for Large Prandtl Numbers

2001

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The Rayleigh-Benard theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] is extended towards very large Prandtl numbers P r. The Nusselt number N u is found here to be independent of P r. However, for fixed Rayleigh numbers Ra > 10 10 a maximum around P r ≈ 2 in the N u(P r)-dependence is predicted which is absent for lower Ra. We moreover offer the full functional dependences of N u(Ra, P r) and Re(Ra, P r) within this extended theory, rather than only giving the limiting power laws as done in ref. [1]. This enables us to more realistically describe the transitions between the various scaling regimes, including their widths.In thermal convection, the control parameters are the Rayleigh number Ra and the Prandtl number P r. The system responds with the Nusselt number Nu (the dimensionless heat flux) and the Reynolds number Re (the dimensionless large scale velocity). The key question is to understand the dependences Nu(Ra, P r) and Re(Ra, P r). In experiments, traditionally the Prandtl number was more or less kept fixed [2][3][4]. However, the recent experiments in the vicinity of the critical point of helium gas [5,6] and of SF 6 [7] or with various alcohols [8] allow to vary both Ra and P r and thus to explore a larger domain of the Ra−P r parameter space of Rayleigh-Benard (RB) convection, in particular that for P r ≫ 1. While the experiments of Steinberg's group [7] suggest a decreasing Nusselt number with increasing P r, namely Nu = 0.22Ra 0.3±0.03 P r −0.2±0.04 in 10 9 ≤ Ra ≤ 10 14 and 1 ≤ P r ≤ 93, the experiments of the Ahlers group suggest a saturation of Nu with increasing P r for fixed Ra, at least up to Ra = 10 10 [9]. The same saturation (at fixed Ra = 6 · 10 5 ) is found in the numerical simulations by Verzicco and Camussi [10] and Herring and Kerr [11].The large P r regime of the latest experiments has not been covered by the recent theory on thermal convection by Grossmann and Lohse (GL, [1]), which otherwise does pretty well in accounting for various measurements. In particular, it explains the low P r measurements of Cioni et al.[4] (P r = 0.025), the low P r numerics which reveal Nu ∼ P r 0.14 for fixed Ra [10, 11], and the above mentioned experiments by Niemela et al. [6] and Xu et al. [8].In the present paper we extend the GL theory in a natural way to the regime of very large P r, on which no statement has been made in the original paper [1]. We find Nu to be independent of P r in that regime. We in addition present the complete functional dependences Nu(Ra, P r) and Re(Ra, P r) within the GL theory, rather than only giving the limiting power laws and superpositions of those as was done in [1]. This enables us to more realistically describe the transitions between the various scaling regimes found already in [1].Approach: To make this paper selfcontained we very briefly recapitulate the key idea of the GL theory, which is to decompose in the volume averages of the energy dissipation rate ǫ u and the thermal dissipation rate ǫ θ into their boundary layer (BL) and bulk contributions...

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“…7, we compare this regime with the Nu ∼ P r 0.072 Ra 0.389 approximate power-law measured by Chavanne et al [28] in Helium, at Ra > 10 11 . Our formula predicts a very weak dependence of in the Prandtl number, like in [28], but with opposite sign. This could be accounted for minute variations of the mean profile around the value ǫ = −1/2.…”

Section: Te Convection and Heat Transfermentioning

confidence: 99%

“…Summary of local exponents of different physical quantities in regime 6: P r > 0.35, fluctuations dominates and λ θ ≫ λ u . The measurements are from [28]. In this table X = ln Ra and Y = ln P r. No dependence on X or Y of the scaling exponent indicates real scaling with respect to Ra or P r respectively.…”

Section: Q(xy)mentioning

confidence: 99%

Logarithmic corrections to scaling in turbulent thermal convection

Dubrulle

2001

Eur. Phys. J. B

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We use an analytic toy model of turbulent convection to show that most of the scaling regimes are spoiled by logarithmic corrections, in a way consistent with the most accurate experimental measurements available nowadays. This sets a need for the search of new measurable quantities which are less prone to dimensional theories. 47.27 -i Turbulent flows, convection and heat transfer 47.27.Eq Turbulence simulation and modeling 47.27.Te Convection and heat transfer When an horizontal layer of fluid is heated from below, a heat exchange from the bottom to the top layer occurs. Dimensional consideration show that the non-dimensional heat exchange Nu depends on the geometry (via for example the aspect ratio), on the boundary conditions, and on the Rayleigh Ra and the Prandtl number P r. Further dimensional analysis of the dynamical equations governing the convective layer suggests that this dependence be a power-law. The exponent of the power law depends on the (dimensional) basic hypothesis. The classical theory predicts that Nu ∼ Ra β with β = 0.33, which seems to be observed in electro-chemical convection [1]. Other theories lead to β = 0.2 or 0.24 [2], β = 2/7 [3,4] or β = 1/2 [5] depending on the regime considered. A unifying synthesis of the possible scaling regimes, including their Prandtl dependence has been recently made by Grossman and Lohse [6,7]. From the experimental point of view, the situation is rather unclear, since almost all the range of exponents between 0.25 and 0.33 has been measured (see [6] for a recent detailed review of the experimental measurements). Furthermore, recent experiments with fluids subject large variation of the Prandtl number have led to a new exploration of the phase parameter, and uncovered new scaling laws as a function of the Prandlt number. This new exploration can be used as a stringent tool to discriminate between the various theories, since two competing theories can provide the same value of β, but different Prandtl number dependence.On the other hand, it is not quite clear whether the present dimensional theories satisfactorily capture the dependence of Nu as a function of P r and Ra. In a recent serie of experiments, conducted with acetone at various aspect ratio, [8] show that their measurements are inconsistent with a single power law Nu(Ra), because the local effective exponent β ef f = d ln Nu/d ln Ra varies continuously with Ra in the range of Rayleigh number they consider. This effect can easily be accounted for in the scaling theory by considering a superposition of scaling laws, as proposed by [6]. There is another solution, connected with the 1

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Observation of the Ultimate Regime in Rayleigh-Bénard Convection

Cited by 264 publications

References 13 publications

Heat transport by coherent Rayleigh-Bénard convection

Heat transport by coherent Rayleigh-Bénard convection

Thermal Convection for Large Prandtl Numbers

Logarithmic corrections to scaling in turbulent thermal convection

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