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Constantin, Peter; Doering, Charles R.

doi:10.1023/a:1004511312885

Cited by 142 publications

(25 citation statements)

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“…Note that in all three very large P r regimes Nu does not depend on P r. Furthermore the ConstantinDoering [15] upper bound Nu ≤ constRa 1/3 (1 + log Ra) 2/3 , holding in the limit P r → ∞, is strictly fulfilled. (a) N u as a function of P r according to theory for Ra = 10 6 , Ra = 10 7 , .…”

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confidence: 80%

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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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confidence: 80%

“…Now eqs. (15) and (16) allow to calculate Nu and Re along such a curve P r(Ra) given by the experimental restrictions, which connect Ra and P r. E.g., figure 3 shows Nu(Ra, P r(Ra)) with P r(Ra) as resulting from the Niemela et al [6] and the Chavanne et al [5] experiments.…”

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confidence: 99%

Thermal Convection for Large Prandtl Numbers

2001

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“…(16)- (25). Then the solutions of the original EulerLagrange equations, which correspond exactly to the steady states of the "time-dependent" Euler-Lagrange equations, can be easily obtained by solving the extended equations numerically using a time-marching method with non-zero initial data for all horizontal modes 1 ≤ n ≤ N .…”

Section: Two-step Algorithmmentioning

confidence: 99%

“…This bound does not contradict the result Nu ∼ Pr 0 Ra 1/3 obtained from marginally stable boundary layer theory, but it does rule out the prediction Nu ∼ Pr 1/2 Ra 1/2 at large Prandtl numbers. And indeed, for convection between no-slip boundaries in the infinite-Pr limit, there exist rigorous upper bounds of the form Nu ≤ CRa 1/3 , where C depends on log Ra [25,26] or even only on log(log Ra) [27]. On the other hand in Rayleigh's original 1916 model, the heat transport satisfies Nu < cRa 5/12 [28,29].…”

Section: Introductionmentioning

confidence: 99%

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

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A new computational procedure for numerically solving a class of variational problems arising from rigorous upper bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number Nu, as a function of the Rayleigh number Ra, Prandtl number Pr, and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and β in the presumed Nu ∼ Pr α Ra β scaling relation. The computations clearly show that for Ra ≤ 10 10 at fixed L = 2 √ 2, Nu ≤ 0.106Pr 0 Ra 5/12 , which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra

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“…which substituted into (26) implies that Q τ [w, θ] ≥ Q τ [w, θ] for all θ satisfying the thermal BCs. The boundary conditions (16) are difficult to incorporate into the variational statements that follow, so we impose a stronger requirement on τ : We say that a background field τ (z) is strongly admissible (for a given control parameter R) if it satisfies the thermal BCs (7) and if Q τ [w, θ] ≥ 0 for all sufficiently smooth scalar fields w and θ satisfying w = w z = 0 at z = 0, 1 and the constraint (20); note that we do not specify BCs on θ at this stage in the development.…”

Section: Admissible Backgrounds: the Spectral Constraintmentioning

confidence: 99%

A rigorous bound on the vertical transport of heat in Rayleigh-Bénard convection at infinite Prandtl number with mixed thermal boundary conditions

Whitehead

Wittenberg

2014

Journal of Mathematical Physics

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A rigorous upper bound on the Nusselt number is derived for infinite Prandtl number Rayleigh-Bénard convection for a fluid constrained between no-slip, mixed thermal vertical boundaries. The result suggests that the thermal boundary condition does not affect the qualitative nature of the heat transport. The bound is obtained with the use of a nonlinear, stably stratified background temperature profile in the bulk, notwithstanding the lack of boundary control of the temperature due to the Robin boundary conditions.

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Abstract: We prove an inequality of the type N CR 1Â3 (1+log + R) 2Â3 for the Nusselt number N in terms of the Rayleigh number R for the equations describing threedimensional Rayleigh Be nard convection in the limit of infinite Prandtl number.

Cited by 142 publications

References 7 publications

Thermal Convection for Large Prandtl Numbers

Thermal Convection for Large Prandtl Numbers

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

A rigorous bound on the vertical transport of heat in Rayleigh-Bénard convection at infinite Prandtl number with mixed thermal boundary conditions

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