Heat transport by coherent Rayleigh-Bénard convection

Waleffe, Fabian; Boonkasame, Anakewit; Smith, Leslie

doi:10.1063/1.4919930

Cited by 52 publications

(72 citation statements)

References 30 publications

(55 reference statements)

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“…Unfortunately and importantly, this means that there is no direct connection between the optimal solution in the background method built around the steady governing equations and a steady solution of the governing equations (here the Boussinesq equations but clearly more generally true). This realisation removes the possibility, for example, that the simple 2D roll solution computed by Waleffe et al (2015) could actually be the optimal solution to the background bounding problem. It now seems clear that it would be spectrally unstable.…”

Section: Discussionmentioning

confidence: 99%

Exhausting the background approach for bounding the heat transport in Rayleigh–Bénard convection

Ding

Kerswell

2020

J. Fluid Mech.

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We revisit the optimal heat transport problem for Rayleigh-Bénard convection in which a rigorous upper bound on the Nusselt number, N u, is sought as a function of the Rayleigh number Ra. Concentrating on the 2-dimensional problem with stress-free boundary conditions, we impose the full heat equation as a constraint for the bound using a novel 2-dimensional background approach thereby complementing the 'wall-to-wall' approach of Hassanzadeh et al. (J. Fluid Mech. 751, 627-662, 2014). Imposing the same symmetry on the problem, we find correspondence with their result for Ra Ra c := 4468.8 but, beyond that, the optimal fields complexify to produce a higher bound. This bound approaches that by a 1-dimensional background field as the length of computational domain L → ∞. On lifting the imposed symmetry, the optimal 2-dimensional temperature background field reverts back to being 1-dimensional giving the best bound N u 0.055Ra 1/2 compared to N u 0.026Ra 1/2 in the non-slip case. We then show via an inductive bifurcation analysis that imposing the full time-averaged Boussinesq equations as constraints (by introducing 2-dimensional temperature and velocity background fields) is also unable to lower this bound. This then exhausts the background approach for the 2-dimensional (and by extension 3-dimensional) Rayleigh-Benard problem with the bound remaining stubbornly Ra 1/2 while data seems more to scale like Ra 1/3 for large Ra. Finally, we show that adding a velocity background field to the formulation of Wen et al. (Phys. Rev. E. 92, 043012, 2015), which is able to use an extra vorticity constraint due to the stress-free condition to lower the bound to N u O(Ra 5/12 ), also fails to improve the bound.

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Section: Discussionmentioning

confidence: 99%

Exhausting the background approach for bounding the heat transport in Rayleigh–Bénard convection

Ding

Kerswell

2020

J. Fluid Mech.

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“…However, theories by Spiegel [20] and Grossman & Lohse [21] suggest that Nu ∼ Pr 1/2 Ra 1/2 , in which case boundary layer effects are negligible and the heat flux becomes independent of the molecular transport coefficients as Ra → ∞. More recently, the study of 2D steady heat-flux-maximizing convective solutions with no-slip boundary conditions at Pr = 7 by Waleffe [22] indicates Nu ∼ 0.115Ra 0.31 for 10 7 < Ra ≤ 10 9 . Rigorous analyses of the three-dimensional (3D) Boussinesq equations governing Rayleigh-Bénard convection show Nu ≤ cRa 1/2 with prefactor 0 < c < ∞ uniformly in Pr for no-slip and isothermal [8] or fixed heat flux [23] or mixed temperature [24] boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2015

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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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“…Much of the current progress in designing industrial heat exchanger devices relies on direct numerical simulations in complex flows and geometries [16]. In a more theoretical context, flow patterns have been optimized to improve Rayleigh-Bénard convection [21,28] or to achieve maximal heat transport in simple 2D geometries [8,25,2,1]. The optimal distribution of sources and sinks has also been investigated for optimal transport of a passive scalar, whether heat or tracer [20,24,22].…”

mentioning

confidence: 99%

Optimal Heat Transfer and Optimal Exit Times

Marcotte¹,

Doering²,

Thiffeault³

et al. 2018

SIAM J. Appl. Math.

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A heat exchanger can be modeled as a closed domain containing an incompressible fluid. The moving fluid has a temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls. Starting from a positive initial temperature distribution in the interior, the goal is to flux the heat through the walls as efficiently as possible. Here we consider a distinct but closely related problem, that of the integrated mean exit time of Brownian particles starting inside the domain. Since flows favorable to rapid heat exchange should lower exit times, we minimize a norm of the exit time. This is a time-independent optimization problem that we solve analytically in some limits, and numerically otherwise. We find an (at least locally) optimal velocity field that cools the domain on a mechanical time scale, in the sense that the integrated mean exit time is independent on molecular diffusivity in the limit of large-energy flows.

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Heat transport by coherent Rayleigh-Bénard convection

Cited by 52 publications

References 30 publications

Exhausting the background approach for bounding the heat transport in Rayleigh–Bénard convection

Exhausting the background approach for bounding the heat transport in Rayleigh–Bénard convection

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

Optimal Heat Transfer and Optimal Exit Times

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