Bounds for convection between rough boundaries

Goluskin, David; Doering, Charles R.

doi:10.1017/jfm.2016.528

Cited by 46 publications

(49 citation statements)

References 38 publications

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“…Our results are consistent with those of Goluskin & Doering [1], who used the background method to compute upper bounds [48] on N u for R-B convection in a domain with rough upper and lower surfaces that have squareintegrable gradients. They prove that N u ≤ CRa 1/2 , where C depends on the geometry of roughness.…”

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confidence: 90%

“…By varying the wavelength λ at a fixed amplitude, we find an optimal wavelength λopt for which the Nusselt-Rayleigh scaling relation is N u − 1 ∝ Ra 0.483 maximizing the heat flux. This is consistent with the upper bound of Goluskin and Doering [1] who prove that N u can grow no faster than O(Ra 1/2 ) as Ra → ∞, and thus the concept that roughness facilitates the attainment of the so-called ultimate regime. Our data nearly achieve the largest growth rate permitted by the bound.…”

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Roughness as a Route to the Ultimate Regime of Thermal Convection

2017

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We use highly resolved numerical simulations to study turbulent Rayleigh-Bénard convection in a cell with sinusoidally rough upper and lower surfaces in two dimensions for P r = 1 and Ra = 4 × 10 6 , 3 × 10 9 . By varying the wavelength λ at a fixed amplitude, we find an optimal wavelength λopt for which the Nusselt-Rayleigh scaling relation is N u − 1 ∝ Ra 0.483 maximizing the heat flux. This is consistent with the upper bound of Goluskin and Doering [1] who prove that N u can grow no faster than O(Ra 1/2 ) as Ra → ∞, and thus the concept that roughness facilitates the attainment of the so-called ultimate regime. Our data nearly achieve the largest growth rate permitted by the bound. When λ λopt and λ λopt, the planar case is recovered, demonstrating how controlling the wall geometry manipulates the interaction between the boundary layers and the core flow. Finally, for each Ra we choose the maximum N u among all λ, and thus optimizing over all λ, to find N uopt − 1 = 0.01 × Ra 0.444 .The ubiquity and importance of thermal convection in many natural and man-made settings is well known [2][3][4]. The simplest scenario that has been used to study the fundamental aspects of thermal convection is the Rayleigh-Bénard system [5]. The flow in this system is governed by three non-dimensional parameters: (1) the Rayleigh number Ra = gα∆T H 3 /νκ, which is the ratio of buoyancy to viscous forces, where g is the acceleration due to gravity, α the thermal expansion coefficient of the fluid, ∆T the temperature difference across a layer of fluid of depth H, ν the kinematic viscosity (or momentum diffusivity) and κ the thermal diffusivity; (2) the Prandtl number, P r = ν/κ; and (3) the aspect ratio of the cell, Γ, defined as the ratio of its width to height.The primary aim of the corpus of studies of turbulent Rayleigh-Bénard convection has been to determine the Nusselt number, N u, defined as the ratio of total heat flux to conductive heat flux (Eq. 1), as a function of the three governing parameters, viz., N u = N u(Ra, P r, Γ). For Ra 1 and fixed P r and Γ, this relation is usually sought in the form of a power law: N u = A(P r, Γ)Ra β , where β has a fundamental significance for the mechanisms underlying the transport of heat.The classical theory of Priestley [6], Malkus [7] and Howard [8] is based on the argument that as Ra → ∞ the dimensional heat flux should become independent of the depth of the cell, resulting in β = 1/3. A consequence of this scaling is that the conductive boundary layers (BLs) at the upper and lower surfaces, which are separated by a well mixed interior, do not interact.However, Kraichnan [9] reasoned that for extremely large Ra the BLs undergo a transition leading to the generation of smaller scales near the boundaries that increase the system's efficiency in transporting the heat, predicting that N u ∼ Ra/ (ln Ra). In this, "KraichnanSpiegel" or "ultimate regime" (β = 1/2), it is argued that the heat flux becomes independent of the molecular properties of the fluid [e.g. and hence values of Ra) t...

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confidence: 90%

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confidence: 90%

Roughness as a Route to the Ultimate Regime of Thermal Convection

2017

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“…Remark 2. Most applications of the background method-that is, of quadratic auxiliary functionals with leading terms proportional to energy-have been to the Navier-Stokes equations and related fluid dynamical systems (e.g., [9,10,19,39,49]). These PDEs have the same type of quadratic nonlinearity as the KSE.…”

Section: Quadratic Auxiliary Functionals For the Ksementioning

confidence: 99%

Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming

Goluskin

Fantuzzi

2019

Nonlinearity

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We present methods for bounding infinite-time averages in dynamical systems governed by nonlinear PDEs. The methods rely on auxiliary functionals, which are similar to Lyapunov functionals but satisfy different inequalities. The inequalities are enforced by requiring certain expressions to be sums of squares of polynomials, and the optimal choice of auxiliary functional is posed as a semidefinite program (SDP) that can be solved computationally. To formulate these SDPs we approximate the PDE by truncated systems of ODEs and proceed in one of two ways. The first approach is to compute bounds for the ODE systems, increasing the truncation order until bounds converge numerically. The second approach incorporates the ODE systems with analytical estimates on their deviation from the PDE, thereby using finite truncations to produce bounds for the full PDE. We apply both methods to the Kuramoto-Sivashinsky equation. In particular, we compute upper bounds on the spatiotemporal average of energy by employing polynomial auxiliary functionals up to degree six. The first approach is used for most computations, but a subset of results are checked using the second approach, and the results agree to high precision. These bounds apply to all odd solutions of period 2πL, where L is varied. Sharp bounds are obtained for L ≤ 10, and trends suggest that more expensive computations would yield sharp bounds at larger L also. The bounds are known to be sharp (to within 0.1% numerical error) because they are saturated by the simplest nonzero steady states, which apparently have the largest mean energy among all odd solutions. Prior authors have conjectured that mean energy remains O(1) for L 1 since no particular solutions with larger energy have been found. Our bounds constitute the first positive evidence for this conjecture, albeit up to finite L, and they offer some guidance for analytical proofs. *

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“…This exponent is also a rigorous upper bound (Goluskin & Doering, 2016), and the corresponding regime is sometimes called ultimate regime of convection, as there could not be a more efficient regime beyond. Several groups have claimed to observe this regime at very large Rayleigh numbers, using cryogenic gaseous helium (Chavanne et al, 1997;Roche et al, 2010), or compressed sulphur hexafluoride (He et al, 2012).…”

Section: Introductionmentioning

confidence: 99%

Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries

et al. 2017

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Several Rayleigh-Bénard experiments in water are performed with smooth or rough boundaries. We present new thermal transfer measurements obtained with large roughness elements arranged in a square lattice. The data are compared to previous ones obtained with smaller elements in the same cell (Tisserand et al., 2011). Experiments in the same apparatus without roughness are fully presented, as reference results, to allow for comparison. In the rough case, several regimes of heat transfer are identified : one similar to the smooth case, an enhanced heat transfer one characterized by a modification of the Nusselt vs Rayleigh numbers relation, and a third part where the relation can be similar to a smooth one with a corrected prefactor.

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Bounds for convection between rough boundaries

Cited by 46 publications

References 38 publications

Roughness as a Route to the Ultimate Regime of Thermal Convection

Roughness as a Route to the Ultimate Regime of Thermal Convection

Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming

Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries

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