Rigid bounds on heat transport by a fluid between slippery boundaries

Whitehead, Jared P.; Doering, Charles R.

doi:10.1017/jfm.2012.274

Cited by 34 publications

(49 citation statements)

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“…Mathematical analysis of the governing Boussinesq equations both complements and informs experiments and DNS by providing provably true bounds on the achievable heat transport. Indeed, Whitehead & Doering [29,31] used rigorous upper bound analysis to analytically prove that Nu ≤ cRa 5/12 as Ra → ∞ for both 2D Rayleigh-Bénard convection at arbitrary Pr and 3D infinite-Pr RayleighBénard convection, both with stress-free isothermal boundary conditions (albeit with slightly different pre-factors c).…”

Section: Discussionmentioning

confidence: 99%

“…Note that homogenous boundary conditions on Ω are not realized for 3D stress-free Rayleigh-Bénard convection; consequently, the quadratic enstrophy constraint can only be imposed in the 2D stress-free convection problem to thereby reduce the upper bounds on the heat transport [28,29,31]. In the CDH upper bound theory the temperature T (x, z, t) is decomposed into a time-independent background profile τ (z) carrying the inhomogeneous boundary conditions plus a nonlinear fluctuation θ(x, z, t) satisfying homogeneous boundary conditions:…”

Section: Problem Formation and Computational Methodologymentioning

confidence: 99%

“…On the other hand in Rayleigh's original 1916 model, the heat transport satisfies Nu < cRa 5/12 [28,29]. This 5/12 scaling exponent for the upper bound also holds for stress-free isothermal boundary conditions in 3D configurations when Pr = ∞ [30,31]. However, since piecewise linear functions were utilized as the background profiles, the upper bounds obtained by Otero [28] and Whitehead & Doering [29] are not the optimal ones within the CDH variational scheme.…”

Section: Introductionmentioning

confidence: 91%

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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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Section: Discussionmentioning

confidence: 99%

Section: Problem Formation and Computational Methodologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 91%

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

et al. 2015

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“…In the light of recent results on slippery convection, 24,25 it is worthwhile to consider the effect that variations from a fixed temperature boundary condition may have in that context. To date the noslip boundary condition has proved essential for bounding the heat transport when the temperature is not fixed at the top and bottom plates.…”

Section: Discussionmentioning

confidence: 99%

“…To date the noslip boundary condition has proved essential for bounding the heat transport when the temperature is not fixed at the top and bottom plates. In addition, the best known bounds for slippery convection have not required a nonlinear stably stratifying bulk background profile, even at infinite Pr, 25 and indeed, the careful numerical and asymptotic calculations of 23 indicate that such stable stratification is not necessary. It is worth considering whether such differences in the implementation of the background method may have physical ramifications beyond the scaling of the Nusselt number with Ra for no-slip or stress-free convection.…”

Section: Discussionmentioning

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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Internally Heated Convection and Rayleigh-Bénard Convection

Goluskin¹

2016

SpringerBriefs in Applied Sciences and Technology

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PrefaceThe purpose of this SpringerBrief is to review heat transfer in layers of convective fluid. Six different configurations are considered-three that are versions of Rayleigh-Bénard (RB) convection, which is driven by differential heating at the boundaries, and three that are driven by uniform internal heating. The essential features of all six models are derived mathematically. The experimental literature is reviewed in depth for the models of internally heated (IH) convection, which are much less studied than their RB counterparts. Experiments on RB convection are treated in less depth, as they have been thoroughly reviewed elsewhere.Along with placing the various convective models within a conceptual framework that brings out their similarities, we give some minor results not published elsewhere. For instance, a few of the linear instability and energy stability thresholds given in tables 2.2 and 2.3 either have not been reported before or have been reported with less precision. One of the bounds proven in §2.3 is also new, as are the visualizations of simulations included as figure 1.2.Chapter 1 provides background and then defines the six configurations under study, the governing equations of our models, and their basic features. Chapter 2 presents results that can be derived mathematically from the governing equations: linear and nonlinear stability thresholds of static states, along with proven bounds on mean temperatures and heat fluxes. For the IH cases only, chapter 3 gives a quantitative survey of heat transport in both laboratory experiments and numerical simulations, followed by suggestions for future work.

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Rigid bounds on heat transport by a fluid between slippery boundaries

Cited by 34 publications

References 44 publications

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

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

Internally Heated Convection and Rayleigh-Bénard Convection

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