Optimal cooling of an internally heated disc

Tobasco, Ian

doi:10.1098/rsta.2021.0040

Cited by 6 publications

(6 citation statements)

References 52 publications

(75 reference statements)

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“…To check if the O(R −2 ) and O(R −4 ) corrections to the asymptotic value of 1 2 are optimal within our bounding framework, one could employ a variation of the computational approach taken in [2] and optimize the tunable parameters τ , β, and λ in full (see also [24] and references therein for more details on the numerical optimization of bounds). A more interesting but also more challenging problem is to identify which convective flows maximize wT and the corresponding optimal scaling of this quantity with R. Considerable insight in this direction can be gained through (i) direct numerical simulations, which to the best of our knowledge are currently lacking; (ii) the calculation of steady but unstable solution of the Boussinesq equations (1.1) that, as recently observed in the context of Rayleigh-Bénard convection [27,28], may transport heat more efficiently than turbulence; and (iii) the explicit design of optimally-cooling flows [6,29,30]. Finally, it would be interesting to investigate if more sophisticated PDE analysis techniques used for Rayleigh-Bénard convection [31] can be extended to IH convection to interpolate between the algebraic bounds on wT proved in this paper for infinite-Pr fluids with the finite-Pr exponential bounds obtained in [5].…”

Section: Discussionmentioning

confidence: 99%

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Rigorous scaling laws for internally heated convection at infinite Prandtl number

Arslan¹,

Fantuzzi²,

Craske³

et al. 2022

Preprint

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New bounds are proven on the mean vertical convective heat transport, wT , for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that wT ≤ 1 2 − cR −2 , where R is a nondimensional 'flux' Rayleigh number quantifying the strength of internal heating and c = 216. Then, wT = 0 corresponds to vertical heat transport by conduction alone, while wT > 0 represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain wT ≤ 1 2 − cR −4 , with c ≈ 0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu R 4 .Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy-Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on R improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with R.

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Section: Discussionmentioning

confidence: 99%

“…Convective flows driven by internal sources of heat have attracted renewed interest in recent years [1][2][3][4][5][6]. Such flows are commonly encountered in geophysics, where atmospheric convection [7] and mantle convection [8,9] are typical examples.…”

Section: Introductionmentioning

confidence: 99%

Rigorous scaling laws for internally heated convection at infinite Prandtl number

Arslan¹,

Fantuzzi²,

Craske³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Therefore, in these situations, considering case 1 can be very useful as one can try solving the Euler-Lagrange equations analytically using the method of matched asymptotics. These ideas can also be of relevance to other variational approaches, such as the wall-to-wall transport problem (Hassanzadeh, Chini & Doering 2014;Tobasco & Doering 2017;Motoki, Kawahara & Shimizu 2018a,b;Doering & Tobasco 2019;Souza et al 2020;Tobasco 2022;Kumar 2022), which asks the question of what is the maximum heat transfer for a fixed energy or enstrophy budget.…”

Section: Summary and Implicationsmentioning

confidence: 99%

Geometrical dependence of optimal bounds in Taylor–Couette flow

Kumar

2022

J. Fluid Mech.

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This paper is concerned with the optimal upper bound on mean quantities (torque, dissipation and the Nusselt number) obtained in the framework of the background method for the Taylor–Couette flow with a stationary outer cylinder. Along the way, we perform the energy stability analysis of the laminar flow, and demonstrate that below radius ratio $0.0556$ , the marginally stable perturbations are not the axisymmetric Taylor vortices but rather a fully three-dimensional flow. The main result of the paper is an analytical expression of the optimal bound as a function of the radius ratio. To obtain this bound, we begin by deriving a suboptimal analytical bound using analysis techniques. We use a definition of the background flow with two boundary layers, whose relative thicknesses are optimized to obtain the bound. In the limit of high Reynolds number, the dependence of this suboptimal bound on the radius ratio (the geometrical scaling) turns out to be the same as that of numerically computed optimal bounds in three different cases: (1) the perturbed flow only satisfies the homogeneous boundary conditions but need not be incompressible; (2) the perturbed flow is three-dimensional and incompressible; (3) the perturbed flow is two-dimensional and incompressible. We compare the geometrical scaling with the observations from the turbulent Taylor–Couette flow, and find that the analytical result indeed agrees well with the available direct numerical simulations data. In this paper, we also dismiss the applicability of the background method to certain flow problems and therefore establish the limitation of this method.

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“…For flows with large spatial scales and/or large flow speeds, the heat transfer is often controlled by thin viscous and thermal layers at the solid boundaries. Consequently, the rate of heat transfer is sensitive to the features of the solid boundaries, and in particular to their shape or geometry (Webb & Kim 2005; Lienhard 2013; Tobasco 2022; Wen, Goluskin & Doering 2022 b ; Wen et al 2022 a ; Song, Fantuzzi & Tobasco 2023).…”

Section: Introductionmentioning

confidence: 99%

Optimal wall shapes and flows for steady planar convection

Alben

2024

J. Fluid Mech.

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We compute steady planar incompressible flows and wall shapes that maximize the rate of heat transfer ( $Nu$ ) between hot and cold walls, for a given rate of viscous dissipation by the flow ( $Pe^2$ ), with no-slip boundary conditions at the walls. In the case of no flow, we show theoretically that the optimal walls are flat and horizontal, at the minimum separation distance. We use a decoupled approximation to show that flat walls remain optimal up to a critical non-zero flow magnitude. Beyond this value, our computed optimal flows and wall shapes converge to a set of forms that are invariant except for a $Pe^{-1/3}$ scaling of horizontal lengths. The corresponding rate of heat transfer $Nu \sim Pe^{2/3}$ . We show that these scalings result from flows at the interface between the diffusion-dominated and convection-dominated regimes. We also show that the separation distance of the walls remains at its minimum value at large $Pe$ .

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Optimal cooling of an internally heated disc

Cited by 6 publications

References 52 publications

Rigorous scaling laws for internally heated convection at infinite Prandtl number

Rigorous scaling laws for internally heated convection at infinite Prandtl number

Geometrical dependence of optimal bounds in Taylor–Couette flow

Optimal wall shapes and flows for steady planar convection

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