Heat transport bounds for a truncated model of Rayleigh–Bénard convection via polynomial optimization

Olson, Matthew; Goluskin, David; Schultz, William W.; Doering, Charles R.

doi:10.1016/j.physd.2020.132748

Cited by 14 publications

(29 citation statements)

References 37 publications

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“…This observation supports the conjecture that steady states maximize among all stable or unstable flows, as was recently verified for a truncated model of RBC (Olson et al. 2021) using methods that are not yet applicable to the full governing equations. If steady-roll transport continues to dominate turbulent transport as , then our finding of classical scaling for steady rolls would rule out ultimate scaling of turbulent convection.…”

Section: Introductionsupporting

confidence: 88%

Steady Rayleigh–Bénard convection between no-slip boundaries

2021

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The central open question about Rayleigh–Bénard convection – buoyancy-driven flow in a fluid layer heated from below and cooled from above – is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to \infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed ‘classical’ $Nu \sim Ra^{1/3}$ or ‘ultimate’ $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$ , we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to \infty$ , it cannot achieve the ultimate scaling.

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Section: Introductionsupporting

confidence: 88%

Steady Rayleigh–Bénard convection between no-slip boundaries

2021

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“…These ideas culminated in an eight-dimensional model introduced by Gluhovsky et al [11] by adding both the θ 04 and the ψ 03 mode to the Fourier expansions used to construct Howard and Krishnamurti's original model. This system is called the HK8 model because it is the minimal extension of Howard and Krishnamurti's model that restores these basic integral identities of the PDE [37].…”

Section: Model Simplificationmentioning

confidence: 99%

“…An ROM satisfies a given balance law if equality holds after substituting the truncated Fourier series into the balance equation. As proved in Appendix A, a truncated model satisfies ( 35)- (37) if modes are selected according to the following criteria:…”

Section: The Hk Hierarchymentioning

confidence: 99%

“…Introducing a carefully chosen polynomial auxiliary function that depends on the state variables of the ODE allows meaningful global bounds to be obtained without solving the ODEs. Writing these inequalities as sum-of-squares constraints enables the feasible minimization over the class of polynomial auxiliary functions of fixed degree d. This approach has been employed in previous work [13,14,10], including a study of an eight-mode Galerkin-truncated model of Rayleigh-Bénard convection that forms the foundation for this paper [37].…”

Section: Introductionmentioning

confidence: 99%

“…Each model in this hierarchy satisfies a set of energy and vorticity balance laws derived from the PDE and includes horizontal shear modes capable of producing zonal flow. Such models build on the results of previous studies [27,54,37] by forming successively closer approximations to the governing equations. Analyzing the heat transport properties of models in the chosen hierarchy supports the hypothesis that equilibria provide maximal heat transport for Rayleigh-Bénard convection.…”

Section: Introductionmentioning

confidence: 99%

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Heat transport in a hierarchy of reduced-order convection models

Olson¹,

Doering²

2022

Preprint

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Reduced-order models (ROMs) are systems of ordinary differential equations (ODEs) designed to approximate the dynamics of partial differential equations (PDEs). In this work, a distinguished hierarchy of ROMs is constructed for Rayleigh's 1916 model of natural thermal convection. These models are distinguished in the sense that they preserve energy and vorticity balances derived from the governing equations, and each is capable of modeling zonal flow. Various models from the hierarchy are analyzed to determine the maximal heat transport in a given model, measured by the dimensionless Nusselt number, for a given Rayleigh number. Lower bounds on the maximal heat transport are ascertained by computing the Nusselt number among equilibria of the chosen model using numerical continuation. A method known as sum-of-squares optimization is applied to construct upper bounds on the time-averaged Nusselt number. In this case, the sum-of-squares approach involves constructing a polynomial quantity whose global nonnegativity implies the upper bound along all solutions to a chosen ROM. The minimum such bound is determined through a type of convex optimization called semidefinite programming. For the ROMs studied in this work, the Nusselt number is maximized by equilibria whenever the Rayleigh number is sufficiently small. In this range of Rayleigh number, the equilibria maximizing heat transport are those that bifurcate first from the zero state. Analyzing this primary equilibrium branch provides a possible mechanism for the increase in heat transport near the onset of convection.

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Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

Ovsyannikov,

Rademacher,

Welter

et al. 2023

J Nonlinear Sci

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Revisiting the Lorenz ’63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite $$\rho $$ ρ , we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average ‘transport,’ which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the nonzero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the nonzero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

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Heat transport bounds for a truncated model of Rayleigh–Bénard convection via polynomial optimization

Cited by 14 publications

References 37 publications

Steady Rayleigh–Bénard convection between no-slip boundaries

Steady Rayleigh–Bénard convection between no-slip boundaries

Heat transport in a hierarchy of reduced-order convection models

Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models

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