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MiS Preprint
57/2016

Limitations of the background field method applied to Rayleigh-Bénard convection

Camilla Nobili and Felix Otto

Abstract

We consider Rayleigh-B\'enard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number and with no-slip boundary condition.

There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number ${\rm Nu}$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime ${\rm Ra}\gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on ${\rm Nu}$ in terms of ${\rm Ra}$. In these applications, the background field method comes in form of a variational problem where one optimizes a stratified temperature profile subject to a certain stability condition; the method is believed to capture marginal stability of the boundary layer.

The best available upper bound via this method is ${\rm Nu}\lesssim {\rm Ra}^{\frac{1}{3}}(\ln {\rm Ra})^{\frac{1}{15}}$; it proceeds via the construction of a stable temperature background profile that increases logarithmically in the bulk. In this paper, we show that the background temperature field method cannot provide a tighter upper bound in terms of the power of the logarithm. However, by another method one does obtain the tighter upper bound ${\rm Nu}\lesssim {\rm Ra}^{\frac{1}{3}}(\ln\ln {\rm Ra})^{\frac{1}{3}}$, so that the result of this paper implies that the background temperature field method is unphysical in the sense that it cannot provide the optimal bound.

Received:
10.08.2016
Published:
11.08.2016
Keywords:
Rayleigh-Bénard convection, Stokes equations, no-slip boundary condition, infinite Prandtl number, Nusselt number, background field method, variational methods

Related publications

inJournal
2017 Repository Open Access
Camilla Nobili and Felix Otto

Limitations of the background field method applied to Rayleigh-Bénard convection

In: Journal of mathematical physics, 58 (2017) 9, p. 093102