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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.
