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

Abstract

We consider Rayleigh-Bénard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number. There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number $\mathrm{Nu}$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime $\mathrm{Ra}\gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on $\mathrm{Nu}$ in terms of $\mathrm{Ra}$. 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 $\mathrm{Nu}$ $\lesssim {\mathrm{Ra}}^{\frac{1}{3}}(\ln \mathrm{Ra}{)}^{\frac{1}{15}}$; 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 $\mathrm{Nu}\lesssim {\mathrm{Ra}}^{\frac{1}{3}}(\ln \ln \mathrm{Ra}{)}^{\frac{1}{3}}$, 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.

54 pages, 5 figures, 25 references

References in corpus (1)

• Bounds for convection between rough boundaries

Cited by in corpus (7)

• The background method: Theory and computations
• Relationship between the methods of bounding time averages
• On the optimal design of wall-to-wall heat transport
• Energy dissipation rate limits for flow through rough channels and tidal flow across topography- Corrigendum
• Optimal cooling of an internally heated disc
• Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries
• Sharp bounds on the Nusselt number in Rayleigh-Bénard convection and a bilinear estimate via Carleson measures