Ultimate-state scaling in a shell model for homogeneous turbulent convection

TL;DR

This paper investigates the scaling laws of heat transport and flow velocity in turbulent convection using a shell model, revealing that the ultimate state may not be bulk-dominated if buoyancy acts on most scales.

Contribution

It introduces a shell model analysis showing that the ultimate state scaling arises with large-scale drag, challenging the bulk-dominated interpretation.

Findings

Nu scales as Ra^{1/2} Pr^{1/2}

Re scales as Ra^{1/2} Pr^{-1/2}

Large-scale drag influences ultimate-state scaling

Abstract

An interesting question in turbulent convection is how the heat transport depends on the strength of thermal forcing in the limit of very large thermal forcing. Kraichnan predicted [Phys. Fluids {\bf 5}, 1374 (1962)] that the heat transport measured by the Nusselt number (Nu) would depend on the strength of thermal forcing measured by the Rayleigh number (Ra) as Nu $\sim$ Ra$^{1/2}$ with possible logarithmic corrections at very high Ra. This scaling behavior is taken as a signature of the so-called ultimate state of turbulent convection. The ultimate state was interpreted in the Grossmann-Lohse (GL) theory [J. Fluid Mech. {\bf 407}, 27 (2000)] as a bulk-dominated state in which both the kinetic and thermal dissipation are dominated by contributions from the bulk of the flow with the boundary layers either broken down or playing no role in the heat transport. In this paper, we study the dependence of Nu and the Reynolds number (Re) measuring the root-mean-squared velocity fluctuations on Ra and the Prandtl number (Pr) using a shell model for homogeneous turbulent convection where buoyancy is acting directly on most of the scales. We find that Nu$\sim$ Ra$^{1/2}$Pr$^{1/2}$ and Re$\sim$ Ra$^{1/2}$Pr$^{-1/2}$, which resemble the ultimate-state scaling behavior for fluids with moderate Pr, but the presence of a drag acting on the large scales is crucial in giving rise to such scaling. This suggests that if buoyancy acts on most of the scales in the bulk of turbulent convection at very high Ra, then the ultimate state cannot be a bulk-dominated state.