Different scaling behavior has been reported in various shell models proposed for turbulent thermal convection. In this paper, we show that buoyancy is not always relevant to the statistical properties of these shell models even though there is an ex
plicit coupling between velocity and temperature in the equations of motion. When buoyancy is relevant (irrelevant) to the statistical properties, the scaling behavior is Bolgiano-Obukhov (Kolmogorov) plus intermittency corrections. We show that the intermittency corrections of temperature could be solely attributed to fluctuations in the entropy transfer rate when buoyancy is relevant but due to fluctuations in both energy and entropy transfer rates when buoyancy is irrelevant. This difference can be used as a criterion to distinguish whether temperature is behaving as an active or a passive scalar.
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 b
y 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.
