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Register a new userUnderstanding the different scaling behavior in various shell models proposed for turbulent thermal convection
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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 explicit 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.
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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.
Broad theoretical arguments are proposed to show, formally, that the magnitude G of the temperature gradients in turbulent thermal convection at high Rayleigh numbers obeys the same advection-diffusion equation that governs the temperature fluctuation T, except that the velocity field in the new equation is substantially smoothed. This smoothed field leads to a -1 scaling of the spectrum of G in the same range of scales for which the spectral exponent of T lies between -7/5 and -5/3. This result is confirmed by measurements in a confined container with cryogenic helium gas as the working fluid for Rayleigh number Ra=1.5x10^{11}. Also confirmed is the logarithmic form of the autocorrelation function of G. The anomalous scaling of dissipation-like quantities of T and G are identical in the inertial range, showing that the analogy between the two fields is quite deep.
Using experimental data on thermal convection, obtained at a Rayleigh number of 1.5 $times 10^{11}$, it is shown that the temperature structure functions $<Delta T_{r}^p>$, where $Delta T_r$ is the absolute value of the temperature increment over a distance $r$, can be well represented in an intermediate range of scales by $r^{zeta_p} phi (r)^{p}$, where the $zeta_p$ are the scaling exponents appropriate to the passive scalar problem in hydrodynamic turbulence and the function $phi (r) = 1-a(ln r/r_h)^2$. Measurements are made in the midplane of the apparatus near the sidewall, but outside the boundary layer.
Anomalous scaling in the statistics of an active scalar in homogeneous turbulent convection is studied using a dynamical shell model. We extend refined similarity ideas for homogeneous and isotropic turbulence to homogeneous turbulent convection and attribute the origin of the anomalous scaling to variations of the entropy transfer rate. We verify the consequences and thus the validity of our hypothesis by showing that the conditional statistics of the active scalar and the velocity at fixed values of entropy transfer rate are not anomalous but have simple scaling with exponents given by dimensional considerations, and that the intermittency corrections are given by the scaling exponents of the moments of the entropy transfer rate.
The relative dispersion process in two-dimensional free convection turbulence is investigated by direct numerical simulation. In the inertial range, the growth of relative separation, $r$, is expected as $<r^2(t)>propto t^5$ according to the Bolgiano-Obukhov scaling. The result supporting the scaling is obtained with exit-time statistics. Detailed investigation of exit-time PDF shows that the PDF is divided into two regions, the Region-I and -II, reflecting two types of separating processes: persistent expansion and random transitions between expansion and compression of relative separation. This is consistent with the physical picture of the self-similar telegraph model. In addition, a method for estimating the parameters of the model are presented. Comparing two turbulence cases, two-dimensional free convection and inverse cascade turbulence, the relation between the drift term of the model and nature of coherent structures is discussed.
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