We present measurements of the Nusselt number N as a function of the Rayleigh number R in cylindrical cells with aspect ratios 0.5 ≤ Γ ≡ D/d ≤ 12.8 (D is the diameter and d the height). We used acetone with a Prandtl number σ = 4.0 for 10 5 < ∼ R < ∼ 4 × 10 10 . A fit of a powerlaw N = N0R γ ef f over limited ranges of R yielded values of γ ef f from 0.275 near R = 10 7 to 0.300 near R = 10 10 . The data are inconsistent with a single powerlaw for N (R). For R > 10 7 they are consistent with N = aσ −1/12 R 1/4 + bσ −1/7 R 3/7 as proposed by Grossmann and Lohse for σ > ∼ 2. 44.25.+f,47.27.Te Since the pioneering measurements by Libchaber and co-workers [1,2] of heat transport by turbulent gaseous helium heated from below, there has been a revival of interest in the nature of turbulent convection. [3] In addition to the local properties of the flow, one of the central issues has been the global heat transport of the system, as expressed by the Nusselt number N = λ ef f /λ. Here λ ef f = qd/∆T is the effective thermal conductivity of the convecting fluid (q is the heat-current density, d the height of the sample, and ∆T the imposed temperature difference), and λ is the conductivity of the quiescent fluid. Usually a simple powerlawwas an adequate representation of the experimental data.[4] Here R = αgd 3 ∆T /κν is the Rayleigh number, α the thermal expansion coefficient, g the gravitational acceleration, κ the thermal diffusivity, and ν the kinematic viscosity. Various data sets yielded exponent valuesγ from 0.28 to 0.31. [5,6] Most recently, measurements over the unprecedented range 10 6 < ∼ R < ∼ 10 17 were made by Niemela et al., and a fit to them of Eq. 1 gaveγ = 0.309; [5] but even these data did not reveal any deviation from the functional form of Eq. 1. Competing theoretical models involving quite different physical assumptions made predictions of powerlaw behavior with exponents γ in the same narrow range. [2,6-8] Here we mention just two of them. A boundary-layer scaling-theory [2,8] which yielded γ = 2/7 ≃ 0.2857 was an early favorite, at least for the experimentally accessible range R < ∼ 10 12 . It was generally consistent with most of the available experimental results. More recently, a competing model based on the decomposition of the kinetic and the thermal dissipation into boundary-layer and bulk contributions was presented by Grossmann and Lohse (GL) [7] and predicts that the data measure an average exponentγ associated with a crossover from γ = 1/4 at small R to a slightly larger γ at much larger R. In the experimental range the effective exponent γ ef f ≡ d(ln(N ))/d(ln(R)), which should be compared with the experimentally determinedγ , is only very weakly dependent upon R and has values fairly close to 2/7. Thus, it has not been possible before to distinguish between the competing theories on the basis of the experimental data.Here we present new measurements of N (R) over the range 10 5 < ∼ R < ∼ 4 × 10 10 for a Prandtl number σ ≡ ν/κ = 4.0. Our data are of exceptionally high precision and a...

We present experimental results for Rayleigh-Bénard convection with rotation about a vertical axis at dimensionless rotation rates 0 ≤ Ω ≤ 250 and ǫ ≡ ∆T /∆Tc − 1 < ∼ 0.2. Critical Rayleigh numbers and wavenumbers agree with predictions of linear stability analysis. For Ω > ∼ 70 and small ǫ the patterns are cellular with local four-fold coordination and differ from the theoretically expected Küppers-Lortz-unstable state. Stable as well as intermittent defect-free square lattices exist over certain parameter ranges. Over other ranges defects dynamically disrupt the lattice but cellular flow and local four-fold coordination is maintained.

This paper reviews results from and future prospects for experimental studies of Rayleigh-Bénard convection with rotation about a vertical axis. At dimensionless rotation rates 0 ≤ Ω ≤ 20 and for Prandtl numbers σ 1, Küppers-Lortz-unstable patterns offered a unique opportunity to study spatio-temporal chaos immediately above a supercritical bifurcation where weakly-nonlinear theories in the form of Ginzburg-Landau (GL) or Swift-Hohenberg (SH) equations can be expected to be valid. However, the dependence of the time and length scales of the chaotic state on ≡ ∆T /∆T c −1 was found to be different from the expected dependence based on the structure of GL equations. For Ω > ∼ 70 and 0.7 < ∼ σ < ∼ 5 patterns were found to be cellular near onset with local four-fold coordination. They differ from the theoretically expected Küppers-Lortz-unstable state. Stable as well as intermittent defect-free rotating square lattices exist in this parameter range.Smaller Prandtl numbers ( 0.16 < ∼ σ < ∼ 0.7 ) can only be reached in mixtures of gases. These fluids are expected to offer rich future opportunities for the study of a line of tricritical bifurcations, of supercritical Hopf bifurcations to standing waves, of a line of codimension-two points, and of a codimension-three point.