The progress in our understanding of several aspects of turbulent Rayleigh-Bénard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large-scale convection-roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Bénard system.

A systematic theory for the scaling of the Nusselt number N u and of the Reynolds number Re in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large scale convection roll ("wind of turbulence") and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number P r phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra < ∼ 10 11 ) the leading terms are N u ∼ Ra 1/4 P r 1/8 , Re ∼ Ra 1/2 P r −3/4 for P r < ∼ 1 and N u ∼ Ra 1/4 P r −1/12 , Re ∼ Ra 1/2 P r −5/6 for P r > ∼ 1. In most measurements these laws are modified by additive corrections from the neighboring regimes so that the impression of a slightly larger (effective) N u vs Ra scaling exponent can arise. The most important of the neighboring regimes towards large Ra are a regime with scaling N u ∼ Ra 1/2 P r 1/2 , Re ∼ Ra 1/2 P r −1/2 for medium P r ("Kraichnan regime"), a regime with scaling N u ∼ Ra 1/5 P r 1/5 , Re ∼ Ra 2/5 P r −3/5 for small P r, a regime with N u ∼ Ra 1/3 , Re ∼ Ra 4/9 P r −2/3 for larger P r, and a regime with scaling N u ∼ Ra 3/7 P r −1/7 , Re ∼ Ra 4/7 P r −6/7 for even larger P r. In particular, a linear combination of the 1/4 and the 1/3 power laws for N u with Ra, N u = 0.27Ra 1/4 + 0.038Ra 1/3 (the prefactors follow from experiment), mimicks a 2/7 power law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through nonnormal-nonlinear transition to turbulence and another regime emerges. -The presented theory is best summarized in the phase diagram figure 1.

The Rayleigh-Benard theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] is extended towards very large Prandtl numbers P r. The Nusselt number N u is found here to be independent of P r. However, for fixed Rayleigh numbers Ra > 10 10 a maximum around P r ≈ 2 in the N u(P r)-dependence is predicted which is absent for lower Ra. We moreover offer the full functional dependences of N u(Ra, P r) and Re(Ra, P r) within this extended theory, rather than only giving the limiting power laws as done in ref. [1]. This enables us to more realistically describe the transitions between the various scaling regimes, including their widths.In thermal convection, the control parameters are the Rayleigh number Ra and the Prandtl number P r. The system responds with the Nusselt number Nu (the dimensionless heat flux) and the Reynolds number Re (the dimensionless large scale velocity). The key question is to understand the dependences Nu(Ra, P r) and Re(Ra, P r). In experiments, traditionally the Prandtl number was more or less kept fixed [2][3][4]. However, the recent experiments in the vicinity of the critical point of helium gas [5,6] and of SF 6 [7] or with various alcohols [8] allow to vary both Ra and P r and thus to explore a larger domain of the Ra−P r parameter space of Rayleigh-Benard (RB) convection, in particular that for P r ≫ 1. While the experiments of Steinberg's group [7] suggest a decreasing Nusselt number with increasing P r, namely Nu = 0.22Ra 0.3±0.03 P r −0.2±0.04 in 10 9 ≤ Ra ≤ 10 14 and 1 ≤ P r ≤ 93, the experiments of the Ahlers group suggest a saturation of Nu with increasing P r for fixed Ra, at least up to Ra = 10 10 [9]. The same saturation (at fixed Ra = 6 · 10 5 ) is found in the numerical simulations by Verzicco and Camussi [10] and Herring and Kerr [11].The large P r regime of the latest experiments has not been covered by the recent theory on thermal convection by Grossmann and Lohse (GL, [1]), which otherwise does pretty well in accounting for various measurements. In particular, it explains the low P r measurements of Cioni et al.[4] (P r = 0.025), the low P r numerics which reveal Nu ∼ P r 0.14 for fixed Ra [10, 11], and the above mentioned experiments by Niemela et al. [6] and Xu et al. [8].In the present paper we extend the GL theory in a natural way to the regime of very large P r, on which no statement has been made in the original paper [1]. We find Nu to be independent of P r in that regime. We in addition present the complete functional dependences Nu(Ra, P r) and Re(Ra, P r) within the GL theory, rather than only giving the limiting power laws and superpositions of those as was done in [1]. This enables us to more realistically describe the transitions between the various scaling regimes found already in [1].Approach: To make this paper selfcontained we very briefly recapitulate the key idea of the GL theory, which is to decompose in the volume averages of the energy dissipation rate ǫ u and the thermal dissipation rate ǫ θ into their boundary layer (BL) and bulk contributions...

Taylor-Couette flow, the flow between two coaxial co-or counter-rotating cylinders, is one of the paradigmatic systems in the physics of fluids. The (dimensionless) control parameters are the Reynolds numbers of the inner and outer cylinders, the ratio of the cylinder radii, and the aspect ratio. One key response of the system is the torque required to retain constant angular velocities, which can be connected to the angular velocity transport through the gap. Whereas the low-Reynolds number regime was well explored in the 1980s and 1990s of the past century, in the fully turbulent regime major research activity developed only in the past decade. In this article, we review this recent progress in our understanding of fully developed Taylor-Couette turbulence from the experimental, numerical, and theoretical points of view. We focus on the parameter dependence of the global torque and on the local flow organization, including velocity profiles and boundary layers. Next, we discuss transitions between different (turbulent) flow states. We also elaborate on the relevance of this system for astrophysical disks (quasi-Keplerian flows). The review ends with a list of challenges for future research on turbulent Taylor-Couette flow. 53

Very different types of scaling of the Nusselt number Nu with the Rayleigh number Ra have experimentally been found in the very large Ra regime beyond 1011. We understand and interpret these results by extending the unifying theory of thermal convection [Grossmann and Lohse, Phys. Rev. Lett. 86, 3316 (2001)] to the very large Ra regime where the kinetic boundary-layer is turbulent. The central idea is that the spatial extension of this turbulent boundary-layer with a logarithmic velocity profile is comparable to the size of the cell. Depending on whether the thermal transport is plume dominated, dominated by the background thermal fluctuations, or whether also the thermal boundary-layer is fully turbulent (leading to a logarithmic temperature profile), we obtain effective scaling laws of about Nu∝Ra0.14, Nu∝Ra0.22, and Nu∝Ra0.38, respectively. Depending on the initial conditions or random fluctuations, one or the other of these states may be realized. Since the theory is for both the heat flux Nu and the velocity amplitude Re, we can also give the scaling of the latter, namely, Re∝Ra0.42, Re∝Ra0.45, and Re∝Ra0.50 in the respective ranges.

Turbulent Taylor-Couette flow with arbitrary rotation frequencies ω 1 , ω 2 of the two coaxial cylinders with radii r 1 < r 2 is analysed theoretically. The current J ω of the angular velocity ω(x, t) = u ϕ (r, ϕ, z, t)/r across the cylinder gap and and the excess energy dissipation rate ε w due to the turbulent, convective fluctuations (the 'wind') are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor-Couette flow with thermal Rayleigh-Bénard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio η = r 1 /r 2 or the gap width d = r 2 − r 1 between the cylinders. A quantity σ corresponding to the Prandtl number in Rayleigh-Bénard flow can be introduced, σ = ((1 + η)/2)/ √ η) 4 . In Taylor-Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh-Bénard flow. The analogue of the Rayleigh number is the Taylor number, defined as T a ∝ (ω 1 − ω 2 ) 2 times a specific geometrical factor. The experimental data show no pure power law, but the exponent α of the torque versus the rotation frequency ω 1 depends on the driving frequency ω 1 . An explanation for the physical origin of the ω 1 -dependence of the measured local power-law exponents α(ω 1 ) is put forward. Also, the dependence of the torque on the gap width η is discussed and, in particular its strong increase for η → 1.

The unifying theory of scaling in thermal convection (Grossmann & Lohse (2000)) (henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (2001) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (2001) in the regime 3 × 10 7 ≤ Ra ≤ 3 × 10 9 and 4 ≤ P r ≤ 34 and Grossmann & Lohse (2002) used the experimental data point from Qiu & Tong (2001) and the fact that N u(Ra, P r) is independent of the parameter a, which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is on one hand well confirmed through various new experiments and numerical simulations. On the other hand these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well established (and sufficiently distant) Nu(Ra,Pr) data points and show that the resulting N u(Ra, P r) function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra Re(Ra, P r) data point is used to fix Re(Ra, P r). As Re can depend on the definition and the aspect ratio the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping N u(Ra, P r) unchanged.

Our unifying theory of turbulent thermal convection [Grossmann and Lohse, J. Fluid. Mech. 407, 27 (2000); Phys. Rev. Lett. 86, 3316 (2001); Phys. Rev. E 66, 016305 (2002)] is revisited, considering the role of thermal plumes for the thermal dissipation rate and addressing the local distribution of the thermal dissipation rate, which had numerically been calculated by Verzicco and Camussi [J. Fluid Mech. 477, 19 (2003); Eur. Phys. J. B 35, 133 (2003)]. Predictions for the local heat flux and for the temperature and velocity fluctuations as functions of the Rayleigh and Prandtl numbers are offered. We conclude with a list of suggestions for measurements that seem suitable to verify or falsify our present understanding of heat transport and fluctuations in turbulent thermal convection.