Measurements of the Nusselt number and properties of the large-scale circulation (LSC) are presented for turbulent Rayleigh–Bénard convection in water-filled cylindrical containers (Prandtl number Pr = 4.38) with aspect ratio Γ = 0.50. They cover the range 2 × 108 ≲ Ra ≲ 1 × 1011 of the Rayleigh number Ra. We confirm the occurrence of a double-roll state (DRS) of the LSC and focus on the statistics of the transitions between the DRS and a single-roll state (SRS). The fraction of the run time when the SRS existed varied continuously from about 0.12 near Ra = 2 × 108 to about 0.8 near Ra = 1011, while the fraction of the run time when the DRS could be detected changed from about 0.4 to about 0.06 over the same range of Ra. We determined separately the Nusselt number of the SRS and the DRS, and found the former to be larger than the latter by about 1.6% (0.9%) at Ra = 1010 (1011). We report a contribution to the dynamics of the SRS from a torsional oscillation similar to that observed for cylindrical samples with Γ = 1.00. Results for a number of statistical properties of the SRS are reported, and some are compared with the cases Γ = 0.50, Pr = 0.67 and Γ = 1.00, Pr = 4.38. We found that genuine cessations of the SRS were extremely rare and occurred only about 0.3 times per day, which is less frequent than for Γ = 1.00; however, the SRS was disrupted frequently by roll-state transitions and other less well-defined events. We show that the time derivative of the LSC plane orientation is a stochastic variable which, at constant LSC amplitude, is Gaussian distributed. Within the context of the LSC model of Brown & Ahlers (Phys. Fluids, vol. 20, 2008b, art. 075101), this demonstrates that the stochastic force due to the small-scale fluctuations that is driving the LSC dynamics has a Gaussian distribution.

For rapidly rotating turbulent Rayleigh-Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like Ra 1/4 Ek 2/3 where the Ekman number Ek decreases with increasing rotation rate.Turbulent fluid motion driven by buoyancy and influenced by rotation is a common phenomenon in nature and is important in many industrial applications. In the widely studied laboratory realization of turbulent convection, Rayleigh-Bénard convection (RBC) [1, 2], a fluid is confined in a convection cell with a heated bottom, cooled top, and adiabatic vertical walls. For these conditions, a large scale circulation (LSC) arises from cooperative plume motion and is an important feature of turbulent RBC [1]. The addition of rotation about a vertical axis produces a different type of convection as thermal plumes are transformed into thermal vortices, over some regions of parameter space heat transport is enhanced by Ekman pumping [3][4][5][6][7][8][9][10], and statistical measures of vorticity and temperature fluctuations in the bulk are strongly influenced [11][12][13][14][15][16][17]. A crucial aspect of rotation is to suppress, for sufficiently rapid rotation rates, the LSC of non-rotating convection [12,13,18,19], although the diameter-to-height aspect ratio Γ = D/H appears to play some role in the nature of the suppression [20].In RBC geometries with 1/2 ≤ Γ ≤ 2, the LSC usually spans the cell in a roll-like circulation of size H. For rotating convection, the intrinsic linear scale of separation of vortices is reduced with increasing rotation rate [21,22], suggesting that one might reduce the geometric aspect ratio, i.e., Γ < 1 while maintaining a large ratio of lateral cell size to linear scale [5]; such convection cells are being implemented in numerous new experiments [23]. Thus, an important question about rotating convection in slender cylindrical cells is whether there is a global circulation that substantially influences the internal state of the system and carries appreciable global heat transport. Direct numerical simulations (DNS) of rotat-ing convection [24] in cylindrical geometry with Γ = 1, inverse Rossby number 1/Ro = 2.78, Rayleigh number Ra = 10 9 and Prandtl number Pr = 6.4 (Ro, Ra and Pr defined below) revealed a cyclonic azimuthal velocity boundary-layer flow surrounding a core region of anticyclonic circulation and localized near the cylinder sidewall. The results were interpreted in the context of sidewall Stewartson layers driven by active Ekman layers at the top and bottom of the cell [25,26].Here we show through DNS and experimental measurements for a...

In turbulent thermal convection in cylindrical samples of aspect ratio Γ ≡ D/L (D is the diameter and L the height) the Nusselt number Nu is enhanced when the sample is rotated about its vertical axis, because of the formation of Ekman vortices that extract additional fluid out of thermal boundary layers at the top and bottom. We show from experiments and direct numerical simulations that the enhancement occurs only above a bifurcation point at a critical inverse Rossby number 1/Roc, with 1/Roc ∝ 1/Γ. We present a Ginzburg-Landau like model that explains the existence of a bifurcation at finite 1/Roc as a finite-size effect. The model yields the proportionality between 1/Roc and 1/Γ and is consistent with several other measured or computed system properties.PACS numbers: 47.27.te,47.32.Ef,47.20.Bp,47.27.ek Turbulence, by virtue of its vigorous fluctuations, is expected to sample all of phase space over wide parameter ranges. This viewpoint implies that there should not be any bifurcations between different turbulent states. Contrary to this, several cases of discontinuous transitions have been observed recently in turbulent systems [1][2][3]. When they occur, they are likely to be provoked either by changes in boundary conditions or boundarylayer structures, or by discontinuous changes in the largescale structures, as a parameter is varied.Recently some of us [4,5] reported on the effect of rotation about a vertical axis at a rate Ω on turbulent convection in a fluid heated from below and cooled from above (known as Rayleigh-Bénard convection or RBC; for recent reviews, see [6-8]). For a cylindrical sample of aspect ratio Γ ≡ D/L = 1.00 (D is the diameter and L the height) a supercritical bifurcation was found, both from experiments and from direct numerical simulation (DNS) of the Boussinesq equations of motion. At a finite Ω, as expressed by the inverse Rossby number 1/Ro ∝ Ω (to be defined explicitly below), there was a sharp transition from a state of nearly rotation-independent heat transport (as expressed by the Nusselt number Nu to be defined explicitly below) to one in which Nu was enhanced by an amount δNu(1/Ro). This is illustrated by the data shown in Fig. 1. The increase of Nu was attributed to Ekman pumping [4, 5, 9-15], i.e. to the formation of (cyclonic) vertical vortex tubes ("Ekman vortices"), which extract and vertically transport additional fluid from the boundary layers (BLs) and thereby enhance the heat transport. The bifurcation was located at a critical value 1/Ro c 0.40 [5]. The reason for the existence of the bifurcation at 1/Ro c > 0 hitherto had not been understood. While such bifurcations are common near the onset of RBC in the domain of pattern formation [16], their existence in the turbulent regime implies a paradigm shift.In this Letter we report on further experiments for samples with Γ = 2.00, 1.00, and 0.50 which (i) all show bifurcations between different turbulent states and (ii) reveal that 1/Ro c varies approximately in proportion to 1/Γ. We offer an explanation of these ...

We report on the influence of rotation about a vertical axis on heat transport by turbulent Rayleigh-Bénard convection in a cylindrical vessel with an aspect ratio Γ ≡ D/L = 0.50 (D is the diameter and L the height of the sample) and compare the results with those for larger Γ . The working fluid was water at T m = 40 • C where the Prandtl number Pr is 4.38. For rotation rates Ω 1 rad s −1 , corresponding to inverse Rossby numbers 1/Ro between zero and twenty, we measured the Nusselt number Nu for six Rayleigh numbers Ra in the range 2.2 × 10 9 Ra 7.2 × 10 10 . For small rotation rates and at constant Ra, the reduced Nusselt number Nu red ≡ Nu(1/Ro)/Nu(0) initially increased slightly with increasing 1/Ro, but at 1/Ro = 1/Ro 0 0.5 it suddenly became constant or decreased slightly depending on Ra. At 1/Ro c ≈ 0.85 a second sharp transition occurred in Nu red to a state where Nu red increased with increasing 1/Ro. We know from direct numerical simulation that the transition at 1/Ro c corresponds to the onset of Ekman vortex formation reported before for Γ = 1 at 1/Ro c 0.4 and for Γ = 2 at 1/Ro c = 0.18 (Weiss et al., Phys. Rev. Lett., vol. 105, 2010, 224501). The Γ -dependence of 1/Ro c can be explained as a finite-size effect that can be described phenomenologically by a Ginzburg-Landau model; this model is discussed in detail in the present paper. We do not know the origin of the transition at 1/Ro 0 . Above 1/Ro c , Nu red increased with increasing Γ up to ∼1/Ro = 3. We discuss the Γ -dependence of Nu red in this range in terms of the average Ekman vortex density as predicted by the model. At even larger 1/Ro 3 there is a decrease of Nu red that can be attributed to two possible effects. First, the Ekman pumping might become less efficient when the Ekman layer is significantly smaller than the thermal boundary layer, and second, for rather large 1/Ro, the Taylor-Proudman effect in combination with boundary conditions suppresses fluid flow in the vertical direction.