We analyze the global transport properties of turbulent Taylor-Couette flow in the strongly turbulent regime for independently rotating outer and inner cylinders, reaching Reynolds numbers of the inner and outer cylinders of Re(i) = 2×10(6) and Re(o) = ±1.4×10(6), respectively. For all Re(i), Re(o), the dimensionless torque G scales as a function of the Taylor number Ta (which is proportional to the square of the difference between the angular velocities of the inner and outer cylinders) with a universal effective scaling law G ∝ Ta(0.88), corresponding to Nu(ω) ∝ Ta(0.38) for the Nusselt number characterizing the angular velocity transport between the inner and outer cylinders. The exponent 0.38 corresponds to the ultimate regime scaling for the analogous Rayleigh-Bénard system. The transport is most efficient for the counterrotating case along the diagonal in phase space with ω(o) ≈ -0.4ω(i).

The ubiquity of turbulent flows in nature and technology makes it of utmost importance to fundamentally understand turbulence. Kolmogorov's 1941 paradigm suggests that for strongly turbulent flows with many degrees of freedom and its large fluctuations, there would only be one turbulent state as the large fluctuations would explore the entire higher-dimensional phase space. Here we report the first conclusive evidence of multiple turbulent states for large Reynolds number Re = O(10 6 ) (Taylor number Ta = O(10 12 )) Taylor-Couette flow in the regime of ultimate turbulence, by probing the phase space spanned by the rotation rates of the inner and outer cylinder. The manifestation of multiple turbulent states is exemplified by providing combined global torque and local velocity measurements. This result verifies the notion that bifurcations can occur in high-dimensional flows (i.e. very large Re) and questions Kolmogorov's paradigm.

We provide direct measurements of the boundary layer properties in highly turbulent Taylor-Couette flow up to Re=2×106) (Ta=6.2×10(12)) using high-resolution particle image velocimetry and particle tracking velocimetry. We find that the mean azimuthal velocity profile at the inner and outer cylinder can be fitted by the von Kármán log law u+=1/κ lny+ +B. The von Kármán constant κ is found to depend on the driving strength Ta and for large Ta asymptotically approaches κ≈0.40. The variance profiles of the local azimuthal velocity have a universal peak around y+≈12 and collapse when rescaled with the driving velocity (and not with the friction velocity), displaying a log dependence of y+ as also found for channel and pipe flows.

The flow structure of strongly turbulent Taylor-Couette flow with Reynolds numbers up to Rei = 2 • 10 6 of the inner cylinder is experimentally examined with high-speed particle image velocimetry (PIV). The wind Reynolds numbers Rew of the turbulent Taylor-vortex flow is found to scale as Rew ∝ T a 1/2 , exactly as predicted [1] for the ultimate turbulence regime, in which the boundary layers are turbulent. The dimensionless angular velocity flux has an effective scaling of N uω ∝ T a 0.38 , also in correspondence with turbulence in the ultimate regime. The scaling of N uω is confirmed by local angular velocity flux measurements extracted from high-speed PIV measurements: though the flux shows huge fluctuations, its spatial and temporal average nicely agrees with the result from the global torque measurements.

Strongly turbulent Taylor-Couette flow with independently rotating inner and outer cylinders with a radius ratio of η = 0.716 is experimentally studied. From global torque measurements, we analyse the dimensionless angular velocity flux N u ω (T a, a) as a function of the Taylor number T a and the angular velocity ratio a = −ω o /ω i in the large-Taylor-number regime 10 11 T a 10 13 and well off the inviscid stability borders (Rayleigh lines) a = −η 2 for co-rotation and a = ∞ for counter-rotation. We analyse the data with the common power-law ansatz for the dimensionless angular velocity transport flux N u ω (T a, a) = f (a)T a γ , with an amplitude f (a) and an exponent γ. The data are consistent with one effective exponent γ = 0.39 ± 0.03 for all a, but we discuss a possible a dependence in the co-and weakly counter-rotating regimes. The amplitude of the angular velocity flux f (a) ≡ N u ω (T a, a)/T a 0.39 is measured to be maximal at slight counter-rotation, namely at an angular velocity ratio of a opt = 0.33 ± 0.04, i.e. along the line ω o = −0.33ω i . This value is theoretically interpreted as the result of a competition between the destabilizing inner cylinder rotation and the stabilizing but shear-enhancing outer cylinder counter-rotation. With the help of laser Doppler anemometry, we provide angular velocity profiles and in particular identify the radial position r n of the neutral line, defined by ω(r n ) t = 0 for fixed height z. For these large T a values the ratio a ≈ 0.40, which is close to a opt = 0.33, is distinguished by a zero angular velocity gradient ∂ω/∂r = 0 in the bulk. While for moderate counter-rotation −0.40ω i ω o < 0, the neutral line still remains close to the outer cylinder and the probability distribution function of the bulk angular velocity is observed to be monomodal. For stronger counterrotation the neutral line is pushed inwards towards the inner cylinder; in this regime the probability distribution function of the bulk angular velocity becomes bimodal, reflecting intermittent bursts of turbulent structures beyond the neutral line into the outer flow domain, which otherwise is stabilized by the counter-rotating outer cylinder. Finally, a hypothesis is offered allowing a unifying view and consistent interpretation for all these various results.