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Bi-stability in turbulent, rotating spherical Couette flow
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Image of FIG. 1.
FIG. 1.

A schematic of the apparatus showing the inner and outer sphere and locations of measurement ports in the vessel top lid at 60 cm cylindrical radius (1.18 r i). Data from sensors in the ports are acquired by instrumentation, including an acquisition computer, bolted to the rotating lid and wirelessly transferred to the lab frame. Also shown is the wireless torque sensor on the inner shaft.

Image of FIG. 2.
FIG. 2.

(a) The dimensionless torque G vs. the Reynolds number Re for the case , stationary outer sphere, with a fit to G = A Re + B Re 2 with and B = 0.05. (b) Mean value of the dimensionless torque G vs. Ro at . Ro is varied by increasing the inner sphere speed in steps of Ro = 0.067, waiting 450 rotations per step. Three curves are differentiated by circles, triangles, and squares. In the ranges of Ro where the symbols overlap, the flow exhibits bistable behavior. H and L denote the torque curves of the “high torque” and “low torque” states. There is a second bi-stable regime starting around Ro = 2.75, with LL labeling the lower torque state.

Image of FIG. 3.
FIG. 3.

Time series of G at fixed Ro = 2.13 and , with time made dimensionless by the outer sphere rotation period. The raw torque signal has been numerically low pass filtered (fc  = 0.05 Hz, 15 rotations of the outer sphere).

Image of FIG. 4.
FIG. 4.

Probability density of the dimensionless torque at Ro = 2.13 and . The full, unconditioned distribution is denoted by small points. Solid circles denote conditioning on low torque state, and open circles on the high, with Gaussian solid and dashed curves for the low and the high, respectively. The mean and standard deviation of the low torque state data are and . In the high torque state, they are and . The data were low pass filtered at fc  = 0.5 Hz to remove the high frequency noise caused by mechanical vibration.

Image of FIG. 5.
FIG. 5.

Probability that the flow was in the high torque (open circles) or low torque (closed circles) state as a function of Ro over the first bistable range with fixed . The dashed and solid lines are fits to the exponential form of Eq. (8).

Image of FIG. 6.
FIG. 6.

The angular momentum, inner torque, outer torque, and net torque. Ro = 2.13, . The upper plot shows the dimensionless angular momentum as defined in Eq. (12). The lower plot shows the inner torque G, the outer torque G o, and their sum G net. The bearing and aerodynamic drag on the outer sphere have been subtracted off, and the torques have been low pass filtered as in Fig. 3.

Image of FIG. 7.
FIG. 7.

(Color) Simultaneous time series of velocity, wall shear stress, and torque. A space-time diagram of the low pass filtered velocity (defined in Eq. (15)) is shown at the top. This measurement is dominated by the azimuthal velocity . The velocity is made dimensionless by the outer sphere tangential velocity, and so can be interpreted as a locally measured Rossby number. The dimensionless wall shear stress is shown in the middle, and the dimensionless torque G is shown at the bottom. The wall shear stress and torque have been low pass filtered with fc  = 0.05 Hz as before, which is comparable to the time averaging of the velocimetry.

Image of FIG. 8.
FIG. 8.

(a) The probability density of the dimensionless velocity conditioned on the torque state, with solid circles denoting the low torque state and open circles denoting the high. Dashed and solid lines are Gaussian. Standard deviations are for the high state and for the low state. The high state mean is , and the low state mean is . The means are dominated by azimuthal velocity, though the transducers are equally sensitive to u z and in this measurement. (b) The probability density of the dimensionless wall shear conditioned on torque. Solid circles again denote the low torque state, and open circles the high, with solid and dashed Gaussian curves. The mean and standard deviation in the low torque state are and . In the high torque state, they are and .

Image of FIG. 9.
FIG. 9.

Power spectra of wall shear stress at Ro = 2.13 and , conditioned on state. Angular frequency has been made dimensionless using the outer sphere angular speed. The black curve is the spectrum from the low torque state, and the gray curve is that of the high torque state. The low torque spectrum has prominent peaks at , 0.71, and harmonics. In the high torque state, there are broad peaks at and 0.53.

Image of FIG. 10.
FIG. 10.

(Color) A spectrogram of wall pressure at 23.5° colatitude shows the evidence of several flow transitions as Ro is varied. and Ro , waiting 430 rotations per step with steps of Ro = 0.1. Instead of averaging the power spectra over an entire step of Ro, there are 30 spectra per step in Ro, so some temporal evolution is visible at a given Ro. In the L state, there are two strong waves, the lower at (a) varies only slightly in frequency with Ro. The higher frequency wave at (b) varies more strongly with Ro, suggesting that advection by the mean flow is important in setting its frequency.

Image of FIG. 11.
FIG. 11.

A sketch of two possible mean flow states. The low torque state is labeled L and the high labeled H. The low torque state at (a) is characterized by fast zonal circulation near the core of the experiment and large amplitude waves. In the high torque state, the velocity profile varies more gradually. The zonal circulation has been destroyed by mixing across the transport barrier, so the fluid near the inner sphere is slower and the torque higher.

Image of FIG. 12.
FIG. 12.

Time delay embeddings of the slow velocity fluctuations at for three values of Ro below, in, and above the bistable range. Arrows in (b) show the direction taken by the transitions. The time delay is 4.5 rotations of the outer sphere in all three cases. The dimensionless velocity as defined previously has been scaled by Ro, roughly collapsing the mean velocity and fluctuation levels of each state. These data have been low pass filtered with fc  = 0.1 Hz or a period of 7.5 rotations of the outer sphere.


Generic image for table
Table I.

Statistics of the interval between high torque onsets. is the time interval between two subsequent high torque onsets made dimensionless by , so the time interval is measured in outer sphere rotations. Ro = 2.13, .


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Scitation: Bi-stability in turbulent, rotating spherical Couette flow