Surface switching statistics of rotating fluid: Disk-rim gap effects

We examined the influence of internal noise on the irregular switching of the shape of the free surface of fluids in an open cylindrical vessel driven by a bottom disk rotating at constant speed [Suzuki, Iima, and Hayase, Phys. Fluids 18, 101701 (2006)]. A slight increase in the disk-rim gap (less than 3% of the disk radius) was established experimentally to cause significant changes in this system, specifically, frequent appearance of the surface descending event connecting a nonaxisymmetric shape in strong mixing flow (turbulent flow) and an axisymmetric shape in laminar flow, as well as a shift in critical Reynolds number that define the characteristic states. The physical mechanism underlying the change is analyzed in terms of flow characteristics in the disk-rim gap, which acts as a noise source, and a mathematical model established from measurements of the surface height fluctuations with noise term.

Figure 1

(a) Experimental setup and measurement apparatus (unit mm). (b) Typical photographic sequence of surface switching showing surface shapes obtained with back-light projection.

Figure 2

Time series of the averaged surface height $\langle h\rangle$ for different disks, (a) disk (A) ($\mathrm{\Omega }=13.3$ Hz, Re $=1.47\times {10}^5$), (b) disk (B) ($\mathrm{\Omega }=13.2$ Hz, Re $=1.46\times {10}^5$), and (c) disk (C) ($\mathrm{\Omega }=13.3$ Hz, Re $=1.47\times {10}^5$). The averaged surface height $\langle h\rangle$ is a moving average of the original surface-height data for the period corresponding to around seven rotations of the disk to remove the rapid fluctuations caused mainly by the rotating humps on the ellipsoidal free surface shown in Fig. 1 (iv). The solid arrows in (b) indicate periods including FR and LTS highlighted in Fig. 5. The dashed arrow in (b) indicates an example of the RO state in the time series. The arrows in (c) mark the time intervals of LTS events, $t_s$ (dips extending below the broken line).

Figure 3

(a) Variation of the intensity of the velocity fluctuations, $V$ vs $\mathrm{Re}$, where the open and solid symbols represent the increasing and decreasing stages, respectively, of $\mathrm{Re}$: arrows indicate the range of the hysteresis region for the disks. (b) Photograph of the surface shape for disk (B) at $\mathrm{Re}=1.11\times {10}^5$ ($\mathrm{\Omega }=10.0$ Hz) and (c) at $\mathrm{Re}=1.40\times {10}^5$ ($\mathrm{\Omega }=12.7$ Hz).

Figure 4

Autocorrelation function of the series of time intervals for LTS event, $\left\{t_s\left[i\right]\right\}$.

Figure 5

Probability density distribution of the time interval for the appearance of LTS events, $t_s$, where the panel inside the larger panel expresses the cumulative distribution of $t_s$, Eq. (5.3).

Figure 6

Time variation of root-mean square values of the spatial velocity fluctuation disturbing the flow by inserting a small hexakey (around 5 mm in mean diameter) in the vessel: horizontal arrows indicate the duration of disturbances, (A) a longer duration and (B) a shorter duration: vertical arrows indicate time of detachment of the free surface from the bottom.

Figure 7

Schematic view of the balancing points embedded in the phenomena and relating events for (a) low, (b) moderate, and (c) high noise conditions, where vertical and horizontal extents indicate the degree of surface height and velocity fluctuation; color of an area represents stability or sojourn time of each state: white, low and very short; gray, moderate; and black, high and long.