Taylor column instability in the problem of a vibrational hydrodynamic top

The object of experimental study is a fluid flow generated by differential rotation of a free light spherical body in a rotating cylindrical cavity. The body stays near the axis under the action of centrifugal force. The body rotation is generated by a force field oscillating in the cavity reference system (vibrational hydrodynamic top). It was found that the Taylor-Proudman column that forms undergoes instability, which manifests itself in the formation of a two-dimensional azimuthal wave at the column boundary, in a Stewartson layer. The experimental results are summarized on a plane of dimensionless parameters, i.e., the dimensionless velocity of the cavity rotation and Rossby number. The bounds of the Stewartson layer stability were found and the supercritical structures and transition sequences were studied. Systematic research into that problem in its classical formulation—when a sphere is fixed on the axis and its differential rotation is imposed—was done for comparison. It was demonstrated that in conditions of vibratory differential rotation of a free sphere the stability threshold of the Stewartson layer was reduced by more than one order of magnitude, in comparison with the classical case. A qualitative change was also found in the wave phase velocity which for a free sphere exceeds the lagging differential rotation velocity of the body. It was concluded that the uncovered specifics are related to the difference in the mechanism of the Taylor-Proudman column formation and of the flow generation in it. For a vibrational hydrodynamic top, streams in the column will not be defined by Ekman pumping but by steady streaming, which is also responsible for the free-sphere differential rotation.

Figure 1

Experimental setup scheme. Supplementary elements presented in hatchwork are used in experiments with the sphere fixed on the cuvette axis. Along with that, the rotation rates of the cuvette and the sphere are independently set with two stepper motors.

Figure 2

Working cavity and Stewartson shear layers (dashed lines) at differential rotation of the free sphere (a) and the sphere fixed on the rotation axis (b).

Figure 3

Cuvette photograph.

Figure 4

Dependence of the relative sphere velocity $\Delta \Omega$ (curves I, points 1 and 2) and wave phase velocity $\Delta {\Omega }_w$ (curves II, points 3 and 4) at the boundaries of the Taylor-Proudman column versus cuvette rotation rate for two experiments with different liquids. Vertical arrows mark the centrifugation thresholds of the sphere, ${\Omega }_{\mathrm{rot}}^{*}$; dashed arrows mark transitions from one structure to another. Hereafter, dark points correspond to a monotonic increase of ${\Omega }_{\mathrm{rot}}$, and light ones to its decrease.

Figure 5

Position of the sphere relative to the cavity ends depending on ${\Omega }_{\mathrm{rot}}$ for $\nu =5.0$ cSt (points 1) and $\nu =18.4$ cSt (points 2).

Figure 6

Relative velocity of the differential rotation of a free light sphere in liquids having different viscosities, depending on the parameter ${\Gamma }^{2}r/\delta$.

Figure 7

Photographs of the Taylor-Proudman column in cross section: (a) Axially symmetric shape of the flow, $m=0$ ($\nu =6.7$ cSt, ${\Omega }_{\mathrm{rot}}=100.5{\text{s}}^{-1}$); wave numbers $m=3$ and 4 correspond to cases (b) and (c) ($\nu =6.7$ cSt, ${\Omega }_{\mathrm{rot}}=75.5{\text{s}}^{-1}$, the structures differ in their history). Photograph (d) is obtained at $\nu =5.8$ cSt, ${\Omega }_{\mathrm{rot}}=85.5{\text{s}}^{-1}$.

Figure 8

The map of the system state and flow regimes on the plane ($\nu ,{\Omega }_{\mathrm{rot}}$). Points 1 and 2 mark the threshold of system centrifugation in case of an increase of ${\Omega }_{\mathrm{rot}}$ and collapse in case of a decrease; 3 and 4 the bounds of structure reconfiguration (increase of $m$) in the case of increase of ${\Omega }_{\mathrm{rot}}$; 5 and 6 the bounds of structure change in the case of decrease of ${\Omega }_{\mathrm{rot}}$; 7 the threshold of two-dimensional Taylor-Proudman column destruction; 8 the stability threshold of the axisymmetric form of the column.

Figure 9

Threshold values of the relative velocity of the differential rotation $\left|\Delta \Omega \right|/{\Omega }_{\mathrm{rot}}$ (points 1) and vibrational acceleration $\Gamma$ (points 2) depending on liquid viscosity; $\rho$ varies within the range $\rho =1.13-1.17$.

Figure 10

Side view of the Taylor-Proudman column. Photo-graphs (a) and (b) correspond to the experiment conditions given in Figs. 7 and 7. Photographs (a) and (b) differ in the visualizer type: resin Amberlite particles (a), and aluminum powder (b).

Figure 11

The map of the mean flow structures: the stability curve of the axisymmetric flow and boundaries of transition from one wave number to another in case of an increase (a) and a decrease (b) of differential rotation rate of the body. Points 1 correspond to the critical value of the Rossby number at which the axisymmetric flow becomes nonaxisymmetric. Points 2 and 3 correspond to transitions to $m=3$ and 4 with an increase of $\left|\text{Ro}\right|$ (a) and to $m=4$ and 5 with its decrease (b). Points 4–6 specify areas of appearance of different structures in the experiment with $\nu =6.7$ cSt.

Figure 12

Photographs of the Stewartson shear layer in transverse section when $\text{Ro}<0$, the axis of sphere rotation coincides with the axis of cavity rotation; (a) Axisymmetric flow shape, $m=0$; wave numbers $m=10-7$ correspond to photographs (b)–(e); (f) after destruction of the Taylor-Proudman column.

Figure 13

Boundaries of existence of different flow modes in cases of increase (light symbols) or decrease (dark symbols) of Rossby number ($\text{Ro}<0$). The solid line shows the threshold of transition to the axisymmetric flow; the dashed line is drawn through points where the Taylor-Proudman column is destroyed; the hysteresis areas are between open and filled symbols of the same shape. Points 1 delimit the mode $m=7$; 2, $m=8$; 3, $m=9$; 4 and 5, $m=10$.

Figure 14

Taylor-Proudman column in cross section; the axis of rotation is on the cavity axis; $\text{Ro}>0$.

Figure 15

Boundaries of existence of different modes of the main flow in the cases of an increase (light symbols) and decrease (dark symbols) of Rossby number ($\text{Ro}>0$). Hysteresis areas are between empty and filled symbols of the same shape. Points 1 limit mode $m=3$; 2, $m=4$; 3, $m=5$; 4, $m=6$; 5 and 9, $m=6/2$; 6, $m=7$; 7, $m=8$; 8, $m=9$.

Figure 16

Wave phase velocity in the cases of negative (a) and positive (b) differential rotation of the sphere.

Figure 17

Transitions between flow structures of streams in the case of the sphere fixed on the cavity rotation axis when $\text{Ro}<0$ and $\text{Ro}>0$.

Figure 18

Stability thresholds of the axisymmetric flow for a body fixed on the cavity rotation axis.

Figure 19

Stability thresholds of the axisymmetric flow generated by the free sphere (circles) and by the sphere fixed on the axis of rotation (diamonds). Wave numbers $m$ of the azimuthal waves developing in the supercritical area are indicated with numbers.

Figure 20

Wave phase velocity in the Stewartson layer in the cases of the free sphere and the sphere fixed on the axis of rotation, $\text{Ro}<0$. The dashed line corresponds to equality of velocities ${\Omega }_s={\Omega }_w$. The wave numbers $m$ of structures are indicated by numbers. The value of $\omega$ for the case of the sphere fixed on the rotation axis corresponds to the symbols in Fig. 16.