Flow patterns in a rotating horizontal cylinder partially filled with liquid

The dynamics of an annular layer of low-viscosity liquid inside a rapidly rotating horizontal cylinder is experimentally studied. Under gravity, the liquid performs forced azimuthal oscillations in the cavity frame. We examined the stability of the two-dimensional azimuthal flow and discovered two novel types of axisymmetric liquid flows. First, a large-scale axially symmetric flow is excited near the end walls. The inertial modes generated in the corner regions are proven to be responsible for such a flow. Second, a small-scale flow in the form of the Taylor-Gortler vortices appears due to the centrifugal instability of the oscillatory liquid flow. The spatial period of the vortices is in qualitative agreement with the data obtained in the experimental and numerical studies of cellular flow in librating containers.

Figure 1

(a) The dependence of relative velocity of steady streaming on the dimensionless frequency of vibration: $f_v={\mathrm{\Omega }}_v/2\pi =15$ and 30 Hz, amplitude $b=0.35$ mm, kinematic viscosity $\nu =1$ cSt, relative filling $q=0.23$; (b) domains of the inertial waves existing on the plot of governing parameters $n,{\mathrm{\Gamma }}_v$; $\nu =1$ cSt, $q=0.23$. Empty and semifilled symbols correspond to the increase and decrease of frequency $n$; transitions $a$ and $c$ represent soft and finite-amplitude thresholds for excitation of the inertial wave, and transition $b$ corresponds to the wave disappearance. Adapted from Polezhaev [9].

Figure 2

Scheme of experimental setup.

Figure 3

Visualization of flow patterns in the annular layer of liquid inside a rotating cylinder: $R=2.5$ cm, $L=31.7$ cm, $\nu =1.0$ cSt, $q=0.31,\mathrm{\Omega }=81.6$ (a) and $69.0 {\mathrm{s}}^{-1}$ (b); light areas represent domains of high concentrations of aluminum powder. Distance between vertical markers equals to 25 cm.

Figure 4

Dependence of spatial period $\lambda$ of flow patterns on rotation rate: $\nu =1.1$ cSt (kerosene), $q=0.12$, 0.16, 0.20, 0.25, and 0.30.

Figure 5

Photo of patterns near the end wall and the scheme of inertial wave propagation in the annular liquid layer of thickness $h$; $R$ is the radius of the cylinder; $\lambda$ is the distance between two successive reflections of inertial wave from the solid surface. Dashed lines demonstrate the positions of bands of high concentrations of aluminum flakes. Bands of high concentrations of powder coincide with the regions of inertial waves' reflection at the cylindrical wall.

Figure 6

Dependence of dimensionless spatial period of ring patterns on relative filling $q$. Solid line represents data obtained from (4).

Figure 7

Photograph of fine pattern formation in the central part of the cylinder; relative filling $q=0.23$, kinematic viscosity $\nu =2.3$ cSt, angular velocity $\mathrm{\Omega }=66 {\mathrm{s}}^{-1}$; distance between white vertical markers equals 7.5 cm.

Figure 8

Dependence of wave number $k\equiv 2\pi h/\lambda$ on relative filling $q$: kinematic viscosity $\nu =3.6$ or 1.0 cSt.

Figure 9

Dependence of dimensionless wave number $k_{\delta }$, based on the thickness of the Stokes boundary layer, on dimensionless velocity $\omega$: kinematic viscosity $\nu =3.6$ and 1.0 cSt.

Figure 10

Dependence of dimensionless wave number $k_{\delta }$ on parameter $\mathrm{\Omega }/{\mathrm{\Omega }}_{\mathrm{osc}}$ (log-log scale): Triangles correspond to data adapted from Kozlov and Polezhaev [30], circle corresponds to $k_{\delta }=0.67$, obtained in the limit $\omega >500$ (Fig. 9). The solid line corresponds to $k_{\delta }\sim {(\mathrm{\Omega }/{\mathrm{\Omega }}_{\mathrm{osc}})}^{1/2}$.

Figure 11

Dimensionless wave number $k_r\equiv 2\pi {\delta }_r/\lambda$ versus dimensionless velocity of rotation $\omega$: Gravitational data (circles) correspond to those in Fig. 9 in the limit $\omega >500$; vibrational data (triangles) correspond to that in Fig. 10.

Figure 12

Dependence of Reynolds number on Ekman number at the threshold of the Gortler vortices' appearance. Positive and negative values of $\mathrm{\Delta }\mathrm{\Omega }$ correspond to prograde and retrograde azimuthal flow, respectively.

Figure 13

Dependence of ratio ${\mathrm{Re}}_{\mathrm{in}}/{\mathrm{Re}}_{\mathrm{out}}$ on Reynolds number. Symbols correspond to those in Fig. 12.