The dynamics of a low-viscosity fluid inside a rapidly rotating horizontal cylinder were experimentally studied. In the rotating frame, the force of gravity induces azimuthal fluid oscillations at a frequency equal to the velocity of the cylinder’s rotation. This flow is responsible for a series of phenomena, such as the onset of centrifugal instability in the Stokes layer and the growth of the relief at the interface between the fluid and the granular medium inside the rotating cylinder. The phase inhomogeneity of the oscillatory fluid flow in the viscous boundary layers near the rigid wall and the free surface generates the azimuthal steady streaming. We studied the relative contribution of the viscous boundary layers in the generation of the steady streaming. It is revealed that the velocity of the steady streaming can be calculated using the velocity of the oscillatory fluid motion.

The rotational and oscillatory flows of a fluid with a free surface in a container are of fundamental interest and are interesting for certain applications. When the horizontal cylinder is stationary, the liquid is at rest in a pool at the bottom of the cylinder. Inside a moving cylinder, the fluid may move. Practical examples consist of undesirable slosh in spacecraft storage tanks and in aircrafts, rockets, road, and marine vehicles transporting liquids (for a brief review, see [

We experimentally examined the dynamics of an annular layer of a low-viscosity liquid inside a rapidly rotating horizontal cylinder. Inside a rapidly rotating cylinder, the fluid is observed to be almost in rigid body rotation. The force of gravity prevents the rise of the fluid on the ascending wall of the rotating cylinder, so the angular velocity of the fluid

The annular layer of PMS-20 silicon oil inside the cylinder with radius

Phillips [

The oscillatory motion of the annular layer is of great interest for both fundamental hydrodynamic research and astrophysical applications. Dyakova et al. [

Furthermore, oscillatory fluid motion is centrifugally unstable to the onset of spatially periodic flow similar to Taylor-Gortler vortices [

The two-dimensional oscillatory flow in an annular fluid layer is a source of azimuthal steady streaming. The progressive azimuthal wave produces boundary layers at both the bottom and the free surface. Batchelor [

The additional steady streaming could be an effect of the chemical heterogeneity of the fluid and the subsequent increase of the surface tension gradient on the free surface. This flow is referred to as Marangoni solutocapillary flow and is directed along the gradient of the surface tension [

The other source of the intensification of the azimuthal steady streaming is the resonant excitation of progressive and standing surface waves in the annular layer of the fluid [

For the first time, we report experimental data regarding the velocity of the oscillatory flow and the steady streaming in the annular fluid layer inside a rapidly rotating cylinder. The experiments were conducted with fluids without surfactants and in the absence of surface waves, which implies that the steady streaming was generated only by the viscous boundary layers at the rigid bottom and at the free surface. The results of the analysis allow us to assess the contribution of these sources to the velocity of the azimuthal steady streaming.

The experimental setup is shown in Figure ^{−1} with an accuracy of 0.05%.

Sketch of experimental setup.

The experiments were conducted using silicon oil PMS-20 with a viscosity of 20 cSt and a density of ^{3}. The quantity of the liquid is characterized by the relative filling ^{3} to measure the free-surface velocity. The markers on the free surface were illuminated using stroboscopic light 4, which had frequency equal to the cylinder rotation rate.

Images of the fluid were recorded using a high-speed camera 5, which was positioned at a right angle to the cylinder axis. The camera provided registration of the free surface with frame rates of up to 500 fps at a resolution of 1280 × 1024 pixels.

Each experiment followed a similar protocol. When the cylinder was partially filled with fluid and rotated sufficiently rapidly, the fluid underwent nearly rigid-body rotation about a central air column. At the definite rotation rate, the mass-transport free surface velocity was measured, and images of the free surface were recorded using the high-speed camera after several minutes had elapsed. The rotation rate was then decreased by

Inside a rapidly rotating cylinder, the annular layer thickness varies along the azimuthal coordinate: the layer has the greatest thickness at the upper pole and the least thickness at the bottom of the cylinder (Figure

After centrifugation, the light markers occupied arbitrary positions at the free surface. According to the observations, the angular velocity of the markers (i.e., liquid)

Note that the steady streaming is two-dimensional. In addition to the azimuthal motion, axial drift was also present: markers at the free surface moved slowly to the nearest end-wall. After sticking to the end-wall, the particles never came back to the central part of the cylinder. The origin of the axial drift will be discussed in detail in the “Azimuthal Steady Streaming” section. Here, we simply note that the period of the azimuthal steady streaming was approximately 10^{1}–10^{2} s, and the typical time of the axial drift was approximately 10^{2}–10^{3} s. The azimuthal fluid motion was almost one-dimensional during the several cycles of the cavity rotation, which allows for the measurement of the azimuthal component of the fluid velocity.

Figure

Time evolution of the azimuthal position of the marker at the free surface in the rotating frame; the dotted line indicates the azimuthal steady streaming (a); azimuthal position of the marker

Here, we focus on the oscillatory fluid motion. Let us follow the evolution of the marker during the rotation cycle (Figure

The dynamics of the annular flow of a low-viscosity fluid is governed by the interplay between the force of gravity and the centrifugal force. Following Ivanova et al. [

Additionally, for certain combinations of the rotation rate and the liquid volume, the excitation of the surface waves with various axial and azimuthal wavenumbers is possible [

Time evolution of the azimuthal position of the marker at the free surface in the presence of a surface wave (a); the results of the Fourier analysis:

Fourier analysis shows that the oscillatory fluid motion is a superposition of two harmonic oscillations (Figure

Hereafter, we will consider only forced fluid oscillations. Because the fluid oscillates under the force of gravity, the amplitude of the forced oscillations is determined by the dimensionless acceleration

Time evolution of the azimuthal position of the marker at the free surface, rotation rate

The summarized results are shown in the plane of the dimensionless parameters

Dimensionless velocity of the oscillatory fluid flow versus the dimensionless acceleration at different values of the relative filling

Dimensionless velocity of the oscillatory fluid flow

The mismatch between the theoretical and experimental results can be explained by the fact that (

In addition to the forced oscillations, the steady flow of the annular fluid layer was observed. In the rotating frame, the fluid distribution is considered as a two-dimensional surface wave propagating in the direction opposite to the cylinder’s rotation. The azimuth wave generates a mean vorticity in the viscous boundary layers at both the bottom and the free surface and gives rise to the azimuthal drift just outside the boundary layers in the direction of the wave propagation. The effect of the Stokes boundary layer near the rigid wall in the approximation of the thin annular layer was studied in detail by Ivanova et al. [

According to the observations, the steady streaming was two-dimensional: the velocity of a fluid element had azimuthal and axial components, but the axial drift was much slower than the azimuthal steady flow. The azimuthal fluid drift was uniform along the axis of rotation except for the short distance near the end-walls of the cylinder: the fluid satisfies the no-slip condition at the end-walls. Therefore, the viscous boundary layer (Ekman layer) develops near the end-wall. As previously mentioned, the fluid rotates slower than the cylinder (see, e.g., Figure

It is worth noting that, if the fluid rotates faster than the cylinder and provokes the onset of centrifugal instability, then the axial inhomogeneity of the azimuthal rotation causes the appearance of spiral Taylor-Gortler vortices [

As the steady motion is induced by the forced oscillations, the intensity of the motion will depend on the parameter

Dimensionless velocity of the azimuthal steady flow

This result is shown by the dotted line in Figure

As previously mentioned, the azimuthal steady streaming is generated in the viscous boundary layers at the rigid wall and near the free surface. It is important to determine the relative contribution of each generator to the velocity profile. Based on (

The velocity of the oscillatory flow can be found from the equation

The typical layer thickness

Using (^{2}. After substituting in the equation above, we obtain the formula

Formula (

Figure

Dimensionless velocity of the steady flow

This result gives the opportunity to determine the stability parameters of the centrifugal instability in the viscous boundary layer [

We have studied the dynamics of a low-viscosity fluid in a rapidly rotating horizontal cylinder. We report the first experimental results on the oscillatory flow of an annular fluid layer under gravity. In the rotating frame, the force of gravity induces the azimuthal fluid oscillations with frequency equal to the velocity of the cylinder rotation. If the free surface is unperturbed and gravity waves are excluded then the velocity amplitude is found to be proportional to the parameter

Therefore, in a uniformly rotating cylinder, liquid performs a combined motion of steady rotation and azimuthal oscillations which is qualitatively similar to the libration-driven flow. The obtained results can be useful to evaluate the stability parameters of the centrifugal instability in the Stokes boundary layer and the onset of ripples at the interface between fluid and granular medium in a rotating or librating cylinder.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work is supported by Grant 14-11-00476 of the Russian Scientific Foundation.