Flow patterns in a rotating horizontal cylinder partially filled with liquid

PHYSICAL REVIEW E 92, 013016 (2015)

Flow patterns in a rotating horizontal cylinder partially filled with liquid Victor Kozlov* and Denis Polezhaev† Laboratory of Vibrational Hydromechanics, Perm State Humanitarian Pedagogical University, Perm, Russia (Received 10 December 2014; revised manuscript received 15 May 2015; published 24 July 2015) 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. DOI: 10.1103/PhysRevE.92.013016

PACS number(s): 47.20.Qr, 47.32.Ef

I. INTRODUCTION

The annular liquid layer is an interesting hydrodynamic system that reveals free-surface instabilities and is an important object for studying planetary dynamics due to their ability to sustain inertial oscillations, which are induced by gravitational coupling between planets or between a planet and a satellite. While the horizontal cylinder is stationary, the liquid is at rest in a pool at the bottom of the cylinder. When the cylinder rotates with low to moderate angular velocities, its rising side drags out a thin film of liquid from the pool. This flow regime has been studied in detail, and an entire series of instabilities has been discovered, including hygrocysts, solitary waves, and shark-teeth and fishlike patterns (for a review, see, e.g., Seiden and Thomas [1]). Inside a rapidly rotating cylinder, the annular fluid layer develops and performs almost a solid-body rotation together with the cylinder. This flow regime of an annular low-viscosity liquid layer with the free surface inside a rotating cylinder is studied to a lesser extent. Phillips [2] demonstrated that the existence of gravity and the presence of a free surface together impose an uneven distribution of the liquid inside a rotating cylinder. Additionally, for certain combinations of rotation rate and liquid volume, the excitation of the surface waves with various axial and azimuthal wave numbers is possible. Gans [3] theoretically found that an asymmetric distribution of the fluid inside a rotating cylinder of finite length induces azimuthal steady streaming. Ivanova et al. [4] experimentally discovered the existence of azimuthal steady streaming and studied the influence of centrifugal waves on this flow. We extend the research of the annular low-viscosity liquid layer with a free surface and examine the stability of a twodimensional liquid flow inside a horizontal rapidly rotating cylinder. The liquid performs an almost solid-body rotation together with the cylinder. The slight deviation from the solidbody rotation is induced by the effect of gravity: The velocity of liquid on the rising side of the cylinder is less than the liquid velocity on the descending side. Thus, the annular liquid layer performs oscillations superimposed on steady rotation. The annular liquid layer motion is qualitatively similar to the liquid

* †

[email protected] [email protected]

1539-3755/2015/92(1)/013016(9)

flow in the gap between independently rotating and oscillating cylinders. Such a flow regime could be centrifugally unstable due to the onset of spatially periodic Taylor [5] or Gortler [6] vortices. Furthermore, rotating fluids support the existence of inertial waves, which are anisotropic and propagate because of the restoring nature of the Coriolis force [7]. The oscillatory motion in a horizontal rotating cylinder partially filled with low-viscosity liquid is theoretically studied in Ref. [2]. It is found that the amplitude of the oscillatory flow is proportional to the parameter  = g/ 2 a, which is the ratio of the gravitational force to the centrifugal force (g is the gravitational acceleration,  is the rotation rate, and a is the radius of the air column inside the cylinder). Ivanova et al. [4] studied experimentally the azimuthal steady streaming excited by liquid oscillations and concluded that the steady streaming velocity was proportional to  2 . Ivanova et al. [8] experimentally studied the resonant excitation of inertial azimuth waves and the corresponding mass-transport flows in a horizontal rotating cylinder under transverse vibrations. The amplitude of the fluid oscillations is determined by the vibrational parameter v = b2v / 2 R, where b and v are the amplitude and the frequency of vibrations and R is the radius of the cylinder. Variation of the vibrational parameters b and v allows for managing the amplitude of the fluid oscillations and controlling the phase velocity of the surface wave, which is determined by the dimensionless frequency n = v / . If n < 1, then the azimuthal velocity of the wave propagation is less than the rotation rate. Therefore, the azimuthal mass-transport velocity is less than the rotation rate; thus, it is called retrograde motion. If n > 1, then the fluid performs a prograde motion. If the frequency of vibrations is equal to the rotation rate, then the vibrations do not influence the fluid motion. Figure 1(a) demonstrates the dependence of the dimensionless velocity of mean azimuthal flow /  ( = l −  is the angular velocity of mean flow in the rotating frame and l is the velocity of a fluid in the laboratory frame) on the dimensionless frequency of vibrations n [9]. Note that the dependence of /  on n has critical points to the left and right of the position n = 1. These points correspond to the soft excitation of the surface waves, which intensify mass transport in an overcritical domain. These surface waves exist only in the limited range of the parameter

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©2015 American Physical Society

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PHYSICAL REVIEW E 92, 013016 (2015)

(a)

(b)

FIG. 1. (a) The dependence of relative velocity of steady streaming on the dimensionless frequency of vibration: fv = v /2π = 15 and 30 Hz, amplitude b = 0.35 mm, kinematic viscosity ν = 1 cSt, relative filling q = 0.23; (b) domains of the inertial waves existing on the plot of governing parameters n, v ; ν = 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].

n; the waves disappear at point b, and the steady fluid flow becomes a rigid-body one. However, the progressive wave is re-excited if we continue to point c in Fig. 1(a). The resonant excitation and the disappearance of surface waves do not coincide, i.e., hysteresis occurs. The domains of waves’ existence are demonstrated in Fig. 1(b) [9]. The dynamics of the annular liquid layer inside a rotating cylinder in the field of gravity is qualitatively similar to libration-driven liquid flows. A better knowledge of these flows is of great interest in astrophysics, where libration is driven by gravitational interactions and is used to investigate the interior structure of planets [10,11]. Theoretical and experimental studies documented various regimes of flow driven by longitudinal libration. For small libration amplitudes, the flow is dominated by inertial modes. The origin of inertial oscillations can differ: vibrating bodies [12], fluctuation of axis of rotation [13], librations [14], or oscillations of a container along the axis of rotation [15]. Libration could generate a steady axisymmetric flow in the liquid interior through the Ekman boundary layer [16]. Laboratory and numerical studies [17–19] corroborated the analytically predicted generation of a mainly retrograde stationary zonal flow in the bulk of a liquid. Noir et al. [19] and Rubio et al. [20] report that a magnitude of a mean retrograde zonal flow scales with the modulation amplitude squared. If the rotating flow is limited by boundaries, inertial waves can combine to form inertial modes whose structures depend on the geometry of the container. Once a mode is resonantly excited, its nonlinear self-interaction also generates the zonal flow [21]. The dynamics of the annular liquid layer is related to the Taylor-Couette problem. The theoretical analysis of this flow is simplified when the length-to-gap aspect ratio is infinite. The presence of end walls in a laboratory experiment makes the analysis of the Taylor-Couette flow more complicated.

For a cylinder at rest, the rotating lid forces the fluid to perform a rotating motion around the center axis. Due to centrifugal forces, the fluid close to the lid moves away from the center and then travels along the cylindrical wall and turns inwards near the fixed bottom. Close to the center axis, it rises vertically and returns to the rotating lid. Examining this flow, Sorensen et al. [22] revealed the existence of vortex breakdown states consisting of doublet, triplet, or quadruplet helical modes. The numerical simulations of dynamics of the swirling flow of Lopez [23] show that all of the states examined in the experiments of Sorensen et al. [22] are rotating waves. For independently rotating end walls, Heise et al. [24] performed experiments to study spiral flows in a counter-rotating Taylor-Couette flow. Depending on the Reynolds number, either pure spirals or a more complicated, mixed spiral flow pattern are revealed. For end walls at rest, the no-slip boundary condition generates radially inward Ekman boundary layer flow in the vicinity of each end wall, which induces a large-scale meridional circulation [25]. For strong corotation, Heise et al. [24] found a dominant axisymmetric flow state consisting of two large vortices. At lower outercylinder Reynolds numbers, this state undergoes a sequence of transitions to different axisymmetric multicell states. The paper is organized as follows. In Sec. II, the experimental setup and the protocol of the experiment are discussed. The onset of end wall vortices and the comparison of multicellular structure with theoretical data are presented in Sec. III. The evolution of small-scale axially periodic flows is discussed in Sec. IV. A summary of the main results and the conclusions appears in Sec. V. II. EXPERIMENTAL SETUP

The scheme of the experimental setup is shown in Fig. 2. Cavity one is made of a hollow Plexiglas tube of radius

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R = 2.5 cm and length L = 31.7 cm. The cylinder is supported by roller bearings and is mounted on the massive horizontal platform two. The stepper motor three is coupled to the cylinder and provides rotation in the range from 0 to 150 s−1 with an accuracy of 0.05%. The fluid is illuminated by stroboscopic light four, which has frequency equal to the cylinder rotation rate or fluid rotation rate. Images of the fluid are recorded by photo camera five, which is positioned at a right angle to the cylinder axis. The experiments are carried out with water-glycerol solutions (ν = 1–10 cSt) and kerosene (ν = 1.1 cSt). The fluid volume is characterized by relative filling q = V /V0 , where V is the fluid volume and V0 is the cylinder volume; q varies from 0.1 to 0.4. We use aluminum flake powder with a mean size of 50 μm for visualizing fluid flows. In the experiments, the total mass of the powder flakes is proportional to the liquid volume and does not exceed 0.1% of the liquid mass. In a rapidly rotating cylinder, the aluminum particles uniformly coat the inner surface of the cylinder, forming a layer a few microns thick. Each experiment follows a similar protocol. When the cylinder is partially filled with liquid and rotated sufficiently rapidly, the fluid performs an almost rigid-body rotation about a central air column. At the definite rotation rate , the mass-transport free surface velocity and spatial period of patterns are measured after several minutes have elapsed. The rotation rate is then decreased by ∼0.5 s−1 , and the procedure is repeated. According to the observations, the initially homogeneous layer of aluminum flakes becomes spatially periodic along the axis of rotation. The flow structure has two components: fine patterns in the center of the cylinder and large patterns near the end walls. In the end wall regions, the narrow rings of high concentrations of powder flakes are separated by wide domains of nearly zero concentration of powder. In the center of the cylinder, they are altered by fine patterns, which have a form of spatially periodic rings of a high concentration of flakes. The main scope of the paper is to study the physical mechanisms of the described phenomena.

rotation rate is followed by the generation of steady flows; the direction of the flows can be detected by the redistribution of the aluminum powder [Fig. 3(a)]. Near the end walls, we observe, stationary in the rotating frame and perpendicular to the axis of rotation, stripes of a high concentration of powder separated by domains of almost pure fluid. Initially, one or two periods are observed; the decrease of  can lead to the appearance of additional patterns. Under certain conditions, seven or eight stripes are found. Usually, the depth of penetration to the center of the cylinder does not exceed two calibers (cylinder diameters). Here, in the center of the cylinder, fine patterns dominate [Fig. 3(b)]. The experiments with low viscous fluid demonstrate that the spatial period λ of the large patterns near the end walls is independent of the rotation rate . At the same time, the increase of relative filling causes the monotonous growth of λ (Fig. 4). The specific data are obtained at relative filling q = 0.25 (diamonds in Fig. 4); the spatial period varies with  and rises at  < 100 s−1 . In a rapidly rotating cylinder, fluid performs forced oscillations under gravitational force. Its impact can be estimated 1.5 q = 0.12 q = 0.16 q = 0.20 q = 0.25 q = 0.30

λ (cm)

FIG. 2. Scheme of experimental setup.

FIG. 3. Visualization of flow patterns in the annular layer of liquid inside a rotating cylinder: R = 2.5 cm, L = 31.7 cm, ν = 1.0 cSt, q = 0.31,  = 81.6 (a) and 69.0 s−1 (b); light areas represent domains of high concentrations of aluminum powder. Distance between vertical markers equals to 25 cm.

1

0.5 50

III. PATTERN FORMATION NEAR END WALLS

In a rapidly rotating horizontal cylinder, fluid forms an annular layer and performs an almost rigid-body rotation, while heavy particles of aluminum powder are distributed quite evenly over the cylindrical wall. The decrease of the

100

Ω (s-1)

150

FIG. 4. Dependence of spatial period λ of flow patterns on rotation rate: ν = 1.1 cSt (kerosene), q = 0.12, 0.16, 0.20, 0.25, and 0.30.

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sin θ = osc /2,

h

λ

R

with the use of the dimensionless parameter  = g/ 2 a, where g is the gravitational acceleration; a is the radius of the free surface of the fluid. The free surface of the annular fluid layer is stationary in the laboratory frame and has a form of a cylinder whose axis is below the axis of the rotating cylinder. The amplitude of displacement is proportional to . In the rotating frame, a fluid distribution is considered as a two-dimensional surface wave propagating in the direction opposite to the cylinder’s rotation. The surface wave induces azimuthal retrograde steady streaming, which is axially inhomogeneous in the vicinity of the end walls. Due to differential rotation of the liquid, a meridional circulation arises near the end wall: The fluid close to the end wall moves to the center and then travels along the axis of rotation, turning outwards in the center of the cylinder, where meridional flows from the opposite end walls collide. Then the fluid moves along the cylindrical wall to the end wall. There have been a number of studies on the transition from laminar meridional circulation to turbulent flows within an increasing Reynolds number, e.g., Refs. [22,23]. The experimental results of Sorensen et al. [22] revealed the existence of a vortex breakdown consisting of helical vortices in the limit Re > 1000. In the present study, the azimuthal steady streaming is slow ( is usually less than 1 rad/s) and Re ≡ R 2 /ν is of the order of 102 . Thus, the end wall flow patterns cannot be a consequence of vortex breakdown and therefore we observe a novel type of liquid flow instability. The fluid particles in an azimuth inertial wave perform orbital motion in the plane perpendicular to the axis of rotation. This motion is very special in a rotating boundary layer (Ekman layer) near the end wall of the cylinder. As fluid particles perform orbital motion in the plane that is parallel to the end wall, then for one half period, the azimuthal velocity is less than the rotation rate , while for another one, the velocity is larger than . For one half period, the Coriolis force pushes fluid particles outwards along the end wall (centrifugal motion) and, in the corner, the radial motion becomes the axial one. In the next half period, the Coriolis force changes its direction and the fluid motion becomes opposite. Finally, in the corner between the cylindrical wall and the end wall, the generator of fluid oscillations acts. In the present study, inertial oscillations generate internal waves that propagate through the liquid in the form of a cone, making an angle θ with the rotation axis; the dispersion relation is as follows (e.g., Greenspan [7]):

FIG. 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; λ 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.

As the fluid performs forced oscillations under gravity, its radian frequency equals to the angular velocity of rotation, namely, osc = . Thus, Eq. (3) takes the form √ λ/ h = 2 3. (4) The spatial period λ/ h calculated by the formula (4) demonstrates good agreement with the experimental data (Fig. 6). Thus, the areas of inertial wave reflection coincide with the rings of high concentrations of aluminum flakes. The onset of ring patterns is caused by the axial inhomogeneity of azimuthal steady streaming near the rigid boundary. In the experimental study of the wave attractor, Maas [27] found that the incident wave induces a mean azimuthal flow at the point of reflection. The azimuthal component of the fluid motion is affected by the Coriolis force, and the radial liquid flow is excited. We expect that the combined effect of the axial inhomogeneity of the azimuthal steady streaming and the Coriolis force produces the axially periodic vortical flow. This flow transports heavy particles along the axis of rotation to the reflection points at the rigid boundary. This is also confirmed by the absence of ring patterns in the central part of the cylinder (Fig. 3): The energy of the inertial wave dissipates after numerous reflections from the free and

(1)

where osc is the radian frequency of fluid oscillations. In a low-viscous fluid, the angle of incidence equals to the angle of reflection if the interface is either parallel or perpendicular to the rotation axis [26]. Thus, the spatial period of the patterns near the end walls can be calculated using the formula λ = 2h/ tan θ,

(2)

where h is the thickness of the annular layer (Fig. 5). Combining Eqs. (1) and (2) leads to λ/ h = 2[(2/ osc )2 − 1]1/2 .

(3)

FIG. 6. Dependence of dimensionless spatial period of ring patterns on relative filling q. Solid line represents data obtained from (4).

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rigid boundaries and becomes insufficient to produce Ekman flows away from the source of the oscillations. Let us discuss possible reasons for the spatial period increase at relative filling q = 0.25. Experimental data obtained at q = 0.25 and  > 100 s−1 are in good agreement with the other results, but the wavelength increases as  decreases at lower rotation rates. According to the observations, the three-dimensional surface waves are excited at  < 100 s−1 (diamonds in Fig. 4). The theoretical [2] and experimental [4] studies of surface waves in an annular fluid layer confirm that, standing along the axis of rotation, spiral waves can appear in the studied range of relative filling. The fluid performs three-dimensional oscillations in a standing wave, and the most intensive oscillations are found at q ≈ 0.25. The excitation of a standing inertial wave is followed by the generation of axially nonuniform azimuthal steady streaming. The inhomogeneity of the azimuthal velocity results in the onset of Ekman flows and the further redistribution of heavy particles near the cylindrical wall. Thus, the spatial period of the ring patterns is influenced by the standing wave.

FIG. 8. Dependence of wave number k ≡ 2π h/λ on relative filling q: kinematic viscosity ν = 3.6 or 1.0 cSt.

IV. FINE PATTERN FORMATION

In the process of propagation through the fluid, the inertial oscillations diminish and the ringlike patterns become invisible away from the end walls. Here, in the center of the cylinder, fine patterns dominate. According to the observations, fine patterns have a form of a series of bands of high concentrations of aluminum flakes in the plane perpendicular to the axis of rotation (Fig. 7). The thickness of a band is approximately 1 mm; the spatial period is of the same order of magnitude. We report that the wave number k of the mentioned flow depends both on the thickness of the annular layer and fluid viscosity (Fig. 8). As the wave number varies in the range from 5 to 20 in the experiments with water, the size of pairs of vortices λ is much less than the thickness h (squares in Fig. 8). This, together with the fact that the spatial period λ depends on the fluid viscosity, makes it possible to propose that we indicate the centrifugal instability of the Stokes boundary layer at the cylindrical wall. Because the frequency of the

FIG. 7. Photograph of fine pattern formation in the central part of the cylinder; relative filling q = 0.23, kinematic viscosity ν = 2.3 cSt, angular velocity  = 66 s−1 ; distance between white vertical markers equals 7.5 cm.

forced fluid oscillations is osc = , it produces a viscous Stokes boundary layer of thickness δ = (2ν/ osc )1/2 . Figure 9 demonstrates the results of measurements of the wave number kδ ≡ 2π δ/λ for a wide range of dimensionless velocity of rotation ω = h2 /ν. The data obtained from the experiments with fluids of various viscosities are in good agreement. In the limit of the high dimensionless velocity ω > 500, i.e., thin boundary layers, the dimensionless wave number kδ is independent of ω and equals 2/3. This proves that the onset of spatially periodic flow is caused by the instability of the viscous boundary layer and, in the limit of the high dimensionless velocity, the spatial period λ depends only on the thickness δ. The experimentally observed wave

FIG. 9. Dependence of dimensionless wave number kδ , based on the thickness of the Stokes boundary layer, on dimensionless velocity ω: kinematic viscosity ν = 3.6 and 1.0 cSt.

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kδ

2

1

vibrations forced oscillations under gravity

0.5 1

2

Ω/Ωosc

5

10

FIG. 10. Dependence of dimensionless wave number kδ on parameter / osc (log-log scale): Triangles correspond to data adapted from Kozlov and Polezhaev [30], circle corresponds to kδ = 0.67, obtained in the limit ω > 500 (Fig. 9). The solid line corresponds to kδ ∼ (/ osc )1/2 .

FIG. 11. Dimensionless wave number kr ≡ 2π δr /λ versus dimensionless velocity of rotation ω: Gravitational data (circles) correspond to those in Fig. 9 in the limit ω > 500; vibrational data (triangles) correspond to that in Fig. 10.

equation (5) as kr ≡ 2π δr /λ =

number qualitatively agrees with the theoretically obtained value kδ ≈ 0.5 for the Taylor-Gortler vortex flow between the oscillating inner cylinder and fixed outer cylinder [28] and the experimentally and theoretically obtained value kδ ≈ 0.85 for the pulsed flow in Taylor-Couette geometry [29] in the limit of the thin Stokes boundary layer. The flows similar to the observed Taylor-Gortler vortexlike flows were revealed in the experiments with the annular liquid layer inside a horizontal oscillating cylinder. Kozlov and Polezhaev [30] stated that, in the limit of the thin Stokes boundary layer (δ  h), the experimental data on the spatial period of vortices are in good agreement (Fig. 10). Referring to Fig. 9, in the limit of the thin boundary layer, we obtain the singular value of the wave number kδ = 2/3 for vortices in the gravitational flow. Figure 10 demonstrates that the vibrational (triangles) and gravitational (circle) data are in good agreement and obey the empirical law kδ = (2/3)(/ osc )1/2 .

(5)

It follows from Eq. (5) that the onset of fine patterns is determined by the instability of the oscillatory boundary layer near the cylindrical wall. It is noteworthy that the parameter /osc represents the dimensionless velocity of rotation ωr = δ 2 /ν within a constant factor. According to formula (5), the increase of the dimensionless velocity of rotation is followed by a decrease of the spatial period of observed fine patterns. This phenomenon is an effect of the enhanced action of the Coriolis force. A similar effect is well known in the problem of the onset of forced thermal convection [31,32] and thermal vibrational convection [33,34] in rotating cells. We also introduce the notation δr = (ν/)1/2 , which is the thickness of the Ekman layer. Taking into account this definition and the link between osc and δ, we may rewrite

√

2/3.

(6)

From (6), it follows that the size of the vortices is determined by the thickness of the Ekman boundary layer and is constant in the limit of the thin boundary layer. The experimentally observed value kr is in qualitative agreement with the data obtained for the Taylor-Gortler vortices in librating shells. Calkins et al. [17] detected the onset of centrifugal instability in the equatorial region of a librating sphere during the retrograde phase of libration as long as the core liquid rotates faster than the peripheral region. Calkins et al. [17] showed that the wave number kr ≈ 2 in a wide range of Ekman numbers E = 10−6 –10−4 . Noir et al. [19,35] studied the Taylor-Gortler flow in a cylindrical geometry and the wave number kr was shown to be equal to 0.3 at E = 5 × 10−5 and 10−4 . These data are in qualitative agreement with our results (Fig. 11). The quantitative difference between wave numbers can be explained by the physical nonequivalence of the problems: In a librating shell, the boundary layer develops near the librating wall; in the discussed problem, the liquid itself oscillates near the rotating wall. By analogy with the onset of centrifugal instability in the librating cells, it is possible to distinguish the two phases of Taylor-Gortler vortices’ growth under gravity. First, the fluid near the wall is “spun-up” to faster rates during one half of the period of fluid oscillations, and a portion of fluid displaced from the boundary layer to the interior experiences a greater centrifugal force than the surrounding fluid. This flow configuration is stable. Then fluid near the wall is “spun down” to slower rates during the rest phase of the period, and a portion of fluid displaced from the boundary layer to the interior experiences a weaker centrifugal force than the surrounding fluid. This flow configuration is unstable, and further inward displacement of the fluid follows. Here we apply the idea of the stability analysis of oscillatory fluid motion provided in the

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layer. We introduce Reynolds numbers Reout = vout δ/ν and Rein = vin δ/ν, which distinguish azimuthal fluid motion near the outer and inner boundaries of the viscous layer. The no-slip boundary condition is imposed at the outer boundary, i.e., the fluid velocity vout = R. In the interior, the fluid performs tangential oscillations with an amplitude b in the rotating frame; therefore, the velocity vin = R + bosc cos(osc t). Thus, the Reynolds number Rein varies over the period of rotation. If the velocity uosc = bosc is large enough and the threshold for the onset of centrifugal instability is obtained, then the Gortler vortices occur. The experiments of Andereck et al. [43] demonstrated that, at the threshold of the Taylor-Gortler vortices’ growth, the ratio Rein /Reout remains constant in the limit Reout  1 and depends on the gap width. Hence, in the limit of the thin boundary layer, we get Rein /Reout = const. Otherwise, Rein /Reout = vin /vout = 1 + bosc / R = α,

(7)

where parameter α is constant. We can rewrite Eq. (7) in the form bosc = (α − 1)R. Hence, if we multiply both sides of this equation by R/ν, we obtain Reosc ≡ uosc R/ν = (α − 1)E −1 ,

(8)

where E = ν/ R is the Ekman number. Thus, the quasistatic assumption demonstrates that a thin boundary layer is centrifugally unstable at Reosc ∼ E −1 . It is worth mentioning that the quasisteady assumption should be valid for both forced and wavy disturbances of fluid motion. Then we can compare our results with the experimental findings of Kozlov and Polezhaev [30]. The individual particles of a perfect liquid in a progressive wave do not exactly describe closed paths; in addition to their orbital motion, they also possess a mean flow in the direction of wave propagation. In accordance with [44], the mass-transport velocity uS = 5u2osc /4c, where c is the velocity of propagation of the surface wave. This formula 2

12000 R, cm 2.5 3.1 4.5 2.5 3.1 2.5 3.1

Reosc

quasistatic assumption, i.e., the basic flow is assumed to vary slowly compared with the growth of a disturbance, which can be treated as a steady basic state using an instantaneous frozen profile [36]. Then centrifugal instability rises if the frozen profile in any phase of the oscillations becomes unstable. According the Rayleigh’s inviscid stability criterion, the centrifugal instability of the azimuthal oscillations develops only if the instantaneous fluid velocity is larger than the velocity of the cylinder rotation. For the case of forced gravitational oscillations, the instability develops near the descending wall of the cylinder. During the unstable phase of the cycle, the excited vortices have time to transport aluminum flakes along the cylindrical wall. In the remainder of the cycle, the vortices vanish but the visualizing particles remain at rest. During a period of rotation, the longitudinal vortices rise (and vanish) at each point of the cylinder’s surface. In consequence of repetitive action of vortical flow, the continuous bands of aluminum flakes appear (Fig. 3). The indicated effect of nonpermanent vortices is not unique. Donnelly was the first who observed Taylor-Gortler vortices which appear and disappear during a cycle in the modulated Taylor-Couette flow and called them “transient vortices” [37]. For small values of the dimensionless frequency of modulation γ ≡ d/δ (d is the gap width and δ is the thickness of the Stokes layer) the vortices appear only in a small part of the cycle and at higher values for γ , and they are persistent during almost all of the cycle [29]. Thompson [38] found that at small values of γ the modulated flow becomes unstable as soon as the maximum angular velocity exceeds the critical angular velocity for the corresponding steady flow. At higher frequencies instability occurs when the mean angular velocity exceeds the critical value for the steady flow. This result agrees well with the linear stability analysis of Carmi and Tustaniwskyj [28]. A similar effect is found for the stability problem of the Stokes layer in a stationary planar channel. It is revealed that the instantaneous velocity profile in the flat Stokes boundary layer can be massively unstable [39,40] at Reynolds numbers at which there are no unstable Floquet modes [41]. In that case the intensive vortices rise in a certain phase of the cycle and vanish in the remainder of the cycle. According to the observations, the weak perturbations introduced by small particles or obstacles in the oscillatory fluid flow lead to the explosive growth of the transversal vortices in the unstable phase of the cycle. For the higher level of perturbation, the critical Reynolds number decreases [40] and tends to the value obtained from a quasisteady approach [36]. The threshold of the appearance of vortices drops significantly in comparison with the case in the absence of disturbing factors [42]. It is noteworthy that the wave number ks (based on the thickness of the Stokes layer) of the transversal vortices in a planar channel and the wave number kr of Gortler vortices in a rotating cylinder have roughly the same value: about 0.5. The above-mentioned similarities between various types of instability give us reason to think about the quasistationary origin of the centrifugal instability of the Stokes layer near the cylindrical wall. By analogy with fluid motion in the gap between two rotating cylinders, we estimate the critical Reynolds number for the onset of centrifugal instability of the Stokes boundary

ΔΩ + vibration + vibration + vibration - vibration - vibration - gravity - gravity

6000

0 0

4x10-5

E

8x10-5

FIG. 12. Dependence of Reynolds number on Ekman number at the threshold of the Gortler vortices’ appearance. Positive and negative values of  correspond to prograde and retrograde azimuthal flow, respectively.

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steady streaming proves that steady streaming plays no role in the onset of centrifugal instability. One can find that some experimental data obtained in the gravitational problem deviate from the general law (semifilled triangles in Fig. 12). This might be explained by the dependence of the proportionality factor (α − 1) between Reosc and E on the Reynolds number Reout (Fig. 13). The parameter α is constant as long as Reout > 500. Finally, the appearance of fine patterns under gravity or vibration is driven by centrifugal instability in the Stokes boundary layer. Consequently, in a certain phase of the cycle, the Gortler vortices rise and transport aluminum flakes along the cylindrical wall of the rotating cavity.

Rein /Reout

1.16 R, cm 2.5 3.1 4.5 2.5 3.1 2.5 3.1

ΔΩ + vibration + vibration + vibration - vibration - vibration - gravity - gravity

1.08

V. CONCLUSION 1 0

800

1600

Reout FIG. 13. Dependence of ratio Rein /Reout on Reynolds number. Symbols correspond to those in Fig. 12.

enables the calculation of the oscillation velocity with the use of the experimentally measured mass-transport velocity of the liquid. This was already mentioned in the Introduction as the forced oscillations under gravity that are considered as a two-dimensional surface wave propagating at azimuthal velocity  in the direction opposite to the cylinder rotation. The experimental data agree to each other and confirm the predicted relation Reosc ∼ E −1 (Fig. 12). Empty symbols in Fig. 12 represent the results obtained in the experiments when the dimensionless frequency of vibration n > 1 and the exciting inertial waves generate prograde steady streaming, i.e.,  > 0; filled and semifilled symbols correspond to the experimental data obtained when the inertial waves are excited under vibration in the limit n < 1 or gravity and they generate retrograde steady streaming ( < 0). Agreement between data obtained in the experiments with prograde and retrograde

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In this paper, we discuss two novel types of flow patterns in the annular liquid layer undergoing forced oscillations under gravity inside a horizontal rotating cylinder. We outline that the appearance of flow patterns in the form of largescale axisymmetric flows in the vicinity of the end walls is determined by the onset of inertial waves. The spatial period of the end wall vortices depends on the thickness of the annular liquid layer and agrees well with the distance between successive points of reflections of the inertial wave from the solid surface. Additionally, fine-scale flow patterns are found in the central part of the cylinder. We propose that the onset of this cellular flow is driven by centrifugal instability in the Stokes boundary layer near the cylindrical wall. The spatial size of the Gortler vortices is determined by the thickness of the Ekman boundary layer, which is generated on the cylindrical wall. In further support of viscous boundary layer instability, we find that in the quasistatic assumption, the critical Reynolds number is inversely proportional to the Ekman number. ACKNOWLEDGMENT

The work is supported by Grant No. 14-11-00476 of the Russian Science Foundation.

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