a light cylinder under horizontal vibration in a cavity filled with a fluid

ISSN 0015-4628, Fluid Dynamics, 2010, Vol. 45, No. 6, pp. 889–897. © Pleiades Publishing, Ltd., 2010.Original Russian Text © A. A. Ivanova, V. G. Kozlov, V. D. Shchipitsyn, 2010, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkostii Gaza, 2010, Vol. 45, No. 6, pp. 63–73.

A Light Cylinder under Horizontal Vibrationin a Cavity Filled with a Fluid

A. A. Ivanova, V. G. Kozlov, and V. D. Shchipitsyne-mail: [email protected]

Received March 9, 2010

Abstract—The behavior of a light cylindrical body of circular cross-section under horizontal vibrationin a rectangular cavity filled with a fluid is experimentally investigated. At critical vibration intensity thebody is repelled from the upper side of the cavity and takes up a stable suspended position, in which thegravity field is balanced by the vibrational repulsive force, executing longitudinal oscillations. As thevibrations are intensified, the gap between the cylinder and the wall widens. A new form of instability,namely, the excitation of the tangential motion of the body along the vibration axis, is found to existon the supercritical range. The cylinder is at a finite distance from the upper side of the cavity and thetangential motion is due to the loss of symmetry of the oscillating motion. The transition of the cylinderto the suspended state and its return to the wall, as well as the excitation of the average longitudinalmotion and its cessation, occur thresholdwise and have a hysteresis. The body dynamics are studied asa function of the dimensionless vibration frequency.

DOI: 10.1134/S0015462810060062

Keywords: light cylinder, vibrations, hydrodynamic interaction.

A solid executing high-frequency oscillations in a fluid is subjected to an average lift. This force arisesdue to the inhomogeneity of the oscillating velocity field around the body, that is, as a result of the in-homogeneity of the average pressure distribution over the body surface. In the case of a symmetric body(spherical or cylindrical) the violation of the symmetry of the velocity field and, as a consequence, the liftcan be generated when the body oscillates near the boundary of a cavity or at a certain distance from anotherbody. This average lift, known as the Bjorkness force, was already described by Lamb [1] and studied bymany authors in the inviscid approximation [2, 3]. This force becomes appreciable only at a considerableviolation of the velocity field symmetry, that is, when the body oscillates in a fluid at a small distance fromanother body. Usually, this distance is smaller than the body size. At the same time, both the body size andthe distance between the bodies (or between the body and a wall) must be much greater than the Stokesianboundary layer thickness, since the Bjorkness force is inviscid in nature.

The nature of the lift changes, if, independent of the body, the fluid itself executes high-frequency shearoscillations synchronized with the body oscillations, as in the case of combined translational and rotationaloscillations of a cavity filled with a fluid and having a phase inclusion, for example, a solid [4, 5]. In this case,the lift, vibrational in nature, manifests itself throughout the entire volume of the fluid and is considerablygreater in magnitude, namely, it turns out to be comparable with the gravity and provides quasi-stationarysuspension and displacement of both light and heavy bodies under the conditions of terrestrial gravity.

In theoretically considering the above-mentioned problems the fluid and body oscillations are assumedto be high-frequency, the Stokesian boundary layers to be negligibly thin, and the fluid to be inviscid.

When a body oscillates in a fluid at a distance of viscous interaction from a rigid boundary (at a distancecomparable with the Stokesian layer thickness), the nature of the average interaction qualitatively changes.In the experiments with a spherical body in a vibrating cavity filled with a fluid vibrational repulsion of thebody from the wall was found to exist [6, 7]. It should be noted that the vibrational lift, which manifests

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Fig. 1. Formulation of the problem; d is the cylinder diameter and h is the gap between the cylinder and the upper side ofthe cavity exposed to vibrational action (the direction of vibrations is indicated by the arrows).

itself at the viscous interaction distance, is directed oppositely to the Bjorkness force, that is, the vibrationalattraction of a body to a wall.

In this study, the behavior of a light cylinder in a rectangular cavity filled with a viscous fluid andexecuting horizontal translational vibrations is investigated. The oscillating and average motions of thebody relative to the cavity are studied using high-speed videofilming.

1. FORMULATION OF THE PROBLEM

A rectangular cuvette with a body within it was filled with a fluid and fastened on a table of a mechanicalvibrator. The table executed periodic reciprocating motion in the horizontal plane in accordance with theharmonic X = bcos Ωt law with an amplitude b = 1 to 50 mm at a linear frequency f ≡ Ω/(2π) = 0 to25 Hz. In the experiments the horizontal position of the cavity and the vibration axis was maintained withan accuracy of 0.01 rad.

The plexiglas cuvette was made in the form of a parallelepiped measuring 120× 55× 95 mm3 (Fig. 1).The wide side was horizontally arranged, while the oscillations are executed along the long axis.

A light 80 mm-long cylinder, d in diameter (d = 6.1, 12.3, and 20.0 mm) was made of an ebonitetube whose ends were sealed by a thin dacron film. The mean density of all the bodies was the same,ρs = 0.66 g/cm3. On the cylinder ends there were light-reflecting markers which made it possible to monitorthe rotational motion of the body.

Water-glycerin mixtures were used as working fluids. By means of varying the glycerin concentrationthe kinematic viscosity of the fluid could be varied on the range ν = 0.1–7.8 St (the fluid density varied onthe range ρL = 1–1.26 g/cm3 and the relative body density on the range ρ ≡ ρS/ρL = 0.66–0.52).

The fluid viscosity was measured by a capillary viscosimeter with a relative error not greater than 0.01,the density by an areometer at an accuracy of 0.001 g/cm3, the vibration frequency by a digital tachometerat 0.1 Hz, and the vibration amplitude by an optical cathetometer at 0.1 mm.

In the experiments with a body of a given diameter the kinematic viscosity of the fluid and the vibrationfrequency and amplitude were varied. For given values of ν and b the frequency was smoothly increasedor decreased. The visual observations were performed in usual and stroboscopic illumination and the photorecording in electronic flash light.

The motion of the body was studied using a high-speed high-resolution Basler A402k videocameraconnected directly to a computer. The recording speed was 380 frames per second at the resolution of800× 300 points per a frame. The camera, fixed in the laboratory reference frame, recorded the positionsof the cuvette and the body viewed from the endface of the latter. A frame-by-frame analysis of the videorecording performed using applied computer programs made it possible to determine the time dependenceof the body position in the cavity-fitted coordinate system (the longitudinal x coordinate along the vibrationaxis, the gap width h, and the angle of body rotation ϕ), simultaneously with the position of the cavity itselfin the laboratory system. The accuracy of measuring the spatial coordinate was 0.04 mm.

FLUID DYNAMICS Vol. 45 No. 6 2010

A LIGHT CYLINDER UNDER HORIZONTAL VIBRATION 891

Fig. 2. Dependence of the distance h between the cylinder and the wall on the vibration frequency; d = 20.0 mm;ν = 1.02 St and b = 44.8 mm (points 1), 2.57 and 28.3 (2), and 5.81 and 48.8 (3).

2. EXPERIMENTAL RESULTS

In the absence of vibrations the light cylinder is in the upper part of the cavity, where it is held up againstthe ceiling. Under vibrations the body executes oscillations relative to the cavity and, as a result, is orientedperpendicular to the vibration axis. This is the known orientation effect due to the average moment of theforces acting on oscillating asymmetric bodies. In particular, it was observable in the experiments with acylindrical body in a cavity executing non-translational vibrations [4, 5].

With increase in the vibration frequency (at b = const) the body is repelled from the ceiling thresholdwiseand executes oscillations at a certain distance h (Fig. 1). In the process of oscillations, the gap width hremains almost invariant. As shown below, h increases slightly only at extreme points, where the velocityof the body motion relative to the fluid is near zero. With further increase in the frequency the gap widensup to a certain limiting value. With decrease in the frequency the gap smoothly narrows. The value of hwas determined from the photographs of the body at the extreme points of the cavity displacement, whenthe cavity velocity was zero. The photos were made both in the rightmost and in the leftmost positions.

With increase in the vibration frequency, the body repelled from the ceiling always moves away jump-wise at a finite distance h (in Fig. 2 it is points a that correspond to an increase in the frequency, whiletransitions are shown as broken lines and upward arrows). After the transition to the suspended state thebody executes symmetric oscillations, the values of h at the extreme positions being the same. With de-crease in the frequency (points b) the gap monotonically decreases and then the body returns jumpwise toits original position (downward-directed broken lines). There is hysteresis in the transitions.

In highly viscous fluids the hysteresis in the thresholdwise transitions can amount to considerable val-ues (points 3). With decrease in viscosity the hysteresis reduces, while in low-viscous fluids and at smallvibration amplitudes (at high frequencies) it is not observable at all.

With increase in the vibration amplitude (at given body dimensions and fixed fluid viscosity) the criticalfrequency decreases, together with the limiting value of the gap h.

In the (b, f ) plane the threshold curves of transitions (cylinder detachment from the cavity wall and itsreturn) are in qualitative agreement with the curve obtained for the spherical body in [7]; because of this,here they are not presented in detail.


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At fairly high vibration intensities, on the supercritical range, an average motion of the body along thevibration axis, that is a new threshold phenomenon, occurs. The cylinder is displaced from one of the endsof the cavity toward the other at a constant velocity; near the ends the direction of its motion is reversed.Thus, the cylinder executes a periodic motion along the cavity with a period equal to several tens of seconds.

The excitation of the average motion is due to the change in the nature of the high-frequency bodyoscillations: the oscillating motion trajectory becomes asymmetric and inclined to the vibration axis (theinclination can be both toward the right and to the left). In the course of a single oscillation at the points of thegreatest displacement the cylinder is at different distances h from the wall. In Fig. 2 the branches of the curve(points 1) show the distance between the body and the upper side of the cavity in the extreme body positions.Under the action of vibrations the body executes oscillatory motion simultaneously along the vibration axisand in the perpendicular direction. The branching point coincides with the critical frequency of the excitationof the tangential motion of the body. With decrease in f the cylinder motion stops thresholdwise, togetherwith the restoration of the body oscillation symmetry.

As shown in [6], the vibrational dynamics of a body in the gravity field are determined by the vibrationalparameter W = (bΩ)2/(gd) and the dimensionless vibration frequency ω = Ωd2/ν . The latter characterizesthe ratio of the body diameter d to the Stokesian viscous boundary layer thickness δ =

√2ν/Ω.

The boundaries of the detachment (transition to the suspended state) and of the return to the original stateobtained for bodies of different diameters in fluids of different viscosities (ν = 0.4 to 5.8 St) are in goodagreement with each other in the (ω , W ) plane (Fig. 3). The boundary of the excitation of the longitudinalmotion of the body is above the boundary of body suspension. Points 1 to 3 correspond to different bodydiameters, while points a indicate the cylinder detachment from the ceiling (curve I) and the beginning ofthe motion (II) with increase in the vibration intensity and points b indicate the cessation of the motion (III)and the body return to the original position (IV) with decrease in W .

On the range of high dimensionless frequencies, ω > 300, hysteresis is absent from transitions, whilethe transition boundaries coincide by pairs and are shown by dark points. Clearly, for ω > 300 the thresholdvalues of W depend on the frequency only slightly.

The width of the gap between the oscillating body and the cavity side characterizes the radius of the re-pulsion force action. This force is determined by the viscous interaction between the body and the boundaryand manifests itself on the entire dimensionless frequency range at distances comparable with the Stokesianlayer thickness. In Fig. 4 the dependence of the distance h measured in the units of the boundary layerthickness on the dimensionless parameter W is presented (curve 1). In this case, points a almost coincidewith points b and, for this reason, are omitted. The cylinder detachment and its return to the original positionare associated with the critical parameters W = 13.9 and ω = 351. As in Fig. 2, the branches of curve 1indicate the limits of the gap width variation during a single period.

The velocity V of the average translational motion of the cylinder measured in the bΩ units (curve 2 inFig. 4) increases with the vibrational parameter W , reaches a maximum, and starts to decrease (points acorrespond to an increase in W and b to a decrease). The critical values of the parameters are as follows:W = 20.7 and ω = 425 correspond to the beginning of the motion and W = 19.5 and ω = 413 to its cessation.Clearly, the threshold of the excitation of the longitudinal body motion coincides with that of the loss ofsymmetry of its oscillations (coincides with the branching point on curve 1).

3. MOTION OF THE BODY UNDER VIBRATIONS

The interaction between the body and the upper side of the cavity and its motion relative to the cuvettewas studied by means of high-speed videofilming at a given vibration amplitude b and different values ofthe frequency f .

The experiments were performed with a body of diameter d = 12.3 mm in a fluid with viscosity ν =0.65 St and density ρL = 1.22 g/cm3. The horizontal coordinate of the cuvette X is measured in the lab-oratory reference frame and the horizontal coordinate of the body center x in the cavity-fitted coordinatesystem. In the case in which the body is held against the ceiling (Fig. 5a and 5b, W = 6.8 and ω = 80) its



Fig. 3. Boundaries of body transitions: detachment and beginning of the motion (curves I and II, respectively; points a) andcessation of the motion and body return to the original position (curves III and IV, points b); d = 6.1, 12.3, and 20.0 mm(points 1 to 3).

horizontal oscillations relative to the cavity (b) occur at a comparatively small amplitude and a slight phaseshift relative to the cuvette oscillations (a).

On the supercritical range (Fig. 5c to 5e, W = 21.5 and ω = 136) the harmonic oscillations X of thecuvette (c) lead to in-phase oscillations x of the cylinder (d) at a certain distance from the ceiling. In thecourse of the horizontal displacement of the body its distance from the ceiling remains almost constant (e),increasing only slightly at maximum displacement points. The mean translational velocity of the body isabsent. Together with the translational oscillations, the cylinder executes slight angular oscillations, whichare not presented in the figure. In individual experiments, a comparatively weak average rotation of thecylinder can be observable; however, it does not lead to an average motion.

At high supercriticities the symmetry of cylinder oscillations is violated, thus generating an averagemotion. We will consider the case in which the body moves from left to right. In Fig. 6 we have plotted thetime dependence of the horizontal coordinate of the cavity X (a), the body coordinate x (b) and its oscillatingcomponent ξ (c), and the gap width h (d) for d = 12.3 mm, ν = 0.65 St, W = 34.3, and ω = 167. In thecavity-fitted reference frame the body coordinate varies with time in accordance with the x = Vt + ξ +const law, where V is the average motion velocity and ξ is the oscillating component (the value of ξ averageover a period is zero). In the case under consideration V = 3.8 m/s. As distinct from the case presentedin Fig. 5, the distance between the cylinder and the upper side of the cavity considerably varies during theentire period. When the cylinder is in the leftmost position, the gap between it and the cavity is wider thanwhen it is in the rightmost position. The trajectory of the oscillatory body motion relative to the cavity isshaped like a loop: in the course of oscillations the cylinder moves from left to right farther from the wallthan when it moves from right to left. The phase shift between the horizontal oscillations of the body andthe cuvette oscillations is almost absent.

As to the cylinder rotation about its own axis, it should be noted that, along with the oscillating compo-nent, there is an average angular velocity. In its motion from left to right, the cylinder rotates counterclock-wise, as though it rolls along the upper side. In the case considered the average angular rotation velocity Ψis greater than the velocity of the cylinder rolling over the surface at the linear velocity V : Ψ(d/2)/V = 2.3.When the body moves in the opposite direction, that is, from right to left, the trajectory is mirror-symmetric.The direction of the average rotation also reverses: now the body rotates clockwise.


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Fig. 4. Dependence of the dimensional gap width h/δ (curve 1) and the mean velocity of the body motion (2) on thevibration parameter W ; d = 20.0 mm, ν = 0.64 St, and b = 29.2 mm.

4. DISCUSSION OF THE RESULTS

The vibrational dynamics of a body are determined by the nature of the interaction between the body andthe fluid and the cavity boundaries. At a distance considerably greater than the Stokesian boundary layerthickness the body oscillating in the fluid is attracted to the wall [6]. In the case under consideration thebody (cylinder) is repelled from the wall. This is due to the viscous interaction and manifests itself only atdistances comparable with the Stokesian layer thickness δ =

√2ν/Ω. This allows us to introduce different

measurement units, namely, δ for the distance between the body and the cuvette wall and the vibrationamplitude b for the body displacement along the vibration axis.

The trajectories of the oscillatory motion of the upper point of the body in different regimes are plotted inFig. 7 basing on the results presented in Figs. 5 and 6. The arrows indicate the direction of the motion alonga trajectories. At small vibration intensities the body executes oscillations at a comparatively low amplitudecontinuously touching the upper side of the cavity (curve 1). After the cylinder has detached from the wall,the amplitude of its oscillations increases (curve 2). The trajectory of the motion is near-symmetric, whilethe tangential motion is absent (the variables x and ξ coincide). In the process of oscillations the distancefrom the body to the wall remains almost constant, h/δ ≈ 0.6, increasing only in the extreme positions upto h/δ ≈ 0.7.

In the case, in which the body executes an average motion, the dimensionless distance between the bodyand the upper side of the cavity varies in the course of oscillations (curve 3). In the leftmost position of thecuvette the cylinder is at a greater distance from the upper side (h/δ ≈ 1.5) than in the rightmost position(h/δ ≈ 0.7). The amplitude of the horizontal body oscillations is comparable to that in the case of symmetricoscillations (curve 2). The trajectory is shaped like a loop. When the average motion direction reverses nearthe cavity ends, the oscillating motion trajectory changes for a mirror-symmetric trajectory, the magnitudeof the average motion velocity remaining the same.

The comparison of the body motion trajectories confirms that it is precisely the oscillation asymmetrythat generates the tangential motion: above the threshold of the loss of the oscillation symmetry the cylinderstarts to move from one cuvette end to the other. The shape of the loop determines the direction of thebody motion: if in the leftmost position the gap is wider than in the rightmost position, then the cylinder isdisplaced to the right, and vice versa.



Fig. 5. Time dependence of the horizontal coordinate of the cavity X (a) and the body coordinate x (b) before the detachment(h = 0, f = 5.5 Hz, and b = 26.3 mm) and time dependence of X (c), x (d), and h (e) after the detachment ( f = 9.4 Hz andb = 27.6 mm).

In the average motion (3) the body executes oscillations at a distance twice as large as in the absence ofthe motion (2). In both cases, the amplitude of the horizontal oscillations of the body relative to the cavityis almost the same.

In the experiments, the average lift, vibrational in nature, was directly measured. In the quasi-stationarysuspended state of the body the lift is counterbalanced by the gravity and is equal to fV = (ρL − ρS)gπd2/4per unit length. The repulsion force depends on the vibration amplitude and frequency, increasing with thevibration intensity. For the given vibration parameters the repulsion force depends on the distance fromthe cavity wall, namely, it decreases with the distance. For this reason, vibrations of greater intensity areneeded for the equilibrium state (suspension) of the body be attained at greater distances from the boundary.However, the vibrations are not capable to repel the body and to make its position stable at a distance greaterthan the double thickness of the Stokesian boundary layer. This indicates the determining role played by theviscous boundary layer near the oscillating body in generating the repulsion force, this being in agreementwith the results of the experiments with spherical bodies [7], in which it was shown that for h/δ > 2 thevibrational lift changes the sign: outside the limits of the viscous interaction the repulsion force changes forthe vibrational attraction force.

We will introduce the dimensionless vibrational force acting on unit body length, which turns out to beinverse proportional to the parameter W


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Fig. 6. Time dependence of X (a), x (b), ξ (c), and h (d) in the average motion of the body ( f = 11.4 Hz and b = 28.4 mm).

Fig. 7. Trajectories of the oscillatory motion of the body relative to the cavity before the detachment (curve (1) W = 6.8and ω = 80), after the detachment ((2) W = 21.5 and ω = 136), and in the motion from left to right ((3) W = 34.3 andω = 167).



FV ≡ fVρLb2Ω2d

=π(1 − ρ)

wW−1.

On the high-frequency range (ω > 300), when the body dimensions are considerably greater than theStokesian boundary layer thickness, the vibrational repulsion force is almost independent of the frequency(Fig. 3). However, on the low-frequency range FV sharply reduces with decrease in ω . This can be attributedto a decrease in the amplitude of the body oscillations relative to the cavity with decrease in the bodydimensions as compared with the Stokesian layer thickness. For ω ∼ 150 (Fig. 7) the amplitude of thebody oscillations relative to the cavity turns out to be considerably smaller than in the high-frequency casein which A/b = (ρ − 1)/(ρ + 1) [8]. The last formula, valid for small-amplitude oscillations of a longcylindrical body in an inviscid fluid at a large distance from the cavity walls gives the value A/b = 0.29 forthe relative body density ρ = 0.54 corresponding to the experimental conditions.

A qualitative change in the nature of the interaction between the cylindrical body and the cavity wallon the range of moderate and low dimensionless frequencies (ω < 300) is indicated by the presence of ahysteresis in body transitions from one state to another. With decrease in the dimensionless frequency thehysteresis depth increases.

Summary. The interaction between a light cylinder and the upper side of a cavity filled with a fluidexposed to horizontal vibrations is experimentally investigated. As a result of the hydrodynamic interactionwith the cavity side, an average repulsion force acts on the cylinder. The cylinder suspension in the gravityfield occurs as a critical value of the vibration parameter W ≡ (bΩ)2/(gd) is reached; here, b and Ω are thevibration amplitude and frequency and d is the body diameter. On the high-frequency range the threshold ofthe suspension is almost independent of the dimensionless vibration frequency ω ≡Ωd2/ν (ν is viscosity).On the range of moderate and low frequencies, when the body dimensions are comparable with the Stokesianlayer thickness, a hysteresis is observable in transitions. With decrease in ω the thresholds increase, whilethe hysteresis depth increases.

A new threshold phenomenon, namely, the excitation of an average tangential motion of the body alongthe vibration axis, is found to exist. The boundary of the occurrence of this motion in the (ω , W ) parameterplane is above the boundary of body suspension. It is shown that the motion is due to the loss of the bodyoscillation symmetry.

The authors wish to thank Head of the Laboratory of Hydrodynamic Stability of the Institute of Con-tinuum Mechanics of the Ural Division of the Russian Academy of Sciences K.G. Kostarev for placing thehigh-speed videdocamera at authors’ disposal and the worker of this laboratory A.V. Shmyrov for assistancein adjusting the camera.

The study was carried out with the support of the Russian Foundation for Basic Research (projectNo. 09-01-00665).

REFERENCES1. H. Lamb, Hydrodynamics, Cambridge Univ. Press, Cambridge (1932).2. B.A. Lugovtsov and V.L. Sennitskii, “Motion of a Body in a Vibrating Fluid,” Dokl. Akad. Nauk SSSR 289, 314

(1986).3. D.V. Lyubimov, T.P. Lyubimova, and A.A. Cherepanov, “Motion of a Solid in a Vibrating Fluid,” in: Convective

Flows [in Russian], Perm Pedagogical Inst. Press, Perm (1987), p. 61.4. V.G. Kozlov, “Solid-Body Dynamics in Cavity with Liquid under High-Frequency Rotational Vibration,” Euro-

phys. Letters 36, 651 (1996).5. A.A. Ivanova, V.G. Kozlov, and P. Evesque, “Dynamics of a Cylindrical Body in a Liquid-Filled Sector of a

Cylindrical Layer under Rotational Vibration,” Fluid Dynamics 33(4), 488 (1998).6. A.A. Ivanova, V.G. Kozlov, and A.F. Kuzaev, “Vibrational Lift Acting on a Body in a Fluid near a Rigid Surface,”

Dokl. Ross. Akad. Nauk 402, 488 (2005).7. A.A. Ivanova, V.G. Kozlov, and A.F. Kuzaev, “Vibrational Hydrodynamic Interaction between a Sphere and the

Boundaries of a Cavity,” Fluid Dynamics 43(2), 194 (2008).8. G.K. Batchelor, Introduction to Fluid Dynamics, Cambridge Univ. Press, Cambridge (1967).


a light cylinder under horizontal vibration in a cavity filled with a fluid

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