c Cambridge University Press 2013 J. Fluid Mech. (2013), vol. 732, pp. 77–104. doi:10.1017/jfm.2013.390
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Experimental investigation of freely falling thin disks. Part 2. Transition of three-dimensional motion from zigzag to spiral Cunbiao Lee†, Zhuang Su, Hongjie Zhong, Shiyi Chen, Mingde Zhou and Jiezhi Wu State Key Laboratory for Turbulence and Complex System, College of Engineering, Peking University, Beijing, 100871, China (Received 18 May 2013; revised 18 May 2013; accepted 24 July 2013; first published online 30 August 2013)
The free-fall motion of a thin disk with small dimensionless moments of inertia (I ∗ < 10−3 ) was investigated experimentally. The transition from two-dimensional zigzag motion to three-dimensional spiral motion occurs due to the growth of three-dimensional disturbances. Oscillations in the direction normal to the zigzag plane increase with the development of this instability. At the same time, the oscillation of the nutation angle decreases to zero and the angle remains constant. The effects of initial conditions (release angle) were investigated. Two kinds of transition modes, zigzag–spiral transition and zigzag–spiral–zigzag intermittence transition, were observed to be separated by a critical Reynolds number. In addition, the solution of the generalized Kirchhoff equations shows that the small I ∗ is responsible for the growth of disturbances in the third dimension (perpendicular to the planar motion). Key words: aerodynamics, flow–structure interactions
1. Introduction Symmetry breaking is a fundamental physical phenomenon of great practical interest and is always related to flow instabilities. The most famous example of symmetry breaking is the wake instability behind a two-dimensional cylinder, which generates the K´arm´an vortex street. The shedding vortex may strongly interact with the body, inducing flutter and vibration. For example, the cylinder wake will induce oscillations in the direction transverse to the main flow. Vandenberghe, Zhang & Childress (2004) showed that the flow symmetry was broken behind a vertically oscillating plate which led to a forward flapping motion. A freely rising gas bubble in water also exhibits oscillatory behaviour. When the bubble size exceeds a critical diameter, the rectilinear trajectory becomes unstable with planar zigzag and helical trajectories (Saffman 1956). Numerical simulations of a rising spheroid without mass by Mougin & Magnaudet (2002b) showed that the trajectory type is dependent on the bubble shape (aspect ratio). Mougin & Magnaudet (2006) later found that a pair of counter-rotating streamwise vortices in the bubble wake induced the spiral motion. A model based on experimental investigations by † Email address for correspondence:
[email protected]
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Shew & Pinton (2006) showed that the transition of bubble motion from zigzag to spiral motion was controlled by the wake instability. The evolution of the lift force indicated a subcritical bifurcation. Horowitz & Williamson (2008) showed that the trajectories of freely falling and rising spheres were always in a vertical plane. Three dimensionless ratios (the aspect ratio, λ = h/d, where h is the disk thickness and d is the disk diameter, the dimensionless moment of inertia, I ∗ , and the Reynolds numbers based on the average falling speed, Re) have been proposed to determine disk motion. For thin disks with small aspect ratios, experimental studies by Willmarth, Hawk & Harvey (1964), Stringham, Simons & Guy (1969) and Fields et al. (1997) showed that the motion was dependent on Re and I ∗ . A phase diagram including four types of motion was given by Fields et al. (1997). As a two-dimensional counterpart of disks, similar free-fall behaviour was reported for falling plates. A phase diagram was obtained experimentally by Smith (1971). A quasi-two-dimensional experiment was carried out by Belmonte, Eisenberg & Moses (1998) and a Froude number, Fr, criterion was deduced to characterize the transition from flutter to tumble. A scaling relation between the rotary velocity and the geometry of tumbling plates was developed by Mahadevan, Ryu & Samuel (1999). Numerical simulations on a tumbling ellipse were carried out by Pesavento & Wang (2004). The aerodynamic lift model was proposed to be the product of linear and angular velocities. Andersen, Pesavento & Wang (2005) measured the instantaneous two-dimensional kinematics of freely falling plates and calculated the fluid forces. Stewart & List (1983) measured the translational and rotational motions of two thick disks and verified the existence of a gyration motion. Recently, Fernandes et al. (2007) performed experiments on the free rise of nearly neutral buoyancy cylinders with aspect ratios from 1/20 to 1/2 and Reynolds numbers of 100–320. Five out of the six degrees of freedom were measured, not including the axis of rotation. Both the kinematics and dynamics of the quasi-planar zigzag motion (Fernandes et al. 2008) and wake structures (Ern et al. 2007) were described in detail. Modelled vortical forces agreed well with low-Reynolds-number measurement (Ern et al. 2009). Fernandes et al. (2005) showed that inviscid theory was unable to predict the phase lag between the instantaneous velocity and the body inclination; thus, vortical effects must be taken into account to correctly predict the body motion. Franck, Jacques & David (2013) did direct numerical simulations by solving the Kirchhoff equations with the Navier–Stokes equations and enriched the phase diagram. A recent experimental study by Zhong, Chen & Lee (2011) showed that the three-dimensional motion of disks includes spiral and transitional motion for disks with small I ∗ . Chrust, Bouchet & Duˇsek (2013) reported a similar result obtained by direct numerical simulations and extended the phase diagram. Variations of the nutation angle were directly related to the motion states. However, the conditions for the transition of three-dimensional motion from zigzag to spiral are still not thoroughly understood. The purpose of this paper is to analyse the onset and evolution of three-dimensional disturbances and the related flow patterns in the transition from zigzag to spiral motion. The six degrees of freedom of a disk are measured and analysed for a disk undergoing the transition from planar zigzag to spiral motion. The oscillations in the normal direction for the zigzag motion, the rotation about the axis of revolution and the nutation angle are used to describe the growth of the spiral motion. The detailed flow separation around the disk and the wake structures are shown by using fluorescent dye visualization and particle image velocimetry (PIV) measurements. The results illustrated the role of the wake structure evolution on the body trajectory transition.
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A dynamic model based on the generalized Kirchhoff equations is used to account for the separate effects of the vortical forces and the added mass. 2. Experimental procedures The experiments were conducted in an open tank 1100 mm tall with a 300 mm × 300 mm cross-section. The tank was made of Plexiglass to allow optical access from the sidewalls. The tank was filled with filtered water to a height of 1050 mm. The water density was measured to be 0.996 g cm−3 by a float densimeter. The kinematic viscosity was estimated to be 1.02 × 10−6 m2 s−1 . The facilities were placed in an air-conditioned room with the temperature maintained at 20 ± 1 ◦ C. The water was left undisturbed for at least 30 minutes before each experimental run. Disks were made of Plexiglass with a density of 1.20 g cm−3 ; thus, the density ratio was fixed at ρs /ρf = 1.20. The dimensionless moment of inertia, I ∗ = (π/64)(ρs /ρf )λ, was also fixed for each set of disks so that only the Reynolds number was varied. The disks were made with two different values of λ = 1/40 and 1/80. The error in the disk thickness due to manufacturing was ∼0.05 mm, within an acceptable range for the current study. In the experiment, the disks were first fixed and then released. The release speed was set to zero, and different release angles were used to represent different initial conditions. The tested cases are listed in table 1. The flow structures were studied using dye visualization and PIV measurements. The dye visualization used two types of fluorescence dyes, rhodamine B and fluorescein, uniformly coated on one or both surfaces of the disk. The dye mass was less than 2 % of the disk mass. A 50 mm thick vertical sheet of light was generated by a halogen lamp projecting through the side of the water tank. Photographs were taken normal to the light sheet. When illuminated, the dissolved dye particles formed coloured patterns in the flows which effectively represented the flow structures. PIV measurements were also utilized to investigate the flow fields. The light source was a continuous semiconductor laser with an output power of 300 mW. The laser beam passed through a set of lenses to form a 1 mm thick light sheet. Seeded particles in the fluid were illuminated by the laser sheet. Fluorescent particles were used in the experiment to eliminate the influence of glare and reflection of the laser light from the disk boundary (Lee et al. 2011). The fluorescent dye, rhodamine B, was encased in glass-based particles. When the particles were illuminated by the laser wavelength of 532 nm, they emitted light at a longer wavelength of around 570 nm. A low-pass filter was placed before the cameras to block the glare near the disk surface so that only the fluorescent light could pass through the filter. This set-up removed the disturbances near the boundaries so that the velocity near the disk surface could be measured accurately. The tracer particles were around 10 µm in diameter with a density of 1.07 g cm−3 so that they followed the flow very well when dispersed in water. The image resolution was 1024×1024 pixels. A cross-correlation-based algorithm was used to calculate the displacement field from the photographs. Multi resolution and window deformation iterative multigrid techniques were used to enhance the precision and robustness of the PIV estimates. A quadrangle-shaped mesh was used to interpolate the velocities near the body boundary. 3. Experimental results 3.1. Measurement of three-dimensional instabilities This section compares the oscillatory amplitudes and frequencies of the spiral motion with those of the zigzag motion. Figure 1(a–c) describes the dynamics of the disk
613 675 610 655 7.06 × 10−4
782 831 882 864 7.46 × 10−4
958 1000 1050 1030 6.90 × 10−4
1181 1306 1283 1252 7.80 × 10−4
1382 1498 1398 1456 7.06 × 10−4
Reynolds number 1531 1706 1621 1645 7.06 × 10−4
1761 1929 1967 1934 7.06 × 10−4
2105 2062 2082 2095 7.56 × 10−4
TABLE 1. The cases that we tested, including different initial conditions and different disk diameters. The Reynolds numbers are slightly different for disks with the same diameter, but with different release angles.
0 ± 1◦ 3 ± 1◦ 5 ± 1◦ 10 ± 1◦ I∗
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F IGURE 1. Description of the disk’s non-planar body motion for d/h = 40.0 and ρd /ρf = 1.2. The measured Reynolds number Re = Ud/ν is 1005. The figure shows the degrees of freedom evolving over time. The disk centre coordinates (x, y, z) are given in parts (a) and (b), but in (c) they are given as z − Ut. Parts (d–f ) show the angular motion of the body as ψ, θ and φ representing the three Euler angles.
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F IGURE 2. Disk velocities for periodic transitional motion (a) in stationary coordinates and (b) in body fixed coordinates.
centre for a disk with d/h = 40 and Re = 1005. Spiral motion is seen to occur. Unlike the zigzag motion, the oscillations here are harmonic functions of time in both the x and y directions. For the zigzag case, the phases in the two directions were the same. However, the spiral case here has a phase shift of about π/4. Also, the oscillatory amplitude is small compared with the zigzag case presented in part 1 (Zhong et al. 2012). The oscillatory amplitude of z − Ut is ∼25 % of that in the x and y directions. In the spiral case, the oscillation amplitude of z − Ut is less than 5 % of the amplitude of the horizontal oscillations. For the zigzag case, z − Ut has an oscillatory frequency twice that in the horizontal directions while the spiral case does not seem to have a fixed frequency. This suggests that for the spiral motion, the vertical velocity did not vary significantly due to the slight influence of the angle of attack because the angle of attack did not change as much as it did in the zigzag case. The projection of the trajectory on the horizontal plane is elliptical with the trajectory rotating in the same direction. The angular evolution with time also differs from that of the zigzag case. Figure 1(d–f ) shows the time evolution of the Euler angles. The procession angle, ϕ, increases with time while the gyrational angle, ψ, decreases with the period of ϕ being slightly larger than that of ψ; this results in a slow rotation of the disk about the centre axis. The nutation angle, θ, is nearly constant with small-amplitude oscillations. Figure 2 shows the translational velocities of the disk. The velocities are decomposed into components according to: (a) the stationary coordinates; and (b) the body fixed coordinates. The representations in the body fixed coordinates are much simpler where figure 2(b) shows that the axial velocity, Vζ , is almost constant with
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F IGURE 3. Plane trajectory for I ∗ = 2.53 × 10−4 and Re = 850 showing the evolution of the planar zigzag to spiral motion.
time. This observation agrees well with the results in Fernandes et al. (2007) who did not measure the Vξ component. The spanwise velocities, Vξ and Vη , are both modulated by a slow rotation here with a modulation frequency of about 1/5 of the zigzag frequency. The motion gradually deviates from planar zigzag to the spiral state, as can be clearly seen from the top view trajectory in figure 3. 3.2. Growth of instability in the third dimension Figure 3 gives an overview of the trajectory in the horizontal plane. The evolution from planar zigzag to spiral motion is related to the rotation of the oscillatory direction and the oscillation amplitude growth in the third dimension. The instability causes oscillations perpendicular to the plane of the zigzag motion, with the amplitude growing with time. At the same time, the direction of the planar zigzag rotates and the rotation velocity grows with time. The non-planar cases form a plane trajectory that is similar to a rhodonea curve. The curves can be expressed in general by a polar equation of the form
r = Ax cos(kθ ) + Ay sin(kθ)
(3.1)
where Ax and Ay are the oscillations in the primary and secondary directions. A Hilbert transform can be used to separate the motion into a rotational component and an oscillatory component, with the oscillatory part shown in figure 4. The trajectory was adjusted to a new X 0 –Y 0 plane with the primary direction, X 0 , aligned with the direction of the planar zigzag motion and the normal secondary direction, Y 0 , indicating the direction of the disturbance growth. On this plane, the trajectory forms a series of ellipses with decreasing aspect ratios, as shown quantitatively in figure 3. The oscillation amplitudes in the primary direction, Ax , and the secondary direction, Ay , are shown in figure 4. As the motion transits from zigzag to spiral, Ax decreases and Ay increases. Figure 5 shows the time evolutions of the disk centre coordinates, the three Euler angles and the angular velocities for a spiral case with Re = 850 and I ∗ = 2.53 × 10−4 . As shown in figure 5(a), the variations of the disk centre coordinates follow a sine curve in the X and Y horizontal directions. The two amplitudes were about the same, with a π/2 phase shift. The trajectory on the horizontal plane is
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F IGURE 4. Oscillation amplitudes in the primary direction, Ax , and secondary direction, Ay , in figure 3 as the motion transiting from zigzag to spiral with Ax decreasing and Ay increasing.
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F IGURE 5. Time evolution of (a) the disk centre coordinates, (b) the Euler angles and (c) the angular velocities for Re = 850 and I ∗ = 2.53 × 10−4 .
circular and coiled around the vertical axis. The oscillation amplitude in the vertical direction, z, is relatively small (the plot shows Z − Ut) compared with the zigzag case. The time evolution of the angular motion is shown in figure 5(b). The nutation angle, θ, is nearly constant with only small variations. The gyration angle, φ, increased and
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F IGURE 6. Accelerations of the disk for I ∗ = 2.53 × 10−4 and Re = 850. 0.60
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F IGURE 7. Oscillation amplitudes in the primary direction, Ax , and the secondary direction, Ay , for the conditions listed in table 2.
the self-rotation angle, ψ, decreased at almost constant rates. Figure 5(c) shows the time evolution of the disk angular velocities in the frame rotating with the moving disk. Previous studies (Fernandes et al. 2007; Ern et al. 2009) observed that the rotation about the axis of revolution, Ωζ , was negligible. However, in figure 5(c), Ωζ is not small in the non-planar motions and increases during the transition process. The corresponding accelerations given in figure 6, show that the oscillation amplitudes in the horizontal directions became identical and that the oscillations in the vertical direction vanish. Thus, the motion has evolved into steady falling. The evolution of the oscillatory amplitudes Ax and Ay are given in figure 7 for different initial release angles and different Reynolds numbers. The initial conditions are listed in table 1. The zigzag motion always transitioned to spiral for these conditions. The final amplitudes of Ax and Ay are not sensitive to the initial release angles, but are sensitive to the Reynolds number. figure 8 shows the relationship between R = Ay /Ax and the Reynolds number for different I ∗ . The trend as R approaches 1 indicates that the trajectory becomes a circular helix as I ∗ decreases to 0. This agrees extremely well with the findings of rising bubbles, which can be
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F IGURE 8. Ratio of Ay to Ax for disks with different I ∗ and Reynolds numbers.
seen as an example of I ∗ = 0 due to the large density difference between the bubbles and the water where the trajectory is a circular helix (Magnaudet & Eames 2000). The Reynolds number also affects the trajectories with R increasing as the Reynolds number increases. The horizontal oscillation amplitudes also show dependence on the Reynolds number, as shown in figure 9. The relation between the Strouhal number and the Reynolds number for different initial conditions is given in figure 10. 3.3. Effects of the initial conditions Transition and instability are unsolved issues in fluid mechanics and are always directly related to the disturbance evolution (Lee & Wu 2008). The initial conditions strongly affect the dynamic process of the transition from one state to another. In the case of falling disks, the initial condition is important for the transition from zigzag to spiral. The initial conditions here are very complex, including the disk release angle, the release speed and the residue disturbances in the fluid. Different release angles were used to study the effects of different initial conditions. The release speed is set to zero and the influence of residue disturbances was kept as small as possible by quieting the water for at least half an hour before each experimental run. In the case of freely falling disks, the behaviours of the disk’s Euler angles directly indicate the growth of the instability. Zhong et al. (2011) discussed the transition from planar zigzag to spiral and used the nutation angle to describe the different motion type (zigzag or spiral). Another feature is used here to characterize the motion type in addition to the nutation angle. A typical zigzag motion is characterized by a square-wave-like gyration angle sequence and large-amplitude oscillations of the nutation angle (Zhong et al. 2012). For a typical spiral motion, the gyration angle evolves continuously, and the amplitude of the nutation angle oscillation is relatively small, e.g. nearly zero when the spiral is complete. The motion type can be classified according to these two features. The most obvious indicator of zigzag motion is the
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F IGURE 9. Average horizontal oscillation amplitudes versus Reynolds number for different initial conditions normalized by the disk diameter. Here λ = 1/80. 1.5
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F IGURE 10. Strouhal number versus Reynolds number for different initial conditions. Here λ = 1/80.
two nearly constant gyration angle periods joined by a jump in each circle. As the transition proceeds, lines in both sections become increasingly slanted until, finally, those two sections merge into one continuous line, which means that the transition has completed and that the motion has become spiral (figure 11). The amplitude of the nutation angle oscillation also provides useful insight into the disk motion mode.
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F IGURE 11. Time evolution of gyration and nutation angle of the disk. Here I ∗ = 7.06 × 10−4 , Re = 655 and the release angle is 10◦ . 1.0
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F IGURE 12. Amplitude of the nutation angle oscillation in each circle normalized by the maximum nutation angle in all circles. The amplitude for each circle was obtained by deducting the maximum angle in that circle with the minimum. Here I ∗ = 7.06 × 10−4 , Re = 655 and the release angle is 10◦ .
Figure 12 shows the evolution of the amplitude of nutation angle oscillation cycle by cycle. The amplitude starts at a high level, which means that the motion is zigzag. Then, the transition begins, and the amplitude decreases as the transition proceeds. As the transition proceeds further, the amplitude becomes relatively small and remains constant which means that the motion is now spiral. This is consistent with the gyration angle characteristics. The effects of the initial conditions were investigated by repeatedly testing different release angles. According to the experimental results, for the tested angle range, the release angle did not affect the motion mode of the falling disks, but did change the time for the transition from zigzag to spiral. In all tested cases, disks with the same dimensionless moment of inertia and diameter (or Reynolds number), but different release angles, showed the same motion mode. Specifically, all of the disks started with a zigzag phase, followed by a transition phase and then ended as spiral. The length of the transition phase decreased with increasing release angle and Reynolds number, as shown in table 2. The time scale of symmetry breaking is shown in table 4. Another kind of transition phase was also observed in some cases. For Reynolds number of 600–1000, the transition was quite simple, with the slopes of the gyration
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Spiral motion of freely falling thin disks Angle ◦
0±1 3 ± 1◦ 5 ± 1◦ 10 ± 1◦
d = 25 9.70 6.81 6.63 6.19
d = 30 7.06 6.30 4.65 5.32
d = 35 8.89 6.03 4.57 5.47
d = 40 6.78 5.61 6.07 5.8
d = 45 5.10 6.40 5.40 5.25
d = 50 6.97 5.32 3.67 3.45
d = 55 4.57 3.60 2.73 2.66
d = 60 4.15 3.48 2.56 2.60
TABLE 2. Starting time of the first complete spiral circle in the fall, normalized by the average oscillation period of the disks, for different cases listed in table 1. The complete spiral circle is defined as a circle without any gyration angle jump in the whole circle. Angle ◦
0±1 3 ± 1◦ 5 ± 1◦ 10 ± 1◦
d = 25 2.61 2.42 2.67 2.06
d = 30 2.05 2.38 2.63 1.80
d = 35 2.23 3.25 2.87 1.74
d = 40 2.26 1.87 1.96 1.55
d = 45 2.09 2.92 2.31 1.28
d = 50 1.53 2.01 2.42 1.58
d = 55 1.36 1.62 1.59 1.34
d = 60 1.45 1.96 2.50 1.82
TABLE 3. Time scales of zigzag–spiral transition of each initial condition listed in table 1, normalized by the average period of each case. The typical transition time is defined as the typical time length of the course in which the gyration angular velocity starts to grow and then stops growing. Angle 0 ± 1◦ 3 ± 1◦ 5 ± 1◦ 10 ± 1◦
d = 25 — — — —
d = 30 — — — —
d = 35 2.23 2.26 2.15 2.47
d = 40 1.90 2.23 1.70 2.49
d = 45 1.81 2.12 2.67 2.35
d = 50 1.65 1.70 1.96 2.57
d = 55 1.80 1.80 1.41 2.79
d = 60 2.07 2.33 1.65 1.82
TABLE 4. Time scales of spiral–zigzag transition of each initial condition listed in table 1, normalized by the average period of each case. The typical transition time is defined as the time length of the course in which the gyration angular velocity first starts to decrease and then stops decreasing.
angle in each region of each cycle slowly changing until they joined together, and the amplitudes of the nutation angle oscillations consistently decreasing to a constant as the motion becomes spiral, as shown in figure 11. However, when the Reynolds number was in the range of 1000–2100, after the first full spiral cycle, some incomplete spiral cycles also occurred, characterized by a jump in the gyration angle and an increase of the nutation angle oscillation amplitude. The motion then returned back to spiral again. This intermittence between zigzag and spiral occurred repeatedly. Figure 13 shows typical intermittent transitions. After the first complete spiral cycle in the 6th cycle after release (arrow A), the spiral is interrupted by two cycles of incomplete spirals (arrow B). In the 9th cycle after release, the spiral motion appears again (arrow C). The nutation angle oscillation amplitude also exhibited this phenomenon, as shown in figure 14. Apart from gravity, this system has only fluid forces and inertial forces that can influence the disk motion. All of the different disk motions are the results of the interactions between these two factors. For small Reynolds numbers, the disk diameter is small; thus, the moment of the fluid force in the direction perpendicular to the diameter, which resists the rotation, is small. Hence, the inertia of the rotation can easily exceed the viscosity effects, and the
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F IGURE 13. Time evolution of the gyration and nutation angle of the disk. Here I ∗ = 6.90 × 10−4 , Re = 1030 and the release angle is 10◦ . 1.0 B
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F IGURE 14. (Colour online) Amplitude of the nutation angle oscillation in each circle normalized by the maximum nutation angle in all circles. Here I ∗ = 6.90 × 10−4 , Re = 1030 and the release angle is 10◦ .
disk continues rotating and transits directly to spiral. However, with higher Reynolds numbers, the moment of the fluid force is large enough to compete with the inertial effects due to the larger diameter; thus, the moment of fluid force can sometimes exceed the inertia. When it does, the disk motion reverts back to zigzag with a jump in the gyration angle. When the inertia again increases, the motion transitions to spiral with possible repeated transitions. Figure 15 shows the angular velocity around the ζ axis and the corresponding torque. Figure 16 compares the ζ torque with that of a smaller disk to show that the amplitude of the torque of the larger disk is significantly larger. The angle of attack also behaves differently in these two kinds of transitions. Figure 17 shows the time evolution of the angle of attack for the first kind of transition. The oscillation amplitude decreases as the transition proceeds and reaches a minimum when the transition is complete (arrow A). Then, the amplitude increases back to a level slightly smaller than that during the zigzag stage. Figure 18 shows the time evolution of the angle of attack for the second kind of transition. The amplitude is also intermittent (arrows A, B and C). This can also be shown by the flow visualization. Figure 19 shows the zigzag–spiral–zigzag transition. In figure 19 (9 s) and (10 s), the motion mode is zigzag and the separation occurs on the edge of the disk (arrow A) to form two
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F IGURE 15. Angular velocity ω of the ζ axis and the torque of fluid force exerted on it Γζ . The trends of the torques correspond rigidly to the motion.
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F IGURE 16. Torque of the fluid force exerted on the ζ axis for two disks with different diameters. The amplitude of the torque exerted on the disk with a larger diameter is significantly larger.
counter-rotating vortexes (arrow B). As the transition proceeds, the disk begins to spiral and the counter rotating vortexes become less concentrated (figure 19, 13.5 s), the separation points move toward the centre of the disk (figure 19, 14 s) and the wake changes (figure 19, 14.5 s). When the intermittence happens (figure 19, 18 s), the separation points move back to the edge and the wake again has two counter-rotating vortexes (arrow B1). 3.4. Visualizations and PIV measurements 3.4.1. Flow separation The flow separation near the disk surface was investigated using flow visualization. From a Lagrange viewpoint, a fluid particle moving away from the near-wall region in asymptotic linear time can be viewed as separation. Thus, one can directly detect flow visualization by identifying where the dye particles leave the wall region. For flow near the edge, a curvature discontinuity in the dye trace indicates separation. For flow near a flat surface, material spike structures indicate the separation location. Flow separation occurs on the edge of the disk with a vortex loop elongating in the
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F IGURE 17. (Colour online) Time evolution of the angle of attack (AoA). The oscillation amplitude of the angle of attack decreases significantly as the transition from zigzag to spiral is completed (arrow A). However, it will grow back when the spiral is steady. Here I ∗ = 7.06 × 10−4 and Re = 650.
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F IGURE 18. (Colour online) Time evolution of the angle of attack (AoA). The oscillation amplitude shows intermittency as the spiral–zigzag–spiral intermittence occurs (arrows A, B and C). Here I ∗ = 6.90 × 10−4 and Re = 1030.
direction of the body movement to form typical hairpin structures, similar to those observed in the wakes of bluff bodies. As the planar symmetry is broken, the shear increases on one side and decreases on the other side, which changes the shape of the shed vortices. On the weaker side, the separation location moves towards the disk centre; thus, the separation occurs on the upper surface of the disk. The periodic shedding of the hairpin-like vortices in planar zigzag motion, as described in Zhong et al. (2012), carries less streamwise vorticity than the continuously shed vortex chain here. The vortex chain shed from the outer edge is stronger and forms a helicoidal vortex wrapping around the wake region. The vortex chain shed from the disk surface is relatively weak and remains in the centre of the wake region. The helical vortices induce a strong downwash in the centre of the disk wake region. The downwash velocity is larger than the average disk falling velocity, an area about three times the disk diameter from behind the disk.
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F IGURE 19. Flow visualization of the zigzag–spiral–zigzag intermittence.
As shown in figure 20(a) (at 0T), the separation (arrow A) occurs on the edges of the disk. The curvatures of the dye streak on the two sides are almost symmetric (arrows A1 and B1). Arrows A2 and B2 point out an obvious curvature change. The curvature changing from convex to concave indicates the separation point moving from
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F IGURE 20. For caption see next page.
the edge to the broad surface (arrow C at 5/3T and 11/6T). Figure 20(b) (at 5/3T) shows the movement of the separation point from the edge to the surface of the disk. Arrow C shows that the distance between the separation location and the edge increases after one period. The flow separation structures are significantly different from those in zigzag motion addressed in Part 1 (Zhong et al. 2012).
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F IGURE 20. Visualization of the initial development from zigzag to spiral motion. Here T is the period of zigzag motion.
As shown in figure 21, separation plays an important role in the evolution of the wake structures. For a spiralling disk moving around a vertical axis, the horizontal velocity near the outside edge of the disk is higher than that near the inside edge. Therefore, more vorticity is produced and fed into the wake by separation on the outside edge and, subsequently, scrolled into a roll. Since the separation location
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F IGURE 21. Separation of the vortical structure from near the disk wall. Two different separation locations were found on the disk edge and the upper disk surface.
rotates, the vortex is helicoidal. The flow inside the helicoidal vortex has a large downward velocity, and this velocity causes separation on the disk surface which is connected to the separation on the inside edge. These together generate an upright vortex inside the helicoidal vortex that carries streamwise vorticity with the opposite sign of the helicoidal vortex.
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F IGURE 22. Velocity field in a horizontal plane after the passage of the disk: (a) velocity distribution and (b) vorticity distribution (the circulations for vortex A and B are 99.21 m2 s−1 and −102.01 m2 s−1 respectively.). The dominant vortical structure is a pair of counterrotating asymmetric vortices with unequal intensities.
3.4.2. PIV measurements Figure 22 shows PIV flow velocity field measurements on a horizontal plane after the passage of the disk. The dominant vortical structure is a pair of counter-rotating asymmetric vortices with different intensities. The outer vortex is slightly stronger than the inner one. The helical-like non-planar motion of the disk creates vortices in the wake, as figure 23(a) shows the velocities in the wake structure. Unlike the zigzag motion, each cycle has a vortex ring structure detaching from the disk. Long helical vortices are then shed continuously from the disk edge. Figure 23(b) shows the vorticity field in
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F IGURE 23. Vortices resulting from the helical-like non-planar motion of the disk for d/h = 150 and Re = 640. (a) The velocities in the wake structure. Unlike in the zigzag motion, where a vortex ring structure is shed in each cycle, long helical vortices shed continuously from the disk edge here. (b) The vorticity in the wake region. The measurement plane was a vertical slice across the wake region. The vorticity cores have opposite signs on the two sides of the wake region with the vortices aligned in the wake boundary.
the wake region. The measurement plane is a vertical slice across the wake region. The vorticity cores with opposite signs form two sequences of vortices in the wake region. 3.4.3. Comparison with falling and rising spheres Horowitz & Williamson (2010) studied the effects of the Reynolds number, Re, on the dynamics and vortex formation modes of spheres freely rising or falling through a fluid for Re = 100–15 000. They found a number of changes occurring at a Reynolds number of 1550, and suggested the possibility of a resonance between the shear layer instability and the vortex shedding (loop) instability. Their study with minimal background disturbances was used to create a new regime map of the dynamics and vortex wake modes as a function of the mass ratio and Reynolds number. The flow separation in the present case is different from their work. In zigzag motion, the flow separation point always appears at the edge of the disk as shown in figure 19. For the spiral motion, the separation begins on the disk surface as shown in figures 20 and 21. For the transitional case, flow separation points move from the disk edge
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to the surface. The vortex wake modes are also different from those of Horowitz & Williamson (2010). Figure 23 in their work shows the brief vortex wakes for the same Reynolds number range as in the present experiments. For low Reynolds numbers, the vortex forms are nearly the same (Zhong et al. 2011). In the present case, when the Reynolds number is larger, there is an upward vortex inside the helicoidal vortex; this is the typical flow structure directly related to the spiral motion. 4. Discussion 4.1. Qualitative analysis of the transition to non-planar motion This section describes the disk dynamics based on a set of equations describing the force and torque balances for the coupled fluid body system to analyse the transition from planar to non-planar motion and the role of I ∗ . Consider a non-deformable body (of mass m and inertia tensor J ) moving in an incompressible viscous fluid which is unbounded and at rest at infinity. The translational velocity and the angular velocity of the body are U and Ω . The force and torque balances are governed by the generalized Kirchhoff equation (Howe 1995; Mougin & Magnaudet 2002a). In the body fixed system, the equations are written as
(mI + A)
dU + Ω × ((mI + A)U) = FΩ + (m − ρf V )g dt
(4.1)
dΩ + Ω × ((J + D )Ω ) + U × (AU) = ΓΩ (4.2) dt where A and D are the added mass and added inertia tensors due to the instantaneous fluid motion caused by the translational and angular motion of the body. Here I represents the identity matrix. The terms on the left-hand side of the equations are the inertia terms. The terms FΩ and ΓΩ on the right-hand side are the force and torque caused by the viscous effects. For a thin circular disk, the coefficients can not be written analytically; so, the analysis used the approximations given by Fernandes et al. (2008), and A, D and J were all diagonal. In the coordinate system Oηξ ζ , the diagonal terms of these tensors satisfy " 1/2 # ρs d 3 h 7ρs d3 h 4/7 ; Aζ = 1 + 0.5 (4.3) Aη = Aξ = 12π d 3π d " 1/2 # ρs d 5 h Dη = Dξ = 1 + 0.8 ; Dζ = 0 (4.4) 90 d (J + D )
md2 mh2 md2 + ; Jζ = . (4.5) 16 12 8 Since Dη = Dξ , Jη = Jξ and Aη = Aξ , the two cross-products vanish in the ζ axis direction in (4.2). Thus, the torque balance in this direction can be written as Jη = Jξ =
dΩζ = ΓΩζ . (4.6) dt Therefore, the rotation about the axis of revolution is purely driven by the torque of the vortical effects. More specifically, the fluid torque exerted on the disk is Z ζ Γ = nζ · [r × (−pδij + τij )nj ] dS (4.7) Jζ
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F IGURE 24. (Colour online) Generation of vortical torque about axis ξ by the rotation about the axis of revolution. The rotation about the axis of revolution results in different velocities, U + and U − on the two sides; thus, the lift on the two sides is not equal, which produces the torque ΓΩζ .
where S represents the disk surface area. The integral of the pressure term is zero for the disk. Thus, the torque exerted by the fluid originates from the shear stress on the disk surface which can be expressed as Z ζ ζ (4.8) Γ = ΓΩ = [rη τξ i − rξ τηi ]ni dS. p Using d and u0 = (ρs /ρf − 1)gh for the length and velocity scales, the dimensionless form of (4.6) can be written as I∗
dΩζ = ΓΩζ /2. dt
(4.9)
The influence of the rotation about the axis of revolution on the transition from planar to non-planar motion was analysed for a disk undergoing planar zigzag motion. Initially, the velocity of the disk centre, U, and axis ζ lie in the same vertical plane so that z · (U × ζ ) = 0.
(4.10)
For convenience, the body fixed system is chosen so that axis ζ is perpendicular to the disk surface, ξ lies in the vertical plane and axis η is perpendicular to them. Then we have Uη = 0 under this coordinate system. Assume that ωζ < 0 arises due to some disturbances. An immediate consequence of the rotation is that the disk velocity on one side of the vertical plane becomes greater than that on the other side during the first half of a zigzag period, which means ωη < 0 as shown in figure 24. The different velocities on the two sides cause different lifts denoted as L+ and L− with L+ > L− , which produce a torque about axis ξ . As a result, the disk tilts to the side with the smaller lift. This lift-induced torque then becomes the main force that disrupts the original planar motion.
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(a)
(b)
F IGURE 25. (Colour online) Illustration of how the added inertia moment affects the development of the non-planar motion: (a) I ∗ > (ρf /90ρs )(1 + (h/d)1/2 ), the added inertia torque is in the opposite direction to the rotation-induced vortical torque; (b) I ∗ < (ρf /90ρs )(1 + (h/d)1/2 ), the added inertia torque is in the same direction as the rotationinduced vortical torque.
Since Uη = 0, the cross-product of U in (4.2) vanishes in the ξ direction, and the torque balance on axis ξ can be written as dωξ + (Jζ − Jη − Dη )ωη ωζ = Γξ . (4.11) dt Substituting (4.3)–(4.5) into (4.11) gives the non-dimensional form, 1/2 !! 1/2 !! h dωξ ρf h ρf ∗ ∗ 1+ + I − 1+ ωη ωζ = γξ . (4.12) I + 90ρs d dt 90ρs d (Jξ + Dξ )
∗ For simplicity, let (ρf /90ρs )(1 + (h/d)1/2 ) be referred to as Icrit . Solving for the rate of change of ωξ gives ∗ dωξ γξ − (I ∗ − Icrit )ωη ωζ = . ∗ ∗ dt I + Icrit
(4.13)
Without loss of generality, start from ωξ = 0. A positive γξ , as shown in figure 24, will always amplify ωξ . Since ωη < 0 and ωζ < 0, the second term on the right-hand ∗ side will be positive, which will further destabilize the system if I ∗ < Icrit . However, ∗ ∗ for I > Icrit , the second term will be negative and will stabilize the system. For the limit of an infinitively small disturbance where γξ → 0, the second term is the only ∗ term to determine the sign of the growth rate, so Icrit can be reasonably regarded as the ∗ critical I for transition to occur. ∗ A curve of I ∗ = Icrit is shown in figure 26. This curve is almost a horizontal straight line due to the weak influence of h/d for a given fluid. This line can be seen to almost separate the planar zigzag and spiral motion data.
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F IGURE 26. Various types of falling disk motion in the two-dimensional phase space Re–I ∗ . The circles are the same data shown in Zhong et al. (2011), the black circles fall in to the zigzag zone, while the red ones are spiral. The other data are from Fields et al. (1997).
To complete this view of transition, now assume that the disk is experiencing the second half of a zigzag cycle with swinging from right to left (ωη > 0). Using the same mechanism as before, now γξ and the growth of ωξ are both in the opposite ∗ direction in the case of I ∗ < Icrit compared with the first half. This may give an impression that the above-mentioned disturbance-induced effect will be cancelled after a whole cycle of zigzag motion. However, when the three-dimensional motion is considered, things will be completely different. In fact, during the first half cycle (swing from left to right), the increase of ωξ would result in a tilting of the disk toward the positive direction about axis ξ so that the lift would cause a side force component along the negative direction of η. Thus, in the top view, the trajectory of the disk centre would become a curve rather than a straight line as before the disk was disturbed as shown in figure 3. Then, during the next half cycle, the disk will pass through another curve with curvature in the opposite direction to form an almost closed curve similar to an ellipse. However, it will not be a closed ellipse due to the asymmetrical separation discussed earlier. Thus, after an entire cycle, the zigzag motion would be shifted by some angle from the top view, as shown in figure 3. 5. Conclusions The transition of the free-fall motion of a thin disk from zigzag to spiral has been studied experimentally with accurate measurement of the six degrees of freedom. The results have shown that the rotation about the axis of revolution is critical to the onset of the spiral motion. Without accurate measurements of this angle, the transition process could not be evaluated quantitatively. The visualizations also show that the separation pattern for the onset of transition was quite distinct from that for a typical
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bluff body separation. The separation point moved from the edge to the broad surface and formed complex vortex structures, including a leading-edge vortex, a helicoidal vortex and an upright vortex; these structures dominated the transition process. Two kinds of transition modes were observed with direct zigzag–spiral transitions for Reynolds numbers between 600 and 1000 and intermittent zigzag–spiral–zigzag intermittency transitions for Reynolds numbers between 1000 and 2100. Symmetry breaking of the counter-rotating vortices and movement of the separation point towards the centre of the disk were responsible for the onset of the spiral motion. Viscosity effects were more important for small I ∗ . Furthermore, the critical value of I ∗ appears to be the most important factor for predicting the onset of the spiral motion for small I ∗ . The critical value of I ∗ controls the evolution of the motion from zigzag to spiral. Thus, vorticity stresses sometimes dominate the motion of the disk. Acknowledgements This work was supported by the National Natural Science Foundation of China under grant no. 109103010062. This work was also supported by the National ClimbB Plan under grant no. 2009CB724100 and the National Natural Science Funds for Distinguished Young Scholars group under grant no. 10921202. REFERENCES A NDERSEN, A., P ESAVENTO, U. & WANG, Z. J. 2005 Unsteady dynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 65–90. B ELMONTE, A., E ISENBERG, H. & M OSES, E. 1998 From flutter to tumble: inertial drag and Froude similarity in falling paper. Phys. Rev. Lett. 81, 345–348. C HRUST, M., B OUCHET, G. & D U Sˇ EK, J. 2013 Numerical simulation of the dynamics of freely falling discs. Phys. Fluids 25, 044102. E RN, P., F ERNANDES, P. C., R ISSO, F. & M AGNAUDET, J. 2007 Evolution of the wake structure and wake induced-loads along the path of free rising axisymetric bodies. Phys. Fluids 17, 113302. E RN, P., R ISSO, F., F ERNANDES, P. C. & M AGNAUDET, J. 2009 Dynamical model for the bouyancy-driven zigzag motion of oblate bodies. Phys. Rev. Lett. 102, 134505. F ERNANDES, P. C., E RN, P., R ISSO, F. & M AGNAUDET, J. 2005 On the zigzag dynamics of freely moving axisymetric bodies. Phys. Fluids 17, 098107. F ERNANDES, P. C., E RN, P., R ISSO, F. & M AGNAUDET, J. 2008 Dynamics of axisymetric bodies rising along a zigzag path. J. Fluid Mech. 606, 209–223. F ERNANDES, P. C., R ISSO, F., E RN, P. & M AGNAUDET, J. 2007 Oscillatory motion and wake instability of freely rising axisymmetry bodis. J. Fluid Mech. 573, 479–502. F IELDS, S., K LAUS, M., M OORE, M. & N ORI, F. 1997 Chaotic dynamics of falling disks. Nature 388, 252–254. F RANCK, A., JACQUES, M. & DAVID, F. 2013 Falling styles of disks. J. Fluid Mech. 719, 388–405. H OROWITZ, M. & W ILLIAMSON, C. H. K. 2008 Critical mass and a new periodic four-ring vortex wake mode for freely rising and falling sphere. Phys. Fluids 20, 101701. H OROWITZ, M. & W ILLIAMSON, C. H. K. 2010 The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J. Fluid Mech. 651, 251–294. H OWE, M. 1995 On the force and moment on a body in an imcompressible fluid, with application to rigid bodies and bubbles at high and low Reynolds numbers. Q. J. Mech. Appl. Maths 48, 401–426. L EE, C. B., P ENG, H. W., Y UAN, H. J., W U, J. Z., Z HOU, M. D. & H USSAIN, F. 2011 Experimental studies of surface waves inside a cylindrical container. J. Fluid Mech. 677, 39–62. L EE, C. B. & W U, J. Z. 2008 Transition in wall bounded flow. Appl. Mech. Rev. 61, 030802.
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M AGNAUDET, J. & E AMES, I. 2000 The motion of high-Reynolds-number bubbles in inhomegeneous flow. Annu. Rev. Fluid Mech. 32, 659–708. M AHADEVAN, L., RYU, W. S. & S AMUEL, A. D. T. 1999 Tumbling cards. Phys. Fluids 11, 1–3. M OUGIN, G. & M AGNAUDET, J. 2002a The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Intl J. Multiphase Flow 28, 1837–1851. M OUGIN, G. & M AGNAUDET, J. 2002b Path instability of a rising bubble. Phys. Rev. Lett. 88, 014502. M OUGIN, G. & M AGNAUDET, J. 2006 Wake-induced forces and torques on a zigzag/spiralling bubble. J. Fluid Mech. 567, 185–194. P ESAVENTO, U. & WANG, Z. J. 2004 Falling paper: Navier–Stokes solutions, model of fluid forces, and centre of mass elevation. Phys. Rev. Lett. 93, 144501. S AFFMAN, P. G. 1956 On the rise of small air bubbles in water. J. Fluid Mech. 1, 249–275. S HEW, W. L. & P INTON, J. F. 2006 Dynamical model of bubble path instability. Phys. Rev. Lett. 97, 144508. S MITH, E. H. 1971 Autorotating wings: an experimental investigation. J. Fluid Mech. 50, 513–534. S TEWART, R. E. & L IST, R. 1983 Gyrational motion of disks during free-fall. Phys. Fluids 26, 920–927. S TRINGHAM, G. E., S IMONS, D. B. & G UY, H. P. 1969 The behaviour of large particles falling in quiescent liquids. US Geological Survey. VANDENBERGHE, N., Z HANG, J. & C HILDRESS, S. 2004 Symmetry breaking leads to forward flapping flight. J. Fluid Mech. 506, 147–155. W ILLMARTH, W. W., H AWK, N. E. & H ARVEY, R. L. 1964 Steady and unsteady motions and wakes of freely-falling disks. Phys. Fluids 7, 197–208. Z HONG, H. J., C HEN, S. Y. & L EE, C. B. 2011 Experimental investigation of freely falling thin disks: transition from zigzag to spiral. Phys. Fluids 23, 9–12. Z HONG, H. J., L EE, C. B., S U, Z., C HEN, S. Y., Z HOU, M. D. & W U, J. Z. 2012 Experimental investigation of freely falling thin disks. Part 1. The flow structures and Reynolds number effects on the zigzag motion. J. Fluid Mech. 716, 228–250.
