Abstract
Heavy particles sink straight in water, while buoyant bubbles and spheres may zigzag or spiral as they rise. The precise conditions that trigger such pathinstabilities are still not completely understood. For a buoyant rising sphere, two parameters are believed to govern the development of unsteady dynamics: the particle’s density relative to the fluid, and its Galileo number. Consequently, with these parameters specified, the opportunities for variation in particle dynamics appear limited. In contrast to this picture, here we demonstrate that vigorous pathoscillations can be triggered by modulating a spherical particle’s moment of inertia (MoI). For a buoyant sphere rising in a turbulent flow, MoI reduction triggers a tumble–flutter transition, while in quiescent liquid, it induces a modification of the sphere wake resulting in largeamplitude pathoscillations. The present finding opens the door for control of particle path and wakeinstabilities, with potential for enhanced mixing and heat transfer in particleladen and dispersed multiphase environments.
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Introduction
The buoyancydriven motion of particles rising or falling through a fluid has captured the interest of scientists and engineers for centuries. From the earliest observations of rising bubbles by Da Vinci to the falling hogbladder experiments by Newton, the variability observed in the trajectories of these particles is intriguing^{1,2}. Such variability can have farreaching implications for many natural and industrial particleladen and tetheredbody flows, wherein the movement of the particles can significantly alter the drag and the transport of heat and nutrients in the fluid^{3}. For example, the presence of rising bubbles and particles in the oceans can result in warmer ocean waters and larger positive temperature gradients^{4,5,6}. Similarly, in industrial reaction catalysis, buoyant particles/bubbles are often released to enhance mixing in the fluid^{7,8,9}. From these observations, a link can be inferred between the oscillatory motions and wakes of the buoyant particles and the processes of turbulent diffusion at work. Several studies have emphasized the importance of this^{10}; however, few have established the precise factors that trigger the oscillatory dynamics.
The motion of a buoyant particle in a flow is a complex twoway coupled problem. The particle moves through the fluid in response to the flow fluctuations, and this motion in turn exerts a backreaction on the flow. The pathoscillations that result are often robust to interactions between the neighboring particles and to changes in the flow conditions, making the studies of isolated bodies relevant to multiphase flows^{11,12}. In order to model such systems, often a prototype problem is considered. The particle is modeled as a sphere, and the flow is assumed to be turbulent, homogeneous, and isotropic. For the particle dynamics, two control parameters have been proposed to be of relevance^{13,14}: the particletofluid density ratio Γ ≡ ρ_{p}/ρ_{f}, and the particle Galileo number Ga ≡ \(\sqrt {gD^3\left( {1  {\Gamma }} \right)} {\mathrm{/}}\nu\), where g is the acceleration due to gravity, D is the particle diameter, and ν is the kinematic viscosity of the fluid. Ga governs the development of wakeinstabilities behind the particle, and Γ, the response of the particle to these wakeinduced forces. Once the background turbulence is also considered, two additional parameters need to be included: the particle’s size in relation to the dissipative length scale of the flow (Ξ ≡ D/η), and the Taylor Reynolds number, Re_{ λ }, of the flow^{15}. Hence, knowledge of these four dimensionless groups: [Γ, Ξ, Re_{ λ }, and Ga] should seem to completely define the problem. Expecting these dependences, researchers have performed extensive studies on the dynamics of light and heavy particles in a range of flow environments^{16,17,18}.
A detail that has often been overlooked in the past is the rotational dynamics of buoyant spherical particles. Theoretical and numerical studies have mostly employed the classical Kelvin–Kirchhoff equations, expressing the conservation of linear and angular momentum for the coupled fluidbody problem^{2}:
where Γ is the sphere massdensity ratio, U is the sphere velocity vector, Ω is the angular velocity vector, g is the acceleration due to gravity, I^{*} ≡ I_{p}/I_{f} is the moment of inertia (MoI) ratio, where I_{p} is the particle MoI and I_{f} is the MoI of the fluid volume displaced by the particle, F_{ Q } and T_{ Q } are the fluid force and torque vectors, respectively, m_{f} is the mass of the fluid displaced by the sphere, and D is the sphere diameter. Note that δ = \(\sqrt {\frac{{\nu \tau }}{{\pi D^2}}}\) is the dimensionless Stokes boundary layer that develops in time τ^{19,20}. The prefactors B_{U} = 18 and B_{Ω} = 2 are known analytically from the unsteady viscous contributions^{18,21}.
Equations (1) and (2) point toward the two parameter dependences, namely, the particle's massdensity ratio Γ and its MoI ratio I^{*}. While rotation appears explicitly through Eq. (2), its effect on the particle dynamics has been ignored for isotropic bodies (spheres and cylinders). A few experiments detected the rotation for neutrally buoyant spheres in turbulence^{22,23}; however, its role on the flow dynamics was not revealed. Even for freely rising spheres, the rotational effects have been largely ignored in the past^{13,14}. One possible reason could be the small torque coefficients (C_{τ} ~ 10^{−4}) reported in fixed/falling sphere studies^{24}. From the torque balance for a spherical particle in a viscous fluid: \(\left[ {{I}^ \ast + {k}_{1}\frac{1}{{\sqrt {{\mathrm{Re}}_{\mathrm{p}}} }}} \right]\) \(\frac{{{\mathrm{\Delta }}\zeta }}{{{\mathrm{\Delta }}\tilde t^2}} = {k}_2{C}_\tau\), one could estimate a typical rotational amplitude. Here I^{*} ≡ I_{p}/I_{f} is the MoI ratio, where I_{p} is the particle MoI and I_{f} is the MoI of the fluid volume displaced by the particle. Re_{p} is the mean particle Reynolds number, Δζ is the rotational amplitude, \({\mathrm{\Delta }}\tilde t\) is the characteristic timescale, and \({k}_{1} = 20\sqrt {\frac{2}{\pi }}\) and k_{2} = 7.5 are prefactors that can be determined analytically (Supplementary Discussion). The largest rotation is expected when I^{*} → 0. In this limit, and considering \({\mathrm{Re}}_{\mathrm{p}}\sim {\cal O}(10^3)\), an order of magnitude analysis would suggest an amplitude of rotation Δζ ~ 1° (Supplementary Discussion). Such small rotations were expected to not influence the instability onset^{25}. Therefore, in most experimental studies, Γ was varied by using various combinations of hollow and solid spheres made from a variety of materials^{13,14,17}. In doing so, the particle’s MoI or I* varied erratically, the implications of which have not yet been considered.
In the present work, we investigate the possibility of changing the translational dynamics of spherical particles in flows by tuning their moment of inertia (MoI). We perform experiments on threedimensional (3D) printed spherical particles in a turbulent water flow, and track their translational and rotational motions by following a pattern painted onto the particles. We demonstrate that vigorous pathinstabilities may be triggered by simply reducing a particle’s MoI. A sphere with a low MoI undergoes a fluttering type of motion, while a high MoI sphere displays a tumbling type of motion. We reveal that the coupling between translation and rotation for the low MoI spheres is crucial in triggering these pathinstabilities. Finally, we draw some analogies to the pathinstabilities already observed for anisotropic particles such as disks, strips, and falling cards.
Results
Turbulence experiments
We begin with the case of buoyant spherical particles released in a turbulent flow. Two spheres were considered, both weighing ≈15.2 ± 0.1 g, that corresponds to a massdensity ratio of Γ ≈ 0.88 in water. The spheres have identical mass, diameter, and surface properties, which leave us with identical values for the particle control parameters (Γ ≈ 0.88 and Ga ≈ 6000). This mass ratio is well above the critical mass ratio (Γ_{crit} = 0.6) reported in prior studies^{14,17}. At the same time, we introduce a modification to the sphere’s composition. One sphere was fabricated as a hollow spherical shell, with an MoI, I_{p} ≈ 1.87 × 10^{−6} kg m^{2}, and the other, as a thin spherical shell with a dense metallic core, with I_{p}≈1.05 × 10^{−6} kg m^{2} (Fig. 1b). These resulted in dimensionless MoIs I^{*} ≡ I_{p}/I_{f} ≈ 1.0 and 0.6, respectively. Here, I_{p} is the particle MoI, I_{f} = m_{f}D^{2}/10 is the MoI of a fluid sphere with the same volume as the particle, and m_{f} = ρ_{f}πD^{3}/6 is the mass of the fluid sphere. We note that for a homogeneous sphere I^{*} = Γ = 0.88. Thus the values of I^{*} we chose here (I^{*} = 1.0 and I^{*} = 0.6) are higher and lower, respectively, than the homogeneous case.
The spheres were released in a turbulent flow, which was generated using an active grid located upstream to the measurement section of the Twente Water Tunnel (TWT) (Supplementary Fig. 1a). The water tunnel was configured to have a downward flow in the measurement section, with a Taylorscale Reynolds number Re_{ λ } ≈ 300, and the particle size was a fraction of the integral scale of the turbulence (Ξ ≈ 100). The downward mean flow speed in the measurement volume was adjusted to be comparable to the rise velocity of the particles. This ensured that the buoyant spheres were rising against the flow, enabling us to track long particle trajectories^{26}. The typical duration of a trajectory was around 30 T_{L} (or 100 τ_{viv}), where T_{L} is the integral timescale of turbulence^{27}, and τ_{viv} is the typical vortexshedding timescale. In total, we recorded a duration of approximately 500 T_{L}, which gave wellconverged Lagrangian statistics. The particles were imaged using two highspeed cameras placed at a 90 degree angle between them, and recorded at 500 frames per second. An analytically prescribed pattern was painted on the spheres, which enabled us to track their orientation in 3D (Supplementary Discussion and Supplementary Fig. 1b–d). This method has been validated and tested^{28}, and the measurement error in the detected orientation was <1°. The output of the orientation tracking method in axisangle convention was used to obtain the instantaneous angular velocity and angular acceleration of the spheres. Thus we obtain a complete description of the particle’s dynamics, composed of threetranslational and threerotational degrees of freedom.
Fluttertotumble transition
The above settings lead to a situation where the control parameters: Γ, Ga, Ξ, and Re_{ λ }, are identical for the spheres we consider. At the same time, the spheres have notably different moments of inertia (I^{*} ≈ 1.0 and I^{*} ≈ 0.6). Strikingly, this difference leads to dramatic changes in the translational dynamics. Evidence for this was first seen while tracking the movement of the spheres. The low MoI sphere undergoes zigzag motions and remains in the middle of the measurement section. In contrast, the high MoI sphere shows a tendency to drift horizontally in the flow and approaches the channel walls (Fig. 2a, b). The mechanism behind these preferential movements becomes clear when we look at the particle’s translational and rotational motions simultaneously (Supplementary Movies 1 and 2). The high MoI sphere is in a tumbling state, while the low MoI sphere displays a fluttering type of motion. In the tumbling state, the directions of rotation and translation do not change much during the motion (Fig. 2a). However, for fluttering, both the rotation and translation undergo periodic reversals of direction (Fig. 2b) at a rate comparable to the sphere vortexshedding frequency (Supplementary Discussion and Supplementary Fig. 2).
Further evidence for the tumble–flutter transition can be found by viewing the particle motion from a Lagrangian (TNB) coordinate system, i.e., one that moves with the sphere (Fig. 2c). The orientation of a vector is expressed in terms of the elevation and azimuth θ and ϕ, respectively. In Fig. 2d, e, we show the normalized histogram of the angle between the vector \(\mathbf{ \upomega}\) and its time derivative \({\mathbf{\upalpha}} \equiv {\mathrm{d}} {\mathbf{\upomega}} {\mathrm{/d}}t\). Here \(\mathbf{\upomega}\) and \(\mathbf {\upalpha}\) are the angular velocity and angular acceleration vectors, respectively. An alignment between these vectors would indicate that the particle’s angular velocity increases in magnitude but without a change in the direction of rotation. An antialignment would mean that the angular velocity decreases but without a change in the direction of rotation. For any situation where \(\mathbf{ \upomega}\) changes direction, the two vectors would not be aligned or antialigned. Strikingly, for the high MoI sphere, \(\mathbf{ \upomega}\) is preferentially aligned with \({\mathrm{d}}{\mathbf{ \upomega}} {\mathrm{/d}}t\) (single peaked). \(\mathbf{\upomega}\) also aligns with the mean horizontal drifting direction of the particle (Supplementary Discussion and Supplementary Fig. 3), which is evidence that the sphere statistically tumbles in the direction of its horizontal drift. We believe that this tumbling motion is crucial in establishing the mean horizontal drift for the high MoI sphere. The low MoI sphere, however, shows almost equal probability for \(\mathbf{\upomega}\) and \({\mathrm{d}}{\mathbf{\upomega}}/{\mathrm{d}}t\) to be aligned and antialigned (double peaked). In addition, \(\mathbf{\upomega}\) does not preferentially align with the mean horizontal drifting direction of the particle (Supplementary Discussion and Supplementary Fig. 3). Thus a fluttering type of motion occurs, which stabilizes the low MoI sphere to remain in the bulk of the water channel flow. An analogy may be drawn to falling disks, strips, and paper, where a similar tumble–flutter transition has already been observed owing to MoI reduction^{29,30} (Supplementary Movie 3). However, some form of geometrical anisotropy (of the particle) was considered necessary to induce this transition^{31}. The present finding demonstrates that such transitions are possible even for an isotropic body such as a sphere.
Acceleration statistics
The tumble–flutter transition revealed that the high MoI sphere drifts predominantly along a particular direction, whereas the low MoI sphere (fluttering) undergoes frequent reversals in its direction of motion (Supplementary Fig. 3 and Supplementary Movies 1 and 2). The fluid forces responsible for the differences in the dynamics may be gauged from the particle’s acceleration statistics. Figure 3a shows the probability density function (PDF) of the centripetal acceleration a_{N} for the two spheres under consideration. a_{N} is always directed along \({\mathbf {N}}\) of the TNB coordinate system. A reduction of MoI produces a significant change in the shape of the PDF, with a mean value that is almost double to that of the high MoI sphere. When we compare the angular accelerations, the difference is even more dramatic (inset to Fig. 3a). If the torques acting on the spheres were comparable, one would expect the angular acceleration to be increased by ~66% (\({\cal T} = {{I}}_{\mathrm{p}}{\kern 1pt} {\mathrm{d}}^2\theta {\mathrm{/}}{\mathrm{d}}t^{2}\)). Instead, the root mean square of angular acceleration is increased by ~430%. Such a dramatic increase in the angular acceleration can only be caused by an enhancement in the torque. Clearly, rotation plays a role in changing the torques acting on the particle, which in turn leads to enhanced translational accelerations. This is quantified in Fig. 3b, where we show that the acceleration a_{N} is conditioned on the magnitude of a rotationinduced lift (or Magnus) force^{32} \({F}_{\mathrm{L}}\propto {\mathbf {v}}_{\mathrm{p}} \times {\mathbf{ \upomega}}\). For the high MoI sphere, a_{N} appears to be almost independent of the Magnus force. However, a clear dependence is seen for the low MoI sphere. The origin of this becomes clear once we plot the orientation of \(\mathbf{\upomega}\) in the TNB coordinates, (Fig. 3c, d). For the low MoI sphere (Fig. 3d), \({\mathbf{\upomega}}\) aligns along the \({\mathbf B}\) direction, resulting in an alignment of \( {\mathbf{ v}}_{\mathrm{p}} \times {\mathbf{\upomega}}\) with the direction of a_{N}. Therefore, the acceleration a_{N} may be written as \({a}_{\mathrm{N}} \approx {a}_0\) + \({C}_{\mathrm{L}}\left {{v}_{\mathrm{p}} \times \omega } \right\), as shown by the dashed line in Fig. 3b. A lift coefficient C_{L} ≈ −0.2 provides a fair prediction of the increase in a_{N}. A negative C_{L} of this value is expected^{32}, since we have a buoyant particle that rises and rotates in the flow^{33,34}. For the high MoI sphere (Fig. 3c), \(\mathbf{\upomega}\) lies in the NB plane with no preferential orientation, which explains the absence of correlation in the conditioned acceleration plot, Fig. 3b. Thus tuning the MoI has enabled us to alter both the fluid forces and torques acting on a spherical particle, leading to an overall modification of its dynamics in turbulence. Whether these effects are restricted to a turbulent flow or not is unclear. To address this, we will next explore the dynamics of buoyant spheres rising in an undisturbed flow setting, i.e., in quiescent liquid.
Freerise experiments
Figure 4 shows the freerise trajectories and wake patterns of the high and low MoI spheres rising through still fluid (Ga ≈ 6000 and Ga ≈ 500). For this experiment, the spheres were released in a glass tank of 280 × 280 mm^{2} crosssection and 1500 mm height. We inject a patch of sodium fluorescein dye, which is the sodium salt of fluorescein (C_{20}H_{10}Na_{2}O_{5}). The dye was injected just above the sphere, near its release position at the base of the water tank (Supplementary Discussion and Supplementary Fig. 4). Once the sphere is released, it rises through the dye, entraining a part of the dye with it, and also shedding some dye in its wake as it rises. The dye fluoresces for blue illumination (≈490 nm wavelength), which helps visualize the wake. The intensity of the dye represents the relative concentration. Note that this does not represent the absolute vorticity in the wake but only gives a qualitative picture of the wake pattern, similar to the wake patterns reported in prior studies^{14,17}. At the high Ga (Ga ≈ 6000 in Fig. 4a), the wake is turbulent. The high MoI sphere (left) rises with a small amplitude of oscillation and a nearly vertical wake pattern. When I^{*} is reduced, large amplitude oscillations are triggered, and we witness a spreadout wake pattern behind the particle (right). Similarly, at a lower Ga (Fig. 4b), the oscillation amplitude is enhanced, and the wakes differ in the spread and the number of structures shed per oscillation. A similar effect was recently observed by us in numerical simulations of twodimensional circular cylinders rising in still fluid^{35} (Supplementary Figs. 5 and 6 and Supplementary Movie 6). Presumably, the wake modification due to particle rotation is crucial to these changes in the oscillation amplitude. Interestingly, in the freerise experiments, we do not observe the tumbling motions. It is likely that the flow perturbations from the incident turbulence are necessary to trigger these tumbling events.
Discussion
Comparing the sphere and (twodimensional) cylinder dynamics has shed light on some key aspects of the buoyancydriven motions of isotropic bodies in general. A longstanding debate in this subject relates to the existence a critical massdensity ratio (Γ = 0.6 for sphere^{14}, and Γ = 0.54 for cylinder^{36}), marking the onset of large amplitude vibrations for freely rising spheres and cylinders. While this has been reported in literature, the physical mechanism behind such a sharp transition has remained puzzling. The main question remains as to how a marginal reduction in mass density (Γ) could give rise to a significant enhancement in vibration amplitude, since the effective mass of the system (i.e., actual+added mass) changes by only a small factor^{37}. Our observations suggest that one possible reason for this could lie in the differences in the particle’s rotational dynamics, which manifest through its rotational inertia (since I^{*} changes along with Γ in most situations). This new perspective might also explain the origin of the wide variation that was witnessed in prior experimental studies on freely rising spheres^{13,14} and cylinders^{36}.
On a different note, the insights gained here may also be of relevance to the rising motions of spherical bubbles in water. It is well known that, when the bubble surface is contaminated, it spirals or zigzags^{38}, while the same bubble in pure water rises vertically straight^{39} (up to a certain larger Galileo number). The clean bubble, owing to its freeslip boundary condition, obviously does not rotate. In contrast, the contaminated bubble can rotate due to a combination of noslip at the interface and low rotational inertia (I^{*} → 0). These could be the contributing factors to the observed differences in the onset behavior of their pathinstabilities^{40,41}. For nonspherical particles (and deformable bubbles), the role of MoI might be even more crucial. For a nonspherical particle, the torque on the particle arises from two contributions: from skin friction forces and pressure forces^{2}. Additionally, the particle’s angle of attack changes as it rotates, resulting in significant torques. Thus strong rotational motions can be expected, and hence the MoIs might have a major role in observed particle dynamics. The coupled translational and rotational motions of nonspherical particles (oblate and prolate ellipsoids) will be the focus of a future investigation.
In conclusion, the use of simultaneous 3D particle position and orientation tracking has enabled us to resolve the coupled translational and rotational dynamics of buoyant spherical particles in a range of flow environments. We have shown that the onset of pathinstability can be closely linked to the rotation of the particle and that resonance may be induced (or inhibited) by tuning the particle’s MoI. This radically changes the way one perceives the dynamics of buoyancydriven isotropic bodies, and opens the door for control of path and wakeinstabilities by tuning the rotational inertia. The concept could be exploited in chemical engineering processes, where the mixing and transport of nutrients can be effectively enhanced through the introduction of vigorously vibrating spheres^{7}. Further, the flutter–tumble transition we observed for buoyant spheres in turbulence offers a few flowcontrol opportunities. For instance, particles that migrate toward channel walls could be used to modify the nearwall flow structure^{3}. By tuning the MoI, one could design spheres that accumulate near the walls, with potential for drag/heat transfer modifications in dispersed multiphase flow environments. Other avenues of application could lie in sports ballistics, where zigzag and spiral trajectories add to the unpredictability of the game. While this has historically been achieved by introducing surface heterogeneities and/or spin to the ball^{42}, rotational inertia could be used as an additional lever to trigger such pathinstabilities, thereby lending richer diversity to various ball sports.
Methods
Experimental methods
The experiments were performed in the TWT facility, in which an active grid generates nearly homogeneous and isotropic turbulence in the measurement section. The water tunnel was configured to have a downward flow in the measurement section, and the Taylor Reynolds number of the flow Re_{ λ } ≈ 300. We used a highprecision 3D printer (Rapidshape S30L) to fabricate the hollow spherical particles. The surface roughness was within 25 μm, and the spheres were symmetric for any plane passing through their geometric center. Rolling and floating tests were used to check for any inconsistencies. The particles were imaged using two highspeed cameras placed at a 90 degree angle between them (Supplementary Discussion and Supplementary Fig. 1). An analytically prescribed pattern was painted on the spheres.
Image processing
The particles were detected using the Circular Hough Transform method, which was implemented using the imfindcircles function in MATLAB. The projection of the painted pattern was compared with the synthetic image to retrieve the orientation (Supplementary Fig. 1c,d). Combining the two detection methods, we can track six degrees of freedom for the sphere released in the turbulent flow.
Data availability
The data that support the findings of this study are available from the authors upon reasonable request.
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Acknowledgements
V.M. is grateful to L. Olde Olthof for significant help during the experiments. We thank P. van der Plas, S. Maheshwari, J. Will, S. Wildeman, B. Verhaagen, and M. van Limbeek for useful discussions. We thank G.W. Bruggert, D. van Gils, and M. Bos for the technical support. This work was supported by the Natural Science Foundation of China under the Grant No. 11672156, the STW foundation of the Netherlands, COST action MP1305, and the European HighPerformance Insfrastructures in Turbulence (EuHIT).
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V.M., C.S., and D.L. designed the research. V.M. performed the experiments. V.M. and X.Z. performed the analysis. All authors discussed the results and contributed to the writing of the manuscript.
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Mathai, V., Zhu, X., Sun, C. et al. Flutter to tumble transition of buoyant spheres triggered by rotational inertia changes. Nat Commun 9, 1792 (2018). https://doi.org/10.1038/s4146701804177w
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DOI: https://doi.org/10.1038/s4146701804177w
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